The London, Edinburgh and Dublin philosophical magazine and journal of science

oes

TASER NT RERS

as Be

a

-_—
.

ee ee

‘y*siar es

v

ae

=

Seely

.

RES

tats

CVSP SP MSLAD TE

ase

TEEETES

a

Eutt?

a Sempg = 2s S=

=

oh ok:

=

- = sSs8

>

oP ah tet AF ohet hen

.

ed Pen Fee OO

ae ee

=, we
Ae ee wy are

Li Gr

howe

V Fey OHO?

ee AS TF: sh bv bbad th rg tapege?
pear i iG eras fo hana tity ,
Seite st: stb tec areraw sit fig
A ae eee
aE erie zi} ry if
f sive Tr
. Yin He Ader 1} Sate ae
be ery ie a ee
. re tn

oer ore tray"
vege eapeyels
; te @

fue
orf om) ‘ alent
ae Yolk PEt Ls te UPd
i, lene wie
vee er Ly
a iibei ag Sued
Oe Weel
san oe iw

1 ty Wal
‘i THY

i)

r . S\«s
ra
H
:
*
i
;
d !
‘ ea Pe thlay
Aa, yy avwity
: Poe : ap ew
tet 1" rit: i ‘Pip ies
rt Fate Os Reb LA
PAA F vA Me ye ee i*
j i
y* uv
4
"= j
" or
i i
ral
,
d x
4
:
& y
.
; ‘
. s
; re 4
re |
’
cay 4
7 4 ¥
q :
i ta
4 4 we
, b
‘
i 4
3 *
.
“ r
a J ‘
i rt
i) i) +)
4 >
bea < | '
a
nits ‘}
i .
‘
'
rey i |
i re ‘
eas ;
oy
f ‘ ,
We be? a |
‘ :
. i ti
. é ‘
;
‘
« he jy
q + }
S , Ma
4 x Tels
" ‘
net
{cea
r ¥
bE 4
* oa
ta “a }
q hie
eet os ;
ier dere
Mise S4 ;
vA! t Prt
hs thd Brice oe
worse 7 eas
TH 4
-$fhss wists F
'
:
y
\*
, 2
TF
ia
‘ey a
i
hy
‘ 5
- “.
ny ’
7 * :
rents

fivch werd
Lfyiaathits
ral

A rhea

to}

Ste eat

5

’ Hee

CAN

‘
4 ff . 4 rs
Ta a? a eho Oa
, rt dh tage

hy “iy *
‘

srt tuated wore

eS he

Pl

Yet

7

bi Hig

;

@7'

Ay $e

’
ta
hit
mein
1p Ge l-heqed

wu
ttle

Hit)

fab yo! it

Tenia

I a i
Mt
‘itt hat fy

i] ; st
Hea

Hea J

ihe

sein ar i)

righ A

} abe Nabe hy
eal
mh

She
ie forse
ee a

ab a

AAS
hahaa
Tt

tie

aa ae ey ae;

Py hirs
be

»
ae shad

Bey

44

AP Way

sity

aie Ne

1 edae

aha
L dubehy Hee
ni of Aa?
ili
Be pyph lee
toh ge jhe
te bieie

oot bday
INS nt kt

pet
ene
fide
bale

ithe
Ht

j vey
fbb
ipa ps iN

¢ 0h 4} ¢
Ay erie yy Heit
oo wt

fh
ry

it

Als phaadl

hae od
1 tibee
Ler
alle ha he
wpe

14) 4

’

Uh ta
ie

>

t

we

Mbt

ie

"ye

Wid

hte ano
i

oh ea
rit Pipa ior'
iby

at
Ht
,

Aa grid bey
+irbe
>

: 1227.3
THE a a 4

a
SS

LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

CONDUCTED BY
mR OLIVER JOSHPH LODGE, D.Sc., LL-D., PRS.
SIR JOSEPH JOHN THOMSON, O.M., M.A., Sc.D., LL.D., F.R.S.
JOMN POLY 5. MeAs Disesek A... Gs.
RICHARD TAUNTON FRANCIS

AND

WILLIAM FRANCIS, F.1..8.

“‘ Nec aranearum sane textus ideo melior quia ex Se fila gignunt, nee noster
vilior quia ex alienis libamus ut apes.” Just. Lips. Polit. lib.i. cap. 1. Not,

VOL. XLIV.—SIXTH SERIES.

JULY—DECEMBER 1922.

LONDON:
TAYLOR AND FRANCIS, RED LION COURT. FLEET STREET,

SOLD BY SMITH AND SON, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN ; —-
AND VEUVE J. BOYVEAU, PARIS,

‘‘Meditatiouss est perscrutar] occulta; contemplationis est admirari
perspicua .... Admiratio generat queestionem, queestio investigationem,
investigatio inventionem.”—Hugo de S. Victore.

——“ Our spirent venti, cur terra dehiscat,
Our mare turgescat, pelago cur tantus amaror,
Cur caput obscura Phoebus ferrugine condat,
Quid toties diros cogat flagrare cometas, “
Quid pariat nubes, veniant cur fulmina ccelo,
Quv micet igne Iris, superos quis conciat orbes
Tam vario motu.”

J. B. Pinelli ad Muzontum.

(
ALERE SH FLAMMAM,

BRASS aS

CONTENTS OF VOL. XLIV.

(SIXTH SERIES).
. 7

NUMBER CCLIX.—JULY 1922.

Page

Mr. A. R. McLeod on the Unsteady Motion produced in a Uniformly
Rotating Cylinder of Water by a Sudden Change in the Angular
Ro em Cee ON MUAD YS 71 ocak 9 ig ask Fick scaddo Hten vig ERO NN
Dr. Norman Campbell on the Elements of Geometry ............ 15
Dr. Dorothy Wrinch on the Rotation of Slightly Elastic Bodies ... 30
Mr. G. H. Henderson on the Strageling of « Particles by Matter .. 42

Dr. G. Green on Fluid Motion relative to a Rotating Harth........ 52
Prof. G. N. Antonoff on the Breaking Stress of Crystals of Rock-Salt. 62
Dr. Norman Campbell on the Measurement of Chance ............ 67
Messrs. H. T. Tizard and D, R. Pye: Experiments on the Ignition

of Gases by Sudden Compression. GR VAE eye E so ete Mics 79
Mr. C. Rodgers on the Vibration and Critical Speeds of Rotors.... 122

Mr. P. Cormack on Harmonie Analysis of Motion transmitted by

“So Tig SNS he NOTTS Shee Bip a cota gre ca or Pn ee Teh ae
Messrs. E. W. B. Gill and J. H. Morrell on Short Electric Waves

156

CMR NEV ES etext. Molar Sus sees wh ad ave wR ee epee acne ha: 161
Sir George Greenhill on Pseudo-Regular Precession.............. 179
Dr. J. W. Nicholson on the Binding of Atoms by Electrons ...... 193

Mr. L. St. C. Broughall on Theoretical Aspects of the Neon Spectrum. 204
Dr. F. H. Newman on Absorption of Hydrogen by Hlements in the

Remon Misenaroe Tbe). Fe oem. bls chats tet de eet U oliacs lao Bie asad 215
Mr. B. A. M. Cavanagh on Molecular Thermodynamics. II. ...... 226
Prof. J. G. Gray on the Calculation of @entuoids" Sv acs. hance tae 247

Messrs. A. P. H. Trivelli and L. Righter: Preliminary Investiga-
tions on Silbersteia’s Quantum Theory of Photographic Lxposure. 252
Dr. L. Silberstein on a Quantum Theory of Photographic Exposure. 257
Mr. R. F. Gwyther on an Analytical Discrimination of Ilastic
Sarecses Mn atl SORrU ple: Ody tire Paes. kh ecw cee tl eves 274
Mri, S. Rowell on Damped Vibrations’ . 2.6.0.0 ce Yee Fk. 284
Notices respecting New Books :—
Mr. L. #. Richardson’s Weather Prediction by Numerical
Ma RE ee eresigce earn ie MEMES ONT ach A sriorh Pelee aot helasin oc acaee Os 285
Proceedings of the Geological Society :—
Mr. C. E. Nowill Bromehead on the Influence of Becky on

FHRoEMAMeMVEGE IOUMONS ss eka gets ck ewe... LOO

Intelligence and Miscellaneous Articles :—
On Young’s Modulus and Poissona’s Ratio for Spruce, by Mx. H.
Carrington :

1V CONTENTS OF VOL. XLIV.——-SIXTH SERIES.

NUMBER CCLX.—-AUGUST.

Page
Mr. C. J. Smith on the Viscosity and Molecular Dimensions of
Gaseous Carbon Oxysulphide. (COS) oauioc et ey. olen = eine 239
Prof. A. O. Rankine on thé Molecular Structure of Carbon Oxy-
sulphide and Carbon Bisulphide = - ce . sam arene 292

Mr. F. P. Slater on the Rise of y Ray Activity of Radium Emanation. 300
Profs. J. N. Mukherjee and b. C. Papaconstantinon on an Iixperi-
mental Test of Smoluchowsli’s ‘heory of the Kinetics of the:

Process-of Coagulation: 22. diy os) wedge sok tee eee nee ee 305
Prof J. N, Mukherjee on the Adsorption of lois) 227.2 ae 321
Prof. W. M. Hicks on certain Assumptions in the Quantum-Orbit.

Uheony ol Spectra: jv. 25 cp ao te seco sacs. 2 nto aie gemen tae eee eae 346
Mr. F. C. Toy on the Theory of the Characteristic Curve of a Photo-

erapline WanuilsiOm Gi. tiaie one! « cos. Slane see eee 352
Mr. A. M. Mosharrafa on the Stark Effect for Strong Electric

CNG Siig ce a aes On usuag: a eal tale ah Shae es Cae A ee, er 371
Mr. I. Takagishi on the Damping Coefficients of the Oscillations in

Three-Conpled MiectriesCircuits.. 72.2. ena Sa eee 373
Prof, 8. C. Kar on the Electrodynamic Potentials of Moving

CHar oes es i takes ae nod Gig wate Rae Is eee GAGE eee a te a O

Mr. A. KE. Harward on the Identical Relationsin instein’s Theory. 380
My. H.S. Rowell on Energy Partition in the Double Pendulum... 882
Piof. J. S. Townsend on the Velocity of Electrons in Gases ...... 384
Prof. H. A. McTaggart on the Electrification at the Boundary
between a Miquid-andia Gas .4....5 2.2 4) ee een eee 386
Prof. L. V. Ising on a Lecture-Room Demonstraticn of Atomic
Models: «Plate dd.) acs ea ee eee oe 395
Mr. Hi. D. Murray on the Influence of the Size of Colloid Particles
upon the Adsorption ‘of Mlectrolytes|s a... ... cn 4.2 eee 401
Notices respecting New Books:—
Dr. G. Scott Robertson’s Basie Slags and Rock Phosphates... 416

NUMBER CCLXI—SEPTEMBER.

Prof. Sir EK. Rutherford and Dr. J. Chadwick on the Disintegration
of Klements by o Pagucles: ce cho ce sete been 417
Prof. W. L. Bragg and Messrs. R. W. James and C. H. Bosanquet
onthe Distribution of Electrons around the Nucleus in the Sodium

and Chlorine A toms’ gist ne mesic aces ect tere ee ee 453
Messrs. C. G. Darwin and R. H. Fowler on the Partition of
Dual) 02 ae ea eM HI en rate eras ila) SU Gen ere Sat hsa’n x 450
Mr. M. EH. Belz on the Heterodyne Beat Method and some Appli-
eations. to Physical Measuremientse ese ese eee 479
Mr. R. #. Gwyther on the Conditions for Elastic Equilibrium under
Surface Tractions in a Uniformly Eolotropic Body............. 501
Mr. C. J. Smith on the Viscosity and Molecular Dimensions of
Salles ioxide. 2. iis eae Neetgtee Saeko ean ne 508

Mr. 8. Lees on a Simple Model to Lllustrate Elastic IHysteresis..., 511
Prof. R. W. Wood on Atomic Hydrogen and the Balmer Series

SI CLCIAANLLOD Gp Crete © cit eC MM MISIPREEE ont: asr bn <tolbg «90 een Per ass 008
_.. Ma, D. Coster onthe Spectra of X-rays and the Theory of Atomic
20 @ a SAO ETUC UC MGT (060.9 ais» wie Sea) a 546

CONTENTS OF VOL. XLIV.—SIXTH SERIES. V

Page
Dr. Norman Campbell and Mr. B. P, Dudding on the Measurement =
© RULE Se see 6 Salts Is TS RSS SRS SR A rte 577
Mr. D. L. Hammick on Latent Ileats of Vaporization and
MEARE 5 oh Dee chee tnire ray Vis 6 va eo) art ah ees wiwrece © ehohe 590
The late William Gordon Browv on the Faraday-Tube Theory of
PMR E EST N Rye wy Ataieit wphw aw eos, oe ayen ts * ihe niele a lactase ss 7. OU4.
aor DB. A, M. ‘Cavanagh on Molecular Thermodynamics. IIL....... 610

Prof. A. W. Porter and Mr, J. J. Hedges on the Law of Distribution
of Particles in Colloidal Puspensions, with Special Reference to

Peet SSPE MESIOR IONS y... Als otic yess et eh Haire woe eal, ee win ad pee Ge G4]
Mr. St. Landau-Ziemecki on the Emission Spectrum of Monatomic
MEMEO UN Ry a ect ae he hit Goa Gos kee, oS ae bee ee wale 651

NUMBER CCLXIT.—OCTOBER.

Sir J. J. Tuomson: Further Studies on the Electron Theory of
Solids, The Compressibilities ot a Divalent Metal and of the

Diamond. Electric and Thermal Conductivities of Metals...... 657
Dr. G. H. Henderson on the Decrease of nergy of & Particles on

peermrmmrarT YON MMR GLEE i, toads es cle) cruise wie wield ORG wel ccg ees 680
Mr. x CC, Henry on a Kinetic Theory of Adsorption ............ 689
Prof. S. R. Milner on Electromagnetic Lines and Tubes ......... 705
Myr. A. PAPUA OM: POLAT: cel Ss Socdiaie cidsan ootlie aes Satine’ eo | 720
Mr. G. Breit on the Effective Capacity Oaeancake Coil. 5... 729
Prof. F. E. Hackett on the Relativity-Contraction in a Rotating

Shaft moving with Uniform Speed along its Axis. ............ 740
Dr. T. J. Baker on Breath F OATES, fy eee Gers San 09) ee Mare ae 762
Mr. A. Sellerio on the Repulsive Effect upon the Poles of the

aren ie ete SCs te Ss. Led Bie WP ae EM e.g eee EME ade a Re baer 765
-Mr. B. B, Baker on the Path of an Electron in the Neighbourhood

J SUT AT SEE SA me be Saree varices be A Rn en nea ELE
Prof. A. W. Porter and Mr. R. E. Gibbs on the Theory of Freezing

Mixtures’: .... NE PURI ESSERE ue He OR i a a RE Soe oe 787

NUMBER CCLXIII.—NOVEMBER.

Mr. G. Shearer on the Emission of Electrons by X-Rays........ 793
Mr. A. J. Saxton on Impact Ionization by Low-Speed Positive

BAe Meathey 48l AAG ROOM hte late opm he es cach ee nates a 8 Es 809
Messrs. C. G. Darwin and R. I. Bowler ou the Partition of Energy

—Part il. Statistical Principles and Thermodynamics ........ 825
Mr. J. H. Van Vleck on the normal Helium Atom and its relation

be mite eoanbil ne Ory i. caicc aie wise Sea aie Ores dae ae oo ore 842
Mr. G. A. Tomlinson on the Use of a Triode Valve i in registering

Pree ctea) Ce anigiens. (sno 6 Wee ag ele. eh s Gioia cell heheh ale bse ® 870
Mr. Id. A. Milne on Radiative Iquilibrium: the Insolation of an

2h SEL Satie ae RDI S 5) a. Jat 8 872
Dr. 5. C. Bradford on the Molecular Theory of Solution. II....... 897
Mr. Rh. A. Mallet on the Failure of the Reciprocity Law in

ae SON wee tte a ate PR eI CAT Pcie 'e ayes 0.6 ¥ 0b os wives 904

Me-srs. R. W: Roleris, J. WH. Smith, and 8. S. Richardson on
Magnetic Rotatory Dispers'on of certain Paramagnetic Solutions, 912

vl CONTENTS OF VOL. XLIV.—SIXTH SERIES.

Dr. F. W. Edridge-Green on Colour-Vision Theories in Relation
vo, Colour=Blindmess. Gif so. sur nett cue aie eae eel, on
Mr, A. H. Davis on Natural Convective Cooling in Fluids .......
Mr. A. Hl. Davis on the Cooling Power of a Stream of Viscous
BLT ses atte ead aha. of P38 tele’ » yaa eR RES sige ae Rca ca teae, oce a
Dr. I. H. Newman on a Sodium-Potassium Vapour Arc Lamp.

Caps eae eet tamer come Ath Accs eect eG GitG We 20
My. J. J. Manley on the Protection of Brass Weights............
Mr. H. 8. Rowell on the Analysis of Damped Vibrations.........
Mr. F. M. Lidstone on the lull Effect of the Variable Head in

Viscosity: Determinations... J. ta. ceo ee oe inact eee ee
Dr. L. Silberstein and Mr. A. P. 1. Trivelli on a Quantum Theory

of Photographic Exposure. (Second Paper.) 4. 2... ase

Dr. J.S. G. Thomas on the Discharve of Air through Small Orifices,
and the Entrainment of Air by the Issuing Jet. (Plate VL)...
Dr. J. R. Partington on the Chemical Constants of some Diatomic
CC ah ee Ma ci ai neh aot ds bo ORS oo
Mr. M. I’. Skinner on the Motion of Electrons in Carbon Dioxide. .
Mr. W. N. Bond on a Wide Angle Lens for Cloud Recording.
CBT ates V TR Gh tases Mie care SL HG BA eg ee
The Research Staff of the General Klectrie Company Ltd., London :
A Problem in Viscosity: The thickne-s of liquid films formed cn
solid surfaces under dynamic conditions. (Piate VIII.) ......
Prot. 5. Timoshenko on the Distribution of Stresses in a Circular
Ring compressed by Two Forces acting along a Diameter ..
Prot, A. W..;Porterom-a. Revised Lquation Of State, 0)... ae
Prof. V. Karapetoff: General Mquations of a Balanced Alternating-
Wurrent Bridore.) es 20 i iG ae clas selec eee ee ae
Prof. J. 8. Towusend and Mr. V. A. Bailey on the Motion of Elec-
tronsain “Areon andsimtbliydrocen. 0) eine se ake
Prof. 8. R. Milner: Does an Accelerated Electron necessarily radiate
Bnerey- on the @lassieal: Mheory (2. ont se ahs cee oe en ee
Mr. A. Press on a Simple Model to illustrate Elastic Hysteresis
Mir Sr lees: Note on thelabover|:2."...0.c)) «80s ee a oe
Notices respecting New Books :—
M. H. Ollivier's Cours de physique générale. 3, 72- ee
Dr. J. W. Mellor’s A Comprehensive Treatise on Inorganic
and, Theoretical Cemistuy wate. oes Ste ee ee cit
Mr. 8. G. Starling’s Science in the Service of Man: Electricity.

M. Gustave Mie’s La théorie EKinsteinienne de la Gravitation. . ~

Wave-lengths in the Arc Spectra of Yttrium, Lanthanum, and

Cerium and the preparation of pure Rare arth Elements. .

The Journal of Scientiue Instruments) 2. ee eee ene
Proceedings of the Geological Society :—

Mr. R. D. Oldham on the Cause and Character of Karthqualcs, .

Sir C. J. Holmes on Leonardo da Vinci asa Geologist ......
Intelligence and Miscellaneous Articles :—

On the Buckling of Deep Beams, by Dr. J. Prescott ........
On Damped Vibrations, by Ma. C,H. Wright ............8
On the Magnetic Properties of the Hydrogen - Palladium

System, by Mir A. Th. Oxley. 2 hemi ale frien etme
On Short Electric Waves obtained by Valves, by Prof. R.

Wilhidldiinne Gon iad he onc ee: ree pM eared elites seers

Page

— 916
920

940 |
944
948
951
953
956
969

988
994

999

1002

. 1014

1020
1024
1053
1052.

. 1053

1054

CONTENTS OF VOL. XLIV.— SIXTH SERIES. vil

NUMBER CCLXIV.—DECEMBER.

Page
Mr. R. Hargreaves on Atomic Systems based on Tree [lectrons, *
positive and MOMALVE, AUC THEI OtADIL ye as owe eee yes cas 1065
Prof. R. W. Wood on Selective Reflexion of \ 2536 by Mezcury
BO SACO URN US RTOs ec OU PS a ee ae 1105
Prof. R. W. Wood on Polarized Resonance Radiation of Mercury
Pm BCE ate Ne a tn Gotaceiicn Gch g emietice sista Mtn Le eth aha es 1107
Mr. 8. J. Barnett on Electric Fields due to the Motion of Constant
PAE NOURAO MEI OMOVSGOUIS. (oo jui a scsteele evap caw euiatee hie 8 Baer vad 1112
Prof. Megh Nad Saha on the Temperature [onization of Hlements
of the Higher Groups in the Periodic Classification............ 1128

Prof. F. Horton and Dr. A. C. Davies on the Ionization of Ab-

normal Helium Atoms by Low-Voltage Electronic Bombardment. 1140
Prof. J. 8. Townsend on the Ionizing Potential of Positive Ions ., 1147
Dr. G. Breit on the Propagation of a fan-shaped Group of Waves

Baresi MRESe rae OMG CUIINTI: 5)5 5). eva ent pata netale’ Caer auee hes « Mawes) ape chs 1149
Prof. E. Keightley Rideal on the Flow cf Liquids under Capillary
SMM Seg nh Niet So oe CaN Goshen SU ee et geek ae and ed 1152
Prof. S. Russ and Misi. Clark on a Balance Method of mea-
SMAPS RUNS APs rare ee sae eh ae ee Sait: d ietee teeter wo at 1159
Mr. J. W. T. Walsh on the Measurement of | Lipa Ser Sepa Mea paper 1165
Notices respecting New Books :—
ine Cambridee Colloquimn, 1916: “Part To. oo eee. 1169

Mr. J. Edw ards’s A Treatise on the Integral Calculus: Vol. II. 1169
oh

‘ f
ee

Il.

UL saat ENE
V.

Nil

VIL.
WARDE

PLATES.

. Illustrative of Messrs. H.T. Tizard and D. R. Pye’s Paper on

Iixperiments on the Ienition of Gases by Sudden Com-
pression.

Illustrative of Prof. I. V. King’s Paper on a Lecture-Room
Demonstration of Atomic Models.

Illustrative of Prof. Barton and Dr. Browning’s Paper on
Vibrational Responders under Compound Forcing.

Illustrative of Dr. F. H. Newmau’s Paper on a Sodium-
Potassium Vapour Are Lamp.

Illustrative of Dr. J. 8S. G. Thomas’s Paper on the Discharge
of Air through Small Orifices, and the Entrainment of Air
by the Issuing Jet.

Illustrative of Mr. W. N. Bond’s Paper on a Wide Angle
Lens for Cloud Recording.

Illustrative of The Research Staff of the Geveral Klectric
Company ona Problem in Viscosity.

IX. Illustrative of Prof. R. W. Wood’s Paper on Polarized NReso-

nance Radiation of Mercury Vapour,

THE

LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

[SIXTH SERIES.]
JUL ¥ 1922.

I. The Unsteady Motion produced in a Uniformly Bees
Cylinder of Water by a Sudden Change in the Angular
Velocity of the Boundary. By A. R. McLnop, J.A.,
Fellow of Gonville and Caius College, Cambridge ”*.

N the following paper a comparison is made between
the observed angular velocities in a rotating circular

cylinder of water, and those calculated by two-dimensional

theory in which Fhe effects of the base and the free surface

are neglected, and each particle of water is assumed to move

in a circle about the axis of rotation. The two cases dealt
with are those of unsteady motion in which the cylindrical
containing-wall is suddenly started from rest, or suddenly
stopped when, with the water, it is rotating uniformly. The
Be gnremienis are all of cunfaee velocities, because the use of

lycopodium particles floating on the surface was necessary,
as liquid globules of the same density as the water would

not remain at the same depth for any length of time.
Three cylinders of approximate diameters 5 cm., 15 cm.,
aud 25 cm. were used, and three angular velocities, viz.
36 r.p.m., 10 r.p.m., and 14 r.p.m. The small cylinder
was four diameters, and the other two cylinders were two
diameters long. The observed velocities show a departure
from theory, which increases with angular velocity and
with the size of the cylinder, but which tends to vanish
at very low speeds. The discrepancy is much greater

* Communicated by the Author.
Plul. Mag. 8. 6. Vol. 44. No. 259. July 1922. B

2 Mr. BE R. McLeod on Unsteady Motion produced

for the “‘stopping” than for the “starting ” experiments.
The work was done at the Royal Aircraft Establishment,
Farnborough, during the months Feb.—Sept. 1919.

In some later work, not yet published, the discrepancy
in the case of the “starting’’ experiments is traced to
the effect of the base; and the theory will probably apply
to this motion in very long cylinders, provided eddies ©
do not form owing to initial instability of the water. In
the case of the “‘stopping’’ experiments, the discrepancy
is due to instability and turbulence.

€ 1. Theoretical.

In dealing with a rectilinear two-dimensional eddy in an
incompressible fluid which contains no sources or sinks,
the usual assumption is that particles of the fluid move
in circles about the axis of the eddy. This makes the
problem one of complete symmetry, and the radius vector r
and the time ¢t are the two independent co-ordinates. The
equations of motion, when written in cylindrical co-ordinates
with the axis of z along the axis of symmetry, reduce to the
following forms, in which p denotes pressure, p is density,
vy is the kinetic viscosity, and ¢ is the angular velocity
about the axis :—

OP = rg? 2 wir ey! See Se

for the pressure, and

Or ror Vv

giving the angular velocity. Let us suppose that the
angular velocity ¢ satisfies the conditions . |

bi F (r)sfor t =.0; 1 ae (3)
d= h(t) ,, r= c= radius of cylinder; . (4)

eae
P42 OP — oot 2)

that is, at the initial instant the angular velocity in the
cylinder is known to be F(r) at radius 7, while thereafter
the rotation of the boundary is prescribed to be ¢(¢). The
solution of (2) satisfying the conditions (3) and (4) is, in

by Change in Angular Velocity. 3
terms of a series of Bessel functions:

5 = AnV/ Ce va, *t/c? ; ; :
gj = 3 Petlentlede {Pr (ef) Ie ae

n=l
Qvand (anr/c)e

—-va,2t/c?
ae Sanda | db(r)ernrledr, . (5)

n=) cr Jo(@n)

where J,(a,)=0 and a, is the nth root of this Bessel
function of the first order.
Taking $(¢) =Q=constant and F(7*)=0, we have

by eg ier ig
2 ? n=l An Jo(en) ‘

This is the solution for the case in which the water is
initially at rest and the cylinder suddenly rotates with a
constant angular velocity ©. The solution is given as
a problem in Gray & Mathew’s ‘Treatise on the Bessel
Functions’ (Ex. 38, p. 236, 1st edn.).

Taking $(t)=0 and F(7)=Q=constant, we find the
solution

d rs 2e $ IJi(an7/¢) ena, t/e?

LP gy n=1 Pres : a Ral oO

This is the solution for the case in which the water is
initially rotating with a constant angular velocity Q, and
the cylinder is suddenly stopped. The solution is given by
Stearne, Q.J. Math. xvii. p- as (1881), and Tumlirz, Siéz. d.
k. Akad. in Wien, Ixxxv. (ii.) p. 105 oe

The phenomena which (6) and (7) are supposed to.
represent are at the basis of the formation and dissipation
of eddies by viscous action. ‘To take one example, they
may be of importance in the theory of the aeroplane
compass. The experiments of Part I. were undertaken
to test the validity of these equations.

§ 2. Numerical Solution of Equation (7), and Discussion.

The numerical solution of (7) is given in Table I. for the
values of r/e and vé/c? which are there indicated. The values
are probably accurate throughout to the fifth significant
figure. For small values af the arguments, the value
if $/Q sometimes differed from unity only after the sixth
figure. The values of Jj(,7/c) for arguments greater

i

a

Mr. A. R. McLeod on Unsteady Motion produced

ABR le
Values of b/a.

vt/c*. nie 10. ll O, "3. “4,
‘001 1:00000 1:00000 1:00000 1:00000 1:00000
"002 1:00000 1-00000 1:00000 1:00000 1:00000
‘003 1:00000 1-00000 1:00000 1:00000 1:00000
004 1:00000 1:00000 1:00000 1:00000 1:00000
006 1:00000 1:00000 1:00000 1:00000 1:00000
7010 1:00000 1:00000 1:00000 ‘99999 ‘99992
015 1:000U0 ~~ 100000 "99996 ‘99969 ‘99795
"020 “99991 99982 "99935 ‘99730 ‘98970
025 ‘99907 ‘99860 "99650 "99002 ‘97242
030 ‘99587 ‘99451 "98923 ‘97591 ‘94634
040 ‘97490 ‘97062 "95616 ‘92675 *87504
050 92.923 "92196 *89890 *85665 “T9077
066 °86299 89373 *82526 “T7594 ‘70409
070 “78497 T7482 “T4419 69279 ‘62086
"080 ‘70314 69294 66248 61234 04393
‘090 62300 ‘61326 58440 53746 "47444
"100 *D4T7T6 03877 *O1224 46943 41257
"150 27221 -26753 "25321 23045 ‘20079
°200 13156 "12916 "12216 11104 09661
300 03034 ‘02979 02817 -02560 02226
“400 *-()?6989 076862 026489 0°5897 075128
“500 071610 071581 071495 0713858 071181
“600 093708 °0°3641 "0734.43 03128 092721
‘700 “078542 ‘078386 077930 ‘047206 046267
vt/e° iG —— 1). 6. hs 8. “Q:
‘001 1-00000 1:00000 1:00000 “99999 ‘97034
"002 100000 1-00000 1:00000 99782 86681
003 1:00000 1:00000 "99982 "98630 *76998
‘00+ 1-00000 ‘99998 ‘99865 -96468 69193
006 "99999 99944 ‘98952 ‘90552 57793
‘010 "99886 ‘99004 ‘94263 ‘78057 44051
7015 "98921 ‘95570 *85956 ‘65615 34305
020 ‘96580 ‘90409 ‘77540 56140 ‘28162
"025 93062 84548 ‘69887 ‘48810 °23855
030 "88785 78592 63137 42973 °20630
"040 Oeil ‘67418 520338 *34229 "16189
"050 ‘69768 ‘57707 43399 “27956 *12940
‘060 ‘60940 49443 36541 es PALS "10650
070 "53008 -42430 30983 19499 08892
080 "45994 -36468 -26406 16509 ‘07497
“090 39845 31382 *22589 14057 06265
"100 *344838 ‘27031 "19375 12017 05430
-150 16624 *12909 ‘09175 "05653 (02544
*200 ‘O7985 06190 04393 02704 012155
300 ‘018395 ‘014256 ‘O10114 076223 022797
-400 074227 073284 022330 0214338 0°6443
“500 0°9760 0°7563 095366 0°3301 ‘0? 1484
‘600 032248 ‘O° 1742 071236 ‘017605 043418
700 045178 “074018 "042847 041752 ‘0°7874

* -0°?6989 means ‘C06989.

re Sas

by Change in Angular Velocity. 5

than 15 were obtained by the use of the first three terms

of the asymptotic expansion. For smaller arguments the

values were found by interpolation from the twelve-figure

tables in Gray & Mathew’s ‘ Treatise.’ Curves representing

Table I. are shown in fig. 1, from which figs. 2 and 3
e . my .

have been derived graphically. The calculation was made

a i ie
(jeez

with the aid of Chambers’s seven-figure logarithms. It
was checked throughout, once everywhere and twice or
three times in parts.

For large values of ¢, the angular velocities are pro-
portional to J,(a,7r)/r (0< r<1). Most of the experimental
curves are roughly of this form for a large value of ¢.

The solution, being non-dimensional, applies to all sizes

6 Mr. A. R. McLeod on Unsteady Motion produced

of cylinder, all angular velocities , and all incompressible
fluids. The significance of vt/c? is of interest. According
to (6) and (7), the behaviour of a cylinder of liquid is
exactly similar to that of a cylinder of double the radius
in four times the time. Larger eddies should therefore
be of relatively longer duration than smaller eddies. If
in these same cylinders we have different liquids, the liqaid
in the larger cylinder must have a viscosity four times that
of the liquid in the smaller cylinder, in order that d may
have the same value for the same values of QO, r/c, and t.
For air v=0:14, and for water v=0-011 at ordinary tempe-
rature, the ratio being about 12. Ignoring the compres-
sibility of air, eddies of the assumed kind should die away
much faster in air than in water, the values cf c and Q being
the same. ,

According to (6) and (7), the rates of growth and decay
of ¢/ in the cylinders are independent of ©. We shall
see from the experiments that this is true only when Q is
very small, or when the radius ¢ is a small fraction of the
length of the cylinder.

§ 3. Huperiments.

In the experiments which have been made, a brass
cylinder, bored as accurately as possible and at least
two diameters in length, was rotated about a vertical axis
at constant speed. The cylinder was filled to within 1 cm.
of the top with ordinary tap-water ; and the observations
consisted in timing through a measured angle by stop-
watch, lycopodium particles floating on the surface, and

so deducing the angular velocity ¢ at the radius r/c
selected. The time of the stop-watch observation was also
noted on a watch, in order to get the value of t from the
instant of starting or stopping the cylinder. To time the
lycopodium particles with more accuracy, a horizontal plane
glass plate was mounted just over the cylinder. This was
marked with ink in circles centring on the axis of rotation.
The radii of the circles had the values 7/c=0°3, 0°5, 0°7, and
0:9, and a fifth circle was added at r/c=1:0 to aid in centring
the plate over the cylinder. Straight lines, 45° apart, through
the centre of the circles served to indicate the angles through
which the particles were timed. A large plane mirror,
inclined at 45°, was placed on the glass plate to enable
observations to be made from the side.

To obtain freedom from vibration, the driving-motor and

=

eT ae se

by Change in Angular Velocity. 7

reduction gears were mounted on a separate table. The
cylinders were clamped against a spinning-table which
rotated very freely and easily in ball-bearings. Slight
want of truth in the centring of the upper “end of the
cylinder was corrected by hanging a small weight on
the rim, or by means of adjusting screws which worked
in the rim of the water-bath and pressed against the
cylinder.

Water-baths, providing a 2-inch layer of water around
the sides and base, were fitted to each of the cylinders
to make the temperature changes less. These rotated
with the cylinder. In the case of the larger cylinders
especially, it was found that without a bath it was im-
possible to secure uniform rotation of the water. When
the room temperature was rising in the mornings or early
afternoons, inward convection currents at the surface
carried an excess of angular momentum towards the axis,
and the angular velocity at the centre was sometimes for
several hours 30 per cent. greater than that of the cylinder
itself, the latter being very small. When cooling down,
the effect is reversed and the ‘‘core” rotates slower than
the cylinder. The water-baths made a great improvement,
but with the largest cylinder, especially, the water in the
bath and in the cylinder had to be mixed thoroughly before
each experiment ; and the temperature of the water was
regulated with an electric heater to be about 0°1C. in
advance of the room temperature (when rising). Some
experiments lasted for nearly an hour, and a difference of
0°1 ©. between the temperatures of room and water was
enough to start thermal currents having considerable effects
(cumulative) on the observations.

The cleaned surface of the water was lightly dusted with
lycopodJium, “‘ rafts’ not being allowed to form. When the
temperature was being adjusted with the electric heater,
the water was thoroughly mixed by stirring and bubbling air
through it. The mean temperature was ‘Obtained for each
experiment to 0°01 C.

At the end of each observation the stop-watch reading
(given to 0:1 sec.) was recorded with the angle through
which the lycopodium particle had rotated, and the time of the
end of the observation as indicated by watch was also written
down. All observations in one experiment were made on a
selected circle. The cylinder was timed frequently to verify
the speed, which was constant to within a per cent. or two,
and irregularities were corrected by an adjustable resistance.

8 Mr. A. R. McLeod on Unsteady Motion produced —

The motor was driven from a 200-volt accumulator circuit.
Three sizes of cylinders were used :

Small cylinder......... 6=¢ = 2:40 cme =a ie
Minden a 25 8 ig hacks C= = See.
Large Mite eT ARa ER C=C. = 2 abo

Three angular velocities were selected for each cylinder :
omspeed tran. =O — (1396 — ape
Niddies <7. 08 O=O0= 047 (= 10
doh. ee O=0;,=3°7 10) —ooae

The large and middle cylinders were painted inside.
They were bored with an error of a small fraction of a
millimetre at the upper rim. The small cylinder was true
to 0:05 mm., and its length was four diameters. The others
were slightly more than two diameters long. The bases of
all were plane and smooth on the inside.

Experiments were made when the water began to rotate
from a condition of rest, and also when it was coming to
rest from a condition of uniform rotation, the uniformity
being ascertained before the cylinder was stopped.

In reducing the observations, the watch-reading was
corrected by subtracting half the stop-watch reading to

give the mean time ¢, and the values of vt/c? and 6/Q were
calculated for each observation. ‘The viscosity of the water,
at the mean temperature during the experiment, was taken

from Kaye & Laby’s Tables. The values of $/Q were
plotted on squared paper against the values of vt/c?, euch
observation being represented by a dot. Hence for a
selected circle corresponding to the selected value of *7/c,
for each value of © there corresponds a series of dots
which lie in a narrow band. This band defines a curve
by its median line. Observations were made on the four
circles corresponding to 7/c=0°3, 0:5, 0°7, and 0 9 for each
ef the three cylinders. Thus for each cylinder there were
finally eight sheets of curves—four for ‘‘starting” and
four for ‘‘stopping”’ experiments. Hach of these sheets
contained three curves corresponding to the three selected
velocities, and corresponded to one of the values of r/c.
The theoretical curve from Table I. was also added for
comparison. Several experiments were made for each set
of conditions, to secure a sufficiently dense band of points
for each curve. Apart from a number of preliminary

‘a

by Change in Angular Velocity. 9

experiments, the results given here are based on 215 distinct
experiments.

In taking the stop-watch readings, disturbances of the
circular motion of the lycopodium owing to turbulence
or eddying were avoided as much as possible, and selections
were made of those motions which lay most nearly along
the circle considered. This means that when turbulence
is increased, the angles through which the particles are
timed become less. The effect of the turbulence is merely
to broaden the band of points, but the mean motion, repre-
sented by the median line, is always well-defined.

§ 4. Discussion of Results.

The results obtained are embodied in the accompanying
eurves. These are derived from the observational curves
mentioned in §3. The following reference table explains
the figures :—

Fig.4. o4=2°40cem. ©,=10 r.p.m. Stopping.

ae 2 ORS SG Starting.
e. ra D.=30- ,; Stopping.
Pees — bo cua ey dt, Starting.
8. re a tes Stopping.
g. if OF 10 es Starting.
10. fe Oh Or ors Stopping.
ae ay C=O, Starting.
12. a 0s, Stopping.
Pe) bei bo Cina: ery — it, Starting.
14. ie Oa Stopping.
Ly. a: oe Starting.
16. Ma OF=10 . i, Stopping.
Sir. ms OF 36, Starting.
18. 02236 |, Stopping.

In these figures each curve represents the distribution of
angular velocities ¢/Q over a radius of the cylinder at
the time vt/c? corresponding to the number given alongside
the curve. The times in seconds corresponding to these
numbers may be obtained by use of the following table,
the water being at 17°C. :—

(= 2°40 cm. vt/c?=-01 corresponds to t= 5°30 secs.
(o> 7°48 99 vi/c?="01 9 yy) = d1°8 99
cee Oe aoe) ne (C7 "OL ms t=1484 ,,

10 Mr. A. R. McLeod on Unsteady Motion produced

Comparison with the theoretical curves (figs. 2 & 3)
shows a marked departure in all cases but that of the
slow speed and small cylinder. The agreement with
theory improves as r—>c, but there is still a large departure
at r/c="9 for the large and middle cylinders, except at the
low speed.

& or oF

NW)

eo

‘

—)

ane

Die: —

To)

on

ie)

-f> ee

°
i)
©

Again, it is noticeable that as the radius of the cylinder
increases, the departure from theory becomes more marked.
This might be expected as there is, near the axis, relatively
Jess constraint from the boundaries with the larger cylinders.
Accordingly, if we suppose an eddy of this kind rotating in
a lake of stationary water, and if instead of stopping the
cylinder wall we annihilate it, we expect the eddy to dis-
appear more quickly than if the stationary solid wall had

by Change in Angular Velocity. 1a

been retained, for the constraint will be still further reduced
and greater irregularity is possible.

It is particularly noticeable that as the constant rotation 0.
of the cylinder is increased, the departure from theory becomes
more marked.

—S=—_

r) he 10

Only three figures were obtained for the small cylinder,
because the agreement with the theoretical curves in the
three cases omitted will be practically exact.

The curves for the ‘‘ stopping” experiments skow a much
greater departure from theory than the curves for the
“starting” experiments. ‘This is due to the break-up of
the regular motion, owing to instability at the fixed outer
wall. Except with the high speed and the large cylinder,

12) Mr. A. R. McLeod on Unsteady Motion produced

the motion on starting, on the other hand, appears to the eye
to be without appreciable irregularity, and it is very striking
to see the sharp dividing-line between quiescent liquid in the
centre and rotating liquid on the outside. This dividing-line,
represented by the steep part of the curve in the figures,
slowly moves towards the centre but becomes indistinct some
distance from it. It is best seen at the higher speeds when

the velocity-gradient is greater. Its rate of travel depends
on the value of (2.

In the case of the large cylinder starting at the high speed,
large secondary eddies 3:5 cm. across were often observed
just inside the cylinder wall a few seconds after starting the
motor. These soon died out when the velocity-gradient
became less, and thereafter the motion travelled in towards

by Change in Angular Velocity. 13

the centre regularly. The effect of these eddies was observed
to cause a shifting of the curve for ¢/Q in the direction of

a greater # for ie same value of vt/c?. In the same case,
secondary eddies were observed about the “4 circle when the
motion had reached the centre. Although mean velocities
were recorded (median ling of the band of points), the effect
is shown in fig. 17 by the wavy appearance of tl.e two upper
curves.

When the cylinder is stopped, the water continues to
rotate until the irregular motions, generated near the cylinder
wall, have had time to extend inwards. Small eddies then
travel about, and the central axis of rotation wanders con-
siderably and often seems to disappear temporarily amid
cross-currents. The motion is very irregular except at the
low speed, and even in this case some irregularity always
remains. The lycopodium particles do not follow the circles
for very long, and are usually moving at an angle to them.

With the large cylinder at the high speed, the velocity
immediately after stopping the cylinder seemed to give
stability and to aid in preserving the circular character
of the motion ; but when the kinetic energy had somewhat
diminished, eddying became more noticeable. .

On the curves mentioned in $3 in which ¢/Q is plotted
against tle’, the bands of points are much narrower in the
“ starting’ experiments than in the others, and determine
the position of the median line easily to -001 in the value of
vt/c? in most cases. For the “stopping” curves, the limit
of error may be two or three times this occasionally. One
noticeable effect is that the band is narrow when the velocity-
gradient has a considerable value, z. e. when the curves in the
figures slope steeply. In these cases considerable momentum
is being transferred through the water, and there will be
considerable shearing stress and vorticity, and the stability
might therefore also be considerable. As soon as the
velocity-gradient becomes small, the band of points broadens.
For example, in the “starting” curves .the bands are some-
times very narrow until the value of $/Q has risen to 0°9,
when they broaden ont. Conditions seem to favour irre-
gularity at the centre (axis) of the cylinder where the
velocity-gradient vanishes. On the axis the stability is a
minimum. In the ‘stopping’ curves the bands are nar-
rower the greater the angular velocity Q, 7. e¢. the greater
the vorticity of the water, especially near the cylinder wall
where the instability originates.

The observational curves show that viscosity alone is not

14. Unsteady Motion in a Rotating Cylinder of Water.

sufficient to account for the effects, except for small values
of cand Q, ¢.e. for long, narrow cylinders and: slow speeds.
A little dye introduced into the rotating water shows no
slons of any minute eddying or micro-turbulence ; and so
we must look for currents in the water as the cause of the
discrepancy, which is obviously the case in the “ stopping ”
experiments. The formation of large eddies in large bodies
of fluid seems to be due chiefly to the interaction of two
local currents, or to low pressure caused by an obstacle or a
sink, and not to the slower processes of viscosity.

If we attribute the deviation from theory to an ignored
increase in the kinetic viscosity v, we find that when the
large cylinder is stopped at the high speed, the increase
would have to be represented by a factor exceeding 10 in
value nearly everywhere, while the value would lie between
50 and 100 on r/c=0°3 shortly after stopping. With the
middle cylinder, stopped at the high speed, the factor has
about half these values; and with the small cylinder, stopped
at the same speed, the factor ranges from 1:3 to 3:0. In
the starting experiments the factors are nearly unity, but
they are meaningless here as the motion is not turbulent.

Some earlier experiments illustrate the instability of the
stopping experiments. In these an inner cylinder was
rotated coaxially with a fixed outer one. As is well known,
it was found that at no speed of rotation of the inner cylinder
was it possible to set the water moving in circular paths, owing
to the eddies which were continually thrown off. The slower
the speed of rotation the more conspicuous were the eddies,
especially on the borders of the outer, more slowly-moving
water. Measurements of the angular velocity showed a large
departure from theory, the inner parts rotating more slowly
and the outer parts more rapidly than the theory indicates.
The effect of the travelling eddies is thus to make the angular
velocity more like that of a rigid body. When the speed was
very great (2500 r.p.m.) the kinetic energy seemed to give
stability to the water. A whirlpool formed next the inner
cylinder, and a large oscillation was presently set up in the
form of a wave with its crest along a radius of the outer
cylinder and its trough on the other half of the same diameter.

Some thick, very viscous oil residues, when rotated in a
cylindrical tin about 15 cm. in diameter, acquired the full
velocity on starting (36-r.p.m.) in something less than
4 seconds, and came to rest in the same time when the
cylinder was stopped. Onlya slight displacement of the oil
occurred, the surface being momentarily roughened with fine
lines like cracks.

Rupa? J

II. The Elements of Geometry.
By Norman CaMpBgé.L, Sc.D.”

Summary.

T is maintained that the geometry of Euclid is best
interpreted as an attempt to deduce as many important
propositions as possible from the assumption that length,
angle, area (and perhaps volume) are magnitudes uni-
versally measurable by the methods that are actually
employed in experimental physics. All his chief pro-
positions (in so far as they are true) can be deduced from
that assumption without any other.

This view is supported, not by a detailed analysis of
the Elements, but by a very summary sketch of the laws
that must be true if the assumption is to be acceptable.
In a sequel it is hoped to discuss similarly the foundations
of another branch of experimental geometry with which
BHuclid is not directly concerned—namely the geometry of
position, which involves the concept of ‘‘ space.”

1. There was formerly much discussion whether geometry
was an experimental or a mathematical science. It is now
generally agreed that there are two closely connected
sciences, one mathematical and one experimental. The
former, which has been defined as the study of multi-
dimensional series, consists of a logical development of
ideas which have no necessary dependence on the experience
of the senses. It does not consist of laws and cannot be
proved or disproved by experiment; it can enter into
relation with experimental science only through theories
and by suggesting hypotheses which, interpreted suitably,
predict laws. The formulation of such theories, in which
Minkowski was the pioneer, is one of the most striking
features of modern mathematical physics. The experi-
mental science, on the other hand, is meaningless apart
from experience, and its propositions are true or false
according as they agree or disagree with experiment.
They are the very fundamental laws which involve only
the geometrical magnitudes such as length, angle, or area.
It may be noted in passing that the laws predicted by
geometrical theories are not in general geometrical laws,
but involve electrical, optical, or dynamical concepts.

* Communicated by the Author.

16 Dr. Norman Campbell on the

The mathematicians who have recently taken over from
the philosophers the task of teaching experimenters their
business have decided that only the mathematical science
is properly termed geometry. In support of their claim
they appeal to the authority of the Greeks, and thereby
imply that Greek geometry is mathematical and not experi-
mental. This implication raises questions of scientific
interpretation and not of mere convenience in nomenclature.
For the matter cannot be dezided by inquiring what Euclid
(for example) thought he was writing about: it is admitted
that, as an exponent of mathematical geometry, he was
guilty of errors ; and, if he was capable of error, he may
have been wrong as to the nature of his assumptions and of
his arguments. If we are justified today in confining the
term to one study rather than another, because that term
was used by Euclid, it can only be on the ground that
Huclid’s propositions and his methods of proving them are
closely similar to those employed today in that study.

If this test is applied, geometry is an experimental
science. For whereas the Hlements is utterly different
from anything modern mathematical geometers produce,
it is, judged by modern standards, quite a creditable
attempt at an exposition of experimental geometry. It
can be regarded broadly as an attempt to deduce as many
important laws as possible from the single assumption
that length, area, angle, and (less definitely) volume are
magnitudes, universally measurable by the methods which
are actually employed in experimental physics, or to which
the methods that are actually employed would be referred
if doubt arose concerning their validity. Nothing is assumed
but that every straight line has a length, every pair of strarght
lines an angle, and every plane surface an area. The
definitions, axioms, and postulates should then be state-
ments of the laws by virtue of which measurement is
possible. It is admitted that the attempt is not wholly
successful ; but its faults, or many of them, are readily
explicable: the author has not to be represented (as he
must be if he is an exponent of the mathematical science)
as constantly straining at gnats and swallowing camels.

Such a view can be established only by a detailed and
tedious criticism which, in so far as it concerns Euclid’s
intelligence, is not of scientific interest. In place of it
will be offered a very summary sketch of the fundamental
notions and laws of experimental geometry and sufticient
comparison of them with Euclid’s assumptions to suggest
that on them might be founded a deduction, by methods

on ee

Elements of Geometry. 17

very similar to those that he employs, of the propositions
which he actually states. References are throughout to
Todhunter’s edition.

(2) But two preliminary questions must be asked. First,
can an experimental science be deductive at all? Certainly
it can. A deduction from a law is an application of that
law in particular circumstances which were not examined
when it was formulated. If, after examining the sides of
squares and of triangles, I assert the general law that
all straight lines have measurable lengths, and then, without
further ‘experiment, assert that the diagonals of squares,
which are also straight lines, are also measurable, I am
making a deduction. It may be true that there is some-
thing precarious about the results of such deduction—
that question is not raised here,—but the deduction itself
is quite unexceptionable; the falsity of the conclusion is
definitely inconsistent with the truth of the premises.
If deubt is raised concerning the conclusion, the ultimate
means of resolving it is by experiment; but experimental
science, in the hands of its greatest exponents, consists in
asserting such general laws that doubt does not arise
concerning the results of deduction based on them.

The second question is whether there are truly laws
which make measurement possible. The question is dis-
cussed at length in my ‘ Physics,’ Part II., the results and
nomenclature of which will be used freely in what follows.
But there is one matter which may receive special mention
here, beeanse it is concerned with ‘“ incommensurables,”
which are often (but falsely) believed to be of especial
importance in geometry. Measurement is possible when,
by means of definitions of equality and addition, a standard
series of the property in question can be established, starting
from some arbitrary unit, such that any system having the
property is equal in respect of it to some one member of
the standard series. Now (it might be argued) such mea-
surement is not possible for length, because the diagonal of
a square cannot be equal to any member of a standard series
based on the side as unit; indeed that result is actually
proved by Huclid. Consequently it is patently absurd to
pretend that Euclid’s propositions can be derived from an
assumption, namely that measurement is possible, which is
inconsistent with its conclusions.

One method of escape from this difficulty may be
mentioned, although it will not be adopted. A slight

Phil, Beg S. 6. Vol. 44. No. 259, July 1922. ‘6:

18 Dr. Norman Campbell on the

amendment in the thesis might be made, and it might
be said that Euclid’s assumption is that the laws are true
which would make measurement possible if there were no
incommensurable lengths—for these laws, though necessary
to measurement, may not be sufficient. But the diticulty
vanishes entirely, if it is remembered what is meant by
“equality” in experimental measurement. When it is
said that A is equal to B, it is meant that there is no
possible means of deciding which of the two is the greater.
If then I say that the diagonal of a square is 2 times
the side, I mean that, if I measure the diagonal in terms

of the side as unit, there is no means of deciding whether |

the value obtained, when multiplied by itself according to
the multiplication table, will be greater or less than 2.
That statement is not in the least inconsistent with my
assigning to particular diagonals values of which the square
is not 2; it is only inconsistent if a law can be found
by which I can tell in particular cases whether the square
will be greater or less than 2. My assertion is that. there
is no such law ; and that assertion is true. In its appli-
cation to all magnitudes except number, equality must be
interpreted in this, slightly statistical, sense.

3. There is then no preliminary objection to the view
that Euclid’s propositions are deductions from the laws in
virtue of which the geometrical magnitudes are measurable.
We now proceed to ask what those laws are.

Geometrical conceptions are derived ultimately from
our immediate sensations of muscular movement, just as
dynamical conceptions are derived from our sensations of
muscular exertion and thermal conception from our senge
of hot and cold. We have an instinctive and indescribable
appreciation of differences in dzrection of various movements ;
we appreciate that one direction may be between two others ;
and if other sensations (e. g. those of hot and cold or rough
and smooth) vary with movement along a certain direction,
we appreciate that of the varying sensations some are between
others. The notions of direction and of the two kinds of
betweenness are the foundations of geometry. It is a vitally
important fact that there is an intimate relation connecting
betweenness determined by one kind of muscular motion (e. g.
that of the hand) and that determined by another (e.g. that
of the eye). ‘The relation is much too complex for any
account of it to be attempted here ; but it is only because it

exists that “space” explored visually or by our different limbs
is always the same.

ee

Elements of Geometry. | 19

The fundamental notions give rise to those of surfaces and
lines. Surfaces are connected with the fact that a sensation
may be unaltered by movement in any of a certain group
of directions (which are said to be in a surface and to cha-
racterize it), while it may be altered by any movement in
any direction not in this group (directions away from the
surface). OF lines there are two kinds, which will be termed
respectively ‘edges”’ and ‘ ‘ scratches.” Hdges arise from
the fact that the group of directions characteristic of a
surface may change suddenly at some part of it. It is
a matter of convenience whether the parts characterized by
different directions are spoken of as different surfaces or
as parts of the same surface: we shall adopt the second
alternative. Scratches arise from the fact that, while the
directions characterizing a surface are unaltered, the sen-
sation the occurrence of which distinguishes “in the
surface’ from ‘‘out of the surface” may change suddenly.
Some, but not all lines, are such that the whole of them
hes along a single direction. Points are of little importance
in the earlier stages of geometry ; they arise from the fact
that two lines may have a part in common. Two points,
both on the same line, are termed the ends of the part of
the line between those points.

The recognition of surfaces and lines is the first step
towards geometry. Euclid attempts to give an account
of them in Defs. 1, 2, 3, 5 of Book JI., which are the least
successful part of his treatise. The account given of them
here is no better than Euclid’s for the purpose of conveying
a notion of them to one who does not possess it already ;
but since there are no such persons, the objection is not
serious. But our account is better in drawing attention to the
notions that are fundamentalin geometry and in not assuming
familiarity with conceptions, such as length, which are
necessarily subsequent.

4. Some surfaces, but not all, when subjected to muscular
force undergo only such changes as can be compensated by
a suitable movement of the whole body; if such a movement
is made, the group of directions characterizing the surfaces
is restored. In other words, such surfaces can move without
alteration of form ; they provide the original and erude con-
ception of a rigid body. By means of the motions of rigid
bodies, it is sometimes possible to bring parts of two pre-
viously distinct surfaces into contiguity, so that there is
nothing between those parts. In particular, edges, or parts of
edges, can often be brought into such contiguity. Scratches

C2

20 Dr. Nerman Campbell on the

can be brought into contiguity with edges, and, in a sense,
into contiguity with other scratches ; but the criterion of
contiguity in the last case is much less direct and requires
methods involving something other than the simple per-
ception of nothing between.

The recognition of the possibility of contiguity is the
second step towards geometry and leads immediately to
the third, which consists in the establishment of a definite
criterion for a straight line. A crude criterion is provided
by direct perception : a young child knows the difference
between a straight and a bent line by simply looking at
them; the recognition seems to depend on ihe fact that
a straight line is all in one direction and is symmetrical
with regard to the unsymmetrical directions of left and
right or back and front. The crude criterion is stated as
well as it can be in Huclid’s Def. 4. But contiguity
provides a much more stringent criterion, which in the
first instance is applicable only to edges and not to
scratches. Two edges are straight if, when two portions
of one are brought into contiguity with two portions of
the other, all the portions between these two portions are
also in contiguity, however the contiguity of the first pairs
of portions is. effected. It appears as an experimental
fact, that if A,B and ©, D are two pairs of straight edges
according to this criterion, C is also straight if tested
against A; accordingly an edge can be called straight
independently of the other member of the pair on which
the test is carried out. A seratch is straight if it can be
brought into complete contiguity with a straight edge,
These facts are stated in Axiom 10.

Other definitions of a straight line are sometimes offered :
e.g., (1) an axis of rotation, (2) the shortest distance
between two points, (3) the path of a ray of light. (1) is
almost equivalent to that stated here; (2) will be noticed
presently ; (2) is not accurately true (7. e., if it is adopted,
the familiar propositions about straight lines are not true),
but it is important as an approximation for comparatively
rough measurements.

A plane surface (or, according to our usage, pari of
a surface) is then defined as in Def. 7. It can also be
defined by the complete contiguity of three pairs of sur-
faces ; but the contiguity of surfaces is not easy to describe
accurately. Such a definition is, however, actually used in
making optical flats and surface plates ; if it were adopted,
it would still be necessary to introduce the fact that it
aorees with our definition, in order to measure angle. The

79 F 2

Elements of Geometry. 21

conception of the contiguity of surfaces is not actually
required, except perhaps for the measurement of volume.
(Cf. § 11.)

5. The third step places us in a position to introduce
measurement and the three fundamental magnitudes,
length, angle, and area. For tundamental measurement
we need definitions of equality and addition, such that
the law of equality and the two laws of addition are
true. The choice of unit may be left out of account ; for,
with geometric magnitudes, the laws are true whatever
unit is selected. The law of equality is Axiom 1; the first
law of addition is Axiom 9. Axioms 2-7 are together very
nearly equivalent to the second law of addition (which may
be stated roughly in the form that the magnitude of a sum
_ depends only on the magnitudes of the parts). Axiom 8 is
an attempt to compress the definitions of equality for all
three magnitudes into a single sentence; it is better to
separate them. Huclid fails to give any definition of
addition : he does not tell us how the “ whole” is to be
related to the “ parts” in order that it should be greater.

6. We will now take the magnitudes in turn. For the
length of a straight line the necessary definitions are :—
(1) Two straight lines are equal in length if they can be
placed so that when one end of the first is contiguous with
one end of the second, the other ends are also contiguous.
(2) The length of the straight line AB is equal to the sum
of the lengths of the straight lines CD, EF, if they can be
placed so that C is contiguous with A, F with B, D with E
and with some part of AB between A and B.

These definitions, like all similar definitions of mag-
nitudes, are satisfactory and are subject to the necessary
laws of equality and addition only if certain conditions
are fulfilled. The conditions are described by saying that
the surfaces in which the straight lines lie must be those
of rigid bodies. This is a definition of a rigid body: a
rigid body is something which (like a perfect balance) is
determined by the satisfaction of the conditions for mea-
surement*. Rigid bodies according to this test include
many of those which satisfy the crude test of § 4, though
they include others (e.g., surveyors’ tapes used as surveyors
use them) which do not satisfy that test. In virtue of the
fact that rigid bodies are necessary to measurement, the

* Cf. H. Dingler, Phys. Zeit, xxi. p. 487 (1920).

22 Dr. Norman Campbell on the

branch of geometry with which we (and, according to our
view, Huclid) are concerned may be fitly deseribed as the
study of the surfaces of rigid bodies. It is thus dis-
tinguished from a wholly different branch of geometry,
with which we are not here concerned, ihat is not confined
to rigid bodies ; this is the geometry of position.

It is important to notice that not all pairs of straight
lines can be brought into contiguity, and that the law of
equality cannot therefore be tested universally. It might
have turned out that there was some material difference
between those which can and those which cannot be brought
into contiguity with a given line; and that if we assumed
that the law of equality is universally true, we should be led
to inconsistencies. It is an experimental fact that no such
inconsistencies do arise when we extend our definition of
equality so that lengths are equal when they are equal to
to the same length, although they cannot be brought into
contiguity with each other. This is, of course, one of the
most important laws that make measurement possible. A
similar remark applies to all the geometric magnitudes and
need not be repeated.

7. The length of lines that are not straight can be
measured approximately as fundamental magnitudes by
means of flexible but inextensible strings. But the laws
of such measurement are not strictly true, because (as we
say now) no string is infinitely thin and the surface never
coincides with the neutral axis. Another possible way,
perhaps more accurate but of limited application, would
be to roll curved edges on some standard edge, which ~
need not be straight. But in truth there is no perfectly
satisfactory way of measuring fundamentally the length of
curved lines. All the measurements which we make on them
are derived from measurement of straight lines ; they involve
numerical laws between fundamentally measured magnitudes.
One of these laws is that the perimeters of the circumscribed
and inscribed regular polygons tend to a common limit as
the number of sides is increased. That law is therefore a
law of measurement if curved lines are to be measured.

The question whether curved lines can be measured
fundamentally is important, because, if they could be, it
would be possible to define a straight line as the shortest
distance between two points. (The definition would have
to be put in some other form, since distance, a conception
belonging to the geometry of position, implies the mea-
surement of length.) But since they cannot be, that

Elements of Geometry. 23

definition must be rejected ; it must be regarded merely as
a generalized form of Prop. I. 20.

8. Angle is the measure of the crude conception of
direction. The following are the definitions of equality and
addition for the angle “between two intersecting straight
lines :—The angle between two straight lines A, B is equal
to that between C, D if it is possible to bring "A into con-
tinguity with C and B with D. The angle between A, B is
the sum of the angles between CO, D and | Dp elistr oe ‘when
A is brought into contiguity with C and D with E, D lying
between C and F and in the same plane with them, F can
be brought into contiguity with B. These definitions are
satisfactory only if the straight lines are in rigid bodies;
or, in other words, there are surfaces which satisfy fhe
conditions for the measurement of length and also those
for the measurement of angle.

But even if the surfaces are those of rigid bodies, the
definitions are not wholly satisfactory and the laws of
measurement not entirely true. We must distinguish
angles according as the two straight lines which they
relate are or are not prolonged on both sides of the
common point: the latter class may be termed “ corners,”
the former “‘ crossings.” Angles between edges are always
corners; those between scratches may be either corners or
crossings. If we try to include both corners and crossings
in the same class as a single magnitude, the law of equality
is not true; for two corners which are both, according
to the definition, equal to a crossing may not be equal
to each other; as we say now, one angle may be the
supplement of the other. But if we treat corners and
crossings as separate magnitudes this difficulty disappears ;
the law of equality is true for either taken apart from the
other. Actually we take corners only as magnitudes ;
crossings we measure by the corners with which they can
be made contiguous. Hach crossing then has four angles
(2. e. corners) associated with it. It is an important experi-
mental fact that the “opposite” angles are equal; it is
best taken as a primary law, instead of being proved from
other axioms asin Prop.I.15. It is a law of measurement,
because if it were not known, we should need four and not
two angles to measure a crossing ; it is thus inherent in our
system of measurement.

But though the law of equality is now true, the first law of
addition is ‘false ; ; it is false for both corners and crossings.
The whole which is the sum of the parts may be equal to

24 Dr. Norman Campbell on the

one of the parts: e.g., if both of two parts, being corners,
are what we now call 120°. Some kind of spiral space can
be imagined in which the law would be true; but actually
it is very important that it is false. For, apparently in-
separable from its falsity is the fact that the angle between
two portions of the same straight line can be measured and
given a finite value in terms of a unit which is the angle
between two intersecting lines. This fact is described by
the assertion that there are right angles and that a per-
pendicular can be drawn to any straight line from any
point in it, a right angle being defined as in Def. 1. 10.
(Axiom 1. 11 follows from this definition, regarded as an
existence theorem, and our axiom Prop. I. 15.) Since the
existence of right angles is vital to geometry, we cannot
avoid the falsity of the first law of equality by some
alteration of the definition. We can only recognize that
the law is true in some conditions, and be careful to apply
it in deduction only when it is true. It is true when all the
lines making the added angles lie on the same side of (or
contiguous with) a single straight line passing through
their commen point; this condition can be expressed,
though with some complexity, in terms of the fundamental
notion of between. Thus, in proving Prop. I. 16 we need
to know that OF and CD both lie on the same side of AG.
. This law, and perhaps others of the same nature, are laws of
measurement, defining the conditions in which angle can be
measured uniquely. They require explicit mention.

The ambiguity which the falsity of the first law of
addition introduces into numerical measurement is removed
by certain conventions. These need not be considered here
for we are not assigning numerical values.

If the length of curved lines were measurable funda-
mentally, angle might be measured as a pure derived
magnitude, e.g. by the ratio of the are to the radius of
a circle in virtue of the numerical law, established experi-
mentally, that the are is proportional to the radius. But
since curved lines cannot be so measured, we must take
angle to be fundamental. We cannot use right-angled
triangles with straight sides to measure angle as derived,
because we need fundamental measurement to determine
what angles are right. Of course we might define for
this purpose a right angle as an angle between some two
lines arbitrarily chosen as standard ; but such measurement
would be intolerably artificial and nothing whatever could
be deduced from such a definition. .

pie ee e

Elements of Geometry. 25

9. Euclid’s definition of parallel lines must be rejected
entirely, for, since all plane surfaces are limited, the
criterion suggested is inapplicable. Since the crude de-
finition of parallelism is similarity of direction, we may
try to define parallel lines as those which being in the
same plane make the same angle with any third line.
We thereby imply the axiom of parallels in the form
(Prop. I. 29) that such lines which make the same angle
with one straight line make the same angle with any other ;
we imply also that the angles which are to be equal are the
“exterior” and “interior” opposites or the ‘alternate ”
angles, since if the interior angles are compared the
proposition is not true. But the definition is not very
satisfactory ; for, when the lines are edges, there is not
always an exterior or an alternate angle. It is better
to adopt the substance of Axiom [. 12 as a definition,
and to say that lines in one plane are parallel when the
sum of the interior angles is equal to two right angles.
This much abused axiom seems to me a very ingenious
way out of a real difficulty. We then assert the axiom of
parallels in the form (implied by I. 32) that if any two
straight lines in a plane are cut by any third line, the
sum of the interior angles is the same for all third lines.
The merit of this axiom is that it indicates clearly that the
‘“‘axiom of parallels” is really something concerning all
straight lines in a plane and not only parallel lines, and
that parallel lines are merely a particular case of other
pairs of lines. The propositicns that parallel lines never
do intersect and that the angle between them is zero follow
immediately.

The axiom of parallels is a law of measurement because
it is involved in the measurement of the angle between lines
which do not intersect. Its use for this purpose requires
that at some point of a straight line it should always be
possible to place a straight line parallel to a given straight
line. This proposition is not true for concave surfaces, but
the complexities arising from this failure and the means of
avoiding them may be left for the present ; they are dealt
with more naturally in connexion with “space.” If the
axiom were not used, we could not by our present methods
measure the angle between non-intersecting straight lines :
first, because the definition of equality given above, though
sufficient for such lines, is not necessary : second, because
the definition of addition is wholly unsatisfactory.

There has been so much discussion of the necessity of the

26 Dr. Norman Campbell on the

axiom of parallels that the matter requires rather more con-
sideration. Two questions are involved. First, would it
be possible to measure the angle between non-intersecting
lines without assuming some proposition logically deducible
from the axiom? It would be if, and. only if, some
property, common to all lines between which the angle
is the same, can be found which is determinable by direct
experiment not involving parallel lines. There may be
such a property, but I have not been able to think of it.
Second, if the axiom were not actually true—but we may
stop there. In a pure experimental science, there is no
sense in asking what would happen if the world were other
than it actually is. Theory is necessary to give such a
question a meaning, by suggesting what might remain
unaltered during the change. For our present purpose
the axiom is as necessary as any other of those we are
considering.

10. Area is distinguished from all other fundamental
magnitudes because the definitions of equality and addition
are inseparable. They may be expressed thus. The areas
of two bounded plane surfaces are equal if (but not only if)
their boundaries can be brought into complete contiguity

with each other or with the same third boundary. (A

bounded surface is a part of a surface which includes all
portions which can be traversed without crossing the
boundary line.) The area of A is the sum of the areas
of B and OC, if when parts of the boundaries of B and C
are brought into contiguity with each other, the remaining
parts of the boundary can be brought into contiguity with
the boundary of A. In virtue of the fact that parts of the
boundaries of two surfaces can be brought into contiguity

in many different ways, there may be many different

bounded surfaces, of which the boundaries cannot be made
contiguous, which are the sum of the same bounded sur-
faces. If the measurement of area is to be satisfactory,
these surfaces must also be deemed to have equal area, and
the definition of equality must be extended correspondingly.
With this extension the laws of equality and addition are
true, and the measurement is satisfactory.

In order that all bounded plane surfaces should have
areas, some rule must be found for choosing the shape of
the members of the standard series and for grouping them
in such a way that some sum of them is equal to any area.
We use for this purpose rules based on the axiom of
parallels, and that axiom is therefore again a law of the

.
)

Elements of Geometry. 2T

measurement of area. ‘The rule might possibly be dis-
pensed with, if we were prepared to spend unlimited time
in selecting by trial and error shapes for the members of
the standard series which fulfil the necessary conditions ;
but actually we could never measure area except by making
use of similar figures, the production and properties of
which depend wholly on the axiom of parallels. Further,
it is the use of that axiom which enables us nowadays to
calculate area from the linear dimensions of a surface
without resorting at all to fundamental measurement.
But of course all the numerical laws on which that cal-
culation depends have to be established by means of
fundamental measurement. It is only by defining area
as we have done, and assuming the axiom of parallels,
that we can prove by deduction that the area of a rect-
angle is proportional to the product of its sides, or equal if
the units are suitably chosen.

The areas of surfaces that are not plane cannot be mea-
sured fundamentally, even to the extent that the length of
curved lines can be. For there are no inextensible surfaces
which can be brought into contiguity with surfaces of any
curvature. Weasurement of curved area is always derived
and estimated by the limit of the circumscribed polyhedra
as the number of their sides is increased. But the whole
matter is obscure, because it is much more difficult to
establish experimentally that there is a limit or to say what
the limit is; for there is here no inscribed polyhedron
tending to the same limit. There is singularly little experi-
mental evidence for the assertion that the area of a sphere
is 4777, and there is great difficulty in saying exactly what
we mean by such an assertion ; curved area is almost always
a hypothetical idea and not an experimental magnitude
at all.

11. Volume is a property of complete surfaces. Since
complete surfaces can never be brought into complete con-
tiguity, volume cannot be measured fundamentally by any
process at all similar to those applicable to the magnitudes
we have considered so far. Volume is measured (a) as a
fundamental magnitude by means of incompressible fluids,

or (2) as a derived magnitude by means of the lengths and
Silos characteristic of the surface. The second method
depends upon numerical Jaws established by means of the
first. In certain cases these laws can be related closely
to other geometric laws by means of the following propo-
sitions :—(1) Two complete surfaces with equal dimensions,

28 Dr. Norman Campbell on the

2. e. with equal lengths and equal angles between them, have
equal volumes. (2) If two complete surfaces have each
one ae plane, and the boundary of the plane part of one
can be brought into complete contiguity with the plane part
of ie other, then the complete surface which has dimensions
equal to that of the complete surface so formed has a volume
equal to the sum of the volumes of the original surfaces.
These propositions could be used as definitions of equality
and addition in a system of measurement, which would be
independent of the measurement of length and angle (and
therefore not derived), because it sales only equality, and
not addition, of length and angle. But it is of limited scope
and, in particular, “would not permit the measurement of
the volumes o£ curved surfaces. Since we do undoubtedly
attribute a meaning to the volume of such surfaces, in a
way that we do not to their area, measurement by incom-
pressible fluids, which is not geometric, cannot, be wholly
avoided. But the propositions, which are those on which —
BHuclid bases his treatment of volume, are actually used in
modern practice, and are therefore regarded per missibly as
laws of measurement.

12. In deducing Euclid’s propositions from the laws
of measurement of these magnitudes, subsidiary laws are
required, corresponding roughly to his postulates, expressed
and implied. TWirst, we need ‘existence theorems” corre-
sponding to each of the definitions; for example, the
definition of a plane surface justifies the conclusion that
a straight edge ean be placed contiguously to any two
portions of such a part of a surface. Second, we need the
assumption that we can make an object having a magnitude
equal to that of any object presented to our notice. All
these propositions are laws of measurement : the first group,
because ail definitions in experimental science are nothing
but existence theorems; the second, because it is implied in
the fact that we can make a standard series by which we can
measure any magnitude.

Euclid’s three expressed postulates are all untrue. I
cannot ‘“ draw a straight line” from this room to the next
when the door is closed. Moreover his constructional
propositions, closely connected with the postulates, are
unsatisfactory because they are all directed to the drawing
of scratches, rather than to the making of edges. The
hypothetical experiments by means of which the deductions
are effected are carried out much more easily with edges

Elements of Geometry. 29

than with scratches ; and if any of the propositions were
donbted and put to the test of experiment, it would certainly
be by means of edges ; the extension to scratches would be
by means of the contiguity of edges with them. Huclid’s
methods here undoubtedly indicate that he is leaving,
perhaps consciously, the realities of experimental science
for the pure ideas of mathematics. But he has made
so little progress towards the new peak that, if he is to be
restored to safety, it is far easier to drag him back to that
which he has never left completely than to guide him
through the bog in which the two sciences are confused

to the very distant goal.

13. Only a few disconnected remarks will be offered
here on the process of deducing the Huclidean propositions
from the fundamental laws that have been sketched. Of
course, we should employ the “application” (or contiguity)
method of Prop. I. 4 wherever possible, instead of trying
to avoid it; for it is based directly on the fundamental
notions. Again, we should not commit Huclid’s error of
supposing that strictly similar triangles can be brought
into contiguity; we should apply the mirror image first
to one triangle and then to the other. There would be
no need to introduce area to prove Prop. 1.47. A Greek
writer was forced to do so, because, not being familiar with
the multiplication table, he could describe in no other way
the relation between a number and its product by ‘itself.
We should proceed from Prop. I. 34 to Book VI. and prove
Prop. I. 47 by drawing the perpendicular from the right
angle to the hypoteneuse and using the relations of similar
triangles, treated by algebra. For nowadays, since we
admit no incommensurable magnitudes, we can dispense
altogether with Huclid’s very beautiful and ingenious
subtleties about ratios. A ratio in experimental science
is nothing but a value taken from the multiplication table,
which is established by the measurement of number, i. e. by
counting. The laws of the measurement of number are
involved in those of the measurement of every ‘‘continuous”
magnitude.

April 22, 1922,

Ill. On the Rotation of Slightly Elastic Bodies. By
DorotHy WrincH, D.Se., Fellow of (Girton College,
Cambridge, and Member of Research Staff, University
College, London™.

HE change in dimensions of a slightly elastic body due
to rotation is a question of some practical importance,
and does not appear to have received any systematic treat-
ment. In the theory of elasticity, the displacements of a
point of the body are of course discussed and the displace-
ments of the points of the boundary determine the increase
of dimensions. But the problems of elasticity which are of
interest mainly from the point of view of increase of dimen-
sions, rather than of the distribution of stress in the material,
can rarely be solved by the current methods or appear only
as special cases of a general mode of analysis. ven the
simple problem ofa circular cylinder of finite length, rotating
about its axis, has not yet admitted an exact solution, though
an approximate solution, which becomes valid when the
cylinder is of infinite length, has been given by Chree.
When the cylinder has a finite length, the surface con-
dition of zero traction over the curved surface is violated,
and instead of this traction becoming zero at all points on
the surface, only its average value over the surface is zero.
The results for the case of an infinite cylindrical annulas
do not appear to be on record, and they are interesting on
account of their marked divergence from those which belong
to the complete disk.

In the present paper we group together some of the
simpler and more interesting solutions of problems of
this type, including those of the infinite circular cylinder
and the infinite cylindrical annulus. These specific pro-
blems are solved to any degree of approximation and for a
non-uniform distribution of density. The analysis is simpler
than is usual, for it does not seem necessary to treat these
comparatively simple problems as special cases of general
theory, and it is desirable, at least in the interests of the
engineer or physicist, that a fundamentally simpler treat-
ment should be placed on record. It also seems possible
that such solutions may be of interest with regard to
scientific instruments of great precision, in which some
portion of the apparatus is in rotation, or, on the larger
scale, in problems of practical engineering. Although no

* Communicated by the Author.

—

On the Rotation of Slightly Elastic Bodies. ol

novelty attaches to some of the earlier results, it seems
desirable to include them.

The simplest problem of this nature is, of course, that of
the thin circular hoop rotating about its centre. When such
a hoop of radius a and density p is spun round its centre
with constant angular velocity w the value of T, the tension
per unit length in the hoop, is well known. Tor an element
ds of its length has an acceleration aw? inwards, and the
resultant of the tensions at its endsis Tds/a per unit area
inwards. Hence the equation of motion is

Tds/a=paow’,
me ra et
giving T= pa’.

If, however, the hoop is slightly elastic, and \ the value of
Young’s modulus for the material of which the hoop is
made, and v the radius of the hoop when in motion, the
equation of motion of the stretched element ds becomes

T/r=re? . pa/r.
Applying Hooke’s law to the stretched element, we have,

T=A(r—a)/a.

Hence eliminating T,
praw?=r(7r—a)/a.

In practice X is always large, and if we may neglect 1/X and
higher powers of 1/X the appropriate value of 7/a, which
differs from unity by a quantity of order 1/d, is 1+ pa?w?/X.
The value of the tension to the same order is pa’w’.

The effect of a rotation is therefore to increase the radius a
of the hoop to a(1+ ), where ~=pa?w?/A, a number depending
on the density, the elasticity, and the radius of the hoop, and
on the rate at which it is rotating.

As regards the practical order or magnitude of pa?w?/r
the extension per unit length, we may take a steel wire for
which X is about 2°12x 10” dynes per square centimetre,
and p isabout 7°5. In order that Hooke’s law may hold, the
extension per unit length must not exceed 107%, roughly
speaking. If the velocity of a point on the rim is in the
neighbourhood of 1:°9x10* cm. per second—which is ap-
proximately the case in a twenty-foot flywheel making two
hundred and fifty revolutions a minute—we find that the
extension per unit length is about 7°9x*107*, which comes
within the limits of applicability of Hooke’s law, and that

32 Dr. Dorothy Wrinch on the

the actual increase in the radius is about a fifth of an inch.
In this case the tension is about 1°6 x 10°.

It is further evident that / nd/p is the largest velocity
if an extension of more than n per unit length is to be
avoided. When the elastic limit for the material is known,
this result can be used to give an upper limit to the velocity

it is safe to use if risk of deformation of the hoop is to be

avoided. | F
We may now proceed to the problem of a thin rod rotating
about one end with uniform angular velocity.

Thin Rod Rotating about One End.

Let ay be the unstretched length of the rod, @ the angular
velocity of rotation about one end QO, po the density when it
is unstretched, and » the value of Young’s modulus for the
material of which the rod is made. Let T be the tension
in any section in the rod during the motion. Let the
distances of the same particle at rest and in motion be
wy, and w The density of the moving element dw is
poda,/dx and its acceleration towards O is ww’. The equa-
tion of motion of the element is therefore,

OT = — py bay. 2o’,

where, by applying Hooke’s law to the element originally of
length da) and now of length dz, we have

T=A(dx/da—1).
Hence, eliminating T, we obtain the equation,
dx /day? = — pow] d.

The solution must give the value «=a when #=a) if a is
the length of the rod when in motion. Accordingly it is

a=asin (& Vpyw?/d) / sin (ay ¥ pow?/2).

We may determine the value of a by means of the condition
that the tension vanishes at the free end, which is given
indifferently by e=a or aj=ay. Thus,

QV pyw?/A=tan dg V pow?/d.

The equation relating the two corresponding positions of a

Rotation of Slightly Elastic Bodies. 33

typical element when at rest and when in motion and the
original length of the rod is therefore

® v/pyw?/A=sin (xy Vpow?/A) / cos (ao WV pow?/X).

Neglecting the cube and higher powers of 1/A, we may
replace this by the simpler form,

L=Xq + Lppow? (3a? — xp”) /6X
to the order 1/A. To the same order,
T= po@?a,? [| 1—2y?/a)" |.

The greatest extension is $pow’a,?/d, and this occurs at
the end about which the bar is rotating. The tension is also
greatest at this point and takes there the value p)wa,”.

As an example of the actual magnitudes of the quantities
in practical cases we may take a twenty feet steel bar, which,
when rotating about one end two hundred and fifty or three
hundred times a minute, increases in length about a tenth of.
an inch.

Rotation of an Infimte Elastic Circular Cylinder
about its Aas.

Passing now to a simple problem in three dimensions, we
take the case of an infinite elastic cylinder of circular section
rotating about its axis. We may consider one of the circular
sections of the cylinder and use polar coordinates. At any
point (7, @) let I, and T, be the transverse and radial ten-
sions per unit length, and T; the axiai tension. We shall
consider the motion of the element of volume which when
at rest is bounded by the surfaces (z,<2+6z), (r, r+é6r),
(0,0+60@). By the symmetry of the cylinder, the element
when in motion will continue to be bounded by the surfaces
(0,9+60): and since the cylinder is of infinite length,
the element will continue to be bounded by the surfaces
(c,¢+6:). Let p represent the radial dimension, so that

—r is the radial extension at any point. Let a be radius
of the cylinder and o its density, when at rest ; let w be the
angular velocity of the cylinder about its axis, and A and yw
the elastic constants for the material of which the cylinder
is made.

The element of volume which we are considering is a
parallelepiped of sides dp, pd@, and dz. The forces on our
element of volume consist of (1) transverse tensions each

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. D

34 Dr. Dorothy Wrinch on the

of magnitude T,dpdz—and these are equivalent, in the
usual way, to a radial force towards the centre of magnitude

T, dp dz ds/p,
where ds=pd0; (2) of radial tensions
T,dsdz and T,dsdz+dpd(T. ds dz)/dp,

towards and away from the centre, which together give a
oe dp d@ dz d(pT.)/dp

away from the centre ; and (3) of longitudinal tensions each

Fig. 1.
7, Sp.

6p Ee (7p ds)

of magnitude T;dp ds, in opposite directions. The resultant
force then is simply

dp dz ds (d(pT,)/dr —T;)

away from the axis, and perpendicular to the axis of the
cylinder. The acceleration of the element is pw’ towards
the centre and its mass is ordrd@, since we may, of course,
treat the density as constant over the element of volume.
We therefore have the equation of motion,

d(pT,)/dp-- T,;= — orp? dr/dp.

Rotation of Slightly Elastic Bodies. 30
By Young’s law we can express T, T., and T; in terms of
the extensions of the cylinder, in the well-known equations,
=X(dp/dr—1)+ (A+ 2)(p/r—1),
= (A+ 2u)(dp/dr—1)+A(p/r—1),
Ts; =A(dp/dr + p/r—2).

Putting these values of T, and T, in the above equation we
obtain the result

d dp p
SEP (pea, eae Buea
dy [p(X +n) (S 1) + pr", 1) |

AS Oe Ze -1)- 2 O23) (0-1) =—orpo

dr
or
(pir) dpidr? + (Lir) [(2u dpfaryi-+ 2) + pir) (4 2u)]
| dp/dr—p/r|=—pow?/(A+2y). . (1)
The value of ww or A varies from about 8 x 108 grammes’
weight per square centimetre for steel to about 4x 10°

grammes’ weight for copper. The corresponding densities
are about 7 and 9 respectively. ‘Terms involving

o/(A+ 2)

are therefore of a smaller order than those which involve
coefiicients of the form
2u/(A+ 2u).

Let us write @’o =q(A+2yu). Then putting
p—r= nt eno ans +
we can obtain a value for p to any order of approximation
which is required. Neglecting, first, all terms involving 9’,
we have the equations for nN;
p=r+qm; dpfdr=1+qdnifdr; ad’p/dr?=q d’n,|dr’ ;
g (149 m/r] dm/dr? +1 ]r [1 +9 2p dm/adr)/(\ + 2p)
+ (An,/r)/(A+ 2p) +9 (dmi/dr—m/r)\=—gr(1t+q mr) ;
q d’n,[dr? (1 +4 m/r)+1/r (1+q Qp dn, dr)/(X + 2p)
+ (Am/1)/(A+ 24) ] 9 (dm/dr—,/r) + gr [1+ 9m/r] =0;
and since we are neglecting terms in gq’, the equation for 7, is
re

y d?x,/dr? a+ Us dn,/dr—n, =>
D2

36 Dr. Dorothy Wrinch on the
giving a solution of the form
m= ay + b,/r—$r?

and 6;=0, since m, is not infinite at the axis. The boundary ~
conditions determine the constants a, and b,; for the radial
traction T is given by

: d
T= (A+ 2u) (* -1) +r(2-1)

=¢ | (A+ 2u)dy,/dr +r 9,/7],

and the radial traction must vanish over the two boundary
surfaces. Thus

(A+ 2p) dnyfdr | pea +X m/7| pe =0.
The constant a, is therefore determined by the equation,

(A+ w)a, = (204 34) a?/8,

giving
Ti=¢ [2+ p)ay—7?/4(204 34) |
=9(2r == dm)(a?— r) [40°
and

m=q(ar—r'/8)

= [20+ 3p)(@+0)/A+p)—77],

The Lffect of a Circular Hole in the Cylinder.

If the cylinder at rest has two boundary surfaces r=a,
r=b (b<a) the solution stands in the form,

NY = (yr + 5/7 — 37°,
and the conditions for zero radial traction on both the
bounding surfaces yield
(A+ pw) ay= pb;/a? + (2N4+ 3u) a?/8, :
(A+ pw) a, = ph, /b? + (2A4+-3u 67/8.
These give
ay = (2X 4- 32) (a? + 87) /S(V 4 p),
6) = (2X4 3p) a7b?/84u.
Thus
Ti=q[|2\A+p) a; —2ply/2? — (2.4 8y)27/4 |

= 9 (2A + 3p) (a? —7°) (9? — 0?) /42°.

Rotation of Slightly Elastic Bodies. 37
And the radial extension is
p—r=g[ayr+h,/r—13/8]
= = [(20 + 3) (a? + 0?) /(A4+ pM) + (20 +3) 0767/2? 77].
The maximum radial traction occurs where

(a? — 7?) (9? — Lb?) /7?

is greatest, namely, at 7?=ab. The greatest radial traction
therefore occurs on the cylinder whose radius is the geo-
metric mean of the radii of the bounding surfaces of the
cylinder. Its value here is

po*(2rX.+ 3u)(a— b)?/4.V 4 2p).
The transverse tension is given by
T,=q[ (A+ 2u) m/7 + 2X dy, /dr]
=q/4[ (204+ 3u)(a? +07) + (204 3p) a7b?/7? —(A4 2y)r? |
=(2A+ 3p) i [a+ 6? — ab? [7° — (A+ 2u)7?/(2A+ 3h) |.
The maximum value of this tension occurs on the cylinder
with radius ry given by
re=abV (204 3py)/(A+ 2pn)},
and there its value is

gi 2rn+ 3M)
4

Finally, the tension parallel to the length of the cylinder is

[a2 + b?—2ab / (A+ 2p)/(204 3p) |.

T; =qrX(m/7 +a /dr)
_ QAP 2A+ 3H) 9 72) ona
=#/ ae (+) —2 i

The longitudinal tension T; is therefore greatest on the
inner boundary. Here its value is

qn pee dpu)a? + (w— as)

4 N+ pb 5

In the well-known case of a solid cylinder, the longitudinal
tension has its maximum on the axis, and its value there is

(2X.+ 3p) ga?r/4 (A+ uw).

ee

38 Dr. Dorothy Wrinch on the

The radial and the transverse tensions have also their
maximum value on the axis, where they take the same
value,

po(2rX.+ 3p) /4a?(A+ 2p).

The severest traction the cylinder is called upon to with-
stand is therefore !
po(2r0+ 3u)/4a?(A+ 2y),

and it occurs radially and transversely on the axis of the
cylinder.

In the case of the cylindrical annulus, the facts are entirely
different. The longitudinal tension reaches its maximum
value

po r| (20 + 34 )a?/(A+ po) — gb?/ (A+ pr) |/4(X+- 2p)

on the inner boundary. The transverse tension, in general,
reaches its maximum value

po2(20. +3) | a? +b?—2Qab Jf (N+ 2m) /2 4+ 3y |/4 (04 Qu),

on the cylinder r=r,. And the radial tension reaches its
maximum value
po*(20 + 3u)(a—6)?/4(A + 2u),

on the cylinder r=7,. As the ratio /u varies, the place at
which the maximum tension occurs varies. If the ratio b/a
is sufficiently large, 2. e. if
Ae i He ag
A+ 26” 20+ 3
the maximum transverse tension will occur on the outer
rim ; otherwise it will occur within the cylinder. Thus a
sufficiently thin annulus will have its maximum transverse
tension on the outer boundary. For different values of the
ratio A/u, the maximum traction will occur transversely or
longitudinally, for a cylinder for which the ratio a/b is given.

And further, for a given ratio X/p, the maximum traction
will occur transversely or longitudinally.

Eigher Approximations for Expansion of a Solid
Infinite Cylinder.

The first approximation for the radial extension was of the
form,
p=rt+gq(ar—7?/8)=r+qn,.

Rotation of Slightly Elastic Bodies. 39

To obtain a second approximation we may put

p/r=1+yn/r+ g'na/? ;
and therefore
dp/dr=1 + gdn,/dr + q?dnz/dr

and neglect terms involving q°. Using, again, equation (1)
we obtain, if terms in g’ are again neglected,

d?no/dr? + dys/rdr—np/1" = —,
or rd?no/ dr? + rdng/dr— y= — ar’? +7°/8.
The solution is of the form,
No = Agr—ayr?/8 +7°/192.
The radial tension T, is given by
gL(X+ 2) dyyldr-+ dm/r] +42[(% +2) dng|de + dm/7.

The condition of zero radial traction on the boundary
therefore yields

(A+ 2p) dn,/dr+rm/7 +9 [ (A+ 2) dyg/dr + Ane/r] =0

at r=a, or since the part independent of g already vanishes
at r=a,

(A+ 2) do/dr + Xyo/7 =0

at r=a. Hence

T,=9(2h-+3u)(a? -r)/4

+ @[(20 + 3u)a,(a?—r?)/8 — BX + 5p) (at —r4)/8],

and since
a= (20-4 3p)a2/(d+ p)
T.=9(2043n)(a—r?)/4

+9? [(2d + 3y)?a?(a?— 1) /8(V + w) — (30 + 5p) (at—r*)/8],
and, finally,

nalr=ay/[(2A-+ 3a)a?/(4-+ w) 7/8
—[(3r4 5p )a4/(A+m)—71*] / 24x 8

Nn+3u eo ; a) 1 (TPH att)

ie ale cap te ee
B(A+ pt)” A+ bw } 8.24\ ~X+p

>

40 Dr. Dorothy Wrinch on the

Our second approximation is therefore,
p—r= qn + Fne

ge vietie 2u+3y ae
agr[ Rta

9) y g
+g] 2N+ 3M 5 a(- Ae 18)
N+ ph N+

tea
al cans

Any higher degree of approximation can also be obtained.
Our results stand in the form,

1/7 = 9% + 14,7”,
No|T = ola + 1Ag?” + oo,
and, in general, the form will be
hel Plea. 2 lt ar) eee ei)
We easily see that if (2) be the form for 7/7, then’

Pc Aah me at jar" Suga a
een tny bbe eee isn PEG aaa |!

and the condition of zero radial traction, requiring that

(N+ 2) dnsai/dr+rqo41/7 =0
on r=a, yields

ots Zr+3pm , 9s 3A + Db A
Wels er Kb Neer
ED ss (s+1) A Zsa a,
25+2.2s+4 N+ pe
so that k
ls (2R4 3m ods (ON+9M 4 4)
Ns+i/T= ral eee nant aes a ae
ss (s+ IN+2s+1p qzst2 — oo) a
Ss Sw i Oo
Pesug. Isr a\ N+ pb a 2)

When we have obtained a solution,

p—r= 9m tq'ne+. + +9°Ns,
it is therefore possible to obtain a solution to a higher degree

Rotation of Slightly Elastic Bodies. 41
of approximation by remarking the relation between the co-
efficients in 9/7 and y,4;/7 and putting instead of

s

adi
Linde ? "5
0

as In 9/7,
Ns 41/7 = $ (“ e 1)r a (2n +4 Me gina — pons) oe: not aes’,
n=0 N+ ph (Zn -- 2)(2n+ 4)

Rotation of an Infinite Circular Cylinder of
Non-uniform Density.

We may next deal with the case of a cylinder in which
the density is a function of the distance from the axis.
Treating the case of the solid cylinder, we may put for the
density of the cylinder when at rest,

N
c=f(r)= Da”,
0

and N may have any value from zero (when the density is
uniform) to infinity—in which case the series Ya,7” must be
convergent. |

The equation to be solved for a first approximation to the
value of 7 is as before,

dy[dr? + 1/r dn[dr—y/7? = —ow'r] (A+ 24).

The solution is evidently,
aon is
n=Ar— eae > ann’ **i((n+3)?—1).

The fact that there must be zero radial traction on the
boundary surface r=a, yields the condition,

T,= (A+ 2p) dyn/dr+rn/r=0
on r=a, giving

2(A+p)A
2 N
iF cone [(n43)(X+2u) +2] a,a"*?/(n+4) (n +2)
2

N . +
ieee ee [(n+4)X+42(n+3)p] a,0"*?/(n+4)(n +2).
od 0

i

42 Mr. G. H. Henderson on the
Consequently, the value of T is

Le S : < (Oa Le paar 9
Kb Dye L$ 3) (A+ 2p) +A] (aro) ay] +4) (m4 2),
and

ee
p N+

N a ‘

La,fa"t?(nt4r+2n+3 p)/2(X+ wp) —7*?} /(n +2) (n +4).
. .

Higher approximations to the value of 7 can be obtained
by the method adopted in the case of uniform density.

IV. The Straggling of « Particles by Matter. By G. H.
Hurnperson, M.A., 1851 Exhibition Scholar of Dalhousie
University, Halifax, N.S.* .

SL. Introductory.

7 HEN a parallel beam of & rays passes through matter,
the particles gradually expend their energy in
passing through the atoms of the matter, until all trace of
the particles suddenly seems to vanish at the end of their
range. In passing through the atoms some of the « particles ©
lose more energy than others, so that at any point along
their path some of the particles will be moving more slowly
than others ; also their ranges will not all be the same. The
a particles may be said to be straggled out, and hence the
term straggling has been applied to this phenomenon by
Darwin. |

The theory of the passage of matter by « rays has been
developed on the basis of the nuclear structure of the atoms
of the matter, and from this theory the amount of straggling
to be expected has been deduced from probability. considera-
tions. On the other hand, the straggling can be determined
from experimental data in two ways.

The first method makes use of ionization data. When the
ionization due to a parallel beam of @ rays is measured at
different points along the path of the rays, the well-known
ionization curve is obtained. This curve is shown as the

* Communicated by Prof. Sir E. Rutherford, F.R.S.

Straggling of « Particles by Matter. A’

full curve of fig. 1, where ionization is plotted as ordinate
and distance from the radioactive source as ubscissa. Now
it has been shown experimentally that the (average) velocity
of the @ particles at any point of their path is proportional
to the cube root of their remaining range. Assuming that
the ionization produced is proportional to the energy lost by
the « particle at any point of its path, it can at once be
shown that the ionization should be inversely proportional
to the cube root of the remaining range. Such a theoretical
ionization curve is shown as the dotted curve of fig. 1. It

Rios 1

{ 2 5 cm.

Range

will be seen to bein approximate agreement with experiment
over the first portion of the path of the @ particle, but as the
maximum is approached this agreement fails.

Geiger * has suggested that the ionization curve observed
for a beam of « rays should be different from that of a single
« particle, owing to slight variations in the ranges of the
latter, 7. e. to straggling. -The ionization curve, built up of
a large number of theoretical curves grouped around one of
average range, will thus be modified considerably near the
maximum where the ionization is changing rapidly Hence
the shape of the ionization curve near the end of the range

should give an indication of the amount of straggling.

* Geiger, Proc. Roy. Soc. A. Ixxxiii. p. 505 (1910).

44. Mr. G. H. Henderson on the

Secondly, a more direct measure of the amount of
straggling can be determined by counting the number of
a particles at different points along the path of a parallel beam.

It is proposed in this paper to discuss the theoretical and —
experimental data on straggling, and it will be shown that
the observed amount of straggling is much in excess of that
allowed by theory. [Further experimental evidence bearing
on straggling will also be brought forward.

$2. Lhe Straggling in Air.

It might be thought that the individual @ particles are
emitted with slightly different velocities, thus giving rise to
straggling. It has been shown by Geiger ( (loc. cit.), however,
that the « particles emitted from a thin layer of radioactive
material do not differ by as much as 4 per cent. in initial
velocity. Thus the cause of the strageling must be looked
for in the air itself.

As the « particle passes through the air it gives up its
energy to the electrons and nuclei of the air atoms, and it is
occasionally deflected through a considerable angle by close
encounters with the nuclei. Different a particles will
encounter different numbers and distributions of electrons
and nuclei and accordingly are straggled out. The calcu-
lation of the consequent probability variations in the ranges
of the individual « particles has been carried out by both
Bohr* and Flammf. They agree in showing that the
nuclei produce practically no stragoling. They. also agree
closely in the amount of straggling produced by the electrons.
The straggling of various types Oe rays in alr, calculated
by Flamm’s method, is shown in the second ‘column. of
Table, 5 Vie ln. tabulated is the distance, measured
along the range, over which the number of particles in a
parallel beam falls off from +92 to :08 of the original number.
This corresponds approximately to the method of measuring
the straggling from the experimental curves.

The ionization curves for three types of a rays have
recently been determined with some accuracy by the writer {.
The full curve given in fig. 1 is a reproduction of the
ionization curve found for RaC. It was shown that the
ionization curve from C to B (fig. 1) could be represented
very approximately by a straight line. The slope of this
straight line furnishes infor mation as to the magnitude of

* Bohr, Phil. Mag. xxx. p. 581 (1915).

ay Flamm, Wien. Ber. IL a, exxii. p. 1893 (1918).
t Henderson, Phil. Mag. xlii. p. 588 (1921).

Straggling of « Particles by Matter. 45

the straggling. The. easiest way of considering the matter
is to imagine the straight line produced in both directions
till it meets the axes of zero and maximum ionization at D
and H. Then the projection of the line DE on the axis of
zero ionization (or the reciprocal of its slope) is a direct
measure of the straggling. The greater the straggling the
greater will be the projection referred to, and as a first
approximation the projection may be taken as proportional
to the straggling. The values of the projections taken from
the writer’s curves are given in the third column of Table I.
That for polonium has been determined from the curves
given by Lawson *.

The curves obtained by counting the number of « particles
in a parallel beam at various points along the path show
that this number remains constant till near the end of the
range and then falls off rapidly to zero. Most of this falling
off nist approximately followsa straight line. The reciprocal
slope or projection of this line is a more direct measure of '
the straggling than the corresponding projection of the
ionization curve. Measurement of the str ageling by means
of counting experiments is, however, very slow, as large
numbers of @ particles must be counted. The values
obtained from the most recent and reliable counting experi-
ments are given in the fourth column of Table I. The
result for polonium is taken from a scintillation curve given
by Rothensteiner + ; that for RaC is from a curve obtained
by Makower f by photog raphic counting of the « particles.
All the results in the Table are in millimetres and refer to
air at 0° C. and 760 mm.

TasceE I.
|
Gas ae. Theoretical | Straggling from | Straggling from
eee oe Baye Straggling. —_Lonization ‘Curves. Counting Expts.

| Polontum: .3...:... 88

31 | 40
| Thorium C, ...... 1:02 2°88 |
| Radium C......... 1-44 | 2-83 | 41
2-92 | |

| Thorium C,....... 1-74

This Table shows clearly that the observed values of the

* Lawson, Wien. Ber. il a. exxiv. p. 637 (1915).
+ Rothensteiner, Wien. Ber. IL a. exxv. p. 1237 (1915).
t Makower, Phil. Mag. xxxii. p. 222 (1916).

AG We a Siloceénton oho

straggling are three or four times greater than those predicted
by theory. Furthermore the calculated straggling increases
steadily with increase of range, while that observed is
constant within the limit of error. It should be pointed out
that the projections given in the Table are measured as the
small differences between two larger quantities, and hence
are more difficult to determine withaccuracy. The straggling
deduced from the writer’s experiments has a probable error
of about 2 per cent., and it will be seen that the values for
the three types of rays agree within this limit.

It was shown by the writer (loc. cit.) that the effect of
straggling due to electronic encounters would be a tailing
off of the ionization curve at the extreme end of the range.
Making some simple assumptions it was shown that the
calculated form of the end of the ionization curve agreed
satisfactorily with the form of the curve observed between
Aand B (fig.1). Thus the effect of the calculated straggling
was amply accounted for by AB, leaving the much greater
straggling evidenced by the straight line portion BC quite
unexplained. ‘The curves obtained by counting experiments
also lead to precisely the same conclusion. In view of the
failure of theory to account for this large excess strage gling
it 1s interesting to see what further information regarding it
can be derived from experiment.

It is remarkable that the straggling (as measured by the
projections of the ionization and also of the counting curves)
should be constant for arays differing so widely in range as
those givenin Table I. This can only mean that the excess
strag ogling takes place only in the last two or three centi-
metres of the range. From experiments with gold foils
which will be discussed later, it appears probable that the
straggling is confined to the last few millimetres of the
range.

Referring once more to fig. 1, it could not be expected
that the strageling deduced from the ionization and the
counting data would agree, for the following reasons :—The
ionization curve is revarded as being built up of simple
curves of different ranges grouped about a common mean.
The form of the simple curve is not accurately known ; the
rule that the ionization is inversely proportional to the cube
root of the remaining range can only be an approximation to
a much more complicated law. As the shape of the simple
curve cannot be taken into account, the projection of the
ionization curve which is actually utilized can only give a

Straggling of « Particles by Matter. 47

rough indication of the absolute magnitude of the straggling.
However, comparative values of the straggling under different
conditions should be given fairly accurately by the method
adopted. On the other hand, in the counting experiments
the assumption is made that the zine sulphide screens or
photographie plates used have the same efficiency for « rays
of low speeds as for those of high speed. This assumption
is not altogether justifiable.

§ 3. Straggling in Gases other than Avr.

The ionization curve in hydrogen was determined with
the same apparatus already used for air. The gas was
obtained from a cylinder of compressed hydrogen stated
by the makers to be of more than 98 per cent. purity.
Small impurities are unlikely to affect the straggling
materially.

It was again found that a considerable portion of the end
of the ionization curve could be represented by a straight
line. When the range of the a particles was reduced so as to
give the same range as in air, the projection of the straight
line was 2'05 mm. with a probable error of 3 per cent.

The straggling in air and hydrogen may be deduced from
the ionization curves given by other observers. ‘lhe results
agree in every case within the limits of error, although the
conditions for accuracy were less favourable than in the
present experiments. The value 2°0 mm. for polonium in
hydrogen may be obtained from some ionization curves given
by Taylor*. From the results of Lawson (loc. cit.) for
polonium the straggling was determined as 3°1 mm. in air
and 2:2 mm. in hydrogen.

The straggling in oxygen has also been deduced from
experiments made in that gas with the present apparatus,
using ThC. The value found was 3°36 mm., when the range
was increased to the same value as in air.

The straggling in several other gases may be deduced
from the ionization curves given by Taylor (oc. cit.), although
the error involved is probably of the order of 10 per cent.
The collected results of straggling in gases are given in fig. 2,
which shows the straggling plotted against molecular stopping
power. The values plotted for air, hydrogen, and oxygen
are from the writer’s results; the remainder are taken from
Taylor’s curves.

* Taylor, Phil. Mag. xxi. p. 571 (1911).

48 Mr. G. H. Henderson on the

Fig. 2 can only be considered to give an approximate
idea of the facts, as the points are not well distributed and
some may be seriously in error. It would seem, however,
that the straggling increases very slowly as the stopping
power of the gas is increased. It is unfortunate that the
dearth of suitable gases of high stopping power makes the
checking of this point difficult.

Fig, 2.

Straqgling inmms.

3

1 2
Molecular Stopping Power.

§4. The Straggling due to Solids.

The great difficulty which at once arises in determining

the straggling due to solids is the uneven thickness of the

solid foils used, the effect of which may completely mask |
the true straggling looked for. An attempt to avoid this
difficulty was made by using a large number of the thinnest
beaten foils of the solid obtainable ; with gold, for example,
as many as 128 thicknesses were used. Composite sheets of
gold and other metals were placed immediately over the
radioactive source (ThC) and the ionization curves deter-
mined in air with the same apparatus as before. Although
a rough calculation seemed to show that the irregularities in
the individual foils would be smoothed out enough to avoid
masking any true increase in strageling, this result was not
borne out by experiment. It was finally concluded that the
increase in straggling observed was inainly, if not entirely,
due to unevenness of the foils, and hence need not be gone
into in detail here. In mica the increase in strageling was
much the smallest, as was indeed to be expected.
Fortunately, experiments on the straggling produced by

Straggling of « Particles by Matter. 49

solid foils for low velocities of the « particles gave results
which were not masked by irregularities of the foils. In
these experiments the foils were placed 3 mm. from the
middle of the ionization chamber (itself 1 mm. deep) in air
at a pressure of roughly 17 cm. Reduced to air at normal
pressure the distance from foil to centre of ionization chamber
was therefore about ‘7 mm. Most of the foils used were
made up of a few thicknesses of goldleaf. ‘lhe air equivalent
of a single sheet was about ‘45 mm. when placed directly
over the source ; ; when placed near the ionization chamber
the air equivalent was about ‘28 mm. ‘The straggling of
the « particles after passing through these foils was deter-
mined in the same manner as before from the ionization
curve in air.

The results are given in Table II. The straggling is in
millimetres, and the probable error is about 2 percent. ‘The
third column shows the straggling observed when the foils
are placed directly over the source, the steady increase with
increasing number of leaves being mainly due to unevenness
of foils. ‘The fourth column shows the increase in strageling
at low velocity over that at high. This increased straggling
is real and almost independent of the unevenness of the
foils. Results with aluminium (1:0 mm. air equivalent)
and mica (8°6 mm. air eq.) are also included in the Table,
but with these foils the change observed is scarcely more .
than the experimental error.

Taste II.
_ Straggling Foil | . . ‘
Blo. of Leaves , near lonization Straggling ae Difference.
in Foil Gis hee near Source.

ees de | 2:88 2:88 Fi |

SEED ae nee | 3°30 2°92 38

et Cae emt | 3-65 | 3-01 64
SES ee | 4:00 3:28 72
Aluminium ...... | 3°08 | 3°01 ‘O7
1 SE eee eee 3°16 3°06 "10

These results show quite clearly the rapid increase of
straggling near the end of the range. One gold leaf nearest
the ionization chamber causes nearly twice as much strageling
as three leaves immediately behind it. Hight more eaves

Phil, Mag. 8. 6. Vol. 44. No. 259. July 1922. E

-

50 Mr. G. H. Henderson on the

placed behind these four again only slightly increases the
strageling.

The same result was also demonstrated in a slightly
different manner. A foil made up of the four gold leaves
already used was placed at different distances from the
ionization chamber and the straggling determined from the
ionization curves as before. ‘Phe results are shown in

. Table III. The distances given in the first column are not ~

the actual distances from the foil to the centre of the
ionization chamber, but are reduced to correspond with an
ionization vessel containing air at atmospheric pressure.

Taste ITI.
Distance from Seah eat
Tonization Chamber. ageing.
“7 mm. 3°65 mm.
ob ah 526,
-12°9_,, 295,

Here again it can be seen that the straggling increases
rapidly near the end of the range.

It may also be noted that the straggling (2°95 mm.) when
the foil was 12°9 mm. from the ionization chamber is less
than that (3:01 mm.) obtained when the foil was placed
directly over the source. ‘lhe difference is less than the
possible error, but such a change is to be expected. For
the straggling due to unevenness of the foil should be less
for low velocities of the « particles on account of the decrease
in air equivalent of the foil. For the same reason the differ-
ences given in the fourth column of Table II. are probably
slightly too small.

From the straggling observed with different types of
a particles it was pointed out in § 2 that the strageling must
occur within the last two or three centimetres of the range ;
from the results with gold foils it seems that the straggling
must be confined largely to the last few millimetres of the
range.

The increase of straggling due to gold foils placed near
the end of the range, though clearly marked, is small.
This is quite in accordance with the view expressed in § 3
that the strageling does not increase rapidly with increase
of stopping power of the substance causing the stragoling,
It must be remembered that the increase- observed is the

Straggling of « Particles by Matter. ant

increase over the strageling which would be produced by a
layer of air equiv alent to the gold foil used. The ionization
must be measured in a gas “such as air at a reasonable
pressure, and hence we have the complication of the strageling
due to the air between the solid and the ionization chaml Jer,
and even in the chamber itself. ‘Chis sets a limitation to
the amount of information to be obtained from ionization
data. Accordingly the ionization experiments were not
earried beyond the stage described, but it is hoped to push
further the attack on the problem by more suitable methods.

An isolated experiment with iron foils may be referred to
before closing this section. A sheet of iron of about 4 cm.
air equivalent was placed directly over the source and the
ionization curve measured when the iron was magnetized
parallel with and perpendicular to the direction of travel of
the # particles. ‘The purpose of this experiment was to see
if there was any change in the straggling due possibly to
rearrangement or change of orientation of the electrons
in the iron, a point which has been discussed by Flamm *.
Alternate readings of the ionization with parallel and per-
pendicular fields of about 100 gauss were made at various
points along the range. No appreciable difference could be
detected. It shouid be added that the iron was very uneven
in thickness, and the consequent’ straggling was so large
(about 20 mm.) that a small change in straggling might
well have been masked.

§ 5. Summary and Conclusion.

It has been shown in this paper that the straggling of
a particles, as deduced from both ionization and counting
experiments, is several times greater than that deduced from
theory based on our present views of the mechanism involved
in the passage of « particles through matter. It has been
shown in a previous number of this Magazine that the effect
of the calculated straggling can be adequately accounted for
by the tailing off of the ionization curve at the extreme end
of the range. The large additional straggling observed
behaves quite differently. Hvidence has been given in this
paper to show'that it increases very slowly with increase of
molecular stopping power, and furthermore, that it all takes
place within the last few millimetres of the range. Here we
seem to be confronted with a behaviour of the a part cle

which present theory is unable to explain.

* Flamm, Wien. Ber. Ila. exxiv. p. 597 (1915).
E 2

\

2 ~ Dr. G. Green on Fluid Motion

Evidence obtained from the Wilson photographs also- leads
to the same idea, for it has been shown by Shimizu that
the observed number of ray tracks which break up into two
branches near the end of the range is much greater than the
number deduced from probability considerations based on
our present theory of atomic structure.

It is noteworthy that this anomalous behaviour of the
a particle occurs at low velocities, where practically no
investigation of the scattering of « particles has been carried
aut on account of the experimental difficulties of dealing
with slow « particles. It is at higher velocities, where the
theory of scattering put forward by Sir Ernest Rutherford
has been so fully verified by experiment, that the most of
the theoretical straggling takes place, and this straggling
has apparently been accounted for. |

In conclusion I wish to express my best thands to Professor
Sir Ernest Rutherford for his kind interest and advice.
I also wish to thank Mr. Crowe for the preparation of the
radioactive sources.

V. On Fluid Motion relative to a Rotating Earth. By
Grorcr Green, D.Sc., Lecturer in Natural Philosophy in
the University of Glasgow f.

fi ese subject of this paper is at present one of consider-
able interest to meteorologists. Papers by the late
Dr. Aitken and also by the late Lord Rayleigh on the
dynamics of cyclones and anticyclones have been followed
by more recent papers by Dr. Jeffreys, Sir Napier Shaw,
and others. Very few actual solutions of the equations
defining atmospheric motions have been obtained. In the
late Lord Rayleigh’s paper { attention is drawn to certain
general hydrodynamical principles relating to the properties
of rotating fluid which can be applied to “assist our judg-
ment when an exact analysis seems impracticable.” The
importance of the theorem regarding the circulation of the
fluid in any closed circuit is clearly explained in its applica-
tion to any actual fluid motion. In applying this theorem
to fluid motion in the atmosphere, however, we must bear in
mind that the motions with which we are concerned are not
the actual motions of the particles in space but their motions
relative to the Earth itself at each point of observation.
. * Shimizu, Proc. Roy. Soe. xcix. p. 432 (1921),
+ Communicated by the Author.
t Sc. Papers, vol. vi. p. 447,

relative to a Rotating Karth. 53

One object of the present paper is to investigate the con-
ditions under which the circulation theorem may be applied
to atmospheric motions relative to the Harth’s surface ; or
more generally to motions relative to any three rectangular
axes which are themselves rotating about each other, with a
fixed origin. In the later part of the paper one or two
additional cases of motion of the atmosphere are discussed
and the system of isobars corresponding to each motion
determined.

In view of the problems to be considered, we shall begin
by specifying the system of rotating axes most convenient
in dealing with fluid motion in the neighbourhood of any
point of reference O on the EHarth’s surface. The axis OZ
is drawn upwards along the apparent vertical at O, and
line OZ continued downwards meets the axis of the Earth
at a point QO’ which is taken as origin of coordinates. Then
axes O’X and O’Y are drawn parallel to horizontal lines
through the reference point O in directions due Kast and
due North respectively. In the most general case to be
considered the reference point O may be in motion relative
to the Earth’s surface, and this involves also a motion of the
origin O/ if point O moves either North or South. But
the motion of O' corresponding to any moderate motion
of O is very small, and for our present purpose we may
regard the origin O' as a fixed point, very near to the centre
of the Earth. We shall denote by («, y, z') the coordinates
of any point referred to origin O', and by (a, y, z) the co-
ordinates of the same point referred to parallel axes through
O. This makes z'=z+R, where KR represents approximately
the radius of the Earth. The components of the velocity of
any particle relative to the axes at any instant are repre-
sented by wu, v, w, and the angular velocities of the axes
themselves, that is, of each two axes about the third, are
represented by w,, w,, wz, respectively. We shall introduce
the particular values of w,, wy, @, corresponding to a reference
point O fixed in position on the Harth, or moving relative
to the Earth, when we come to deal with special problems.
Referred to the above system of axes, the equations of motion
of any fluid particle take the form :—

Du - OV 1 Op

tego da SY
Dv nA oes OP ;
Pome a®
Dw oy Lviap
ay, nh =o ae ia: © ee @)

54. Dr. G. Green on Fluid Motion

In these equations, 6), 0,, 8; represent the terms depending

on the rotation of the axes themselves, being given by

equations of the type

0,= —20,v+2o0,w—o.y+0,2' +0,0,) +0,0,2' —(0/ +o) a.
(4)

The function V(x, y, 2) represents the gravitational potential

function. We have also

a move) 0 Oli

Pe CE
ae 7. ae ee Ap Pe (9)
The equation of igi of the fluid is then
Ou Ov $)
= == ().) ee
pe te( Se + Se 4 Se (6)

In applying these equations we treat the atmosphere as a
perfect gas in which viscosity may be neglected.

Circulation Theorem for Relative Motion.

Consider now the theorem relating to the relative circula-
tion. We have

py (nde vdy+ waz) = By “dx + pi + yp e+),

Dt
(7)

where g=w?+v?+w?, the square of the resultant relative
velocity. By means of equations (1), (2), (3), the above
ee may be rewritten in the form:

py (udev dy +wd: )
=—(O,dx+, sdy+0,d2)—"P —aV + AG; Gaye (8)
We can now integrate each term of this equation along any

curve within the fluid from any point A to any point B.
This integration gives the result,

‘B

ay ie dp
De (uda +vdy+wdz)=— nae dx + 0,dy +0@3;dz)— } —
JA oe

hei 7s —3q,° ; (9)

and, if the integrations are applied to a closed curve

relative to a Rotating Earth. 5D

beginning and ending at the point A, we obtain
= { (ude+vdy+wdz)=— \ (0,dx+O@,dy+O@3dz), (10)
v5 Ǥ

where the suffix 8 indicates that the integration is to be
taken along a definite curve S. We have assumed in
obtaining (10) from (9) that V is a single valued function
of (wz, y, ¢), and that p isa function of p. It now appears
that the rate of change of the relative circulation in any
closed circuit which consists of the same fluid particles at

all times is not zero unless, in addition to the above con-
ditions, we have

00: 00;, 302301, 9% dz

Boule ds. |) O66, LOM ha OF y. OF a a)
When these conditions are not fulfilled, the relative vorticity
does not move with the fluid itself, and if a velocity potential
exists for a certain portion of fluid at a given instant, a
velocity potential will not exist for that portion of fluid at
a later instant.

The first case of importance of the above conditions in
relation to problems relating to the atmosphere is that
in which the angular velocities w,, wy, @, of the axes are

constants. Jn this case, the conditions given above take the
form

Pet ae meee Ou ov ow 12
fre Ores Oe ee Dz? seis ( )

where 6 represents an operator defined by
Sao eee) 13)
= Rig et ere . Pe ai ( y
These equations have a solution of the form

Uu (0)

—— - =f (w,«+@,y+o,2), rae os (14)

@, @y @,

where f denotes any arbitrary function. If we draw an axis
to coincide with the axis defined by the resultant of the

three component rotations @,, ay, @:, then (@,v + wy + @22)

is equal to ORcos¢@, where 2 is the resultant of (@;, @y, 2)
and R is the line joining the origin to the point 2, y, 2.
That is, wu, v, w are functions of p the perpendicular from
the origin to a plane through the point (a, y, 2) perpen-
dicular to the axis of the resultant rotation.

56 Dr. G. Green on Fluid Motion

When the fluid is incompressible, and when a compressible

fluid is moving in such a way that ou + SS + a IS Zero,
a solution of a different type obtains. The solution in this
case may be written in the form

Uu=fi { (@yx—ary), (@24 — @,2) ei
v=J2 {(wv—wzy), (w2%—@r2)},6. . « (15)
PN RE Ayer in |

where /;, /o, /s are arbitrary functions subject only to the

condition oY + Oe ee =0(0. This solution includes as a
ae’ 09° Oz |

particular case any motion of rotation of the atmosphere as

a solid about the axis of the Earth.

The solutions which we have above obtained make it clear
that the fluid motions relative to rotating axes in which the
relative circulation moves with the fluid belong to a very
restricted type. A relative motion, for instance, similar to
that taking place in a free vortex, does not fulfil the con-
ditions required for permanence of the velocity potential,
and therefore no steady motion of this type could take place
in the atmosphere—as has been assumed to be the case.

The conditions which we have found to be necessary for
the validity of the circulation theorem when the fluid motion
is relative to rotating axes, may be obtained in a manner
different from that employed above. Taking 3, H, Z to
denote the components of angular velocity of a tuid ele-
ment, and U, V, W to denote components of linear velocity
of the element, each referred to fixed axes which coincide at
instant ¢ with the instantaneous positions of the moving axes,
we may derive the conditions from the equations employed
by von Helmholtz in his papers on vortex motion :—

Dei aU ol en So
Dir = Oe oo ae 0s =(S- . OY 4s 02 73 a
with two other similar equations. With &, , ¢ to represent
the components of relative angular velocity of an element of
fluid, referred to the moving axes, we have,

B=E+o,; H=n+oe,; Z=l+o,;

U=u-ayto,z; V=v—a,7+0,0; W=w-o0,«+ ory ;

~

and Din 2 = Oo 11@,6,.

relative to a Rotating Earth. a7

By means of these relations we can readily transform (16)
and obtain the corresponding equations for the rates of
change of the circulation components of an element of fluid
referred to the rotating axes ; in this way we find

Dé Ou Ou Ou Ou Ov or)
‘am ae (5. + oy | 2

My. OU Ou One, Ov a Aw <
tort bays) +8252 wo, (So 4 an (17)
with the corresponding equations in 9 and ¢. Now the
hydrodynamical theorem that relates to the permanence
of a velocity potential for the motion of a given portion of
fluid and the theorem of the permanence of the circulation
of an element of fluid depend on equations (16). The
equations which we have obtained for the relative circula-
tions reduce to these equations exactly when the conditions
expressed in (12) are fulfilled ; and these conditions must
accordingly be fulfilled in order that the theorems referred
to may apply to the relative motion, in the same way as they
apply to the actual motion.

Particular Cases of Motion Relative to the Earth,

We shall now discuss one or two particular cases of fluid
motion relative to the Earth, and we shall, to begin with,
take the reference point O as a fixed point on the surface of
the Earth at latitude ¢ degrees North. The angular velocity
components @,, @,,@, have then the values 0, Ocosd,
sind, respectively, where © represents the angular
velocity of rotation of the Earth about its axis. If we now
let a represent the perpendicular distance from any point
z, y, < to the axis of the Karth, we can write the equations
of motion (1), (2), (3), in the form

Du ° yi eS OV! 1 0p >)
ee Cet ee ae alae (138)
BM sae bx Wane «yal: Op

pp +22 sing -u ae ey a (19)
Dw BeGuel: ap
Seen — e t, == Sarees ———— e- = Z )
Dp 222 cos G - ¥ i 82? (20)

where V'=V—1025%, V’ is, in fact, the potential function

eee -

58 Dr. G. Green on Fluid Motion

!
corresponding to apparent gravity, so that -< at point
(v=0, y=0, z=0) is the value of —g at the reference
point O. In applying the above equations to motion of the
atmosphere, we may take .

cea ee OV! |
jn By) ee

in the immediate neighbourhood of O in the region within
which the value of apparent gravity may be regarded as
constant in direction and amount. If we neglect a change
of direction of one degree in g, our equations (18) to (21)
would then represent conditions of motion within a radius
of about seventy miles from point O. In order to render our
equations suitable to represent circulations of air of diameter

exceeding, say, 150 miles, we might employ the approximate
values

OMG Gt re 2 4ON a coterie 5
04,0 ROO have Be. oe ae (22)

wherein we neglect the variation of g with height.

If we exclude certain cases of motion relating specially to
the tides, very few solutions of the above equations have been
recorded. In order to make ourselves familiar with the
types of fluid motion possible in the atmosphere it is of
interest to examine all solutions which can be obtained
having a bearing on meteorological problems. We, accord-
ingly, take first the steady rotational motions of incom-
pressible fluid under the force of gravity alone.

We may take the boundary condition w=0 to apply at
the surface of the Earth. A simple rotation of all the fluid
about a vertical axis through O, with a uniform angular
velocity w, would be represented by

w=—=oy > t=+or; w=0. eee

With these values in equations (18) to (2Q), it would be
impossible to satisfy (19) and (20) simultaneously ; but
the motion represented by

Uu=—@(¥-—6z); v=o#r; “o—0,.: ee

fulfils all the conditions contained in our equations, pro-
vided 8 is given by
a2 2Q, cos d

2 Bees D
To! }-2 OFsima gh ote, Ce eee =)

relative to a Rotating Earth. bY
and the pressure p is given by
P = —V' +4(w? + 20 sin p) {a2 + (y—B2)?} +0 (26)

fe Nie
The approximate value of V’ is + a(° a
the pressure at the reference point O. In the motion indi-
cated above each particle of fluid moves in a horizontal
circle whose centre lies in the line (y=8z, x=0). This line
lies in the meridian plane through point O and is inclined to
the vertical at O at an angle @ towards the North, where
tan@d=8. When a is very small in relation to © this in-
clined axis is almost parallel to the axis of the Earth; and
when o is large this axis comes almost to coincidence with
the apparent vertical at O. With very large the motion
described above corresponds very closely with a uniform
rotation of the fluid about a vertical axis, as in the case of a
simple forced vortex.
Another case of motion of incompressible uid of interest
in the same connexion is that represented by

+2), and 9 1s

oa i — oy toe; w= 20 cos. dh. 2, . (27)

ae —V'+ 4 (o? + 200 sin ¢) (a? +y*) + 20? cos’ p . a? + a
(28)

as the equation showing the distribution of pressure. This
motion differs from that first discussed in not being exactly
horizontal. The plane of motion of each particle of fluid
passes through the line OX, and is inclined to the horizontal
plane XOY at an angle @ given by tan6=20 cos ¢/o.
When the angular velocity of rotation w is very large com-
pared with © cos@, the plane of motion of each particle is
practically horizontal, and the motion then corresponds very
closely with that of simple rotation of all the fluid as a solid
about a vertical axis. When @ becomes small, on the other
hand, the inclination of the plane of motion of each particle
of fluid to the horizontal increases. The two motions, repre-
sented by (24) and (27) respectively, are almost identical
when @ is very large, and they differ entirely when @ is
very small. It would be interesting to investigate the
manner in which a fluid, such as water, subsides to rest
from an initial condition of steady rotation about a vertical
axis. The solution represented by (24), (26), would appear
to be the exact solution for steady rotation of water in small
scale experiments.

60 Dr. G. Green on Fluid Motion

The motions considered above are motions of any incom-
pressible fluid and do not indicate, except as approximations,
conditions of motion possible in the atmosphere. One or
two solutions of a similar type can be obtained which refer
to incompressible fluid and accordingly represent motions
possible in the atmosphere. Consider now the motion repre-
sented by

u=—o(y—62)3; v= 0 =O) ae

The equations of motion of any element of fluid in this
case are

p28 242555, ee

R Ox
—20 sing .o(y—A2)=—% — $ log Pengo)
SO eae eo e= g —h2-log p, Mice

and these equations are satisfied provided the pressure
system throughout the fluid is that indicated by

; d v+y?

k log p=Qsin d. o(y—Bz)? —9 ap +2) +hlog po (33)
where B=cot.¢, and po is the density of the fluid at the
reference point O. The continuity equation is also satisfied
provided we can neglect the term (ga/R)a(y— Bz). This
condition limits considerably the extent of the region
around O to which our solution is applicable, as stated
earlier. Within the region to which the above applies the ©
isobars at the surface of the Earth run East and West,
being determined by

klog p=Qsing.ay?+klog po, . . . (84)
which indicates a system symmetrical on the two sides of
the Hast and West line drawn through reference point O.
In this case the isobars become closer as we proceed North
or South from point O. They are also parallel to the lines
of flow of the fluid.

The coefficient = has the value 1:5x 107° with the foot

and the second as units; while 20sing has the value
1:03 x 1074 at latitude 45°. Certain cases of interest arise
in which the terms containing may be neglected. For

R

relative to a Rotating Earth, 61

example, we may take the case of a uniform east or west
wind over a considerable region. In this case

eee Bae Me Oya tn tr. aN (3D)

and the pressure distribution consistent with this motion is
represented by
klog p=2Qc (cos 6 z—sin g y) —gz +h log po, . (36)
where py refers to the density of air at the reference point O.
The isobars at the Earth’s surface are in this case a uniform
system running due Hast and West.
A similar case is that given by

u=0, v=cer+d, w=0, .. . (87)
which represents a wind towards the North, while the corre-
sponding pressure distribution is that represented by

klog p=Q sin ¢ (ca? + 2dx)—gz+klog py. . (38)

The isobars are again a system of straight lines, but running
north and south, and uniformly spaced when c=0.
In a similar manner we find that

U=c, V=c, we=0, . - . . (39)
corresponds to a system of straight isobars represented by
k log p=20 sin $ (qx—cyy) + 20 cosh .cye—gz + k log po. (40)

The isobars are again lines of flow of the air, as in each case
considered above.

The case of motion corresponding most closely to a
cyclonic or anticyclonic circulation is that discussed in an
earlier paper *, represented by

u=—o(y—Pz); v=er; w=0..., (41)
In this case
20 cos

wpeOsaran ce e Y AD)
and the pressure distribution is that represented by
klog p=3(@" + 2@Q sin $) {2 + (y—Bz)?} —gz +h log py.
(43)
J

The term oe must be small in order that the continuity

C=

equation may be fulfilled.
* Phil. Mag. vol. xli. April 1921 ; vol. xlii. July 1921.

62 Prof. G. N. Antonoff on the

In each case considered we have only to replace 0 by an
increased value ©' to obtain a motion for which the system
of isobars travels eastward at a uniform speed (Q'’>92).

Hach of the above solutions has been given to apply to an
isothermal atmosphere, and in every case considered the
fluid moves so that no element of fluid undergoes change of

; ee + ou = oy =( ateach point. Provided
OL Oy, Oe
this condition is fulfilled, any solution obtained for motion of
an atmosphere, all at one temperature, can readily be trans-
formed to suit an atmosphere in convective equilibrium
(p=kp’), or one in which pressure is any given function of
density. Thus, taking the motion represented by (41) above
in an atmosphere in convective equilibrium, we have merely

density. That is

to replace log p and log py in (43) by re and wee
respectively, all other conditions being unchanged.

VI. The Breaking Stress of Crystals of Rock-Salt.
By Prot. G. N. Antonorr, D.Sc.(Manch.) *.

i a paper published in Phil. Mag. vol. xxxvi. Nov. 1918,

I have developed a theory of surface tension under the
assumption that the attraction of molecules is due to electrical
or magnetic forces, or both. Instead of assuming a uniform
field round the molecules as it is generally accepted acccording
to Laplace, I accepted the view that the molecules act as
electrical doublets, and from the theory of potential I
deduced that the attraction between them must be inversely
proportional to the 4th power of the distance, provided the
distance between the doublets is large compared with their
respective lengths.

It was shown that the attraction between the doublets can
be represented by an expression of the type

ke
dt ?

where & is a constant, | the length of the doublet, and d the
distance between them. In these calculations the magnetic
forces were disregarded altogether, as the law of attraction
between small magnets would be just the same, so that they
could only have an effect on the value of &,

* Communicated by Dr. J. W. Nicholson, F.RS.

Breaking Stress of Crystals of Rock-Sult. 63

For the surface tension, the expression was given as

k? 1
— | ¢
Cie tis
or assuming qe =P where p is the number of molecules per
; ]
unit volume, the expression for the surface tension « becomes
N18 25/8
al ‘pp! 5
61 — 5s
Ma
the liquid, 6, that of the saturated vapour, and M the
molecular weight.

LOS NEE
Thus wait (“5°*) re eRe iter oC)

It was also shown that the internal pressure P can be
calculated by the formula

Bee ame ae oc ene)

instead of p we may put where 6, is the density of

= 1/3 ;
or P=22 a5) ae aru dh wie) ae

In other words, the intrinsic pressure can be calculated
from the surface tension if the molecular weight of the
liqnid is known. Thus for normal or non-associated liquids
it should be possible to calculate the normal pressure from
the value of the surface tension,

It should be pointed out that the assumption made in the
above theory, that the length of the doublet is small compared
with the intramolecular distance, is not necessarily the right
one.

In the expression (2) this law is, however, eliminated, and
the same expression is obtained for any law of molecular
attraction. ‘The figures for P obtained from the above
expression agree with those from indirect evidence. How-
ever, experimentally it is not possible to determine P
directly, owing to the mobility of the particles of liquids
which always adjust themselves so that the molecular
pressure is inappreciable.

It is not so in the solid or crystalline state, in which the
particles have a definite orientation, and where the internal
pressure can be determined by a direct experiment. It is
sufficient to apply to a crystalline body such a weight as
would overcome the attraction of the molecular forces and
cause the disruption of the body. The force applied is not

64 Prot. G. N. Antonoff€ on the

necessarily the same in all directions, and it is therefore
necessary to specify the direction in which it is to be applied.

The question arises now whether it is possible to calculate
the surface tension of a solid body.

For the solid state there is no direct method of determining
the surface tension, all methods used for liquids being
inapplicable in this case. Some attempts were made to
estimate the surface tension of solids from indirect evidence. ©
For example, Ostwald * and Hulett t calculated the surface
tension of some calcium and barium salts on a basis of a
certain theory from the solubility data. The figure given
for the latter is about 4000 dynes percm. From the point of
view of our theory, it seems possible to calculate the surface
tension by the use of formula (2) by determining experi-
mentally the internal pressure per square cm. of the cross-
section, if the molecular structure of the substance in the
crystalline state is known.

At the present time the X-ray analysis throws a light on
the above question. |
- For example, according to W. H. and W. L. Bragg, the
erystal of rock-salt consists of charged ions situated at regular
distances from one another.

Such a case is somewhat different from the one discussed
in my paper (loc. cit.). Here it is necessary to assume that
l=d, where / is the length of the doublet and d the distance
between them, under which conditions the ordinary inverse
square law must hold true. The attraction between the
charges in a row is equal to

e*k

dq?
where ¢ is the elementary charge, and the value of & is the
sum of a series L—$+4—}43—...=0°6931.

Assuming that the adjacent rows have no effect upon the
charges, the expression for the surface tension is of the form

2
[a=ekp,
where p=number of particles per unit volume.
For the normal pressure the expression will be
: P=khe*p*,

* Zeit. Phys. Chem. xxxiv. p. 503 (1900).
+ Zeit. Phys. Chem, xxxvii. p. 386 (1901).

Breaking Stress of Crystals of Rock-Salt. 65

For rock-salt
the density = 2°15,
M164 ¥5S'o: x. 10-73,
— 2 2 yA x TO",
Assuming that & is approximately=0°7, the tension is
BS cOrt aed 10% 29 x 10-28 x, OT x 2°24. « 107?
=3500 dynes per cm. (approximately).

The figure obtained is of the same order of magnitude as
figures derived by Ostwald and Hulett (loc. cit.) for barium

salts.

The normal pressure P would be accordingly

P=98'7 x 10° dynes per square cm.

It is interesting to see now how far the above results
agree with the experimental evidence.

An experiment was performed as follows :—

I took a good specimen of rock-salt crystal and I cut a
prism of the section about 15 square mm. and about 2-3 cm,
long. I used a suitable cement to hold the piece from both
ends, and by applying a suitable weight produced a rupture
of the crystal into two halves. Measuring the cross-
section of the rupture accurately, I calculated the weight
required to produce the rupture per square cm. I have
repeated the experiment many times with different samples
of rock-salt. I£ the crystal is well formed, the agreement
between individual experiments is fairly good. In one
series of experiments, I cut the prisms so as to have several

samples cut parallel to the three principal axes. I have done
the experiment with three pieces for each direction. For
one direction I obtained :

89 lb. per cm,
BOS 45 1155
83 39 33

In the other two directions the results were identical.

In some cases it happened that the rupture took place
under a much smaller weight. This, however, could be
attributed either to some faults in the structure of the
crystal, or to some other disturbances. Such measurements
were simply disregarded.

Taking as the average value 91°7 lb., or 41°5 kegrs. per
square cm., one can calculate the inward pull per row of unit
length.

Phil, Mag. 8. 6. Vol. 44. No. 259. July 1922. F

66 Breaking Stress of Crystals of Rock-Salt.
This is obtained by dividing P by p'*, and becomes

P _ 415000 x 981 aa ayn Bap
Bert Fert ee

If the attraction between the charges in a row is not
appreciably influenced by the adjacent rows, this value will
represent the accurate value of surface tension in the direction
coinciding with the vertical axis. If the field is symmetrical
in all three directions parallel to the main axis, this figure
will characterize the surface tension of rock-salt in all
three directions.

The symmetrical structure indicated by W. H. and W. L.
Bragg is in accord with the experiment in this sense, but on
the other hand, the above figure is about 1000 times less than
one would expect.

One could expect such a small value if the salt consisted
of molecules with a very small polarity situated at large
distances from one another. But in such a case the force

in the direction coinciding with that of the doublet would

have to be twice that in the perpendicular direction. ‘This,
however, is not the ease.

In my paper (loc. cit.) it was shown that in a case of small
doublets the adjacent rows have practically no influence on
the attraction between the doublets.

However, in a case of charges situated at regular distances
from one another, such seems not to be the case.

If the charges influence one another, one can expect the
forces to be weaker in the middle of the substance, and much
bigger at the surface where the above effect is only one-
sided.

It is therefore probable that the above figure 1:4 dynes

per cm., although quite characteristic for the substance, is

not the actual value of the force in the surface layer. The
calculation of these effects is not easy, owing to the fact that
one has to deal with a very slowly converging series. I[
satisfied myself that these influences may be appreciable, but
I do not see clearly at the present time whether they can
account for the weakening of the forces about 1000 times,
or even more.
6 Featherstone Buildings,

High Holborn,
London, W.C. 1.

VIL. The Measurement of Chance.
By Norman Campsett, Sc. D.*

Summar Ys

T is maintained that the chance of an event happening

is always a physical property of a system, measured

by a process of derived measurement involving the two

fundamental magnitudes—number of events and number
of trials.

Chances are not measurable by a process of fundamental
measurement. But the calculation of chances is analogous
to fundamental measurement. It is usually theoretical, and
is valuable only in so far as the calculated chances are
confirmed by measurement.

When a proposition concerns a system characterized by a
chance, it may sometimes (but by no means always) be
regarded as having a definite probability determined by that
chance. The probability of propositions which do not
concern systems characterized by chances has nothing to do
with chance.

Ll. It is generally recognized that there are two kinds of

“probability.” There is (1) the probability (of the

happening) of events, and (2) the probability (of the truth)

of propositions. tymologically the term belongs more
properly to the kind second of probability, and it will be
confined to that kind in this paper. For the first kind, the
term “chance,” often used in some connexions as a synonym
of probability, is available. Accordingly we shall speak
throughout of the chance of an event happening and of the
probability of a proposition being true.

Various opinions have been entertained concerning the
relation between chance and probability and between the
methods of measuring them. Some have held that chance,
some that probability, is the more fundamental conception,
and that the measurement of the less fundamental depends
on that of the more fundamental conception. Others have
held that only one of the two, or neither, is measurable.
The conclusion towards which this paper is directed is that
chance, in the sense primarily important to physics, is a phy-
sical property measurable by ordinary physical measurement.
This view is similar to that held by Venn; indeed, it is

* Communicated by the Author.
2

68 Dr. Norman Campbell on the

probably the view that Venn would have held if he had ever
considered the nature of physical measurement. But the
further view often attributed to Venn, though it is doubtful
whether he actually held it, that probability is always
measurable in terms of chance—this view will not be upheld,
but, so far as it is discussed at all, will be combated.

Many of the ideas and terms used in the discussion are
explained more fuliy in my ‘ Physics,’ to which references are
made by the letter P. In fact, this paper may be regarded as a
substitute for pp. 168-183 of that book, some of the difficulties
of which are avoided by the alternative method of treatment.
However, I should like to add that I do not accept any of
the criticisms that have been directed against those pages by
others. |

2. Suppose we are presented with a pair of dice, and asked
what is the chance that when one of them is thrown it will
turn up six. The answer may be different for the two dice.
If one of them is accurately cubical with its centre of mass
accuraiely at the centre of the cube, while the other has
corners and edges variously rounded and is loaded so that
the centre of mass is appreciably nearer one face than
another, then the answer will be different. On the other
hand, if in all respects the dice are the same, then the
answer will be the same; even if they are both inaccurate
in form and both loaded—the inaccuracy of form and the
loading being the same—the chance that they will turn up
six will be the same. This chance is something uniformly
associated with and changing with the structure of the die,
just as is (say) the electrical resistance. This uniform
association of the resistance with the other characteristics
of the die is what we assert when we say that the resis-
tance is a physical property of the die, and accordingly
the chance of turning up six is a physical property as much
as tle resistance. 7 |

Moreover, the chance is measured by essentially the same
process as that by which the resistance is measured. Resist-
ance is measured (in its original meaning) as a derived
magnitude by means of a numerical law (P. Ch. xiii.), We
place two electrodes in contact with opposite faces of the
die, and measure the current which flows through it when
measured potential differences are maintained between the
electrodes. We then plot current against P.D., and find we
ean draw a straight line through (or more accurately
among) the resulting points. The fact that the graph isa

Measurement of Chance. 69

straight line passing through the origin shows that a
numerical law of a certain form holds, and therefore that the
die is characterized by a single definite magnitude, which is
what we mean by resistance ; the slope of the line tells us
the numerical value of this magnitude. When we proceed
to measure the chance of turning up six, we make several
groups of trials, measure in each group the number of trials
and the number of those in which six turns up. We plot
these two fundamental magnitudes against each other, and
find that a straight line can be drawn through (or among)
the points. The fact that the graph is straight and passes
through the origin tells us that the die is characterized
by a definite magnitude, which is what we mean by chance ;
the slope of the line tells us the numerical value of this
magnitude.

3. The resemblance is exact in all essentials. But as the
conclusion that chance is an ordinary physical magnitude
does not seem to be universally accepted, some objections
may be considered.

The first may be (though I am not sure that it will be)
raised by those who denounce the “frequency theory” of
probability. They might say that, though the derived
magnitude, estimated in the manner described, isa true or
approximate measure of the chance, yet it is not what is
meant by the chance—that is something much more abstruse.
Such an objection can only be met by stating more clearly
what is asserted, and recognizing any difference of opinion
that remains as insoluble. What I assert is (1) that all
chances determined by experiment are determined by a
relation between frequencies, and (2) that chances are
important for physics only in so far as they represent
relations between frequencies. Few examples can be cited
in support of (1), for chance in physics is usually a theo-
retical and not an experimental conception ; but it may be
suggested that anyone who proposed to attribute to the
chance of a given deflexion of an a-ray in passing through
a given film any value other than that determined by fre-
quency, could convince us of nothing but his ignorance of
physics. In support of (2) it may be pointed out that
the chance, which is such an important conception in the
statistical theories of physics, enters into the laws predicted
by those theories only because it represents a relative
frequency.

70 Dr. Norman Campbell on the

4, A second objection may be based on the fact that the
straight line has to be drawn among and not through the
experimental points. It may be readily admitted that this
fact shows that the chance cannot be estimated with perfect
accuracy. but there is also some uncertainty in determining
the resistance ; and since I am concerned only to enforce the
analogy between chance and resistance, the admission is
innocuous. If it is urged that this uncertainty shows that
the derived magnitude cannot be the chance, because chance
is something to which a numeral may be attached with
mathematical accuracy, then it is replied (as in answering
the first objection) that such a chance, to which no experi-
mental error is attached, is something totally irrelevant to
physies. . |

But the objection may be put in a less crude form. It
may be urged that, in the matter of experimental error, there
is a fundamental difference between resistance and chance.
For in the latter, but not in the former, the error is something
essential to the magnitude ; we can conceive of a resistance
measured without error, but not of a chance measured with-
out error ; if all the points lay accurately on the line, then
the magnitude measured by its slope would not be a chance.
Again, there is a simple relation between the average error
about a point on the ‘‘chance”’ line and the co-ordinates of
that point ; while in the “resistance” line the relation is
much more complex, and depends on the exact method of
measuring the current and potential. All this is quite true,
and would be important if we were considering the theory of
chance or of resistance. There is a great difference in those
theories; we suppose that the ‘‘real” Ohm’s law holds between
the real and not the measured magnitudes of the current and
potential, while there is no real magnitude involved in the
chance relation. But we are not considering theory but
experiment; I am only asserting that chance is an exper!l-
mentally measured magnitude. The fact that the errors in
the two cases are differently explained does not affect the
fact that there are errors in both cases, and that the problem
of determining the derived magnitude in spite of these errors
is precisely the same.

5. Asa third objection it might be urged that the two
measurements are not really similar, because the chance is
not really determined by the slope of the line, but by the
ratio of the two numbers when they are sufficiently great.
Here is a misconception which it is important to correct.
If we know that the happening of the events is determined

Measurement of Chance. th

by chance, then it is true that we need only plot one point
on the line ; and the distribution of the ‘errors is such that
the relative error of a determination from a single point is
less the greater the number of trials involved. We shall
group all our observations together, so as to imake their
totai number as greatas possible. But similarly, if we know
that the material of the die obeys Ohm’s law, one observa-
tion is sufficient to determine its resistance ; and the accuracy
of the determination will be greatest if we choose the
measuring current within a certain range. An even closer
parallel would be obtained if we took in place of resistance
the derived magnitude, uniform velocity. If we knew that
the velocity was uniform, we should choose our time and
distance as great as possible, and determine the velocity from
this single pair of values without troubling to plot smaller
values.

But in order that determination by a single point should
be legitimate, we must know that the events really are
determined by chance, and the only test of chance is that,
when a series of points are plotted in the manner described,
the only regularity discoverable in them is that they lie
about the straight line. Their distribution about that line
must be random. Thus, to take Poincaré’s excellent example,
if the trials were made by selecting the first figure of the
numerals in a table of logarithms in the conventional order,
and the events were the occurrence of the figure 1, the
plotted points would lie on the whole about a line with
a slope of 1/10. But a regularity of distribution about
that line would be apparent; we should have a series of
points all lying above the line followed by a set all lying
below. If, on the other hand, we took the last figure of
the numerals, no such regularity would be apparent; the
distribution of the points about the line would be random ;
the events would be dictated by chance.

It is of the first importance to insist that in measuring a
chance we are picking out the only regularity that we can
find in some sequence of phenomena, leaving a residuum
which is purely random. Randomness is a primary con-
ception, incapable of further definition; it cannot be explained
to anyone who does not possess it. It is based, I believe, on
observation of the actions of beings acting consciously under
free volition ; and it is subjective in the sense that what is
random to one person may not be random to another with
fuller knowledge (P. p. 203). There are certain forms
of distributions that are random to everybody ; it is this
common randomness, objective in the sense in which all the

72 Dr. Norman Campbell on the

subject-matter of science is objective, that is the characteristic
of the objective chance which is physically measurable.
Chance is applicable only to events which contain an element
which is wholly and completely random to everybody *.

6. We shall then base our further discussion on the
assumption that any physically significant chance (of the
happening of an event) is a measurable derived magnitude,
a property of the system concerned in that event, determined
by a linear numerical law relating the fundamental magni-
tudes, number of events and number of trials. It is thereby
implied that the “errors” from the law are random, for
otherwise the law would not be linear. The definition of
chance as the limiting ratio of the fundamental magnitudes
as they tend to infinity is identical with that given, if it is
known as an experimental fact that the magnitudes of the
errors fulfil certain conditions which need not be discussed
in detail here; these conditions are not inconsistent with
the randomness of the errors.

Chanee as a fundamental magnitude.

7. Another important question may be raised, again
suggested by the analogy with resistance, Resistance means
the derived magnitude defined by Ohm’s law. But actually
resistance is measured nowadays, not as a derived, but asa
fundamental magnitude, in virtue of the Kirchhoff laws for
the combination of resistances in series and parallel f. Can
chance, though meaning the derived magnitude, be measured
independently as fundamental ?

In order that a property may be measured as a funda-
mental magnitude, it is necessary that satisfactory definitions
of equality and of addition should be found (P. Ch. x.).
In addition, some numerical value must be assigned arbi-
trarily to some one property, which with all others can

* In this sense the last figure of the logarithm is not wholly dictated
by chance; for we know that there must be some regularity in the
distribution of the points about the straight line, even if we cannot say
exactly what it is. In the strictest sense, therefore, there is no such
thing as the chance of the last figure being 1. But there are events
which are, at present at least, wholly dictated by chance in this sense,
e.g. the distintegrations of a radioactive atom. Here I do not think
anyone has imagined what kind of regularity there can be, except the
falling of the plotted points about the straight line which determines
the chance.

t+ Ultimately measured, that is to say, hy the makers who calibrate
our resistance boxes. In the laboratory we use a method which is
essentially that of judgment of equality with a graduated instrument.

Measurement of Chance. 73

be compared by means of these definitions. Since we take
the meaning of chance to be that of the derived magnitude,
the definitions will be satisfactory if they are in accord-
ance with the derived process of measurement; but we
shall not succeed in establishing an independent system of
fundamental measurement, unless the definitions are such
that they can be applied without resort to that process.

The arbitrary assignment is usually made by attributing
the value 1 to the chance of an event which always happens
as the result of a trial. The only question that can arise
here, namely whether all other chances can be connected
with this chance by addition and equality, will be considered
presently. The definition of addition presents no difficulty.
The chance of A happening is the sum of the chances of
x, y, 2,... happening if, wv, y, z, ... being mutually exclusive
alternatives, A is the event which consists in the happening
of either z or y, or z,..... This proposition is introduced in
all discussions of chance, but it is often introduced as a
deduction and not as a definition. The inconsistencies which
result from such a procedure are discussed in P. pp. 174, 184,
185. As a definition, it is satisfactory in our sense, for
measurements by the derived method would show that, in
such conditions, the chance of A is the sum of the chances of
, y, 2,...,and yet it does not presuppose such derived
measurements. If the points in the derived measurement
lay on the straight line, this result would be a direct
consequence of the definition of the derived magnitude ;
but since they do not, it can be deduced from that
definition only if some assumption about the distribution
of the errors is made. The assumption that the errors are
random would probably suffice if randomness could be
strictly defined; since it cannot, the agreement of the
proposed definition of addition with the results of the derived
process of measurement must be regarded as an experimental
fact. The definition is thus precisely analogous to that used
in the fundamental measurement of resistance, namely that
resistances are added when the bodies are placed in series.

The definition of equality is much more difficult ; in fact,
it is the stumbling block of many expositions of the measure-
ment of chance. [For resistances we can say that bodies are
equal if, when one is substituted for another in any circuit,
the current and potentials in that circuit are unchanged ;
that definition does not involve a knowledge of Ohm’s "law
and of the derived measurement. The only attempt at an
analogous definition for chance, of which I am aware, is
that based on the principle of sufficient reason; chances

74 Dr. Norman Campbell on the

are said to be equal when there is no reason to believe
that one rather than the other will happen as the result
of any trial. But what reason could there be for such
a belief based on experiment? No a priori principle can
determine the property of a system, which is an experi-
mental fact ; we cannot tell whether a die is fair or loaded
without examining it through our senses. The only experi-
mental reason I can conceive for believing that one event
is more likely to happen than another is that it has
happened more frequently in the past. But if an attempt
is made to define ‘“‘ more frequently ” precisely, the judg-
ment of equality is inevitably made to depend on the
derived measurement, and the fundamental process ceases
to be independent of it. This dependence is often con-
cealed by the use of question-begging words. Thus, the
principle of sufficient reason may be reasonably held to
decide that, in a perfectly shuffled pack of cards, the chance
that the card next after a heart is another heart is equal to
the chance that it isa club. But if inquiry is made what is
meant by a perfectly shuffled pack and how we are to know
whether a pack is or is not perfectly shuffled, I can seen no
answer except that it is one in which a club occurs after a
heart as often as another heart. But, of course, to define
perfect shuffling in that way is to admit that the criterion of
equality is based upon the derived measurement of
‘frequency’ *. I can find no proposed definition of the
equality of chances that is both applicable to experimental
facts and independent of frequency ; and I conclude, therefore,
that there is not for chance, as there is for resistance, a
fundamental process of measurement independent of the
derived.

8. But there is a further difference to be considered.
Even if equality of chance could be defined independently,
there would still be many chances (and those some of the
most important) which could not be connected with the unit
by the relations of equality and addition. Any resistance is
equal to the sum of some set of resistances such that the sum
of another set of them is equal to the unit or to the sum of
some set of units. The analogous proposition about chances

* It is not always realized by those who calculate card chances in
great detail that in actual play, even among experienced players, the
shuffling is so imperfect as to distort very seriously the chances of such
events as the holding of a very long suit.

Measurement of Chance. 75

is not true, if the chances are always experimentally deter-
mined. Consider, for example, the disintegration of a
radioactive atom within a stated period. There are only
two alternatives: the atom does distintegrate, or it does not.
The sum of the chances of these two events is 1, but the
chances of the two are not in general equal. And neither
of them can be shown experimentally to be the sum of the
equal chances of other events such that the sum of some
other set of those chances is equal to the unit. The definition
of unit chance together with definitions of equality and
addition would never permit us to determine such chances ;
they can only be determined by derived measurement.

9. Chance is therefore not capable of fundamental
measurement. Nevertheless the principles of fundamental
measurement are important in connexion with chance,
because they are involved in the calculation of chances.
When we calculate a chance we always assume that it is
measurable by the fundamental process. Thusif we calculate
the chance of drawing a heart from a pack, we ¢ argue thus :—
The chance of drawing any one of the 52 cards is equal to
that of drawing any other. The chance of drawing one of
the 52 cards is, by the definition of addition, the sum of the
chanees of drawing the individual cards, and, by the
definition of unit chance, itis 1. Consequently the chance
of drawing any one card is 1/52. Butthe chance of drawing
a heart is the sum of the chances of drawing 13 individual
cards ; it is therefore the sum.of 13 chances each equal to
13/52, i 2.e.1/4. The calculation is perfectly legitimate, so
long as we know (1) of how many individual events tlie
event under consideration (and any other event introduced into
the argument) is the sum, and (2) that the chances of these
individual events are equal. (1) does not depend on the
derived system of measurement, but it does involve a very
complete knowledge of the event under consideration; (2), if
it is an experimental proposition at all, must depend upon
derived measurement. The calculation is often made when
(2) is not experimental, and when there is no direct know-
ledge of (1); it is then purely theoretical, and the only
legitimate use that can be made of it is to confirm or reject
the theory by means of a comparison of the calculated chance
with that determined experimentally by the derived measure-
ment. The fact remains that true chance, the property of
the system, is always and inevitably measur ed by the derived
process and not by the fundamental.

76 Dr. Norman Campbell on the
Chance and Probability.

10. It remains to consider very briefly what connexion, if
any, there is between the chance of events and the proba-
bilities of propositions.

Probability is usually admitted to be an indefinable
conception, applicable to propositions concerning which there
is no complete certainty, and roughly describable as the
degree of their certainty. It appears to me one of those
conceptions which are the more elusive the more they are
studied ; | am quite certain that I do not understand what
some other writers mean by the term, and am not at all
certain that I can attach a perfectly definite meaning to it
myself. The observations that I can offer are therefore
necessarily tentative. But it is clear, at any rate, that
probability is not a property of a system and is not physically
measurable: any propositions connecting it with chance
must depend ultimately on fundamental judgments which
can be offered for acceptance, but cannot be the subject of
scientific proof.

There are two kinds of propositions the probability of
which may plausibly be connected with chance ; and they
naturally can apply only to systems that are characterized by
chances. Of the first kind the following is typical :—This
die will turn up six the next time it is thrown (or on some
other single and definite occasion). Here (cf. P. pp. 192-200)
it seems that, if the proposition is really applied to a single
occasion only, the probability of the proposition must be that
characteristic of absolute ignorance ; for the assumption that
anything whatever is known of the result of a single trial is
inconsistent with the experimental fact that the result of any
one trialis random. The only exception occurs when the
event is one of which the chance is so small (or so great) that
the happening of it (or failure of it) would force us to revise
our estimate of the chance or to deny that there was a chance
at all. Of such coincidences, in systems of which the chance
has been well ascertained, the assertion that they will not
(or will) occur may be made with the certainty that is
characteristic of any scientific statement. There is no
probability.

On the other hand, it is very difficult to be sure that only
a single trial is contemplated. For when such statements
are important, there is alwaysa clear possibility of a consider-
able number of repetitions of the trial. If this number is
so great as to permit a dermination, by derived measurement,
of the chance of the event within some limits relevant to the

Measurement of Chance. “a

problem, then it will be found by examination of the use of
such propositions that their importance depends simply on
the value of that chance. If that chance is greater than a
certain value, the proposition will be true for the purposes
concerned ; if it is less, it will be false. I cannot myself ever
find in such propositions any meaning which is not contained
in the proposition :—The chance that the die will turn up
six is greater or less than some other chance. Accordingly
again, there seems no room for a probability which is distinct
from chance.

11. Of the second kind of proposition an answer to the
following question may be taken as typical :—I have two
dice, of which the chances of turning up six are unequal.
I throw one, but I do not know which. It turns up six.
Which of the two dice have I thrown?

Here again (P. pp. 185-192), if the question is asked
of a single throw, it seems to me that the only possible
answer is simply, I do not know; except, as before, if the
throw would be a “ coincidence ’’ with one die and not with
the other. For, once more, if the events concerned are
really characterized by chances, it is inconsistent with the
statement that they are so characterized to assert that, at a
single trial, the result, if compatiole with either of the two
““causes,’ may not happen as the result of either of them.
If, on the other hand, the throws are repeated (while it is
certain that the same die is always used), and if they deter-
mine the chance of one die rather than that of the other, it
is clearly certain that this die, and not the other, is being
used ; «a die can be identified by its chance as certainly as
by its resistance or any other physical property. But
intermediate between these extremes, there certainly seem
to be cases in which, though the evidence is not sufficient
to enable us to assert definitely which die is being thrown,
we begin to suspect that it is one and not the other. ‘Lhe
possibility of such a state of mind arises from the fact that
there is necessarily a finite period during which the
evidence is accumulating; it does not arise when, as in
the usual determination of resistance, the evidence is obtained
all at the same time. And our suspicion will increase
generally with the “ probability ”’ as estimated by the well-
known Bayes’s formula for the probability of causes, In
this case it appears to me that there is such a thing as
probability, determined by but distinguishable from chance,
and applying to a proposition, and not to au event. But
I can find no reason to believe that this probability is

18 The Measurement of Chanee.

numerically measurable in accordance with Bayes’ or any
other formula.

But in most cases where an attempt is made to apply a
probability of causes, the condition is not fulfilled that it is
known that the same die is always used. If that condition
is not fulfilled, the probability, according to orthodox theory,
depends on certain a prior probabilities which are not chances.
The problem then ceases to be one of the connexion between
chance and probability, and thus falls without the strict
limits of our discussion.

12. But itis necessary to transgress those limits for one
purpose. It has been often urged by philosophers that
probability is characteristically applicable to scientific
propositions, which are to be regarded, not as certain, but
only as more or less probable. If this be so, the con-
ception of chance, being a scientific conception deriving
its meaning from scientific propositions, must be subsequent
to the conception of probability, and the order of our
discussion should have been reversed. Of course I do not
accept the philosoph:eal view, and perhaps it will be well to
explain very briefly why I reject it.

Doubtless there is a sense in which scientific propositions
are not certain ; but in that sense no proposition is certain,
so long as its contrary is comprehensible. For if I can
understand what is meant by a proposition, | can conceive
myself believing it. Iam not perfectly certain ae that
Ohm’s law is true, or that (o+a)?=a?+2ax+a?: I can
conceive myself disbelieving either. If I were Focestl to say
which I believe mere certainly, I should choose Ohmn’s law ;
for I could give a much better account of the evidence on
which I believe it. A mathematician, of course, would mike
the opposite choice. But it appears to me useless to com-
pare the “certainties”’ of two propositions when they are of
so different a nature that the source of the uncertainty is
perfectly different. If a proposition 1s as certain as any
proposition of that nature can be, and if nothing whatever
could make it more certain, then it seems to me misieading |
to distinguish its pr obability from certainty.

Now, fully established scientitie propositions are ecrtain in
this sense. They are uncertain only in so far as they predict
If in asserting Olim’s law, I mean (and I think this is my
chief meaning) that it appears to me a perfectly complete
and satisfying interpretation of all past experience and that
other prone appear to share my opinion, then Ohm’s law is

Ignition of Gases by Sudden Compression. 79

perfectly certain, or at least as certain as any mathematical
or logical proposition. Onthe other hand, if it is meant that
the law will be to me and to others an equally satisfying
interpretation of all future experience, then [I am _ not
absolutely certain ; I am only as certain as I can be about
anything in the future. And it must be noticed that nothing
can make me more certain. If I were predicting something
about a single future occasion, I might in the course of time
become more certain; for that future occasion might some day
become past. But if, as in the case of a scientific law, I am
predicting something about all future experience, then, since
the future is indefinite, no amount of additional experience,
converting finite portions of the future into the past, can
make me more certain; for there will always remain as much
future as before. Such uncertainty as there is in the
proposition is inherent in its nature; if it were absolutely
certain, it would not be the same proposition,

VIII. EHvperiments on the Ignition of Gases by Sudden
Compression. By H. T. Tizarp and D. Rh. Pye*.

[Plate I.]

i. a a previous paper J, it was shown that when a mixture

of a combustible gas or vapour with air was suddenly
compressed, explosion might take place after an interval the
duration of which depended on the temperature reached by
the compression. It is known that below a certain tempera-
ture, called the ignition temperature, no explosion, and no
very appreciable reaction, takes place under these conditions ;
and the experiments referred to showed that just above the
ignition temperature, the delay before explosion occurs may
be of the order of one second in certain cases, while—in the
case of hydrocarbons and air—the delay at a temperature
some 50° above the ignition temperature was very small.
It was pointed out that the observed ignition temperature
must not only depend on the properties of the combustible
substances, but also on the conditions of experiment, and
particularly on the rate of loss of heat from the gas at the

* Communicated by the Authors.

+ H. T. Tizard, “The Causes of Detonation in Internal Combustion

zines.” Proceedings of the N.E. Coast Institution of Engineers and
Shipbuilders, May 1921,

80 Messrs. H. T. Tizard and D. R. Pye on the

ignition temperature. The fact that this has not been fully
taken into account previously seems to account in some
measure for the differences in the results obtained by other
workers. It was further shown that the period of slow
combustion before explosion took place also depends on the
properties of the combustible substance, and a theory was
briefly developed connecting the “delays” observed at
different temperatures with the effect of a risein temperature
on the rate of combustion, 7. e. with the so-called tempera-
ture coefficient of the reaction. The object of the experi-
ments described in this paper was to test these theories
quantitatively, and to attempt to deduce from the results the
temperature coefficients in certain typical cases. The mea-
surement of the temperature coefficients of simple gaseous
reactions is of considerable importance in connexion with
the theory of chemical reactions, for one of the great
difficulties in the development of theory hitherto has been
the fact that most gaseous reactions have to be investigated
under conditions which are complicated by the disturbing
influence of solid catalysts or of the walls of the containing
vessel. Gaseous reactions which occur on sudden com-
pression are free from this complication, for the walls of the
containing vessel are much lower in temperature than the
gas; by quantitative measurements of the rate of loss of
heat near the ignition temperature, and of the delay before
explosion occurs, it therefore seems possible to gain some
real insight into the mechanism of homogeneous gas reactions.
Experiments of this nature also havea considerable practical
interest for the development of internal combustion engines,
for, according to our views, the tendency of a fuel to
detonate at high temperatures depends not only on its
ignition temperature but also on the temperature coefficient
of its reaction with oxygen.

Previous experiments on the ignition of gases by sudden
compression have been made by Falk, at Nernst’s suggestion
(J. Amer. Chem. Soe. xviii. p. 1517 (1906), xxix. p. 1536
(1907) ), and by Dixon and his co-workers (see Dixon, Brad-
shaw & Campbell, Journ. Chem. Soc. 1914, p. 2027; and
Dixon & Crofts, p. 2036). Nernst first put forward the view
that at the ignition temperature the evolution of heat due to
the reaction was just greater than that lost to the sur-
roundings; but this suggestion has not hitherto been carried
further, since no previous workers have attempted. to
measure the rate of loss of heat near the ignition tempera-
ture. Further, previous work has been mainly confined to
the measurement of ignition temperatures of mixtures of

Ignition of Gases by Sudden Compression. 81

hydrogen, oxygen, and an indifferent gas. In such experi-
ments the interval which occurs at the lowest ignition
temperature between the end of the compression and the
occurrence of ignition is very small; under these conditions
an apparatus of the kind used by Nernst and Dixon gives
fairly satisfactory results. It is not well suited, however,
for experiments with other gases, such as the hydrocarbons,
when there may be an appreciable delay before ignition
occurs. In such cases it is of great importance to ensure
that the cylinder in which the compression is effected is as
gas-tight as possible, and that the piston is held rigidly in
position at the end of the compression stroke.

II. The apparatus used for our experiments was originally
designed and built by Messrs. Ricardo & Co. with a view
to determining the temperatures of spontaneous ignition of
various fuels used in internal combustion engines under
conditions which correspond closely with those obtaining in
an engine cylinder.

Fig. 1 shows diagrammaticaily the arrangement of the
mechanism. A very heavy flywheel A rotates quite freely
on the shaft B, and is kept spinning by an electric motor at
about 360 R.P.M. The shaft B carries between bearings
the crank D, and outside one bearing, the internal expanding
clutch ©, which can engage with the flywheel rim.

The piston E moves vertically in the jacketed cylinder F,
which has an internal diameter of 44 inches and can be
raised or lowered bodily in the heavy cast-iron casing of the
apparatus when the compression ratio is to be altered. The
length of stroke of the piston is 8 inches, and its motion 1s
controlled by the two hinged rods G and H of which the
latter is carried on a fixed bearing at K. The hinge L is

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. G

-

82 Messrs. H. T. Tizard and D. R. Pye on the

linked up with the crank pin by the compound connecting
rod N. That part of the connecting rod attached to the
crank pin is tubular and contains the sliding rod M attached
to the hinge L. A clip O carried on the sleeve can engage
with a collar on the inner rod and hold the latter rigid in
the tube. With the connecting rod I-cked as one link, the
crank is rotated by hand for setting the piston in its lowest
position. When a compression has to be made, tle clutch is
suddenly expanded by a hand lever while the flywheel is
running at high speed, clutch and crank are carried round
with the flywheel, and the toggle joint ELK is straightened
until the hinge L lies on the vertical line between the piston
centre and hinge K. At the moment when L is vertically
over K it comes up against a leather pad, and a clip comes
into action which holds it in this position. At the same
moment, too, the clip O releases the two parts of the com-
pound connecting rod, so that while the two rods G and H
are held in the vertical position to take the large downward
thrust of the piston when explosion of the compressed
mixture occurs, the crank, clutch, and flywheel are free to
go on rotating, and the shock due to destruction of the
momentum of the moving parts is reduced to a minimum.
The initial temperature of the gases in the cylinder can be
varied by means of a water jacket round the cylinder, and
the variation of pressure during and after compression is
recorded by means of an optical pressure indicator of the
Hopkinson type.

For the purposes of the present experiments it was
necessary :—

(1) To make the compression space above the piston
absolutely air-tight so as to eliminate all pressure
drop due to leakage.

(2) To arrange an accurate timing gear upon the
indicator so that pressure-time records correct to
about one per cent. could be obtained,

(3) To insert some kind of fan in the compression space
so that the effect of varying turbulence in the
compressed gas could be determined.

(1) Azr-teghtness of the Compression Space.
There were three points at which appreciable leakage was
liable to take place :—

(a) Round the sides of the main piston.
(b) Past the indicator piston.
(c) Round the fan spindle.

Ignition of Gases by Sudden Compression. 83

Of these, the first two were completely eliminated and the
third reduced to something quite negligible. Figures of
the actual leakage are given below.

The method of eliminating piston leak will be understood
from fig. 2. The piston is made in two parts. Below the

FIG 2

cast-iron top, which carries a single piston ring of the usual
type, there is an aluminium body which for some distance is
of smaller diameter than the cylinder. Round this waist
are two cup-leathers, C and D, each with the periphery
turned upwards, which are separated by a cast-iron ring of
square section (H, fig. 2). The whole space above and
between the cup-leathers was filled with castor oil so that
no air could leak down past the piston head until it had first
made a space for itself by forcing castor oil down past the
two cup-leathers in series.

To eliminate lek past the indicator piston, the plan was
hit upon of turning the whole indicator upside down and
then pouring in a little castor vil into the space above the
piston, as illustrated in fig. 3. Here A represents the piston
of the inverted indicator with the space above it filled with

G 2

84 Messrs. H. T. Tizard and D. R. Pye on the

castor oil. The duct BC in the bracket which carries the
indicator is based on the slope to prevent the possibility of
any oil getting back along it and into the combustion space
at C, This arrangement was very satisfactory, for, besides
forming a perfect seal, it served to keep the piston well
lubricated ; it was found, moreover, that only a very small
quantity of oil was forced down past the piston, even after
continued exposure to high pressure.

The arrangement of fan and fan-spindle is shown in
fig. 4. Here the plug A, shown in section, was made to
screw vertically* downwards in the centre of the cylinder

PACKING

— CYLINDER —

head, its lower end finishing flush with the top of the com-
pression space. At the highest compression ratios this space
is 3 inch deep, so that the piston head under these conditions
is close up to the under side of the fan. The fan was driven
at varying speed by an electric motor through the usual
form of flexible drive used for speedometers.

To prevent leakage up round the fan-spindle, this was
made long and thin, and was also provided with a cup-leather
of which the tightness could be adjusted by the screw B in
fig. 4. By using a motor with ample reserve of power for
driving the fan, it was possible to keep the cup-leather so
tight as to reduce the leakage to a negligible amount. —

Ignition of Gases by Sudden Compression. 85
Experiments on Air Leakage from the Cylinder.

Observations were made of the rate of fall of pressure
beginning 10 seconds after compression had occurred. By
this time all pressure-fall due to heat-loss has ceased, and the
observed fall represents leakage only.

Experiments were made—

(a) With fan stationary. Compression ratio 9:1.
Contents of cylinder were the products of com-
bustion of a benzene and air mixture.

se Deflexion on Indicator
Time.

Screen.
10 secs. after compression and combustion ...... 0°63 inches,
20 r : Satahy Bisse OGL) 3:
30 : - mA We EL Ulsan airs (CRE ee
45 : fe any ae Ain ya oe O56?)
60 B A Stil Bia nes Ooar

The indicator calibration was 1 inch deflexion= 188 lb.
per sq. in., so that the above gives a rate of fall of
0-1 x 188

50
(6) With fan rotating at full speed, 2000 r.p.m. Com-
pression ratio 10:1. Contents of cylinder pure

= *38 lb. per sq. in. per sec.

hydrogen.
Ti Deflexion on Indicator
ime. 5
creen.
10 secs. after CoOMpressiON ............seeseseeeserees 0°66 inches.
20 ” oT Oo 0°53 Pe
30 9 59 HAR tee twee eter eeereeres 0°40 a

This, owing to the low viscosity of hydrogen and the high
compression ratio, was a very severe test. The rate of fall was
26 x 188
20
riment the rate of fall of pressure due to cooling was as high
as 600 lb. per sq. inch per second immediately after com-
pression, so that the effect of.gas leakage on the apparent

rate of cooling was clearly negligible.

= 2°45 lb. per sq. in. per sec. Now in this expe-

The Indicator.

It will be convenient at this point, before going on to
describe the timing apparatus, to give some further data as
to the indicator used. As stated above, this was of the

86 Messrs. H. T. Tizard and D. R. Pye on the

standard Hopkinson optical type *, and need not be described
in detail. The piston used throughout the experiments was
one of 0:125 sq. in. area. Pressure on the piston deflects
a spring attached to it, and thus tilts a small mirror which
reflects a point of light from a fixed lamp. The magnitude
of the pressure is thus arranged to correspond to the vertical
(downward) deflexion of the image of the light on a 44 x 33
photographic plate carried in a camera fixed to the indicator.
The calibration of the pressure scale was made in two.
ways. Firstly, by subjecting the piston to known oil
pressures and measuring deflexions on the camera screen,
and, secondly, as a check on the piston area, by direct dead-
weight loading of the spring and measurement of the
deflexion. The oil pressures in the first calibration were
produced by a carefully gauged vertical plunger loaded with
weights. The calibration by this method was carried out at
the beginning of the experiments, and it was found that the
relation between pressure and deflexion was given very
closely by a straight line at a slope corresponding to a
pressure rise of 188 lb. per sq. in. per inch deflexion. At
the conclusion of the experiments this catibration was
checked by the dead-weight calibration as follows :—

Reading on Camera Screen.

Load on Spring. Loading. Unloading.

| 0 aes 0°39. in. 0:40 in.
Weight carrier = 3°25 Ib. ...... ODA: 0-54 ,,
Carrier - 10) = 13°257 ees: 0-975 ,, 0:99 ,,
PEO OBI a Men ik 1:40, 1:43 ,,
O30 == 8825) 6 a IESE habe 1:81

This also shows a satisfactorily straight-line relationship
which checks remarkably well with the previous calibration,
for D : 33°2 93-4 |b

ead weight per inch deflexion = ao 3 :
corresponding to a pressure on a piston of area 0°125 sq. in:
O18 x 23:4 = 15722 ib

The previous figure of 188 lb. per sq. in. per inch de-

flexion has been used throughout the calculations.

(2) The Timing Gear.

The pressure in the cylinder being given by the vertical
motion of a point of light on a photographic. plate, it was
necessary for the accurate measurement of the lengths of
the delay period to give the point of light at the same time

* Hopkinson, Proc. Inst. Mech. Eng. Oct. 1907, p. 863.

Ignition of Gases by Sudden Compression. 87

a uniform and exactly known motion across the plate
horizontally. To obtain this uniform horizontal velocity, the
arrangement in fig. 5 was adopted.

A is a vertical cylinder closed at the bottom and carrying
at the top a guide for the piston rod B. At the lower end
of the rod B is a light loosely fitting piston in which are
drilled one or more small holes. The rod B has a collar C
above the guide, and can be loaded with weights, up to
50 lb. if need be. The cylinder is filled with paraffin
which, when the piston and weights have been raised and
then released by a trigger arrangement, is forced at high
velocity through the holes in the piston from the under to
the upper side. This arrangement gives a velocity of fall of

the piston which is uniform to within 1 or 2 per cent., and
it was found, moreover, that this velocity varied directly as
the square root of the weight carried: a result which shows
that the viscosity of the oil is a negligible factor in deter-
mining the rate, and that the latter will therefore not be
attected by any small changes of temperature which might
occur from day to day in the oil.

This point was checked experimentally, and it was found
that the maximum change in the rate of fall produced by
heating the oil from 19° C. to 65° C. was 14 per cent. As
the oil temperature during the experiments never varied by
more than 2° or 3°, the effects of temperature changes were
quite negligible.

Piston and weights are suspended by a steel wire from an
arrangement of pulleys (F) carried on spindles attached to

\
iii

88 Messrs. H. T. Tizard and D. R. Pye on the

the cylinder head. D and E are balance weights to keep
the wires taut, and F is a compound pulley which reduces
the horizontal motion of the wire G to about one-half the
vertical movement of the piston. At H the wire G is
divided, and each half wraps round the periphery of a sector
attached to the arm K of the indicator. The barrel of the
indicator is thus uniformly rotated through a sector about a
vertical axis, and the speed of the point of light horizontally
across the photographic plate in the indicator camera can
be varied between wide limits by alteration of the weights
and the number of holes in the loaded piston through which
the paraffin is forced. Actually it was found in these
experiments that only two speeds were required, of 5-77 and
4-30 cm. per second. These were obtained when the piston
carried weights of 40 and 24 lb. respectively, and was
pierced by a single hole } inch diameter. :

Both the speed and its uniformity were measured by in-
terrupting the light at. known intervais while it traced a
straight horizontal line across the plate. To obtain the
uniform interruptions, 60 equally spaced contacts were
arranged round the periphery of a disk about 6 inches in
diameter. An electric motor was used to rotate the disk
through a friction clutch, the speed being kept steady by
the operation of a governor, which caused the disengagement
of the clutch when the speed tended to increase. The
arrangement was very similar to that of a gramophone
motor, except that the speed of the latter is kept steady by
means of a little brake which is operated by a governor,
whereas in our experiments the governor operated the clutch.
This apparatus, which was made at the Royal Aircraft
Establishment, Farnborough, was lent to us for these experi-
ments by kind permission of the Director of Research, Air
Ministry.

To calibrate the falling weight apparatus, the speed of
revolution of the disk was first adjusted roughly to one com-
plete revolution in 2 seconds. It was then timed repeatedly
over twenty complete revolutions by stop-watch ; the mean
time for this number of revolutions was found to be 36:5
seconds, the variations in successive timings not exceeding
‘4 second. Hence, since sixty contacts were made each
revolution, the time between the beginnings of successive

36°5
contacts Was 90 = 0:0304 second. The 6-volt lamp of
the indicator was then connected up, through the contacts
on the disk, to a 12-volt battery; it was thus greatly

“overrun” whenever contact was made. The effect of

Ignition of Gases by Sudden Compression. 89

overrunning the lamp in this way was to make it flicker
brightly, so that when the indicator mirror was rotated by
the dropping weight, the reflected light made a horizontal
line of dashes across the photographic plate, the interval
between the middle of two consecutive dashes corresponding
to the interval between two contacts on the uniformly
rotating disk. The magnitude and uniformity of the velocity
of rotation of the indicator mirror by the falling weight
could thus be measured. Records were taken with falling
weights of 8, 24, and 40 lb.

To show the uniformity of motion during descent, the
results of measuring two plates taken with the 24-lb. load
are given in full. The apparent variations in any single
experiment are due rather to the difficulty of estimating the
centre of a flicker to 01 mm. than to any real variation in
the rate of fall of the loaded piston.

Puate A. Puate B.
5 flickers in 0°70 cm. 0°65 cm.
10 1°34 1:30
15 2:0 1:95
20 2°66 2°6
25 3°30 3°28
30 3°98 3°95
35 4°63 4°58
40 5:29 5:22
45 5°93 585
50 6:59 6°51
55 7:22 Fale,
60 7:90 7°78
65 8:53 8:40
70 9°21 9:06
75 9°89 9-70
Mean 1 flicker in 0:132 em. Mean 1 in 0°129 cm.”
1 flicker = '0304 second.
.’. 1 second = 4:34 em. on plate. 1 second = 4°26 cm.

Mean 4°30 cm. = 1 second.

Similar records with loads of 8 and 40 lb. gave time
scales of

2°22 cm. = 1 second
and 5°77 em. = 1 second.

The means for the two plates with 40-lb. load were 5:75 and
2°79 em. per second, so that it may safely be assumed that
the time scale was known correctly to within 2 per cent., and
probably less. This is a degree of accuracy as great as
that with which, as a rule, it is possible to measure the
records.

The vertical movement of the weights was about 7 inches,

90 = Messrs. H. T. Tizard and D. R. Pye on the

which produced a horizontal movement of the point of light
more than twice as great as was necessary to traverse the
photographic plate, so that ample time was provided for the
speed to have become uniform before a pressure record was
taken. The fact that the indicator and all gear except the
actual falling weight was carried on the cylinder head, made
it possible to raise or lower this when changing the com-
pression ratio without affecting the timing gear. The only —
difference was an alteration of about an inch in the distance
fallen by the weights between the position for maximum and
minimum compressions.

Figs. 6 and 7 are prints taken from two typical records.

Fig. 6.

Pressure . lbs. per S?. in. above abpospherce.

—

— > Jime in Seconds

In the first there is no explosion, and the record shows simply
the rise of pressure due to compression, and subsequent fall
as the compressed mixture cools. Fig. 7 shows the first
rise of pressure due to compression, the “‘ delay” period of
nearly constant pressure followed by the practically instan-
taneous rise on explosion, and finally the rapid cooling of
the intensely hot products of combustion. It muy be
mentioned that the spring of the indicator was protected from ~
the force of the explosion pressure by a stop which prevented
the deflexion from ever being greater than just to the edge
of the photographic plate.

III. If a gas at a known temperature and pressure is
suddenly compressed in a gas-tight cylinder, we can calculate
from a measurement of the maximum pressure reached the
average temperature of the gas at the moment of maximum

Ignition of Gases by Sudden Compression. oy

compression. If the volume then remains constant, measure-
ment of the rate of fall of pressure with time gives the rate
of loss of heat, if the specific heat of the gas is known. One
would expect this rate to be closely proportional at any
moment to the difference in the average temperature of the

fh
°o
°o

w
oO
ao

ny
oO
{o)

. | BAL) Ignition of Heotane and Air \
Maximum compression Termp. 10°C.

100

Pressure, 108, Per $9. 177, 2bove atmospherie pressyre,

i¢] 0-25 6-5 0-75 1G
Time 17 Secoro’s.

gas and that of the cylinder walls. If the temperature of the
gas is T° absolute, and that of the walls is 0, then

aT
— >, =a(T—6), . bein 8y (CE)

where “a” is a constant which we call the cooling factor,
and which depends on the nature of the gas, and its degree
of turbulence in the cylinder. The results given later will
be seen to justify this equation.
Integration ot (1) gives
log. Te 0 OE RS 2

where T is the average temperature when ¢ seconds has
’ 1
elapsed from the moment the average temperature was T).

92 Messrs. H. T. Tizard and D. R. Pye on the

Since the total volume remains constant, this equation can be
written
eer =
09, =e y
Jel

if the simple gas laws hold. Pyas, is here the maximum
compression pressure, P the observed pressure ¢ seconds after
the maximum compression pressure is reached, and Py the
final pressure of the gas when its temperature is the same as
that of the walls. Pyis therefore equal to 7X Pi, where 7 is
the’ compression ratio and P, the initial pressure (in these
experiments one atmosphere) before compression.

Fig. 6 is a typical cooling curve obtained when air at an
initial pressure of 14°73 |b. sq. in. and temperature of
23° C. was compressed in the ratio of 7°02 to 1. The time of
compression was 0°08 second, and the values of the cooling
factor a, obtained from observations of the fall in pressure,
are given in the following table :—

spat. iw al yee

TABLE I,
Time after Observed | naan Eye 2°30
max. pressure }, pressure= P | P—P» | log 3p =o. = ae
(seconds). lb./sq. in. ice,
0 1907 Fi S6sam
aie 185°7 | 82:3 _ 068 ‘90
B47 174°6 taal an ASK ‘87
“H2 | 165 616 194 "86
695 158°3 | 54:9 "244 “81
‘87 152°4 | 49 "294 ‘78
| Mean=:85
@ (initial temp.) =
P,=14'78 lb./ 23°C. Pp=Py x 7'02=103°4.
sq. in. Max. temp. (cale.) =
299°C.

It was found as a rule that the calculated cooling factor
tends to diminish as “‘¢”’ increases. This may be duein part
to errors of observation, for the errors are big when the time
interval after the attainment of maximum pressure is small ;
but experimental error will not wholly account for it, and it
may be explained on the reasonable assumption that there is
a fair degree of turbulence in the gas just after a sudden
compression, which dies down after a short time. Rotation
of the fan at a high speed increases the rate of cooling con-
siderably ; experiments with air gave a cooling factor about

Ignition of Gases by Sudden Compression. 93

three times as great when the fan was running as when the
gas was “stagnant,” This will be referred to later when
discussing the results of the experiments of ignition. Raising
the compression ratio also increases the cooling factor for
air. ‘his is also to be expected since the distance between
the top of the piston and the head of the cylinder is lessened.
The cooling factor for “stagnant” air in the apparatus was
found to increase from about 0°76 at a compression ratio of
6 to 1 to about 1:0 at a compression ratio of 10 to 1, the
distance between the piston and cylinder head being approxi-
mately 4 cm. (1°6 in.) at the lower and 2°3 cm. (‘9 in.) at the
higher compression ratio. The experimental error in the
cooling factors, obtained by experiments similar to that
quoted above, is probably about 5 percent. It should be noted
that the rate of loss of heat under the above conditions is
considerable. For instance, in the experiment quoted the
maximum difference in temperature between the gas and the
walls after compression is 276° ; and since the specific heat
Cy is about 0°18, the air is losing heat at the maximum
temperature at the rate of

0°18 x °85 x 276=42 calories per second per gram.

It follows that if an explosive mixture of gases is suddenly
compressed to its ignition temperature, in such an apparatus
as that described above, the initial rate of the chemical
reaction at the lowest temperature at which ignition is ob-
served must be considerable, for the evolution of heat due to
the reaction must equal approximately the rate at which heat
is lost to the walls. For example, the total heat of com-
bustion of a mixture of a paraffin hydrocarbon and air, in the
correct proportions to burn to CQ, and water, is approximately
700 calories per gram of mixture. If it ignites when suddenly
compressed to such a temperature that the rate of loss of heat
is 35 calories per gram, the reaction, if it continued uniformly
at the initial rate, would be complete in 20 seconds. This
illustration may serve to show the general nature of the
reactions that occur on sudden compression ; what occurs in
practice is that the gas, or part of the gas, reacts so that the
evolution of heat takes place at a somewhat higher rate than
the loss of heat by conduction etc.; hence the temperature
of the reacting gases must automatically increase, and with
it the rate of reaction, until the gas ‘‘ explodes.” The interval
between the end of compression and the explosion must clearly
depend mainly on three factors: (a) the compression tempera-
ture, (b) the temperature coefficient of the reaction, (c) the
rate of loss of heat to the walls.

94 Messrs. H. T. Tizard and D. R. Pye on the

IV. It is now generally recognized that the rate of an
ordinary chemical reaction varies with the temperature in a
way which may be empirically expressed by an equation of

the form
B

bee TC) ee

where k is the velocity constant, and A and B are constants,
B being the temperature coefficient.
In the case of reactions evolving heat, we can write this :
B

Q= Ae Ty uc eo aie ee
where ( is the initial rate of evolution of heat when a definite
mixture is suddenly compressed to the temperature T. At
the lowest ignition temperature Ty we have,

B

Qader
where Q) can be measured from observation of the rate of
cooling of the gases at a temperature slightly below the
ignition point: 2. ¢€

Qo= ae, (Ly 8), ee)

where @ is the cooling factor, c, the specific heat of the
mixture, and @ the temperature of the walls.

Now suppose that the gases are compressed initially to a
higher temperature T: the initial rate of loss of heat will
then be higher than Qo, namely

Qo — 6?

but since the effect of temperature on the rate of evolution
of heat due to the reaction is so much greater than that on
the rate of loss of heat, it is sufficiently accurate at present to
assume for our purposes a constant initial rate of loss of heat
—(,. The initial rate of reaction at the higher temperature
will be given by

ae
From (5) and (6) we have
Qr hs b iy Ty
loge =7(2 ae ae

where 6=0'4343 B.

Now under these circumstances the net rate of gain of heat

is (Qr—Q) calories per second.

Ignition of Gases by Sudden Compression. 95

The initial rise of temperature is therefore given by
aT 4 T
0 = A-W=G(G7—1}- - -

Beet) gp a
eo

or

i iE Qo
or (o a d (7) =ore

Qo
But since Qo=2e,(To— 8),
we have

] T T,—@

et): a() — a{ dé. ‘ é (10)
Qo

V. Can this equation be used to determine the delay, or
period of slow combustion, that occurs before the temperature
suddenly rises very rapidly, i.e. before explosion takes
place? Strictly speaking, this could only be done if Q, the
rate of evolution of heat during the initial reaction, depended
only on temperature and not on the concentration of the
reacting substances. According to all ordinary theories of
reaction, this would not be true; the rate of reaction should
depend in some way on the concentration of the reactants.
If we consider such a reaction as the combustion of heptane

CHie — 11 = rs CO, + 8 HO,

we can hardly suppose it is necessary, before the initial re-
action—whatever is its nature—can occur, for 1 molecule of
heptane and 11 moleculeg of oxygen to collide; but it is
reasonable to assume that the rate of reactiun must at least
be proportional either to the concentration of the heptane or
to that of the oxygen, or to the product of the two. It is
necessary, however, to point out that there is little, if any,
satisfactory evidence that homogeneous reactions in gases
obey what is ordinarily understood by the law of mass action.
In fact, evidence from the ignition of gaseous mixtures by
sudden compression points rather to the reverse. Dixon
and Crofts’ experiments* on the ignition of mixtures of
hydrogen and oxygen are difficult to explain by any reasonable
assumption as to the mechanism of the reaction based on the

* Dixon & Crofts, Trans. Chem. Soc. 1914, p. 2036.

96 Messrs. H. T. Tizard and D. R. Pye on the

law of mass action, even after taking into account the possible
effects of a different rate of loss of heat in the mixtures with
which they experimented ; the authors state, in fact, that they
‘“‘can offer no satisfactory explanation of the phenomena
observed.” Recent experiments by one of us * have shown
that the ignition temperatures of hydrocarbon-air mixtures
are independent, within the errors of measurement, of the
proportion of the hydrocarbon within quite wide limits. For
instance, when the proportion of heptane to air was varied in
the ratio of 1 to 10, the ignition temperature was only lowered
apparently by some 8°C. from 293°C. to 285°C. As the
error of observation may certainly amount to 4°C. in such
experiments, there is here no evidence that the alteration of
the concentration of heptane has any effect on the ignition
temperature. If, as might appear reasonable on the law of
mass action, the rate of reaction depended directly on the

concentration of the heptane, we should expect it to be 10

times as great in one case ag in the other. Now, it is shown

later in this paper that the temperature coefficient B, in the

case of heptane, is about 13,000, from which it follows from

equation (5) or (8) that if the initial rate of reaction on com-

pression were really 10 times as great in one case as in the
other, the ignition temperature of the richer mixture should

be over 50° lower than that of the weaker mixture. One

could not fail to detect with certainty a difference of this

order: hence from the experimental results we come to the

conclusion that the rate of combustion under these conditions

does not depend on the concentration of the heptane vapour
within wide limits. Experiments with other similar sub-

stances support this view.

VI. If this be correct, the only probable alternative is that
the rate depends essentially on the concentration of the
oxygen. We have not yet attempted to put this to a direct

_ test in the apparatus used by us, since it would be necessary,

to do this completely, to explode detonating mixtures of
hydrocarbons and oxygen; and we were anxious to avoid
the danger of breaking the apparatus before other important
experiments were carried out. We intend, however, to
examine this point in the near future.

It remains to consider whether, if the rate of reaction is
directly proportional to the concentration of oxygen, the
effect of the automatic decrease in oxygen content on ignition
can be safely left out of account in calculating “delays” by

* Tizard (oe. cit.).°

Iynition of Gases by Sudden Compression. 97

means of equation (10). For this purpose we have to estimate
to what extent the reaction occurs during the slow period ot
combustion before the explosion occurs, and the pressure-
time curve becomes almost perpendicular to the time axis
(see fig. 7). Taking the experiments on heptane as an
example, we find (Table III.) that the biggest delay observed
was about 0°6 second ; that the ignition temperature under
these conditions is 250° C.=553 absolute, and that the differ-
ence of temperature between the gas and the walls was about
240°C. The observed cooling factor was 0°51. Since the
specific heat of the mixture experimented with is approxi-
mately =0°2, the rate of reaction at the ignition temperature
must correspond to a heat evolution of

0:51 x 0°2 x 240 = 24 calories per gram per second.

The total heat of combustion of 1 gram of the mixture
(containing about 75 per cent. of the theoretical amount of
heptane for complete combustion) is about 510 calories, so
that it is evident that during the period when the temperature
is only rising slowly, which is always less than half a second,
the amount of reaction and therefore the changes in con-
centration must be small. Once the temperature begins to
rise quickly, it is evident that the disappearance of oxygen
can have only a secondary effect on the rate of the reaction
compared with that of the rise of temperature until the com-
bustion is nearly complete, so that the error involved in the
calculation from equation (10) of the total time of combus-
tion, by ignoring the effect of changing concentration, must
be small. In view of the unavoidable experimental errors in
carrying out experiments of this kind and of our still in-
complete knowledge of the mechanism of combustion, we do
not, in fact, think that any attempt to take fuliy into account
such secondary effects is Justified at present.

VII. We therefore arrive at the conclusion that the time
for complete combustion of an explosive mixture of gases
when suddenly compressed to a temperature above its ignition
temperaiure is closely given by the integration of equation
(10), and is therefore

l we 1 ( ty
Se ———d | ar My yi 1
naa & (i) (11)
a ( ak
To T/T Qo
where T, is the lowest ignition temperature under the con-

ditions of the experiment, and T is the initial compression
temperature.

Phil. May. Ser. 6. Vol. 44. No. 259. July 1922. H

Wn

98 Messrs. H. T. Tizard and D. R. Pye on the
In this equation, e is given by equation (8); its value
, .

depends on the magnitude of the temperature coefficient of
the reaction. It is not possible to integrate the equation
completely, but the integration can be carried out by approxi-

mate methods for any value of T° In the table below are
0 penis “
given values of the expression 2 (ee nN at different
values of —,. and for various values of —. The
ae a,

corresponding curves are shown in fig. 8.

o[To-6]t
; To

0-1 G-2 0:3

Tf our views are correct, these should form standard curves
representing the delays which should be observed under
different conditions when any explosive mixture of gases is
compressed to a temperature above its lowest ignition
temperature. ‘The application of the theory to any specific
case should enable the temperature coefficient of the reaction
to be determined.

Ignition of Gases by Sudden Compression. 99
TABLE II.
T)—@ age T l
Values of «a(~° “ji for different values of = and , ross
a5 Lo 1

T/T,. | o/T,=12.| =10. | =8 | =6. =4,

1004 | -086 Oe gee beg ae.

VOL ‘056 "OTD "108 ‘169 321

1:03 "0245 036 ‘056 ‘098 *209

L-OF ‘OOS1 O42 4.) “O25 onset eke

rPk ‘0033 ‘0065 | Ola, sue ‘095

1-27 “0002 ‘0007 | 0022 ‘0083 | ‘0415

The curves bring out clearly the effect of the two main
factors which determine the characteristics of an explosion
by sudden compression ; namely, the initial rate of loss of
heat, and the temperature coefficient. If two gases have the
same lgnition temperature under the same conditions of loss
of heat, the sharpest explosion will occur in the case of the
gas with the highest.temperature coefficient, and the greater
in this case will be the effect, on the magnitude of the delay
before explosion, of a higher temperature of compression.
On the other hand, in any one case, the ignition temperature
will be raised by carrying out the experiment under con-
ditions which involve an increased rate of loss of heat; at the
same time the sharpness of the explosion will also be in-
creased,

VIII. To test the above theory, and to use it to obtain a
measure of the temperature coefficient in certain cases, we
chose three substances: heptane C;Hj,, ether C,H;.0.C,H;,
and carbon bisulphide CS,. These substances were chosen
for the following reasons: (a) they could be obtained in a
sufficiently pure state; (b) they all have low ignition tem-
peratures, which lessens the practical difficulties of the
experiments ; (c) they are known to behave very differently
from the point of view of detonation when used as fuels for
internal combustion engines; and (d) their difference in
chemical and physical properties makes the comparison of
their behaviour on combustion particularly interesting. To
test the theory adequately we considered it absolutely
necessary, particularly in view of the simplifying assump-
tions made, not to be content with one set of conditions for
the ignition experiments. Two series of experiments were
therefore made with each substance; in the first series the

H 2

100 Messrs. H. T. Tizard and D. R. Pye on the

gaseous mixtures with air were compressed in a non-turbulent
condition, while in the second a high-speed fan was kept
running throughout the period of compression and subsequent

ignition. The use of the fan increased the rate of heat-loss

at the compression temperature by about three times ; hence
the difference in ignition temperatures observed with and
without the fan running gives an important and necessary
check on the value of the temperature coefficient which is
calculated from the “ delay”? curve obtained when the non-
turbulent gases are compressed. The temperatures given
below represent the average temperature of the gas at the
instant of maximum compression. By measuring the com-
pression pressures in each experiment, a value of “vy,” the
apparent ratio of the specific heats, is obtained from the
expression :

aes aE pt
le

2)

where 7 is the compression ratio.

The average value of y is taken for the series, and the
compression temperatures then calculated for each case from
the expression

Ae Ag! ipl —!
@ e

In each set of experiments the initial mixture of gases was
of the same composition throughout, the proportion of air
being somewhat greater than that required for complete
combustion. The intention of using a weak mixture was to
avoid as far as possible the deposition of carbon ; as stated
above, the absolute ignition temperature is not affected
appreciably by fairly wide changes in the original strength
of the mixture.

IX. The first results with heptane gave a very satisfactory
confirmation of the theory developed above. The results of
the experiments are given in the following tables and
diagrams, which include measurements from all the records
made under each set of conditions. No unsatisfactory
records have been discarded.

Ignition of Gases by Sudden Compression. 101
TaBueE III.

Ignition of mixtures of heptane and air.

Mean apparent value of “y” observed = 1°313.

Fan stationary. Initial pressure (atmospheric) = 14°8 Ib./
sq. in. in expts. A; to Ayo, and 14:9 |b./sq. in. in
expts. Ay, to Aj;.

Strength of mixture = 1 grm. heptane: 20 grms. air.

|

Max. Avge. temp. Delay Cooling

No. of | Compn. | Initial Max. Compn. See ponte
expt. ratio. | temp. pressure. y=1°313. obs. Ley
! eee Boo | | 51°C. 405 1b./sq. in. | 280° C. No ignition. 0:49
ee 6:03 50°5 ~~ | 156 295 0:19 sec. _
aS 702 | 505 192 323 0-04 =
Bape] 802 | 49°5 225 346 0-007 —
Perec Oo. .| 46 | 141 273 No ignition. 0°53
Benes. | 582 46 | 151°5 281 0°56 _
: PEERS | 623 | 46 163 293 0:21 _~
Oe 802 | 465 227 340 Very small.
). ARS 7:02 4 189'5 316 0-06
aay fascees 702 =|. 42 «166 307 0:07 ——
:. Sanne | 656 58 174 324 0:05 —
ORE es 6°56 53 Dae 315 0:06 _
1. es | 6°56 48 176 306 0-12 —
aa exteie - 6°56 44 pales 298 0-18 —
|: See 6°56 41 | 176 | 293 0°28 —
, eee | 6°56 38 178 288 0:25 =
Ce 6°56 39 176 282 | 0°58 —

The cooling factors in Table III. were obtained from the
results of those experiments where no ignition occurred by
the application of equation (3). The results for A, were as
follows :—

TaBLeE LV.
Calculation of cooling factor.
wa i 3 | e
| Time from Obs. pressure. | Lr lays 0.
‘max. pressure.| 1b./sq. in. Pde | log P—Pr = 0. ae 2°35,
0 140°5 58°5 aa —
0°23 sec. 134 a2 0512 | 0-51
0:47 127°5 45°5 | "109 | 0°53
0-7 | TD i saa | 149 | 0:49
0:93 119 toe 199 | 0°49
1°16 116 | 34 ‘236 | 0°47
1°39 | 112°5 30°5 "283 0°47
Mean=0'49
P= 5°50 x 148 = 821 lb./sq. in.

102 Messrs. H. T. Tizard and D. R. Pye on the

The results for plate A; were similar, the mean being
a = 0°53. In the calculations the figure 0°51 has been

taken.

TABLE V.

Tgnition of mixtures of heptane and air. [an full speed.

Initial pressure = 14°86 lb./sq. in. Strength of mixture as
before. Mean a value of y = 1°310.

No. of | Compn. | Initial |Max. Compn.) Max. Aver. Cooling |
| expt. | ratio. temp. | are | temp. cale. Delay ees | Coning |
B, 6:03 \G) he OF 157°5 310° C. 0°16 sec. —-
B, 5:52 59 139 291 No ignition. | 1:43
| Bs D738 595 148 298 i 1-40
| B, 5°91 595 154 304 4 1°36
B; 6:13 59-5 161 310 ob 1-47
| B, 6°33 59'5 165 313 0°13 =
1s 6°56 o9 175 321 0:08 —
B, 701 60 188 336 0:05 —
ndBy, 8:02 60 e 362 tO: 00 —

Mean= pose

* Tonited before top of compression.

X. Considering firstly the results of the experiments
without the fan running, we find that at the lowest ignition
temperature of about 280° C., the ae of the walls

Gye)

was about 40° C. The expression « an ”) (equate ahale)
| Lo

has therefore the value—

0-51 x 240
D938

Assume that the true ignition temperature under these

conditions is 280° C. = 553 absolute, and that ~ =10:
0

then by integration of the theoretical equation (see Table II.)
we obtain the following results :—

= Pa,

Ignition of Gases by Sudden Compression. 103
TasBieE VI.
Ignition of heptane by air. No fan.
Lowest ignition temperature Ty taken as 553 absolute.

7 assumed = 10.

0
iy lh ie 1° QC, a (eae ) ¢ (theor.). t(calc.). |
0

1°004 555 282 0-112 0°51 sec. |
1:01 558°5 285°5 0-075 OBE as >|
1:08 569°5 296°5 0-036 O1G4... | ..|
1:07 592 319 0-014 OuGsi |

ues Ga 614 341 00065 Ce 0
1:27 702 429 0:0007 0-003 |

J

The last column contains the theoretical ‘“ delays” that
should occur at compression temperatures given in the third
column, if the temperature coefficient of the reaction

L 5
corresponds to a value of ~ = 10.

0

Fig..9

400
~)
G Heptane and Aji,
ae!
& Y
350

No ignition.

2 03 0:

Seconds delay

The corresponding theoretical curve is shown in fig. 9,
the experimental points taken from ‘lable III. being marked
with a cross. The general agreement is all that could be
desired.

No ignition.
4 0

O°! O-

5

of a ke

104 Messrs. H. T. Tizard and D. R. Pye on the

XI. If the value for the temperature coefficient so deduced
is correct, it should be possible to use it to calculate the
higher ignition temperature when the fan is used, and also
the shape of the new delay curve.

Now, the mean cooling factor with the fan has been shown
to be 1°42. The ignition temperature under these conditions
is evidently about 310° C. (see Table V.), the temperature
of the walls being 60° C. Hence the ratio of the rate of
loss of heat with and without fan at the respective ignition
temperatures is—

Qo 142 (810—60)

Q) 0°51 (280= 40)
Hence the new (theoretical) ignition temperature Ty’ (with
fan) should be elven by (see equation (8))—

b Ae
log Q- = log 290 — Te = a

= PU.

ise. 0-462 = 10 (1 — Ty
To
Diao a
or Le = 0°954

i = Sol) c= B07? Ok
This is close to the observed value.

bi ail
The new value of Ad will be eek Te = 9:5; while the value
0
Ti a) _ 1:42 x 247
580
The corresponding theoretical values of the delay are
shown in the following table.

TABLE VII.

Ignition of heptane by air. Fan full speed.
Ignition temperature calculated from previous results, 580°C.

= 0°60.

b
==9°
ae
| y , a:
nya .. 10 ml ==") Haheor)o| tesla
\ ‘Lp |

10045) Wy yr582 349 0:12 | 0:20 see,
100 |) 4 586 313 | 0:08 a Ons
105 (M0 597 324 0-04 007
Oy GL 348 0-016 0-027
LI | 644 371 00076 0-013

Ignition of Gases by Sudden Compression. 105

The theoretical values given in this table are represented
by the dotted curve in fig. 9, the experimental results
(Table V.) being shown by circles. The close agreement
between experiment and theory is obvious; it is, indeed,
closer than could reasonably have been expected in view of
the fact that the temperature errors must be estimated as 3 or
4 degrees, while the cooling factors are subject to an error
of about 5 per cent. The results can, however, leave little
doubt of the substantial accuracy of the simple theory
worked out above, and the temperature coefficient deduced
must be very near the truth. It is of great interest to note
that it is of the same order as that of chemical reactions in
liquids at ordinary temperature ; for the reaction velocity is
approximately doubled for a 3 per cent. rise in absolute
temperature.

XII. The only experimental values given in Tables III.
and V. which seem to call for any special comment are
those corresponding to experiments A, and A,g. The
‘delays’ found in these experiments were considerably
smaller than those expected theoretically. This may be due
to the fact that the measurement of very small delays is
necessarily somewhat inaccurate with the apparatus used,
since the speed of the piston falls off as the compression
approaches its maximum. In such cases the lowest ignition
temperature is, of course, reached before the piston reaches
the top of compression, so that the measured ‘‘ delays ”
which are measured from the time of maximum compression
tend to be too small. But there is also a curious effect,
which is invariably observed in these experiments on the
self-ignition of carbon compounds, when the initial tem-
perature is high, and the time of explosion short. It is
always found that the explosion, though apparently sharp,
is not complete, but that a fluffy deposit of carbon is thrown
down. This deposition of carbon in an explosion has often
been noticed by other workers when ignition is effected by
a spark, but it is usually thought to be a consequence of
having too little oxygen for complete combustion; in our
experiments, however, the oxygen was always in considerable
excess. When the minimum ignition temperature is not
greatly exceeded, and when therefore the explosion is
comparatively slow, combustion is complete, and no carbon
deposit is formed. At higher initial temperature, however,
one cannot escape the conclusion that the hydrogen is burnt
preferentially to the carbon, and that the rate of combination
of carbon atoms can be greater than the rate of combination

106 Messrs. H. T. Tizard and D. R. Pye on the

of carbon with oxygen. The exact conditions when this
occurs seems well worthy of further investigation.

It is always necessary to clean out the cylinder carefully
after such a deposit has been formed, and before the next
experiment is made; for, if not, abnormal results will be
obtained, and the “delay” before ignition occurs will be
very much shorter than is expected. To explain this it
does not appear to be necessary to attribute any special
“catalytic” activity to the carbon; a simple physical ex-
planation seems to be sufficient. Such a deposit is known
to be a very bad conductor of heat. If left on the walls of
the piston and cylinder, we shall therefore have, on the next
compression, large portions of gas from which the beat
cannot get away quickly. Hence the ignition temperature
is lowered, and the explosion takes place more rapidly.

XIII. Experiments on the self-ignition of mixtures of the
vapour of ethyl ether C,H;.O.C,H; gave very similar
results. The results of the experiments on the compression
of non-turbulent mixtures are shown in Table VIII. ; while

AW Nssicoe WOEUE.

Ignition of mixtures of ether and air. Fan stationary.
Initial pressure (atmospher ic) = 14°77 lb./sq. in.

Mean apparent value of “y” = 1°309.

Strength of mixture = 1 Le ether to 15 of air by wee

No. of coe. Initial | Max. Compn. ee oe nee Cooking
expt. | ratio. temp. pressure, y= 1309. s Ens
Ci | ol 25°C} 105 1b./sq. in. 201°C. | No. ign.| 0°47
C, 4°83 24. 116 4 211 Ms 0-47 |
OF 5:02 23 APS) as 214 O41 | — |
Cy 5:21 23 128 bs 220 0:30 —
C,. 5:42 (23 BORD Sy, 226. >| 10-205 —
Or; 5°63 23 143 sp 232 N12 —
CP 5°82 23 149 :; 237 0093 | —
C, 6:03 23 155 4 242°5 0:071 “=
C, 656 | 238 172 43 256 0 035 —

Table X. shows the results when the fan was rotating at
full speed. No ignition was observed in experiments C,
and ©, in the first series, and in experiments D,, D., and D,
in the second series. The details of the calculation of the
cooling factor corresponding to experiment C, are shown in
Table [X. The cooling curve was not so regular in the case
of ©,, but the mean value of the constant was the same,

Ignition of Gases by Sudden Compression.

TABLE LX.

Jaleulation of cooling factor corresponding to C.

10

Time from |
maximum en Pe log
pressure. — 7
0:0 116:2 449
0232 sec ELST 40'6
“A GD), 6: 107'3 36:0
<i ae 103° + 32°35
ay) gy ah LOG 29°3
1G! 4 «SS 25°6
Paes gg eh SET a y28r4
£65. x 92°5 Pe

P,=4:83X14-77=71'8.

TABLE X.

Tenition of mixtures of ether and air.

Mean = 0°47

Fan full speed.

Initial pressure (atmospheric) = 14°77 lb./sq. in.
Mean apparent value of y=1°308.

Strength of mixture as in Tab

{ |
| No. of | Compn. | Initial | Max. Comp.
expt. ratio. temp. pressure.

ee ee ee OM ee
yee. D738 24 | _146°5

i, 6°03 24. 155°5
De | CSaO hk OA } Wes
fh Ds |) 663 24 Via
foe. r02 24 186
| Dry: | 5°82 22 149

Ds 5°94 22 1505
bey. |, GOS 22 155 |

DB 6:13 22 161

le VIII.

Max. temp. 5 Coolin

cale. fen | jee ashe /

y=1:308. | °* 27
226°5°C.| No.ign.| 1°33
235 : 1-28
2435 0:10 _-
251 0047 | —
259 0038 | —
268°5 0026 5 —_
255 No.ign.| 1:25
238 0°13 —
240 GOST 4. —
243 0078 £=—

Mean = 1:29

namel y @
culations

=0-47. Table XI. shows the details of the cal-
for D,; it will be noticed that the cooling factor

appears to diminish fairly steadily as the time increases.
D, and D,; showed a similar effect ; in eaeh case the mean

has been taken, and the mean value for the three experi-
ments, namely « = 1°29, is probably fairly accurate.

The

108 Messrs. H. I’. Tizard and D. R. Pye on the

values for the cooling factors so obtained are lower than
those found in the heptane experiments; this would be
expected, for, although the specific heats of the mixtures are
about the same, the compression ratios used were lower in
the case of ether mixture than in the case of heptane, since
ether has a considerably lower ignition temperature.

TABLE XI.
Calculation of cooling factor corresponding to Dy.
| Time from ae Pedy eh te ae |
| maximum Observed P—P,.| tog nee Least) a= 230)
WORE pressure. f . DD ep | |
pressul e
0 146°5 Ib./sq.in.| 61:8 oe fabs
"232 sec. | 129°5 5 44-8 "140 1:39
"465 _s,, 118-4 Vy, 337 263 | 1°30
TON 2 | 100% ee 25-8 379 | 195
Oa. Mui: 103°6 F 19:9 492 1°22
1G" 55° | Oneiaaamme 13:0 | ‘677 1:34
Oe: 96:2 is lid | "730 1-21
| Pp=9'783 Xx 14-°77=84-7 Ib. /sq. in. Mean =1:28

XIV. Taking the ignition temperature Ty (without fan) as
212° V.=485 abs., the experimental observations of the delay
before ignition at higher temperatures are reproduced closely

if the value of ir =11. In this case, we have
0
Pe 0s D123

The theoretical values for “t,’ calculated according to
equation (11), are shown in the following table.

TABLE XII.
|
| ON :
| T/T». T t2C a ( 5 ) t (theor.). t (cale.).
1-004 487 214 098 0:535 sec.
| 1:01 490 217 ‘065 "B50
| 1:03 499°5 226'5 ‘030 "164
1:07 519 246 "0105 057
Neil 538°5 265°5 0046 "025

1:27 616 343 “0004 002

Ignition of Gases by Sudden Compression. 109

The lower curve in fig. 10 is the theoretical curve derived
in this way; the experimental points are shown by crosses.

In the series of experiments with the fan running at full
speed, “a” = 1°29, and the new ignition temperature Ty 48
ae iis 235° C.= 508 abs.

Fig. 10.

| Ethyl ether Jand air.

250

ae No ignition. expts)
| |

lo 02 o3 O4
Seconds delay.

T,/—0 235 —23
Biches a(™ 1 )= 1:29 x x(a = ) =0°54,

Op) 1:29 (255223)
Qo 0-47 (212 —23)

and

—allter

ae gD :
Hence if T. =11, T,’ should be given by
To

log 3 08=0'489= 11 ( Pe a)

To
D5 2
or T =(0°956; .°. TT) =507°5 (abs.),

which is evidently very close to the observed value.
Taking this value for T,’, we have

110 Messrs. H. T. Tizard and D, R. Pye on the

Table XIII. gives the theoretical values for the time of
ignition at various temperatures, calculated as already

described, taking Ty’ =507°'5 ; ! =10°5.

ie
L

TABLE XIII.

| 9
Jj 0p liege £°.C. | re t(theor.).| (cale.),
1004 | 509°5 236°5 0-104 0°19 sec.
WOM |) SLAs 239°5 0-069 013 ,,
1:03 522-5 249-5 | 0:0325 006 ,,
1:07 543 270° 0-012 0-022 ,,
1) i a68i5 290°5 0-0055 0:010,,

The corresponding curve is the upper dotted curve in
fig. 10, the experimental values being shown by circles.
The general agreement is again all that could be desired.
Although the ignition temperature of ether is very much
lower than that of heptane, the temperature coefficient of
the combustion reaction is the same within the experimental
errors involved.

XV. The experiments on carbon bisulphide were expected
to be of particular interest, on account of the anomalous
behaviour of this substance if used as a fuel in internal
combustion engines. It is known that for any given fuel
the highest thermal efficiency obtainable is limited mainly by
the tendency to “‘ detonation ”’ at high compression ratios. It
is usually assumed that the tendency of any fuel to detonate
depends upon its ignition temperature; the lower the
ignition temperature, the greater will be the tendency to
detonation. According to our views, the ignition temperature
is not a safe criterion ot the tendency to detonation ; the
temperature coefficient is also an important tactor which
must be taken into account. The use of carbon bisulphide
as a fuel illustrates this point very well; although it has a
lower ignition temperature than heptane, it detonates less
easily in internal combustion engines, and not more easily,
as might be supposed. We expected, therefore, to find, by the
experiments described in this paper, that the temperature
coefficient of its reaction with oxygen was very distinctly
lower than that of heptane and similar substances. The
experiments fully confirmed this, although the results do not
appear to be so satisfactory in all respects as those carried
out with heptane and ether. |

Ignition of Gases by Sudden Compression. tt

Tables XIV. and XV. contain all the experimental results
of the ignition at various temperatures of mixtures of CS,
and air (a) when initially stagnant, and (b) when the fan was
rotating at full speed.

TABLE XIV.
Ignition of mixtures of CS, and air. Fan stationary.
Initial pressure (atmospheric) = 14°60 Ib./sq. in.
Mean apparent value of “vy” = 1:332.
Strength of mixture = 1 part CS, to 8 parts air by weight.

|

No. of | Compn. | Initial | Max. Compn. | etl ee Delay | Cooling
exp. ratio. temp. pressure. — 1-339 obs. factor.
VS dale
E....| 603 | 495°O.) 1591b./sq. in. 313°C. | 0-086 |
Be...) 5:63 | 475 14 98 296 0-115
E;...| 563 | 42 Lage" - 7; 286 0-18
Pe...) 5°63) | 39 fea ag 281 0-26 |
Bs ..|- 5:63 | 36 uD hy eee 2755 0°35
PHig..| 563! | 32. i 2 aa ee 268°5 0-42 |
LE, ...) 502 | 32 WES. ,, 248 No ign.| 0:46 |
Bes) abo? | 31-5 13275 5, 254 0-71
ee.) oa) 31-5 Oa... 261 0-59 |
Pies) .b62 | 315 1545, 2735 0-34 |
Big.) 7-02) -| 31 193°, 308 0-087
TABLE XV.

Tgnition of mixtures of CS, and air. Fan full speed.
Initial pressure (atmospheric) =14°6 lb./sq. in. expts. F;-F,.
eed i hy F,-F
Mean apparent value of y = 1°323.
Strength of mixture as before.

10-

| fis | |
No.of |Compn. Initial | Max. Compn. pee tee Delay Cooling |
| : cale. from | aa
| expt. | ratio. | temp. | pressure. y= 1323 obs. | factor. |
leno aera ge ms nas 2:8 ae 4
Led MP 43°C. | 169 lb./sa.in. 300° C.| No ign. | 1:30 |
FH. ...) .644. | 43 171 5 804 Ose ys |
BF .| 656 | 48 77 ey. 307 |Noign.(?)) 1-38 |
PE. ...| 663 | 42°5 ce aan 309 0-13 ee
| F; ...| 6°84 45 185 is 315 0:12 =
| Be ...| 755 | 43 208 3 334 0°05 —
| Bet 656 43 175 3 307 (Ud es fe a
bBo vet 7g 43 191 i 320 0-09 ==
iBF> ...| 802 | 48 224 = 346 0:03 — |
ce ee ie ec We ane > ? ‘ 391 [Ignit’d be 9
tore top of
j compn.
Mean = 1°34

The cooling curves were very uniform in the above cases.

112 Messrs. H. T. Tizard and D, R. Pye on the

XVI. It is clear, from the shape of the curve connect-
ing time of ignition with temperature, from the results of
experiments without the fan, and from the difference in
ignition temperature observed with and without the fan,
that the temperature coefficient of the reaction is low.
If we take Ty, the ignition temperature without the fan,

to bem eiovO.= 526 ‘absaltandan =
To

theoretical values for the time of ignition from equation (11)

which are given in Table XVI.

= 7, we obtain the

TSO 253 —32
The value of a—°,~ is 0°46 X —___— =0:193.
Te 526
TaBLE XVI.
T,=526 abs. }/T,=7-0.
T,—0
T/T,. | T(abs.). | °C. a( T. Ve (theor.). ¢ (cale.),
1004 | 5228 255 0-187 98 sec
1:01 531 258 0133 704
1:03 5423 269 0-072 38) 1s
1:07 563 290 0 036 RG shee
tala 584 oll 0:020 (Gar
NLA 66S 395 0-004 021 ,,
Fig. 11
at and aif.
. 350 ma
S [IN B. Time scale half that jof Figs. |IO and it.J |
S
R
300b- = = No ianitlion.
4 Gea ‘aan = aes

O-1 02 03 04 0-5 0-6 0:7 0:8 0-9
Seconds delay,

The theoretical values are shown by the lower curve in
foe A,

ao

Lgnition of Gases by Sudden Compression. 113

The cooling factor when the fan is used is 1°34. The ratio
of the loss of heat (or reaction velocity) at the ignition
temperatures with and without fan is consequently

Qo __ 1:34 (296 —42)
Qo 0°46 (253 —32)
=O:

Hence the new ignition temperature T,' should be given by
the expression

log 3°35=0°525 =6b/T» (1 - =

0

Tye

=7(1- 7?)
Shire yeaa
or T,’ =1—0:075

=e
Ty’ =568°5 = 2955 OC,

The new value of <”- corresponding to this is therefore

To’
b 526
mate ( 555) =6°5.
The calculation of the corresponding delay curve is shown
Ui a aa
in Table XVII, tho value of «(= *) being
0
204
1°34 x 569 == 06:

TaBLeE XVII.

|
| ° a (=| t (theor.). 1
Li ea iy t°C, T,! t (eale.).
| 1008 | 571 | 298 0-205 34 sec._|
Ot 574 301 0°15 rt
1:03 585°5 312°5 0:08 Sys |
1:07 6085 | 335°5 0:042 OF cl
Ill 631 | 3858 0:025 gic sewer |

This is the dotted curve shown in fig. 11, the experimental

points being marked as before. The agreement between

Phil Mag. 8. 6. Vol. 44. No. 259. July 1922. I

114 Messrs. H. T. Tizard and D. R. Pye on the

theory and experiment is in this case only moderate. In
particular, it will be noticed that although the ignition
temperature with the fan running is calculated, from the
results without the fan, to be 295°°5 C., actually no ignition
was observed at 300°C., and even in one experiment at 307°C.,
although this is extremely doubtful, since in two other
experiments ignition was observed to take place at 307°C.
and 304° C., with the comparatively short delays of 0°11 and
0:14 second respectively. It is possible that when the loss
of heat is considerable, and the temperature coefficient small,
there is an appreciable error introduced in neglecting
changes of concentration when calculating the time of ignition
at temperatures near the ignition temperature. This would
account for no ignition being observed when a long delay
was expected. The value of 6/T,) given above cannot, how-
ever, be very far wrong. For suppose we take the ignition
temperature Ty’ of the mixture when the fan is running as
300° C. instead of 295°°5, and calculate the temperature co-
efficient solely from the difference in Tp and To without
regard to the “delay” curve when the fan is stationary ;
then we shall have

Tpe526 hs
D7 = 573 70918
and 20, = 1:34 (300-42) Lae

Qn.) 0162538 232)

ek ee Ne ticg
log 3:40 ="531 = a (1— ry

b
mp SUE

e =6°'5 instead of 7:0.
Lo

This value for < would, in fact, fit the lower part of the
0

delay curve without the fan rather better than the value 7-0,

but the calculated delay curve when the fan is running

would then be some way from the experimental points. If
we take a mean value

6) T= 6-7,

we shall be very unlikely to be as much as 10 per cent. from

Ignition of Gases by Sudden Compression. 115

the true value, even when all possible sources of error are
taken fully into account.

XVII. It has already been mentioned that when the time
of ignition by compression of hydrocarbons (and of ether) is
small, 2. e. when the gases are suddenly compressed to a
temperature well above the ignition temperature, carbon is
thrown down, even though excess of oxygen is present. In
the experiments with CS, an even more curious phenomenon
was noticed. In this case, whereas the sulphur burns to
SO, when the initial temperature of compression does not
exceed very greatly the lowest ignition temperature, when
the initial temperature is high the products of combustion
smell strongly of H,S. For instance, the products of com-
bustion in experiments E, and EH,, above both smelt strongly
of HS, although in H,—Ej) inclusive only SO, could be
detected by smell. It was also possible to detect H,S after
experiments F, and Fy) (with the fan), the smell being
particularly strong in the case of Fy). The smell of H,S
could also just be detected along with SO, in experiment Fy,
whereas in the remainder only SO, could be detected.

The H.S could only have come by combination with water-
vapour present in the air, which was not dried. This
occurrence of H.S is all the more interesting since it is
known that a perfectly dry mixture of CS, and oxygen can
be exploded by a spark, whereas perfectly dry mixtures of
other gases, e. g. carbon monoxide with oxygen, cannot. It
is possible that some such reaction as

CS, + 2H,0=C0,+2H,8

takes place, followed by the combustion of H.S ; but even if
this is the case, it would be expected that the H,S would be
quickly burnt in presence of excess of oxygen under the
conditions of these experiments. Further experiments on
the ignition of H,S itself will probably throw some light on
these observations.

XVIII. In Table XVIII. are summarized the chief results
of the experiments described above. The ignition tempera-
tures represent the lowest average temperatures at which
the non-turbulent mixture could be caused to ignite. The
rates of evolution of heat at these temperatures for the three
cases are calculated from the cooling factors and the specific
heats of the mixtures.

i?

116 «=«©Messrs. H: T. Tizard and D. R. Pye on the
TasLe XVIII.

| Carton
Heptane Ether :
ESCs: ol) CEO de, ie
Composition of gas by weight... 1: 20 of air. i: 15 Lis
| ee
|
T,=ignition temperature ...... ee 200. a DAC 253° ©.
Cy at ignition temperature ...... e020 | 0:20 0-18
| |
Rate of evolution of heat due to, 25 calories |
: i | | 18 18:5
reaction per gram of mixture per second.
ERO IU TIES OE EDL DDE eae calories. | 510 calories. | 386 calories.
gram Ol MIXVURC © yee sass cen | |
ASL Meee A Sat ENO Bele x zs] pa aeEY
Waltie of 6/Ese £2). AS o4 100+5°/, | 110+5°%, | 67+10°/,
| |

XIX. In order to calculate the true temperature co-
efficient B (see equation 5) from the values of 6/To, it is
necessary to examine the significance of Ty a little more
closely. As already stated, Ty is a measure of the lowest
average temperature of the gas at which ignition takes
place. Now the actual temperature of the gas after sudden
compression can hardly be uniform throughout ; in fact,
when the gas ignites after a considerable delay, it is always
found that the pressure, and therefore the average temper-
ature, falls, in some cases quite considerably, before ignition
takes place throughout the mass. This shows clearly that
that portion of the gas which ignites at first has initially
a higher temperature than the average, thus confirming
Dixon’s experiments. Absence of information as to the
temperature gradients which may exist under these conditions
has no doubt led Nernst and Dixon in their experiments to
calculate the ignition temperature as if the compression were
adiabatic, and to ignore the influence of loss of heat during
compression and before ignition. They assume, in fact, that
that portion of the gas which does ignite is at the adiabatic
temperature.

It is hardly likely, however, that big differences in
temperature exist after compression when the gases are in a
turbulent state ; and the fact that the temperature coefficients,
calculated from the differences in “average” ignition
temperature between turbulent and non-turbulent mixture,

Ignition of Gases by Sudden Compression. Ly

agree so well with those calculated from measurements of
time of ignition at various temperatures with non-turbulent
mixture, confirms the views taken in the previous paper
(Tizard, loc. cit.) that it is unlikely also that any big
differences of temperature exist in the non-turbulent mixture
after compression. In the absence of direct evidence on
this point, however, it is important to calculate the ‘ adia-
batic ” temperatures also in the above cases.
The mean specific heats C, per gram molecule are :

Heptane (room temperature-300°C.) =50 calories
approx.
Ether (65°-230° C.)=33°6 calories (Regnault).
CS, (70°-194° C.) =10°0 (Regnault).
Taking these values, and C,=5-0 for air, we obtain the

fioures for the mean true value of “ry,” and the corresponding
adiabatic temperatures, given in the following table.

TABLE XIX.

Heptane. | Ether. CS...

}
|
Mean apparent value of |
rye TODSELVEU) si... 1°313 | 1-309 1°332
Mean true value of y ... 1353 | 1:347 1:384
“« Average” ignition tem-
| 2 2 a a eee 553 abs. 485 abs. 526 abs.
| ‘“* Adiabatic” ignition |
| temperature ............ 594 516 572
PUI EROMCEY rods. ceca se ANE C4 Sof ole ©. 6749), | 46°. 0.==8:°7.°/,,
|

|

The average specific heat for CS, taken in the above
calculations is probably too low, since it refers only to a
range of temperature up to 194° C., whereas the ignition
temperature was 250° C. |

It will be observed that the difference between the average
observed and the theoretical adiabatic temperatures is not
very great ; we consider that the “average”? temperature is
probably closer to the true ignition temperature than is the
“adiabatic” temperature, but for the purpose of estimating
every possible source of error in the temperature coefficients,
it is better at this stage to recognize the uncertainty, and
take for the true values of the ignition temperatures the
values

Heptane 573° absolute
thers = 500?" +4 per cent.
CS, 549° 39

118 Messrs. H. T. Tizard and D. R. Pye on the

Hence, from the values of = we get finally for the true
temperature coefficient B : p

Temperature coefficient B
(equation 5).
Heptane-air... 13,200+ 9 per cent.

Hiher- pk 20 oule2 eG (0c eo ae
CS.- fp 4 COW ORE ees

The significance of these figures will perhaps be better
appreciated by the statement, that the percentage rise in
absolute temperature necessary to treble the reaction velocity
is 4 per cent. in the case of heptane and ether and 7 per
cent. in the case of CS».

XX. Of recent years, considerable attention has been
directed to the “radiation” theory of chemical reactions.
According to this theory, the ultimate cause of any chemical
reaction is to be found in the absorption of radiation of a
frequency which depends upon the nature of the reactants.
In the case of the majority of chemical reactions, namely
those which are not “‘ photochemical ”’ in nature, this radiation
will belong either to the visible, or more usually in the short
infra-red part of the spectrum. The supporters of the theory
hold the view that it is only through the absorption of such
radiation that a molecule is able to acquire that excess of
energy, over the average at any temperature, which enables
it to decompose or to react with another molecule. The
frequency of the radiation is therefore known as_ the
‘activating ”’ frequency.

This reasoning leads to the conclusion that the temperature
coefficient B of a mono-molecular reaction is determined by
the relation

hy

where “v” is the activating frequency, which should
correspond to an absorption band in the reacting species.
No reliable experimental evidence has yet been brought
forward in support of this theory, but in view of the scanti-
ness of the data existing on homogeneous gas reactions, it is
of particular interest to apply it to the results of the
experiments described above.

In attempting to apply the theory, a difficulty at once
arises. Hvidence has been brought forward to show that
the ignition temperature of substances with oxygen is

Ignition of Gases by Sudden-Compression. FY

practically independent of the concentration of the com-
bustible substance. If the rate of the reaction were deter-
mined solely by the amount of oxygen present, we might
expect the temperature coefficient also to depend solely on
the oxygen, and therefore to be the same in all cases. This
is clearly not true. Nor does the temperature coefficient, in
the case of the heptane explosion which has been most closely
investigated, correspond at all closely to that calculated by
means of equation (12) from the infra-red absorption of
oxygen. Oxygen has an absorption band corresponding to

N=3'2 mw, or v="94x10"; hence, since 7 = 486 LO.
we should have

B=4550 (cale.) instead of 13,200+9 per cent. (obs.).

It is clear that equation (12) cannot be applied. On the
other hand, if the rate of reaction depended on the product
of the number of active molecules, both of oxygen and the
other reactant, we should expect, on the same theory, to
find the temperature coefficient given by

/
B= + (+),

where v,, v2 correspond to absorption bands in the reacting
substances.

Now, all hydrocarbons have a weak absorption band at
A=2'4 w, and a fairly strong one at 3:43 pw. Taking
X=2°4 uw, which is most favourable to the theory, we have
v= 1°25 x 10" and vy, (oxygen) ='94 x 10.

Hence B= *-s600.10 2-2 10 = 10; 700:
This approaches more closely the experimental value
B=13,200+9 per cent.

It must be pointed out, however, that this approximate
agreement is only obtained by an assumption as to the actual
mechanism of the reaction which does not agree with the
existing experimental results.

The failure of the “radiation theory” to account for the
results obtained in these experiments is more significant
when we regard it in a different way. The theory requires
that the rate of a chemical reaction should be proportional
to the density in the reacting system of the radiation which
is absorbed by the reacting substances. Now, in the case
of gases which are caused to react by a rise in temperature
due to sudden compression, the radiation density must
remain practically unchanged, for the temperature of the
walls remains constant. It may be momentarily increased

120 Messrs. H. T. Tizard and D. R. Pye on the

owing to the sudden compression, but such an increase
cannot persist during the period of delay, and in any case is
negligible compared with the increase in radiation density
which would occur if the temperature of the walls of the
vessel were raised to the compression temperature of the
gas. Again, the emission and absorption of radiation by
the gas itself at the compression temperature of 500-600
absolute is negligible compared with that of the solid walls.
Hence we arrive at the conclusion that, although the density
of radiation in the system is not appreciably changed, the
gases react ata high rate. This fact appears to us to prove
conclusively that the radiation theory cannot be accepted
either in its original form, or as modified to meet its failure
to account quantitatively for the temperature coefficients of
chemical reactions in liquids under steady conditions of
temperature.

It must be pointed out, however, that in spite of this con-
clusion, there does seem io be an indirect connexion between
the effect of temperature on the rate of combustion of many
substances and their absorption of infra-red radiation. For
example, Coblentz has shown that all paraffin hydrocarbons
have very similar absorption spectra, with a weak band at
A=about 2°4 uw, and strong bands at X= 3°43, 6°86 yp, etc.
Now, Ricardo hag shown that the tendency of hydrocarbon
fuels to detonate in an internal combustion engine depends
consistently on their ignition temperatures as determined in
the manner described above. According to our views this
is strong evidence that the temperature coefficients are
practically the same throughout. Again, it has been shown
that ethyl ether has approximately the same temperature
coefficient of combustion as that of heptane; while Coblentz
has found that its absorption spectrum is also nearly
identical, with bands at 2°4 wand 3°45». Carbon bisulphide,
on the other hand, has a much smaller tendency to detonate
in an internal combustion engine than heptane, although it
has a lower ignition temperature; corresponding to this we
find that the temperature coefficient is low, and that the first
strong absorption band in the infra-red occurs at 7X=4°6 p.
Finally, hydrogen “detonates” easily in spite of its high
ignition temperature ; its temperature coefficient must
therefore also be high, a deduction which is confirmed by
some preliminary experiments we have made on the delay
before the ignition of a non-turbulent mixture of hydrogen
and air, We should expect from this point of view to find
an absorption band in the short infra-red region (say about
1:0 w); actually no absorption is observed, but that the

Ignition of Gases by Sudden Compression. 121

frequency of atomic oscillation is high is in agreement with
our general knowledge of the hydrogen molecule. In spite,
therefore, of the strong arguments that have been brought
forward against the radiation theory of chemical reactions,
these results support the view that there is a connexion,
even though an indirect one, between the temperature co-
efficients of gaseous reactions and the infra-red spectra of
the reacting substances.

XXI. The results of this investigation may be summarized
as follows :—

(a) Quantitative experiments confirm the view that at the
lowest ignition temperature the heat evolved by the
combustion of a gas just exceeds that lost to the
surroundings.

(>) From measurements of the rate of loss of heat just
below the ignition temperature, and of the intervals
between the end of compression and the occurrence
of ignition at different temperatures, it is possible to
deduce the temperature coefficient of the gaseous
reaction.

(c) The temperature coefficients so obtained are confirmed
by the increase in the minimum ignition temperature
which is observed when the gas is in a turbulent |
state. |

(d) The results show that the temperature coefficient of
the combustion of carbon bisulphide is much lower
than that of heptane or ether. This is in agreement
with the relative tendencies of these fuels to detonate
in an internal combustion engine.

(ec) The results do not agree with the radiation theory of
chemical reaction.

(7) Some evidence is put forward to show that the rate
of reaction on sudden compression is independent
within wide limits of the concentration of the com-
bustible gas, but only depends on the amount of
oxygen present. ‘This evidence is, however, incom-
plete.

Weare greatly indebted to Messrs. Ricardo & Co. for the
loan of their apparatus and for much additionil assistance ;
also to the Department of Scientific and Industrial Research
for a grant towards the expenses of the investigation. We
also take this opportunity of thanking Mr. C. ‘I. Travers for
his help in carrying out some of the experiments.

an sss ——_—

a

SSS SS

SS eee ee
SSS SS SSS

ae

[#10859

IX. On the Vibration and Critical Speeds of Rotors.
By C. Rongers, 0.6:2., B.Sc., B.iing., ML b ee

ING eee US papers have been written on the question

of the whirling and vibration of loaded shafts and
kindred subjects, and the calculation of the first critical
speed—the lowest speed at which the vibration shows a
maximum value, is now a matter of daily routine in
designing offices.

This critical speed can be calculated with sufficient accuracy
for practical purposes and as a rule the running at speeds
not in the neighbourhood of that indicated by the calculation
is free from vibration. But cases occasionally arise where
troubles from vibration occur at speeds above or below the
calculated critical speed, the reason for which is obscure and
the remedy correspondingly difficult to find.

It is the object of this paper to discuss various subsidiary
causes which might conceivably lead to unsatisfactory run-
ning at other than the usual calculated critical speed, but
while these are indicated as possible causes of disturbance,
it is not to be assumed that these causes always exist or that
they will always induce disturbed running. The object is
rather to indicate reasons why vibration might possibly arise
and thus if an actual case occurs, to suggest a clue to the
cause.

Although the basis of the paper is a physical or mechanical
one, the treatment is largely mathematical, as it is only
by this means that formule can be obtained from which
numerical results can be worked out.

The phenomena when a rotor vibrates are complicated, as
the shaft is supported in the bearings on a film of oil, the
thickness of which is continually changing, the bearings and
foundations are not themselves perfectly rigid, and there is
a certain amount of initial bending of the shaft (and toa
much smaller extent of the rotor body) due to gravity. If
the rotor consists of a number of disks as in a steam turbine,
there is also the inter-action of the forces of each disk on
the others.

For the sake of simplicity, we shall confine our attention
to the case of a single part rotor, either a disk or a cylinder,
rigid as regards bending and mounted on an elastic shaft
running in rigid bearings. Some effects of non-rigidity of
the rotor and bearings and of alterations in the thickness
of the oil film in the bearings will be indicated.

* Communicated by the Author.

On the Vibration and Critical Speeds of Rotors. 123

A single part rotor can vibrate in either of two ways, as
shown in figures 1 and 2, or in a manner which is a combi-
nation of the two motions :—

Fig. 1 shows a purely transverse vibration, while in fig. 2
the motion is solely one of oscillation about the centre of
eravity. In the transverse vibration the conditions are
clearly the same whether the rotor body is a disk or is
cylindrical ; but in the case of the oscillation, the motion,
owing to the gyrostatic effects called into play, depends
both on the proportions of the rotor and the speed at which
it is running.

The speed at which transverse vibration becomes a maxi-
mum we shall call the “first critical speed,’ and that at
which the oscillation becomes a maximum, the “second
eritical speed,” as the latter is in all practical cases con-
siderably higher than the former.

The following is a general outline of the treatment adopted
and the conclusions reached :—

Section 1 deals with the vibration of a rotor when not
running, and a relationship is deduced between the fre-
quencies for the transverse motion and for the oscillation
which we shall call respectively the “stationary first critical
speed,” and the “ stationary second critical speeds.”

The second section deals with the transverse vibration,
frictional resistance being ignored. It is first shown that
there appears to be no foundation for the frequently ex-
pressed view that there is a possible region of marked

vibration at +, times the first critical speed, as such a
conclusion can only be reached through an incorrect assump-
tion with regard to the conditions. It is then shown that
the motion or vibration is a circular whirl about the statically
deflected position of the shaft, and that this motion reaches
a maximum at a speed equal to the stationary first critical
speed. The magnitude of the whirl is proportional to the
amount by which the machine is out of balance, so that
the main vibration here dealt with should disappear with
good balancing.

The action of gravity is then gone into more fully, and it

124 Mr. G. Rodgers on the Vibration

is shown that in addition to producing the ordinary static
deflexion, the action of gravity is such as to cause a double
frequency ripple in the whirl which would tend to reach a
maximum at half the first critical speed. The magnitude of
this ripple is, however, proportional to the square of the
amount by which the rotor is out of balance, and would
therefore fail to appear in a well-balanced machine. In any
case the effect is very small.

It is then shown that a rotor with bi-polar asymmetry,
such as exists in a rotor slotted for a two-pole winding, may
show a double frequency vibration at half the critical speed
even when the rotor is perfectly balanced, so that such a
machine might vibrate at half the critical speed even when
it would ren perfectly at the full critical speed. Vibration
arising from this cause could not, theretore, be rectified by
balancing, and this is the only case met with where improved
balancing would not effect an improvement in the running.
This case is gone into in some detail, and it 1s shown that
the motion here also is a circular whirl of double frequency,
that is, of twice the speed of rotation of the machine. If, in
addition, the machine is out of balance, a triple frequency
effect might appear, but is not likely to do so.

The effect is then discussed of lack of proportionality in
the deflexion of the shaft and again the possibility appears
of vibration appearing at half the critical speed, but only if
the machine is not properly balanced. The effect is then
gone into of fluctuations in the angular velocity through
variations in the driving torque, and of resonance between
the rotor and the foundations or other masses outside the
machine, from which it appears that marked vibration might
appear at almost any speed through either of these causes.

The effect of friction on the transverse vibration is then
discussed, and the results are given for the case where the
frictional resistance varies as the first power of the speed,
and also where it varies as the second power of the speed,
the latter being more probably in accordance with the facts
than the former. It is shown that the maximum vibration
appears in both cases at a speed equal to the stationary
critical speed, also that the phase difference between the
force due to the out-of-balance and the displacement
depends on the amount of friction, and also on the speed.
If the frictional forces vary as the square of the speed, as
is probably the case, the angle varies also with the amount
by which the rotor is out-of-balance.

Some effects of viscosity of oil in the bearings, and of
different bearing clearances are then gone into.

be i as

and Critical Speeds of Rotors. 125

In Section 3, the oscillatory vibration is dealt with,
taking into account the gyrostatic effects when the machine
is rotating, but ignoring the friction in order to keep the
expressions as simple as possible. It is there shown that
the gyrostatic effect causes the point of marked vibration to
occur ata higher speed than would be the case if the machine
were not rotating, and simple rules are given for calculating
this vibrating speed. An example is added to illustrate the
method of working the rules given.

Much of the work on the main transverse vibration and
the main oscillatory vibration bas been dealt with in various
forms by Chree, Stodola, Morley and others, and the solution
for the transverse vibration with friction depending on the
first power of the speed has been given by H. H. Jeftcott
(Phil. Mag. March 1919), but the ground covered by the
remainder of the paper, particularly the question of sub-
sidiary critical speeds, does not appear to have received
much attention ; there is, however, in ‘ Hngineering’ a dis-
cussion where subsidiary critical speeds are touched on,
arising out of a paper by W. Kerr in that journal
(Feb. 18th, 1916).

Srcrion 1.—STarioNARY VIBRATIONS.

A. Transverse Vibrations.

1. If M is the mass of the rotor body (the mass of the
shaft being being neglected), and we assume the rotor to be
perfectly balanced, the shaft will, when not rotating, show a
deflexion measured at the centre of gravity of the rotor of

Pee eC ee eee 8 CL)

where a is the force required to produce unit deflexion. The
method of working out the static deflexion of the rotor for
~ actual cases is well understood and the value of o can be got
from the deflexion diagram.

2. If now the rotor is set in vibration in a vertical plane,
the motion is represented by the following equation (using

d?
fluxional notation, where # is written for “7 and y for

at?
dy etc.)

de? Mij+éy+My20:- 0: 2. (2)

126. Mr. ©. Rodgers on the Vibration

The solution is

yatiain(s/Zeom) M2,

where N, and y; are constants the values of which depend
on the initial conditions. The vibration therefore takes
place about the statically deflected position as a centre, and
with a frequency of vibration of a, where

2a

a=a/ x, Ms spi ol ee

This vibration takes place in a vertical plane and may be
considered as the resultant of two vectors rotating in oppo-

site directions, each with an angular velocity of fe 2 aE

M
o and M are expressed in c.g s. or f.p.s. units, this angular
velocity will be in radians per second and since from (1)

U

is numerically equal to a the speed of either of these
0

M
60 g

vectors in revs. per minute will be ie

9

If, further,
g and yo are in c.g.s. units we have the conn

ost 2 300
Yo Yo
where yp is the static dellenionn in cm.

3. It will be seen afterwards that, as is well known, this
formula gives the first critical speed in R.P.M. ; this is to
be expected, as the out-of-balance forces will then resonate
with the natural free vibrations, with the result that the
latter will become of considerable magnitude.

Jide t= = 5 = -approximately, . (5)

B. Oscillatory Vibrations.

1. If the rotor is twisted about its centre of gravity so
that the deflexion is in a vertical plane, and is allowed to
oscillate freely, the motion is represented by

Babich =O; ene i an

where B is the cross moment of inertia, that is, the moment
of inertia of the rotor about a line through the centre of
gravity at right angles to the shaft, yr is the angle through

and Critical Speeds of Rotors. 127

which the axis of the rotor at its centre of gravity is
deflected from the stationary position, and « is the torque
required to produce unit angular deflexion.

The solution is

p=Nosin(a/Ke—m) +r Neate ener Ge)

where N, and y2 are constants the values of which depend
on the initial conditions. The frequency of the oscillation
is therefore
itiof ie hiongl
Onn Bo on where

a=al *. ee eee =. (8)

As already indicated, we cannot at once deduce from this
what will be the actual second critical speed, owing to the
gyrostatic effects, but the result is of importance, as it
simplifies the calculation of the actual second critical speed,
as will be shown later. We shall in what follows call ¢, the
stationary second critical speed.

2. It should be pointed out that there is a simple relation
between c, and c, which greatly facilitates the calculation of
the stationary second critical speed in those cases where the
centre of gravity of the rotor is midway between the bearings.
If 2] is the distance between the bearing centres and P, the
force exerted by the deflected shaft on either bearing,

np = ZBL

The angle is very small so that the force P, is the same as
would be required to depress the shaft through a distance
al if the rotor were held rigidly. Now we have seen that
the force Mg at the centre of gravity causes a transverse

deflexion of Yo=r— eS and as is small,
yo=vl,
so that Mg=ocowl,
also kyp=2P,l and P,=3Mg;
therefore ky = Mol,
so that =o vier),

nt
and (8) becomes mia. as SO aren pee a)

123 | Mr. C. Rodgers on the Vibration
and at B=Mzk,’, a am

= hp M ’
and comparing with (4) we thus get:

(a

a 9
oe gf 12)
We thus find that |
First critical speed (transverse vibration)
Stationary Second critical speed (oscillation) ”

__ Radius of Gyration for the cross moment of inertia
Half the distance between the bearing centres

This is a useful formula for calculating the stationary
second critical speed when the first is known, for cases
where the centre of gravity is midway between the
bearings.

It shows that with cylindrical rotors of this type the
second critical speed must always be considerably above
the first, and the only instance in normal designs in which
the second critical speed could be lower than the first would
be that of a flywheel mounted on a short shaft.

Secrion I1.—TRANSVERSE VIBRATIONS—FIRstT CrITICAL
SPEED. ~

A. Neglecting Frictional Resistance.

1. It will simplify the treatment of this question if we
first consider the case of a rotor unimpeded by frictional
resistances set up by the air and then treat separately the
effects produced by friction.

The conditions obtaining when a rotor is not perfectly
balanced and is rotating are illustrated in fig. 3, where O
represents the position of the centre line «f the bearings,
and C the deflected position of the centre line of the shaft,
while G shows the position of the centre of gravity of the
rotor. O thus gives the undeflected position of the shaft
centre line and OC=r the shaft deflexion at any instant,
while CG=e is the error in the centering of the rotor ;
Mg is the weight of the rotor acting vertically downwards.

The rotation of the rotor about its centre line, 2. e. the
rotation imparted by the prime mover, is represented by.
the motion of G around G, ¢. e. by the rate of change of 0.

and Critical Speeds of Rotors. 129

The whirling of the rotor is represented by the motion
of C about the undisturbed position of the shaft centre line,
2. e. by the rate of change of a.

The “ vibration” of the rotor is judged in a general way
by the vibration of the bearings as felt when the hand is
applied to them. The force on the bearings is that applied
along OC by the deflexion r of the shaft, and vibration of

Fig. 3.

the bearings arises through the varying position and magni-
tude of OC; these in turn are due to the motion of the
ceritre of gravity G.

2. If the machine is steadily rotating it might at first
sight be thought that OC and CG would be in the same
straight line, so that G would be steadily revolving together
with C about the undisturbed position O of the shaft centre
line with an angular velocity n say. At the same time the
deflexion OC=r might be changing its value and (neglecting
the weight of the rotor) the motion would thus be given by

Mr—Mn?(r+e)+or=0,
or putting o/M=c,’,
P+ (ce —n?)r=n’e,
the solution of which is

: Te eee ne
r= N,sin (./ = Seer eRe?
: cy? —n? 11 cy —n*’

/

where N, and y; are constants.
We should thus conclude that » would become unlimited

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. K

130 Mr. C. Rodgers on the Vibration

in magnitade when the angular velocity n is equal to ¢ or

4), ap and, further, that the variation in the magnitude

of 7 consists of a free vibration having a periodicity of
1 V4 A
Ser, N ef —n?’

This result for the periodicity of the free vibration would
lead to the conclusions that when the machine is not rotating
(n=0) the periodicity is c, the same as for the stationary
transverse vibration, and that when running at the critical
speed (n=c,) the periodicity is zero. There would thus be
some intermediate speed where the periodicity of the free
vibration corresponds with the running speed, and resonance
might take place. This would occur when

¢2— nan?
n “1

or SS
V2

We should thus be led to expect marked vibration when
the running speed is 45 x the critical speed.

This conclusion and the argument on which it is based are,
however, erroneous. In the first place, the assumption is
made that w or @ is constant and further the condition
is omitted that, as all the forces pass through O, the angular
velocity about O must be constant, or °>9=h, say. The
correct equations for the free vibrations are thus :

Ga eR Ie 0,
PA=h. |

This does not admit of direct solution *, and it is simpler
to use rectangular co-ordinates, as we shall now proceed
to do.

* The solution is, however, well known and is given in books on
Dynamics dealing with Central Forces :—-

If p is the length of the perpendicular from the centre of force on the
tangent to the path, it is known that

h? dp
PON eae
ps dr ‘
ae h? ;
giving — =a constant— or",
pi

which is the pedal equation of a central ellipse.

and Critical Speeds of Rotors. 131

3. With the notation given in fig. 3, it will be seen that
the position of the rotor is completely defined by the co-
ordinates x, y, and @ (wand y being the co-ordinates of G)
and only three equations are required to express the motion
fully ; the value of the whirling angle a follows from the
magnitude of the other co-ordinates.

The force exerted by the deflected shaft is or, the com-
ponents of which are

—o(v—ecos@) along OX and
—o(y—esin@) along OY.

Resolving along OX and OY and taking moments about
G, we thus have:

Meso (ce (Cosiy JON yh ieee, (a8 7 (13)
Mi+o(y—esin@)+Mg=0, . . . . (14)
Mi.20+ce(asin@—ycosO)=0, . . . (15)

where k, is the radius of gyration of the rotor about the
longitudinal axis through the centre of gravity.

In practice the rotor is driven at an average angular
velocity » say, which will vary from constancy only by
small amounts which we shall find later are negligible.

Assuming as a first approximation that the angular
velocity is constant=@, so that @=qt, (13) and (14) then
become, writing ¢/M=c,’ as before,

HO eye COBIG ais a) He Me CLG)
Uti yy OSU WG.) tiie duels) GL)

while (15) becomes an identity.
The solutions are

€
red

C)7e
w=N, sin (cé—y1)+——5coswt, . . . (18)
Cea)
<N cosite ‘ ¢,7e 5 9
y=Nocos (eyt—y1) + as a sin wt—y/e?, . (19)
- l — :

where N,, No, and y,; are constants.

4. The above results give the motion of the centre ot
gravity G. The motion of the centre of the shaft C is to be
found in a similar way, still on the assumption that the

K 2

132 Mr. C. Rodgers on the Vibration

angular velocity of rotation is constant. If a’ y' are the co-
ordinates of C,
v=u' +ecosat,
y=y' +e sin of,
giving the equations :
i’ 4 ofa =07e€0S ol). 5 «eee

yj +e7y =e sin wl—g (21)

4
vu

the solution of which is:

v' =N,sin (C\t—y,) +

9

OMe c
gn ge wt, ance : (22)
7

t ‘
y =N> COS (Cyt — 1) + ae wt —g/cy’. . (23)
1 —

5. These equations are the same as for a perfectly balanced
rotor with a weight attached to it of such small value as not
to affect the position of the centre of gravity, or the down-
ward pull due to gravity, but producing a force of Mo’e.
In other words, we can treat the unbalanced rotor as if it
were a perfectly balanced rotor with a force Mw’e attached
to it, and as this mode of presentation is easier to follow than
the former, we shall employ it in the remainder of the paper.

6. It will be seen that the solutions for the motions of the
centre of gravity and the centre line of the shaft are the

C,7e

same, except that the former has an amplitude —, — ,, and
Cm @

the latter an amplitude of = so that they differ by the
C7 == }

amount e; this shows that OC and CG are in the same
straight line when not running at the critical speed.

7. The solutions show firstly that the motion takes place
about the position

Mig ig

99
(oR Cy™

which we have seen is the statically deflected position of the
centre of gravity of a perfectly balanced machine. It is
sometimes contended that as the speed increases the rotor
shaft tends to straighten out, but there is no indication in the
present treatment that this is the case.

&. The free vibration

“«=N,sin (¢,¢-y;),
y= Ny. cos (q¢— 1),
is the same as for a perfectly balanced rotor and has a

and Critical Speeds of Rotors. 133

frequency the same as the frequency for stationary vibra-
tions, and, as we shall presently see, if expressed in R.P.M.,
is the same as the first critical speed. It is, therefore, inde-
pendent of the speed of rotation of the rotor, and there is no

possibility of resonance occurring at V9 x the critical speed

as suggested by the erroneous method mentioned earlier in
this section. ‘The free vibration itself is thus represented by
two components having the same frequency, but different
amplitudes; it is therefore a central ellipse, the centre
Mg
Sar

It will be shown later that the free vibration is damped
out by friction, so that it has no importance in practice.

9. The forced vibration for the centre C of the shaft is
given by

being at the point y= —

we
i c2—o Cos at, ‘ e 5 = . (24)
Hy Dre Os i. On
= ae sin ot — 9] cy 5 A : c (2: )

This naturally has the frequency corresponding to the
angular velocity of rotation w, and has a maximum value of

9
@~e .
for each axis.

6 2

The motion of the centre is thus a whirl, the radius or

amplitude of which is proportional to the out-of-balance and

is zero when the rotor is perfectly balanced. A perfectly

balanced rotor, therefore, cannot whirl in the manner ex-
pressed by equations (24) and (25). 1

The amplitude of the whirl is also proportional to —;—,

e - ! : Cy —-@®

and it thus becomes a maximum when

oO = Se Oya

([he sign + merely indicates that the rotation may be in
either direction.)

This value of w gives the first critical speed, which is thus
the same as the stationary frequency for transverse vibra-
tions. Reasons will be given later why the radius of whirl
does not become infinite at the critical speed, ¢.¢., why the
shaft does not break when the rotor reaches this speed.

°

134 Mr. C. Rodgers on the Vibration

10. At the critical speed where w?=c,? equations (20)
and (21) become :
di + ci?" = ¢,"e cos yt,

YtoPy=cesin oyt—g,

which admit of a solution not involving infinite values,
namely,

(oe
v=t— sin Cyt
2

C1é F TT
= cos( ot - Ae os
C1eé

cos yt —g/cey

Cc .
=1Fsin( et 5) gore ere

Equaticns (26) and (27) thus give the motion at the first
critical speed when friction is ignored. They show that the
component along each axis has an amplitude which con-
tinually increases in proportion to the duration of the
motion, in other words the motion at the critical speed is
a spiral of continually increasing radius.

11. From (24) and (25) it will be seen that the phase
difference of the motion with respect to wt changes from
zero to 180° as w passes through the value ¢, 7. ¢., as the
speed passes through the critical ; (26) and (27) show that
at that speed the motion lags behind wt by 90°. It will be
seen later that when friction is taken into account, the lag
increases gradually as the speed increases, being still 90°
when o=(}. |

12. Up to this point we have treated the angular velocity @
as a constant=q@, as on the average it will he in practice.
Suppose now that it varies slightly from constancy, so that
the angular position wt becomes wt+u, where u is so small
that its square can be neglected, and we may write sinu=w
and cosu=Il. We then have cos@=cos@t—usinwé and

sin 0=sin wt +u cos at, also O=ii. Substituting these values

in equations (20), (21), and (15), and writing i =O ee
before, we get

é+o’e=ew7%ecos@t—o*eusinwt, . . . (28)
YtocP=yow'e sin ot+@*cucoswt—g, . . (29)

Mit oe(w sin wt —y cos wt + xu cos at +yusin@t)=0. (30)

and Critical Speeds of Rotors. 135

As a first approximation we substitute in (30) the values
already found for # and y, as given in (24) and (25), for the
forced vibration and thus obtain (neglecting wu in comparison
with 1 radian) :

2
ee @
1 —

giving for the forced vibration (the free vibration may be
ignored as it will be damped out) :

. e(¢,?— 0”)
ae wee)? — wk (cr— w”)

a9 COM@in Ne « - (32)

At the critical speed w=c,, and u=0, 2.¢., the angular
velocity of rotation is constant ; at other speeds than those
in the neighbourhood of the critical, the term we?c,? may be
neglected, as e? is much smaller than /,?, so that
eb)
ae ee é Boron wt. . . a a (33)
The variation in the angular velocity has thus the same
frequency as that of the rotation itself, but is very small in
magnitude, as will be seen from the following figures. For
a turbo-generator rotor balanced to about 1 oz. at radius
ky per ton weight of rotor, e/k, will be about 3x 107°, and
for a machine to run at 3000 R.P.M., &, will be about
90 cm., and @ is, say, 277 x 50,
U= 3x10 mia teks eos at
~ 50. ~*~ 4m? x 50x 50
=6 x 10-* cos at,

that is the vibration is very small, the amplitude being only
6X 10~ of one radian.

It should be noted that w is sao genoa to eand to q;
this variation therefore arises through the action of gravity
on the rotor when not perfectly balanced, and the variation
will be absent if the balance is perfect.

Substituting the -value of wu in (28) and (29) in order to
find the effect of the irregularity on the displacement :—

al
ui+ec?r=w’ecos wot + 7.29 sin wt cos wt,
|

2
Ne e
¥ + ¢7y=o7e sin wt — Ea 9 cos? wt —g,
‘ Uy

136 Mr. C. Rodgers on the Vibration
or

d+ ¢x=w'e coswi+ <— g sin 2at,

€

2 ky
ee ; 1 1 @2

Yto-y= we SID w Seg 082 Zot — 9 Wo ;

The solution is (for the forced vibration)

w’ecoswt 1 é i 4 ;
a= moog ar 9 hee hey? SIN 2@t, 9 S.. e aeoe)
wesinat 1 é 1 a / 5
= = Ses os 2 Jae il 35
Pow? 2 be eee +s 2 fe =). (35)

The irregularity has thus two effects on the main whirl :
firstly, the static deflexion is increased by a small amount,
and secondly, there is superimposed on the main whirl a
ripple of double frequency, which rises to a maximum at
half the critical speed. But the effect is very small, and may
not be noticeable ; in any case, as the double frequency effect
depends on e it cannot appear when the machine is well
balanced. .

There is, however, the possibility that a rotor which is not
perfectly balanced may show vibration at half the critical
speed due to the action of gravity, although gravity would
produce no such effect at the full critical speed. A vertical
_ spindle rotor is not of course subject to the action of gravity
in this sense, and if it vibrated specially at half the eritical
speed, the cause must be sought elsewhere.

13. We shall now consider some further possible causes of
subsidiary critical speeds or speeds where marked vibration
may appear other than the normal calculated critical speed.

14. An important case is that of a rotor slotted for a 2-pole
winding or with a shaft in which a key-way is cut, where
the rigidity of the rotor is greater in one direction sham in a
direction 90° away, so that if the shaft is rotating, the stiff-
ness in the direction of any one of the axes is not a constant o
but o +¢€ cos 2wt, where ¢ is small in comparison withe. We
shall assume that the rotor is perfectly balanced (e=0) and
to simplify the examination shall first consider the vertical
motion only. The equation is:

My+oy=—Mg— poe 2ot.
The first approximation is :

and Critical Speeds of Rotors. 137

Inserting this on the right-hand side of the equation and
solving, we get
Mg ; €
a = ———— - 2 t
y o 1 mea Wipe

ey he jvc/Dls yn) 1
=—g/c {1+ oa gp 008 Bat BP es ei {ee Os)

There is thus a double frequency vibration about the
statically flexed position, which has a maximum when
@ =), that is, at half the critical speed.

It is evident that such a motion must have a tendency to
arise if a rotor is unsymmetrical as regards its rigidity, for
in such a case when the shaft rotates the deflexion will be a
maximum or minimum twice every revolution, and if the
frequency of the consequent up and down motion is equal to
the critical speed there will be resonance ; this will be the
case whether the rotor is perfectly balanced or not.

Tt is thus possible for a perfectly balanced rotor which
would be quite steady at the critical speed to show marked
vibration at half that speed. If the normal running speed is
above the critical the forces called into play at half the
critical speed will be very small and may give no appreciable
effect, but if the running speed is in the neighbourhood of
half the critical speed vibration might arise.

Hig. 4.

15. It is worth while to examine the motion a little more
fully as there will evidently be some vibration in the hori-
zontal plane also. Let C (fig. 4) be the position of the
centre line of the shaft and OA, OB two axes at right
angles rotating about O with the same angular velocity w as
rotor. Let the co-ordinates of © be a and 6 with respect to
OA and OB and wu and v the corresponding velocities along
those axes.

ee

138 Mr. C. Rodgers on the Vibration
Then u=a—bo,
v=btao,
and the accelerations are
along OA: %t—va=a— 2he —aw’,
along OB: v+ue =b + 2a0—be?.

If the force required to produce unit deflexion in the shaft
is o+e along OA and o—e along OB, and we Tee: along
OA and OB, the equations are:

M(a — 2bw —aw?) + (o+¢)a= —Mgsin ot,

M(b + 2a@ —bw*) + (c—e)b = — Mg cos ot ;
that is,

{M(D?—o?)+ (¢+¢€)}a—2MeDh=— Mg sin ot,
{M(D?—o?) + («—e)} 6+ 2MoDa=— Mg cos ot,
giving
[ {M(D?—o?) +0}?—e? +4M?w?D?] a
| = — Mo(o —e— 4”) sin of,
[{ M(D?—@?) +0}? —e? +4M?w?D? | |
=— Mg(o+e—4o’) cos at,
The solution is (neglecting e? in comparison with o)
c= = thie) sin at,
b= - =o (aee ee ,) eos wt.

This gives the position with respect to the rotating axes ;
the position with respect to the fixed axes is

v=acos wt—b sin at,

y=b cos wf+asin of,

that is: |
tect VEG € ah
pe Jaa | SIM De ese (37)
iis. «ela €
yao fis =e? 008 200 | ial SNR, pee

This result is the same for y as obtained in (34) by the

and Critical Speeds 07 Rotors. 139

method of approximations ; it shows that there is a similar
motion of equal magnitude and 90° out of phase along the
horizontal axis, so that the motion is a circular whirl of
double frequency, which rises to a maximum at half the
critical speed.

16. When the rotor is out of balance the equation for the
vertical motion is

Mi + oy=Mo’e sin at —Mg—ye cos 2at.

The first approximation is given by (25) and inserting this
on the right-hand side of the above equation we get:

Mij+oy=Mo’e sin ot —Mg+ = € COS 2wt

e Moe

+ 3 SMe? Lin et sin Bat}. (39)

The first three terms on the right correspond to the main
whirl and the double frequency whirl already dealt with.
The last term on the right will give in the solution a triple
frequency vibration, viz. :

1 Moe € as
i Me eeoMae oe

which has a maximum value at 4 the critical speed. This
vibration is, however, proportional to e, the out-of-balance
force, and cannot arise in a perfectly balanced machine.
The remarks made as to the limited conditions under which
the double frequency vibration might arise apply with even
greater force to the triple frequency vibration as the damping
effect of friction will be correspondingly greater.

17. Another case of interest is that in which covers or
sleeves are mounted on the rotor, or the rotor has slots in
the periphery for an exciting winding, closed by pressed-
in keys; the closeness of these force fits will vary with the
deflexion, and the deflexion of the shaft may therefore be
not quite proportional to the force applied, 2. ¢., the force to
produce a deflexion x will not be ox, but say o(a+ea2*),
where ¢ is small in comparison with unity. (The expression
for the force must contain odd powers of a only as the
rigidity is symmetrical, that is, the same numerically for the
same numerical value of « whether « is positive or nega-
tive; if even powers were included this could not be the
case as an even power of .v is always positive even if w itself
is negative.)

140 Mr. C. Rodgers on the Vibration

The equations then are, putting the small quantities on

the right-hand side:
a+ ¢,?a4 = we cos wt — ¢,7exr’,
YT cy = we sin wt —g—¢,7ey?.
Neglecting the small quantities, the forced vibration is as

betore given by

®)

we |

ff == ¢;°— @ COS ot, ° ° ° e ° . (40)
ORE |

— aa sin wt —9/aq2..° 2 ae)

Inserting these values on the right-hand side of the
original equation, we get after some reduction :—

a + 7a = {we — 3c?ep?} cos wt — 1c,2ep* cos 3at,
2 2 Daye eee Dee
9 + ery = —9 + 61° €yo(p? + yor) + {we + cep (2yo" + Zp”) f sin wt

—c¢yep7y) cos 2wt + +¢,7ep? sin 3,
c,7e

where ~ Pome ;

ea)
and You ger
Solving these equations we get:

2
ee) p

L=p | 1-40 OF aH cos wt —1¢/?

COs dwt,

eae
C7 -- Q@?

Y= — yo{ 1—eyo(p? + Yo") }

a = Ne

ce —@"

(2y9? + 3p”) ; sin at

2
GCOS Wt
cy — 407 pPYo

Soneane pemrhee p> sin det.
Examining these terms in turn we find that the centre of
motion is now at the point
e= 0,
y= —Yyotl—eyo(p +4?) 5,

instead of the point e=0, y=— yo. This indicates that the
centre of motion rises, 7. e., the shaft straightens out slightly,
as the vibration increases.

and Critical Speeds of Rotors. 141

The main vibration, represented by coswt and sin af,
shows a slight change in amplitude, but as before the
maximum oecurs at the critical speed.

The term in cos2t indicates there is a double fre-
quency ripple in the vertical motion y (but not in the

, 26
° . . C7
horizontal component S) having an amplitude aa a 5 P'Yos
ae

>

C) € wo €

1 a) Yo This rises to a maximum
G2 —

that is, a des
at half the first critical speed (when the amplitude changes
sion) and again at the first critical speed (when the ampli-
tude does not change sign). Noticeable vibration may thus
occur at half the critical speed, but it will take” place
principally in the vertical plane.

Both components show a triple frequency vibration ex-
pressed by cos3@é and sin 3et, which reaches a maximum
at one-third the critical speed, and the amplitude of the
vibration changes sign at that point. This vibration also
has a maximum value at the critical speed.

Points of marked vibration due to lack of proportionality
in the deflexion can thus only show themselves when the
machine is out-of-balance, and if they become appreciable
at all will only occur at half or one-third, ete., of the critical
speed. If, however, these fractions of the critical speed
correspond to low running speeds, the forces may be so
small as not to produce any noticeable effect.

18. It thus appears that subsidiary critical speeds are only
to be expected at half or possibly one-third of the calculated
first critical speed, and only then when the subsidiary critical
‘speed is high enough to make the forces appreciable—for
example, in the case of a turbo alternator when the speed
indicated by the calculation approaches the running speed.
All these effects should disappear with perfect balancing,
excepting that due to lack of uniformity in the resistance
of the shaft or rotor to bending in directions perpendicular
to its axis, such as might arise “through two-pole slotting of
the rotor or through a key-way in the shaft.

The forces tending to produce vibration are small, and the
vibrations arise through a kind of resonance ; as there is a
good deal of damping due to air friction and to the move-
ment of the shaft in the bearing where the oil exercises a
strong damping action, the vibrations may not arise at all.
This question is gone into more fully in a later section.

19. We have now to consider some cases where resonance
may arise from causes outside the machine itself, and two

142 Mr. . Rodgers on the Vibration

classes may be noted, firstly where there is an irregularity
in the torque applied to the shatt, and secondly where there
is resonance with masses outside the machine.

20. Irregularities in the torque driving the machine may
arise, for example, through variation in the steam admission
or through a fluctuating electrical load.

The result of fluctuation in the torque will be a corre-
sponding fluctuation in the angular velocity of rotation so
that the angular position, instead of being o?, will be
wt+esinpt, where € is a small angle and p is an angular
velocity corresponding to the frequency of the disturbance.
Then cos (esin pt) =1 and sin (esin pt} =esin pt. The equa-
tions then become, taking the small quantities on to the

right-hand side :
CONG 9 9 " ° .s
% + ¢°u =e (cos wt —esin pt sin at),

jto?y=o’e (sin wt +e sin pt cos at)—g;
that is,

u+tcPa=ore{coswt+e/2(coso+p.t—cosw—p.t)},. (42)
. (48)

The main vibration is the same as before, but there are
two small vibrations superimposed ; the one has a frequency
corresponding to a +p and a maximum when w=c,—»p, the
other a frequency corresponding to w—p and a maximum
when @=¢,+ 7p.

This shows that the vibration may have a maximum at
speeds corresponding to the sum of and to the difference
between the critical speed and the speed corresponding to
the frequency of the disturbing fluctuation. So that if dis-
turbed running show itself at such a speed that it cannot
be otherwise explained, a cause may be sought for in this

)y
yj + cyy = @e{sin @t + e/2(sin op (esos o- p ; t) i

_direction.

21. The other variety of resonance mentioned is that
where, for example, the foundations are not sufficiently
rigid and the machine as a whole is vibrating so that there
is resonance between the rotor on its shaft and the machine
on its foundations. A similar case would be that of a machine
rotating in or near a building which itself shows marked
vibration, possibly in certain parts only, corresponding to
the vibration of the machine. Both these cases are similar
in principle and may be illustrated by supposing the whole
machine to be mounted on foundations having some elas-
ticity. If then M, is the mass of the rotor and M, the

and Critical Speeds of Rotors. 143

effective mass of the machine and that part of the founda-
tions which moves with it, and the forces required to give
unity deflexion are in the two cases o, and ay respectively,
the equations of motion are as follows :

My, + om (yy = Y2) 2s M,o7e sin pt,
Moijo + O2Y2—01(Y1 — Y2) = 0.
giving for the amplitude of the forced vibration

we (¢1?m + 9? — w”)

y= es a Wied” CAA
Tee wo? (Cy? + Co? + C47) + C475” —
‘ O71 9 Oo M,
mhere ¢¢=———, ¢-=—,.) m= —
M,’ M,’ M,

Points of marked vibration may thus occur at either of
two frequencies given by putting the denominator = 0 ;

these frequencies will therefore depend on the ratio 2 as
3 2
well as on c, and c., and may thus have almost any values.
For example, if cp=c,; and m=0°2,12.e., the mass of the
machine and foundations is five times the mass of the
rotor—

ype ed Be Se)

w* — WoC? X 2°2 + ¢;*’

and a maximum occurs when
w=c, X12) or 6, x 0°80,

that is, at speeds 25 per cent. above and 20 per cent. below
the calculated critical speed.

If M,, the mass of the machine and foundations, is very
large in comparison with Mj, the mass of the rotor, the
denominator is very nearly equal to (w—c,) X (o—c.), which
shows that in such a case the two speeds where marked
vibration may occur nearly correspond to the natural fre-
quency of the rotor and of the machine and foundations
respectively. But as the numerator is also small the vibra-
tion might not appear if considerable friction is present.

If vibration should occur when w=c,//2, which is, as
mentioned above, sometimes thought to be a critical speed,
this might indicate that there was resonance with the founda-
tions or some structure outside the machine, in which case,

144 Mr. C. Rodgers on the Vibration
putting w?==¢,7/2,
67 —2 {¢,7(L+m) +7} +4¢2=0,

that is CA, ae

Vi 2D

so that if, for example, M, were large in comparison with
M, and m is therefore small,

Pao A= 6, <U00T,
but if, as in the former example, M,=5M,,

C= C1 x Oroabr

B. Transverse Vibrations with Friction.

1. The frictional resistance opposing the motion of the
rotor may be considered to consist of two parts. The first
part opposes the rotation of the rotor about the centre
line, C, of the shaft, and this is counteracted by the torque
supplied by the turbine, or, if the turbine is cut off from the
steam supply, it tends to bring the set to rest; it has no
retarding effect on the whirling. The second part opposes
the whirling only, and it is with this that we have to deal.

2. It is not known how the frictional resistances opposing
the whirling vary with the speed, but it seems likely that
they vary with the square of the speed at least. We shall,
however, first consider the case where the resistance is
assumed. proportional to the first power of the speed, as the
motion is then simpler to work out, and there is an inter-
esting electrical analogy, which enables the motion to be
more readily followed.

The resistance to whirling is in opposition to the path of

1g ; :
the rotor centre, so that if = is the speed of the centre in
any direction, the frictional resistance is My (S). and the

eas along (OX ard ONG ane Mu() > i and

ds

; :
Me (G; yo 7,7 that is, Mus‘ and Mys"~’y, where mw is a

constant.
3. In the particular case we are about to consider, n=1,
and the components are therefore Mud and Mpg.

and Critical Speeds of Rotors. 145
The equations of motion are therefore
Mé+Mypet+or= Moe cos at,
My + Mpy+oy=Mo’e sin wt —- My.

Electrical engineers will notice the similarity between
these equations and

Lg+Rg+ cq=E sin wt + Ko,

I:
which holds for a circuit comprising an inductance L, a
resistance R, a capacity K, an alternating E.M.F. of
maximum value H, and a steady H.M.F. Hy, ¢ being the
charge in the condenser at any time. Thus, mass is equiva-
lent to inductance, capacity to deflexion per unit force, and
applied H.M.F. to applied mechanical force.

The solutions of these equations are, as is well-known :—

g= Ne *" sin (pt— $)

+ z par gets hs )
Ft /Kon tary ( 1/Ko—Lo,
EN ah mn emEnS Ay ca ic tnt ky eng Weim ho ey Cede)
y=Ne ‘sin (pi—¢@)
Mo’e _, Meo
sin (ot —tan Fane),

= WV { (o— Mo’)? + M?u?w?}

—Mglc, (46)

that is,

SS

y=Ne #T sin pt—o

we : % @
Ti Veo)? 4 we? sin (wt —tan : aS — gfe’,
47
where ey OL, i 2M (47)
art: Mp’
p=v 1/LK—1/T? or Vo/M—1/T?,
Cy = stk or VW o/M.

There is therefore in both cases a free vibration having a
frequency slightly less than the natural frequency of the
system, but independent of the frequency of the applied
W.M.F. or of the speed of the rotor. This vibration is
damped out by friction.

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922, L

146 Mr. C. Rodgers on the Vibration

The forced vibration is permanent and has a value which
is proportional to the applied H.M.F. or out-of-balance force.
The vibration is a maximum when o=«, that is the critical
speed is the same as the stationary critical, as in the case
where friction is ignored; the amplitude of the vibration at
the critical speed is : or a that is, is equal to the applied
E.M.F. or out-of-balance force and inversely proportional
to the resistance.

The frequency of the forced vibration is the same as the
frequency of the E.M.F. or of the rotation, and the charge
or displacement lags behind the E.M.i’. by an amount
depending on the frequency or speed and on the capacity
and inductance or elasticity of the shaft and mass of the
rotor.

The lag is zero when the frequency is low, but increases
to 90° at the critical speed, which, as will be seen, is that
corresponding to the natural frequency of the system, while
at very high speeds the lag increases to 180°, in other words
the force is in opposition to the displacement. The change
is similar to that occurring when there is no friction except
that in the present case the change is gradual instead of
taking place suddenly at the critical speed.

It will be noted that in both cases the vibration takes
place about the statically deflected position as a centre.
It is evident in both cases that a large static deflexion
would increase the tendency to break down, in the one case
by puncture or flashing over of the condenser, and in the
other by fracture of the shaft.

4, If the resistance to whirling is proportional to the
square of the speed, that is n=2, the equations-are :

+ pst + c2u=w'e cos wt,
i + psy +¢°y=o'e sin wt — 9.

The free vibration (7. e. the vibration when e=O or the
rotor is perfectly balanced) cannot be expressed in simple
terms, but as it will be damped out, as before, it is not of
interest.

The forced vibration (2. e. the vibration due to the rotor
being out-of-balance) is given by

e—Rieos(@t—O).4 o) os) - deem
y=R sin (@t—d)—g/c",. - - . (49)

and Critical Speeds of Rotors. 147

where R is the radius of whirl of the value :

ae | pan eee 50
Vi V (cr — 07)! + 4e?otpw' + (¢7—w?)”} aM,
and tan ¢6= Le Pe 2S (51)

The lag of the displacement behind the force is in this case
proportional to the actual deflexion, and in this respect differs

from the result obtained in (47). This is of interest as it

shows that since the radius of whirling is for a given speed
dependent on the out-of-balance, the phase lag will be smaller
the more perfectly the machine is balanced ; in the former
case where the friction varied as the first power of the speed,
the lag was independent of the amount of out-otf-balance.

The maximum deflexion occurs, as befcre, when a= ¢,, that
is, when the speed is equal to the stationary critical speed.

5. It is impossible to draw any conclusion from these
formule as to the real angular advance corresponding to a
given speed, as it is not known how the frictional resistance
varies with the speed. We can, however, say that if the
machine is rotated first in one direction and then in
the other, the position corresponding to the out-of-balance
will be mid-way between the points of maximum deflexion.
When balancing a machine in the running condition it is
usual to hold a pencil or chalk against the shaft so that a
mark is made on the shaft at a point corresponding to the
maximum deflexion. If there were no friction and the speed
were not the critical speed, this mark would be in phase with
the heavy side of the rotor below the critical speed, and 180°
out of phase with it if above the critical speed. But it will
be seen from (51) above that the actual position of the mark
depends both on the amount of friction and on the amount
of out-of-balance. At the critical speed the heavy side of
the rotor should be 90° out of phase with the mark on the
shaft, but the actual position will be uncertain, as the angle
varies rapidly with departure from the critical speed, and it
is not usually possible to judge exactly when the machine is
running at the critical speed.

6. There is another reason why the position of the mark
on the shaft is somewhat uncertain. Referring to fig. 3, if
we ignore all other vibrations than that corresponding to the
variation in ¢, we get by taking moments about G:

Mked+oresind=0. . . . . (52)
L2

148 Mr. C. Rodgers on the Vibration
If ¢ is a small angle this becomes :
Mh 2g +orep=0,

the solution of which indicates a periodic motion having a
time of vibration of

Mk?

hore

ky
= 2or oe bene (53)

In an actual machine for 3000 R.P.M. we shall have
figures, of the order of:-%=90 cm, Lje=— ae
¢;= 27 x 30, say, while » may be of the order of 1 mm.,
so that

P= 27;

ah soar 50x3x 104
~ Qer x 30 O-1
== 1-410? see. or about 2 mins:

As the time of vibration is very long in comparison with
that of the other vibrations occurring, it will be almost.
unaffected by the latter, and the assumption that the other
vibrations can be ignored, which was made in deducing
(52), is therefore justified.

For lar ger values of ¢ corresponding to less perfect balance
and for deflexions of greater magnitude, T will be corre-
spondingly less, and will be oreater the more perfect the
balance. If friction is ignored, r becomes infinitely great
at the critical speed and I’ becomes zero, and although this
can never be the case in practice, it is lore that T’ may have
a value of two mins. or more down to something considerably
smaller.

In other words, if the rotor is disturbed from its position
of equilibrium by any chance external cause, it may take a
considerable time to settle down, and during that period the
position of the mark on the shaft. will vary considerably
from its normal position.

7. At the critical speed the lag is 90°, and the vibration
is also a maximum, but the sharpness of this maximum will,
as indicated above, depend on the frictional resistance to
whirling. In addition to this it will also be influenced by
the condition and design of the bearings, as the oil in the
bearings exercises a considerable damping influence and also
introduces a further complication as follows :

When the speed is low and the vibration therefore small

and Critical Speeds 07 Rotors. 149

in magnitude, the film of oil in the bearings will allow the
shaft a certain amount of play ; this will increase the effec-
tive length of the shaft and ion er the critical speed; with
increasing speed the vibration will therefore start up fairly
smartly. As, however, the speed increases and the vibration
becomes greater, the shaft may bed hard up against the
bearing bush, and increased deflexion will decrease the effec-
tive length of the shaft, and so raise the critical speed. As
the speed is further increased a similar state of things is
gone through, so that at a certain point the vibration will
die down more quickly than if there had been no film of oil.

The effect of the oil in the bearings is thus to give an
added amount of friction to whirling, and at the same time
flatten the maximum peak of the vibration, that is, the
vibration will start up and cease fairly smartly and remain
more or less constant throughout a fair range of speed.
If, however, the film of oil is sufficiently thick or the
balance sufficiently good, the vibration may not show itself
at all, although it might do so with the same out-of-balance,
if the film of oil were thinner.

Section LI1.—OsciLLAToRY VIBRATIONS—SECOND
CRITICAL SPEED.

1. Oscillatory vibration may arise in two ways, either
through lack of balance or through vibration transferred
from the transverse motion.

2. The lack of balance referred to is of the skew type, that
is, is equivalent to a pair of weights at opposite ends of the
machine, and 180° apart, giving an out-of-balance couple
when the machine rotates; such an out-of-balance will not
show itself when the machine is being statically balanced on
knife edges, and can only be corrected through observations
when the machine is running.

3. Vibration can be transferred from the transverse motion
only when the machine is unsymmetrical in the sense that a
force applied to the centre of gravity at right angles to the
shaft gives a displacement which is not parallel to the centre
line, that is, in those cases where, on the static deflexion
diagram, the shaft in the deflected position is not parallel to
the centre line of the bearings.

4. The form of out-of-balance mentioned will produce a
couple rotating with the machine, that is, a couple alter-
nating with the frequency corresponding to the running
speed. Vibration transferred from the transverse motion
may, however, be of the frequency corresponding to the

150 Mr. C. Rodgers on the Vibration

speed of the machine, but may also be of double frequency
arising through any of the causes we have discussed.
Further, when the normal frequency oscillation has estab-
lished itself, a double frequency oscillation may start up
owing to bipolar asymmetry or some of the other causes
mentioned in connexion with the transverse vibration. It
is therefore necessary to consider in the oscillatory motion
forces both of the actual frequency of rotation and of
double frequency.

5. We have found that the transverse motion can, with
sufficient accuracy, be considered the same as for a perfectly
balanced machine with an ‘out-of-balance force attached to
it. In the same way we shall treat the oscillatory motion as
being due to an out-of-balance couple of the frequency corre-
sponding to that of rotation or a multiple of that frequency
acting on an otherwise perfectly balanced machine.

6. In the diagram fig. 5 let G be the centre of gravity of
tne rotor and GL the direction of the centre line of the rotor

Fig. 5.

| ¥

twisted from its normal position by an angle w= LGZ, where |
GZ is the direction of the centre line when not vibrating.

The direction cosines of the centre line GL with the axes
GX, GY, and GZ are respectively £, 7, and ¢; if L is a
point at unit distance along the shaft from the centre G,
then &, 9, and € are also the co-ordinates of the point L, as
shown on the diagram.

If the moments of inertia of the rotor about the shaft
centre line and about the line at right angles to it, through

and Critical Speeds of Rotors. LoL

the centre of gray ity, are A and B respectively, the angular
momenta are *

about GX, hy=B(nf—f) + Aw€,
about GY, h,=B(cE—£¢)+ Aon,
about GZ, h;=B(£)—7né) + Aol.

In the actual case the angle y is very small, so that we

can put siny=w, cosy=1, and ¢=1, €=0, also the

preducts £7 and n£ are both negligibly small.
We thus get :

k= —Bn + Ao€,
ho=BE+ Aon,
h;= Ao.

* Another and perhaps more legitimate way of deducing these
equations is as follows :—

If 0, ws, and 3 are the instantaneous angular velocities about moving
axes +X’, GY’, and GL fixed in the rotor and moving with it: consider
the stant when GX’ is perpendicular to GL and GZ (ef. fig. 5), and
let 9 be the angle between the planes LGZ and YGZ,

Then o,= =H. w,=8 sinw, and w,=w.

The angular momenta about GX’, GY', and GZ are Bw,, Bw», and

Aw, The angular momenta about the fixed axes GX, GY, and
GZ ae:

2i=Bi w,cos 0+, cos W sin 6++ Aw, sin sin 8,
h,=B{ —o,.sin 9+w, cos cos 0} + Aw; sin W cos 8,
hs = Bi — we sind}+Aw,cosw;

that 1
h\= B{—v cos 9-+6 sin Y cosW sin 8} + Aw; sin ~p sin 8,
ho= Bf wbsine+ésin WJ cosw cos 9 + Aw, sin Y cos 4,
h,=—B6 sin? J+ Aw, cos v.
Aljo
sinwsin@ and c= W cos W sin 0+0 sin W cos 50,
n= BUS ON? and y= By conn eon Oi: @ sin Wsin @,
g=cos y. and g=—wdsinw;
so thit

no—ln=—w cos @+6sin cosysin@.
gé—tZ= p sin 0-+-8 sin w cos w cos 9,
| in—ni= —@ sin? vy.

Fy substituting these values in the equations for Ii, h,, and h,, we
obain the relations given above.

———————eEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeeEeeEeEe ———

152 Mr. C. Rodgers on the Vibration
The force of restitution of the shaft is :
about GX=xn,
ott GY=— «6,
about GZ =0.

If the out-of-balance couple is tw’, the moment of the

couple is:

about GX= —7w’ sin pot,
about GY =7o? cos pat,
about GZ =nil,

where p= =1 or 2.
The equations of motion are thus :

—Bij—«n+ Awé=—rtw’sin pot, . . (54)
BE+KE+Aon=Twcos pot, . . . (55)

Ae Ch) ae
or putting D for «/dt,

(D? + ¢?)n —moDE=Tw?/B. sin pot,
(D? + 62) £ + mw Dn = re0?/B . cos Ue

Aw=0,
K A
where ¢,?= B and m= B:
- The last equation A®=0 gives .«=const., that is, there i is

no Huctuation in the angular velocity of rotation.
The other equations give

D? +0)? + m?w?D?! E=7o?/B §{ (co? — pw”) —mpw? coi pot,
s ie PO 5 COvP

and a similar equation for ¢ in terms of sin pat.
7. The free vibration is not of importance, as it wil be
damped out as before, but it is of interest to exami its
value as illustrating the effect of gyrostatic action o1 the
motion.
The free vibration is of the form

“x=N,sin (gt—¢;); y=N.cos (gi—¢do),
where Nj, dy, No, o2 are constants and

=—lLime-+ J mo b4e7to oe 2 167)

and Critical Speeds of Rotors. 153

There are thus two natural frequencies of whirling,
depending on the direction in which whirling takes place.
For example, if the machine is running at what we shall

; fea
presently see is the critical speed, namely, mo=co= ve :

there are two possible frequencies of whirling for the free
vibration, viz.: g=1°618 C., or 0°618 C,.
8. The forced vibration is given by

o ta" /B ‘ ;
~ pw? (p—m) — 2” ie aa

g

2
TW

ae hiner ees a ey

with a similar equation for 7 in terms of sin pot.
The amplitude.of the vibration is a maximum when

po?( pB—A) — Be? =0,
or
1 eae age
ei oe

2

*~ p@B=B) ae

|e

9. This enables the second critical speed to be calculated
without difficulty, and fig. 6 gives the necessary curves for
reading off the proportional values directly.

The ordinates give the values of w/c, and the abscissee the
values of the ratio A/B. The method of using the curves is
as follows :—

1. Work out the radius of gyration /, about the shaft centre

line.

2. Work out the radius of gyration ky about a line perpen-
dicular to this through the centre of gravity.

. Work out the ratio of A/B or k,/k,; for turbo-generator
rotors its value is usually between *2 and ‘4 and for
flywheels up to 2:0. (This gives the working point on
the horizontal axis of-the curve.)

. Work out the “ first” critical speed in the usual way and
multiply by l/h, so as to obtain the stationary “‘second ”
critical (where /=half the distance between bearing
centres).

5. To obtain the second critical speeds read off from the

curves the figures given on the vertical axis and
multiply by the stationary critical found from (4).

io)

is

ill usually be

ents represented by the

bee
oe a
os =
= 6
i g
ee a
2
a ae lo)
Sis,
Ss og ;
a oS a8
D ae Fy
5
= on a
S) eo
oH
PS RB
BS
@ oo
HO Ee
= a
Oo on
o 4
o~-
1
Eio
BH
>
+ 4 &
“ed cL
a jo)

colicin

Lo
z
8

Ne

uae

right-hand side of equations

I

Thus

It is, however,

(04) and (55).
due to some external cause.

18

e that one of these components may be absent
ple

bl
turb
if there is a couple round the « axis only, equations (54) and

ing cou

concelva
the d

1s

and Critical Speeds of Rotors. 155
(55) become
—By—«n+AwE=—Te’ sin pot, . . (59)
Bees kam Oe viel istmer + (60)
giving the forced vibration

‘ 2 9.2
Ta)” Co’ — po

n= pe (es —po?)?— pat sin pot. - (61)
ones mpe BEE (62)
~~ B (e2—p?w?)?— mi" p?o* LSet EN ‘

The oscillatory motion is thus a whirl the nature of which
depends on the speed. The whirl is a maximum when the
denominator is zero, that is, at two speeds, one on each side
of the stationary critical, and given by

1
i
| ole =p Em) ik

There is thus a possible further second critical speed,
corresponding to the + sign, lower than that already found
corresponding to the — sign.

For the ordinary rotating couple the direction of whirling
is, of course, always in the direction of rotation, whether the
speed is above or below the critical, and this is indicated by
the fact that, as will be seen from equations (58), the sign of
the amplitude in both planes changes, showing also that the
phase of the motion has changed by 180°. But for an
alternating couple about the « axis only, as will be seen from

(61) and (62), the motion in the horizontal plane changes.

sion at each of the critical speeds indicated by (63), while
the motion in the vertical plane has a further change of sign

when p’e?=c,”. It will be seen, if these changes are fol-
lowed out, that the whirling is in one direction below the

stationary second critical and in the opposite direction above
that speed, while at the stationary second critical the motion
is in the horizontal plane only, that is, at right angles to the
applied torque.

11. The following example is given to illustrate the appli-

cation of the above curves and formule.

A rotor consists of a solid cylinder 30 ins. diameter and
60 ins. long, running in bearings, 107 ins. between centres.
From the deflexion diagram, the deflexion at the centre of
gravity is, say, ‘0087 in. or °0221 cm.

156 Mr. P. Cormack on Harmonic Analysis of
We then have:

».
The first eritical speed ¢; is |) =2020 R.P.M.

Bat
"148
The ratio of the radii of gyration h/t) is 6/19=0°316.
Half the distance between bearing centres=53°5 ins.

The radius of gyration for the cross moment of inertia
ko 18°37 ins.

The stationary second critical is, therefore,

=0730 K.P. MM.

aS D3'd
i alee
From the curves, fig. 6, it will be seen that the main
oscillation will occur at ;
9720 x 1:21=6940 R.P.M.
and that a double frequency vibration may possibly show
itself at -
5730 x 0°545= 3120 R.P.M.
Thus a turbo-generator designed for a speed of 3000 R.P.M.

and having the above mechanical constants might show
marked vibration on the overspeed test.

X. Harmonie Analysis of Motion transmitted by Hooke’s
Joint. By P. Cormack, A.A.C.Sel., Lingineering Dept.,
feyal College of Science for Ireland *.

is Wie the growth of high-speed machinery, the

determination of the accelerations of machine
pieces becomes of increasing importance. These deter-
minations are considerably simplified by expressing in the
form of a Fourier Series the displacement of the piece
under investigation. The value of this method in the

_ analysis of the various phases of the motion of the mechanism

of the direct-acting engine is well known. It is here
proposed to investigate the coefficients of a Fourier Series
for the angular displacement of the driven shaft of a Hooke’s
Joint. The method being applicable to certain inversions of
the slider crank chain, these are also included. ‘The ease
with which the coefficients can be determined, and calculations
made from the resulting series, make the study of these
mechanisms from this aspect one of considerable interest.

* Communicated by the Author.

Motion transmitted by Hooke’s Joint. 157

2. In Hooke’s Joint the point B moves in the great circle
CBN and the point A in the great circle CAN (fig. 1). The
arc AB is constant and equal to a quadrant of the great
circle. The point A receives its motion from an arm OA

Fig. 1,

set at right angles to the driving shaft OX, while B transmits
motion to the driven shaft OY. Let the angle between the
shafts be y; this will be the angle between the planes of CAN
and CBN. In the spherical triangle ABC we have

cos¢ = cosa cos b+sinasin b cosy,
Since c=7/2, this becomes
cosa@cosb+sinasinbcosy=0. . . . (1)

Plainly when B is at C, A will be at T; B will therefore
move through the angle a while A moves through the angle
b—/2.

Writing ¢ for a and @ for b—7/2, equation (1) becomes

i —cos dsin 8+singdcosfcosy = 0.
Put cos y=(1—n)/(1+72), and we have
(1+n) cos ¢ sin @ = (1—n) sind cos 0.
*, n(cos d sin 8+sin d cos @) = sin dcos 6—cos ¢$ sin 0.
sin(@—@) = nsin(d+@).
e(o—8) p99) — yeild +) ne-Ho+8),

158 Mr. P. Cormack on Harmonic Analysis of
Multiplying both sides by ¢“?—® gives
ye) 0) ata teen new"? —ne—228.
eto 9) 11 — ne?]. — ee

e2(o—%) = (1 ne (nee

Since d—O lies between +e and —S and n» is less

than unity, we may write
2i(@ —0) = log (1—ne~*) —log (1—ne?"®)

: = —ne— 29 —Inre— MO _ nig 69 __

tne 4 birch 4 132618 + , Ae
= n.2isin 20+ $n*.2isin4944n3.2isin60+...

-. &—6 = nsin 20-57 sin 40 + an sin 60>... a enemen

Tt will be evident that (2) gives the displacement of the
driven shaft relatively to the driving shaft. In practice
the angle between shafts joined by a, Hooke’s coupling
=

we have
1l+yn

rarely exceeds 15°. Since cos a=
a
Oe
For «=15° we get n=°0173, so that we can without

appreciable error neglect the terms containing the square
and higher powers of n in (2) and put

@— 0 == nsinZ6, 0 ee a
For the above value of a, the maximum value of o—O
given by (3) is ‘0173 radian or nearly one degree.
From (2) we have
@ = 0+nsin 20+ $n’ sin 40+4n? sin60+.... . (A)

a aoe 2n cos 24+ 2n* cos 46 + 2n? cos 60+ ...).(5)

n = (1--cos«)/(1+cos a) = tan?

2 2
a = (5) (—4n sin 20 —8n? sin 40—12n3 sin 60—...).
(6)

In obtaining (5) and (6) we assume the series formed by
the term-by-term derivative of the member on the right
in (4) and (5) to be convergent and to converge to the
differential coefficient of the member on the left. In

Motion transmitted by Hooke’s Joint. 159

obtaining (6) we take the case in which the speed of
the driving shaft is uniform.

From (5) the maximum speed of the driven shaft
is readily seen to be w{1+2n/(1—n)} or wseca; the
minimum speed is seen to be w{1—2n/(1l+n)} or wcosa,
wo being the angular velocity of the driving shaft.

When the anole between the shafts is not lar ge, (6) may

be written
8 _ansinv0(@Y
ThE CTY a i a ee

Thus the maximum angular acceleration of the driven
shaft is very approximately 4nw’. If «=15° and w=
60 radians per sec., the maximum angular acceleration
is almost 250 radians per sec. per second.

Fig, 2.

ee
WS
N

\

Ys N

BAR
RW

z

3. In the mechanism of the pe caliaanae cylinder engine,

and the quick return crank and slotted-lever mechanism
(fig. 2), we have

sing siny_ sin (8+¢)
ae QA Whe eel Chant
sind = ~sin (6+6)=nsin(@+¢), where n= -
2 —e—1 = nett) — ne +9),
e246 —1 = nell +29) ne 2,

eb (1 —ne’”) = 1—ne-,
2 = (L—ne—) /(1—ne’*).

TT ee ee ee ek omy pap!

160 Harmonic Analysis of Motion.

Since ¢ lies between +5 and =37 and n<1,

2id = log (1—ne-) —log (1 — ne”)

? van: Way
0 Lr 220 SUS ho

= — ne anre
4 nc 4 dnret® + ins
=n.2i1sind+4n?.2i sin 204 4n? . 21 sin 30+....
o = nsin 0+4n? sin 20+ 4n' sin 30+..... . . (8)

= (n.00s +n? cos 26 +n cos 30+...) “Hines (9)

j 2
- = (—n sin 0—2n’? sin 20—3n? sin 30—...) (“) ,

4. In the Pin and Slot mechanism (fig. 3) we have
Vi = a—O—d¢..
The angle ¢ is given by (8), so that

n?

2 |
we = w—O—nsin p— sin 20— 3 sin SO ates ea alata)

a = (1+ncos @+n? cos 20 +n cos 30+...),(12)
2 2
ju a (Fi ) (n sin @-+ 2n? sin 26-+ Bn? sin 36-+-.,.). (13)

It will generally be found that we need to consider but
the first few terms of these series in making numerical

calculations.

Lee AGRE 4

XI. Short Electric Waves obtained by Valves. By E. W.
B. Gi, W.A., B.Sc., Fellow of Merton College, Oaford,
and J. H. Morreut, M.A., Magdalen College, Oxford *.

x. HERE have recently been discovered methods for the
generation of continuous oscillations of short wave
length (of the order of aboat a metre) by means of three
eect: ode valves. In January 1920, Barkhausen and Kurz +
found that with hard valves—. e., valves at extremely low
pressure, if the filament and the plate were approximately
the same potential, or, indeed, if the plate were at a potential
considerably lower than the filament, provided that the grid
was. kept at a high potential with regard to them, continuous
oscillations could be maintained in a cirenit of the Lecher
Wire type connected to the grid and plate. The wave length
depended primarily on the grid voltage, but also on tlie
emission from the filament and on the plate voltae.
Whiddington f had previously described another method
of getting oscillations of lower frequencies using a soft valve,
1.é@., a valve containing gas at low but appreciable pressure.
He employed more usual circuits for a valve, in that the
plate was at a high positive potential with regard to the
filament and the orid at a few volts above the filament. In
this case longer waves were emitted, and he noticed that if
V was the orid potential and »X the wave-length emitted,
then >7V was constant §.
There appear to be other arrangements not hitherto re-
corded which will also give these waves. With a hard valve

and with the grid at a positive potential, oscillations can be.

obtained if the Lecher Wire system is connected across the
filament and grid; the plate may be positive, negative, or at

. the same potential as the filament, or it may be insulated.

Further, the third electrode—the plate—is unnecessary, for
aalincione can be sustained by means of a valve consisting
of a filament and an anode formed as a spiral of wire con-
centric with the filament, when these two are connected to
the Lecher wires. An intermediate arrangement has been
worked successfully in which the wave- length of the diode
connected as above is modified by a cylinder concentric with

* Communicated by Prof. J. S. Townsend, F.R.S.

+ Physikalischer Zeitschrift, Jan. 1920.

{t Whiddington, * Radio Review, Nov. 1919.

§ For a general account of these experiments see ‘ Radio Review,’
June 1920.

Phil. Mag. S. 6. Vol. 44. No. 259. July 1922, M

oe nail

erence, high
Ps Xue

TT Ee eS

162 Messrs. E. W. B. Gill and J. H. Morrell on Short

the anode, but placed outside the valve and set at various
potentials. The best conditions for these cases are still
under investigation.

2. Barkhausen and Kurz were apparently unable to give
any explanation of the way in which the oscillations were
sustained, while Whiddington assumed that the emission of
ions from the filament was discontinuous and occurred in
bursts. The authors, on the other hand, do not think that
any special assumptions are necessary, and that the ordinary
conditions for the maintenance of oscillations by continuous
emission will account for all the tacts they have observed,
provided that the time taken by the electrons to pass between
the electrodes is taken into consideration, as this time is of
the same order as the period of the short waves.

In the present paper only oscillations of the Barkhausen
type are considered in detail, but the theory can be extended
to cover all the types, and an account of some experiments
on the last type (with a diode) will be published later.

It is worth noting that certain writers give the impression
that the seat of the oscillations is in the gas or in the
electrons in the valve, and that the Lecher wires connected
to the valve serve only to demonstrate their existence*. It
appears from our experiments that the wires or conductors
attached to the electrodes are a necessary part of the
oscillatory system. Hven with the Lecher wires removed,
there will always be some circuit composed of the connecting
wires to the batteries or even the valve leads up from the
sockets, which will have natural periods of a suitable order
for short wave oscillation. This fact seems to have been
overlooked in some recent determinations of ionizing poten-
tials, where large emissions from a heated filament were
used as a source of electrons. Oscillatio:s will take place
even when the valves contain a small amount of gas, but in
all the experiments described in this paper gas-free valves
were used.

3. It will probably be most convenient first to describe the
experiments in detail, and then to set out the theory and
apply it to the observed facts.

Various valves were used, but mostly the Marconi M.T.5
valves, which were very kindly given to us by the Marconi
Company. These valves consist of a straight filament FF
held in the centre of the valve by springy arms. The ad-
vantage of the spring is that when the filament is heated and

* Whiddington’s theory is independent of there being any external
tuned circuit.

ao. aia 524

——
:

Electric Waves obtained by Valves. 163

expands, the spring prevents sagging. Surrounding the
filament is a cylindrical wire grid, GG, composed of thin
wire of square mesh, each square having a side of about
15mm. The lead to the grid goes out at the bottom near
the filament leads. A cylindrical plate, PP, surrounds the
whole with its lead going out through the top of the bulb.
These valves being used for transmitting purposes are very
thoroughly “glowed out”? and pumped tas avery high vacuum.
The filament emission is very high when heated w with 6 volts
direct, and for the low emissions that were generally used it
was very constant. As the plate lead passes through the top
of the bulb, instead of through the bottom and the sealed

socket, very high insulation is obtained, and, if a strip of

tinfoil ‘connected to earth is placed cama the outside of the
glass, very small anode currents may be measured by an
dieeiiombice without any disturbance due to leakage.

It is not necessary for ordinary wireless purposes that the
valves should be constructed with the grid and plates either
accurately circular in section or accurately centred with

regard to the filament ; but for the purpose of calculation a
symmetrical system of electrodes is necessary and the M.T.5
valve used in most of the experiments was specially selected.
All the numerical results to be quoted were obtained from
this valve. There is no difficulty in getting the slort-wave
oscillations with many types of hard valve, Whe French type
produces them quite easily, but the chief reason for selecting
the Marconi M.T.5 type was that the electrostatic field
between the square-mesh grid and the plate approximates

M 2

164 Messrs. BE. W. B. Gill and J. H. Morrell on Short

much more closely to the calculable field between two co-
axial cylinders than does the Held in the French type, where
the grid is a spiral coil of fine wire. The diameter of grid
- used was 1 em. and that of the plate was 2°5 cm. to an
accuracy of about 5 per cent. |

4. The preliminary experiments were made with the
apparatus arranged as in fig. 2. The valve is shown dia-
grammatically: F is the filament, G the grid, P the plate,
LL the Lecher wires, which were of copper wire each about
850 cm. long and spaced 5 cm. apart. ‘They were suspended
about 200 em. above the floor from insulators secured to the
walls at each end, and from one end were leads about 70 cm.
long to the grid and plate respectively. The bridze consisted
of two equal condensers, ©, C, joined through the heater-coil
of a Paul thermo-junction, T. The outer plates were fitted |

with contacts to slide along the Lecher wires. The capacity
of these condensers is unimportant, provided it is large com-
pared with the capacity of the valve. In practice, the
capacities were of the order of 1 milli-microfarad. The
terminals of the thermo-junction were connected to a gal-
vanometer by two long leads, which are not shown. The
sliding contacts were also connected to the negative side of
the filament-heating battery B, that on the grid-wire through
a high-tension battery V,and that on the plate-wire through
a potentiometer S. which could raise the potential of the
plate +6 volts above the negative end of the filament.

Two sensitive milliammeters, A, A, gave the steady currents
through the valve to the grid and filament respectively. A
rheostat, R, controlled the filament-heating. In all cases
potentials are measured with regard to the negative end of
the filament.

With this arrangement the electrons set free at the

Electric Waves obtained by Valves.: 165

filament move outwards under the positive voltage, V, of the
grid, and a certain number go direct to the grid and are
collected there, the remainder pass through the grid, and, if
the potential of the plate is just less than that of the filament,
they return to, and are finally collected on the grid. If, on
the other hand. the plate potential is a little above that of the
filament, a certain proportion of those getting through the
grid reach the plate. If the plate potentiometer is now ad-
justed till the plate current is just zero, and the bridge is
moved along the wires, it will be found that with the bridge
in certain regions a plate current appears. It was the
appearance of this plate current which led Barkhausen to the
discovery of the short waves. With the present apparatus
these oscillations are also made apparent by the deflexion of
the galvanometer attached to the thermo-junction. The
positions of the bridge at which the galvanometer gave a
maximum deflexion were fairly sharply defined, and did.
not always coincide with the positions for maximum plate
current.

It is not necessary for the plate potential to be so adjusted
that the plate current is just zero when oscillations are not
occurring. ‘The plate may be set at a considerable negative
potential, or the plate voltage may be positive. It was found
that for 'a given grid potential there isa certain plate potential
at which the oscillating current through the thermo-junction
is a maximum. Also as the potential of the plate was
increased, for plate potentials only slightly positive, if oscil-
lations commence the plate current increases ; at a certain
plate potential no change is noticed in the plate current ;
and at higher potentials the plate current decreases. For
the M.T.5 valve this critical potential was about +2 volts,
when the voltage drop down the filament due to the heating
current was about 4 volts.

In the first experiments with this apparatus the position of
the bridge was varied and the current in the thermo-couple
observed when the grid voltage V, the heating current, and
the plate potential were all kept constant.

The oscillating circuit consists of a condenser formed by
the plate and grid of the valve, the distributed inductance
and capacity of the Lecher wires up to the bridge, and the
capacities C, C in series with the wires and with the short
resistance of the thermal heater which connects them. Hence,
if there is an optimum wave-length A corresponding to the
grid voltage V, and if, starting near the valve, the bridge is
pushed along the wires, maximum amplitude of oscillation

eS ee

466 Messrs. E. W-B. Gill and J. H: Mortell-on Short

should occur when the above circuit is tuned to A, 2A, 3r
etc., these positions being indicated by the deflexions of the
galvanometer connected to the thermo-junction. Moreover,
the distances measured along the wires between successive
positions of maximum oscillations should be equal to S and
all therefore should be equal. It was soon found that this
simplicity was not attained, in certain cases equi-spaced
positions were found, but in the majority of cases there were
millet two sets of positions forming two series of equal
spaces, which, as the spacing distance of the two sets was
different, appeared to indicate two optimum wave-lengths.

These effects are due to the different modes of oscillation
of the system, and, according to the theory which we give
below, a grid voltage V will, under suitable conditions, sustain
oscillations of short wave-length between certain limits. Any
mode of oscillation corresponding to a wave-length between
these limits will be maintained. It was therefore desirable
to arrange the apparatus so as to avoid these complications.

5. The most obvious improvement was to give up the idea
of finding the wave-lengths by moving the bridge, and to put
the br idge and its leads at the far end of the parallel wires
joined to the valve, and to measure the wave-lengths of the
oscillations by means of a loosely coupled secondary circuit.
The system of wires connected to the valve is thus fixed. A
second pair of long Lecher wires were set up with a loop
joining one end, and this loop was brought near the valve
circuit. When the secondary is in tune with an oscillation
in the primary the current in the primary is reduced. The
deflexion cf the galvanometer connected to the thermo-
junction in the primary cireuit may be reduced by 50 per
cent. when the bridge in the secondary circuit is in the tuned
position, and a movement of 0:5 em. either way will restore
the deflexion to its original value. The distances between
the successive positions of the bridge on the secondary circuit,
for which the deflexions of the galvanometer attached to the
primary circuit are a minimum, are the same, and are equal
to half the wave-length of the oscillation in. the primary
circuit. All the wave-lengths quoted were measured on
this form of wave-meter and may be taken as accurate to
0°5 per cent. *

W:th the condenser bridge and thermo-couple at the far end
of the Lecher wires the filament was heated to give an
emission of a few milliamperes (this is low heating for an

* Townsend and Morrell, Phil. Mag. Aug. 1921, pp. 266-268.

Electric Waves obtained by Valves, 167

M.T.5 valve) and the grid voltage was raised by two volts at
a time by means of batteries of small accumulators from 16
volts to 120 volts, while the plate was kept about 2 volts
positive, as this gave large deflexions. The corresponding
galvanometer deflexions are shown in fig. 3. !

The deflexions are plotted against grid volts ; as a thermal
detector was being used, the deflexions are proportional to
the mean square of the oscillating current.

The curve shows that oscillations are occurring over nearly
the whole range, but that there are maxima for certain
voltages—viz., 16, 24, 42, 58, 82, 114, approximately. ‘The
wave-lengths measured as above give from 16 to just below
24 volts 1586 cm., from 24 to 40 volts 1451, and so on, the
wave-lengths for successive portions of the curve being 366,
307, 262, 233. These correspond to the free oscillations of

Fig. 5.

82 Vof.

the system, the wires of which were 850 cm. long with
leads to the valve about 70 cm. long, with a slight addition
for the leads within the valve itself.

The system of wires connected to the valve therefore
present a selection of various modes of oscillation with wave-
lengths 586, 451, 366, 307, etc., cm., from which the valve
chooses the one suitable for the particular voltage V between
the grid and plate—the sharp rises just before the various
maxima showing that the system oscillates on the longer
wave-lengths by preference. For each particular wave-
length there is a certain grid voltage which gives the
strongest oscillations when the heating current in the filament
and the plate voltage is constant ; but the heating current
and the potential of the plate relative to the filament both
affect the optimum voltage for a given wave-length. In-
creased emission has the same effect, but this effect depends
on the degree of saturation of the emission current.

In the preceding experiments the wave-lengths of the
oscillations were measured with a constant heating current

168 Messrs. E. W.B. Gill and J.H. Morrell on Short —

in the filament, but the current from the filament varied
with the grid voltage. For the lower voltages all the
electrons leaving the filament do not reach the ‘grid space,
some returning to the filament. or theoretical reasons it 1s
more convenient to find the grid voltages which give the
maximum amplitudes of oscillation on the various wave-
lengths when the heating current is so adjusted that the
same current flows from the filameut to the grid space for all
the voltages, the plate voltage being kept constant as before.

The table below gives a set of experiments done under such
conditions with an emission current of 6 milliamperes, and
the plate at 1:3 volts positive to the filament. In column 1
are given the wave-lengths Xin cms., in column 2 the grid
volts V, which excite these wave-lengths most strongly, ‘and
in oldie d the product A?V :—

Xr. Vv. rv.
208 em. 156°5 68 x 10°
233 L225 66
A 262 92°5 64
307 68°5 ” 64°5
366 50°5 67:5
451 36°5 74

All these results, with the exception of the last, agree well
with the relation \?V = const.

It is not difficult to see whv this agreement should be less
exact as V decreases. The electrons concerned are not all
moving under similar conditions. Owing to the voltage drop
of the heating current down the filament, the field between
filament and grid differs by about 4 volts for electrons starting
from the extreme ends of the filament. And when V be-
comes comparable to this 4 volts a disturbing factor is
introduced.

6. These experiments thus give the grid voltages which
produce the strongest oaurlle Wome on coria donnie wave-
lengths determined by the particular length of wire used.

To find the range of wave-lengths fieiniaiaed by a given
erid voltage a slightly different apparatus (fig. 4) was “used.
An adjustable circuit was constructed of two rods, and two
telescopic tubes fitted over the rods, so that the effective
lengths of the system could be varied by sliding the tubes
over the rods.

The condensers and thermo-junctions were attached at the
ends X, X! of the rods, and the ends Y, Y?! of the tubes were
connected to the plate and grid of the valve respectively, the
other connexions being as before.

Electric Waves obtained by Valves. 169

For brevity, the adjustable circuit will be referred to as
the rods. It is not possible to graduate the rods in wave-
lengths as against extension of the arms, as this wave-length

depends on the emission and on the plate volts. Thus with
a fixed length of the arms and 44 volts between grid and
plate:

(1) With plate potential fixed.

Emission 2°2 m.a. XN=311 cm.
6°8 306
9°38 300°

(2) With emission constant at 5°2 m.a.

Plate potential 1°2 volts. A=O0S em:
2°4 314

- Hence for a given setting A decreases as the emission rises,
and increases as the plate voltage is increased. This is due
to the fact that the plate and grid are not a potential node of
the oscillating system, but are a variable distance from it
depending on the alternating voltage necessary to sustain
the oscillation, and this in turn depends on the emission and
plate voltage. It is vot, however, necessary to go further
into this, as the wave-lengths were always found directly by
a secondary circuit as in Paragraph 6, the rods being used
as 2 convenient way of varying continuously the wave-length
of the system connected to the valve. With all the other
factors fixed, the rods were. pulled out a centimetre at a time
and the oscillating current and wave-length recorded for
each position. In one experiment the emission was 1°5 m.a..
the grid potential 44 volts, the plate potential 1°8 volts, and
oscillations were maintained from }=320 em. to X=451 em.
with a maximum oscillation about X= 323cm. It was
always found that the maximum oscillation was close to the
short-wave end of the range.

170 Messrs. ©. W. B. Gill and J. H. Morrell on Short

The effect of (A) varying the emission current keeping
the grid and plate voltages constant, and (B) varying the
plate voltage keeping the emission current and potential
between grid and plate constant was investigated with this
apparatus.

In (A) increased emission broadened the range and de-
creased the wave-length of maximum oscillation.

For example, with V,,=44 volts and V,,=1:2 volts the
wave-length for best oscillation with total emission 7:0 m.a.
was 295 cm. With total emission 10°6 m.a. it was 274 em.

In (B) increase of plate voltage increased the wave-length
and also broadened the range. Thus with V,,=44 volts,
and total emission 3°8 m.a. with V,,-=1°2 volts X=298 cm., ©
and for V,r=3°0 volts X=3821 cm. This last observation |
must not be confused with the case in which the potential of
the plate is increased and that of the grid kept constant. In
that case also, increase of plate potential increases the length
of the strongest wave, as was observed by Barkhausen, who
attributed all the difference in wave-length to the alteration
in potential difference between plate and grid. This cannot
be the whole of the explanation, for, as stated above, similar
results may be obtained by raising both plate and grid
equally with respect to the filament.

7. A simple theory to account for the maintenance of the
oscillations can be worked out by making some simplifying
assumptions ; but a general theory will not be attempted,
partly because the resistance of the oscillating circuits used
was unknown and partly because if the assumptions are not —
made the calculations become extremely complicated.

These assumptions are :—

A. That the grid and plate ean be regarded as forming a
parallel plate condenser. — -

B. That, of the electrons which leave the filament, a fixed
small proportion pass through the grid in a uniform
stream, and that each electron on passing through the
orid has the same velocity.

C. That the electrons which return to the grid from the
plate side are nearly all collected directly on it, 7. e.,
only a few pass through on the return journey.

D. That the oscillating potential differences are small com-
pared with the fixed potential differences employed.

It is also assumed that the pressure of the gas inside the
valve is so low that the number of collisions between electrons
and gas molecules is negligible—this is certainly true for the
valves used.

Klectric Waves obtained by Valves. Lea

With these assumptions we shall only attempt to show that
an oscillation can be maintained of about the right order of
wave-length.

I'he principle involved is the fellowing :—

Suppose the filament and plate are at zero potential and
the grid at + V ; then the electrons from the filament which
pass “thr ough the erid with a velocity v due to the potential
V come to rest at the surface of the plate and return to the
grid, which they again reach with velocity v. In the space
between the erid and the plate the total work done by the
fixed potential V on the electrons which move in this space
is zero, all the work having been done between the filament
and grid.

If now superposed on the fixed potentials there is an
alternating potential V, sin pt between grid and plate due to
oscillations, the work done by the potential V)sin pt on the
electrons is not necessarily zero. If the work is positive the
electrons are abstracting energy from the oscillating system,
and the average velocity with which the electrons hit the
grid is increased; the oscillations cannot in this case be
sustained by the movement of the electrons. But if the work
is negative the electrons are giving energy to the oscillating
system, and if the rate at which this energy is given is at
least equal to the rate of dissipation of energy in the oscil-
latory circuit by resistance, radiation, or dielectric loss, the
valve will maintain the oscillations. The average velocity
with which the electrons hit the grid is in this case less than
the velocity v due to the potential V which they acquire
between filament and grid, and hence the energy put into the
system from the battery V is not all used in heating the grid
but part is turned into energy of oscillations.

The above argument is not affected if, in consequence of
the oscillation, some of the electrons are collected on the
plate. In all cases, provided the total work dune per oscilla-
tion by the alternating field is negative, an oscillation can
be sustained if the dissipation of energy in the oscillatory
circuit is not large.

8. The particular ease in which the filament and the plate
are at the same potential when there are no oscillations may
be considered first. Let V be the potential above the plate
of the grid and d the distance between them.

When there are no oscillations the electron passes the grid

Wy
with velocity v |= py a ne v] and is then subject to a constant

m
retardation f, which brings it to rest just at the plate. If T

172 Messrs. E. W. B. Gill and J. H. Morrell on Short

is the time the electron takes to pass from grid to plate,
v=fT. A further interval of time T brings the electron back
to the grid with velocity v.

Assume now that superposed on the fixed potentials is an
alternating potential Vosin pt between plate and grid; the
electric force due to this in the space between plate and grid

Tr

is qin pt, and if —eisthe charge onan electron the corre-
Pay tect 4 HON ane
sponding force on it is —7 sin pt towards the plate.

Since Vy is taken to be very:small compared with V the
motion of the electron may to a first approximation be taken
as determined solely by V, 7.¢., its time across is T and
retardation f. | |

The work done by Vosin pt depends upon the time fp at
which the electron passes the grid, and for a particular value
| deV o

of t the work is equal to 7 sin ptdx. The axis of #

0
being perpendicular to plate and grid and #=0 being on the
atc
But the velocity at time ¢ is

ada “
Te Od Mi Oat Need eto)

and the above work reduees to
ott

fe “ (T+ to—t) sin pt dt,

to
which finally gives :
Work on electron going from grid to plate =

2eVo/T cos pto sin pto—sinp Tt +.ty\_ l

ey p di i py? SRE L Oe (1)
similarly, the work done on the same electron as it returns
from plate to grid comes out as

2eVo ak aE PPR TN
“pe (cos to + 2'1 aaa

sn pt,+T—sin pto+2T A
; a) ), (2)
; P

Thus the velocities of the electrons on their arrival at the
plate or on their return to the grid depends on ¢o, that is, on
the value of Vosin pt at the instant they pass through the
grid. Assuming a constant stream of electrons through the
grid, it is easily seen by integrating (1) for values of

between O and = that the total work done per period is 0

Electric Waves obtained by Valves. Rae SC

and similarly for (2). Hence, if all the electrons returned
to the grid an oscillation would not be maintained. The
possibility of a maintained oscillation depends in this case
on the fact that in each oscillation a certain group of the
electrons are collected on the plate and the integral of (2)
does not in consequence include all the values of t) between

2

2Qar

0 and — and its value is not therefore zero, but may be
Pp eee,

negative.

The first step is, ietctore: to find which electrons reach
the plate. When there are no oscillations the electrons have
sufficient energy on passing the grid to just take them to the
plate against ‘the potential V, uel if therefore any extra
work is done on them they will be collected on the plate,
but if the work is negative they will fall short of the plate
and return to the grid. Expression (1) shows that ali the
electrons which pass through the grid at times ¢), such that
T cos pty if sin ptp—sin p(T + to) —

) 2
plate, while those for which it is negative just fail to reach
the plate and return to the grid.

OF the electrons then which pass the grid hale go on to
the plate and half return to the grid, the electrons. running
to the plate for a time equal to |p (half the periodic din
of the oscillation) and then running back to the grid for
time z/p and so on. But the total work done “by the
oseillating potential on the two halves as they go from
grid to plate is zero; and therefore the net work done
is the work done on the return journey on the half which
returns to the grid.

To find therefore if an oscillation whose periodic time is
277/p and amplitude V, can be sustained by a grid voltage V
it is necessary first to find the time T w hich the electrons

take to pass from the grid to the plate under the field due

to V alone, next to find. from equation (1) the values of ¢,
for those electrons which return to the grid when the
system is oscillating, and, finally, by taking the mean value
of expression (2) \ for fies values of fo and, knowing the
emission current, to find the total work done per second by
the oscillating potential. If this work is negative and at
least equal to “the dissipation loss per second, the oscillation
will be maintained.

A table of calculated approximate results is given below
for various values of the ratio T:1/p.. The second column
gives the values of pt) for the electrons which return.

is positive, will reach the

174 .Messrs. E. W. B. Gill and J. H. Morrell on Short

to the grid in each oscillation, the third gives the total
work done per second by the oscillating potential in
arbitrary units for a fixed value of V, and of emission,
and the fourth gives the corresponding wave-lengths for
the particular value, V=44, for which T==4°3 x 107° second
(see next paragraph) :— |

Values of pty
for electrons which Work. Vo
return to grid.

165° to 345° negligible. 1040 cm.

ny
Do AT
Tw
= 50° ,, 330° —" 200
P= oy 150° ,, 330 47 5
pee 135° ,, 315° —-85 Suge
ay
B= ap ve TOe | e00R ~"36 260,
ad © ,, 180° 32 130
o=e 90° ,, 1 oe

In all these cases the work is negative and oscillations
can be theoretically sustained, though in practice the
dissipation losses in the oscillatory circuits are such that
generally only those wave-lengths corresponding to the
larger values of the work exist.

For all values of the ratio T:1/p outside the range of
the table the work is small. The larger the work the
greater the amplitude of oscillation that will be sustained,
and the periodic time 27/p of the oscillation of maximum
amplitude for a given value of ‘I’ is seen to be in the region
of 8T. In general, the wave-length corresponding to this

will be sustained, and aiso a certain range of wave-lengths ~

on both sides of it, the limits of the range being determined
by the dissipation losses in the. oscillatory circuit. The
theoretical result is in good agreement with several of the
experimental results of paragraph 6.

In the particular case recorded there for V=44 the
range of wave-lengths sustained was from 320 to 451 em.,
with a maximum amplitude for 323 cm. Increase of

emission broadened the range of wave-lengths sustained,

which is in accordance with the fact that for a given Vo
the work put into the oscillatory system is proportional
to the emission current.

The fact that the wave-length of maximum amplitude
of oscillation was near to the short wave-length end of

the range also agrees with the calculated fact that the

Electric Waves obtained by Valves. 175

work done falls off much more rapidly on the short-wave
side of the maximum than on the long-wave side.

9. To calculate the time T an electron from the filament
takes to go from the grid to the plate when the grid potential
is V volts above both filament and plate it 1s not necessary
to assume the grid and plate to be parallel, but they may be
taken, as they actually are, to be concentric cylinders of
radii a, 0.

The retarding force on the electron when it is at distance v

from the axis is 2 where k=V/log.~
The equation of motion is therefore

: d*r —ek
i ae AS
dt? yr?

which gives when integrated twice, remembering that

dr
i 0, when r= 0,

bE ee a ey gee
ae ee (ae af oda
eV Seo i é aN.

In the actual valve used a=*5 cm., b=1'25 cm., and, taking

é = 7 _ . e
“+= 5:3 x 10" E.S. units and measuring V in volts,

qe 200 x 107°
V/V

the accuracy of this being limited by the accuracy to
which a and 6 are known and probably from 5 to 10 per cent.

The wave-length for any relation between T and 1/p can
now be at once calculated. If p=nz/T, the time of one
oscillation is 2a/p or 2T/n and the wave-length in cm. is
6x 10° T/n. oie

The simple theory shows that for the oscillations of
maximum amplitude pT has a certain value about 37/4. But

second,

1 1
T x Fs andrA« a and hence the connexion between the
Vv

grid voltage V and the wave-length » of maximum
oscillation is A7V =constant.

10. The theory is thus in good general agreement with
the experimental results, but there is one fact unaccounted
for—that being the variation in the wave-length of the

176 Messrs. BE. W. B. Gill and J. H. Morrell on Short

oscillation of maximum amplitude, for a fixed potential
between grid and plate, when either the emission is altered
or the plate potential is slightly altered with respect to the
filament.

There is also a special case, which is forming the subject
of a separate investigation, in which, when ‘the plate is
very negative (40 volts or so} with respect to the filament,
oscillations can still be produced, but without any current
reaching the plate at all. These oscillations are, however,

_ very much weaker and more difficult to produce than those

dealt with in our experiments.

The simple theory which depends on the collection on the
plate in each oscillation of a group of electrons will ob
not account for this special case.

The explanation of the above considerations is to be looked
for in the assumptions made in the simple theory. The first
assumption that the grid and plate could be regarded as
parallel is not important, as the field between cylinders of
the size of the grid and plate used is not far from uniform.
(It will be noted that the value of T was calculated for the
valve used by taking the field between cylinders.) The only
difference between cylinders and parallel plates on the simple
theory would be to make the ratio T to 1/p for maximum
oscillation slightly different. But the second assumption
that the electrons pass the grid in a constant stream all
having the same velocity requires more careful examination.
This velocity is not actually the same for two reasons :

A. Because there will be alternating potentials between
the filainent and the grid which set up. a velocity
distribution at the orid.

B. Because of the voltage drop of the heating current
down the filament.

In the usual methods of producing oscillations by means
of valves the alternating potential of (A) is most important,
as it controls the whole action of the valve, but in our
experiments it is only of secondary importance. The
alternating potentials induced between filament and grid
are smaller than those between erid and plate, and the
major part of the work done by “the alternating field on
the electrons, which is what determines whether the
electrons reach the plate or not, is done between grid
and plate, and it is therefore nearly correct to say that
all the electrons passing the grid at times ¢, such that
expression (1) is positive, reach the plate.

In the extreme case, however, when the plate is. so

Electric Waves obtained by Valves, 177

negative that the system is unable to oscillate by the method

of driving groups of electrons on to the plate, the oscillation
is almost certainly due to a velocity distribution at the
erid, as this means that the electrons do not pass the grid in
a uniform stream, and allows the integral of expression (2)
to be finite and not zero, as it normally is when all the
electrons return to the grid.

The comparative weakness of the oscillations in this case
shows that the electrons have all nearly the same velocity
when passing the grid.

The simple theory should therefore be in agreement with
the observed facts, as it is when the oscillations are mainly
due to the collection by the plate, but as the plate is made
more negative with respect to the filament the velocity dis-
tribution at the grid becomes more important and the simple
theory is less accurate.

The velocity distribution at the grid will also bo affected
by the emission, as this varies the space charge round the
filament—this affecting the time the electrons take to pass
from the filament to grid,—and this in turn varies the small
effect of the alternating field in this space.

The effect of the voltage drop of about 4 volts down the
filament is that, instead of dealing with one stream in
the field due to the grid being charged to V volts, there
are a series of streams moving under potentials varying
from V to V—4 (V being the potential difference between
the grid and the negative end of the flament). The number
of electrons in the various streams varies from a maximum
number corresponding to V—2, the middle of the filament

being the hottest. The emission falls off equally on both

sides of this middle point.

In the general case, when the plate is slightly positive with
regard to the negative end of the filament when there are no
oscillations, some of the streams reach the plate and the
remainder approach it closely, but to varying distances.

IE oscillations commence some of these latter streams are
periodically diverted to the plate, while in the other half
oscillations some of the former are diverted off.

Thus all the streams concerned maintain the oscillation as
in the simple theory, and unless V is small the wave-lengths
they each maintain best are nearly the same, so that the
combined effect ditfers little from that of a single stream
moving under potential V.

The ¢ question of whether the mean plate current rises or

falls when oscillations begin depends on whether the average

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. N

178 Short Electric Waves obtained by Valves.

density of the streams diverted to the plate exceeds or is less
than that of those diverted from it.

If the plate is at the same potential as the centre of the
filament, no, change should therefore occur. If it is above
this the current should drop, and if below the current should
rise, which is in agreement with the results in paragraph 4.

In conclusion, we should like to express our thanks to
Professor Townsend, who has assisted us with much valuable
advice and criticism.

Note on the Determination of Ionizing Potentials.

In the experiments above described, the electric fields in
the spaces between the grid, filament, and plate are similar
to those used in experiments on the determination of critical
potentials when a small quantity of gas is introduced.

In the latter experiments variations in the plate current
are observed as the grid potential is raised, and at certain
potentials of the grid abnormal variations in the plate current
are observed which are interpreted as indicating certain
critical potentials, characteristic of the molecules of the
as.

: The experiments which are here described show that
abnormal variations in the plate current are to be expected,
due to oscillations which may be maintained where large
currents of the order of a milliampere flow from the filament
towards the grid when the gas is ata very low pressure. The
effect of varying the potential of the grid is clearly shown by
the curve of fig. 3.

In all the ordinary methods of wiring the valve to the
cells and galvanometers or electrometers, the system seems
to be as often in a state of oscillation as not.

Hven if the plate is so negative that the oscillations are of
the weaker type which do not affect the plate current, the
difficulty arises that the oscillations superimpose an unknown
potential difference between filament and grid, and the
electrons are not moving under the fixed field alone.

It is necessary therefore, in order to obtain reliable values
of the critical potential, to take precautions to prevent
oscillation, which is best done by using emission currents
much less than a milliampere.

pesk?g:.-]

XII. Pseudo-Regular Precession.
By Sir G. GREENHILL*.

a. is the gyroscopic motion described and illustrated

in Klein-Sommerfeld’s Kreisel- Theorie, p. 209, where
a spinnving top appears at first sight to be moving steadily in
uniform precession at a constant angle with the vertical, but
on closer scrutiny the axle is seen to be describing a crinkled
curve of small loops or waves; so that in this penultimate
state a realisation is obtained of a motion expressible by a
function which does not possess a differential coefticient,
a paradox fascinating to a certain school of pure mathe-
maticians,

A pseudo-regular precession, although invisible, would
not feel impalpable to the analytical thumb passed over it,
which would detect a roughness. But in regular precession
the curve would feel quite smooth.

In the investigation the axle may first be supposed moving
in perfect steadiness with no trepidation or nutation ; and
then to receive a small impulse, blow or couple, giving rise
to the pseudo-regular precessiun visible to the eye.

1. Begin witha rapid spin about the axle, held fixed at a
constant inclination, taken at first as horizontal for simplicity,
in fig. 1.

Bie. 1.

For visible experimental illustration, it is convenient to
take a large (52-inch) bicycle wheel, mounted on a spindle
with ball bearings, and to prolong the spindle by screwing
on a stalk, at one end or both.

The end of the stalk may be supported on the hand and
the wheel set in rotation by a swirl; the hand accompanies
the wheel in the precession ; or else the point may be placed
in a cup fixed on the floor.

When the axle is released from rest, it will start from a
cusp and sink down, then rise up again in a regular series of
loops or festoons ; so that to secure the uniform precession

* Communicated by the Author.

N 2

:
3

180 Sir G. Greenhill on

an impulse couple must be applied, given by a horizontal
tap of appropriate amount.

The word moment or momentum is of such frequent
occurrence in dynamical theory that we prefer to replace it
by zmpulse instead of momentum, linear or rotational.

Representing then the impulse (rotational) CR due to the
rotation R about the axle by the vector OC, the impulse
required to start the rotation from rest, or reversed to stop
it again, the axle OC in steady motion will move round the
vertical OG at a constant (horizontal) inclination, with pre-
cession mw, such that the vector velocity of K, the end of the
resultant impulse OK, is equal to the impressed couple of
gravity ; then CRu= gMh, MA denoting the preponderance
or first momeni about O.

This result, true accurately when the axle OC is _hori-
zontal (fig. 1), is obtained at any other inclination @ of the
axle with the vertical (nadir or zenith, figs. 2, 3) in
the elementary Kindergarten treatment, where the top is
supposed spinning so fast that the deviation is insensible
of axial impulse OC and resultant impulse OK, and then
the velocity of C may be equated to the gravity couple,
making CRw sin @=gMh sin @, as before, when the axle was
horizontal, on dividing out sin @.

2. Hanging down inert, vertically from O in fig. 2, the top
forms a compound pendulum of S8.H.P.L. (simple-equivalent-
pendulum-length) OP=l/=A/Mh, A denoting. the trans-
verse-diametral M.I. (moment of inertia) at O, and © as
above the axial M.I.; and in small invisible oscillation in a
plane, the top will swing as a pendulum, and beat n/m times
a second, where n?=g/l, An?=gMA, or make a swing or
beat in min seconds ; Mh may be called the Regen ne
about O.

Falling down from rest from the upward vertical position,
the to» will have acquired the angular velocity @ in the
lowest position, such that by the Energy-Principle,

4 Aw?=2qMA,

and the equivalent rotational impulse Aw=2An, while
An?=gMh is the equivalent of P in the Krezsel-Theore.

The impulse 2An=2/(gMAA) is a dynamical constant of
the top, and to a geometrical scale may be represented by a
length &, in addition to the 8.H.P.L. OP=/ measured along
OC; and then if in any assigned top motion the constant
impulse component G and CR, about the vertical OG and

Pseudo-Regular Precession. 181
the axle OC, is represented to the same scale by 0 and 0’,
0; 0! ie G, CR TA h, h!
RSS ODA eos 8
in Darboux’s notation (a different use of hk from that
employed above).

Time can be reckoned in the pendulum beat, m/n seconds ;
and the relation, CRu=gMh=An?, can be written

He An
EB

or expressed in words, the number of beats per circuit of the
axle is C/A times the number of revolutions of the top per
double beat.

The resultant impulse vector being OK, the component
perpendicular to the axle, if horizontal, as in fig. 1, is

gMhA _ A?n?
CB er COR?

or to the geometrical scale, OC. CK=i4?, in the steady,
regular precession.

For brevity we are allowed to assume tacitly the geo-
metrical scale, and to replace any dynamical quantity in an
equation by its vector length, such as the axial impulse CR
by the vector length OC, or 0’.

3. To change this steady motion of the. axle into a
penultimate pseudo-regular precession, another impulse is
applied about a vertical axis, supplied by a hase ts tap
on the axle perpendicular to the plane OCK, in fig. 1

This will cause CK to grow to CKs, and the resultant
‘impulse to change from OK to OK,;; and to make the
pseudo-regular precession advance through a series of cusps,
we find that KK;,=CK, and the axle rises from OC to OC,
at an angle @, with the upward vertical, zenith ; where (,
reaches the level GK; of K,; and here 63, the inclination in
the lowest position, is $7.

By a general dynamical principle

OK,?—OK,?=4k? (cos 8; —cos 63)
=20C. CK (cos 6,—cos 63).

ee

For in the general unsteady motion of the axle of a top,
where the inclination @ is varying, a new component KH is
added to the impulse OK perpendicular to the vertical plane

182 Sir G. Greenhill on

GOU, of magnitude Ane and then the resultant impulse
vector OH describes a curve in a horizontal plane GHK,
with velocity equal to the gravily couple gMhsin 0.

Fig. 2.

: The radial velocity of H in the horizontal plane is then

j

: eae = gMh sin 0 cos GHK = gMhsin ae
| dGH £7 eae

GH aie == gMhA sin 0 AE 5

: and integrating,
4 GH? =g9MhA(E — cos @) = A’n?( EK —cos 8)
| =ik (H—cosé), 40H? =i (F—cos9),
| to a geometrical scale, with HK, F' dynamical constants.
: The component HK is zero in the upper and lower position,

where = (65, 6s.
In the general unsteady motion of the top, the impulse

vector moves from OK to OH with KH=A% and with

2
KH*= OH? - OK’=OH? — ae
2 Y
az Uh annie ebapyes Sa ee OU cont ee

dt sin? 0 2

Pseudo- Regular Precession. 183

or with cos @=z, and replacing OG, OC for dynamical
homogeneity by Darboux’s 2A, 2AN’,

(S) = 2n®(F—2)(1— 22) —4(? — 20M cos 0+ W'2) = 2n2Z,
thus defining z as an elliptic function of ¢,

Resolved into factors, we write

L=21— 2% .29—2 2 — 23,
in the sequence © >2;>1>2.>z2>¢3> —1; and then
z=c,sn*imt+e,cn?bmt, m?=2n?(2;—23).
4, Here with OC horizontal in fig. 1, cos 6;=0,
OK,?— OC,?=G;3K,?— G,K,?=CK,?
—2Z9UC 20 Weast,—20K..CK, | CK.=29CK.

A greater impulse would make the cusps open out into
loops in the pseudo-regular precession; but the cusps would
be blunted into waves if the impulse was reduced.

Reverse this tap, and K is brought back again to C, and
the axle would fall as at first from a cusp and rise again.

In the first cusp motion where the axle rises to a series of
cusps and sinks again to the horizontal, the motion is found
to be pseudo-elliptic and can be expressed in a finite form,

sin 8 exp (w—ht) 2
= V (1—cos 8, cos 8) +2/(cos 4 cos d—cos? 8),

connecting azimuth yr with @, the inclination to the zenith.
The verification is left as an exercise. Here h=h’ cos 0,.

In the second cusp motion, where the axle is horizontal
and falls from a cusp, and then sinks down to an angle @;
with the downward vertical, nadir, the (4, @) motion is not
pseudo-elliptic ; but azimuth W and hour angle ¢ change
place (0, ¢, w the Eulerian angles), and

sin 6 exp (6 —A't) 1= ¥ (sec 03—cos 83 . cos @)
+7,/ (cos 0;— cos 6. cos @+sec 83),
CR

where Darboux’s h'= a V/ (4 .sec 0;—cos @;) changes

place with /, or.9' with 9 ; h=h' cos 6; and 9 now zero.
But an interchange again of ¢,W will give the (wy, @)

motion of a non-spinning gyroscopic wheel, or spherical

| 184 Sir G. Greenhill on

pendulum, with h', CR and 9’ zero, in which the axle is
projected horizontally, with angular velocity ah=sin? OL,
and sinks down to an angle 03; with the nadir, rising up
again to the horizontal, and this makes

h=n¥v (4 .sec 6;—cos 03).

The motion can be shown with a plummet on a thread, say
about 10 inches long, to beat as a pendulum twice a second,
a double beat period of one second ; whirled round swiftly,
the thread rising to the horizontal position, and sinking down
again periodically.

Then we find

U9? = AIP? = 2gl(sec O,—cos 03), v3?= 2gl sec Os,

and in the conical pendulum, at angle 03, v?= 4,2.
The apsidal angle is found to be

Wa har+K V(1— 2x2) > m(1—§ cos? 85)

as the plummet is whirled round faster.

5. But next suppose the axle OC is held at an angle 0
with the zenith, the wheel spun with impulse CR=OC, and
then released, in fig. 3.

The axle will start from a cusp, at (==02, and the motion
in general is not expressible in finite terms as pseudo-elliptic ;
but it will represent a gravity brachistochrone on a sphere.

To make the axle move steadily at the inclination @ with
constant precession pw, the impulse vector CK is applied
perpendicular to the axle, such that wsin@ being the com-.
ponent rotation of the wheel about the axis OK’ perpen-
dicular to OC, CK=Apsin @ (the inertia of the stalk being
ignored), MC=Aymcos 0, MK=Ap, KM drawn vertical to
meet OC in M, with the condition

GK . w= gravity couple=gMh sin 0= An’ sin 0,
2
M= ass = ae :
STO
then the geometrical relation CM + MO=OC becomes
2
Ap cos 8+ ne =CR,
the condition for steady motion. Also
OM .MC=A°n? cos9=}k? cos 8, OM.MK=2 27,
so that K lies on this hyperbola with asymptotes OC, OG.

Pseudo-Regular Precession. 185

Or the position of M is determined by drawing QQ’
parallel to OC, to cut the circle on the diameter OC in
Q, Q’, when MQ?=+k? cos 0.

This is for the small value of yw in fig. 3, and the quiet
precession when M is taken close to C, the other point Q’
will determine a motion where the precession pw is swift and
the motion violent.

Fig. 3.

OC, = 20M. MK’ = 2MK.,
2 \ :
Spinning upright with cos@=1, w+ eh =<R ; and the
. mn :

two values of w will give the independent normal invisible

2
circling of the axle round the vertical; there will be as
rapid beats for one slow beat of the axle. fe

The general slight oscillation will be compounded of these
two circlings, adding up to an epicyclic motion of the axle,
a result obtained in this manner without any appeal to
approximation.

In fig. 2, the axle is pointing downward at an angle @
with the nadir ; here the slow precession of K is retrograde,
but in the swift violent motion of the associated K’', the
precession is direct.

186 Sir G. Greenhill on

6. Next to make the axle rise to a cusp on 0=0, from
d=6, in the penultimate pseudo-regular precession, the
impulse applied is KK3, to make the axle come to rest on
the horizontal G,K; in K,, C,, in fig. 3.

Then in the general formula, or on the figure, with CD
the perpendicular on K,M,,

G,K.?— G,K,?=OK,?—OK.?=CK,? = 3 k? (cos 8, — cos 0s) ,

or : CK,?=K,;D.K,M,;=2 OM.MK ae

and producing MK double length to K’,
K3,M;.0C=20M.MK=OM.MK’,

implying that if OCLN’ is the parallelogram on OO, the
diagonal OL will cut MK’ in Lg such that ML;=M;Kg, and
K,; 1s determined by drawing L;K, parallel to OC, cutting
off the length CK; on OK, in fig. 3.

7. If the impulse is applied about the axle of the top, to
increase OC to OC;, and make the axle rise to a cusp in
fig. 3,

OC; ° K;M, = 20M 6 MK, with MK = K3Ma,,
so that OC;,=20M.

Thus the axle will rise from the horizontal in fig. 1 toa
cusp by the application of an axial impulse CC;=OC.

8. The impulse might be applied about a vertical axis to
the steady motion, making K rise vertically to K,; and
then in a cusped motion, with OC changed to OCs, and rising
to OC, at K, on the level of G3K; in fig. 4,

G@ Ke 6 Ke 0K 0k? 0. Kika. Ke
=20M . MK (cos 6.—cos 6;)=20M .MK
MK,. 0C,=20M.MK=OM. MK’,
MK, MC, OM

NK. MC! OCG

<<
OC,’

dropping the perpendiculars K,C 3, K’C’ on OC; and drawing
the circle on the diameter OC’, with ordinate MQ’,

G,M.C,0=OM.MC’=MQ”.

Pseudo-Regular Precession. 187
Then if L is the midpoint of OM,
C;M.0;,0 = LC?—LM?=LQ”, LC; = LQ’,
to determine (3.
Fig. 4.

K E

: 7
2 Tc NO

j a;

M \
i.
/
ON
MK’ = 2MK. ip mid-point of OM.
MC’ = 2MC. LO, =

9. Applied about a horizontal axis in the vertical plane
GOC, the impulse will make K move horizontally to K,, and
the axle rise to a cusp at Kg, Cg, in fig. 4, if 3

C; 2=M.K,.K,D=20M.MK7,
M,K;.0C,;=20M. MK, aoe = We =e

FC,.C,0=20M .FC-2GK .KF=G/K'.K’F’,

thence a geometrical construction may be devised for the
determination of C, and Kz.

10. When the impulse is applied in the vertical plane
GOC, the impulse vector starts out of the plane, from OK
to OH, and K moves to H perpendicular to the plane GOC.

188 Sir G. Greenhill on

The axle then oscillates between 0, and 63,

6,<6<0;, KH-A™,
dt

—_— 2

GH?=KH?+GK?=KH?+ (Sy)
sin 0

=20M. MK (K—cos @),

— 2

CH? =K H?+ CK?:= -KE4 (4 OC cos a
sin 0

=20M.MK (D—cos @).

Fig. 5.

When the axle rises to a cusp, C risos to Cy, Ky at the
level of G, where in fig.5

KH=0, OG—OCcosé,=0, D=cos 6,
CH?=20M . MK (cos @,—cos 8)

—20M. MK AG = oo EO
KH?=CH? — CK?= (aa —1) CK’,

- giving KH the impulse.

Pseudo-Regular Precession. 189

This can be applied by hammering the rim of the bicycle
wheel with a stick in a vertical low at its highest point.

Then

OK2—0K2=0K,?—0C,?=C,K,?
=20OM. MK (cos 0,—cos 63),

KX3D
OG >, M,K,.0C=20M . MK,

and this determines the level of AM; in fig. 5, and provides
a geometrical construction for the position of C3.
For
OM;.M;0,;=0OA sec 6;. M;K; cos 0;=OA. AG,
so that, if B is taken in OG where AB=OA,

40A .AG=OG?— BG’?=OK,?— K,B? ;

M.K, . K.D.=2 OM ° MK

and the circle centre K, and radius K,B will cut off a length
OE on OK,=OF =20Msz, and so determine the direction of
OMCs, the axle in the lowest position.

When OC is horizontal, as in fig. 1,

cos 9,=0, CH?= 200. CKeos 652 CK? KH KC.
fae ron; one, OC — OM.- NK, — 2MK. 0,-+ 0,—= rr,

the axle oscillating to an equal angle above and below the
horizontal.

11. When the cusp motion is pseudo-regular and in small
loops, it can be projected on the tangent cone of a sphere in
a series of small hypocycloidal branches, and the motion is
realised as discussed in the Principia, Book I, section X,
when the tangent cone is developed into a plane and oravity

radiates from a centre.

With the axle horizontal, a necklace of Peeiiaroeneond
ceycloids is formed round the equator, with mean regular
precession yw, fluctuating in azimuth between 0 and 2uy, with
azimuth interval 2¥= 2 Ke, tending as the rotation R and
axial impulse CR is increased to 7 cos J,, in a.zone above
the equator of angular width $7—0,, and area 27a’ cos 0,
on a sphere of radius a, and the number of cycloids in the
necklace would be about 2 sec 0y.

12. Even in the steadiest smoothest Regular Precession a
close scrutiny will reveal under the slightest disturbance an
almost invisible deviation from a perfect circular motion, in
the shape of a progressive motion of an apse line, realised

190 Sir G. Greenhill on

easily with the thread and plummet; utilised by Newton to
illustrate the Evection of the Moon.

Returning to the general unsteady motion of a top in § 8,
and its vector impulse OH, the velocity of H imparted by
the gravily couple An’sin@ is horizontal and along KH
perpendicular to the plane GOC. This velocity is the rate
of growth of KH, added to the velocity of K carried round

by the plane GOC with angular velocity Be so that, with
KC = Asin go and putting a = (),

Cate da eee, dp -,dQ “GR IKE

Ant so 0 Aaa ann

nae GK = OC = 0G cosa KC 2 ees
sin @ sin @

J dGK KC ak. Gk
a dé sin@’ dé sin 0%

obvious geometrically on fig. 3.
Differentiating with respect to 0, with dd=Qdt,

LdQe ae g_ K+ KO*—GEK . KC cos 0
Oat ge eee
Bee) ale GC?—3GK. KC cos 0
ace A? sin? @
/ 2
= G + oo 2) sé@— a ‘

an exact equation.
In a state of perfect Steady Motion of Regular Precession,

() and a are zero, and, in dynamical units, from § 5,

OM 7» OM m2 eX

NOE ah aie Natasa ang

OM. ON= A?n?, A eee ON ee
where A is the height of the equivalent conical pendulum,

Ne= 0) ONG occ — a

OG. n m OC on
Oy 9. ee

GO OK? sin? 8 ne (3

+ cos ie,
n

2
At = aa 2 +2cos0+ H )sin® C.

Pseudo-Regular Precession. 191
Then in this fed Motion,

2
Q ee +m?=0, where
xs OK? AP?
has —4cos0= * 2 0080+ 5 = OA OP?
Lame OP?
m2 AP? Oe:

This result is exact and reached without any approxima-
tion: and the slightest disturbance will give a nutation
Q=Q) cos (mt+e), beating m2 times a second, and the
apsidal angle, from node to node, is

on Ae |
In Darboux’s representation of top-motion by a deformable
articulated hyperboloid of the generating lines, the model is
flattened into a rigid framework for Steady Motion ; and

KM, KN produced to double length at 8,8’ will make the —

focal line SS’ parallel to MN; this will be revolved about
the vertical line ON with constant angular velocity uw. The
small nutation will be due to a slight play or backlash in
the frame.

13. The same argument can be applied to the invisible
oscillation of a Simple or Spherical Pendulum, or to the
apsidal angle of a particle describing a horizontal circle on a
smooth surface of revolution about a vertical axis.

Taken as the axis Oy, the general equations of motion of
the particle are

Lae) Lay L of
Saat 5 Sats +5 ee +gy=H (energy),

and 2? oY =K (impulse) ;
so that, with 4 — - =Q, and ae ee

dy?\ | 1K?
=) === — :
50 ee )+5a +gy=H;

and Sg, with respect to x, dx=Q dt, qt = a)
BO Fee .
(1+ 95) + gine d?y eae 2G

da dx? at Tax

Ag 0 Aes 4 9 AQ dy dy d’y Vy 3
Q dt? oe dt da dx? ( )

exact equations.

—_

192 Pseudo-Regular Precession.

In a state of Steady Motion in a horizontal circle,

in ao ee
| Q=0, dt =(0, K=pr ’
da . d.
g=wag=w NG, vt apatmgos” =9 NV,
if the normal and tangent at P meet the axis Oy in G and V;

and then with a =tan 0,
da

D) 2 : 2
; a +m’=0, F = (84+ NGS) cos? 0;
9 oscillates between close limits a+,

O= (a+) sin? $mt + (a—BP) cos? 4mt, Q=mB sin mt ;

and the particle beats m/2a nutations per second, syn-
chronizing with the beat of a simple pendulum of length
r=g/m?, where | }

Lh or ay ane 1
a Giver is a foaey fcio ty
where PR is the semi-vertical chord of curvature upward of
the profile curve of the surface.

Then on a cone X=4GV ; and on a sphere or spherical
pendulum, PR=NG, and to radius a,

~=3 cos 6+sec 8.

For a profile given by y=cz”,

a)
dy n2, NG= Be

; v=¢g. NV our

dn cae dy my
dy nm—lyy , reas Ay 0G,
da? i eae TNG ise Sailing n+2°

Thus on the surface of a free vortex, where the angular
impulse wv is constant of all annular elements of liquid of
the same volume, 2’y is constant, and n=—2,rA=a0.

On a motor or bicycle track of this shape, the steering
will be easy, and a change of place can be made without
difficulty or danger, as with the annular elements of liquids
in the vortex volume.

A start is made with moderate velocity from the circum-
ference of the track where the slope is slight, and the car

The Binding of Electrons by Atoms. 193

is steered with increasing velocity down towards the middle,
where the cars can pass and repass without difficulty.

To avoid a deep hole in the sink in the middle, the profile
ean change to the parabola of a forced vortex, where

v=pr, n=2, y= Ip NG=p, A=4GV=ISP.

On a horizontal circle of this track of one lap to the mile,
NP=840 feet ; described in two minutes at 30 miles an
hour, NV=60 feet, and cot 0= pes = 14, a slope of 4°.

Raise the speed to 60 miles an hour on this track,
NP=420, NV=270, feet, and the slope is nearly 30°, the
circuit of two laps to the mile made in 30 seconds.

At a speed limit of 90 miles an hour, NP=280, NV =540,
feet; round a circle of three laps to the mile, on a slope of
over 62°. The surface could then change to a paraboloid,
with a flat area in the middle, where a car could come to rest.

XIII. The Binding of Electrons by Atoms. By J. W.
Nicuoxson, F.2.8., Fellow of Balliol College, Oxford *.

CCORDING to the quantum theory of atomic structure
and of the emission of line spectra, the paths of the
electron in the atom vary according to the particular co-
ordinates used in the process of quantizing the separate
momenta. ‘Thus in the simple case of a hydrogen atom,
containing a nucleus and one electrun, we may use either
spherical polar or parabolic coordinates, and the admissible
orbits are entirely different in the two cases. Yet the final
values of the atomic energy are the same, and consequently
each method yields the same theoretical spectrum. It has
been suggested that there is in fact, in every case, only one
type of coordinates which can be used, when all the modi-
fying circumstances, such as the variation of the mass of the
electron with speed, are taken into account. ‘The only pro-
blems yet solved are those in which the separation of
variables, after the manner of Jacobi, can be effected, and
the contention is in fact that there is, in every case, only one
set of coordinates which allows this separation, when non-
degenerate cases of the motion are discussed.
But it is generally believed that the atomic energy is in
all cases determinate and definite. We shall show, in the

* Communicated by the Author.

Phil. Mag. Ser. 6. Vol. 44. No. 259. July 1922. O

194 Drs: W. Nicholson on the

first place, that this conclusion requires modification when
the path extends to infinity. The hyperbolic orbits of
Epstein, which have been used extensively in the inter-
pretation of certain groups of y rays associated with many
of the chemical atoms, constitute an instance, and we shall
show that they rest on a mathematical-error, and that in fact
it is not possible to preserve finite phase-integrals in the
process of quantizing the momenta. In fact, it appears that
the whole process is only applicable to finite paths, and gives
no clue to the phenomena taking place during the binding
of an electron which comes from a considerable distance.

In another form, the question we propose is as to whether
a hyperholic path is possible in the same way as an elliptic
one. Such would, of course, be characterized by a posotive
energy W. Certain available evidence of a simple kind,
apparently not hitherto noticed, is in existence. For the
existence of such paths involves the existence of parabolic
paths, with W=0. In passage from a stationary state of
energy W, (negative) to a parabolic path taking the electron
outside the atom altogether, a quantity of energy W, should
be involved. Spectral lines given by

hy = W,,

where W,, corresponds to any one of the stationary states,
should thus exist. In other words, the ‘limits’ of spectral
series should themselves be spectral lines. But there are
two reasons why evidence on these lines cannot be decisive,
especially when it is negative evidence. For in the first
place, the values of W, determining the limits of series are
of such magnitude that only for two or three, in any case,
can the corresponding lines come into the visible spectrum,
and with only hydrogen atoms and charged helium atoms
to test, and enormous band spectra for both elements, the
test cannot readily be applied. Moreover, the probability of
_an electron entering the atom in a parabolic rather than a
hyperbolic path is so small that any resulting lines could
hardly be expected to be of visible intensity under ordinary
conditions. We consider, therefore, that the question
whether limits of series are themselves spectral lines, on
the principles of the quantum theory, cannot, at least at this
juncture, be examined in the light of experiment, and
that it must remain a matter of deduction from other
phenomena.

We find it necessary, as stated, to disagree with the
hypothesis, explicitly indicated several times by Sommerfeld
and others, and implicitly assumed at least by the remaining

Binding of Electrons by Atoms. A195

writers on the quantum theory of spectra, that the energy W
is always completely determinate when all the momenta are
quantized. This can be disproved not only for fictitious laws
of force in an atom, but for laws which must actually occur
in systems with an existence, if only a temporary one.
Consider, for example, a simple doublet and an electron
in orbital motion about it. Regarding the doubiet as
stationary, and of moment M, its external potential is

when it is situated at the origin, with its axis along the axis
of <, using spherical polar coordinates. The equation of
energy for an electron moving in its presence is
Me cos 0
lm {i+ 16? +r? sin? ee a —W.

The momenta are, in the usual notation

») >]
fo hh .
EU yp cee, sce ON Dliaig: SUNT:
Da Vial mr,

or
ot Pee,
ee | mr sin Od,
He ae, b

so that

um | pe sp Be igo ae eee —W.

eS sin
Now ¢ is a speed coordinate as usual, so that
Ps = const. = mh/27

when subjected to the quantum relation, n, being an integer.
For the Jacobi solution, we must also take, in separating
variables,

2
te A + 2mMecos? =8

where £ is constant, and

5m | pt Et =—
n=a/—Imw— 8,

Thus

QO 2

196 Dr. J. W. Nicholson on the

With a positive W, the motion is not real. Thus W must be
negative and the path necessarily extends to infinity. A

critical value of 7 is \/ ee and the other is infinity.

The phase-integral for p, 1s

2 pdr
B

—2mW

which is infinite, but nevertheless independent of W. For
writing

— ts i =;.6, it becomes
—2mW

sts a) He / =
¢ Ve aes ti
2 { V —2mW (1 =) ~amw 0%

== 2 vp{ da(1— 2)

A finite integral is secured,—Hpstein’s procedure, :for
instance,—by, using the phase-integral not for p,, but for
Pi—(P1)r=0, Which in the same way yields

2 va ae f(i-4) 1} = 2v7 (1-5):

again independent of W. Now 8 is quantized, or expressed
definitely in terms of integers already, from the phase-
integral for the momentum p,. The phase integral for py
can only, in this case, lead to another expression of similar
type for 8, but to no expression for W. Itis not at all clear
that the two expressions for 8, also, can both be valid
simultaneously.

This possibility has hitherto apparently been overlooked
by authors in this subject.

No case has, however, been noticed in which W is inde-
terminate for a finite path. One very important conclusion
is that the whole investigation is valid for a negatively
charged atom with a distant electron.

We proceed now to discuss the possible existence of
definite paths with a positive total energy and infinite
extent, for a single electron around a nucleus of charge ve,
situated at the origin. This is Epstein’s problem, which he
treats as only two-dimensional. The energy equation is

7 Lr

Binding of [lectrons by Atoms. 197

where W is positive, and represents the total energy, and
the p’s are the momenta.

We have thus
h

p= const.=
2a?

ate Da = 9?
P2 sin? 6 ;

being clearly positive,
2 2
4m {pit = } ps ee
; ,

The phase-integral for py, is

aA pedo dé ‘des aE
all Pa y p sin? 0”

the limits being the suitable values of @ for which p.=0.
The factor 2 represents the double journey in this co-
ordinate,

sin = 3
where wy is one of the limits, and the other admissible value,
for a real integral, is 7—y. Thus

mY
Rah = 2 { dé / 8’ —p7/sin? 6

B
— ap ( wl { with sin 6 =3 vo).
1

fox
w = sin? w+ —5C0s’ a,
P3
and we have
fs —n3" 1/2
nh = 48 -+———
YE Dana

4

cos? w dw
B? ’
sin? wm + —; cos? wo
2,
. P
or with tanw =f,

4 3 Rey at
ngh = 75 (8?—p,?) (| ——4, —
P3 J0 a+e)(", +2)

= 48 { tan i— Pian}

0

= 2084 1-72 | = 2n(@—ps),

198 | Dr. J. W. Nicholson on the

whence
h ]
C= se a P23. (mitms) 5

these integers being thus additive, in the usual way.
The phase-integral for jp, is

: i 2m ve? 2
3 —— dae 2m W + ie
3 7 72?

if we seek to quantize p, as it stands. The limits would then
be a positive value of r and infinity, for half the path, and
the integral would be infinite. But it is clearly necessary
to suppose that when the electron is at infinity, out of range
of action of the nucleus, it should not be subject to a quantum
relation, so that (p;)-=« 1s not affected by the rule, and only
the variable part

Pie Pie

is so affected. Yet this question of quantizing p, presents
some difficulties in whatever way it is suggcsted that it
should be effected, and we consider that Hpstein’s discussion
of the matter is very incomplete and not logically justifiable
in its mathematical procedure. We shall thus consider
various alternatives which may give a finite phase-integral.

Now the actual r-path is not a passage from r=« (say) to
y=oo and back, and the phase-integral is not twice the
defiuite integral between these limits. The electron goes
from a limiting radius to infinity, and back to the same
radius elsewhere, and the passage through infinity distin-
guishes this phase-integral from those which occur in the
other coordinates.

We must, of course, also remember that the sign of p,
depends upon the part of the path concerned ,--whether the
electron is departing or returning. The critical value of r
is the positive root of

Pave ea

p<

2mW + ——
, P

1 _ mye? + /mv?e4+ 2mW PB? 1
ig ig PCE Lee NGA MITE AC ie re (say).

or

Writing, a an with a new variable 4,

_ me? mye! + 2m WB? + oe
= cos qh,

Binding of Electrons by Atoms. ivd

we have @=0 in the critical position (perihelion, in the
usual terminology), and

mve eee nW B?
sega ETEONB uy ay)

when r=

What is required for the correct evaluation of the phase-
integral is a continuous variable which shall change in one
direction,—and thus give a definite integral,—as r goes to
infinity and returns, the sign of p; being automatically taken
into account,—or the sign of py;—(pi), when (p),, is not
zero as in a parabolic path. The new variable ¢ has this
property, and ranges from zero to 2m as r goes through its
changes. We have denoted its value, when r=m, by 7
above, where 7 is evidently an obtuse angle.

The phase-integral for p; alone would be

PE evils Wee ery De
ngh =| a dr x / 2mW + —— TE —

(the square root being properly interpreted in different
regions) where

1 = mve?

e 8 V/ mpv*et + 2mW 8? cos d

0= ee + = V mvet + 2m WB? COS 0,
and we find
‘ 5 AMUN ee,
dr= — uy ae » g= VIMW 4+ me,

(cos@— cos)” yg

2
a/2nW + . — = = gain dp.

If the integration were continuous throughout,—as as-
sumed by Epstein, —we should thus have

hae ae meen ap

(cos @— cos 7)?

=26(" _ sivtddd _

(cos @— cos 9)?’

which is an infinite integral, as would be expected.

200 Dr. J. W Nicholson on the

If we merely quantized over the finite part of the hyper-
bola,—another possible suggestion,—we should have

oe sin?
ng 8 he - (a (cos @—cos 7)? a
7 sin? ddd
9 (cos d— cos 7)?’
which is again infinite.

The nature of the first infinity merits a remark, however,
for it is independent of » and therefore of W. For

72) ie sim oddb __ sin a ee cosh db
Jo (cosd— cosy)? Paee ah 9 CosSp— cosy

= [ an | — 7 — COS af” ode

cos d6— cos cos }— cos n-

—= 7)

em

The principal value of the last integral is well known to
be zero, for all values of 7, so that the last term is zero.
Our equation would be

nh= —278 + | any

cos d— cos

where the principal value of the bracket must be taken,
2. €. it is to be interpreted as

fee

This becomes

sin sin) Le a
ina Se oe ir Wen

which, though infinite, is an infinity independent of 7 and
therefore of W. We have another aspect of the indeter-
minateness of W for such paths.

Our fundamental objection to Epstein’s mode of integration
may now be introduced. He integrates p,;—(p;),,, and not
p1, but this fact does not affect the question. Foras ¢ ranges
between 0 and 2m, if p=/($), we have p, varylng con-
tinuously with 6, and remaining positive, till@=7. Then p,
becomes —/( (dy) when ¢=27—d, on the return Journey
after 6=2r—n. Between d= and d=27—7, the value
of r should be infinite, and p, changes from 2mW_ to
— /2mW, as in the figure.

Binding of Electrons by Atoms. 201
The variation of p, between + “2mW at infinity is the

source of trouble, and it takes place while : =A)

O= 27-7

Epstein takes twice the integral from ¢=0 to d=z7, but
according to the substitution formula, r is negative when $
goes from 7 to 7, and negative values of r are clearly not
permissible. A suitable integration for the infinite region
cannot in fact be effected, and any supposition of a suitable
variable in place of ¢, for the change of p, at oo from
/2mW to — V¥2mW, would be entirely arbitrary,—but as
it could not lead to a finite phase-integral, we pursue the
matter no further.

‘These considerations, nevertheless, have considerable force
when, thrown back as we now are upon the necessity, if the
quantum theory is applicable, of using p;—(p,),,, we attempt
to quantize this.

We have, when 0=n

(pi). = ¥2mW

— 2 sin N;

8
where g= Vmv?e' + 2mWB? as before.
And when ¢=27— 7,

® (pr). = sin (24 —m) = — 4 sin 7.

From ¢=0 to d¢=7,
Pi (Pr) = B (sin $— sin).
From ¢=27—7 to $6=2r7,
Pi- (Pi). = + A sin p+ sin),
and from ¢=7 to $6=27—7,
Pin Pu. O-

2020 Dr. J. W. Nicholson on the
With the value of dr, the phase-integral

2a
ngi=( dr( py — (P1).)

breaks into three parts, thus

nh =p)" sin ¢ (sin @— sin 7 ae 0.d¢

_(cos @— cos 7)”
ue sin @ (sing + sinn\dd
+0(." He (cos @ — cos 7)’

ee S07) a 5 og\ 0. dd

(cos @ ~ cos)’
by a simple transformation.
Finally. the only accurate phase-integral is

nah ee ve sin @ (sin @— sin ”) 4 ae,

(cos 6— cos)”

while Epstein gives, iu our notation,

| a “® sin d (sin d— sin 7)

nah = 26) (cos d— cos)? dp,
the part of his range from to 7 involving a meaningless
negative value of r, and violating p;=(p,),, though the
moving electron is at infinity. The principal value of
Hpstein’s integral is, using the indefinite integral for the
function in the form, readily obtained by parts,

{= @ (sin @— sin Le

(cos d— cos n)?

_ sing@— sin
~ cos @— cosy

Je a cotn. ac

sin;
of the type
9 °
nzh=28q { ols ee
yy (ny + No a 13)
or 7o aoa wT
T sin 9 Ny + No
and ultimately
Qian ody if
we sag fae
q (mine +3)? — (my +79)?

—generalized from his value which relates only to a plane

Binding of Electrons by Atoms. 203

hyperbola. We have the sum 2+, of the angular quanta
in place of his single integer.

But this formula, with all the applications he makes to
characteristic y radiation, is not tenable, as resting ona
mathematical error. Its apparent success appeared at one
time to the writer to justify it as an empirical formula, in
spite of his independent investigation, outlined above, indi-
cating the impossibility of quantizing such orbits. Close
examination, however, of the calculations of y radiation and
so forth made it clear that they were in several cases
illusory, and determined more by order of magnitude than
by the nature of the formula.

There is one convincing argument against the formula,
however. It should give an emission spectrum for all values
of ny, m2, nz and m4, mo, m3 making

© — W(m, mo, m3) + W124, 2, 73)
positive. This can be tested in great numerical detail on
the spectrum of a hydrogen atom, and the test fails entirely.
No spectrum line is found,—in the secondary hydrogen
spectrum,—in any of the assigned positions. Thus the
formula really fails as an empirical one.

We have seen above that it must be replaced by

nah = 28 ik sin (sin f— sin 9) dd
0

(cos ¢ — cos 7)’

sin gd— sin | 2
= 28 | —",-——_ — $+ cotnlog,< —
cos @— cos n ane
2 0

which is logarithmically infinite.

The attempt to obtain a finite phase-integral, in this
manner, in fact fails, and we must give up the hypothesis
that even the variable part of p, can be quantized for the
infinite path.

It is not difficult to see that this conclusion is general for
any infinite path which is possible for an electron about a
physically existent atom, whose nucleus can always be
regarded, for the present purpose, as a superposition of free
charges and a set of doublets. We have demonstrated the
result for a single free charge, and previously for sets of
doublets. Further analysis of the more general case does not
seem necessary, and could readily be supplied by the reader.

Our conclusion must be as follows :—

A determinate and finite value of W cannot be obtained
for an electron moving about any atomic nucleus, if the path
involved takes the electron to infinity.

Lio e.]

XIV. Theoretical Aspects of the Neon Spectrum.
By Laurence St. C. BRouGHatt *

\HE object of this paper is to attempt to explain the

- spectrum of neon in a manner somewhat similar to
that used by Bohr f in his explanation of the reason for the
existence of the Balmer series in the hydrogen spectrum.

The principle on which this hypothesis rests is that when
an electron rotates in a fixed orbit it does not radiate energy,
although the principles of electrodynamics state that it
should; if, however, the electron changes from one orbit
to another, then energy is emitted, provided that the kinetic
energy of ‘the electron is less in the second orbit than in the
first.

In order to account for the spectrum, it is assumed that
the energy emitted is numerically equal to the product
of the frequency of the spectral line produced and the
quantum constant. We thus obtain the equation

K=nh,

where Hi = energy emitted, n = frequency of the resulting
radiation, and 4 = quantum constant. c

In the case of hydrogen, it was assumed that the orbit of
the electron was circular, and then the attractive force
between nucleus and the electron due to their equal and
opposite charges was balanced by the centrifugal force of
the electron due to its rotation about an axis passing through
the nucleus.

The energy of the electron can thus be found for any
radius of orbit. When the electron changes its orbit, it
moves to one with a radius which is an exact multiple of the
radius of the original orbit. In this manner the change of
energy due to a change of orbit can be found, and then,
using the equation given above, it was shown by Bohr how
the constant of the Balmer series could be found; and the
value so obtained agreed extremely well with that found by
experiment.

In the case of neon, we are dealing with an atom which
contains more than one electron; and since the atomic
number is 10, it follows that if the atom is to be neutral,
then there must be 10 electrons present to annul the excess.
of 10 positive charges in the nucleus.

* Communicated by the Author,
tT Phil. Mag. vol. xxix, p. 382.

Theoretical Aspects of the Neon Spectrum. 205

In order, therefore, to study the atom, it is essential that
the electrons be given definite position relative to one
another. This has been undertaken by Langmuir”, and
there is considerable evidence in favour of the postulate that
eight of the electrons arrange themselves at the corners of a
cube at the centre of which the nucleus is situated. The
other two electrons are imagined to lie within this cube,
probably on a line joining the mid-points of any pair of
opposite sides. If we make use of this hypothesis, and
further are in possession of data which will allow us to
find the length of the diagonal of the electron cube,
then it was shown by the author{ that it is possible to
calculate the angular velocities of the electrons about the
nucleus.

Since the determination of the spectral lines is an ex-
tension of the matter given in that paper, it will be advisable
here to state the principles on which the calculations of the
electron frequencies depend.

It has already been stated that the two inner electrons will
probably lie on a line joining the mid-points of any pair of
opposite sides. If this be the case, then the electrical forces
acting on the outer electrons due to the other electrons in
the outer shell, and to the two inner electrons, will be the
same whichever electron we take, provided that the two
inner electrons are equidistant from and on opposite sides of
the nucleus.

The next consideration was the axes of revolution. As
before, it was desired if possible to get the forces acting on
the outer electrons due to centrifugal action the same for all
of the electrons. If we take as axes the three lines which
pass through the mid-points of the three opposite pairs of
sides of the electron cube respectively, then the above con-
dition will be satisfied. In the diagram the axes of revolution
are illustrated by XX’, YY’, and ZZ’. The inner electrons
being on the axis XX! will only rotate about two axes.
It is, of course, quite immaterial which axis the inner
electrons lie upon. The forces acting on any outer electron
were then considered, and were taken along the three sides
of the cube which meet at the point where the electron is
situated. Now, since the electron must be in equilibrium,
so the force along each of these lines due to electrical
attraction and repulsion and also due to the motion in a
circular orbit must be equal to zero,

* General Electric Review, 1919.
*, Phil. Mag. Feb. 1922,

206 Mr. L. St. C. Broughall on Theoretical

In this manner three equations were obtained, namely :

gel) 1a ene e*(r +l) e(r—lL)

4s — 4 t 98 + [n+l +s2]22 ~ [(r—1)? + s?}9?
+mo t+ mel, . . db)

3981 oP el Le

Ae aap. woe 6 (o+ lets 3? tT (leer :
me 71+ mes le.) nll)

DOE 7.) e7U e7] e7|

Wet tos + (eae (Gee
yas + mast, s 2 eaten

Another equation can be obtained by considering the
forces acting on either of the inner electrons along the line
joining the two inner electrons. Equating the forces to
zero, we found that

Mr ee ne) een ee
e — fer)? +s 22 t free? t ae
+mor+mo’r, . . (LV.)

In the above equations, e= charge on an electron ;

Aspects of the Neon Spectrum. 207

ry = radius of orbit described by the two inner electrons ;
@, =angular velocity about YY’; w,=angular velocity
about ZZ'; w; = angular velocity about XX'. |= 43 length
side of electron cube; s = $ surface diagonal of the electron
cube ; c = 4 diagonal of cube. ‘s’ and ‘l’ are of course
functions of ‘c, and if the latter is known, then ‘s’ and
‘1? can be found. m = mass of an electron when its velocity
is small compared with that of light. -

From these equations it was shown by the author that
an equation involving only ‘7, ‘ce,’ ‘1, and ‘s’ could be
found. In the paper mentioned, ‘/’ and ‘s’ were not
expressed as functions of ‘c,’ but expressing them as such,

since /= nee and, s= ee. we obtain the equation
oe r+°d7Tc f;_ 2}
pom Neco ie)?-- *66fe? |" r
r—‘d77Te f 2°308¢ D°63¢
~ [ (r—-577c)? + -667c? |*? vai r } caer ele

(V.)

In a recent article it was shown by Prof. W. L. Bragg *
that the diameter of the neon atom could be found by an
inspection of the diameters of the atoms of elements whose
atomic numbers were near that of neon. It is impossible to
measure the radius of the neon atom directly, since it forms
no chemical compounds. ‘The value obtained was very much
smaller than that found by gas measurements, and the former
is considered by Bragg to be the distance between the elec-
trons in the atom—that is to say, is equal to ‘ 2c.” The value
obtained by Chapman 7 from gas measurements is, however,
the diameter presented by the molecule when in collision
with other molecules. ‘The difference is due to the fact
that in molecular collisions in the gaseous state the outer
electrons of the molecules do not come into contact tf.
Using Bragg’s value we have 2c=1°30 x 107° cm., and on
substituting in equation (V.) we have a means of obtaining
the fundamental value of ‘7.’

An inspection of equations (II.) and (III.) shows at once
that mj=@, and equations may be obtained for @, and a3.

* Phil. Mag. vol. xl. p. 169.
ft Trans. Roy. Soc. A. vol. 216, p 279.
t Rankine, Proc. Roy. Soc. vol. xeviil. p. 360.

208 Mr. L. St. C. Broughall on Theoretical

These aa take the form

es rt r—l
Imr {i a ktces) a SLB T Fp S(r =e] t = 04" ;
(VI.)
er ih it
a. hese 1G Hae =,'—o;. (VIT.)

Using a slightly different value for ‘m’ from that used in ~
the previous paper, we obtain the following values :—

i, w,(=,).

ol9x 10-° em. 6°034 x 10 radj/sec.. 4290 x 10™ madi zee

‘m’ being equal to 9°005 x 107-3 orm. and e=4°774 x 107
ES.

These figures refer to the neon atom when in its normal
state. There is some donbt as to whether they apply without
modification in the gaseous state, but certain assumptions
are made later in this paper which leads one to the conclusion
that if the atom is larger under natural conditions, then the
only result will be the elimination of certain spectral lines
in the ultra-violet. When the atoms of the neighbouring
elements were submitted to measurement, they constituted
a solid body ; it is, therefore, quite conceivable that modi-
fication will occur if the element becomes gaseous.

In order to explain the nature of the spectral lines, we
have to consider the change of energy due to a change of
orbit, energy being emitted when the orbit increases in
diameter.

Bohr, as already stated, imagined in the case of hydrogen
that the radius of the orbit increased by constant multiples
of the radius of the initial orbit. To adopt such a plan in
the case of neon would lead to the emission of spectral lines
of a frequency which would only give ultra-violet lines under
reasonable circumstances. Further, there is no reason why
the increment should be of such a nature, and the hypothesis
used in our case is that the spherical shell formed by the
inner electrons increases in radius until the shell has a
radius equal to that of the initial outer shell of electrons.
In order that equilibrium may remain, it is essential that
the outer shell also expands to an extent which can be
calculated from equation (V.). |

The initial increment is of the nature of 3x1071 cm.
This process of expansion continues again and again, the

inner electrons always occupying the orbit previously

occupied by the outer electrons.

Aspects of the Neon Spectrum, 209

Having thus found a definite value for ‘r’ corresponding
to a detinite value for ‘c,’? we are now in a position to
calculate the values of the angular velocities about the
several axes. To do this, equation (VI.) is first applied ;
and having found ‘@,’, the value obtained is substituted in
equation (VII.), and @;? obtained. In calculating the value
of ‘¢c,’ use is made of the fact that the ratio ¢: 7 is practically
constant; such ratio values are used in Table II. The
closeness of the ratio figures and the absolute figures is
shown in the appended table.

TABLE I,
r. ¢ (ratio method). e (from eqn. V.).
650x10~° cm. 683X107? cm. 683x107? cm
1233). 1295. 1295,
yh) eae 1577) 045 Lite ols

It wili be seen that the agreement of the figures is so
nearly exact as to warrant their use, remembering that our
fundamental value of ‘c’ has not been obtained experi-
mentally.

Table II. shows in columns II. and III. the values -
c? and ‘r,’ and in columns IV. and V. the values of w,’
and w;” respectively.

c

TABLE II.
|
| I. | II. III. IV. V. |
Radius Radius 5 2 |
No. | Outer Orbit = c. | Inner Orbit =r. na ty it |
1... 650x10-%cm.| 619x10-%cm. | 36-41 1082 | 18-40 x 10°? |
SS Sam ae) 3141 ,, | 15°84
apt Was (oe 68s) 4); TOS A WISER
eget Ve es Tiger, 2 tae a ey f=
a ay gs ate foi cae BAG | TOR os
A? eee 7 Vesicle Ses”,
ONG? >. Cie aos 1500 ,, 7-568,
8 = is. 3. S74, 19-05 6528,
x 964; i ae tics. E20", . |
es Soe ee 964 9625 ;, | 4860 ,, |
H1,..| 1084" IGS! |, 8-299 ,, 4-182 ,,
Gi a aes i 7-160 ,, 3613. ,,
aa a er iris’ |, 6170 ,, 3109 ,, |
A 18S oy £YFAO iy 5326 ,, 2684 ,, |
Bees 1295.1, 12:43...) sy, 4°602 ,, 2-322,
1s en” | 12:95, ty ae F002, |
yin i > 1360 _,, 3:428 ,, 1s et! a
163) GeO ©, 1429, 9-954 ,, 1489 ,, |
ee G7 ae 1501, 2°550 ,, 1-286. ,,
aS > saree” 5, ey, 2-201 ,, 1113
21 | Cine Oe 1657, 1897 ,, 9581 ,,

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. P

210 Mr. L. St. C. Broughall on Theoretical

Let us now consider the energy of an electron in the
outer shell when the diameter of the shell =2¢.

Let the angular velocities about the axes YY’ and ZZ’
= VW.

Let the angular velocity about axis XX’= Ws.

Since the diameter of the shell is equal to ‘ 2c¢,,’ it follows
that the radius of the electron orbit =8, where S;=¢, V/ 6/2.

Using the above notation and remembering further that
the kinetic energy of a particle describing a circular path
of radius ‘R’ with an angular velocity ‘W’ is equal to
4MR?W?, where M is the mass of the particle, we find that
the kinetic energy due to rotation about the axis XX’

=imS,?W,’,

and the kinetic energy due to rotation about the axis YY’
plus that due to rotation about ZZ’ |

=mS,;7W,? since W.=W2.

Therefore the total kinetic energy of the particle is
equal to

3mS77f2W,?+ Wt.
In the case of an electron in the inner shell where the

radius of the orbit =R,, we have the kinetic energy of

the electron due to its rotation about the axes YY’ and
ZZ’ =K, where

Ky = mR,’ W,?.

Now let the inner shell expand until it occupies the space
previously occupied by the outer shell—that is to say, until
R,=C), then kinetic energy in new orbit = Ey,

Where He=mO/?W,?, ‘Wy,’ being the new angular
velocity about the axes YY’ and ZZ’, the change of energy,

Tm Ky, ne K,,
= m(RYW — CW") °

Meanwhile the outer electrons have moved further away
from the nucleus, and now the outer shell has a radius =C,,
and the orbit of the eleetrons is now S,. 3

Therefore the energy in the new position is equal to
im,” [2W4,? + W37 | °

Where ‘W,,’ is the new angular velocity about the axis

dlspects of the Neon Spectrum. 211

XX', the change of energy due to change of orbit is
therefore equal to

tm{S?[ 2 WY + V ical — S.?[ 2W,? + W37 | Pi:
Now, by Bohr’s assumptions we have the equation
Energy Hmitted = Frequency x h.

Now, the frequency of a light-wave =c/X where ‘¢’ is
the velocity of light and ‘2? is ‘the wav e-length.
Therefore

Hinergy Emitted = e= a
or —— ch
€

Now, in the case of an inner electron
eE= m(R,? WY — C?W;,,”) 3

Ss ch
oe m(h,? W As ae Uy? W 11’) :

giving a series of lines for different values of R.
In the case of the outer electrons,

2ch

giving a second series of spectral lines.
Table ILI. shows the energies corresponding to definite

radii. Column II. shows ie energy content of an inner

electron on the left, and that of an outer electron on the
right. Column III. shows the change of energy, and
column 1V. shows the wave-lengths of the spectral lines
produced. It should be stated here that the energy under
consideration is the energy of one electron and not of the
whole shell. It has been stated that there is a possibility
of the atom not being in its normal condition to begin with,
owing to its gaseous condition. If, however, it has expanded,
then instead of starting with an atom whose diameter is
1°30x10-8 cm., we start with one whose radius is in all
probability equal to one of the radi given in Table IJ.
If this is so, then the only change produced will con-
sist of the elimination of some of the lines of higher
frequency.
P2

212 Mr. L. St. C. Broughall on Theoretical

Taste III.
I Ila. IL 3, IIL a. TIL 2, | Wao aes
No. Energy per electron. ace ate nee oe
Inner. | Outer. Inner, Outer. Inner. | Outer.
12255 X10 1-157 x 10-0 |
2.11195 , |1101 , |80X1077* 56x10~™) 3272 A° 3505 A®
g..j1137,, [1048 ,, (58 » [58 4» | 3886, | 8704 ,,
ie eo) 970 ,, |25 » [51 | 3570 ,, | 3850 ,,
cree 49], |52 » [479 4, | 8776 ,, | 4100 ,,
6 | 9801 9035. [496 » (456 ., | 3959 ,, | 4305 ,,
Mr Ve iggay 604 ,, [£67 » |481°,, | 4205 ,, | 4555 ,,
8 ...| -8886 gis7 ,, [#51 » {417 ., | 4853 ,, | 4709 ,,
Oe einy 7793. [#29 3, [394 4, | 4576 ,, | 4983,
LON "8055 __,, 7423 fs 4-02 Ai 3°70 35 4884 5 5307 y
igo) -veGo. ‘7063, DOOR AE BGO" Se 50384 ,, | 5454 |
12 | -7297 , | e723, [868 » [840 ., | 5885 ,, |-5775 ,.
13 .../ 6940, | 6396 ,, [857 » (327 | 5499 ,, | 6005 .,
14 ...| 6609 ,, | -eos9 ,, |88l » [307 ., | 5982.,, | 6396 .,
ib ..| 6298 ., | 5903, Jetl » |286 ., | Sale), ieee
16 ...| 5995 ., | -55g5 , [3038 ». {278 .. | 6480 ,, | 7063 ,,
17 ...| -5709 ... -|--59g0. «|288 » =) 285s, 9) 6864 | Oa
Jig ...| 5430 ,, | 3003 ., |279 » |257 » | 7037 ,, | 76205,
19 ...| 5172 ,, | 4766 ,, [258 » {287 ., | 7610 ,, | 8285 ,,
20 ....| 4997, | 4540. [245 » [226 », | 8013, | 8688 ,,
91 ...| *4689 nf 439] vi 2°38 25 2°19 a9 §249 a 8566 Bout

Owing to the complexity of the neon spectrum, it would
be useless to attempt to compare our calculated lines with
those found by experiment ; indeed, it would be deleterious
to attempt such a comparison, since the impression would be
given that there is a definite line in the spectrum which
corresponds to one of our calculated lines. Hmphasis may
only be laid upon the fact that our series produce lines
in the visible part of the spectrum, which do not compare
unfavourably with those obtained by experiment. Reference
to Table III. will show at once that only forty lines have
been determined between 7X=3272 A and 7A=8966 A,

Aspects of the Neon Spectrum. rp

whereas there are many more lines in existence. These lines
can only be explained by the fact that when the spectrum is
obtained, large numbers of ionized atoms exist, and under
such circumstances our fundamental equations no longer

hold.

II. Jonization in the Neon Atom.

The discussion of the properties of ionized atoms is very
complex, since the possible degrees and modes of ionization
are very numerous. The first case which comes under con-
sideration is the atom which has lost one electron, thus
leaving an excess of one positive charge.

It is very probable that one of the outer electrons will be
removed, thus leaving seven electrons in the outer shell.
Now, it seems probable that the angular momentums of the
remaining electrons will suffer no change, the light pro-
duced during ionization being due solely to the change of
energy of the electron suffering removal. We are thus left
with seven electrons, each possessing the same angular
velocity. It is a matter of considerable difficulty to arrange
these electrons, and it is impossible to arrange them ona
spherical surface without the force acting on an electron
varying with the electron taken. We are therefore obliged
to separate them into different shells.

Now, since the angular momentums of our seven outer
electrons are the same, it follows that, if they are not on the
the same spherical surface, then they must be in motion
relative to one another. Under such conditions the positions
of the electrons will vary with time.

Owing to the complexity of such a case, it seems impossible
to treat the case mathematically without more experimental
evidence. There are further atoms present which have been
ionized to a greater extent, thus losing several electrons.
Similar difficulties are met with in the cases of atoms with
five or six electrons in the outer shell as in the case of seven
electrons. The cases in which four or six electrons have
been removed are, however, considerably simpler, since the
electrons may then be given positions on a spherical surface
such that the force acting on an electron is not dependent
on the electron taken. In general, the atoms ionized to so
great an extent will be comparatively few. In all our cases
of ionized atoms, it must be remembered that it is not only
the normal atom that is ionized; an atom may have given
out several spectral lines before it becomes ionized. Soa

214 Theoretical Aspects of the Neon Spectrum.

large number of lines will be obtained depending upon the
state of the atom when one or more electrons are removed
from it. There is another form of ionization which is worthy
of consideration. That is the case in which an electron has
succeeded in penetrating the atom and reached the nucleus,
thus temporarily reducing the positive charge and therefore
giving a negative ion.

The fundamental mathematical expressions for such a case
are found by extending our formule for the oe atom
for the case where the char ge on the nucleus is ‘n’ instead
of ten. |

The angular velocities about the several axes will remain
unchanged ; and so only two equations will be required to
determine the new values assumed by ‘c’ and ‘7,’ the radii
of the outer and inner shells respectively.

We have three available equations ; and expressing them
in the notation previously used, we obtain :

(4n—1)e"l _ eee eu e(r-+1) e*(r —l)
4e8 A]? ails eo ob) (r+)? +e] ap i (r— 1)? + 922?
+ 2mo?l: oc 2 el
Ciel eo el e7l e7|
4 AP T28T (elPee pet (Gb? +8?
%

+mol+mo,l, . . (Ila.)
ney Ioan |) Ae? (r—l) ip
P= Toth +e 2 ct (oD? +9? 1 Ape

+ Qo? 2.4) 3) ele

which are obtained from equations (I.), (II.), and (IV.),
replacing the nucleic charge of ‘10e’ by ‘ne,’ and remem-
bering that w;=@».

The result of the alteration will he that the electrons will
move further out from the nucleus, since ‘n’ is of necessity
less than ten. In consequence of this, the frequency of the
spectral lines produced by such ionized atoms will be of a
lower frequency than those produced by the neutral atom.
There will in consequence be a larger number of lines in the
part of the spectrum of greater wave-length. It is for this
reason that there are so many lines in the orange, and red
in the case of neon.

Feb. 18, 1922.

C’ \

Eaviiaes

XV. Absorption of Hydrogen by Elements in the Electric
Discharge Tube. By F. H. Newman, Ph.D., F. Inst..P.,
A.R.C.S., Head of the Physics Department, University
College, Eweter *.

1. Introduction.

‘ia phenomenon of the disappearance of gas in the

electric discharge-tube, and in the presence of incan-
descent filaments, has received much attention recently owing
to its importance in technical applications. Langmuir + has
shown that hydrogen disappears from a vacuum tube in
which a tungsten filament is heated above 1000° C., This
fact has been utilized by him in the removal of the last traces
of gas in valves,and the effect has been termed a “ cleaning
up’ one. The pressures at which he worked were very low ;
for example, he found that the pressure in a tube was lowered
to 0°00002 mm. of Hg. Other gases, including nitrogen and
carbon monoxide, are removed in a similar manner, and
molybdenum, when incandescent, has the same effect as
tungsten. In all cases Langmuir found that the cooling of
part of the apparatus by means of an enclosure at liquid-
air temperature greatly accelerated the rate of disappearance
of the gases. In addition he noted an electro-chemical
“clean up,” which occurred at much lower temperatures of
the filament, when potentials of over 40 volts were used
in a way that caused a perceptible discharge through the
gas.

More recently Campbell, conducting work for the General
Electric Company f{ and using incandescent filament cathodes
in electric discharge-tubes, has made an exhaustive study of
the ‘clean up” effect, and has come to the conclusion that
there is much evidence for believing there exists an electrical
action which is quite independent of the thermal action, and,
providing the temperature of the filament is kept below that
at which the chemical ‘‘ clean up” occurs, the effect appears
to be one dependent only on the electrical discharge. Inthe
case of the disappearance of carbon monoxide there is proof
of the conversion of this gas into carbon dioxide, and the
action takes place more rapidly when part of the apparatus
is cooled to liquid-air temperature. This has the effect of
removing the carbon dioxide by condensation as quickly

* Communicated by the Author.
+ Am. Chem. Soc. Journ. vol. xxxvii. (1915).
t Phil. Mag. vol. xl. (1920); vol. xli. (1921) & vol. xlii, (1921).

216 Dr. F. H. Newman on Absorption of Hydrogen

as it is formed. The presence of phosphorus vapout
accelerates the rate of disappearance of all gases except
the inert ones, and much lower final pressures are attained.
This, the author believes, is due to the deposition of the gas
on the walls of the vessel, this deposit then being covered
with a layer of red phosphorus formed by the electric
discharge passing through the phosphorus vapour. The
covering of red phosphorus prevents liberation of the
hydrogen by bombardment of the ions, and at the same
time provides a new surface on which further gas can be
deposited.

The problem of the disappearance of the gas is a very
complicated one, owing to the many factors to be considered.
The walls of the vessel and the electrodes will certainly receive
some of the gas, although the latter may not disappear in its
original state. There will be chemical changes occurring in
the volume of the gas, such as the conversion of carbon
monoxide into carbon dioxide, and, in addition, any other
elements present in the discharge-tube, either in the form
of vapour or solid, will affect materially the rate of dis-
appearance of the gas and the final pressure reached.

The author * has shown previously that various substances
present on the electrodes of a discharge-tube alter considerably
the amount of gas that can be caused to disappear when an
electric discharge is passing. In particular, phosphorus,
sulphur, and iodine cause both hydrogen and nitrogen to be
absorbed ata very great rate, and a high vacuum is quickly
produced as a result. This action of phosphorus has been
used for many years to obtain and maintain very low
pressuresin valves. These three elements stand out as being
very effective even at high pressures, but other substances
which were tested in a similar manner did not appear to
absorb hydrogen. On the contrary, gas appeared to be
liberated. This effect can be explained as follows. At
pressures above 1 mm. of Hg.a certain amount of the gas
in a discharge-tube becomes occluded within the walls.
This gas will be liberated when the walls are bombarded
by the ions produced by an electric discharge. This effect
will mask any disappearance. If, however, the tube is
heated almost to the softening point of glass and highly
exhausted, then on admitting hydrogen at a small pressure
such as 0°l mm. of Hg., very little occlusion of the gas
within the walls will take place, and on passing the electric

* Newman, Proc. Roy. Soc., A. vol. xe. (1914); Proc. Phys. Soc.
vol, xxxii. (1920) & vol. xxxiii. (1921).

by Elements in the Electric Discharge- Tube. 217

discharge practically no hydrogen will be liberated from the
walls by bombardment with the ions. If there is any
absorption of the gas, this effect will not be masked by the
liberation of the gas from the walls or electrodes.

The object of the present work was the study of the
behaviour of hydrogen in the presence of various elements in
a discharge-tube when a current was passing through it.
The pressures of the gas in these experiments were much
lower than those used by the author in the papers quoted
above, but they were much greater than those used by the
previous investigators—Langmuir and the General Electric

Campany.
2. Description of Apparatus.

At gas-pressures below 0°1 mm. Hg. it is difficult to obtain
a current through a discharge-tube unless very high potentials
are used. <A valve also must be placed in the circuit to make
the discharge unidirectional. This entails further diminution

Pie

MCLEOD GAUGE
i

PUMP | HYDROGEN

1

of the current. By using a Wehnelt cathode the poteutial
required was greatly reduced. The apparatus employed is
shown in fig. 1. The incandescent filament was a strip of
platinum foil 5 mm. long and 3 mm. wide. As the
discharge-tube had to be thoroughly cleaned after each
experiment, the cathode was sealed in a glass stopper which
could be removed when the tube was cleaned. This
necessitated the use of tap-grease, but the vapour arising from
it did not appear to aftect the results at the pressures used.

i

218 Dr. F. H. Newman on Absorption of Hydrogen

Previous experiments had shown that elements such as
sodium and potassium are only effective in causing the
disappearance of gas in the electric discharge-tube if the
surface of the element is clean, and if it has been prepared
in vacuum. Accordingly the substance under test was
placed on the platinum foil forming the cathode, and after
the tube had been heated almost to the softening point of
glass and exhausted, the element on the foil was vaporized
by passing an electric current through the latter. In this
way the substance was then deposited on the inner surface of
the anode D and an uncontaminated surface obtained. The
anode was of aluminium and was cylindrical in shape, fitting
very closely to the glass walls. Enclosing the cathode in
this way, the effect of the surface of the glass on the
absorption of the gas was minimized. A side tube B was
used to contain the easily volatile elements such as phosphorus,
sulphur, and iodine. An aperture was made in the anode
opposite the mouth of B so that the vapour of the substance
from B could pass through and be deposited on the inner
surface of D. The pressures of the gas were measured with
a McLeod gauge. The hydrogen was prepared by the
electrolysis of barium hydroxide and stored in a reservoir.
This method of preparing the gas ensues great purity. Any
oxygen present was removed by passing the gas through a
bulb containing sodium-potassium alloy. Phosphorus pent-
oxide in F removed any water-vapour, and of course the alloy
was effective in this respect also. The gas could be admitted to
G, which was a known volume (0:051 ¢.c.) enclosed between
two taps. A definite volume of gas at a known pressure
could thus be admitted to the discharge-tube. From obser-
vations of the pressure in the tube before and after a discharge
had passed, the actual volume of gas—at atmospheric
pressure—which had disappeared could be calculated.

The current through the discharge-tube was kept constant
by altering the filament current, and was measured with a
galvanometer. In previous experiments the quantity of
electricity passing through the tube while absorption was
taking place had been measured with a water voltameter,
but in the present work this method was not sensitive
enough. . | |

After deposition of the substance on the anode D, the
tube was again highly exhausted to remove any gases
liberated from the volatized substance. The tube was placed
in an enclosure maintained at —40° C. while absorption
of gas was in progress.

by Elements in the Electric Discharge- Tube. 219

Observations were then taken of the changes in pressure
due to the disappearance of the hydrogen when an electric
discharge passed through the gas. The results obtained are
shown in the accompanying table.

The accelerating potential was 94 volts, obtained by using
small accumulators. The current through the tube was kept
constant, and was 546 micro-amps.

Each set of readings corresponds to an electric discharge
for ten minutes, except in the cases of sulphur, phosphorus,
and iodine, where the observations were taken at intervals
of two minutes—2. ¢., with sodium the pressure changed
from 743 x10-*. mm. of Hg. to 336x10-? mm. of Hg. in
ten minutes, while with sulphur the pressure was lowered
from 740% 10-* mm. of He. to 329x10"° mm. of He. in
two minutes.

The amount of hydrogen which would be liberated from
a water voltameter in 10 mins. by the same current is
39 x 1073 ¢.c. at atmospheric pressure. —

As the gas may disappear into the walls of the anode
even in the absence of any substance on the anode, and
as the glowing filament may affect the rate of disappearance,
preliminary observations were always made when an electric
discharge passed through the tube without the substance
present on the anode. The volume of gas which disappeared
owing to these two effects was always very small compared
with that which was absorbed when the element under test
was on the anode.

3. Heperimental Results. |

After each element had been tested, the tube was heated
to 300° C., and the volume of gas reliberated was calculated
from the observed change of pressure. The amount thus
recovered varied considerably, but was always less than that
which had disappeared. This evolved gas was again absorbed
when a discharge was passed, and it is evidently hydrogen
in the same condition as it was before disappearance.

If, after the gas had disappeared, a fresh amount of
hydrogen was admitted, the volume which disappeared on
discharge was reduced. For example, with sodium and the
gas pressure at 743x107? mm. of Hg., the vacuum was
reduced to 96x10-? mm. Hg. before the action ceased.
Admitting a further supply of gas to the tube, the pressure
fell “roms Ma l0= imme He: to 233 107% mm. Hg.
Repeating the process again, the pressure fell from
743 x 10~* mm. Hg. to 486x107? mm. Hg., and then the

(220) Dr. F. H. Newman on Absorption of Hydrogen

Initial Gas Final Gas Volume of Gas a nee pian
Element. Pressure. Pressure. absorbed. of & a ae
mm.H¢e,xl0-3 mm.Hg.x 10-3. Ge) <1 Ome rome 10-3.
. 743 336 28
SOU as. eee 4) 336 123 14 96
a 726 349 26 84
Potassium .....-... { 349 163 13 ,
Sodium-Potassium 738 392 23 110
Alloy. 392 188 lo
( 740 829 28
Sil plaice eee 4 329 145 fabs, 26
\ 145 58 6
(748 352 Zt
Phosphorus :...:: 4 352 147 14 14
| 200 44 @
s (7638 338 28 5
Todine OCUC ADIGA AER it 338 150 13 124
f 750 394 24
IATSCING Rete. o2 2s. { 304 906 13 108
4 j 744 463 19 ‘
Cadmium dicoickoio ce | 463 321 10 284
Galena gece 1 Be ie me 152
rs 750 386 Dy
cee at a | a Be : 131
; 758 Has: 99
BNO ee ae ea 433] 306 8 276
Seam 732 560 12 |
Mba. ee ae’. { 560 399: 1 297
ee \ Se aetced Hydrogen was liberated and not absorbed.

absorption ceased. There appears to be a fatigue effect
whereby the actual amount of gas which can be absorbed by
any surface is limited. This fatigue effect may be due to
three causes. If the disappearance of the gas depends on
chemical action, the latter will occur mainly at the surface
otf the element. The formation of a chemical compound will.
thus protect the rest of the substance from the action, and
the process will gradually cease. If, on the other hand,
the effect is due to a deposition of the gaseous atoms on

by Elements in the Electric Discharge- Tube. 221

the surface, as Langmuir suggests, these atoms will diffuse
slowly into the substance. ‘The atoms arriving later will
have less area on which deposition can take place. A limit
to the action will be reached when the number of atoms
deposited is equal to the number set free by the bombard-
ment of the surface by the ions.

After absorption, the proportion of the hydrogen re-
liberated when the tube was heated to 300° C. varied
considerably in different cases, not only with different
elements, but also with the same element. This is to be
expected when it is remembered that the thickness of the
substance deposited on the anode varied with different
substances.

The accelerating potential affected to some extent the rate
of disappearance of the gas and also the final pressure
attained. Owing to liberation of the gas by the bombardment
with the ions, the final pressure reached must depend on this
reverse action, and the greater the accumulation of the gas on
the surface of the anode, the greater will be the amount of
gas evolved.

With sodium on the anode a potential of 94 volts reduced |
the gas-pressure from 743 x 107° mm. Hg. to 123 x 107? mm.
Hg. in the course of 20 minutes. When the potential was
lowered to 54 volts, the pressure fell from 743 x 107? mm.
Hg, to 476x10-**mm. Hg. in the same time-interval.
The final pressures reached before absorption ceased were
96x10-* mm. Hg. and 202x107-* mm. Hg. respectively.
The current through the discharge-tube was kept constant
threughont.

The principles of the disappearance of the gas will be
discussed later, but there are certain features of the
phenomenon which can be traced to chemical actions.

Many of the elements tested combine with bydrogen at
high temperatures to form chemical compounds which are
very stable. Any chemical action occurring in the present
experiments cannot be due to the heat, as the discharge-tube
was maintained at —40° C., and the incandescent filament
was always at a lower temperature than that at which
Langmuir found chemical action occurred with hydrogen.
The effect may be caused by “activation”’ of the gas, the
latter assuming some modification under the action of the
electric discharge. In the above experiments the amounts of
gas absorbed were so small that it would be extremely
difficult to detect the existence of any chemical compound in
the tube. In order to increase the amount of gas absorbed,

222 Dr. F. H. Newman on Absorption of Hydrogen

and test for any chemical compounds formed, a modified form
of the apparatus was used, as shown in fig. 2.

Pure hydrogen could b: admitted to the discharge-tube A
in small volumes by manipulation of the taps 1, and ‘Ty.
Two strips of platinum foil, about 10 cms. long, were sealed
in the tube EH. These strips fitted closely to the glass
surface. A potential difference of 600 volts was applied
between these strips by means of small accumulators. In
this way the ions actually present in H were removed while
the discharge was proceeding in A. The tube EK communi-
cated with a mercury cut-off K, and a U tube immersed in an
enclosure maintained at —40° C. Sodium-potassium alloy
was prepared in D, and after the whole of the apparatus had
been evacuated, the alloy was run into C. In this way a
bright and clean*surface was obtained on the alloy in C. A

HYDROGEN

small volume of hydrogen was then admitted to A, the
mercury cut-off preventing the gas from entering C and D.
While the electric discharge was passing in A, the hydrogen
was allowed to enter C by manipulation of the cut-off,
Admitting successive volumes of hydrogen into A in this
way, and each time allowing communication with C while
the discharge was proceeding, an increasing amount of active
gas entered ©, and an effect was observed on the surface of
the alloy. At first 1t appeared to be covered with a thin
white crystalline compound when observed through a
microscope. ‘This white layer slowly changed, on the
admission of more active gas, to a dark grey-coloured
deposit. To show that this surface effect was not due to
impurities in the hydrogen, previous experiments were
made, the gas being admitted to C without the electric
discharge proceeding. There were no surface effects then,

by Elements in the Electric Discharge- Tube: 223

so it was concluded that some of the hydrogen assumes an
active modification under the action of the electric dischar; ge,
and in this form it is able to form chemical compounds w ith
the sodium and the potassium present in the alloy. The U
tube in the enclosure at —40° C. excluded the possibility of
the action being due to the heat from the discharge-tube.
The white erystalline compound which first appears is a
mixture of the hydrides of sodium and potassium. ‘The exact
nature of the greyish-coloured product formed afterwards is
unknown, but it is probably a solution of the hydrides in the
alloy.

tess is evolved by an electric discharge when passed
through any vacuum vessel. It comes from the glass, and
would not be kept back by the trap cooled to —40° C.; for
at that temperature water substance has a vapour-pressure of
about 0'l mm. He. The presence of water-vapour “fouls ”’
the surface of the alloy, but this fouling gives a black
deposit on the surface which is quite different from the
white crystalline layer observed in the present experiments.
The black deposit consists of sub-oxides of sodium and
potassium, and its appearance has been noted previously
by the author*, although in ,that paper it was attributed to
the hydrides. It has now been proved by chemical analysis
that this black deposit does consist of the sub-oxides.

Sulphur was tested in the following manner:—A small
piece of filter-paper, soaked in lead-acetate solution, was
placed together with a small amount of the solution in D.
The rest of the apparatus was separated from D by a mercury
cut-off not shown in the figure. C contained sulphur which
had been deposited in a thin film over the interior. After
exbausting the whole of the apparatus to a pressure of about
5 mm. of Hg., the mercury cut-off between C and D was
closed and the rest of the apparatus highly exhausted.
Hydrogen was then admitted to A until the pressure was
about 7mm. Hg. While the electric discharge was passing,
the mercury cut-off was opened. This was repeated many
times, the pressure of the gas in A being gradually increased.
Each time communication with D was established, any
gaseous product formed in C was admitted to D. In the
course of a few minutes the paper soaked with the lead-
acetate solution turned black, showing the presence of a
sulphide of hydrogen. This chemical compound must have
been produced by the action of an active form of hydrogen.
on the sulphur. The surface of the mercury at the cut-off

* Proc. Roy. Soc. A. vol. xc. (1914).

224 Dr. F. H. Newman on Absorption of Hydrogen

also lost its bright appearance. This was due to the action on
it of the sulphide of hydrogen. ‘The mercury surface
remained quite clear when the sodium-potassium alloy was
tested.

Sulphur and the alloy were selected for tests because the
chemical actions in these cases give rise to compounds whose
effects can be noted easily. It is extremely difficult to
examine phosphorus and iodine in this way owing to their
high vapour-pressures. A possible test would be “the com-
parison of the vapour-pressures before and after absorption
of hydrogen had taken place.

These two experiments indicate that the chemical action is
not due directly to the ions in the discharge-tube, as they
were all eliminated by the charged platinum strips before
reaching either the alloy or the sulphur.

Wendt * showed that hydrogen can be activated by the
passage of e-rays through the gas, and it has been shown by
the author f that the active modification so produced is able
to react chemically with sulphur and the alloy of sodium and
potassium.

A. Discussion of Results.

The disappearance of gas in a vacuum-tube is due probably
to several principles, some of which may be fundamental.
It is certain, however, that any attempt to explain the
principles by the same theory would lead to conflicting
results, but the processes occurring can be divided into
two classes, chemical and mechanical. There is much
evidence that the gas can be caused by the electric
discharge to adhere to the solid parts of the discharge-tube
in some manner which is at present unknown. In man
cases a portion of the gas can be reliberated by heating
the vessel, but no reason can be advanced for the non- ~
reliberation of the whole of the gas which has disappeared.

Langmuir assumes in the paper previeusly quoted that the
hydrogen in the presence of an incandescent filament under-
goes dissociation. The gas shows abnormal thermal
conductivity at high temperatures, due to its atomic nature.
The dissociation does not occur apparently in the space round
the wire, and is not due to the impacts of the gas molecules.
against its sur face, but takes place only among ‘the hydrogen
molecules which have been absorbed by the metal of the wire.
Some of the atoms leaving the wire do not meet other atoms,

* Nat. Aead. Sci. Proc. vol. v. (1919).
t+ Phil. Mag. vol. xliii. (192%).

by Elements in the Electric Mischarge- Tube. pays)

owing to the low pressure, but diffuse into the tube cooled by
liquid air, or become absorbed by the glass, and thus remain
in the atomic condition. They retain all the chemical activity
of the atoms. Langmuir also found that when the liquid air
was removed, some of the atoms would come off the glass
and recombine with other atoms to form molecules. These
molecules could not be recondensed by replacing the liquid
air. This gas which would not again disappear he termed a
‘“‘non-recondensible’’ gas.

This hypothesis, which is applicable to very low pressures,
cannot hold at the pressures used in the present work. The
gas in the atomic condition can scarcely move from the
discharge-tube for a considerable distance and still retain
its atomic nature. The “non-recondensible” gas found by
Langmuir is probably hydrogen in its normal state.

When nitrogen gas disappears in the discharge-tube,
practically none of it can be reliberated, even when the tube
is heated to the softening point. This fact indicates a
striking difference between the disappearance of hydrogen
and nitrogen. If chemical compounds are formed by the
absorption of the gases, this difference can be explained
in terms of the difference in the stability of the hydrides and
nitrides produced.

The chemical action may take place between hydrogen and
the vapour of the element, and also it may occur at the
surface of the solid. The majority of the elements studied
have such small vapour-pressures that a very small propor-
tion of the action is due to the vapour. The active condition
of the gas must be caused by the ions, although results seem
to indicate that the number of active atoms or molecules in
the gas is of a much higher order than the number of ions
present in the gas at the instant of recombination.

The absorption is not due entirely to chemical action, as the
law of constant proportions does not seem to be followed.
It is of significance, however, that the rate of disappear-
ance of the gas increases, and the final pressure attained
decreases, as the temperature of the discharge vessel is
lowered. This arises from the lowering of the vapour-
pressure of the compounds produced, with the result that
the final pressure reached is lowered.

Although the formation of hydrogen sulphide in the
discharge-tube by the action of the activated hydrogen
on the sulphur will not explain the disappearance of the
gas, it does indicate the production of a modified form of
the gas which is able, possibly, to form other compounds
with sulphur in addition to hydrogen sulphide.

Phil. Mag. 8. 6. Vol 44. No. 259. July 1922. Q

226 Mr. Bernard Cavanagh on

That the mechanical deposition of gas on the walls of the
discharge vessel will not account entirely for the disappear-
ance of the gas is shown by the difference in the behaviour
of nitrogen and hydrogen with phosphorus, sulphur, and
iodine. Practically none of the nitrogen can be reliberated
by heating, but a large proportion of the hydrogen is evolved.

There is reason for believing that the modification of
hydrogen is triatomic in nature. Wendt has shown in the
paper previously quoted that hydrogen drawn from a tube
through which an electric discharge is passing contains a
small quantity of H;. Probably monatomic hydrogen is first
formed, and owing to collision with neutral molecules of the
gas, H; then appears. The monatomic gas may be produced
originally by the action of the swift-moving electrons on the
molecules. Wendt and Grubb * have also shown that Ns; is
produced when an electric discharge passes through nitrogen.
Thomson + found evidence of H, in his positive-ray experi-
ments. It is this triatomic form of hydrogen which is
effective in the production of chemical compounds in the
electric discharge-tube.

XVI. Molecular Thermodynamics. Ul. By BERNARD
A. M. Cavanaau, B.A., Balliol College, Oxford t.

I. Morecuss, THERMODYNAMICS, AND
(JUANTUM THEORY. © |

| le developing a molecular treatment of the thermodynamics

of dilute solutions in simple solvents, Planck § deter-
mined the form of the integration constants in the entropy
function by a method which was at the time the subject of
some controversy.

M. Cantor || objected that the hypothetical transition to
the gaseous state without change of the molecular composition
was not even theoretically possible, since there probably
existed in the liquid state, complex molecules whose existence
was inseparably connected with the condensed state of the
phase, and entirely incompatible with a state of high
temperature and low pressure.

* Science, vol. lii. (1920).

+ Proc. Roy. Soc. A. vol. Ixxxix. (1918).

t Communicated by Dr. J. W. Nicholson, F.R.S.

§ ‘Thermodynamics,’ 1917 (Trans. Ogg), pp. 225-226. Or see Phil.
Mag. xlii. p. 608 (1922).

| Ann. der Phys. x. p. 205 (1908).

bo

Molecular Thermodynamics. 27

In reply, Planck * pointed out that the theoretical pos-
sibility of the ideal transition depended only on the fact that
the numbers of the various molecular species were, together
with temperature and pressure, the independent. ‘variables
which determined the phase.

Now the present author would suggest that Planck’s reply
can be construed (and, to be unanswerable, must be construed)
as a rider to the definition of the terms “ molecule” and

** chemical compound,” for the purposes of molecular
thermodynamics, viz. :—‘** That the numbers of the various
molecular species can be considered, together with temperature
and pressure, as the independent variables determining the
phase,” or, in other words, ‘that it shall be theoretically
sound to conceive any desired change of temperature and
pressure of the system as taking place without change in the
numbers present of the various “molecular species f.”

Only with this rider ses our definition can it be laid down,
for instance, that the “ mass-action”’ equilibrium law must
be obeyed in sufficiently dilute solution, for it is to be
observed that purely ‘ general” thermodynamics has no
cognisance of molecules, but takes for its independent
variables, besides temperature and pressure, the masses of
the “components.” [See next section of this paper. |

The misconceptions which have so long stood in the way
of a satisfactory general theory of electrolytic or “ con-
ducting ” solutions seem sufficiently to illustrate the indis-
pensability of this postulate.

A parallel illustrating its significance may be drawn from
the dynamical theory of chemical combination and dis-
sociation.

The classical dynamical conception of a binary molecule
(for example) was a pair of simpler molecules (or atoms)
moving relatively to one another in closed orbits, and the
principle of the conservation on energy forbade the spon-
taneous dissociation of such a ‘‘ molecule,” requiring that its
disruption should depend on collision with another molecule.

The well-known fact that dissociation is (at constant
temperature) independent of collision-frequency, showed the
inadequacy of this conception, and pointed to some property,
in the “forces” producing and maintaining a molecule,
altogether incompatible with the older or ‘‘ continuous ”
dynamics,

* Ann. der Phys. x. p. 436 (1903).
+ Intermediate and final states being unstable, of course, in general.

Q 2

228. Mr. Bernard Cavanagh on

The same difficulty arose when a dynamical explanation of
the law of mass-action was attempted, the essential continuity
of action of “physical” forces * standing in the way, and
Boltzmann had to assume—with conscious artificiality—dis-
continuity in a field of force in order to arrive at the desired
result.

It seems indeed that, besides the more obvious and less
peculiar properties of shortness of range, “‘ specificity,” and
saturability, there is a quality of discontinwity of action (in
time or in space or in both) which distinguishes ‘ chemical
forces” from “ physical,” the distinction being sharp so far
as we can yet see.

We find then an absence of direct dependence of
dissociation upon the thermal motion closely connected with
the possibility of accounting for the law of mass-action
dynamically.

The parallel with our “ rider ” and its indispensability as a
basis for the deductions of molecular thermodynamics is
significant.

The transitory orbital system which was the older
“physical” conception of a molecule, and which quite
probably occurs in all dense gases and liquids, is clearly
quite directly dependent-on the thermal motion, is, in fact,
itself merely an “episode”? in that motion, and cannot in
any sense be regarded as fulfilling the requirement of our
rider. We cannot, therefore, predict from thermodynamics
the ‘“‘ mass-action ”’ equilibrium law for the “reactions” of
such “ molecules” under any circumstances, for we cannot
treat them ‘as molecules for the purposes of molecular
thermodynamics. And, in parallel, we find that dynamical
theory is unable to predict the law of mass-action for such
‘“‘ molecules.”

The electrolyte question provides an important application
of these considerations.

It has frequently been supposed that a pair of ions, closely
linked by their electrostatic fields alone, must be regarded as
2 molecule, and should behave thermodynamically as such.
leenrostatic forces ‘as we know them, however, are typical

* That is, “forces”? within the conception of the older physics. It
seems convenient to use, in contradistinction, the term ‘“ chemical forces ”
for the “ forces” or means by which a molecule is formed or “bound”
[see end of this section] and held together, and the expression may
find some further justification in the fact that in the nature of these
latter ‘‘ forces” lies, probably, the key to all the facts and phenomena of
chemistry.

+ Compare here Cantoz’s objection, mentioned above.

| Molecular Thermodynamics. 229
“physical” forces, and it is our present conclusion that such
forces are capable of forming only transitory associations—
‘‘episodes in the thermal motion ””—essentially different from
what we regard as molecules.

When, as in the weaker acids and bases almost certainly,
we really have partial ionization, or rather partial association
of the ions to form “ undissociated molecules,” the latter
must be regarded as produced and held together not by such
ordinary electrostatic forces, but by ‘“* chemical forces ” with
the peculiar property already discussed.

With regard to “strong electrolytes,” the work of Debye
and Brage gives good reason to believe that the molecule in
the salt crystal is the ion. If this be so, it appears necessary
to admit that the solid salt is essentially a mixed crystal whose
special simplicity and homogeneity is due simply to the
polarity of the electrostatic forces which dominate its
** orowth.” |

Now, it would seem altogether inconsistent to suppose that
the chemical “ association” which does not take place in the
intimate contact of the solid state, ensues when the ions are
dispersed in a solvent, so that until the calculated effect of
the electrostatic forces between the ions upon their thermo-
dynamic behaviour can be shown to be inadequate when
compared with experiment, the ‘‘ complete-ionization ” theory
seems the only rational theory for strong electrolytes.

Another application of the above general conclusions is to
be found in the important question of “ solvation ” of solutes,
which is treated in a paper to follow this.

There occur in the literature of this subject such state-
ments or suggestions as that “the solvates need not be
definite chemical compounds,” and vague theories of the
“solvate molecule’? as a mere indefinite conglomerate.
From the preceding, at any rate, it is our conclusion that,
unless the “solvate molecule” is produced and maintained
by ‘‘ chemical forces” in the sense already considered, so that
it fulfils the requirement of our ‘ rider,” it will not, for
the purposes of molecular thermodynamics, be a molecule
at all.

With regard to this remarkable characteristic of ‘‘ chemical
forces’ which appears to be reduced to its lowest terms in
the expression “ discontinuity of action,” this seems to mark
out the problem of molecule formation (including, be it noted,
reaction-velocity) as one of those many whose solution may
be hoped for from the new quantum-dynamics of phenomena
on the atomic scale. Indeed, it is tempting to believe that
Bohr’s conception of ‘‘ electron-binding ” may be the solution

230 | Mr. Bernard Cavanagh on

in embryo of the larger and more complex problem of “ atom-
volo and that in his distinction between “bound ” and

“unbound ” electrons in the atom, we may have in its
simplest aspect the distinction berwecn “chemical”? and

‘¢ physical ”’ forces.

Il. MoLecuLar THERMODYNAMICS.

In general thermodynamics, which is based on, and applies
to experience, the independent variables, besides temperature
and pressure, are the masses of the ‘“ components,” and
these are reduced to the minimum necessary to define the
system under all circumstances not conventionally or prac-
tically excluded from consideration.

Yo take a familiar example, hydrogen and oxygen will
suffice as the components of a system containing in addition
water, provided low temperatures are excluded from
consideration, or the presence of efficient catalysts is assumed.
In so far as we may suppose that the decomposition and
formation of water do proceed even at low temperatures in
the absence of catalysts, though at an immeasurably small
rate, it is clear that theoretically the two components would
always suffice for this system @f sufficient time were allowed.

And conversely they would never suffice if the rate of experi-
mentation were sufficiently increased.

Striking examples of the practical reality of this entry of
the tame factor into the question of the necessary number of
components, have been given in recent years by the work of
A. Smith and of A. Smits, who by increasing the rate of
experimentation, have increased the number of components
necessary to describe certain systems, the latter author having
propounded an interesting theory of allotropy on the basis
of his experiments.

Now we can conceive this carried far beyond the bounds ot
purely practical limitations, and the question arises, “ How
fair, 7

The atomic or elementary theory of matter is introduced
when we say that at one extreme, when unlimited time is
available, the elementary atomic species will be necessary as
well as sufficient as the components of any system.

Starting from this extreme and increasing the rate of
experimentation we can imagine one complex arter another
of these elementary atoms (as its rate of formation and
decomposition ceases to be great in comparison with the rate
of experimentation) taking its place in the list of ‘ com-
ponents necessary to describe the system.” Remembering,

Molecular Thermodynamics. 231

however, that the masses of the components are together with
temperature and pressure the independent variables, we see
that there may be a limit, for it must be theor etically possible
to alter temperature and pressure so quickly that the
numbers present of these complexes which we are admitting
as components remain sensibly unaltered during the change.
According to the view put forward in the previous section,
it involves us in a definite postulate bearing upon the nature
of molecules and of chemical change, when we say that the
limit is reached when and only when every molecular species
which can be formed in the system has taken its place in the
list of ‘* components.”

Proceeding in theory to this limit, we obtain the general
expression for y% which is referred to in the sequel as the
*‘ molecular expression for W,” and as Planck showed, we can
determine it completely when it is linear by ‘‘ connecting-up ”
with the known properties of the low-pressure gaseous
mixture.

When the expression is not linear, the higher or “ general ”
terms are subject only to a single limitation inherent in
Planck’s method, as pointed out in the previous paper *
Observing that the “ general”? terms in the corresponding
expression for U ¢ do not involve ‘chemical’? energy, the
present author also suggested and illustrated f the interesting
possibility of employing ordinary dynamical theory, at least
as a valuable aid, in determining and interpreting the form
of these ‘‘ general” terms.

This ‘molecular expression for a,” however, will clearly
not in general correspond with our experiments carried out
under ordinary conditions. They will correspond with an
expression of the ‘‘ general thermodynamic ”’ type in which
the components are appropriate to the conditions of ex-
periment.

The theoretical problem then presents itself of connecting
this ‘‘ experimental ” expression for Wy, in a manner at once
rigorous—that 1 1s trustworthy—and practically effective, with
the “ molecular ” expression and its possibilities of theoretical
interpretation.

The treatment of two important problems of this kind has
been attempted.

Planck pointed out that when a single molecular species

* Phil. Mag. xlini. p. 606 (1922).

+ Entirely “analogous considerations apply to V, but ordinarily owing
to the low pressures used, V figures relatively negligibly in the determi-
nation of ¥ [e¢f. footnote, p. 630, Phil. Mag. xiii, “(192 2)}.

t Loe. cit. p. 625.

232 Mr Bernard Cavanagh on

greatly preponderates, it is a matter of mathematical
necessity that the “ molecular expression for y”’ should take
(in the limit) a linear form, and it was to this type of -
‘dilute solution”? that Planck confined himself, arriving
readily at the Raoult-van’t Hoff “laws of dilute solution.”
Dilute solutions in a liquid paraffin would be of this
type.

Van Laar * took the linear expression as the criterion of
** perfect solution ”’ in general, and not making the approxi-
mations which Planck, considering very low concentrations,
had made, was able to show that the Raoult-van’t Hoff
laws formed too restricted a criterion when the solution was
very dilute.

He, however, considered only solutions in which the
solvent was of the same type as that of Planck, viz.: a
single molecular species.

In view of the fact that the Raoult-van’t Hoff laws have
been found to hold for dilute solution in our common and
useful solvents, which are certainly not of the type considered
by Planck and Van Laar, the present author was led to the
problem of ‘complex solvents,” which will be the first illus-
tration of the theoretical problem outlined above. A
preliminary treatment appeared in the first of these papers,
but a more complete and rigorous treatment is now presented.

The second illustration will be the problem of partially
‘‘ solvated ”’ solutes, a discussion of which will follow that of
““ complex solvents.”

The first result is that the Raoult-van’t Hoff laws have
been rigorously predicted for extreme dilutions in such
solutions. It is shown, in fact, how the ‘experimental ”
expression for yw simulates, in the limit, the “ molecular ”
expression in form.

But further the way is prepared for the thorough investi-

gation of middle and high concentrations in such solutions.

To this end the ‘linear’? terms in the experimental
expression for y, which simulate and replace the simple
linear terms in the molecular expression, have been treated
with some thoroughness and rigour, these being the terms to
which the expression reduces when the solution is ‘‘ perfect.”’

When the quite practical criteria thus provided are applied,
the belief that “‘ perfect solution” always ceases in these
“complex” solutions at quite low concentrations may be
largely dispelled.

In simple solutions of the kind considered by Planck and

* Z. f. Phys. Chem., several papers, 1903 etc.

Molecular Thermodynamics. 233

Van Laar we find “perfect solution’ persisting up to very
high concentrations—sometimes over the whole range.

The ‘ general” terms of which (excepting the case of
electrolytes) little or nothing is yet known, have been touched
on cnly in so far as the treatment of the “linear” terms
involves {in general) a certain very slight alteration in the
division into “linear” and “ general”? terms, which may
sometimes have to be taken into account when dynamical
theory is employed. .

Experimental determination of the ‘“ general” terms for
comparison with theory must of course be preceded by
knowledge of the ‘‘linear” terms,—hence again the need

. >) ;
for rigorous and thorough treatment of the latter.

66

III. CompLex SOLVENTs.

The importance of this question of ‘ complex” (polymer-
ized and mixed*) solvents is sufficiently obvious when it is
considered that of this type are most of our best solvents,
probably all our “ ionizing” solvents, and, chief of all, water.

We have to consider a solution consisting of the solute-
molecular species, ™m ,, mo, .....- ite... slim adaiions to. the
Varlous species 791, Mp, .-.., Which constitute the solvent.

Concentrations being expressed in gram-molecules per gram
of solvent, the ‘“‘ molecular” expression for y is

Jr
bh = Sng ($u Rlog re =

+3n.(g,—R lege es) BT Ye (ares 4), (GIL

which is of the form

Se OY an OY
Vv 7 >No) N01 eed a a somes 2 (2)

the several solvent-molecular species appearing as separate

* Mived solvents, while submitting to the same theoretical treatment
as the merely polymerized solvents, present certain peculiar difficulties
and some interesting possibilities with which it is hoped to deal at length
in a later paper.

+ A suffix outside a bracket is used to indicate, in less obvious cases,
independent variables which are held constant in a partial differentiation,
—a well-known usage. A single suffix may be used briefly for a whole
series, aS 7; here standing for m1, %o2,.... -

234 Mr, Bernard Cavanagh on

components, whereas the “

experimental *”’ expression must
have the form

pela oy
wr = My OM, +n, au. ii ulate ee Ue eae (3)
the solvent appearing as one component only.

Clearly

Bh Gen BI

Ons, T NORe Moi ON: — Osi My

but the relation we have to use in making the change of
variables is that given by the condition for chemical equi-
librium among the molecules of the solvent, viz. :—

[Se dun] = 0, . eae

@.-.

and comparing then (2) and (3),

Os ov
OM, deo Ai Sea ea pel shy been ea aun (7) |

which could be regarded as physically obvious, as was done
in the preliminary treatment (previous paper).

so that

We shall abbreviate (1) by writing m for = C for de,,
01
and M,G' for the “ general” terms; so that,

pr = Dino ($u—Rlog il 0
MCs ;
+3n.($.—Rlog +6

It will be convenient also to write Go’ for =o (M,G'),
U1

the “general” terms in OWE. similarly G,’ for those in
On

0

Ons

Now, in the first place, we have to show that it is permis-

sible to assume that (8) has already been so arranged that °
G', Gs’, Go’, etc., all vanish in the limit when © becomes

, ete.

Molecular Thermodynamics. 235

very small. This is essential to the rigour of the demon-
stration that the Raoult-van’t Hoff laws are still the limiting
laws of dilute solution when the solvent is complex.

Now G’ is, at constant temperature aud pressure, a func-
tion of ¢ 3, Cog) «.«+; a8 well as of ¢y, Co,..-.. Owing to the
chemical equilibrium controlling cv, coz, ...., however, these
quantities have, in the pure solvent, values depending only
on temperature and pressure, and the departures from these
limiting values, caused by the presence of solutes, will clearly
decrease with the concentrations of those solutes. Thus as C
diminishes, the ranges of variation to be considered of the
variables co1, co, .... are progressively limited, the same
being obvious in the case of ¢, cg, .....

But clearly any finite, continuous, differentiable function
of several variables must behave as a linear function if the

range of variation considered of every variable is sufficiently
limited *.

Thus
Lt Geilo le . . E : . ° (3)
C>0
0.
Lt (MyG’) = Snoyloy + Vrrsls, 2 : : - (10)
C>0
where Io), ...... Rin tie , depend only on temperature and
pressure, being, in fact, limiting values of Go’, ....:. :
5 eee , respectively.
But clearly (9, ...... a ape te , can be transferred to, and
included in the “ linear ” terms,— in go, ..-.-. » di, re-

spectively—whereupon the residual “ general” terms wiil .
satisfy the requirement that G’, G,’, (Go,’, ete., should all
vanish as C becomes very small. We shall assume that
in (8) this adjustment has already been carried out.
Returning now to (7) and comparing with (8), we see that

Oy te fo
eS ee eae,
C>0 OM C>0 ul a

= 4 | Seon os _ R log m cot) | = ou (say), ° (11)
o>0 ;

since, in the pure solvent, cq, ....-. , and ™ assume values
dependent only on temperature and pressure.

* Merely the obvious property of tangency in n-dimensions. The
theorem quoted by Planck in treating simple solvents [‘Thermo-
dynamics’ (Trans. Oge), 1917, p. 225] appears to be the particular case
of this, when the ranges of variation of the variables are all located (as
here in the case of ¢1, ¢,,...-) close to zere.

236 Mr. Bernard Cavanagh on

Then we have

oy = out | aon 3 a ov) eh ta ae

but again remembering :

5 OY den =0, . oa
01

we get

oy = =ou+ ( Xyi a(2*), ean (1.24)
Sa. 0
and so from (8)

: M ee :
or =ou+ \e(— Rd log rae a\ + ie vei jel
0

% C=0 NC=

=¢uth ac (1+ mO+ {; ScudGan. =~ (16)

C=0 “C=0
Now (6) with (8) gives

or) =o mC, / 17
(se Mo [ Sled P ae pe 1+ l+ mC 2 ue (
And if we write (16) in the feaedias :

oe =o + R (- dlog (1 + mC) is Gu, 9 (18)

C=0
it is at once clear that the “ general” terms thus adopted
(and therefore, of course, the “ linear” terms similarly) are
connected by the Gibbs fundamental relation [see note at
end of this paper], for (MyG') being a function of 1q......

eee , homogeneous and of the first degree,
Sy eG ed Ge ON
WO: Mp2; €Go;’ + an, dG. =0,
1. é. M,dGy: +n, AGE =O" hoe ty Ce

which means that [MoGu + 2nsGs'] or (say) M,G is, as a

* The Gibbs relation might indeed have been used to obtain (16) direct
from (17), dy, appearing as the integration constant. The above treat-
ment [(12) to (16)] appeared, ihonree to be more interesting, and to
introduce yy in a more natural and illuminating manner, in its relation
to the original “ molecular ” expression for w.

Molecular Thermodynamics. 237
function of Mg and nyng...... homogeneous and of the first

degree. ‘And the same will hold for sihe ‘* linear ”’ terms, (if

M,G) be accepted as the new “‘ general terms”), a cari of
0 I

obvious importance since the “ linear” terms must alone

remain when the solution is “ perfect.” ov retains the
Ns

simple form (17), which may now be written

fo ae Mes
oY = ($.—Rlog =a) RGA, Hs (20)
since
_/O(M,G)\ 3G .
G=(Ss. Mo OG 2d)
G being a function of cc, ...... only (besides T and p).
Also, of course, s
o(M, ')
— J= SS GS a ° . e . hy
G,=G aan (22)
and
MG = IM Ge srs ath et on (23)

We observe, however, that this convenient arrangement
involves theoretically a certain definite (though probably
always very small) change in the division into “ linear” and
“ general ” terms, since

MG. = SarpiGor 2 rs Ge! :
G! == ey, Sal Gee + Ye.G
==G a5 ( > Gio 2 en eel Bal Wee Wet ge (24)

a

c=0
Now it is readily shown that

0G’ 0G’ 0G’ ;
G,=—— +m | G’—Se =~ + De, |; 25)
ae oe gy Ge : Oe |e (25)
and since, mop, being unity, Lmp dey, is always zero, we
have
!
G'—G= {> Oia By de oy ir 2G)
O¢o1
C=0

To say that any such modification of the division into “ linear ”
and “‘ general” terms is due to, and represents, a departure
on the part of the solvent molecules from ‘“ perfect”
behaviour would be to make a qualitative statement of no

238 | Mr. Bernard Cavanagh on

practical significance. (24) is a quantitative statement, and
the form to which it is reduced in (26) has practical meaning
and value, as will be shown by means of a simple illustration
in a short appendix to this section. From (26) it is clear that,
owing to the relative smallness of the changes in ¢q, ¢p,
ae , produced by the presence of solutes, the difference
between the “ general” terms in the original “ molecular ”
expression for wand those in the “ experimental ” expression
we have obtained will generally be very smali. But it may
have to be taken into account in making use of dynamical
theory (at high concentrations).

Of course, until and except when the “ general” terms
can be given more definite form, we cannot say anything
about the way in which they will depend cn the constitution
of the solvent and its variation. For the present we have to
suppose that Co1Co9 ..---- are eliminated in terms of T, p, ¢, ¢2,
, from G, which takes some form

GSR or 77 (cits <5. ) oa Sea
Theoretically, and in the general case, the application of
dynamical theory will precede this elimination of ¢9,¢9...... }
will deal in fact with the original general terms M,)G’, so
that, in greater or less degree, knowledge of the. constitution
of the solvent and its variation will be necessary before such
theory can join issue with practice, but in some cases, as the
appendix will illustrate, this may not be necessary, even
though the constitution of the solvent does affect the
“ general” terms.

It should be noticed that the value and convenience of our
“experimental” expression for yis by no means entirely
dependent on a deficiency of knowledge of the constitution
of the solvent, though the latter makes it practically
indispensable.

In the “linear ” terms the effect of the constitution of the
solvent is concentrated in the quantity m, the mean molecular
weight of the solvent.

In the pure solvent this will have a limiting value i,
dependent only on temperature and pressure, and we can
therefore regard the quantity (—R log mj) as included in the
quantity $s, when our “‘ experimental” expression for w will

finally take the form
r= Mo | but R fie dlog (1. +m) |
“o=0

c Bes (OER ete
$3n[ e—B4 loges—log (7 +70 |] + Mot me (23)

Molecular Thermodynamics. 239

when m is constant (at constant temperature and pressure)
and therefore equal to mo we get

R _—
= My | éxc+ — log (J +m,C) |
Mlo 4
+3ndd g.—R4 tog e,—iog (1+ 70) \] HONOR 0, (20)

which is equation (52) of the first of these papers, from
which the “second approximation ”’ equations were obtained.

It is, of course, not possible to say how, in the most
general case, m will depend on the concentrations of the
various solutes, but an interesting case, of probably very wide
application, may be treated and will at the same time serve
as an illustration.

This is the case where m can be written, with sufficient
approximation, as a series of ascending integral powers of C,
the total solute concentration.

This can be shown to be the case, for instance, when the
various solvent-molecular species behave as perfect solutes
(in the true sense,—not in the sense of the Raoult-van’t
Hoff laws).

Some simple cases have been investigated, but the detail
need not be given here. It will suffice to say that in the
simplest case, for example where the solvent consists of two
molecular species, mj and (2 mo), the one the doublet of the
other, we find that ™ can be expressed as

ifi= Tol 1 + (migC) + nCOC)?],

where the values of @ and 7 depend, of course, on the
proportions in which the two species are present in the pure
solvent, but 7m any case cannot exceed % and 3/5 respectively
(these maxima not being simultaneous).

We may carry this expansion of m, which is formally
convenient for our purpose, to one further term of which
only the order of magnitude will matter,

— =1+4 ATC) + n(imgC)? + EMC)... (30)
0

mm

the last term being, as we shall find, altogether negligible if £
is no greater than about 7}.
Approximating on the assumption that @ and » are of the

| 240 Mr. Bernard Cavanagh on

order of magnitude ;1,, we obtain :

log (= +m) =aC0—JaC?+ 4,0", BI)

= d log (1+ mC) =C(L—4a,C + 3ag0?—}a30%), (32)
a0
where a, =m)(1—8),

dg=mM,"(1—2(0—n) J, ° 5 ° (33)
dz= my 1—3(0—+ &)]
it being clear that, as stated above, & can be neglected

altogether if no greater than about 74.
We get then the equations

| St Fart ROLL — 3a + Jaya] + Cry |
ENS: Su (34)
Aide = ¢,—R|[ log ¢, —aC + fay 7 —3a3C? | +Gs |

\

and thence the successive stages :—

| ov =$,, + RO[1—}ito(1—6) 0 -+.37%7C"] + Gy |
TL. ge | (35)
| ov =; — R[log ¢e;—77,(1—8)C + 4m,?C?] + G, |
and |
: SM =o ROLL 30") + Gye |
Dae 2) es
i =6;—R[loge,—mC]+G, |
\ Sy !
and
\ aa =$y+RC+ Gy |
lay ; | ° e ° . . ° e (37)
an, Tia log és +G, }

‘J. and II. being the first and second approximations
obtained by the preliminary treatment in the previous paper.
Taking } per cent. as the “ probable experimental error,

Molecular Thermodynamues. 241

we get roughly the following upper limits (of total concen-
tration C) for the applicability of the four successive stages
of approximation :

Approximation I.......... = |
55 el et 5 a 2M (38)
een ee ers |
A SO van ee 8 M. J

—that is, considering aqueous solution, and assuming my to be
about 40. The limits would be considerably different, par-
ticularly in the case of III. and IV., if mp were given a very
different value, as can readily be seen.

The practical criterion of a perfect solute in a complex
solvent is now that its behaviour should be expressed by that
one of the above succesive approximations appropriate to the
total-solute concentration of the solution, with the “* general”
terms omitted.

If the assumption of perfect behaviour in the case of a
particular solute be made, an experimental determination of
the quantities mp, 9, 7, ete., can be made and concordance in
several such determinations made upon different solutes
would tend to justify the assumption that perfect behaviour
persisted up to the concentration at which concordance was
found.

According to the concentration reached (with concordance)
some of the quantities my, 0, 7, etc. would then be known
with some approximation (closest in 7m, next in 0, and so on).

Then, on the assumption of “ perfect” behaviour on the
part of the solvent-molecular species—that is, a sufficient
approximation thereto,—these quantities would. suffice to
discover something about the constitution of the solvent.

Thus two solvent-molecular species would be completely
determined by a knowledge of mp alone (that is, the propor-
tions of the two kinds present would be determined),—and
in this case the question of “ perfect”? behaviour of the two
species would not enter. m, and @ wonld suffice to deter-
mine completely three species, or would provide a check if
only two species were present, and so on.

The problem, however, as a practical problem is compli-
cated by considerations which are the subject of a succeeding
paper, viz. the question of solvation and partial solvation of
solutes.

Finally, it is proposed to consider one point with regard

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. R

242 Mr. Bernard Cavanagh on

to the expansions of U and V, the total energy and volume
of the solution. (Fuller consideration is postponed, as this
paper is already rather long.)

In “ perfect”’ solution we have

U = Too, + Drsus
0101 | t (39)
v= =N1Vo1 oh LNsVs
and
a) —— 791901 + 359s . . Q d . (40)

where, according to the usage of the previous paper, Q is
(U+pV), qs 18 (us + pus), etc., and we can write this

Q = Mod co1901 + 3,5. . ¢ : A (41)

>o1go1, however, depends (through ¢ 9, ¢.; ....) upon the
concentrations of solutes present, but has, in the pure
solvent, a limiting value gu depending only on T and p, and

201901 = au + 2Go1 | dey. eee
2 C=0
= gu + SqoiAcy. « 5... Seen

And so
Q == M,gu+ Snsde+ My dq Aco. <+ jae (44)

Dilute now such a solution “infinitely” by adding a large
mass M,’ of pure solvent at the same T and p, for which

QQ’ aa M'gm.
The united heat will be
(Q ar Q’) aa (My 7F Mo gm 2 DNss a Mo>go1Acor:

But since the solution is now “infinitely ” dilute, we shall
have on bringing it to the original temperature and pressure

Q’= (Mo+ My Jam + 2resg: = (Q + Q) — Mo Xo Aco.

In other words, My3¢qo,:Aco, was the heat developed (evolved)
on diluting the Xn, molecules of solute.
That is, there is a heat of dilution of

ih
G2daAcn eh aiten item Ih Bak became (45)

per gram-molecule of solute, in “perfect”? solution in a
““complex”’ solvent. The explanation of this apparent

a ee gee Se Se awa!

se

ths Didi

=

Molecular Thermodynamics. 943

anomaly is that in a “complex” solvent the process of

dilution is not quite so simple as in a “simple” solvent,
being accompanied by a change in the constitution of the
solvent—a reversion, in fact, to the constitution of the pure
solvent. It is, of course, plain that MyXqo:Acoi, or ZquAnor,
is the (“ chemical a} heat evolution accompanying this
reversion (at constant temperature and pressure).

With (44) analogously we have

U = Moum 4 Srstte+ Mode, - . - (46)
V = Moom + 2nsvs + ModvpAcn, - - - (47)

and it may easily be verified from (L1) that

Ur = m2 ae -
(48)
wuw= ph oom
id Op. c

APPENDIX TO SEcTion III.

A simple example, which is essentially merely illustrative,
but may possibly be something more, will serve to make clear
the practical significance of equation (26) concerning the
slight modification of the ‘‘ general ”’ terms.

In section V. of the first of these papers was obtained for
the general term in the case of a dilute solution of a binary
strong electrolyte

M,G'= RM,¢/e32, ... . ... (49)

where g’ depended, in some way, upon a certain “ effective ”
dielectric constant (D), which, at sufficient dilution, would
be that of the pure solvent.

Let us suppose that D depends on ¢, in such a way that ¢’
is a linear function of ¢,

we — Ol -raeis)) a & . s (50)

and also, in the first instance, that this effect of ce, on D is
entirely independent of its effect on the constitution of the
solvent, that is, that the slight change in the latter produced
by the ‘addition of the electr olyte would alone [if produced,
for instance, by some different solute, ¢,, the concentration

R 2

244 Mr. Bernard Cavanagh on

of the electrolyte being made quite small] have but negli-
ible effect on D, then G/ is independent of the constitution
of the solvent (practically), 2. e.

(x8 aeu=0; . ae
C=0

so that,
eG one; -C + acs)

}
G, = @/ = 2Re'e7(l-t eee) es
Gue= Bco1 Goi’ = —ERG,'c,°7(1 + 3ae,) } .

If, on the other hand, we suppose that the effect of ¢, on D
is entirely dependent on the change produced in the consti-
tution of the solvent, and would be fully obtained if the
latter could be brought about in some other way, while c,
was made very small, then we should have

> OG! Oen _ Res? oP |

Oc Oa 06s (53)
= Rostge |

and

ze

l f
{Se dey = Roe | cc toys eB)
C=0 c,=90

so that
Gia G 22d, oere, see (95)

2. é. in this second case,

2
I

cone -| soe dey = Rdo’c2?(1+ sac.)
C=0
G 2 = 8Rdy'c*(1+0e) |. (56)
Gu = oi Gor’ =| S Den dey, = — 5 Rd 6°71 + 3aCs)
C=0

In both cases, of course,

Gs isl G,’,

but they are not the same in the two cases because in the
second case is a constant when 19, n2.... are held con-
stant, but not in the first case. (for the same reaso
>co1Goy’ is also different in the two cases.)
In the second case we note that although the constitution

Molecular Thermodynamics. 945

of the solvent enters into the ‘‘ general’? terms it does so

only through the quantity D, and if this is the dielectric’

constant of the solution in bulk it can be measured and so
determined as a function of c, without considering the corre-
sponding variation in the constitution of the solvent, or the
way in which the latter exerts its effect on D.

If, as is probable at the less extreme dilutions, D is not
the experimental or “‘ bulk” dielectric constant, but a certain
statistical-average quantity of a peculiar kind, then its varia-
tion is, at least partially, not due to a variation in the solvent,
but directly to variation of c, as in the first case above.

Clearly from the preceding it would theoretically be pos-
sible in such a case to determine from comparison of theory
with experiment whether the effect of c;on D was direct or
indirect or in what proportion both, but it might not be
practicable owing to the smallness of the effects to be
measured,

Notes ON THE “ Gipss FUNDAMENTAL RELATION.”

Consider any property 7 of a homogeneous substance or
phase, which is determined in magnitude by the composition
of the phase and the quantity of substance considered.
In view of the homogeneity it must then be proportional to
the quantity of substance when the composition is fixed.

Such properties are (at constant temperature and pressure)
U and V, the total-energy and volume, Q or (U+ pV),
which might be called the “‘reversible heat content,” and
any thermodynamic potential such as entropy, free energy,
Gibbs’ ‘‘ chemical potential,” or Planck’s yy, which may all
be expressed (at constant temperature and pressure) as func-
tions of the quantities M, M,.... of the constituents which
suffice, under the conditions considered, to produce the
phase.

We can show, as Planck does in the case of y, that for
any such property 7, )

T= sm, 07 ° . ° . ° (on)
for if e be some infinitesimal fraction and we remove (ej)
of the first constituent, clearly 7 is diminished by eM ar

1

Removing simultaneously the same fraction of the total
quantity of each constituent we diminish 7z, in all, by

Sei, SF, or e>M kom But, in so doing, we have

246 On Molecular Thermodynamics.

simply removed the fraction e of the whole phase, without
altering its composition, so that 7 must have diminished by
err, that is

whence (57) follows.
When 7 is either Wy, or Gibbs’ “chemical potential,”
whole system of ee in equilibrium can be congienee

together, since then Sai , ete., are the same in every phase.

1
Without actually quoting Huler’s theorem, Planck remarks,
in regard to ae that this relation means that vr, as a function
of M, M,.. , ls homogeneous and of the first degree, though
not, of course, in general linear, and the same remark applies
to our typical property 7.
From (57) we can at once get a more practically useful
relation by differentiating both sides fully :—

OT OT
SME dM, = 257 aM, + 3M, a( or ‘

that is,
OT
=ud (37) = 0...) er

Now equation (97) of Gibbs’ classical paper reduces at

constant temperature and pressure to

>m,du, = 0
and is then simply (58) applied to Gibbs’ ‘“ chemical
potential.”

(58) is therefore referred to in these papers as the
“Gibbs fundamental relation,” but its general applicability
to any property of the type of a (for a single homogeneous
phase) is to be borne in mind.

It is to be observed that while the constituents whose
masses are M,M,.... must be sufficient to produce the
phase under the conditions considered, they need not he all
nepeesary——1hiey need not be the “ general-thermodynamic ”
components. And since also (obviously from the form of
(58)) M,M,.... need not be expressed in the same units we
see that equally valid is the form

End ($7) = 0, ee

the “ molecular-thermodynamic” form, in which Ny Np «
are the numbers present of the various molecular species.

The Caleulation of Centrords. 247

It is an important point in the treatment of the two
problems, “complex solvents” and “ solvation,” presented
in this and a succeeding paper, that in the “ practical” or
“experimental” expression for w finally obtained the
“linear terms” by themselves satisfy the Gibbs fundamental
relation, for in perfect solution these terms alone remain.
And this is preserved in the successive approximations.
The relation also serves as a useful check upon the correct-
ness of the detail.

Balliol College,
March 1922.

XVII. The Calculation of Centroids. By J. G. Gray, D.Sc.,
Cargill Professor of Applied Physics in the University of
Glasgow *.

| Bae position of the centroid of a plane are or area is

usually determined by the application of one or other
of the two theorems of Pappus. The methods described and
illustrated below seem, however, to be novel ; they are useful
in a great number of cases, including many to which the
theorems of Pappus do not apply.

Fig. 1. | Fig. 2.

0

A BP

Consider a system made up of two masses M and m
(fig. 1). Let the centroids of m and of the system M and m
be at a and G respectively. Now suppose the mass m moved
so that its centroid is brought to a’. G moves to G’, where
GG! is parallel to aa’; and we have (M+m) GG’= maa’.

As a first example, consider the case of a circular are AB
mass per unit of length m (say). Let O be.the centre of the
circle of which the are forms part. Now suppose the are

* Communicated by the Author.

248 Prof. J. G. Gray on the

rotated about O in its own plane through a small angle @,
so that A is brought to A’ and B to B’. In effect the small
portion AA! of the arc is transferred from one end of the are
to the other. The mass of this element is mr, and it has
been translated (virtually) through the distance 27 sin a/2,
where « is the angle AOB. If G is the centroid of the arc,
we have obviously

mré 2r sin «/2 = mra OG 8,

or : sake
oe = 2r = [2

As a second example we take the case of a sector of a
circle OADC (fig. 3). Let the sector be turned in its own
plane through a small angle @ about an axis through O, so
that A comes to A’, B to B’. The effect of the rotation has
been (virtually) to transfer the triangle OAA' to OOC’.

Fig. 3. Fig. 4.

/

The centroid G of the sector has moved parailel to gg

through a distance OG 0. The mass of the sector is 47x0,
and that of the triangle OAA' is 47°6c, where a is the mass

of the sector per unit area. Since gq’= rr sing, we have

1 eee eae eee
gP Ooarsing = 5 rac O0G8,
ae 4 sin a/2

OG= 5 5

Again, let it be required to find the position of the centroid
of a segment of a circle ABC (fig. 4). The segment is
turned in its own plane, about O, through a small angle 0.
A is thus brought to A’ and CV to ©’. If the mass per unit
of area of the segment is o, the mass of the triangle DAA’

Calculation of Centroids. 249

is cOtrsindarsinda, or 4c0r*sin? ta. The area of the
segment is $7’«—7? sin «cos 4a, and its mass is

or*(4a—sin $e cos $e).
Si rae eee ae ]
ince gg’ is #7 sin $a, we have
or (3a—sin $a cos $a)v0 = 2.067% sin’ 4a,
or
2

vT=s

3 3%4—sin $a cos ta’

gin? L
rsin® 5a

where z is the distance of the centroid of the segment from O.

Consider next the solids obtained by dividing a right
circular cylinder into two parts by means of the plane abcd
(fig. 5). Let it be required to find the position of the

centroid of the lower solid. We suppose the solid rotated
through a small angle @ about the axis OO! (the axis of |
figure of the complete cylinder); ais thus brought to a’,
bto b',ctoc’,andd tod’. In effect the wedge ebb'e'cc' has
been removed from the solid and replaced in the position
eaa'edd’. If AA denotes an element of area in abcd ata
distance z from ee’, the volume swept out by this element
due to the turning of the solid is AAaw@. The mass of this
element of volume is pAA2@, and since the element is moved
(virtually) through a distance 2, we have, if V is the volume
of the solid, Le 7
VpO0GO = 298 > AAz’,

where the summation is made over the complete area abed.
Hence

Vx OG = AK?,
where A is the area abcd, and K is its radius of gyration

250 Prof. J. G. Gray on the

about ee’. If « is the angle bOa, and / the length of the
solid, we have for the sectional area

47°(27 —a) +7? sin $4 cos 4a.

And since A = 2/rsin 4a, and K? = 47’ sin? 4a, we have

Ge 2r sin? 4a
a—ta+sin $a cos $a"

Similarly, if G' is the centroid of the upper solid, we have

oc $r sin’

ta—sin } ta cos 4 da!

The positions of the centroids of the solids obtained by
dividing a right circular cylinder into two portions by means
of planes Oa, Ob (fig. 6) are easily determined. If G is the

centroid of the larger solid, we obtain at once, by supposing

Fig. 6. Fig. 7.

-—
-— ~

the solid turned through a small angle @ about the axis of
figure of the complete “cylinder, so that a arrives at a’, and
b at bY,

Vx0OG = 2AK?sin de,

where A is the area represented by Oa, and K is its radius
of gyration about the axis of figure. We have, if / is the
length of the cylinder,

‘_ 2lrdr’? sin $a
his 47?(Qar —a)l

4 rsin ta

3, Oe.

[f the cylinder is divided into two equal parts, we obtain

Calculation of Centroids. 2901

the position of the centroid of each by putting «=7. Thus

For the portion of a sphere shown by the firm lines of
figure 7, we have, if OG is the distance of the centroid from
the centre of the complete sphere,

AK?

0G = =

where A is the area abcd, and K is its radius of gyration
about its diameter. Thus

mr sin? a4? sin? da
4r°(1+ cos 4a) X 2rr
3 r

Ty SE Gos ca

OG =

For the portion of a sphere enclosed by the firm lines of
figure 8 we have

2 le gin L
i Slt

2 Qa
3 wrgin da
~& Yor—a
where wz is the distance of the centroid from OO’. For a
hemisphere «=77, and 5
= 8 / ae

Finally, for the portion of an anchor ring shown in figure 9
we have, if ris the radius of the ring and a the radius of
the section,

Gos: ma? (ta? +77) 2 sin ta

2a —o

*°

Ta xX 2arx

2 _ 2(7r?+ 4a’) sin $a

r (27 — a) :
which reduces to 2r/m when «= and a is very small in
comparison with 7.

If a body is floating partly immersed in a liquid, the

252 Messrs. Trivelli and Righter on Silberstein’ s

distance of the centre of buoyancy B (the centroid of
the displaced liquid) from the metacentre M is given by
BM = AK’/V, where V is the volume of the displaced liquid
and AK? is the moment of inertia of the water-line area
about the intersection of the wedges of emersion and im-
mersion. The equilibrium of the floating body is stable if

Fig. 9.

M is above G, the centroid of the floating body, that is
if BM> BG; it is unstable if BM <BG;; and it is neutral if
BM=BG, that is when M coincides with G. In figure 5 we
may suppose the complete cylinder floating in water so that
abed is the water-line area. The cylinder is obviously in
neutral equilibrium so far as turning about the axis OO’ is
concerned. ‘Thus M lies in the line OO’, and we have

BIS

OOS:

University of Glasgow,
_ May 1, 1922.

XVIII. Preliminary Investigations on Silberstein’s Quantum
Theory of Photographic Exposure. By A. P. H. TRIvELLI
and LL. RigHrTEr *,

| Introductory.
fie ae paper is the first of a number of investigations

now being conducted in this laboratory to test experi-

mentally the light-quantum theory of photographic exposure
recently proposed by Dr. Silberstein before the Toronto
meeting of the American Physical Society.

* Communication No. 141 from the Research Laboratory of the
Kastman Kodak Company.

Quantum Theory of Photographic Kxposure. 253

Originally, these experiments were started independently
of that theory, our intention being primarily to study the
effect of the clumping, or clustering together in groups, of
silver halide grains ina photographic emulsion *. According
to Slade and Higson+, ‘lt seems reasonable to assume
that each grain acts quite independently and that one grain
which has become developable is unable to make a grain,
situated in close proximity to it, developable unless the
latter grain is developable in itself.” From the table
(p. 256) it is readily seen that this statement is not true, but
the grains when clumped together act as one grain for
development to the limit.

Since each clump acts as one grain a very much broader
range of grain sizes, or their equivalent, is obtained extending
from the smallest single grains up to the largest clumps
(containing 30 or more grains) in a given emulsion.

Tt was thought that these results afforded a rigorous test
for Silberstein’s theory, and it seemed therefore worth while
to compare them with the implications of that theory.

Silberstein’s fundamental formula is essentially, 7. e. apart
from chromatic complications, and disregarding the lateral
dimensions of the light quanta,

where N is the original number, per unit of the p'ate, of the
class of grains of size (area) a, n the number of incident
light-quanta, again per unit area of plate, and & the number
of grains affected {. If the finite cross section o of the
light-quantum is taken into account, a is to be replaced by

d'=al1—a/2]’.

At first, the rapid increase of & with the size a as required
by that formula seemed (qualitatively speaking) not only
attained but even exceeded by the experimental results.
This seemed to us to indicate that the sizes (areas) of the
clumps of grains were under-estimated by us. In fact, for

* Extensive experiments are being conducted in this laboratory to
study the clumping of grains in different concentrations of the same
emulsion. These investigations will be published later.

t Slade and Higson, “ Photochemical Investigations of the Photo-
graphic Plate.” Proc. Roy. Soc. xeviii. p. 154 (1920).

t Silberstein, L., “Quantum Theory of Photographic Exposure,”
infra, p. 257.

254 Messrs. Trivelli and Righter on Silberstein’s

an estimate of the areas of all clumps one and the same
average grain size (area) had been assumed throughout.
Upon recalculating the results, however, and assigning the
correct average grain size (area) to the single grains and to
the different clumps, the very interesting fact was observed
that the average grain size (area) increases from the single
grains to the clumps of two, three, etc. The corrected results
conform, even better than expected, to the above formula,
with a finite o.

Heperimental.

These experiments, although of an extremely tedious and
trying nature, were performed with the utmost care and, te
the best of our knowledge, ali sources of error were either
eliminated or reduced to a ininimum. Only a brief descrip-
tion of the experimental procedure will be given at this
time, as a more extensive paper containing further experi-
mental results is to be published in the near future with a
detailed account of our methods of photomicrography, and
in which all errors will be discussed fully.

A simple silver bromide emulsion was used for these
experiments having a speed of 112 and gamma 0°8 for six
minutes development in an ordinary pyro-soda developer
at 17° ©. The average size of the grain is about 0-9
diameter, :

The method of preparation of strips for sensitometric
exposure is, briefly,as follows :—One 5 in. x 7 in. plate of the
original emulsion is soaked in distilled water for one half-hour
at 0° C. to 8° C. (AIL work for sensitometric exposure was
done in a dark room by the aid of a dull red safelight,
Wratten Series 2). The water is removed and a warm
solution of gelatine, alcohol, and water is added and the
whole solution heated in an oven for 20 minutes, while care
is being taken not to heat over 40° C., because above this
temperature much fogging takes place. With several such
applications of the aforesaid solution the emulsion is entirely
removed from the plate and the resulting solution made up
to such a volume that it will give one laver of grains upon
coating and drying. Some of the slides are used at once to
get the clump frequency data. Those for exposure are
backed with an opaque substance to prevent reflexions, then
exposed in a sensitometer, developed to gamma infinity with
a pyro-soda developer at 17° C., washed, and the developed
silver removed with a dilute solution of chromic acid and

Quantum Theory of Photographic Lxposure. 255

sulphuric acid. The strips thus obtained contain the unde-
veloped grains, and by taking the difference the number of
developed grains is calculated.

The data given below cover only the first or highest
density step of a Hurter and Driffield sensitometric strip *.
20 fields on each of 3 strips are employed to determine the
developed grains. To determine the number of grains and
clumps in the original one grain layer plate before sensi-
tometric exposure and development, 10 fields on each of
4 strips were used. By taking this large number of fields
on several strips we obtained a much better average. ‘The
results in both the above cases are reduced to a number of
grains or clumps per square centimetre of one grain layer
plate. Then as the dilution is known, one may, with certain
restrictions, refer back to the original plate.

All photomicrographs were made at a magnification of
2500 diameters and these negatives enlarged 4 times in
printing. On the prints the grains and clumps are measured
and counted, and then classified in class sizes (areas). The
class sizes (areas) are 0 to 0°2 w?, 0°2 to 0°4 p?, 0°4 to 0°6 p?,
etc. ‘The light source is a point source from a Pointolite
lamp which is screened with a Wratten (H) blue filter to
restrict the wave-length range and therefore increase the
resolving power of the microscope. A cell containing
copper sulphate solution absorbs heat rays and a cell con-
taining a solution of quinine bisulphate excludes the ultra-
violet light. The optical system is built up as follows:
Cedar oil immersion condenser and objective, aplanatic
condenser of numerical aperture 1°4, and Bausch and Lomb
objective 1°9 mm. numerical aperture 1°3 in combination
with a No. 6 compensating ocular.

In the following is a table of our results. Column 1
contains the number of grains in each clump, column 2 the
average area of yvrains in corresponding clumps, column 3
the number of grains times 107% per square centimetre of
original one-grain layer plate, column 4 the number of
grains times 107? per square centimetre of developed one-
grain layer plate, and column 5 gives the proportionate

number = of clumps affected.

* The five remaining steps each corresponding to one-half the exposure
of the preceding are now being counted and mapped out, and we hope
to be able to publish the results obtained with them in the course of
one or two months,

256 Quantum Theory of Photographic Exposure.

Number of N. 107" per &. 107° per
ents a in p. sq. cm. 1 grain | sq. em. 1 grain k
in Clump. layer plate layer plate N°
(original). (developed).
1a 0-754 6577-0 1086'0 0°165
Dae We 1-92° 1322-0 5940 0449 |
Ae ese 3°03 664°0 508°6 0-766
BP eds shee ae 4°83! 3280 286°3 0°3871
BN ot oa 18 se 249°9 240-0 0:960
Se 743 f Unreliable 157°8 155-0 0-982
he ane 86 105°2 1052 1:000
Sine Gait ee 9°8 52'6 52°6 1-000
alae oc oe 11-0 52°6 52°6 . 1000
MO Ber 12:0 39°5 39°3 1-000
MOB ong Ss a oe) 26°3 26°3 1-000
Me Reisas uae e 39° 39°). |} 1-008
Perera ive | ie 19:8 19°8 1-000
Ceara fee 19°8 19°83 1-000
PO ars gee ve 6°5 6°6 1-000
LO Reta He 13-2 13:2 1:000
Wi pias ene 13°2 13°2 1-000
LSet aes 13-2 6°5 1:000
1 i a ea 6°5 33 1-000
OU Lt ra ae | 33 66 1:000
Ly Vay gaan 6°6 13°2 1-000
LOANS Een) 13-2 33 | «090
Pos a Pea 3°3 13 1-000
Oe sean 13 6°6 1-000
Dey st eee | 66 1:3 1.000
TBR EAA Ce| cB 13 0 1-000
2G Riche at eat 0 39 1-000
TCT aA Hs 33 0 1-000
OO aks ae < EME 0 0 1-000
OO ou Boece ao 0 1:3 1-600
ail ae a Bat 13 13 1-000
BO hail 24-0 1°3 1:3 1-000
Ges aec anes | 25° 13 Ho 1-000

The agreement of the numbers of the last column with
those calculated by Silberstein’s formula (with N=0°572
and o=0:0973), as given in his paper, is manifestly a very
pronounced one. The differences between the observed and
calculated values are even in the case of the 3 grain clumps
entirely within the limits of experimental error, particularly
concerning the area measurements.

In continuance of this work the theoretical formula will
be subject to further tests, the results of which will be
published shortly.

Rochester, N.Y.
January 24, 1922.

XIX. Quantum Theory of Photographic Exposure.
By L. Sirperstew, Ph.D.*

r. HE purpose of the present paper is to describe a first

attempt at a light-quantum theory of photographic
exposure, or of the production of the so-called latent image,
with the immediate consequences of such a theory and some
of its experimental tests.

The silver-halide grains of an emulsion spread over a plate
or a film base may be considered (apart from the smallest
grains) as small flat plates, of comparatively small thickness,
which in a dry emulsion lie almost parallel to the base. The
sizes a (areas) of these plates range from submicroscopic ones
up to 18 or 20 square microns. Hven the most uniform
emulsions obtainable in practice consist of grains of different
sizes, the distribution of sizes among the grains being in
each case characterized by what is technically called “the
frequency curve” of an emulsion. In what follows the
number, per unit area of the photographic plate, of grains
whose areas range from ato a+da will be denoted by /(a)da,
and a photographic emulsion will be shortly referred to as
being of the type f(a). For certain emulsions f(a) is, with
good approximation, an exponential, for others a Gaussian
error-function of the area a, and so on.

2. Without, for the present, dwelling any longer upon
details of this kind, we may pass at once to our main
subject.

According to Hinstein’s well-known hypothesis of 1905
light does not consist in a continucus distribution of energy, as
in the classical theory, but is entirely split up into light quanta
or discrete parcels of very concentrated monochromatic light,
each parcel containing a quantum of energy, hv=Ad/c,
in obvious symbols. Somewhat more generally we may
assume that enly a fraction aH of the total light energy
FE is thus split into concentrated parcels, the remainder
Ey=(1—a«)H being distributed continuously f, without
however prejudicing the possibility of Hy) being zero.
Then, if EL be the energetic value of a monochromatic

* Paper read December 28, 1921, at the Toronto meeting of the
American Physical Society in affiliation with Section B of the American
Association for the Advancement of Science. Communication No. 139
from the Research Laboratory of the Eastman Kodak Company.

_ + Somewhat as in E. Marx’s theory of “ concentration places” or of
“light specks” as suggested by Sir J. J. Thomson.

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. 8

258 Dr. L. Silberstein on a Quantum

“exposure ” of wave-length , the number of light-quanta
contained in it will be

al
Melee ee 6 (t

From a recent conversation with Einstein, there are
weighty reasons for making «=1. But since we do not
prejudice its value, there is no harm in retaining this
coefficient in the formula.

Now, let us assume that the necessary and sufficient con-
dition for a silver-halide grain to be affected, 1. e., to be made
developable (entirely or in part) is that it should absorb one
leght-quantum.

Moreover, let us assume that a grain *~ does absorb a light
quantum whenever it is fully hit by one, of a sufficiently
high frequency vy, or of a wave-length not exceeding a
certain value Ag.

There are perhaps some experimental hints or more or less good
reasons for making these two assumptions, but we need not stop to
consider them here. It will be time to reject or to modify them when
they are contradicted by photographic experiments. Nor is it necessary
to enter here upon the mechanism by means of which a silver-halide
grain is affected by a light-quantum, whether it be the knocking out of
an electron, as suggested by Joly, or something entirely different. For
none of such details will influence our main argument, to be treated in
the next section. Only when we come to consider the dependence of the
photographic effect upon the wave-length will it be interesting to con-
sider the photoelectric hypothesis and necessary to take account of the
fact that a photo-electron is not liberated unless the frequency exceeds
a certain, the so-called critical value. Under these circumstances
Ae appearing in our second assumption will stand for the critical
wave-length as known from Photo-Electricity.

Again, whether a grain being “ affected” is made developable in part
only or throughout its whole area (no matter how large) is, in view of
the kind of the contemplated experimental tests, of no great importance.
As a matter of fact, however, there is good evidence that a grain is
always made developable as a whole, no matter what its size, and this
seems even to hold for “clumps” or aggregates of several smaller
grains, as will be explained hereafter. If so, then our formule, to be
developed presently, will give not merely the number of affected grains
but also, by integration, the total ‘‘ mass” made developable and hence
also the photographic density. But we may as well remain content with ~
the formule for the number (%) of grains affected, and count these in all
experimental tests. This, far from being a disadvantage, will enable us
to subject the proposed theory to more precise, though at the same time
more severe tests.

One more remark. It will be understood that when we come to adopt
the photo-electric hypothesis, a grain “affected ” will stand for a grain

* Or perhaps, more generally, one of every p grains hit, where p
is a number to be determined by experiment, but presumably equal
to unity.

Theory of Photographic Exposure. 259

which as a whole has been deprived of even a single electron only.
It need not lose more than one electron in order to be made entirely
developable. According to Professor Joly’s original hypothesis * the
latent image “is built up of ionised atoms or molecules.” In our con-
nexion, this does not mean that for every pair of atoms, say Ag Br,
there is one electron liberated. Since every grain of silver bromide
(as well as of silver chloride) is a crystalline, to wit a simple cubic
space-lattice arrangement of Ag and Br atoms +t, we may as well con-
sider the whole grain as a single molecule. Such a crystalline structure
being hit by a light-quantum and deprived of but a single electron,
may well become susceptible throughout to the subsequent action of
a developer.

3. With the assumptions just made the question is reduced
to a mere problem in probabilities.

Consider first the ideal case of equal grains. Let there
be upon an area S of the photographic plate (say unit area)
NV grains. Let a be the size (area) of each of them divided
by S, and let » light-quanta impinge upon S, due allowance
having been made for those which may be reflected or
absorbed by the gelatine. The problem consists in finding
the number & of grains hit and (if p=1) affected by this
light exposure.

Roughly, this simple problem can be treated as if NV, k
were continuous quantities, in the following way, familiar
from many other instances (‘“ mass-law”?). At any stage
the number of unaffected grains is N—k, representing an
available fraction a(N—k) of the total area S. Thus, if
further dn light-quanta be thrown upon S, and if their trans-
versal dimensions be negligible as compared with those of
the grains f, the corresponding increment of k will be

dk=a(N—k) dn,
and since k=0 for n=0, this gives at once
ee Ome yet casi saat ote. a Gel)

which in fact will presently appear to be correct enough
except for the practically unimportant cases of small NV or

small VW—k.
More rigorously, but provided always that V at least is a
large number §, the required formula can be obtained by the

* Nature, 1905, p. 308.

+ R. B. Wilsey, Phil. Mag. xlii. p. 262 (1921).

{ An assumption which will be given up in the sequel.

§ Which will be the case if S is taken large enough. Since plates
and films in actual use contain as many as 10° grains per cm.’, S can be
made as smal] as one-hundredth mm.’, and even less.

S 2

260 Dr. L. Silberstein on a Quantum

following reasoning. The total area of silver halide being
a fraction Na of the area S of the plate, each of the nNa
light-quanta will fall upon some grain. Of these nNa

N
N—-1
will hit

quanta one, say the first, will hit one grain, the next

quanta will fall upon another grain, the next Wo
yet another, and so on, up to Wopy oan for the kth
grain *, Thus the required relation between & and n
will be

1 1 1

iL
INT NL ee oe
Now by a well-known theorem of Analysis

feed 1
lag dah eh See leg mee elm) ets eRe

where C is Euler’s constant and 0 <e(m) =1/m. Thus in
our case

=

N
log w= SSS iG
where € lies between —1/N and 1/(N—k) or practically,

since N is at any rate a large number,

i 1
Whence, |
k= N(1l—e77-S$). A 3 4 ‘ s ‘ (3)

In practicable experimental tests (counts of affected grains
of a given, narrow size-class) the réle of the correction €
whose value €=e(V—k) can at any time be found by (2),
may become perceptible only when the contemplated grain
class is near exhaustion.

Thus, apart from such extreme cases, we have again,
as in (A), the simple formula

b= IN Oe ye tai a ee (4)

which, though approximate only, will turn out to be
accurate enough even for moderate values of JV.

A thoroughly rigorous treatment of the probability
problem valid for any numbers JV, n, seems to be the

* It is scarcely necessary to say that statements such as “ W/N—1
quanta hit another grain” are to be taken statistically as relating to
averages over a group of many trials.

Theory of Photographic Exposure. 3 261

following one. Of the n quanta the total number falling
each upon some of the WV grains will be *

n= ING, G,

Va being the total area of silver halide, with S as unit area.
Thus the problem is reduced to finding the distribution of m
quanta among WV grains. Now, let pn(z) be the probability
of affecting, in a single trial, a number 7 of the WV grains, by
the m quanta. By a combinatorial discussion which may
be omitted here, I find
7 ’ N! Ami
Plt) ae N™(N—?) ee : ° : : 4 (5)

where a»; may be most shortly described as identical with
the number of ways in which a product of m different
primes can be decomposed into 2 factors. These numbers,
which will be known to many readers from combinatorial
algebra, have the obvious properties

iby gic 4. LOL. any: 1,
&,2=0 fer +> m,

and satisfy the general recurrency formula
Omi =Am—1,i-1 + Wri, iy

which enables us to write down successively without trouble
any number of them.
Thus, up to m=10, we have the following table which the reader

may continue to extend at his leisure. Columns correspond to con-
stant m, and rows to constant 2.

1 1 E iE 1 1 i 1 it 1
0 1 3 ii 15 ol 63 127 259 511
0 1 65-525 90 301 9686 3025 9330

0 iL 10 65 350 1701 7770 34105

0 i 15 140 2401 138706 76300

0 i 21 266 3997 37688

0 1 28 462 7231

0 1 oo, ':):) FeO

0 1 45

0 ]

We way mention in passing that any am; can be represented by +

imi | o— (1) @—D™+ (5) G—2ym—......+ (3) J.

But for any numerical applications the table will be found more
convenient.

* In a large number of trials of the same experiment.
+ Cf. e. g. Lehrbuch d. Kombinatorik by E. Netto, Leipsig, 1901,
p. 170.

262 Dr. L. Silberstein on a Quantum

By formula (5) we have for instance the probability of
hitting but one grain

1
Aye Nm-1?

which is an obvious result. The probability of hitting two
grains will be

pn 2) =e — a yee

which is (approximately) 2”V times as largeas the preceding
probability, and so on. If m be kept constant, pn»(i) will
increase with growing 7 up to its largest value for 1=m,
if m=, or fori=N, if m> NV. Butit would be futile to
expect, on an average, that distribution to which corresponds
the greatest probability *. For all other distributions have
some generally non-negligible probabilities and these are by
no means symmetrically spaced with respect to the largest
one. The only reasonable way of determining the number k
is to define it as the average of 7 taken over a large number
M of trials. Out of these M trials a number M »p,,(2) f of
trials will give each i grains affected, and the total number
of grains affected in all M trials will be 2Mzpn(2), to be
summed fromz=1 toz=m, if m= Nand tor=Nif m> NV;
but since a,,=0 fori >m, we can as well extend the sum
in each case from 1=1 toi= J. |

Dividing this sum by M we shall have the average number
of grains hit in one trial, 7. e., by (5)

INA N ean
= nae ao oe (6)
This, with m=Nan, is the required rigorous formula for
the number of grains affected, 7. e., hit once or more. In
order to see how this complicated formula degenerates into
(4), which, of course, will be our working formula, develop
the sum in (6). Collecting the terms in V4, WV? etc., and
taking account of the values of ami, it will be found that

ee a 1 es (- ls jie
ees N 2 iP WN? 3 —.eet N y)

* Tf, say, m=, the most probable distribution is the equipartition
as
m) !
would give as the number of grains affected k=m=WNna, or just the
first term in the series development of (4), which would be hopelessly

wrong unless m WN were very small.
+ With a deviation ruled by Bernoulli’s law.

ke

(a quantum per grain), corresponding to pm(m)= Re

Theory of Photographic Exposure. 263

or, dividing by WN and subtracting from unity,

k m 1 m 1\n
1~ wait (T)(—H) +--+ CR) (— 3)”

2. €. ultimately

hk £4 1 \m F
fu = (1%) hy ee ama
This is rigorously equivalent to or identical with (6) for any
mand N. Now, for large N, and any m, equation (6’) in
which (1—1/N)*=1/e, asymptotically, gives at once
k _m

ee —l—e W=1—e7™,
This is the connexion between the rigorous arithmetical
formula and the exponential one.

It is needless to insist that under the conditions prevailing
in all practicable experimental cases there is more than sufi-
cient mathematical accuracy in formula (4)

h=N(1—e-"),

which, apart from minor modifications, will henceforth be
used in what follows. |

This formula is of the familiar type proposed (1893) by
Elder, with the notable difference, however, that while his
exponent contained a free “‘ parameter ” or coefficient to be
evaluated empirically and principally depending upon “ grain-
sensitivity ’’ and wave-length, both coefficients in (4) are
completely determined, and the exponent moreover shows
an explicit and most essential dependence upon the size (a)
of the grain, and in the right sense too, 7.¢., giving an
increase of the “speed” with grain size. The comparison
with experimental facts of Elder’s and of a number of other
formulz, constructed empirically, is too well-known to be
discussed here *. Suffice it to say that, although it repre-
sents to a certain extent the phetographic behaviour (the
“characteristic ” curve) of some emulsions, and particularly
those with what is termed an “extended toe,” it cer-
tainly shows considerable deviations from the observed
characteristic curves. Yet it will not be forgotten that all
these comparisons bore upon the resultant total photographic
densities, containing or integrating the effects of grains
belonging to a broad range of sizes (a), instead of equal

* Cf. for instance, a paper by Dr. F. E. Ross, Journ. Opt. Soc. Amer.
vol. iv. p. 255 (1920).

264 Dr. L. Silberstein on a Quantum

grains, so that no better agreement could be expected.
The refined experimental tests which are now in progress
in this laboratory, and by means of which it is hoped to
corroborate the proposed theory, deal, as they should, with
separate size-classes of grains. 7
But questions concerning the comparison of the theory

with experiment will be treated in a later part of the present
and in subsequent papers.

4. Passing next te the case of an emulsion of any type
f(a), it can be easily proved that the approximate formula
(4) will hold for each class of grains separately. In order
to see this it is enough to consider the case of two distinct
classes of grains. Thus, let there be NN, grains of size ay
and WV, grains of size a, spread over the (unit) area S of the
plate, and let n light-quanta be thrown upon S. Of these a
number m=nNja, will fall on the a,-grains and a number
mz=nNoa, upon the dg-grains. It remains only to be found
how many a,-grains will be hit by the m, quanta, and how
many of the a,-grains will be hit by the m, quanta. Now
each of these is a problem of the kind we have already
treated. The number /, of a,-grains hit will be given by

1 eee
Dc ee ea eee

and similarly for the a -grains, so that k, and k, will each be
determined by the previous formula for & with JV, a replaced
by Yj, a1, and Ng, ag respectively. Similarly for an emulsion
consisting of three or more classes of grains.

Thus, also, for an emulsion of any type f(a), the number
of grains ranging from a to a+da hit and (if p=1) affected
by n impinging light-quanta will be 4, da, where, apart from

the correction €
kg=((a)[t—e-"). 6 er

The total area * of silver halide affected or made developable
will be found by extending the integral

K-= akede gE faa: 30 as

over the whole range of sizes, say from a, to dy.
If, for instance, f(a)=Ce-**, say from a=ay, to a2, where
C, » are constants, as in the case of some films and plates

= NA,

* It will be kept in mind that a stands for the “ efficient ” area of a

grain (plate), 7.¢. for the orthogonal projection of the grain upon the
film base.

Theory of Photographie Exposure. 265

investigated in this laboratory for their frequency curves,
then, with A written for the total area of silver halide,

K=A— eos { fa + (n+ p)a,le7# tm

(n+)?
—[1+(n+p)a,je"H+},

But this only by way of illustration. The fundamentally
important thing will be formula (7), applicable to each
size-class of grains separately. In fact the experimental
verification of the theory now in progress in this laboratory
deals, not somuch with X but, as it should, microscopically,
with k=k, for each class of grains separately, including
clumps of grains.

Before passing to a further discussion and development of
the elementary formula (7), but one more remark concerning
the presence of more than one layer of grains. The case of
two or more layers will at once be seen to be reducible to
that of a single layer. In fact, either a grain of, say, the
second layer and of size a is not shielded by any of the first
layer or else it is thus screened off and only a part 6 of it
remains uncovered. In the former case the grain in question
will simply be classified among those of size a of. the first
layer, and in the latter case among those of size b. This will
hold with respect to the exposure to the impinging light-
quanta, and 6 will also be the contribution of the grain in
question to the photographic density; for its covered part
will remain inoperative. Similarly for three or more layers.
In fine, the presence of a plurality of layers of grains will
modify only the frequency curve N,=/(a) which would
otherwise belong to a single layer. We shall henceforth
assume that this factor has already been taken into account
in constructing the function f(a) or in microscopic counts of
the grains within every particular size-class. We disregard
bere, of course, such factors as a possible absorption of light
in additional strata of gelatine. |

5. Dependence upon wave-length—Once more return to
the elementary formula (7) or (4). Denote by s the ex-
ponent so that

== |] —eg- 5,

N

Under the more or less implicit assumption that the trans-
versal dimensions of a light-quantum are negligible in
comparison to those of a grain, we had s=na. But it will

266 Dr. L. Silberstein on a Quantum

be seen presently that such an assumption is too narrow and
unnecessarily so.
In fact, substituting the number n of light-auauie from

equation (1), the exponent s* will become s=7 ED, Or, it
we put for brevity

s= a/he, oi eer Ng Sacra ea (9)
which may be considered as a constant,
S= BE. ON.) i.e eae

Here F is the incident light energy (exposure) and A the
wave-length of the light assumed to be monochromatic. |
Thus the sensitivity exponent would be directly proportional
to the wave-length, and the number & would, for constant
Fi and a, increase steadily with the wave-length of the
incident light up to the photoelectric critical value A, and
then drop suddenty to zero,

i=l cae QS xX < oe
k ==(() N> Ne

Now, such a sensitivity curve does not seem to resemble the
familiar experimental sensitivity curves which show a more
or less gentle maximum followed by a gradual decrease
down to zero. It is true that such experimental curves Tf
represent the resultant effect due to grains of a whole range
of sizes, so that the k—) curve belonging to a single class
of (equal) grains may well be of the said abrupt type,—a
question to be decided only by micro-spectrographic experi-
ments and counts now in progress. Yet it seems advisable
even at this stage to provide for the possibility of smooth
maxima preceding the critical wave-length 2..

This might be obtained by attributing to grains of different
sizes different values of X. For then, although the curve
of each grain class would end abruptly, the superposition of
such curves ending over a range of different abscissee, might
properly displace and smooth out the resultant maximum.
The correctness of such an assumption (A, a function of a)
can at any rate be tested by direct experiments f.

* s/H can be referred to as the “ sensitivity exponent.”

' + Apart from the fact that they are not taken for H=const.

{ Preparations for such experiments are now being made 1 in this
laboratory.

Theory of Photographic Exposure. 267

Another way is to take account of the possibly finite trans-
versal dimensions of the light parcels, which may perhaps be
comparable with those of the lesser silver halide grains.

Let us assume, therefore, that a light-quantum, of suffi-
ciently high frequency, becomes efficient in affecting a grain
only if it strikes it fully, or almost so. To fix the ideas, let
r be the equivalent radius of a grain, 7. e., such that

=r",

and similarly let p be the average radius of the transversal
section of a light parcel (so to call the space occupied by a
quantum of energy). Then the efficient area of a grain to be
substituted instead of a, will be

2
a’ =1(r—p,/?=a E e A
es
and we shall have for the exponent s, instead of (10),

s=na =6 La | 4 x. Longo 7, See (10 a)
amd s—() for p> 7 or A> Xe.

It remains to assume in a general way that p, which may
be the average of section-radii different even for parcels of
the same wave-length, is itself a function of the wave-length
increasing with A, without prejudicing, however, the parti-
cular form of this function. Certain easily ascertainable broad
features of such a function and thence also of the resulting
factor in s,

poryarn[1—P]",. Portia BP Ghai)

will suffice to ensure a maximum of the sensitivity exponent
between X=0 and A=A.. The value of A, itself may still
turn out to depend on the size a of the grain and on its
physical conditions as well. Every process which will make
the liberation of a photoelectron trom the grain (crystal)
easier will lengthen A,. Part of the effect of sensitizing
may arise in this manner. But questions of this kind must
necessarily be postponed until some further experimental
data are gathered. Of such a kind is also the question
whether p (which, for a given X, may also extend over a
whole range of values) attains at all the semi-diameter 7 of
even the smallest of the actual grains, and whether the
corresponding wave-length , entailing the vanishing of s,
exceeds or is smaller than 2, as derived from direct photo-
electric experiments. In absence of all knowledge concerning

268 Dr, L. Silberstein on a Quantum

the spatial properties of light-quanta it would be utterly
unjustified either to deny or to assert that their lateral
dimensions are at all comparable with those of a silver
halide grain (of the order of one-tenth up to several
microns) *. If, by way of illustration only, p is propor-
tional to a power of A, say p=dA*, the only condition for the
existence of a maximum of ¢(Q), and therefore of sensitivity,
wil easily be found to be «>0. If this be satisfied, the
maximum will occur at a wave-length 2,, given by

(2e+1)brA, ="

increasing with the diameter of the grain and bearing to Ao
the fixed ratio

Xm _ Ve
x == Zea ue
Asa matter of fact, the maximum sensitivity is known to shift
(by two or three hundred A.U.) towards the red by making
the grain coarser. But thus far too little is known of the
quantitative aspect of such an effect to entitle one to con-
sider the above equation as anything more than an illustrative
example. The precise form of the function p=p({d) can only
be derived from spectrographic experiments followed by
microscopic grain counts, or if arrived at by a guess, has
to be verified by them. Such experiments are now in
preparation in this laboratory, and their results will be
reported in due time. A shift of the maximum sensitivity
towards the red or the infra-red ean, of course, be brought
about by a function form more general than a mere positive
power of the wave-length.

6. Generalities, and preliminary account of experimental —
tests.—The chief and most immediate consequence of the
proposed theory is the essential dependence of the propor-
tionate number of grains affected, k/V, on the size a of the
grain, viz., the rapid increase of the former and, therefore,
of “the speed ” of an emulsion with the latter. Now, it has
been known for a long time that (cvteris paribus) the speed
increases notably with the size of the grain, and we shall
see from the experiments to be described presently to how

* According to E. Marx, Annalen der Physik, xli. pp. 161-190 (1918),
the volume of a light-parcel, which according to him is only a ‘“‘ concen-
tration place ” within a continuous distribution of energy, is proportional
to 4 and amounts for D-light to almost 8.1077 cm.*, which even with
a leneth of 10 cm. (200,000 D-waves) would still give a section area
8.107 3 cm.?, just of the order of about the average grain area. ‘here is,
of course, nothing cogent about Marx’s estimate, yet the matter is not
without interest.

Theory of Photographic Hxposure. 269

large an extent this is actually the case. But perhaps
the most tangible proof of the essential correctness of the
assumption of spatially discrete as against continuous action”,
seems to be the mere fact, disclosed by microscopic counts,
that out of a number of apparently equal grains subjected
to a sufficiently weak exposure one or two are affected while
the others, nay their next neighbours, remain perfectly
intact. It would be in vain to ascribe to these survivors a
greater immunity or indifference to light. For it is enough
to protract the exposure a little to make them succumb in
their turn. Now such a behaviour is most typical of rain
as contrasted with flood action, and the discrete light-quanta,
hitting now this and now that grain, appear to be a most
natural inference, while all attempts to bring into play the
individual “sensitiveness” of the units seem to involve
considerable difficulties.

As to the dependence of the number of grains affected
upon the wave-length, little more of interest in the present
connexion is known than the qualitative fact of a shilt
towards the red of the maximum sensitivity with increasing
size of the grain. Moreover, the available curves repre-
senting the sensitivity across the spectrum concern the
emulsion as a whole and not the separate a-classes of grain
with which we are primarily concerned. Spectrographic
and spectrophotometric experiments of such a kind, to be
aided by direct photo-electric measurements, are now in
progress in this laboratory, and all discussions involving
wave-length will best be postponed until the results of
these experiments and of laborious microscopic counts are
forthcoming.

Before passing to the mentioned quantitative test of the
dependence on size, but one more general remark. ‘The
reader will have noticed the complete absence of the time-
variable in all our formule, the exposure entering only
through the total number n of light-quanta or through the
energy / which, in obvious symbols, is (2 dt. The pro-
posed theory, therefore, as thus far developed, does not take
any account of the little infringements against the reciprocity
law t, in short, of the so-called “failure of the reciprocity
law.” Now, it is by no means my intention to deny the

* A rain as against a flood, of light, that is.

t+ This early law asserted the dependence of the photographie effect
(density) upon J and ¢ only through the total incident energy or ex-

posure { Lat. For constant intensity this is Zt, whence the name of the

law, relating to intensity and exposure-time as factors of a constant
product.

270 Dr. L. Silberstein on a Quantum

reality of these infringements which have been extensively
studied and condensed into empirical formule by Abney,
Schwarzschild, Kron and others. But it has seemed inad-
visable to encumber the very beginnings of the proposed
theory by complicated details of such a kind*.

The failure of the reciprocity law can more profitably be
taken up later on, after the fundamentals of the theory have
been somewhat solidified and extensively tested, and the
prospects of mastering the “failure’’ theoretically are by
no means averse, a very promising scheme seeming to lie
in the possibility (suggested by Joly and taken up by
H. 8S. Allen) of the liberated photo-electrons being regained
by some of the grains which were deprived of them by
previous impacts of light-quanta. In fine, the failure of the
reciprocity law as well as the facts known under the head of
“reversal” have at first to be neglected and considered
as future problems for the light-quantum theory united with
Joly’s photo-electric theory, problems to which these com-
bined theories seem well equal.

To pass to numerical facts, a short description will now be
given of the results of certain experiments undertaken in
this laboratory by A. P. H. Trivelli and Lester Righter f
which seem to corroborate the proposed theory most
emphatically. In order to have a much wider range of
sizes a than is usually afforded by the single grains,
Trivelli and Righter applied their counts and area measure-
ments to clumps of from one to as many as 33 grains,
basing themselves upon the well-supported assumption that
if one of the component grains be affected, the whole clump
is made developable. (This, at any rate, is the behaviour

* R. E. Slade and G. I. Higson, “‘ Photochemical Investigations of the
Photographic Plate,” Proc. Roy. Soc. xeviii. pp. 154-170 (1920), on the
contrary, make the failure of the reciprocity law their point of departure.
They mention at the very beginning (p. 156) the possibility of a light-
quantum theory and write the Elder-type of formula in J, ¢, remarking
even that its coefficient would have a different value for each size of
erain, but being discouraged by or rather preoccupied with the failure
of the reciprocity law do not enter into the details of the probability
problem, which would have disclosed them the structure of that co-
efficient, and without much ado dismiss the quantum theory as
“impossible.” Independently of Slade and Higson the possibility of
a discrete theory (radiation in “ filaments”) is mentioned by I. EK. Ross,
Astrophys. Journ. vol. lii. p. 95 (1920). Dr. Ross, without being
prejudiced against such a theory, notes even that it would lead ration-
ally to a mass-law, but does not enter into the details of the probability
problem and does not develop the theory.

+ For technical details of these laborious experiments, see Trivelli and
Righter’s own note in this issue of the Phil. Mag. p. 252.

Theory of Photographic Exposure. 271

of the larger, flat grains piled upon each other in part,
although the smaller, spherical grains, in less intimate
contact, may perhaps behave differently.) Such being the
ease, their experimental results should be covered by our
formula with a written for the area of the whole clump, no
matter how large and how numerous its components. This
has seemed a rather severe test but the more so tempting and
instructive. Since all the classes of clumps were given, in
each trial, a unique exposure (through a blue filter specified
loc. cit.) and there was no question of varying 2, it will be
most convenient to retain in the corresponding formula the
original light- quantum number n as the parameter common
io all clumps. Thus the formula to be tested becomes

h 2
| eal =nal =na(1— p ;
or somewhat more conveniently for computations, if c=’
be the (average) area of the transversal section of a light

parcel,
k a}?
log (1-4) =-ra [14/2] . eis (12)

In the following table the first column gives the number
of grains in a clump, the second the average area a of a
clump in square microns, and the third column the per-
centage of clumps affected out of all (JV) clumps of each
kind originally present, 7. e.

100 &
Tee eRe 2
as deduced by Trivelli and Righter from their observations,
Clumps of @ in p?, Cone. Oeics Ay.

PRI GEMIND o.cscsec.-6 0-754 165 16°2 +03
foeraims, fhcti.... 1-925 44:9 48°4 —3'5
(2 ae eee 3°03 76:6 68:9 +8°3
Ca ee A'88 8771 87°3 —02
yl eee 6-18 96-0 93°3 27
By Bee Be ee Ne 742 98°2 96-4 1:8
OSS A is seae (8°6) 100 98:0 2°0
= alpeitiey RSA (9°) 100 99:6 0-4
Des eet grt’) 100 99°38 Or2
‘pa ah Tet (12° ) 100 100 0:0
ete., ete, etc., ete. idem, idem. idem.
A PTAA | cope oan >24 100 100 0:0
<eeeh iiane ee >25 100 100 0:0

The most reliable a-values are those for the clumps of one
and of two grains, being averages of the largest numbers

272 Dr. L. Silberstein on a Quantum

of individual clumps ; the following a’s are vradually less
reliable ; from the 7-grain clumps onward the areas
(bracketed) are only extrapolated, but since here y has
practically reached 100, no greater accuracy is needed.
The fourth column contains the theoretical values of y
following from formula (12) with the constants, determined
from observations 1, 2, 4,

n=0°5724 per a
c= 0091 ae

or p=0'176u. The fifth eolumn gives Ay= ose venee
The agreement is certainly very pronounced, the differences
being, perhaps with the exception of the third, well within
the limits of experimental error chiefly in the a-estimates.
The fitting could be made even closer by retouching the
constants n, a, but this is scarcely worth while, the formula
itself being of a statistical nature, and the agreement being
good enough as it stands. The same is manifest also from
the figure giving a graphical representation of the last
columns of the table.

(12 a)

|

1001 E og 2 0 2282008 8-89 8-8 88-8 8 5-886 OO 6
i :
YU 60) — CALCULATED |
4 e OBSERVED

49 3

26

8 5 10 15 20 25

a

The reader might think that the finite section o, or
whatever this parameter may stand for, has been forced
upon the light-quantum and that the observations might
perhaps be as well represented with e=0, and another value
of n. But actually, just the contrary has been the case,
inasmuch as the author first tried the simpler formula
log (1—4/N )=—na, and then only found himself compelled
to take in the correction factor as given in (12). In fact,

Theory of Photographic Exposure. 273

dividing the observed values of log (L—4/NV) by the areas
the reader will find that the quotient increases considerably
and systematically, apart from a casual drop at the fourth
clump, throughout the whole series of the clumps. Thus,
the correction factor seems to come in quite spontaneously.

On the other hand, there is nothing unlikely about the
values of either of the constants (12a). Our units of area
being here square microns, we should have 57 millions light-
quanta per cm.” (about which judgment has to be suspended
until absolute energy measurements are available), and as
the cross section of the space occupied by each of them
(on an average) a little less than one-tenth of a square
micron or a diameter of about 0°35 micron. Since each is
presumably of about the order of a million wave-length
long, they are rather slender at that cross section, and,
instead of light parcels, as they were called above, would
perhaps deserve rather the name of light darts. In Hinstein’s
own theory there is nothing on which to base an estimate of
the volume occupied by a light-quantum, but on Marx’s
less radical views this is about 8.10~-‘ em.® for D-light and
proportional to A4, and therefore in our case (narrow blue
spectrum region with maximum at A=0°470u) about
3.10-* em.’, which with the said cross section would mean
a length of 3.10% or over six million wave-lengths. But
this by the way only. The important thing is to see whether
the above numerical value of the average cross section of
blue light darts will continue to fulfil its function with regard
to the remaining “‘steps’? (weaker and weaker exposures)
of plates coated with the same emulsion, the above being the
highest “step.” These have just been completed in this
laboratory, ceteris paribus with the above one, and are now
being subjected to counts and area measurements. ‘This
material will also serve to test the constancy of k/N if,
varying n and a’, their product is kept constant.

An account of the results of these and of several other
experiments now in progress will be given in future papers.

I gladly take this opportunity to express my best thanks
to Dr. C. i. K. Mees for having proposed to me the problem
of “discriminating, if possible, between the consequences of
a diserete and a continuous exposure theory,” and to my col-
leagues Trivelli and Righter for furnishing the results of
their experiments.

Rochester, N.Y.,

January 19, 1922.

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. dh

ya

XX. An Analytical Discrimination of Elastic Stresses in an
Isotropic Body. By Rh. F. Gwytuer, M.A.*

Gy G. B, Ary obtained from mechanical considerations

(British Association Reports, Cambridge, 1862) a solu-
tion in Cartesian coordinates of the mechanical stress-
equations, but he ignored all elastic requirements. In one
sense this paper may be regarded as an extension of Airy’s
scheme, though it has nothing in common with that scheme
either in general plan or detail.

The method is purely analytical, depending upon general
solutions of the mechanical stress-equations and upon the
development of a scheme for the selection of those stress-
systems which satisfy the stress-strain relations, briefly
called Hooke’s law, from the general mechanical stress-
systems.

It is shown that the elements of a mechanical stress
depend upon an arbitrary primary stress-system, and, to
form a connexion with the stress-strain relations, I introduce
a subsidiary, but allied, stress-system which is such that the
vector system naturally deduced from it possesses inherent
qualities distinctive of the displacement conrespondie to an
elastic stress-system.

The main body of the paper consists in dhysteene the
requirements necessary to ensure that the stress-system
should be an elastic stress-system.

The displacement, according to this method, becomes
somewhat incidental, however necessary, and the elements
of stress are given prominence. ‘There are no displacement
equations.

“In the first instance I deal with a body under tractions
only, and extend the scope of the results later.

I. INTRODUCTION.

1. By first treating certain ancillary matters as lemmas,
the steps in the final stages can progress more
steadily.

The mechanical equations of stress in a body under

* Communicated by the Author.

Analytical Discrimination of Elastic Stresses. 275

tractions only are, in Cartesian coordinates,

Obeseu - ho
Oey eee Ot
Saar Cee
aU 3, 38 _
Sc a
or oS. oR
— SSE) ie nce ee
and they are identically satisfied by the values
be 0° 0°; | 9 OV
a7 a2 Oy? **ayoe
Q=- 078) 0°03 ae Cae
iets te 037 Oxo2
0°O, 078; Obs
+ eee Oa +30
oa ow Oh ola"
Oy 0¢ Ox" Ovoy Or02
pa 020, * Ob; 4 or; ai O23
0202 ez0z 8OY 0y02’

070s Oy O's

Ordy Oxdz OyYdZ O27"

These contain six arbitrary an1 general functions, and
form the general solution of (1).

Lemma A.

These six arbitrary functions have the same mode of
resolution on transformation of axes as the elements of
stress.

For proof I use the method employed in the subject
of differential-invariants.

Imagine the axes of coordinates to be rotated about their
own positions by the infinitesimal amounts @,, w,, w,, and
consider the consequent changes in whatever quantities
we may be considering—components of a vector, elements
of a stress, etc. These changes will be linear functions of
Wr, Wy, Wz.

T 2

276 Mr. R. F. Gwyther on an Analytical Discrimination

For example, in the case of the components of a vector
u, v; w, the changes will be :

aa —Oyw+ozv,
in’ v, —O,U+o,W,
In WwW, —@,0+@y,l.

These changes will now be represented by partial dif-
ferential-operators ©,, Q,, QO; arranged to produce the ~
coefficients of @,, @, @, in the quantities considered
respectively.

_ Thus, for vectors generally,

aL fo)
QO, ee a
Oo os we,
S

NOs a ee Sie

This covers direction-cosines (/, m,n), and may be made
the basis of most operators. For example, we may deduce
for stresses generally :

40) if r) Oe
28(s—-sn)+®-O SUS 47 9,

oO
R
Vian eintig aig aes
2, = 21(S.—yp)+@-R) sy aU oe
aie

|

OQ,

oT UES Ls tbe! ce. On e
QO, 20 ( a oO P) sagt Se

The differential coefficients ce oO ce) resolve as com-
O® OYy102 5)
ponents of a vector, but, for simplicity, we must introduce a
different notation. — al
Write d; for 0/O2, d, for 0/dy, ds for 3/dz, and d,, for
Ms / 9 ky o 11 TOT
07/02", di. for 07/Ox0y, doo for 07/dy?, and so on; then

for first differential coefficients

ao fe)
O, = Ti ge Od,’ etc.,

of Elastic Stresses in an Isotropic Body. 277

as for vectors, and we deduce for second differential co-
efficients

PMO TMs NOU pig Deus e 3
D1 = dis dia sr, +2 (So 5g) 7 (dose) at
‘cede 3 ee eae.
aga OO Sa, + 2d,3 Soe | (d33 dis

a 0 fo fe 0 0
QO; 7 dos => Odis — Oa Odo a7 2d» (.5--9-}- (dy1— dep) Ody,"

It is required to establish that in (2) 6;, 0, @3, Wi, ro, Ws
act as stresses. Actually I shall assume that the operators
QO), Qe, 3; act upon them as upon the elements of stress,
and first examine this hypothesis when the three operators
separately act on the six equalities in (2).

Selecting the first equality in (2) and applying the
operator ©, on each of the two sides of the equality,

then
Oe = ®) 3
and 20s eo 0°63 tonal
ROY ey etal ONC ees dal
1 32 O, PY ots “Oy Oz
becomes
Or 5 ON , 5 07(Os—F2) 9 0° (0,—63)
9 2
ate ay + 3: RE

On.
EiKg
(3 -32) my
which is null, and the hypothesis is not negated.

In fact, the hypothesis is not negated in any case.

The argument is then as follows :—If we had written
040, 028, ete. in the equations, and had proceeded to find
these 18 quantities, we should have 18 linear equations
from which to find them. The solution is therefore unique,
and cannot differ from that employed by hypothesis and
found to satisfy the 18 tests.

We shall therefore regard 6, 02, @3, Wi, Wo, Wr3 as acting
as elements of a stress, though they have not the proper
dimensions.

It is proposed to find the conditions which must exist
between these primary arbitrary stresses in order that the
elements P, Q, R, 8, T, U may be elements of an elastic
stress.

278 Mr. R. F. Gwyther on an Analytical Discrimination

Note.—If, in the ordinary notation for strains, we give
a, b,c each one-half of the usual value given to it, strains
would follow the same laws of composition and resolution as
stresses, and would therefore have the same differential
operators. In this paper, I shall use a, b, ¢ in this sense—
that is, one-half of their usual.value.

Lemma B.

Hxcept when we have reasons for keeping the expressions
quite general, it will suffice to limit the arbitrary stress-
system to such stresses as have the co-ordinate axes as their

‘principal axes.

If in equation (2) the elements of stress, P, Q, ete.,
are made zero, the set of equations will then be recognized
as indicating that 0,=e, 06,.=f, O:=9, Wi=a, We=), W3=6,
where e, f, g, etc. are elements of strain arising from some
arbitrary displacement.

Hence on the right-hand side of equations (2) we may
always replace 0, by 0;—e, by 0.—f, 03 by 6;—9, Wi by
Via, Wo by we—b, 3 by W3—¢. Consequently we may
eliminate several sets of three functions, such as Wi, We.
and 3, when some displacement is possible which makes,
say, Wi=a, Wr2=, W3=e.

Hence

which is the form given by Airy, is a quite general form of
solution, although for the purpose of this paper the full form
given in (2) is requisite until we have decided upon some
particular set of axes.

2. The choice of a vector to represent the displacement,
and the descriptive criterion of elastic stress.

The mechanical stress has been represented in terms of an
arbitrary stress-system, and it is possible and desirable to
represent the displacement in terms of a similar stress-
system.

For this purpose I form a subsidiary stress-system,

of Elastic Stresses in an Isotropic Body. 279
indicated by

0, eS O,-+ Hinail Eee 0, 4- EAL

—wW, oe ro, — 3.

This subsidiary system may be described as comple-
mentary to the primary stress-system, in the sense that
the two together form a hydrostatic pressure whose intensity
is one-half of the sum of the principal stresses or one-half
of the First Invariant of the primary stress-system.

I shall form the assumed components of the displacement
from the elements of this subsidiary stress in the manner of
forming a force-system from a stress-system.

Thus, I shall write

fe Fee:
2nu = © (02+8;— 6) 2 022

OY Oz’
ESpoes |, @ howe Sea
2nv =—2 Sener (0;+ 0; 0.) 2 55°
ab 9g Ove toh. 0.
2nw= —?2 aye 2 Oy oe (0,+ 6,— 3), e 6 (3)

Forming the values of 8, T, U from these on the elastic
stress-strain hypothesis we have

g — 01 Oh _ O's O_O"
OY 02: .Otoy) Or 0e Oy (102

ete.

On equating these to the values for the same elements given
in (2), we find they require

Vi = 9, Vr2=0, V3=0. . . (4)

Since {0), 42, 03, Wi, Wa, Ws} act on transformation

of coordinates as elements of stress, it follows that the

system must consist of a hydrostatic pressure and a general
stress-system, each of the elements of which is a Spherical
Harmonic. That is,

A=P+xX, B=h+X, O=h+Xs
V?X1 = 0, V Xe — 0, VX == ae (5)

This is the descriptive criterion of an elastic stress-
system.

where

280 Mr. R. F. Gwyther on an Analytical Discrimination

3. Completion of the discrimination. The metric criterion.

The remaining requirements of the stress-strain relations
may be written

Ne OY
Ov Ow
Ow
P+Q+R= (3m—n) ($ page Sy tae) «2k eae)

which are to be completed from the values in (2) and (8).
The first two only confirm the descriptive criteria of (4)
and (5). The last leads to

2 2
—V?(91+02+ 03) + {<% eres +S

Oy O's ohne
ee i

O21 10105 One,
—" 1 V(8: +0+85)— ae ae ot S38

Ov , 9 O's | 9 Obs
Poo one 2seanh

==" —n

and therefore to
(3m+n)V7(O; + 0, + 85)

0°6; a O70; 5 OW Oe | ON
= 6m mS Se ee a oe

or

(m+n) V :

ae 0x1 , O'X2_, O'X3 Oy oO dome 9 OVS
= 2m Da? 4 5 By? + 523 erry ae i,
(7)

Owde "Ow oy
which is the metric criterion and completes the dis-
crimination sought for.

This can be integrated, and gives @ in terms of the y’s
and w’s, of which we can always take the w’s to be null,
when desirable.

This completes the investigation for Cartesian coordinates
under normal tractions only and with no inertia terms.
With the conditions in (4), (5), and (6), the equations (2)
give the general elastic stress-system, and these con-
ditions discriminate an elastic stress-system from any other
mechanical stress-system.

of Elastic Stresses in an Isotropic Body. 281

4, The inclusion of inertia terms.

We must now modify equations (1) by writing pii, pv,
pw on the right-hand side, where wu, v, w are to have the
values given in (3).

We consequently replace P by P—p(@,+6;—0,)/2n,
Q by Q- p(O;+6,— G.)/2n, R by R— p(9, + O,— 63) | 2n,
S by S+ pyr/n, T by T + pho/n, U by U+ prrs/n.

With these alterations the equations (2) still hold good.

In forming our criteria, we equate values found from (2)
to values given by the stress-strain relations deduced
from (3). In these latter P, Q, R, 8, T, U are to have
the original values of these quantities and not those which
replace them as above.

Consequently, as our first step in the criteria, in place of
V?r,=0 ete. we obtain

nV] Nr, = evi, nV bro = pro, nV/ rs = pws,
and similarly
mx = PX 2V?X2 = PX» MV?Xs = px — (8)

In place of the last stage which gave the metric criterion,
we find

Rui Th) 0°; , 0°: , 0°0:
3 (81 + 82+ 8s) — VA, +0, +95) +} Seat gat Gat
snip OLX, ok Cee eal

Se Once ean.

070; 20 ee 20)
Ox: O'xe) 45 0°%3 a
2 x2, I°X3 :
ei Oy Oz 22422)

— ee

and finally,

po— —(mtn)V2h4+ 2m {54+ 2 Ox: ae us

Os Yel

ae a
These are the modified form of (5) and (7).
3. Inclusion of bodily forces, with particular reference

to gravity on the surface of. the Earth, and to
centrifugal forces.

In any case we shall have to consider the alteration made
in equations (1), and their solution in (2) by the introduction

282. Mr. R. F. Gwyther on an Analytical Discrimination

on the left-hand side of (1) of terms representing the com-
ponents of the force per unit volume.

These components can always be represented in a form
similar to that given for the displacement in (3), but I shall
suppose that the force per unit volume can be represented
by the simpler forms

pe TOE st Si) O(E +f.) OCH +73)
OE uae ae ce os
and I shail suppose that we have selected the axes and
that the W’s are null. Then in (2) we must replace P
by P+p(F +f), Q by Q+p(E +s), R by R+ pf tp)
leaving 8S, T, U unchanged.
In the melee found from the stress-strain relations there

are no such changes to be made.
We shall thus obtain from (6)

VO: phi ora V0: — pfs = V?03— pj's
2n{3pF + p( fitsfo+fs)} + (3m +n)V?(O1+ Os + 4s)

0°0, . O'b7 OU:
= 6m} =i at ye ime i, te)

and

Tf we ee
rt ea = ++ +xX25
5 oe

where $, ¥1, Xa, X3 the values in (5) and (7) and may
be regarded as Complementary Functions, then we remain

with
V?°xX1' =a phis We: ae pi 2s V 2x3" == p73

2npk + (m+n)p(fitfetfs) + (m+nV7¢'

2
= 6m(2% 48 oe ox (11)
which may be regarded as giving ‘s Soda Integral
corresponding to the particular force acting.
There are not many cases of interest. In the case of
gravity on the surface of the Earth, as under natural forces
generally, we have

A=hr=p=0 and (m+n)V?d' + 2npk=0.
If we suppose (—A, —p, —v) to be the direction-cosines
of the attraction of gravitation,
BP = —g(\a+ wy + vz)

n
Nee ee n) Ca!

and

and

of Elastie Stresses in an Isotropic Body. 283

and if we write P’, Q’, R’ for the Particular Integral portion
of P, Q, R—7.e. the terms which depend explicitly on g—
we find

m—n

— gp(mytve), . « . (12)

P’= gpxr
JP tig a

with similar values for Q' and R’, the Complementary
Function part of P, Q, and R and the values of 8, T, U
being those given in the earlier part of this paper.

The other case which I propose to consider is that of
a body moving with angular velocities @;, @,, @, about
the axes of coordinates which must be axes fixed in the
body. It is implied either that the question is purely
kinematical, or that a problem in Rigid Dynamics has
been previousiy solved.

The expressions for the acceleration of a point in the
body are well known, and give for the effect of the reversed
effective forces

F=
Ho +02)0? + (@2+ o2)y'+(o2 +o)
— —20,0,Ly — 20,0202 —20,0,y2},
with

fi=2(yo:—2@,), fo=y(2@r—2oz), fs=clvay—yoz).

The form of the forces f, /,, fs indicates that they will
cause no strain in the body, and consequently cause no
stress. If we proceed to find the effect which they have on
the values of the stresses, they will be seen to disappear from
the stress-equations. I shall therefore omit them for this
purpose, and treat yy’, x2’, x3 as null.

We then find

n
g = Ce {(@,? =5 ais" ya" + (@,” =f w.”)y* + (,? a3 w,")z*
—Leyinyely?+2!)—2a,0.00(@ +2)

—2owyxy(u? +y")}
and pie pen Omn Oe)
en A-(Sptga)? Seep
at i Pe ke. (1B
Ss war (13)

with Complementary Function terms as before.

These give no solution of any specific question. They
only give a skeleton ot the general form which a solution
will take.

XXI. Note on Damped Vibrations.
By H. 8. Rowe *.

T is well known that the space time curve for free un-
damped vibrations may be derived from the projection
of a rotating vector, the end of which describes a circle,
and it is fairly well known that for vibrations which are
resisted by fluid friction proportional to the velocity, the
space time curve may be projected (as remarked by P. G. Tait)
from a rotating vector, the end of which describes an equi-

augular or logarithmic spiral.

The vibration of bodies when roseedl by a oonsteral
frictional force-—say solid friction—is of great importance
in practical work and does not appear to have been adequately
treated. The results obtainable are, moreover, in themselves
of much interest.

If F is the constant force of friction the equation of
motion is

ma+c?e+F=0,

wherein the sign of F depends on the direction of motion.

Substituting z= X + F/c’, we have
ae a= cos ( e

== 55 e))
/m
which gives a series of harmonic vibrations about alternating
centres distant F/c? from the equilibrium position when
~* Communicated by the Author.

ag

Notices respecting New Books. 285

friction is absent. The motion can be obtained by pro-
jection from a spiral which is composed of semicircles as
shown in the diagram.

It will be seen that the amplitudes are in arithmetical
progression and the difference fur a complete period is 4I'/c’,
which may be called the arithmetic decrement. There is
little purpose in using the ordinary definition of decrement,
but it may be remarked that on this definition (2. e. ratio
of successive amplitudes) the decrement ranges from unity
for infinite amplitudes to infinity for zero amplitudes.

The spiral curve described here does not appear to have
been used before in scientific work, and it might be con-
veniently called the arithmetic spiral or the spiral of semi-
circles.

XXIT. Notices respecting New Books.

Weather Prediction by Numerical Process. By Luwis F. Ricuarp-
son, B.A., F.Inst.P. 4to, pp. xii+236. Cambridge Univer-
sity Press. 30s. net.

= usual method employed in weather forecasting is a

development of that of Abercromby. Distributions of
pressure are classified aceording to standard types, and the vari-
ation on any occasion is predicted according to the behaviour of
the atmosphere on previous occasions when conditions of the
same type occurred. The method is therefore one of sampling
inference in which the information utilized is all of one kind.
Mr. Richardson believes that other information is relevant to the
behaviour of the atmosphere; and in this book he shows how to
make use of the known results expressed in the hydrodynamical
equations of motion and the equations of emission, transference,
and absorption of heat and water. The method adopted is to
work with equations each containing only one partial differential
coefficient with regard to the time, so that this can be determined
by means of the equation when the other quantities involved are
known; they include, of course, partial derivatives with regard to
the position on the map and the height. These are to be found
by observation at stations distributed according to a regular
pattern, and the rate of change of each meteorological element at
each station is to be calculated from them. Complications arise
from the facts that the observations must be made at finite
intervals both of time and of position, but these are allowed for.
The stations required are more numerous than those at present
in operation, and observations should be made every three hours
to obtain the best results. Observations of upper-air Winds and
temperatures are required.

286 Geological Society :—

The method is one that appeals strongly to the mathematical
physicist. It is necessarily laborious in its present form, and
probably could not be worked with sufficient speed to make it a
practical method of forecasting; but when forecasters have
acquired experience in its use, they will probably find that a
sufficient number of the quantities allowed for are comparatively
small to make it possible to expedite the calculation considerably
without great sacrifice of accuracy.

The value of the work is not confined to the application to
forecasting, though the possibility of predicting the disturbing
occasions when cyclones cause merriment in the daily press by
moving in the wrong direction makes this the feature of most
general interest. Its discussion of the physical properties of the
atmosphere is so thorough that it constitutes a text-book of the
subject. Copious references to original literature are piven, and -
any meteorologist requiring serious information on any topic will
do well to look first in this book. The section on evaporation
suggests that the only limitation on the evaporation from vege-
tation is imposed bv the dithculty of passing along the stomata
tubes; this is not always true even for an isolated leaf, and is
certainly wrong for a carpet of grass, on account of the obstruc-
tion offered by the vapour from one stoma to evaporation from
another. The numerical data actually given, however, eliminate
this source of error.

Concerning the printing and style of the book, it is only
necessary to say that it is published by the Cambridge University
Press. The index is good. Hd

XXUI. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from vol. xlili. p. 1186.]

February 1st, 1922.—Mr. R. D. Oldham, F.R.S., President,
in the Chair.

Mr. Cyrit Enywarp Nowrit BromeneaD, B.A., F.G.S., de-
livered a leeture on the Influence of Geolog sy (On “nae
History of London.

The 6-inch Geological Survey maps constructed by the Lecturer
were exhibited, and some of the new features pointed out. The
small streams now ‘buried’ are indicated on the maps, and the
historical research involved in tracing them led to an appreciation
of the connexion between the geology and topography on the one
hand, and the original settlement and gradual growth of London
on the other. The reasons for the first selection of the site have
been dealt with by several writers: below London the wide allu-
vial marshes formed an impassable obstacle; traffic from the

Influence of Geology on the History of London. 287

Continent came by the ports of Kent, and, if destined for the
north or east of Britain, sought the lowest possible crossing of
the Thames. This was near old London Bridge, where the low-
level gravel on the south and the Middle Terrace deposits on the
north approached close to the river-bank. A settlement was
obviously required here, and the northern side was chosen as the
higher ground. The gravels provided a dry healthy soil and an
easily accessible water-supply; they crowned twin hills separated
by the deep valley of the Walbrook, bounded on the east by the
low ground near the Tower and the Lea with its marshes, and on
the west by the steep descent to the Fleet; the site was, therefore,
easily defensible. ‘The river-face of the hills was naturally more
abrupt than it is now, owing to the reclamation of ground from
the river; the most ancient embankment lay 60 feet north of the
northern side of Thames Street.

The first definite evidence ef a permanent settlement was the
reference in Tacitus. The early Roman encampment lay east of
the Walbrook, and the brickearth on the west around St. Paul’s
was worked. Later the city expanded, until the St. Paul’s hill
was included, the wall being built in the second half of the 4th
century. The great Roman road from Kent (Watling Street)
aveided London, and utilized the next ford upstream—at West-
minster—on its way to Verulamium and the north-west. The
earliest Westminster was a Roman settlement beside the ford,
built on a small island of gravel and sand between two mouths of
the Tyburn. This settlement could not grow, as did London,
since the area of the island, known to the Saxons as Thorney, was
small. The road from London to the west joined the St. Alban’s
road at Hyde Park Corner, running along the ‘ Strand,’ where the
gravel came close to the river; a spring thrown out from this
gravel by the London Clay was utilized for the Roman Bath in
Strand Lane.

Throughout Medizval times London was practically confined to
the walled city, a defensible position being essential. The forests
of the London-Clay belt on the north are indicated in Domesday
Book and referred to by several writers, notably Fitzstephen,
whose Chronicle also mentions many of the springs and wells
and the marsh of Moorfields, produced largely by the damming
of the Walbrook by the Wall. The same writer mentions that
London and Westminster are ‘connected by a suburb.’ This
was along the ‘Strand,’ and consisted first of great noblemen’s
houses facing the river and a row of cottages along the north
side of the road; this link grew northwards, at first slowly,
but in the second half of the 17th century with great rapidity.
By the end of that period the whole of the area covered by the
Middle-Terrace Gravel was built over, but the northern margin of
the gravel was also that of the town for 100 years, the London-
Clay belt remaining unoccupied.

The reason for this arrested development was that the gravel

288 Intelligence and Miscellaneous Articles.

provided the water-supply. In early days the City was dependent
on many wells sunk through the gravel, some of which were famous,
such as Clerkenwell, Holywell, and St. Clement’s. In the same
way the outlying hamlets (for instance, Putney, Roehampton,
Clapham, Brixton, Haling, Acton, Paddington, Kensington,
Islington, etc.) started on the gravel, but later outgrew it, as
pointed out by Prestwich in his Presidential Address of 1878. In
the City the supply soon became inadequate, or as Stow says
‘decayed,’ and sundry means were adopted to supplement it. The
conduit system, bringing water in pipes from distant springs, began
in 1236; London-Bridge Waterworks pumped water from the
Thames by water-wheels from 1582 to 1817; the New River was
constructed in 16138, and is still in use. It was not until the
19th century that steam-pumps and iron pipes made it possible
for the clay area to be occupied, thus linking together the various
hamlets that are now the Metropolitan Boroughs.

Some of the ways in which Geology affects London to-day were
briefly indicated, and the lecture was illustrated by a number of
lantern-slides, reproduced mainly from old maps and prints.

XXIV. Intelligence and Miscellaneous Articles.

YOUNG'S MODULUS AND POISSON’S RATIO FOR SPRUCE.

To the Kditors of the Philosophical Magazine.
Dear Sirs,—

(A my recent paper in the Philosophical Magazine for May 1922
there is an error on page 877. It is there stated that

Sve Ser es Se eae gel eae
ee hee ADoser
S
are equal respectively to 7008,, 700 8, and 5:

This should read

are equal respectively to 700 S,, 700 8, and 3:

The error becomes evident on reading the paper, but I very
much regret that it has crept in.
Yours faithfully,

The College of Technology, H. Carrineron,

Manchester.
May 24th, 1922.

Tizanp & Pyn.

lanGele

Corresponding to experiment A,..

iG, B.

Corresponding to experiment A,,.

In these photographs the lower horizontal line is the line
of atmospheric pressure. The ordinates represent pressure,
and the abscisse time. A is the beginning of compression,
B the point of maximum compression, and C the explosion.
The curve in the top left-hand corner is the cooling curve
of the products of combustion.

Phil. Mag. Ser. 6, Vol, 44, PI.

Corresponding to experiment A,,.

ine

Corresponding to experiment Ds.

[Qnree ye

Corresponding to experiment Ds.

THE

LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCE

ASO

[SIXTH SERI

AGPGT ST A922:

XXV. On the Viscosity and Molecular Dimensions of Gaseous
Carbon Oaysulphide (COS). By C. J. Smirz, B.Sce.,
A.R.CS., DIC., Research Student, Imperial College of
Science and Technology™. |

HE present research is a continuation of the work on
the measurements of the viscosities of gases, for the
purpose of elucidating the structure of the molecules
constituting them. Some measure of success has attended
this investigation in many cases where the necessary data
are known, and suggests that an accumulation of further
similar data may be fruitful. A case in point js that of the
molecule of carbon oxysulphide, and this paper describes the
measurements of the viscous properties of this substance,
which is ordinarily gaseous. The data, hitherto unknown,
which have been obtained, have been applied to calculate the
molecular dimensions in the ordinary way.

Apparatus and Method of Observation.

The apparatus and method, which have been used to
determine the viscosity of carbon oxysulphide, have recently

been fully described Tf.
: Method of Haperiment.

The viscometer was carefully standardized with a new
mercury pellet in the manner indicated in previous papers.

* Communicated by Prof. A. O. Rankine, D.Sc.
+ A. QO. Rankine and C. J. Smith, Phil. Mag. vol. xlii. p. 601, Nov.
1921, and C. J. Smith, Proc. Phys. Soc. vol. xxxiv. p. 155, June 1922.

Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. U

290 Mr. C.J. Smith on the Viscosity and Molecular

and the corrected time of fall proved to be 104-70 sees., a
value which is probably correct to 0O'1 sec. With this time
of fall the corresponding time of fall for carbon oxysulphide
has been compared, and with appropriate corrections gives
the relative viscosity, from which the absolute viscosity has
been obtained by assuming the viscosity of air at 15°-0 C.
to be 1°799 x 1074 C.G.S. units. In addition, the variation
of viscosity with temperature has been derived from com-
parisons of the corrected times of fall at atmospheric and
steam temperature.

Preparation and Purification of the Carbon Oxysulphide.

The carbon oxysulphide was prepared by the action of
sulphuric acid (five vols. acid, four vols. water) on pure
potassium thiocyanate in the cold (room temperature). At
the same time hydrocyanic acid, formic acid, and carbon
bisulphide are formed. To remove these impurities the
method recommended by Moissan* was used. ‘This consists
in passing the gas through a strong solution of caustic
potash to remove the hydrogen cyanide and then over wood
charcoal to remove the carbon: bisulphide. The gas was
dried by being passed over calcium chloride, and then
solidified at liquid air temperature. All permanent gases
were pumped out of the U-tube containing the solid COS by
means of a mercury pump. The liquid air was then replaced
by a mixture of solid CO, and alcohol at —80° C., when it
was observed that the vapour pressure of the liquid COS was
about 30 cm, of mercury. The CO, mixture was then
removed and samples of the gas collected over mercury. It
was further purified before being introduced into the
viscometer by fractional distillation at liquid air temperature.
The liquid air having been replaced by CO, and alcohol at
—80° ©., it was possible on account of the comparatively
high vapour pressure of COS at this temperature to pump
off successive quantities of dry COS sufficient to fill the
viscometer at atmospheric pressure.

Heperimental Results. (Table I.)

We have ¢,;;=69°96 sees., and ty)=90'64 sees.
The ratio of these times of fall gives the ratio of the
viscosities at the corresponding temperatures.
Thus
moo = too _ 90°64 _ 1 5x6
pis ate OOOO) GF

* Moissan, Zraeté de Chime, vol. 1. p. 318,

i” P

Dimensions of Gaseous Carbon Ovysulphide. 291

TABLE I.

Each time recorded is the mean of four observations in each
direction for the whole pellet, and of three for the pellet when
divided into two segments. The letters in parentheses indicate the
order in which the observations were made.

ma : ee of ay (segs. ). | Capillary Corrected | Time Ate
(deg. C.). | Whole | Two | Ae a 15°00. 100°-0C. |
pellet. | segments. | | | |
(a) ipag...| 70L | 7685 | 00416 | <7001 | 6992) —
(0) 15-44... | 7308 | 7625 | 00399 | 70-16 | 7005) —
(e) 15°68 ... 73-02 76°22 0:0403 | 70-07 | 69°91 | —
| | Mean | 69:96) —
forage ce. |, OT51 92°43 | 0:0099 | 90°60 -— 90°63
a) deo...) | 91-54 9248 | 00101 | 9062 — 90°65
Mean — 90°64

Assuming Sutherland’s law to hold for this gas, the value
of Sutherland’s constant obtained is C=330.
miso at 15-0 C.,
tcos id 69°96
fe ATT)
Correcting for difference of slipping of COS and air, we
obtain

=O 0G 2.

JESS Ug ane

lair

On the assumption that the viscosity of air at 15°-0 C. is
1:729 x 1074 C.G.S. units, the values for COS are

7 1200 x10E* CGS funits
and Nig = L504 x 107* C.G.S. units:
and by extrapolation using Sutherland’s formula,
n= lel 3a x10 C.G. Saamits:

Calculation of Molecular Dimensions.

The particular dimension calculated from the above
results is the mean area A which the molecule presents in
mutual collision with others. The basis of this calculation
is Chapman’s formula (doc. cit.) modified in its interpretation
in the manner suggested by Rankine. The value obtained
is A=1:06 x 10~-” cm.? which may be subject to an error of
2 or 3 per cent.

U2

292 -Prof. A. O. Rankine on the Molecular Structure
Summary of Lesults.

tein als
Viscosity in C.G.S. units x 10-4. | |
-Sutherland’s 1 Mean
“| constant. |°° ene
15°:0 C. 160°:0 C. 0°:0 CG. cm. X i
1-200 1°554 TeI35)) on 1-06

Tn bdnein ee the author gladly acknowledges the grant
for this research, which was made by the ore ee Grant
Committee of the Royal Society, and also wishes to thank
Professor Rankine for his continued help and advice.

Imperial College of

Science and Technology, S.W. 7.
Ist May, 1922.

XXXVI. On the Molecular Structure of Carbon Oxysulphide
and Carbon Bisulphide. By A. O. Ranking, D.Se.,
Professor of Physics in the Imperial College of Science and
Technology *.

Ie HERE are at the present day in the process of
development several theories of atomic and mole-
cular structure which are in many respects discordant.
They have, however, at least one feature of general agree-
ment namely, the common view that the aieme of the
inert gases occupy unique positions in the various schemes.
The distribution of the electrons with respect to the nuclei
in these atoms is regarded as having the characteristic of
completeness, so that there is displayed no marked tendency
to lose electrons or to capture additional ones. Moreover,
atoms other than those mentioned are believed to have in
varying degrees what may be called deficiencies and
redundancies of extra-nuclear electrons, which they endeavour
to adjust by entering into suitable combinations with one
another ; so that either by the process of give and take, or
by common use of the same electrons, contigur ations corre-
sponding closely to those of the inert atoms are attained by
the individual atoms forming the compound.
2. These views of chemical combination find site most

* Communicated by the Author.

of Carbon Oxysulphide and Carbon Bisulphide. 293

complete expression in the theory of Lewis and Langmuir *,
particularly in relation to the type of compound with which
this paper is concerned—namely, that in which atoms,
deficient in electrons, are regarded as sharing them in euler
to reach the completeness of inert configurations. The main
purpose of this paper is to apply the principles of this theory
to the special case of the molecule of carbon oxysulphide,
and to show that the molecular dimensions of this compound,
as derived from viscosity data, are consistent with the Lewis-
Langmuir view of its constitution. This test of the validity
of the theory is made possible by the recent measurements
by ©. J. Smith t of the viscous properties of the gas in
question. Similar calculations for the molecule of carbon
bisulphide have been made, and these await verification or
otherwise when the necessary viscosity data are availabie.
3. Carbon oxysulphide belongs to a family of three
ola having the chemical constitutions GO, COS, and
CS,. The two former are gaseous at ordinary temperatures,
and the latter a highly volatile liquid. Inall of them carbon
is a constituent, and COS can he regarded as the molecule
obtained by the substitution of a sulphur atom for one of the
oxygen atoms in COs, or by the reverse substitution in CS,.
It is probable that the carbon atom occupies the central
position in each molecule, and that the nuclei of the three
atoms lie in each case upon a straight line.
4. According to the Lewis-Langmuir theory (loc. cit.), the
atoms in these molecules are linked together by sharing
external electrons in such a manner that each atom approxi-
mates to the configuration of the inertatom at the end of the
corresponding row in the periodic table. Thus, in CO, the
ceutral carbon atom shares altogether eight electrons, four on
each side with an oxygen atom. The electron configuration
thus formed is that of three neon atoms in a row, for the
inert atom corresponding to both carbon and oxygen is neon.
In the molecule COS there are again eight electrons shared
by the carbon atom, four on one side with the oxygen atom,
and four on the other side with the sulphur atom. The
electron arrangement thus attained is that of two neon atoms
(corresponding to the oxygen and carbon) and one argon
atom (corresponding to the sulphur). Applying a similar
argument to the CS, molecule, we are led to regard it as
resembling closely the electron distribution of inert atoms
in the sequence argon-neon-argon ‘in a line. In other
words, we can treat each carbon or oxygen atom in a

* IT. Langmuir, Journ. Amer. Chem. Soe. vol. xli. p. 868.
+ C.J. Smith, supra, p. 289.

294 Prof. A. O. Rankine en the Molecular Structure

molecule as having nearly the same dimensions as a neon
atom, and each sulphur atom in combination as approxi-
mating to the dimensions of an atom of argon.

5. The remaining question of how far apart are the nuclei
of the atoms in the molecule finds a satisfactory answer in
the work of W. L. Bragg *, whose X-ray -crystal measure-
ments have enabled him to assign probable values for the
radii of the-outer electron shells of the atoms of the inert
gases. The only values with which we are at the moment
concerned are those of neon, and argon, which are given
respectively as 0°65 and 1:03 Angstré jm units. In cases like
those under consideration, where outer electrons are playing
a double part, the sharing is equivalent to contiguity of the
outer shells, so that the distance apart of the nuclei is the
sum of the radii of the appropriate inert atom shells. Thus
for CO,, which is pictured as three neon atoms in line, the
three nuclei are equally spaced and separated by distances
equal to twice the radius of the neon outer “shell, 7. e.
-2x0:65 A=—1:30 A. In COS the distance between the
carbon and oxygen nuclei is the same, namely 1°30 A,
but the distance between the carbon and sulphur nuclei
is the sum of the radii of the outer, electron, shells of
the neon and argon atoms, 2. e. 0°65 A+1:03 A=1°68 A.
The three nuclei in COS are thus unequally spaced on _
account of the greater size of the argon atom. In CS, the
distance between the, carbon nucleus and each sulphur
nucleus is also 1°68 A, and the three | nuclei are again
spaced symmetrically.

6. It is evident that none of the three molecules under
consideration, if their configurations are as indicated, can be
expected to display spherical symmetry. In these circum-
stances it is necessary to interpret in a special way the
results of the well recognized method of caleulating
molecular dimensions from viscosity data. The quantity
which is actually derivable from the formula is the mean
value of the area which the molecule presents, for all
possible orientations, as a target for mutual collision with
other molecules in the gas. This area the author + has
ventured to call the mean collision area, and its value for
COS is given by C.J. Smith (loc. cté.) as 1°06 x 107 em.?
‘The immediate problem before us is to find how nearly the
tentative model of this molecule described above would
exhibit this value for its mean collision area. The values of

* W. L. Brage, Phil. Mag. vol. xl. p. 169.

7 AgeO, Rankine, Proe. Roy. Soc. A, vol. Sead p. 360, and Proc.
Phys. Soc. vol, xxxili. p. 362.

of Carbon Oxysulphide and Carbon Bisulphide. 295

the mean collision areas of the constituent configurations
(which we are taking to be those of neon and argon) are
known, and it is usual to. regard these symmetrical inert
atoms as behaving as elastic spheres for purposes of collision.
The radii of these collision spheres, as we may: call them
for the sake of precision, are 1°15 A and 1:44 A respectively,
and they are considerably larger than those of the corre-
sponding outer electron shells, so that they overlap when

Fig. 1.—Molecular Dimensions from the point of view
of the Kinetic Theory.

os

The Carbon Dioxide Molecule: equivalent to three linked atoms of Neon.

ena §

The Carbon Oxysulphide Molecule: equivalent to two Neon atonis and
one Argon atom linked together.

The Carbon Bisulphide Molecule: equivalent to two Argon atoms
linked together by one Neon atom,

the nuclei are separated by the distances demanded by
electron sharing. Fig. 1 shows three models, drawn to scale,
representing what we may conceive CO,, COS, and OS, to
be like for purposes of intermolecular encounters. CO, may
be regarded as three overlapping spheres, each of the neon

296 Prof. A. O. Rankine on. the WMoleould® Seruature

collision size, with centres separated by the distances already
specified. In COS we take instead of one of the extreme
neon spheres an argon collision sphere ; while in CS, both
the outer spheres are of the argon size. In all three cases
ithe diagram represents all the nuclei in the plane of the
paper, and the line joining them is evidently an axis of
symmetry. If these symmetrical axes are variously oriented,
the area presented by the model assumes different values,
and our problem is to caleulate the mean value of this pro-
jected area for comparison with that deduced from viscosity
data. The author (loc. cit.) has already derived the necessary
formule for this purpose, and has shown that the result
obtained by application to the first model in fig. 1, namely
CO., is very nearly equal to the actual mean collision area of
the carbon dioxide molecule. In other words,. a carbon
dioxide molecule behaves in collision as though it had the
configuration of three neon atoms in a straight line and with
outer electron shells contiguous.

Calculation for the COS Model.—In the model which
we are taking to represent the COS molecule, the calculation
in the strictest sense is greatly complicated by reason of the
particular distribution of the spheres. The exact formule
which have been obtained (loc. cit.) for equal and unequal
spheres only apply rigidly to cases where a special relation
exists between the diameters of the spheres and the distances
apart of their centres; and the model under consideration
does not fulfil this condition. But by regarding the problem
from two different points of view, we can obtain, by means
of the comparatively simple formule: already availakle, upper

and lower limits which are so close together as ic render

unnecessary the laborious exact calculation. This course is
all the more justifiable because it is fully recognized that
the general treatment of the problem itself can only be taken
as a first approximation to the truth.

8. Let us consider the effect on the area of projection of
the model (reproduced in the full lines of fig. 2, a) caused
by variations of orientation of the symmetrical axis joining
the centres O,, O., and O; of the constituent spheres. It
will be convenient to speak of the sphere with centre O,
simply as sphere 1, and so on, and of the projections of the
spheres, which will of course be circles, as projection 1
etc. «As the axis O, O3 approaches the line of sight, the
projections of the centres approach one another, and the
eclipsing of the spheres becomes more and more marked. Up
to a certain point the total projected area is equal to the sum

of Carbon Oxysulphide and Carion Bisulphide. 297

of the areas of the whole of projection 3, the crescent formed
by the overlapping of projection 3 over projection 2, and the
crescent formed similarly by the eclipse of projection 1 by
projection 2. Before the eclipsing of 2 by 3 is complete,
however, projection 3 begins to encroach upon regions of
projection 1 which are not already covered by projection 2.
It is this fact that introduces into the exact treatment of the
problem the complications to which reference has already
been made. Thus in fig. 2,6, which shows the projected
area for that orientation of the axis for which the eclipse of
2 by 3 is just complete, the crescent formed by 2 and 1 still

-4- mest oes ag fs -}+---4--

i / /
0, 03 0; /

survives, but parts of it (as indicated by the shading) are
covered by projection 3. The projected centres are O,’, O,’,
and QO,’ respectively, and this particular state of affairs occurs
when the angle between O, O; and the direction of projection
is 9° 47’ for the spheres having the dimensions and distribu-
tion already specified.

9. Overlapping of the type just indicated, like all
overlapping, has the effect of reducing the projected area ;
it is therefore clear that if we neglect it we shall obtain too
large a value for the mean area of projection—that is, an
upper limit will be obtained by taking the mean collision
area as the sum of the three parts : (a) the area of the circle

3, (b) the mean value of the area of the crescent formed by
ites 3 and 2 (c)the mean value of the area of the crescent

298 Prof. A. O. Rankine on the Molecular Structure .

formed by circles 2 and 1. The first of these quantities is.
the area of the central cross-section of the argon sphere
itselr, viz. 0°648x107-" cm.?; the two latter are readily
obtained from the graph in the paper already mentioned *

They prove to be 0-217 x 10° ™ em? and 0;226> 109 Senn
respectively. The total is 1:09 x 10~” em.’, and this provides.
our upper limit.

10. With regard to the lower limit, we can obtain a.
satisfactory value by contemplating a variation of our model,
which avoids the special type of overlapping responsible for
complications. A suitable change for this purpose is to
substitute for the sphere 1 a smaller sphere having the same
centre but of such magnitude that its projection becomes
just eclipsed by projection 2 at the same orientation of the
symmetrical axis for which projection 2 is just eclipsed by
projection 1, as shown by the dotted circles in fig. 2. The

radius of the necessary sphere is found to be 0:93 A as
compared with the original value 1:15 A. [Examination of
the projection of a sphere of this size, in relation to the other:
two projections, shows that for no orientation does eclipsing
of the shaded type appear, and the formule already available
enable the mean area of projection to be calculated exactly.
The value so obtained will, however, obviously be less than
the true value aimed at, on account of the reduction of size:
assumed for sphere 1. Using the graph already mentioned,
the lower limit thus derived is

0°648 x 105” cm.? 4:0°217 x 10> © em? -+.0°138 x 105 mire
= 1°00 x 10 get iemme
11. The foregoing justifies the assertion that a molecular
model having the dimensions of an argon atom succeeded
by two neon atoms in line and spaced according to the

demands of outer electron contiguity may be expected to.
have a mean collision area intermediate between

J)S) 5 1 ean.
and LeOO se O72 oin2

The actual value of the mean collision area of the COS
molecule, as determined from viscosity is

DO << WO ena

with a possible error of 2 or 3 per cent. It falls definitely
between the upper and lower limits obtained from our
calculations, and seems to provide striking corroboration of

7 XO vankine, Proc wehys, Soc; vol, aacxai. app ole

of Carbon Oxysulphide and Carbon Bisulphide. 299

the theory upon which the estimates are based. But we
must be content with the conservative remark that the
dimensions of the carbon oxysulphide molecule, as found by
the application of the kinetic theory, are consistent within
the limits of experimental accuracy with the view that the
three atoms of the molecule, by sharing external electrons,
assume the electron configurations and ee wounk in collision

of particular groupings of the neighbouring inert atoms.

12. Calculation for the CS, Model. —Although there exist
at present no data for carbon bisulphide which suffice to
calculate the mean collision area of the molecule in the
gaseous state, the success of the previous comparison
would appear to justify a prediction of its value by con-
sideration of the appropriate model. This has been repro-
duced in the full lines of fig. 3,a. Here again the mode! is

Fig. 3.
(2)

one which does not lend itself to exact solution without
laborious calculation; but again, also, we can obtain
satisfactorily close upper and lower limits. The area
of projection will clearly be less than that corresponding to
the mode! in which the dotted sphere is substituted for the
small central one, so that we have three equal spheres of
the argon size in line ; ; it will, on the other hand, be greater
than if the central sphere i is entirely dispensed with , so that
ae are two equal argon spheres only, as represented in

fic. 3,b. The dimensions of the spheres and the distances

300 Mr. Hf. P. Slater on the: Ruse of

apart of their centres have already been stated ; and both
modified models have mean areas of projection which are
very easily calculated. The upper limit thus determined
proves to be 2°12 times the collision area of the argon atom ;
the lower limit is 1°90 times the same area. Using the
known value 0°648 x 107)? em.? for. the collision area of the
argon atom, we find that the mean area of projection of the
model consisting of two argon atoms with an intermediate
neon hes between
13 Creeks

and 1223 Al? ema

We may venture to predict with some confidence that the
mean collision area of the CS, molecule, when determined,
will be found to be between the above vaiues. A more
exact estimate could of course be made, but the degree of
accuracy at present attainable in determining molecular
dimensions from viscosity measurements is not sufficient to
render the additional calculation worth while.

Summary.

On the assumption of the validity of the Lewis-Langmuir
view of molecular constitution, the probable betaviour during
encounters has been examined for the molecules of carbon
oxysulphide and carbon bisulphide. In the former case it is
shown that the molecular dimensions as derived from the
application of the kinetic theory to the viscosity measure-
ments of C. J. Smith, are in striking accordance with the
results of the above examination. In the latter case
comparison is not yet possible, on account of the absence of
necessary data.

Imperial College of Science and Technology,
May 11th, 1922.

XXVIL. The Rise of y-Ray Activity of Radium Emanation.
iy. P. SLATER WUE SCay ica aso. Canmaue ae

N a previous paper f it has been shown how the initial
rise of y-ray activity, starting from pure radium
emanation, depended on the nature of the walls of the tube
containing the gas, the reason .being that a small but

* Communicated by Prof. Sir E. Rutherford, F.R.S.
+ Slater, Phil. Mae. vol. xi. p. 904 (1921).

y-Ray Activity of Radium Emanation. 301

detectable y-radiation was excited in the walls by the impact
of the & particles emitted by the emanation. The amount of
this excited radiation was, however, very small when the
walls of the tube were composed of atoms of low atomic
weight, and for a lining of pure paper the y-ray aetivity of
the emanation and its products was found to rise practically
from zero. Under such conditions the y radiations from
the tube are due only to the products radium B and
radium C.

Taking the number of emanation atoms disintegrating per
second at initial time as unity, the number of radium-B atoms
disintegrating per second at any subsequent time ¢ is

Aor
>a (Ay—Ay)(Ag—Ay)’
X=1, 233

where A,, Ay, As are the transformation constants of the
emanation and the products A, B, and © aay aN This
quantity is tabulated for various times up to 220 minutes at
the end of this paper (Table II.).

Similarly, the number of radium-C atoms disintegrating
per second at time ¢ is

e-Ale
eg: 2, (Ag — Aq) (A3—Ay} (Ag Ay)’

A= 1, 2,3, 4

Tables for this quantity for various times up to 258
minutes have been given by Moseley and Makower * and by
Rutherford f.

The rise in y-ray activity of a tube filled initially with
pure emanation can therefore be represented by

e-Aié
KA.A3> (A2— Ay) (As — r1)
N=1,2,3
e Ait
Su meee (ga) s— WJM)

where K is the fraction of the ionization, measured under
given absorption conditions, due to radium B when in radio-
active equilibrium with radium C.

Thus it is necessary to determine “K.” Since the y rays

* Moseley and Makower, Phil. Mag. vol. Xxili. p. 302 (1912).
1 Rutherford, ‘R adioactive Substances s, p. 499.

302 Mr. F. P. Slater on the Rise of

from radium B are less penetrating than those from radium
C, “ K” depends on the thickness of matter through which
the radiations pass before entering the ionization chamber.
Rise curves have been experimentally determined for
different thicknesses of absorption material, both lead and
aluminium being used. The values of K for various thick-
nesses have been deduced by trial, and are shown in fig. 1.

ie

15 20
—-~ Moms. c fead.

O
i)

A comparison of the experimental and calculated rise curves
of the y-ray activity through 12°0 mm. of lead is given
in fig. 2, After six minutes from the introduction of pure
emanation, the calculated and experimental curves agree
very closely.

From these curves the absorption coefficient of the
radium B-y rays can be deduced, and the values found are
given in Table I. along with comparative determinations
by Makower and Moseley (loc. cit.) and Rutherford and

Richardson *

The values of the absorption coefficients for the thick-
nesses of aluminium are somewhat doubtful, since the
supposition of homogeneity of the radium-C y rays is not
justifiable through such small thicknesses. The increasing

* Rutherford and Richardson, Phil. Mag. vol. xxv. p. 722 (1913).

y-Ray Activity of Radium Emanation. 303

value of w (cm.~!) with decreasing thickness of absorption
material (see Table I.) is to be expected, since Rutherford
and Richardson (loc. cit.) showed that radium B emits
certainly two types of radiation having absorption coefficients

in aluminium of 0°51 em.~! and 40:0 em.~’, and possibly a

third type (w= 230°07" in aluminium).

Fig, 2.—Rise of y activity from Radium emanation through
12:0 mm. of lead.

a % Max.

Activity

\

0-60
0-50
0-40
0:30
0:2

0-10

The absorption coefficients in lead only, given in Table I.,
are corrected for obliquity of the rays entering the electro-
scope, and King’ s correction is used as given in Case II.
of his paper *

_f (wt)—cosOf[mtsec@]
t= 1—cos 0

where I; and I, are the intensities of the radiation emerging
through a plate of thickness ¢ cm. and incident radian
respectiv ely, w the absorption coefficient expressed in cm.™’,
and @ the semi- -angle of the cone of rays entering the

electroscope.
* King, Phil. Mag. vol. xxiii, p. 248 (1912).

304 Rise of y-Ray Activity of Radium Emanation.

TARE aeY 3)

Absorbing medium is Lead, except where otherwise shown.

Value of p (cm.—1) Moseley

Thickness of Rutherford
Absorbing Plate. ae ate one
e Radium-B rays. Makower. Richardson.
1G O26: Ohm mis eis. 2. ees. 2) 27 tem) 1 — Varying from
TOO SSO. Oe oe A ew Ga a. 11-0 cm.—1
QE ROS) pc” empl 3 sbss a Brars 4-1 em,-1 4:0 em.-1 to
(lead). 2°8 cm.—1 in
ISS Se. 2 hol aan 62 cm.—1 6:0 em.—1 lead.
(lead),
3°0- 4:0 ,, (Aluminium), i (ema CAlye

mm. (Aluminium)

TO:Otenn al (Al) &

ae hie ele
Rise of Radium B from Radium Emanation.

Maximum = 0-97480 is taken as unity.

Calculated

Calculated

Calculated

Time ‘Time
in rise of 1D rise of in ‘rise of
mins Radium B. mins. Radium B. mins Radium B.
1 PEs Cn 0:00269 Vale ee 0:2260 TO Pee 0:9502
De EOC Nea 0-O1016 NG teste toeeee 0:2649 120) a oe 0:9643
Sie an 0:02129 lowe ae 03023 Ould eee 0:9750
Ais 1 aes 0:035138 Ie Oca ceehcte: 0:3379 PAG) no Ae ee 0°9835
Lee tae 0:05115 12 (SSK Nreeaitle e 0°4942 TSO a eee 0:9886
Gouna 0:06862 he SN 06185 160 eee 0:9929
"Tne 5 aoe 0:08718 A te 0:7066 Ree 0°9956
So. ee 0 10647 IN SC 6 Hautes 0'7780 [S02 ee 09979
GF, ee 126929 Moe ee 0:8329 19025 Rue 0:9992
LQ 2k eee 0'14624 SOie) alae 08748 ZOO eee 0:9998
emertee ols 0°16637 OU) eee anae 0:9058 DLO* a eecaaee 0:9999
Oy me 018641 TOO a ae 0:9315 Die eee 1:0000
DEA) aikeee ae 0:9999
Summary.

Curves showing the rise of y-ray activity from pure
radium emanation measured through a wide range of
absorption thickness of matter have been determined and
utilized in deducing the absorption coefficients of the hetero-
geneous y¥ radiation from radium B.

My thanks are due to Professor Sir KH. Rutherford for his
invaluable help in carrying out this research, and to
Mr. G. A. R. Crowe for the preparation of the radioactive
material,

Eo B0aa5q

XXVIIT. An Experimental Test of Smoluchowski’s Theory of
the Kinetics of the Process of Coagulation. By JNANENDRA
Nara MouxuerJen, /).Sc., Professor of Physical Chemistry,
University of Caleutta, and B. CoNSTANTINE PaAPacon-
STANTINOU, D.Sc., Assiscant Professor of Chemistry, Uni-
versity of Athens ™*.

A short account of the Theory.

aT some experiments on the degree of dispersion of

colloidal arsenious sulphide on the rate of coagulation,
it has been shown (J. Amer. Chem. Soc. vol. xxxvil. p. 2026,
1915; and Sen, Trans. Chem. Soc. vol. cxv. pp. 467-8,
1919) that the finer sol is less stable. In 1915 one of
us pointed out the obvious connexion with the increased
facilities of coalescence. The smaller particles have a
more vigorous Brownian movement due to the smalier
frictional resistance of the medium. ‘This would be clear
from the well-known equation of Hinstein. The diminution
in the mean distance between the particles also increases
the rate of collisions. It was stated that the adsorption
theory does not take these factors into consideration.
Recently Smoluchowski (Zeit. Phys. Chem. vol. xcii.
p- 129, 1917) has been able to formulate the progress
ot the coalescence with time. His attention was drawn
to the subject by Zsigmondy. Bredig (Anorganische
Fermente, 1901, p. 15) suggested as the cause of coalescence
an increase in surface tension with a decrease in the
electric density on the particles. Zsigmondy (Zertsch.
Physikal. Chem. vol. xcii. p. 500, 1918) medified this idea
in the sense that there is an attraction, between the particles
which increases with decrease in the electric charge. As
a result of this attraction he assumes that when one particle
coines within a certain distance of another, the two coalesce.
This distance is taken as a measure of the force of attraction
and is called the radius of the sphere of action. I[t has
been shown by Zsigmondy that the time required for a
definite colour-change in a gold sol gradually decreases
with rise in electrolyte concentration till it reaches a
minimum +, which does not change any further with higher

* Communicated by Prof. F. G. Donnan, F.R.S.

+ Similar minimum times have been observed with cupric sulphide
and mercuric sulphide sols by the writers. A copper sulphide sol gave
two minutes as the time necessary for the appearance of visible clots
when the concentration of the precipitating electrolyte (barium chloride)
was varied from N/300 to N/20. At dilutions higher than N/300 the
time was observed to increase as usual (Mukherjee and Sen, Joc. cit.).

Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. X

306 Profs. J. N. Mukherjee and B. C. Papaconstantinou on

concentrations. This was assumed to prove that the radius
of attraction reached a maximum value.

Smoluchowski utilized this idea of a sphere of action
to avoid a consideration of the forces that influence the
coalescence. He considers the probability of particles
coming within their mutual sphere of action when the
radius of the sphere has a constant value determined by
the conditions.- It is assumed that as soon as a particle
comes within the sphere of attraction by virtue of its
Brownian movement the two particles coalesce. This dis-
continuous view of the obviously continuous process of
coalescence was assumed to avoid a consideration of the
nature and distribution of the forces that are present.

Considering the effect of the motion of each particle
and also that each of the aggregates acts as a condensation
centre, he derives the following equations :

n=, (1)
I+ ip
y= ae (2)
(145) )
(a, ngt)”

eae

w here “2.” denotes the total number of particles originally
present per unit volume before coalescence begins. They
are all assumed to be-spherical and equal in size. ‘“¢” is
the time in seconds that has elapsed since the electrolyte
and the sol have been mixed. “TT” is a constant charac-
teristic of the rate of coagulation and is given by

1

Ta aa) Site nioke 1 ane eomlede

where ‘‘D”’* is the diffusion constant as given by Hinstein’s
‘equation; #«=4.a7.D.Ra, and Ra is the radius of the
sphere of uate.

HH, ule ]
* BD) MN SEE) te where H=the gas constant,
0 Hie 6=the absolute temperature,
N)=Avogadro’s number,
n=the viscosity,

and =radius of the particle.

the Kinetics of the Process of Coagulation. 307

yn denotes the total number of particles in all stages
of coalescence in unit volume when the time is <4? ; ;
n, denotes the number of the primary particles whose
original number was 7 at the time “t” ; nz denotes the
number of particles of the kth stage of coalescence—that is,
the number of aggregates each of which consists of ‘ h”
of the primary particles. ‘“‘k” is evidently an integer.
In 1918 Zsigmondy published the results of an investi-
gation to test this theory. He restricted his investigation
to the rate of decrease in the primary particles (green in
the ultramicroscope) in a colloidal gold sol when the
minimum time of coagulation has been reached. He found
that Ra=2°2 times r, the radius of the particles. Similar
values were obtained by Westgren and Reitstétter (Zeitschr.
Phys. Chem. vol. xcii. p. 600, 1918) with more coarsely
dispersed gold sols. The value of Ra/r, however, varied
in one experiment from 1-4 to 3°8. The recent experiments
of Kruyt and Van Arkel (Rec. Trav. Chim. Pays-Bas,
vol, sextrx, [4] p. 656, vol. xl. p. 169, 1920) show greater
variations. hey are of opinion that there is some regularity
in these variations. They could not observe a maximum
value of Ra/r equal to 2. They found a maximum value
equal to 0°73.

Smoluchowski, assuming from the data of Zsigmondy
available at that time that Ra/r=2, points out that the
maximum rate of coagulation is reached when each collision
between two particles is successful in bringing about a
coalescence. When the rate of coagulation is slower,
all the collisions are not successful in bringing about a
coalescence of the particles. If ‘“‘e” is the fraction of the
collisions that are successful in Meets about coalescence,
then “ T” in equations (1) and (2) takes the form

a No.” >

ire ACRE c ti) ‘ o

where No, Ra, 7, 0; and » have the same meaning as in
equations (1) and (2).

Putting
3 : Ng - 7) 1 (6)
4 Ra.€.n 8’
we have
No _ eee ; 7
area ee. t (1)
Leip

eee OeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeeeeeEeEOEOeeeeEeeEeEeEeEeEeEeEeEeEeEeEeEe_oeG_ eee

308 Profs. J.N. Mukherjee and B.C. Papaconstantinou on

Since only ‘‘e’’ is variable, a comparison of the coagn-

lation time “‘t” for the same change in the sol makes it
possible to determine the variation in the percentage of
successful collisions and its dependence on the conditions
of experiment. When the maximum rate is reached, e=1
and hence a measure of the absolute value of € is possible.

Problems awaiting solution.— A 3zlance through the experi-
mental work would show that the assumption of the constancy
of “TT” is not well justified. The simplicity of Smolu-
chowski’s equations consist in that there is only one constant.
The experimental limitations are great, and it is quite possible
that the discrepancies are due to the defects of the ultra-
microscopic method. The other possibility is that the
simplifying assumptions of Smoluchowski—for example,
the constancy of “T”’ independent of the stage of co-
alescence—are not true within narrow limits. It is of
great interest to know the limits within which these
equations are valid.

The important questions that await solution in this
connexion are :

(a) the limits within which the above equations are
valid ; and

(b) if the above equations are valid, the variation of ¢
with concentration of electrolyte ;

(c) the dependence of ¢ on the electric charge ;

(d) the variation of e with temperature.

In the following an account of an attempt to examine
these factors, with the exception of (c), is recorded.

Indirect Methods.—Variations in physical properties that
occur simultaneously with the process of coagulation can be
utilized to measure the rate of coalescence.

Smoluchowski pointed out that the viscosity measure-
ments of Gann (Koll. Chem. Beithefte, vol. vill. p. 67 (1916) )
do not satisfy the main requirements of his equations—
namely, a similarity in the form of the curves (showing the
variation in viscosity with time) independent of the nature
of the electrolyte. He concludes that viscosity changes do
not form a measure of the coagulation process. Yet he
considers that the method is suitable for a quantitative
comparison of the effect of various concentrations on the
values of e when the curves are similar.

The variation in physical properties, however, is likely

the Kinetics of the Process of Coagulation. 309

to show the validity of the fundamental equations of
Sinoluchowski. The fact that the curves showing the
change in viscosity with time are dissimilar shows that
these assumptions are not justified, and Smoluchowski
thinks that “T” is dependent on the magnitude of the
aggregates.

Since as yet it is not possible to express physical pro-
perties—e. g., the viscosity or the absorption of heht—in
terms of definite functions of the number and size of
particles, a quantitative comparison of different sols is
not possible by indirect methods. We have, therefore, to
restrict ourselves to the same sol.

Experiments with Gold Sols.—An examination of the
changes in the colour of gold sols on the addition of an
electrolyte showed (Mukherjee and Papaconstantinou, Trans.
Chem. Soe. vol. exvii. p. 1563 (1920)) that the variation in
the absorption of light of old sols affords an easy and
accurate method suitable for this purpose. The gold sols
prepared by the nucieus method of Zsigmondy ‘conform
very nearly to the requirements of equations (1) to (3)
in so far as the particles are fairly uniform in size. It
would be very convenient to work with a sol with re-
producible properties, as data obtained on different dates
with different preparations could be rigorously compared.

It was found that a sol on standing for some time under-
goes somewhat irregular changes, which may in part be due
to dust particles cetting in accidentally. In spite of all
precautions, one cannot be sure that there is no such variation
in a particular sample. This variation is not wholly due to
the fungus that grows in these sols. For this reason it is
necessary to vary one factor only at a time and compare
its effects.

The comparison was therefore restricted to the same sol
so long as it showed no variation in its properties.

The Constancy of “T” in Equations (1) to (8) during
the Process of Coalescence.

According to the simple assumptiens of Smoluchowski,
the progress of coalescence should be uniformly the same
for various electrolytes and for their different concentrations.
The constancy of * T” implies that if we assume a series of
consecutive stages of coalescence of a sol—under a definite
set of conditions, namely a definite electrolyte concentration

310 Profs. J.N. Mukherjee and B. C. Papaconstantinou on

and temperature—following each other by intervals of time
equal to “dt,” they are each characterized by a definite
number and manner of distribution of particles of each
category (primary, secondary, etc.). Let us indicate the
stage of coalescence corresponding to the time ‘“t” seconds
(since the sol and the electrolyte were mixed) under the
given conditions by the numbers

IN, No NANG ONG eee

where the subscripts refer to the number of primary particles
by the union of which the aggregate is composed. Thus
N; denotes the number of aggregates, each of which is
composed of “” primary particles. ‘“k” is evidently
an integer.

Similarly let us denote the stage of coalescence corre-
sponding to the time ¢’ (=t+At) by

> iN Ny’, Ne NGS ar ary N,,’ oe © e

These stages of coalescence are independent of external
conditions so long as equations (1) to (3) are valid. ‘Lhe
only change that external conditions can bring about is
a variation-in the value of T—that is, if the external
conditions are varied the sol will always pass through the
same consecutive stages of coalescence and only the rapidity
of succession of these stages will be determined by them.
Any property which varies continuously with the progress of
coalescence without having any maxima or minima can be
utilized to characterize the stages of coalescence ; for each
value of this property is characteristic of the time that has
passed since the mixing of electrolyte and sol. According to
the equations of Smoluchowski, the times taken to reach any
particular stage depend only on the value of ‘° T,”’ which is
constant under a definite set of conditions. Let us compare
two different electrolytes, A and B, of concentrations C,
and Cy, Let us suppose that after the time ‘‘t”’ the stage
ot coalescence indicated by

> Nj NEN Gee yes he N ates

has been reached when the electrolyte is “A” of con-
centration C,. This stage of coalescence has a definite
value for the physical property we are considering, and
is independent of the value of T. Let us assumé@ that “T, ”
and * T,” are the corresponding values of “‘'l’”’ for the two
cases. ‘To be definite, we shall consider the variation in the

the Kinetics of the Process of Coagulation. oii

total number of particles of all categories, which varies
continually with the progress of coalescence. Let us
assume that at the times “?¢,” and “¢t,” both electrolytes
have reached a state at which the total number of particles
is the same. From equation (1) we have, therefore,

No No Kk
= = PN a ee AM 2)
Sn ty ty (¢ )
I Ss m 1 ole mn
4 1 Dy
t, e .
nv. A . . . . . . 3)
I’ Is ( )

The general equation (3) may be written as

vw (BP
Tei

- Fae
Since m and & are constants, if 7 is constant, n; has a

(10)

2d

4 a ‘ag which is deduced
I, I;

from the condition that nn has a fixed value, also implies
that the values of 71, no, nz, ... 7, are the same in both cases.
This means, in other words, that a definite value of Sn fixes
unequivocally the stage of coalescence. ‘Therefore, from
the deduction that the successive stages of coalescence are
always the same and depend only on the time, any property
of the sol that varies continuously can be utilized to re-
present a fixed value of Sn or n, or a definite stage of
coalescence. A definite vaiue of this property is thus
characteristic of the stage of. coalescence. It also follows
from the above considerations that all curves showing a
variation of this property with time should be similar. A
deviation from this similarity, in itself, would mean that
equations (1) to (3) do not represent the facts,

The absorption coefficients of gold sols for different wave-
lengths change on addition of an electrelyte in a complex
manner. The theories of the colour of these sols as
advanced by Maxwell Garnett (Phil. Trans. vel. cci. A,
p. 385, 1904; vol. cev. A, p. 237, 1906) and by Mie
(Ann. der Phys. [4] vol. xxv. p. 377) would lead one
to expect that-any change in the number and manner of

fixed value—that is, the condition

ad

312 Profs. J.N. Mukherjee and B.C. Papaconstantinou on

distribution of the particles n,, m», etc. will produce a
great change in the optical properties of the sol. This
is in agreement with observations. Now, if the successive
stages of coalescence were independent of the nature and
concentration of the electrolyte, then the manner of
variation of the complex absorption would be the same
in each case. The absorption in the red region of the
spectrum varies continuously, corresponding to each value
of the absorption coefficient for a particular wave-length in
this region ; the values in the other parts should be fixed.
If the contrary holds good, then the conclusion is obvious
that the successive stages of coalescence are not inde-
pendent of the nature and concentration of the electrolyte
as assumed by Smoluchowski.

It has been found that for the stage indicated by the
value of the coefficient of absorption for 683 pu=0°4985,
the values of the coefficient for the other wave-lengths
given in the following table in column II. are independent
of the nature of the electrolyte.

The concentrations of the electrolytes were such as to
produce rapid congulation. In columns III. and IV. the
coefficients of the “nucleus sol”? have been given for
the original sol and for the stage of coagulation cha-
racterized by the value of the coefficient for 683 ppe=0°4156
(Mukherjee and Papaconstantinou, loc. cit.).

TaBue I.
Absorption coefficients (4).
Wave-length, a Se oe ieee ae
in pp I II. III IV
O83 onuease: 0:0453 04985 0:0376 0°4156
OOD ye ares 0-1055 0°3679 O-1131 0°409
DSOs eee 0-3518 0:3388 0-1595 03986
DOS Ae 0:2076 0°3294 0°2076 0°336
OAT. haa 0:2512 0:3238 0°2867 03732
O23 3.0... | '0:3780 0°3780 0:3780 0°3882
DOG IER. 0:4647 0°3581 0°3882 0°3780
AO ee ree 0°3581 0°3198 03581 0°3780

Comparison of the Values of “T” as a Test of
Smoluchowski’s Theory.

Since the absorption coefficient in the red region varies
continuously with the coagulation and its magnitude is

the Kinetics of the Process of Coagulation. 313

sufficiently great, a definite value of the absorption co-
efficient for a fixed wave-length (683 uw) can be taken
as representing a definite stage of the coalescence.

In the following tables the absorption coefficients at
different times are given for the wave-length 683 yy.
The tables are taken from the paper by Mukherjee and
Papaconstantinou, loc. cit.

TABLE TE

Electrolyte: Potassium Chloride.

Time in minutes, Absorption coefficients
after mixing for various concentrations.
equal volumes of - ——_—- aan
electrolytes and sol. N/24. N/26. N/28.
—- 0:0453 0°0453 0:0453
(CG See yee 0°3732 0°2867 0 1683
ee Sar es 0°438 0°3650 0:2257
fA ieee ane 0°4497 0°4046 —
1? ee eee — . 0488 0:2777
ete cia. -—— 04497 03431
EH ee ptt _ — 0°3836
+ ae RE -- -= 0:4263
11) Ace ec — — 0°438
Ai 4 Seat — — 0:4497
TaBueE III.

Potassium Nitrate.

Concentrations.
Times. — “~ ~
N/24. N/26. N/30.
Se Ph aaek 0:0453 0:0453 0:0455
OPA ne 0°3336 — =
i LAA She es 2 0°4263 0:2866 0269
ee eee 0:4497 0°3271 031438
Pe ag Ye ee — 0°3629 0°3356
Ee ae — 0°4156 0°3732
(A eee — 0°438 0°394
Tee? ~- 0:4497 —
SMa ae — — 0°4263
WO ce ano es -- -—— 0:438

16 -— — 14497

314 Profs. J.N. Mukherjee and B.C. Papaconstantinou on

TABLE IV.
rb Chloride.
Concentrations.
Times ————= aS : =
0°852N/900. 0°852N/1000. 0°852N/1100.
Se) 5. eae 0:0453 0°0453 0:0453
Ma aaicene 8 2 O°2257 — 0-1603
Ay Se MN ae 0°2867 — 0:2007
ZO ae ee 0°3529 — 0:2687
fy Re 0°3836 0°3051 0°3051
To ees 0438 0:3431 0°3237
Si aa 0:4497 — 0°3336
Oy Gian 0:4497 0°3836 0°3529
a: Gy Pees ee — 0°4263 0°363
[Sia eae = 0:4497 0:363
Gita dee ce: — — 0°3732

The limits within which the rate of coalescence could be
varied were restricted by the fact that when the rate is
slow the particles begin to settle, leaving a clear layer
at the top, and the measurements are not reliable. Also,
with time, some of the particles stick to the sides of the
vessel. Lastly, it is difficult to avoid dust particles for
a long time.

The values given in Tables II.-V. were plotted graphi-
cally, and the time intervals given in Tables V.-VII. below
were determined from these curves.

Hach of these curves is characterized by a definite value
of T (ore). Corresponding to the three concentrations of
any one of these electrolytes, there are three intervals which
must pass in order that the absorption coefficient may have
the same value. These intervals are co-related by the
following relation according to equations (1) to (3)
or (6) :— ain
Oy ace aes
ea SD, gee ie). (a)

or

bots: t; = 1): 2 le ey ees (12)

Dalene
Eee

Since T,, T,, and T; are constant, the ratio of the time-
parameters corresponding to the same absorption coefficient
should be independent of the absolute value of the absorp-
tion coefficient. Corresponding to different values of the
absorption coefficient we get different values of ¢), ¢:, and fs.

All these values should show a constant ratio. In the

the Kinetics of the Process of Coagulation. 315

following three tables this comparison is made for the three

electrolytes mentioned in Tables II.-IV.

Values of
absorption
cvetlicient.

Absorption
coefficients.

Absorption
coeflicients.

TABLE V.

Electrolyte : Potassium Chloride.

Time in seconds,

—-oO;

N/24.

N/26.

— —————

di
N/28.
a
255

sete er eeseoe

TABLE VI,

Hlectrolyte : Potassium Nitrate.

Times.
— pe
N/24 N/2s.
ie a
a ogee 105
30 j or a
45 165
60 210
90 300

Extreme variation from average

N/24.
~
135
165
225
265

Extreme variation from average

Average

ey

TABLE VII.
Electrolyte: Barium Chloride.

Times.

—_

N/26.

277
345
430
480

Average

eeeeee

Peewee

i 2°47 9:44
1 wei 2°) O's
1 Oe 3 13:0
il 274.--> 12-0

eee 11-4
14% 19°

Ratios.
Dune ey eee Dee
We rae O42 4
he SSG) es Sain Nal

Reese cor | se sOr2
Ieee. oor
haeicao! | (OS.

Bre 40 83,77

Ratios.

OS hes Le
il 2 POS,
l 7g) eae.
1 IOs Dt
1 ist oO

It will be seen from Tables V. to VII. that the agreement
is as good as can be expected. The variation in T is as great

as 1] times, but the ratios are constant.

The agreement

316 Profs. J. N,. Mukherjee and B. C. Papaconstantinou on

shows that the ratios of the values of T are independent of
the time or the stage of coalescence. The ultramicroscopic
measurements so far made show even during one experiment
a much greater variation in ‘I’, as will be evident from the
following tables :—

Tasue VIII. (a).
(Observer: Zsigmondy.)

Values of #’= -

Series D. Series H. Series F.
0-083 0°105 0-040
0-028 0-058 0:0195
00302 0-049 0:0183
0:0309 0:0475 0°0153

-—— 0:0403 0:0187

= — 0:0126

Zsigmondy used high concentrations of electrolyte for
securing a rapid rate. When the rate of coagulation is
slow and the duration of experiment is greater than a
few minutes, he found that impossible values of ' are
obtained. He thinks that the presence of impurities in
the water used in diluting the sol for ultramicroscopic
observations is the cause of this irregularity. In his case
the miximum time covered by the experiments is 80 secs.
Similarly, Westgren and Reitstétter, working with coarse
gold sols, find the following range of variation in the
constant :—

Tasrn VITO).
(Observers: Westgren and Reitstotter.)

Values of a

Series I. Series II. Series LILI. Series IV.

O74 2:56 2°75 3°41
247 2°81 2°60 2°80
2°07 2°33 217 2°60
2:10 2°31 2°40 2°48
2°09 2°31 2°12 2°14
1°62 —- — —
1-41 2°16 2°15 2°15
a wy) — 2:05
VONAGE apehen opens 2-2 2°38 2°36 2°19

Extreme variation... 75 7% 10 % li % 55 7%

the Kineties of the Process of Coagulation. 317
Kruyt and Arkel *, working with selenium sol and very
slow rate of coagulation, find extr emely wide variations in T’
in the same exper ut
PAsnE IX:
(Observers: Kruyt and Arkel.)

Values of T (in hours).

if Ti al IV.
28 260 131 1:3
51 390 55 3-4

44 270 52 2:2
(43) 320 54 43
(157) 600 68 10°5
200 sey (0 55 40
a 510 a *
= 440 48 37
ae = = 52
: af a 38

The above few instances will suffice to show the range of
variations in ** T” during the course of one experiment that
has been observed in the ultramicroscopic measurements.

Considering that in Tables V. to VII. the ratios between
the different values of T are taken, the range of variation
is extremely small. The actual deviations in the value of T
in any one experiment must be much less than the extreme
variations given. ‘This comparison leaves no room for doubt
that “T” is a constant in the case of gold sols and within
the limits of the rate of coagulation that have been studied.
In fact, these data constitute the best evidence so far recorded
in favour of the theory of Smoluchowski.

The Dependence of € on the Concentration.

Tables V. to VII. show clearly how rapidly e, the
percentage of successful collisions, increases with con-
centration. A change of concentration in the ratio 24
to 28 increases the rate in the ratio 1:11 or 1:6 as the
case may be. It wouid be extremely interesting to work

with a sol which is less susceptible to impurities tan these
gold sols.

* Ree. Trav. Chim. Pays-Bas, vol. xxxix. [4] p. 656 (1920); [4]
vol. xl. p. 169: (1921).

318 Profs. J.N. Mukherjee and B. ©. Papaconstantinou on

Variation of T or € with Temperature. )
Similarly, by determining the times required to produce a
definite change in the colour of the sol for the same electrolyte
concentration but different temperatures, we can determine
the variation in e with temperature.
From equation (3),

pail No
. ae we Nt
we get
No pe
~=1 Oct. i Se
Sune (13)
Since a definite change of colour is being used, a is
constant, or . 8
14+ 8 .¢,.¢ = %), 2 constant. “2..7) aaen,
Substituting the value of 6 in (14), we get
4 Ra ° d - No ~
I+3 No.7 INS 0 : ° . > (Gi,

Since Ra, No, and zp» are constants, we have

fb . G
1)

The viscosity of colloidal gold solutions has been found to
be practically equal to that of water, and the variation with
temperature can be assumed to be equal to that of water.
For different temperatures we have

= k'5 a Constant, (A... 6); oo eal

iy JOnees (tty Ce ves

hg a eee
1) UP Sie

Since ¢, is experimentally determined and @ and 7 are
known, variations in € can be compared.

The experimental data are given below. They are taken
from the same paper (pp. 1570-71).

TABLE X. |
Temperatures.
— aS —
Electrolyte. Standards *, Le. 30°. 50°.
N/30 Potassium chloride... V Sol.C. Simin. 10 min. 8 min. 80 see.
N/30 Potassium sulphate. _,, 3 WL 6 dO'sec.’.. 10 see: 10 sec.
N/30 Potassium nitrate .. ,, NG Dae: any Wh 55 Leas

* These refer to the protected gold sols used as standards for
comparison of colour. See Zoc. cit.

the Kinetics of the Process of Coagulation. 319

TABLE XI.
Electrolyte: Barium Chloride. Sol. E.

Temperatures.

Con- — on eS
centrations. Standards, 15°. 30°. 40°. 50°,
0°852 N/1000 ve Tmin. 6min. 4min.50ser. 4 mir.:20 sec.
0°852.N/1000_ ~—s: BB, 34 ,, BS\. a> — Tati oO

0°852 N/1200 Mie Lae US ee 12 min. 30 sec. Gee? Lo. |;

0°852N/1200 3B, 124 ,, Oe) 333 62 min. or
TasLe XII.
Electrolyte : Strontium Nitrate. Sol. F.
Temperatures.

Con- = i eis ve
centrations, Standards. Lae. 30°. 50°,
N/1000 V; 1 min, 10 see, 20 sec. 8 sec.

N/1000 By S.xie nib be 1 min. 40 see. 45

”

At 15°, 30°, 40°, and 50°, 7/0 has the values 3°96 x 10~°,
wert), 2-1 10-5, and “1°7 x 10~ ‘respectively. The
values for the viscosity are taken from the tables in Kaye

and Laby’s book on Physical and Chemical Constants,
peau; 1919.

Krom equation (17) we have
Ey50 + Exq0 * Ego * €509
= (/t@) 15° s (7/t@) 30° (7/tO) 40° 5 (/t®) 50°.
TasLEe XIII.

em peratures.
Electrolyte. 5°. 39°. 50°.
NSO KG) Yas.'ss083 N/t@ x 107 1-32 0°50 0°33
Bi a0 WSO) 9 5.22. aN 13:2 35 ly
W/o) KNO, 2.32: % 9:43 184 14:0
TasLe XIV.
Electrolyte: Barium Chloride.
Temperatures.
» Con- pa eee
centrations. Standards. LOTS eee Oe.” BAS:
0°852 N/1000-...... VE n/t0 x 108 9°45 972 7:24 6:54
ea Suet. B, ‘3 1°94 2:4 -- 2]
Ratio between { ae | eer pepe Ieee TT i 70
ID os caiccerey al ihe se dak 100 52). : 108
0°852 N/1200 .. v. n/t@ x 10° 2°87 4:24 2:8 4:5
53 Par tot By; rt 53°22 745 64 —
Ratio between { 2 Bera eee 1cGne> ie : 98 : 158
TL a, Oe BEEGy ‘bigs ecesccus Wes 140; 106 : —

320 The Kinetics of the Process of Coagulation.
TABLE XV.

Hlectrolyte : Strontium Nitrate.

Temperatures.
Concentrations. Standards. 150. aan 50°.
INT NOOO, cocen ces V, /t@x 106 "965 1°65 271
SS: ae re B, /t0x 107 “80 3°93 O77
atio waiween: (vie eeeenee 1007 2 290 ou
iy COMM eee | Boece ee 100 : 410 : 480

Since 7=¢6 is a constant for a definite electrolyte con-
centration and temperature according to Smoluchowski’s
equation, the ratios should be independent of the standard
used. This is true within the limits of experimental error
with -852 N/1200 Barium Chloride. In the other two cases
the variations are not great considering that we are com-
paring the ratios. A slight variation in each value will be
magnified in the ratio. Taking into account the probable
experimental error, it can be said that eis roughly constant
in each experiment.

On the other hand, the variation in e with temperature is
considerable. We have already seen that the irreguiarity
in the variation of e means that the precipitating power of ©
the ions changes with the temperature (Mukherjee, Trans.
Chem. Soc. vol. cxvii. p. 358, 1920).

Further experiments with arsenious sulphide are in
progress on similar lines.

Summary.

(1) It has been shown that the equations of Smoluchowski
on the rate of coalescence of the particles of gold sols agree
with the results obtained by the writers.

(2) It has been suggested that the disagreement of the
ultramicroscopic measurements with this theory may in part
be due to the difficulties inherent in them.

Our best thanks are due to Professor F. G. Donnan for
his kind interest and encouragement, and also to our friend,

Professor J. ©. Ghosh.

Physical Chemistry Department,
University College, London,

XXIX. The Adsorption of fons. By JNANENDRA Natu
MUKHERJEE, D.&c., Professor of Physical Chemistry in
the University of ¢ jale GED >.

+ a paper in the Transactions of the Faraday Sceciety

(Far. Soc. Dise. Oct. 1921) an attempt has been made to
define the nature of the adsorption of ions to which the
origin and the neutralization of the charge of a colloidal
particle are due. The origin of the charge was assumed to
be due to the adsorption of tons by the atoms in the surface
as a result of their chemical affinity.

It was pointed out that the adsorption of one kind of ions
will impart a charge to the surface, in virtue of which ions
of opposite sign will be drawn near the surface. In the
liquid there remains an equivalent amount of ions.of opposite
sign. The electrical energy will be a minimum when these
ions are held near the surface so that the distance between
the oppositely charged ions has the minimum value possible
under the conditions, and they will be held opposite to the
ions chemically adsorbed. An “ion” so held will not be
“free” to move if its kinetic energy is less than “ W ” the
energy required to separate the ion from the oppositely
charged surface. The number of such “‘ bound ” ions deter-
mines the diminution ia the charge of the surface. When
the concentration of ions of opposite charge in the liquid is
small the number of ions “ held” to the surface by electrical
attraction will be small.

If the chemically adsorbed ions have a valency equal to
€ N,,” and if ‘ N,” is the valency of the oppositely charged
ions in the liquid in contact with the surface, then

where H = the electronic charge, « = the distance between
the centres of the ions at the position of minimum distance,
and ‘* D)”’ is the dielectric constant of water.

Depending on the concentration of the oppositely charged
ions in the liquid near the surface, at any instant a certain
number of the “chemically adsorbed” ions are ‘* covered ”
by ions of opposite charge. In the liquid near the surface
there are always a number of free ions equivalent in amount
to the “ uncovered’’ chemically adsorbed ions on the surface.
The total amount of ions of opposite sign both “ bound”’ and

* Communicated by Prof. F. G. Donnan, F.R.S.

Phil. Mag, Ser. 6. Vol. 44. No. 260. Aug. 1922. od

322 Prot. J. N. Mukherjee on

‘“‘free”’ is equivalent to the amount of ions ‘ chemically.
adsorbed.” These “‘ free”? ions form the second sheet of the
double laver. It is evident that as a result of their thermal
motion the mean distance between the two layers will be
greater than “2.”

The charge of the surface was treated as due to discrete
charged particles widely separated from each other compared
with molecular dimensions. [It was shown in the previous
paper that this view gives a rational explanation of the fact
that a reversal of the charge of a surface can be brought out
only by polyvalent ions of opposite charge.

The equilibrium conditions were iieenesed and the equa-
tions deduced were shown to be in agreement with the
valency rule, the influence of the mobility of the oppositely
charged ion, and with the influence of concentration on the
charge of the surface. Oniv the theoretically simplest case
has been discussed in the earlier paper. In the present paper
the more important facts connected with the adsorption of
ions are discussed from this point of view, and it will be
seen that this view gives a simple explanation of most of
the general conclusions already arrived at on experimental
grounds.

Theories of Adsorption.

Before proceeding to discuss the adsorption of ions it
will be convenient to deal briefly with the different views
advanced to account for adsorption in general. The with-
drawal -of a solute from a solution by a solid may be the
result of the formation, of definite chemical compounds, of
solid solutions, of mixed crystals and surface-condensation.
In many cases all these changes are simultaneously present.
In this paper the word “adsorption ” denotes condensation
or combination, at the surface only, without the interpenetra-
tion of the adsorbed substance throughout the mass of the
adsorbent (Mecklenburg’s criterion, Ze Phys. Chem. \xxxuii.
p. 609 (Lona ae: also the sense in which the term is used
in deriving Gibbs’s equation).

Faraday (Phil. Trans. exiv. p. 55 (1834)) in his well-known
explanation of the catalytic combination of hydrogen and
oxygen on platinum surfaces, remarks “‘that they are de-
pendent upon the natural condition of gaseous elasticity
combined with the exertion of that attractive force, possessed
by many bodies, especially those which are solid, in an
eminent degree, and probably belonging to all, by which they
are drawn ks aaaooian fan quore on. lee close, sulinemi at ae
same time undergoing chemical combination though often

the Adsorption of Lons. Sea

assuming the condition of adhesion, and which occasionally

leads under very favourable Genre as in the present
instance, to the combination of bodies simultaneously sub-
jected to this attraction.” It is remarked further “that the
sphere of action of particles extends beyond those other
particles with which they are immediately and evidently in
union, and in many cases produces effects rising into con-
siderable importance.” These remarks of Faraday mean, in
modern terminology, that there is a sort of combination at
the surface and that the transitional layer is more than one
molecule thick. The subsequent views are in a way develop-
ments of this conception. |

Gibbs treated adsorption from the standpoint of thermo-
dynamics. A number of important investigations has been
earried on by Milner (Phil. Mag. [6] xiii. p. 96 (1907)),
Lewis (Phil. Mag. [6] xv. p. 506 (1908)), sbed. xvii,
p. 466 (1909)), and Donnan and Barker (Proc. Roy. Soe.
Ixxxy. A. p. 552 (1911)). The present position is that the
amount adsorbed is often considerably greater than what
could be expected from Gibbs’s equation.

J.J. Thomson (‘ Applications of Dynamics to Physics and
Chemistry’) showed that it follows from Laplace’s theory of
capillarity that in the surface layer between two liquids,
chemical actions may take place which are absent in the
bulk of the liquids.

Lagergren (Bihang K. Svenska Vet. Hand. xxiv. p. 11,
No. 415 (1898)) considers that adsor ption in the surface of
solids in contact with aqueous solutions is due to the com-
pressed state of the water in the surface layer.

On the experimental side the work of Freundlich and his
collaborators—|[ Kapillar-Chemie, 1909; Z. Phys. Chem.
lix. p. 284 (1907); Ixvii. p.538 (1909); Ixxiii. peogs) (1910) ;
ixxxill. p. 97 (1913); Ixxxv. p. 398 (1913); men py bol (1915);
Koll.-Chem. Beihefte, vi. p. 297 Rees) : see also Schmidt,
Z. Phys. Chem. \xxiv. p. 689 (1910) ; Ixxvii. p. 641 CEPEL). 3
fxxviil. p. 667 (19%); lxxxiil. p. 674 ( (1913); xci. p. 103
(1916). In the last-mentioned paper Schmidt and Hinteler
conclude that Freundlich’s equation represents their experi-
mental data better than that of Schmidt |—and of others, have
shown that adsorption-equilibria can be generally expressed
in terms of the well-known equation of Freundlich:

fy ee 1/ 2
Bim Ce 2! ew Ce)

Freundlich expressed the opinion that adsorption is mainly
due to a decrease in surface tension as suggested by Gibbs.

¥ 2

324 Prot. J. N. Mukherjee on

In the case of adsorption of gases by solids, Arrhenius
(Medd. f. k. Vef. Nobelinstitut, ii. N. 7 (1911); Theories of
Solution, 1912, pp. 55-71) has drawn attention to the
parallelism between the van der Waals’s coefficient “a” for
the different gases and the amounts of these gases dee
by charcoal, and he believes that this is deans evidence of

the compressed state of the surface layer. At the same time .

he lays stress on the chemical aspect—namely, that in
addition to the attractions between the molecules of the gas
in the surface layer, one has to consider the chemical
attraction of the surface atoms and the molecules of the gas.

Recently, Williams (Proce. Roy. Soe. xevi. A. p. 287 (1919) ;
xevil A. p. 223, (1920); also Trans. Mar. Soc. x.) glee
(1914), in which complete references to the literature on
negative adsorption are given) has treated adsorption from
the points of view of Lagergren and of Arrhenius in a number
of interesting communications.

It may be mentioned here that the disagreement of ob-
servations with calculations from Gibbs’s equation is at least.
in part due to the fact that only one source of change in the
free energy of the surface layer is taken into account. In
the simplest case ot the interface, liquid-saturated vapour
(one component system), it is open to objection whether “vy”
denotes the total change in free energy of an isothermal and
reversible-formation of unit surface. Bakker (7. Phys. Chem.
Ixvui. p. 684 (1910)) has pointed out that if the density of
the surface layer is different from that of the liquid in bulk
a second term is necessary to represent the change in free
energy.

It is possible that in this particular case this second term is
negligible in comparison with “‘y,” the tension os unit length
at low temberatures, but at hai temperatures « ” has a low
value and the saturation pressure is very pee so that the
second term may be even more important.

Williams (Proc. Roy. Soc. (Edinburgh), xxxviii. p. 23
(1917-18) ) has drawn attention to the effect of the variation
of the ee of an adsorbent when adsorbing—a factor
which is very often neglected.

Lewis (Z. Phys. Chem. |xxui. p. 129 (1910) ; also Par-
tington, ‘Text-book of Thermodynamics,’ p. 473 (1943)
has discussed the influence of a variation in the electric
density on the surface on the form of Gibbs’s equation.

These may be called the physical theories of adsorption.
The difficulty in accepting them as general theories of ad-
sorption is that they attempt to explain adsorption in terms
of a single physical pce e.g. diminution in surface energy

the Adsorption of Lons. 329

or a layer under great internal pressure. The necessity for
recognizing the existence of a sort of chemical interaction (as
Arrhenius has suggested) becomes evident when one con-
siders the specific nature of adsorption processes. This point
has been justly emphasized by Bancroft in recent years.
Besides his papers in the ‘Journal of Physical Chemistry,’
compare ‘ Applied Colloid Chemistry,’ 1921, p. 111).

The chemical point of view has been pe clearly by Lang-
muir (J. Amer. Chem. Soc. xxxvili. p. 2221 (1916) ; RIK
p. 1848 (1917)). He believes that adsorption is due to the
chemical affinities of the surface atoms. Considering the
thermodynamic equilibrium between molecules of a gas at
the surface and those in the surrounding gas he has deduced
the following equations correlating the variation of the ad-
sorbed amount with its pressure,

where ‘‘ @,”’ is the fraction of the solid surface covered and is
a measure of the amount adsorbed, v, is the rate at which
the vas would evaporate if unit area of the surface were
completely covered, “‘w” is the number of gas molecules
striking unit area of the surface per second and is given by

o=—43'15 x10 aT A and ‘“‘»’’ denotes the pressure of
the gas, ‘‘T” its absolute temperature, and ‘“‘M” its molecular
weight. « denotes the fraction of the total number of
collisions of the molecules of the gas that leads to a condensa-
tion on the surface; it is usually close to unity and evidently
can never exceed unity. Some Interesting applications of
his theory to catalysis of gaseous reactions by solid surfaces
are given, This theory explains many phenomena which are
otherwise difficult to understand.

Michaelis and: Rona (Bio-Chem. Zeitsch. xevil. pp. 56, 85
(1919)) conclude from the investigations of Michaelis and his
co-workers that the assumption of special forces at the sur-
face fails to account for the facts and that adsorption is the
result of chemical affinity.

I. TheAdsorption of a Constituent Ion by a Precipitate.

The adsorption of ions is different from the adsorption of
neutral molecules or groups in that it introduces a new
factor—an electrically charged surface. The variation in
the electric charge enables us to follow the net effect of the
adsorption of the two ions, as the electric charge depends

326 Prof. J. N. Mukherjee on

only on the total number of ions (of both signs) fixed per
unit area of the surface. Kataphoretic and electro-endosmotic
experiments give us a quantitative idea of the relative ad-
sorption of both ions.

_ The electric charge helps to peptize the adsorbent, and a
qualitative idea of the adsorption of ions can be formed
from peptization by electrolytes. An insoluble precipitate
formed by the union of two oppositely charged ions has a
marked tendency to adsorb its component ions. In many
cases the connexion between the adsorbed ion and the
electrical charge has been established. These instances have
been given in the earlier paper. The nature of the chemical]

’ forces responsible for this adsorption has also been defined.

Instances of adsorption of ions as judged from peptization by
electrolytes are given below.

Bancroft (Rep. Brit. Assoc. p. 2 (1918)) remarks :—‘ It
seems to be a general rule that insoluble electrolytes adsorb
their own ions markedly, consequently a soluble salt having
ene lon in common with a sparingly soluble electrolyte will
tend to peptize the latter. Freshly precipitated silver halides
are peptized by dilute silver nitrate or the corresponding
potassium halide, the silver and the halide ions being ad-
sorbed strongly. Many oxides are peptized by their chlorides
and nitrates, forming so-called basic salts. Sulphides are
peptized by hydrogen sulphide. .... The peptization of
hydrous oxides by caustic alkali can be considered as a case
of adsorption of a common ion or as the preferential adsorption
of hydroxylion. Hydrous chromic oxide gives an apparently
clear green solution when treated with an excess ot caustic
potash ; but the green oxide can be filtered out completely
by means of a collodion filter, a colourless solution passing
through.”

“Hanztsch considers that hydrous beryllium oxide is
peptized by caustic alkali, copper oxide is peptized by con-
centrated alkali, and so is cubalt oxide. In ammoniacal
copper solutions part of the copper oxide is apparently colloidal
and part is dissolved. Freshly precipitated zine oxide is
peptized by alkali, but the solution is very unstable ” (cp. also ~
negative hydroxide sols— Freundlich and Leonhardt, Koll.
Chem. Bethefte, vii. p. 172 (1915)).

At least in some of these cases the formation of new com-
plex anions is possible, and it is not definitely known to what
ion the peptization is due. Regarding the peptization of
stannic acid gel by small quantities of alkali, Zsigmondy
(Kolloidchemie, p. es et seq. (1920); also Varga, Koll.
Chem. Bethefie, xi. p. 26 (1919)) remarks: “Dieses kann

the Adsorption of Lons. 327

sowohl auf Adsorption des gebildeten Katiumstannats wie
auch darauf ziiruckfiihren sein, das Kaliumhydrat mit
den Oberflichenmolekeiilen der Zinnsiureprimiirteilchen in
Reaktion tritt, wobei diese von der Oberfliiche der Primiir-
teilchen festgehalten werden.”

The view suggested by the writer to account for the ad-
sorption of a common ion, leads one to expect that ions
which can displace one of the constituent ions in the crystal
lattice should also be adsorbed. Mare (Z. Phys. Chem. Ixxx1.
p- 641 (1913)) has observed that crystalline adsorbents adsorb
erystalloids to any marked degree only when they can form
mixed erystals with them and are isomorphous with them.
Paneth and Horrowitz (Physik. Zeitsch. xv. p. 924 (1914))
have noticed that of the radio elements those only will be
adsorbed that can form insoluble salts with the common
ion of the adsorbent and can also form mixed crystals with
the adsorbent. This kind of adsorption is somewhat different
from the type we have considered for, as Paneth has pointed
out in his case, an actual interpenetration of the two non-
common ions is occurring in the crystal lattice. Thus
radium is taken up by barium sulphate giving out to the
solution barium ions in exchange. Such an interthange will
not impart a charge to the surface.

Attention may also be drawn to the explanation advanced
by Bradford (Biochem. J.x. p. 169 (1916); xi. p. 14 (1917) )
to account for zonal precipitations, first studied by Liese-
gang. Bradford thinks that the adsorption of a constituent
ion is responsible for their formation. From the numerous
instances given above, this conception seems to be quite
plausible. It is probable that other factors have also an

influence on the process (Hatschek, Brit. Assoc. Rep. p. 24
(1918)).

Il. The Variation of the Density of the Electric Charge
with the Concentration of an Hlectrolyte.

In the previous paper the particular case when the charge
of the surface is due to strong chemical adsorption of ions of
one kind and when the added electrolytes have not any ions,
subject to the chémical affinity of the surface atoms, has been
fully treated. In this case it was assumed that the number
of ions adsorbed at the surface by chemical affinity remains
constant. The experimental data of KElissafoff on glass
and quartz agree well with equations deduced from these
assumptions, on the basis of the theory of . electricat
adsorption.

328 Prof. J. N. Mukherjee on

The general case, however, is that :

(a) At low concentrations the density of the charge on
the surface at first increases to a maximum and at higher
concentrations falls gradually towards a null value when the
oppositely charged ions are monovalent.

(b) On the other hand, when the oppositely charged ions
are multivalent or complex organic lons the charge passes
through a null value, becomes reversed in sign, and again
reaches a second maximum, after which it falls slowly
(Hillis, 7. Phys. Chem. Ixxvit p. 621 (A911) > lxxxo pea
(GEO ix x aie P 145 (1914); Powis, Z. Phys. Chem.
Ixxxix., pp. 9, 179 (1904) 3" Riety, ‘Compi.” wena netme
pp- 1411, 1215°(1912)5 ela p. 1368" (1913) > Youre ame
Neal, -/. Pigs. Chem: xxiep. a UGT is akGrny te Verst. Kom
Akad. v. Wetensch. Amsterdam, 27th Juin, 1914, also Koll.-
Aevesch. XXile p. wed (VILS)).

The usual explanation is as follows :—

The adsorption-isotherms for the two ions can be written
as

| Ds
ee and aye, we
where the subscripts A and K refer to the anion and the
cation respectively. To explain the increase in the charge
at low concentrations it has to be assumed that

as OAK and Ba < foure aT sto de ° (4)
Thus in a paper read at the Discussion on Colloids arranged

by the Faraday and the Physical Societies of London,
Svedberg remarks: “* Now as a rule, it happens that for the
two ions of a salt both a and 8 have different values, e.g.

x (cation) <« (anion)

B (cation) > B(anion).”

It is clear that the equation of the adsorption-isotherm can
be reconciled with the first increase in the charge. But two
objections can be raised against this empirical point of view.
In the first place, no reason is given why the constants « and
B shall have generally the relative values assumed above for -
the cation and the anion. Secondly, these assumptions can-
not explain the second maximum charge and the subsequent

decrease. observed with multivalent ions of opposite charge.
It will now be necessary to assume that

CL ier and Ba>Bx, . ahi ae . (5)

in direct contradiction to the assumptions already made

the Adsorption of Lons. 329

(ep. (4)). Besides, one cannot get any idea as to why the
anion is generally more strongly adsorbed at low con-

centrations.

The facts ean, however, be explained as follows :—

The negative charge of surfaces in contact with water is to
be sought | for in the chemical natures of the anions and the
cations. The simpler electrolytes (excluding dyes and
complex organic ions) have cations whose chemical behaviour
can be referred simply to the tendency of the component
atom (eg. of the alkali and alkaline earth metals) to pass
into the ionic state. These ions do not form any complex
ions. They form only one type of compounds that are stable
in aqueous solutions, namely, electrolytes with the atem
existing as a positively charged ion through the loss of one

o
or more electrons. On the other hand, the anions in general

co)

form types of compounds other than electrolytes, and also
_form complex ions. It is, therefore, possible to imagine that
anions are subject to the chemical affinity of the surface
atoms and that the chemical action on the cations is relatively
small. Complex cations like those of the basie dyes should,
for the same reason, be easily adsorbable. ‘This isa wok
known fact. ;

If now, the assumption is made that the chemical affinity
acting on the anion of the electrolyte added is stronger than
the electrostatic attraction of the surface on the cation, the
observed variation of the charge with the concentration of the
electrolyte is easily accounted for. This case corresponds to
a strongly marked maximum of a negative charge at a low
concentration of the electrolyte.

The initial charge of a surface in contact with pure water
can be due either :

(a) to the strong adsorption of an ion of a minute quantity
of suitable ‘electrolyte associated with the solid,
(b) or to the adsorption of hydroxy] ions from water.

On the addition of an electrolyte the density of the electric
charge will increase at low concentrations because of the
chemical adsorption of the anion. The electrical adsorption
of the cation is smaller as the chemical adsorption has been
assumed to be stronger. Besides, the electric charge of the
surface is also not at its maximum. As the surface becomes
more and more covered by the anions the rate of adsorption
da/de—where ‘‘ dx” is the increase in the amount adsorbed
per unit surface due to an increase in the concentration
‘‘de”’—rapidly decreases. Also, the electric charge repels
the anions, and those only can strike on it that have sufficient

330 Prof. J. N. Mukherjee on

kinetic energy to overcome the potential of the double layer.
The number of collisions is thus not proportional to the
concentration but rises more slowly. Near about the point
where the surface becomes saturated the value of da/de will
be almost zero (cp. the shape of the adsorption-isotherms.
of Freundlich, Arrhenius, and Langmuir). On the other
hand, the electrical adsorption increases continually with
the concentration and the increase of the charge. It is.
apparent that soon a balance will be reached between the
chemical adsorption of the anion and the electrical adsorption
of the cation. The minimum charge will correspond to the
stage when da/dc for the cation is just equal to da/de for the
anion.

Beyond this concentration the charge will decrease rapidly,
and when the surface has been saturated with the anion the
subsequent variation in the charge is simply due to electrical
adsorption. ‘The reversal of the charge by electrical adsorp-
tion has been discussed in the earlier paper. It is necessary
to add that as the electrically adsorbed polyvalent cations
impart a positive charge to the surface, the atsorption of the
cation decreases and the electrical adsorption of the anion —
becomes possible. As long as there is a positively charged
surface the adsorption of the anion will increase more rapidly
with the concentration than that of the cation. A second
maximum will thus be reached and a decrease in the charge
will follow. The electrical adsorption of the anion is small
because of the smallness of the positive charge and an
initially existing negatively charged surface. A further
reversal of the charge is not possible, and, in fact, has never
been observed.

TI. The Action of Acids and Alkalies.

The works of Perrin and of others (J. Chim. Phys. i.
p- 601 (1904); ii. p. 50 (1905); Haber and Klemensie-
wiez, Z. Phys. Chem. \xvii. p. 385 (1909); Cameron and
Oecttinger, Phil. Mag. [vi.] xviii. p. 586 (1909)) have shown
that hydrogen and “hydroxyl ions behave exceptionally in
that they impart to the surface a charge of the same sign as
they carry. This behaviour is in contrast to that of the
other univalent ions.

Perrin attributes their singular activity to the smallness of
their radii. In order to explain the presence of these ions,
in excess, in the surface layer, it is necessary to assume some
sort of a restraining force acting on them at the surface.

Haber and Klemensiewicz consider that there is an ad-
sorbed Jayer of water in the surface by virtue of which the

the Adsorption of Ions. 331

solid acts as a sort of combined hydrogen and oxygen
electrode. They treat the subject from the points of view of
thermodynamics and Nernst’s theory of electrolytic solution
tension. It has been pointed out by Freundlich (and
Rlissafoff, 7. Phys. Chem. Ixxix. p. 407 (1912)) that
hydrogen and hydroxyl ions are not the only ions which
impart a charge to the surface. In many cases, acids have
been observed not to reverse the charge at all. Many sub-
stances have a negative charge in contact with pure water.
These facts show that selective adsorption of hydroxyl ions
has also to be considered.

This thermodynamic treatment from the point of view of
Nernst’s theory does not attempt to explain electro-endosmosis.
For this purpose it is necessary to conceive of an electrical
double layer, of which the layer imparting « charge to the
surface is fixed relative to the mobile second layer.

Freundlich, and Freundlich and Rona (Koll. Zeit. xxviil.
5, p- 240 (1921); Kol. Preuss. Akad. Wiss. Berlin, 1920,
p- 397, C. 1920, ii. p. 26) have shown that the potential
measurements by Haber’s method are not in agreement with
those measured by electro-endosmotic experiments. They
therefore suggest that there are two distinct drops in
potential as one passes from the solid to the liquid (glass to
water). The first drop is wholly in the solid and is probably
of the nature associated with the Nernst theory of electrolytic
solution-tensions.

The second drop is in the liquid and composes the Helm-
holtzian double layer which it is necessary to assume to
explain electro-osmosis and cataphoresis.

At the same time the characteristic effects of hydrogen
and hydroxyl ions on neutral substances like barium sulphate,
silver chloride, naphthalene, etc., point strongly to the
correctness of Haber’s fundamental assumption that the

explanation is to he sought in the equilibrium between the
hydrogen and hydroxyl ions in the adsorbed layer of water
and those in the bulk of the liquid.

Williams (Proc. Roy. Soe. xeviii. A. pp. 223 (1920)) has
recently suggested that the layer of water adsorbed on a
charcoal surface is under great internal pressure (about
10,000 atmospheres). Applying Planck’s equation he shows
that the effect of this pressure will be to increase the con-
centration of hydrogen and hydroxyi ions in this layer.
This increased concentration will set up a diffusion potential.
He draws attention to the difficulties in accepting this view
of the origin of the potential difference at the surface. In
the cera considered by Haber and Perrin, the solid has

a32 Prof. J. N. Mukherjee on

little or na potential difference in contact with pure water,
and the considerations developed by Williams are not
applicable.

Case 1.—The surface 1s inert.

We shall assume that the atoms in the surface do not —
exert any chemical affinity on hydrogen and hydroxyl ions
as such, or on the dissolved acid (or alkali) with which it may
be in contact. The adsorbed water molecules behave as a
solid layer, being held by strong chemical forces (Haber,
loc. cit.; Hardy, Proc. Roy. Soc. lxxxiv. B. »..217 (aia
It is clear that the surface will be neutral in contact with
pure water. The molecules of water in the adsorbed layer
are in thermodynamic equilibrium with those in the bulk of |
the liquid. It is reasonable to imagine that a transfer of an
electron is taking place between the hydrogen atom and the
hydroxyl group in the water molecules in the surface layer,
as it does in the molecules in the liquid. That is, the water
jolene are dissociating into ions at a. definite rate. Let
“ne? be the number of water molecules (in the adsorbed
ye passing into the ionized phase per unit area per
second. For equilibrium, as many hydrogen and hydroxyl
ions are uniting to form neutral water molecules. Since the
adsorbed water molecules behave asa solid layer, recombina-
tions would take place mostly between adjacent hydrogen
and hydroxyl ions. The recombination will be extremely
rapid. It can be assumed that at any instant the number of
hydrogen or hydroxyl ions actually remaining free in the
surface will be a negligible fraction of the total number of
water molecules.

The neutralization of the ions being formed in the surface
layer can also be brought abont by impinging hydrogen or
hydroxyl ions present in the liquid. In contact with pure
water the probahility of such collisions is small, for the
concentration of hydrogen and hydroxyl! ions is extremely —
small. Thus neutralization of the ions being formed in the
surface layer is possible in two ways :

(1) H,° + OH,!—» HOH—the subscript “s” refers

to ions in the surface layer ;

(2))a(@ elo oe Ott el Ot,

(>) HP + OH,—+ HOH—the subscript “f” refers
. to the freely moving ions in the liquid.

In contact with pure water, neutralizations according
to scheme 2 are small in number. Also 2 (a) and 2(b) are
equally probable. Consequently the numbers of H,° and
OH," remaining in the surface at any instant will be equal,
and the surface will be neutral.

the Adsorption of Ions. 333

When an acid is added to the water the neutralizations
according to scheme 2 (a) will be Sener negligible, but
those according to scheme 2(6) will not be so. The total
number of neutral molecules of water formed in the surface
is still equal to “a,” but a number of them is now being
formed according to 2 (6). Corresponding to the number of
neutralizations according to 2(b), a number of hydrogen
ions will remain in the surface layer in excess of the number
of hydroxyl ions. The rate at which 2(b) proceeds thus
determines the free charge on the surface. An equivalent
number of anions remain unneutralized in the liquid and
form the second mobile sheet of the double layer.

The free charge on the surface will evidently increase with
rise in the concentration of hydrogen ions in the solution.
There are, however, two factors opposing this increase in the
charge of the surface.

A. The proportion of hydrogen ions striking on the surface
diminishes as the positive charge of the surface increases.
Only those ions which have sufficient kinetic energy to over-
come the electrical repulsion can reach it. If e€ be the
potential of the double layer in C.G.S. units, then the number
of collisions of the ions per unit surface per second is pro-
portional to

Uno : Cro - e-€-B/KT, Po Vile aie tiene (6)

where Cyo denotes the concentration of free hydrogen
ions in the liquid, ‘‘ Hi” is the electronic charge in C. GS.
units, T is the absolute ae ture, Ugo is the mobility of
the hydrogen i ions in water, ana K =R/No, where “ R” is the
gas constant and No the Avogadro number.

- B. The other factor that tends to diminish the charge of
the surface is the electrical adsorption of the anion of the
acid added to the solution. That this plays an important
part will be evident from the following examples taken from

the observations of Perrin :—
Rate of Electro--

Substance. Electrolyte. endosmotic outflow.

PASO, coca t<~ M/1000 HCl. + 110
2 RAGAN Ae M/1000 citrie acid + 5
Pee Perey M/1000 HNO; (or HCl) + 100
APEC enere M/1000 H.SO, + 15
CO re us vii 0 M/1000 HNO; + 85
jee, s. M/1000 H.SO, + 2l
jin Rees M/500 HCl + 90
Pete M/1000 H.C.0, + 30
as tea Feebly acid with HCl + 79

Pibege he: Solution of KH. (POx:) )
with approximately the | ete

4
same number of free
hydrogen ions as above

—— Oe SS eee ee

;
;

O34 Prof. J. N. Mukherjee on

Both these factors tend to diminish the rate of increase of
the charge with rise in the concentration of hydrogen ions.
For acids with simple univalent anions, the electrical ad-.
sorption at low concentrations can be left out of account in
view of the excessive mobilty of the hydrogen ions.

A quantitative relationship can now be obtained between
the charge on the surface and the concentration of the acid.
Let w! be the rate of neutralization according to 2 (b) above.

We have then
wi=K ia. Cy. 76: EAT, oy ea (7)

where & is a constant.

‘The density of the charge on the surface is proportional to
«'—which is a measure of the number of hydrogen ions
remaining in excess in the surface. If the thickness of the
double layer remains constant then the potential of the
double layer is proportional to the density of the charge:
HOVENG Ik NO) aban

When all the hydroxyl] ions in the surface layer are being
neutralized according to 2 (6) the surface will have a maximum
charge determined by ‘‘ w.”’

Putting «/z=@, since “x” is a constant, we have

ea 3.

and @ represents the ratio of the hydrogen ions present ‘in
excess at any instant in the surface layer to the maximum
number possible when the neutralization takes place only
according to 2(b). The potential of the double layer can be
written as

exh. a'=h,.0=ks.Cqo.e78-%?. Uno, . . (9)
or) OS ky: Ono.€ 70 YU 4 kr
where hy, ko, ks, ko and 8 denote constants.
Siailarly, for alkali solutions we have
O=hy.Com.e 8-9/2, Ucn. A ee hc (hay

The maximum charge, being determined by wz, will be the
same with alkali as with acid. Of course, the influence of
the oppositely charged ion in the acid or ie base is being
neglected.

Sa

CasE 2.—The surface is not chemically inert : preferential |
adsorption of one ion 1s possible.

A review of the literature shows that surfaces in contact
with water are seldom neutral. They are generally more or:
less negatively charged. This is intelligible in view of the

the Adsorption of Lons. 335

chemical! reactivity of the hydroxyl group. The presence of
the potentially tetravalent oxygen atom possibly leads to a
selective adsorption of hydroxyl i ions by most surfaces. Thus
glass and quartz have'a “marked negative charge in contact
with water (cp. Elissafoff). On the addition of an acid the

electrostatic forces will produce a diminution of the charge.
‘The electrical adsorption of hydrogen ions by hydroxyl ions
cannot be distinguished from the recombination of hydrogen
and hydroxy! ions to form neutral molecules of water. This
is confirmed by the fact that the equation of electrical ad-
sorption (cp. previous paper) satisfactorily represents the
diminution of the charge.

Perrin (loc. cié) found that, excepting alumina and chromium
chloride, all other substances (naphthalene, silver chloride,
boric acid, sulphur, salol, carborundum, gelatine, and cellulose)
show a preferential adsorption of hydroxy! ions. The sur-
faces have a negative charge even in contact with acid
solutions. He also found that at higher concentrations of
the acid the surface acquired a positive charge. LElissafoff,
McTaggart, Ellis, Powis, and others could not observe this
reversal in their investigations. Hlectrical adsorption of
hydrogen ions cannot lead to a reversal of the charge. The
reversal (or the non-reversal) of the charge becomes intelli-
gible if it is assumed that the considerations set forth in
deducing equations (8) or (9) are correct.

In contact with pure water the surface has a layer of
adsorbed water and a number of hydroxyl ions. The amount
of hydroxyl ions adsorbed by the surface will, in general, be
small, as the concentration of the hydroxyl lons is very
small in pure water. If, however, the adsorption is very
strong the surface will have a considerable negative charge.
On the addition of an alkali the negative charge of the
surface will increase, due to two reasons:

(1) the preferential adsorption of hydroxyl ions will
Bec. and

(2) the number of hydrogen ions being formed at the
surface will be more and more neutralized by hydroxyl ions
in the liquid (cp. scheme 2(a)). A maximum will be
reached when the surface is saturated by preferential ad-
sorption and when @=1 in equation (9). The maximum
charge per unit area can be written as

ewe ie (Lor mean, 2) CRY

where “wz” corresponds to the charge when 6=1 in

equation (8) and ‘y” is proportional to the number of

|
;
|

OEE

Doo Prof. J. N. Mukherjee on

hydroxyl ions the surface can adsorb per unit area when it is
saturated.

Since the chemical adsorption of hydrogen ions is assumed to
be absent, on the addition of an acid the; negative charge will
decrease owing to electrical adsorption till the surface becomes
neutral. At this concentration ot the acid, the surface has
an adsorbed layer of water, and an equal number of hydrogen
and hydroxyl ions. An increase in the positive charge
eannot be due to electrical adsorption of the univalent
hydrogen ions (cp. previous paper). The increase in the
charge is due to the neutralization of the hydroxyl ions
being formed in the surface by impinging hydrogen ions, as
represented in scheme 2 (6) above.

The maximum charge E,, for an acid will, therefore, be
equal to “wz.” The maximum charge due to acids thus gives
a measure of the hydration of the surface. The difference
between the maximum charge observed with acid and with |
alkali gives a measure of the amount of hydroxyl ions that
is required to saturate the surface.

In tne preceding discussion, the chemical and electrical
adsorption of the anion of the acid has been left out of
account for the sake of simplicity. If the initial negative
charge of the surface in contact with pure water is consider-
able the electrical adsorption can be complete only at high
concentrations of the acid, 2. e., the surface will be neutral at
a high concentration of the acid. The electrical adsorption
of the anionis no longer negligible. A reversal of the charge,
though theoretically possible, may not be actually observed
owing to the great concentration of the anion.

The reversal is thus dependent on :—

(1) a large value of w, and
(2) a small value of y.

A non-reyersal is to be expected when the opposite is the
ease, 7. é.,

(1) a small value of *w,” and
(2) a large value of “y.”

A regular transition from marked reversal to non-reversal
can be observed in Perrin’s work. With cellulose he also
does not record a reversal of the charge. It is to be expected
from the preceding considerations that non-reversal will not
be observed when the concentration of the acid required to
render the surface neutral is comparatively high, 7. ¢., the
anion concentration is high. The concentration of the acid
in the case of cellulose is the greatest recorded by Perrin.

the Adsorption of Ions. d3%

The chemical adsorption of the anion is also not to be
neglected. The experimental data on this subject are meagre.
The various points raised here can be experimentally eluci-
dated. As shown above, the standpoint developed in this
paper can correlate all the observed facts. Besides, it gives
a definite idea of the electrical double jayer.

Adsorption of electrolytes.

In the preceding sections the adsorption of ions has been
considered with reference to the electrical charge of surfaces
in contact with aqueous solutions of a single electrolyte.
The electric effects accompanying the adsorption of ions have
enabled us to follow the total adsorption of ions of both signs.
In considering the adsorption of ions measured by chemical
means it is important to remember the influence of the ad-
sorption of the solvent pointed out by Arrhenius, Bancroft,
Williams, and others.

The amount adsorbed is small and the analytical measure-
ment is difficult. For this reason, investigations have centred
round adsorbents with great adsorbing power and substances
which are strongly adsorbed. Often it happens, that if a sub-
stance is used in a satisfactorily pure state it does not have
the necessary specific surface to make the estimation of the
adsorbed amount possible. Asa result adsorbents generally
contain small amounts of other substances. The importance
of these impurities has been pointed out by some investigators.

Michaelis and Freundlich and their co-workers have done
systematic work in this field. Their investigations have
brought out the following regularities :— |

(a) The electric charge of the solid influences the ad-
sorption, Thus Michaelis and Lachs (Z. Hlektro-Chem. xvii.
poe vi CITE); Biochem Zeresch. xxv ap. 309. (L910)...
and Davidsohn, Biochem. Zeitsch. liv. p. 323 (1913)) found
that in contact with acid solutions charcoal adsorbs anions
strongly and does not adsorb cations. The reverse happens
in the case of cations. Freundlich and Poser (Koil. Chem.
Beihefte, vi. p..297 (1914) ) undertook an extensive investiga-
tion, and they agree with Michaelis as to the electro-chemical
nature of the adsorption. ;

(b) The chemical nature of the adsorbent has a specific
action.

Michaelis and Rona (biochem. Zeitsch. xevii. pp. 57, 85
(1919)) believe that adsorption is due to chemical affinity.
They mention that charcoal has a great capacity for adsorbing
substances containing a chain of carbon atoms. (Cp. Abder-
halden and Fodor, Fermentforschung, ii. p. 74 (1917).)

Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. Z

338 Prof. J. N. Mukherjee on

Freundlich and Poser (loc. cit.) found that the nature of the
adsorbent plays an important part in determining the ad-
sorbability of a dye.

Both Michaelis and Freundlich agree that at least two
types of adsorption of ions can be recognized.

(c). Exchange or displacement of ions alr eady adsorbed by
ions of a second electrolyte (cp. Freundlich, “ Verdriingende
Tonenadsorption ” and Michaelis, “ Austausch- Adsorption ”’).
Michaelis (Z. Electrochem. xiv. p. 353 (1918) ) considers that
a substance like mastic, or kaolin (bolus), acts as a “‘ zweler
electrode’’ (a binary electrode). Thus kaolin has a slow-
moving anion (silicate ion) anchored on its surface and tends
to send hydrogen ions into the solution under a definite
electrolytic solution tension. Freundlich points out (and
Poser, oc. cit.) that other cations can displace the hydrogen
ions and form undissociated complexes (and HElissafoff, Z.
Phys: (Chem 0: 33) 1912).

(d) An adsorbent which contains some adsorbed electro-
lytes need not be necessarily saturated. In this case, besides
an exchange of ions, primary adsorption of ions is possible.
This also applies to substances which act as binary electrodes
in the sense in which the word has been used by Michaelis.
He considers that, besides adsorption through exchange of
ions, there is only one other type of adsorption, namely,
adsorption of both ions in equivalent amounts (“ Aquivalent
Adsorption ”’).

One other fact has been emphasized ie these authors.

(ec) It is the irreversible nature of electro-chemical ad-
sorption. The well known instance of the adsorption of
hydrogen sulphide by metal sulphides studied by Linder and
Picton (T. Ixvii. p. 163 (1895) ; Whitney and Ober, J. Amer.
Chem. Soc. xxii. p. 842 (1901)) can be mentioned. The
adsorbed substance does not come out in solution when the
adsorbent is brought in contact with pure water.

(f) Lastly there is no clearly established instance in
which Rael alte splitting up of neutral salts such as
potassium chloride has been observed through adsorption.

Theories regarding the Hachange of Ions.

The conception of an adsorbent acting as a binary electrode,
suggested by Michaelis, is not of much help in explaining the
exchange of ions and other peculiarities of the adsorption of
electrolytes. The relationship between the adsorption of
ions, electro-endosmotic cataphoresis, and precipitation of
colloids has been established beyond doubt. The only theory

—
.

the Adsorption of Ions. o0U

that attempts to correlate them is that due to Freundlich.
This view is an extension of Michaelis’s idea referred to above.
The adsorbent (or colloidal particle) is regarded as a great
multivalent ion (cp. Billiter, Z. Phys. Chem. xlv. p. 307
(1903) 3 Duclaux, J. Cham. Phis. vi p: 29 (1907)). . The
following extract shows clearly their standpoint (Freundlich
and HElissatoff, loc. cit. p. 411) :—

“Die Ladung soll nun durch die verschieden grosse
Lidsungstension der lonen des schwerldslichen festen Stofts
zustande kommen, aus dem das suspendierte Teilchen, bzw.
die Wand besteht. Nimmt man als Beispiel das Glas, so hat
man an der Oberfliiche desselben eine Schicht von gelostem,
oder bzw. in wasser gequollenem Silkat; die K- and Na-
Ionen haben eine grosse Lisungstension und bilden eine
ftussere Schicht, die ‘schwerldslichen, langsam diffundierenden
(vielleicht auch stark absorbierbaren) Silikationen bilden
eine innere Schicht, die mit dem festen Stoff verbunden wie
ein vielwertiges Ton sich verhalt. Der wesentliche Unter-
schied gegen ein gewohnliches Ion liext darin, dass wegen
der Groésse Grenafliichenwirkungen eintreten, die Konzentra-

tion ist in der Umgebung dpe so homogen, sondern es sind

durch Adsorption hervorgerufene Konzentrationsunterschiede

vornanden.

“Fiir zwei ionen gilt nach der Massenwirkungs gesetzat
(Anion) . (Kation) = i (nichtdissociertes Salz), deshalb auch
fiir das aaa vielwertige Anion des als Beispiel
betrachteten Glases.

‘(Vielwertiges Anion) . (Kation) = K (nichtdissocierter
Stoff). Es wird also von der Konzentration der Kationen die
‘ Konzentration des vielwertigen Anions,’ d.h. auch die Zahl
der auf der Grenzfliche vorhandenen Ladungen abhingen.

“ Die Kationenkonzentration, um die es sich hier handelt,
wird aber in erster Linie die dee nichsten Umgebung der
Grenzfliche, d.h. die Adsorptionschicht sein. Die adsorbierte
Menge Kation wird also fur die ‘ Konzentration des viel-
wertigen Anions’ d.h. fiir die Ladung der Grensfliiche mass-
eebend sein. Dies ist eine andere Verkniipfung von
Adsorption und Potentialdifferenz an der Grenzfliiche.
Genau das Gleiche gilt naturlich fiir ein vielwertiges Kation
und die adsorbierten Anionen.”

There are several difficulties in accepting this theory.
Salts of alkali metals can neutralize. charged surfaces at
moderate concentrations (N/10 or N/20). One has to con-
clude that the alkali salts of these ‘* multivalent anions” have
a low solubility product. The effect of the valency of the

ZL 2

340 Prof. J. N. Mukherjee on

oppositely charged ion cannot be accounted for. The activity
of the cations is generally in the following order :—

fh > Al > Ba > Sr Cae > Cs > Rb > he Nae

The postulates that alkali metal salts become undissociated
at low concentration of the cation and that their solubility
products are of the above order for a large number of diverse
chemical snbstances, are contrary to experience. Regarded
from the chemical point of view the generality of these ob-
servations cannot be explained. Besides, the conception of
the suppression of the dissociation of a salt cannot explain the
reversal of the charge which is met with when the oppositely
charged ion is polyvalent.

The view of electrical adsorption put forth by the writer
gives a definite correlated account of these various facts.

The Role of Llectrostatic Forces in the Absorption of Tons.

(a) Adsorbent in contact with a single electrolyte :—
Let us consider an adsorbent, P,in contact with an electro-

lyte A’ B. It is assumed that the substance P only adsorbs

the anion B’ by chemical affinity. For simplicity it is also
assumed that “‘P”’ is a pure chemical substance of definite

composition. The amount of B- adsorbed per unit area will
depend on the concentration of A* B™ and on the streneth

of the chemical affinities acting on B . Corresponding to
the number of anions adsorbed an equivalent number of

cations A” remain in the solution. These are held near the
surface by electrostatic forces, and form the second mobile
sheet of the double layer (cp. the earlier paper referred to).
If the concentration of the electrolyte is sufficient, some of
them will be fixed on the surface by electrostatic forces.
These ions of opposite charge fixed on the surface by electro-
static forces will be spoken of as electrically adsorbed in
the sequel. The chemical adsorption of an ion thus concen-
trates both ions at the surface in equal amounts. That is,
the primary adsorption is an equivalent adsorption of both
ions. Analytical methods cannot differentiate between the
two adsorptions, but electro-osmotic and cataphoretic experi-
ments can (cp. (d) above).

If the adsorption of the anion is due to strong chemical
forces, perceptible amounts of the electrolyte A*B°™ will be
adsorbed at very low concentrations. Hven saturation may
be reached at low concentrations. In such cases, if the

the Adsorption of Lons. 341

adsorbent with adsorbed electrolyte is suspended in pure
water the adsorbed electrolyte will not be set free (e).

Since the primary adsorption of the ions is due to chemical
affinity, the influences of the nature of the adsorbent and of
the electrolyte (b) are intelligible.

(6) The addition of a second electrolyte :—

The general case when both electrolytes, A” B™ and
OT D7, are present in all possible concentrations will be too
complex. It will be assumed for the sake of simplicity
that

(1) the substance P adsorbs chemically the anion B™
strongly, and that the concentration of the electrolyte A’ B™
in the liquid is negligible. We are thus dealing with an
adsorbent with an amount of adsorbed electrolyte in contact
with a second electrolyte solution ;

(2) the atoms on the surface of the adsorbent P do not
exert any chemical affinity on the ions C* and D™.

This particular case corresponds with most actual systems,
and the electrolyte A‘ B” plays the part of the “ Aktiver
Hlectrolyt” of Michaelis.

Let us now consider the effects of the electrostatic forces
on the ions OC’ and D.. A cation ©", when it diffuses into
the double layer owing to thermal energy, will be attracted
to the surface. Considering the kinetic equilibrium between
the ions in the second sheet of the doyble layer (Ae and C")
and those in the liquid, it is evident that the relative propor-
tion of A‘ and (‘" ions in the double layer will depend on
(i.) their respective concentrations in the bulk of the liquid,
and (i1.) their valency. The same consideration applies to
the electricaily-adsorbed ions A* or O°. At. sufficiently
large concentrations the whole of the mobile second layer
and electrically-adsorbed ions will be formed by the ions
C*. There will thus be an exchange of ions, and the amount
of exchange will depend on the concentration of the second
electrolyte. When the displacement is complete the amounts
exchanged will be equivalent to the amount of B- ions
primarily adsorbed and independent of the nature of the
replacing ion C’—a fact often observed (cp. (Linder and
Picton, loc. cit.; Whitney and Ober, loc. cit. etc.).

The ions CG will be positively adsorbed.

The relationship between the charge of the surface and
the positive adsorption of the oppositely-charged ion is also
obvious. The amount of (” ions absorbed depends on the

342 Prof. J. N. Mukherjee on

amount of the negative ions chemically adsorbed. es
that increases he total amount of adsorbed negative ions

will increase the positive adsorption of cr.
The reverse case, when the positively-charged ions A” are

adsorbed chemically instead of the ions B’, and no other

ions are chemically acted on by the surface, is obvious.

Negative ions will now be positively adsorbed and exchanged.

This state of affairs corresponds with the statements made in
(a) and (c) above.

Taking the same case again, we shall consider the effect

of the electrostatic forces on the ion D - Anion D , diffusing
into the double layer, will be driven out of it. So long as
the potential of the double layer is sufficiently strong, a
volume of the liquid equal to “S/,” where ‘‘S ” is the extent
of the surface and “1” is the thickness of the double layer—

will be free from the ion D . In other words, the concen-

tration of D” increases in the bulk of the liquid and a
negative adsorption will takeplace. This will increase with
the concentration of the electrolyte so long as the potential
of the double layer is sufficiently strong. Since with in-
crease in concentration the potential falls, the negative
adsorption will reach a maximum. At concentrations when
the surface becomes electrically neutral, there should be no
negative adsorption due to electric forces. It is difficult to
determine negative adsorption at high concentrations as the
osmotic pressure opposes it. Also, the variations in concen-
tration due to negative adsorption become relatively small.
The experimental ‘difficulties lie in the analytical estimation
of small amounts. Only ions which ean be estimated in
extremely small amounts are suitable for experiment.

Hstrup, (Moll. Zeitsch. xi. p. 3 (1911)) thas Vacnuallig
observed a negative adsorption of the oppositely-charged
ion. He estimated the adsorption of the iodate, dichromate,
and chromate of ammonium. Michaelis and Lachs (Koll.
Zeitsch. ix. p. 275 (1911)) did not observe a negative
-adsorption with potassium chloride.

Eixactly similar observations have recently been made by
Bethe (Wiener Mediz. Wochsch. 1916, Nr. 14; Koll. Zeitsch.
xxi. p. 47 (1917)). He worked with gelatine gel, gelatine
sol, and a number of animal cells. The adsorption of a basie
dye is greater in weak alkaline solutions tae in neutral
solutions. The same is the case for an.acidic dye in weak
acid solutions. In alkaline solutions the adsorption of acid
dyes is negative, and the same is the case with basic dyes
in acid solutions. Examples of the role of: the electrical

the Adsorption of Ions. 343

force in the adsorption of ions can be multiplied (cp. Baur,
Z. Phys. Chem. xcii. p. 81 (1916) ; Michaelis and Davidsohn,
foc. Cit.) ;

Exchange of Bases in Soil and Soil-Acidity

It is now easy to understand the nature of the exchange of
bases in soil-analysis and the cause of soil-acidity. A com-
plete reference to the older literature is given in the following
papers :—

(1) McCall, Hildebrandt, and Johnson, J. Phys. Chem.
$9265 SK.\ps O1.

(2) Ruce, ibid. p. 214; (3) Truog, ibid. p. 457.

Russell (Brit. Assoc. Rep. 1918, p. 70) has given an
excellent summary of the present position of the subject.
The facts are that—

(a) Neutral solutions of salts like potassium chloride, if
treated with samples of soil, give acid extracts though the
extract with pure water is neutral,

(>) In a large number of cases it has been shown that
there is a definite exchange of the cations. Equivalent
amounts of bases are exchanged in many cases. |

Two different views have been advanced to explain the |
facts. The older chemical view regards the process as a
chemical interaction between definite acids (e. g., humus |

:

acid) or complex salts (e.g., silicates) and salt solutions.
The other view begins with von Bemmelen, and regards it |
as an adsorption process. Cameron suggested (cp. Russell’s |
Report) that the soil adsorbs the base more strongly than it |
adsorbs the acid. ;

The objections against the chemical view can be sum- |
marized as follows:—The extract with pure water being )
neutral, the soil-acids must be insoluble. ‘The acids must be :
unusually strong, as they evideutly decompose a neutral |
salt solution combining with the base, liberating the strongest
known acids, like hydrochloric acid.

Evidently such acids are unknown, and it is difficult to
conceive of such reactions. Regarding the exchange of
bases, the difficulty lies in the assumption that the basic ion
is taken up to form an insoluble salt. It is necessary to
postulate the existence of insoluble salts of alkali metals in a
: large number of cases (cp. the remarks on Freundlich’s
' theory).
: That adsorption plays an important part is also evident

from the works of Russell and Prescott (J. Agric. Sci. vill.
p. 65 (1916)) on the interaction of dilute acids and phos-

phates present in the soil. But the view of Cameron does

o44 Prof. J. N. Mukherjee on

not seem to be tenable. The preferential adsorption of an
ion by the soil does not mean hydrolytic decomposition of
the salt. It appears from the summary given by Russell
that the equivalent exchange of bases lies in the way of
regarding the reaction as an adsorption process (loc. cit.
pp. 71, 75, 76). It would be apparent from the previous
discussion that this, in itself, does not contradict the adsorp-
tion hypothesis.

Soil can be regarded as a complex colloidal system. It is
a complex gel consisting of aluminium and other silicates,
free silica, ferric hydroxide, ete. The gel is mixed with
insoluble crystalloids. It also contains small quantities of
adsorbed electrolytes and organic matter in indefinite and
varying proportions. The gel adsorbs anions by chemical
affinity. These anions may be :—

(1) of organic acids, such as humus acid ;

(2) of simple electrolytes like chlorides, sulphates, car-
bonates, ete. ;

(3) hydroxyl] ions rom ater,

Owing to the complex chemical nature of the gel and the
enormous specific surface of gels, large quantities of anions
may be adsorbed. An equivalent number of cations remain
near the surface as the mobile second sheet or as electrically
adsorbed. The exchange of bases is simply due to the dis-
placement of these ions. When the displacementis quantitative
equivalent amounts are exchanged. ‘The anions primarily
adsorbed or the cations in the second sheet are not of one
kind. The relative numbers and chemical natures of these
ions will evidently vary with the different soils.

An extract with pure water will be neutral unless the soil
contains free acids. An extract with a neutral salt can only
be acid when the cations displaced from the second sheet (or
electrically adsorbed) contain hydrogen ions or such ions as
aluminium, which~ hydrolyse in dilute aqueous ‘solutions.
The role of the aluminium ions in determining the acidity of
the soil extract has been pointed out ioe Daikuhara (Bull.
Imp. Central Agri. Expt. Station, Tokio, ii. pp. 1-40 (1914)),
and has been fully confirmed by Rice (Loe. cit.). The function
of organic acids has constituted a oreat objection against the
adsorption hypothesis. The hydrogen ions in the second
sheet have probably, in most cases, their origin in these
acids. This view thus correlates the exchange of bases
observed with soil with such exchanges as have been observed
in the adsorption of electrolytes (ep. Michaelis).

That sometimes considerable quantities of bases are ex-
changed should be referred to the enormous surface of these

the Adsorption of Lons. 345

gels, and that probably the surface is saturated with anions.
As erystalloids (insoluble) are also present, the type of
exchange considered by Paneth (loc. cit.) is also possible.

It is needless to point out that in this discussion only the
theoretically simple case has been considered. Complications
due to simultaneous primary adsorption of different ions and
their mutual displacement are not always negligible. Besides,
the changes may not be restricted to the surface ; for faiion
of solid ‘solutions, etc., are not excluded. Considering all
these complex influences, it is interesting to note that most
ot the observed regularities correspond to the theoretically
simple case.

Adsorption of Ions in its Relation to Permeability of
Membranes and to Negative Osmosis.

In conclusion, a few remarks will be made on the funda-
mental interest that a study of the adsorption of ions has for
biological phenomena. Cell activity 1 is greatly conditioned
by the permeability of its ‘‘ walls” or the cell-substance to
the contents of the liquid with which it is in contact. The
connexion between the rate of osmotic flow through mem-
branes and even the direction of the flow, and the potential
differences existing on the two sides of the membranes, has
been clearly established (Girard, C. &. cxlvi. p. 927 (1908),
and following authors: Bartell, J. Amer. Chem. Soc.
Xxxvl. p. 646 (1914) ; Hamburger, Z. Phys. Chem. xcii.
p- 885 (1917)). The origin of the potential difference
is generally assumed to be due to the fact that the rate of
diffusion of the electr olytic 1 ions in the membrane substances
is different from that in water. That the membrane potential
is due to a selective permeability of ions was first suggested
by Ostwald (7. Phys. Chem. vi. p. 71 (1890)) ; Donnan (Z.
Elektrochem. xvii. p. 572 (1911)) has discussed the origin
of the potential differences theoretically, and has given it a
quantitative form based on thermodynamic considerations.
In collaboration with others he has carried out a number of
investigations which have established the validity of this
view.

The simpler case of a potential difference between two
interfaces when an immiscible liquid is placed between
two aqueous solutions has also attracted a good deal of
attention. The work of Loeb and bis co-workers on cell-
permeability and origin of the membrane potential is of
fundamental importance (Loch and Beutner, Biochem. Zeit.
li. p. 295 (1913) ; Beutner, 7. Phys. Chem. |xxxvil. p. 385

346 Prof. W. M. Hicks on certain Assumptions wn the

(1914), 2. Hlektrochem. xix. pp. 329)/473 (1913); Loep
J. Gen. Phys. xx. po 113 (1909). a1.) i273, 290510 eee
563, 673, 659). The part played by the adsorption of ions
in these phenomena is twofold. The origin of the potential
is In many instances due to the adsorption of ions (cp. Baur,
Z. Elektrochem. xix. p. 590 (19138); Z. Phys. Chem. xcii-
PONG co Sly

Secondly, the electrostatic forces of the surface probably
determine the relative permeabilities of the two ions. To
this the semi-permeability of an ion can be referred.

eee negative osmosis, attention may be drawn
to the suggestion of Freundlich (Koll. Zeitsch. xviii. p. 1
(1916)) that the thin walls of the membrane substance
conduct electricity, and electro-osmotic flow of the liquid
occurs. A necessary condition is that one ion is permeable
and the other relatively impermeable. This explanation
meets thermodynamical requirements, and is the only satis-
factory one hitherto put forward.

In all these cases the same influences of polyvalent ions
and ions of opposite charge are noticeable.

The change in the collodial properties of the membrane is
an important additional factor which has to be remembered.
The influence of the electrostatic forces is unmistakable.

Physical Chemistry Department,
University College, London.

XXX. On certain Assumptions in the Quantum-Orbit Theory
of Spectra. By W. M. Hicks, &R.S.*

(ee practically complete success of the quantum-orbit
theory in describing all the known facts of spectra,
in cases where we know experimentally that the source
consists of a single nucleus and a single electron, must
give assurance that the same procedure must also be capable
of application to more complicated atoms than those of the
hydrogen and enhanced helium types. Unfortunately,
however, mathematical difficulties have so far prevented
any rigorous application of the theory to definite cases,
even of the next simplest atomic configuration of a single
nucleus and two electrons. The attempt of Sommerfeld
at an approximate solution shows, on the one hand, how
hopeful we may be of a description of spectra on this ‘basis,
and at the same time how far we are at present from its

* Communicated by the Author.

Quantum-Orbit Theory of Spectra. “B47

achievement. In the present. note I wish to illustrate this
by drawing attention to certain assumptions as to actual
spectral data, which have been made and which do not
appear to be justified. The criticisms may not affect
essential points, but they would appear to require some
modification in the presentment of the theory. References
will be made to Sommerfeld’s ‘Atombau und Spektrallinien,’
2nd edition (1921).

L. Sommerfeld (pp. 276, 506) takes a configuration of
a central nucleus, surrounded by a ring of equally-spaced
electrons, and at a considerable distance furiher out one
electron revolving in a quantized orbit. On the assumption
—here justified—that the ring can be treated as if the
whole charge of the electrons on it were continuously
distributed along it, he obtains as an approximation the
same form for a sagan function (or term) p as that’
suggested by re VIZ.

= N/{m+ptap}?.

He says that silva, is the actual true form, as already deter-
mined by observation. This is, pee by no means the
ease. No form has yet been found which will fit in for all
series, and indeed the form N/(m+p+a/m)* is in general
rather superior to that of Ritz. It is to be noted that the
assumption made above leads to the same result ag if the
force to the centre depended only on forces inyersely as
even powers of the distance, and forces depending on odd
powers—say 1/r*—are excluded. It may also be noted in
passing that the theory so developed applies only to the
case of a single external electron and one internal ring,
that is, on the usually assumed configuration of eight-
electron rings, only to the spectra of the fluorine group,
or the ionized rare gases, or the doubly-ionized alkalies, ete.
By taking his E as (k—s;’e in place of ke, the formula
would meet the more general case. This modification,
however, would only slightly affect the order of magnitude
of the quantities pw, a.

In the formula m=n-+n’', where n, n' are respectively
azimuthal and radial quantum numbers, and yp, e& are
functions of n and not of n’.

* Asa result of successive approximation, « being small,:this means
for a complete approximation the form

pH=N/{m+ pta/(m+p)?+B/(mtp)i+...P,

which, «8 is well known, is capable of reproducing practically all cases
if p,a, 8... are all at disposal, and are not related necessarily to
one another, as here.

348 Prof. W. M. Hicks on certain Assumptions in the

It is not to be expected that the numerical values of the
constants 4, « on this special theory should accord with any
determined by experiment, but they should be of a suitable
order of magnitude and general character. It may be
interesting to test this. The expressions for the constants:
fs, a may be written

> 3 k—s,
p= Ey i4 PS (1-3 St) 92 boa

n 4. DL—
where
(277)4m?e? E(Z— k) 5
(3 = A} : = 8°9(Z—k)(k—s;)10!%r?
and

ine 2° k—se
If p be measured in wave number instead of frequency,
the a must be multiplied by the velocity of light. Then

ee Ope ee eae) 1010p?

ee ‘i n? Zi k— sp, :

Here v denotes the radius of the internal ring in em., Z the
atomic number of the element, k the number of external
electrons, and s;, depends on the mutual action of the external
electrons on‘one of them. For Li, Z=11, k=3, 5% =:577,
and

a 37r?m € L— k ) \
ee he Deh on

9: ; e
ms =-(1 +22 10048) 1018,
ele
as 13 10192
Voinn (ihe

In actual cases, for wave numbers of about p=10°, pac/p
lies between about -9 and ‘01. Hence the second equation
requires r<10-7°>10-*°. Since «<1, the first equation
requires 7 to be about 10-®, but as the second term in the
bracket is determined by an approximation it must be a
small fraction, whence r<10-8 >. The fact that both give
values of the same order of magnitude, even if they cannot.
exactly agree, and not far from what might be expected for
an 8-electron ring, is certainly satisfactory.

2. It is deduced that the different types of sequences
correspond to azimuthal numbers n=1, 2, 3, 4, the different
orders of the same type to radial quanta n'=0, 1,.... These
are then co-ordinated with the s, p, d, f types because it is
stated that these types have their lowest orders respectively
of 1, 2,3, 4. It is difficult to see how this statement has
been arrived at, as it is quite incorrect. For the sake of

Quantum-Orbit Theory of Spectra. 349

readers who may not be familiar with spectral data, it may
be well to consider them here.
(s, p.) For the s, p the lowest order are:

s p. Seal (ps
Rare gases ......... 1 1 II { Alkaline earths... 2 1
(The alkalies ...... l 2 * ais Cabin, bho. (ae il
eta i Ane. [ib Ave Mer cet eimany 1 it ARR Bray

If it were not for the cases of the rare gases and the
Cu subgroup, the assumption might be explained by an
interchange of the nature of the sequences which produce
P, S series (for which in Groups I. and IJ. indeed there is
also direct evidence). But that two groups make s, p both
have unity for their first order is fatal.

(d.) The assumption of 3 as the first order for d(m}
no doubt is based on the fact that Ritz made it in dealing
with the D series in the alkalies. The denominators of the
first orders in this group are comparable with 2°9, which
Ritz wrote as 3—‘1 and called the first order 3. But this
procedure is inadmissible either on the side of the formula or
from what we know of the constitution of the d sequence.
In Sommerfeld’s formula w is positive, and it is only by
treating the fraction as positive that we find a detinite
dependence of it on certain spectral constants. But even so,
the first order is not 2 for all groups. The law of the first
order of the d sequence is a quite simple and definite one,
and is given on p. 188 of my recently published ‘ Analysis of
Spectra. It is that in each group of the periodic series,
the subgroup of elements whose melting-points increase with
atomic weight take as their first order m=1, whilst the sub-
group with decreasing melting-points take m=2.

(f.) In the case of / the 3+fraction has again clearly
been written 4—/, and the assumption has been made that
the lowest orders of the f type take m=4. But here also,
for the same reasons as in d, the fraction must be taken as
positive. In the alkalies certainly the lowest order observed
is F(3), but F(2) would lie far up in the ultra-red, beyond
even Paschen’s longest lines. In the Cu subgroup there is
evidence for m=2 and indications for m=1. The alkaline
earths have m=2 both in triplets and doublets. In the
Zn subgroup only F(3) has been observed, but F(2) would
lie in the extreme ultra-red. In Group III. there is no
evidence, whilst in the rare gases there are examples of F(1)
and F(2). |

It would thus appear that the theoretical deduction that
different types depend on successive changes of azimuthal
quanta by unity is not tenable.

350 Prof. W. M. Hicks on certain Assumptions in the

3. In dealing with the Zeeman effect on p. 422, Sommer-
feld adopts Puaschen and Back’s interpretation of their
experiments on the Zeeman effect in the case of close
multiple lines.. This interpretation was based on precon-
ceptions as to the nature of the series types in He and
Li, which they investigated. I have given * reasons why
this interpretation should be modified. On either inter-
pretation, however, a consequence follows which appears
difficult to explain on the quantum-orbit theory. Take, for
example, the case of the helium doublet at 4713 A. Hach
cemponent in weak fields shows special Zeeman patterns.
Wath increasing fields and consequent approximation of
certain constituents from each pattern, an interaction occurs
of one on the other. Such an effect can only be produced
if the two patterns are produced in the same source. Hence
the original components of such a doublet must be produced
simultaneously in each atomic configuration, whether a

magnetic field is present or not. It follows that in radiation .
there must be simultaneous passages of two electrons, each

from its original orbit to its final one. But as the effect
takes place at one operation, the total change of energy
is passed on to the radiator and emitted as a single mono-
chromatic radiation, 7. e. no doublet. It might be suggested
that the effect could be explained on the hy pothesis that the
magnetic field affects the mutual possible orbits, and that
sometimes one passage occurs and sometimes the other. It
is difficult to see, however, how an orbit can be modified by
another supposed one which is non-existent, 7. e. not being
described at the same time.

4. This consideration does not affect evidence for the
quantum theory, but will serve to illustrate a habit which is
somewhat exasperating in reading the writings of many
exponents of the quantum theory—viz., the picking up of
small and often irrelevant points as charming results of the
theory. On p. 300 ff. it 1s expected that each doublet
separation on passage from arc to spark conditions should
be magnified in a measure corresponding to the ratio 4N:N,
and satisfaction is expressed that in data adduced from
corresponding elements in the doublets of group I. and the
enhanced doublets of group JI. this expectation is fulfilled.
The ratios of the separations are reproduced (with Hg: Au
added) in the first line of the following :

Mg. Ca. Sr. IB) tae PAK Cd. Eu. Hg.
53 3°9 ot Bll aS) ir ? 2°5
2°24 2:10 205 1:92 Deal 2°12 2°03 2:12

* ‘Analysis of Spectra,’ § 7, p. 96.

Quantum-Orlit Theory of Spectra. oer

But surely these numbers show that the comparison is not
justified. As is known, a correspondence actually lies
between the enhanced doublets and the triplets in the
same element. Thus the ratios of the doublets to the first
separation of the triplets are given in the second line of
figures above, where the agreement is remarkably close.
In this latter case, however, the correlation is not direct.
It is due to three foncurrent facts: (1) ratio 4N:N;
(2) the oun multiples in the doublets and vy, of the triplets
are very nearly the same in each element ; and (3) the
denominators in the doublets and triplets have nearly the

same ratio in all (see below). ‘There is, however, a close
correspondence between the mantisse of the doublets in the
two groups I. and II., especially as between the alkalies and
the alkaline earths, fleas of the latter being about double the
first. Correlation is also shown between the denominators
of the triplet and doublet sets in all the group IL. elements.
These statements are illustrated by the following data :—

IL. i

ee Tae aot mri too

Tripl. Doubi. Ratio. Doubl. Ratio.
1 ee 1-660 2:265 1:36 DN) ha 2117 aya)
a es Poo) 249 peat Wy Biss. 4. 1, 2-935 9-12
Sh eee 1:880 2611 1034 aaa EO etc 2-292 2:09
eT Hs os 1:957 2139 1:39 Ca. ee 2°361 2°03
ee 1°5$9 2-098 1S ie | Ck ee 1:869
he 1641 2144 1°30 ve tee 1:892
Ho’ fe. 1:648 9-190 132 vas 5S
ba teats 1°653 £ — ANU aaa 1-929

Here under Ll. the third column gives the ratio of the
denominators ; under I. the second column gives the ratio o7
mantissie in LI, to those in I.

On the other hand, there appears very little correlation
between the oun multiples which give the separations in
corresponding elements of groups I. and Il. A much
closer one is found between those of the triplets and doublets
of the same element in II. Thus, in the following, the first
line gives the ratios of the oun multiples of doublets in
group I. to those of the enhanced doublet of the corre-
sponding element in group II. The second line gives the

ratios of these multiples for the first triplet separation and
the enhanced doublet in each element.

Mg. Ca. Sr. Ba. Zu. Cd. Eu. He.
683 179 "843 903 ‘847 [71 ae ?
“691 *709 "726 ‘761 ‘789 846 "842 ?

B52.)

XXXI. On the Theory of the Characteristic Curve of a Photo-
graplic Emulsion. (Communication No. 22 from the
British Photograpnic Research Association Laboratory.)
ag. C.. oy MSc ainsi Ec Me nega seme

li the most recent investigations on the relation between
the photographic effect and the light-exposure, special
plates containing only a single layer of grains have usually
been employed. With such plates the photographic effect
is determined by counting the percentage of grains made
developabie. The curve expressing the relation between this
percentage (wv) and the logarithm of the exposure may be
called the characteristic curve of a single-layer emulsion,
corresponding to the ordinary curve of a commercial
emulsion, in which, instead of w, values of the density (in the
photographic sense) are plotted.

In a recent paper (Phot. Jour. 1921, Ixi. p. 417) the author
has shown that such a curve, fora set of grains which are
geometrically identical, is of ang usual S-shaped type, 2. e. a
difference in size or shape does not account tor the fact that
all the grains do not become developable with the same ex-
posure. Now, a set of geometrically identical grains, all in
a single layer and similarly orientated to the incident light,
represents the simplest possible emulsion which we can in-
vestigate experimentally. It also corresponds to the simplest
theoretical case, eliminating many complicating factors
which, though greatly affecting the form of the characteristic
curve, have “nothing to do with the primary mechanism of
the photograph process. In other words, with this emulsion
the curve is reduced to its “‘ purest” form, and is determined
almost solely by the photochemical process which takes

lace.

: It is now generally believed that the primary action of
light on the grains is ce form in or on the surfaces of them
certain “ centres” “»yoints of infection”? which act as
starting-points for Hee reduction by the developer. This
view has for some time had considerable evidence in its
favour. Chapman Jones" (Phot. Jour W911, lr pai)
showed that by stopping development at a very early stage
it is possible to get particles of silver tco small to be visible

microscopically, but which can be shown to be present by the
colour imparted to the film, and by enlargement to visible
dimensions by the deposition on them of mercury. Hodgson
(Brit. Jour. Phot. 1917, p. 532) carried development a little

* Communicated by Prof. A. W. Porter, F.R.S.

Characteristic Curve of a Photographic Emulsion. 353

further, and showed it possible to observe the silver reduced
by the developer only around certain centres in the grain,
and a recent paper of Svedberg’s (Phot. Jour. April 1922)
leaves little room for doubt that the possibility of a grain
being made developable depends on the existence in it of
some kind of reduction centre.

Opinion as to the nature of these centres seems at present
to be divided. There are those who assert that they are
formed by the light-action, and that they do not exist before
exposure is made. Such, for example, is the case if the
centre is really a molecule of silver halide which has lost an
electron, as is believed by H. 8. Allen (Phot. Journ. 1914,
liv. p.175). On the other hand, there are those who believe
that the centres are actual particles other than silver halide
formed in the grains during precipitation and subsequent
ripening, and that these only become susceptible to the action
of developer after exposure to light.

There certainly is considerable evidence to show thuit
silver halide is not the only substance in the grains. Luppo-
Cramer (Kolloidchemie und Photographie) was led, asa result
of his work, to the conclusion that, at any rate in the most
sensitive emulsions, nuclei are present which probably consist
of a colloidal solution of silver in the halide. Renwick
(J. S.C. I. 1920, xxxix. No. 12, 156T.) extends this idea,
and says: ‘In our most highly sensitive photographic plates
we are dealing with crystalline silver bromide in which,
_ besides gelatin, some highly unstable form of colloidal silver
exists in solid solution, and it is this dissolved silver which
first undergoes change on exposure to light.’ These silver
particles are negatively charged, and Renwick believes that
the action of light is to discharge, and hence to coagulate
into larger groups, those particles of colloidal silver which
existed in the grain before exposure ; it is these groups of
coagulated electrically neutral particles which are the re-
duction centres. This view is supported by the ultra-micro-
scopic observations of Galecki (Koll. Zeit. 1912, x. pp. 149-
150), who showed that X-rays have a coagulating effect on
the particles in gold sols; by Svedberg (Koll. Zeit. 1909,
iy. p. 238), who has similarly shown that ultra-violet light
agglomerates ultra-microns to larger aggregates; by Spear,
Jones, Neave, and Shlager (J. Amer. Chem. Soc. 1921, xliii.
p. 1385), who have observed the same kind of effect with
colloidal platinum; and by recent experiments of Weiger
and Scholler (Sitz. Preuss. Akad. Wiss. Berlin, 1921,
pp. 641-650).

Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. 2 A

———————— ——

354 Mr. F. C. Toy on the Theory of the

These facts are at any rate sufficient to justify an attempt
to explain the relation of the number of grains changed to
the light-intensity on the basis of the existence in the
grains of actual particles which are not silver halide. These
are not necessarily all changed to reduction centres from in-
active particles by the same light-energy as they would be
if they were single molecules of the halide. We shall make
no assumptions as to the composition of these centres, and
the theory does not depend on their being composed of
colloidal silver. We shall use the term “nucleus” rather
than centre to indicate the presence in the grains of actual
particles before exposure.

The Characteristic Curve of a Set of Geometrically
Identical Grains.

Theoretical.

Our first object is to consider the case of a set of grains of
identical size and shape, and to determine the relation we
should expect to find between the percentage of these which
are made developable and the light-intensity. The time of
exposure is kept constant throughout. —

If we consider a volume V of the silver halide which is
very large compared with that of a single grain, we may
assume that the total number of nuclei in any such volume
of the emulsion is the same, though the number contained in
individual grains in this volume may vary. We will define
the sensitivity of a single nucleus as the minimum intensity
which must be incident upon it in order to make it ‘ active ”
in the presence of the developer. For a given intensity of
the incident light there will be a definite number of such
active nuclei in every volume V, and they will be distributed
amongst the grains entirely haphazard, according to the laws
of chance. Every gram which happens to have at least one
active nucleus will be developable.

When the intensity of the light increases, more grains are
changed. On any “nucleus” “theory this happens because.
more nuclei are present, so that a single grain has a greater
chance of having at least one of them. This may be ex-
plained in one of two ways. Firstly, all nuclei may have the
same sensitivity, say I, but owing to the rapid absorption of
light, those nuclei which are situated in the grain at some
distance from the surface on which the light is incident, do ©
not receive an intensity of [ when the incident intensity is
small. As the latter becomes greater, the volume of silver
halide, throughout which the intensity is at least I, increases,

Characteristic Curve of a Photographie Hmulsion. 355

so that the number of active nuclei increases also. Secondly,
the sensitivity of every nucleus may not be the same, so that
as the intensity of the light is increased, nuclei become
operative which are unaffected by lower intensities, and
again the total number of active nuclei increases with the
intensity.

We will consider only the case of grains in the form of
thin plates as they oceur in high-speed emulsions. Hggert
and Noddack (Preuss. Akad. Wiss. Berlin. Ber. 1921, xxxix.
p- 631) have recently measured photometrically the fraction
of the incident light which is absorbed by an ordinary
commercial photographic plate, and have found it to vary
with the different plates from about 4 to 12 per cent. for
violet light, for which the amount of light absorbed is near

the maximum. Now, these plates contain several layers of

grains, so that a very extreme upper limit to the fraction of

light absorbed by a single grain is, say, 20 per cent. hus, if

there is an increase in the incident intensity of the order of
20 per cent., the intensity of the light transmitted through a
grain will be equal to the intensity incident before the
increase took place. Thus, if all nuclei are equally sensitive,
a change in the incident intensity of the order of 20 per
cent. will cause a difference in the number of active nuclei
from zero to some fixed maximum, so that the characteristic
curve can only function over a range of intensity such that
the ratio of its extremes is of the order of 1°2:1. As will
be shown later, for the steepest characteristic curve plotted
this ratio is about 25 times as much as this, so that as an
appreciable factor in determining the increase of nuclei
with intensity the first assumption is untenable. We have,
therefore, to assume that all the nuclei are not equally
sensitive.

Since these nuclei are all formed in the same emulsion,
most of them will have a sensitivity near the average value
for the whole, and there will be a few which are very
sensitive and a few which are very insensitive. ‘There will
be none which will respond to zero intensity, and none so in-
sensitive that it takes an infinite intensity to affect them.
We therefore expect the curve showing the relative number
of nuclei R having any given sensitivity I to be of the
general form shown in fig.1. The exact mathematical form
of this curve is immaterial at present, but it will be similar
in general form to that obtained by Clerk Maxwell for the
distribution of velocities between the molecules of a gas. By
similar reasoning to his, the number of nuclei (N,) which

2A 2

SS eee

356 ~ Mr. F.C. Toy on the Theory fo the

have sensitivities between zero and I, (which is the number
operative when the intensity of the light is I,) is given by

Fig. 1.

n—-

e)

the area OAB, 7. e.
N=) /@Dd. on. a ee
where f (1) gives the Ae of the ordinates in terms of I; or

if No is the average number of nuclei per grain and a the
number of grains in volume V,

No= GQ) -al. oes

The total number of nuclei is given by

N= | Qa,

t
N,

0 es

The curve showing the relation between No and I, shown in
fig. 2, is characterized by its unsymmetrical-shaped § form,

Characteristic Curve of a Photographic Emulsion. 357

When the a@ grains in the volume V of silver halide are
subjected to an intensity 1,, every grain which happens to
have at least one of these N, nuclei will be made developable.
We have, therefore, to find the chance of a grain containing
at least one of the N; nuclei when they are distributed hap-
hazard amongst a grains. This can easily be obtained from
the theory of probability.

If p denotes the very small probability that an event will
happen on a single trial, the probability P,. that it will happen
r times in a very great number, say 7 trials, is (Mellor,
‘Higher Mathematics,’ p. 502)

ee) omer a) eee 8)

Let the volume of a single grain be v, then since the volume
of every grain is the same the total volume V is av. Let p
be the very small probability that a volume dv will contain a
nucleus, then

PN Geeta «Mme Sei al yal AL)

To obtain the probability of the volume v containing a nucleus,
we may suppose each dv to be a trial, so that the number of
trials n is |

eM eee |. menace 0") Ged)

Therefore the value of np in equation (3) is N/a, which is
equal to No. If in this number of trials the event (i.e. v
containing a nucleus) happens once, a grain will contain one
nucleus; if it happens r times it will contain 7 nuclei, so
that from (3), (4), and (5) we see that the probability of a
grain containing 7 nuclei is

PES te irl) aL ughe (ae)

which is the same equation as was obtained independently
and first published by Svedberg. The probability of a grain
containing ne nuelei is the value of this expression when
e— 0, 2.2.
: Paeo

Now, since it is certain that a grain must contain either zero
or at least one nucleus, the probability P, that a grain will
have at least one is

Pi be No, ° > ° . ° . ° (7)

ew ee

ee eee

~~

]

358 Mr. F.C. Toy on the Theory of the

But if w/a is the fraction of grains which are changed,
Pi =aa ;
ala=1—e%o,
or, denoting log a/(a—a) by A, we have
ANG
Thus the same form of curve should be obtained when No is
plotted against the intensity as is obtained when A is plotted
against the same variable. The form of this curve should
be an unsymmetrical 8, as shown in fig. 2.

Haperimental.

The first experiment carried out was to determine the
relation between A and the light-intensity I for a set of
geometrically identical grains, every grain counted being
measured, as described in a previous paper (Phot. Jour.
192 xa. 41 e

TABT EOI

Cross-section of grain = 0°98’.

Pane ce fou Be recs oe (dee
values.) . | values.)

0 92°0 91:5 2°45 —0°895 | 47-7 466 0-62
—0:162 | 89°5 90-0 2°30 —1-215 52 52 0:05
—0°310 | 87°7 870 2°04 —1:487 0:0 0-0 0:00
—0°572 | 74:8 75:0 1°39 —1°788 0-0 0:0 0-00

In the first two columns of Table I. are shown the values
of log I and # determined experimentally by exposure behind
a neutral wedge, and these are plotted in fig. 3. The values
of # given in column 3 are read off the curve in fig. 3, and it
is these values which are used in calculating A in column 4.
This is the best way of obtaining the A values, since when «
is large a very small error in its determination means a very
large error in A. The A, JI curve, shown by the solid line
in fig. 4, is exactly as predicted by the theory.

We must note here that this is not in agreement with the
results of Slade and Higson (Proce. Roy. Soc. 1921, A, xeviii.
p. 154) and a previous experiment of the author (ibid. 1921,
c. p. 109) to confirm their result. Slade and Higson stated
that the relation between A and I can be expressed by the
equation

Azal(1—e7*!),

Characteristic Curve of a Photographic Emulsion. 359

where « and 8 are constants. A comparison of the form of
this curve (fig. 5) and the curve in fig. 4 shows a difference

Fig. 5.

DOO) Ie” Mr. F. C. Toy on the Theory of the

when I is large, but this can be explained. Firstly, the
grains used in Slade and Higson’s experiments were not all
of one size, the variation being about 30 times that in the
present case. Secondly, the best curve given in Slade and
Higson’s paper has actually the same form as that in fig. 4
if equal weight is given to each point plotted. Also, in the
author’s contirmatory experiment, the main point was to show
that at low intensities A varied, at any rate approximately,
as I? ; at high intensities the work was not nearly as accurate

Fig. 5.

i

Bl

as in the present case. To be certain of the form of the
curve in fig. 4 the upper part of it was plotted for another
size of grain, and that the same result was obtained is shown
by the dotted line.

The position of a nucleus can be detected by Hodgson’s
method of partial development of the exposed grains (ibid.).
The developer used was made up as follows :—

200 ¢.c. saturated Na,SQsz,
8c¢.c. 10 per cent. KBr,
0°3 gm. Amidol.

This is a weak, slow developer, and is best for this purpose
because there is a bigger latitude than if the developer is
strong in the time of development necessary to render the
position of the nuclei visible and yet distinct from one
another. The best development time was found by trial and
examination of the grains under the microscope. After
exposure the plate was plunged into the developer for a
known time, then quickly and thoroughiy washed, and dried
without fixing. The flat triangular grains used were so thin
that. the silver deposit was visible without dissolving away

Characteristic Curve of a Photographic Emulsion, 361

the silver bromide. In fig. 6 are given some examples of
grains in which these nuclei appear ; they are formed more

on the edges of the grains than anywhere else, though quite

a number appear either inside or on the flat surfaces.

Positions of nuclei numbered.

The next experiment was to show if equation (6) holds
good. A plate was exposed to a uniform intensity, partially
developed, and the number of nuclei occuring on each of
150 grains was counted. Hence the average number per
grain was known and also the number of grains having 0, 1,
2, 3, etc. nuclei each.

In Table II. are given the theoretical and observed values
of P,. for two equal-sized sets of grains, in one and the same
emulsion, having widely different values of No.

‘TABLE EA:

No=0°480. Se was |

Value N®- grains Probability ‘Value No. grains Probability |

| of 7. | camel Ounren. | Soe | moe Obs. Cale.
0 91 |0607 0619 | 0 43 |0-287 0-303.
1 47 10313 /0297 || 1 |° 55 {0367 0-362 |
2 | i |o073:\0071 | 2 | 36 [0-240 0-216
3 1 {0007 |0001 || 3 | 12 |0080 0-086
4 4 0-027 0-026

|

0 |0:000 - 0-001

The observed values of P, were determined by the fact that
the probability of a grain having 7 nuclei is equal to the
fraction obtained by dividing the number of grains which
have r nuclei by the total number, 2.e.150. In fig. 7 the
theoretical values are represented by the smooth curves, and

ae

SS Se

Sa

362 Mr. F. C. Toy on the Theory of the

those observed by the plotted points. The agreement is very
good, and proves the validity of equation (6) in the case of a
fast emulsion. :

To find the relation between the average number of nuclei
per grain and the intensity, a plate was exposed behind a
step wedge and partially developed. The size of grain
selected was the same as used for determining the A, I curve
in fig. 4, the plate being exposed for approximately the same
time behind the same wedge. At each intensity (1) the total
number of nuclei on 200 grains was counted (except at
I=0:044, where 100 grains were considered sufficient), and
hence the average number per grain found. The values are
given in Table ITI., and it will be seen that the curve in
fig. 8 is of the same general form as the A,I curve in fig. 4
as is predicted by the theory.

?

Tasue III.
Cross-section of grain = 0°98,?.

I. No. I No.
1-000 0:98 0270 | 0-24
0689 0°88 | 0-180 005
0490 0°63 0128 0:02
0356 0:43 0:044 0-00

Characteristic Curve of a Photographic Emulsion. 363

The highest value of No is about 1, which corresponds to
less than 70 per cent. of grains changed, whereas actually
the percentage changed corresponding to this value of Ny was
about 90. This is because the partial development has not
been sufficient to show up all the nuclei, and it is very
difficult to do this, since before this stage is reached, nuclei
which initially were distinguishable from one another have
grown together into a single mass of silver. It is, however,
very unlikely that even if every nucleus could be observed
the general form of the curve in fig. 8 would be changed.

025:

ee ERE

to) 25 a5
a

The most natural assumption to make is that longer develop-
ment would merely result in an increase of the number of
visible nuclei in proportion to the number already observable,
and that this is the case is shown by the following experi-
ment :—Two plates were given the same exposure under the
wedge and partially developed, one for 15, and the other for
18 seconds. The values of No were then found for widely
different intensities, with the following results :—

(1) T= 1-00, No for 15 seconds development = 0°613,
No for 18 seconds=0-980, whence (No) 18/(N,)15= 1°59.

(2) I = 0°27, (No)15 = 0°153, (N,)18 = 0°240, whence
(No) 18/{No)15=1°57 ; so that this ratio is practically
constant, and the general form of the curve is indepen-
dent of the development.

|:00

a a

364 Mr. F. C. Toy on the Theory of the

Variation of Grain Size.
Heperimental.

It will be convenient to deal first with the experimental
curves. When the values of 2 were being found for the
curve in fig. 4, the corresponding values for three larger
sizes of grain were determined at the same time and in the
same way. The characteristic curves for the four sizes are
shown in fig. 9. |

100

80

60

3 —>

40

CROSS SECTION.
a= 0-48 2.

b = 1°75 ar.

C =2:73hK.

d = 3-934.

20

—|-0 —0-5 0-0
LOG Eom

The important points in regard to these curves are that for
one and the same emulsion :—(1) a set of large grains is
more sensitive than a set of small ones, which confirms
Svedberg and Anderson’s result (Phot. Jour. 1921, ixi.
p- 325) ; (2) the characteristic curve for small grains has a
greater maximum slope than that for large ones, 2. e., the
ratio of the intensity which just changes all the grains to
' that which just causes the smallest possible change is larger
the larger the grain size. As will be seen from the figure,
the logarithm of this ratio for the smallest size grain is about
1:5, which is equal to an intensity ratio of 30:1, whilst for
the largest size a ratio of 100: 1 is necessary to give half the
curve.

In Table IV. are given values of a, as read off the experi-

Characteristic Curve of a Photographic Emulsion. 365

mental curves in fig. 9, corresponding to known relative
intensities, and in the third column the values of A are

TaBLe IV.
Cross-section of (a) =0°98p’,
: (0) = 175 pe
. (c) =a (ome.
“4 (2) == S305".
| a (curve values). A.
Relative |
Intensity.
(a) = =(6. =). @) (a). (0). (e). © @)
1-000 Oh 95'S; 9iae 980 24 o LGoO) ook
0-689 900 945 96:8 97:8 ou, 294, | 3°80
0-490 S607, 93077960 (97'0 204 266° 3:22 851
0-270 TO 6866 925. 4.04°5 Bo) 2°08 2:60) 2790
0-128 46:0. 7a:0.- SiO ~ S85 O62 loi 667 2-16
0-061 P24 525 O66 780 0D.) OFZ AAO FS
0-033 0:0. 26:0. 540. -676 G00 OFa0e.0°78 *) dei
0-016 C0. ,207., 405. 57-0 0:00 O22 052 0°84
0-008 0-0 TO. . 28:5. 465 0:00 O07 O83 90°62

0 0:25 0:50 0-75 1-0
I

calculated. The A, J curves for the four sizes of grains are
given in fig. 10.

366 Mr. F. C. Toy on the Theory of the

Theoretical.

Consider what is the effect of a variation in grain size on
the nuclei distribution curve shown in fig. 1.

We will first assume that the sensitivity of a nucleus is
quite independent of the size of the grain in which it chances
to be, i.e. once a nucleus is formed in a grain, its sensitivity
does not change as the grain grows. This is apparently
Svedberg’s assumption, for he says : ‘* the small and the larger
grains in one and the same emulsion are built up of the same
kind of light-sensitive material—just as if they were frag-
ments of different size from one homogeneous silver bromide
crystal.” If this is the case, then the only result of in-
creasing the size of grain is to increase the total number of
nuclei, and these will be distributed amongst the different
sensitivities in the same proportion as before. Thisis shown
in fig. 11, where the distribution curves for two sizes of

Fig. 11.

iy ie

grain are given. We have made no assumption regarding
the relation between total number of nuclei and grain size
exeept that large grains have more than small ones.

The curves relating I and No (average number of nuclei
per grain) which will be obtained from distribution curves
such as those in fig. 11 are shown in fig. 12. We have already
shown that the No, I curve is identical in form with the A, [
curve, so that those in figs. 12 and 10 should be of the same
form. As a matter of fact, there is a striking difference.
The experimental curves in fig. 10 le practically parallel to
one another at the higher intensities, and the point of in-
flexion (which corresponds to the maximum ordinate in the
nuclei distribution curves in fig. 11) moves towards the origin
as the grain size increases. In curves (0), (c), and (d),
fig. 10, which are for exceedingly sensitive grains, the point

Characteristic Curve of a Photographic Emulsion. 367

of inflexion has moved so near the origin that the part of the
curve to the left of this point does not show on the scale to
which the curves are plotted. On the other hand, the
theoretical curves in fig. 12 are characterized by the fact

Fig. 12.

Ls ie me

that the ratio of the ordinates for different sizes of grain is
independent of the intensity, and the value of I at the points
of inflexion, I; and the average sensitivity do not change as
the grain size is varied. Thus we cannot explain the effect
of a variation of grain size on Svedberg’s assumption.

Now let us assume that the sensitivity of a nucleus depends
on the size of the grain in which it is contained, and that if

Fig. 13.

I oer 4

La

it is in a large grain it is more sensitive than it would have
been ina small one. The effect of this on the distribution
curve for the larger grain in fig. 11 is to shift it bodily
nearer the zero, thus decreasing the value of I; and increasing
the average sensitivity, as in fig. 13. The Nol curves

|

368 Mr. I’. C. Toy on the Theory of the

plotted from these distribution curves are shown in fig. 14,
and it will be seen that they are similar to the experimental
curves in fig. 10. The reason why, for very sensitive grains,
the lower half of the S-shaped curve appears to vanish (0),

(ce), and (d), fig. 10, is that the value of I; is very nearly

zero, but it would be shown if the points were plotted on a
bigger scale. :
Fig. 14,

|=

The evidence thus points to there being two reasons why
large grains are more sensitive than small ones. Firstly,
there are more nuclei present in the larger grains, so that a
single grain has a greater chance of having at least one; and
secondly, the average sensitivity of the nuclei increases with
the size of grain.

Svedberg in his most recent paper (2id.) discusses the
relation between the average number of nuclei per grain and
the grain size. He says:—“ The rapidity of the increase of
the average number of nuclei per grain No with size of grain
would depend on two factors: the ability of the developer
to penetrate into the grain, and the homogeneity of the field
of light in the grain. If the developer is not able to get
into the interior of the grain, but only attacks the surface
layer, then No would mean the number of centres in that
surface layer, and therefore would increase in approximate
proportion to the grain surface even in cases where the field
of light in the grain was not homogeneous (because of strong
ligbt absorption). On the other hand, if the developer is to
penetrate the grain, No would depend upon the field of light
in the grain. If the absorption of light were feeble, No would
increase in proportion to the volume of the grain; if the
absorption were very strong, No would increase approximately
proportionally to the cross-section of the grain.” Later in

Characteristic Curve of a Photographic Emulsion. 369

the paper he compares the variation of Ny with grain size
for grains which have been exposed to light with the variation
when the exposure is to X-rays, and suggests certain deduc-
tions as regards the absorption of light ‘and X-rays by the
silver halide from the difference which he finds.

Now, from fig. 10 we see (since A= Ny) that the manner
in which No varies with grain size depends on the intensity
to which the grains have been exposed ; we can select an
intensity such that No varies in almost any manner we please.
Thus, unless the difference between Svedberg’s results and
those found here is due to the different emulsion used, there
seems to be no justification for making deductions from the
relation which is found between No and the size of grain at
one fixed arbitrary exposure.

The theory which has been advanced here is capable of
explaining an important fact which appears quite inexplicable
on such a theory as Allen’s (2bzd.). Itis well known that the
sensitivities of the grains in an emulsion depend to a great
extent on the conditions of precipitation and ripening; and
that, in different emulsions, sets of equal-sized grains may
have quite different sensitivities, and even different maximum
slopes for their characteristic curves. If, as Allen suggests,
the nucleus is really a simple molecule of silver halide which
has lost an electron, its characteristics will be the same
whatever the emulsion, and it is difficult to see why grains in
one emulsion should be more sensitive than those of the same
size in any other emulsion. If, however, the nucleus is not
silver halide, it is very probable that the conditions of pre-
cipitation and ripening do play an important part in deter-
mining its characteristics.

Thus, on Renwick’s theory, the condition of the colloidal
silver which is produced will certainly depend on such factors
as the kind of gelatin, conditions and time of ripening, etc.,
and the ease with which colloidal silver particles can be
coagulated will be affected by the amount of gelatin present,
since this is a protective colloid. The creat difficulty
in accepting Renwick’s theory as it stands is this :—It is
known that an unprotected silver sol is very stable to the
action of light. Therefore, if a protective colloid is present,
it will be still more difficult to effect its coagulation and
precipitation by light, whereas in the case of our most
sensitive silver halide grains the energy necessary to make
them developable is exceedingly small.

Liippo- -Cramer (ibid.) believes that the mechanism of the
formation of the latent image is not the same for the most
sensitive and very insensitive emulsions, and he claims that

Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. 2B

rs 2 one.

370 =Characteristic Curve of a Photographic Emulsion.

this is supported by his experiments. He found that the
sensitivity of a very fast emulsion was decreased considerably
by treatment with chromic acid, but that the sensitivity of a
very slow emulsion remained unchanged. He explained
_ this by the existence on the surface of the sensitive grains of
colloidal silver, formed during the ripening process, which
was not present in the insensitive grains, and which was
removed by the chromic acid.

It is very difficult to imagine that the Fudacreneal hight
action varies with the kind of emulsion, and that considering
a whole series of emulsions, from the most sensitive to the
most insensitive, there is a transition region where an entire
change of mechanism takes place. Strong evidence against
Liippo-Cramer’s view is that Svedberg (zbid.) has shown that
in one of the slowest emulsions the reduction centres are
distributed amongst the different grains according to the
same law as has been shown here to hold for their distribution
in the case of one of the tastest commercial emulsions.
This is in favour of the view that for all kinds of emulsions
the process of the formation of the latent image is the same.

The existence of this chance distribution of developable
“centres” in the grains does not conclusively prove that
they are the kind we have considered in this paper, and there
are at least three other possibilities. Assuming a discrete
structure of the radiation, the centres may, as suggested by
some, be the points of impact of light quanta on the grains,
but the fact that the majority of these centres are located on
the edges of the grain is strongly against this view. Also
within the crystal there may be a chance concentration of the
hight energy at certain points, and both these possibilities are
being tested in this laboratory. Again, this chance dis-
tribution may be due merely to the fact that the grain as a
whole is changed by the light, but the developer reaches
some points of it sooner than others. If this is so, there
appears to be no reason why the average number of centres
per grain, considering only developable grains, should in-
Increase, as it does, in a regular manner with the light
intensity. The author believes that the evidence so far
obtained is mainly in support of the theory discussed in this

paper.

In conclusion, the author wishes to express his thanks to
Dr. T. Slater Price, Director of Research of the British
Photographic Research Association for much _ valuable
criticism and advice.

_ On the Stark Eqect for Strong Electric Fields. 371
Summary.

A theory is advanced which explains the relation found
experimentally between the number of geometrically identical
silver halide grains.made developable and the lght in-
tensity. It is assumed that there exist in the grains particles
which are not silver halide, and which are formed during
precipitation and subsequent ripening. With any normal
exposure (?.e. one which gives a value between 0 and 100
for the percentage of developable grains), it is these particles
which form the reduction nuclei, the only action of the light
being to change their condition in such a way that they
become susceptible to the action of the developer. Each
nucleus does not necessarily require the same intensity to
changeit. The nuclei are scattered haphazard amongst the
grains according to the laws of chance, and only grains
which have at least one will be developable. The sensitivity
of a grain is the sensitivity of its most sensitive nucleus.

The effect of a variation of grain size is explained, and it
is shown that Svedberg’s assumption regarding the similarity
of the light-sensitive material in large and small grains is
not in agreement with the experimental facts in the case of a
fast emulsion.

XXXII. On the Stark Hyfect for Strong Electric Fields,

To the Editors of the Philosophical Magazine.
GENTLEMEN,-~—
M* attention has been drawn to the results of experi-
ments by Takamine and Kokubu™* in which an effect
of the nature indicated in a recently published paper t of
mine was detected, namely, a shift of the central line in the
perpendicular component of Hy in a strong electric field.
Before comparing the experimental amount of this shift
with the theoretical value it would have on the Quantum
theory of spectral lines, it is necessary, however, to point
out a slip in my paper referred to above: thus on p. 945
a term is missing from the value of the contour integral (4),
instead of (6) the full value should be
a os Diatval? SBD? /, ras
* “The Effect of an Electric Field on the Spectrum Lines of
Hydrogen,’ Part III. Memoirs of the College of Science, Kyoto
Imperial University, vol. ili, p. 271 (1919).
+ Phil. Mag, xliii. May 1922, P. ae this will be referred to freely.

a

- —

372 On the Stark Liffect for Strong Electric Fields.
Consequently the third term on the right-hand side of

equation (25) p. 948 should be

ain WY? N’ N’i (ji
1024m°H'n,ie * (n)—N'(m)}, » 0. 2 Cap

where N’ is now given by

N’(n) = (ny +7243)! 35 (m+1413)°
N(n)

MW |e oh Se Pe Soi 2 i ee
15n3° — 21 (n,g—7) OF ee

N(n} being still given by equation {10). In view of the.
identity

N(n) = (21 +13) (629? + 6rgn3 + 13”)
+ (2ng + nz) (62)? + 6nyn3+ N57)
= 3(ny +n +73)? —3(my + Ng +73)(Ng— 0, )?

—n3?(ny +ng+nz),
N’ can be reduced to the form

N'(n) = (1 +724+7n3)4 £17 (ny + not ng)? — Ing? —3(»—n1)?},

Gin).
which shows in conjunction with (i.) that the remarks in
the paper about the symmetry of the components are not
affected by this correction. In order to calculate the amount
of shift of the middle n-component of H, we observe that

this component can arise from any of three possible transi-
tions corresponding to

(ms, Mg, My 3 3, Ng, M4)
=i (3, 1,15 2;05-0) sor s252 52,10, 0) ori 2 2a

respectively, the values of {N’(n)—N'(m)} corresponding
to these combinations being — 2°14 x 10°, —2°59 x 10°, and
again —2°59x 10° respectively. And on substituting the
values of the universal constants in (1.) for hydrogen (H=e)
the following expression is obtained for the wave-length
shift }
2 2,12
An =—™ Ay =—1-42™"

{N’(n) —N'(m)}.

C :

Damping Coefficients of Electric Circuits. ato

This gives for H, (A =4°34x 107°) and the value of F
used by Takamine and Kokubu *, namely,

P4833 * 10° ¢.¢.s.e.s..units |-=1°3.x 10° volt.x ¢m.~"],
An = °36A or *43.A respectively.

The experimental value observed by Takamine and Kokubu
is about 1A, which is larger than that predicted by the
Quantum theory. It is, however, possible that part of
the experimental shift is due to a Doppler effect, and in
any case the experiments could hardly be considered accurate
enough to exclude a possible experimental error of what is
only about 4 an Angstrém unit. On the other hand, the
photographs of the shift [plate 1. fig. 1] pownt decidedly to a
general displacement of all the components in the direction pre-
dicted by the theory, namely towards the red, and may be
taken as corroborative of at least this qualitative aspect of it.
It is seen from (i.) and (ii.) that this lack of symmetry in a
strong field would be expected on theoretical grounds to be
more pronounced for the higher members of the Balmer
Series (e.g. H; or H.), and it would be highly desirable ta
obtain measurements relating to these lines as a further test
for the quantitative aspect of the theory.

In conclusion I wish to thank Mr. W. E. Curtis for
drawing my attention to the experimental results already
referred to.

King’s College, London, Yours faithfully,

May 12th, 1922. A. M. MosHarRaFa.

XXXII. On the Damping Coefficients of the Oscillations
in Three-Coupled Electric Circuits. By i. TAXaGIsHt,
Electro-technical Laboratory, Department of Communica-
tions, Tokyo, Japan tf.

HOUGH the importance of the problem of three-coupled
electric circuits has arisen with reference to radio-
telegraphy, it does not seem to have been attacked with any
great amount of attention except by B. Mackt, EH. Bellini,
_and very recently L. C. Jackson{. The valuable paper of
the latter made me feel very much interested, especially as
it will make an important contribution to radio fields, but
¥ 1. &.

+ Communicated by the Author.
t Phil. Mag. vol. xlii. No. 247, July 1921, p. 35.

o74 Mr. E. Takagishi on Damping Coefficients of

unfortunately I found there slight errors concerning the
damping coefficients of the circuits.

Now, let us proceed to correct them, using the same
notation and abbreviations for the sake of simplicity.
Comparing coefficients in the. equations (5) and (6) in his
original paper, we obtain, instead of (7),

—2(q+r+s)
= (Riles +L, ReLs+ LL.B,
— R,M.,;? Ih, oM 33? ar Jal es? = D, : G. ) |

(wo? + @;?+ @3”) 7
as J Lyle = Mie" at LL; — M3? a LL, = Moo |

i 13 | C, "2
+TTQR,+ RRs +R Rls b=D, . Gi)
—2{o(r+s) +0°(s+9) +03°(q+7)}
y { alt Le a AE Ey ane LF,

“ll

4 | |
seesoatelg he Go

— (@1°@9" + @9°@;," + @3°@ 1")
dey eee we Ee f
ce Co OC,
Eerie iS ee) tau A
a2 ; alin Gy a GC, ) —D, . (iv.) |

\
Ls
— 2( 0)?" 8+ @,"03"9 + @;°01"7 )

Ry lay ai \ |
= alee + — GC, 4- Ee, —)D, 5 ‘ rah 5 6 (v.)

(7')

~@, 773 |
IL

Se > 3 (Wile)

where
D = My» Mos Ms; ara L,L, Ls
+ L,M,.?+ LoMs,?+ LsM,.? . (vii.)

J

From equations (7") li., 1v., v1., vil. we obtain the same
equation for w, as (8) and (9) i in his original paper. For the
damping coefficients g, r, and s, however, we find the
following values, different from those in (LO je |

Oscillations in Three-Coupled Electric Circuits. 375

Now, making use of the abbreviations given below and in
the original paper, we get the following equation :

A
—2g+r+) =X,

2 2 : B
— 2 { g(a? + 3”) + 7(@3? + @,") + 8(@)" + @,.”)} = ae
C

—2{o02o/7tofo"rt+oa,"s} = Xx?

where

A =k (1-8) +4o(l—y?) $hs(1—2),
B= n?(ky thy) +P (hy + hy) + m? (ky + hes) + ky hohs,
C= khym?n? + kl?n? + kyl? m?,

X = — (1-2? — 8? —9? —2aBy).

Solving these simultaneous equations, we have

Q

= YY?

ea:
=y

aS

Q Tey?

in which
A
ee) Lee)
x 9 2 A 1 it

4 2
—2(w; + @, ay — (w +o.) nm B (37+ ,”) (a? + 2”)

— 2(@3"@,’) = 2(@ >a") CG W3°@ 1" W?W9"
z “ fo) i hae Bae’ CY,

similarly

ge © (oP -o2) haa Seem PEC},

se = (o2=o,) [4a — Bay + 0},

and Y = 8(@,2?—@,”) (w2?— 37) (@3? — @)2)

376 Prof. 8. C. Kar on the Electrodynamic

That is,
q = [ thym?n? + kan? P + hl?) — @? {PF (he + kg) +m? (kg + ky)
+ ?(ky + hy) + keykoks} + or" {ky (1 — 8?) + ko(1—y’)
+ ke3(1—a?) } | |
=- 2 (1—2«?— B89? — 2a Bry) (w1? — 0,7) (@;7—@3"),
== | (kym?n? + kon?l? + ksl?m?) — wo? {0 (ke + ks) -+ m? (kz + Ay)
+n? (ky + ke) + kykoks} + wo*{ky(1—B?) + ky (1—y¥’)
sii) |
21 = 27 — 6° — 9? — 2a By) (os 03 | es oe
= | (kym?n? + kon?l? + kgl?m?) — ws? LP? (ky + kg) + m?(hg + fy)
+n? (hy + ke) + kyhkoks} + @3* {ky (1—?) + ko(L—y?)
hela") ta
22a payee By) eae) oe eee

~—

(7)

On inspecting these equations for damping coefficients it
is noticed, at once, they are also correct with respect to the
dimensions.

XXXIV. On the Electrodynamic Potentials of Moving
Charges. By 8. C. Kar, M.A., Professor of Mathe-

matics, Bangabast College, Calcutta *.

VENUE electrodynamic potentials of a moving charge or

the electron have been the subject of several in-
vestigations and the earliest were those of Liénardt and
Wiechert f. Among recent writers who have found the
potentials on a relativity basis may be named Sommerfeld §
and M. N. Sahal|. Both of these writers performed a four-
dimensional integration in the Minkowski space-time mani-
fold and have obtained results which are quite general.
It appears to the present writer that the Liénard and Wie-
chert result—and the method admits of easy extension to
the case of a straight linear current—may be obtained easily
enough by a Lorentz transformation to a rest-system and
back without resort being had to four-dimensional integration.

* Communicated by the Author.

+ L’ Eclawrage électrique, vol. xvi. pp. 5, 53, 106 <1898},
{ Arch. Néerl. vol. v. p. 549 (1900).

§ Ann. d. Phys. vols. li. and liu.

|| Phil. Mag. vol. xxxvii. p. 847 (1919).

Potentials of Moving Charges. 377
The equations for the potentials may be written

BE). (iceb, F, G, BH) =pox(ic,u, .v, w)

where eal, yes ae +10?

0° cM ees. role

and UO = Pane an + Oy Tee
It is well io point out that this mode of writing the
equations is Henne different from the customary mode

where OF’= =pok so that our F is cK’. This deviation

from usage 18 amited by the greater symmetry and homo-
geneity of form resulting. The equations for (magnetic
intensity) and d (electric intensity) will on account of this
change assume the forms

Hee ene EL),
c

So ae

d=— alh, G, i) + (55 Oy? dz

(2)

It is evident that the operator 1 is an invariant under

a Lorentz transformation. It will therefore follow that -

(ic®, F, G, H) is a four-vector, because x(ic, wu, v, w) is a
four-vector. Therefore DSi — Fx — Gdy—H82 which
represents the scalar product of the four-vectors (ic®, Ff, G,H)
and (icdt, da, dy, dz) is invariant under a Lorentz trans-
formation. Therefore,

e’@dt — Fbx— Gov — H$z=c?@' bt! — F'82’ — G'dy' — H'62",
where the dashes refer to a system of axes moving with
velocity v along the axis of «.

But bu=x(du' + vot’),
déy=6y’', eck

and bts a( de + = pt z =} where c=1fy/ 1-2. =

Substituting and equating coefficients of 842’, dy', dz’, and
ot' we have

F'=<(F —»®), G'=G, H’=H,

and o'=(@-"2),

378 Prof. 8. C. Kar on the Electrodynamic

These formulee are exactly similar to the usual formule:
for 6x, dy, 6z, dt and connect the potentials for any system

of axes with those of another moving with velocity v along —
the axis of 2.

The reversing formule are
H=«nci’ fo’) Ga G7, A= HH’,
and O— (Oe vl! =}

(3)

Let us suppose an electron moving with velocity wv
along the axis of w and let us take a system of axes moving
with the electron. It is apparent that for the latter system
of axes the electron is i. rest. The vector potentials.

CE", GE’) =0and O'= i

r! is the distance of the point P at which the potentials are
considered.

For the original system of axes, therefore, we should have
according to the formulee of transformation given above,

B= xy - —, G= 0, ia) and = eer.
An Agr
r’ is, however, expressed in terms of the coordinates of the
rest system and it will be necessary to transform it to a form
involving the coordinates of the original system.
But if the time-difference between the point P and the
electron is Ai’, then

be = cAt’

=f rE ar| 1 ee
C Cin arial

due toa static charge e where

€
and Da a

which are Liénard’s results.

‘é

Potentials of Moving Charges. oto

(4)

Let us suppose a straight linear uniform current to
arise from continuous and uniform rush of electrons in the
conducting wire in the direction of the current. Viewed
from a system of axes moving with the common velocity of
the electrons the phenomena reduce, as far as the rushing
electrons are concerned, to the case of a linear and uniform
distribution of electric charge. If N electrons each with
charge —e be supposed to rush with velocity v to the
observer in the rest-system the linear density of static charge
is —Ne.

From the ordinary theory of potential, the potential ®' for
such a distribution is —2Nelogr’ where r?=y"+2” and
(F", G', H’)=0. Transforming to a moving system according
to our formule we should have

Poe ——26Nevu loo'*) “G—G'=—0,; B= 0,
G= 70! = —2eNelogr [n= |.

The magnetic field therefore would be given by h,=0,.

InNevz 2nNev
hy=— oi ;h=+ aE / and the electric field would be
Cr 79
InNez
amen by d_— 0, d,—— ees dj=— at weet the veon-

ducting wire, however, there is also a linear distribution of

positive nuclei at rest of which the potential would be

+2Nelog r.
The electric field due to these would be given by d,’=0,
IN Vine
d= + ~ eeu d= Pith
7a r
The resultant electric field would therefore have the

2Ney 2Nez
2 a Ch a) D) (

The magnetic field is of finite magnitude and cir-

1—«), andisof the order

of —.
ec?

2nNev ae
cular round the wire, the resultant being ———— which is
: cr

ae : ee (eurrent):..

quite in accord with the expression eneonreny) if we put
fe

kNev

Nev . cM
current =——— or —— neglecting quantities of the order
c c

9
“

.. : ;
, In comparison with unity.
c

ee

EE SS Ee

F 380:

XXXV. The Identical Relations in Hinstein’s Theory.
By A. KE. Harwarp*.

VQYHE March number of the Philosophical. Magazine con-
tains an interesting proof of the identity

Gi eee

pve Otp

by Dr. G. B. Jeffery.

Apparently it is not generally known that this identity is
a special case of a more general theorem which can be very
easily proved. I discovered the general theorem for myself,
but I can hardly believe that it has not been discovered
before.

The theorem is

(Buvo®)r + (Buor®)y + (Burr?)o= 0. . . C1)

This identity can be verified in a rather laborious manner
by forming the covariant derivative of Byys?, but it can be
more easily proved as follows :—

The identity
Au, Vo = Jai ov — Buvo® Ap e e ° ° (2)

can be easily generalized so as to apply to the case where
instead of the vector Ay we have a tensor of any order ; thus

Aw, OTe Any, r= Buyer? A pv + Byer? Anup:

This is proved in the same way as (2).
Now, if Ay be any covariant vector, then

(Ap, vor — Ap, vro) + (Ag, ory — Ap, ovr) + (Ap, rxo— Ap, rev)
= (Ay, ve—Ap, ov)r + (Ap, or —Ap, ro) v + (Ap, ro — Ag, v7) 53
~. Byor? Ap,» t+ Byer? Ap, p+ Bury? Ap,o+ Bory? Ap, p
+ Buvo® Ap, r+ Brro® Ap, p
= (Buyo? Ap)rt(Buor? Ap)y + (Bury? Ap)o-

* Communicated by the Author.

The Identical Relations in Hinstein’s Theory. 38h
Now,
(Byvo® Ap)r = (Bure?) Ap + Bure? Ap, z ;
so after cancellation we get
(Byor? + Bory’ + Broo”) App
= [(Buvo)r + (Busr?)» + (Burv®)o} Ap.

The expression in brackets on the left vanishes identically.

Since Ap is arbitrary, the expression in brackets on the right

must also vanish. Q.
The identity

Byer? a Dery’ + Brve® =.)

follows at once from the well-known identical relations:
between the Riemann symbols. The three-term identity is.

usually stated in the form
(utov)+(yorvr) +(pvtc) = 0,
or in the modern notation

Buyvor+ Burvo + Buory =i) 3

here Buror denotes gre Buyos = (tov). Since Buryvo=Byoyr

and Buory = Bopyr,
0= Buyer ap Byour rE Boor
= Gre (Buro® “la Bropf aT Bopr*).

We assume that the determinant | gy,| = g does not vanish
in the region under consideration; therefore the expression
in brackets must vanish. -

This identity can also be proved by observing that the
expression

ee ee NRL e tee e
vanishes if A, is the derivative of a scalar; for in that case
Ay.or = Ac,vr, Ao,ry = Ar,ov, and Az, ve = Av, ro.

If we contract (1) by putting T=p, we get
(Biot at Ops,v—Ge,c = 0... 2°. (8)

082 Mr. H. 8S. Rowell on Hnergy Phrtition

If we contract this again by multiplying by ne we get the
familiar identity

2G. ee ogi iy
for since (g'”)o = 0,
9” (Bune? )p = (gt? Bura?)p = (gH gP" Buvor)p
= (9? og!” Brovu)p = (7°" Brow”) p
== (O° Gras — Gee ;
‘similarly

gt? (Crug, v) =P? Gael = Ge,
and

cas (Guy, 9) Py (gh Guy) o = (G)o ==
‘since G is a scalar.

Jersey,
18th May, 1922.

XXXVI. Energy Partition in the Double Pendulum.
By H. 8. Rowe. *

N a letter to ‘ Nature’ (July 28,1921) the present writer
gave a theorem on the double pendulum which is capable
-of interesting extension.
If the masses of the bobs are m and M and the respective
amplitudes are a and A with suffixes.to denote the normal
modes, then the theorem states that

AOR ans
If this equation is squared and Boae sides multiplied by
am?/M?, we have

M172, MAg?N9"

MA,?n 1 MA,?n,?

=

where 7; and ny, are the radian frequencies of the two moles.
‘This equation may be readily interpreted thus :—
“The ratio of the kinetic energy of one bob to that of the

* Communicated by the Author,

in the Double Pendulum. 383

other bob in one mode is the reciprocal of the corresponding
ratio in the other mode,”’

Proceeding to the general case of an elastic system with
, . 5 Ls .
two degrees of freedom, using Professor Lamb’s notation,

2T = Aé?+2HOd + BE»,
2V = 2V)+a0? + 2h0gp+ bd’ ;
so that with a time factor = in
(N04 Hh) = ad +hd,
n*(HO+ Bd) = h0+b¢ 5
' bH—AB
whence the product of the roots in @/¢ is — RAS
If H=0 so that T is a function of squares of velocities,
the product of the amplitude ratios is —B/A, or, in the
double pendulum, —M/m.
If h=O0 so that the potential energy is a function of
squares of displacements, the product of the amplitude
ratios is b/a, i.¢. the ratio of the two stability coefficients.

‘Thus in either case we have an energy relation. For the
kinetic energy take

ho and). —— = — =

square and multiply by A’/B?, and insert the frequencies.
For the potential energy take

A=0 and —1?=—-

which, when squared as before, yields a similar relation.

The two results may be expressed in words thus :—

When the Kinetic or Potential Energy is written as a
function of squares only, the ratio of the Kinetic or
Potential Energy expressed in one co-ordinate to that
expressed in the other co-ordinate for one normal mode is
the reciprocal of the corresponding ratio for the other
normal mode.

This investigation gives an insight in certain cases into the
indeterminateness of the normal modes with equal periods.

[ 8eiee

XXXVII. Velocity of Electrons in Gases.
To the Lditors of the Philosophical Magazine.
GENTLEMEN,—

|e a paper in the Jahrbuch der Radioactivitidt und Electronik

(vol. xvill. p. 201, April 1922) H. F. Mayer gives an
account of some of the formule obtained by different
physicists for the velocity of ions or electrons in gases due to
un ‘electric force, and concludes that a formula recently
given by Lenard is more correct than the others.

Among the other formule which are discussed, the author
gives what purports to be an account of a formula for the
velocity of an ion which I published in the ‘ Proceedings of
the Royal Society’ (A. vol. Ixxxvi. p. 197, 1912), and states
that this formula is so incorrect that it does not even give
the right order of the velocity. JI should like to draw
attention to the way in which Mayer has misinterpreted the
matter, and to quote the formule as I gave them for the
different cases in which the mass of the ion is small or large
compared with the mass of a molecule of the gas through
which it moves.

On pages 199, 204, and 206 of my paper, three formulee
are given for the velocity U of an ion in the direction of the
electric force X in terms of the mean free path / of the ion,
its mass m, charge e, and velocity of agitation uw which is
supposed to be uniform and large compared with U.

The first of these is

UHXlimu,, . . .

and applies only to cases in which the mass of the ion is
small compared with that-of a molecule of the gas (an electron
for example), since it is here assumed that after a collision
with a molecule all directions of motion of the ion are equally
probable.

I pointed out that when the mass of the ion is larger than
that of a molecule of the gas, all directions of motion of the ion
after a collision are not equally probable, and that in this
case an ion travels a considerable distance (having an average
value 2X) after a collision in the direction in which it was
moving before a collision. A more general formula for the -
velocity was given, which is

Us XeG@Pymu. 258 4 een

If the mass m of the ion is so large compared with the

Velocity of Llectrons in Gases. 385

mass m’ of a molecule of the gas that all directions of
motion of a molecule become equally probable after a
collision with an ion, it was shown that formula (2) reduces to

MAU Wey a We Mien! area ose, (3)
as in this case it may be seen that

t+ he
Three . & . e ° . (4)

It will be observed that formula (2) reduces to (1) when
Xr is zero, that is when m is small compared with m’, so that
either of these two formuls: may be applied to the case of an
electron moving ina gas. Mayer, however, selects formula
(3) to find the velocity of ions of small mass or electrons,
although it is definitely stated in my paper that formula (3)
refers to large ions, and the relation (4) on which it depends
ean only hold when m is greater than m’. As the correct
formula (1) for electrons differs by the factor m’/m from
formula (3), it is unreasonable to expect the latter formula to
give the velocity. of an electron.

The above formule, obtained by simple considerations
when the velocities of agitation of all the ions are taken as
being the same, are of course not absolutely exact. There
is a numerical factor by which the expressions should be
multiplied in order to allow for the variations of the velocity
of agitation about the mean velocity. In the most interesting
cease, which is that of electrons moving in a uniform electric
field, the value of the numerical factor is about °9, but it has
not been determined exactly. The determination of this
factor is very difficult, as the distribution of the velocities of
agitation of the electrons depends on the energy of an
electron which is lost in. a collision, and experiments show
that the proportion of the total energy of an electron which
is thus lost depends on the velocity. This problem has been
fully considered by F. B. Pidduck (Proceedings of the
London Mathematical Society, ser. 2, vol. xv. pt. 2, 1915),
who shows that under certain conditions the proportion of
the velocities which differ largely from the mean velocity
of agitation is much less than the proportion indicated by
Maxwell’s formula for the distribution.

It appears that the error introduced by taking the velocities
of agitation as being all equal to the mean velocity may be no
greater than when the velocity distribution is taken as being
the same as that given by Maxwell’s formula.

In order to obtain an exact formula for the velocity U it

Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. 2C

386 Prof. H. A. McTaggart on the Electrification

would be necessary to take into consideration the variation
of the mean free path of an electron with its velocity of
agitation, and the large reduction of the energy of an electron
when ionization by collision takes place.

These points in connexion with the motion of electrons in
gases have not been taken into consideration by Lenard, and
it does not appear that his formula is more correct than
others which have been proposed.

Yours faithfully,
8rd May, 1922. ) JOHN S. TOWNSEND.

XXXVI. On the Electrification at the Boundary between a
faquidand a Gas. By Professor H. A. McTaccart, M.A.,

University of Toronto”. |

ANY years ago, in the course of some experiments on
4 the effect of an electric current on the motion of small
particles in a liquid, Quincke (Ann. d. Phys. exiii. p. 513,
1861) observed that small gas-bubbles in water moved as
though negatively charged. Although a good deal of atten-
tion has been paid to the movement of solid and of liquid
particles in such cases, very little effort has been devoted to
the study of small spheres of gas suspended in a liquid—one

obvious reason being the difficulty of controlling them while

under observation. A systematic examination of their elec-
trical properties ought, however, to yield further information
as to the physics—and chemistry too—of surface layers. |

Before the war experiments in this field were begun by
the author in the Cavendish Laboratory under Sir J. J.
Thomson, and some results were obtained. Measurements
were made (Phil. Mag. Feb. 1914, p. 297) of the velocity,
uuder a fall of potential, of small spheres of air in distilled
water and their electrical charges were estimated. ‘The
effects on the charge of the addition of minute amounts of
various inorganic electrolytes were studied. Results were
obtained (Phil. Mag. Sept. 1914, p. 367) showing how the
charge varies with the presence in the water of certain
alcohols and organic acids, and a parallel was shown to exist
between the variation of the electric charge and the surface
tension.

The present paper deals with some further experiments
carried out in the University of Toronto, and describes the
variation observed in the electric charge on small spheres

* Communicated by Professor J. C. McLennan, F.R:S.

:

at the Boundary between a Liquid anda Gas. 387

of air when a particular electrolyte, Thorium Nitrate
[Th(NO,),], was dissolved in water. This salt was selected
for special study because it had been found to be unusually
active in charging these surface layers.
The apparatus used was similar to that referred to in a
former paper, one or two changes being. made in i ton
greater convenience. The arrangement is shown in fig. 1.

A isa small cylindrical glass cell rotating about its axis
on pivots and driven by a belt of thread from a pulley F on
a Rayleigh motor. This motor was made in the laboratory
workshop, und has, instead of the usual fly-wheel with a
hollow rim filled with water, a solid brass wheel H—a modi-
fication suggested by Professor Wilberforce of Liverpool.
The wheel, although loose on the shaft, has enough friction,
when a heavy oil is used for lubricant, to keep the shaft in
steady motion after synchronism with the tuning-fork is
attained.

D is a timing device consisting of a vertical post carrying
a pointer and made to rotate by a toothed wheel working in

202

388 Prof. H. A. McTaggart on the Electrification

the worm HE. The pointer rests by its own weight on the
top of the post, but at any instant in its motion over the fixed
dial D it may be raised and stopped by a small electromagnet
controlled by the key B. When released it falls back on the _
post and begins to record time with the same regularity as
the tuning-fork. It forms a very convenient stop- -watch if
velocities are to be measured.

A travelling microscope M measures the distance travelled
by any bubble on the axis of the rotating cell.

The water used was twice distilled—the second time in
‘“¢ Pyrex” glass and condensed in a silver coil.

The thorium nitrate was by Merck, and-was assumed to
have 12 H,O—water of crystallization.

A stock solution was made up containing 4 x 10° equiva-
lents per c.c. (1/250 normal), and from this other solutions
were made by successive dilution.

A first series of readings was taken with various concen-
trations of the salt, but with bubbles of nearly the same size
in order to reproduce the effects previously observed—the
method of working being to fill the cell A with the desired
solution, introduce a single bubble of air with the gas pipette,
and set the cell in rotation. The bubble very soon takes up
a steady position on the axis, and its motion under any fall
of potential L. may be examined.

Very small concentrations sufficed to reduce to zero the
natural negative charge found in pure water and to give
the small sphere of air a positive charge.

The following readings are typical :—

Fall of potential ......... 34 volts per cm.
Diameter of bubble...... 0-3 mm.
Concentration. 4: Velocity of
Equivalents sea bubble.
per C.c. aoe ems./sec./volt./em.
ASC TO Ot Mak se teats + 5x10-4
SLOT Oy ohare phate a - slower
8510-9 | Lae + very slow
(CEES MO)
DT VOT?) 5s eens ~ slow
ACTEM O! 1: A ee | — faster
ASN Ogre 5. le — faster

Pure’ waver .........0e% _— 4x1074

'

at the Boundary between a Liquid and a Gas. 389

The zero point was reached at a concentration of about
7x 10~, a result rather higher than that given in a former
paper. The salt was an entirely different sample, and may
not have contained the same proportion of water of erystal-
lization. (See Abegg and Auerbach, ‘ Inorganic Chemistry.”

A series of readings was then taken for spheres of air of
different sizes, one ‘object being to observe the charge on
very small spheres. It is very difficult, by the use of any
kind of pipette, to introduce into the rotating cell bubbles
smaller than 1/5 mm. in diameter. To avoid this difficulty
the following mode of working was adopted. ‘The solution
was first placed in a partial vacuum to remove as much
dissolved air as possible, and afterwards poured inte the cell.
A bubble into this gas-free solution slowly decreased in size
by absorption until it vanished, while the electric charge
could be observed at any stage.

Under these circumstances it was found that for a suitable
concentration of solution a sphere of air which began with a
small negative charge almost invariably and in a regular
way reduced its charge to zero, and gradually took on a
positive charge. ;

The following readings illustrate this point —

No:
SRE oy cee ethers - = apne
per c.c. In mm, =:

WO Sea tees eee 0:26 —
0717 =
0°14 —
0°10 0
0:08

No. 2.
i ae ae a oO cee
per C.c. in mm.

LO a st oe 0-44 -
0:35 me
0°26 —
0-17 0

0°14

390 Prof. H. A. McTaggart on the Electrification

Novia:
Concentration. Diameter of giasvae
Equivalents sphere ase x
per’e.e. in mm.

107? bo Le ee 0°62 —
0°53 —
0°39 ap
Oat7, +
0:08 +

No. 4.
, Concentration. Diameter of Sign of
Equivalents sphere charge.
per c.c. in mm.

LOR < OMe ese: 0-71 =
0°53 —
0°44 —
0°35 0)
0°32 a
0:23 +

No. 5.
yore ere ae
per c.c. in mm.

LO? OMe clues see 0:28 =
0-17 —
0:08 _
0:05 —

It will be seen from the first four examples given that at
a concentration of 10-°x 5:7 the change of sign occurs in
every case. Rarely, as in No. 5, and then only when the
original sphere was small, did the sign remain the same.
Even then the charge grew steadily less. In_ practically
every case the negative charge slowly decreases as the
bubble gets smaller, passes through zero, and increases to a
small positive value.

at the Boundary between a Liquid and a Gas. a9]

Three examples are given for slightly greater concentra-

tions :—
No. 6.
Concentration. Diameter of gi f
Equivalents sphere raved
: charge.
per c.c. in mm. %
TOF GG ek 0°53 =
0-41 0
0°35 at
0°17 +
0:08 +-
Nada hh
Concentration. Diameter of ; :
Hquivalents sphere me o
per c.c. in mm. Be.
TO? 660.2! i 0°35 =
0:28 0
0:26 ae
O17 +
0°14 +
No. 8.
Concentration. Diameter of d
Equivalents sphere Bee a
per ¢.c, in mm, ean
We? iS ees te ita 0°44 Almost zero.
0°35 —-
0-17 +

Above a concentration of 10~* x 8 the bubbles were always
positive.

The examples given show that the spheres do not all have
the same size when they reach the zero—isoelectric—point
in a given solution. The larger a sphere is at the beginning
the larger itis when its charge becomes zero. This suggests,
as the cause of the change in sign, a kind of coagulation of
something in the free surface.

It is known that, in a solution of thorium nitrate in water,

392 Prof. H. A. McTaggart on the Electrification

hydrolysis occurs with the formation of thorium hydroxide
thus—

Th(NOs),+4 HOH—>Th(OH),+4 HNOs.

There is present in the solution some of the original salt,
some acid, and the hydroxide in colloidal form. The pre-
sence of the last-mentioned was suspected as one of the
causes producing the reversal of sign, and experiments were
then made to test its activity in altering the charge.

A colloidal solution of thorium hydroxide as free as
possible from salt and acid was prepared by dialysis (Burton,
‘Physical Properties of Colloidal Solutions,’ 2nd Ed., p. 16).
A dialysing “‘ sleeve’? shaped in the form of a test-tube was
made of “ parlodion”’ (sold by the Du Pont Chemical Co.,
New York}. A solution of the parledion in ether and
alcohol was used to coat the inside of a test-tube of suitable
size. After the solvent had evaporated the parlodion re-
mained as a thin but strong film which when detached from
the glass served very well as a dialysing vessel.

For this experiment a solution containing about 2 gm. of
salt in 50 c.c. of water was dialysed for a period of three
weeks, after which an estimate was made of the colloid pre-
sent. A sample of 10 c.c. evaporated over sulphuric acid
gave a residue of -0034 9m. The residue formed a thin
layer of gelatinous material on the bottom of the evaporating
dish, with drying cracks across it in all directions.

The effect of this colloid on the charge on small spheres
of air in water was then examined, the dialysed solution
above mentioned being diluted as shown in the following
examples :—

C.c. colloid Diameter of

No. solution in sphere pier of
100 ¢ c. water. in mm, rae
Ate Siac Bas 10 0-21 te
0-12 +
0:07 ae
DN ge Borhihioure 5 0-17 aK
Beene ss ral 25 0-17 +
0:07 +

It is seen that the surface is charged positively by the
presence of very small amounts of the colloid.
The following examples show the gradual reversal of the

at the Boundary between a Liquid and a Gas. 393

sign of the charge accompanying the absorption of the

bubble :—

C.c. colloidal Diameter of :
No. solution in sphere ele of
100 ¢,c. water. in mm, mare ise
CT Al A a8 1:0 0-17 =
0-14 +
DMiereetecs 0-5 0°35 —
0°26 --
0:12 “ins
0:05 se
Oi A erceahd 0:25 0°32 _
0-17 —
0:08 —
0:05 se

The experiments show that the colloidal thorium hydroxide
gives both the effects observed with the ordinary solution.
It not only charges the surface positively if present in sufhi-
cient amount. but it also exhibits the reversal of charge with
diminishing size of the bubble, and this, too, in concentrations
of thorium of about the same order as in the case of the salt.

TNiscussion.

The ‘state of the matier and the nature of the electric
forces in surface layers of liquids is still a subject on which
no very clear ideas exist. Hxperiments on electro-endosmosis
all point to a selective action in such layers so far as the
ions in the solution are concerned. But the observations
are always complicated by the presence in contact with the
liquid surface of a solid whose role in the selecting we are
ignorant of. The same is true of cataphoresis experiments
with solids, as, for example, in the study of the electrical
‘charge on colloidal particles. This difficulty is avoided,
however, in similar experiments with small spheres of air—
or any gas—and in such cases we can safely regard any
effects observed as due largely to the properties of the liquid
and its free surface. In particular, the electrical charge
existing at any air-liquid surface may be considered as the
result of forces residing altogether in the liquid. It ought
to be possible, then, in considering potential differences at
solid-liquid junctions to isolate the contribution of the liquid.

In the case of thorium nitrate in solution the selective

394 Electrification at Boundary between Liquid and Gas.

action of the air-water surface is very marked, a positive
charge being acquired by the surface with very minute
concentrations of the salt. The positive ions available for
selection are Tht and H™, but neither of these separately
can be responsible for the unusual activity of the salt. ‘The
mere presence of H™ ions, as, for example, in the form of an
acid, does net produce so great an influence on the surface
charge. “Nor can free Tht ions have much effect, for they
disappear in the dialysis and yet leave the pure colloidal
solution practically as active’ as before. The real agent
must be the particles of colloidal thorium hydroxide which
gather about them groups of H* ions and carry them into
the surface in larger numbers than would be possible for the
H* ions alone.

The nature of this selective action must be connected with
the shape of the surface, or, to put it in another way, a
particle must reach a certain size before it can be regarded
as liaving a surface-layer about it with a tension and an
electric charge. We have at present in order of size—ions,
ionic micelles (Prof. McBain, ‘Soap Solutions,” Nature,
March 10, 1921), ultra-microscopic colloidal particles, micro-
“scopic and macroscopic particles including gas-bubbles. At
what stage a surface-layer is formed it is difficult to say, but
it seems reasonable to suppose that the curvature of such a
surface would have an effect on the charge adsorbed. The
change of sign with decreasing size of air-sphere shown in
these experiments seems to bear out this idea.

The information obtained regarding the effect of thorium
nitrate on the electrification of air-water surface layers may
be summarized as follows :—

1. Thorium nitrate in aqueous solution and in concentra-
tions as small as 8x10-® normal gives a positive electric
charge to the surface of a sphere of air immersed in it. (In
distilled water the charge is always negative.)

2. For concentrations in the neighbourhood of 6 x10
normal a sphere initially negative becomes gradually positive
as the sphere diminishes in size.

3. Colloidal thorium hydrexide in small concentrations of
the same order also gives a positive electric charge to a
sphere of air immersed in it. |

4. Colloidal thorium hydroxide also exhibits the reversal
of the sign of the charge with a decrease in the size of the
bubble.

5. It is suggested that this reversal of sign is experimental
evidence of a relation between the curvature of the surface
and its adsorptive power.

Lecture-Room Demonstration of Atomic Models. 395

The experiments are being continued as time permits in
the hope of obtaining some new information regarding these
free surfaces. Is is the intention to compare with thorium
the effects of one or two other tetravalent and_ trivalent
metals in the colloidal state.

I wish to thank Professor J. C. McLennan for his kind
and encouraging interest in the work.

XXXIX. Note on a Lecture-Room Demonstration of Atomic
Models. By Louts V. Kine, D.Sc., Macdonald Professor
of Physics, McGill University *.

| Plate IT. }
Section 1.

EVERAL mechanical models illustrating various types
of atomic structure have been proposed from time to
time. Among these we may mention Mayer’s classical
experiments with floating and suspended magnets, illus-
trating the action of atomic forces t.

Many modifications of these classical experiments have
been suggested. In particular, a paper by R. Ramsey de-
scribes interesting modifications of the original apparatus f.

Actual apparatus illustrating the supposed structure of
atoms can now be obtained ready for use from scientific
instrument makers §.

All these methods involve the repulsive forces between
steel elements (needles or spheres) in a permanent magnetic
field, together with the central attraction set up by a per-
manent magnet. An important point contributing to the
success of the experiment is that all the magnets, repre-
senting electrons, have as nearly as possible equal pole
strengths. Owing to magnetic reluctance and effects of
demagnetization, these conditions are difficult to realize in
practice without a considerable amount of care and ex-
penditure of time.

* Communicated by the Author.

¢ J. J. Thomson, ‘ Corpuscular Theory of Matter’ (1907), Chapter 6,
pages 105 et seq.

t R. R. Ramsey, ‘The Kinetic Theory of the Electron Atom.” Pro-
ceedings of the Indian Academy of Sciences, 1918. Phil. Mag. vol. xxxiii.
Feb. 1917, pp. 207-211.

§ W.M. Welch, Scientific Company, Chicago.

396 Prof. L. V. King on a Lecture-Room
Section 2.

The magnetic elements which form the essential feature
of the apparatus to be described consist of a number of steel
spheres or small soft-iron rods magnetized in a strong
alternating field.

One evel model is shown diagrammatically in Pl. I. fig. 1,
while fig. 2 shows the actual apparatus. The coil A consists
of 340 turns of number 12B. «8. copper wire (2 mm. diam.);
inside radius of winding 8°8 cm., outside radius 13°5 em.,
width of coil 3°9 em. Such a coil has a resistance of
approximately 1°3 ohms and self-inductance of about 32
millihenries. It may be connected directly to a 110-volt
60-cycle A.C©. circuit without overheating. In such cir-
cumstances it draws a current of about ¥ amperes. It is
approximately of such dimensiors as to. give a maximum
field strength at the centre of the coil.

Placed over the opening of the coil is a large watch-glass
B whose radius of curvature is approximately 25 cm. If
available, an accurately ground concave glass mirror may
be used to advantage. If, now, a supply of steel ball- bearings
about 3 mm. in diameter is available, these may be placed
on the concave surface B, where they will experience an
attraction towards the lowest point approximately pro-
portional to the distance. When the maximum current is
passed through the magnetizing coii, the steel spheres will
become A.C. “magnetic doublets of very uniform magnetic
moments. It will be noticed that the magnetic axis will
always be very accurately along the direction of the mag-
netic field, independently of the rolling motion of the balls.
Furthermore, if the spheres are of fairly uniform quality and
the field strength sufficiently great, the instantaneous mag-
netic moments of these doublets will be equal in magnitude
and phase. In these circumstances the steel spheres will
repel each other with a force varying as the inverse fourth
power of the distance, the constant of proportionality being
accurately the same for all the spheres. With the attraction
to the centre varying as the distance, it may be expected that
the magnetic elements will form remarkably symmetrical
stable groupings. One such grouping is illustrated in
Pea enos 3) (a).

It is obvious that by a very simple arrangement of lenses
and mirrors this model atom may be projected on a screen.
The concave surface B may, if desired, be mounted so as to
allow of rotation, thus increasing the interest of the “atomic ”
arrangements. “This experiment is extremely convenient for
lecture-room purposes, as it requires no preparation and is

Demonstration of Atomic Models. hs ah

_always certain to give results which never fail to delight an
audience.

An interesting variant of this experiment is to make use of
the arrangement of two coils described in Section 5 (figs.
5 & 6). A surface of clean mercury is placed midway between
the two coils. A number of steel balls floating on this surface
will repel each other as already described, and will all tend
towards the centre, owing to the greater intensity of field.
The remarkably regular + arrangement taken up under these
conditions is shown in ne 3 (b). The damping is so slight
that the system may be set into oscillation in various ways
by means of external magnets, giving a good illustration of
internal vibrations in the atom. It would, moreover, be
possible with no very great expenditure of labour to deter-
mine the frequency of various modes and compare the results
with theoretical calculations.

Section 3.

The same apparatus may also be used to illustrate the
motion of the molecules of gas or the Brownian movements.
For this purpose an elongated piece of iron is employed,
e.g.a short cylinder of iron or steel wire about 1 em. in
leneth by 1 mm. in diameter. In the alternating field of the
coil such a magnet experiences a very strong torque, which
vanishes when the axis lies along the direction ot the
resultant A.C. field. If sucha magnet is placed in a flat
eylindrical glass vessel occupying the centre of the coil, ai.d
the field suddenly applied, violent movements of the little
iron rod will be observed. The instantaneous moments set
up by the field will be sufficient to make the rod leave the
surface on which it is resting and describe a trajectory
under the combined efféct of oravity and the magnetic field.
At the termination of the flight, it will again strike the glass
plate and will then receive an additional impulse made up of
the magnetic torque and the elastic reaction at contact with
the glass. This will start it on a new trajectory, and the
process will be continued indefinitely until the rod makes
contact with the plate at the termination of its flight in such
a way that the instantaneous torque is zero. Then it stops
dead with the axis pointing along the direction of the field.
This is an event which happens very rarely. Several such
rods enclosed within a glass vessel will keep in constant
motion in a manner resembling the motion of molecules in a
rarefied yas. An interesting variant of this experiment is to
insert short steel wires along the diameters of small pith balls
which hop around, describing flights in the glass vessel as if

|

398 Prof. L. V. King on a Lecture- Room

they were animated with life. As before, the glass vessel
and its contents may be projected on a screen, the resulting
effect being illustrative of molecular movements.

Section 4.—Haperiments on Electrodynamic Repulsion.

Owing to the distribution of the magnetic field around the
coil employed j in chis experiment, the same apparatus is well
suited to the demonstration of electrodynamie repulsion. For
this purpose several plates of aluminium or copper should be
cut with a radius approximately equal to the outer radius
of the coil. Such a disk may be anchored by three strings
fastened at equidistant points of the circumference so as
to allow it to move vertically, with its centre over the
axis of the coil, which is laid in a horizontal position.
On applying A.C. circuit, the plate will float three or
four centimetres above the coil. By placing a light iron
rod (3 cm. x 1mm.) on the plate, the direction of the
A.C. field is easily demonstrated, as shown in Pl. IT. fio. 4
It will be noticed that over an annular region bounded by
the outer edge of the plate and a circle of half its radius, the
lines of force are inclined at approximately 45° to the vertical.
It is the reaction of the horizontal component of the A.C.
field with the induced current due to the vertical component
which causes the repulsion referredto. To demonstrate this,
a circular plate may be cut up into several concentric rings
and laid on a sheet of glass. When current is applied it is
only the outer rings which are repelled, the force on the

inner rings gradually becoming less, until that on the central.

disk in a practically uniform field perpendicular to its plane
is practically al.

Tron filings poured on a glass plate laid horizontally over
the coil assume an interesting laminar distribution, which
again may be projected on a screen. The iron filings tend
to arrange themselves in a series of vertical planes about
iene high arranged radially. It is easily seen that this
arrangement i is nee to the fact that under the influence of the
alternating field, each of the radial planes represents a series
of vertical A.C. magnets which repel each other. Their
height is limited by the vertical stability of the plates under
the combined effect of gravity and of the alternating field.

Section 5.— Experimental Model of the Rutherford Atom.

By using two coils of the dimensions already described,
arranged with their planes horizontal at a distance apart
equi al to the mean radius (Helmholtz arrangement), it is

Demonstration of Atomic Models. 399

possible to secure a fairly uniform field over a considerable
area midway between the coils. Such arrangement (Pl. If.
figs. 5 & 6) allows of interesting experiments ona model atom
approximating more closely ‘to modern ideas. A shallow
circular basin of mercury is placed on an adjustable stand
between the two coils. A number of steel pins with glass
heads serve as the elements (electrons) for the model. It
one of these is placed with the glass head on the mereury
surface, it will float in a vertical “position and tend to move
towards the centre ot the field, owing to the greater concen-
tration of lines of force. This force towards the centre may
be varied at will by adjusting the height of the mercury
surface, or by placing rods of soft iron along the axis of the
coils at adjustable distances above or below the mercury
surface. Ifa second pin be floated on the mercury surface,
it will repel the first with a force varying nearly as the
inverse square law when the distance apart is not too great.
A third pin may be added, when a triangular arrangm lent will
be formed. Successive pins give the familiar series of regular
polygons arranged in Senet nie rings. It is evident that
the vreat advantage of the A.C. field is to make the mag-
netical polarity of each of the pins very nearly equal, thus
giving rise to a remarkable symmetry in the arrangements
formed, as illustrated by figs. 7 (a) and 7(b} (Pl. IL).
As before, the experiment can be carried out in such a way
that the various stable arrangements may be projected on a
screen. It isextremely simple to demonstrate the apparatus
at a moment’s notice, the only precaution necessary being to
use clean mercury so as to allow a great mobility of the
floating pins on an uncontaminated surface.

It is interesting to notice that rotation of the basin con-
taining mercury does not disturb any particular stable
arrangement, owing to the fact that the centrifugal force
is accurately balanced by the change of slope of the para-
boloidal mereury surface.

The use of an A.C. field allows of the possibility of realizing
positive electrons and a central nucleus, the law of forces
between them being very nearly that of the inverse square
and at the same time very exactly that corresponding to
charges of te, +2e, +3¢, etc. It is evident from fig. 5

(PT: I1,), illust: rating the model under consideration, that
ie ons may be represented by lengths of soft-iron wire of
the same diameter arranged to move with both ends in the
same plane at distances not too far apart compared with their
length. In these circumstances we have repulsion according
to the inverse square law, the charge —e being represented

Cae ees TH

Tb Senna ee

i

400 Lecture-Room Demonstration of Atomic Models.

by the average pole strength -+-m of each rod, which is
extremely uniform. A nucleus of positive charge ne may be
made up by taking 2n lengths of the same wire and inserting
them in a small glass or aluminium tube, as shown in fig. 5,
illustrating a nucleus of charge + 2e. In these circumstances,
each of the rods representing electrons is attracted to the
nucleus with a force varying nearly as the inverse square of
the distance and proportional to nm xm, the average pole
strength of each end of the rod being bo,

In order to realize this arrangement, the rods (about
7 cm.x 1 mm. diameter), representing negative electrons,
should be suspended from silk fibres about 1 metre or more
in length. By adjusting the position of the rods in the
space between the coils, a position of neutral equilibrium
may be found in which there is practically no tendency for
the rods to move either towards the centre or radially
outwards. Under the combined effect of gravity and of
the magnetic field they seem to float in any position. When
this adjustment has been made, the rods representing the
nucleus should be set in position along the axis of the coils.
The suspended rod representing the electron may then be
projected so as to describe a path about the fixed nucleus,
and a damped elliptic orbit will be observed, the nucleus
being at one focus.

If 1 two lengths of. wire are ued to make up a nucleus +e
in the manner illustrated by fig. 5 (a), we obtain a model of
the hydrogen atom which is dynamically stable.

If we make up positive nucleus of charge 2¢e, represented
by two pairs of iron rods, we obtain a model (fig. 5 (d))
of the ionized helium atom which is dynamically stable.
If we introduce an additional iron rod representing an
electron (fig. 5), and therefore a complete helium atom, it
seems impossible to obtain a dynamically stable arrangement
by any circumstances of projection. For instance, any
attempt to reproduce the symmetrical oscillation suggested
by Langmuir meets with failure, owing to the dynamical
instability cf this arrangement.

It is obvious that further experiments along these lines,
leading possibly to results of great interest, might be carried
out by constructing large solenoidal coils to give a uniform
A.C. field, in which circumstances the inverse square law of
attraction and repalein between electrons and nuclear
charges ne (n=1, 2, 3, etc.) would be faithfully reproduced.

ee ee ee ae

| 401]

XL. The Influence of the Size of Colloid Particles upon the
Adsorption of Electrolytes. By Humpnrey D. Murray,
Exhibitioner of Christ Church, Ouford*.

we, EVERAL workers have examined the influence of con-
K- centration upon the coagulation of colloidal solutions,
but references to the effect produced by alteration in the
degree of dispersion are few and not very definite. Kruyt
and Spek f examined the coagulation of colloidal arsenious
sulphide, and found that the coagulative value of univalent
ions increased with increasing dilution; in the case of a
divalent ion there was a slight decrease ; whilst for a ter-
valent ion there was a rapid decrease in the coagulative
value. Burton and Bishop t examined the coagulative
values of various ions upon colloidal solutions of arsenious
sulphide, copper, and gum mastic, and as the result of their
experiments found that with univalent ions the concentration
of the ion required for coagulation increased with decreasing
concentration of the colloid, for divalent ions the-concentra-
tion of the ion was nearly constant, for trivalent ions the
concentration of the ion varied almost directly with that of
the colloid. More recently Weiser and Nicholas§ have
extended these researches to colloidal solutions of hydrous
chromic oxide, prussian blue, hydrous ferric oxide, and
arsenious sulphide. They found in the case of the first three
that the coagulative values of electrolytes tended to
increase with dilution of the colloid, but the increase was
less marked with electrolytes having univalent precipitating
ions, and became more marked as the valency rose. Odén
found that sols with ultramicroscopic particles are more
sensitive to electrolytes than those containing amicrons.

The object of these experiments was to examine the
influence of the size and uniformity of colloid particles upon
the adsorption of electrolytes as measured by the minimal
concentration for coagulation. For this it was necessary to
obtain solutions of the same colloid prepared under identical
conditions, but containing particles of different mean size.
It was decided to employ Odén’s method of fractional
coagulation. The most suitable colloid to use, therefore, is
one which, when first made, contains particles of markedly

* Communicated by the Author.

+ Kruyt and Spek, Koll. Zeit. xxv. p. 1 (1919).

t Burton and Bishop, Jour. Phys. Chem: xxiv. p. 703 (1920).
§ Weiser and Nicholas, Jour. Phys. Chem, xxv. 742 (1921).

Phil. Mag. 8. 6. Vol. 44. No. 260. dug. 1922. 2D

402 Mr. H. D. Murray on Influence of Size of Colloid

different size, and is stable when precipitated, redispersed,
and dialysed. Gum mastic was found best to meet the
requirements, and was used in the subsequent experiments.
To show that the solutions employed were comparatively
stable, the concentration of NaCl required to precipitate one —
of the fractions at the beginning and end of the experiments
was measured and found to be :—

Bebwibth, ex i ae ae 433 millemols.
Manig2Otle\) anaes, 439 ii
Fractionation.

One gram of finely-powdered picked gum mastic was
dissolved in about 20 c.c. of alcohol, and poured slowly with
vigorous stirring into one litre of distilled water. By this
method seven litres of mastic solution were prepared. Odén
recommends that in all cases NaCl should be used for the
precipitation. With mastic this necessitates a very large
concentration of salt, which appears to be strongly adsorbed,
and comes slowly through the dialyser. It was thought
better to employ HCl, which precipitates in smaller concen-
tration. It was found convenient to separate the mastic into
seven fractions with these concentrations of HC] :—

Concentration of HCl Condition of
Fraction. in millemols. Precipitate.
Hae eS ooh as 0-11 trace
1 Brae Arcee 11-14 gooa
1 BE Dee 1°4-1°7
Vota wnat 1:7-2:0 -
RE AA 2°0-2°3 »
S\ i ley tae 2°3-2'6 small
TA El De bh eee 2°6-2'°9 trace

The procedure was as follows :—200 c.c. of the’ mastic
solution were mixed with a quantity of 7 HCl to give the

required concentration, and then poured into the centrifuge
vessels and allowed to stand for 60 minutes from the moment
of mixing. It was then centrifuged at 3000 r.p.m. for
30 minutes. At the end of this time the supernatant liquid
was poured off and the precipitate carefully shaken up with
about 100 c.c. of distilled water. I ractions II. and VI. were
retained until about 1500 c.c. of each had accumulated; the
other fractions were rejected. At the same time 1500 c.c.

Particles upon the Adsorption of Klectrolytes. 403

of mastic were completely coagulated with a concentration of
3°0 millemols. of HCl, and redispersed i in an equal quantity
of water. It appears below as solution B. It is to be
expected that Fr. II. will contain particles of an average size
greater than those in Fr, VI. and both will contain particles
of more uniform size than those in solution B. |

The solutions after dispersion were kept in dialysers of
parchment paper until the dialysate was uncontaminated
with HCl. They were then placed in perfectly clean vessels
of resistance glass fitted with a siphon, and a soda-lime tube
attached to the air-inlet. The siphon pipes were closed by
short pieces of rubber tubing and pinch cocks.

Basis of Comparison.

Any method of comparison between two or more solutions
based upon the total masses of the disperse phase in unit
volume is useless when applied to data due to adsorption. It
is possible to take as a basis the number of particles in unit
volume, or, what is probably more characteristic and capable
of giving more directly comparable results, the total inter-
facial surface in unit volume. The former may 1n most cases
be ascertained by a direct count under the ultramicroscope.
To evaluate the latter it is necessary, beyond this, to know
the total mass of the disperse phase, which can be effected by
weighing after evaporation to dryness, or by the methods of
volumetric analysis. In addition it demands a knowledge of
the density of the disperse phase, or of the specific gravity
of the solution and of the dispersion medium.

Perrin in his researches upon Brownian movement.
obtained the density of the mastic with which he was working
by evaporating a portion of his suspension to dryness and
estimating the density of the solid mastic. This value
(1:064) he found to agree admirably with the density as
determined from specific gravity measurements. It seems
uncertain, however, as Burton * has pointed out, whether it
is justifiable to assume that the density of the particles in the
ordinary colloidal solution of gum mastic is the same as that
of the solid substances. Perrin, as a matter of fact, used a
suspension of mastic which had bean obtained by centrifuging
the larger particles from a solution of mastic and rejecting
the remainder. In the case of the present solution, it
seemed desirable to determine the density of the particles
directly, with a pyknometer.

* Burton, ‘Physical Properties of Colloidal Solutions,’ 2nd Edition,

p. 125
ah Dae:

404 Mr. H. D. Murray on Influence of Size of Colloid
Concentration of Mastic.

Thirty c.c. of the three solutions were evaporated slowly
to dryness in a steam oven, and, as a mean of several deter-
minations, gave the following weights of mastic in 10 c.c. of
solution :—

Weight found.

SoluttvomGB: ere ncscseseseesee seer ‘00463 gms.
GU egh ER Re ies ange tes cc sO Olas.
Hine Le Ae cctee sero ne ta eeeee eet 00339 __,,

Number of Particles.

A true ultramicroscope was not used to count the particles,
but a cardioid condenser, fitted to an ordinary microscope.
The chief difficulty in work of this nature is to ascertain
accurately the volume of the liquid within the field of view.
A cell was made according to the recommendations of
Siedentoft *, the only alteration made being the substitution
of heavy glass for fused quartz. Fluorescence due to the
glass was not sufficient to render difficult the counting of the
comparatively large mastic particles. The cell consists of a
glass plate, 5 cm. in diameter and 1:0 mm. in thickness,
provided with a circular groove. The portion enclosed by
the groove, 1 cm. in diameter, was polished exactly 2 pw
‘deeper than the surface of the plate. This was used with a
cover slip about ‘25 mm. in thickness. The cell was soaked
in concentrated sulphuric and chromic acids, washed with
water, and then passed through two solutions of re-distilled
alcohol. It was finally famed. The source of illumination
was a Pointolight lamp, fitted with a condenser. All the
solutions examined were diluted with water which had been
carefully distilled and allowed to stand for a month undis-
turbed. It contained on an average 1 particle in 20
counts in a volume of 14°1 x 107° cu. mm. and could, there-.
fore, be considered optically pure to the degree of accuracy
to which work was carried. All the solutions were contained
in vessels of resistance glass, closed with corks covered with
tinfoil. The method of procedure was to transfer, by
means of a clean platinum loop, a very small drop of the
solution to be examined to the central portion of the cell.
The cover slip was laid on and pressed down until the
Newton interference rings appeared at the edges. The
dilutions were such that, when viewed with a convenient
stop in the eyepiece, about three or four particles appeared

* Siedentoff, Verhd. Deut. Phys. Ges. xii. p. 6 (1910).

Particles upon the Adsorption of Electrolytes. 405

in the field of view. One hundred counts were taken at half-
minute intervals, and the average number deduced from this.
A few of the particles, especially in the case of Ir. II.,
tended to adhere to the walls of the cell, and to prevent
any error due to this, the field of view was shifted five
times during each count.

The results obtained were as follows :—

une |

iT)ie } | if
Solution. Dilution. . Obe Hye- aa o Volume of Field oes
‘ ‘| jective. | piece. | _. of view. ns
| Vlew. Particles,

|-- _- SC es

|Soln.B..) 726 |4mm.f.]. x12 -30 mm.| 14:1«10-° mm.? 41
IFr. II... 396 aioe) AD). (SO 141% 10-5 . 3:0
i | |

iFr.VL..) x3896 | 43 ee ee LS ee eae xX LOT 5 4-0
ee i

|

Density of the Particles.

The density of the solutions was determined with an
accurate pyknometer in a thermostat at 17:2°C. The
weighings agreed to ‘0002 gm. Fr. VI. was too dilute to
give accurate results. The specific gravity of each solution
rose slightly during dialysis, owing probably to the removal
of adsorbed or dissolved alcohol. ‘This rise continued for
about five days. The weighings were made at the end of
ten days. The dialysis was then continued in more efficient
dialvsers made by Soxhlet thimbles impregnated with
collodion, but the specific gravity remained constant. Asa
mean of four weighings for each solution, the following
values for the density of the particles were obtained :—

Holm. gh:— 14195.
Fe MY3=1'186:

As a mean the density of the mastic was taken to be 1°190.
This value is considerably higher than the density of the
mastic in bulk, owing possibly to changes occurring either
on dispersion, or coagulation. Perrin* states that he
observed the densitv of his carefully washed granules
apparently to rise in salt solutions, and this may account in
part for the difference.

Size of Particles.
From these three sets of data—the number of particies in
unit volume, the total mass of mastic in unit volume, and the
density of the particles—it is possible to calculate the mean

* Perrin, Ann. Chim. Phys. xviii. p. 5 (1909).

406 Mr. H. D. Murray on Influence of Size of Colloid

radius 7 of the particles in each liquid. Of these three
measurements that of the density seemed possibly least
accurate, but, as it occurs in each calculation, the relative
sizes remain unchanged.

Pe ‘Total Mass; Mean volume Radius .
Solution.) -~ "3 ‘jof Particles of one CK (72
LOL VT i hiss (7s |
ap in 10\cie,\), * particle.
Soln.B..| 211 10° | 0046 gm. | 1:83x10-7 yp? | 164 wp 564
Hr oEL. 842 <0 0034 eas Oe ea 2 Oli 340 |
By, V1. |.|-252 <x 12) 0017 «1 57 10s 2 2 er las 507) 4H

7 xn is a measure of the interfacial surface in unit
volume.

Borjeson * has successfully combined the principle of
gilding metal particles with observation of the rate of sedi-
mentation of the particles so gilded, to measure the size of
the original particles. He failed to obtain successful results
with gelatine and gum arabic sols. An attempt to apply
this method to the mastic solutions as a check on the results
obtained also met with failure. It appears therefore unsuit-
able for organic colloids.

Rate of Coagulation.

It has been customary to fix an arbitrary time during
which the colloid solution is allowed to stand after the
addition of the electrolyte and before the amount of
coagulation is measured. Burton { allowed the solutions ~
which he examined to stand ten hours ; and again f the more
dilute solutions which he examined were left for ‘‘ some days.”
Weiser and Nicholas § allowed the solutions under examina-
tion to stand for twenty-four hours.

In some preliminary experiments the writer found that
abnormal results were obtained with a dilute solution owing
to the fact that the time elapsing before examination was too
short to permit of coagulation with the minimal quantity of
electrolyte. This led to an examination of the actual rate of
coagulation. Two solutions were employed, one being ten
times more dilute than the other. The results were as
follows :—

* Borjeson, Koll. Zeit. xxvii. p. 18 (1920).

+ Burton and MacInnes, Jour. Phys. Chem. xxv. p. 517 (1921).
t Burton & Bishop, Jour. Phys. Chem. xxiv. p.. 7103 (1920).

§ Weiser and Nicholas, Jour. Phys. Chem. xxv. p. 742 (1921).

Serizs [.

tht
4 10 c.c. of Mastic Dilution 100 per cent. Precipitant Al,(SO,)s.
; | Time after Mixing...... 6 hours. 24 hours. 144 hours.
S oe BU See ais pene =
> |
"S 19:00 millemols./litre Slight settling Clear with large flocks Clear
= We 22 . | Clear with large flocks % ‘:
L
Aa 114 ” ” | ” ” ”
<a ‘76 ” ” ” ” ”
‘61 5 Not completely clear _ %
"46 ” ” o) ” a)
| 30 re 45 No change No change No change
| “15 |
| 9 ” ” ” ”
| | |

Time after Mixing...

19°00 millemols./litre

Particles upon the Adsorption of

1:90 Bs 2
‘76 ” ”
‘61 ” oe
"46 : ” ”
"38 re) ”
*B0 3 99
319)
ed td ” ”

10 c.c. of Mastic Dilution 10 per cent. Precipitant Al,(SO,);.

6 hours.

No change

7

99

24 hours.

Not completely clear

”

Shght settling

”?

9
No change

Sreriss II.

48 hours. |

Clear with large flocks |

)

Not completely clear
Slight settling

rel

No change

72 hours.

Clear

?

9

Not completely clear
Slight settling

”
No change

96 hours.

Clear

?
Slight settling

79
| No change

144 hours.

Clear
)
”)
9?
9?

”)

Slight settling

408 Mr. H. D. Murray on Influence of Size of Colloid

It is obvious that the rate of coagulation decreases con-
siderably with decreasing concentration of the mastic.. The
point of complete coagulation was taken to be ‘clear with
large flocks.” These flocks do not necessarily settle to the
bottom of the vessel ; some adhere to the side. The interior
of the liquid, however, appears quite clear. It is noticeable
that the flocks adhering to the sides are more numerous
upon, if not confined to, the side of the vessel away from
the source of daylight illumination. |

It is apparent that, in order to arrive at the point of
complete coagulation in the case of the more dilute solution,
it is necessary to leave it undisturbed for a good many days,
a course of action to which there are several objections,
apart from that of mere convenience, in carrying out a series
of numerous determinations. During this time external
influences, such as chemical action at the surface of the
particle, have more time to show themselves.

These objections can be obviated by centrifuging the
solution after a definite time at a constant speed. The
rate of coagulation is made up of two factors, the rate of
aggregation of the particles and that of settling of the aggre-
gates so formed. By centrifuging, the influence of the
latter is reduced to a minimum, and we arrive at a truer
measure of the former. An examination of this method
shows that it is possible to obtain complete coagulation after
a reasonable length of time.

The solutions were treated in the way to be described, and
the following tables (and figs. 1 and 2) show the minimal
concentrations of Alo(SOz)3 and NaCl at various intervals
after the moment of addition for complete coagulation.
Similar results were obtained with BaCly.

It will be seen that the minimal concentration of electro-
lyte decreases rapidly with time until from 12 to 22 hours
after mixing, thereafter, it remains fairly constant. In
order, however, to ensure reaching the true end-point, the
solutions in the subsequent experiments were allowed to stand
48 hours after mixing and before centrifuging.

Several workers have pointed out that the rate and
method of addition of the electrolyte affect the end-point.
Weiser and Middleton * devised an apparatus, by the use of
which they obtained concordant results, and a modification
of it made by the writer was found to give equally good
results. In order to ensure perfect cleanliness, it was made
of glass throughout. The modified apparatus consisted of

* Weiser and Middleton, Jour. Phys. Chem. xxiv. p. 30 (1920).

409

lectrolytes.

yh

4
all

Particles upon the Adsorption of

| Liminal
' Concentration

| of Al,(SO,),.

Time after Mixing ........

Dilution 100 p. c.|>20 millemols, | >20 millemols.

9)

Srerizs ITT,

Rate of Coagulation of Mastie with Al,(SO,)3.

15 minutes. 60 minutes. 5 hours.

15 hours.

10

9)

>20 3

1:04 millemols,

>20 ‘5 >20 ry, ‘23

Series LV.
Rate of Coagulation of Mastic with NaCl.

38 millemols.

)

25 hours.

°38 millemols.

"15 "

48 hours.

°38 millemols.

Time after Mixing ..........0.00

30 minutes. 65 minutes. - 2 hours,

Liminal

Concentration

of NaCl.

Dilution 100 p, c./Ca, 1500 millemols.

2)

10

?

630 millemols.! 450 millemols,

>1500 SIiG00) > 3, 900, 810

5 hours.

450 millemols,

9?

24 hours.

360 millemols.

720s,

, 48 hours.

bars

360 millemols.
720 3

410 Mr. H. D. Murray on Influence of Size of Colloid

two vessels, one slightly smaller than the other, and fitting
by a ground-in joint inside the larger.

The smaller vessel has a slightly higher inner cylindrical
vessel, the base of which is concentric with that of the
outer vessel and fused to it. The electrolyte solution is
placed in the inner vessel and the colloidal solution in the
annular space, both having been previously rinsed out with
their respective solutions. The larger vessel is placed over
the smaller, and the whole inverted and left for 30 seconds

millemols

oncentralion of NaCl in

L
=
Sa

2k 36 48

Time in hours: after mixing

to drain. By this means a sudden and complete mixing of
the two solutions is obtained: The mixed solution is then
poured into a vessel of hard glass and corked. The whole
apparatus, as were all the vessels with which the mastic
solution came in contact, is made of hard glass and was
steamed out between each series.

The experiments were conducted with 10 c.c. of the mastic
solution at the required dilution. Into the inner vessel
was poured enough water to make the volume of the elec-
trolyte solution up to 9 c.c,, and then the latter solution
was added at a convenient concentration. To determine

Particles upon the Adsorption of Electrolytes. 411

one end point for a given concentration of mastic anda given
electrolyte, four solutions were made up with a fairly wide
difference of concentration in each solution, so-as to give a
large bracket. After standing 48 hours and centrifuging
for half an hour at 2000 r.p.m., at which speed there was no
sedimentation of the pure mastic solution, four more
solutions were made up, in which the concentrations of
electrolyte were such as to cover the interval between the

r ra
= yo

in millemols,

Concentration of Al,(S0,),

Time in hours after mixing

two concentrations in the first determination, within the
limits of which the end peint was observed to lie. The
process was repeated until the limit of observation was
reached, and eventually gave two concentrations of which it
was possible to say that one definitely caused complete
coagulation, and one did not. The end point was taken as
the mean of these two concentrations. The observation of
the solutions was made by daylight against a black back-
ground. The size of the final bracket of concentration
varied directly with the concentration of electrolyte necessary
for coagulation, and the results were therefore more accurate
with trivalent ions than with monovalent. The results were
as follows (Series 5, 6 and 7, and figs. 3, 4 and 5).

IP 19 G8 GOT S6l Srl PIT FOG ay oe Sua ie
GP 89 16 SLT 96T 61 TSI 96 ee SUNG eee) ee
GL éII IGT 88I yGG> te | uve 106 OLE ee edie

Oraeen| cordgz, | 9 dion | a dees, | od Oy | od 1.97 | 0 '°d¢.gg | ‘0d 1.99 Josey Jo WOlZerjUe0 KOH [Truly

‘OOvTING [VlovyiopUy

e of Colloid

~

~

a

06L 069 Ole | Sane 06h 09 TA“) -sromertuu af (Oey
0&9 084 OIG O&P ee OOP Je eu veaquaou0/ r
| 009 OFe O1¢ OIF a OF are -uqog | JO UOr}eVAZUBOUOH [TVULUNTD
‘a 'd 0% od 7.94 | ° ‘d ¢.e¢ Od .O7 2 “d @.eG ‘ad 1.99 ‘OLISVI JO UOTFVAJUEDMOH [eULT

TORN WUL[NSvoy —]] A saTUAG

Mr. H. D. Murray on Influence of S&:

|

au 16 re aes STAM] gorau wt "igva
8.9 @.9 6 0.6 Hees cer cutog JO u01je.1yUI0U0D eure
‘od ET "ad 1.97 | ‘od OF ‘od €.6G ‘a “d 1.99 ‘OIISV] JO UOIFVAPUOOUOD [VUTY |

Vg yuRRoBVo)—'] A SUIYAS

a CFI- GLI Wa ee a. ee occas are es :
Oi | Ob. (One Woe lees | Se re ra) ee ee
OLT. COG. cee in wo Li Sa a ‘ujog J Lyey O) [LAE a

‘od get | o-doz |) 'd 49g | 9 'deeg | ‘o-dop 0 'd¢g.eg | od 1.99 “OIISLIW[ JO WOTyVAzUeOUOD [BULA

412

FOO )IV que|n.ovory— A SHTYAS

of A1,(S0,), in @illemols

Cofitentration

Concentration of BaCt, in millemols,

tM
|| FEF

Concentration of NaCl in millemols.

414 Size of Colloid Particles and Adsorption of Electrolytes.

Discussion of Results.

It will be seen that, under the conditions imposed and
within the limits of the experiments, a comparison of the
. data obtained, upon the basis of the total interfacial surface
_ in unit volume, leads to uniformity in the curves. Such
uniformity is not to be observed when the comparison is
based upon the mass of the disperse phase, or the number
of particles, in unit volume. It appears that adsorption is
very largely conditioned by the amount of interfacial surface
exposed. It is to be noticed, however, that the minimal
concentration of electrolyte is higher throughout for the
fraction containing small particles than for that containing
large particles. This may be brought about in two ways.
The smaller particles may bear a higher charge per unit
area of their surface, or the critical value to which their
charge must be reduced before coagulation begins may be
lower than in the case of the larger particles. The latter
explanation is more probably correct, since it is known that
the surface tension of larye particles is greater than that of
small ones. It seems probable, if the existence of a critical
potential difference for coagulation between a particle and
the dispersion medium be admitted, that this should be
lower in the case of small particles which have Jess tendency
to adhere, and should thus permit of a greater freedom of
approach between the particles. If the former explanation
were correct, we should expect a separation of the particles
according to size in an electric field; but this is contrary to
experience, the particles move at the same rate independently
of their size. According to the Helmholtz theory of the
electrical double layer, this effect is due to equal density of
the charge upon unit area of the surface. It appears
probable, therefore, that the smaller particles have a lower
critical potential difference for coagulation. The behaviour
of the solution containing mixed particles of different size is
in some respects curious. With both Al,(SQs)3 and NaCl
the curve representing the coagulation of this solution
is more flattened relatively than the other two curves. A
lack of uniformity in the size of the particles appears to
render the solution less sensitive to change in concentration,
in the case of coagulation by univalent and trivalent ions.

Notices respecting New Books. 415

Summary.

(a) A separation of the particles present in a suspension
of gum mastic has been effected by Odén’s method
of Ee actional coagulation.

(b) The density of che particles, and the mass of mastic
and the number of particles in unit volume have
been measured, and from them the interfacial
surface in unit Salanie calculated.

(c) The variation of the minimal concentrations of
Al,(SO,4)3, BaCl,, and NaCl to coagulate solutions
containing particles of different mean size with
change in concentration of the solutions has been
inv esti gated.

(d) It has heen shown that uniformity in comparison of
the results can be obtained upon the basis of the
interfacial surface in unit volume. Ithas also been
shown that, upon this basis of comparison, small
particles require a higher minimal concentration of
electrolyte than large particles.

In conclusion I should like to thank Dr. A. 8S. Russell for
his valuable advice and assistance. and Mr. H. M. Carleton
tor kindly putting at my disposal the microscopical apparatus
required.

Christ Church Laboratory, Oxford,
May 15th, 1922

XLI. Notices respecting New Books.

Basic Slags and Rock Phosphates. By G. Scorr RopeErtson.

Pp. xiv+112, 8 plates. 1922. Cambridge Agric. Monographs.
Cambridge University Press. 14s. net.

Os value of sciertific investigation of the results accruing from

the use of phosphatic dressings on crop-production is obvious to
all, but it gains in emphasis when, as Sir E. J. Russell points out
in a preface to the above book, agriculturists have to realize that
the composition of basic slag has undergone much change in
consequence of the enforced modifications in the processes of “steel
manufacture. We would go farther than Sir E. J. Russell and

416. Notices respecting New Books.

say that even if the war had not given an impetus to the change
over from the basic Bessemer and acid open-hearth processes,
economic considerations would none the less have demanded the
development of the basic open-hearth production of steel from
low-grade iron-ores. ‘This result” (to quote from the preface)
‘“‘is, of course, distinctly awkward for the agriculturist who sees a
valuable fertilizer disappearing, and being replaced by one which
is more costly and at first sight seems to be nothing like as
ood.”

r After a review of the various scattered experiments on the use
of rock phosphates and basic slags hitherto undertaken, Dr. Scott
Robertson describes in detail the Essex experiments carried out
in the winters of 1915, 1916, 1918, and 1919 under the auspices
of the Hast Anglian Institute of Agriculture. The soils treated
were those of the Chalk, London Clay, and Boulder Clay, and
varied considerably in mechanical and chemical composition. The
yields of hay and clover were correlated with the rainfall, and it
was found that the drier the season, the greater was the increase
in production due to the use of phosphates. The botanical
results are also given, the crowding-out of the weeds and the
covering of bare areas with grass being noteworthy. Dr. Robert-
son’s main conclusion is that for root crops and late harvests with
high rainfall, rock phosphates will prove a suitable substitute for
the high-grade Bessemer basic slags. The careful records and
correlations were made personally by Dr. Robertson at consider-
able inconvenience and discomfort, and under most difficult
circumstances. They are therefore the more valuable, and do him
the greater credit.

The latter part of the book is concerned with investigations of
the large yields resulting from the use of basic phosphates. From
botanical analyses it is evident that the open-hearth fluor-spar
slags of low solubility are less effective than the non-fluor-bearing
and therefore more highly soluble slags. The effects of the
temperature and texture of the soil on the accumulation of -
nitrates, on the soil bacteria, and on the acidity and lime-require-
ment are clearly expounded, and the deductions emphasized by
means of abundant statistics.

Altogether, the work constitutes a most valuable contribution
to agricultural knowledge. It is a pity that the publishers
cannot retail this book of 112 pages and 8 plates for less than 14s.

P. G. ae:

Phil. Mag. Ser. 6, Vol. 44, Pl. II.

lig. 7 (a).

Fre. 7 (8).

INING.

Iie, 2.

I'tg. 6.

+m
-m

-m

Fie. 3 (a),

fap) (80) (an) '

a) ay gy ay
eS as ad an an
)

) (av) (8)

Fre.

Tie. 3 (0).

Fie. 5
+2m
+m -2m
+2m
-m
g
e
3
3
-2m

+2m

+m
-2m
Lia +2m

-em

+m

Phil. Mag. Ser. 6, Vol. 44, Pl. II.

ia. 7 (a).

Pia. 7 (0).

‘ rare
mini a

oer
: {
Pore ay?

Tita ee
LONDON, KDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

\
[SIXTH SERIES‘.

S —_
St ate

SAP TM Bie Bi ag22.

XLII. The Disintegration of Elements by « Particles. By
Sir E. Rourgserrorp, £.4.S., Cavendish Professor of
Experimental Physics, and J. CHapwick, Ph.D., Clerk
Maxwell Scholar, University of Cambridge *.

7 a former papert we have shown that long-range
particles, which can be detected by their scintillations
on a zine-sulphide screen, are liberated from. the elements
boron, nitrogen, fluorine, sodium, aluminium, and_phos-
phorus under the bombardment of « rays. The range ol
these particles in air was greater than that of free hydrogen
nuclei set in motion by @ particles. Using radium C as a
source of a rays, the range of the particles varied from
40 cm. for nitrogen to 90 em. for aluminium, while the
range of free hydrogen nuclei under similar conditions
was about 29 cm.
Previous experiments { by one of us had indicated that
the long-range particles from nitrogen were deflected in
a magnetic field to the extent to be expected if they were
swift hydrogen nuclei ejected from the nitrogen nucleus by
the impinging « particle. The nature of the particles from
the other five elements was not tested, but it seemed very
probable that the particles were in all cases H nuclei which
were released at different speeds depending on the nature of

* Communicated by the Authors.
- + Rutherford and Chadwick, Phil. Mag. vol. xlii. p. 809 (1921).

t Rutherford, Bakerian Lecture, Proc. Roy. Soc. A, vol. xevii. p.374
(1920).

Phil. Mag. Ser. 6. Vol. 44. No. 261. Sept. 1922. 2E

418 Sir E. Rutherford and Dr. J. Chadwick on the

the element and on the velocity of the incident a particle.
Under the conditions of the experiment, these H nuclei
could only arise from a disruption of the atomic nucleus by
the action of the « particles. | :

Attention was also drawn to the remarkable fact that
in the case of the one element examined, viz. aluminium,
the particles were liberated in all directions relative to the
incident « particles. |

In the present paper we Shel give an account of ex-
periments to throw further light on these points and to
test whether any evidence of artificial disintegration can
be observed in the case of other light elements.

Magnetic Deflexion of the Particles.

In the course of this work, the microscope used for the
counting of scintillations has been further improved. For
the present experiments it was essential, in order to obtain
a sufficient number of scintillations per minute, that the area
of zinc-sulphide screen under observation should be greatly
increased without diminution of the light-gathering power
of the microscope system. Following the suggestion of
Dr. Hartridge, a modified form of Kellner eyepiece was
constructed. A planoconvex lens of about 7 cm. focal
length was placed so as to render the rays of light from the
objective approximately parallel, and the image so formed
was viewed through an eyepiece consisting of a similar lens
and an eye-lens of 4 cm. focal length. Used in conjunction
with the old objective, Watson’s Holoscopic of 16 mm.
focal length and *45 numerical aperture, this system gave a
field of view of a little more than 6 mm. diameter. A
rectangular diaphragm was placed in the eyepiece, limiting
the field of view to an area 6 mm.x4°9 mm. Our previous
system had a field of view of 8-3 sq. mm. area, so that the
new microscope, under similar conditions, gave about three
times the number of scintillations of the old,

The precautions adopted in counting were similar to those
described in our previous paper. |

The method of measuring the magnetic deflexion of the
long particles was very similar to that described by one*
of us in the Bakerian Lecture of 1920. The pa
arrangement is shown in fig. 1.

The source of & rays was placed at R and was inclined at
an angle of 20° to the horizontal. The lower edge was
level with the face of a brass plate S which acted as a slit,

* Rutherford, loc, eit.

Disintegration of Elements by « Particles. 419

The distance from the centre of the source to the farther
edge of the slit was 2°95 cm. The carrier of the source and
slit was placed in a rectangular brass box between the poles
of an electromagnet, the field being perpendicular to the
length of the slit. A current of dry oxygen was circulated
through the box during the experiment.

Pie A

An extension piece L, projecting 1°7 cm. beyond the edge
of the slit, was fixed to the carrier in order to increase the
amount of deflexion of the particles issuing from the slit.
In the end of the box was a hole 1 cm. wide and 2 cm. long
covered with a sheet of mica of 3°62 cm. stopping-power.
The ZnS screen was fixed on the face of the box, leaving a
slot of 1 mm. depth in which absorbing screens could be
inserted.

The source R was a brass disk of 1°2 cm. diameter coated
with the active deposit of radium. Its initial y-ray activity
was usually equivalent to about 40 mgm. Ra.

The material, the particles from which were to be investi-
gated, was laid directly on the source if in the form of foil,
or if in the form of powder dusted over its face.

The experiment consists in obtaining an estimate of the
deflexion of the particles falling on the screen by observing
the effect of a magnetic field on the number of scintillations
near the line HE, the edge of the undeflected beam of particles.
The position of the microscope was fixed in the following
way :—After placing the source of « rays in position,
hydrogen was passed through the box. The « rays could
then strike the ZnS screen, and the edge EK of the beam was
ciearly defined. The microscope was adjusted so that the

2H 2

420 Sir E. Rutherford and Dr. J. Chadwick on the

edge of the beam of scintillations appeared a little above a
horizontal cross-wire in the eyepiece of the microscope,
marking the centre of the field of view.

When the magnetic field was applied in such a way as to
bend the & particles upwards (called the positive direction
of the field), the edge of the beam is deflected downwards in
the field of the microscope and the scintillations appear only
in the lower half. When the field was applied in the opposite
direction (negative field), the edge of the heam moved
upwards in the field of view. The strength of the magnetic
fields used in the experiments was always such that the
whole field of view was covered with scintillations when
the negative magnetic field was applied. In the experiments
on the magnetic deflexion of the long-range particles, the
number of particles is far too small to give a band of scintil-
lations with a definite edge. It is clear, however, that if
the particles are positively charged, the number of scintil-
lations observed with the negative magnetic field will be
greater than the number observed with the positive field,
and that the ratio of these numbers will give a measure
of the amount of deflexion of the particles. By determining
this ratio for the long-range particles and comparing it with
that for projected H particles of known velocity, we can
obtain an approximate value for the magnetic deflexion
of the long-range particles. The general method of the
reduction of the observations is perhaps best shown by an
account of the experiments on the particles from aluminium.

EKaperiments on Particles from Alumimum.

After fixing the position of the microscope in the way
described above, an aluminium foil of 3°37 cm. stopping-
power was placed over the source. Dry oxygen was passed
through the box, and a mica sheet of 10 cm. stopping-power
was inserted in front of the ZnS screen. The total absorption
between the source and screen was then equivalent to 30cm.
of air. The scintillations observed were consequently due to
long-range particles from the bombarded aluminium; the
ranges of the particles under observation varied from 30 cm.
to 90 em., the average range being about 45 cm.

Counts of the numbers of scintillations observed with
positive and negative fields due to an exciting current of
6 amps. were then made. The mean ratio of the numbers
with a — field to those with a + field obtained from several
experiments was 3°7. The observations were repeated with

a1 a

Disintegration of Hlements by « Particles. 421

a field due to an exciting current of 4 amps.; the corre-
sponding ratio was 2°1.

When the source had decayed to a small fraction of its
initial value, the aluminium foil over the source was removed
and a thin sheet of paraffin wax put in its place. The mica
sheet in front of the ZnS screen was replaced by a sheet of
34 cm. stopping-power, making the total absorption equi-
valent to 16 cm. of air. The scintillations observed on the
screen were now due to H particles ejected from the paraffin
wax of ranges between 16 cm. and 29 cm., the average range
being about 22cm. ‘The ratio of thenumbers of scintillations
for — and + fields was determined for an exciting current of
4 amps. and found to be 3:2.

It appears from these results that the long-range particles
from aluminium of average range 45 cm. were less deflected
by the same magnetic field than H particles of average range
22 cm.; and that in the magnetic field due to a current of
6 amps., which was 1°34 times the intensity of the field due
to 4amps., they were more deflected than were the H particles
in the latter field. To a first approximation we may say that
the value of mv/e tor particles from aluminium of range 45 cm.
is 1°23 times greater than that for H particles of range 22 cm.
This result is clearly consistent with the view that the particles
from aluminium are H nuclei moving with high velocity ;
for, assuming that the range of the H particle is pro-
portional to the cube of its velocity, the velocity of a particle
of range 45 cm. is 1°27 times that of a particle of 22 cm.
range.

These experiments show, therefore, that the particles from
aluminium carry a positive charge and are deflected in a
magnetic field to the degree to be anticipated if they are
hydrogen nuclei moving with a velocity estimated from their
range. While there can be little doubt that the particles
are hydrogen nuclei, it is very difficult to prove this point
definitely without an actual determination of the velocity and
value of e/m of the particles. Our knowledge of the relation
between the range and velocity of complex charged particles
is too indefinite for purposes of calculation. On the other
hand, if we assume, as seems a priori probable, that the
ejected particle is the free nucleus of an atom, it is possible
to show with some confidence that only a particle of mass 1
and charge 1 can fit the experimental results.

Additional evidence as to the value of mv/e of the particles
from aluminium was obtained by comparing their magnetic
deflexion with that of the « particles of &°6 cm. range emitted
by thorium C. In this experiment the source R was a very

422 Sir ®. Ratherford and Dr. J. Chadwick on the

weak source of the thorium active deposit obtained by ex-
posing a disk to thorium emanation. Hydrogen was passed
through the box, and sufficient absorbing screens were inserted
in front of the ZnS screen to cut out the 5 cm. a particles of
thorium C. The numbers of a particles falling on the screen
for — and + fields due toan exciting current of 6 amps. were
counted, and the ratio of these numbers was found to be 2°4.
Comparing this ratio with those found for the long-range
particles, we see that the value of mv/e for the latter is about
0°8 of that for « particles of 8°6 cm. range, 2. e. about
3°4x10° e.m. units. The calculated value, assuming that
the particles are H nuclei and that their velocity is pro-
portional to the cube root of the range, is about 3°7 x 10°,e.m.
units. Considering the difficulty of the experiments, the
agreement is satisfactory.

Haperiments on Phosphorus and Fluorine.

Measurements similar to the above have also been made on
phosphorus and fluorine.

In the case of phosphorus, a thin layer of red phosphorus
was dusted over the face of the source. The total absorption
in the path of the particles was about 35 cm.; the range of
the particles under observation varied therefore from 35 cm.
to the maximum range of 65 cm., the average being about
45cm. The ratio of the numbers of scintillations for — and
+ fields due to current of 4 amps. was 2:0.

In the case of fluorine finely powdered calcium fluoride was
dusted over the source. Previous experiments have shown
that no long-range particles are emitted fromcalcium. The
total absorption in the path of the particles was about 30 em.
The maximum range of the particles from fluorine is approxi-
mately 65 cm., and the average range of the particles falling
on the screen was around 40 cm. The ratio of the numbers
of particles observed for — and + fields due to an exciting
current of 4 amperes was 2°95.

It is clear from these results that, within the error of
experiment, the particles liberated from phosphorus and from
fluorine are bent in a magnetic field to approximately the
same extent as the particles from aluminium. We may
conclude, therefore, that these particles also are H nuclei
moving with high speed.

We have not examined the particles from boron and
sodium in this way, but there seems no reason to doubt that
they also consist of H nuclei.

Disintegration of Elements by « Particles. 423

The Ranges of the H Particles.

In the experiments described in our previous paper only
two elements, nitrogen and aluminium, were investigated
in any detail. The other elements were examined in a
qualitative manner, but it was shown that the ranges of
the liberated H particles were in every case greater than
40 cm. of air. The ranges of the particles from these
elements—viz. boron, fluorine, sodium, and phosphorus—
have now been determined more accurately.

Attention has been drawn to the remarkable fact that the
H particles liberated from aluminium appeared not only in
the direction of the incident « particles but also in the reverse
direction. The number of particles emitted in the backward
direction was of the same order of magnitude as for the
forward, but the maximum range in the backward direction
was smaller, being 67 cm. as against the 90 cm. range of the
forward particles, for « particles of 7 cm. range. Some
experiments with nitrogen showed that the number of
H particles emitted in the backward direction was very
small at absorptions of more than 18 ecm. of air.

We have repeated these experiments and extended them
to include the other elements boron, fluorine, sodium, and
phosphorus, with the result that we find that in every case
the H particles emitted on disintegration of the nucleus
escape in all directions, the maximum range in the backward
being less than in the forward direction.

Fig. 2.

I
pea

The experimental arrangement for the measurement of
the ranges of the forward particles was the same as that
described in our previous paper. The apparatus used in
the investigation of the particles in the reverse direction
differed from this in the arrangement of the source, and is
shown in the diagram (fig. 2).

The source of a particles was carried on a rod passing with
a sliding fit through a stopper which fitted tightly into the

494 Sir E. Rutherford and Dr. J. Gitar on the

brass tube T of-3 cm. diameter. The end of this tube was
provided with a hole 7°5 mm. in diameter, closed by a silver
Foil of 3°75 em. air equivalent. The zinc sulphide screen S
was fixed on the face of the vessel leaving a slot in which
absorbing screens could be inserted. The apparatus was
placed between the poles of an electromagnet to reduce the
luminosity produced in the screen by the B rays.

The source R was a silver foil of 4:15 cm. stopping-power
coated on one side only with the active deposit of radium.
Its initial y-ray activity was in most experiments equiva-
lent to about 30 mg. Ra. The inactive side of the silver
foil faced towards the ZnS screen. The distance of the source
from the screen was generally about 3°5 cm., but could be
varied, and its position read off on ascale. The elements
to be examined could in most cases only be obtained in
the combined state. The powdered compound was heated
in vacuo, and a film prepared by dusting on to a foil smeared
with alcohol. The screen thus prepared was placed immedi-
ately behind the source. As in our previous experiments, a
stream of dry oxygen was circulated through the apparatus.

In all cases, except that of nitrogen, the maximum range
of the particles emitted in the backward direction was
greater than the range of free hydrogen particles, so that no
complication arises from the presence of hydrogen in the
silver foils or other materials in the path of the @ particles.

In the case of nitrogen, however, as our previous experi-
ments had shown, the range of the backward particles is
much less than that of free hydrogen particles, and it was
consequently necessary to allow for the “ natural” effect, 7. e.
for the H particles arising from hydrogen contamination of
the source and screens in the path of the a rays. It was
found inconvenient to use gaseous nitrogen for these experi-
ments, and a suitable screen was prepared by sifting a thin
layer of powdered paracyanogen, C,N,, on to a gold foil.
The scintillations observed on the ZnS screen when the film
of paracyanogen was placed against the source were due to
the “natural” particles from the source and screens, together
with those which came from the nitrogen in the paracyanogen.
On taking away the film of C,N, the natural particles alone
were counted. In some experiments a film of paraffin wax
was placed against the source. The natural effect remained
the same, showing that even if the film of paracyanogen
contained a large amount of hydrogen the number of free H
particles scattered to. the ZnS screen by the walls of the
vessel was negligible.

Figure 3 shows the type of results obtained in these

Number per Milgram.

Disintegration of Elements by « Particles. 425

experiments. The ordinates represent the number of scintil-
lations observed per minute per milligram of activity of radium
C, measured by y rays; the abscissze, the stopping power
for a rays of the absorbing screens, expressed in terms of
centimetres of air. The dotted curve A gives the natural
effect observed when the screen of paracyanogen was absent
or replaced by a film of paraffin wax ; the full curve B the
effect when present. The difference of these curves there-
fore represents the effect due to the nitrogen in the para-
cyanogen. It will be seen that the maximum range of the

Fig. 3.

Forward and Backward Particles
trom Mitrogen.

Absorption in cms. of air

backward particles from nitrogen is about 15 cm. Curve C
is the absorption curve for the particles emitted by nitrogen
in the forward direction. |

In the following table are given the maximum ranges of
the particles liberated from the elements which show the
disintegration effect, for both forward and backward

directions. —

Element. Forward range. Backward range.
Cr: cm,
Borer Se Rates « 58 38
Brian 7 Le Aree 40 18
Mhrarme 1253)! 65 48
ecu Me BU lw et, 58 36
Alaremiony 92:22. 28.0.0 90 67

Phospherwua? 2:05. Mol 25: 65 49

426 Sir E. Rutherford and Dr. J. Chadwick on the

_ It should be pointed out that the ranges of the forward
particles from boron, fluorine, sodium, and phosphorus may
be subject to considerable error, owing to the use of a film of
powder as the bombarded material. The particles of
maximum range are produced on the surface of the grains of
powder, and theretore to find the true range the size and air
equivalent of the grains of powder must be known. For the
ranges given above it has been assumed. that the grains were
uniform in size and an average value of the air equivalent of
the film of powder has been calculated from its weight per
sq.cm. ‘The ranges so determined are obviously somewhat
less than the true ranges. The ranges of the backward
particles are, of course, not subject to this source of error.
It was observed that the number of particles liberated
from the different elements appeared all to be of the same
order of magnitude when allowance is made for the differ-
ence in range. In our original experiments we found that
the number of particles from boron was somewhat smaller
than the numbers from the other elements, but this was
due to the use of an irregular film. Using a film of more
finely powdered boron it was found that the number of
particles from boron was about the same as from the cther |
elements.

Haamination of other Elements.

In our former experiments we examined all the light
elements, with the exception of the rare gases, as far as
calcium. Of these only the six elements of the above table
were found to emit H particles in detectable amount under
the bombardment of ¢ rays. As was pointed out in that
paper, the atomic masses of these elements can be represented
by 4n+a where n is a whole number, a result which
receives a simple explanation on the assumption that the.
nuclei of these elements are composed of helium nuclei of
mass 4 and hydrogen nuclei. On the other hand, some
of the light elements which gave no detectable number of H
particles also had atomic masses given by 4n+a. It was
thus a point of great importance to repeat the examination of
these elements with the improved microscope, and to search,
if possible, for the emission of particles of shorter range than
free H nuclei. In some cases it was only possible, on account
of hydrogen contamination of the materials, to observe at
absorptions yreater than 30 cm. of air, while in others the
observations were carried well within this range.

Lithium was examined as oxide and as metal, a thin sheet
of the latter being obtained by pressing molten lithium

Disintegration of Elements by « Particles. A427

between two steel plates in an atmosphere of carbon dioxide.
No evidence was found of any particles of range greater than
30 cm. Owing to hydrogen contamination of the Li and
Li,O the observations at smaller ranges were not decisive.
Observations in the backward direction revealed no detectable
number of particles of range greater than 14 cm.

Beryllium was examined as the powdered oxide, and there
was again no evidence of the emission of particles of longer
range than 30 em. in the forward direction or 15 cm. in the
backward.

Magnesium was examined with a sheet of the metal and
also with a screen of powdered magnesium. There was no
evidence of long-range particles.

For silicon a screen of powdered silicon and a thin sheet
of quartz were used. With the sheet of quartz it was
possible to make observations in the forward direction at
absorptions as low as 17 em. The scintillations observed
were due entirely to the natural H' particles.

Fig. 4,
To Pump.

Chlorine had been previously examined in the form of
various chlorides. These observations were repeated, and the
results confirmed the conclusion that particles of greater
range than 30 cm. were not liberated in any detectable
amount. In order to pursue the observations within the
range of free H particles a special series of experiments was

428 Sir E. Rutherford and Dr. J. Chadwick on the

carried out. A glass apparatus, similar in design to the
standard ae) was used. The details will be clear from
the diagram (fig. 4

In order to oid ‘the bombardment of the glass walla and

consequent liberation of H particles the inside of the tube

was lined with platinum foil. The surfaces of the brass

plate B and of the rod carrying the source were protected

from the action of the chlorine by a coating of hard pitch.
The stopcocks and ground-joint were lubricated with a
brominated grease. The source of «rays wasa platinum foil
coated with radium active deposit. Pure dry chlorine was
prepared by heating gold chloride, AuCl;, contained in the tube
A, and was passed over P.O; before entering the vessel T.
As an additional precaution a little P 205, was placed in the
vessel itself.

When the source was placed in position the air was
removed by pumping and washing with dry carbon dioxide.
Carbon dioxide was then let in to atmospheric pressure and
the natural H particles were counted at absorptions varying
from 16 cm. to 30 cm.- The carbon dioxide was then
replaced by chlorine, and tbs scintillations at similar absorp-
tions were observed. The chlorine was then allowed to be
reabsorbed by the gold chloride and carbon dioxide let in
again. In this way counts on the chlorine were included
between counts of the natural particles, and any traces of
adventitious hydrogen could be allowed for. The results
showed no evidence of the liberation of H particles from
chlorine in the range examined, 7. e. at absorptions more hes
1oxcm. oF alr,

Discussion of Results.

For convenience of discussion the atomic numbers and the
masses of the isotopes of the elements from hydrogen to
potassium are given in the following table. Of these
elements aluminium is the only one which has not yet been
examined for isotopes, but it appears likely that it is a pure
element of atomic mass 27. With the exception of helium,
neon, and argon, all the elements in the table have been
tested to see whether H nuclei are ejected by the action of a
particles. The six active elements, as they may be termed
for convenience, are underlined.

sintegration of Elements by « Particles. 429

TABLE J;

, Atomic Atomic iy ; Atomic Atomi

Element. Number. Masses. Klement. N eae Mikes,
fe cn. tte [LL 1 POS IN Oe 1) 23

| —_—

1 Sg Sn 2 4-00 | be 12 24,25, 26
| 2 Sees 3 6,7 1 gece lea el oe 13 27
ae ee 4 9 lick iabradaal 14 28. 29
“ geliaat NEUE. NORA sable 15 31
7 edema st bedi: ee AL 16 32
N Packet ox... 4 14 Sly Cea ee Ly, 30, 37
EE epee 8 16 tae geet 18 36, 40
Fl Ht ee 9 19 | Re Suse. 19 39, 41
A cess bs 10 20,22

An examination of the table shows that the active elements
may be classified in different ways :—

(1) Active elements are odd-numbered elements in a

regular sequence of numbers, viz., 5, 7, 9, 11, 13, 15.

(2) The atomic masses of the active elements are given by
4n+a where nis a whole number ; a=3 forall the
elements except nitrogen, for which it is 2.

(3) With the exception of boron, which has two isotopes
10, 11), the active elements are all pure elements.
P

We have seen that no evidence has been obtained that the
preceding element lithium (3), and the succeeding elements,
chlorine (17) and potassium (19), show any trace of activity
under a-ray bombardment, although they are odd-numbered
elements and the masses of their isotopes are given by 4n+a.
Magnesium and silicon, which are even-numbered, but which
contain isotopes of mass 4n+1 or 4n+2, show no sign of
activity.

There thus appears to be no obvious general relation which
differentiates active from inactive elements. The activity
starts sharply with boron and ends abruptly with phosphorus.
It is a very unexpected observation that neither lithium nor
chlorine shows any certain evidence of activity in the emission
of either long-range or short-range particles. It is of
interest to consider whether any deduction can be made as to
the structure of these nuclei in the light of these experimental
facts.

In our previous paper it was pointed out that the H nuclei

A430 ~=6Sir E. Rutherford and Dr. J. Chadwick on the |

liberated from the active elements probably existed as
satellites circulating in orbits round the main nucleus, In
the case of an effective collision of an a@ particle with such
a nucleus, part of the momentum of the « particle is com-
municated to the central nucleus, but the satellite is
sufficiently distant from the latter to acquire enough
momentum and energy to escape from the system. It was
shown that such a point of view offers a general explana-
tion of the variation of the velocity of the expelled H
nuclei with the speed of the « particle and also of the escape
of the H nuclei in all directions, The chance of ejecting
an H satellite at high speed from a nucleus is much smaller
(for nitrogen, for example, about 1/20) than the chance of
setting a free H nucleus in correspondingly rapid motion,
It appears therefore that the release of the satellite only
takes place under certain restricted conditions of the
collision of the « particle with the nucleus. If the H
satellites were present in lithium and chlorine and were very
lightly bound to the nucleus, it is to be anticipated that
the number released by the « rays would be of the same
order of magnitude as if the H nuclei were free. As this
is found not to be the case, we may conclude that neither
lithium nor chlorine has any lightly bound satellites in its
nuclear structure. The complete absence of long-range
particles from these elements shows that the H satellites, if
they are present at all, are strongly bound to the main
nucleus. If, for example, the satellite revolves very close to
the nucleus, the « particle may only be able to give such a
small part of its momentum to the satellite that it is unable
to release it from the system. It does not, however, seem
likely that the forces binding a satellite would vary greatly
in passing from phosphorus to chlorine. It seems more
probable that the general structure of the chlorine nucleus
differs in some marked way from that of the group of active
elements. The H nuclei may perhaps be definitely incor-
porated into the main nuclear system, so that the a particle
has no chance of concentrating its energy upon a single unit
of the nuclear structure. In a similar way it seems probable
that lithium must differ widely in structure from the suc-
ceeding element boron. ‘The facts brought to light in these
experiments indicate that the nuclei even of light elements
are very complex systems and illustrate how difficult it will
be to find any simple and general rule to account for the
variation in structure of successive elements.

It has been pointed out that, with the exception of the first

Disintegration of Hlements by a Particles, 431

element boron, all the active elements are.“ pure” elements,
2. €., have no isotopes. This may be of some significance in
differentiating between the structure of active and inactive
elements, ‘The absence of isotopes indicates that, as regards
mass, there is only a narrow range of stability of the nucleus
for a given nuclear charge; the addition or subtraction of
an equal number of H nuclei and electrons leads presumably
to an instability of the nuclear system. In the case of
lithium and of chlorine, which form isotopes, the forces
binding the nuclei together may consequently be very
different from those in the case of the pure active elements.
If there is any significance in this point of view, it would
indicate that H_ satellites are only present in pure odd-
numbered elements; but, as we have seen, boron is an
exception to this rule.

In comparing the phenomena shown by the six active
elements, it seems at once clear that nitrogen occupies an
exceptional position in the group. Not only is the range of
the expelled H nuclei the smallest of all the group, but the
ratio of the ranges in the two directions is markedly different
from those shown by the other elements. It is natural to
connect this anomalous behaviour with the fact that the mass
of the nitrogen nucleus is given by 4n+ 2, while the rest of
the group are of the class 4n+3. The slower speed of
ejection of the particles from nitrogen at first sight suggests
that the H satellite is more hghtly bound than in the case of
the other elements. This suggestion is, however, not borne
out by calculation of the distribution of momentum among
the three bodies involved in the collision, viz., the « particle,
the H satellite, and the residual nucleus. In our previous
paper, we showed that the distribution of momentum could
be calculated on certain assumptions from the observations
of the ranges of the expelled nuclei in the forward and
reverse directions of the a particle. It was supposed that
the law of conservation of momentum holds, and that the sum
of the energies of the H particle and the residual nucleus
was the same whether the H particle was liberated in the
forward or backward direction. It follows from these
assumptions that the relative velocity of the H nucleus and
the residual nucleus is the same in thetwo cases, The results
of this calculation for the group of active elements are
collected in the following table (Table IT.).

432 The Disintegration of Elements by « Particles.
TABLE IT.

Distribution of Momentum.

H particle. Residual Nucleus. a Gainin
Hlement. forward. Backward. Forward. Backward. particle. Energy-
Borow peepee... 202V —-175V —d4V. S23V 252V 427%
Nitrogen ...... 78 V —1:32V Pa Ve 4b Vi ‘78 V -13 %
Fluorine ...... 210V* -189V —J0V 389V 200V 35%
NOGIUM As... ..5: 202V —172V 1A 2V 7) OG, ‘56 V 6 %

Aluminium... 234V —211V “8V + 5:23.V. 788 V wet
Phosphorus... 210V —189V 114V 513V -76V 15%

The momenta are expressed in terms of the initial velocity
V of the « particle. ‘ne initial momentum of the a particle,
and consequently the sum of the momenta of the three bodies
after collision, is therefore 4V. Momenta in the direction of
the incident « particle are taken as positive, momenta in the
opposite direction as negative. The percentage energy,
gained from the nucleus as a result of the disintegration, is
given in the last column, in terms of the initial energy of the
« particle.

It wiil be seen that in the case of nitrogen a considerable
part of the momentum of the @ particle is communicated to
the main nucleus, a much greater part than in the cases of
the adjacent elements boron and fluorine. This indicates
that the H satellite of nitrogen is in relatively close
proximity to the main nucleus. It will also be noted that
while for the other elements there is a gain of energy from
the disruption varying from 6 per cent. for sodium to 42 per
cent. for. boron and aluminium, for nitrogen there is a loss of
energy of 13 per cent.

It is apparent from the above table that the distribution
of momentum among the three bodies varies considerably for
the different elements, but, in the absence of any definite
evidence of the validity of the theory on which the calcula-
tions are based, it seems inadvisable to discuss these differences
in any detail at the present stage.

Cavendish Laboratory,
June 20, 1922.

XLII. The Distribution of Hlectrons around the Nucleus in the
Sodium and Chlorine Atoms. By W. LAwRENcE Braga,
M.A., F.R.S., Langworthy Professor of Physics, The
University of Manchester; R. W. James, J/.A., Senior
Lecturer in Physics, The University of Manchester ; and
C. H. Bosanquet, Jf.A., Balliol College, Oxford*.

1. J N two recent papers f in the Philosophical Magazine

the authors have published the results of measure-
ments made on the intensity of reflexion of X-rays by
rock-salt. The mathematical formula for the intensity of
reflexion, as calculated by Darwin {, involves as one of
its factors the amount of radiant energy scattered in various
directions by a single atom when X-rays of given amplitude
fall upon it. The other factors in the formula can be
evaluated. By measnring the intensity of reflexion experi-
mentally we can therefore obtain an absolute measurement
of the amplitude of the wave, scattered by a single atom,
in terms of the amplitude of the incident radiation.

This measurement is of considerable interest, because it
may throw some light on the distribution of the electrons
around the nucleus of the atom. We regard the wave scat-
tered by the atom, as a whole, as the resultant of a number of
waves, each scattered independently by the electrons in the
atom. A formula first evaluated by J. J. Thomson is used
in order to calculate the amplitude of the wave scattered by
a single electron. If an incident beam of plane polarized
X-rays consists of waves of amplitude A, then the amplitude
A' at a distance R from the electron in a plane containing
the direction of the incident radiation, and at right angles
to the electric displacement, is given by

Pel Mahe weet
AY > Rone’ ° . ° e ° . (1)

Here e and m are the charge and mass of the electron in
electromagnetic units, and ¢ is the velocity of light.

What we measure experimentally is the resultant ampli-
tude of the wave-train scattered in various directions by a
number Z of electrons in the atom. If all the electrons were

* Communicated by the Authors.

+ Phil. Mag. vol. xli. March 1921; vol. xlii. July 1921.

ft C. G. Darwin, Phil. Mag. vol. xxvii. pp. 315-675 (Feb. and April
1914).

Plul. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2F

434 Prof. W. L. Bragg and Messrs. James and Bosanquet :

concentrated in a region whose dimensions were small com-
pared with the wave- “length of the rays, then the resultant

2
amplitude would be equal to a. since the scattered

wavelets would be in phase with each other in all directions.
It is found experimentally that the measured amplitude
tends to a value which is in agreement with the formula |
at small angles of scattering, but that at greater angles it
falls to a very much smaller value. ‘This is to be accounted
for by the action of interference between the waves scattered
by the electrons in an atom, which are distributed throughout
a region whose dimensions are large compared with the
X-ray wave-length.

It is an easy matter to calculate the average Be His
scattered in any direction by a given distribution of electrons
around the nucleus. Here we are attempting to solve the
reverse of this problem. The experimental results tell the
amplitude of the wave scattered by the sodium and chlorine
atoms through angles between 10° and 60°. We wish to
use these results in order to get some idea of the manner
in which the electrons are distributed.

2. In addition to Darwin’s original mathematical treat-
ment, the question of the effect on X-ray reflexion of the
distribution of electrons around the atom has been dealt
with by W. H. Bragg *, A. H. Compton f, and P. Debye
aG yey “Scherrer f.

W.H. Bragg considered the inter one wien of the diminu-
tion in the intensities of reflexion by a crystal as the
glancing angle is increased, due allowance being made for
the arrangement of the atoms. He concluded that “an
ample explanation of the rapid diminution of intensities
is to be found in the highly probable hypothesis that the
scattering power of the atom is not localized at one central
point in each, but is distributed through the volume of the
atom.” He did not regard the experimental data then
available as sufficient to justify making an estimate of the
distribution of the electrons. These data indicated that the

intensity of reflexion fell off roughly as a (0 being

the glancing angle), and he showed that a density of
distribution of the electrons could be postulated which

* W. H. Bragg, Phil. Trans. Roy. Soc. Series A, vol. cexv.
pp. 253-274, July 1915.

+ A. H. Compton, Phys. Rev. vol. ix. no. 1, Jan. 1917.

{ P. Debye and P. Scherrer, Phys, Zeit. pp. 474-4838, July 1918.

Distribution of Electrons in Na and Cl Atoms. 435

accounted for this law, just as an illustration of the appli-
cation of the principle involved in considering spatial
distribution.

A. H. Compton used the experimental results obtained by
W. H. Bragg in order to calculate the electron distribution.
W. H. Bragg showed that the intensity of reflexion is a
function of the angle of reflexion alone, when allowance has
been made for the arrangement of the atoms in the crystal,
and he determined the relative intensity of reflexion by a
number of planes in rock-salt and calcite. Compton cal-
culated from these values the relative amplitudes of the
waves scattered by the atoms in different directions, by
means of the reflexion formula of Darwin, and proceeded
to test various arrangements of electrons in order to find
one which gave a scattering curve agreeing with that found
experimentally. He supposed that the electrons were
rotating in rings, governed by Bohr quantum relationships
In sodium, for example, he placed four electrons on an
inner ring, six on the next ring, and a single valency electron
on an outer ring. In chlorine the rings contained four, six,
and seven electrons respectively. Compton found that these
atomic models gave a fair agreement with W. H. Bragg’s
results.

Debye and Scherrer came to the same conclusion as to the
significance of intensities as regards electron distribution
which was implied in W. H. Bragg’s work and stated more
Fully by Compton. They considered two interesting cases.
The first was that of the lithium fluoride crystal. They
compared the intensity of reflexion by planes where the
fluorine and lithium atoms reflected waves in phase with
each other, with that by planes where these atoms acted
in opposition to each other. The relative amplitudes at any
F+i
ov
where F and Liare the amplitudes contributed by the fluorine
and lithium atoms respectively. Their figures indicated

angle for such planes may be expressed by the ratio

that the limiting values of su at zero angle of scattering
is 1*5, signifying that a valency electron has passed from the
lithium to the ae atom (F925 = 15) :

Their intensities of reflexion ey) measured by the

darkening of a photographic plate in the powder method
of analysis which these authors initiated. In view of the

22

436 Prof. W. L. Bragg and Messrs. James and Bosanquet :

difficulties of estimating intensities in this way, of the few
points which they obtained on the curve for the oa ratio,
of the difficulties in interpreting intensities which we have
discussed in our papers, and of the large extrapolation
which they had to make in order to get the limiting value
if F+hi i
eat
proving that the transference of the valency electron has
taken place. The fact of the transference is supported by
much indirect evidence, and their conclusion is probably
correct.

Debye and Scherrer also compared the intensities reflected
by various planes of the diamond, and concluded that the
electrons in the carbon,atoms were contained within a
sphere of diameter 0°43 A, assuming a uniform distribution
throughout this sphere. |

In all the above cases, the results were obtained by com-

paring the relative intensities of reflexion by various faces.
The results which we have obtained, and which will be used
to calculate the distribution of electrons in sodium and
chlorine, are, on the other hand, absolute determinations.
The intensity of reflexion was compared in each case with
the strength of the primary beam of X-rays, so that the
absolute efficiency of the atom as a scattering agent could
be deduced.
_ In a paper on “The Reflection Coefficient of Monochro-
matic X Rays from Rock Salt and Calcite” *, Compton
made comparisons of the incident and reflected beam, for
the first order reflexion from cleavage faces of these'erystals.
He obtained results for rock-salt which were rather less than
those which we afterwards obtained for a ground face, but
he noted that the effect was increased by grinding the face.
In our notation the results were

we feel that their results cannot be regarded as

Compton - = 00044 + :00002 ;
NaCl‘(100).
Ko. i

B.J.and B. “7 -= 00055

As Compton surmised, and as we have found experimentally,
this figure for the efficiency of reflexion has to be modified
considerably to allow for the extinction factor. The difference

* A, H. Compton, Phys. Rev. vol. x. p. 95, July 1917.

Distribution of Electrons in Na and Cl Atoms. 437

between his results and ours is accounted for by the extinction
or increased absorption of the rays at the reflecting angle.
Compton pointed out that the reflexion factor was of the
order to be expected from Darwin’s formula, but did not
use the value he obtained to solve the electron-distribution
problem.

3. For the sake of convenience of reference, the formula
which forms the basis of all the calculations is quoted below.
Let the intensity I) of a beam of homogeneous X-rays, at a
given point, be defined as the total energy of radiation falling
per second on an area of one square centimetre at right
angles to the direction of the beam. Ifa crystal element of
volume dV, supposed to be so small that absorption of the
rays by the crystal is inappreciable, be placed so that it is
bathed by the X-rays, and if it is turned with angular
velocity w through the angle at which some plane in it
reflects the X-rays about an axis parallel to that plane, the
theoretical expression for the total quantity of energy of
radiation reflected states that

a0: IN? Xx? 4 1+4cos? 20
san Oe oe ae Og « (2)
resin 20 mc 2

=a V.
In this expression

N= Number of diffracting units per unit volume*,
X% = Wave-length of X-rays.

? = Glancing angle at which reflexion takes place.
e = Electronic charge.

m= Electronic mass.

c = Velocity of light.

The factor e—Bsi""? (the Debye factor) represents the effect
of the thermal agitation of the atoms in reducing the
intensity of reflexion.

The factor F depends on the number and arrangement of
the electrons in the diffracting unit. At @=0 it would have
a maximum value equal to the total number of electrons in
the unit, and it falls off owing to interference as @ increases.

The experimental observations have as their object the
determination of @ in absolute units. In practice we cannot
use a single perfect crystal so small that absorption is

* No account is taken here of the “structure factor.” The diffracting
units are supposed to be spherically symmetrical as regards their
diffraction effects.

438 Prof. W. L. Bragg and Messrs. James and Bosanquet : .

inappreciable. We use a large crystal consisting of num-
bers of such homogeneous units and deduce, froin its
reflecting power, the reflecting power Q per unit volume
of the units of which it is composed. The assumptions made
in doing this are by no means free from objections, and will
be discussed later in this paper. Taking this to be justifiable,
however, our experimental results yield the value of Q for
rock-salt over a wide range of angles, and from them the

values of Fo, and Fy, follow directly. These values are
shown in fig. 1.

Fig. 1.

Values of F

(4) Fo) corrected for Debye factor. (c) Fy, corrected for Debye factor.
(6) Fy uncorrected __,, 5 (d) Fy, uncorrected __,, ”

4. We must now consider more closely the significance of
the factor F. The most simple case is that of a crystal con-
taining atoms of one kind only. Parallel to any face of the

Distribution of Electrons in Na and Cl Atoms. 439

crystal we can suppose the atoms all to lie in a series of
planes, successive planes being separated by a distance d.
We get the nth order spectrum formed at a glancing angle
by the reflexion from such a set of planes if

9d sin @=n\X.

This spectrum represents the radiation diffracted by the
atoms in a direction making an angle 20 with the incident
beam, and it is formed because in this particular direction
the radiation scattered by any pair of atoms lying in suc-
cessive planes differs in phase by 2n7. Thus the amplitude
of the beam scattered in this direction is the swm of the
amplitudes scattered by all the neighbouring atoms taking
part in the reflexion.

Let us consider the contribution to the reflected beam of
a group of atoms lying in a reflecting plane. To obtain the
amplitudes of the reflected wave, we sum up the amplitudes
contributed by the electrons in ail the atoms, taking due
account of the fact that the electrons do not in general lie
exactly in the reflecting plane and so contribute waves
which are not in phase with the resultant reflected wave.
By symmetry, the phase of the resultant wave will be the
same as that reflected by electrons lying exactly in the
geometrical plane passing through the mean positions of all
these atomic centres. The phase of the wave scattered in a
direction @ by an electron at a distance x from the plane
differs from that of the resultant wave by an amount

Agr

4% sin @.

We will suppose that there is in every atom an electron
which is at a distance a from the centre, and that all direc-
tions of the radius joining the electron to the atomic centre
are equally likely to occur in the crystal. In finding the
effect of these electrons for all atoms (M in number) of the
group, we may take it as equivalent to that of M electrons
distributed equally over a sphere of radius a. It can easily
be shown that, if z is the distance of an electron from the
plane, all values of 2 between +a and —a are equally likely
for both cases. Such a shell scatters a wave which is less
than that scattered by M electrons in the plane in the ratio
sin |

p

, where

o¢= =) sin 0.

440 Prof. W. L. Bragg and Messrs. James and Bosanquet :

The average contribution of the electron in each atom to the
fF factor is therefore gut and not unity as it would be

if the electron were at the centre of the atom.
If there are n electrons at a distance a from the centre of
the atom, their contribution to the F factor would be

oe 5 5
gt ls

Any arrangement of n_electrons at a distance a from the
centre of the atom, provided that all orientations of the
arrangement were equally probable, would make the same
contribution to the F factor. For example, eight electrons
arranged in a ring about the nucleus would give the same
value for F as eight electrons arranged at the corners of a
cube, or eight electrons rotating in orbits lying on a sphere
of radins a. This illustrates the limitations of our analysis,
which cannot distinguish between these cases. We can only
expect to get information from our experimental results as
to the average distance of the electrons from the atomic
centre, and this for the average atom.

Suppose now that any atom contains a electrons at a
distance 7, from the nucleus, 6} at a distance 72, ¢ at a dis-
tance 73... ata distance 7,, then the value of F tor the
average atom would be given by

pee phe foe +.. oe a (4)

Thus, given the distribution of the electrons on a series of
shells or rings, we can calculate the value of F for any value
of 8. The problem we have to solve here, however, is the
converse of this. We have measured the value of F fora
series of values of 6, and wish to determine from the results
the distribution of the electrons. We have seen above that
there is no unique solution of this problem, but we can get
some idea of the type of distribution which will fit the
experimental curves.

In order to do this, we suppose the electrons to lie on a
series of sheils, of definite radii 7), 7), .... and determine
the number of electrons a, 0, ¢ on the various shells which
will give values of F corresponding to those observed
experimentally. Suppose, for example, we take six shells
uniformly spaced over a distance somewhat greater than

Distribution of Electrons in Na and Cl Atoms. 441

the atomic radius is expected to be. For any given value
of 6 we have |

mee sin d; 4 pein ba Zan sin 3 b )3in py ie sin bs
$1 $2 $3 di bs

sin
on ae Syl aes (5)

8
We chose from the experimental curve six values of 0
evenly spaced over the range of values at our disposal, and
for each of these values read from the curve the value of F.
Since definite radii have been assumed for the shells, the

nae ete., can be calculated for each value of @.

Hence, for sone value of 0, we have an equation involving
numerical coefficients and the quantities a, b, ¢, d, e, 7, so that
if six such equations are formed we may calculate these
quantities.

If Z is the total number of electrons in the atom we have

BOGE OG ad Pet ct. 2. fe an 1 (6)

and this will be taken as one of our equations (corresponding
to @=0). In calculating the results for sodium and chlorine
we have assumed the atom to be ionized, and bave taken
Zo = 18 and _Zy, = 10.

It wil be evident that this method of solution is somewhat
arbitrary, and that the results we get will depend on the
particular radii assumed for the shells. By assuming various
radii for the shells, however, and solving the simultaneous
equations for the number of electrons on each, we find that
the solutions agree in the number of electrons assigned to
various regions of the atom.

As a test of the method of analysis, a model atom was
taken which was supposed to have electrons arranged as
follows :—

2 ona shell 0:05 A radius.
DP Men rs Oren)

Se ae ORL ON hates

The F curve for this model was calculated. Then the simul-
taneous equations for the electron distribution were solved,
just as if this curve had been one found experimentally.
This was done for two arbitrarily chosen sets of radii, taken

out to well beyond the shell at 0°70 ye

EE a

442 Prof. W. L. Bragg and Messrs. James and Bosanquet :

The comparison between the two analyses (dotted curves)
and the atom model we started with (continuous curve) is
shown in fig. 2. The abscissee represent the radii of the
shells in A, the ordinates the total number of electrons
inside a shell of that radius. When the limits of the atomic
structure are reached, the curve becomes horizontal at the
value 10, corresponding to the ten electrons. The analyses
not only indicate with considerable accuracy the way in

Fig, 2.

) 0-5 1-5 2:0 25
Radius #4 of sptere, measured in Angstrom units.

Number of electrons inside a sphere of radius 7.

which the electron-content grows as we pass to spheres of
larger radii, but also tell definitely the outer boundary
of the atomic structure. Both give a number of electrons
very nearly equal to zero in the ahote outside 0°70 A.

5. The F curves for sodium and chlorine can be solved
in the same manner. We have expressed our results in
two ways.

First, we have supposed the electrons to be grouped on
shells. The numbers of electrons on each shell, and the
radii of the shells, have been so adjusted as to give the best
possible fit to the experimental curves. In the case of
sodium it is found that a fit can be obtained with two
shells, and in the case of chlorine with three shells. The

Distribution of Electrons in Na and Cl Atoms. 443

numbers of electrons on each shell, and the radii of the
shells, are as follows :-—

Sodium.

7 electrons on a shell of radius 0°29 A.
3 be i SO Or,s
Chlorine.

10 electrons on a shell of radius 0°25 A.
5 mt ‘3 ) UTGG5 5
3 uF a Be ci | nee

Secondly, we have solved the simultaneous equations for
the distribution in shells with several sets of radii, and
drawn a smooth curve through the points so obtained in
such a way as to represent the density of distribution of the
electrons as a continuous function of the distance from

Fig. 3.

1 (£lectron density per Angstrom unit).

Oistance From centre of atom in Angstrom waits,

the atomic centre. The density P is so defined thatZPdr
is the number of electrons whose distance from the centre
lies between r and r+dr. The curves which we obtain for
sodium and for chlorine are shown in figs. 3 and 4. The

444 Prof. W.L. Bragg and Messrs. James and Bosanquet:

total number of electrons in the atom is represented by the
area included between the curves and the axis.

Fig. 4,

ee. a

P (Electron density per Angstrom unre).

. e fe) oy °
Distance from centre of atom in Angstrom units.

The following table shows the agreement between the
F curves found experimentally and those calculated from
the electron distributions :—

TasLE I.—Sodium.

Sin 6, 0-1. 0-2, 0:3. 0-4, 0-5.
(Observed: on... 832 B40. 887) 202) aman
F shells {234 \ 8°56 5:59 3:33 219 0-98
(Smooth Ourve .. 837 540 3290 1-91 1-00

TaBLE II.—Chlorine.

Sin 9. 0-1. 0:2. 0:3. 0-4. 0:5.
/ Observed Wag se ah: 12°72 7°85 5:79 4°40 3:16
fe)
| (025A)
F 4 Shells ee i 13°53 (ule 5:90 4°61 2°69
| | 1:46
| Smooth Curve ... 12°70 7°80 B55) 410 3°20

6. We have also made an approximate calculation ef the
F curve to be expected from an atom of the type pro-
posed by Bohr*. In the ionized sodium atom containing

* Nature, cvil. p. 104 (1921).

Distribution of Electrons in Na and Cl Atoms. 445

10 electrons, two are supposed to describe circular one-
quantum orbits about the nucleus, while, of the remaining
eight, four describe two-quantum circular orbits and four
two-quantum elliptical orbits. We have calculated the size
of these orbits from the quantum relationship and the
charges ; this can only be done very approximately, owing
to the impossibility of allowing for the interaction of the
electrons. We take the following numbers :—

Radius of 1 quantum ring ...... 0:05 A.
$5 2 AA aN aes te 0°34 ,,
Semi-major axis of ellipses * ... 0°42 ,,

To get a rough idea of the diffracting power of such an
atom, we suppose, first, that the orientation of the orbits is
random so that the average atom has a spherical symmetry,
and also that the periods of the electrons in their orbits are
so large compared with the period of the X-rays that we
need not consider the effect of their movements.

The calculation of the effect of the circular orbits offers
no difficulties. To allow for the effect of the ellipses, the
following method was used. The elliptical orbit was divided
into four segments, through each of which the electron would
travel in equal times. It was then assumed that, on the
average, one of the four electrons describing ellipses would
be in the middle of one of these segments. This gives four
different values of the radius vector, corresponding in the
average atom to four spherical shells of these radii.

We thus calculate the value of F for an atom having

2 electrons on a shell of radius 0°05 A.U.

relate £ Aen egeei arc.
1 i Maer Op oa.
fe Piling ; a Se gens Ou
be fbi, i aac OSTEO 2
Re ! m Owe ky

* The elliptical two-quantum orbit of a single electron about the:
sodium nucleus would have a semi-major axis equal to the radius of
the two-quantum circle. We have used the larger value 0°42 to make-
some allowance for the fact that part of the orbit lies outside the inner
electrons, so that the effective nuclear charge is reduced.

446 Prof. W. L. Bragg and Messrs. James and Bosanquet:
This gives the following figure for Fy, :—

Sin @. 0-1. 0:2. 0:3. 04. 0°5.
F calculated ............ 8:73 5-04 3-76 2°58 1:80
IPO HSELVEC ) iii). icieosctuease 8°32 5:40 3'37 2°02 0:76

The agreement, of course, is not perfect, but one must
remember that no attempt has been made to adjust the size

of the orbits to fit the curve. The method of calculation

too is very rough, although it must give results of the right
order. ‘The point to be noticed is that the curve is quite of
the right type, and there is no doubt that an average distri-
bution of electrons of the nature given by such an atom
model could be made to fit the observed value of F quite
satisfactorily.

7. The points which appear to us to be most doubtful in
the above analysis of our results are the following :—

(a) We have assumed that each electron scatters inde-
pendently, and that the amount of scattered radiation is that
calculated for a free electron in space according to the

classical electromagnetic theory. It is known that for very

short waves this cannot be so, since the absorption of y rays
by matter is much smaller than scattering would account
for, if it took place according to this law. On the other
hand, the evidence points towards the truth of the classical
formula, in the region of wave-lengths we have used
(0°615 A).

(6) We have used certain formule (given in our previous
papers, to which reference has been | made) in order to
calculate the quantity we have called @ in equation (2)
from the observed intensity of reflexion of a large crystal.
Darwin * has recently discussed the validity of these
formule. The difficulty lies entirely in the allowance which
has to be made for “extinction” in the crystal. X-rays
passing through at the angle for reflexion suffer an increased
absorption owing to loss of energy by reflexion.

Darwin has shown that this extinction._is of two kinds,
which he has called primary and secondary. If the crys-
talline mass is composed of a number of nearly- parallel
homogeneous crystals, each so small that absorption in it
is inappreciable even at the reflecting angle, then secondary
extinction alone takes place. At the reflecting angle the

* Phil. Mag. vol. xliii. p. 800; May 1922.

Distribution of Electrons in Na and Cl Atoms. 447

X-rays suifer an increased absorption, because a certain
fraction of the particles are so set as to reflect them and
divert their energy. We made allowance for this type of
extinction in our work, and Darwin concludes that our
method of allowance, while not rigorously accurate mathe-
matically, was sufficiently so for practical purposes.

Primary extinction arises in another way. The homo-
geneous crystals may be so large that, when set at the
reflecting angle, extinction in each crystal element shelters
the lower laye ers of that element from the X-rays. Darwin
has calculated that this will take place to an appreciable
extent for the (100) reflexion if the homogeneous element
is more than a few thousand planes in depth. A large
homogeneous element such as this does not produce an
effect proportional to its volume, since its lower layers are
ineffective, and a crystal composed of such elements would
give too weak a reflexion. Our method of allowing for
extinction will not obviate this effect.

We cannot be sure, therefore, that we have obtained a
true measure of Q for the strong reflexions. The F curve
may be too low at small angles. It is just here that its form
is of the highest importance in making deductions as to
atomic structure. Until this important question of the size
of the homogeneous elements has been settled, we must
regard our results as provisional.

(c) The allowance for the thermal agitation of the atom
{the Debye factor) is only approximate; it depends on a few
measurements made by W. H. Bragg in 1914. In order to
see how much error is caused by our lack of knowledge of
the Debye factor, we have calculated the electron distribution
without making any allowance for it. The result may appear
at first rather surprising ; the electron distribution so calcu-
lated is almost indistinguishable from that which we found
before, when allowance for the Debye factor had been made.
This is so, although the factor is very appreciable for the
higher orders of spectra, reducing them at ordinary tem-
peratures to less than half the theoretical value at absolute
zero. The difference which the factor makes can best be
shown by comparing the radii of the shells which give the
best fit with (1) the F curve deduced directly from the expe-
rimental results, (2) the F curve to which the Debye factor
has been applied.

448 Prof. W. L. Bragg and Messrs. James and Bosanquet :

(1). | (2).
Radius Radius
(without allowance (with allowance
for thermal for thermal
agitation). agitation).

Sodium.—Seven electrons . ... Oa 0°29
Three electrons...... 0-79 0-76
Chlorine.—Ten electrons ...... 0°28 0°25
Seven electrons...... 0’81 0:86
Three electrons.. ... 1:46 1-46

A little consideration shows the reason for this. The form
of the F curve at large angles is almost entirely decided by
the arrangement of the electrons near the centre of the atom.
A slight expansion of the grouping in this region causes a
large falling off in the intensity of reflexion. This is shown

in the analysis by the slight increase (0:02 to 0:03 A) in the
radius of the shell which gives the best fit to the uncorrected
curve. ‘!he effect of ihe thermal agitation is to make the
electron distribution appear more widely diffused ; however,
the average displacement of the atom from the reflecting
plane owing to its thermal movements is only two or three
hundredths of an Angstrém unit at ordinary temperatures,
and so we get very hiiie alteration in our estimate of the
electron distribution. The uncertainty as to the Debye
factor, therefore, does not introduce any appreciable error in
our analy sis of electron distribution.

8. It is interesting to see whether any avaclenee can be
obtained as to whether a valency electron has been trans-
ferred from one atom to the other or not. This may be put
in another way: can we tell from the form of the F curves
in fig. 1 whether their maxima are at 10 and 18 or at
11 and 17 respectively? It appears impossible to do this ;
and, when we come to consider the problem more closely,
it seems that crystal analysis must be pushed to a far greater
degree of refinement before it can settle the point. If all
the electrons were grouped close to the atomic centres, and
if the transference of an electron meant that one electron
passed from the Na group to the Cl group, then a solution
along the lines of that attempted by Debye and Scherrer
for LiF might be possible. The electron distributions we
find extend, on the other hand, right through the volume
of the orystal. The distance between Na and Cl centres is
253 iy and we find electron distributions 1 A from the centre

in sodium and 1°8 A from the centre in chlorine. lf the

——-

Distribution of Electrons in Na and Cl Atoms. 449

valency electron is transferred from the outer region of one
atom to that of the other, it will still be in the region between
the two atoms for the greater part of the time, since each
atom touches six neighbours, and the difference in the
diffraction effects will be exceedingly small. It is for this
reason that we think Debye and Scherrer’s results for LiF,
which were not absolute measurements such as the above,
were not adequate to decide whether the transference of a
valency electron has taken place.

We have assumed that the atoms are ionized in calculating

-vur distribution curves. If, on the other hand, we had

assigned 11 electrons to sodium and 17 to chlorine, we
should have obtained curves of much the same shape but
with an additional electron in the outermost shells of sodium
and one less in those of chlorine.

9. Summary.—We have attempted to analyse the distri-
bution of electrons in the atoms of sodium and chlorine by
means of our experiments on the diffraction of X-rays by
these atoms. The results of the analysis are shown in
figs. 3 and 4.

The principal source of error in our conclusions appears to
be our ignorance as to the part played by “extinction” in
affecting the intensity of X-ray spectra. The distributions
of the electrons are deduced from the F curves (fig. 1).
The most important parts of these curves are the initial
regions at small angles, for errors made in absolute values
in this region alter very considerably the deductions as to
electron distribution. The exact form of the curve at large
angles is of much less interest. Now, it is in this initial
region, corresponding to strong reflexions such as (100),
(110), (222), that extinction is so uncertain a factor. Until
the question of extinction is satisfactorily dealt with, the
results cannot be regarded as soundly established..

If our results are even approximately correct, they prove
an important point. There cannot be, either a sodium or
chlorine, an outer “shell” containing a group of eight
electrons, or eight electrons describing orbits lying on an
outer sphere. Such an arrangement would give a diffraction
eurve which could not be reconciled with the experimental
results. Hight electrons revolving in circular orbits of the
same radius would give the same diffraction curve as eight
electrons on a spherical shell, and are equally inadmissible.
On the other hand, it does seem possible that a combination
of circular and elliptical orbits will give F curves agreeing
with the observations.

Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2G

[ 450 ]

XLIV. On the Partition of Energy. By C..G. Darwin, I.A.,
F.RS., Fellow and Lecturer in Christ’s College, Camb.,
and R. H. Fowier, M.A., Fellow and Lecturer in Trinity
College, Camb.*

§1. Introduction.
A N important branch of atomic theory is the study of

the way in -which energy is partitioned among an
assembly of a large number of systems—molecules, Planck
vibrators, etc. This study is based on the use of the principles
of probability which show that one type of arrangement is
much more common than any other. The most usual method
is to obtain an expression for the probability of any state
described statistically and then to make this probability a
maximum. This always involves a use of Stirling’s approxi-
mation for factorials, which in many cases is illegitimate at
first sight, and though it is possible to justify it subsequently,
this justification is quite troublesome. It is also usually
required to find the relation of the partition to the temperature
- rather than to the total energy of the assembly, and this is
done by means of Boltzmann’s theorem relating entropy to
prebability—a process entailing the same unjustified approxi-
mations.

The object of the present paper is to show that these
calculations can all be much simplified by examining the
average state of the assembly instead of its most probable
state. The two are actually the same, but whereas the most
probable state is only found by the use of Stirling’s formula,

the average state can be found rigorously by the help of the

multinomial theorem, together with a certain not very
difficult theorem in the theory of the complex variable. . By
this process it is possible to evaluate the average energy of
any group in the assembly, and hence to deduce the relation
of the partition to temperature, without the intermediary of
entropy. The temperature here is measured on a special
scale, which can be most simply related to the absolute scale
by the use of the theorem of equipartition, and we shall also
establish the same relationship directly by connecting it with
the scale of a gas thermometer. Throughout the paper the
analysis is presented with some attempt at rigour, but it will
be found that apart from this rigour it is exceedingly easy to
apply the method of calculation. Most of the results are not

* Communicated by the Authors.

ae

On the Partition of Energy. 451

new; it is the point of view and the method which, we
think, differ from previous treatments

No discussion of the question of partition would be com-
plete without consideration of its relation to thermodynamic
principles. We shall leave this view of the subject to a
future paper ; for the increased light thrown on the statistical
nature of entropy raises many interesting points which could
not be discussed here properly without making the present
work run to inordinate length.

§ 2. Statistical Principles and Weight.

Before proceeding to the problem it will be well to review,
in general outline, the principles of the theory of the
partition of energy, though we have nothing new to say in
this connexion. We shall be concerned with collections of
molecules, Planck vibrators, etec.—each individual unit will
be called a system, and we shall call the whole collection an
assembly. We shall be dealing mainly with assemblies com-
posed of groups of systems, the individuals in each group
being identical in nature. In order to make the problem
definite it is necessary to assume that each system has.
a definite assignable energy, and yet can interact with the
others. This requires that the time of interactions, during
which there will be energy which cannot be assigned to a
definite single system, is negligibly small compared with
the time during which each system describes its own
motion,

For such an assembly we are to calculate various average
properties of its state, when it describes its natural motion
according to whatever laws it may obey. There will at any
rate be an energy integral, and we have therefore to calcu-
late these averages subject to the condition of constant
energy. To determine the basis on which these averages
are to be calculated we are to apply the principles of proba-
bility ; and the calculation of itself falls into two stages, the
prior and the statistical. The prior stage aims at establishing
what are the states which are to be taken. as of equal
probability. In the statistical stage we have simply to
enumerate the states specified in the prior stage, allow for
the fact that the systems are macroscopically indistin-
guishable, and evaluate the averages taken over these
states.

It is not here our purpose to enter into a full discussion
of the fundamental questions that arise in connexion with
the determination of what states ought to be taken as equally

2G 2

452 Messrs. C. G. Darwin and R. H. Fowler on

probable. It will suffice to recall tnat for assemblies obeying
the laws of classical mechanics the theorem of Liouville
shows that the elements of equal probability may be taken
to be equal elements of volume in Gibbs’ “ phase space.”
It follows out of this, for example, in the case of an assembly
of a number of identical systems—say simple free mole-
cules—that the elements of equal probability can be simpli-
fied down into 6-dimensional cells dq,dq.dq3dp,dp.dp3 of
equal extension, where 4, q, g3 are the coordinates, and
Pi, Px» pz the conjugated momenta, of a single molecule.
We shall describe this by saying that the weght of every
equal element dg, ... dp; is the same, and by a slight
generalization, that the weights of unequal cells are pro-
portional to their 6-dimensional extension. The word weight
is here used in exactly the sense of the term a priori
probability, as used by Bohr and others.

But when we come to the quantum theory, mechanical
principles cease to hold, and we require a new basis for
assigning the equally probable elements. Such a basis is
provided by Hhrenfest’s * Adiabatic Hypothesis and Bohr’s +
Correspondence Principle. These show how the theorem of
Lionville is to be extended, and allow us to assign a weight
for each quantized state of a system. It is found that we
must assign an equal weight to every permissible state in
each quantized degree of freedom. At first sight this is a
little surprising, for it would seem natural to suppose that a
vibrator which could only take energy in large units would
be less likely to have a unit than one which could take it in
small; but this is to confuse the two stages of the problem.
It is only by the supposition of equal weights that we can
obtain consistency with classical mechanics by the Corre-
spondence Principle. It is customary { in assigning a definite
weight to every quantized state to give it the value h, so as to
bring the result to the same dimensions as those of the
element dq dp in the classical case. But there is considerable
advantage in reversing this, and taking the quantized weights
as unity and the weight of the element in the phase space as
dqdp/h; for if this is done, the arguments about entropy are
simplified by the absence of logarithms of dimensional
quantities. We shall adopt this convention here, though in

* Ehrenfest, Proc. Acad. Amst. xvi, p. 591; Phil. Mag. xxxiil. p. 500
(LOM arene:

{ Bohr, “The Quantum Theory of Line Spectra,” Dan. Acad. iv. p, 1
(1918).

{ Ehrenfest & Trkal, Proc. Amst. Acad. Se, xxiii. p. 162. See in par-
ticular p. 165 and Additional Notes, No. 1.

the Partition of Energy. 453

all our results it is immaterial—indeed, until such questions
as dissociation are considered it makes no difference to adopt
different conventions for different types of system. The
convention has the advantage of shortening a good many
formulee and freeing them from factors which are without
effect on the final results.

An exception to the above rule for assigning weights to
quantized motions occurs in the case of degenerate systems,
where there are two degrees of freedom possessing the same
or commensurable frequencies. In this case there is only
one quantum number, and the state of the system is partly
arbitrary. Bohr* shows that the rational generalization is
to assign to such a state a weight factor which can be
evaluated by treating the system as the limit of a non-
degenerate system, and quantizing it according to any pair
of variables in which it is possible to do so. The number of
the permissible states which possess the same total quantum
number will give the weight of the state. A corresponding
rule holds for systems degenerate in three or more degrees
of freedom.

The meaning of weight can perhaps be made clearer by
considering its introduction the other way round—beginning
with an assembly of simple quantized systems of various
frequencies. Given the energy, there is a definite number
of possible states, which are fully specified by the energy
assigned to each system. We then make the hypothesis that
it is right to assign an equal probability to each such state
in the calculation of averages. Thisis now the fundamental
postulate. The generalization to degenerate systems goes as
before, by introducing weight factors. Finally, passing
over to mechanical systems, such as free molecules, we are
led by an appeal to the converse of the Correspondence
Principle to attach weight dg, ...dp3/h? to each 6-dimen-
sional cell which specifies completely the state of a single
molecule.

The second, statistical, half of the problem consists in
enumerating the various complexions possible to the assembly.
By a complexion we mean every arrangement of the assembly,
in which we are supposed to be able to distinguish the in-
dividuality of the separate systems. We count up the total
number of complexions which conform to any specified
statistical state of the assembly, and attach to each the
appropriate weight factor. Thus the probability of this state
is the ratio of the number of its weighted complexions to the

* Bohr, loc. cit. p. 26.

454 Messrs. C. G. Darwin and R. H. Fowler on

total number of all possible weighted complexions. This
part of the problem depends on the nature of the particular
assembly considered, and so must be treated separately in
each case.

We start in §3 with a problem which concerns not the
partition of energy, but the distribution of molecules in a
volume. It illustrates the method in its simplest aspect and
has the advantage of being purely algebraic. Next, in $4,
we take the distribution of energy among a set of similar
Planck vibrators, which is again a purely algebraic process,
and then proceed in § 5 to introduce the main theme of this
paper by dealing with the partition of energy between two
sets of Planck vibrators of different type. ‘This is most
conveniently treated by using the complex variable, and in
§ 6 there is a discussion of the required theorem. The par-
tition of itself introduces the temperature, and in §7 the
special scale is compared with the absolute. In $$9, 10, 11
the partition law is generalized to more complicated types of
system, such as the quantized rotations of molecules. In
§$ 12, 13 the method is extended so as to deal with the free
motion of monatomic molecules, intermixed with vibrators.
The work leads to a rather neat method of establishing the
Maxwell distribution law.

§ 3. The Distribution of Molecules in Space.

The first example we shall take is not one of a partition of
energy, but of the distribution of small molecules in a vessel.
It illustrates in its simplest form the averaging process,
and has the advantage of depending only on elementary
algebra. |

Let there be M molecules, and divide the vessel into m
cells of equal or unequal volumes 1, v2...Um, which may

each be as large or as small as we like. Then

Oy hte hy Um SVs Tee

By well-known arguments which we need not consider, it
follows that any one molecule is as likely to be in any element
of volume as in any other equal one. So by a slight ex-
tension of the idea of weight we attach weights vj, v2, ... Um to
the cells. To specify the statistical state we say that the
first cells has a, molecules, the second a2, and soon. Then

Ay+ Ag+ Ae +a, M. wie ey het eta te (3°2)

By the theory of permutations the number of complexions.

the Partition of Energy. 455
which conform to the specification is
M!
NG Ogee

and each of these must be weighted with a factor
V1 V9"? eee Vin ™.

The total number of all the weighted complexions is
1 | :

faye hs
Qi Api...

= (vy + (op) =a paste a
by the multinominal theorem. This could have been deduced
at once by working direct with probabilities v,/V instead of
weights v,, but the argument has been given in detail to
illustrate the method for more complicated cases.
We next find the average value of a,. This is given by

Ca— > .G,

Di pe? wae |s
ay ! Ag ! ware

To sum this expression we only have to cancel a, with the
first factor in a,! in the denominator, and then it is seen to
be equal to

Mov,(v, + 2 +...)"74,
and so, as is implicit in our assumptions,
a> = Mv,/V. e e ° ° e . (3°3)

But we can now go further and find the range over which
a, will be likely to fluctuate. ‘This is estimated by averaging
the square of the difference of a, from its mean value. We
shall throughout this paper describe such a mean square
departure as the fluctuation of the corresponding quantity.

Thus the fluctuation of a, is (ar—a,)?. Now

(a,— dy)? =a,(a,—1) +a,—2a a,+a,",

and averaging the separate terms by the multinominal
theorem, we have

———=, . M(M—1)r Mv, 2M», Mv, Mov, 2
co Sar ae OTe ae aa,

Mv, Uy _ = ay F
=> (1-7) =4. (1-H). De gaan

This result represents the fluctuation however large or small

456 Messrs. C. G. Darwin and R. H. Fowler on

v, may be. In all cases we have the result that (a,—q,)? is
less than a,, and therefore that the average departure of.
a, from a, is of order (a,)?. We can also interpret this fact
by saying that departures of a, from a. which are much
greater than (a,.)? will be relatively rare ; as M is large and
(a,)? small compared with a,, this is precisely equivalent to
saying that the possession of: the average value of a, is a
normal property of the assembly in the sense used by Jeans *
We have thus a simple and. complete proof that uniform
density is a normal property of this assembly.

§4. The Distribution of Energy among a Set of
Planck Vibrators.

Another case where the treatment can be almost entirely
algebraic is that of the partition of energy among a set of
Planck vibrators which all have the same frequency. Let e
be the unit of energy so that every vibrator can have any
multiple of e. As we saw in § 2, the weight attached to every
state is to be taken as unity.

Let there be M vibrators and let there be Pe of energy
(P is an integer) to be partitioned among them. To specify

a statistical state, let a, be the number of vibrators with no
energy, a, with e, a, with 2e, etc. Then we have

dod +aota,+:..= My. « 1 ee
Oy + 2ao 4+ 3G3 4 wes = Ps . e e . (4°2)

and anv set of a’s which satisfy these equations corresponds
to a possible state of the assembly. By the principles of § 2
each of the complexions will have unit weight. Now count
up the number of complexions corresponding to the speci-
fication. By considering the various permutations of the
vibrators, it is seen to be

M!

a)! a! Gs

(4°3)

We must next find © the total number of all possible
complexions. Let Sy segobe summation over all possible
values of the a’s which satisfy (4:1) and (4:2). Then

Os M!

oh Cig icy | te

Consider the infinite series
(l+e+74294 ...)™

* Jeans, ‘Dynamical Theory of Gases,’ passim.

the Partition of Energy. 457
expanded by the multinominal theorem. The typical term is
M!

Ty big Wey lew

~24+2do+ 343...
oe 5)

where the a’s take any values consistent with (4:1). Then
if we pick out the coefficient of 2’, we have the sum of all the
expressions for whieh the a’s satisfy both (4:1) and (4:2).
Observe that we may take the whole infinite series because
the later terms are automatically excluded. ,
Now this will be the coefticient of z? in (1—z)~™, and so

gE Ee
(M—1)! P!’
which is the ordinary expression for the number of homo-

geneous products as formerly used by Planck *.
We next evaluate the average of a,;

(4-4)

eat ao! al ae
ERS VA
= Me tates
where >, denotes summation over all values satisfying
Ay +a;' +a’ +a;'... =M—l,
a,’ +2a,/+ 3a,’ ... =P—r.

The sum is thus
pl +P—r—2)!

‘ (M—2)!(P—r)!’

and we have
M+ P—r—9?)! P!
Ar — M(M 1! : :

Meri (Pant

This is exact, and holds for all values of r ; now 7 can have
any value up to P and the majority of the a,’s will be zero.
The ordinary method of proof applies Stirling’s formula for
a,! to these zero values. In the important case where both
M and P are large, it will be only necessary to consider
values of + which are small compared with P. Now, if 7? is
small compared with P, P!/(P—r)! has the asymptotic
value P”. Using relations of this type and also disregarding
the difference between M and M—1, we have

JA We eee 3
(M+ prt ne Oe ee (4°5)

* M. Planck in the earlier editions of his book on Radiation.

ie

458 Messrs. C. G. Darwin and R. H. Fowler on
The same methods give the fluctuations of a, For

Cas — Ay)? = Ay Ap — 1) + Up — (Ge),
and a process similar to the above gives
———= M+ P—2r—8)!
NG _y@
Pitas ble Se) (M—23)h(P— art
The exact expression for the fluctuation can be at once put
down. When M and P are taken large the leading term
cuts out, and so it is necessary to carry the approximation to
the next order. If we substitute |
P!/(P—r)! ~Pr—4$r(r—1)P"7},
we find that
(4G) = |
a4 Sar epee Ore \ (4:6)
The formula for a, can be put into a more familiar form by
the substitution P=M/(e*—1), which gives
p= Mera — 65%), i.)

and leads to a corresponding but more complicated ex-
pression for the fluctuation. Here, as we shall see later, «
can be identified with the familiar ée/kT. Equation (4'6)
establishes at once that the statistical state specified by (4°5)
or (4°7) is a normal property of the assembly.

$95. The Partition of Energy among two Sets
of Planck Vibrators.

After these preliminary examples we now apply our method
to a problem which will bring out its distinctive character,
that of the partition of energy in an assembly composed of
different types of system. We shall consider first the simplest
of such cases—an assembly consisting of a large number of
Planck vibrators of two types A and B. The number of A’s.
is M, and the energy unit of an A is eas before. There are
now also N B’s with energy unit 7. To make exchanges of
energy possible we have to suppose, say, that there are
present a few gas molecules, but that the latter never possess
any sensible amount of energy. (Later on in § 12 we develop.
a method by which we shall be able to include any number
of such molecules in our assembly.) We also require for
the purposes of the proof to assume that e and 7 are com-

mensurable, but it does not matter how large the numbers
“may be which are required to express the ratio e/n in its

the Partition of Energy. 4.59

lowest terms. To avoid introducing new symbols, we may
suppose that the unit of energy is so chosen that e and » are
themselves integers without a common factor.

We have already introduced the idea of weight, and seen
that we must assign the weight unity to every permissible
state of a linear vibrator. To calculate the number of com-
plexions of the assembly of any given sort, we have merely
to calculate the number of ways in which the energy may
be distributed among the vibrators, subject to the given
statistical specification. A simple example will make the
process clear. . |

Let there be two A’s and two B’s; let n=2e, H=4e.
Then the possible complexions are :—

eae aab
/
| Anat ed ee | aab
a'a'a'a
aa’b bb
Yoel A AY
pad aa b b'b'
aay Oe) /
aa’a’a
aalb
aaa'a’ aa’ b/ bb!

Here, for example, aaaa’ means that there is 3e of energy
on the first of the A’s, e on the second, and none on the B’s.
Hach of the fourteen complexions is, by definition, of equal
weight, and is therefore to be reckoned as of equal probability
in the calculation of averages. Observe how with the small
amount of energy available a good deal more goes into the
smaller than into the larger quanta; for the pair of A’s have
on the average +e, as against +e for the pair of B’s.

We pass to the general case. The statistical state of the
assembly is specified by sets of numbers a,, b; where a, is the
number of vibrators of type A which have energy re, and 6,
the number of B’s with energy sn. All weights are unity
and the number of complexions representing this statistical
state is the number of indistinguishable ways (combinations)
in which M vibrators can be divided into sets of a, a,... and
at the same time N into sets b), b, .... As illustrated by
the example, it is therefore given by the formula

M! N!

Ay! ay! ay! Sins bWO;! fila:

(5:1)

In (5:1) a, and 6, may have any zero or positive values
consistent with the conditions
Ors M, 3.b=N, &,rear+S.snbs=H, .. (52)

where E is the total energy of the system—necessarily an

460 Messrs. C. G. Darwin and R, H. Fowler on

integer in the units we employ. The total number C of all
complexions is therefore

M! N! y
Ore? a Vata, Ni, ea Pe)
where the summation &,,, is to be carried out over all positive
or zero values of a, and 6, which satisfy (5:2).

By using (5°1) and (5°3) we can at once obtain an expression
for the average value, taken over all complexions, of any
quantity in which we are interested. We have already
studied a, in §4. The main interest centres in Ba, the
average energy on the A’s. We have at once

(S,0e0,)M! N!

Ay! ay! ay! wat by! by! be! eee ;

CE,= 2.5 (5:4)

The following process leads to simple integrals to express
the quantities C, CE, etc. Consider the infinite series
(L428+2% + 0... ya
expanded in powers of z by the multinominal theorem. The
general term is
M! =p EA,

Gehl 6

~

It follows that if we select from the expansion of
(L-pett 204.) ML 4284 et a)

the coefficient of z®, we shall obtain the sum of all possible
terms such as (5'1) subject to the conditions (5:2), that is to
say C. Similarly, if we form the expression

{oo (+s habe jc) el aoa

the general term in the first bracket must be

(Syreas)M! yyy
a, ! ay! eee c

and by the same reasoning the coefficient of 2* in (5°6) must
be CHa. :
Expressions (5°5) and (5:6) are easily simplified— they are
respectively |
it
(1 —2*)-M(1—27)-9, {oF (1 —2)x| (L—27)-%,

the Partition of Energy. 461

The latter can also be written as

{ —Mz“log (1—<2*) \ (1—2§) -M(1—21)-N,

If these expressions are now expanded in powers of < by
the binomial theorem, they give asum of products of factorials
which are, of course, the “‘ homogeneous product ” expressions
used by Planck. It is possible to approximate to these by a
legitimate use of Stirling’s theorem and to replace the sums

ae : Ee
by integrals without much difficulty. It would, indeed, have
been possibig to start from these expressions, but we have
not done so because in the general case to be discussed later
that method would not be available. To make further
progress’ by a method of general utility, we discard Stirling’s
theorem and express these coefficients of <c® by contour
integrals taken round a circle y with centre at the origin and
radius less than unity. By well-known theorems on in-

tegration * we have at once

Peis se di

2 oe ee
d
ee _ —Mez-, log (1—2*)
on,-.( & dz (5°72)
2m yee ts (l—2*)M(1—2)8

We can no longer hope for the single-term formule of
§§3, 4. But (5°71) (5°72) are exact, and when M, N, E are
all large in any definite fixed ratios, we can make use of the
method of steepest descents to obtain simple adequate approxi-
mations. The method is very powerful and can be applied
in a great number of cases without difficulty. Moreover, it
is comparatively easy to use it with mathematical rigour if
that is desired ; and thus the somewhat clumsy calculations
in the usual proofs of partition theorems are entirely avoided.

In general terms the process is this. Consider the in-
tegrand on the real positive axis. It becomes infinite at z=
and again at z=1, and somewhere between at <= there is a
minimum which is easily shown to be unique. Take as the
contour the circle with centre at the origin and radius $
passing through this minimum. Then we find that, for

* For those not familiar with these theorems we may remark that

200 \ ae

== 4 2dz=0 -w i dz
as | dz=0 when r is any integer other than —1, while ri scent
¥

these equations at once give (5'71) and (5°72).

462 Messrs. C. G. Darwin and R. H. Fowler on

values of z on the contour, z=S corresponds to a strong maxi-
mum, and when M, N, E are large, such a strong maximum
that practically the whole value of the integral is contributed
by the contour in the neighbourhood of this point. Hence
in the integrals it is legitimate to substitute the value at this
point for any factors which do not themselves show strong
maxima here or. elsewhere. On this general principle we

can remove the term —Mz © log (1—z*) from under the

integral sign, provided that ¢is given the value S determined
by the maximum condition. The part of the integrand in-
volving the large numbers M, N, Hi, determines the value of
S as being the unique real positive fractional root of the
equation :

That is, 9 satisfies the equation :

Jot Mills / No yes;
ea he ae (5°8)

The remaining integrands in C and Cli, are identical, and we
therefore have

aah d
K,= —Ms selog Glee)

Me
= S-6—1 ° e . ° e . ° e (5°9)
Jf a similar process is carried out for the B’s, we have
ee Nn
Kp = 3-7] 5 (5°91)
in agreement with the necessary relation
E,+E3=E.

Equations (5°9) (5°91) determine the partition of energy and
take their familiar form if we replace 3 by e~'*7, We shall
return to this point later.

§6. Application of the Method of Steepest Descents.

After this sketch it will be well to establish the validity of
the ar uments used. ‘This section is put in for mathematical
completeness, and is not concerned at all with physical
questions. We treat of a more general case than that of § 5.

the Partition of Energy. - 463

Arguments of this type—asymptotic expansions by steepest
descents—are, of course, well known in pure mathematics.
Consider a contour integral of the form

s dz
eee? (6-2)
pA

subject to the following conditions :—

(i.) d(¢) is an analytic function of z, which can be ex-
panded in a series of ascending powers of z.

(ii.) This series starts with some negative power.

(iii.) Every coefficient is real and positive.
(iv.) Its circle of convergence is of radius unity. (This
condition is quite unessential to the mathematics,
but makes the statement simpler, and is physically
true. )

(v.) The powers that occur in the series cannot all be put
in the form «+7 where « and 8 are any given
integers and 7 takes all integer values.

(vi.) F(¢) is an analytic function with no poles in the unit
circle, except perhaps at the origin.

(vii.) y is a closed contour going once counterclockwise
round the origin.

The problem is to obtain the asymptotic value as EH tends
through integer values to infinity.

We shall first study the properties of ¢(z). Considering
real values, it must have one and only one minimum between
Qand1. For it is continuous and tends to +0 at both 0
and 1, and so must have at least one minimum between.
Further, to find minima we differentiate, and then all the
negative powers will have negutive coefficients and all the
positive positive. It follows that minima are given where
two curves cut, one of which decreases steadily between 0
and 1, while the other increases steadily. These curves can
only cut in one point, and so there is only one minimum.
Next, for the complex values, consider a circle of any radius
r less than 1. As the modulus of a sum is never greater
than the sum of the moduli, it follows that at no point on this
circle can |¢(z)| be greater than ¢(r). Moreover, it can only
equal ¢(7) provided that condition (v.) is broken, and in that
case there will be 8 points at each of which |¢(<)|=¢(r).
We can thus see that on account of the large exponent it is
only the part of y near the real axis that contributes effec-
tively to the asymptotic value of the integral. This suggests

464 Messrs. C. G. Darwin and R. H. Fowler on

the substitution z=re, with a as the new variable of integra-
tion. Expanding near the real axis we have

[p(2)]*=[o() ]* exp. {ralird!/+ O(Ha*)}. (6°11)

This function contains a periodic term of high frequency
which cuts down the contributions for small values of a, so
that the value of the integral will in general depend on more
distant parts of the contour than those to which this approxi-
mation will apply. But if we choose for r the special value
3S corresponding to the unique minimum ¢’=0, then the
oscillating term in the exponential vanishes and the contribu-
tions for small values of « dominate the whole integral. For
this special value of r the exponential becomes

exp. {—4Ha?3’p"/o+ O(Ha’)}, . . (6°12)

and by (ii.) 6”>0. We see at once that we can suppose
that H?a ranges effectively over all values from —o to +00
while all other terms, such as a,..., Ea3,... remain small.
We then obtain for (6°1) on putting z=Se” the asymptotic
expression

5 Lo(s)]® ( (FS) +i9E (8) + O(c?) + Oat) jet,

For most purposes the first term in the expansion will
suffice, but if the precise values of the fluctuations are re-
quired, the second also is necessary. As it is in general
rather complicated, we shall content ourselves here with
pointing out its order of magnitude.

On carrying through the necessary calculations we find

ge (9, ¢, F} |
onnseTa lL — smstgrig| «+ 6®

The argument of F and @ is everywhere 3; the term
{3, h, F} denotes a complicated expression of 3, and its
first four derivatives, and F and its first two derivatives, but,
is independent of H. If condition (v.) is dropped, we shall
have 8 equal maxima arranged round the circle y, and, pro-
vided F' has the same value at each of them, the integral will
have a value equal to (6:2) multiplied by @.

Now consider the probiem of § 5, to which our work applies
immediately with

tba 27 (dee) ee et) Nog LY ets ea
We may suppose that E tends to infinity and also M and N

the Partition of nergy. 465

in such a way that M/E and N/E are constant. This func-
tion satisfies all the conditions of this section—the fact that
it is in general many-valued is irrelevant, for we are only
concerned with that particular branch which is real when z
is real and 0<z<1 and this branch is one-valued in the unit
eircle. The unique minimum 9 is determined by the equa-
tion ¢'=0 or
Me Ny

K= S625] ae 5-71] iy rey ia) ae (6°4)
Fhe value of the integral (5°71) is then by (6°2) (omitting
the second term the form of which is only required in
calculating fluctuations),

§-2(1 — $*)-M(1—$1)-N
(27 hs"p"'/$ }?

If, contrary to hypothesis, we had taken ¢ and 7 as having
a common factor 8, condition (v.) would have been violated.
In this case C would be 6 times as great as before, but so
would CEy,, so that EH, and all other averages would be
unaffected. The use of (6°2) to evaluate CH, (5°72) etc.
leads at once to expressions similar to (6°5), and so to the
results given formally in the last section and to others to be
given later.

As we shall see, 3 is the temperature measured on a special
scale, and there is great advantage in regarding 4, rather
than EH, as the independent variable which determines the
state of the assembly. If this is done the expression
ES’$"'/@ can be put into simpler form. For it is easily
verified that

C= (6'5)

Me?3-s Ny?3-2
BS§'/6= ay t Coma

een See di i
=57, (Hat Hs)=370, eee (a6!)

if E is regarded as a function of 3, given by (5°8).

It may be remarked that the constant occurrence of the
operator 3d/d3 suggests a change of variable to log S.
Though this has some advantages we have not adopted it,
partly because it makes the initial argument about the
multinomial theorem a little harder to follow and complicates
the contour of the integration, and also because log $ is not
itself the absolute temperature—if it had been, the physical
simplicity might have outweighed the other objections—but
only a quantity proportional to its reciprocal.

Phil, Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2H

A66 Messrs. C. G. Darwin and R. H. Fowler on

§ 7. The Meaning of §.

Returning to the subject of §5, we see that the partition
of energy between the A’s and the B’s depends on a para-
meter which can be determined in terms of the energy by
means of the equation (5°8). It is fairly evident that 3 is
connected with the temperature, though it requires more
general considerations to prove this properly. But if we
assume this connexion and are content to replace the thermo- |
dynamic definition of absolute temperature by one based on
the law of equipartition of energy for systems obeying the
laws of classical mechanics, then we can at once identify:
the meaning of 3.

For let us suppose that the B’s are vibrators of very low
frequency. They will then obey the classical laws, and the
ayes energy of each will be AT. But

i
Mat ie oie ai.
which shows that
: Serr ah) AE Chay
Substituting this in (59), we obtain the well-known form

us Me

N= °
Bete i

Observe that while 7 is tending to zero there need be no
difficulty about the condition that 7» and e are to be com-
mensurable. We shall later return to this question of

temperature and establish it for much more general types of
system.

§ 8. Other mean values.

Hxactly the same methods can be applied to evaluate any
ether mean value besides Ey. For example :—

aye M! N!
U0, = Fatt en ee Baa Bee
i (M—1)! N!
be cre pun se Sue

summed over all zero and positive values such that
2,4, =M—1, 30,=N, >,rea,'+%,snb,=H—re.
Applying the multinomial theorem and reducing the

the Partition of Energy. 467
expression to a complex integral, we have
A) Ce
amie (ley ri —2")"
which, by virtue of the value of C from (5'71) and the
argument of § 6, at once yields

a,=Ms"(1—$°),
=Me-"k2(1—e-ekT), | 2. . (83)

which is the formula of § 4 over again, the presence of the
B’s being immaterial.

When we come to evaluate fluctuations the matter is a
little more complicated, because the leading terms cut out,
and so the second term of the asymptotic expansion will in
general play a part. For example, consider the fluctuations
of a,: |

(4,—4,)°=4,(4,— 1) +4a,— (a,)*.
By arguments exactly similar to those above, we have

‘ee 1 7 ae Bee
Ca,(a,—1)=M(M—1)5— eg ee se

and so by (6°2) ;
a,(a,,— 1) =M(M—1)3""*(1—3*)’$14 O(1/E)}.

Thus the fluctuation is

a, —M3$*’*(1—3*)? + O(M?/E).
This is sufficient to show that the possession of a, (8°3) is a
normal property of the assembly.

The complete calculation of the O-term is rather com-
plicated; the result is given at the end of this section. But
a great simplification arises if we suppose that there are
many wore B’s than A’s, while E is so adjusted that $ the
temperature is unchanged. In this case the term O(M?/E)
becomes small and may be neglected. We shall describe
this case by saying that A is in a bath of temperature 4S.
Then, provided this is so, we have

(a) =e ae VEy 2 |. (84)
A much more important quantity is the fluctuation of E,.
This is found by evaluating H,”. Now, just as CH, was

given by operating with zd/dz on the first factor in
2 2

oa fd

468 Messrs. O- G. Darwin and R. H. Fowler on
(eae jo le en tee CE,2 is easily seen to be given

2
by operating with (:5) in the same way. Thus

9 1 @ dz a \e €\— =
CE) na) te

271 ve

If we again suppose an infinite bath of we 3, we
can omit the second term of the asymptotic expansion G 2)
and obtain

Be=(1-9°)* Gas) (i—sy™},

= (1— ae = BE, (1—9*)-™},

=(E,)?+3—5)

and so the fluctuation is

(E,—B,)?=E,?—(H,)?,4
ean
= erage : (8°5)
Ege 4

This is a result of which Einstein * made use in his work
on fluctuations of radiation. It should be emphasized that
these results are only accurate in a temperature bath, and
not when the number of systems A is a finite fraction of the
assembly. ; a

In all cases (6:2) shows that the possession of Hy, is a
normal property of the assembly.

If we work out exactly the second terms in the asymptotic
formuls of § 6 and apply them to the fluctuations of a, and
Ei, we find

3 re — Hi, /M
(a,—4,)° =a, [1-9 {1+ Me En y |. (8°6)

(n= oe soe 94 lh . 6 gee

Formula (4°6) above is a special case of (8°6).

* A. Einstein, Phys. Zettschr. vol. x. p. 185 (1909).

the Partition of Energy. 469

Finally it is of some interest to point out that we can

obtain a formula for (H,—H,)** of general validity. We
have in fact

(Ba—Ha)*=1.3...(2s—1){(Ea—Ba)?}*, . (8°8)

where (E,—E,)? is given by (8°7). We retain of course
only the kighest order term *, which is thus O(E,)*.

§ 9. Generalization to any number of types of system, and
to systems of any quantized character.

It is clear that the present method of treating partitions
is of a much more general character than has so far been
exhibited. Consider an assembly composed of two types,
A and B, of quantized systems more complicated than Planck
vibrators. We suppose generally that the systems of type A,
M in number, can take energies to the extents €9, €, €, ...,
and these states have weight factors py, 1, Ps, ... in conformity
with the discussion in § 2. Similarly, the B’s, N in number,
can take energies , 7, M2, ... with weights qo, q1, Ya, --.-
We have to suppose that it is possible to determine a basal
unit of energy such that all the e’s and 7’s can be expressed
as integers. Further, it simplifies the work if we suppose
that there is no factor common to all of them. Proceeding
exactly as before, we set down the weighted number of com-
plexions which correspond to the specification that, of the A’s,
a, have energy e,; of the B’s, b, have energy n,. This
number is

M! ts Us N! pate
Elance a ee Cao ey Thies
and the a’s and 0’s are able to take all values consistent with

2 ¢—=M, 2ba=Ne peat Sy b= Hh. ©. (92)

Tengen ts

Now form the functions
ie) ae + py2" + p92? Seer a tre (oo)
Wage (hai hee + Ys. (9°31)
These will be called the partition functions f of the types of

* Cf. Gibbs’ ‘Statistical Mechanics, p. 78. But (8°8) is generally
valid, while Gibbs’ formula really refer only to a group of systems in a
temperature bath.

tT They are practically the ‘‘ Zustandsumme” of Planck, ‘ Radiation
Theory,’ p. 127.

470 Messrs. C. G. Darwin and R. H. Fowler on

system A and B. The application of the multinomial
theorem then leads to the consideration of the expression

[FE l9@T"s

and pursuing exactly the same course as in § 5, we find

| en VOMU@T . - > @8

~ Qari

=; | Safe orl wor. @9)

Assume for the moment that we can choose a (2) con-
forming to the requirements of §6. The whole calculation _
then goes on as before. The radius of the circle to be taken
as contour is given by the equation

B=Ms 4 log/(3) +N9 Z log g(3). . - (96)

This equation has one and only one root. We thus can
at once put down

=[f(8)]-" 3, 1 5 LA8)]":

; (9°7)

=Ms 5 log f(S).

In exactly the same way we have
a, = Mp 3/73), 0 00) Dn ee

and we can also verify that in the case of an infinite bath
the fluctuations are again given by (8°4), (8°5), and that
equation (6°6) is still true. ‘The exact forms of the fluctua-
tions (8°6), (8°7) are aiso valid if we replace re by e,.

e have now to examine whether ¢(z) can be properly
chosen. It is natural to take :

o(2)=e Efe) [gO . . . (9°81)
By its definition it must satisfy (i.). For (ii.) to be true,
we must have

E>Me.+ Na,

which is the trivial condition that there must be enough
energy to provide each system with the Jeast amount of
energy it is permitted to have. Condition (iv.) does not
appear at first sight inevitable, but must follow from Bohr’s
Correspondence Principle *, for the convergence of the series
f(z) and g(z) depends on their later terms—that is, those of

* Bohr, doe. cit.

the Partition of Energy. AT1

large quantum numbers. Condition (v.) is satisfied if not
all the e’s and m’s have a common factor. There remains
(iii.), and here there are trivial analytical difficulties when,
as in general, M/E and N/E are fractional.
B
It is, however, easy to generalize § 6 by replacing [#(:) |

by

~Er ¢7.\7Mr 474)

& L/(2)] Lot) ] 2

and letting EK, M, N all tend to infinity independently.
Condition (iii.) is then satisfied, as can be seen by multiplying
out, and so all the conditions are satisfied, and the final
results stated above are unaffected.

Finally, we may observe that all our results can be
extended at once to an assembly containing any number of
types of system. ‘If there are M, systems of type C, for
which the partition function is £,(3), then

- d
B= M$ clog f.(9),

where $ is determined by
d

The formal validity of the proof will require all the
quantities e, to be commensurable. It will be shown in § 12
how this restriction may be removed.

§ 10. Vibrators of two and three degrees of freedom.

As a first example we take a set of vibrators each of which
is free to vibrate ina plane under a central force proportional
to the distance. The sequence of energies is again 0, e,
2e, ..., but the weights are no longer unity, as the system is
degenerate. Following the principle laid down in § 2, we
may evaluate the weights by treating the system as non-de-
generate and counting the number of different motions
which have the same total quantum number. «Now we can
quantize the plane vibrators in directions w and y, and as an
example for the case 2e, we have three alternatives (2e, 0),
(ce, €), (0, 2c). This is easily generalized, and gives to re
the weight r+1. The partition function for these vibrators
is thus :

fc) =14+ 27432? 4427 +...,
Ed

472 Messrs. C. G. Darwin and R. H. Fowler on

From the general theorem (9°7) we at once have

= 2Me

A Sele

so that such vibrators have just twice as much energy as
the line vibrators.

In exactly the same way we can treat the case of three
dimensions. To illustrate the weights we again take the
case of 2e and quantize the system in w, y,z. There are
six alternatives (2e, 0,0), (0, 2e,0), (0, 0, 2e), (e, €, 0),
(e, 0, €), (0, €, e). The general form for ve is 4(r+1)(r+2).
The partition function is now

Ke) =14+324 627+ 1027+...,
= (1-2-4,
which leads at once to the expected result
= 3Me

WS gee

§ 11. Rotating Molecules.

Another interesting example to which the calculations at
once apply is that part of the specific heat of a gas due to
the rotations of the molecules. Various writers* have
quantized the motions of a rigid body, and it is found that
the system has at most two instead of three periods, so that
it is partly degenerate. We may consider for simplicity a
diatomic molecule. Then, on account of the small moment
of inertia about the line of centres, the third degree of
freedom may be omitted altogether—its quantum of energy
is too large. A simple calculation then leads to energies of
rotation €. given by

h?

= Sr"? ay) Watch me neu nn pro Ut hve @bit)
where I is the moment of inertia about a transverse axis,
which we shall assume to be independent of vr. This is a
degenerate system, and considerations of the number of cases
which occur if it is quantized for the two degrees shows
that the weight to be attached is 2r+1. This is on the
principles suggested by Bohr f with a simplifying modifica-
tion; for Bohr had to suppose that certain quantized motions
were excluded for other reasons which are not operative

* Among others, Khrenfest, Verh. Deutsch. Phys. Ges. xv. p. 401
(1913). Epstein, Phys. Zeitschr. xx. p. 289 (1919). F. Reiche, Ann.
der Physik, liv. p. 421 (1917).

t Loe, cit. p. 26.

ee

the Partition of Energy. 473

here *. There can, we think, be no question as to the correct-
ness us the weight 27+ 1, put most recent writers have used
a factor x; our formula for the specific heat has therefore
a rather different value.

We may now apply our general formule to this case with

he
ies WL Wo.§) ste ze :
p,=2r+l1, ¢,=9r'e, e= 331° (11-2)
Then {a= 1 ae a ee (LSS.

E, =Ms, “0g f(3). ee cee aE ST)

The contribution of i rotations to the molecular specific
heat, C.,,, is dH,/dT, where M must be taken as the number

of molecules in one gramme-molecule of gas. Thus, using
(7°1), we have

he deat d e :
C= Tege eo lo g f(s Df : : (11 4)
and Mk=R, the usual gas constant. If we write
he
~ 8 Lk” uP)

then
d?
C,4¢=Ro? + log (1+ 3e Sawer Men te (lot)

Equation (11°5) shows that, when [>«,oa->0. It can
be shown by the application of standard theorems on series f
that when o—>0,

ies ee teenta Piers NU) wapes ABE)

which is the correct limiting value as required by classical
dynamics.

In the general case of any body we have three degrees
of rotational freedom, the motion is simply degenerate f,
and the energy enters as a sum of square numbers multi-
plying two units of energy. The motion of the axis of
symmetry and the motion about the axis of symmetry are
not independent, and it is impossible therefore for the parti-
tion function to split up into the product of two partition
functions which represent the separate contributions of the
two motions. The result is a double series of the same
general type as (11°3).

* Assuming that no extraneous considerations rule out any of these
states,

+ Bromwich, Infinite Series, p. 132. The theorem is due to Cesaro.
t Epstein, Phys. Zeit xx. p. 289 (1919).

474 Messrs. C. G. Darwin and R. H. Fowler on

It does not appear profitable to examine these expressions
further here, since the agreement with experiment is not
very good at all temperatures. It is to be presumed that
the assumption of constant moments of inertia is at fault,
and this is supported by some of the evidence from band
spectra ; further, it is probable that the case of no rotation
must be excluded, involving the omission of the first term
in the partition functions. The discussion of the practical
applications of these formulee cannot be entered into here.

$12. Assemblies containing free molecules.

The problems we have so far discussed have all possessed
the distinguishing characteristic that the temperature is the

only independent variable. As soon as we treat of free
molecules this is no longer the case, for now the volume
must be another independent variable. Nevertheless, as we
shall see, the same methods of calculation are available.
The partition is no longer represented exactly by integrals,
as it was for the quantized motions, but from the nature of
the case some form of limiting process is required. The
free molecules cannot of course be regarded as the limit of
three-dimensional vibrators of low frequency, for they have
no potential energy to share in the partition. We must
proceed by the method common to most discussions of the
distribution laws of classical assemblies—divide up into cells
the six-dimensional space in which the state of any molecule
is represented, associate with each cell a certain constant
value of the energy, and in the limit make all the dimensions
of all the cells tend to zero*.

We take an assembly composed of M systems of the type
A of § 9 and P free-moving monatomic molecules of mass m
and of small size, the whole enclosed in a vessel of volume V.
The energy of the molecules is solely their energy of trans-
lation; they are supposed to obey the laws of. classical
mechanics (except during their collisions with the A’s). In
order to specify the state of the assembly, we™take a six-
dimensional space of co-ordinates 91, g2,+++ 3, the three
rectangular co-ordinates and momenta of a molecule in the
vessel. We divide up this space into small cells, 1, 2, 3,...,
¢..., of extensions (dq,...dp3); which may or may not be

* That the limit of the distribution laws worked out for the cells is
the true distribution law for the actual assembly is an assumption
implicit in all such discussions.

the Partition of Energy. A75

equal. Then by the principles of § 2 the weight factor for
the ¢th cell is

= ee ay Mee i ah aCk)

h

provided of course that the cell is relevant to our assembly.
Only those cells have a weight for which the q’s lie in the
vessel; but the p’s may range over all values from —« to
+a, for the method of summation will automatically
exclude values which could not be allowed. Associated
with the ¢th cell there is energy given by

1 e !
C.= te, (py + po? + p37). Renae pili (i227 bl)

The state of the molecules in the assembly is specified by
the numbers ¢, c., ... of molecules in cells 1,2,.... The
specification of the A’sisas before. The number of weighted
eomplexions corresponding to the specification is then

APN ocrinn, pak
. Ge ee RLS

where 2 aoe > ae oc. th...” | (12°13)

(ame Pa

See et (1292)

In proceeding thus we are constructing an artificial
assembly in which the energy is taken to have the same
value €, in all parts of the ¢th cell, and in which all the @s
and all the e’s can be expressed as multiples of some basal
unit, without a factor common to them all.

This assembly can be made to resemble the real one to
any standard of approximation required. For such an
artificial assembly we can make use of the whole of our
machinery. ‘The results all depend on integrals such as

=a \anlOMB@r, . . 022)

Qart
where the partition function of the artificial molecules is
Ree ead: Wheiyet m)s.(12-21)

and the formule of § 8 follow at once for Ey, (Eyg—EH,)’, ¢
and (c,—¢,)”. These results give completely the exact partition
laws for any artificial assembly of the type considered. To
obtain the actual distribution law for the real assembly, we
must make all the dimensions of all the cells tend to zero,
and obtain the limit of the partition function. Now, by the

476 Messrs. C. G. Darwin and R. H. Fowler on
definition of an integral, in the limit A(3)->H(S), where

1(© _ 10818

H@)= 2) gam PY Pe dg * dp, ae

The integration is over the volume V and over all values of
the p’s from —o to +0. This gives at once

(27m)3??V
Plog i/s)\72
In the formule the functions dh/d3 and d?h/d% also occur,
and it is easily shown directly that their limits are dH/ds
and d’H/dS°. We may therefore use (12°4) throughout for

the real assembly, and at once obtain the following ex-
pressions :

H(s)= (12-4)

E.= bas _ sper, (12:5)
Teas RE CH. | eM Wiarton ee:
(ey 3/2 1 j 9 1)
C= =< S28) Dee ee aa ) dx dy dz du dv dw,
» ((T2eon)
3/2 m ‘
= Sea eae On dw. eae
The temperature 3 is determined by
13 qo « 3 i ).
E= Ms J log 7(S)+$P Rese (do sa0)

and the fluctuation of energy of the molecules in a bath of
temperature 3 is given by

(Bp—E,jind SE,=3PeT . . (1254)

If the fluctuation of ¢, is evaluated, it takes the simple value
(Gi 6) 6s «i a et seu, eae lee

whether it is in a bath or not; for the second factor,
analogous to that in (8°4), can be omitted when the cell is ©
taken to be of small size. Thus in all cases the possession
of ¢,1s a normal property of the assembly. These results
can be readily extended to cases where there is an external —
field of force acting on the molecules.

By means of this assembly we can establish the meaning
of 3 in terms of T, by observing that the gas itself constitutes
a constant volume gas thermometer. It is easy to show that
the pressure of a gas must be $ of the mean kinetic energy

the Partition of Energy. ATT

in unit volume, that is to say, p=P/V log (1/3). Since the
gas temperature is measured ; the relation pV =PAT, we
are again led to the relation 3=e7"",

We may observe that it is now possible to drop the
assumption of commensurability, which was necessary in
the sections which dealt with quantized systems. It was
there essential, physically speaking, in order that it should
be possible that the whole of the energy should be held
somewhere; but as we now have molecules which can hold
energy in any amounts, it may be dispensed with, the modifi-
cation being justified on the same assumptions. and by
the same sort of limiting process as have been used in this
section. Again, we can see that the correct results are
obtained if H(<) replaces h(z) in (12°2) and all the other
integrals, even though the interpretation as coefficient in a
power series is no longer possible, and though the integrand
is no longer single valued. In such many-valued integrands
the limiting process shows that we simply require to take
that value which is real on the positive side of the real axis.

§13. The Mazwell Distribution Law.

We have carried out the whole process so tar with quantized
systems included in the assembly, but it may be observed
that it is immediately applicable to an assembly composed
solely of molecules. If this is done the value of ¢; in (12°51)
establishes at once the Maxwell distribution law, and its.
fluctuation in (12°55) proves that it is a normal property of
the assembly. This is probably the simplest complete proof
of the ordinary distribution law ; its special advantage is
that by means of the fluctuations it is easily established that
the actual distribution will hardly ever be far from the
average.

The method can also be made to establish the distribution
law for a mixture of gases*, and indeed for a mixture of
any kind, provided that the systems can be considered to
have separate energies.

It is also possible to extend the method to cases in which
the total momentum or angular momentum is conserved, by
constructing partition functions in more than one independent
variable. In fact, there will be as many independent
variables as there are uniform integrals of the dynamical
equations of the assembly. For simplicity we shall suppose
that the linear momentum in a given direction is conserved,

* The effects of the semi-permeable membranes of thermodynamics.
can be conveniently treated by the partition function.

478 On the Partition of Energy.

and let its total amount be G. The method now requires
the averaging process to be applied to expressions depending
on

P!

c,! Cate

CHOC
Oj Oo es

where we now have not only
Lic, =P, 70,510,
but also 2 C= Ge

where #, is the momentum in the given direction of a mole-
cule in the tth cell. To sum the appropriate expressions we
must take as our partition function

hii i)\== Dd 25ta"t,

With this function C will be the coefficient of 22° in
[A(c,v)]*, and this can be expressed as a double contour
integral. So can the other averages, and the usual asymptotic
expansions can be found. ‘The correct distribution law
follows on replacing h(z, x) by the integral which is its limit
when the sizes of the cells tend to zero. This subject lies
rather outside the theme of the present paper and need not
be elaborated further. |

$16. Summary.

The whole paper is concerned with a method of calculating
partitions of energy by replacing the usual calculation, whieh
obtains the most probable state, and is mathematicaily un-
satisfactory, by a calculation of the average state, which is
the quantity that is actually required and which can be found
with rigour by the use of the multinomial theorem together
with a certain theorem in complex variable theory. 1

After a review of principles and two preliminary examples
the real point of the method is illustrated in §5. Here
there are two groups of interacting Planck vibrators of
different types. It is shown that the partition can be found
by evaluating the coefficient of a certain power of z in an
expression which is the product of power series in z. This
coefficient can be expressed as a contour integral and can be
evaluated by a well-known method, the “‘ method of steepest
descents.” The result expresses itself naturally in terms of
a parameter 3 which is identified with temperature measured
on a scale given by $=e7™,

The work is extended to cover the partition among more

The Heterodyne Beat Method. 479

general quantized systems in $9, and examples are given.
In $12 it is shown how it may be made to deal with as-
semblies composed partly of free molecules and partly of
quantized systems. In $13 we deal with extensions possible
when only molecules are present.

The methods we have described can also be made to throw
an interesting light on the statistical foundations of thermo-
dynamics ; Gat in that connexion many points have arisen
which require rather careful discussion, and in order not to
make the present paper too long, we have deferred them to
a future communication.

Cambridge,
May, 1922.

XLV. The Heterodyne Beat Method and some Applications
to Physical Measurements. By Maurice H. Bruz, M.Sc.
(Cantab.), Barker Graduate Scholar of the University of
Sydney *

N a recent paper f, a preliminary account was given of
the application of the heterodyne beat method to the
measurement of magnetic susceptibilities. In virtue of the
importance of the method as a sensitive measure of physical
quantities, it seems desirable to give a more complete account
of the principle and of some of the difficulties encountered
in its application.

Hssentially the method consists of the following arrange-
ment shown in fig. 1.

Two oscillating circuits, Set 1 and Set 2, are set up side
by side and ar ranged so as to have approximately the same
frequency. The two sets are loosely coupled so that in the
telephone included in one of the circuits a resultant beat
frequency is maintained equal to the difference between the
frequencies of the fundamentals or overtones in the two
circuits. If symmetry in the two circuits is essential, direct
coupling can be replaced by indirect coupling by means of a
third circuit in which the telephone is placed. In either
case, when the beat frequency is low enough, an audible note
will be heard in the telephone, and any changes in the
constants of either circuit will cause the frequency of
the audible note to alter by an amount equal to the change
in frequency of the responsible circuit. This at once provides

* Communicated by Professor Sir E, Rutherford, F.R.S.
+ Belz, Proc. Camb. Phil. Soe. vol. xxi. part 2 (1922).

480 Mr. M. H. Belz on the Heterodyne Beat Method

avery sensitive method. It is now easily possible to maintain
oscillations of frequencies up to 10’ per second. ‘Taking the
case when Set 1 has a frequency of 1,000,000 per second,
Set 2 a frequency of 1,001,000 per second, the audible note
will have a frequency of 1,000 per second. If the frequency
of Set 1 is changed to 1,000,001 per second, the frequency of
the audible note will now be 999 per second, and this change
in pitch can readily be observed by comparison with a note
of standard pitch.
Bigs

yee all) 7 een

Sena

com

See

This sensitive method has been successfully employed by
Herweg *, Whiddington ft, Pungs and Preuner ft, Falcken-
burg §, and several others in physical researches, but the
precautions necessary for steadiness in the beat note have
never been completely specified.

Precautions.

With high frequency oscillations of the order 3 x 10° per
second to 5 x 10° per second such as were used in the present
investigations, electrostatic shielding from all external in-
fluence was of the first importance. This was ensured by
placing all the elements of the circuits in earthed metal-
lined boxes, one of the variable capacities, by means of which
final small adjustments were made, being provided with a
long ebonite spindle which projected beyond the containing
box. With the box closed the note from the telephone IT
was considerably reduced in intensity, and in order to obtain
the maximum loudness, a small section was removed from
the box, shielding being maintained by means of a piece of
fine metal gauze.

* Herweg, Zeit. f. Phys. vol. 111. p. 86 (1920).

+ Whiddington, Phil. Mag. vol. xi. p. 634 (1920).

t Pungs and Preuner, Phys. Zeit. vol. xx. p. 543 (1919),
§ Falckenburg, Ann. d. Phys. vol. |xi. 2, p. 167 (1920).

and some Applications to Physical Measurements. 481

Solidity of foundation is a most important requirement.
In the experiments of Whiddington*, although the apparatus
was set up on a solid base, vibrations of the building even
at 2 a.M. proved troublesome. A somewhat similar trouble
was experienced in some of the earlier experiments when the
apparatus was installed on the top floor of the laboratory.
It was found that the vibration of the building caused by
people walking about the corridors, and by the passage of
heavy motor traffic, appreciably affected the steadiness of the
note. Although some of the work was done during the night
and over the week end, the trouble always persisted.

Finally the apparatus was transferred to a room on the
ground floor and supported on stone pillars by means of solid
rubber pads. The trouble was now completely removed so
that successful observations could be made during the day
despite the fact that people were continually walking beside
the apparatus.

After lighting the valves, a certain amount of time must
elapse before the oscillating system has settled down to a
steady state. This initial variation is due to the heating and
expansion of the elements of the valves, causing changes in
the whole capacity linked with the oscillating systems. In
order to save time, thus sparing the high tension batteries
and prolonging the life of the valves, the latter were contained
in small tin boxes, placed outside the large box, which were
lagged with asbestos and cotton-wool. In this way the heat
conduction was minimized, and the valves settled down much
more rapidly. Other conditions being the same, it was found
that certain valves were less satisfactory than others. For
some types the settling down process was very long, and by
the time the valve was set, other things began to vary.
After long trials with “ R,” “ Fotos,” and ‘“ A. T.” types,
it was found that “B” type valves manufactured by the
General Electric Company gave most satisfactory results,
settling down most rapidly and remaining steadiest.

The effects of the changes in the elements of an oscillating
system on the frequency have beeu examined by Eccles and
Vincent f in the case of wave-lengths of 3000 metres. They
determined that between certain limits for each value of the
coupling between the plate and grid coils there was a
particular value of the filament current for which the wave-
length was a maximum. Working at this value of the
current it was found possible to hold the beat note steady to

* Whiddington, loc. cit.
+ Eccles and Vincent, Proc. Roy. Soc. A. vol. xcvi. p. 455 (1919).

Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. ria |

482 Mr. M. H. Belz on the Heterodyne Beat Method

one part in 100,000 for several minutes in spite of small
unavoidable variations. With the frequencies employed in
the present experiments, however, such a condition could
not be established. The heating of the valve parts and the
consequent change in capacity in the system resulting from
changes in the filament current cause changes in wave-length
which certainly far outweigh any real change due to increased
thermionic emission alone. In order then to secure a constant
filament current, accumulators of 100 ampere-hours capacity
were employed. These were charged regularly after about
three days’ use, and after the valve had settled down, the
current from them showed no variation during a single run.

Faulty contacts of wires joining the elements of the
circuits were avoided by soldering, the only sources of
uncertainty being the sliding resistances in the filament
circuits. These, however, were good types with bright
surfaces and stiff springs so that the chance of error due to
change of contact was small.

The principal cause of variation in the frequencies of the
circuits was found to be due to variations in the high tension
batteries. This trouble has been mentioned by Hecles and
Vincent *. In the present work the plate voltage was
obtained from trays of portable accumulators of fairly low
capacity, each tray providing 40 volts. After the valves
had been burning for an hour or so, taking a current of
about 10 milliamperes, this voltage began to vary and the
beat note consequently drifted. However, giving the valves
time to settle down, a matter of 15 to 20 minutes, it was
found possible to hold the heterodyne note quite steady for
intervals of 30 to 60 seconds, and this is ample time in which
to make a single observation. After about 90 minutes
burning the variation was too rapid and the batteries had to
be recharged. The size and consequent capacity of the cells
of these batteries is limited by the fact that they have to be
contained in a metal box, and thus this source of variation
ean only be provided for in special cases.

Technique.

It is essential to maintain the oscillations generated in the
circuits at frequencies considerably different trom the natural
frequencies of the coils alone, that is to say with an
appreciable capacity in the system, and under these conditions
the frequency, n, of the oscillations in such a circuit containing

* Eecles and Vincent, doe. cit.
+ Cf. Townsend, Phil. Mag. vol. xlii. August (1921).

and some Applications to Physical Measurements. 483
inductance L and capacity C, is given very approximately by
n=1/(27,/LV).

Changes in » can thus be brought about by changes in
( or L. In the experiments to be described below, the
changes in n were brought about by variations in L, and in
this case, with C constant, a small variation, dL, in the
inductance produces a corresponding change, dn, in the
frequency given by

Ory SMa oem iy ay fy ael me (L, )

The experimental part thus reduces itself to a determination
of dn. This is accomplished by obtaining beats between the
heterodyne note and a note of constant pitch, and then
counting the change in the number of beats per second
caused by the change ininductance. A. considerable amount
of practice in listening is required in order readily to be able
to adjust the heterodyne note to the pitch of the constant
note. This note can be very conveniently obtained by means
of a third set, some distance away from the other sets,
oscillating with audible frequency, in the plate circuit of
which a telephone is placed. The intensity of the note
heard can be altered by adjusting the filament current, and
in this respect the note is very much more convenient than
that obtained from a tuning-fork. For it was found that
the heterodyne note could be more easily brought to tune
with the standard note, and false beats more readily recognized
when this latter could be altered so that both notes had
approximately the same intensity.

In some of the experiments * it was found impossible to
obtain a beat note of convenient audible frequency when the
fundamental frequencies of the oscillations were approxi-
mately the same. It was observed that, as the capacity of |
one of the sets was altered, only very shrill notes could he
heard on either side of the very large region of silence.
This synchronization effect appears to depend on several
factors, but chiefly on the coupling between the circuits.
Cn account of the limited size of the box containing the
coils, the coupling could not be reduced beyond a certain
lower limit, and reducing the strengths of the oscillatory
currents merely reduced the intensity of the limited note.
In ali these cases it was possible to obtain the heterodyne
note between the fundamental of Set 2 and the first overtone
of Set 1, which was quite steady and possessed the normal
region of silence. Since the changes in 7 were produced by

* Those in which the determinations of the magnetic susceptibilities
of certain salts were made, see below.

2.1 2

484 Mr, M.H. Belz on the Heterodyne Beat Method

variations in the inductance of Set 1, this arrangement
increases the sensitiveness of the method. For let N be the
frequency, determined at the centre of the region of silence,
of the fundamental oscillation in Set 2, the Trequency of
the fundamental oscillation in Set J, then since the tirst
overtone of Set 1 is employed to produce the note, N=2n.
Let the frequency of the audible note from the third circuit
be m. Then when the heterodyne note is adjusted, by
slightly varying the capacity of Set 2, so that q beats per
second are counted, the frequency of Set 2 is (N4+m2q)
=(2n+m+q). If now the frequency of the fundamental
oscillation of Set 1 is altered by dn per second, the frequency
of the first overtone is altered by an amount 2dn per
second, so that the frequency of the heterodyne note is
now (2n+m+q)—(2n+42dn)=(m+q+2dn), whence if a
change of p beats per second is observed when the induc-
tance change is accomplished, p=2dn. The sensitiveness
is thus doubled, and could similarly be increased by em-
ploying higher overtones of Set 1. Against this, however,
is the fact that the notes so obtained are very feeble, and
counting becomes increasingly difficult.

From equation (i.) we see that the sensitiveness depends
onn. Itis now possible to maintain oscillations of frequencies
up to 10‘ per second, but in cases where the change in in-
ductance is caused by inserting a specimen within the coil,
there is an upper limit to determined by the form and
function of the coil L. It is necessary to divide this coil
into two parts between which there is no mutual inductance,
one part L, being coupled to the grid circuit in order to
maintain the oscillations, the other part Ly serving as the
coil in which the inductance changes occur. This latter
part must be a fairly long coil in order that there may be an
appreciable region within it through which the magnetic
field is constant, in which region the specimen is placed.
On account of the dimensions of this coil, the first part has
to possess a fairly large inductance in order to get sufficient
mutual inductance with the grid coil: further, a certain
amount of coupling is required with Set 2 to produce the
heterodyne note.

Haperimental.
In the present experimental arrangements the details of
the coils are as follows :—

Coil 10pe
The coil was 10 cm. long, and consisted of 100 turns of
copper wire, no. 22 s.w.g., double cotton covered. It was

and some Applications to Physical Measurements. 485

wound on a short length of glass tubing and had an effective
diameter of 2°10 cm. The self-inductance, employing the
exact formula of Nagaoka™*, viz.,

where L, is the self-inductance of a current sheet of the
same dimensions as the coil, n,; the number of turns per em.,
a the effective radius, 6 the total length, and K a factor
depending on the ratio of the diameter of the coil to the
length, to which was applied the correction for spacing, was
calculated to be 39,160 em. The small frequency correction
was nevlected.

Coil Le.

The coil was 30°70 cm. long. and consisted of 541 turns
of copper wire, no. 24 s.w.g., silk covered. It was wound
on a long glass tube, of external diameter 1°00 cm., and
separated therefrom by means of a layer of paraftined paper.
The effective diameter (2a) of the coil was 1:105 cm., and
self-inductance, calculated as above, was 92,430 cm.

The total inductance L (= L,+ I) is thus 131,600 em.

The coil L, was outside the box containing the rest of
the circuits, and was shielded from external electrostatic
influences by means of an enveloping earthed metal cylinder.

Coil L;.

The length was 9 cm., and the coil consisted of 90 turns
of copper wire, no. 22 s.w.g., double cotton covered, It was
wound on a short length of glass tubing and had an effective
diameter 4:13 cm. ‘The self-inductance was similarly cal-
culated to be 124,600 cm.

The capacities employed had a range of 100 to 1200
microfarads and were provided with a slow movement.

Changes in the frequency of the oscillations of Set 1
brought about by the insertion of a specimen within the
coil L, may be due to three causes :—

(a) In the first place, if the coil is not shielded from the
electrostatic effect of the specimen, the self capacity of
the coil will be changed. In order to observe changes
of inductance alone, it is necessary to guard against this
possibility. This was done by depositing a thin layer of
platinum ¢ on the outside of the glass tube on which the
coil L. was wound, and earthing. The thickness of the deposit,
obtained by weighing, was 7x107-& cm. It is necessary to

* Nagaoka, Jour. Coll. Sci. Tokyo, xxvii. art. 6, p. 18 (1909).
+ The function of the paraffined paper was to prevent any possible
short-circuiting of the coil through the layer of platinum.

486 Mr. M. Gi. Belz on the Heterodyne Beat Method

determine the effect of this shield on the strength of the
magnetic field within. The magnetic force, H;, at a depth ¢
in a mass of metal is related to the force, Hy, at the surface
by the equation

2arupt

Tone ow cor.

in which pw, o represent the permeability and_ specific
resistance respectively of the metal, and p=2n, n being
the frequency. Taking n=4°84 x 10° per second, the largest
frequency used, and for platinum, w=1, c=11,000 c.g.s.
e.m.u., we find that within the shield for t=7 x 10° em.,

BL EL) =e, ON
= 0°9997,

The effect of the field on the changes of inductance can thus
be neglected. |
- With the shield, the remaining causes of the change of
frequency are due entirely to changes of inductance, being
(6) an eddy current effect within the specimen, and
(c) in the case of the magnetic substances, a susceptibility
efrect.

The Eddy Current Effect.

The magnetic field, H, within the coil will be of a
harmonic type, and on this account circular eddy currents
will be induced in the specimen in planes perpendicular to
the axis of the coil and in sucha direction that the magnetic
forces arising from them oppose and consequently diminish
the value of H. This virtually means a diminution in the
inductance of the coil, and the frequency change will be in
the direction of nm increasing. ‘This result can also be
predicted mathematically by regarding the specimen as
equivalent to a coil with self-inductance and resistance,
coupled to the main coil. The analysis, however, is com-
plicated and its development was not proceeded with. Some
experiments were made, however, to determine the order of
the change in inductance and its dependence on the charac-
teristics of the specimen.

In the first place, as bearing on the results obtained in
the magnetic measurements, sulphuric acid was examined.
This was contained in a long glass tube, the effect of which
had previously been determined to be zero, and was lowered
into the oscillating coil L,. Any change in inductance due
to an eddy current effect should certainly depend on the

and some Applications to Physical Measurements. 487

conductivity of the specimen. This solution had a con-
ductivity far greater than any of the magnetic specimens,
which were examined in liquid, crystal, and powder forms,
but, counting over periods ranging beyond ten seconds, no
variation in the beat note was observed. Accordingly it
was assumed that the effect in the magnetic substances was
negligible.

Some tests were then made with different lengths and
sizes of graphite taken from ordinary pencils. The con-
ductivities of these specimens were much greater than that
of the solution of acid, but there were no changes in the beat
note indicating an increase of frequency. In the case of
some specimens changes occurred which correspouded to an
increase of inductance, but some iron must have been present
in them.

Finally, some tests were made on ditferant specimens of
copper, brass, manganin, platinoid, and constantan wires.
In these experiments, which were carried out after the
susceptibility determinations, some alterations in the elements
of the circuits enabled the heterodyne note to be obtained
by beats produced between the fundamental oscillations.

It is interesting, at this stage, to give the results obtained
for the variation in frequency change with the conductivities
of the specimens. In this connexion two tests were made,
the first employing small specimens and counting the beats,
the second employing large specimens and computing the
frequency change from the alteration in capacity necessary
to bring the heterodyne note to coincidence with the auxiliary
note.

(a) The specimens used were wires of copper, brass,
platinoid, and manganin, each of length 0°32 cm., and
diameter 0'711 mm. The wave-length was 710 metres.
The results are included in Table I.

TaBLe LI.
Frequency Change
Specimen. per second.

dn.
BUAMIICES oe fay diss cad Bond eckh an oh 9°54
PSO yeas Like Ree SK encek ds o> 4°88
DN RONIN Rera she a0 nce x20 0°33
LN ae ee 0:28

(b) The specimens used were wires of copper, brass, and
constantan, each of length 8 em. and diameter 2°03 mm.

488 Mr. M. H. Belz on the Heterodyne Beat Method

The wave-length was 725 metres. The results are included

in Table II.

Tasze II.
Capacity Change Frequency Change
Specimen. ne as per second.
(arbitrary units). Fs
Copper ists devas ee eeeee 201 1027
EB ASS CEPA E wie. ok tame Sho 190 961
Monstamtan oka essa ne tones 98 501

The wires of platinoid, manganin, and constantan used in
the above tests contained 15 per cent., 4 per cent., and
40 per cent. nickel respectively. The magnetic suscepti-
bilities of copper-nickel alloys have been determined by
Gans and Forseca*, who found that when the nickel
content was 40°4 per cent. the susceptibility was only
0°189x107-® cg.s., being much smaller for lower per-
centages. The effect of the susceptibilities of the specimens
used on the frequency changes can thus be neglected.

Fig. 2.

oCurve B

nN
°

COIS.

—— > frequency change, 07, pers
ES
°
to)
SB

o 0
Oxio> 1 2 3 4 5 6x 10°
——> Specific Conductivity, KR, mhos. ¢cn.-’.

The specific conductivities, K, in mhos cm7}., of the
materials used, taken, with the exception of brass, from
Kaye and Laby’s tables, are:—for copper, K=6-29 x 10°,
for platinoid, K=0:291 x 10°, for manganin, K=0-233 x 10°,
and for constantan, K=0:204x 10°. The specific conduc-
tivity for the brass used was determined to be K =1'182 x 10°.

Mica? represents the relations obtained by plotting dn
against K, curve (a) referring to test (a), curve (0) to test (0).

* Gans u. Forseca, Ann. d. Phys. vol. lxi. p. 742 (1920).

and some Applications to Physical Measurements. 489

The forms of both curves indicate a relation of the type

K2
dlL=constant x K?4B°
the term B varying for different sizes of specimen, being
relatively less important the greater the whole conductivity
of the specimen, as shown in curve (b). A similar type of
dependence was calculated by Mr. Kapitza, of the Cavendish
Laboratory, for the case of a thin spherical shell.

For solid cylinders, the dependence of the frequency
change on other characteristics of the specimens was
examined, but it is hoped to give a complete account of the
results together with further tests on different forms of
specimens, in a subsequent paper.

THE APPLICATION OF THE METHOD TO THE DETERMINATION
OF THE MAGNETIC SUSCEPTIBILITIES OF CERTAIN SALTS.

The general principle of the method has been given in
the previous paper*. It has since been applied to the
determination of the magnetic susceptibilities of many salts,
the results obtained being given below. In addition it seems
desirable to indicate more carefully the nature and magnitude
of the corrections to be applied.

Considering the insertion of a magnetic substance in the
form of a cylinder of cross section A’ and length l', the
volume susceptibility of which is K,, within the coil L of
Set 1, the cross section and length of which are A and /
respectively, thus causing the frequency, n, of the oscillations
to alter by an amount dn, we find, subject to the corrections
to be given below,

K i dn Al L, ae L, :

sider Wa ae Wn Wn OS

In all the experiments for determining K,, the heterodyne
note was obtained by beats between the first overtone of the
oscillations in Set 1 and the fundamental oscillation of
Set 2, as already mentioned on pages 483, 484 supra. In this
cease, if N is the frequency of the oscillations in Set 2, and a
change of p beats per second is observed on inserting the
specimen, N=2n, and p=2dn. We thus obtain

an 1 P Al I+ L, ee
— dN AT i- ° ° ° e (11.)

* Belz, loc. cit.

490 Mr. M. H. Belz on the Heterodyne Beat Method

Before this formula can be applied it is necessary to—
investigate several corrections. The absorption in the
platinum shield which covered the tube on which L, was
wound, and the eddy current effect, have been discussed and

shown to produce no correction terms. But there remain

two further corrections to be investigated, (i.) due to the
demagnetizing effect of the specimen, and (ii.) due to the end
correction term in the calculation of the self-inductance of
the coil and the effect on this due to the position of the
specimen.

(i.) The demagnetizing effect on the magnetic field due
to the magnetism induced in the specimen.

By treating the specimen as an elongated ellipsoid of
revolution, of semi-minor and semi-major axes @ and ¢
respectively, with its long axis parallel to the field, Ewing *
has shown that if H is the value of the magnetic force.
within the specimen, and H’ the original value of the
magnetic force before the specimen was inserted, then

H=H’'(1—NK;),

where N is given by
Ne4r(l/e—1) J © slog. (1 +ey/(1—e—1},

e being the eccentricity and equal to ,/1—a?/c’.

In a typical ease, that of cobalt chloride in solution,
which had a value of Ky, approximately 20x 107° c.g.s.,
2a=0°507 c.m., 2e=8'0c.m. Hence e=0°998. This gives
N=0'126, and hence

H= H’(1—0°126 x 20 x 107°)
= 292 x 101)

so that this correction is negligible.

To get the effect on the external field, consider the
resultant magnetism induced on the ends and sides of
the specimen. Ewing} shows that the free magnetism,
although densest at the ends, extends towards the middle,
and it is only on the equatorial line that there is none.
Also the total quantity of free magnetism on any narrow
zone taken perpendicular to the direction of magnetization

* Ewing, ‘ Magnetic Induction in Iron and other Metals,’ pp. 23-25.
+ Ewing, loc. cit. pp. 25-27.

and some Applications to Physical Measurements. 491

is proportional to the width of the zone and to its distance
from the equatorial line.

Consider such a zone at distance # from the equatorial
line, x being measured in the positive direction of the field,
width dx. Then the total quantity of free magnetism on
it is equal to +Cxda, where © is determined from’ the fact
that the total quantity of positive (or negative) magnetism,

a. ef Cate is equal to zal, I being the intensity of
magnetization. This gives C= rs ii and hence the free

magnetism on the strip is + Qa yes wala.

Let us consider the effect: of this quantity of positive
magnetism at the distance x, and an equal quantity of
negative magnetism at —# ata point on the equatorial plane
outside the specimen. Suppose the free magnetism on the
strip to act, in regard to external points, as if it were
concentrated at its centre of gravity, and let d be the distance
of the point in question from the axis of the specimen.
Then the magnetic force in a direction parallel to the axis

at the point 0, d will be
a? ada Hh
—_ FTN I s e —— — — 5
47 > a + dP +

c
and the effect due to the whole specimen will be the integral
of this from 0 to c, which becomes

—4n 3 ea ;1.(- Pers

If H is the value of the verte force within the specimen,
which we have seen is also the original value of the field
within the coil before the specimen was inserted, we may
write [=K,H, and the field at the point 0, d is reduced
from H to

14a eee higinih - 1
H {1-4 ce Ae i Taz ae + sinh a
Taking d=0°5 em., i. e. just within the winding of the coil,
and with the same numerical data as before, this becomes
H(1—1°79 x 10-8).
Ata point on the surface of the specimen, for d=0°25 cm.
we obtain for the reduced field,
H(1—2°5 x 10-).

The correction in this case is also negligible, and we are

— at sini a).

492 Mr. M. H. Belz on the Heterodyne Beat Method

thus justified in completely neglecting the effects of de-
magnetization. |

(i1.) The effect of the end correction term and the position

of the specimen.

In a long coil there is an appreciable length over which
the magnetic force is constant and given by H=47ny, ny,
being the number of turns per unit length. At the ends the
force falls to half value, and this diminution in H is
responsible for the correction term K in Nagaoka’s formula

Jy aed) 6 i

When this correction is small and so can be neglected, the
expression (il.) becomes exact. In the present case, the value
of K was equal to 09873 and the error involved in neglecting
this end correction is about 1°3 per cent. We can allow for
the correction in the following way :

The shortspecimen (about cm. as compared with the length
of the coil, 36°70 cm.) was suspended in approximately the
centre of the coil, which can thus be supposed to consist of
three coils in series, one in the centre of length equal to that
of the specimen and without end correction, and two approxi-
mately equal coils on either side, to the open ends of which
the correction is to be applied. Let /’ be the length of the
specimen, / the length of the coil, then the total self-
inductance is L=C/K, while the self-inductance of the coil
of length J’ is L,=Cl’, where C=4m’a’n,’. The area of
cross section of the coil being A, that of the specimen A’,
the self-inductance of this portion is altered to Ly’ where

L,’=1L)(1+47rK,A‘/A),
so that the whole change in inductance of the coil, dL, is
given by
du=1)47K,A'/A,
= L)47K,A'l'/(AIK),
and thus the expression (ii.) in the corrected form becomes
m2 pep K .volume of coil L,+ L,

Se , NS Gime
2a N volume of specimen’ Ly Gn

EHeperimental Errors.

The accuracy of the estimations depends almost entirely
on the determinations of the change in the number of beats
per second and of the frequencies. The counting was done
by means of a stop-watch guaranteed to read tenths of a
second. ‘lhe heterodyne note was adjusted by means of the
fine movement on the condenser of Set 2 so that a reasonable

and some Applications to Physical Measurements. 493

number of beats per second, sometimes zero, was heard.
The time for 20 beats was noted, the specimen then inserted
the new time for 20 beats being again observed, and finally
a third count made when the specimen was removed. ‘The
note was then slightly varied and the same procedure
repeated. In this way from 6 to 10 readings were obtained
for each particular frequency used. A review of all the
measurements so made under conditions when the heterodyne
note was steady show that the greatest error incurred was
about 2°2 per cent.

The wave-lengths were measured with a calibrated Towns-
end wave-meter, and the resonance point could be fixed to
within 2 metres. ‘he shortest wave-length measured was
about 350 metres corresponding to a fundamental oscillation
in Set 2, so that the maximum error involved here is about
0-6 per cent.

From all sources, then, the maximum error involved in
the estimation of the susceptibilities is about 3 per cent.

The Experimental Results.

The specimens were all prepared by Kahlbaum, Berlin,
and were contained in short lengths of glass tubing thus
permitting of quick insertion into the coil. The effect of
the tubing alone was found to be zero in every case. The
results are given below.

I. Ferric Chloride—The salt was examined in the form
of a solution in air-free water. It was contained in a closed
glass tube, occupied a volume 1°321c.c.. and had an approxi-
mate length 7°5 cm. The density was determined to be
1152 gm. per e.c. at 15° C., and a volumetric estimation
showed that the solution contained 0:189 om. FeCl, per c.c.
The observations are shown in Table III.

Taste III.
| Frequency | Frequency | Change in &
pe re of Set 2 of Set 1 number of rade
ae eee per sec. per sec. | beats persec.| p/N. Fae say CG
in metres. N. Ns p=2dn. egrees C.
x 10° Sc16? sein?

O72 8:07 4:03 2°59 3°21 15

385 7°80 3°90 2°49 3°19 15

400 | 7°50 3°75 2°39 3°19 5)

418 - foe erin 259 2-29 3:19 15

Mean p/N=3'19x 107° at 15° O.

494 Mr. M. H. Belz on the Heterodyne Beat Method

The values of p/N show that for the range of frequencies
used the susceptibility is constant. This was found to be
the case for all the substances examined.

Formula (iii.) gives

° > \2 e :
Kot 319 x 10-8 (7/4 CL 105)? x 36°70 x 09873
20 te a2
,. 131600
= ILRI Sek s C.2.5S. 99420

The mass susceptibility of the solution K,,, is thus

Kini LOK LO ne ail 2
— hoon eis

For a solution of a salt in water, the mass susceptibility is
given by

Ip = pK, a5 @ —Po VK es
where p)=mass of salt per c.c.,

K,,=mass susceptibility of the water-free salt,
K,,,= susceptibility of water, which may be taken as
-—0°75 x 107° c.g.s.

We thus obtain
= 9007 x LOT Sic och at ion C:

The values for K,, determined previously by balance
methods are* 92x10*, 91 «x10°°, 33x10°" O2Zegiias
88 x L0-8, and 103 x 10~¢ ¢.g.s. with a mean value 91 » 107
c.g.s. The present result is in good agreement, and thus
shows that up to a frequency of 4:03 x 10° per second the
susceptibility is not altered.

II. Ferrous Sulphate-——Tests were made on the salt in
forms of crystals and powder. The crystals were good ones
of the monoclinic type, and on estimation proved to have
the composition, FeSO, 7H,O almost exactly. Hqual masses
of crystals and powder were used, the powder being obtained
by grinding up the crystalsin a glass mortar. The substances
were enclosed in glass tubes, the transference after weighing
being rapid to avoid absorption of water from the atmosphere.
The mass of each specimen was 1:00 gm. at 15°C. At this
temperature the density of FeSO,, 7H,O crystals is 1°899 om.
per c.c.,and hence the volume of each specimen was 0°527 c.c.
In the form of powder this volume occupied a length 5°6 cm.,
while for the crystals the length was about 8 cm.

* Landolt and Bernstein Zabellen (1912).

and some Applications to Physical Measurements. 495

The observations are shown in Table LV.

TABLE LV.
| 3) F Cl i |
| Brequency requency lange in ‘ 7
ie | of Set 2 of Set 1 number of yen?
ee |: par sec. per sec. beats per sec. p/N. ee
in metres. | N. n=N/2. p=2dn. | degrees C,
_ Cryst. |Powder.| Cryst. | Powder. |
AS eHOR at) oe 10° 10-5" | «107
352 Le: S58 L426. 4-40 4:40 515 SN Pana
a eee | 4:00 4:10 | 412 5:13 ioe 9.16
392 | 765 3°83 3°91 3°90 SE 510 | 16
410 | 7°32 3°66 O74 371 5:11 5'07 16
. 414 beitceaitas | S63 3°70 3°69 510 509. -|.- 16
a. T20 1 VSO S002 bla us ts Be aN ey names a 16

Mean p/N for both erystals and powder=5'12x107® at 16° C.

The susceptibility of the salt is thus the same in the solid
as in the powder form. A similar identity in the sucepti-
bilities of different forms of the same material was observed
by Wilson * using a balance method in the investigation of
certain iron ores.

The volume susceptibility is

K,=76°6 x 10-® ¢,9.5.,

whence the mass susceptibility of the complete salt is

40-4x10-* c.g.s. The percentage mass of FeSO, in one

eram of salt is 54°6, so that, neglecting the contribution to

the susceptibility of the water of crystallization, we find
Ki, = 120 X10 7".¢.5,5. at 16°C.

The mean of the values for K,», all determined in the form
of a solution, given in the Tabellen of Landolt and Bornstein
is 75x 107% c.g.s. Finke+ found for K, for the complete salt
a value 80x 107° ¢.g.s., and the present result agrees very
well with these. ;

Ill. Ferrous Ammonium Sulphate.—The salt was examined
in the form of monoclinic crystals which proved, on estima-
tion, to have the composition FeSO,(NH,).SO,, 6H,O to an
accuracy of 0°3 per cent. The density of this salt.is
: 1°813 gm. per c.c. at 15°C. A mass of 1 gm, was taken and
: placed in a glass tube, occupying a length of about 8 em.,

* Wilson, Proc. Roy. Soc. A. vol. xcvi. p. 429 (1919).
1 Finke, Ann. d. Phys. (4) xxxi. p. 149 (1910). /

496 Mr. M. H. Belz on the Heterodyne Beat Method

and a volume 0°551 ec.e.

We find hence

29°8 x 1076 c.g.s.

The observations are shown in

Mean p/N=3'79X10~® at 16°C.

Ke 0450 x10 etic ers:
which gives for the mass susceptibility of the complete salt

in one gram of salt is 72°5, and hence we obtain
pO Se LO eas, aie ILE Oe
The values given in the Tabellen of Landolt and Bornstein

Table V.
; TABLE V.
Change in
Wave-length| Frequency | Frequency S | Tempera-
of Set 2 of Set 2. of Set l. number of Ae
in metres beats per sec.| p/N. degrees C
; N. w= N72. p=2dn. 8 :
105 - x 10° «10-8
335 8:95 4:47 3°39 3719 16
340 8°83 4°41 3°34 3°78 16
363 8°72 4:13 ili 3°79 16
400 7:50 3°75 2°84 3°79 16

The percentage mass of FeSO,(NH,).SO,

of K,, for the salt when examined in the form of a solution
at 18°C are 44x 1076, and 45 x 10-6 c.g.s.

IV. Mickel Chloride.—The salt was examined in the form
of a solution in air-free water. The volume of the specimen
was 1°572 c.c. and occupied a length in the glass tube of
approximately 7-4cm. The density was determined to be
1°332 gm. per c.c., and an electrolytic estimation yielded
0°255 gm. NiCl, per c.c. The observations are shown in

Table VI.
Tasie V1.
Wave-length| Frequency | Frequency See a Tempera-
of Set 2 of Set 2. of Set 1. hears »/N ' ture
i tres. SEN Dem ees TS) degrees C.
Tae IN n=N/2. p=2dn. @
x 10° x 10° Same
, 309 8:45 4:22 2°12 2°31 15
380 7°90 3°95 1:99 2°52 15
404 7:43 371 1:88 2°52 15

Mean p/N=2'52X107 at 15°C.

and some Applications to Physical Measurements, 497
_ This gives |
ee bebe «10 el ps:
The mass susceptibility of the solution is
B= 949x107 c.g.s.,
and the mass susceptibility of the water-free salt becomes
Ke pSod'o X:LOT? c.o-s) ab, 15°C.

This is in good agreement with the values given in the
Tabellen of Landolt and Bornstein, the mean of which is
39°66 x 1078, but is considerably different from the value
recently given by Théodoridés *, viz. £47°6x107° c.g.s.
at 15°-2 C.

V. Nickel Sulphate—The specimen was in the form of
the green rhombic prism crystals which, on estimation,
were shown to have the composition NiSO,, 7H,O; the
density of these crystals is (Thorpe and Watts) 1:950 gm. —
perc.c. A mass of 1°438 gm. occupying a volume 0°737 c.c.
was used, having a length in the glass tube of approximately
7°-2cm. The observations are shown in Table VII.

TaBLeE VII.

_ Wave-length| Frequency | Frequency |— ek od a Tempera-
of Set 2 of Set 2. of Set 1. es N ture
; ance eats persec.| p/N. degrees O
ae Ne n=N/2. p=2dn.
x 105 x 105 alms

342 8°77 4°38 2°63 3°00 16

370 812 4°06 2°44 3°00 16

412 7:29 3°64 2°16 2°96 16

Mean p/N=2:99 x 10~° at 16°C.
This gives |
K»=32'1x 107° c.g.s., |
from which we get the mass susceptibility of the complete
salt to be 16°43 x 107%c.g.s. The percentage mass of NiSO,
in one gram of salt is 55, and hence we obtain
29-9 x 10 Scre.s; at L6> ©.
The mean of the values in the Jabellen is 30X10~° c.g.s.,
while Finke + finds for K, the value 29°1 x 10~$ c.g.s.
* Théodoridés, /ouwrn. d. Phys. III. i. p. 1 (1922).
t Finke, loc. cit.
Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2K

498 Mr. M. H. Belz on the Heterodyne Beat Method

VI. Cobalt Sulphate.—Tests were made with the salt in
the form of crystals and powder.

(a) Tests on crystals.

The salt proved on estimation to have the composition
CoSO,, 7H,O ; the density of this salt is 1°950 gm. per c.c.
A mass of 1:1375 gm. was used which thus occupied a volume
0-583 c.c. The length of the crystal column was 6 cm.
The observations are shown in Table VIII.

TABLE VIII.

Wave-length | Frequency | Frequency nee o Tempera-
of Set 2 of Set 2. of Set 1. ture
in metres beats per sec.) p/N. degrees C

N. n=N/2. p=2dn. 8
x 10° x 10° <iGme
349 8°60 4:30 4°28 4°97 16
385 7°80 3°90 3°85 4°94 16
410 7-32 3°66 3°59 49} | 16

Mean p/N=4-94x 10-6 at 16° C.
This gives |
Ke — 660 <0 gcic ocr.

and thus the mass susceptibility of the complete salt is
33°85 x 1078 c.g.s. We thus obtain

KK, = 61-4 x 105 Scle.s.at loa:

(b) Tests on powder.

A mass of 1°3150 gm. was used having a volume 0°674 c.c.
and occupying a length in the glass tube 5°5 cm. The
observations are shown in Table IX.

TaBLE IX.
Wave-length| Frequency | Frequency Creme) a Tempera-
abet 2's:| | of Set 2)./| | of Set i.) Gmueher @ ture
‘ t ‘ ’ | beats per sec.) p/N. a C
ihe N. n=N/2. p=2dn. ta
x 10° x 10° x 1076

350 8°57 4:28 4°88 5°70 16

385 7°80 3°90 4°45 571 16

414 7°25 3°62 4:14 5°72 16

Mean p/N=5'71 10-8 at 16°C.

and some Applications to Physical Measurements. 499
This gives
K,=67:0 x 10-6 c.g.s.,
and finally
K,,=62°4x10~ ¢.¢.8..at 16°C.
The results of the two tests thus give as mean values
K— 6679 < 10-* cas;
Re Old 10> %e.o-g: at 16° GC.

For the complete salt Finke* finds K,=68 x 107° c.g.s.
The mean values for K,, in the Tabellen is 66 x 10~¢ c.g.s.,
while Théodorideés } gives for K,, the value 64°5 x 10~®c.g.s.
aE EG.

VII. Cobalt Chioride.—The salt was examined in the
form of a solution in air-free water. It occupied a volume
1°620 c.c. and had a length in the glass tube 8 em. The
density was determined to be 1:186 om. per c.c., and an
electrolytic estimation yielded 0°2078 gm. CoCl, per c.c.
The observations are shown in Table X.

EABILE X.
eo Change in
Wave-length| Frequency | Frequency Tempera-
of Set2 | — of Set 2. of Set 1. | abe ture
in metres. | 2 jbeats per sec.) p/N. degrees C
N. n=N/2. p=2dn. 8
«10° x 10° S10"
340 8°82 4-4] 3°48 3°94 20
385 7°80 3°90 | 3:09 3°96 20
|
414 | 7°25 | 3°61 | 2°85 3°94 20

Mean p/N=3-95 x10—® at 20°C.
This gives |
E30 105 eres.
from which we find for the mass susceptibility of the solution
Km, = 16°28 x 10-® c.g.s.,
and finally for the mass susceptibility of the water-free salt
ie dix 10—* e.a-s: ab) 20°C.

* Finke, Joc. cit.
+ Théodoridés, loc. ctt.
Fag | Sb

500 On the Heterodyne Beat Method.

The values in the Tabellen for K,, are 81 x 1076, 101 x 108,
and 82 x 10~&ce.g.s., while Théodoridés* gives 97°9 x 10-®c.g.s.
The present result thus indicates that the lower values are
the more accurate.

Summary of Results.
The results obtained are grouped together in Table XI.

TaBLe XI.
; Form in Mass susceptibility
Substance. sv hiok tesla: Sf ea Temp. C.
Ferric Chloride, FeGl,.| Solution ...... 90°7 x 10—° ¢.g.8. 15°
Ferrous Sulphate, Crystals and 74:0 x 107° ¢.g.8. 16° sem
FeSO,,7H,O. Powder.
Ferrous Ammonium §ul- | Crystals......... 41-1 x 1076 &g-8. 16°
phate, FeSO,, (NH,),
SO,, 6H,O. .
Nickel Chloride, NiCl,.| Solution ...... 39:5 x 10-6 ¢-g.8. 15°
Nickel Sulphate, Crystals 29-9 x 106 &-g-s. 1162
NisO,,7H,0. ;
Cobalt Sulphate, Crystals and 61 910-6 «.g.s. 16°
CoS8O,,7H,0. Powder.
Cobalt Chloride, CoCl,.| Solution ...... 81:°3x 10-6 ¢.g:s. 22
Summary.

The precautions necessary for steadinessin the heterodyne
beat method are described, employing frequencies greater
than 10° per second, and it is shown that under normal
laboratory conditions a beat note can be obtained perfectly
steady for large periods.

Some applications of the method to physical measurements.
are then considered, the changes in frequency being conse-
quent on changes in. the inductance of one of the circuits.
These latter changes were brought about by inserting a
specimen within the coil, and were of two kinds :—

(i.) An eddy current effect, the nature of which was
examined for cylindrical specimens of different materials.

(ii.) In the case of magnetic substances, a susceptibility
effect. This latter variation was employed to measure the
magnetic susceptibility of several salts at frequencies ranging

* Théodoridés, Joe. cit.

On Elastic Equilibrium under Tractions. 501

from 3 x 10° per second to 4 x 10° per second, and the results
obtained are in good agreement with those previously
obtained by static methods.

My best thanks are due to Sir Ernest Rutherford for
many suggestions connected with the problems and for his
helpful criticism and encouragement during their progress.

Cavendish Laboratory, Cambridge.
May 20, 1922.

XLVI. On the Conditions for Elastic Hquilibrium under
Surface Tractions in a Uniformly Eolotropic Body. By
R. F. Gwytuer, 1/.A.*

ie a paper “On an Analytical Discrimination of Hlastic

Stresses in an Isotropic Body,’+ I have expressed the
elements of mechanical stress under tractions, and also
the elements of strain, in terms of quantities which resolve
on transformation of orthogonal axes in the same manner as
elements of stress resolve. In this paper I propose to adapt
the same method to uniformly eolotropic bodies.

Briefly, we shall have under tractions,

= ma 0’d3 nhs O°ds Ox!
P= a2 AE +2 RE
n= 10"; Or er > ON
ae oz 02 ie Ox 52
es, 07d es 0°¢, : 0x3
a Ox? Oy’ ieee Oy’
S= OK ee OP y x SO7N3 01) 1DENs
on Ode | Ozrd2 ‘Or0y’
T= O°X2 ee 0x3 Ms 0’d2 as 0x1
ol Oyorm O02 Oz dy’
U= Bx Dye BK , De
Oe Oy O02 Nn 0e > O07 07’

(1)
where {,, do, $3, ¥1 X2 X3} resolve as elements of a stress
resolve.

* Communicated by the Author.
+ Phil. Mag. July 1922, p. 274.

502 Mr. R. F. Gwyther on Conditions for
If {01, 05, 63, Wi, We, 3} form another set of quantities,

acting also as elements of stress, we may take as the com-
ponents of an arbitrary displacement

_ 0h: | Os Shh,
7, Sa. Oe Oe:
_ Ov; | 082 | Ov
- 2 Oy? oer
on tis, Mw
ai ae aye
and therefore for the elements or strain
3°, ahs os
e= Pye 45 Oa dz + 02 OY 9
Me 0°02 oy o's
a oy” ¥ Oy Oz + Bray ”
a 3%; Oth Deyo
Lae 02 y OY Oz ui 02 02 ;
ie O' , Ob . 0702+ 42) Os Os
eis ey’ v 02? ne Oy 02 i OL 02 i Oroy ”
_ Os "bro Os 0°03 + A) Ovi
fie 02? + 02 f Oy 0 € 02 02 OL0y —~
ay Obs Os 07h, Oy 07(9; + Ae)
Sera ae tS Ouiee + 3202, @

(2)
Before proceeding, I will recapitulate that if the co-
ordinate axes are rotated about their own positions through
the small angles ,, wy, w,, and if 0,, O., Os (being partial
differential operators) give the consequent coefficients of

®,, @y, ®, in {é, fg, a, 6, cor nm { P,Q) Rae. Uy aen
0 . OO
Bie a(S 5) | ee eet
in the case of elements of strain, and
fe)
o,= 28($ - 2) ao, uv. 4 78

in the case of elements of stress.

Elastic Equilibrium under Surface Tractions. 503

Both {, do, ds, Xl» X2> x3 and {6,, 0», Os, Whi, Wro, vrs |
are to follow the latter type.

The relations between these two sets in elastic equilibrium
must be linear.
I shall take the invariant function (2V), which gives the
potential strain-function per unit of volume in the form,
2V = Aye? + 2A + 2X13 eg + 2r44ea + 2X15 eb + 216 €C
+ Nog J? + 2ro3 fg +2 fa +25 fb + 2ro¢fe
+ 33 J? + 2r34. ga + 2r3590 + 2r36.9¢
+ Ay a?+2r0yzab +2rggac
+ Nsz5 b? + 2rs6 be
+ Nes in
(3)
Since V is to be an invariant function, that is, since

Q2,V=0, O.V=0, 0;,V=0, we deduce that, acting on the
coefficients,

ig fe fo) fe fo fo)
a= Drus( 35 — —_ 5) = (i3— Az) Sree means ley ae

fe) fe) fe)
+ 4r<o4 aoe + 2(Az4—eoq) ae + (24yp— Age + das)

0
Ove

fe)
+ (2r453— Dog) ang + (2rgg + Vos)

Pa) WO
= Days — (2N44—Agz + Aes) aie. (245 + Ase) Ailes

7 fe)
— (2yg— <5) Ds

+ 2(rsy —)eu)

fe) fo)
ak + (Ags — Aes — Vag) a:

- + (Asg— Ave #45) xo
fo)
Be ee so + (ss Aee) 50 + eNom Men ca)! ce (2)

From (3) we find the elastic values for P, Q, R, 8, T, U
by differentiation in e, f, g, a, b, c respectively. Thus
LB = Met Aye f+ N39 + Aga + Arsb + Are, etc.,

and we may form an apparently suitable set of values for

504 Mr. R. F. Gwyther on Conditions for
bi, ho, $35 X15 X20 X3 in terms of 4, Oo, O3, Wi, Wo, Wr similarly

hus —
Py = yy + Aye 2 +3393 + 2rqavhy + 2ZrisYro + 2ZrreWrs, ete.,
and bere rai a NosOs a Asa0s an ray oi 2rasWr3 alia 2ragWs; etc.,

which include all of the 21 constants.

The next step would be to substitute these values in the
mechanical values of the elements of stress in (1) and equate
to the elastic values found as above from (3) and (2).

We should thus obtain six independent differential equa-
tions in A, Oo, 03, Wi, ro, ws. But these quantities are not
independent, and we are at liberty to put each of the
ar-functions equal to zero, and thus get six independent
equations in three quantities, leading to conditions which
I do not pursue.

Instead, I shall follow the method used in my earlier
paper and be guided by the form of equations (1).
Accordingly I determine the values of dy, ds, d3 by
selecting the terms affected by 97/dyd0< in 8, by 07/d# dz
in T, and by 07/02 dy in U, both in their mechanical and
elastic expressions.

We thus obtain

py = (Agog + Asa) Wi + Agi(Go + O3) + AssWr3 + Ase Wo,
es (Ais + Ags) We + Asses + A55(83 + 01) + Ase,
= (Aye +Ave) Ws + AseYro+ Assi +AGe(O1 +2), . (9)

thus pe connecting the two sets of functions. A
first condition to be applied is that Q, 6;=0, and therefore

(Asa + Noa) (05 = 0.) SF 2(A34 = Nos) (0, a> 0s)

+ (Ag3— a2) + (Ase — Ave + 2A45 ro + (Ass — X25 — 2A4e r3 = 0,
| (6)

with two other identities, whose consequences we can infer

by symmetry.

Thus

Ay) = Age =Agg=A (an invariant),
Noa = Ags = Ais = Avs = Me= Avs = 0,
Nya ae 256 = 25 + 246 = 36 ote 245 = 0. ° . (7)

Also, as there is no term in fa or ga in 2V, the coefficients
of 0/OA, and of 0/ds4 In Q, are separately zero, and

A2g=A—2Ayu, Ats=A—2A55, Arg=A—2Abge. . (8)

Elastic Hquilibrium under Surface Tractions. 505
We therefore find

OV =X(o+f+9)? + ru(a?—4/y) + Aso(b?—4eg) + Ass(c?—4e/)
+ 2r56 (be— 2ae) + 246 (ac— 2b/) + 205(ab — 2cq) ° ()

and
0 fe) fe) fe)
O = —2rso( aaa N55 — Age) 3. — — Au55 —
} "\OAss — OA6s ( Mckee lis Bas
fe)
a
© OAs
I shall now write, for convenience,
Agg=V1, A55=V2, Ase=V3, ASE= — V4, Ats= —V5, Ass= — V5

and
2V = rle+f+y)?+ r1(a—47y) +2(08—4eg) + 04(e—4ef)

+ 2v4(2ae —be) + 2v5(2bf—ae) + 2vg(2cg—ab), . (10)
where

es 0 fe
2 = 2n (3 a = 2) 4 (4- Pee Ome

The requirement that the solid os act as an elastic
solid under tractions has reduced the 21 eolotropic constants
in (3) to 7 constants of elastic equilibrium in (10).

Accordingly, we find frum (5),

oi = V\(O9+ 03) —vstro— Vows,

fz = Vo( B24 41) —vyiri — ves,

d= v3(0,; +05) —vyhr — Vso. . . : (TE)
The value of 2y, is deduced from 0,¢,=

2, = v4( 20; + 02+ 83) —(ve+ v3) Wi —VeWo—vs3, with
2X = V5(41 + 202+ O3)— veri — (11 + V2) Wo— aps,
2y3 = ve( 0; 9 0, + 203) —v5y i Varo — (v, i Vo) Pr. ( 12)

From the value of 2V given by (10), we tind the elements
of elastic stress, as usual, by differentiation,

P = re+ (A— 203) f+ (A— 22) 9+ 2y,a,
Q = (A—2vz)e+ Af + (A—2)g + 2050,
= (A— 2y.)e+ (A— 21 f+rAG + 2vVec,
S = 2vye+vja—v,b—vsc,
T= 2v3f—vea t+ vob —v4c,
U=2y,.9—v,a—vwb+vsc. .*. . . . ~ (18)

306 Mor. i. 3 Gwyther on Conditions for

To complete the equations, we substitute for {e, f,g, a, 5, c}
from (2), equate to the mechanical elements of stress given
by ( 1), and substitute for pis bo, Pe X1> X23} to find the
S1x equations for condition in {6; 2, 03, Wi, Wo, 3}. This
general process will be now simplified.

In order to carry out the scheme of this paper it has been
necessary that the axes should be arbitrary and the stress-
elements should be written in full. But this stage having
been completed we may now select a special set of axes, and
also simplify the arbitrary stress-elements.

ee {e, f, g, 4, b, c} act on transformation of axes as
ue Up ry AOS Qe, Qayt act, as do also {a’—4fg, b?—4eg,

—Aef, Awe —2be, Ab f — Zac, 4cq—2ab}, and since

a CEOs io 3 3
o— Dia ty sate) Serene ays + Vaae

we find that
V2? + Voy? + v3e" + 2vgyz+ 2vsve + 2veny = 1

- is an invariantal ellipsoid, and the principal planes of this
ellipsoid are planes of elastic symmetry in the body.

I shall take these principal planes as the coordinate planes,
and consequently v,=0, v;5=0, v,=0, while 1, v2, v3 now
stand for the roots of the Discriminating Cubic.

The equations (11), (12), and (13), become for these axes,

do, = V1( A, +E 03), 2M) = = (V9 v3)Wi1,
do = v2( 03+ 0), 25 = — (¥3+11)ho,
d3 = v¥3(0,+ 85), 23 = — (1. + 2)s,

P = re+ (A — 203) f+ (A— 202) 9,

Q = (A—2vz)e+rAf+(A—2n;)g,

= (A— 2, )e+ (A— 2”) f+ AQ,

S = vas TS 95b, WW ac ee
It will now suffice to put ~,=0, ~.=0, 3=0, so that
.. OOF “OU: _ 063

3) ee eae tS) Ue p)
Den ee OZ

and it follows, from above, that y,;=0, ¥.=0, y3=0 at the
same time. We shall therefore remain with only three
equations in 0,, 0, 03, when these simplifications are made.

Elastic Equilibrium under Surface Tractions. DOT

These three equations take the form

B81, B,D) __, BULB) _ B%G—8
Se + OT 32 )= hoa a 3
=, BUG =61) _, 9°(0.— 8)
ay 3 Ou? 022 ?
_ BOO) _ 3° —4,)
Vo Bu? 1 Oy? ’
so that ety ct era HELD.)
v1V7?(@,—03) = 2 (6, — 43) + v,(A3—0,) +v3(0;—O.) },

with two similar ee and

4.02 .v20y"

: =) fy1(8—03) +¥9(85—0,)
erg TS LUN AUS

Hence v,(@.—63) + v2(@2— 01) + ¥2(@,—92) is an Ellipsoidal
Harmonic, from which 6,—6, 0,— 63 can be deduced.

In a previous paper, I have dealt with the simpler case
for an isotropic body, when vy; =y,=v3=n and 6,—63, 6.—93
are Spherical Harmonics.

The present equations apply to crystals having three
orthogonal planes of elastic symmetry.

The investigation seems interesting because it passes some-
what outside the range of elastic equilibrium, even if it is
ultimately confined within that range. If we regard a piece
of sound material intended to serve as a test-piece we cannot
consider its potential energy of strain to be precisely in the
form given by {10}, although by judicious working it tends
to approach thatform. The actual potential energy has pro-
bably a form such as that in (3), until it has been worked.
The effect of “working” simplifies the form of V in a
manner perhaps comparable with the algebraic discarding
of constants inconsistent with elastic equilibrium. If this
comparison is not unreasonable, we may venture to extend
the idea, aud to imagine that an excessive exertion of trac-
tion may again alter the form of the potential energy, and
introduce into V terms inconsistent with elastic equilibrium
and may, if continued, lead to rupture. At any rate, the
theory of rupture must lie outside the range of elastic equi-
librium, though not necessarily outside the elastic stress-
strain relations.

| 908 1

XLVI. On the Viscosity and Molecular Dimensions of Sul-
phur Diomde.” By C.J. Suive, B.Sc; ALkICsS, aes
Research Student, Imperial College of Science and Tech-
nology, London ”*.

ECENT work on the viscous properties of compounds
which are ordinarily gaseous having been successful
in elucidating the molecular structure of these compounds, it
was thought that it would be interesting to apply similar
methods in the case of sulphur dioxide, especially as. Lang-
muir f has already suggested a possible arrangement of the
atoms which constitute this particular molecule. This paper
describes the necessary viscosity measurements for sulphur
dioxide. Previously the data regarding the viscosity of
this vas were very scanty and did not extend over a sufficient
range of temperature to determine Sutherland’s constant—a’
factor of almost as great an importance as that of the viscosity
itself in determining the mean collision area of a molecule.

Apparatus and Method of Observation.

The apparatus and method of observation which have been
used for the purpose of measuring the viscosity of sulphur
dioxide have recently been fully described {.

Method of Experiment.

The mercury pellet, which was used to drive the gas through
a capillary tube which forms part of a complete circuit con-
sisting of this tube and a fall tube in which the pellet moves
between specified marks, is the same as that which was used
by the author in his experiments on carbon oxysulphide §.
The time of fall for air proved to be 105°53 secs. at 18°0° C.
With this time of fall the corresponding time of fall for
sulphur dioxide has been compared, and, with appropriate
corrections, this gives the relative viscosity of air and sulphur
dioxide. From this relative value the absolute viscosity has
been obtained by assuming that the viscosity of air at
18°:0 ©. is 1°814 x 1074C.G.8. units. In addition, the varia-
tion of viscosity with temperature has been derived from
comparisons of the corrected times of fall at atmospheric and
steam temperatures.

* Communicated by Prof. Rankine.

+ Langmuir, Journ. Amer. Chem. Soe. vol. xli. p. 868 (1919).

t A. O. Rankine and C. J. Smith, Phil. Mag. vol. xli. p. 601 (1921) ;
and C. J. Smith, Proc. Phys, Soc. vol. xxxiv. p. 155, June 1922.

§ C.J. Smith, Phil. Mag. vol. xliv. p. 289 (1922).

. See eee eS eo
> re

—

Molecular Dimensions of Sulphur Lnowide. 509
Preparation and Purification of the Sulphur Diowide.

The sulphur dioxide was generated by the action of dilute
sulphuric acid on sodium sulphite. It was dried by being
passed through several wash bottles containing concentrated
sulphuric acid, and then solidified in a U-tube surrounded by
liquid air. The U-tube was then cut off from the generating
apparatus, and all permanent gases removed by means of a
pump. The gas was made to evaporate by removing the

liquid air, and samples were collected over mercury. The.

chief difficulty in using this gas is to dry it sufficiently well
that the motion of the pellet of mercury in the fall tube shall
be smooth. It is difficult because sulphur dioxide boils at
—10°C., and at this temperature water has a small vapour-
pressure. It was finally purified and dried in the following
way, before introduction into the viscometer :—The gas was
solidified in a tube maintained at —80°C. by means of a
mixture of solid carbon dioxide and alcohol, and all per-
manent gases and possible traces of carbon dioxide removed
by means of a Toepler pump. Sufticient alcohol was then
added to the carbon dioxide mixture to raise the temperature
to —60°C. At this temperature water has a negligible
vapour-pressure, while sulphur dioxide is liquid and has an
appreciable vapour-pressure. This enabled successive small
quantities of the dry gas to be pumped off and introduced
into the viscometer, which was previously exhausted, until
the pressure therein was atmospheric.

Heperimental Results.

TABLE I.

Each time recorded in this table is the mean of four observations in each
direction for the whole pellet, and of three when the pellet is divided into two

segments. The letters in parentheses indicate the order in which observations.

were made.

| Time of fall (secs. }.. ‘Time at
| ‘Temp. Capillary | Corrected
| (deg, C.). | Whole | Two | correction. time | 18°:0C.|100°-0C.
| pellet. |segments.| (a). (7).
fe ee eee eS a EC
(a) trie se16 | 79°67 | 00422 72:95 73:03
|(6) 17°80 | 7648 | 80°38 | 00463 72°95 73°01
| (ce). 17°35.) 76°82 81:00 | 0:0491 73°05 73°09
(Ff) 17°38 | 7640 | 80°37 | 00471 72°80 72:97
| | Mean|...73°03
1(¢@)1000 | 97:10 | 99:32 | 0:0219 94:97 ai, 94°97
(e) 99°96 | 97°13 99°37 |. 0°0220 94°99 Sey 1} ye eee

| Mean ...94°99

o10 Molecular Dimensions of Sulphur Dioxide.

We have t)3= 73°03 sec., and t)9)= 94°99 sec.
The ratio of the viscosities at 18°-0 C. and 100°:0C. is
given by the ratio of these times ; thus
Mion 0p ee

ms tig 13°03
Assuming Sutherland’s law to hold over the range of
temperature used in these experiments, the value of Suther-
Jand’s constant is 416. The validity of Sutherland’s law for
this gas over the range of temperature investigated cannot
be expected to be great on account of the probable large
deviations from Boyle’s law which this gas may exhibit,
since the temperatures at which measurements have been
made are not very far removed from the boiling-point of
liquid sulphur dioxide. The value of C given, and subsequent
deductions depending thereon, should therefore be accepted
with some reserve.

al) le

oe ts0. __ 73°03 _0:f09
Also at 18°70 C., i = 10553 =() 6923,

Correcting for slip in the usual way, we obtain

Nair

Assuming that the viscosity of air at 18°0C. is 1°814~x
1074 C.G.S. units, the values for SO, are
Me 1°253 x 107-4 C.G.S. units,
Mion = L630 x 10%" CxG. 5. anits,
and by extrapolation, using Sutherland’s law,
no lose L0e. CaG.Sammits,

According to the usual works of reference and published
papers, Vogel * is the only modern worker on this subject,
and he found that m)=1:183 x 1074 C.G.S. units.

Calculation of Molecular Dimensions.

The above results enable us to calculate for sulphur dioxide
that mean area which is interpreted by Professor Rankine Tf
as the area which the molecule presents in mutual collision
with others. Chapman’st formula, modified in its interpreta-
ction, as indicated above, is ‘the basis of this calculation. The

* H. Vogel, Berlin Diss. p. 46, 1914.

+ Proc, Faraday Soe. vol. xvii. part 3 (1922).
{ Chapman, Phil. Trans. A. vol. cexvi. p. 347.

Simple Model to lllustrate Elastic Hysteresis. oll

value obtained is A=0°94 x 10~” cm.?, which may be subject
to an experimental error of 2 or 3 per cent. It is difficult to
estimate the degree of precision with which this figure
represents the real dimensions of the molecule. The mea-
surements of viscosity, owing to the comparatively small
temperature range over which they extend, provide no. proof
that, for this gas, Sutherland’s law holds. Indeed, as men-
tioned earlier, it is improbable that the sulphur dioxide in
the cireumstances of the experiments was sufficiently super-
heated to give the true value of Sutherland’s constant. It is
not unlikely that the actual mean collision area differs from
that calculated by an amount appreciably greater than that
attributable to experimental error.

Summary of Results.

TABLE TI.
| Viscosity in C.G.S. units x 1074, Mean col-
| Sutherland’s | lision area
18°00. 100°°0 C. 0°-0 0. constant. |(em.?x 10-15).
1:253 1°630 1168 . 416 | 0-94

In conclusion, the author would like to record his apprecia-
tion of the continued help and advice received from Professor
Rankine, and also to thank the Goverment Grant Committee
of the Royal Society for a grant which enabled the research
to be undertaken.

Imperial College of Science
and Technology, London, S.W.7.
10th June, 1922.

XLVI. On a Simple Model to Illustrate Elastic Hysteresis.
By 8. Luss, M.A., St. John’s College, Cambridge *.

; § 1. Introduction.
\ | UCH material has accumulated + in recent years con-

cerning the behaviour of metals when taken through
either a series of cycles of alternate compressions and
tensions, or a series of periodic shear stresses. In the
main, the experimental results here utilized are those of
* Communicated by the Author.
T See e. g. ‘ Dictionary of Applied Physics,’ yol. i. p. 178.

5 Mr. 8. Lees on a Simple Model

Messrs. Smith & Wedgwood, ‘Journal of Iron and Steel
Institute,’ vol. xci. p. 374. It must be admitted that in one
important respect, the results of these authors appear to
differ from those of other experimenters; in that, according
to the cited authors, elastic hysteresis would appear only to
make itself evident under certain conditions, whereas other
experimenters have been of opinion that elastic hysteresis *

always occurs with stress change, even with small range of -

stress. Without expressing any opinion on the existence or
not of elastic hysteresis with small ranges of stress, the
author has taken the results of Smith & Wedgwood, and
attempted to construct a simple model illustrating these
results. Whilst it is not contended that the model to be
described is the best possible, it does to some extent satisfy
a desire to reduce to simple mathematical treatment many
of the well-known elastic phenomena.

It may be here noted that the model described below
in § 3, and the modification of § 9, between them illustrate a
whole series of well-known phenomena, such as:

(j.) Existence of elastic hysteresis loops only under certain
conditions. 3

(ii.) Small variation of area of loop with speed of
description.

(iii.) General shape of loop.

(iv.) General character of mean-stress-strain loops obtained
when slow speed periodic stress is combined with
rapidly alternating stress.

(v.) Existence of two points in loop at which (limits of
Hooke’s law) the loop commences to depart from
the straight line law.

(vi.) The production of permanent set in a material and

the existence of a true elastic limit beyond which

elastic recovery 1s infpossible.

(vii.) The existence of a yield point.

(vii.) The general relationship between elastic hysteresis
and the three conditions ‘referred to in (v.), (vi.),
and (vil.)). |

(ix.) The phenomena of slip bands.
(x.) The effect of overstrain on the two points of (v.).

So far as the author knows, such a model with its illustra-
tions as above, is original.

Before Aeecoib in the model, some reasons for ruling out
ordinary viscous fluid effects as the predominating cause of
elastic hysteresis will be given.

* Diminishing, of course, with the range of stress.

to Lllustrate. Klastic Hysteresis. 513
Ȥ 2. Discussion of the Problem.

In connexion with elastic hysteresis, it seems natural. to
invoke viscous effects analogous to those found in viscous
fluids. A very obvious idea. is to introduce fora material
undergoing cyclical variations of stress, a stress term always
depending upon the rate of change of the corresponding
strain. Thus, e.g., if f denote stress, s denote the corre-
sponding strain, we may try

ji Ks As) Wace (1)
where K and XY are constants. For the type of stress con-

sidered, K would be the ordinary modulus of elasticity. If
we now make s go through a cycle given by (¢=time)

Sr ESHCOS Doth tad ati sis. CD)
we get different values of f which can be plotted against s,
giving rise to a stress-strain loop*. This loop is clearly an

Fig. 1.

ellipse, asin fig. 1. For on eliminating ¢ between

f= s(K cos pi—Apsin pi) ... - . (8)
and (2), an ellipse arises. It is not difficult to verify that
the area of the ellipse is proportional to both p and s).

Thus with such an assumption, the area of the loop, for a
given 8, will diminish indefinitely as the speed of fluctuation

* In plotting stress-strain loops, we can always make the straight line
f= Ks have any slope we please by suitably choosing the scale for
fands; but in general this will give us a loop of minute proportions,
for breadth. Having chosen a suitable slope for the above line, the loop
can be magnified by representing the divergence ( f—Ks), of a point of
the loop from the straight line corresponding to no hysteresis, on a scale
any number of times that of f. This will be assumed to have been done
for the stress-strain diagrams shown in this paper.

Phil. Mag. 8.°6. Vol. 44. No. 261. Sept. 1922. 2 L

514 Mr. 8. Lees on a Simple Model

is caused to diminish indefinitely. But this is quite contrary
to the facts obtained by experiment.
The results obtained by using a formula * of the type

f= Kets + pets + cc, et ae eee

are quite analogous. On putting s=s)cos pt as before, we
shall get

Sf = 5 {cos pt (K—pp?+ ....)—psin pt (A—vp?+ ....)3, (5)

and on plotting f against s, we shall again get an ellipse.
Further, whilst the area of the ellipse will not follow quite
so simple a law as before, it will easily be seen that as p is
made to approach zero as limit, the area (for a given 59) will
do likewise. This, again, is contrary to experience.

It may be said, in passing, that elastic hysteresis loops are
not found in exact experiment to be ellipses at all. A nearer
approximation to actual shapes can be found by assuming
the loops to be lenticular. Such a shape can be got, e.g. by
taking

fH=Kst BF, 003. Co

where the sign of the 8? term is always so to be taken as to.

make the frictional stress term oppose the change in strain.
ic y2

Taking s = s) cos pt as before, it will be found that we get a
diagram which is the result of eliminating ¢ between this
relation and

f = Ks cos pi bsp" sin? pts.) 2) 1
If the loop be considered to be described in the clockwise sense,
the plus sign will be taken from t=—7/2p tot=a/2p. The
loop is then seen to be lenticular, as in fig. 2. Here, again,

* See Maxwell, ‘Collected Papers,’ vol. ii. p. 628.

to Illustrate Elastic Hysteresis. o15

the area of the loop vanishes with p, and so violates the
facts. A combination of formule (4) and (6) will suffer a
like disability.

In view of the undoubted fact that when the strain range
is large enough, hysteresis exists, whether the speed of
describing the “eycle of strain change be quick or slow, the
author has felt it necessary to fall back on the rather primi-
tive notions of solid friction. It is not contended that the
ordinary ideas of fluid friction do not enter into the produc-

tion of elastic hysteresis at high speeds, but it is asserted »

that with low speeds of describing a cyclical Guana? fluid
friction effects are negligible.

The author has found it possible, using the notions of solid
friction, to get results in good qualitative agreement with
many of the facts, and these notions Spelt mathematical
analysis of a simple character to be applied. The chief
factor in solid friction that is made use of in this paper is
the property according to which there is a marked discon-
tinuity between the limiting tangential force just required
to produce sliding (of one surface over another pressed
against it) in one direction as compared with the opposite
direction. If the tangential force available lies between
these limits, no slipping takes place. In fluid friction, as
usually understood, the frictional force will vary continuously
as the direction of relative motion or sliding is altered, and
at the instant at which the two surfaces have no relative
motion, the tangential force exerted by one on the other will
momentarily vanish. This behaviour is totally different to
that obtained with the agency of solid friction.

§ 3. Description of a Simple Type of Model.

In the model now to be described, the author conceives
that in a metal under stress there are groups of molecules
(or possibly crystals) which in some way are capable of
recelving and transmitting directly a portion of the stress
applied. There are also other groups of molecules which
may take up a portion of the stress, but this portion is deter-
mined by considerations of solid friction. The solid friction
is supposed to arise owing to the pressure of the first set of
groups acting normally on the second set. ‘This pressure
may be considered as molecular in origin, and of the same
character as cohesive effects. Such pressures will doubtless
be large, and as a first approximation we may suppose that
this pressure is not materially altered even if slipping takes
place between members of the two sets of groups. It may

212

516 Mr. 8. Lees on a Semple Model

further tentatively be assumed that whether or not slipping
takes place, the two sets of groups will behave elastically, so
far as each set is concerned. These points are involved in
the model shown in fig. 3. In this diagram, A, and A,
represent two groups of molecules of the first kind referred
to. A,and A, are directly connected up to the agency pro-
ducing the stress (represented in the diagram by forces
I’, F), and are shown joined together elastically by a spring
(marked with tension T, in the diagram). As typical of the
groups of the second kind referred to, we have the items
B,, B,, Cy, and C,. Here B, and C, are between them
squeezing A, with a pressure N. B, and C, are performing

Fig. 3.

Springs to produce
compressio7 VV

a like office for A,. B, and B, are shown joined together
elastically by a spring (represented as being in tension T,).
C, and C, are similarly connected together by a spring (also
shown in tension T,). The diagram is intended to represent
a state of affairs such that when F is zero, the tensions T,
and the tension T, all vanish. Such a state of affairs
may be called the neutral state. For such a state, the dis-
placements of A, B,, and C, from their equilibrium positions
are all equal, and (measured with OO, taken as unaffected)
may be denoted by w. Similar remarks apply to A», Bg, and
C,. Since we shall have under these circumstances

MN, = AL, MM = Now, e ° . e e (8)
where A, and A, are appropriate elastic constants, it follows
that

= (A, at 2X2) a. siueiie tal tte let hagite ° (9)

to Lilustrate Elastic Hysteresis. 517

Thus regarding F as representing externally applied stress,
and w as the corresponding strain, we get the usual Hooke’s
law holding for stresses which do not disturb the neutral
state. This remark, of course, holds for either tension or
compression.

If w denote the coefficient of solid friction between A, and
both B, and C,, the same value holding for A, and both B,
and Cp, slipping will take place when

a OLENA Lat solr iweaea” ab ae vO)
This will correspond to a value of a given by
pens Fis NG

Thus the value of F at which the linear proportional relation-
ship between F and z breaks down will be

2
F = 0,42a4)2 = Cot 2h) py, ah (Gay

2

if F be increased beyond this limit, slipping will take
place, of amount (say) y, between A, and B, and C,; also
(see fig. 3a) between A, and B, and O,. The tensions T,
will remain constant during such an increase of F, and we
shall now have |

F = Ayw+ 2uN = Aye + 2r(av—y). ° ° (12)

This, though a linear relationship between F and #, has a
different slope from (9), and F is no longer proportional to w.

318 Mr. S. Lees on a Simple Model
§ 4. Hlastic Hysteresis Loop for Model.

After slipping has occurred as just explained, the moment F
is caused to diminish, slipping will cease, and A,, B,, and Q,
(also A,, B,, and C.) will move together. Thus if Fj, 2, be
the values of F and 2 at the instant that F is caused to
decrease, we shall have at first

F,—F =, 42))\Geyee)) (ee

During any change of the type given by (13), there will
be a constant amount of slip (reckoned from the neutral
state) given by
ie) 2 ee
YN a ons oe Uy SOLS ° e (14)

It will be noticed that during the change given by (13), the
slope of the I’, # curve is exactly the same as that given
by (9), 2. e. as that for the neutral condition, also T, is now
at any instant given by

T,—pN = Ao 4% — 21), ° . ; ° (15)

since the slip does not alter, and dT./dx = 2g. |

As F goes on diminishing, ultimately becoming a force of
compression, a time will arrive when slipping will once more
occur. This will clearly be the case when

To —yN, 2.
or from (15), when
p=— (7 —«,). al laPa a at Chi}
Ae

This corresponds to a value of F' given by
F = ye 2uN 2a (= +1) Mee
2

This result may be compared with (11). It will be noticed
that the value of F given by (18) is not in general the nega-
tive of that given by (11). It is easy to see that further
compression beyond the value given by (16) will result in a
straight line law for F and a, such that

ee) Make aw 4 a its) he

For such further compressions, of course, sliding takes place.
If now the compression be gradually diminished, 2. e. F be

to Illustrate Elastie Hysteresis. 519

increased algebraically, we shall again go through a series of
operations in which first

dk

: ae Neanane ane (aieisu by (20)
after which (17) will hold.

By performing such reversals of stress, between limits
+F,, sufficient to cause slipping to occur in both directions,
we can ultimately reach a cyclical condition; the stress-
strain loop, 2. e. the F—w loop, being as shown in fig. 4.

Fig. 4,

- Xp7F)

The diagram is such that the limits of F are +F), the
corresponding limits for # being +2}.

The typical feature of the cyclic condition is, of course,
that the F—.z loop is symmetrical with respect to the
origin. In our case, the diagram is a _ parallelogram
JKLM, with dF/dx=X, for the lines KL and MJ, whilst
d F/d«=(d,+2A,) for the lines JK and LM.

To get the F—-2 coordinates of K, the intersection of the
lines JK and KL, we have for the line JK 3

F—(—F)) = (A, + 2A) [a—(—2,) }, ae (21)
whilst for the line KL we have
FL—-F = Ay4(a,—2). ee an) aM Baer (22)
These two equations give for the point K :—

ESI 220 00 (
rg
ee pts (24)

«

\
|
j

520. Mr. 8. Lees on a Simple Model

For the point M, the values of x and F are clearly the eee
tives of (23) and (24) respectively.

The value of the maximum slip (measured from ‘the
neutral state) can easily be obtained, for this must vary
between (say) y; and —y,. The range of slip is accordingly
2y,, and this range of slip is incurred during the description
of both the lines KL and MJ. Thus 2y, must equal the

difference between the values of # at Land K, ?.e.

Bed us Az)@ A, +2As)e—Fy (25)
2 Ae

This result can also be obtained from equations (12).

Yet another expression can be obtained by remembering
that at the point L, where the slip is ¥;, we must have
from (12)

2y; = 14 —

Ao(a1— 71) = WN.
Substituting the value of y given by (25), we get *
Fy— 2A, = 2yN. ue (26)

§ 5. Area of Hysteresis Loop.

The area of the hysteresis loop can now be obtained. Itis
clearly twice the area of the triangle JKL, the coordinates
(2, F) of the vertices of which are respectively (—2a#, —F)),

Ce NE, aa ae Se

No Ae
and (a, Fy). The area + is therefore |
ae ee
Fy (Ay # Ae) 21 FiOg +2) —Mi(Ay + 2Ap) ay s

Bik hii tu ler Voaumaa. GT
Ag re

zy ) 7 F, ah | {tall

2(F,—ayn gg

ws ane Doig cy Fy] = 8¢#Ny1, + (27)

on using (25) and (26).

- This result might also have bese obtained by remarking
that the work lost in a complete cycle can be accounted for
as due to a force of friction 24N overcome twice (during

* When hysteresis exists, the locus of points L is therefore a straight
line.
+ The negative sign is used so as to make the area positive.

to Illustrate Elastic Hysteresis. 521

the stages KL and MJ) through a slip on each occasion of
amount 27.
The area of the hysteresis loop clearly vanishes when
y,= 0,7. e. when

F, = Oy a 2). . . . e ° (28)

Using (26), we get for the corresponding limiting values of
&y and F, ——

pN
XY = ies

" 1 29)
Fi= NOY +- 2r2) 1 ; } ? 5

Xe

The limiting loop for which the area just vanishes is shown
in fig. 4 as the straight line PON. It is easy to verify
from. (23) and (24) that N is the middle point of KL, and
similarly that P is the middle point of JM *.

It is to be understood that for values of 2, aad F, less than
those given by (29), the F'—w relationship is always ex-
pressed by a portion of the straight line NP, whose slope
is the same as that of both JK and LM, and is that cor-
responding to the neutral state. This statement arises
from the fact that for such values of # and F, there will be
no sli

F ae (26) and (27) we can get two oe expressions for
the area of the hysteresis loop (when it exists) in the forms:

N\ _ 8uN N(Qro+2
uN (x.— BA) = KS [pe ). . (30)

From these we see that the area has a value proportional] to
the excess of 2, over the limiting value just referred to;
alternatively, it is proportional to the excess of He over. its
corresponding limiting value.

§ 6. Comparison of Results obtained with Experiment.

A large amount of data is available concerning elastic
hysteresis, but, for the moment, reference will only be made
to the paper by Smith & Wedgwood, loc. cit. figs. 2-5. The
static stress-strain loops obtained by these authors for tension
and compression of a material in the cyclic state agree in
several particulars with the theory just outlined. Thus
elastic hysteresis does not occur till the limits of stress

* To get the model to the neutral state from the cyclical state,
we arrange for it te be put into either of the conditions corresponding
to the points N or P, and then take off the load F.

522 Mr. 8. Lees on a Simple Model

exceed numerically a certain amount, and when this does
occur, the area is proportional, for small-sized loops, to the
excess of I, over the critical amount. Also, immediately
after a change in sign of di'/dt, where F represents stress
and ¢ the time, the stress-strain curve is always straight and
parallel to the ‘strai ght line which represents. elastic change
without hysteresis.

On the other hand, the author’s theory fails to explain two
noticeable points indicated in Messrs. Smith & Wedgwood’s

aper :—

. (1) When a point corresponding to K in fig. 4 has been
reached, the stress-strain curve is actually found to become

Fig. 5.

curved, as shown in fig. 5, instead of following a straight
line like KL of fig. 4.

(2) For large areas of loop, it is actually found that the
proportionality between area of loop and excess of F over
the critical amount above referred to breaks down.

Other divergences between experiment and the theory at
present outlined will be indicated below (§ 9).

§ 7. Steady Hysteresis Loop for Unsymmetrical Stress Limits.

We shall now discuss the nature of the steady hysteresis
loop when the limits of stress are not equal and opposite.
The appropriate stress-strain (in our case, '—2) diagram 1s
indicated in figs.6 and 6a. From the considerations out-
lined in § 4, it will readily be seen that just previous to F
reaching its lowest value (as at J), the F—vw relation will be
(on the theory outlined) dF/da=X. The corresponding line

to Illustrate Elastic Hysteresis. 523

in the diagram is MJ. . After F has begun to increase, the
F—zx relationship will correspond to the straight line J, K,,
whose slope is given by dF /dzv=2,+ 22. It will readily be
seen that the steady loop ultimately described will be a
parallelogram like J,K,L,M, of figs. 6 or 6a, the .slopes

Fig. 6.

of whose sides have been indicated, but the corners of
which have to be found. Referring to the model shown
in fig. 3, it will be understood that in describing the loop,
the slip y will change in one direction and the other alter-
nately. When slip is occurring with increase of F, the

524 Mr. 8. Lees on a Simple Model

H—z relation will always be given by a portion of the line
KL of fig. 4. Similarly when slip occurs with diminution
of F, the F—z relation is given by a portion of the line MJ.
The actual loop for unsymmetrical stress limits will therefore
be obtained by choosing the points L, and K, on these lines
respectively, so that the stress at L, is Ei+F, and at J, 1s
F)— F,, where F)+ F, are the unsymmetrical stress limits.

It will readily be perceived from the diagram that the
area of the loop will only depend ‘on the range of variation
of stress, 7.e. on F; ; and fora given value of I, the diagram
is got by taking F,=0 (i.e. for the cyclic condition), and
displacing it parallel to the line KL of fig. 4 until the stress
limits come right.

In particular, the results of §§ 4 and 5 for the area of the
loop in terms of F will hold, provided we interpret IF’, as
being one-half the greatest variation in stress,

We may conveniently term the state of affairs here referred
to the asymmetric cyclic state.

For such a state we may briefly indicate the results
involved.

_ From equation (29), provided our assumptions hold, we
see that for no hysteresis to occur, }' must always lie
between

FEN Qu hs) ee
2
where, of course, F, may have any fixed value.
When this condition (31) is not satisfied, the area of the
loop is given by (30), where F, has the meaning of half
the greatest variation of stress.

§ 8. Particular Case of Fo= Fj.

If the lower limit of stress for the asymmetric cyclic con-
dition be zero, we get an interesting case. We have here
to take Fy>=F,, and hence Fyay,=2F,. From (31) we see
that the greatest value of F for no hysteresis to occur will
accordingly be

2MN(Ay oP 2r2)/Ao- o Misre VT (32)

This is exactly double the maximum stress for the cyclic

condition, which just fails to produce hysteresis.

Tf the stress varies between O and F, and hysteresis does
occur, equation (30) shows that the area of the steady loop
will be given by

AuwN 2uN(2r,.+Az)
<|F- i. i eS

It is thus proportional to the excess of F over the value (32).

to Illustrate Elastic Hysteresis. D925
§9. Effect of Rapid Periodic Changes of Stress combined

with Slow Variation of Mean Stress.

If in $7, we imagine that F) changes slowly and _ periodi-
cally whilst the F—- loop is being rapidly described under
the influence of a constant Fy, we get an important case in
experimental work. In such a case, we may imagine that
the variation in Fy is so slow and the periodic change of
superimposed stress (between limits +f) so rapid, that at
any instant the material is always in the asymmetric cyclic
state. It is interesting to discuss the loop obtained (if any),
by plotting the different values of I, against the corre-
sponding mean values of «x for the rapidly described
hysteresis loops. We may call such a diagram a mean-
stress-strain loop (see Smith & Wedgwood, loc. cit. p. 318).

There are three cases to consider :—

(a) If the limits of stress F)+F,, Fo—Fy, are such that
the corresponding points in the F—wz diagram lie between
the lines KL and MJ of fig. 4, then the mean-stress-strain

Fi

a. 7.
F L

loop reduces to astraight line. This straight line will either
be a portion of the line PON, or a portion of some line
parallel thereto. For the neutral condition, the points
P, and N, which limit this type of mean-stress-strain curve,
will clearly have values of I) given by minus and plus
respectively an amount given by subtracting the amplitude
of F, from the value of F at N (see fig. 7).

(b) Using | F,| and | F,| as amplitudes or positive limiting

values, we may still consider the case of |F,! less than the

526 - Mr. §. Lees on a Simple Model

value of F corresponding to the point N ; call this value Fy
(given by equation (11)). Keeping |F,| constant and slowly
allowing F, to increase from zero, we ultimately get (on the
mean-stress-strain curve) to the point Nj, corresponding to
the maximum stress Fy. Further increase of the maximum
stress can only be obtained with slip, and such maximum
stresses must correspond to points lying on the line KL.
During the rapid variation of F,, whenever the F—« point
on the diagram leaves the line KL, it will do so to travel
along a line (like RS) parallel to PON, and must return to
the line KL at the same point that it left it (since during this
travel there is no slip). Hence the line of the mean-stress-
strain curve corresponding to the description of the line KL
will be Kgl, a line parallel to KL (see again fig. 7). The
complete mean-stress-strain diagram for the complete cyclic
variation of I, is therefore given by J,K,L,M,. It will be -
noticed that it is of similar type to the static stress-strain
loop of fig. 4, 2. e. its sides are parallel to the sides of the
parallelogram JKLM of fig. 4.

If we draw in the static hysteresis loop J;K3;L3M; cor-
responding to the range +(|F,| + |Fi|), we see at once
that the area of the mean-stress-strain loop J,K,L,M, is
(Fy—|F,|)/ Fn times that of J;K,L;M;3, and is therefore
by (380) given by

. Cee | (Fol+1 Fil) — aos |:
(34)

(c) We lastly consider the case of | F,| greater than Fy.
In this case, the rapid variations in | F,| will always cause
the F—z2 or state point of the model to move from the line
KL to the line MJ (or vice versa). A point on the mean-
stress-strain curve is therefore always to be regarded as the
geometrical centre (or centre of gravity) of a loop in
the form of a parallelogram like J'K’L'M’ of fig. 7a, such
loop corresponding to some asymmetric cyclic condition.
The locus of such centre of area is clearly a straight line
UOV, passing through O and lying parallel to KL or JM.
In such a case, therefore, the area of the mean-stress-strain
loop vanishes. For a given |Fo| and |F,|, it is easily
verified that the x coordinate in the diagram (fig. 7a) of
the end point V of this line is given by | F,|/2,, the corre-
sponding value of F being | F9|.

We may compare these results for mean-stress-strain loops .
with those given by Smith & Wedgwood (loe. cit.).

to Illustrate Elastic Hysteresis. 527

The two sets of results are in many ways in agreement.
Smith & Wedgwood found that, provided the amplitude of F,

was not too great, the mean-stress-strain curve was a straight

Fig. 7 a.

line for values of | F,| which did not exceed certain limits
depending on |F,|. These limits were sharply defined,
and when |F,,| was caused to go outside these limits, a
mean-stress-strain loop was formed. A typical diagram is
shown in fig. 76. Such a loop, which increased in size with

528 Mr. 8. Lees on a Simple Model

inerease of | Fo], was generally quite similar-in character to
the static stress-strain loops referred to in § 6. ,

The limits. just referred to correspond to P, and N, , of our
fig. 7. If we take the static elastic limits for zero hysteresis
loops given by Smith & Wedgwood’s fig. 14 as 9°5 tons
per in.’, and consider the case indicated in the same authors’
fig. 15, wiz. | #;| = 7°65, we ought by (a) of this section to
get for the point N, of our fig. 7 a value of

0-0) — oo) toms Mer dina:
The actual value indicated in Smith & Wedgwood’s fig. 15

would appear to be about 1°6 tons per in.’.

A difficulty arises in connexion with the differences in the
stress limits for zero hysteresis loops as shown in Smith &
Wedgwood’s figs. 2 and 14. The two results of these figures
can be to some extent reconciled by assuming that repeated
slip causes some change in the value of mw in the model,
e.g. by temperature change, etc. Ultimately, of course, the
explanation is molecular, but the fact that rest causes the
material to go from a state corresponding to Smith &
Wedgwood’ s fig. 2 toa state corresponding to their fig. 14,
is a good reason for taking this simple explanation.

There are certain outstanding differences between the
results obtained from the model of § 3, and the actual results
of Smith & Wedgwood. The first is that the mean-stress-
strain loop is not a parallelogram in actual experiment.
This has been already noted. Further, when the loop does
not reduce to a straight line, it is found experimentally that
for a given | F,|, variation of | F)| causes the points of maxi-
mum F', (corresponding also to points of maximum strain)
to lie on a curve like N’‘L’ of fig. 7b. According to the
present theory, this curve should be a straight line (parallel
to KL of fig. 4).

In an attempt to meet the objections just raised to the
model of § 3, and those mentioned at the end of § 6,
the author now puts forward a slightly modified theory
and model.

§ 10. Modified Model.

Fig. 8:shows a modification of the model described in § 3.
It will be noticed that the contacts for A, with C, and B,

are point (or line) contacts ; similarly for A, with C, and B,.

The rubbing surfaces of A, and A, are now taken as curved, ,
instead of straight (or plane).

to Illustrate Elastic Hysteresis. 529

Fig. 8 shows the model in a configuration corresponding
to the neutral condition, whilst fig. 9 shows a state of

eo, S,

affairs when slip y has taken place. Fig. 10 gives an
enlarged view of one of the two rubbing surfaces of <A.
It will be seen that the rubbing surface is taken as syin-
metrical about an axis Oz perpendicular to the direction of

Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2M

530 Mr. S. Lees on a Simple Model

application of the force F, the origin O corresponding to
the point (or line) of contact for the neutral condition.
Any equation for the surface (assuming Oy to be the
tangent at O) must give z as an even function of y. To
simplify the algebra, we shall assume that

CS OY Ee ere

where a is constant and small. Thus for such displacements
as we have to consider, the slope of the rubbing surface for
any given y is given by
dz |
ene em ne 35 (8S)
which is also small. 3

Referring back to fig. 8, it will be noticed that the tensions
(or compressions) T, are drawn so as to pass through the
corresponding points of rubbing. This is done so as to
avoid the consideration of tilting effects of the tensions or
compressions T, on the pieces By, By, C;, Cz. Tf such tilting
effects are taken into account, it is easy to see that even with
no slipping taking place, the F—vw relationship will not be
exactly linear, but F will involve small terms in wz? and
higher powers of 2.

Although the magnitudes of z to be considered are small,
it is conceivable that they will exert appreciable influence on
the pressure N between (say) A, and By, normal to the direc-
tion of F. We shall accordingly assume a linear relation-
ship between N and ¢, and take

N = N,—az = N.—By’, ) (31)
where « and @ are constants, and 8 = 2aa. }

For the general case of slip, the value of T, will be given
by +T,= Ntan(¢+@), with » = tan @.

If we make the further assumption that p is so small *
that tan @ is negligible compared with unity, we get the
following expressions for T, :—

a and T, positive, T,=(N,—8y’)(u—2ay), - (38)

“ and Ty negative, T= —(No—Ay*)(u + 2ay). (89)

* This is justifiable if the cohesive forces in metals have the large
values usually attributed to them as compared with the stresses here
dealt with.

to Illustrate Elastic Hysteresis. ddl

Starting with the model in the condition corresponding
to the neutral state, and applying a gradually increasing
force F, we have at first

Peace eater ty a.) (9)
exactly as in § 3. Slip movement will take place when
Not — PN, = TS e ° e - . (40)

After this, we shall have for further increased values of F':
T, = A(a—y) = (No—By?)(u—2ay). . . (41)
The relation between y and w is therefore the cubic equation

mM (Ay — 2aN,) (Apw— No) _
y"— oat + 2a8 EO Chae aan De Nee)
Lf we consider such displacements that (Ny—#y’) is always
positive, it is physically obvious that there must be a real
positive root of y for any given value of # (which value must
of course be such that >A»v>N uw). The real positive value
of y can also be shown to exist by putting successively y=0
and y= in the left-hand side of (42), when there appears
a change of sign. Bie
As y goes on increasing, a time will come when T, will
vanish. This, from (41), will clearly be the case when

y=p/2a, and P= wp/2a ... . (48)

Further increase of y will result in T, becoming negative,
i.e. becoming a compression. If after this bas occurred,
F is caused to diminish, ultimately becoming a compression,
it is easily seen that T, will always remain negative, as
friction and slope @ will now be assisting each other. No
matter what compression may correspond to 'T’,, y cannot he
caused to diminish. Thus the value of y given in (43) really
corresponds to the beginning of permanent set. For values
of F beyond that given by (43), complete recovery cannot
be afterwards attained. We may therefore call this value of
F the elastic limit. It is to be distinguished from the value
of F obtained from (9) and (40), viz.

he (Ay + Zn») UN, ,
Xo

This latter limit corresponds to the limit of Hooke’s law.
For values of F less than this, Hooke’s law will he ac-
curately followed, and there will not be any hysteresis
effects.
If F be increased continually beyond the value given by
2M 2

(44)

532 Mr. 8S. Lees on a Simple Model

(43), not only will T, become negative, but dT./da will
become negative and numerically increase with @ or y.
This may be seen at once physically, or may be verified by
the algebra of (41) and (42). Thus a time will in general

arrive when dI',/da= —4d,, and when this is the case,
dec an ann

This would correspond to yield point, as ordinarily under-
stood in the testing of materials.

With regard to the term (No—y’) which has been fre-_
quently utilized in this section, we can suggest a rough
interpretation. It is essentially to be regarded as a force
of cohesion. Hence the value y = N,/8 must be regarded
as a slip sufficient to break down this cohesion. Thus it
may roughly be identified with the slip producing /racture
or failure. In this connexion, it should be borne in mind
that the model is only a rough representation of what is
essentially a statistical problem, involving a multiplicity of
surfaces, whose normals and corresponding cohesive forces
act in all directions.

§ 11. Cyclic State for Modified Model.
Teliee., UL.

We proceed to discuss the nature of the hysteresis loop
for the model when in the cyclic condition. The appropriate
diagram is shown in fig. 11. The points L and J refer to

to Illustrate Elastic Hysteresis. 533

the states (Fj, x), (—F,, —.x,) respectively ; whilst the cor-
responding slips are y, and —y, respectively. We shall
suppose that y; is less than y/2a, so that the cyclic condition
is possible. By (42), we shall have

ral: — 2aNo) (Agvi—PNo) _
yy — aut Soper are : 2u8 —— 0 (45)

F, is then given by
BE, = Aye} + 2ro (vy — 71) = (Ay ue 2p) ty = 2roY1- (46)

Now imagine F diminished, corresponding to the path LM.
At first, we shall have

a Se asdeswele en (20)

and this will go on till slip occurs at M. If the value of «
at M be ay, the value of T, there will be

(Ts)ar= (No— By2”) (4 2ay1) —Ae(4i1—2m), - (47)

which must be a compression sufficient to cause slip. Hence
also

(Ts) = (No— Ay?) (w+ 2a)... (48)
From (47) and (48) we get
Ao(@:—em) = w(No—By’). - « + (49)

For a given a, (45) determines y;, and then (46) gives Fy.
From (49) we can find xy, and the appropriate value of EF at
M is easily obtained, since the point lL and the slope of LM
are both known. The points J and K are got by symmetry
from L and M respectively.

To get the shape of the curve KL (and therefore by sym-
metry JM), let at any point of the curve the slip be y._ Then
(42) holds and gives y in terms of 2, say at Also

hence on eliminating x between f(x) and I’, we get the
required '—w relation for the curve KL. Without going
into the algebra, it is quite clear physically that the shape
must be something like that shown.

The area of the. complete loop is most easily obtained by
observing that energy is lost on the whole, only during the

534 Mr. 8. Lees on a Sumple Model
process of slip. Along KL the loss by friction is therefore

Yi yy
2(" tty = 2)" By —Rady - GL)
a aya!

Y1

whilst along the path MJ, the loss is

Twi Tu

2 Tody =— 24 (N,—By?)(wt+2ay)dy. (52)
4 yn

By changing y into (say) —u in (52), we see that (51) and

(52) give the same result, as might be expected. Hence the

total loss by friction in describing the loop is

Yi :
a4? Ay? \(u—2ay)dy = Bun (Me— Fue). (63)
WN

This is the required area of the loop. It clearly vanishes
when the slip y; vanishes.

When the slip y; is small, a first approximation for yj is
from (45),

= (Agv,— No) / (Ag —ZaNo). 5 ‘ 3 (54)
A nearer approximation is therefore

le (Apt —pNo)?
J a8 No = 2aNo av HB (Ag == 2aNo)° 7 ; : (55)

Hence from (53) and (55), as a, and y, increase, the area of
the loop increases at a greater rate than (Asv,—pNo). This
result may be compared with the results of $5, and the
comments in § 6.

§12. Asymmetric Cyclic condition for Modified Model.

We can work out the case of a hysteresis loop produced
with unsymmetrical stress limits, following the procedure
of $7. A typical resulting loop is shown in fig. 12, which
shows a case with the mean stress Fy positive. The point L
will still lie on the dotted curve NL of fig. 11, whilst the
point J will also lie on the dotted curve JP of the same ©
figure. The new curves KL and MJ will not, however, be
symmetrical about the origin O. As FE, is increased, the
point J will ultimately coincide with P, and further increase
of F will result in J lying on the curve PT, which is the
continuation to the right of P, of the curve JP. A limiting
case of elastic hysteresis will arise when Fy is increased to
such an extent that JK just touches the curve JPT. This

to Illustrate Elastic Hysteresis. 535

will, of course, correspond to an amount of slip given by
equation (43). Further increase of Fy will clearly involve
permanent set.

In the above argument, it is supposed that the variations
of F) have been made with the range 2| f,| of the alternating
stress kept constant.

Fig. 12.
F

The expression for a typical area of loop in the fgeneral
case is complicated, and is not given here.

For small slips y, it is clear that the results of calculation
for our original model can be used.

§ 13. Mean-stress-strain Loop for Modified Model.

Following the argument of §9, we can consider for the
modified model the effects of superposing a periodic slow
variation of mean stress on a rapidly alternating stress.

There are three cases to consider —

(a) Starting from the neutral condition, provided | Fo|
and | I,| are small enough, the mean-stress-strain curve will
be a portion of the straight line PON of fig. 11. The
limiting points of this line will be given exactly as in
(a) of § 9. :

(6) With | F,| less than Fy, but | Fo} + |F,|> Fn, we shall
get hysteresis loops of the mean-stress-strain type arising.
Referring to fig. 13, a typical loop like J,K,L.M, will
consist of lines J.K, and LM, parallel to the straight line
PON, and two curves K,N,L, and M,PiJ>. The curve
K,N,L, is clearly obtained as the locus of points G@ such
that GR drawn upwards and parallel to PON from G to R
(a point on the curve NL) is exactly equal to |F,|. The

BAG ie): Mr. §. Lees on a Simple Model

curve M,P,J », which is symmetrical with K,N,L, about the
origin, is similarly obtained from the curve PJ (compare

with fig. 7b) *.

Fig, 13.

(c) With |F,|>Fy, we get hysteresis loops whatever be
the value of ||. The mean-stress-strain-curve, 7. e. the
locus of the point given by the mean of the extreme stresses
and the mean of the extreme strains for an instantaneous
loop with the given | F',| corresponding to any values of Fo,
is therefere a curve like UOV, of limited length (see fig. 12).
Provided permanent set does not take place, the Jength of
UOV will be determined by both of the values of Fol and
|F';|, and not merely by |Fo|, as in §9 (c). A limitation is
set up to the value of |F)|+/F,| by the restriction that no
permanent set shall take place. The algebraic details are
not given here.

§ 14. Application of Model to Shear Stresses.

The model of §3 and its modification of §10 can also
be used to illustrate elastic hysteresis for shear stresses.
Referring, e.g. to fig. 3, we have only to imagine the
force F now applied to B, instead of A,;, and the equal and
opposite force applied to C, instead of As, to get a model
showing shearing action. Weare here neglecting the con-
sequent tilting of the portions of the model ; this tilting may
be avoided by supposing the component parts to be con-
strained to move parallel to the forces F by suitable friction-
less guides. Whilst it is not intended that any comparison

* Notice, however, that our theory would make the dotted line N’L’
of fig. 76 always coincide with a curve of the mean-stress-strain
diagram, which apparently is not the case.

a5

to Illustrate Klastic Hysteresis. 537

between the cases of direct and shear stresses shall be made
numerically, it is quite clear that such a model will lead to
exactly the same type of results in the two cases. A com-
parison of the preceding theory with the results of Mr. F.
H. Rowett’s experiments is satisfactory in many ways.
These experiments dealt solely with shear stress hysteresis
(see Proc. Roy. Soc. A. vol. Ixxxix. p. 528 et seq.).

$15. Time Effects.

It is quite clear that the model and its accompanying
theory do not explain what may be called teme effects in
elastic solids. By this phrase is meant here the changes
which take place in a solid with lapse of time, with or
without the application of external stress (kept coastant).
Some account can be rendered of these effects by assuming
that when slip takes place, the frictional force of slipping is
not a constant as in the model of § 3, or a function of the
slip as in the modification of § 10, but depends also on
the velocity of slip, not on the rate of change of the strain.
The difficulties arising out of time effects have been avoided
in the above discussions as far as possible. They are re-
garded as arising out of change of mw with velocity. ‘This
“coefficient of friction” will doubtless depend not only on
the velocity of rubbing, but also on temperature, which in
turn is a function of the number of hysteresis cycles
described, ete. In this connexion, it should be observed
that the introduction of w into the argument of the paper
is essentially an artifice which brings out the striking
analogy which seems to hold between the every-day phe-
nomena of solid friction, and the more subtle friction
obtained in elastic hysteresis experiments.

Conclusion.

In this paper the author has described.a simple model
which would seem in many ways to bring out, at least quali-
tatively, many of the general effects observed in connexion
with elastic hysteresis. The ultimate standpoint is that such
effects are due to something analogous to the ordinary solid
friction of every-day life. ‘he model enables some simple
deductions to be drawn of a mathematical character, but no
attempt is made here to stress these deductions unduly, as
the model (involving one dimension of displacement) can
only but roughly represent what is essentially a question of
statistics, involving slippings in all conceivable directions.

XLIX. Atomic Hydrogen and the Balmer Series Spectrum.
By R. W. Woon, Professor of Experimental Physics, Johns
Hopkins University *. ;

Part I. Atomic Hydrogen.

ie the present paper an explanation will be given of
practically all of the very curious spectroscopic phe-
nomena observed with very long vacuum tubes containing
hydrogen, which [described in two previous communications
(Proc. Roy. Soc. vol. xevil. ; Phil. Mag. vol. xlii. p. 729).

The explanation has apparently been tound of the
necessity for water vapour or oxygen for the development
of the Balmer spectrum—a matter which has always been
a mystery. Very remarkable effects have been obtained,
which show how important is the réle played by the wall of
the tube and the gas film adsorbed on it. It has been found
possible to pump practically pure atomic hydrogen gas out of
the discharge tube and study its chemical and physical pro-
perties. Wires of certain metals and certain oxides, when
introduced into the stream of atomic hydrogen, come
spontaneously to incandescence and cause the formation
of molecular hydrogen.

To make the present paper intelligible, it will be
necessary to recapitulate very briefly the subjects taken up
in the two earlier communications.

It was found that if a hydrogen vacuum tube of moderate
bore (4 to 6 mm.), and a metre or two in length, was
excited by the discharge of a high-potential transformer
(or the direct current from a battery: of dynamos), the
central portion of the tube showed only the lines of the
Balmer series, the secondary spectrum appearing only in
the vicinity of the electrode bulbs at the ends of the tube.
By using the central portion (suitably bent) ‘ end-on,”
the series was photographed to the 20th line—a gain of
eight lines over previous laboratory records.

The hydrogen was introduced moist from an electrolytic
generator through a long fine capillary, and continuously
withdrawn from the tube by a mercury pump.

With small currents (0°5 to 1:5 amps.) in the primary of
the transformer, the luminosity of the tube was low, and
the secondary spectrum predominated, the Balmer lines
being very weak. As the current was increased, the
Balmer lines increased, and the secondary spectrum de-
creased in intensity, reaching its minimum value with a

* Communicated by the Author.

Atomic Hydrogen and the Balmer Series Spectrum. 539

current of 15 or 20 amperes in the transformer. The
actual intensity of the secondary spectrum (at its minimum)
was about 1/50 of the intensity which it had at the ends of
the tube.

As we shall see later on, the probable explanation of
the peculiarities thus far outlined is as follows :—The
secondary spectrum is emitted by the hydrogen molecules ;
the Balmer lines by the atoms.

With a heavy current in the tube, the dissociation into
atoms is nearly complete and permanent, no appreciable
recombination occurring during the very brief current-
pauses which occur when the transformer potentials are
near the zero value, the duration of which can be observed
by viewing the discharge in a revolving or ‘ wabbled”
mirror. This time is of the order of 1/1500 sec., and,
as we shall see presently, about 1/5 of a second is required
for the recombination of the atomic hydrogen.

At the ends of the tube, molecular hydrogen is con-
tinuously supplied by the bulbs, the metallic electrodes
acting ascatalyzers causing the instantaneous recombination
of the dissociated hydrogen. The probability of the truth
of this explanation will appear when we come to the subject
of the action of metallic wires on the discharge.

With a feeble current, however, atomic hydrogen is not
formed rapidly enough to permit of high concentrations,
and the secondary spectrum predominates in consequence.
_With the heavy current, about 1/50 sec. is required for
the dissociation of all of the molecular hydrogen in the
central portion of the tube. This we know from the study
ne oe duration of what I called the “‘ secondary spectrum

ash.”

It was found that with the tube operating under such
conditions a direct-vision prism showed only the Balmer lines
in the central portion, the intervening regions being quite
black (the “black stage’’) ; if the current was interrupted
for a moment and then turned on again, the secondary
spectrum appeared as a brilliant flash, which lasted
from one to three or four half-cycles of the current,
according to the pressure of the gas in the tube. At
high pressures the duration ef the flash was longer, as
was to be expected, more time being required for the
complete breakdown of the gas into atoms.

The further extension of the Balmer spectrum depends
upon ascertaining the cause of, and abolishing, the secondary
spectrum and the faint continuous background, and giving
a sufficiently long exposure. The most promising line of

540 Prof. R. W. Wood on Atomic Hydrogen

attack appeared to be a study of the “Infected spots”
which I spoke of in the earlier papers. These are portions
of the tube in which the discharge appears white or pink, in
contrast to the fiery purple exhibited by the remainder of
the tube. The spectrum of these spots shows the Balmer
series only to the 12th or 14th member, and a fairly strong
secondary spectrum. I suspected that they were due to a.
contamination of the wall of the tube; and if the cause
could be found, it might be possible to improve in some way
the condition of the rest of the tube, and so obtain a more
complete series of Balmer lines.

The clue was obtained, as I have shown in a recent paper
in the Proc. Roy. Soc., by the accidental entrance of a
speck of sealing-wax into the discharge tube. This was
speedily changed to a spot of stannous oxide by the heat
of the discharge, and examination of the spot with a lens
showed that it was covered with minute globules of metallic
tin.

A section of tube was now fine-ground on the inside with
carberundum, and this ground portion made a part of the
long hydrogen tube; the portions to the right and left of
the ground portion were purple, and showed the pure Balmer
spectrum, while in the ground portion the discharge was
white, and showed a very strong secondary spectrum.

The ground-glass tube was tried as a result of an experi-
ment with a tube of unglazed porcelain, inserted at the
middle of the tube, which was then bent at two right-angles,
so that the discharge in the porcelain tube could be viewed
or photographed “end-on.”’ The porcelain tube gave only
secondary spectrum with the first four or five Balmer lines.

A tungsten wire was then inserted in the tube at a spot
which showed the pure Balmer spectrum. ‘The wire was
raised to a white heat by the discharge, though a fine thread
of soft glass inserted in the same way was not even softened.
In the vicinity of the wire the secondary spectrum came out
strong, and further experimenting showed that the secondary
spectrum appeared a second or so before the wire became
incandescent. Addition of oxygen to the hydrogen suppressed
the heating of the wire.

As I have shown in the paper previously alluded to, the
action of the wall of tie tube at an infected spot, the speck
of stannous oxide, the tungsten wire, and the ground-glass
surface appears to be a catalytic one, these surfaces causing
a recombination of the atomic hydrogen, thus furnishing
molecular hydrogen at a rapid rate, the breakdown of which
by the current causing the secondary spectrum.

and the Balmer Series Spectrum. 541

In all of the recent work, tubes of pyrex glass, carefully
cleaned with hot chromic acid, have been used. The tubes
usually get into good condition after a few minutes’ operation,
and show less luminosity through a green ray-filter (which
is opaque to the Balmer lines) than the tubes of soft glass
used in the earlier work.

One of the most discussed problems in spectroscopy is
why the presence of water vapour in the hydrogen enhances
the Balmer series and suppresses to a great degree the
secondary spectrum. As I showed in the earlier papers,
if dry hydrogen is employed in the long tube, fed in at
intervals through a palladium tube and pumped out with the
tube in operation, the discharge eventually becomes white,
and all of the Balmer lines disappear except H., which is
so faint that it appears of adull brick-red colour, in contrast
with the secondary spectrum. In view of what we now
know, it appears as if the glass wall of the tube, when
thoroughly freed from adsorbed water vapour or oxygen,
acts as a powerful catalyzer of the atomic hydrogen, which
never reaches a sufficient concentration to cause the Balmer
spectrum to appear.

Dr. Irving Langmuir, with whom I discussed these
results, made the very valuable suggestion that the glass
surface might be “poisoned” by the oxygen. He has
found, in the course of an extended study of the production
of atomic hydrogen by an incandescent tungsten wire, that
the presence of small traces of oxygen prevented the
formation of atomic hydrogen by “ poisoning” the cata-
lyzing surface of the metal. This being the case, the
oxygen must also render the surface of the (comparatively)
cold tungsten incapable of bringing about the recombination
of the atomic hydrogen, as had been found to be the case.

This makes it appear extremely probable that the part
played by water vapour in bringing out the Balmer series
is merely that of supplying a “ poison” (oxygen) for the
catalyzing wall of tube, thus permitting a high concentration
of atomic hydrogen in the tube under the action of the
heavy discharge. With a feeble discharge, as I showed in
the earlier paper, the secondary spectrum predominates and
the Balmer lines are weak. This is probably due to the fact
that the atomic hydrogen is not formed fast enough to get
ahead of the catalyzing power of the tube wall.

If dry hydrogen is admitted through palladium, and the
tube brought to the white stage by long operation, it is
found that if a condenser is put in parallel with the tube,
the discharge becomes red and the Balmer lines appear.

542 Prof. R. W. Wood on Atomic Hydrogen

The action of a condenser is to pass currents of enormous
magnitude but of very brief duration through the tube.

With these very heavy currents (hundreds or even
thousands of amperes) we have a sufficient momentary
concentration of atomic hydrogen to bring out its charac-
teristic lines, even with the tube wall thoroughly freed
from water vapour—the condition which gives the secondary
spectrum only, owing to the powerful catalyzing action of
the walls.

Merton’s observation that the discharge became white
when the hydrogen tube was immersed in liquid air, is
at once explained by the greater catalyzing power possessed
by the wall at a low temperature. Langmuir found that
the atomic hydrogen produced by incandescent tungsten
would not pass through a tube cooled by liquid air, though
it passed to a considerable distance down a tube at room
temperature. This observation I have confirmed in the
study of the properties of atomic hydrogen, pumped from
a discharge tube operated by a heavy current, small
particles of thorium oxide being brought to incandescence
at a distance of 20 cm. from the discharge tube. Touching
the wall of the tube leading to the pump with a pad of
cotton wet with liquid air, at once extinguished the
specks of thoria in the tube beyond the cooled spot. A
fine tungsten wire, inserted in the tube leading to the
pump at a distance of 4 or 5 cm. from the discharge
tube, is brought to a red heat by the current of atomic
hydrogen pumped from the tube. Clean aluminium foil of
the thickness of writing-paper, when introduced into the
discharge tube, at first caused the appearance of the
secondary spectrum in its vicinity, but after several
minutes’ operation the deep purple colour returned and
the secondary spectrum disappeared. If the hydrogen
current was now shut off and air admitted, and the tube
operated with air at about 0°5 mm. for a few minutes,
it was found that, on again operating it with hydrogen,
the aluminium had regained its catalyzing power, and the
white discharge appeared in its vicinity ; in a few minutes
this disappeared, however, as before. This makes it seem
probable that slightly oxidized aluminium will catalyze
the atomic hydrogen, but that the clean metal will not.

The mystery of why the long tube gives a pure Balmer
spectrum at the centre now appears to be explained. The
more or less oxidized aluminium electrodes act as catalyzers
supplying molecular hydrogen continuously from the atomic
hydrogen formed by the discharge, and the concentration of
the atomic gas never reaches a high value at the ends of the

and the Balmer Series Spectrum. 543

tube near the electrode bulbs. In one tube of pyrex glass
which had been very carefully cleaned with chromic acid,
with especially clean bright electrodes, it was found that the
white discharge extended to a distance of only 3 or 4 cm.
from the bulbs, while in the earlier tubes it often reached
to a distance of 20 or 30 cm.

In the earlier papers I have drawn attention to the circum-
stance that if the hydrogen tube is brought to the white
stage, and then highly exhausted, with the current shut off,
if a small amount of air or nitrogen is admitted, the discharge
is of a most beautiful golden-yellow colour, resembling the
discharge in pure helium. Photographs of the spectrum
showed that the second positive spectrum (violet and ultra-
violet bands) was nearly absent, the yellow colour being due
to the first positive spectrum, consisting of red, yellow, and
green bands.

Applying the catalysis theory to this result, the indications
are that the first positive spectrum is due to the nitrogen
molecule, the second to the atom. It was found that the
yellow discharge was obtained only if a very small amount
of air was admitted, doubtless due to the fact that if too much
air was introduced there was enough oxygen to poison the
walls of thetube. J have not yet been able to get the second
positive spectrum free from the first, but no very great amount
of work has been done in this direction. Possibly by intro-
ducing an excess of oxygen it can be accomplished.

It was found that the tungsten wire was not heated to
visible luminosity in the discharge in air, while a platinum
wire of the same size was raised to a white heat. Platinum
therefore seems to be a catalytic agent for atomic nitrogen,
while tungsten is inoperative. Ihave not yet tried platinum
in pure nitrogen, and it may be that the oxygen plays a part
in the surface reaction which heats the platinum.

It seems to be now clear why a more complete series of
Balmer lines is obtained in the solar corona and probably
in nebulee (provided sufficient exposure were given) than in
vacuum tubes. The luminous gases are in these cases not
in proximity to catalyzing surfaces, and consequently atomic
hydrogen of 100 per cent. concentration can exist. Of
course the possibility of a different type of excitation still
remains.

To further extend the series in the laboratory, it will
probably be necessary to devise a method of more com-
pletely poisoning the walls of the tube, or abolish the wall
entirely, as can be done perhaps by means of a very powerful
discharge of the ring type excited by high-frequency in-
ductive effects, in an electrodeless tube.

D944 | Prof. R. W. Wood on Atomic Hydrogen

Pyrex glass has been found better than soft glass, but
quartz appears to be no better than pyrex. i

The fact that we have only atomic hydrogen in the central
part of a long spectrum tube, even during the brief moments
when no current is passing (between the half-cycles), makes
it appear probable that the refractive index of the gas in the
atomic condition can be determined by introducing the tube
into one path of an interferometer, and illuminating the
instrument with flashes of hght, during the moments when
the current is not passing, by means of a disk perforated
with two slots rotated by a synchronous motor. This
experiment will be tried in the autumn.

Part il. The Balmer Series.

In continuing the work on the Balmer spectrum of

hydrogen, tubes of pyrex glass have been used exclusively. ©

The aluminium-foil electrodes were attached to tungsten
wires, which fuse easily into pyrex. As there is apt to bea
slow capillary leak along the tungsten wire, a drop of sealing-

wax was always melted around the wire on the outside of the

bulb. These tubes, if made of carefully cleaned glass, will
come into the ‘‘ black stage’ (showing the Balmer lines on
a black background when viewed through a direct-vision
prism) after ten or fifteen minutes’ operation.

The lines of the Balmer series were photographed in the
2nd- and 3rd-order spectrum of a very perfect 7-inch plane
grating (temperature controlled to 0° 1 by a thermostat) with
an objective of 20-foot focus, H,, Hs, and H, all showing as
clearly separated doublets. The series was recorded with
this apparatus as far down as the 18th line with an exposure
of only 12 hours. Only a few minutes were required for the
plates showing H,—H,.

Ourtis, using a concave grating of rather short focus, with
an exposure of 5 hours obtained only the first six lines of
the series. When we consider that the sixth line has an
intensity about 4000 times as great as that of the 18th line,
the enormous intensity and efficiency of the long end-on tube
is at once apparent. The tube was excited by a large 6000-
volt transformer. |

Photographs were also made in the 5th-order spectrum of
a plane grating, in combination with a collimator and Cooke
portrait lens of 1-metre focus. A screen of glass coloured
by nickel oxide was used to cut out the overlapping green of
the 3rd order and violet of 5th order, which covered the
region of the last lines of the Balmer spectrum. This glass
is opaque to all visible light except the extreme red, and is

<4

and the Balmer Series Spectrum. 549

highly transparent to the ultra-violet region in the vicinity
of the end of the Balmer series.

The grating was selected by illuminating it with the light
froma quartz-mercury arc, mounted in a closed box provided
with a window of dense nickel-oxide glass. This arrange-
ment gives a powerful beam of radiation of wave-length 3660,
which was rendered convergent by a lens and reflected from
the grating to a screen of barium platinocyanide.

In this way a grating was found which was enormously
bright in the 5th-order spectrum for the region of the end of
the Balmer series (W\=3676).

With this grating and the new tubes of pyrex glass, the

20th line was photographed with certainty, and probably the |

22nd line. The 21st is so nearly in coincidence with a strong
line of the secondary spectrum that it cannot be identified
under present conditions.

H, and Hg were not photographed, as we have Curtis’s
values, which are of the same order of accuracy as those
about to be given.

The plates were measured and the wave-lengths computed
by Mr. Arthur E. Ruark, one of my students, to whom I am
indebted for a large amount of very faithful and accurate
work. The reference iron-lines were the tertiary standards
determined by St. John and Babcock ™*, lines showing no
pole effect being used in most cases.

Reduction to vacuum was done by the table of Meggers

and Foote fT.

Difference between these values

ai hier Ni tvels and these of Curtis.
eee *6562°793 0

eee *4861°326 0

: 4340-465 4341-681 —0:001 a
ee 4101-731 4102-884 aU Vi Sab get ce
[oe 3970078 =: 3971-192 SeOOR (Ske see an,
Cees 3889-064 3890°161 TSANG A he eect
ies osu. 3835°397 3336°481 “ea O10)

2 eee 3797-910 3798°984 + 010

- pose note 3770°634 3771°701 22-00 |

1 37507152 751-214 + 902

oe 3734371 3735°429 ‘000 |

es: 3721-948 3723:008 + :007 From

Le Spigenenee 3711-980 3713-032 a OUR | eet luted
Were). 3703°861 3704-911 re CLG Gun ai.
Le eS... 3697°159 3698-197 + -005 | Ve ett
1G pies 3691°553 3692-600 + 004 |

heen 3686°833 3687°878 2008

rt Partita 3682825 3683-869 + 015 |

19 Foes 3679°372 + O15 |

eee 3676'378 “01s |

* Values given by Curtis.

* Astrophys. J. vol. liii. (1921), or Mt. Wilson Contrib. No. 202.
+ Bur. of Standards Sci. Papers, No. 327 (1918).

Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922.

2N

546 Mr. D. Coster on the Spectra of X-rays

Ov one of the plates, taken with an exposure of only
) minutes, H, appeared beautifully separated, the distance
between the components being fully three times the width
of either component. This makes it appear probable that
we can resolve also the 4th and possibly the 5th line, by
operating the tube with a current of comparatively small
intensity. |

Two series of measurements, each consisting of fourteen
settings on the components of Hy, gave the following values
for the doublet separation :

TSGHSORTCS 5 ey cee ey ahead "0568
Pinduseries: | fri (76 Wane Toittee 0601
Mioain jae aateeee 0584 J.A.,

which is in exact agreement with the value obtained by,
Gehreke and Lau and is ‘004 I.A. less than the value given
by McLennan and Lowe.

The actual wave-lengths of the components of the doublet
H, are 4340°494 and 4340°435, the value given in the table
being, of course, the centre of the doublet.

I have been aided in this investigation by a grant from
the Rumford Fund of the American Academy.

Baltimore, U.S.A.,
June Ist, 1922.

L. On the Spectra of X-rays and the Theory of
Atomic Structure. By D. Costsr*.

Part LV.f

New Measurements in the L-series of the Elements

La (57)—Lu (71).

iL N Part II. of this paper I gave the results of my
measurements of the elements Rb—Ba, in this part

I will deal with those of the rare-earth metals. As is known,
the latter elements are very similar in chemical properties,

*

* Communicated by Prof. Sir E. Rutherford, F.R.S.
+ See Phil. Mag. xlii. p. 1070 (1922), Parts I., IL., and TIT,

aoa

and the Theory of Atomic Structure. 547

and it is rather difficult to get them in a very pure state.
Some preliminary investigations with rather impure speci-
mens of these elements showed clearly that from these no
certain information could be obtained about the weaker lines,
since the great number of ‘“ foreign ” lines appearing on the
plates make the interpretation of the photographs nearly
impossible.

The present investigation has been rendered possible by
the kindness of Mr. Auer von Welsbach, who offered to this
laboratory a beautiful collection of very pure salts of these
elements. Ou the photographs taken with these salts usually
no lines were observed belonging to any other rare-earth
metal. As the presence of about 0°1 per cent. of another
element would be sufficient to give the stronger lines, we
may conclude that the salts used were extremely pure. In
the ene ease, however, of Yb (70) several lines were observed
belonging to Lu (71).

§2. The apparatus was the same as that used for the
former work. In this region of wave-lengths the spectro-
graph need not be exhausted. The slit of the X-ray tube
was covered with an aluminium sheet of 104. After being
glowed in a Bunsen flame the salts (sulphates) were pressed
ona roughened copper or silver plate, which was soldered
on the anticathode. Imperial Hclipse plates were used,
which appeared to be much more adapted to X-ray work
than the technical X-ray plates previously used in this
laboratory. These plates are extremely sensitive, especially.
for wave-lengths of more than 3A.U., and even for wave-
lengths of 1-2 A.U. they give betier images than the
technical X-ray plates in half the time of exposure.

For the rather small glancing angles in question (10-20
degrees) the apparatus seemed in the beginning to be very
inconstant, giving sometimes fairly good and sometimes very
bad plates. This phenomenon appeared to be connected
with the reflexion on different parts of the crystal; very
good plates were only obtained if the radiation was reflected
by the middle part of the crystal. As it seems to be impos-
sible that this phenomenon should depend on the geometrical
conditions of the apparatus we must assume that the reflect-
ing power of the crystal was greatest at the middle part.
Once the cause of this phenomenon was known, it was only
a matter of time to obtain rather good plates for all the
elements, as by a change of the position of the focus spot on

2N 2

hy

548 Mr. D. Coster on the Spectra of X-rays

the anticathode the radiation can be directed on any desired
part of the crystal. This change of the position of the
anticathode spot was easily obtained by a change of the rela-
tive position of the anticathode and the hot-wire cathode.
The hot wire must be renewed about every 8 hours, and
frequently after this operation it was necessary to readjust
for the most favourable conditions of reflexion.

The tension on the tube was about 25 k.v., the current
was not more than 15 m.a., the time of exposure varying
from 10 to 45 minutes. Calcite was used as analysing
crystal. Some trouble was caused by the copper K lines
and the tungsten L iines, which cannot be avoided when
working with a tube of brass and a tungsten hot wire. In
a few cases these foreign lines made it impossible to measure
some weak lines belonging to the element under inves-
tigation.

§ 3. As regards the accuracy of the measurements I may

refer to Part II. $4 of this paper. In general, the lines for
the elements La—Lu are much sharper than those for the
elements with lower atomic number. On the other hand,
most of the stronger lines are accompanied by fairly intense
satellites which often lie very close to them, thus diminishing
the accuracy of the measurements. Usually the errors in
the wave-lengths of the satellites themselves are larger than
those in the wave-lengths of the diagram-lines, as the satel-
lites are more diffuse.
. With the exception of the lines 6, and yg, which are dealt
with in Part IV. § 10, all the diagram-lines (see diagram IV.
Part V. §2) are given in Tables XIII., XIV., and XV.
The lines appearing with two decimal places have ‘been taken
from Hjalmar’s precision measurements*, the other lines
having been determined in the present investigation relatively
to these lines or to the copper, tungsten, or zinc lines, which
also appeared on the plates. For the elements Hu, Gd, and
Tb the line 7 could not be separated from a, or #, which are
much sharper and more intense.

* Zeitschr. f. Physik, iii. p. 262 (1920) and vii. p. 841 (1921).

O49

Structure.

?
4

ic

and the Theory of Atom

L-1PIL
0-681T
G-ELGT
L-6161
FILET
6-6¢F1
8-181
6-€091
8-0PLT
6-G181
G-C68T
L-8L61

G-LLUT
8-616T
8-TTéT
6-198
6-SLFL
€-89F1
6-Gé¢T
L-L8ST
L-T99T
¢-G6LT
6:698T
6-096T
9-9606

fh

G-E8IT
9-S66T
P-8IE1
L- LOST
6-0GF1
8-ELF1
0-TE¢T
6-€641
6-S¢9]
P-L6LT
0-GLZ81
6-61
9-TF0G

zd

€.0881
8-591
| ¢-z9et
| &PIPI
L.69FI
9.9281
29-881
FOOT
60-8LI
88-8181
18-961
88-FF02
06-LE1Z

_—_—_—_—

th

9961
0-S061
SOFT
6971
GEIST
G-FLGT
9-LE9T
GOLT
L-GL41
§-1&61
T-9106

9-GOTS

8-0066

ch

G:LOSl | &PIFl
S-GIFl | L-GOFL
9-011 | 9-89¢T
L-89S1 | §-819T
R-6191T | L-LL9T
06491 | G-LELT
6-1PL1T | 1-S0ST
GS808T | G-OL8T
T-8181 | G-GP6I
F-180G | §-660G
8-F11Z | 6-S816
T-F0GZ | 6-916GG
0:8666 | 6-ELE6
“| ag

68681
v-6PF1
6-L¢¢T
0-9T91
L-LLOT
¢-GPLI
6-O18T
L-88T
0-846T
GGGIG
b:G1GG
64086
6-G0PG

tg

L-0GF 1
G-GLFI

PP-E8ST

GS-F91
8¢-90L1
89-BLL1
OF-GPSI
18-9161

1G-£661
16-29%
06-8922
00-TS8%
O8-ESFZ

9)

G-LEFL
G-88Fl
F-96971
§-99T
L-OTLT
F-18LT
€-6F81
1-661
F-9661
G:G9IG
1-0966
GFPEG
8-EPFG

rg)

*("WO 17_OT) “Q'X Ul Syycuey-oar AA
“TITX @Tav

T¢-CT9T
6L-L99T
OF-0841
86-0F8T
09-F06L
6F- TL6L
66-1F0G
66-9116
10-G61Z
T§-C9&6
OL-LEFS
00-9996
89-6996

96-9691

6-8L91
OF-16LT
90-6931
P9-ST6L
1€-G861
69-6206
SE-LGIG
89-406
€9-GLE6
89-LOFG
IT-¢9¢¢
66-8996

l

8-1§8T
0 068T
1-G10@
1-6806
0-F916
0-6666
T-LO€G
$0686
0-LLFG
€-0L96
L-8LLé

0008

OTE
"OK OL |
1 89
“OH 19
aL o9
“PD FO)
nT 69 |
US GQ |
“PN 09 |
Id 69
“*aQ 8g
“eT La

——

)

Mr. D. Coster on the Spectra of X-rays

550

L3-ELL
GO-LPL
19-069
GF-699
6F-FV9
£9-069
G6- L6G
G6-ELG
TL-TS¢
1-804

6E-L8F |
OL-LOF |

PP-LOP

eX

FL-OFL
G0-06L
6-899
98-FF9
20-029
€6:966
09-8L¢
8.046
98-82¢
08-98F
89-COF
PL-ObP
18-96

he

69-261
G€-669
OL-6F9
L9-¥69
6P-109
LYV-FGG
8P-9G¢
69-FE¢
9-E1¢
G8-1LP
66-197
LL:CEP
90-F IP

ch

64-999
00-479
VG-E09
SL-68¢
62-29G
PL-GhG
PL-6G¢
L6:€0¢
GG: S8P

69-87F |

“¢)

G8-FF9

ZB-28¢
86-298
Q1-EhS
LP-EEE
6E-209
8T-L8P
_ 61-69%
60-F&P

6G-00P
| 18-686

*g)

66-669 ©

68-91F |

6L-199
FL-869
66-784
06:9¢
O1-EF¢
96-469
1G-80¢
€0-F8P
1f-G9P
68-6¢F
68:1 1F
S1-46E
18-816

ed

GP: 1P9
G8-819
8P-L¢
GP-FgG
96-6€¢
¢0-F1¢
89-F6P
GG-GLP
60:L¢F
PP-1GP
O€ FOP
09-186
PV-ILE

‘d

10-F89
E2319
G8-0LE
1¢-0¢¢
8.089
PG-11¢
9L-G6F
LLPLP
CPF-9GF
PF ICP
66-FOF
GL-88E

68-718

rel

i

90-799
8E-9F¢
G8: 11S
86-F6P
PP-8LP
1Z-G9F
LG-OFF
8G-08F
PI-GIP
GG-C8E
LL-OLE
1G-99¢
19:29§

107)

0¢-09¢
9)-GFE
89-80¢
0-26F
69-GLF
69-6¢F
V6-EFF
GE-8GF
FL-SIP
8G.E8e
83-698
GS.GG8
CFT FE

79

"(,"WO 7e)GOT) toquinu Srsqpdy oy} Jo sopdryynut ut serouenbo.s jy
AMOS BEES,

SP-L6F

CL-G8P
GG-GGP
89-LEF
LO-86F
68:80F
GT-96€
66-186
68-L9€
96-1FE
10-866

LL-S0€

ney
Cbs 0)
"* IT 89
OH 19
“£99
at 9
py £9
"MT 89
“Ug Z9
PN 09
dd 6G
7 00 8¢
"ery LG

Yon)

and the Theory of Atomic Structure.

a a Na ETE CGT © GT SG Lan y GPS T m e

690-82
991-12
EGL-9%
816-92
LLL
868-82
661-42
| 0F8-86
| 618-26
GOF-2G
826-12
09F-1Z

bh

618-16
GSE-LE
LOE-9G
EL8-G6
L8E-CG
€16-F6
SEF-FG
LG6-&G
88F-EG
LEGG
910-46
619-16
SST-1G

gh

CGLLZ LEE-LE
LOT-LZ FE8-9B
163-93 $98.62
ZIS-GZ G8E.cz
O88-6Z 006.4%
998-4 GEF-FS
LEEFZ — 0G6-EE
11686 OLF-8%
6CF-E% 166-22
LIS-2G © 60-26 |
9F0-2 O8C-12
G8G-1Z SIL-TZ |
LLG — 6F9.06 |
oh iL

9€6-92
CPV-96
68F-S6
666-46
F6G-FG
090-F¢
069-€G
IGI-@
899-66
G6L-16G
096-1G
608-06
8hE-06

118.6%
168-96
19¢-FZ
IFL-FZ
6148S
168-82
ZL8-BE
6FF-2E
860-26
O8L-1Z
8¢1.02
GE8.06
FI16-61

a

F8E-GG
096-F6
CVL-FG
961-8
90E-EG
106-26
I8F-2G
GLO-CG
199-1¢
610-16
8-06
900-06
€6¢-61

oe!

669-96
GLO-9¢
GST-FE
LEL-EG
908-86
898-66
CEP-GG
100-66
PLG-1G
GEL-06
€66-06
618-61
COF-6T

i

9CE-SE
LL8-¥6
686-6
6F9-€G
801-€%
€19-CG
666-66

908-16 |

O8E-1Z
669-06
LOT-06
189-61
€126.61

tg

9) me AR: lp

I8T.&G
GP L-FG
C68:€6
COF-ES
OFO-6
LI9-66
861-66
FLL1G
P9E-1G
669-06
FGL-0G
9TL-6T
O1S-61
re

OGL-€2 | 119-82
G1E-8% | 168-82
HOLE | FOLEE
ShS-2Z | CST-ES
G18.1Z O18-2
66F-1G | OFF-TZ
GZI-1% | 020-12
0GL-0% | 169-02
C18-0% |- 988.06
129-61 | ¢8¢.61
GGZ-6T | LIZ-61
C8E-8T | SFS.ST
OIG-8T | LLF-ST
Ix S29

‘(se1ouonbe.ay nj Jo s}004 oaenbs) is

€90-4G | FOE-2z
LE9-8% | 896-12
S8L-2% | 96-12
F9E-2Z | 126.02
CHG-TZ | 699-02
“| 028-02
818-61
“| GeG-61
886-02 | 181-61
GOF-GT | ELF-ST
190-61 | ITL-81
g99-81 |

996-81 | 6¢F-LT

li 2

gael 2

"aK OL
"AT 89
"OH 19
~~ £99
* q7, @9
"PO #9 |

“Ml €9

PN O9
1d 64
"09 Bg
BT 1G |

Dd2 Mr. D. Coster on the Spectra of X-rays

§ 4. The classification of the lines proposed in this paper
is strongly supported by Tables XVI. and XVII. Table
XVI. gives the relativity L-doublets. From this table we

TABLE XVI.

Relativity L-doublets.

|
| ge! By-%, | Ys-Be %1-B2 | Yo-Pis | Lp- Ly
gules ne | 29°51 | 30:01 | 80:19 | 29-86 ie yes
58 Ca lee | 14 ME 82:85 9825" | 82:20 | So AoNmeaD
/59 Pr.......... 85°53 | 35:02 | 35:10 | 3479 | 34:80 | 35:24
60 Nd......... 8177. | «87°86 |) ST 76 |. B71 1 Br6T | aemee
62'Sim 4. | 48°70 | 48°95 | 44:17 | 48°64 | 43-62 ~
68 Blanes: e 47-17 | 47-41 | 4688 | 47-10 ie
64 Gd.....2. ne 50°64 | 51-11 | 50-46 | 50:83 x
Goby. ana Ke 54°36 | 5448 | 5419 | 54:32 a
66 Dy......... 5852 | 5827 | 5827 | 57-43 rf
G7 AKon gil 62.48 | 62:43 | 61:74 | 61:58 | 62-08
68 Eis: 67:09 | 66:80 | 66:88 | 65°68 F
70. Nii. 7658 | 76:09 | 7636 | 75:05 oe
Tilman ee | 8156 | 81:12 | 8088 | 80-22
7OVRaya ee. | 93°33 | 99°68 | 91:99 | 91-64
TAN ee | 98°76 | 98:54 | 99:00 | 98-05
TaBLE XVII.
Differences in A
R:
OS [Sr Oy Were eR
Bila eet 1054 | 0-192 | 0779 | 0:504
FASS OB ck soca 1:048 | +192 "782 500
5OuPr so 4 1:059 | 192 ‘786 -496
60Nd eon) 1:060 | +193 ‘795 495
62iSmmes 1076 | -194 ‘801 49)
63Eu..... ... As 195 ‘790 ‘487
64 Gd... 193 ‘807 -488
Gorin - 195 ‘806 ‘481
66 Dy......... 1:095 | +198 806 ‘487
67 Ho. 1099 | -198 819 -488
68rd ee 1104 | :196 “802 494
70 Yb. 1:108 | +198 822 | -498
FAT oe 1118 | 202 ‘816 499
(alata 1119 | +204 -848 “494.
GAN ta 1:136 | -202 "829 483
BBeR iy. Kan 1-200 | 224 910 ‘495

may conclude that the frequency difference y,—, is essen-

tially smaller than the other doublet differences
* Compare Part II, § 5 of this paper.

Physik, vi. p. 185 (1921), table 3.

*

As the

See also D. Coster, Zevtschr. jf:

~

and the Theory of Atomic Structure. D998

lines y; and 8, were usually measured only by reference to
other lines this table is not very well adapted for an accurate
calculation of the energy difference between the N, and N,
level (see diagram IV.). The lines y) and @,, are dealt with
in §11 of Part IV. The last column gives the frequency
difference between the absorption discontinuities L, and L,
as measured by the author.

Table XVII. gives those differences of the square roots of
the frequencies, which according to Part II. $5 are nearly
constant. As the absolute errors in the frequencies are
about inversely proportional to the second power of the
corresponding wave-length, it is scarcely surprising that
there are more irregularities in this table than in the corre-
sponding Table V.

§5. In Table XVIII. the L absorption discontinuities of
some of the elements are given. They were found on the
same plates where the emission-lines were photographed

TABLE XVIII.

Absorption discontinuities.

| Wave-lengths. Frequencies.
| | |
Mg. Vy ils Baan) 2 due
2159°7 | 2005 || 421-94 | 454-44
HOWe Ae oe: |
(2158) | (2007) || (4223) | (454-0)
20727 1919-7 || 439-44 | 474-68
59 Pr Spi
(2071) | (1922) || (440-0) | (4741)
1990°3 | 1887°6 | 457°86 495-90
GOING 65.3. |
(1992) | (1842) | (457°5) | (494°7)
Sn eee 18409 | ... || 495-02 |
Ge Bun i 1773 Pies BLOG
I} |

(compare Part II. § 9), and were measured relatively to 8;
or y; of the same element. MHertz’s values are added in
parentheses ; they agree better with ours for the elements
Ce, Pr, and Nd than for Ba and Cs* (see Tables XI. and XII.
Part II. § 9). Absorption-discontinuities may be measured
very easily in this way, but the method is only applicable if

* It may be stated that my values for the latter elements agree very
well with those of Mr. Lindsay (compare Part III. § 8).

D04 Mr. D. Coster on the Spectra of X-rays

no emission-lines are found in the same place, and, again, it
seems to be only suitable for a definite range cf wave-
lengths.

§ 6. The faint lines lying on the short wave-length side of
B2 gave a great deal of trouble. The photographs of the
elements W, Ta, and Lu, as well as those of the elements
Sb-Ag, showed that we have to expect 4 or 5 lines in this
region. For the elements Er—Sb, however, these lines lie —
very near to one another (they all fall in a length of less

TaBLeE XIX.
Wave-lengths.
Bx By | Pro Bir By»

AN ee ee I’ (Bs) b 8620) 3680 "|" 366331) Sena
PAS OES siecle eau ge ve 3477°5 | 8468-4
49 Mra see, $817 «| 3259°8 | 3265°8 | 8304-0 | 3295-9
5O'Sm one. 3149 | 31081 3114-4) 31426 | 3134-7
GU SIor gy an- | (B,,)° | 2965°8 | 29725 | 2993-4 | 2985-8
SO ees. 2480 | 2478 By | 2483

S6aB ane rae 23756 | 2371-2 a 2381-7 |
Bi Way ees 2270) 2297 |. )2285 (Bo. Bao)

58 Cente. 2176°3 | 2184-0 2191-6 | “4

HOUR st ale 2087:-+'| 2095°8 21025 z
60'Nd......... 2004;3))| 2010-7: | 2019:3)| ,

162 Simin 1852°3 | 18581. 1865-7 (840)

OS Mu eee MSS) Y

M6LGid ee 71976: 1 By AI “

WHS D year at a — 1655°8 cy Gan | 4
166,Dy...... 51) 15957 es oe uf |
(68 Bie. | 14892 | 14823" (8, | 1501-4 |
PAU sess 1345°9 | 18380 , 1339°8 | 1359

Go any poe | 12600 | 12429 | 1250°6 | 1273'8
PAWN elo | 12208 | 12021 | 1209-4 | 1235°4 |
| | |

than 1 mm. of the plate), so that we may assume that they
must coincide or cross in several cases. This made the
classification of the lines a very complicated puzzle. Several
solutions of it have been tried, that which seems to be the
best is given in Tables XIX. and XX.

The following facts support this classification. It has
been assumed that the line 8; does not occur in this region
under the conditions of the experiments, for the following
reasons. For the elements in the neighbourhood of W it
was shown * that 8; and the L, discontinuity nearly coincide.
For the elements from Eu (63) to Cs (55) the L, absorption
discontinuity was obtained on the plate as a white line on a

* D. Coster, Phys. Rev. xix. p. 20 (1922).

--— =

and the Theory of Atomic Structure. ddd

dark background. No indication of an emission-line which
could be 8; was found for these elements. Interpolating
between the values of 8; for W and Ta and the L, discon-
tinuities of Eu—Ba we get values which do not coincide with
a measured line for any of the elements Lu-Gd. Only in
the case of Tb (65) a very faint discontinuity in the blacken-
ing of the plate was observed at about 1647 X.U., a value
which agrees very well with the interpolated value ; but
this discontinuity in the blackening looked more like an

TABLE XX.

Frequencies.
| | |
B, | By Pro By By» Bo-Byo | %- %. |
47 Ag... (8,) | 251-67 | 25105 | 248-76 | 249-40 | 0:62 | 0-43
AROdeat.. ., | H re 262:05 | 26273 |... 0:50
49In ... 274-73 | 279:55 | 279:03 | 275°81 | 276-48 | 0:52 | 0:57
50 Sn... 289°37 293:19 | 292-60 | 289-97 | 29069 | 0°59 | 0-62
51 8b... (B,,) | 307-26 | 30657 | 304-43 | 305°20 | 0°69 | 0-69
55 Os...| 367°38 | 36858 | °B, | 367-00 1:20 | 1:04
56 Ba... 383°60 | 384:31 en 382-62 i 111
57 La... 401:41 | 400-23 | 398:87 (B5, Bro) 1:36 | 1:18
58 Ce... 418-73 | 417°25 | 415-80 ty 1:45 | 1:26
59Pr... 48656 | 43480 | 433-42 a 1:38 | 1:49
60Nd... 454°65 | 45299 | 451-29 ” t- 70s cer
62Sm... 491-95 | 490°44 | 488°42 | (Bx) 202 | 2:00
63 Eu... 510°76 | 509°78 | 507-31 ob Ap 208
64Gd... 53022} (B,) | 527°32 . Bel 29:38
a 550°34 * 547-63 | : Sie 252
66 Dy... 571-09 i MM re ns 2°75
68 Er... 61191 | 61469 | (G,) 606-96 278 | 3:14
71Iu... 677-07 | 68363 | 680715 670°61 348 | 3-76
73 Ta... 72330 | 733:22 | 72865 715°60 4°57 | 4:37
74W... 74645 | 758-06 | 753-50 | 737-60 4:56 | 4:60
| | |

absorption edge than a line, whereas the line 8;, though very
faint, for W and Ta is still a very sharp line.

§ 7. The line 8; follows Moseley’s law closely. Further-
more, the behaviour of this line is what we should expect
from theoretical considerations (compare Part V. § 2). For
Cd and Ag it could not be separated from the broadening on
the short wave-length side of 8, but, since for Ag and espe-
cially for Cd this broadening extends to a larger region and
is more sharply limited than for the other elements in the
neighbourhood, we may assume that 8; exists also for these
elements.

§ 8. The lines 8, and B,) were studied by the author in a
former paper *. They are both very faint lines, Aj is still

* Zeitschr. f. Physik, vi. p. 185 (1921).

D956 Mr. D. Coster on the Spectra of X-rays

weaker than @,. They seem to correspond to transitions
which are inconsistent with the rules of selection given in
Part I. § 3. , should correspond to the transition M,—Ls,
By) to the transition M,-L;. From this we must expect that
the frequency difference of these lines is equal to the fre-
quency difference La,—La,. Both differences are given in
the last columns of Table XX. They agree within the limits
of experimental error. Further, we must expect the follow-
ing relation to hold for the frequencies :

LB, = L2,+ L;—L,,
where L, and L; are the first and third absorption discon-
tinuities in the L-series. In Table XXI. the frequencies
TaBLE X XI.
Frequency of Bo.

Calculated. Measured. |
AT Ne MER Ss 25195 | 26167 |
AS iian nee PPB || Ban HES) +
BOSH posse 99312 | 293-19
BSD vate: 30739 | 307-26
OS ane acs 36836 | 36858
56. Ba ye. 38482 | 384-31
Bele: ie: 417-4 Pel
HO Pr ie. 435°3 We ASA: S 0m ae
60 Nd......... 453 3 hi 42-995 ip
G2 Sma ee 4905 | 49044 |
TA We, 75752 | 758-06 |

calculated in this way are compared with the experimental
values. The absorption edges of Ag have been taken from
Part I. $9, Table XII., the edges for Sb, Cs, and Ba have
been taken from measurements by Mr. Lindsay, those for W
from Duane and Patterson. For In and Sn the frequency
difference y,—8; has been taken, which for these elements
must be nearly the same'as L;-L, ; for Cd, Pr, Nd, and Sm,
L, has been taken from the author’s measurements (see
Table XVIII.) ; whereas the value for L; for these elements
has been calculated from y, (it has been assumed that for
these elements the frequency difference L3;-y, is 2 units).
As is seen from the table the agreement between the measured
and calculated values is very good.

§ 9. The lines 8;; and Bi, have recently been measured by
the author for the elements Nb-Sb (see Part II. Table X.).

and the Theory of Atomic Structure. DDT

For the elements Mo-In they are fairly intense lines, but
they are not very sharp; their intensity seems to decrease
rather rapidly for elements with higher atomic number than
In. As has been pointed out in Part IT. § 8, the frequency
differences 8,,—83; and {.-8, vary linearly with the atomic
number. If we assume, as has been done by Dauvillier *
and Wentzel f, that the line 3, found for a great number
of the elements Ta—U f is a double line, it is very probable
that this line is the same as the lines formerly denoted as
By, and fj, for the elements with lower atomic number.
The variation in intensity of these lines, however, is very
remarkable. As has been stated above, their intensity rela-
tive to that of the other lines is decreasing for the elements
with higher atomic number than In ; for the elements in the
neighbourhood of W these lines are extremely faint, whereas
their intensity seems to increase again for the elements in
the neighbourhood of Au. Besides, their frequency difference

with the lines 8, seems to vary for the heaviest elements
-more rapidly than with the first power of the atomic number.
This made it somewhat difficult at first to recognize the line
8, of the heaviest elements in the lines 8, and fj, for the
elements Nb-Sb.

Where two lines were expected nearly to coincide, in
most cases a broadening or darkening of the measured line
was actually observed. In this case the wave-length given
is only a mean value. In Tables XIX. and XX. this value
stands in the column of the line which was considered to be
the most intense of the two coinciding lines. For Ho (67)
and Yb (70) the lines in question could not be measured, as
in this region copper K lines appear. It may be pointed
out that in general for the elements of Tables XIX.and XX.,
with the exception of some copper and tungsten lines, no
other lines were found on the short wave-length side of f,
other than the lines appearing in the tables. |

§ 10. Table XXII. gives two very weak lines, which were
only found for some elements where very good plates were
obtained. y; is a non-diagram line, formerly given for the
elements Mo—Ba (Table X., Part II.) ; yg is a diagram line,
representing the transition O;-L,. As is seen from the
table the frequency difference Ly,-L8, is equal to L8,-Le,

within the limits of experimental error. The line yg was

* Dauvillier, Comptes Rendus, clxxiii. p. 647 (1921).

+ Wentzel, Ann. d. Physik, \xvi. p. 487 (1921).

t D. Coster, Zertschr. f. Physik. 1. c.; Dauvillier, 1. c, (Dauvillier calls
this line f,"').

598 Mr. D. Coster on the Spectra of X-rays
Taste XXII.

Wave-lengths. Frequencies.

Vas Ys Ue Y8 Ya—By B,-2,
DO BAG ances: 2218 2218 410°82 | 410°82 | 27-22 | 27-74
Dic) UCIe Aa eae 2029 2019 449°22 | 451-24 | 32°51 32°35
Owe Tee tt. 1942°2 | 1932-2 || 469-20 | 471°62 | 35:06 | 35°02
OO INidiseeie 6: 1859 ies 490-20 ee ok Le
Coe De ae 1644 1629 564:18 | 559°23 | 48:47 | 47-17
Gey ee. 1211 1202 10232 | "70802 | 80°96 Sera
Hie A aie WAL ER a 1117 ee 815:83 | 92°53 | 92°68
WA Wir a cape ee TOPO aul) Pte: 84456 | 98-11 98°54

never observed by the author for elements with higher

atomic number than 74. yy; is not only an extremely weak,

but also a rather diffuse line, which for the heavier elements
lies rather close to the very intense line y;. ‘This makes it
nearly impossible to be sure of the existence or non-existence
of this line for the heaviest elements.

$11. Tables XXIII. and XXIV. give some lines denoted
as B14, Yo, and yo, which were only observed in the region
of the rare earth metals. Their appearance seems to be
connected with the change in the N-shell of the atom, which
we must expect in this region according to Bohr’s theory
(compare Part III. $8). For Ba the lines By, and yo do not

exist, neither were they observed for La, but here the plates

TABLE XXIII.
Wave-lengths.

Bua Y9 Yio By3 (?)
50 (Cattanine | 2236°9
06 Ba ...,02..2 2140-2 |
Bf ua eae Ne us 2048°1
58 Cel. Ste 2212°1 | 2051 1962°3 |
SO Pisa ; 21220) 1962°2 ; 1881°1
BOON dd icinacase 20388 | 18804 | 18022...
Oe SiMe. 1885°1 | 17285 | 1659°3 | 1987°1
02), eee 1781-4 | 1659°3 oe 1909°2
G4, Gall,....%... 17481 | 1593°6 | | 1835°5
69 Tb 6.0.50; 1685°1 | 1531*4 | 1765°5
GO Wy oo... ad. 162571 ae 1699-2
67 Ho 1567 1416 1635°5
06 Hitec. A 1512 a ama || | 1575°6

| |

and the Theory of Atomic Structure. 559
TABLE XXIV.

Frequencies.

Bis Ys Yio Bo-Bis | Y17Yo Y27Yio
Bete * Sa areoe) Uiten eht a O84
56 Ba......... trys (hh, SBBRBHb ® cea, ba: 1-25
Be Ledisi... ats ea MAAAGS D8! ou Tel coc: 1-44
58 Ce ........., 411:95 44437 , 46439 | 1:50 | 1:37 1:44
PEP 0. 42943 46442 48444) 145 | 1:26 15
GOWri...3..... 44696 48463 50564] 1°63 167 1:58
tS 483-40 527-20 54920] 1:82 1:66 1-12
SE | 502708 549718 its 1°89 1:67
JS ae | 521:25 . 572-08 1°89 1°52
BOR ic... | 540°77 | 595-09 1:97 1:84
oo en 560°76 AG 1:83 “i
tO... <5. ...| 581-48 | 643°56 1:50 OeOr,
6S Mies... 60270. 0:54. or

| |
were not especially good. For Ce they are very weak, lying
on the long wave-length side of 8B, and y,. From Ce down
to Tb they get more intense (for Tb they are of about the
same intensity as §,) at the same time as their frequency
difference with the lines 8, or y, slowly increases. For the
elements with higher atomic number than Tb, these differ-
ences rapidly decrease, whereas their intensity relatively to
the other lines remains nearly the same. Tor the elements
Ho and Er they could not wholly be separated from @, and
y, ; for these elements the values of their wave-lengths were
roughly calculated from the broadening of @, and y,. For
Yb (70) and Lu (71)-8y, and yy had wholly disappeared.
For Dy 8:4 could not be observed, as for this element the
tungsten La, line was appearing on the long wave-length
side of B,. As is seen from Table XXIV., the frequency
difference y;—y9 seems to be somewhat smaller than the
difference 8,-9 44.

A very remarkable line is yo, which appeared for the first
time for Cs at the long wave-length side of y.. Itisa rather
intense line, being more sharply limited on its short wave-
length side than on the other side. As is shown in
Table XXIV. its frequency difference from vy. increases first
with increasing atomic number, but for elements with higher
atomic number than Nd the same difference rapidly decreases,
so that this line could not be separated from y, for elements
with higher atomic number than Sm. But up to Er (68)
Y2 was somewhat broadened and diffuse on the long wave-
length side, indicating that also for these elements 79 is still
present. For the elements Nd-Cs, where yj, was wholly
separated from yp, the latter line appeared to be a faint, but

560 Mr. Ty Coster on the Spectra of X-rays

fairly sharp line. Since in the elements Ta—Er, for which
Yio 18 not present, a change in the relative intensity of y2 was
already observed, it appears that this change in intensity
cannot be interpreted as being connected with the presence
of 49. For Te and the elements with lower atomic number
Yy2 can no longer be separated from v3. It is difficult to say
whether 49 still exists for these elements or not, but it seems
to be hardly probable. |

For the elements Sm—Eu a faint line was observed on the
short wave-length side of 6, The presence of @, in this
place made it impossible to study this line for the elements
with lower atomic number than Sm, and in the same way
the presence of $8, made measurements impossible for
elements with higher atomic number than 68. It may
be that this line is the same as 8,3 previously measured
for the elements Rb-Sb—its wave-lengths are given in
Table XXIII.

Part V.

Comparison of the new Haperimental Results with
Bohr’s Theory.

§ 1. The L-series of most of the elements from Rb (87) to
U (92) has now been thoroughly investigated by the author™,
working in the laboratory ot Prof. Siegbahn ; and from this
work it appears that the same simple laws which have been
dealt with in Part I. govern the emission of the X-ray
spectra of the elements from U down to Rb. There are still
some lacune in my tables, but these have no particular
meaning, as it should only be a matter of patience and time
to fill them. Some rather faint lines were overlapped by
sharper and more intense lines of the same or another
element, making measurement very difficult. Of some ele-
ments no suitable specimens were available, some elements
gave difficulties due to their high vapour pressure. In this
counexion it may be pointed out that the L-spectrum of
mercury has been measured in a very ingenious way by
Mr. Miller}; most of his results agree very well with ours.

The L-spectrum of tungsten has been measured by several
authors with essentially the same results. In addition,
Dauvillier | has measured the L-spectrum of U, Au, Pt, Ir,

* As has been stated above, for the measurements of the elements
Rb-Ta, use was made of the precision measurements of the strongest lines
of these elements done by Hjalmar, and for W Siegbahn’s precision
measurements were used. For the wave-length tables of Ta—U, see
D. Coster, Zectschr. f. Phys. vi, p. 185 (1921).

+ Miller, Phil. Mag. xli. p. 419 (1921).

{t Dauvillier, Comptes Rendus, clxxiii. pp. 647, 1458 (1921),

and the Theory of Atomie Structure, 561

and Os, and in the lighter elements Sb and part of the
Ce-spectrum. In many respects his results agree very well
with ours ; there are, however, also some important diver-
gences *. Tt seoms to me that, in general, Mr. Dauvillier
attaches too much importance to some lines, which he could
only establish for one or for very few elements.

Diagram [V.—Niton, XENON, and Krypton,

%, j
ON ean c
ot ee ee ee 8 hy
a Semmes Doom ae
Ny. eke a a te ne Soe
oF TUL 2 a aoa Spee ele eS ae
a ee ba a es
pr so a . AE es ee
i 0 See SP one a a
M, iMIAIAIAIBAAMINIAIAIAT ee
ea L BAK KH 172,44 Bi neg lsh

Pr — oat

ST Rimini awe ae

Le : a ky
poeta) ee

J /

A a2, 5h, ky
A b4

§ 2. In Parts I. and III. the level diagrams for niton,
xenon, and krypton have been given. These constitute a
summary of the measurements, giving at the same time an
idea of the successive development of the shells of electrons
according to Bohr’s theory.

Diagram IV. represents a combination of these former

diagrams. The line yg, which now has been established for

* Compare D. Coster, Comptes Rendus, clxxiv. p. 378 (1922).
Phil. Mag. 8. 6. Vol. 44. No. 261. are 1922. 20

562 Mr. D, Coster on the Spectra of X-rays
7 elements (see Table XXII.) has been added, besides some

of the lines belonging to the N-series which has recently
been detected by Mr. Dolejsek * in this laboratory have been
added. The leyels falling out between niton and xenen are
denoted in the figure with one dash, those which fall out
between xenon and krypton with two dashes. The diagrams
I, II., and III. have been thoroughly discussed in Part III.,
where a comparison of the experimental results and the
theory has been given. Here will only be added something
in connexion with the line L8;, which has now been measured
for a great number of elements. This line is for all elements
rather faint, and becomes still fainter for the elements with
lower atomic number than Ba (56). As has been stated in
Part IV. § 7, for the elements with lower atomic number
than Ag, it could not be separated from §>, which for these
elements has nearly the same wave-length as the L,-absorp-
tion edge.

TABLE XXV,
Frequencies.
Lp, |La,+Ma,| 1, | i Xs
| C6aiyee 2 | 571-09 | 574-27 (74:56) 3:18 | (0-29)
Esai 611-91 | 615-73 | (617-03)| 3:81 | (1-30)
rag ete 677-07 | 68062 (68330)| 355 | (2-68)
rai ee | 72330 | 72612 | 72865) 282 | 258
ayy are | 746-45 | 74914 | 75085! 269 | 1-71 |
he Bb 844-93 | 846-76 | 851-281 183 | 4:52 |
| are Made od ae 870°77 | 872513 | 877-64] 136 | 551 |
CH! eee es 922-64 | 92363 | 93216! 099 | 853 |
59 Ph cf. us 95022 | 95026 | 95954 | O04 | 998 |
9a) 123814 | 123680 1263-20 | —1:34 | 26-40 |

Table XXYV. gives the frequencies of the line L@; for the
elements Dy-—U, and in the second column the sum of the
frequencies of the La; and the Ma, line. The third column
gives the absorption-edge L,; For W to U Duane and
Patterson’s values are used, for Ta I have used my value of
8;, which must have nearly the same value as 1,, for Lu,
Er, and Dy the values for L; have been interpolated. Of
course, we cannot draw any certain conclusions from the last
values, but still it seems to me very improbable that the
frequencies of L; for Lu, Er, and Dy should be appreciably

* Dolejsek, ‘ Nature,’ cix. p. 582 (1922).

and the Theory of Atomic Structure. 563

larger than the values given in the table. The last columns
give the differences :—

A, =(La,+ Me,)— L@;, and
A,= L, — (Le; + Ma).

As may be seen from diagram [V., A, represents the energy
difference between the O; and the N, level. For the ele-
ments with lower atomic number than Pb the N, level
appears to lie higher than the O; level, the difference
between these levels increasing with decreasing atomic
number. As one unit of the frequency, i.€., the Rydberg
number, corresponds to 13°45 volt, we may ‘conclude that
for Lu, ae and Dy the N, level lies about 45 volt higher
than the Q; level.

A, represents the binding energy of the 4, electron (see
diagram IV.), The discontinuity appearing in the table for
the value of A, for W is not essential. Using the author’s
value of 8; for L, of this element, instead of Duane’s and
Patterson’s value for Ly, we get i W : A,=2°42, a value
which agrees fairly well with the values for the other
elements standing in the table. It appears from the table
thatit is most probable that for Dy the binding energy of
the 4, electron at least must be Jess than 20 volt, thus being
of about the same magnitude as the binding energy of the
valency electrons.

As to the binding energy of the 5, electrons, we have to
consider two values: one corresponding to the QO; level and
one corresponding to the O,level. As may be seen from the
table, the first value is at least more than 45 volts for Dy,
the second value is given approximately by the frequency
difference of the L; absorption-edge and the line Ly,. From
a discussion in Part III. § 8 it follows that the latter
difference is likely to be more than 25 volts for Dy. From
this we see that, where the 4, electrons appear for the first
time in the periodic table, they are more loosely bound than
the 5, electrons; for the elements with higher atomic
number than Pb, however, the 4, electrons are definitely
bound more firmly than the 5, electrons.

§ 3. It is of interest to consider other regions in the
periodic system where we might expect to meet with pheno-
mena analogous to those discussed in the last paragraph. In
the neighbourhood of the iron group the M-shell develops
from a shell containing four 3, electrons and four 3, elec-
trons into a shell containing six 3), six 3,, and six 3, electrons.
When the 33; electrons appear for the first ble some
electrons of the N-shell are already present. Numerical

202

564 Mr. D. Coster on the Spectra of X-rays

data about the binding of the 3; electrons in this region can
only be obtained for Cu and Zn, for which elements the
M-shell even has been completed. For this purpose we may
compare the frequency of the line K®, with the sum of the
frequencies of Ka, and Le. They are given in Table XXVI.

TasLe XXVI.
Frequencies.
Kp. Ves pili:
Cah a Oa 661°30 661:24

ZT 709°87 | 710°88

The values for the K-lines have been taken from the new
measurements of Siegbahn and Dolejsek *, the Le, line has
been determined by Hjalmar. This table shows that the
binding energy of the 3, electron for these elements is not
much different from that of tlfe electrons which are present
in the N-shell.

For Ag the N-shell has been completed for the first time
into a shell of three subgroups of 6 electrons each and
respectively with the quantum symbols 4), 4,, and 4,. From
a comparison of the L,-absorption edge and the line LB, it
appears that for this element the binding energy of the
4,-electron is.not more than 10 volts, thus being of about
the same magnitude as the energy of the 5-electron which is
already present. For the elements with higher atomic
number, however, the binding energy of the 4,-electron
increases more rapidly than that of the 5,-electron.

In the same way it appears from a comparison of the
frequencies of the line L@; and the absorption edge L, for
Au, where according to Bohr. the O-shell even has been
completed into a shell containing six 5,, six 5,, and six
5,-electrons, that for this element the d3-electrons are bound
with nearly. the same energy as the 6,-electron. |

In this connexion it may be pointed out that the rare-
earth metals form a particular group of elements which in
certain respects has no analogy in the periodic system. In
fact, as the N-shell develops from a shell of three sub-
ie each of six electrons, with the quantum symbols 4,,

4,, and 4, into a shell of four subgroups, each of eight
electrons, with the quantum symbols 44, 45, 4,, and 44, the
O-shell has already a certain completion, which was first

* Siegbahn and Dolejsek, Zeitschrift fiir Physik, x. p. 159 (1922).

and the Theory of Atomic Structure. 565

reached for the rare-gas Xe containing a subgroup of four
5, electrons and.a subgroup of four 5, electrons. Tor the
other regions of the periodic system, however, where, accord-
ing to Bohr, an inner shell of electrons is being completed
(in the neighbourhood ofthe iron group, of the palladium
group, and of the platinum group) no other shell with higher
total quantum number than the shell in question has reached
a stage of completion.

§4. The new measurements of Siegbahn and Dolejsek in
the K-series * show that for the elements with lower atomic
number than 18 there seems to be some irregularity in the
relation of the frequency of the line Kf, to the atomic
number. This irregularity has been connected with the
development of the M-shell. For the elements in the neigh-
‘bourhood of the iron group, however, where the second stage
of this development takes place, no such irregularity was
observed. It could perhaps be supposed that there should
exist such irregularities for the lines L8,, Ly, Ly2, and Ly,
in the region of the rare-earths, and that the appearance of
the satellites on the long wave-length side of these lines
should be connected with this fact. Extrapolating these
lines, however, from the values for the elements which either
precede or follow the rare-earths in the periodic table, we
get in both cases values which agree fairly well with those
of the measured lines, and at any rate agree much better
with those than with the values for the satellites. If there
are some irregularities in this region—as we should expect
from what has been said about the binding energy of the
4,-electrons (Part III. § 8)—they must lie within the
limits of experimental error. In this connexion, however,
it would be of interest to measure the lines Bs, ¥1, y2, and
y; for the rare-earth metals and for the elements in the
neighbourhood of this group with the method of high pre-
cision recently developed by Prof. Siegbahn. Unfortunately,
this would be very expensive work.

§5. As has already been stated in Part III. § 8, it may
be assumed that the remarkable satellites, Qy4, yo, and yo;
dealt with in Part IV. § 11, are connected with the comple-
tion of the inner N-shell. A possible explanation of these
lines is suggested in Part V. § 9. Here I shall only con-
sider certain special points. Though the satellites on the
short wave-length side of 8, and y, (2.e., By, and By, and ¥7)
are very faint for the region Ba—Lu, and though they are
partly overlapped by other lines, we may conclude, from a
thorough examination of the plates, that there is not the

* Siegbahn and Dolejsek, doc. cit.

566 Mr. D. Coster on the Spectra of X-rays

slightest indication of anomalies in this region for these
satellites. Therefore, we may assume that there exists no
simple numerical relation between the satellites on the: long
wave-length side and those on the short wave-length side of
8, and y¥,.

It is difficult to say whether the line denoted as yo should
be connected with y, or with y3. As yo gets closer to yo for
elements with higher atomic number than Nd, and finally
coincides with this line without crossing it, we are inclined
to assume that yo is only connected with yp.

We might expect to meet such anomalous satellites as
those described in this paragraph in other regions of the
periodic system where an inner shell of electrons is being
completed. Thus, in the case of the elements in the region
of the palladium group, we should expect anomalies for
Liy2,3, LB,, and Ly, and for K®,. The authors who have
investigated the K-lines in this region do not mention any
anomaly for K@,. This line, however, is very faint. As to
Lyp,3, it might be that the nes eet and 3631 for Rh
and 7=3450 and 3433 for Pd are such anomalous satellites of
Yo,3 (compare Part III. § 5); but they lie on the short wave-
length side of this line. For the lines L@, and Ly, in this
region no new satellites have been found up to the present.
It should, however, be worth while investigating this region
again with quartz * as analysing crystal, making use of the
Imperial Eclipse photographic plates.

Further, we might expect anomalous satellites in the
region of ‘the iron group, where, according to Bohr, the
M-shell develops from a shell containing four 3, electrons
into a shell containing six 3), six 3,, and six 3, electrons.
In this region Hjalmar { actually observed a satellite on the
long wave-length side of K,, which he denoted as f’.
Wentzel t has already suggested that the appearance of this
line should be connected with the development of the
M-shell. In their recent paper Siegbahn and Dolejsek
observe that this line is much broader than the breadth of
the slit of the spectograph, and that it was not possible to

‘separate this line from K;. From this and from the

theoretical interpretation of this line (see Part V. § 9) we
might conclude that it is hardly probable that this line is

* As has been pointed out by Prof. Siegbahn, the grating constant of
this crystal lies between those of gypsum and calcite, making it especially
adapted for the region of wave-length in question.

+ Hjalmar, Phil. Mag. xli. p. 675 (1921).

t Wentzel, Annalen “a. Pnysik, \xvi. p. 437 ee

and the Theory of Atomic Structure. 567

identical with the line denoted by Hjalmar as @, for the
elements Mg-S, as has been supposed by Sommerfeld *.

Anomalies in the region of the platinum group, which we
might expect for the lines LA;, Lys, Ly,, and perhaps also
for the line L§;, have as yet not been established. Only the
line L8; seems to be a double line in this region. As, how-
ever, for the corresponding line Lf, in the region of the
rare-earth metals no anomalies were observed, it is hardly
probable that this structure of 8; should be connected with
the development of the O-shell.

§ 6. As has been stated at the end of Part ITI., most of
the non-diagram lines lie on the short wave-length side of an
intense diagram line, their frequency difference with this
line being approximately proportional to the atomic number.
Moreover, it was suggested that these lines might be emitted
by an atom which had lost more than one electron.

In the meantime, a very interesting paper has been pub-
lished by Mr. Wentzel f, in which he treats the non-diagram
lines of the X-ray spectrum. In this paper Mr. Wentzel
was able to show that the lines measured by Hjalmar on the
short wave-length side of the Ka, line for the lightest ele-
ments are emitted by atoms which have lost more than one
electron, and he could account in a very suggestive way for
the simple numerical relations which hold for the frequencies
of these satellites, and for the order of magnitude of the
frequency differences of the satellites and the diagram-lines
with which they are connected.

In the further elaboration of his theory, however, and
especially in the part of his paper dealing with the fine
structure of the absorption discontinuities, Mr. Wentzel has
made some assumptions which seem to be rather unsatisfac-
tory from a theoretical point of view t. In Part ILI. § 9 of
this paper I suggested that under certain conditions the
atom might lose more than one electron at the same time by
an impact with one single high-speed §-particle, and that the
regeneration of such an atom should be accompanied by the
emission of a line, which should lie at the short wave-length
side of an ordinary diagram-line. Mr. Wentzel, however,
supposes that the electrons of double or threefold “ ionized ”
atoms have been removed one by one, thus assuming that the
atom, which has already lost one or more electrons, remains

* Sommerfeld, Zeitschrift f. Physik, v. p. 1 (1921).

+ Annalen d. Physik, 1.c. :

t Mr. Rosseland, who first called my attention to this fact, will soon
give a theoretical discussion of the problem.

—— = -

568 Mr. D. Coster on the Spectra of X-rays

in this state long enough to be deprived of one electron
more, before its regeneration takes place.

§ 7. As this point is of fundamental importance for the
understanding of the laws which govern the constitution of
the atom, I have tried to get some experimental information
about this question. Though the experiments have only
recently been started, the results hitherto obtained seem to
be inconsistent with the assumption of the successive ioniza-
tion of the inner shells of the atom. As has already been
pointed out by Mr. Wentzel himself, from his theory we
might expect that the intensity of the lines which are emitted
by double ionized atoms should increase with the second
_ power of the intensity of the corresponding ordinary diagram-
lines.

In Part II. § 7 the satellites of the lines La, and LA, have
been discussed. It was shown that for Ag and the elements
with lower atomic number, the satellites of a, have a rather
complicated structure. This line seems to be accompanied
on the short wave-length side by an emission-band and two
rather sharp lines. With the new photographic plate I was
able recently to establish a third very faint line of still
shorter wave-length than the line denoted as a.

After some preliminary examinations five photographs
were taken of the Le lines of silver under the same condition
as regards position of the crystal, but under the following
different conditions :—

| Tension. | Current. Time of exposure.
| Plate Ro: Tove spark gap. h i9) m.a. 3 bt

| qi FLT: 8000 volts. 40; 40

[iis teers 8000 _,, Oe 80-

| ee Ws eae a 4700 ,, 20." 23 hours.

| aN eae etki 9 mm. spark ee | Oise, 15 minutes.

For the Plates II., III., and IV. the tension was read
with a Braun electrometer giving the mean tension on the
tube.

No appreciable difference was found for the Plates I., IL.,
and III. This means that in the case of Plate II., the radia-
tion being about 2 times as strong as in the case of Plate III.,
no variation in the relative intensity of the diagram-lines and

and the Theory of Atomic Structure. 569

the satellites was observed*. It is very difficult to draw
any definite conclusion from a comparison of Plate I. on the
one side and Plates II., III. on the other side, as these
plates have been taken with different tensions. But at any
rate it is very remarkabie that no appreciable difference
between these plates could be observed. A great difference,
however, was observed between Plate IV. and the other
plates.

On this plate only one line could be observed on the short
wave-length side of «,. This line was extremely faint, and
seemed to have about the same wave-length as a. As the
diagram-lines on this plate seemed not to be quite as strong
as on the other plates, a fifth plate was taken under the same
conditions as Plate I., but with a time of exposure of
13 minute. On this plate the lines a, and a were certainly
not stronger than on Plate IV., whereas the satellites could
very well be seen.

The great difference between Plate IV. and the other plates
is easily explained if we assume that by an impact with one
‘single @-particle more than one electron at the same time
may be removed from the atom. The energy connected

with the removing of an electron from the Ly-level corre-—

sponds for Ag to 3350 volts. Thus, if two electrons
should be removed at the same time from the L-shell, the
tension on the tube must be at least more than 6700 volts.
It is hardly probable that the maximum tension on the tube
should have this value in the case of Plate 1V.; therefore
the lines corresponding to a double ionization of the L-shell
could not appear on this plate. The one satellite which still
was present on Plate IV. may be due to atoms which have
lost at the same time one electron from the L-shell and one
electron from the M-shell. :

Further, these experiments indicate that it should be
possible not only to obtain information about the state of the
atom in which a certain satellite is emitted, but also to gain
more insight into the laws governing the disturbance of an
atom by an impact with a @-particle. This was shown by
some photographs taken of the fA, satellites (8), and 8,.) of
silver with different tensions. These lines are very broad
and diffuse. Under conditions differing about the same as
those of Plates I. and III. in the above table, no change in

* From a comparison of Plates II. and III. it appears that the intensity
of the spectral lines is not proportional to the current through the tube.
There is no real difficulty in this, as we have to expect that the intensity
of the spectral lines is proportional te the current density in the focus-
spot on the anticathode rather than to the total current.

570 Mr. D. Coster on the Spectra of X-rays

the relative intensity of these lines in regard to the line 6,
was observed. Photographed with the higher tension, how-
ever, they seem to be much more intense on their long
wave-length side than on the other side. This difference
was not observed on the plates taken with the lower tension.

I intend to continue these researches, using a direct-
current source of high tension. 7

§ 8. From these results and the considerations of the
former paragraphs it is evident that we cannot agree with
the theoretical interpretation given by Mr. Wentzel of the
fine structure of the absorption discontinuities. It is well
known that the researches by Fricke*, Stenstrom f, and
Hertz t showed that at a short distance from the principal
discontinuity another discontinuity may be found. The
photographs give the impression that there are two white
lines; the most intense of which usually lies on the long
wave-length side. From this we may conclude that two
different, rather definite, frequencies are selectively absorbed.
According to Wentzel, the principal discontinuity should be
connected with the removal of the first electron from the’
corresponding shell to the outside of the atom, the second
with the removal of the second electron. As it seems very
improbable that this successive ionization really takes place,
it seems to be more likely that this structure of the absorp-
tion-edge is connected with the conditions at the outer side
of the atom, as has been suggested by Kossel §. In this
connexion it may be pointed out that in this laboratory
Mr. Lindh has found rather great differences for the wave-
lengths of the principal discontinuities of the same element
in different chemical compounds.

§ 9. For the elements Ti (22), V (23), and Cr (24) Fricke
also found a second discontinuity lying on the long wave-
length side of the principal discontinuity. Itis very probable
that this anomalous discontinuity, as already has been sug-
gested by Wentzel, is connected with the completion of the
inner M-shell, which, according to Bohr, takes place in the
neighbourhood of the iron group. I shall give an explana-
tion of the appearance of this anomalous discontinuity, which
differs from that given by Wentzel in some essential details.
Wentzel suggests that in the region of the iron group the
M-shell may occur in different modifications in different
atoms of the same element, and especially that there are

* Fricke, Phys. Rev. xvi. p. 202 (1920).

+ Stenstrom, Dissertation Lund, 1919.

t Hertz, Zeitschrift f. Physik, iii. p. 19 (1920).

§ Kossel, Verhandlungen d. D. Phys. Ges. xviil. p. 339 (1916).

and the Theory of Atomic Structure. o71

found some atoms for which the M-shell has one electron
more than usual. The principal discontinuity should be
connected with the removal of an electron from the K-shell
of a “normal” atom, the anomalous discontinuity with the
removal of an electron from the K-shell of an atom whose
M-shell has one electron more than usual.

I, however, suggest the following explanation. We will
assume that the atoms of the same element have all the same
initial state. ‘The removal of one electron from the K-shell
by absorption of X-ray energy may, however, happen in
different ways. The electron may be removed wholly to the
outside of the atom, a process which should correspond to
the principal discontinuity, or the electron may be removed
from the K- and enter the M-shell, which for the elements
in question is in a state of development*. The latter process
should correspond to an anomalous absorption-line lying on
the long wave-length side of the normal absorption-edge.
Besides, we may assume that, if in the latter case the re-
generation of the K-shell takes place by an electron of the
M-shell, this should give rise to a line which should have
exactly the same wave-length as the corresponding absorp-
tion-line. Such a line has actually been found by Hjalmar ft
(compare Part V. §5). If this explanation is correct, we
should have in this line an example of a line which appears
at the same time as emission and as absorption-line. A line
of this type is only possible in a region of the periodic
system where the initial level corresponds to a shell which is
in a state of formation.

Now we may assume that in the rare-earth group where
the N-shell is in a state of formation we should have the
same phenomena. [For these elements an electron may be
removed from the L-shell under two different conditions : it
may be removed wholly to the outside of the atom, or it may
be transferred from the L-shell to the N-shell. The first
transition should correspond to the normal absorption-edge,
the second to an absorption-line lying on the short wave-
length side of the edge. Besides, the inverse process of the
second transition should give rise to an emission-line lying
on the Jong wave-length side of a diagram-line, and having
exactly the same wave-length as the corresponding absorp-
tion-line.

Anomalous satellites lying on the long wave-length side
of diagram-lines have actually been found in the emission

* Compare N. Bohr, Zeztschr. f. Physik, vi. p. 1(1922). See especially
p- 60, where the possibility of such a transition has been suggested.
t Hjalmar, Phil. Mag. xli. p.675 (1921). Hjalmar calls this line Kp’.

572 Spectra of X-rays and Theory of Atomic Structure.

spectrum of the rare-earth metals (compare Part IV. § 11
and Part V. § 5). If these lines may be explained in the
above way, we must expect that they also appear as absorp-
tion-lines. In this case, however, we should have the
remarkable fact that the difference between these absorption-
lines and the corresponding normal absorption-edges should
be quite considerable (é.¢., in the case of L, for Tb about
4() X.U. corresponding to 250 volts). I intend to start an
experimental investigation about this matter.

Summary.

This paper has been divided into 5 Parts: Parts I., I1.,
and III. have already been published in Phil. Mag. xliii.
p. 1070 (1922).

In Part I. the general laws governing the emission of the
characteristic X-ray spectrum have been dealt with and
the relation between these laws and Bohr’s theory of atomic
structure has been discussed.

Parts II. and IV. contain the new experimental results,
Part II. gives the new measurements in the L-series of the
elements Rb-Ba, Part IV. those of the elements La—Lu.
These measurements comprise: 1°, a great number of lines
which may be arranged in a simple diagram, these are
denoted as diagram lines; 2°, some non-diagram lines ;
3°, some absorption discontinuities.

- Parts III. and V. give a theoretical discussion of the new
measurements. From this discussion it appears that the
new results as regards the diagram lines are in beautiful
agreement with Bohr’s theory as regards the successive
development of the shells of electrons in the atom (Part ILI.

ney

eee to Bohr, at different stages of the periodic table
we meet with atoms for which an inner shell of electrons is
completed. Thus the M-shell is completed in the neigh-
bourhood of the iron group, the inner N-shell is partly
completed in the region of the Pd group and again definitely
for the rare-earth metals. The appearance of the Pt metals
is connected with the partial completion of the inner
O-shell. This conception is found to be in agreement with
the experimental results. Especially for the completion of
the N-shell in the region of the rare-earth metals several
experimental proofs have been given (Part III. § 8).

Again, according to Bohr, we must expect that, where an
inner shell is. being completed, the most loosely bound elec-
trons of this shell are bound not more firmly than the

9

Vibrational Responders under Compound Forcing. 573

valency electrons belonging to the outermost shell. From
the experimental data of this paper it could be proved that
this is really the case (Part V. $$ 2 and 3).

In Part V. §§ 6-9 a theoretical discussion has been given
of the non-diagram lines and of the fine structure of the
absorption discontinuities. An experimental proof has been
given that part of the non-diagram lines are emitted by
atoms which have lost more than one electron at the same
time by an impact with one single high-speed §-particle
(Part V.§7). Furthermore, an explanation has been given
of some non-diagram lines heh lie at the long wave-length
side of diagram lines, and it has been suggested that these
lines should | appear at ieee same time in the emission and in
the absorptien spectrum (Part V. ¢ 9).

I am much indebted to the kindness and interest of

Prof. M. Siegbahn and Prof. N. Bohr.

LI. Vibrational Responders under Compound Foreng. By
Prof. EH. H. Barron, F.A.S., and H. M. Brownine,
(SU a .

[Plates III. & IV.]

‘a previous papers ¢, experiments were described in which

sets of responders were under double forcing, but these
forces were quite independent of each other. The present
paper deals with cases in which the vibrational responders
in use are under forcing, either (1) from the compound
harmonic motion of a single-pendulum driver, or (2) from
the associated motions of two coupled pendulums.

In the first case they illustrate the analysis of a musical

tone by the mechanism of the ear on the resonance theory of

audition. In the second case they show the double resonance
sometimes observed when dealing with two electrical circuits
closely coupled, or two communicating resonating chambers
used with thermophones.

Compound Harmonie Vibration.—To illustrate by a set of
responders the analysis of a compound harmonic vibration,
the apparatus was arranged as shown in fig. 1. A stout
cord ACB is fixed at A and B and set in motion by the
swinging of the pendulum CD. The mass of the bob D is
made paramount, in order that it may be used as the driver f.

* Communicated by the Authors.

+ Phil. Mag. vol. xxxvii. pp. 453-455, April 1919, and vol. xxxviii.

pp. 163-173, July 1919.

{ From this it follows that ABCD keeps the shape shown during
vibration of D; hence the virtual length of this driving pendulum is ED.

> —

— rn

d74 Prof, Barton and Dr. Browning on
The lengths of the twenty-five responders (RI....I7...III..8)

range from one to sixteen, and are in geometrical progression.
The distances from A of their suspension points are pro-
portional to their lengths, so as to ensure that each shall
receive an equal forcing from the driver. The advantage of
this geometrical progression of length lies in the fact that
the “intervals” (to use the musical expression) between
adjacent responders are then all equal. Indeed, the whole
set of twenty-five responders forms two octaves of the

Fig. 1.—Apparatus for Harmonic Analysis.

es

4
{
Ra

chromatic scale. It is to be noticed that these light vibra-
tional responders have virtual lengths from their bobs to the
cord AC and not to AEB. Their bobs consist of paper cones,
with the addition of a ring of copper to diminish the damping
to a suitable extent, and so make the resonance or response
just sharp enough. The bobswere made precisely alike, and
this ensured the damping being the same for each responder.

When first the bob D was allowed to swing freely, only
one resonance maximum occurred in the vibration of the
responders at the: part marked I1 in the diagram, thus
showing that the motion was simple harmonic of the
frequency thus indicated.

Next, the bob D was moved by hand in a compound
harmonic motion of the same fundamental frequency as was
natural to it. Then the responders showed the fundamental

Vibrational Responders under Compound Forcing. 575

frequency as before, but showed also by higher maxima the
quicker vibrations which are the harmonic components of
the compound motions now executed by D. Three cases
of such harmonic analysis were carried out with this
apparatus and the vibrations and results are shown in figs. 1,
2, and 3 of Pl. III. The curves shown below each indicate
the motions given to the bob by hand, and the photographic
reproductions above give time-exposures of the responders.
In fig. 1 (Pl, III.) the motion is compounded of vibrations of
relative frequencies one and two, or tone and octave, to adopt
musical language. In fig. 2 the motion of the bob was com-
pounded of vibrations of frequencies one and three, and the
result of analysis is seen to be that of a tone and its twelfth.
In fig. 3 the motion of the bob, as shown below, is com-
pounded of frequencies one, two, and three, and the responders
give maxima at the corresponding places, which the musician
would call tone, octave, and twelfth.

Coupled Vibrations.—The arrangement now adopted is
shown by the reproduction in fig. 4, Pl. III. There it may be
seen that the two pendulums of nearly equal mass (each
suspended by a bridle and vertical cord) are coupled by the
bridge across the near part of the bridle. On the far part
of the bridle of one pendulum a set of thirteen responders
are in use. ‘These are of precisely the same type and
arrangement as the twenty-five used for the harmonic
vibrations, but here only one octave of the chromatic scale
is provided instead of two octaves. (The white cones which
constitute the bobs are clearly visible, but the black suspension
threads do not show.) For convenience, the responders may
be uamed according to the chromatic scale of C. It was
recently found to be far more convenient to have the bridge
adjustable along the bridle instead of fixed at the junction of
the bridle and vertical suspension for the heavy bob. If a
line is drawn from the bridge across either of the coupled
pendulums with heavy bobs to the distant end of the bridle,
and the suspension cord imagined to stretch vertically upwards
to meet this line, then the completed length of the suspension
gives the vibrating length of the coupled pendulum when the
bridge is held at rest, and therefore this length defines
the quick period peculiar to the coupled system with the
coupling in question.

Fig. 5 shows the effect of starting the vibration by burning
a thread which held the bobs near together. In this case the
bridge (as shown in the photograph) remains at rest, and
the bobs execute the quick vibration alone. This is exhibited
by the responders, which show a maximum amplitude at a

976 Vibrational Responders under Compound Forcing.

frequency between those of the fifth and sixth responders
counting from the shortest (or in musical terms between
Ap and 6).

In fig. 6, on the contrary, the effect is due to starting the
pendulums simultaneously and in the same phase, so that
each swings with bridle and suspension remaining in a plane.
This isolates the slow motion of which the coupled pendulums
are capable, and the responders now show a maximum ampli-
tude at a frequency between D and DD), to use the musical
terms. The longest responder, C, does not show in the
photograph.

In Pl. IV. are seen the effects of starting one pendulum
while the other hangs free. This results in the quick and
slow vibrations being performed simultaneously by each
pendulum. Fig. 7 begins with about 10 per cent. coupling.
This leads to the execution of frequencies so near alike that
it is difficult to discriminate between them in the phetograph.
The response is here seen to be spread upwards as compared
with fig. 6. In actually watching the responders, the beats
helaeen the two rates of forcing were clearly visible.

For a coupling of tifteen per cent., as shown in fig. 8, two
maxima are. distinctly visible. Musically speaking, the
notes D and EH are responding best, and the interval between
them is one tone.

In passing from fig. 8 to 9, fig. 10 to 11, and fig. 11 to 12,
the position of the upper maxima rises by one responder at
a time, and it may be said musically that the pitch is raised
by a semitone each time.

Between figs. 9 and 10 the pitch rises by a tone. Thus
in fig. 12 the responders called D and A will be found to be
the maxima, and the musical interval to be a perfect fitth.
The couplings required for the various responses are shown
on the Plate for each figure.

In order to show that the responders are responding
accurately to the two vibrations of the coupled pendulums,
traces might be taken on a board moving perpendicularly to
the pendulums. But as this has been done for the pendulums
alone, comparisons can be made between the figures on
Pl. IV. of the present paper and Plate V. of ‘ Vibrations
under Variable Couplings,” Phil. Mag. vol. xxxiv. Oct. 1917.
It will be seen that fig. 12 of the present paper and fig. 7. of
the Oct. 1917 paper have the same coupling, and that the

ratio of the ad: of the two component vibrations is
approximately 3: 2 in each case, though exhibited in entirely
different ways.

Nottingham,

May 31, 1922.

yhSamy J

LI. The Measurement of Light.
By NorMAN CAMPBELL, Sc.J)., and BERNARD P. DupDING”.
y ) ;

Summary.

T is maintained that in order to establish a scientifically
or legally satisfactory system of measuring any physical
magnitude it is not sufficient to define the units to be em-
ployed; it is necessary also to state the laws of measurement
involved. Photometry provides an exceptionally favourable
illustration of this necessity ; the laws of measurement
underlying photometry are therefore considered in’ some
detail.

1. International congresses have from time to time fixed
with great elaboration the units of certain practically impor-
tant physical magnitudes. The results of their labours are
embodied, not only in scientific treatises, but also in much
national legislation. It appears to be thought widely that,
when the units have been fixed, an entire system of measure-
ment has been established and that no further question can
arise concerning the value to be attributed to any magnitude.
Weare of the contrary opinion. It appears to us, for example,
that when the unit of resistance is fixed, we know certainly
that one definite body has a resistance of 1 ohm ; but we do
not know certainly what other bodies have a resistance of
1 ohm or what bodies, if any, have a resistance of 2 ohms.
And knowledge on these matters, which are not decided, is .
of much greater importance from every point of view than
knowledge of those which are decided.

Let us put the matter practically. An Act of Parliament
_ has laid down what is one ampere and one volt, and decided that
one watt-second is the energy required to maintain one ampere
under a potential of one volt for one second. Well and good.
But when we are presented with our electric-light bills, we
are asked to pay for a good many kilowatt-hours. . And the
Act has never told us what kind of current, under what kind
of potential, for what kind of time, is the precise number of
kilowatt-hours for which we are asked to pay. It appears
to us that the Courts ought to decide that we cannot be
legally forced to pay for anything but one watt-second,
because otherwise we are paying for something of which the
nature is wholly unknown to the law.

Doubtless counsel would advise us that, if we tried to

* Communicated by the Authors.
Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2.7

578 Dr. N. Campbell and Mr. B. P. Dudding on

maintain such a position, the judicial mind might suffer one
of its occasional lapses into common-sense. But while we
would accept his view of the probable results of the action,
we would protest that common-sense has nothing to do with
the matter. It is not common-sense that tells us how to
measure watts; it is the wisdom of the giants of physical
science accumulated for more than a century. If we know
quite definitely how to measure watts, we know all the
fundamentals of electrical engineering and much of experi-
mental physics. All measurement depends upon laws, and
these laws usually include most of the important experimental
facts of the science concerned with the magnitudes in ques-
tion. Knowledge of these laws is not common property,
and can no more be assumed than a knowledge of the
customary units. |

In opposition to our contention that a statement of the
laws of measurement should be included with a description
of the units in any defining legislation (by which we refer
now to the decisions of scientific committees rather than to
those of national legislatures), two arguments may be used.
First, it may be said that the choice of units, being arbitrary,
is suitable for legal decision, while the truth of laws, being
wholly free from arbitrariness, is not; the first, therefore,
but not the second, is suitable for inclusion in legislation.
But surely, even in legislation, relevance is of some import-
ance. Our only objection to present legislation on such
matters is that it is totally irrelevant to all cases of practical
importance ; all that we suggest is that something should be
added which will make it relevant. Secondly, it might be
said that to meet our suggestion would be to convert the
legislation into a complete treatise on science. But such a
treatise would be either too large or too small for the purpose.
It would be too small if we were urging that the impossible
task should be performed of stating all the experimental laws
that are concerned ; it would be too large if we are urging
(as we do) that the laws directly concerned should be indi-.
cated with a reasonable amount of precision. If we are asked
what is meant by “directly.” and “ reasonable,” we can only
reply by giving an example. And that is what we propose
to do.

2. Photometry seems a suitable system of measurement to
choose; its measurements (even so far as units are concerned)
have not yet been fixed completely with international con-
‘sent; the laws of measurement concerned present some
features of intrinsic interest; and some of the attempts at
establishing systems of measurement that have been made

the Measurement of Light. d79

appear to us exceptionally unfortunate. Thus the inverse-
square law, upon which most practical photometry depends,
is treated very inadequately. The units cannot possibly be
defined without reference to the law. It is usually implied,
and not openly stated, as we hold it should be: and state-
ment by implication is peculiarly objectionable here, because
the law is only true in very special conditions which cannot
be sufficiently described. But it is more serious that it is
treated as a primary and necessary law for the measurement
of all photometric magnitudes, so that the only form in
which it can be stated with full precision is that photometric
measurement is possible. This view seems to us utterly
mistaken ; one photometric magnitude at least can be defined
without reference to it; and when that magnitude is de-
fined the proposition can be stated as an independent law
concerning it, and can be subjected to direct experimental
proof. Closely connected with this defect is the total
absence of any clear statement of the meaning of the addition
of illumination, although in the practical use of light that
law is much more important than the inverse-square law. It
is more important to know that we can double generally the
illumination by doubling the number of lamps than that we
ean multiply it four times by halving their distance. Lastly,
the significance of the magnitudes is made to depend wholly
upon a theory of illumination—the theory which may be
baldly stated in the terms that illumination is due to the
incidence of something called light. That theory is wholly
unnecessary; and since, like all theories, it might be doubted,
it should be studiously avoided and all definitions framed in
terms of experimental concepts only.

With these preliminary remarks we will proceed to busi-
ness. Perhaps the reader should be warned that what follows
is not put forward as a model of an actual formulation, but
merely as a sketch of the principles to be followed. ‘The
actual formulation would require much greater verbal pre-
cision and much less explanation of reasons for the scheme
adopted.

The nomenclature of concepts concerned with measure-
ment adopted here is that employed by one of us in ‘ Physics,’
Part II.

3. Light-measurements are based on judgments of equality
of brightness of photometric surfaces. These judgments
are made by the direct perception of normal persons—
normality in this matter meaning simply agreement with
the great majority of mankind.

a gM

ad nd

580 Dr. N. Campbell and Mr. B. P. Dudding on

Photometric surfaces (P.S.) are members of pairs. A pair
of photometric surfaces is defined by the condition that, if
their positions are interchanged while everything else remains
unaltered, equality of brightness is undisturbed whatever
the nature of the illumination. P.S. which form a pair with
the same P.S. form a pair with each other. The condition
laid down ensures that the surfaces shall be of the same
shape, including radius of curvature, of the same colour, of
the same reflexion and diffusing coefficients, and of the same
intrinsic luminosity. Ifit is fulfilled, it would be possible to
use the pair of P.S. for light-measurement—unless, perhaps,
if they were absolutely black—for all or for some colours.
But it is convenient to choose surfaces which fulfil as nearly
as possible other conditions which ensure that the surfaces
are white, matt, and non-luminous. White is here used to
include grey. A white surface is one such that light reflected
from it is of the same colour as the direct light whatever the
nature of that light; similarity of colour is judged by direct
perception. A pair of matt surfaces are such that, if they
are equally bright when viewed from one direction, they are
equally bright if viewed from any other. No actual surfaces
fulfil these conditions perfectly ; none are perfectly white or
perfectly matt. But some surfaces fulfil the condition very
nearly, and these are best adopted for photometry. Further,
it is desirable to choose surfaces which are white (in the
ordinary sense) rather than grey ; that is to say, if B, sub-
stituted for A, is less bright than A, A should be preferred
to B. And, lastly, it is desirable to choose surfaces free

from intrinsic luminosity, that is, such as can be made to

appear perfectly dark by placing suitable screens round
them. dy

4. The fundamental photometric magnitude is ilumina-
tion. A pair of perfectly matt P.S. have equal illumination
(or are equally illuminated) when they are equally bright.
But since, as we have remarked, there are no perfectly matt
surfaces, the law of equality, interpreted according to this
definition, is not strictly true; the P.S. may be equally
bright if viewed from one position, but unequally bright if
viewed from another. If, however, a considerable change
in the directions from which the surfaces are viewed does
not change their relative brightness, the law of equality will
be true for observations made within this range, and the
definition so far will be satisfactory. The question remains,
however, what is a considerable range for this purpose ; it
can only be answered perfectly definitely when the further
definitions of measurement have been added ; it can then be |

the Measurement of Light. 581

stated that the range must be such that the entire system
of measurement established is satisfactory. But meantime
conditions can be prescribed more nearly in which the defi-
nition will be satisfactory even if the P.S. are not perfectly
matt.

There are two such conditions to be fulfilled: (1) The
lines joining the two P.S. to the observing eye must make
equal angles with the normals to those surfaces. (2) The
lines joining the illuminating source (or sources) to the P.S.
must make equal angles with the normals to those surfaces
and with the lines joining the surfaces to the eye. The
assumption that these angles are unique implies that the
surfaces and sources are “points.” Again, it is impossible
to prescribe exactly what are points, except by reference to
the completed system of measurement that we are about
to establish. But the proposition which is important for
our purpose is that, if the dimensions of the surfaces and
sources are made sufiiciently small compared with the dis-
tance between them, the definition .applied subject to the
conditions just stated will prove satisfactory, even if actual
surfaces, not perfectly matt, are employed as P.S. We can
therefore give instructions whereby satisfactory measure-
ments can be made; and when once such measurements
have been made, the range of permissible variation of the
conditions is determined by the agreement of other measure-
ments with those made under these standard conditions.
Difficulties such as these in describing the precise condi-
tions for satisfactory measurement occur in all branches of
physics; but they are probably more acute in photometry
than in most other branches.

It will be observed that the absence of perfectly matt
surfaces has forced us to introduce at this stage the ideas of
““a source” and illumination by a sonrce, which properly
belong to a later stage. We may, therefore, describe them
rather more closely. A surface is said to be illuminated by
a source when its brightness can be changed by changing
the physical condition of some body (e.9., by changing the
current through an incandescent lamp) or by interposing
suitable media between that body and the surface. The
body, the condition of which thus affects the brightness of
the surface, is a source illuminating it. “

We can now state the first of the important laws of illumi-
nation which are not necessary laws of measurement of that
magnitude. If a pair of equally illuminated P.S. are re-
placed in position by another pair of P.S., the second pair
will also be equally illuminated, even if a member of the

582 Dr. N. Campbell and Mr. B, P. Dudding on

first pair does not form a photometric pair with a member of
the second. Since any surface (except an absolutely black
one) can be used as a P.S. with a suitable pair, this law
naturally leads us to regard illumination as something
characteristic, not of the surface illuminated, but of the
circumstances in which it is placed. We find, further, that
among the most important of these circumstances are the
positions of the surfaces relative to the sources and the nature
of these sources. It is these laws which give us the first
clues to a theory of illumination ; but such a theory, though
it is a very useful guide in seeking a satisfactory system of
measurement, is best excluded entirely from any deserip-
tion of it.

5. We must now define Pee The illumination on a
surface X from the sources A and B is equal to the sum of
the illumination of X from A and the illumination of X from
B, if A and B, when they are illuminating the surface, are
always in the same physical state and in the same position
relative to the surface whether they are acting together or
singly.

With this definition, the first law of addition is true ;
cutting off the illumination from a source always decreases
brightness. But the second law is not true in all cireum-
stances ; it is not true, for example, when the Purkinje
effect is apparent. For, if R, and R, are red sources, B, and
B, blue sources, and if the illumination from R, is equal to
that from B, and that from R, to that from Bg, the illumina-
tion from R, and R, will not be always equal to that from B,
and B,. On the other hand, if all the sources are of the same
colour, the second law is true; and it is true apparently if
the sources, though of different colour, give sufficiently great
illumination and fulfil some other minor conditions. The
Purkinje and allied effects are simply failures of the neces-
sary laws of photometric measurement, and any complete
statement concerning such measurement must include the
proviso that the conditions are such that these effects do not
enter and that the laws of measurement are true. It must
be insisted that the only logical way to describe these effects
is in terms of the failures of the laws of measurement which
they involve; to describe them in terms of the measurements
which they make impossible, though it may be convenient
and conduce to brevity for general purposes, is utterly
ludicrous if precision is important.

6. We have now defined completely the magnitude illu-
mination, and can proceed to measure it and to state signi-
ficantly and prove experimentally the following important

the Measurement of Light. 583

numerical law concerning it :—-The illumination on a surface
A froma source X is inversely proportional to the square
of the distance of X from A, so long as (1) X and A are
“points? ; (2) the angles between the line XA and any
lines characteristic of X and A remain the same; (8) the
medium between X and A (or rather the variable part of it)
is perfectly uniform and transparent ; (4) that the bodies sur-
rounding X and A are perfectly black. Of course, as usual,
these four conditions are really in part definitions of points,
perfectly transparent media and perfectly black bodies, the
significant proposition being that there are conditions in
which the numerical law is true and that they are indicated
by the crude meaning.attributed to the terms used.

Though the principles involved in the fundamental mea-
surement of illumination by means of the definitions of
equality and addition are the same as those involved in any
other fundamental measurement, it may be well to describe
in some detail one form of experiment by which the inverse-
square law might be proved.

Two plane P.S. A and B are taken. A is viewed along
its normal, and any sources X (all of which must be points)
illuminating it are disposed in fixed positions on the surface
of a circular cone of any apical angle with this normal as
axis. B is viewed at any convenient angle, and the sources Y
illuminating it are placed in any convenient positions with
regard to it. Some one constant source Y, is chosen arbi-
trarily and placed arbitrarily in some constant position
relative to B. The illumination of B by Yo is arbitrarily
chosen as unit. We proceed, then, to find two sources, X,
and X,' such that either of them, acting alone, makes the
illumination of A equal to that of B. Hach of these sources
then gives unit illumination on A. X, and X,'are then made
to illuminate A together; Y, is extinguished and some other
source Y,is found which, placed in a certain position, makes
the illumination of B equal to that of A. X, and X,' are
now extinguished, and a source X, found which, in some
position on the cone, makes the illumination of A equal to
that of B. The illumination of A by X, is then 2. And so
on for the other positive integral values of the illumination
of A. Experiment shows that it is impossible to find sources
which give negative values for the illumination. But frac-
tional illumination can be obtained. In order to make the
members of the standard series of illumination the value of
which is 1/n, we have to find n sources placed on the cone
such that any one of them illuminating A makes the illumi-
nation of A equal to that of B when illuminated by some

584 Dr. N. Campbell and Mr. B. P. Dudding on

source (to be discovered by trial and error) while all n of
them acting at the same time make the illumination of A
equal to unit illumination. And so on for the values m/n,
and the completion of the standard series.

We now illuminate B by some point source which can be
moved along a straight line passing through B. When the
distance of Y from B is 7, we find what member of the
standard series gives the same illumination on A as Y on B.
We then multiply 7? by the value assigned to this member,
and find, after a large number of trials, that there is some
value of this product such that we cannot find any law to
predict whether any value of it resulting from single obser-
vation will be greater than or less than this value. We have
then proved the numerical law.

7. This numerical Jaw, like almost any other enables us
to define a derived magnitude, namely, the constant product.
I7r?. Hxperiment shows that this constant is indeed a
magnitude, an ordered propertv of the system under con-
sideration, determined by and in general variable with
(1) the nature of the point source, (2) the constant imper-
fectly transparent medium (if any) intervening between it
and the P.S., (3) the direction of the line joining the source
to the P.S. relative to lines characteristic of the source,
(4) the angle 6 which that line makes with the normal to the
P.S. We shall suppose that (2) is not important because all
the medium is perfectly transparent. Hxperiment shows
that Ir? is approximately proportional to cos 8. We might
therefore define a new derived magnitude I7? sec 0, which
would depend only on the source and on the direction of the
P.S. relative to it: we might term it the intensity of the light
emitted by the source in the direction of the P.S. and denote
it by ®. But since the cos @ law is not accurately true, ®
would not be truly a magnitude ; it is better to define ® as
the value of Ir? when cos 0=1 ; it is then totally indepen-
dent of the cos 0 law, and the limitation to.one value of @ is
not practically tr oublesome.

8. ® is a function, not only of the nature of the source,
but also of the direction of the P.S. relative to it.. We can
eliminate the direction and obtain a magnitude depending
only on the nature of the source, if we can form the sum
\Bdo, where wo is the solid angle subtended at the source by
the P.S.‘in any direction. The formation of this sum will
be legitimate if the Inverse-square law is obeyed whatever
the direction, so that there is a ® in every direction, and if
® is independent of w, the direction being the same; these

the Measurement of Light. 585

conditions are fulfilled if the P.S. and the source are suffi-
ciently small. JPdo when the sum is taken over all directions

from the source is called the flux of light from the source ; if
it is taken over a limited range of directions, it is called the
flux emitted within the limit of those directions (fF). The
name that we (and everyone else) have chosen for this
magnitude is, of course, suggested by a theory of illumina-
tion; but it is essential to notice that it can be defined
wholly independently of any theory. The magnitude is very
important, because it is closely connected with the energy
lost by the source and, through that energy, with the
magnitudes of other branches of physics. Accordingly, it
is often useful to invert the relations at which we have
arrived and to express intensity and illumination in terms of
flux. It is apparent that intensity is the flux per unit solid
angle subtended by the P.S. at the source, and illumination
the flux per unit area of the P.S.; but it must always be
remembered that these definitions are inverted and _ that
really we know nothing about flux till we have measured
illumination. Again, once we have arrived at the connexion
between flux and illumination we may use this connexion to
measure flux when the conditions which we have hitherto
supposed necessary for its measurements are not fulfilled.
Thus, even when a surface is illuminated by a source or
sources which are not a single point obeying the inverse-
square law, we may say that the flux incident upon it is the
product of the illumination and the area. If we make this
purely verbal definition of flux, we then find (in virtue of
the law of addition of illumination) that the flux from many
point-sources incident on a surface is the sum of that inci-
dent from each of them ; when we can use a definition to
state a law, the definition ceases to be purely verbal and
becomes an expression of fact ; it may be admitted to our
scheme on a parity with the other definitions of measurement.

9. One important photometric magnitude remains for
discussion, brightness. We have used the conception of
brightness before as something directly perceived, but we
have not framed any scheme for measuring it. The magni-
tudes already established enable us to measure brightness as
a derived magnitude. Let us take several surfaces which
are unequally bright, though each of them is uniformly
bright, and allow them in turn to illuminate a P.S., and
measure the illumination of that P.S. (from which we
deduce the intensity ® of the light falling on it), the area S
of the bright surface and the angle @ which the line joining

586 Dr. N. Campbell and Mr. B. P. Dudding on

the P.S. to the surface makes with its normal. Then we
find as a matter of experimental tact that, when the inverse-
square law is true, ® is proportional to 8; and that if we

hee , the order of this expression for
S cos @

the different sources is the order of their brightnesses
directly perceived ; the source for which the expression is
the greater is always the brighter. That is equivalent to
saying that this expression is a measure of the brightness ;
and accordingly we define the brightness of the surface, now
a magnitude, as the intensity of the light emitted by it in a
given direction divided by the area of the projection of the
surface on a plane normal to the direction of viewing.

In general brightness so defined is a function of the
angle «, in accordance with the fact that surfaces in general
alter in brightness when the direction of viewing is changed.
But there are certain surfaces, which are those most nearly
matt, the brightness of which does not depend greatly on
that angle. For such surfaces, the intensity of the light
emitted at angle @ is proportional to cos a; 7.e¢., Lambert’s
law is true. When Lambert’s law is not true, if brightness
is to be defined uniquely, some convention concerning the
relevant values of « must be introduced. It is usual* to
define brightness only for «=0; another course, in some
ways more satisfactory, would be to take the mean brightness
over the whole range of a.

So far we have regarded the measurement of the intensity
of the light emitted by a known area of a bright surface as a
mere means of measuring the brightness directly perceived.
But for some purposes this intensity is important on its own
account, e.g., when we are considering the illumination that
the bright surface would produce. For the purpose of this
illumination it is immaterial whether the bright surface is or
is not uniformly bright ; so long as the intensity of the light
emitted by unit area is the same, it does not matter whether
the light comes from all parts of the surface or only from a
few specks on it. Accordingly, it has become customary
(though we think the custom unfortunate) to speak of all
surfaces as equally bright so long as the intensity of the light
from them per unit area (or possibly unit projected area) is
the same, regardless of the fact that the apparent brightness
of the surfaces, the quality directly perceived, is utterly
different.

10. There remain to be considered certain subsidiary laws,

form the expression

* Cf. Winkelman, Handbuch der Physik, Optik, p. 747 (2nd Ed.).

the Measurement of Light. ENE

which, though not absolutely necessary to photometry, are
useful in its more elaborate developments. It is doubtful
whether they should be included in any official statement
concerning photometric measurement ; for it is difficult to
describe precisely the circumstances in which they are
accurately true, and in the last resort measurements should
always be checked by the basic methods that have been
described so far. But three of them may be noted. The
first arises when reflecting or refracting surfaces are placed
hetween the P.S. and the source or (more often) the eye.
We have then to note that the line joining the P.S. to the
source or eye is to be taken as the optical path between
the two, and all statements concerning direction or distance
interpreted accordingly. The second arises:when a sphere-
photometer is used to measure average flux or average
candle-power. The jaw involved is then that the iUlumina-
tion of the P.S. used in the measurement is determined
wholly by the average intensity of the source in the
sphere. This proposition is never true universally, and only
experiment with each photometer can tell within what limits
it is true. The last is Talbot’s law, employed in rotating
sector methods or when asymmetrical sources are rotated to
obtain average intensity. This law is apparently accurately
true, and might therefore be included on an equality with
the other laws of photometry.

11. Having stated the facts, we may proceed to conven-
tions. We have to define the units of the various magnitudes.
Since we have one fundamental magnitude (illumination)
and three derived magnitudes (intensity, flux, brightness),
we shall need n units arbitrarily assigned to some specified
physical systems or substances and 4—n formal constants for
the three numerical laws defining the derived magnitudes
(n<4). We also need units of the other fundamental (or quasi-
fundamental) magnitudes involved in the laws of derivation,
namely, distance, solid angle,and surface. For many reasons
it is convenient tomaken=1. But it is important to observe
that it is not necessary to assign to the arbitrarily selected
system unit value of the fundamental magnitude; we can
assign to it unit value of any of the connected magnitudes
and define the unit value of the fundamental magnitude as if
it were measured as quasi-derived.

This procedure is actually the most convenient ; we assign
arbitrarily a value of intensity and not of illumination. We
describe a physical system which we call a standard candle.
We then assert that the intensity of the light emitted by it

588 Dr. N. Campbell and Mr. B. P. Dudding on

in some direction, specified by its relation’ to lines charac-
teristic of that system, is 1. (Intensity of standard candle,
or 1 candle-power.)

We choose the centimetre as unit of distance, square
centimetre of area, and steradian of solid angle. We have
now to choose the formal constants «, 8, y in the laws

o=21y7)
=B\ Pde,
@D
Hanne ane

We have hitherto treated these constants as all being
unity, but there is no reason, if we see fit, why we should
not assign to them other values; and the choice of unity is as
arbitrary as any other. Asa matter of fact, «, 8, y are all
usually chosen to be 1, and there is no objection to that
practice. The unit of illumination (phot) is then that of a
P.S. distant 1 cm. from the (point) standard candle in the
prescribed direction ; the unit of flux (lumen) that emitted
by the standard candle within unit solid angle surrounding
the prescribed direction, if the condition is fulfilled that the
intensity is the same within all parts of that solid angle.
The unit of brightness (lambert) is that of a perfectly matt
surface which emits light of unit intensity when its area
projected on the plane normal to the direction of viewing is
one square centimetre. It is to be noted that these derived
units (unlike the fundamental unit, which is not that of the
fundamental magnitude) cannot all be actually realized.
But such a feature of derived units is quite usual ; e.g., unit
Young’s modulus cannot be realized.

12. It may possibly be useful to compare our nomencla-
ture and notation with that adopted by some official body.
We choose for this purpose that of the Standards Committee
of the Optical a of America, described by its Chair-
man, P. R. Nutting *

Our Notation. American.
Aiioinmatvon | oye) Flux density. .;.. D
Intensity... p Iimpensity, os be C
ID vox ae sag eva Cat F Lite SPOR ais Ba. F
Brightness —...... B Brie nimess ee B

Of course, notation is a matter of mere convenience ; but
we have departed from the American scheme because we

* Journ. of the Opt. Soc. of America, iv. p. 230, July 1920.

the Measurement of Light. 089

think it is most highly inconvenient to denote the trul
fundamental magnitude by a name (flux density) so highly
suggestive of a derived magnitude. And we have preferred
® to C in order to indicate the close connexion between
intensity and flux.

But we have more serious differences with the American
Committee. On p. 232 a lambert is defined as the bright-
ness of a substance emitting one lumen per square centimetre
of projected area in the direction considered. But to speak
of a lumen emitted in a direction is to talk nonsense; the
flux emitted within an infinitesimal angle is always infini-
tesimal. In place of ‘Sone lumen” should be substituted
‘one lumen per steradian.” Since on p. 231 brightness is
defined as flux per steradian per square centimetre, it is
probable that the omission of the words ‘per steradian ”
is a mere oversight—the more unfortunate because it is
twice repeated.

But the statement, also on p, 232, that “one candle per
square centimetre equals 3°1416 lamberts” is less easily
comprehensible. In the first place, lamberts have only been
defined for surfaces which obey Lambert’s law ; for with
other surfaces the lumens per steradian per unit of projected
area will vary with the direction considered. Since a surface
made up of “‘ candles per square centimetre” does not obey
that law, it is meaningless to attribute to it any number of
lamberts. To remove this objection two alterations can be
made :—(1) For ‘candles’ can be substituted ‘ candle-
power’; for candle-power is often used as a synonym for
what we and the American committee call intensity *.
(2) Some specification can be added concerning the direction
in which the candle-power or intensity is to be measured.
For (2) it is most natural to take the direction normal to the
surface ; but if we take this direction, a surface of one square
centimetre emitting an intensity of one candle-power (2. e.,
one lumen per steradian) has a brightness of 1 lambert, not
a lamberts as the American committee affirm. We might
also take average intensity over a hemisphere based on the
surface. If the surface is plane and obeys Lambert’s law
the average intensity over this hemisphere is 4 candle-power
per square centimetre, and a surface emitting 1 mean
candle-power per square centimetre would be equal to

* E. g., average candle-power or horizontal candle-power means
average or horizontal intensity. The term ‘‘candles”’ is often here, too,
used in place of candle-power. We believe that this confusion between
a thing and its properties is one of the main sources of the obscurity of
photometric definitions.

590 Mr. D. L. Hammick on Latent Hea

2 lamberts. Doubtless some function of the intensity and
the angle could be devised such that, in a corresponding
sense, one candle-power per square centimetre would be
equal to w lamberts, but we cannot discover a function
which has the least intrinsic plausibility. Doubtless the
American committee have succeeded where we have failed,
but they have been unduly modest in concealing their
ingenuity ; they may fairly be asked to explain how they
have arrived at their surprising result and why they have
preferred it to the simple and obvious convention which
gives the result that 1 candle-power per square centimetre
equals 1 lambert.

May 27, 1922.

LI. Latent Heats of Vaporization and Expansion.
3 By D. L. Hamoick*.

A CCORDING to van der Waals, the pressure in a
homogeneous fluid system is p so, When the

‘system expands, work is done against the pressure. The
value of this work for the expansion of 1 gram of a liquid
(volume x c¢.c.) until its volume is wv? c.c., the specific
volume of the liquid when completely vaporized, is

given by if |

V2 LC bB) a
= pdv+ { aye
UV
e ¥1 al
Ney nee oe _
SSS, flict Ohta aie = 8
Bia a te

When v, is large when compared with v1, this expression
becomes
a
Jak — ae UT ais
Uy

\}

or, putting pr, equal approximately to Mw

REY a
a a hac ae xe . ° ° ° ° (1)

* Communicated by the Author.

of Vaporization and Expansion. aon

From (1) we have, at constant volume,

dA 1nd Ne Beemer
4y fe LEAs ag a Te? ir
On v Moin ag) Es :
elated
eae aT
(= can)
aah h
Hence, substituting in the equation
& dA
A-U= dee >
E dA) a wee Hoe
— = a id pata >= — — () e e . . 2
8 A+ Tom pike al) (2)

= Nex, the latent heat of the expansion.
Now Davies (Phil. Mag. [6] xxiv. p. 421) has obtained the
following expression for 7, the initial pressure in a fluid:—
aes = (2t,—T)
(v, and T, are eal volume and temperature).
il Aan ae 1 peal
Davies has also shown (Phil. Mag. [6] xxiii. p. 415) that

1 2 e . .
Sav = 4, the coefficient of cubical expansion, so that
2T;=T

1 (da -
a = =—=— —— O. 6 é . * . (3)

But, if += = we get from (3):
Lda
a (Gr),= a

her, = —(1 +a). Seer sn (a)

ay

Hence 1 (32) 1

Substituting in (2)

When T=0, we find Nex = <3 in other words, at the

absolute zero the expression becomes identical with Bakker’s
expression for the latent heat of vaporization.

——————ESE

i

592 Mr. D. L. Hammick on Latent Heats

The question now arises as to the connexion between the
value for the latent heat of expansion as given by (4) and the
latent heat of vaporization at ordinary temperatures. The
two latent heats would be equal, provided that no change in
the internal energy of the substance occurred during the
transition, at constant volume, from the liquid to the vapour
phase. In other words, the condition for equality is:

{ cat ( cat =0. ey

e/ I
(C,, ¢ are the specific heats at constant volume
respectively.)
If (5) does not hold, then the difference between the

latent heat of vaporization and Xex, of equation (4) will be

oT

{ © 2 jar a

Again, if during the passage from the liquid to the
gaseous state a change in molecular aggregation occurs,
a further quantity of heat, h, representing heat of asso-
ciation, must be taken into consideration. Hence we
aed Me = Me PHO

In the Table values of > x, calculated according to
equation (4) at the boiling-point, are compared with the
observed values of Aya, the latent heat of vaporization.
Values of “a” are given as atmospheres pressure+ (volume
of 1 gramme molecule of gas at N.T.P.)? (Guye and Frederich,
Arch, Sci. phys. et natur. Geneve, ix. p. 22, 1900). Specific
volumes and a, the coefficients of expansion, are taken from
Young (Sci. Proc. Roy. Dubl. Soc. xii. p. 414, 1910) and
Tyrer (Trans. Chem. Soc. 1914, p. 2534). The latent heats
of vaporization refer to 1 gramme of liquid, and are mately
Young’s values (loc. cit.).

It will be seen from the Table that the values of Nex, at.
the boiling-point agree very well with the values of the
latent heat of vaporization. At the boiling-point, there-
fore, we have, in equation (6),

H+hk = 0.

For ‘“ normal” or unassociated liquids h=0, and hence

17
H= | (C,-c,)dT = 0.
0

a

Substance.

Nitrogen
Oxygen

ey

ee a

Methyl acetate

Fluor-benzene......
Propyl formate ..

Propyl acetate
Ethy! propionate..
Mlethy! butyrate...

MMOTIC ............

es

EMEC) 2 owas eae cle d

eee eeesec eas]

EH wass.--......., |
ul im SMTUTA, 5.50.20

Ethyl acetate ...... |
hexane ............ Nae

Methyl oeesug
»

ee

Im-heptane ......... |

Methy! isobutyrate| :

Chlor-benzene......
Bromo-benzene ...|
Todo-benzene ......

Mcetic acid... ......
BMMEAMEYS 2 cevclecccce =)

Ethyl alcohol ......
Methy}! alcohol ...

1 Dewar.

3 Keesom and Onnes (1913).

a.
ats.

(22°4 litres)”

00042?

002762
00272
00260
027982
0750

03688
03915
02355
02412
03302
03316
04383
05383
04272
04388
04330
04148
03945 (11).
‘06876
05659
05582
‘05577
05339
05240
05926
08351
‘05580
(05692
06592

03732
‘0118

‘02617
02512
01959

TABLE.
a.
Cok :
e.c. per| cals.
gram. |per gram.
1429: | 89°47
1:230+ | 3482
*857* | 37°75
"7124 | 27°95
1-4766 | 227-7
1:00 47°15
1-431 56°08
1°622 59°96
1-059 75°22
"822 61:8
17133 64°64
1:136' 61°88
1-20 57°39
1-633 54:12
| 1-053 53°61
1-199 57-46
1-189 57°17
‘676 316
1:233 64:03
1633 51-26
1:257 52°67
1°258 519
1-240 52-55
1:239 50°3
1-283 58°75
5021 | 21°12
1621 48:2
983 47-54
‘769 36°5
‘633 30°5
1068 | 118-2
1044 | 424-7
1321 71°56
1:360 | 106°3
1324 | 1760

0,142

0,150
‘0,166
I 0,.35
0,138
‘0,156
0,155
0,159
| 0,156
0.155
09125
| “Og 141
0,154
-0,113
0,114

0,128
| -0;,766
0,149
0,125
0,121

2 Kaye and Laby’s Tables.

0,11162

‘0569+
04264
‘03854
‘0,176
‘0,186
‘0,169
‘0,163
Onl ag
‘0,126
‘0,159
‘O, E58
0,152
02159

1:227

1:438
1-362
1°385
1-419
1-472
1-52

1-511
1-491
1-407
15138
1-528
1-521
1544
1-503
1-530
1-586
1-470
1-487
1-582
1-578
1593
1°58]
1-565
1-478
1-542
1606
1-506
1-490
1-516

1-484
1-286
1:492
1:44]
1-402

+ Baly and Donnan, Jour. Chem. Soe, lxxxi. p. 907 (1902).

5 Eucken, Ber. deut. phys. Geselisch, xviii. p. 4 (1916).
° Lunge. :
® Marshal and Ramsay, Phil. Mag. (5) xli. p. 28 (1896).
® Brown, Jour. Chem. Soe. Ixxxiil. p. 987 (19
10 ‘Tyrer.

7 Estreicher.

03), Ixxxvii. p. 265 (1905).

Phil. Mag. Ser. 6. Vol. 44. No. 261. Sept. 1922.

r

vap.
observed.

cals. per gr.

108°5-

113-23
49°83
53.8
37°55

3217

594 The late W. Gordon Brown on the

At the end of the Table are given some results for liquids
ordinarily regarded as associated. It will be noticed that in
the case of water the value of H+h is approximately zero ;
in other words, h+H must be nearly equal but opposite in
slon.

EN nt between the values of Aex, and Avap. 1s not
found at temperatures other than the poiling-point. Thus
tor ether|at O° C., Acx = $5'D cals..s -Avap, = 9 470 Calcap aan
from equation (4) it is easy to show that the slope of the
Nex. Curve is given by

ht Nee a bal aT
ae a ee Bi

Now at the. boiling-point, «T is approximately *5 (wde
Table above. Hence

Ee a <i
2 1 ANaan. e °
The average value of \ aT the neighbourhood

of the boiling-point is 2a—3a.

The Dyson Perrins Laboratory, Oxford.
May 12th, 1922.

LIV. On the Faraday- Tube Theory of Electro-Magnetism.
By the late Wit1i1am GorDOoN Brown™*.

Is ae method of describing a field of force by means of

lines or tubes of induction, which originated with
Faraday, was given a quantitative form by Sir J.J. Thomson f,
and further discussed by N. Campbell in his book ‘ Modern
Hlectrical Theory.’ Since Maxwell himself looked on his
work as a mathematical theory of Faraday’s lines of force, one
is tempted to examine the original physical theory for hints

* Communicated by Dr. C. G. Knott, F.R.S., General Secretary, R.S.E.

The young author had just finished his school life at George Watson’s
College, Edinburgh, when the outbreak of war in 1914 called him to .
the service of King and Country. He met his death in France on
November 18, 1916, at the age of 21. The paper was put in final form
about a year earlier when W. G. Brown, after serving in Gallipoli, was
in hospital. A short sketch of his life and of his other mathematical
notes will be found in the Proceedings of the Roy. Soc. Edin. vol. xlii.
1921.

+ ‘Recent Researches, chap. i.; ‘Electricity and Matter,’ chap. i.

‘araday-Tube Theory of Electro-Magnetism. D995

as to the modification of the Maxwellian theory to suit
certain modern requirements.

What is attempted in the present paper is a reconstruction
of the quantitative theory of Faraday tubes on a dynamical
basis from the minimum of hypotheses: partly to enable the
electromagnetic consequences of altering the Principle of
Action to be estimated, and partly to suggest plausible
directions for modification of the electromagnetic relations
themselves. It will incidentally be shown that the stress
which may be supposed to act in the electromagnetic field
requires certain modifications if the theory of lines of force
is adopted.

2. The first assumption required is as follows :—A tube of
induction, or Faraday tube, may be defined as a continuous
line having certain physical properties. Any tube may either
be a closed curve, or its ends be connected to a positive and
a negative electric particle respectively ; the positive direction
will then be from the positive to the negative particle. It
would be superfluous at present to specify any further
properties of the electric particles.

The tubes at any point may be divided into sets dis-
tinguished by each set having a common direction and a
common velocity of translation.

In what follows the vectorial notation of Heaviside is
employed *, and electrical quantities are measured in rational
units. Let the density of the tubes of the mth set and their
direction, at any point, be represented by the magnitude and
direction of the vector d,, ; then the number of tubes of that
set passing through unit area normal to the unit vector N
will be Nd.,. 7

Let d= a, SO ei Sats ia = Yak Sua elie (1)

the summation including all the sets present at the point ;

* [Heaviside’s vector notation is a modification of Hamilton’s quater-
nion notation, the main difference being that the quaternion product
of two vectors AB is not used in Hamilton’s sense but is used to mean
the scalar of the complete product—that is, Heaviside’s AB is equivalent
to Hamilton’s —SAB, and may be defined geometrically as equal to
ABeos 9, where A, B are the lengths of A, B, and @ the angle between
them. As in other non-associative vector algebras, the square of a
vector is equal to the square of its length; in quaternions A*=—A?,
The notation introduced by Gordon Brown in equations (9), (10), etc.,
has been suggested by others but generally discarded. Burali-Forti and
Marcolongo, however, make it a feature of their system of vector analysis.
As a notation it is misleading; as an operator it is inferior to the
- quaternion A.—C. G. K.]}

2Q2

ed

596 The late W. Gordon Brown on the
then the total of all sets passing through the same unit area, is
SNA, = ND,

where tubes passing through the area in the direction of N
are reckoned positive, and “the algebraic total is intended.
Thus D represents vectorially the total flux of tubes ; it is to
be identified with the D of Heaviside, and, except for the
question of units, with the (f, g, h) of Maxwell.

Let Gm be the (vector) velocity of the tubes of the mth set
at the point in question, and let

H=SVaiden |. ee

The quantity thus defined will be shown to have the PRR eE
of the magnetic force.

This completes the geometrical and kinematic specification
of the properties of the tubes. It is not difficult to see that
if we define the charge of an electric particle as the number
of tubes leaving it, in the sense that the direction of the
tubes at a positive particle i is outwards, then the density of
electric charge will be given by

p= div D. Ae ian (3)

If we take the curl of (2) and expand the right member
fully, interpreting the terms kinematically, we obtain the
equation

curlH= at Om div dn
=<Dep up; fos) 6

where D is the time rate of change of D at a fixed point, and
u is the mean velocity of translation of the electric particles
calculated so as to make up the convection current.

The second assumption made is dynamical. Let us write

D :
p= K? ° ° . ° ° . (5)
137) en REE i) (6)

where » and K are constants, and E and B are new vectors,
the electric intensity and magnetic induction.

Then we assume that the alan densities of kinetic and
potential energy are given by

Cf 7) ee
THORB. 0.) oie

Faraday-Tube Theory of Electro-Magnetism. oot

The meaning attached to the above quantities is that if

we write
ie (rte,

where the volume integral is extended throughout all space,
then L may be used as the Lagrangian function in equations
of motion of the usual form. For the sake of brevity,
vectorial general coordinates will be employed. In order to
preserve the form of the equation -

it is sufficient to write, in the case of a vector coordinate r
(equivalent to the three scalar coordinates 2, y, z),

DAG EVO Ws 3 OUT
ar 9 Ge ay ta, | 4
Pa) rye ae) i io . . °

This notation in vectorial analysis is of course not generally
applicable, but is convenient for the purposes of the present
paper. ‘The general results of differentiation which will be
required are 7

a ae em ah ee (10)

2 sys=2¥8, meh} SA ea SS Glew)

where s is any vector variable, a is a constant vector, and
is a constant self-conjugate linear and vector operator.

4. To define the general coordinates, let all tubes at a
given moment be divided into small unit lengths ; and let r
be the vector from a fixed origin to the centre of one such
unit segment, which forms part of a tube of the mth set, then
the Lagrangian equation corresponding to r will be

& Olas iol
Ee) | ae Re am (al
dt Or or ® )
Now, when a unit length of a tube of the mth set is added
to, or removed from, an element of volume, the increase or
decrease of the whole Lagrangian function due to this

598 The late W. Gordon Brown on the
element will be
ol
dlL= Od, od,
=— 6d,,(E + Vq,B) se r A (13) °
for
io (os U) ee a [su(2Vand ye —+4D?/ Oe
= 2 [2>e pe — —d,VqnVqsd )—4( (2d,7 /(K)]
aaa (E+ Vq,,B), oy dite. bel Pp eek vo aerial (14)

where the summations include all values of the suffixes n, s,
the differentiation of terms such as (—dpVQnVQndn) being
performed by means of (11), since (—VqnVQm) is a self-
conjugate ‘operator ; ; and that of cross-products, such as :—

(—d,,VqnVq,d;) by means of (10), writing a= —Vq,Vq.ds.
(| GC.

A D

Thus, if in the figure the unit segment is removed from
the position AD (at which (14) has the value —(E+ Vq,,B) )
to the parallel position BC (at which (14) has the value
—(1+6ry) (E+ Vq,,B), AB=6r), then the total increase in
L is given by

6L=—éry.dd,,(E+ Vq,,B).

It will now be convenient to suppose (as we may without
loss of generality) that the mth set consists of but one tube,
so that 6d,,=4,, and is in fact a unit vector.

Then

6,L=—ory.d,(E+Vq,B), . . . (15)

and in applying the axial differentiator 6ry we must re-
member that neither d,, nor q,, as they occur explicitly are
to be considered variable.

But to preserve the continuity of the tube we require to
introduce the segments AB, CD, as shown in the figure, so

Faraday-Tube Theory of Electro-Magnetism. 599
that, again applying (13), we have the change of L due to
this cause

ole0, 7 Or Va_B),) 2). (16)

in which q, is variable (but not d,).
Hence

§L=6,L+6,L=oer[ day. (E+ VgnB)—V .dn(E+ VanB) |,

q, Varying in the first term only, and d,, not at all, and
finally

on = aay (E+ VqnB)—V .dn(E+Vq,B) . (17)

with the same convention.

In calculating the momentum term ob we have r=qp.-

or
Then by the method employed above in aa (14),
since T is symmetrical in q,, and d,,,

ox

yy = mage ° ° ° ° . 18
Aah Vd,,B (18) |
. . oL e e
This will be the value of F when d,, is a unit length of

tube, but in performing the complete differentiation to time
in (7) we must remember that any length of tube will in
general be continually varying in direction and magnitude.
It is clear that

d
dt i oaegpemen | A - Ams a Nae eaae MO peeeie (19)

since the rate of change of a segment of a straight line, as
AD in the figure, will be the relative velocity of its ends
(vectorially) ; “while, of course, if q, expresses the velocity
of any point of the tubs, as A, the velocity at D will be
(1+ADvy. )Qm, where AD is the vector element.

Thus

dol d
aan oa
= VdnV -Gm)B+ VdnBt+Vdn(dnv-B), - - (20)

where B is the rate of chan ge of B at a fixed point coincident
with the moving centre of the segment, qnV-B being of
course the term in the rate of change due to motion of the
segment with velocity qn.

600 UWP Tate Worden Brown on the
Equation (12) is therefore by (17) and (20),
Vn -Im)B+ VdnB—Vdn(dnV -B)
—d,V - (E+ Vq,B) +7 -dn(E+ VqnB)=0, . (21)
d,, and q», being constant in the last term, and 7 operating
forwards only. i or
In carrying out the simplifying transformations we may
drop for the moment the suffix m.
From the last two terms we have, in part,

—dy.E+y.dE=+VdVvE
= + Vd curl Ey eee)
From the remainder we find

VdB+ V (dy .q)B+ Va(qy.B)—dv. VgB+ v,.dVqQB,
= VdB+ V(dy .q)B+V .d(qv.B)
(—V(ay .q)B—V .a(dy.B)+y1.dVqB,
=VdB+V.d(qvy.B)—V.q(dv.B)
—V.d(qv.B)+V.q(dvy.B)+ Vdq. vB
=VdB+ VdadivB,. 0 4

where the suffix restricts the action of vy to the vector
carrying the same sufhx.
Equation (21) then reduces to

Vd,;(curl B+ B+ div Bi). =) Dates)

Now d,, will have different values according to the different
directions of the various sets of tubes; hence (unless ail.the
tubes are parallel) we may write

‘curl }B+q,,divB=0.. 1 Yo 907 Qe)

From this, since q,, is the velocity of any set of tubes,
unless all the sets have a common velocity, we must have

div B= 0} hee a 2s Sea er
and thus
“enrl R= Bie oe eer ee (27)

We have now shown that the first four laws of the ordinary
theory of electromagnetism are consequences of the assump-
tions which have been made. It may be observed that
whereas, in the proof of the first two laws (3) and (4), no
departure of importance is made from the method of ‘ Recent
Researches,’ the proof just given of the laws (26) and (27) is
quite different from that adopted in that work. This is

Faraday-Tube Theory of Electro-Magnetism. 601

rendered necessary by the purpose of the present paper,
which is not to deduce the properties of the tubes from the
known laws of electromagnetism, but to show that, given
the tubes with the (essential) properties assigned to them by
Sir J. J. Thomson, the laws of electromagnetism follow.

5. It remains to discuss the forces acting on the electric
particles. Referring to the figure on p. 598, let B be a
particle at the end of the tube B,C, D. ‘Then the change in
L due to the displacement of the end of the tube from B to
A (introducing a new segment BA), is by (13)

Boe Ee VOB ere sy fe. ues C28)
since

éd,,=AB= — or,

B being the positive end of the tube, and thus equivalent to
a positive unit of electricity. Hence the foree acting per
unit charge moving with velocity q is ©

| ie OTE Mate 3, Aa era a Ek 45)

the Fifth Law of Electromagnetism.

6. The definite dynamical assumptions of this theory
enable us to examine very thoroughly such questions as the
stress in the field and the mechanism of radiation.

Heaviside * has given a general discussion of the problem
of stresses from which it is not difficult to deduce the
following general result :—

Let yr be the operator of Maxwell’s stress,

yy=E.D+H.B—3(ED+HB), . . . (30)

where any vector operand forms with D and Bscalar products
in the first and second terms. When this operand is a unit
vector N, WN is the stress on the plane perpendicular to N.

Let be the stress derived from yy, by putting for E,
E+ VqgB, and for H, H— VqD, namely

w= + VqB.D—VqD.B—43(VqB)D+4(VqD)B
=yW+ VqB.D+ VDq.B—DVqB
=Wo+VDB.q. Z : ; ee 2 é . (31)

by mere vector transformation.

Then if Nis unit normal toa surface moving with a velocity
q at any point, YN is the flux of momentum through the
surface in the direction opposite to the positive direction of
N, per unit surface per unit time.

* ‘Electrical Papers,’ vol. ii. pp. 521 et seg.; also Phil. Trans, A. 1892

|.
|
|
i:
|

602 The late W. Gordon Brown on the

To see that this is true we have only to apply the theorem
of divergence; in the first place we note that since

ol. — Vd,,B, c 0 iY ot ° : (18)

summing for all values of m we have VDB equal to the
momentum per unit volume. Bat

yv= 2 VoB, Me

a result easily deduced (Heaviside, loc. cit.) from the circuital
laws, and usually expressed in words by stating that Maxwell’s
stress gives rise to a translational force per unit volume
equal to the rate of change at a fixed point of the momentum
per unit volume (the absence of electrification being assumed).
We are thus entitled to say that YN is the flux of momentum
per unit area of a fixed surface. Now it is clear that
VDB.QqN is the flux per unit area due to the motion of the
surface with velocity gq. Hence w is the general operator
giving the flux of momentum. The equation of rate of
change of momentum per unit volume at a point whose
velocity is q is 7

yo= 2 VDB+ vq. VDB

a 2 VDB+ay _DB+VDB.divg, . . (83)
the first wo terms giving the rate of change of density of
momentum at the moving point, and the last term the rate of
change due to expansion at the rate div q.

This flux of momentum > is partly due to convection, and
partly to be ascribed to a stress. It is interesting to note
that if all the tubes were of one set, we could determine the
stress by simply putting q equal to this velocity. We should
then have H= VqD, and the stress would be

=(E+ VqB).D—3(E+VqB)D
=F.D—LFD
=F. D+L(B=—ED). a 2) 3

In general the stress operator will be obtained by sub-
tracting from yo the operator —3(Vd,,B.qn) which gives the
convective flux of momentum relative to a fixed point ; thus

a

Faraday-Tube Theory of Electro-Magnetism. 603

the stress is

p= Wot+d(Vd,B. oer (35)'
=E.D+H. B—1ED—1HB+SV a ee eis Va. ai: B
+3(Vq,,dm)B
=E.D+H.B—iED—+HB+Vq,,B.d,—H.B+HE
= {E+ Vq,B.d,} —4ED+iHB. FEM Leen eres (DD)

From (35)' we see that the stress coincides with Maxwell’s
stress when there is no convection of momentum relative to
the (so-called) fixed reference frame ; and from (35) that it
consists in general. of a quasi-tension equal to E+ Vq,,B per
tube of the mth set together with a hydrostatic pressure
4(ED—HB). The torque per unit volume is seen to be

i) a gp’ =s=— +V(E + VqmB) cee

= + > VdinVqmB
=-+ >VanVd,,B ai >V (Vqmdm)B
SSeS MeN Bt a ad ats ts Moe (3B)

the last expression being the rate of change of moment of
momentum about a fixed point due to component of velocity
perpendicular to the momentum, familiar in the hydro-
dynamics of the motion of bodies in a fluid.

7. The flux of energy also consists of two parts: the

convective flux due to the motion of the tubes, and the flux
due to the activity of the stress. To find the convective flux
we require to localize the energy in a manner rather difficult
to justify. The whole energy per unit volume may be
written
INB+1ED
=+4%3d,,(E— Vq,B). Poe tke Th 3 (37)
Then we may suppose the part d,,(E—Vq,,B) of the energy
to be moving with velocity q,, and so on. The total con-
vection of energy will therefore be
$3d,.(E— —VqnB). Qm- e. berRiise) Wurm (38)
To find the stress- pies flux from (35), consider first

the term (E+ Vq,,B) .d,,; the appropriate velocity is clearly
Qm, and the flux (by Heaviside’s method)

—Qn(E+ VgmB) .dn= —QnE. dn.
Again, we may write the second term
— LED + SHB=—34(54,,)E—(SVandn)B
= —42d,,(E+ VanB),

and it seems permissible to write the activity flux due to the
term —34d,,(E-+ Vq,,B) as +4q,.d,,(E+Vq,,B). Hence the

604 "The late W. Gordon Brown on the
total activity flux will be a
: —S4qnE.d,—$dn(E+ VanB)}, . . . (39)
and the whole flux, adding (38) and (89), ne.
W =324,,(E— VanB) «Gm —2OmE «dn +320,,(E + VanB)n
dee Qn —AmE . din) .
= VES Vq,,d,
== aR RT erie Le te al

8. Since we have By ‘that this theory leads to the
ordinary equations of the electromagnetic field, it is un-
necessary to give a separate proof of the uniform propagation
of disturbances with velocity 1/ mK. It is perhaps as well,
however, to examine shortly the mechanism of propagation,
particularly since the mental picture of electromagnetic
radiation afforded by the theory is in many respects very
satisfactory.

N. Campbell gives a short discussion of the question, and
shows that a tube at rest may be compared to a flexible cord
of linear density »D under a tension D/K ; the square of
the velocity of propagation of transverse disturbances being
then 1/uK by the elementary dynamics of cords. To extend
this result to the case of a tube having a general velocity v
perpendicular to its own direction, we have only to remember
that, by equation (39) above, the stress to which the restoring
force i is due will now be the. quasi-tension E+ VqB, where q
is the velocity of the tubes, of which we ‘shall suppose that
only one set need be taken into account ; and with this last
assumption we may drop the suffix m and so write

. d

B =pVqD, k= K :

The d component of E+ VqD is the only effective part of the
stress, and its magnitude is given by

d
(E+ VqB)d, = (x + pVqVad ) ie

where d, is the unit vector parallel to d, or d=dd,. This
equals

2 (1+ wKd,VqVqd)

d
= x (l-wK(Va.a)"}

vw? !
lo ee aie, il)

Ale

eukK=1.

Faraday-Tube Theory of Electro-Magnetism. 605

The linear density will remain pd, so that the velocity of
propagation along the tube will be 4/c?—v?. Since the tube
itself is in motion with velocity v in a perpendicular direction,
the propagation of the disturbance in space will be with

. . . . . . UV .
velocity ¢ in a direction making an angle sin~"- with the

tube. When v=c the disturbance will not be propagated at
all along the tube, which will lie in the wave-front ; and the
traction (E+VqVqD) will vanish.

9. To take into account a general velocity of the tube in
the direction of its length, let us restrict ourselves to plane-
polarized radiation. We shall take the w-axis in the direction
of propagation, and the y-axis in that of the disturbance.
Since we are dealing only with transverse vibrations, the
velocity of the tubes in the direction of the ray will be
constant from point to point along a tube. Let wu be this
x-component of velocity. Also let (#,y) be the coordinates
of a point on some particular tube at time ¢t, so that y is a
function of z andt. Then the whole y-component of velocity
of the point will be

se eee OF ?
oS pea Nin Ey eee See (42)

It is obvious that the shearing motion perpendicular to the
a-axis of the tubes in their vibration will not affect the
number of tubes per unit area passing through a plane normal
to the x-axis. Thus the quantity d,, the x“-component of
electric displacement, will be constant at a point on the

tube, or
fe Or oa
(2 +u <.\d-=0.

Also, if d, is the y-component, we shall have

dy _ OY
de OF
And thus ips oy,
P= dy + dy?
DE | Oy ie ;
pe ia las) ; ; Sere 2 os (43)

The momentum per unit length along the tube is
Vd,B=,Vd,Vqd
4 | . d—d ° qd;) .

606 The late W. Gordon Brown on the
Multiply this by $ to find the value appropriate to unit

length along the #-axis, and, taking the y-component, we
have

war Ge F530) 45a ate (SE 44g) ef
pd,

mua.| (2 +e a ae (a y ut) OU |

Ot Ox Ov ae Ox

[Se ++(S2) -(82) } +32 +(82) 2-2) f
Side ol

Hence the rate of change of momentum in the y-direction
per unit length along the #-axis is

O20) By Oy O°” D
(5: tego )adedy eee Set aapef CS)
The force to be equated to this arises from the quasi-

tension

B+VoB= — +yVaVaD

d 2
= ted. ad—pd.a°,

of which the y-component is

cd SY 4 w(t 4) Sud, + (SY +See)

Kei ow fo Ot. 0 foe
—ud.2" 4 2+(8 a (22 4a out
fe OF oy, OY Oy
mie 3p HS; Tees ud os 2
il
= hel + pd OU wan ie beam ae MMT fics ma » (46)

Differentiating with respect to # we have the force per
unit length

a,| oh tweed ay a

Faraday- Tube Theory of [lectro-Magnetism. 607

Equating therefore expressions (45) and (47) and dividing
by dz, we have

10% By | Oy | OY
K 32 THY 30 ot aco | Ot”
or
O’y 2oy e
Ae ok wating joy C2)

exhibiting the uniform propagation with velocity ¢ inde-
pendent of the general motion of the tube.

The relation between the electromagnetic disturbance and
the displacement y of the tube is easily seen to be given by

eyo OY
By= 3 = Kd, Se enc
Ov
ee at
= . . (60)

But while in plane-polarized radiation the displacement of
the tube from its normal position is thus perpendicular to the
plane of polarization, in circularly polarized rays it is easy
to see that the reverse is the case.

10. The intention in presenting the theory of Faraday
tubes in*the present form was to suggest possibilites of
modification which might explain various phenomena of
which no entirely satisfactory electrical explanation has been
given so far.

In making attempts of this kind we may, for instance, take
advantage in various ways of the fact that the electric dis-
placement has been considered as a mean value taken over a
small, but not infinitesimal, area. From this point of view
the Maxwellian theory is microscopical, and a more micro-
scopical theory may be what is required in various regions
of modern physics.

Again, the present theory rests on the fecalieaiion) of
electric and magnetic energy as functions of D and H and on
the derivation from these of equations of motion. Hence it
would be comparatively simple to estimate the effects either
of a modified distribution of energy, or of substituting any
different hypothesis for the principle of action.

Lastly, quite a variety of hypotheses are possible as to the
exact nature of the electric particles.

11. It will be observed that in describing the properties of

608 The late W. Gordon Brown on the

the tubes of force we have so far assumed that two oppositely
directed tubes at the same point exactly cancel each other in
their effects, if they are moving with the same velocity.
Now, just as the electrical theory of matter explains all the
phenomena of neutral bodies as due to the existence of the
equal mixture of positive and negative electricity, which on
the two-fluid theory was supposed to have no recognizable
physical properties, so on the lines of force theory we may
perhaps speculate with advantage on the possibility of ex-
plaining by means of properties of equal mixtures of
oppositely directed tubes the phenomenon of gravitation,
which seems for many reasons to be on a different level from
the ordinary electrical phenomena. Let us consider the
potential energy of such a mixture of tubes. So long as we
choose an element of area large enough to include many
tubes, the density of energy 4ED must always vanish ; but
as we take smaller and smaller elements of area, there will
be an increasing probability of the number of tubes passing
through it in one direction being not quite equal to the number
passing through it in the opposite direction : in other words,
what to ordinary microscupic electrical measurements is a mni-
form absence of electric displacement may consist of alternate
regions of opposite displacement so smali that only the mean
field of a considerable number of regionsis measured. Such
a field would have positive potential energy ; but since the
more closely the tubes are packed, the smaller is the element
of area we can take without considering this effect, it seems
reasonable to suppose that the effect will become smaller the
more numerous are the tubes of either sign. Not improbably
a mathematical form might be given to this hypothesis which
would explain and locate the energy of gravitation. Let
de,, —deé,; deg, —de,, be pairs of opposite charges ; 11, 72
the (small) distances apart of the components of each pair ;
and R the distance between the pairs. Then if the hypothesis
could be so formulated that the potential energy of the
system would include a term of the form

—yde,*de,*
3 Tro
where ¥ is a positive constant, the law of gravitation would
be completely satisfied, and gravitational mass would be
identified exactly with electromagnetic mass ; for
Vy

is proportional to the element of electromagnetic mass due
to two elements of de,, — de.

Faraday-Tube Theory of Electro-Magnetism. 609

This last question is of some interest in the theory of
atomic structure ; a number of writers have laid stress on the
importance of mutual electromagnetic mass, and in particular
Harkins and E. D. Wilson * have used this phenomenon to
explain the departure of atomic wezghts from whole numbers.
It appears, however, that such an explanation could alone be
valid if mutual mass were ponderable.

12. The theory of Faraday tubes might possibly be em-
ployed with advantage in other investigations connected with
atom theory. Sir J. J. Thomson + has made several sug-
gestions of this nature; his conception of the electron as
possibly simply the end of a single Faraday tube would, of
course, have very important consequences if adhered to in any
theory of atomic structure.

Again, if we suppose that electrons and positive nuclei
have the property of excluding the tubes of other electrons
and nuclei, the attractions between particles of opposite
sign would become a repulsion at very small distances. Or
we may suppose that some or all of the tubes of an eiectron
in an atom simply end at a nucleus, instead of spreading
equally outwards in all directions ; and different states of an
atom, with different periods of vibration, might arise according
to the number of tubes so connected. ‘Suggestions have also
been made as to the application of the theory in connexion
with a possible discrete structure in radiation f.

CONCLUSION.

13. It has been shown that the general equation of the
Maxwell-Lorentz-Heaviside theory of electromagnetism can
be derived as macroscopic consequences of a simple dynamical
theory of Faraday tubes.

This theory also gives explicit and non-contradictory
expression to the ideas of electromagnetic stress, momentum,
and flux of energy, and an electromechanical picture of
radiation explaining the law of uniform propagation in spite
of the motion of the source.

A number of suggestions are made as to applications to
the theory of gravitation and other problems.

Hawke Battalion,
Royal Naval Division.

* Phil. Mag. Nov. 1915, p. 72

Tt Phil. Mag. (6) xxvi. p. 792,

t Jeans, “Report on Quantum Theories,’ Proc. Lond. Phys. Soc.,
1915.

Phil. Mag. Ser. 6. Vol. 44. No. 261. Sept. 1922. 2R

) 610 J

LV. Molecular Thermodynamics. III. By Bernarp
A. M. Cavanacu, B.A., Balliol College, Oxford *.

| SOLVATION OF SOLUTES. .
ee partial solvation of a solute—that is, the. com-

bination of some fraction of it with the substance
which, in the free state, constitutes the solvent—to form
a ‘‘solvate” or “solvates”? 1s a phenomenon probably of
the very widest prevalence in solutions, particularly in our
more common and valtable solvents. It has unfortunately
been too readily ignored, because in dilute solutions its
effects are not of the first order of magnitude. They are
nevertheless considerable, in many cases much more con-
siderable than has generally been allowed for ; and, in any
attempt to pass beyond the region of the most dilute

solutions (say, 55), the question of solvation of the solutes

must receive very serious consideration. 5
In the present treatment, it is to be understood, the
“solvates ’ considered are true chemical compounds, at
any rate in the one sense required by molecular thermo-
dynamics f, and the conclusions reached are not to be
expected to apply in any degree to any other “ associations ”
or “complexes” to which the name “solvate” might be

given. .

The following are the general circumstances considered
here :——

Any number of “solutes” dissolved in a complex ft
solvent are indefinitely solvated, without dissociation or
association—that is to say, each forms one unsolvated
molecular species and any number of solvate-species by
combination of one unsolvated molecule with different
amounts of solvent.

The residual solvent not so “absorbed” by combination
with the solutes may be called the “ free solvent ”’ §.

The separable or experimental solutes have been called
the ‘“unsolvated” solutes, and it would appear a little
paradoxical to say that these may be solvates ; but, in fact,

* Communicated by Dr. J. W. Nicholson, F.R.S.

+ See second paper, Phii. Mag. xliv. p. 229 (1922), 1st section.

+ See second paper, Joc. cit. The case where the solvent is simple is
included as a limiting case.

§ In the previous paper the term “solvent” was used to signify
‘‘ free solvent,” since the combined or ‘‘ bound” solvent was not there
under consideration.

¢

: we
a"

On Molecular Thermodynamics. 611

the treatment below will apply in full and without alteration -
in such a cuse, if a lower solvate be regardedas a “ negative”
solvate of a higher.

The following alternative and more general statement of
the problem will make this clearer :—

Yhe molecular species present in the phase or solution
are the several solvent-molecular species which ‘together
constitute the ‘free solvent”? and a number of solute-
molecular species. Now the latter can be grouped in such
a way that members of a group differ in composition only
in the amount of solvent per molecule, and can be regarded
as positive or negative solvates of one another. One member
of each group appears in the present discussion as a separable
or ‘‘ experimental ”’ solute-—a component under the conditions
of experiment ; and the (positive or negative) quantities of
solvent by which the other members differ from this one
are, and are alone, regarded as ‘“‘bound”’ solvent, which
together with the “ free ” comprises the “ total solvent.”

In the “molecular” expression for yw, from which we
started in the previous paper, every molecular species
appears as a component. In the “ experimental ” expression
obtained in that paper the free solvent appears as one com-
ponent only, and in the present paper, starting from this
result, which we may refer to as our “ original “ expression
for wp, we shall obtain an expression in which the “ total
solvent ” and the ‘‘ unsolvated solutes”’ are the components.

It is clear that problems, such as dissociation and com-
bination of the solutes, excluded from present consideration,
can be treated as further stages in this gradual reduction
of the components.

Let M, be the mass of free solvent considered, and M,’'
that of the total solvent, the latter but not the former being
an experimental (known and controllable) quantity.

Concentrations referred to the latter may be called
““ experimental” concentrations,

DY, Peas)

feet te. Re es ae
=p? C MiGr ane (1)

while those referred to the former,

mee eS Site N
cs = M,’ Sie M, = M,’ . = oy Wie (2)

may be ealled ‘‘ true”’ concentrations. The latter alone were
used in the consideration of “ complex solvents.”
It will be a convenience making for clarity, to obtain

2K 2

612 Mr. Bernard Cavanagh on

our results in the first place in terms of the “true con-
centrations,’ introducing the “experimental” concentrations
afterwards.

Of the total number n, of gm.-molecules of the solute, (s),
put into the solution, let the fractions ws,, as,,...... , go to
form solvates containing respectively M,,, M,.,...... > grams
of solvent per gram-molecule, and let us write X, for 22;,,,
and a; for 2s, Ms, ; then clearly (from previous paper) our
‘ original ” expression for sf is

ee Mo[gxtR| = ae (140) |
c=

vim f a -X,)[.—R log gaa

m

—
~

Css vod
+ Das, [@.—R log aera } +M,Gs 2)
MoC

and is of the form
fo) fo) fo)
ap = My Su, tet Xe) se ai BEng Ss, s (4)

the independent variables being My, n.(1—X,), nets,
Ns& 504 eoceces

The ‘“‘experimental”’ expression for yw, now required,
must have the form

De,
vr = M,’ oM,! +3n aay ae? (5)

the independent variables being My’, n,, ......

In changing the variables we make use of the equations
given by the chemical equilibrium governing the various
stages of solvation, viz. :

OV Pot Ob

Ons, < ONsy jas ill; ne
Ons. Ones 29 OM,
etc.,

whence, of course,

OY uy OM ov

Se =a Sis “SOM, =) Bi ar tade KONE AC (7)

Molecular Thermodynamics. 613

and similarly

ON teres
Es, tnt, A1—X)

whereby the (somewhat lengthy) full differentiation of (3)
with respect to the new variables reduces to

ow gege, er (8)

Ov _ ov (9)

OM,’ OM,’

Ov _y, OE Ch oh a

Sn, 7g, 1 KS tes -. (10)
= 2, (11)

equations of obvious physical significance.
Now (11) with (3) gives

B3 +R log | = ¢,,—R log (1— Xs)
+R log (= Hf mig()) BGH G2)
and

Lt [S¥+ Rog cs | = ¢y—B log (1—X,,) = $s (say), (13)

since X,,, or ie X,;, depends only on temperature and
pressure.

Then
B¥ + Riga] = dt | a(R tog],

1. é.

On the one hand, of course,

Sie ela
in. as MgO
m

i Hx] +Ge! =, (15)

but, on the other hand, from (8) and (10) we get

a(3t) = 28 (Se) +a—Ko4 (SF) —aa( Sa, 9

ov be— —R loge Cs + +f a3 oF R log cs]. (14)

ee el

614 Mr. Bernard Cavanagh on
and, referring to (3), while remembering
La, +(1—X,) = 1,
Ya,,d log #,, +(1—X,)d log (1— Xs) = 0,

we obtain

d [SY + R log «, |

mH [a ee (S+mc)—Sa aie +m) |

+ Xw,,dG,,’ + (1— Xs) dG,,/ —a#sdGm'; (17)

whence (14) gives, as alternative to (15),

Ons Mow
= + Te) C=0 (18)

where

Gr=\ 10x dG./ 42nd,’ = .dGa Jon

C=0
Comparing (15) and (18), we obtain
— Pulivees Ea: te

R ee Ke = R| : —d log (1+ mC)—(G.—G,, ), 20)
or 4

Rd log (L—X,) = R Sd log (1+mC)
=
+ [a,dGmw’ +X,dG,,'—d#,,dG,,'], (21)

which could be obtained independently from (7) and (8)
with (3), and then used to get (18) from (15) ; but the
above derivation from the physically significant equation (10)
appeared more interesting, and brings out an analogy with
the previous problem of “ complex solvents.”

When there is present only one solute (s) (but any number
of solvates thereof), (18) can also be obtained easily, by

means of the Gibbs fundamental relation, from oF .
0

The general terms being G, in (18) and G,,’ in (i5)

we see that the division into “linear” and ‘* general 7”

Molecular Thermodynamics. 615

terms is different in the two expressions for a the
s

difference being the quantity
Co 1 { Bee ae eM Cac). « (22)
C=0

The form (18) will probably be the more generally useful,
possessing the advantage that the “ linear ” and ‘‘ general”’
terms separately are connected by the Gibbs fundamental
relation with those (respectively) in the simple form

ov
f 5M,”
a= 2 st = [otk | <dlog (1+m0) | +Gy'. (23)
C=0

For since (cf. (3), (4), and previous paper)

M,dGy' + ¥n,(1—X,)dGz' +Sn.2,,dG,’= 0°. (24)
and .

ge eee eae een ey
clearly
M,'dGy' + Sn, [ (1 — X,) dG,,’ + S2,,dG,,'—a,dGu']=0, (26)
2. e. Mion Sones = Oe te (28)
so that, if G is a function of ¢’, ¢9',...... (‘“* experimental ”
concentrations), such that
M,'G = M,’ Gm’ + SnsG, : - : = (28)
we have
Sys GSS) a ee :
Ge = (SS = Se ee <8)
OM,'G as
Gar = (Sar 7). [= Gu (say)] =G—Se/&*.. (30)
Now G’ is a function of ¢, ¢, ...... and also of tlle various
fractions #},, 21,,...... 5 Mg eed hs If we take the quantity
eee ae
ML,’ G', and eliminate the “true” concentrations (C, ¢,) by

* This common usage of the suffix outside the bracket was mentioned
in the previous paper. x here indicates constancy of all the n’s and #
(later) constancy of all the 2’s.

616 Mr. Bernard Cavanagh on

means of the relations

!
1— dc, as = wa: => as =¢ sh eee ae ote Gy

we obtain a function (say, ai of the “ experimental ”
concentrations (O’,cs’) and of the various a’s, as inde-
pendent variables, and clearly

Se eee

= 6, PME) 430, e) +05 BH),

Ons
= (l—X)G,’ +32.,G,,/—a:Gu',... >. 1) 2 aes
for which we may therefore write G,’', so that
M,'G" = M,'Gu' + dn.G.". 2 2 ees)

On the other hand,
a ba me Se le ae
= M/ naz Gey oe (see aa

| eG

es [ Gs, — Gs, —Mi,Gau'] = Gz,” (say). = (84)
(19), (32), and (34) then give

Gs = G,!’- Gao rs

C=0
mG: p) oGe |
=Sa Sle aie me SS.
C=0

and therefore from (28)
G= ode 2 Ga, xs,

ele

= Ows,

dz, iGo
ox

(35) and (36) being plainly in accord, as they must be,

with (29).

Molecular Thermodynamics. 617

Our final “ experimental” expression for yy, in terms of
“ true’’ concentrations, is

ce 1 pS
wr = M, [out Rf —d los (1+: nC) |
C=0
Cs

+ ins [ oR flog =———

mM =

+ (Ba log (1+ 0) } +e, (37)
C=0
where G has some form

Gree (ales earn one cat tae)

The “general” terms in the original equation (3) were
M,G'—that is, M,'G"’; so that the quantity
; reins OG!
v M Vins ati SSE
Sn. | 3G., dfs, or Ene 71> ae Te GD)

C=0

has passed from “‘ general ” to “ linear ” terms in w& ; or, in
b)

cy, the quantity
$s

Gs'—Gs= | Ge, "dt, . + + - (40)
C=0
We may recall at this point equation (22), which is now
Gs—Gs, = | 2asdGe,", - - 2 + (41)
C=0 :

and note that, disregarding (more or less fortuitous) can-
cellation, the condition for the vanishing of (40) and (41) is
qualitatively the same, viz., that G’” shall be independent
of the extent of solvation of the solute (s)—an obvious
particular case being ‘Perfect Solution,” where G"
vanishes.

The lack of practical significance in such a qualitative
statement is illustrated and emphasized when we observe
that quantitatively the condition is very different in the two
cases. The two differences (40) and (41) are, in fact,
complementary parts of the quantity

eee eee fe! (aay

618 Mr. Bernard Cavanagh on
and since sith
{ Co Gan
te dix ce

whereas ( dz,, will be only some fraction—generally, as
wic=0

we shall see, not large—of «;,, a simple graphical considera-
tion will suffice to show that (41) will generally be much
greater than (40).

The practical significance and value of the quantitative
statement (39) is further illustrated in Appendix II.

The analogy between the modification of the “ general”
terms, expressed in (39), and that found necessary in the
previous treatment of ‘‘ complex solvents” is plain. In
the latter we kept

fA ;
Gres G; ’

whereas in the present case we have kept
Gu mae Gy’,

but, of course, since we started in the present case from the
results of the previous paper (and G’ in this paper corre-
sponds to G in the previous paper), the two modifications of
the ‘‘ general” terms in the ‘‘ molecular” expression are, as
it were, superimposed.

More detailed treatment than the above of the “ general ”
terms will be possible only when something is known about
their form (see, for instance, Appendix IT.).

For the evaluation of the linear terms we observe that.
_ a; must be expressed as a function of the concentrations of
the solutes.

An Expansion of as.

Consider, first, a solute which forms only one solvate. In
this case we can write as, M,, etc., instead of a,,, M,,, ete.,
and, moreover, we have

ag = Ma,
X; = Ws : ° 0 ' 5 (43 )
ee

and equation (21) assumes the form,

Rd log (1—a,) =as Es log (1+mC) —dG,, ale (44)

ee

Molecular Thermodynamics. 619
or if we write
== ; 4 ; aay 1 | B\
y= [uf = log (1+ mC) — Bors Sat (AOD
then

d
— am Sa fy —— wo). e . . ° 4

If now we write w, as a series of ascending integral
powers of y, we can show that a few terms only need be
taken for our practical purposes *, so long as y is little
greater than unity—that is, we obtain a practical expression
for ws which will serve up to very high concentrations.

Putting then

Us= As, t+ by t+ boy? + bsy? + 2... +Ony”...

by straightforward substitution in (45) (and equating co-
efficients) we obtain 0, b,, ...... in terms of a@s,.

A simple relation between these coefficients enables them

to be written down very easily, viz. :

1b, = by ao
the first coefficient ), being zs,(as5,—1), and being a factor of
all the others. The alternate coefficients are also divisible
by (a’s,—34). :

The numerical values of the coefficients depend, of course,
on that of ws, which may be any proper fraction (positive).
Their maximal (absolute) values can be obtained and shown
to decrease rather rapidly. The higher members of the
series have, of course, several maxima, hut the greatest
maximal (absolute) values diminish by alternate long and
short steps, the. following round values being sufficient for
present consideration :

b,. be. bs. ve De. De b..
1 i i i 1 uh er
+ 22 48 300 480 4400 -

Taking } per cent. as our standard of experimental accuracy,
we see that the series as far as 7°,

Vs Xo + by + boy? + bsy?, : . : ° ° (4 7)

will suffice until y closely approaches unity, when the 7 term
will be just appreciable.

* That is, the remainder after a few terms is negligible experimentally
up to high concentrations. We may conveniently call this “‘ converging
practically up to high concentrations.”

620 Mr. Bernard Cavanagh on

We have then a as a practical function of y, which is a
function of the concentrations of the solutes (see (44)), and
if we know the form of Gz, we can readily complete the

as

evaluation of ie dlog (1+mC).

Mh
C=0
Us

R

First, let us assume that =; G,," can be neglected and write

y=M| J dog Gmc):
C=0 3 .
From the previous paper we can then write,

y= M,C[1—$a,C + 4a,C? —443C*],
and from (43) and (47) we then get

t= a+ B,C y,C? +6,05%.. . 5.) eecean

Where
as Mien (3S (b.M,? mae $y b, M,”) 9 49
Be= M275), onan Osan ie eee eS
which may be expected to hold up to a value of C rather
higher than ir (since y is considerably less then M,C at

high concentrations), that is, in aqueous solution for example,
up to a total concentration (“ true’’) of at least twelve-molar
(12M.) when the solvate is a pentahydrate, or six-molar
(6M.) when it is a decahydrate.

When ae! is not negligible the above will cease to hold

exactly, but it may be possible to express =0Q5 (with a

R
negligible residual error) as a function of C, the total solute
concentration, and if this function can approximately take
the form of a short series of integral powers we shall get a
result in the above form, (48), but with departures in the
values of the coefticients y, and 6,.
When the solute considered forms several solvates, the
problem of obtaining «, as a function of the concentrations
is more complex. Hquation (21) takes the form

d log (1—X,) = 2a, [= dlog (1 + mC) —

but we now have to use instead the series of equations (6)

dGe,,” 3 (90)

Molecular Thermodynamics. 621

from which, as already remarked, (21) might have been
directly obtained, and which can be written

= Mint h — Lay 2 4a,C? —ta;,0*)dC— cen
1X, Hit (BU

vs

d log

=M,1—},0 + Ja,(?#—3a,0%)dO— 5 dGe,,

(and so on).

Again, we pass over the most general ease, where it is
necessary to consider Ge"; etc., as depending in specific
ways upon the particular solutes present, and we suppose

a . . ° al
(except for negligible residues) G,, ", etc., can be accounted
for as functions of C. If these functions can be written
approximately as short series of integral powers, then the
above equations will take the general forms,

Fee. NB Y
= [got mC + 920? + g3C® Poses |dC |

d log

En
fee Re eee ante , (52
dog —* = fg! + 9/C-+g/0r+g/0......Ja0 (7 ©”
ZS |
ete. J
WHERCE 25, 25,5: .2-5-: (and so e,), can all be obtained in the

form of ascending series of integral powers of C, by a
procedure entirely analogous to that used in the simpler
case of one solvate, viz., by assuming such forms for a@,,,
, and then determining the coefficients in terms of
Gos Gy. Pas7- ee , by substituting in the above equations used
of course simultaneously. |

The interesting simple case where there are two solvates
ae Gog. ean, be neglected, has been investigated and
it has been found, as might be anticipated, that in the
expression for a,

a=a,+8,C+70C?+6,0?, . . . (53)
finally obtained, the coefficients are quite of the same order

of magnitude, as regards maximal values, as in the case of
one solvate. They are conveniently obtained in terms of a

a

622 Mr. Bernard Cavanagh on

-series of symmetrical expressions of the form

(My @e4- Ms), CMs, 7 2s, + Mig eM, 8a. tM eee
where %, and w,, stand for the limiting values Lt ,,
and Lt a,.,. C+0
C>0

The same procedure exactly, however, can be applied to
the more general case (52), but becomes, of course, more
complex and laborious as the number of solvates is increased.
There can be little doubt that the maximal values of the
coefficients remain of the same order of magnitude so long as
Guo,", ete., are minor quantities as they will be in general ; so
it will be assumed in what follows that «, can be expressed
in the form (53), and (49) will be used in roughly estimating
the ranges of validity of the results we shall ebtain.

THE INTEGRATED LINEAR TBRMS.

We can now evaluate the quantity

{¥atog (L+mC) or f a.(1 a0! + an? —agC*)dC,
c=0 : c=0
and, subtracting log (= + To (as evaluated in previous

paper), obtain

— log = at mig) + {2c log (1+mC)

c=0

, =¢,C + $2,074 42,0? + 12,0) = (oa)

‘ where
é=:(4s,—41), €3 = (Ys — 41 Os + Ag as, — a3), } (55)
09 = (Bs— ay As, + Gy), €y= (05 — A Ys + 428s — 434s) (

and (37) now appears in the more practical form ; -

Lon |

OM

oY = $;—R [log ¢, + e,C + $e0? + de303 + ¢,C4] + Gs

=u + RO (1—$a,0 + Jao? —30,0%) + Gu
. (56)

INTRODUCTION oF “ EXPERIMENTAL ” CONCENTRATIONS.

We have now to introduce the “ experimental” concentra-
tions (¢c,', C’) in place of the “ true ” concentrations (c,, C)
in the expressions we have obtained.

Molecular Thermodynamics. 623
Since MS= Mo a DNges,
let us write (as we have already done) N for }n,, and also

N2=Dd nas

= >n; [a,,+BsC +ysC? + §,C? ’

and T= + BC+ 7C? 4 80%,

so that oh zn = , B= et, Sly cy day (OO)
Besa © a M,’ ML:

and G,! = Cc = M, =i | se)

=142,0+8C?+yO? +804. (57)

If %; 8, 7. 6 are not to be determined, but are already
accurately known, equations (56) can be used as they are, for
the ‘ experimental ” concentrations can be translated into
“true”? concentrations by means of a curve plotted from
(57).

We are dealing now with the more common case where
Zs By ete., are not known.

From (57), in a fairly obvious manner, we obtain

3 =1+7,C' + 7,0? + r3C'? 4-740", ° : (58)
where
pee r= (293+ 3%8+7), | 59)
ro= (Hy? +B), y= (Got + 6%°B+ 4ayy + 26?+5
and
ie we =1,C' +40? +140%+41,0"%, . . (60)
where
1, =a, l;=(1a°+22,84+7), (61)
=(2%7+8), 1y=(24'+ 3%)78 + 3%07 4+ 38°45)

It is now a straightforward matter to eliminate C and
c, trom the “ linear” terms in (56), obtaining

Ov

OM, ou + RO 1+4,0' + #0? + t3C'* + 4,04] + Gu, i
0 3

( 9
(62)

=~ =,—R [log e,' +#,'C' + t,'C/? + t,'0'8 + t,'0"4] +G,

624 | Mr. Bernard Cavanagh on

where
t=(71—$4)), (= [ 73 ay (1, + $13") + aor — tas |,
to = (7, ee Oy: + 4 a9), ‘i= [%4—- ay (73 + 117) + Ao + 1”) — As?
and |
ty)’ =(4+h), ts = | e179 + €o7 + 403 + I], i (64)
to’ =(eyri tdeotl,), t4 =[ errs t+ eo(72 +4717) +e: + te, +1.) )

The Gibbs fundamental relation can readily be applied at
this point as a check on the detail.

3)

A SPECIAL PROCEDURE, SOMETIMES NECESSARY.

A difficulty arises, however, in the practical application of
this result, as it stands, at high concentrations.

We have seen that the series obtained for a, may be
expected, up to sufficiently high concentrations, to “‘ converge
practically” in the few terms given, and the same clearly
applies to the e-series of (54), (55), and (56).

In passing to “ experimental ”’ concentrations, however, we
I /
0

have introduced the 7v- and I-series for M and log——
My M,

respectively, which, when 2, and perhaps also 8 assume
large values, will frequently fail at high concentrations to
“converge practically ” in so few terms. A glance at the
values of the r- and /-coefficients will show this, and also that
the difficulty would vanish if % and @ were small, since
y and 6 occur by themselves only in the first power.

Now since (62) can be used at lower concentrations (with
the ¢ and ¢’ series cut down to two terms, see below) to
determine «) and 6 experimentally, we are at liberty to assume
on turning our attention to the higher concentrations that a
close estimate of a and at least a rough estimate of ® can
be made, though neither, of course, will be accurate enough
for these higher concentrations. Our difficulty can then be
overcome by the following procedure.

Let %', 2’, 7’, 6! be estimated values of Zp, 8,7, 5. (The
procedure is stated in a general form—if y and 5 are not
estimated 7 and 8! will be zero.) hen there will be an
estimated value (M)"’) of Mo, and ‘‘estimated”’ concentrations
(c'’, C'’) referred to it, such that

o/ORey ML)!
sl ce C’ Es My”
=[1+4%)'0" + BO" +y/C'F450"]. . . (65)

Molecular Thermodynamics. 625

Comparing this with (57) and the remarks which immediately
follow (57), we see that we can easily translate ‘“ experi-
mental” into “estimated”? concentrations and express our
experimental data in terms of the latter, so that it is now
only necessary to express (56) in terms of these “ estimated ”
concentrations.

Now &@ being really expressible as some series

a=8, + BC+7O? +508, . . . . (66)

of which we do not know the coefficients accurately, let this
be equivalent to some series

%=a,+ BO" +y"C'72+5"C"8, 2 2. (67)
Then my a
M,= M,'—N(@+80"+7'"C'?4+8"C'?). . (68)
But from (65)

M,’= M,'—N (a Peay 02 £9), KG)
so that

My= My’ —N [(Aa) + {AB)C" + (Ay') C0"? + (A6)C"7], (70)

where Lag as
(Aa) = (%—%'), (Ay!) = =F), ete.» (71)
And then we have ©
en ° M sage i!
=o = 9y,7 =1—[(Aa)0" + (a8) 0"
+ (Ay’)O"8+(A8\C"], (72)

which can readily be shown to be equivalent to

br le 2 ,
a =or= TL, =1+(Ae)C+As)C+ (Ay) C% + (A6)C%,
(73)

where

(Ay) =[(Ay) + (Aa)(A8)] 5
(Ad') =[ (Ac) + 2(A2)(Ay) + (AB)? + (A«)?(A8)]. (74)

On comparing this with (57), it is at once obvious that
the result of introducing (¢,"’, C’’) the “ estimated ” concen-
trations into (56) can be written down at once from
equations (58) to (64) by simply writing ¢,’’, C” in place
of ¢,/ and ©’ respectively, and (Aa), (AB), (Ay), (Ad) in
as of %, B, ¥ y, and 5 respectively.

(A8) being small, and (Az) very small, it is clear that the
new ¢ and ¢' series will converge with the necessary rapidity.

Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 258

626 Mr. Bernard Cavanagh on

Comparison with experimental data me in. ‘ esti-
mated” concentrations will thus give (Ax), (AB), (Ay),
and (46d), and hence also (Ay') and (Aé') [from (74) ].
This will give a, 2, 7'', and 8", and we can then at ¢ once
get ¥ and 6, for it is sisi shown lat

7" =17+B(Aa)} ; ce
5 = {8+ 25(Aa) + BL (AB)?+ (Aa)’]}. . . (75)

This procedure being available wens 2 a and # are. large,
the highest concentrations, for which four-term expansions
of « and @, suffice, fall machin the scope of the results given
here and it has been seen that, these concentrations are high.

In considering (62) in the sequel, it is understood that
this special procedure will be resorted to where necessary, in
which case (Aa) etc. will replace a etc. in the eee Hees
dasnty SeUCe

MAIN RESULT OF THE ANALYSIS.

Having now completed the analysi s we may conveniently
use Mo, C, cs, in place of My’, C’, ¢;', for the “ experimental ’ i
quantities, since the corresponding “true” quantities will
but seldom have to be considered.

If, then, we write

Jog = RO? (t+ 20 + tC? + 140°)
JRO +E0 “CLOW
(62) takes the form

(76)

0 7
OY = byt ROTI + Gye |
r (77)
oF = 9,— —Rloge+J,+G, j
and, in fact, if, corresponding to G, we write
b=Gy F 2605, (iol ok bode) ore
Jada Pe es) Lae ha aon

= —RC?(t, + $40 + 4430? + 44,0*%), . . (80)
(77) can be condensed to

= O+R3q(1—loga)+I+6, . . (81)
which is a the simplest and most concise expression
of the main result of the analysis of this and the preceding

papers.

Molecular Thermodynamics. 627

PRACTICAL SIGNIFICANCE AND APPLICATION.
Now the equation

HL = $+ RXe(L— log «), BMOD RO}

or less concisely, but more practically,
(83)

is equivalent to the Raoult-van’t Hoff limiting laws of dilute
solution, the older criterion of “ perfect’ behaviour.

In the circumstances considered in this paper (as stated at
the beginning) C and ¢, are “experimental ”—that is, directly
determinable—quantities, and to the experimenter, as such,
J and G will appear together simply as the measurable
departure (J +G) from “perfect” behaviour according to
the older (van’t Hoff) criterion. 3

In fact, in terms of the “activity coefficient” (y) of
G. N. Lewis, and the osmotic coefficient (l1—g) of Bronsted,
expressions pow much used in practice for the observed de-
partures from (van’t Hoff’s) “ perfect” behaviour, we have

logy= ~7(.+G)

oS

(84)

LU
@—-1= But Gy)

where @ is a mean quantity for the whole solution,
characteristic, therefore, of the given solutes, mixed in given
proportions.

It may be convenient therefore to call (J+G) the
“‘apparent” general terms, G the “true” general terms,
and J the “ pseudo-general”’ terms. It is, of course, only G
which represents real departure from perfect behaviour (as
defined by the linearity ot the full ‘ molecular” expression
for % ; see second paper, Section IT.).

It is clear that any physical interpretation of the
‘“‘apparent”’ general terms based upon the ignoring of either
J or G, without due consideration and adequate grounds,
must be unsound, and that in general neither will be
negligible, so that a separation of J from G must be
attempted.

This will only be possible by a critical application of the

28 2

|
;

628 Mr. Bernard Cavanagh on

limitations imposed upon J and G by their respective physical
significances.

It seems reasonable, however, that when all possible has
been done in this direction, if doubt still remains as to
whether all or some portion of the observed effects should
be assigned to J or to G, a “casting-vote ” should be
given (temporarily) in favour of J, since J does not re-
present real departure from perfect behaviour, properly
defined,

It is possible that G may be completely determined on
theoretical grounds alone in certain cases, the nearest
approach to this, so far, being in the case of electrolytes, or
rather of a mixture of ions all of the same valency, for
which the (probable) form and order of magnitude of G, in
(sufficiently) dilute solution, was obtained in the first of
these papers, on the basis of a calculation of Milner’s.

With regard to J, besides the fact that it takes the form
of a series of ascending integral powers of O, the total solute
concentration, a good deal can be deduced from the form of
the coefficients as t,, etc., and the physical significance
of the three series of quantities, the ‘“solvent-constants ”
(a;, M2, a3), the a-series (2, 8, F % 5), and the a,-series (cs,,

Bsy Ys9 bs)

THe Form OF THE COEFFICIENTS ¢,, ¢,', ETC.

(59), (61), (63), (64) give us these coefficients as functions,
at first sight rather complex, of these three series of quantities,
but for practical purposes this complexity is only apparent,
for each successive term of the J;-series introduces Just one
further term from each of the three series of physically
significant quantities, and introduces it in the first degree,
while in J,, the same is the case except that the a,-series
does not enter.

These coefficients can be obtained in a much simpler and
more practical form by introducing the quantities fs, hs, ks, ...,
and the corresponding mean quantities t, h, k, ..., such that

ky—Ys— Lah =h— y = Sayh

|
LU aa Th Cena RR i (as
= — 40; +4a,a,—30,°~ — ya,° J

Molecular Thermodynamies. . 629°

anlso on, for we then obtain

t=t; ty’ =t+t, )
=P +h; ty! = (tty + Bt?) + (h+ Bhs) | re
tz=t+3th+h; te’ =(@t, +40) +(2th+ tat ih.) hs )
+ (b+ 3A) J
and so on.

Clearly the coefficients of J,, alone, determined term by
term (with increasing concentration) will yield the quantities
t, h, etc., one by one, while, if the coefticients of both J,, and
3 are BiGRibd the pairs (¢,, t;'), (ts, &'), ete., will eaatd
in succession the pairs of quantities (, bx) yu Rs, Ng )n, CLG

It must be observed, however, that while ¢;, h,, etc., are
constants peculiar to the solute s (in a given solvent), on the
other hand, ¢, h, etc., are the corresponding average quantities
for the whole solution, and so depend upon the relative pro-
portion in which these solutes are present [see (56a)|. Some
cases where t, h, etc., vary under the conditions of experi-
ment can in fact be dealt with, without difficulty, as will be
shown later, but, of course, it is simpler if they are constant.

Two important cases where this is so present themselves,
viz., the case where there is only one solute and the case
where there are two, but the concentration of the one is
negligible in comparison with that of the “second,” the
former being the solute s. ‘hese two cases gain further
importance from the fact that the form of G, the “true”
general terms, will here probably be simplest, depending in
the second case almost entirely on the ‘‘ second”’ solute.

The last point suggests, however, that in the matter of
separating J from G, cases where ¢, h, etc., vary under the
conditions of experiment may prove more useful.

THE QUANTITIES 255) %, @y, ETC.

(85) shows that the evaluation of t, ts, ete., would nee give
us these quantities (in general), but since the differences
shown in (85) will generally be much smaller than the

(corresponding) quantities a, @,,, etc., a rough estimate of

these differences (all fairly nearly expressible in terms of a,
which is something slightly greater than the mean molecular
weight of the pure solvent) will suffice tu determine 2, &,,,
etc., with a relative error much less than that of the estimate.

Actually, to determine the solvent constants experimentally

630 Mr. Bernard Cavanagh on

(and so permit accurate evaluation of @,,, %, ete.) we shall
require an entirely unsolvated solute (such as an inert gas, in
water), for which t, will be (—4a,), and so on *

That only the solute s need be unsolvated is an iaroari
point, since the solubility of the more inert gases (for
SA is small.

PuHysicaL SIGNIFICANCE AND LIMITATIONS OF THE
(JUANTITIES,

The quantities a, as, 8s, etc., though more difficult to”
estimate than the solvent-constants, are not, of course, merely
arbitrary parameters by any means

(49), for instance, shows that if only one solvate is formed
by the solute s, then

(.—S)=M, mee ia

so that the composition of the solvate is determinable from
a,, and 8s, and, of course, M, will have to be an integral
number of times the molecular weight of the solvent,
Similarly, if two solvates are formed and the amount of (s)
remaining unsolvated is very small, then, on the assumption
of perfect solution a, 8;,,and ys =reil| ee to determine the
composition of both solvates, according to the analysis men-
tioned, but not inserted, above. If the unsolvated residue is
not negligible, 8, will be required. ae:

A more general limitation is that so long as only * ‘positive’
solvation is in question (see first pages of this paper) a,
cannot be negative, and ~, cannot be positive, the latter
being demonstrable from ‘“ Le Chatelier’s Principle.”

In other words, t, may be a fairly large positive quantity,
but one a quite ‘small negative quantity, ranging, in fact,
from —+a, to the highest probable value for M,. Similarly,
hs; can range from (very nearly) j4,a,? to the highest probable
value for 1M,?.. When the solvent is water a, may be 60,
possibly even higher near 0° C., and M, might easily be 180
(decahydrate), while, of course, hydrates much higher than
this are known even in the solid state with low aqueous,
vapour-pressure, so that a value of M, as high as 300 would
not be improbable in certain cases. :

* Compare here 2nd paper loe. crt., p. 241.

Molecular Thermodynamics. 631

RANGES oF APPLICABILITY OF APPROXIMATIONS.

-Our four-term expansion of a, leads to (62) or (76) or
(80) which may be called a “ fifth-approximation” and
from which, by curtailing the series one term at a time,
the fourth, third, second, and first approximations are
obtained,

Owing to the diversity of possible magnitudes of the
quantities as, %, Qs, etc., precise statement of the ranges
of applicability of these successive approximations is not
possible, particularly in the case of the higher ones, but some
indication of probable range of the lower, for aqueous
solution, may be given,

The third (as far as ¢,) will commonly suffice as far as
Molar (M.) or 2M. solution (but will often go further), the
second (as far as ¢,) probably up to $M. or M. (sometimes
further), while the first in which the pseudo-general terms J
are omitted altogether cannot be assumed to be accurate to -

0-2 per cent. above hundredth-molar (00) concentration.

When hydrates higher than the decahydrate are not im-
probable, the upper limit for the first approximation must
be set still lower.

EXAMPLES OF APPLICATION.

Detailed application of the results of these papers to
existing data must be postponed, but it may be of interest
to cite one or two of the simpler instances. Consider, first, a
perfect solution ( vanishing) of middle concentration, for
which our “ second approximation” will suffice, so that

oH gy t ROL +AU). . Se pik)

If P be the osmotic pressure, defined and measured so
that the pressure on the pure solvent is relatively negligible,
and @ the coefficient of compressibility of the pure solvent,

then writing (with van’t Hoff) Vo for enn and considering

poU
the smallness of ¢,C relative to unity, we easily obtain
P [vo- 2 ee pART) | are 2 eG
0 ad

where, if we can neglect the effect of the pressures used on
the solyation of the solute, the quantity in round brackets is
a constant at constant temperature.

632 Mr. Bernard Cavanagh on

Since, further, at these middle concentrations the density
of the solution is expressible as a linear function of the con-
centration, so that the ‘“partial-molal volume” » of the
solute is a constant (at constant temperature),

VeVi oo ee

where V is the volume of solution containing one gm.-molecule
of solute, and (89) can take the form

PIV es eR a

where 6 is constant at constant {emperature, and
ba [e+ fe Ss etallp Mh i)

Equation (91) was put forward by O. Sackur *, in analogy
with van der Waal’s equation for imperfect gases, and was
shown to represent the data for middle concentrations. We
see that for a perfect solute this equation can be predicted
for such concentrations.

Now consider a slightly soluble solute s in a solution con-
taining also a “second” solute in concentration ©, relative
to ane? c; is negligible.

If G, is negligible, we have

oY = $,—Rlog tend... ae

In solubility measurement we have the solution in equi-
librium with a phase consisting entirely of the solute s at
constant temperature and pressure, so that

log ¢,— = log cYs)=constant, . . (94)

where ¢, is the solubility in presence of the concentration C
of the “second” solute. If ¢,, be the solubility in absence
of a ‘second ” solute, we have

OG 2, = 02 ge as nee recemee | (5) 55)
80 IE
or log (= logy.) =— sds. |). ey
For concentrations at which the ‘second approximation ”
suffices, we expect then to find

log =4'O= (¢+4,)C, i eS a

where ¢;’ is a constant, since ¢ depends practically entirely
* Zeitschr. f. Phys. Chem. (1910).

Molecular Thermodynamics. 633

on the “second” solute. We thus obtain a simple first-
approximation law for the solubility lowering of wu slightly
soluble perfect solute produced by the addition of other
solutes.

If the “second” solute be an electrolyte G,, will not
vanish, but since at the very moderate concentrations to
which (97) would apply, G will probably be practically
entirely due to the electrostatic forces between the ions,
G, may be expected to vanish if (s) is a non-electrolyte.
Thus (97) may be expected to apply to the solubility lowering
of slightly soluble non-electrolytes—the more inert gases,
for instance—produced by the addition of small quantities
of salts, providing the complete ionization theory is accepted.
(97) therefore constitutes a prediction, compatible with
perfect behaviour on the part of the non-electrolyte of the
empirical law of Setchenow, which latter should, however, be
re-written in terms of solvent-weight (instead of solution-
volume) concentrations.

For binary electrolytes we see that 1000C (roughly)
equals the equivalent or twice the equivalent concentration
according as the ions are bivalent or univalent respectively,
and that

Zi Nie Pee totsy. re 6! @ fee (98)

where ¢, is peculiar to the cation, ¢, to the anion, and ¢, to
the slightly soluble non-electrolyte.

The data tabulated by Rothmund seem on investigation
to be of somewhat doabttul accuracy, and the concentrations
examined too high. Moreover, they are expressed (as per-
centages) in terms of Setchenow’s law, that is, in terms of
solution-volume concentrations, and the recalculation is thus
necessary. ;

They serve at once, however, to show that our prediction is
quantitatively quite of the right order of magnitude to suffice.
In particular, the smallness of the few negative values should
be observed. 3

Among the more reliable figures there is also some evidence
of the parallelisms which the additive form of (98) would
predict for the recalculated values, while the greater degree
of solvation clearly assigned to the ions of higher valency,
(Mgtt, SO,--, CO;-~), is quite what we should expect.
Closer examination and more accurate measurement will be
necessary to determine how far and for what range of con-
centration G, can be neglected here.

Finally, the single case (as yet) of (probably) accurately
known ‘‘ true” general terms may be introduced.

634 Mr. Bernard Cavanagh on

It was shown in the first of these papers that in the case
of a mixture of ions, all of the same valency, theory alone is
able to predict, for (sufficiently) dilute solution, a simple
limiting form |

G =—R¢'C??,. hie oad ee ee
where 20, is the total ion-concentration*. The range of
validity of the ‘ point-charge’’ assumption, on which this
form depends, should be greatest in the case of the univalent
ions of simple structure, and might here extend to, or even
above, tenth-normal concentration.

Assuming this, and supposing that, while other solutes
(typified by s) besides the ions (typified by 7) are present,
there are no other “true” general terms at these low
eoncentrations, we have ;

Nh o+R[C—2Xe, log ¢,— Xe; log ¢;| + J —Rd'C,3?, (100)

or

1 eOe
@-l=podut2e C=
se 1 ae en
og Ve ae Ro» 9 ° ° (101)

log Wan ea al, ale 3h'CO.'?

where 2c; is 2C;, and C is Se,+ Xe;.
At the concentrations considered, J, of course, reduces to
one term, and we have

\

gate page |
log ys = (t+t.)C, | 5 uence lly
log yj = ¢+1)C + 39'C?”, | ,

In practice, logy; will always: be a mean quantity for the
two ions of the electrolyte J aes so that

log yj(=4 log Yje- Yja) = (EF Atje + Etja)C+34'C:", (103)

where ¢;, and tj are peculiar to cation and anion respectively.
For a single binary electrolyte by itself

log yj = 2c + ta) Oi + 36' Ci ie
G1 = Whe ttya)Cit 3's? J”
* See equations (88), (89), (92) of 1st paper.

(104)

Molecular Thermodynamics. 635

CO; being 2c;. Now tj and tj, being constants peculiar to
the particular ions constituting the. electrolyte, it is plain
that we have here a possible explanation of ‘the specific
divergences of the simple uni-uni-valent electrolytes at and
below (say) tenth-normal concentration, not involving the
rejection of the “ point-charge” assumption at these concen-
trations.

As might be expected, the limiting “law” Uc!” for @ or
logy, found by G. N. Lewis to apply, below: (about)
hundredth normal solution, to uni-uni-valent electrolytes,
ean be extended to cover the data to ligher concentrations
in the form (bc?+ ac), where a, unlike 6, is specific, or
peculiar, to the particular electrolyte. But, further, the
values of a required are quite of the order of magnitude to
accord with the interpretation of (104).

One other particular case of (103) may be cited, viz. the
case of two electrolytes together, the concentration of
the one, j, being negligible compared with that of the
other, 2, so that C,;is practically 2¢;, and f depending on 2
only can be written (4¢;.+4tia), whence

log y; = (tie +tyat ticttia)C; +36'Cit?, . (105)

in which, it is seen, the specific properties (as regards solva-
tion) of the jourions present enter very simply, sy mmetr ically
and additively.

~ These few brief and very*limited illustrations must suffice
for the present rather lengthy paper.

APPENDIX ].—THE Expansions oF U anp V.

- As in the previous treatment of “complex solvents” the
full consideration of the expansions of U and V, the total
energy and volume of the solution, is postponed, but one
point concerning the simple linear forms applying to
“ perfect solution” is briefly considered.

We have then the “ molecular” expressions

cae Oe ee
oe ee Us, + Dats,Us, 19 >, - (106)
Q = 20,90, + 2Ms[ (1 —Xs) Qo + 291]
and, starting from the results of the previous paper, we have
U = Moum + duo, Ano, + in,[ (1— X,) us, + Lars,Us, |
= My'uy + Su, Ano, + Sn; [ (1 — Xs) us, + Dare,e, “2 ot sUyg | -

636 Mr. Bernard Cavanagh on
Writing |

Lt l(1—X,)us, + Sats, — asm | = Us, « (107)
C>0 7

and, of course,

PNT \ 9 ON i
we have \ 0s)
U = Mo’ um t+ Sngtte + {So Ano, + 75d (ls, — Us, — Mewar) Axs,},
(108)

and, similarly,

V = My om + angus + {Sw,Ano, + rsd (Vs, — Vs — Ms,vm) Avs};
; (109)

Q = Moau t+ Srgs + {Eo Ano, + 2Ns=(ds,— Io — Ms, qu) Ate}
(110)

It is plain (cf simple demonstration in previous paper)
that the bracketed quantity in (110) is the heat developed
on “infinite” dilution of the solution, and it also clearly
represents the heat of the chemical action involved in the
change of the free solvent and of the solvation equilibria
to their limiting states pertaining to “infinite” dilution.
When only one solute, (s), is present, this heat of dilution
assumes the simple form—per gm. molecule of solute—

[| SaoAno + 2(Gs— qu Magn) Ae een. Ok

Similarly, the bracketed quantity in (109) is the “contraction
of dilution.” )

We saw in the previous paper that

\

uM = TS, OM;

(112)
um =—T 0 out
Op
The demonstration that
a
hg == ae a.
auees (113)

; ey
Us mer ee

Molecular Thermodynamics. 637

is not quite so obvious, but will be obtained without difficulty
if (13), (10), (6), and (3) are used in the order named.
Finally, we have the interesting relations

1? ea se {uy Acy,’ Lg ce,’ > (u,, .e Us, Ate M,%) Aes |
oe GOES
_ ay Op se {Ziv Ay,’ a XC, (v,, ik, ur Yay M, vy) on J |

APPENDIX II.

The practical significance of (39) may be illustrated in a
simple way, by means of a suppositious case in connexion
with strong electrolytes, entirely analogous to that given in
the appendix to the previous (second) paper.

The ‘general terms” for a dilute solution of a binary
strong electrolyte being

M,'G'"'= RM)'¢'c,'2?, - . iy ce wie Gena)

we suppose that the “effective” dielectric constant (D)
depends on ¢,' in such a way that @¢’ is a linear function
of ¢,',

i Sela ag opis nsw C16)

If this effect of c,’ on D is independent of its effect on the
solvation-equilibria, then we have

G!!
sp Oa 0,
so that
eS Ge ee a. lean

Gee. eho 62S 3Rdo'e./ (1+ ac,'), §. (117)
Gar= Gye" = —ER¢'e,! 92 = —ARGole,!3°U + a0!) {

If, on the other hand, the effect of ¢,’ on D is entirely
dependent on its effect on the solvation-equilibrium, then

OG da,

04s, deg!

fs
— Re, 3/2 o¢
og

= Rady'e,’9?

638 Mr. Bernard Cavanagh on ~

and
OG" | ee 5G" Oe
lie 0 Cs 1526"4, te eae
= 2Rado ¢s. ae

so that
eee \ 7 da, =Rey'e (1+ face’)

C=0 Cs" ; : |
G; = G,!’ a fr ae ke | oles = 3Rdolcs. Het a5 Bucs’)
Gy Gy" re moun ‘es. 3/2(1 + ac) J

(118)

- Asin the previous paper, it is to be remarked that in such
a case, if D were the ordinary .measurable or “ bulk ”
dielectric constant, the necessary modification of the general
terms (if not negligible) might be introduced without know-
ledge of the way in which the degree of solvation of the
‘solute was able to affect D.

In both cases also it may be observed that something
might be learned by the introduction of other solutes, since
both the constitution of the solvent and degree of solvation
depend mainly, if not entirely, on the total concentration
‘C’—not on the concentration c,’ of the solute (e ie)
considered.

The case of electrolytes provides an interesting and
important example of the (probable) fulfilment of the con-
dition for the vanishing of (40), though the solution is not
perfect.

It seems quite likely that—at least in the more dilute
solutions—the ions will behave essentially and approxi-
mately as point-charges, whether solvated or not.

The general terms for a solution of ions obtained in the
last section of the first of these papers would then hold
equally for solvated and unsolvated ions. That is, we should
have

i ! i
G =. = Sa ee ey

and since M;Gy;’ will be a much smaller quantity in dilute
solution, Gra," ', ete., will probably be quite small.

For Gz; ete., ie vanish altogether, we should require
that the “ bound” or “absorbed” solvent should continue to
function in the same way as the free solvent, as separating
dielectric between the point-charges.

Molecular Thermodynamecs. 639

APPENDIX III.

The form of the cofficients in (86) suggests an alternative
method of meeting the practical difficulty of slow convergence
of the J series, which will be preferable in many cases
‘though not always) to that already given on account of its
very attractive simplicity—both formal and practical.

Extending (85) to the fourth term

(1, —8,—a,ks+ hagh,) =((—8 —a,k + Ja,h)
=(— yeast + 4aj7a, —taya3— fay’ +a.) ~ Jgay*, . (119)
we have for the coefficients of the Jy,-series,
t; tz=(t?+ 3th+k
ain aay aga sbrs Meat t ee
and on comparing this with (59) it is at once seen that if

1 .

we write CU’ for (© a pat) then

agiu= fa —1)=0' +f +20" +10". bah)
Also (but less obviously), if
Ene,
a Os
we find
Rlog¢—J,=Kilog¢, —J;, .,: + (122)
where

—J, =RO (Gs +h + KC? +1,0%),.. .° .(1238)

These ‘“ concentrations ” (c;', C’) have no simple physical
significance, though they will approximate roughly to the
“true” concentrations of the earlier analysis. They are,
however, a great practical convenience, since the series in
(121) and (123) will clearly converge “ practically”? in few
terms so long as the expansions of a, and @ (in ascending
integral powers of the total “true” concentration) do so,
that is, as we have seen, up to very high concentrations.

Now, supposing either that we can neglect G (thus
making, tentatively, the assumption of perfect solution), or
that we can separate G from J in our measurements, then
solvent-separation data will give us Jy and solute-separation

640 On Molecular Thermodynamics. |
data will determine (differences in) Rlogc,—J,, that is,

if
Rlogc,'—Js', In the former, therefore, = as well as C
is a measured quantity, and when obtained (graphically) as
a series of integral powers of C' gives the quantities ¢, h, ...,
directly. |
But without (and independently of the precision of) this
(ie

Ae rains: C 3
determination of ¢t, h, .,., the measurement of cq makes ¢,! a

measured quantity, so that a combination of solvent-separa-
tion data and solute-separation data gives us J,' directly as a
measured quantity, and this when obtained (graphically) as
a series of integral powers of O’ gives us ¢,, hs, -.., referring
to the solute “separated,” at once.

It is, of course, in the obtaining of the series to express,
as accurately as possible, the data, that the essential ad-
vantage, the rapid convergency, attained by the use of the
‘¢ convenience-concentrations ” (c,', C’), appears. The sim-
plification, however, of the coefficients of the series (as
compared with (62)) will probably mean greater accuracy

in the evaluation of t, ME Oe ipeeal Hee

SumMARY OF Papers II, anp ITI,

Paper II., Section 1. A postulate, a rider to the definition
of “a molecule,” is proposed as the basis of rigorous
molecular thermodynamics, and its significance is discussed,

Section 11. The theoretical basis of the method and the
nature of the general problem are outlined.

Section 1. Analysis of the problem of ‘complex sol-:
vemlise yy |

Paper III. (the present) contains some analysis of the
problem of “solvation of solutes,” to which is attached a
discussion of the results with some brief preliminary
illustrations.

Balliol College,
March 1922,

[64a J

LVI. The Law of Distribution of Particles in Colloidal
Suspensions, with Special Reference to Perrin’s Investi-
gations. By AuFreD W. Porter, D.Sc., F.R.S., FInst.P.,
President of the Faraday Society, and J. J. Hepass,
B.Sc. *

iO the Proceedings of the Royal Society for January last
(A, vol. 100, No. 705) KE. F. Burton criticises Perrin’s
work on the law of distribution of particles in colloidal
solutions. In the course of his work Perrin shows that, at
any rate when the solution is dilute, we should expect a law
of distribution of the particles analogous to that which
characterises the distribution of the molecules in an
atmosphere. By balancing the osmotic pressure of the
particles against the effect of gravity he deduces the
equation
RT n
Ty loge No = V (pi—ps)g(h— ho),
where n and zy are the numbers of particles present per unit
volume (i.e., the numerical concentration) at depths h and ho,
V is the volume of each particle, p, and py, the densities of
the material of the particles and of the medium respectively.
Experimentally, for example, Perrin obtains for particles of
gamboge, 2°12 x 10~° em. diameter, the number of particles
at four depths differing successively by 3x 1073 cm. and
finds values proportional to the numbers 12, 22-6, 47, and
100, which numbers are almost in the same proportion as the
members of the geometric progression 11'1, 23, 48, 100.
Burton’s criticism amounts to saying that if this law of
doubling continued as the depth increases then at 3 cms.
depth the concentration should become 21°” times the: first
value given (viz., 12). Now this ratio isa number containing
more than 300 digits; and even casual observation shows
that no such increase occurs. In faet, Burton makes experi-
ments to show that throughout most of a tall column of
suspensoid there is no sensible change in the concentration.
Perrin’s own observations were confined to very small
ranges near the top of the suspensoid where the concentration
was very small, in order that his theoretical and experimental
work might correspond to one another. He says in “ Les
Atémes,” § 61, “Ce n’est pas sur une hauteur de quelques
centimetres ou méme de quelques millimétres, mais sur des

* Communicated by the Authors. A paper read before the Faraday
Society, Monday, June 26th, 1922.

Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2T

642 Prof. Porter and Mr. Hedges on the Law of

hauteurs inférieures au dixieme de millimetre, que l’on peut
étudier utilement les émulsions que j’ai employées.”

We have thought it important not only to prove that the
concentration tends to uniformity as the depth increases, but
also to find the law of change when the concentration ceases
to be sufficiently dilute for the perfect gas equations to hold
good.

Haperimental,

A suspension of gamboge was prepared by rubbing the
solid under distilled water with a soit brush. From this,
one in which the particles were practically of the same size
was obtained by means of fractional centrifuging. The
process actually adopted was that worked out and described
by H. Talbot Paris *.

To count the number of particles at various depths, a
modified arrangement of the Zeiss ultra-microscope was
used. The cell in which the susnensoid was contained was
viewed by the microscope placed with its axis horizontal,
while it was illuminated by a vertical beam passing down-
wards through the cover slip which served as a lid to the
cell. This vertical beam was obtained from the horizontal
beam passing from the lantern through the usual lens system
by means of suitably placed reflexion prisms.

The final condensing lens with its screw adjustment was
placed with its axis vertical, and the cell was attached to and
movable with this lens, so that various depths could be
examined in turn. An iris diaphragm in the eyepiece per-
mitted the field to be cut down until the number of particles
visible at any moment was sufficiently small to be estimated
at a glance. ‘lo diminish convective effects the light was
passed through a water-cell, and a shutter was employed so
that the light only passed through the observation-cell for
a time sufficient to enabie the particles to be counted. With
the exterior of the cell blackened and all stray light screened
off, it was found possible to count the particles in the field of
view over a range of several millimetres from the surface.
This arrangement was not convenient for making obser-.
vations so close to the surface as those made by Perrin
which were all confined within the range of less than
Q:1 mm. ; but this was no drawback to the object we had in
view.

The method of obtaining the concentration at any depth
was to count the number of particles in the field of view
from twenty to forty times at intervals of ten seconds, and

* Phil. Mag. xxx., October 1915.

Distribution of Particles in Colloidal Suspensions. 643

then to take the mean of these counts. The volume to which
these counts correspond is governed by the diameter of the
circular field and by the depth of focus of the objective, both
of which were determined ; its value was 2°:1x107‘ cm.?*,

‘The depths were measured (by means of the micrometer

screw which moved the cell) from an arbitrary level, which
was found to be about ‘023 cm. from the surface. The

depth reckoned trom the surface we denote by y, the number
-of particles per unit volume (7. e., the numerical concentration )

by n.

Several complete sets of observations were taken in order

to practice the method. The results for the final sets are

given below :—

y. -023.|-033.|-043.|-063.| -083.|-103./-123,/-143.
Number of counts ............ ee 20 20 | 20. 20 40— “20. 20 “40°
Total number of parlinles cacaecd a % 1 s- 30 72 | 40. AL 82
Number per citi Ie eee ons. ol "20 “45° ‘75 1:50 [1-825 2°00 |2:05 2:05
MXIOM6 oo ceveeesesessee veseeeees| ‘9B [214 8-61 [714 869 [952 9-76 [9-76

Professor Perrin’s calculations are explicitly based upon
the assumption of the application of the laws of perfect

gases to dilute solutions.

It is easy formally to extend them to solutions of any

concentration. This was done by one of us for true solutions
nie COLT *.

Solufion Solvent”

Imagine a column of solution to be put into connexion
‘with a column of pure solvent at two points through semi-
permeable membranes, the difference of depths of these points

* Porter, Faraday Society. Discussion on Osmostic Pressure, 1917.

2T 2

644 Prof. Porter and Mr. Hedges on the Law of

being dy (reckoned positive when downwards). The concen-
tration of the solution at any depth is c where c=nm, m
the mass of a particle and n the number of particles per c.c.
(2. é., the numerical concentration). Considering the osmotic
pressure P, 2.¢. p—po, as being a function both of the hydro-
static pressure of the solution, p, and of the numerical
concentration, we have the mathematical identity

rapt ea

dp \Op Jn” \On/p dy dp’
Now eo —“—* where s is the shrinkage*, and simple
uM a utichale dpy u-—o
hydrostatic considerations give — =1l———-= ; also:
dp dp dip U
ae 3
Therefore Sve OF an

oF pe ae
This formula is exact, and is independent of any particular
hypothesis of the mechanism of the variation.

Now it was shown by Sackur and by one of us that the
variation of P with concentration in the case of a sugar

solution can be represented very nearly, up to high concen-
trations, by the formula

nrT

= (1—bn)’

where 6 is a constant which is of the same order of size as,
but is larger than, a molecule of sugar. If we assume the
applicability of the same type of formula to a suspension of
gamboge :
OF organ et
on (1—bn)?°
In these formule 7, which applies to an actual particle, is.

connected with the usual molecular gas constant, Ry (which
refers to one gram-molecule), by the equation

R :
t= nN? where N is Avogadro’s number.
dn __s—oa gN

Hence dy = “uo Rol

(1—bn)?.

* Porter, Proc. Roy. Soc., A. 1907, p. 522.

Distribution of Particles in Colloidal Suspensions. 645

If we assume that there is no contraction when gamboge
and water are intermingled

o=cy+(l—c)u and s=u,
where y=density of solid gamboge. Hence
o—s=c(y—u) =nm(y—u).

Further writing K= a ”

and, putting o in denominator as equal to u [which is justi-
fiable, because even the strongest suspensions of gamboge
are fairly dilute],

dn they,
Pe = Kn(1—én)?.
The integral of this equation is
n i
CS spear uty aia maT

where A is the constant of integration which can be expressed
in terms of (the unknown) concentration when y=0. This
is a curve which tends asymptotically for large values of v
to the value n, =1/b; and which has a point of inflexion for
a. |.

A ais of this kind can be fitted to the experimental
curve within the limits of experimental accuracy (fig. 2).

The following values are obtained by taking

b=10°2 4% 105" em:-:and:-K = 1291.

y (in ems.)...... 024-082, 0375: 047 “061 089 ‘115
ca A Tae 1 2 3 5 7 9 9:7

A closer fit could, of course, be obtained by allowing 6 to
vary with the concentration, as was done in examining the
osmotic pressure of sugar solutions*. It is difficult, however,
in the present problem, to give anything but an empirical
significance to this constant. It enters into the osmotic
pressure in the same way as the least volume of the liquid
enters into the gas equation. But in this case it would
mean that even in the fairly dilute concentrated suspension of
gamboge the effective volume of the particles is the volume
of the solution itself—that is to say, that the suspensoid

* Porter, loc. cit.

tame ws

——

646 Prof. Porter and Mr. Hedges on the Law of

plus the atmosphere (or ocean) of solvent (or second phase)
which surrounded it, and to which it was attached (so
that motion of one entailed similar motion of the other),
constituted the whole of the substance present. The radius
calculated by considering b ag the effective volume is
28x 10-+cm. Now it is hard to think that the whole of
the liquid in the domain of a particle should be so attached
to the particle as to form practically part of it, for, if this
were so, the whole suspension would be rigid. It might
be conceivable in the case of even a dilute concentration of
gelatine, for this can form a rigid gel; no such rigidity is
observed in the case of gamboge. Itis noteworthy, however,
that if 6 be regarded as an effective volume, then b and K.
are not independent constants, for K is proportional to the
effective volume of the particle. When the radius is calcu-
lated from K, it is found to be 1:6 x 10~° em., which is very
much less than when calculated from 6, but is certainly a
measure of the effective radius of the particles that were
examined. We return to this point later. |

It is safer, however, to regard 6 as merely an empirical
constant. It may be pointed out that Callendar’s. equation
for unsaturated steam is of the form

p(w—b) =RT,

where, however, b is even negative and represents the com-
bined effects arising from the finite size of the molecules
and the attractive forces which tend to “ co-aggregate” the
molecules. It is noteworthy, however, that, if 5 1s to become
large and remain positive, repulsive forces are required.

Burton, in the second edition of his monograph on the
physical properties of colloidal solutions, page 87, attempts
to explain the present problem by considering repulsion
arising from the electrical charges in the particles. He
concludes that the forces on any layer due to the rest of the
solution will be of the form kne per unit charge on the
layer dh, where k, he considers, may be taken as a constant,
at any rate for regions near the surface, and e is the charge
in electrostatic units on each particle. Consequently the
total force per unit area is kn7e*"dh. A term of this kind is
taken to represent the excess electrical repulsion of all the
particles below dh over that due to those above dh. Perrin’s
equation then becomes, for the forces balancing per unit
area, :

= dn + kn*e?dh=nV (d —w) gdh.

Mistribution of Particles in Colloidal Suspensions. 647

Writing this in the form Adn=n(C—Bn)dh, and solving,

he gets
C
= B+ Ke- Oa?

in which K=(C— Bno)/np and mp is the concentration when
h=0. To obtain Perrin’s formula B must be put equal to
zero. ‘The ratio C/A can therefore be obtained from Perrin’s
experiments. Calculation shows that e~/4 tends rapidly to
zero.as h increases, and ultimately n becomes

YON (G—w)¢

rT a)
2 es) ke?

The depth at which this uniform concentration is practi-
cally attained will depend upon the relative values of K and
B; it will be nearer the surface the larger the electrical
forces are compared with the gravitational.

Now there are serious objections to the theory as thus
stated.

In the first place, if the particles really contained charges
all of one sign only they would tend to move toward the
boundary. This is the equivalent of the fundamental
electrical fact that statical charges reside close to the surface
of conductors. When we are dealing with large particles
instead of electrons, there is no doubt that they would occupy
a larger region, instead of a thin superficial area, but still
there would be an accumulation at the boundary. This is the
opposite to what is observed.

But the charges in the solution are not only ef one sign.
The solution, as a whole, is uncharged; consequently an equal
opposite charge is to be looked for. This opposite charge is
the second member of the double layer close to the surface
of each particle. When the existence of this double layer
is recognized, the electric forces between the particles become
zero, except in so far as relative displacement takes place by
induction between two members of a layer so as to give it an
electrical movement. In this case the force between two
such doublets in the equilibrium state will, on the average,
be an attraction and not a repulsion.

Now we have fitted a curve calculated from Burton's
equation to the experimental points. ‘hey are shown by
large circles on the figure and are seen fo fit the experiments
remarkably well. In view of the above objections to the

648 Prof. Porter and Mr. Hedges on the Law of

theory, this cannot be taken as indicating more than that in
the differential equation for the distribution there is a term.
proportional to n? in addition to the one depending on n.

Digeez
ee ees Soe.
7 (stern Mn

AGRE REN ERORROS DS See ORe
USOT TEE Na SIC ca aa el
A LL ene no L OOo
87 eet eal htt al
7) SE Caen eae eee
ee eT 970 ae a oa
BECO R Ue
ee ee a oe Sem
Te ea VST eal
A
ig SAREE See |
> Gennes Zee
7 CRE a se eee eee
= A ae
1 fT Ae ea aa a

wae im me

wae eee ares eae

Bee cane eee eee eee

Ltt | ee eel Cr

‘Ol -02 :03 :04 -05 :06 -07 :08 -09 -10 <i) “!2 +13 4 -15
Depth m Centimetres.

SE SO Ie Perfect Gas Law ©) @ Burtons Equation
+--+ = Experiment a/ Points ———*-Porter and Hedges

From the form which Burton gives to his force term there
appears to be some confusion in his mind between force
and pressure. If his statements were to be taken as correct,
there would be a uniform force on any layer (counteracting
the uniform force of gravity) in the greater part of the
solution, for his force term is proportional to n’ and n becomes
practically uniform ; whereas it is clear that in the middle
where a layer wart be driven as much up as down the
force would really vanish.

Now, although 6 in a van der Waals’ formula represents
the volume of a particle when it is treated as a rigid body, it
is in general the coefficient of the internal virial arising from
molecular repulsions during “ collisions.” The large value
obtained with gamboge would indicate therefore that, even
when particles are at a distance from one another much
greater than that which denotes their size when it is estimated
by optical examination with a microscope, the particles begin
to experience a mutual repulsion. To put it in other words,
the actual visible particle is connected in a semi-rigid way
with a very large quantity of the surrounding fluid. What
the nature of this connexion may be can only be a matter

Distribution of Particles in Colloidal Suspensions. 649

for speculation. A sugar molecule is tied in a more or less
rigid way with at most a single layer of water molecules *.
Molecules, however, attract one another with forces which
vary according to a high power of their distance apart.
But the very large particles of gamboge have a far greater
range of action. Large masses attract according to the law
of gravitation. The particles in question may be assumed to
attract molecules of water with a force varying according
to some intermediate law, and have in consequence an inter-
mediate range of action. Thus they can form a loose com-
bination with comparatively large masses of water, each
such agregate moving as a single molecule, so far as kinetic
theory is concerned. Hence the large value of b, which is
determined by the mean radius at which such units bounce
off one another.

Nothing that is said here must be taken as excluding the
action of other intermolecular forces besides those concerned
with collisions. If analogy with a van der Waals’ substance
is made use of, the osmotic pressure could be written

p+an?=nrT/(1— dn),

‘where it may be left an open question as to whether a is
positive (attraction) or negative (repulsion).
The equation for dn/dy then becomes

dn 1 2a
dy { n(1 — bn)? He ai, ae

t
\

whence
n A 2an

EP DN SR A a a
one Me > oy ea ee
The limiting volume of n is still determined, not by a, but
by 6, and this is so whether a is positive or negative.
Further light is thrown upon the phenomenon by taking
‘suspensions of various limiting concentrations. In fig. 3
are shown experimental curves for two cases; in curve 2 the
limiting concentration (?.¢., the concentration in all except
the upper layers) is twice the value for curve 1. Hence,
according to the equation given, the value of b in the latter
(i. e., the dilute solution) is twice the value it has for the
stronger solution. ‘lhis, again, is quite in accordance with
the behaviour of sugar, for which substance more molecules
of water were found to be attached to each molecule of
sugar in the case of a dilute solution than in the case of a
strong one.
* Porter, loc. cit.

650 Distribution of Particles in Colloidal Suspensions.

These last experiments show that care should be taken to:
avoid thinking of the limiting concentration as necessarily
giving a saturation value. The same kind of curve is-
obtained when more dilute suspensions are investigated.
The value of K is also different for the two curves. For
curve 1 (fig. 3) it is about 29, while for curve 2 itis 55..
It would seem ion the size of the solid nucleus diminishes
to one-half when the suspension is reduced to half strength.

Fig. 3.

if BEERS Bama
suseeccaneeeee

HE
—p tty

me
Pea
Vane
Fai RY
aanes
en la
faves

ABH
axe

vi
2a)
ela aie

aE a

Although Burton’s spivabilbay2 assumption of the existence:
of an electrical repulsion between the particles has been here
criticised, it might still seem an open question as to whether
some other force between them might not be the operative:
cause. The chief difficulty applicabie to all possible laws of
force is that such forces will be exerted in all directions and
must act outwards at every surface. Hence, though near
the top of the fluid they might be assumed of such a charac
ter as to oppose gravitation, at the bottom they should
assist it in producing a change of concentration, which might
be expected to be exceedingly large in consequence. Again,
at the sides where the forces would act laterally they would
produce a change of concentration independent of gravi- |
tation. What really happens can be seen from fig. 4,.
which represents the approximate ‘relative values of the
numerical concentrations actually observed at various points
of a cell 5 cm. deep and} cm. broad. It is obvious from the .
numbers given in the figure that the only variation of
concentration is near the top. It ought to be mentioned that,.
since the suspension was covered with a microscope cover-
glass, the suspension was surrounded by glass on all sides ;.
the peculiar effect at the top is therefore not due to a
transition from liquid to air.

It would appear from these preliminary determinations
that there are in reality three regions in a suspension to-

Emission Spectrum of Monatomic Iodine Vapour. 651

which particular names might be given. In the first place,
there is a very thin layer close to any surface in which there
may be a special value for the mean concentration. This is
the layer discussed by Willard Gibbs, and has been called
after him. Its thickness is of the order of the range of

Fig. 4.

intermolecular forces. Secondly, there is the layer studied by
Perrin, “inférieure au dixieme de millimetre,” in which the
change of concentration of a suspensoid can be calculated
from an application of the laws of gases in an analogous way
to that in which the change of pressure of an atmosphere is.
calculated. This is the Perrin layer. Thirdly, there is a
layer of one or two millimetres’ thickness (in the particular
cases studied in this paper), in which a further change of
concentration occurs which cannot be calculated in the way
adopted by Perrin. This gradually merges into the main
body of the suspension, throughout which the concentration is:
sensibly uniform.

LVIL. Onthe Emission Spectrum of Monatomic Iodine Vapour...
By Sr. Laypav-Zisemecni, M.Sc., Lecturer in Physics,
High School of Agriculture, Warsaw”.

I. The aim of this work. .
M ANY of the phenomena of the multiple spectra were

ascribed to the dissociation of the molecules or even
of the atoms (the work of Lockyer may be only mentioned),
but there does not seem to be direct experimental evidence
for such a theory. On the other hand, the models of atoms
and molecules, now generally adopted, lead us to the idea
that an atom should give a spectrum quite different from the

* Communicated by the Author.

Presented to the Polish Academy of Sciences by Prof. L. Natanson,
March 7th, 1921, and published in the Bulletin de [Académie Polonaise,
série A, 1921. Since that time the experiments have been verified by
the Author.

652 Mr. St. Landau-Ziemecki on the Emission

spectrum of the molecule. To verify it experimentally the
writer chose iodine, the dissociation of which has been well
studied and which dissociates at a relatively low temperature.

Il. The Experiments.

Bodenstein and Starck * have shown that the dissociation
equilibrium can be expressed by the formule :
py K (C2 ae : We
per log K= — pg +1 75 log T—0:000416 T+ 0°422,
where p,; is the pressure of the monatomic iodine vapour,
pe the pressure of the diatomic vapour, K the equilibrium
constant, T the absolute temperature. This expression agrees
well with the experiments. Applying that formula, we obtain
the following table for the degree of dissociation—that is, the
ratio of dissociated molecules to the total number of the
molecules which would be present were there not any dis-
sociation.

Degree of dissociation as a function of pressure and temperature.

t=500°C. 600°. 700°. 800°. 900°. 9608:

p=imm. 01 0:36 0-76 0:95 0:99 1
ole 0:05 0°19 0°52 0°83 0:94 if
f= 10) 5; 0°02 0:06 0:19 0°42 0°68 0-81

Taking these data as the starting point for my experiments,
I prepared a Geissler tube from quartz of the form shown
ray siaes Ihr.

The iodine crystals were at C, and that part of the tube
was immersed in a water-bath, the temperature of which
varied in the different experiments from 15°-20° C.; thus the
corresponding vapour-pressure was always less than 0°25 mm.
of mercury (for 20°C.). The central part of the Geissler
tube was placed in an electric oven, and the temperature
raised from 960°-1000° C. It is seen from the above table
that in these conditions the dissociation was fairly complete.

* Zeitschrift fiir Elektrochemie, vol. xvi. p. 961 (1910).

+ Some months after this work was published in the Bulletin of the
Pol. Acad., W. Steubing described in the Physikalische Zeitschr. (1921,
p. 507) his experiments touching the influence of the temperature on the
band spectrum of iodine. The experimental arrangement of this author
was similar to mine. Glass tubes were used by him, and thus he was
unable to go further than 450° C., when practically there is no possibility
of observing any trace of the line spectrum of monatomic iodine, the
percentage of dissociated molecules being quite insignificant. The ex-
‘periments of W. Steubing gave interesting results touching the weakening
of the band spectrum with the rise of the temperature. His observations
are limited to the visible part of the spectrum. This part of the
Spectrum was inaccessible to me in the experiments touching the
spectrum of monatomic iodine on account of the radiation of the electric

oven.

Spectrum of Monatomie Iodine Vapour. 653.

It is obvious that in the case considered there exists a circu-
lation of the vapour, but, for the diameter of the joining
quartz tubes, no serious pressure differences could exist ;
besides, it was shown by Nernst * and his collaborators that
the dissociation formule can be applied to a gas flowing
through a pipe, if the flow is not too quick a one.

Fig. 1.

Two main iodine spectra are known—a band spectrum and _
a line spectrum. The most complete study of these spectra
was made by H. Konenf. Condensed, strong discharges in
narrow parts of the Geissler tubes give generally the line
spectrum ; with wide tubes and feeble discharges we obtain
the band spectrum. The first experiments to show the in-
fluence of the dissociation on the emission spectrum were
made by the author, using Geissler tubes of ordinary shape
with vertical capillary tube. No interesting results were
obtained. A new spectrum was observed when using a tube
“end on” with large central part, as is seen in fig. 1; the
central part B had 2 length of about 12 cm. and a diameter
of 1 cm.; the parts A, covered with tinfoil, had a diameter
of about 3 cm.and a length of about 8 cm. This tube, not

* W. Nernst, Theoretische Chemie, 7th ed. p. 709. W. Nernst and
H. vy. Wartenberg, Zeitschrift f. physik. Chemie, vol. lvi. p. 535 (1906).

+ Annalen der Physik, vol. lxy. p. 265 (1898).

r654 Mr. St. Landau-Ziemecki on the Emission

heated, showed the ordinary band spectrum. The temperature

being raised to 960°-1000° C., the conditions of the electric
excitation remaining the same, a new line spectrum replaced
the band spectrum. The appearance of that spectrum is
quite different from the appearance of the spectrum obtained
in a narrow tube with condensed discharges. It consists of
relatively few lines; two lines in the ultra-violet are especially
strong and characteristic ; the others are faint.

Some details of the experiments may now be given. The
iodine was introduced into the tube by sublimation in the
following way. Pure iodine of commerce (Kahlbaum) was
re-sublimated in vacuo; the crystals were then put into the
glass tube R, (fig. 2), which was to the right connected with
‘the mercury Gaede pump and to the left with the quartz
apparatus. Quartz and glass were joined together with

Fig. 2.

ground : surfaces ; sealing-wax made that joint perfectly
‘tight (W, fig. 2): The apparatus was exhausted during two
hours. During that time the U-tube was immersed in solid
CO, to prevent the iodine vapour from penetrating into the
pump; at the same time the quartz part of the apparatus,
~which was previously chemically cleaned, was now heated in
‘the most energetic manner with a Bunsen burner and finally
with a blow-pipe. The apparatus was then cut away from
the pump at P,, and the iodine sublimated from R, to Ry,
‘this part being immersed in liquid air. After some hours,
R; was in turn immersed in liquid air, and a crust of iodine
crystals was formed there. Hvery precaution against con-
tamination having been taken, the quartz part of the
apparatus was again put in communication with the pump,
and after an exhaustion of half-an-hour’s duration the quartz
:apparatus with the iodine crystals at R; was cut away from —
the pump at P,. The Geissler tube obtained was put in the
electric oven shown in fig. 1, and finally bent with the coal-
gas oxygen flame in the desired manner. The tube was
provided with external electrodes formed by tinfoil cemented
to the quartz with a mixture of graphite powder and water.

Spectrum of Monatonuc Lodine Vapour. 655

The luminescence of the tube was excited either by con-
necting it directly with the secondary of a Ruhmkorff
induction coil (15 em. spark), or by using in connexion with
the induction coil a high-frequency transformer of the Oudin
type, and joining the tube to the circuit as shown schematically

in fig. 3; in the latter case a Wehnelt interruptor was used.
Fig. 5.

The temperature of the electric oven was measured with a
‘thermo-couple Pt—Pt/Rh. By means of two quartz lenses
(L in fig. 1) the light was concentrated on the slides of a
-quartz spectrograph.

At low temperatures the tube has shown a band spectrum.
H. Konen * finds that the band spectrum is almost continuous
in the ultra-violet part, having only few diffused maxima at
A= 3300, 4270, 4520, 4760 A. On the contrary, I found
that this band spectrum has a very regular structure with
many well-defined maxima. There seems to be no discrepancy
between our results: this author has taken photographs
with long exposures, and in these circumstances, as I find,
all details disappear; on the contrary, my exposures never
lasted longer than a few minutes. For a more detailed
‘study of the band spectrum, I used a special quartz tube
with external electrodes, having a capillary part in the
middle. This tube being connected with the secondary of a
Ruhmkorff coil, I found distinct maxima at ~\=422, 404,
a99, 301, 300, 3/3, 368, 363, 340, 325, 323, 320; 319, 312,
310, 306, 304, 302, 299 uu; for shorter wave-lengths the
maxima were too feeble to be measured.

Raising the temperature of the oven, I found that the
transformation of the band spectrum into a line spectrum

occurred gradually. At550° C. we have yet a band spectrum;

* Loc. cit,

696. Emission Spectrum of Monatomic Lodine Vapour.

at 650° C. one of the two characteristic lines 3281 A appeared ;
the second strong line 3384 A appeared later; at 800°C. it
was yet faint. ‘The manner of producing the spectra seemed
to be of no influence: I observed the same phenomena
whether applying uncondensed discharges of a Ruhmkorft
coil provided with the hammer interruptor, or with the
Oudin high-frequency transformer, AGUS by an electrolytic:
interruptor.

I determined the wave-lengths of the main lines of the
spectrum observed; they were found to be 4868, .4769,
4680, 4324, 4132, 4100, 3384, 3281, 3081, 2879, 2593, 2583,
2566 A. My spectrograph had a dispersion insufficient for
accurate work. The error of the determinations should not
exceed 2- 3A for the fainter lines, and 1A for the two
stronger ones™

It would Ihe difficult to tell whether all these lines are
absolutely new, or whether some of them can be found in
the ordinary line spectrum of iodine: the exactness of my
measurements is insufficient, taking into account that the
lines of iodine are exceedingly numerous; happily there are
no lines of the ordinary spectrum in the proximity of
3281 A; that characteristic line is thus certainly a new one.

Ill. Results.

A new line spectrum of iodine was observed, and wave-
lengths of the main lines approximately determined. This.
spectrum belongs evidently to the atom of iodine ; it appeared
more and more distinctly with increase of the dissociation
of iodine molecules; at the same time the band spectrum
disappeared. The band spectrum of iodine seems thus to be
inherent in the molecule. It was found that the band
spectrum possessed in the ultra-violet a very regular structure,.
contrary to that which had hitherto been imagined.

It may be permitted to the Author to express here his best
thanks to Prof. 8. Pienkowski, who in a most generous way
has supplied him with a quartz spectrograph.

A great part of the expense was covered by the Mianowski
Foundation.

Warsaw, Physical Laboratory of the

State Technical School, founded
by H. Wawelberg and 8S. Retwand.

May 1922.

* In the violet part of the spectrum much greater errors are possible,
the dispersion of the quartz prisms being there very small, While this
paper was in the press some other lines were measured and the following
wave-lengths found: 4640, 4414, 4217, 38576. 2528, 2524, 2516, 2507.

Barton & BROWNING.

Phil. Mag. Ser. 6, Vol. 44. Pl. TIT.

SSS

Barron & BROWNING.

Phil. Mag. Ser. 6, Vol. 44, Pl. IV.

) i
D0

i Phe, wn

ah

Ve

fy,

THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCI ENG Eis “A

/ > ne Qi L )
19 1
[SIXTH ona Ceri sous
NS

ny,
OCTOBER 1922.“~~

LVIII. Further Studies on the Electron Theory of Solids.
The Compressibilities of a Divalent Metal and of the
Diamond. Electric and Thermal Conductivities of Metals.
By Sir J. J. THomson, O.M., F.RS.*

- a paper published in the Philosophical Magazine, April

1922, I calculated on the Electron Theory of Solids
the compressibility of monovalent and trivalent elements
crystallizing in the regular system. In this paper I propose
to do the same fora divalent element. The simplest case
is that of calcium, for Hull has shown that it crystallizes
in the regular system and that the atoms are arranged in
face-centred cubes. Thus, as far as the atoms are concerned,
the metal may be supposed to be built up of cubical units,
each unit having 4 of an atom at each corner and 4 of an
atom at the centré of each of its six faces. Thus each unit
contains four atoms; and as calcium is a divalent element,
there must be twice as many disposable electrons in the
unit as there are atoms, so that each unit must contain
8 electrons.

A symmetrical way of arranging these 8 electrons is to
put 4 of an electron at the middle point of each side of the
unit cube, one electron at the centre of this cube, and one
electron at the centres of 4 out of the 8 small cubes into
which the unit cube is divided by planes bisecting its sides
at right angles. These 4 cubes are chosen so that if we

* Communicated by the Author.
Phil. Mag. 8. 6. Vol. 44. No. 262. Oct. 1922. 2U

658 Sir J. J . Thomson: Murther Studies on -

move parallel to any side of the unit cube, the empty cubes
and those containing eJectrons occur alternately.

Assuming this to be the constitution of the metal, we can
easily calculate the electrostatic potential energy by the
method given in the paper referred to. Let H be the charge
on the calcium atom, e the charge on an electron, and 2d
the side of the cube taken asthe unit. Then the electrostatic
potential energy for a single atom 1s

sB(-37 +35),

where 7 is the distance of an atom and 7’ that of an electron
from an atom under consideration.
The potential energy of an electron is
e si s e
heat > Rai}
where 7’ is the distance of an electron from the one under
consideration.

By the method described in the former paper, I find for
the electrostatic potential energy of an atom the expression

16°23 36°85
Lehi — oti are
which, since H=2e, is equal to
e”
7 4,

The potential energy of a single electron, if it is one at the
middle point of a side of the unit cube or at its centre, I
find to be

H.———e.-4-)= 5, 83,

d
while if the electron is one of those at the centre of the -
small cubes, the potential energy is

° O 2 5
ye(B. A We 2) = 5 59°65,
Since the neutral calcium atom consists of one positive
nucleus and two electrons, one of each type, the potential
energy per normal atom will be

Fr Le 97 Seay het

9 2
= (444 1°325+°415) = 7 6°15.

*

the Electron Theory of Solids. 659

If A is the density of calcium, M the mass of an atom,
since our unit cube with side 2d contains 4 atoms,
4M
8d*

1 =o 1)
ae o ‘

Hence the electrostatic potential energy per normal

atom is
2a [A \¥8
2 ° 1/3 a
2.615 2 (sr)

and the electrostatic potential energy per unit volume is

#615 B8(5)
M

A 4/3
ee mat |
=e.715x (i)
Hence, by page 736 of the former paper, & the bulk

modulus for calcium will be given by

FTP
bay cn Ex
ae 9 (iu)

For calcium | ed ed
M=40 x 1°64x 107%,
This makes 1// the compressibility equal to
5705< Lvs

the value found by Richards is 5°5x107, so that the
agreement between the calculated and observed values is
again quite close. The other divalent metals Mg, Zn, Cd
crystallize in the hexagonal system; in this system the elastic
properties vary in different directions, and a uniform pressure
would produce a change in shape as well as in volume.
The arrangement of electrons and atoms appropriate to this
case is when the electrons are at the corners of a hexagonal
prism and the atom at the centre. To fix the shape and
size of the prism we require two lengths, the radius of the
base and the height of the prism instead of the one which
sufficed for crystals in the regular system. The necessity
for taking two variables instead of one makes the calculations
more lengthy than those for the regular system, and I shall
defer their consideration for the present.

2U 2

A,

660 Sir J. J. Thomson: Further Studies on

Compressibility of the Diamond.

In the diamond we have a quadrivalent element orale
lizing in the regular system. The arrangement of the carbon
atom in the diamond has been shown by Sir W. H. Bragg
and Professor W. L. Bragg to be given by the following ~
scheme. They occupy

a, the corners of a cube;

b, the centres of its faces :

c, 4 of the centres of the 8 ies into which the large
cube is divided by planes bisecting its sides at right
angles.

We shall take this cube as our unit; it contains eight

carbon atoms. Since carbon is quadrivalent, it must contain
32 electrons ; these electrons will be situated

a, at the middle points of the edges of the cubical unit:
this accounts for 3;
b, at the centres of each of the faces of the 8 small cubes:

this accounts for 24;
c, at the centres of the four small cubes not occupied by
the carbon atoms: this accounts for 4 ;
d, one at the centre of the large cube.
Making use of this unit, we can calculate the electrostatic
potential energy due to ‘the charges on the atoms and
electrons. Jet E be the charge on a carbon atom, e that on

an electron.
The electrostatic potential energy of a carbon atom

LE (= =
r
I find to be equal to :
1B ee Sica gee
9 7 (149 346.e—35'13.5),
where 2d is the side of a unit cube. Since E=4e, this reduces
to
e2
IO) 71°

The electrostatic potential energy of an electron I find to

be
se “42 119°340—147-59¢ |
‘ e?

Seal
2d p:

the Electron Theory of Solids. 661

Hence the electrostatic potential energy for the atom and
2

its four associated electrons is 21°15 _

Since there are eight atoms in the cube whose edge is 2d,
if A is the density of the diamond and M the mass of a
carbon atom,

af A 1/3
or a = (a) °
Thus the electrostatic potential energy per one atom and
four electrons is
Av 13
Q115e (a)

and the energy per unit volume is
4/3
21-15 (5)

Hence, by page 736 of the former paper, & the bulk
modulus of the diamond is given by the equation

ee ED: SA oe
— 9 é(5) :

for the diamond A=3°52,. M=12x1'64x107-*4; hence
jp ole Wir Ais x 10-2.

This value for 1// is much less than that, °5 x 10~*, found
by Richards. It is, however, in close agreement with
"16 x 10~”, the value recently found by Adams (Washington
Acad. Sc. xi. p. 45, 1921).

The properties of solids formed by elements whose atoms
have more than four disposable electrons are quite different
from those of solids formed by the elements with one, two,
or three disposable electrons. The latter are, with the
exception of boron, metallic and good conductors of
electricity and heat. The former, for instance sulphur and
phosphorus, are insulators. Not only do they insulate in the
solid state, but they do so after they are fused. They differ
in this respect from solid salts which, though they may
insulate when in the solid state, generally conduct when
melted. This suggests that in the salts there are positively
and negatively electrified systems which are fixed when the
substance is in the solid state, but can move about when it
is liquefied. In such elements as sulphur or phosphorus
there does not seem to be any evidence of the existence of

662 Sir J. J. Thomson: Further Studies on

anything but neutral systems ; in other words, the solid may
be regarded as built up of units, each of which contains as
much positive as negative electricity. It is noteworthy that
according to the Hlectron Theory of Chemical Combination,
two similar atoms if they have each more than four dis-
posable electrons, like the atoms of sulphur and phosphorus,

can combine and form a saturated molecule, which is
electrically neutral. ?

Thus we are led to distinguish three types of solids :—

a. A type where the atoms are arranged in lattices,
and the electrons in other lattices coordinated with
the atomic ones. In this type each electron has no
closer connexion with a particular atom than it has
with several others. Thus, for example, when the
electrons form a simple cubical lattice with the atoms
at the centres of the cubes, each electron has 8 atoms
as equally near neighbours ; so that an electron is not
bound to a particular atom. This type includes the
metals ; it also includes boron and carbon in the form
of diamond, which are insulators.

b. A type represented by the salts ; here the atoms are
again arranged in lattices, but each electron has much
closer relation with one particular atom than it has
with any other. Thus to take the case of Na. Ol,

where the Braggs have shown the atoms to be
arranged according to the following scheme :—

Na Cl Na C]
Cl Na Cl Na
Na Cl Na Cl

We suppose that each sodium atom has lost an
electron, while each chlorine atom has gained one;
thus each chlorine atom has eight electrons around it,
and each electron is much more closely bound to one
particular chlorine atom than to any other. It is so
closely associated that it is not dissociated from it;
partner in either the solid or liquid state of the
substance. Thus the chlorine system always has a
negative charge, the sodium one a positive. These
atoms do not move when the substance is in a solid
state, though they may do so when it is liquefied.

If the distance of the electrons from the chlorine
atoms were to increase until it was not far from half
the distance between the sodium and chlorine nuclei
this type would approximate to type a.

the Electron Theory of Solids. 663

c. A type where the lattices are built up of units which
are not electrified ; such units are probably molecules
containing two or more atoms, though in certain
cases they may be single atoms. The characteristic of
the type is that each unit has sufficient electrons
bound to it to make it electrically neutral, and that
each electron remains attached to a particular atom.
Thus where an electric force acts on the system there
is no tendency to make the unit move in one direction
rather than the opposite, so that the substance cannot
conduct electricity.

Metallic Conduction.

We now pass on to consider why it is that the arrange-
ment of atoms and electrons in type @ is in many cases,
though not in all, connected with the property of metallic
conduction. ‘The consideration of the frequencies of the
vibrations of the electrons in a lattice will, I think, throw
light on this connexion. I showed (Phil. Mag. April 1922,
p. 721) that these frequencies may extend over a very wide
range of values as the type of displacement of the electrons
is altered. Thus, if ali the electrons in a region whose linear
dimensions are large compared with 2d, the distance between
two electrons, have the same displacement relatively to the
atoms, the frequency n of the vibrations for the alkali metals
is given by the equation

eC SC. ed aes OL)

This frequency, even in the case of the univalent element,
corresponds to that of light in the visible part of the
spectrum ; for elements of greater valency it is far in the
ultra-violet. This is also the frequency with which a single
electron vibrates if the surrounding atoms and electrons are
fixed. As those frequencies are so great very little energy
will go into them at ordinary temperatures, and they will
have little or no effect on the specific heat of the solid.

There are, however, other types of vibration for which the
periods may be very long. Thus if all the electrons ona
certain line of the lattice are displaced along the lattice by
the same amount, while those on adjacent lattices are
displaced in the opposite direction, the frequency is given by

the equation
a eOt eee Bene" (2)
mp Sr, ie = ga eae oc bt aie

thus we see that only under certain conditions is the ex-
pression for p? positive, and it is only under these that the

OO

664 Sir J. J. Thomson: Further Studies on

equilibrium is stable for this mode of displacement. A
negative term will occur in the expression for p? if the
electrons along one line of the lattice are displaced relatively
to those on adjace:.t lines, even though the displacements
are not equal and opposite. The view I wish to put forward
is that in metals the frequency of this type of vibration is so
low that the equilibrium for such a displacement is practically
neutral, and therefore that a system vibrating in this way
can absorb at any temperature the full amount of energy
which at that temperature corresponds to each degree of
freedom.

It may be desirable to illustrate the argument by a
particular case (fig. 1). Let us take that where the electrons

Fig. 1.

A -A A A

are arranged in a simple cubical space lattice with the atoms
at the centres of the cubes. Then, if an electron were dis-
placed independently of the others, the frequency of its
vibration would be very great and it would absorb very
little energy; while if a chain of electrons along a lattice
like AA’ A'’A'’ were displaced along the line of the lattice,
the time of vibration of the chain might be comparatively
~ infinite, so that the chain would absorb the full amount of
energy corresponding to one degree of freedom. However
many electrons there may be in the chain, it has only one
degree of freedom, for the nature of the displacement
supposes that they move as a rigid body along a definite
line. To sum up, the study of the frequencies of vibrations
of the lattice of electrons shows that while at ordinary
temperatures little energy could go into vibrations corre-
sponding to the motion of an electron as a separate individual,
yet groups of electrons along a lattice forming a rigid chain
and moving in the direction of the length of the chain might
absorb a full quantum of energy.

Thus in a solid with the constitution we have sketched,
chains of electrons lying along a line of a lattice may be
travelling along that line carrying energy and electricity
from one part of the solid to another; the frequency of

the Electron Theory of Solids. 665

vibrations of these chains is so low that they readily absorb
energy even at low temperatures, so that the average energy
of the chains at the absolute temperature is KO, where
k represents the factor corresponding to one degree of
freedom.

Thus, though the electrons in the solid are not free, and
are in a very different condition from those of an electron
gas diffused through the solid, yet like those in the gas thev
can carry energy and electricity from one place to another.
In the gas, howev er, each electron is supposed to be moving
independently of its neighbour, and also to possess energy
3k@ corresponding to three degrees of freedom; in our case
the agents which carry heat and electricity are not isolated
electrons, but chains of electrons moving as if the electrons
which compose them were rigidly connected together ; thus,
however many electrons there may be in the chain, the
average energy of a chain will only be £0, i. e. one-third
of that of each electron on the gas theory. Thus on this
view the contributions of the electrons to the specific heat of
the solid will be a very small fraction of the contribution
of the same number of electrons on the gas theory.

Professor Lindemann has given (Phil. Mag. xxix. p. 127,
1915) a theory of Metallic Conduction which, though on
quite different lines to the present one, agrees with it in
making the electrons which carry the current move along
the lines of the lattices, and in the view that the electrons
make no appreciable contribution to the specific heat.

The existence of these chains requires that the frequency
of this vibration should be exceedingly small ; if the dimen-
sions and arrangements of the lattice are such that the
frequencies given by equation (2), are not less than 10”
or so, the chains will not absorb energy at moderate tem-
peratures, and at these temperatures the solid will act as
an insulator. Thus it requires special conditions for the
lattices of electrons to give rise to conductivity, so that the
fact that neither boron nor the diamond is a conductor is
not inconsistent with the theory. The motion of the chains
need not necessarily be a reciprocating motion, for if the
amplitude of excursion of an electron in the chain exceeds
half the distance between two electrons, an électron such as
A’ would shoot past another position of equilibrium; the
forces acting on it would change sign and would tend to
increase the distance still further; thus the chain would
continue to move on in one direction and would not oscillate
backwards and forwards.

666 Sir J. J. Thomson: Further Studies on
On the Origin of the Chains.

if we consider the state of things inside a solid, we can
I think see reasons for believing that the existence of moving
chains of electrons is probable. The solid is traversed by
the radiation corresponding to the radiation from a black
body at the temperature of the solid. This radiation consists
of a series of discrete pulses, each pulse being the seat of
intense electrical forces. When the effect of these pulses is
represented by a Fourier series of waves, the wave-length
for which the intensity of the light is a maximum is inversely
proportional to the absolute temperature, and at 0° ©. is
about 107? cm.

We may suppose that the linear dimensions of the regions
occupied by individual pulses are grouped about a mean
which varies inversely as the absolute temperature, and
which is large compared with the distance between two
electrons. Thus the radiation will furnish fields of electric |
force which have a high and fairly constant value over a
length which includes a good many electrons; and the
electrons in a lattice will from time to time be exposed to
electric forces extending over a considerable length, and
thus a chain of electrons will be started in motion as a
whole. We should expect the average length of the chain
to be inversely proportional to the absolute temperature.
Moreover, such chains of electrons moving past the atoms
would themselves tend to set up pulses of radiation, the
dimensions of the pulse being commensurate with the length
of the chain. Thus there would bea kind of regenerative
action ; the radiation would tend to produce the chains, while

the chains would tend to produce the radiation. When the
two processes got into equilibrium the radiation would be
that corresponding to the black body radiation at the
temperature of the solid, while the average kinetic energy
of the chains would be proportionai to the absolute temper-
ature.

When the solid is not acted upon by electric forces, there _
will be as many of these chains moving in any one direction
as in the two opposite, so that there will be no current of
electricity through the solid as a whole. The motion of the
chains will give rise to “local currents” whose distribution
might be affected by magnetic forces. .

We shall now consider the effect of an electrics force on
the motion of the chains.

On the old theory that the electrons moved freely through
the metal and kept striking against its atoms, the result of an
electric force X was to give to the electrons an average

the Electron Theory of Solids. 667

velocity in the direction of the electric force equal to XeA/2mv,
where A is the mean free path of an electron and v the mean
velocity. This result is obtained as follows: in a collision
between an electron and an atom, since the mass of the electron
is infinitesimal in comparison with that of the atom, there will
be no “persistence” of the velocity of the electron. The
velocity communicated by the electric force to the electron
before it came into collision with an atom will, as it were,
be completely wiped out by the collision, and the electron
will make an entirely fresh start. Thus if ¢ be the interval
between two collisions, the average velocity of the electron
in the direction of the electric force will be

tT iXe DER eX
Bones 2m v

On the theory we are now discussing, the carriers of
electricity are not free electrons, but chains of electrons
rigidly connected moving along a line of the lattice; since
the chain has only one degree of freedom, the average energy
of a chain at the temperature @ is R@/2 ; hence

ees ee ss (8)
where n is the number of electrons in the chain and v its

velocity. Thus the average energy of a single electron in
the chain is R@/2n. On the old theory when each electron

was supposed to be free, its average energy was 36/2. The

energy and velocity of an electron on the new theory are
smaller than on the old.

The ‘collisions’? between the electrons and atoms are also
different. On the new theory an electron in a chain is
moving past a row of atoms arranged at equal intervals 2c

along a line parallel to the path of an electron; the time it

takes for an electron to pass from closest proximity to one
atom to closest proximity to the next is 2c/v. If the inter-
change of energy between the electron and the atom were
limited to the time when the electron was closest to the atom,
the electron for a time 2c/v would not be losing any energy,
and so could, under the electric force, acquire a velocity
equal to Xe.2c/mv. The loss of energy by the electrons
will not, however, be confined to the positions of closest
proximity, but will extend some way on either side. The
result of this will be that in part of the interval 2c/v the
electron will be losing velocity, so that the velocity it will
acquire under the electric force will be less than Xe . 2¢/mv,
and the average velocity will be less than half this value.

668 Sir J. J. Thomson: Further prune. on

We shall suppose that the average velocity due to the
electric force is

OE
Ge as rae . ° ° ° ° ° . (4)
where g is a fraction. This by equation (3) is equal to
Xencv eXlv

JRE —79Re°?
where / is the length of the chain.

If g is the number of chains parallel to w per unit volume,
the number crossing unit area in unit line is equal to
Xely
g oRA 2?
and since each chain carries ne units of electricity, the current
across unit area is
Xe’nly —
7 ORG
Hence a, the specific electrical conductivity, is given by
the equation f
elung _e’lufp

ls Rea OR ais. Late o: 0Re. oatee (5)

where p is the number of electrons per unit volume and /
the fraction of them formed into chains.

On the theory we are considering, these moving chains are
responsible not only for the electrical conductivity of metals,
but also for the production and absorption of the radiation
which fills the space occupied by the metal. They may be
regarded as in some ways analogous to Planck’s oscillators,
the slowly moviny ones corresponding to oscillators with a
long period of vibration, producing mainly the long-wave
radiation while the chains with high velocities give out the
radiation corresponding to the shorter wave-lengths. We
see from equation (3) that, at the same temperature, the
chains which have a high velocity contain a small number
of electrons and are therefore short, the chains which have a
small velocity contain a large number of electrons and are
long. Thus the long chains produce the long wave-length
radiation, the short chains the short waves. We should
expect on this view that the lengths of the various chains in
a metal should be distributed according to a law analogous
to that which governs the distribution of the energy corre-
sponding to waves of different wave-lengths in the radiation
from a black body. But according to Wien’s Displacement
Law, the length-scale of the radiation varies inversely as the
absolute temperature; 1,0=aconstant. Hence we conclude

the Electron Theory of Solids. 669

that the average number of electrons in a chain varies
inversely as the absolute temperature.
If cn=£/@, where @ is a constant, then by equation (3)

v= (3°) 0.

Bm

Thus nev is independent of the temperature except for the
variation in c, due to the alteration in the volume of the
metal caused by a change of temperature. jp, the number of
electrons per unit volume, will only change with the temper-
ature through thermal expansion. Hence we see from the
ex pression (5) for the conductivity that if the number of
electrons concerned in carrying the current does not vary
with the temperature, the specific conductivity will vary
inversely as the absolute temperature, which is sey approxi-
mately true for pure metals.

Resistance under rapidly Alternating F\ orces.

We can get an estimate of the average velocity of the
chains in the following way :—In the preceding investigation
we have supposed that the electric force acting on the metal
was steady. The argument will evidently not hold when
the force is alternating so rapidly that while the electron is
passing through the distance 2gc the force changes its
direction ; tor in that case the effect of the electric field in
altering the motion of the chains will be much less than that
expressed by equation (4). When the force is reversed
many times during this period there will be very little
alteration, and therefore very little conductivity. Thus the

resistance of metals under alternating forces should begin to.

increase when the period of alternation becomes comparable

with the time taken by an electron in a chain to travel over

a distance equal to g times that between two electrons in the
chain. When the period of alternation is considerably

greater than this time we should not expect the resistance to.

vary with the period.

Rubens and Hagen determined the conductivity of metals

under alternating “forces by measuring the amount of light
of very long wave-length reflected from the surface of the

metals. They found that the electrical conductivity of
certain metals at room temperature under electrical waves.

whose wave-length was 2°5x 107° cm. was the same as the
conductivity under steady electrical forces, and that even

when the wave-length was as short as 4x107* cm. the
electrical conductivity was within about 20 per cent. of that.

670 Sir J. J. Thomson: Further Studies on

for steady forces. As the period of the longer waves is
8:3 x 10714 second, we may conclude that the time taken for
an electron in a chain to pass over g times the distance which
separates it from its next neighbour in the chain cannot be
greater than about 107 second. If we take the distance
between 2 electrons as 2xX107%8, this would make the
minimum velocity of the chains about 29x 10°. This refers
to the temperature at which Rubens and Hagen made their
- experiments—presumably about 15° C. As the velocity of

the chains decreases as the temperature falls, the reflexion
from a metallic surface should become at very low -temper-
atures abnormal at longer wave-lengths than those determined
by Rubens and Hagen.

We can get in another way an estimate of the magnitude
of the time taken by an electron in a chain to pass over a
distance equal to half the distance between two neighbouring
electrons in the chain. At the temperature of 15° C. the
wave-length of the light of maximum intensity in the black
body radiation is 107? cm.; the time of vibration of this
light is }x 107% see. We should expect from the way we
have supposed the black body radiation to arise, that this
time would be of the same order as that taken on the average
by an electron in the chain to pass over g times the distance
between two electrons, and so again we arrive at 107 sec.,
as being a time of this order.

On the supposition that c/v is proportional to the time of
vibration of the light of greatest intensity, we have

eens
Vins
where y is a constant which does not depend on the metal.
nv
Hence N= ae
but imnv? =4R8,

m being the mass of an electron ; thus

nv’= #9= 15x 10" x.
7

Thus Ei —leodusicy x Ores
The specific conductivity
e*ncv
= ofp RO

| sO) oe WO hy oipee.
es RO ;

the Electron Theory of Solids. 671

in this expression the only factor which varies from one metal
to another is fp, the number of electrons made up into
chains ; the conductivities of metals at the same temperature
are directly proportional to the number of electrons in unit
volume which take part in carrying the current.

We can put the expression for the conductivity in the
form
ec

a mv
If we take ge/v at 15° C. to be 107", since
é?/m=2°8 x 10738,
the conductivity at this temperature is equal to
hp eo SLO

The values of f calculated from this expression for some
metals are given in Table I.

TABLE I,

Metal. sap ap Dp. f.
Babitunn, h,, dads: 1:1x107* 25 x 10? ‘16
Sodium ...... ce ee! aoa Mont tg ‘47
Potassium ..... Rees tye SO 7s ‘70
Bybidium. ..... 03 ..08 “See *O6:..,, 7510)
Ch ante 048 ,, 85
olen net. 9283 ES. 25 19
Magnesium ......... DSi ty aS “1%
Shin ee ae CER IGDs: & ho. ,: 08
Oadmmme-......-... PS: ee 33) ) ee ‘09
POMP <...3.... 3 ds Tas HE Oe =. a 4

Thus on this theory, potassium has a much larger per-
centage of its electrons moving about in chains than any
other metal.

To form an estimate of the average number of electrons
in a chain and the velocities of the chains, we may proceed
as follows :—If we suppose that at 15° C., gc/v=107, then
if c=10~*, which is about right for sodium, v=10° xg.
When v is known, we can get n from the equation

tmnv?=tRO.
If v=109, this equation gives
n=4°4 x 10/9’,

As g must be less than unity, the chains at this temperature

672 Sir J. J. Thomson: Further Studies on

will on the average probably contain more than 10,000
electrons, and their average length would be greater than
2.10-4em. As the average length of the chains varies
inversely as the temperature, the average length at 3° Ab.
would be greater than 02 cm. and their average velocity
less than 10° cm./sec.

Super-Conductivity.

The expression for the specific conductivity given by
equation (5) is based on the assumption that in a “ collision ”
between an electron and an atom, the energy imparted to
the electron by the electric field is given up to the atom
during this “ collision,” so that the electron starts as it were
afresh after each collision. For this to happen there must
during the collision be a considerable transference of energy
from the electron to the atom. The energy of the atoms is
due to their vibrations about positions of equilibrium, and
the frequencies of these vibrations, according to the experi-
ments of Nernst and Lindemann on the variation of the
specific heats with temperature, range from 10” to 10% for
the different metals. Now, it follows from general dynamical
principles that a collision lasting for a time which is long
compared with the time of vibration of a system, will excite
very little vibration in the system and communicate very
little energy to it. The amount of energy communicated |
will fall off very rapidly as the ratio of the duration of the
collision to the time of vibration increases Ina case con-
sidered by Jeans, ‘ Kinetic Theory of Gases,’ § 481, the
energy communicated to the system was proportional to
e *?, where c is the duration of the collision and p the fre=
quency of the free vibration of the system. It follows from
this that when the chains of electrons are moving so slowly
that the time of a collision is long compared with the time
of vibration of the atom, very little energy will be trans-
ferred. Our expression for the electrical conductivity was,
however, obtained on the assumption that at each collision
the excess energy due to the electric field was given up.
If, however, the transference of energy is not sufficient to
allow of this, the average velocity of the electrons will be
greater than that calculated, and the conductivity greater to
a corresponding extent. If there were no transference of
energy, the average velocity of the electrons and the electrical
conductivity would both be infinite. We see then that when
the temperature gets so low that the time taken by an

the Electron Theory of Solids. 673

electron to pass over a distance 2¢ is comparable with the
time of vibration of the atom, any diminution in the temper-
ature will produce an abnormally large increase in the
conductivity, and thus the metal would show the super-con-
ductivity discovered by Kammerlingh-Onnes.

The numbers we have just obtained for sodium show that
at a temperature of 3° Ab. the time taken by a collision
would be greater than 2x 107", and this is very long com-
pared with the time of vibration of the sodium atoms, which
have a frequency of 3°96 x10". There would be very little
transference of energy at this or even considerably higher
temperatures, so that the conductivity would be very
oreat.

” We have associated the time taken by a chain to pass over
the distance 2c at any temperature with the time of vibra-_
tion of the light of predominant energy at that temperature.
On the theories of the variation of specific heat with temper-
atures given by Nernst, Einstein, and Debye, this variation
isa function of the ratio of the time of vibration of this
light to the time of vibration of the atom. Thus on the view
that the average time of a collision is about that of the time
of vibration of this light, the variation of the specific heat
with temperature and the communication of energy from the
electron to the atom depend upon exactly the same quantity,
and thus the variation of the specific heat with temperature
ought to be closely connected with the super-electrical
conductivity. The product 0c of the temperature and the
electrical conductivity ought to change rapidly with the
temperature when the specific heat does so. The product 0c
will increase as the specific heat diminishes ; if, however,
we were to plot the reciprocal of 0c against the tempe-
rature, we should expect to get a graph very similar to the
one representing the connexion between specific heat and
temperature.

That a connexion of this kind does exist between 1/0c and
the specific heat is, I think, shown by Table II., which
‘contains the values of 1/@o for lead and silver calculated
from the values of the resistances given by Kammerlingh-
Onnes (communications from the Physical Laboratory of
Leiden, cxix. 1911); the third column contains the values
of 6/0 when @=hy/R, where N is the time of vibration of
the atom ; the fourth column gives the value of the specific
heat calculated by Debye’s theory (Jeans, ‘ Kinetic Theory
of Gases,’ §§ 553) ; and the fifth column the ratio of 1/00 to
the specific heat.

Phil. Mag. 8. 6. Vol. 44. No. 262. Oct, 1922. 2X

674 Sir J. J. Thomson: Further Studies on

TaBLeE II.
Lead, @=95.
Specific Ratio of 1/0¢

6, 1/00. 6/0. heat. to specific heat.
273 366 2:88 993 868
169°3 351 1°78 “984 357

779 325 +82 928 350

20°18 150 ‘215 ‘41 365

13°88 87 145 2 435

Silver, @=215.

278 366 1:27 ‘965 380
169°3 3438 ‘79 924 Be

77-9 252 362 ‘691 365

20:18 45 ‘095 ‘073 615

Thus_except at the lowest temperatures the ratio of 1/A¢
to the specific heat is fairly constant ; and inasmuch as
Kammerlingh-Onnes and Clay have shown that when a small
amount of impurity is present, the resistance at very low
temperatures approaches a finite value instead of continually —
diminishing as the temperature falls, it is evident that at
these temperatures a trace of impurity would produce a large
increase in the value of 1/@c. The higher the value of ©,
the higher will be the temperature at which an abnormally
large increase of the conductivity with fall of temperature
sets in. Of all metals, beryllium has the smallest atomic
value, and so we should expect it to have the greatest value
of vy and ©; it seems probable that the temperature coefficient
of this metal may be abnormal even at room temperatures.

Thermal Conductivity.

The motion of the chains of electrons along the lines of
the lattices will in an unequally heated conductor tend to
equalize the temperature, for much the same reason as on
the Kinetic Theory of Gases the conduction of heat is
brought about by the motion of the molecules of a gas.
There are, however, several points of difference which require .
discussion before we can proceed to find an expression for
the thermal conductivity on the chain electron theory.
When the temperature is uniform, there is no ambiguity in
the statement that the average kinetic energy of the chain is
that corresponding to one degree of freedom. A chain of
electrons, however, stretches over a distance large compared
with the distance between two atoms, and when the temper-
ature is not uniform the temperature at one end of the
chain may not be the same as that at the other, As the

the Electron Theory of’ Solids. 675

electrons in the chain move like a rigid body, each electron
has the same kinetic energy; we shall suppose that this
energy is the same as if the whole of the chain were at the
temperature of its middle point, so that the kinetic energy
of the whole chain is that corresponding to one degree of
freedom at the temperature of the middle point of the
chain.

Another important point is that the energy carried across
a plane by a chain of electrons passing right across it may,
when the temperature is not uniform, be much greater than
the actual kinetic energy in the chain when it first reaches
the plane. This is important because if it were not so the
transport of energy due to the motion of the chains would
not be great enough to account for the observed thermal
conductivity even it every disposable electron were utilized
to make up the chain. It must be remembered that on this
theory the number of disposable electrons in unit volume is
known; for example, in the alkali metals it is equal to the
number of atoms, and cannot be regarded as a quantity
whiclr can be adjusted so as to give the right value to the
thermal conductivi'y.

To see how this additional transport of energy is brought
about, consider what happens when a chain “of electrons
ABCDE crosses the plane ZZ, moving past the atoms in
its neighbourhood and exchanging energy with them. ~ If
2¢ be the distance between neighbouring electrons or atoms,
we shall define a collision between an atom and an electron
to be the passage of an electron past its shortest distance
from the atom. If we take the axis of x parallel to the
chain, then when the head A of the chain reaches 7Z
each of the electrons in the chain has 1/n of the energy
corresponding to one degree of freedom at-the temperature

l dé
tid 2 dx’
its electrons, and @ the temperature of the plane ZZ. When
A makes a collision with the atoms just to the left of the
plane ZZ, it will momentarily lose an amount of energy

1/dé

proportional to = qa This will lower its energy below that
n

which must be possessed by every electron in a chain whose
middle point is now at a place where the temperature is

where / is the length of the chain, n the number of

6+4(1—2c) = this energy only differs from that before

. 2c dd '
impact by onda? the electron has, however, since / is much
uv

ae 2

676 Sir J. J. Thomson: Further Studies on

greater than 2c, lost far more than this, so that energy must
be transmitted along the chain to A to bring its energy up to
its proper value. Thus when the electrons are connected
together in chains, the transference of energy is not confined
to the energy carried by the electrons when they are crossing
the plane ; each collision made by an electron in the chain
will, until the whole of the chain has passed the plane, result
in the transference of energy across the plane; if the chain
is long this second type of transference may far exceed in
magnitude that which would occur if there were no
collisions.

We shall now proceed to find an estimate of the trans-
ference due to the collisions.

Let us take the electrons in the chain in pairs, the con-
stituents of a pair being equally distant from the centre; let
this distance be y. Then, as the chain moves along, one of
the constituents of the pair has energy corresponding to a
temperature 7 =

e dé

corresponding to the temperature Use below the temperature
of its position. If $R@ is the energy corresponding to one
degree of freedom at the temperature 0, this excess or defect
nee We shall suppose
that at each collision of an electron with an atom the energy
of the electron is restored to the value corresponding to its
position. |

Let us begin with the electrons at the beginning and end
of the chain. We have seen that the first collision of the
front electron after passing the plane results in the trans-

above, the other constituent the energy

of energy of an electron will be

l
ference of a units of energy across the plane. The

collision made by the electron in the rear will result in its
gaining Tors units of energy ; this will have to be given

up by the chain, but inasmuch as all the chain is on the right
of the plane ZZ, the energy will be given off in this region
and will not be transferred across the plane.

Thus the first collision of this pair of electrons transfers

R1 dé
An dx

units of energy across the plane.
Let us now consider the next collision. The front electron

the Electron Theory of Solids. 677

will lose - os units of energy, and this will have to be

n dx 5

supplied from the chain ; since part of the chain is now on
the left-hand side of the plane, some of this energy will come
from this part, and will not be transferred across the plane.
The energy coming from the part of the chain to the right
will be transferred across the plane ; the ratio of the length
of chain to the right of the plane to the length of the chain
is (l1—2c)/l. We suppose that this fraction of the whole
energy comes from the part of the chain to the right, and so
is transferred across the plane. ‘Thus the transference of
energy due to the second collision of the front electron is

Rl dé [—Ie
4ndx °
Now consider the second eollision of the electron at the
rear of the chain.

This electron will by the collision receive a = units of

energy, and as there are a large number of electrons in the
chain practically the whole of this must be given out again
by the chain. If it is given out uniformly from all parts of
the chain, since the length of the portion to the left of the
plane ZZ is 2c, the amount of energy given out in this region,
which is the amount transferred across the plane ZZ, is

IR dé 2c
Anda’ I
Thus at the second collision of this pair of electrons the
energy transferred across the plane is equal to

IRd@ 1—2c IR dO 2%
Lee Lede. |
IR dé

~ An dx’

the same as that transferred at the first collision. We can
see that this must be true of all the collisions; and as there
are n of these before the chain gets right across the plane,
the total amount of energy transferred across the plane by
the collision of this pair of electrons is equal to

IR dé

4 dx

For a pair of electrons at a distance y from the centre of
the chain, the interchange of energy at each collision with an

atom is 2 = and the number of collisions with one

Qn dx

678 Sir J. J. Thomson: Further Studies on

member of the pair in front of the plane is equal to 2ny/l;
alt

hence the energy transferred by this pair is = ue Thus,

giving y all possible values, we find that the total amount of

energy transferred across the plane ZZ through the collisions |

of all the electrons in the chain is

dG rl. \7 l : l \7
My eG) +G-a) +)

<= =. —,— when x Is large.

24. dea

their average velocity, the energy transferred across unit
area per second is | 7

7 nl dé
24 ai dx’

In making this rough estimate of the transference of
energy, we have supposed that the transference occurred only
when the electron was in closest proximity to the atom.
The process by which the electron first loses energy to the
atom and regains it again by a transference of energy
along the chain will begin before the electron reaches its
shortest distance from the atom and go on after it has passed
it ; the result of this will be that at each passage of an electron
past an atom the transference of energy may be very con-
siderably greater than that in the case we have considered.
We must therefore suppose that the transference of energy

y Rdé | !

at each collision is not 5 —— but a multiple of this, viz.
Al

Ax
ey Rd@

2n dx’
where eis a number greater than unity which depends on
the law of force between the electron and the atom. This
will make the transference of energy across unit area per
second equal to

IL dé
ae ee

where ¢ is the number of chains per unit volume and » the
velocity of a chain. Hence K, the thermal conductivity of
the metal, is given by the equation

1
| ae

i enlguR

igre
54 efplvh,

the Electron Theory of Solids. 679

The ratio of K to o, the electrical conductivity, is given
by the equation

oir Geyer?

i beg aes

The right-hand side of this equation does not involve any
quantity peculiar to the metal; hence the ratio of the thermal
to the electrical conductivity should at the same temperature
be the same for all metals, and at different temperatures
should be proportional to the absolute temperature. This is
the well-known law of Wiedemann and Franz, which is obeyed
with fair accuracy by many metals.

Summary.

This paper contains a calculation of the compressibility of
a divalent element, calcium, and also that of the diamond by
the method given in my paper on the Hlectron Theory of
Solids (Phil. Mag. April 1921). The results obtained are
in good agreement with those found by experiment. The
same theory is then applied to the consideration of metallic
conduction, electrical and thermal. It follows from the
theory that when an individuai electron is displaced relatively
to its neighbours, the frequency of the vibration is that cor-
responding to the visible or ultra-violet part of the spectrum ;
these vibrations would not, unless at extremely high temper-
atures, absorb an appreciable amount of energy. When,
however, instead of a single electron being displaced, a chain
of electrons lying along one of the lines of the lattice is
displaced as a rigid body relatively to the neighbouring
atoms and electrons, the time of vibration of this chain may
be very long, so long that even at very low temperatures the
chain may acquire the full quantum of kinetic energy cor-
responding to one degree of freedom at its temperature.
Thus chains of electrons moving like rigid bodies may travel
along the lines of the lattices, and carry electricity and energy
from one part of the metal to another. The theory that
electric and thermal conductivity is due to the movement of
these chains is worked out, and is shown to account for the
variation of electrical resistance with temperature, for the
super-conductivity of metals at very low temperatures dis-
covered by Kammerlingh-Onnes, and for Wiedemann and
Franz’s law of the proportion between electrical and thermal
conductivity.

[= 680.4

LIX. The Decrease of Energy of « Particles on passing
through Matter. By G. H. Henperson, Ph.D.*

~§ 1. Introduction.

FXHE general laws governing the passage of @ particles
through matter have been discussed theoretically by
both Darwin + and Bohr f.

If E, M, and V be the charge, mass, and velocity of the a
particle and e and m be the charge and mags of an electron,
then, when the « particle approaches an electron along a line
at a distance p from it, the energy given to the electron is,
by the ordinary laws of dynamics,

2H? :
Q= mV? (pe a a?) 5 ° ° ° ° . (1)
a2 He(M-+m) is: Ke

where . =

MmV2) > mye thie

if the electron is free.
In passing through a thickness Aw of matter, the number
of encounters in which p lies between p and p+ dp is

2a NnAxp dp,

where N=the number of atoms in 1 em.’
and n=the number of electrons in one atom.
Then, if T is the energy of the « particle,

a iy 4 H?e? Nn | pdp
De NO Neth iae

If the limits of pin this integral be taken as 0 and w, the |
integral becomes infinite, 7. e. an a particle could not pass
through an appreciable thickness of matter atall. Hvidently
some upper limit to the radius of action of the & particle
must be taken.

In the first paper dealing with the motion of « particles,
Darwin made the assumption that the effect of the @ particle
at any instant was confined to the electrons of the atom
through which it was passing. He was able to calculate the
motion of the @ particles through matter for various arrange-
ments of electrons within the atom. Theoretical velocity
curves showing the yariation of velocity with distance

* Communicated by Sir EK. Rutherford, F.R.S.
+ Darwin, Phil. Mag. xxiii. p. 901 (1912).
t Bohr, Phil. Mag. xxv. p. 10 (1913), and xxx. p. 581 (1915).

(2)

Energy Decrease of « Particles passing through Matter. 681

travelled were obtained which showed the same general form
as the experimental curves, and from these an estimate of the
number of electrons in the atom was made.

On the other hand, Bohr considered the time of passage of
the 2 particles past an electron to be the determining factor,
and assumed that as long as this time of passage was small,
compared with the period characteristic of the electron in
dispersion phenomena, the electron could be considered as
free. When the time of passage was comparable with this,
however, the electron could no longer be considered as free,
aid in this way an upper limit to p was introduced. The
calculated velocity curves showed good agreement with
experiment over most of the range of the 2 particle.

§ 2. Method adopted.

In this paper the law of decrease of energy of an a
particle is developed along lines somewhat different from
those of the writers mentioned. |

According to the modern ideas of atomic structure, due
to Bohr, the electrons are thought to be arranged in various
stationary states or energy levels. An electron leaving one
of these stationary states can only move to another such
state or completely out of the atom (to infinity). Thus
the energy which an electron can take up is limited toa
number of finite amounts characteristic of the atom.

These views furnish a simple method of fixing the upper
limit to the radius of action of an @ particle upon the
electrons of matter. If the electron is to be moved from
one stationary state to another by the passage of an «
particle near it, then with the finite amount of energy
which the electron must take up in order to effect the
change there may be associated an upper limit to the
radius of an & particle upon the electron.

The mechanism which is involved in this transfer of energy
from « particle to electron may be difficult to conceive.
However, when an electron is moved from its stationary
state to infinity by the action of light, the frequency v of

the light must be such that the quantum hy is greater |

than the finite difference of energy between the initial and
the final states of the electron. Here the rule governing
the transfer of energy is known, although the mechanism
involved is not. A similar statement holds for the case
when the transfer of electrons from one state to another
is caused by electron impacts (elastic and inelastic impacts).
Similarly, in the case of « particle impacts the transfer of

682 Dr. G. H. Henderson on the Decrease of

energy a) be determined by appeal to experiment
before the mechanism is understood.

Accordingly, it seemed of interest to apply the classical
theory of the exchange of energy as given by (1) to the case
of the « particle and electron, having regard to the limited
number of stationary states which the electron can occupy
within the atom, and to compare the resulting law of motion
of the « particle with experiment.

This has been done in the following paper, taking asa basis
the following assumptions.

Interchange of energy with an electron takes place
according to (1) provided that the energy transferable,
according to (1), is greater than the ionization potential of

that electron.
hus for any given V a definite upper limit is placed upon
p by (1), where Q is equal to the ionization potential. For
values of p less than this limiting value po, the excess of
energy over that required to remove the electron from
the atom may be in the form of kinetic energy of the
electron.

The existence of resonance potentials is taken into account
by assuming that when the energy available according to (1)
lies between the ionization and resonance potentials, or
between two resonance potentials, the energy transferred is
constant and equal to the lower resonance potential.

For encounters where p is greater than po, given by (1)
for the lowest resonance potential, it is assumed that practi-
cally no energy is transferred to the electron, the latter con-
tinuing to move in its stable orbit and behaving as if rigidly
bound to the atom.

§ 3. Calculation of the Law of Motion of « Particles.

Consider a substance in each of the atoms of which there
AVe Ny, Ny... ny electrons with the ionization potentials Q,, Qe
. Q, respectively. The total number of electrons

N=Ny+Ng+...0Nn

Then, for the n,; electrons having the ionization potential
1»

P|
AT _ 2An 4 pdp
Av =V? pita’

= > ) ‘ ‘ : 4 206
Energy of « Particles on passing through Matter. 683
where p; is given by
2 Hie?
9 9
+o
/ mV" 2),

and

' paar
Hence —-= —,- log y PY
Aw VA a

An, 2mV? |
——— 10D

72
‘Saison AO ag

Summing for all types of electrons,

AN Gia ie ImnV?
Rey ays Gk

= Ey log 2mV2—X,n; log Q: |
is the rate of loss of energy due to ionization potentials.

To take into account the effect of resonance potentials,
consider, first, the n,; electrons with the ionization potential.
Q,. Let there be resonance potentials Q,', Q,’..., all less
than Q,, and let the corresponding upper limits, given by (1),
for the p’s be p,', p;''.... We assume that for values of p
lying between p; and pj’ (t.e., when the energy available,
according to (1), lies between the ionization and resonance
potentials) ite energy transferred to the electron is constant
and equal to Q,’. Similarly, for all values of p between Pr
and p,'', the energy transterred is constant and equal to Qu" :

Then the total loss of energy by the « particle passing
through a distance Ax, which is due to the presence of
resonance potentials, will be

pt Pes
AT=27Nn, Ax [a fray + Qu" {ody + a | ;
: Py P,’
ae aNn [Q K 12 aD) Q a TED S ay HY ae |
aE 11 Qi (p1°— pi") + 1 (Pi a a i
2a

AT 27NE’e’n, ae A Pe 1 ) |
Bet mv L&(Go-g) +"(Q7- gt)
An ; Cer)
=ViFq1- Qo f

684 Dr. G. H. Henderson on the Decrease of
Summing this expression for all the types of electrons

Ny, Ng, etc., we have

ey
oe = V3, a 3e4 1 = ar}

Thus the e, expression for the loss of energy is

(é+1)
a == 7 [ leg V? 4 log2m—3.— oa _ log Q:+ 2s 3,41— ars |

(+1)
Put log b= log 2m —¥, “log Q,+> oS te 12 (3)

Then Ap = v2

dV An fae

the negative sign entering because AT is a loss of energy.
MV2dV
Thus dz= — An log V3
ION OND)
2Anl? log bV?
M e-Ydy Me Si. ao
SME PME ae where ha — 2 log dV’.
Let the velocity of the a particle initially be V) and the
velocity of the « particle after doing a distance # be V, then

es Mee
Mee DAsib? \ omy eee
Yo
where Y = -—2 log 6V?= — log b?V4,
Y= — log b?V,'.

M
Hence C= 9Anb? [ Ee — Yo) — Hu(— Y) {| Sie (4)
Hi(x) is the exponential integral, defined by

aby

Real {=

. @

numerical values of which have been tabulated by various

writers.

Energy of « Particles on passing through Matter. 685

It is very interesting to observe that this equation is of
the same type as that derived by Bohr in his second paper
on the motion of a particles through matter, although derived
on quite different assumptions. ‘The meaning of some of the
constants is, of course, quite different.

§4. Comparison with Hxperiment.

Substituting accepted values of the physical constants we
have from (4)

3
=e = Vay bat ¥) |

for air at 15°C. and 760 mm. pressure, assuming the number
of electrons in the fictitious air molecule to be 14:4.

We substitute numerical values in expression (4) for log 6,
term by term,

log 2m= 2°30 logy) 2 x 9°0 x 10-2 = — 61°58.

The remaining two terms are more difficult to evaluate, as
the values of the ionization and resonance potentials are
not completely known, and we are treating with average
values for air. The order of magnitude of these quantities
is, however, fairly well established. We will choose values
of this order of magnitude which give the best agreement
with experiment.

Assuming 4 electrons for which Q=200 volts=3°18x
10-” erg and 10-4 electrons for which Q=15 volts
aero LQ 10: ergy,

Be tis 10°4 ii
—3,"" log Q.=—23 | rq login 2°38 x 10-

+ 7qqlog 3-18 x 10- we) = 28°78.
For values of the resonance potential which are near the
ionization potential the terms (1—Q&*)/Q(”) will be practi-
cally zero. Hence we need only concern ourselves with
those few resonance potentials which are considerably lower
than the ionization potential. We shall probably not be far
wrong if we set X, (1—Q¢+)/Q=2 for each set of electrons.
Then log b= —61°58 + 23°75 + 2:0 = — 35°83,
log b?= —71°66,
corresponding to a value of b?=7°5 x 10-* approximately.
Thus the velocity equation of the « particle becomes

#=7°79 x 10-5 [Hi( — Y,) — Ei(—Y)].

686 Dr. G. H. Henderson on the Decrease of

It should be pointed out that since } appears as well in the
exponential integrals, the value of wis not very sensitive to
changes in 0.

Values of the range given by the formula have been
calculated for various velocities of the @ particle, and the
results are shown in the second column of Table I. The
evaluation of K7(—Y) has been carried out By interpolation
from the tables given by Jahnke and Emde*. In column
three are given the experimental results of Mavsden and
Taylor +. Column four shows the ranges as calculated by
Bohrt. The Table refers to RaC in air vat 15° 0,

MW ocimogs &
V/V. Cale. Experimental. Cale. Bohr.
20 1°87 1:90 i ako)
"8 3°43 aoe OO
=i 4°50 AA 4°45
(5) Dea Doe oo
0 5°67 Daag 57

From the table it will be seen that the calculated values
agree well with experiment. Both series of calculations fail
for low velocities of the @ particles. Bohr’s theory holds
down to values of V/V, equal to about -5; the present
theory has not quite the same range of applicability, failing
below values of V/V of about ‘6.

One or two points of interest should be noticed here.
First, we shall see what is the actual size of the radius
of action of the a particle called for.: Taking an a
particle moving with the initial velocity of radium C
(1°92 x 10° cm./sec.) and a resonance potential of 10 volts,
when numerical values are substituted in (1), we obtain

Po Pe BO C107 eg. oo Ome
and hence
Po = 1°83 x 107%,

Thus, ~o= 2°80 x 10-° cm. is the distance from the electron
within which an « particle must come in order to transfer
to it energy corresponding to 10 volts. This distance is of
the order of one-tenth of the diameter of an atom.

Secondly, there wiil be a velocity below which the @ particle
will be unable to ionize, however close the collision. Asthe
velocity of the « particle decreases the value of yo increases.

* Jahnke u. Emde, Funktionentafeln, p. 19.

|} Marsden and Taylor, Proc. Roy. Soc. A. 88, p. 4438 (1913).
+ Bohr, Phil. Mag. xxx. p. 597 (1915).

Energy of « Particles on passing through Matter. 687.

at first in inverse proportion to the velocity. A time will
come, however, when the term in a? will become important,
the value of po will then begin to fall off rapidly to zero ata
finite value cE V, which may be called the critical velocity
V.. For the ratio (9 + a*) /a2=2mV?/Q must not be less
than unity. When it 1s equal to 1, 7% ban

Then V2=Q/2m=8'8 x 10%,

The critical velocity V-=9°4 x 107 em /sec.

The same results should hold for positively charged
hydrogen atoms, which should cease io ionize a gas of 10 yolts
ionization potential at velocities less than about LO*em./see.,
equivalent to about 5000 volts. ‘This point has been discus<ed
by Sir J. J. Thomson *, who has dealt with the problem of
the ionization produced by moving electrified particles along
somewhat the same lines as that followed in this paper, by
assuming that a definite amount of energy is necessary to
remove an electron from an atom. It appears from experi-
ment that positive rays cease to ionize only when their
energy is less than 1000 volts. ‘This is not surprising, since
at these low velocities the velocity of the electron itself
probably plays an important part.

§ 5. Conclusion.

From the figures given in Table I. it will be seen that the
theory developed in this paper gives good agreement with
experiment in air for e particle velocities which are not too
low. It has already been pointed out that both Bohr’s and
the present theories lead to equations of the same type (4).
In both cases constants of the proper order cf magnitude
give good agreement. These constants are adjustable, but it
should be pointed out that their orders of magnitude are
fairly well known, and hence they are adjustable only within
narrow limits. It seems a little surprising that practically
the same results should be arrived at, starting from such
different assumptions. It is possible that these views might
be assimilated when more is known about the actual
mechanism of the transfer of energy from « particle to
electron.

Unfortunately the exponential integral which occurs in the
final equation of motion is of such a character that a certain
amount of variation In the values of the constants employ ed
does not materially affect the agreement with experiment.
It is thus impossible to decide definitely by appeal to experi-
ment which of the two points of view adopted is the more
correct.

* Thomson, 2nd Solvay Congress, 1913,

688 Knergy Decrease of a Particles passing through Matter.

For the same reason calculations have not at present been
carried out for substances other than air. In this connexion
the remarkable agreement obtained by Bohr in the case of
hydrogen, making use only of known data, should not be
lost sight of. Unfortunately, data regarding resonance and
ionization potentials for gases in the molecular state are still
incomplete.

It is regretted that results of a more decisive character
have not as yet developed from this application of the con-
cepts of resonance and ionization potentials. However, it is
felt that the possibility of explaining, along the lines followed
here, much of the experimental data on the motion of a
particles should be pointed out. Tlie equation of motion is
obtained comparatively simply, as will be seen from § 3.

The above remarks concerning agreement with experiment
referred to velocities not lower than ‘5Vo, half the initial
velocity of RaC. For velocities lower than this the agree-
ment breaks down completely. This is only to be expected,
for experimental data of other kinds show that the previous
homogeneity of the beam of « particles begins to disappear
at about this velocity.

In the foregoing calculation no account has been taken of
probability variations in the beam, while from a velocity of
-5V> downwards these variations become marked. Neither
has account been taken of the orbital velocities of the
electrons which may become appreciable for low & particle
velocities. Experiment shows that the -beam becomes
anhomogeneous in velocity. Straggling becomes very large
at this point. Further, recent results obtained by the writer
give evidence to show that the charge on the a particle does
not remain invariable for low velocities, though further
discussion of this point must be reserved. All things
considered it seems clear that the behaviour of a beam of a
particles becomes much too complicated at low velocities to
be dealt with by simple treatment.

Summary.

In this paper the equation of motion of an «@ particle
passing through matter is developed, making use of the
concepts of resonance and ionization potentials.

The equation found is shown to give good agreement with
experiment in the case of air, but does not furnish a decisive
test when compared with other solutions which have been
proposed.

Cavendish Laboratory,
July 1922.

Bee

LX. A Kinetie Theory of Adsorption.
By D. C. Henry *.

6a to the present time no theory of adsorption has been
developed which leads to equations valid over the
whole range from low to high concentrations. The adsorption,
both of gases and of solutes from solution, is well expressed
for low concentrations by the empirical ‘ exponential

formula ”
1

Timely a
where « denotes the quantity adsorbed, ¢ the exterior con-
1
centration, and & and 5) are constants. As soon as moderately
U

high concentrations of adsorbate are reached, this formula
gives results greatly in excess of the values observed, which
appear to tend to an upper limit.

In the present paper a theory of adsorption is developed
based on the conceptions of surface action introduced by
Hardy and Langmuir. An adsorption equilibrium is con-
sidered as involving a balance between the rate at which
molecules of adsorbate condense on the surface of the
adsorbent and the rate at which molecules leave, or evaporate
from the same surface. The fundamental assumptions made
are two, for both of which Langmuir has produced much
evidence. In the first place, it is assumed that the range of
action of the forces which bind molecules of adsorbate on to
the adsorbing surface is comparable with the diameter of an
atom, so that the layer of adsorbate molecules bound by the
field of force of the adsorbent will be only one molecule
thick. Secondly, it is assumed that the impact of a molecule
on a surface is completely inelastic, so that every impinging
molecule will condense.

General Adsorption Equation for n Gaseous components.

Consider an adsorbing surface, of area w, brought into
contact with a homogeneous gaseous phase containing
components §j, Sg,...S,. Whether the surface be crystalline
or liquid, it will present a more or less regular arrangement
of points of unsaturated field of force, where molecules of
adsorbate can condense; it the surface is crystalline, the
arrangement will be related to the crystal lattice, if it is
liquid, to the packing of the oriented surface molecules.

* Communicated by Prof. S. Chapman, F.R.S.
Phil. Mag. 8. 6. Vol. 44. No. 262. Oct..1922, 4 ¥

690 Mr. D. C. Henry on a

Suppose there are Ny, such points of attachment. If a
molecule from the gaseous phase impinges on a point of
attachment unoccupied by any other adsorbed molecule, it
will condense, forming a single ‘adsorbed layer’; if it
impinges on a point already occupied, it will also condense,
forming a second layer. But the relative life of a molecule
on the surface will depend on the attractive force exerted on
it by the surface, and if, as is usually the case, the attractive
force between molecules of adsorbent and adsorbate is much
greater than that between two molecules of adsorbate, the
relative life of a molecule in the second layer will be so small
compared with that of a molecule in the first layer, that we
may treat molecules impinging on points aiready occupied
as if they were immediately reflected and never condensed. —
At any moment let a fraction @ of the points of attach-
ment be vacant, and fractions 6), 0,,... 9, be covered with
monomolecular layers of 8;, 82, ... 8, respectively; then

Oo O)4 0, +>... Fez 1c. oo eee

Let a single molecule of 8), Ss,...8, occupy respectively
a1, Ag, ... dm points of attachment *; the number of molecules

of component 8S, adsorbed will be ea, and the adsorbed
quantity in gram-molecules will be f
: 0,No
oD)
LN? eo

x=

where N is Avogadro’s constant (6°06 x 1075).

Now the rate of evaporation from the surface will be
determined by Maxwell’s distribution law as the number of
molecules which reach, per second, a state of agitation
sufficient to break free from the force field, and is given,
for §,, in gram-molecules per second, by the expression Tt

Ay
AGN Bee RE. X= (ay) v.Xaoe ee ee (3)

where 2X, is the internal heat of evaporation of §, from the
surface, A, is a constant depending on the field of force,
and R and a have their usual meanings; vy, is written
for the expression
Agee ae) | BT
* Since the number of points of attachment occupied by a molecule is

not subject to a merely geometrical restriction, but is determined by the
field of force, it does not appear essentia} that the quantities a, should
be integers. ‘

+ Langmuir, Jour. Amer. Chem. Soc. xxxy. p. 122(1913). Ri ]
Phil. Trans. A. cei. p. 501 (1908), Pe ae

691

Now if p, is the partial pressure of §, in the gaseous
phase, the rate of impact of molecules of this component is *

Kinetic Theory of Adsorption.

]
———__ Xx p. om.-molecules per second per sq. cm.
oo Pr p per sq.em,,

where m, is the inolecular weight of §,; hence the rate of
impact on the surface of area w is w times this quantity,
which may be written
(4)
For a molecule of 8, to condense, ‘+ must impinge on a
spot where there are a, vacant adjacent points of attachment.
The chance that a given molecule will find one point vacant
is 0), and the chance that it will find a, adjacent vacant
points will be 0), if we neglect the possibility that the
molecule may need to impinge in an orientation related to
the configuration of the vacant points+. Hence the rate of
eondensation of molecules of S, is

@u,p, gm.-molecules per second. .

(9)
For equilibrium, the expression (4) must equal the
expression (5), and we get the n equations

@,p,9)” gm.-molecules per second. .

ae vere 1

VX, = Opp prOy”
or, from (1)

| X,= op(1— (a ee ae a
which from (2)
Sa aes ae |.
= pol N, Xy ane Ny Des
xy Sin tie
= 67rP- f= xa! ea Danis p) (6)
where € is written for
APO ae
fir os Oy, apnea Len
A) ae aR: yl) ome Se aera Od)
N
and = aN’ (8)

and is therefore the saturation capacity of the surface for S,,
supposing it to be completely covered by a monomolecular
layer of that component. :

* Jeans, ‘Dynamical Theory of Gases,’ 1916, p. 133.
Phys. Rey. i. p. 831 (1918).

+ This possibility could probably be allowed for by multiplying Oy
by a constant depending on @,, which could then be included in ¢...

AX 2

Langmut,

692 Mr. D. C. Henry on a

The n relations (6) determine the equilibrium adsorptions
of the n components.
In a similar manner we can write for the peau Ny of
adsorption of S$, in gm.-molecules per second
At, Ln

rr = Wy Py dee X, aT eae EON xe a ° 1)

where #, is the instantaneous value of the adsorption of 8, at
time ¢, and p, is the instantaneous value of the corresponding
partial pressure. A similar relation holds for each of the
nm components.

The Temperature Coefficient of the Isotherm.

The effect of temperature on the equilibrium adsorption
follows from equation (6), the only constant of which that
involves the temperature being &, which from equation (7)

is given by

A

fata -e8,

where ¢ is a constant ee of the temperature. The
qualitative conclusion that adsorption decreases with rise of
temperature follows immediately.
The relations (6) and (9) do not admit of general solutions.
Solutions must therefore be obtained for special cases.
One Component only— Adsorption of a single Gas.

_ Equation (6) reduces to
X=tp[1— ei) on ol

which can be expressed

x x
In = Ing-+ a. In| 0 (12)

NR od ewe
=Inf—a| 5) +5 c te |
For moderately small adsorptions we can use the approxi-
mation |
Xx i
In - =Inf— xi

xX cg ON ae | :
or lee = log €—0°4343 . XT° D Pap cme 0) (115
The relation (13) is of the same form as that obtained by

Kinetie Theory of Adsorption.

693

Williams * from entirely different assumptions, and can be
tested with the help of Williams’s calculations of the

measurements of Titoff, Homfray, and Chappuis.

Reducing

the equation to his units (adsorption a in c.c. of gas at
N.T.P., pressures in em. of mercury), we obtain

log « = log (2°988 . 108. £) —0°4348 . “ a

= A )— Ay. .

(14)

The agreement found is very good, as is shown in the

following table :—

"PARR

| Number of
Observations.

Pitrogen (Vitel) <3... .....<..00.5.-| 8
Methane (Homfray) ............... | 8
| Carbon monoxide (Homfray) Z| 10
| Oarbon dioxide (Chappuis) up to} :
| an absorption of 40 per cent.
/ saturation, (excluding one
EI dn io nwt ont dese, benawaes 49

a ee
| |

Mean divergence of observed |
results from those calculated
by “‘exponen-
tial formula.”

by equation (14).

|
|

0°9 per cent. | 7:0 per cerit.
OG 5; [44 ,,
DSO ret aam

|

|
0:8 ry) | ae

Above 40 per cent. saturation, as might be expected, the
approximation (14) ceases to hold exactly.

I have also applied relation (14) to some measurements of
Schmidt + on the adsorption of vapours of charcoal ;. these
observations are not very precise (they do not lie evenly
on any smooth curve), but for fairly low concentrations
reasonable agreement is found. In fig. 1, log «/p is plotted

against « for three series.

From the experimentally determined values of the
constants Ay and A,, values can be obtained for €, a, and No.

These are shown in Table II.

The values for a are positive

small quantities, and, moreover, for the four gases the relative
magnitudes are as would be expected, small for the inactive
nitrogen, which cannot saturate much of the field of force,
intermediate for methane and carbon dioxide, and largest

for the unsaturated carbon monoxide.

The values for &,

which is a measure of the relative stability of the molecules
on the surface, also follow the same sequence, the inactive

gases having the shortest life on the surface.

Finally, the

* A. M. Williams, Proc. Roy. Soc. A. xevi. p. 287 (1919).
+ Schmidsz, Zeit. f. Phys. Chem. xci. p. 115 (1916).

694 Mr. D. C. Henry on a

values obtained for Ny per gram of charcoal are of the same,
and that a reasonable, order of magnitude.

iow: :
Log
¢
1:0
©
(t)
0-5
oe 3 4 Ob me EE GEO)
3 4 d 6 (2)
3 4 5 ge)

Schmidt's Observations on the Adsorption of Vapours,
(1) Carbon disulphide. (2) Benzene. (8) Chloroform,

Yasue II.

| | : N,

Gas. T° C., Observer. a. pergm. per gm.

| charcoal. charcoal.
INMIUGTROTEE, | Wee eenooc Oo Raitott WeHO. |SU2Z5105 §\ 8-22 scites
WMiethhamey 2 25.2 sod: 0 | Homfray.) 372 | 489x10-9 71D aos
Carbon Dioxide...... | 0 | Chappuis | 3°32 | 1:47x107° 6°04 x 10°
Carbon Monoxide...| —82 Homfray 474 | 317x107° 5:37x10°°
Benzene. cos. a8. ccs. 6 | -15 Schmidt | 22 60s <OmsS 8:90 x 107°
Hexane os fies s0cs ce 15 A ioe) W220 Oa sae 5:99 « 1070
Carbon Disulphide.) 15 s | 528 | 1:86x10-* 10:36 10”
Chloroform ......... 15 i | 579 | 630x10-§ 11:59 10”°
Acetone. 4. ieeereren es LD.) Woman e213 xaik ae bea 2<Oz?

Kinetic Theory of Adsorption. 695

In order to obtain more direct evidence of the validity of
the un-approximated formula (11), I have applied it to
observations of Travers * on the adsorption of carbon dioxide
by charcoal, in which an adsorption 68 per cent. of satur-
ation is reached, and to some measurements by Langmuir ft
on the adsorption of methane and carbon monoxide at low
temperatures, in which adsorptions were reached respectively
85 per cent., 77 per cent., and 59 per cent. of the saturation
values. Fig. 2, in which loge/p is plotted against log
(1—a/a’), represents Travers’s measurements at —78°2 C.

Fig. 2.

0-0 10 | 2-0 log

Trayers’s Observations on the Adsorption of Carbon Dioxide
by Charcoal at —78°2 C,

In equation (11), if a is put equal to unity, we obtain an
equation of the same form as one used by Langmuir in the
paper quoted for most of his experiments, and with which
he finds good agreement. For three experiments, however
(Tables 9, 13, and 16 of the paper quoted), he has to use a
more. complicated formula based on the assumption of two
distinct kinds of points of attachment. Equation (11),
which makes no such assumption, is found to fit these
results about as well as Langmuir’s equation, as is shown in

* Travers, Proc. Roy. Soc, A. lxxviii. p. 9 (1906).
+ Langmuir, Jour. Amer. Chem. Soc. xl. p. 1882 (1918).

696 Mr. D. C. Henry on a |

the following Tables, where « is calculated both by the latter
equation, and by equation (11).

Tae III.
Calculated by Langmuir. Calculated by equation (11).

z a. Divergence, a. | Divergence.

Methane by Mica at 90° A.

1220 | 1040 101°6 2A Die te3 b =05

83:0 986 98°5 01s ale 1989 +0:3

| 45:0 90:2 O16 — oe Sah oe Brocsop al 0:1
| Dee | BOD 82:2 0:08 0 80:1 — 2)
| [rae ea ge eee 735 Hupp 72:0 +08
128 | 606 65:9 +53 | 65-0 as

80 52-7 53°6 +0°9 54:2 +16

Se As 7 41:8 —19 44-0 +0°3

3°7 36°3 33:8 — 26 36:3 0-0

27 | 30-6 27-0 —36 30:9 +0°3

Carbon Monowide by Glass at 90° A.

61:6 20-0 20-2 +0:2 90:4 = ae

315 18°5 18-4 —01 18:3 —0:2

17°3 16-2 16-2 0-0 16-2 0-0

9:3 140 13-9 = (il 13°7 --0°3

Soe Oe) 11:9 +0:1 11-6 —02

27) 49:6 9°6 0-0 7-9 =F

14 | 38:3 8-2 —Or1 4:7 — 36

Methane by Glass at 90° A.

67-0 20°3 202 iil 20 ie 208 0-0
34:8 17:3 175 +0°2 Lebok oleae
19:3 14:3 14-4 +01 14:2 —0'1
116 113 115 02) ee aC +0°7

7-0 9-0 BOE ei 20. 8-9 —0:1
3:4 614 5:98 —0:16 6°14 0-00
1:9 4:35 4-49 +0:07 4-20 —0:16

The Influence of Temperature on the Adsorption
Kquilibrium.

From (10) and (11) we obtain, if X is maintained constant,
a

pe Po Ts

where & is independent of the temperature. Transforming

=e

Kinetic Theory of Adsorption. 697

the unit of pressure to cm. of mercury, and taking logarithms.

loo? =log (7°52 x 10-* x &) —0°4343 fa

ec, Lk eS (15)

r, and therefore B,, is a function of the temperature, but
may probably be taken as constant over a fairly limited
range of temperature. Over a wider range, it can probably
be expressed |

A=A,— BI,

which leaves (15) of the same form, B, being then equal to

0-43.43, An isostere of the form (15) is not peculiar to
the present theory ; Williams (Joc. cit.) has found the same
relation from his assumptions, and a similar form can be
derived from Perrin’s radiation hypothesis.

Fig. 3.

log =
10

50
0-0002

The Isostere, from Travers’s Observations.

Williams has shown excellent agreement with (15) fora
large number of observations of Chappuis, Homfray, and
Richardson ; argon and ammonia deviate at low temper-

atures. I have found fair agreement with the measure-

ments of Travers (fig. 3), and excellent agreement with the

698 Mr. D. C. Henry on a

observations of Brown* on the adsorption of water vapour

(fig. 4).

Fig. 4.

00025 0:0026 g0027 £

The Isostere, from Brown’s Observations.

(1) «=3°35 gms. (2) a=2'70 gms. (3) a=2-00 ems.
(4) a=0°57 gm. |

From the empirically found values of B,, X can be deduced.
Table IV. shows the values so found for various adsorbing
systems, and for comparison, where it is known, the value
of Q—RT, where Q is the total heat of adsorption expe-
rimentally determined.

Two Components— Adsorption of Mixed Gases.

For moderately small concentrations, relations can be
deduced from equation (6) which express the mutual influence
of two gases on each other’s adsorption. Since no experi-
mental results are available to test these adequately, they are
merely quoted for reference :—

I. X,, X,; represent the actual adsorptions of the two
components under partial pressures p, and po. yo denotes the

* Brown, Phys. Rev. xvii. p. 700 (1921).

Kinetic Theory of Adsorption. 699

pressure of component 1 which would praaee the same
adsorption X, in the absence of component 2, and similarly
for Po. Then

Pi = pene *?'
an d Po = Pye ™r,
Taste IV.
| Gas. | Absorbent. Observer. d cal. | Q—RT cal.
ae Charcoal | Homfray 3690 3090 (Dewar)
| Methane............ ‘ x 4800 — |
| Carbon Dioxide... Me Chappuis 6390 | |
_ Carbon Dioxide... Pe Travers
ao 12 Gc. ... 7140
ee TAT ec... | 7340 |
Mean of the three...... 6950 6700 (Chappuis) |
Ammonia ........- Charcoal | Richardson 7410 7800 (Chappuis) —
Water Vapour ... a Brown
a=3'35 ems. eK! one 10)30 | |
a—=2'70 gms. st ae 9580
a—=2°00 gms. — ae 9460
=0°57 gm. x x 9305 |
PPPS oss ous cc. see Mica Langmuir Osler
geen. =... 61.6... a ss _ 651
INGROPeN..\ cease. - | { ed
Sates |
1897 |
[ APetHAne...........: eS uA 1618 |
1690 |
Carbon Monoxide Hf i: 2155

II. X,) denotes the adsornecn of component 1 in the
absence of component 2, component | being at pressure 7 ;
similarly for Xo. X, and x, denote as before the adsorp-
tions in a mixed system of partial pressures p, and py.

Then *
= ge aD
Xy — Xie @,Xo/Xo

and Nee

IIl. Combining the above, we have for constant Xz,

Xp = XioPr0,
and for constant X,,
X2p2 = Xaopo0-

* These relations have been roughly verified on observations of Bakr
& King (Jour. Chem. Soe. exix. p. 453 (1921)).

700 Mr. D. C. Henry on a
Adsorption from Solution.

The derivation of the adsorption relations used above
depends on the kinetic theory of gases, and cannot be applied
_to adsorption from solution. It appears, however, to be a
thermodynamic necessity that an adsorbent which is in
equilibrium with a (mixed) liquid phase, should also be in
equilibrium with the saturated vapour phase above the liquid.
Williams * has found experimental confirmation of this
assumption, and I am at present carrying out further experi-
ments to test it. If it is valid, the adsorption equilibrium in
solution is determined by a combination of equations (6) with
the vapour pressure relations of the solution. Duhem’s
equation and Margule’s integration of it both lead to ex-
pressions too complicated to test experimentally, but the
special cases of (1)a dilute binary mixture, and (2) a mixture
giving a linear vapour pressure-concentration curve (Dole-
zalek’s normal mixtures) may be amenable to experimental
verification.

Adsorption from dilute solutions,

For a dilute solution, we have for the solute, by Henry’s
law, |

Przke, (ce; denotes volume concentration),

and for the solvent, by Raoult’s law, if P, is the vapour
pressure of pure solvent,

P iste : °
ee Pe =O, (C, denotes molecular concentration)
2

=1— C,,
s0 that py )=Po€;-

Now in dilute solution, both ¢ and C are practically pro-
portional to 7, the concentration in grams per gram solution.
Combining with equation (6), transforming the unit of
amount adsorbed to grams, and collecting the constants, we
get

Uy Ug 7
w= hy | 1

1
Uy Ug a2

where uj, uw, represent the true adsorptions of solute and
solvent respectively, uw,’ and vu,’ the corresponding saturation

I
Ug

* Williams Medd. 7. K. Vet.-Akad. Nobelinstitut, 11. No. 27, p. 2.

Kinetic Theory of Adsorption. 701

adsorptions. The apparent or observed adsorption of the
solute will be (Williams *)

Uy =U Ty,
a) am See
Ne

For small concentrations 7, vp is practically identical
with w,, and the apparent adsorption may be used for the
true adsorption. For higher concentrations, the true ad-
sorptions can be calculated from the apparent adsorption by
use of equation (17) together with the empirical relation of
Williams (loc. cat.),

Bac Re aa en

Uy U25
where w,,, and wu, denote the adsorptions from pure solute
and solvent respectively (determined by weighing after
adsorption from the vapour phase).

Fig. 5.

z | 5 (2)
135 T-40 45 1-50 T55 T-60 1-65 (3)
u“
Aibers

Williams’s Observations on the Adsorption of Acetic Acid.
(1) © Williams’s Table 11; (2) x Table 12; (8) @ Table 13.

From equations (16) we deduce
log “! — 4 Joo 2 = log h,— “log hp= constant,
om a2 ° 2 a2 on 3
so that plotting log u,/n, against log w,/n, should give a
straight line. Figs. 5 and 6 represent the results of plotting

* Loe. eit,

702 Mr. D. C. Henry on a

in this way some results of Williams and of Gustafson *
Though the agreement is not perfect, the straight line fits
the Sheer anne for low concentrations as well as any line.

Fig. 6.

log =

Od

1-9 0-0 Ol u 0-2

log +

Gustafson’s Observations on the Adsorption of a
Phenol-Alcohol Mixture.

No observations are available for testing these relations
with a mixture of linear vapour pressure-concentration
curve.

A Derivation of the “ Exponential Formula” for the
Adsorption ono

The relation — a= he? E

has hitherto had no theoretical basis, but represents experi-
mental results very well for low concentrations. By a
method based on the same assumptions as those employed in
the earlier part of this paper, this formula can be put on a
theoretical basis.

Consider the adsorption of a gas. Lek the free surface
energy of the bare surface (7. e. the surface in contact with
a vacuum) be oo ergs per sq. cm., and let the free surface .
energy of the same surface covered with a monomolecular
layer of gas be a). In adsorption equilibrium, let a fraction
6 of surface be covered, and let the free energy of the whole
surface be o ergs per sq. cm.

Then, since the range of molecular action is assumed
small,

og =0)(1—0) + o,0=09— (09-0) 0.

* Gustafson, Zeit. f. Phys. Chem. xci. p. 885 (1916) ; Zeit. f. Electro-

chem. XX]. p. 459 (1915).

Kinetie Theory of Adsorption. 7038

But @=X/X’, and

But X is identical with Gibbs’s “surface excess,” I, and
o is numerically identical with the surface tension; by
Gibbs’s equation, for small concentrations,

x Cae: c dp—a, dX
DON a en ° mare ir

Redes -0 i X! de’
which on integration gives

Dx

InX = a .Ine+lIn&,

09-91

where In & is an integration constant.

RRS it

Omen Ont eR nN’
1

If we write

this becomes Nae

which is the “ exponential formula.”’

The same argument applies to adsorption from a dilute
solution, where op is taken as the free surface energy of the
surface in contact with pure solvent, and the apparent ad-
sorption (to which Gibbs’s relation applies) is taken equal to
the true adsorption, which is permissible in dilute solution.

The exponent

1 _ osmotic work in adsorption of X' gm.mol.
n total work in adsorption of X' gm.mol.

Fe a ane oe al
~ RIX'+WX'” RT+W’
where W is the non-osmotic work in the adsorption of 1 gm.

mol. ; this is probably accounted for by the work done in
the orientation of tke surface molecules, and is not likely to

vary much with temperature. It follows that : isa quantity
nr

always less than unity, which tends towards unity with rise
of temperature. This is in accord with experience.

W
Re

so that, at a given temperature, the greater the orientational

Again, n—1l=

a A Kinetic Theory of Adsorption.

work the larger will n be. This is in accord with experi-
ment, for we find that the larger the molecule of adsorbate,
the larger is n, as exemplified by the following Table for
various Setloviemees adsorbed on charcoal.

TABLE VY.
Gases. N. Solutes. 10
pt |
a Horne NCI wee eae 2°22
Hydrogen ...... ACC CUACId! reenee ne ee 2°35
Nitrogen ...... 1 to 2 Propionic Acid... 2°54
Oxypenyy ee Butynle Acidian ce em 3°32
Carbon Dioxide...| 2°47 (Chappuis) | Benzoic Acid............ 2°72
Sulphur Dioxide .| 3:09 53 Pieri Acid 22.27: aeenes 4:17
Chlormethy] ...... 8:2 as Benzenesulphonic Acid] 4:48
New Buchsin. 07.225... 5°38
Crystal Ponceau ...... 6°67
Came sugars 9. -. es 8:2 (Bancroft)

(Data, except where specified, from
Freundlich, Kapillarchemie, p. 159.)

We should expect that the value of W would increase in
the same sequence for a series of adsorbates on different ad-
sorbents, since the orientational work will be chiefly due to
orientation of the adsorbate molecules. This is approxi-
mately the case (Freundlich, Kapillarchemie, p. 155).

It would be interesting to compare the values of W
obtained from the exponential formula with the work done
when two unit surfaces of two liquids come together
(Harkins *), but as far as [ am aware, there are no experi-
mental data for the adsorption isotherm on liquid adsorbents.

Summary.

1. A theory of adsorption of gases has been developed on
a Kinetic basis by means of assumptions derived from Lang-
muir’s conception of a monomolecular layer.

2. Kquations for the adsorption isotherm and isostere
are deduced which are in satisfactory agreement with
experiment.

3. A method is suggested for applying these equations to
adsorption from solution, and the results compared with
experimental data.

* Harkins, Jour. Amer. Chem. Soc. xxxix. p. 541 (1917).

On Electromagnetic Lines and Tubes. 705

4. A theoretical derivation for the ‘‘ exponential formula ”
is given, which attributes to the exponent 1/n a theo-
retical significance which is in qualitative agreement with
experience.

It is not claimed that the theory advanced is a com-
plete solution of the problem, or that it is valid in all
cases, but the agreement obtained is sufficient to indicate
that the mechanism of adsorption suggested may be an
approximation to the truth.

The work contained in this paper was done while the
author was “ Coutts Trotter’? student of Trinity College,
Cambridge.

LXI. On Electromagnetic Lines and Tubes. By 8. R.
Miner, D.Sc., F.RS., Professor of Physics, The Uni-
versity, Sheffield *.

N a recent paper Professor Whittaker ¢ has shown that it
is possible to extend the conception of the tubes of force
of electrostatic and magnetostatic fields to the general electro-
magnetic field, when this is considered as a four-dimensional
entity. The differential equations which express the pro-
perties of the calamoids, or surfaces from which the tubes
are formed, are rather complex, and it is not easy to see from
them in their genoral form a clear picture of what the tubes
really are. Itis hoped that the treatment of the question
contained in the following paper may be of use, as it not
only enables such a picture of the position and direction in
hyperspace of the tubes to be formed, but it also extends
Whittaker’s results in certain respects.

§1. The Construction of Electromagnetic Lines.

In an electrostatic field in three dimensions the course of
a line of force can be traced out from a given point by first
orienting the axes so that 2 lies along the direction of the
electric force at the point, and then continually rotating
the axes so that this condition is still obeyed at successive
points infinitesimally distant from each other along the
curved line which the z-axis thus forms. The properties
of the tubes can be expressed in terms of the curvatures of

* Communicated by the Author.
+ Proc. Roy. Soc, Edin. xlii. p. 1 (Nov. 1921).

Phil. Mag. 8. 6. Vol. 44. No. 262. Oct. 1922. 2 Z

706 v. Brots., i. Milner ow

the lines, or in other words in terms of the infinitesimal
rotations which are given to the set of axes as the linés are
traced out. Although a precisely similar method can be
applied to the electromagnetic field, it is not at first sight
clear how the axes are to be oriented at any given point
which may be chosen as a starting point. In the electro-
magnetic field the directions of the electric and magnetic
forces are in general neither along the same line nor at right
angles to each other, and there is no more justification for
putting the x-axis along either of these lines than along the
other. It is, however, always possible to choose the axes of
w«yand z at any point such that their directions enjoy a
unique symmetry with respect to e and h, in that no pre-
dominance is given to either over the other. To effect this
choose them so that the following equations are satisfied :—

A= 0. p= On erty thhy= 0.2 Seely

This makes z perpendicular to the plane in which e and h lie
at the point, « and y lie in it, their directions, shown in
fiy. 1, being such as to make the dotted rectangles equal.
in area. :

Fig. 1.

x

Considering the field as a four-dimensional entity there is
the time axis to be oriented also. This les at right angles
to the hyperplane, or instantaneous. space in which the axes
of wv yand-z are drawn. The consideration of the field in’
hyperspace is greatly simplified by adopting the formal
representation of it introduced by Minkowski in which the
time axis ¢ is replaced by an “imaginary time” axis /=ict,
or, taking c=1, /=it. The Minkowski world substitutes a

Hlectromagnetic Lines and Tubes. 707

hyperspace with Huclidian geometry in place of the difficult
hyperbolic geometry connected with real time. In it e andh
are vectors which obey at each point (where there is no
charge) the formal equivalent of the Maxwellian equa-
tions * ;

Bi-Bsthan Be Bes)
‘S32 tae7" 6S <i =
anata (ihe 25 _ Bee 9, | . (2)
Sitio Beth Be!

From the point of view of a super-observer surveying the
four-dimensional field, or for that matter of a person of
ordinary mentality who attempts to form a conception of the
underlying reality which shows itself to observers in different
hyperplanes as electric and magnetic forces of varying
strengths and directions, the hyperplane «yz in which the
values of e and h are originally specified must be looked
upon as ap ar bitrary one; the l-axis perpendicular to it !s
therefore also in an arbitrary direction. By means of the
‘ Lorentz transformation the hyperplane may be readily
changed. If the observer of e and h at the point is in
motion relatively to it with the velocity v along the axis
of z, the observed constitution of the field is changed, and
the new electric and magnetic forces are given by the well-
known equations

é; = B(éz—vhy),... hy = B(hr+ vey),
ey = B(e,+vhz), hy = Bhy— Gor ae its (3)
a) = e,,, | i he |
where B= (1—v?)-#,
On the Minkowski representation this transformation is

* The mathematical results can always be re-expressed in terms of

: Sexe j fo)
real time by substituting 7 for J, tse for =

2Z 2

708 Profs. RB. Milner on

equivalent to rotating the whole system of axes in the plane
of zl through an angle @ given by

tan 0 = iv.

If now starting from an arbitrary hyperplane and with the
axes of x yz at a given point as origin oriented as in (1), we
rotate the axes through the angle 0.,, where

Sipe)

tan O1= 1%, ah ee re

av

we find for the field as observed in the new hyperplane
eJ=E, ¢/=e/)=0, ky'=H, h,'=h'=0, . (5)

where } and H are related to the usual invariants of the
field by the evident relations

W?—H’=e’—h?, EH=(eh). . . . (6)

E and H are thus themselves also invariants, and the field at
the point has been simplified by the orientation of the axes
into a combined electric and magnetic force acting in the
same direction along the (unchanged) axis of x *.

At an infinitesimal distance along the 2-axis from the
origin e and h will no longer be collinear, but they can be
made so again by a suitable orientation of the axes through
infinitesimal angles ; and it is evident that in this way a
continuous line, at every point characterized by the col-
linearity along it of the transformed electric and magnetic
forces, may be constructed in the four-dimensional space.
This procedure is precisely analogous to the method of

* When hy>er the transformation (4) involves a velocity greater than
that of light for the observer. To get over the difficulty we might use
in this case the strictly legitimate transformation |

Le
tan 0.;= iz

hy to a re
collinearity is again produced, but now along the y-axis. There is. how-
ever, no need to deal with it separately, it can be included with the
other in the transformation (4) by imagining that the Lorentz equations
are valid for values ot v greater than 1. It may be noted here that if
we apply transformations (4) and (4a) to the same field, while (4) gives
collinearity along « with er’ =H, h;’=H, (4a) gives collinearity along y
with ey’=—7H, hy'=+7H. It will be evident in § 5 that these are the
yz and 21 components of the electromagnetic five-vector (R, 7R), and
that the two cases consequently form merely different aspects of the
same field. ;

Electromagnetic Lines and Tubes. 709

constructing a line of force in an electrostatic field, and
the lines so traced out may be ealled ‘‘ electromagnetic
lines.”’

Close to the origin the line passing through it lies along
the w-direction satisfyi ing equation (1) in the arbitrary
hyperplane in which the field is initially specified. This is
the case because the rotation of the axes in the plane </ does
not change the axes w and y. Thus the length dz of the line
lies in the original hyperplane. For its next infinitesimal
portion, however, the rotation to produce collinearity is in
general in a plane infinitesimally inclined to the original al,
so that the line bends out of the hyperplane. It is thus in
general impossible to draw continuous electromagnetic lines
in a fixed hyperplane ; at all points, however, the tangents
to the lines passing through them may be drawn.

§ 2. Transformation of the Electromagnetic Equations into
terms of the Lines.

The axes wycl having been chosen so that at a given
point
é-= KH, ee H, e, =e, = = li OQ,

at a neighbouring point of the four-dimensional field e and h
will have the values

Bitde.,* de,» de, H+dh,, dh, -dh,.

We can determine the infinitesimal angles through which
the axes have to be rotated to produce collinearity again
by the following process. The rotations necessitated by
the variations de,, dh, are independent of de,, de,, etc., and
we can treat them as existing se tely Suppose therefore
that the new field consists simply of E and H along a2, and
de, and dh, along y. Rotate the axes through the angle dO,
in the plane of zy until the relation (1) is again obeyed.

We shall have for the components of e and h along the new
axes vy

e n= H, fi Hi.
ey= —Hdé,,+de,, hy=—Hdo,,+dhy,
and, since they must satisfy the relation

éxéy thah, = 0,

TAO Prof. S. R. Milner on

we have for the angle of rotation

Ede,+Hdh
ee
Hence. - |
ee — H(E dhy —H de,) poe E(t dhy— JEL dey)
oY ime i? 4 H? ) yaar E24 H?

We now further rotate the axes in the plane of <l (this
rotation does not affect directions in the plane xy) through
the angle

slap ,Edhy— H de,

Ab i= 1 ra, EK? + He a

The axes are now oriented so that e and h are collinear,
and w lies along the electromagnetic line.

Variations de, and dh, are similarly transformed by rota-
tions in the planes of wz and y/ through angles

_ Edez+ Hdh, 1 Wid heed dez

es 5 uae m+?

ii a

Variations dex and dh» do not involve any rotations of the
axes, but we have simply —

dB = dex, dH = dhe.

Solving these six equations for dex, dey, etc., we obtain
den= dE, dhy= dH, |
dey= Bd@xzy+tH dO, dhy= Hd@z,—1H déz, (7)
déez= Hd@xz2—iH d0y, dhz= Hd0z,+1H dOy.

These form a set of transformation equations by which we
can express the infinitesimal variations of e and h between
the origin and a neighbouring point in terms of the collinear
invariants 4 and H and of the infinitesimal angles through
which the system of axes has to be rotated in order to lay
the axes at the neighbouring point along the appropriate |
directions for observing the collinearity.

Let the values (7) be substituted in the electromagnetic
equations (2); if we multiply the resulting equations by iE
and H, add or subtract them in suitable pairs, and observe
that the sign of 6 is reversed by changing the order of its
suffixes, we get the following set of eight equations which

Electromagnetic Lines and Tubes. 711
forms a complete equivalent of the original set (2) :

228

3 +Ho" 4 (r+ H) (& it 4. Set) = 0, ti at
Ex +H&>+ (E+ H) (Se + £2) 20, hs
mee Hoe. (E? +H) (Se + oe eee Ca)

eee Hoey (E? + H?) C: om) = Ohta (al) | “A

Boe o + i(B? + H?) = - oa) = 0,27 (t!) |
po aS +i +B) (Se 4 Oe) =0,.. @)

— = + i(E? + H?) (S24 a)=0, oc)
Boe a (HP + H?) (Qe4 oe) = 0... @')

§3. A Flux Theorem for each of four Electromagnetic
— Lubes.

To interpret the equations (8) we observe that through
any point of the four-dimensional field not one only but
jour electromagnetic lines may be constructed, each of which
is uniquely determined when the initial hyperplane in which
the field is specified i is given. Starting from this hyperplane
the four axes’ directions required to produce collinearity at a
given point are uniquely fixed. We can thus proceed to
a neighbouring point lying on any of these axes, and rotate
the system to. give collinearity. at this point, and ‘by con-
tinuing this process obtain a continuous curved line in
hyperspace which at the origin coincides with the given
axis. We will call these curved lines the 2, y, z, /-electro-
magnetic lines respectively; botnded by a set of each of
them an &, y, 2, l-electromagnetic tube can be constructed
in the usual way.

Consider now an infinitely thin a#-tube, which we may
take as having a rectangular cross-section at the origin.

AZ Prof. 8. R. Milner on

Let the adjacent edges of the section which contain the
origin be infinitesimal lengths OY=y,, OZ=z (fig. 2) of
the y and ¢-lines through the origin respectively, and let
YY’, ZZ' be the 2-lines through Y and Z. The area of the

section at the origin is y, 2, The cross-section at an infi-
nitesimal distance along the z-line of OO'=a, will be a
quadrilateral whose sides are

) 2
OY = nts, O'Z! = Atta ae

and which will be altered in size, shape, and orientation from
the original rectangle. If Y; and Z, are the projections of
Y and Z by lines parallel to OO’, and we draw Y’A, Z/B
perpendicular to O'Y, and O’Z,, we can readily see that

004 ZNiYA_ 1 id Jom
OY oo YN , vce vy "1 (Oude

and similarly Gd. Be

ol

Oz 21 O02
On substituting these in (8a) that equation becomes

OF OH pay 8 (ye) =0
mie ct a

or

& ¢ JERE? . 4,2,) 00

Electromagnetic Lines and Tubes. (pk,

This equation expresses the fact that the flux of the
invariant function E?+H* over the cross-section 7,2, of
an infinitely thin #-tube is constant throughout the whole
tube ; since any tube may be formed by the juxtaposition of
infinitely thin elementary tubes, the statement is true for a
tube of finite cross-section also. In this case the corre-
sponding cross-section may be defined as any continuous
surface at any point of which the yz plane corresponding to
the x-line passing through the point differs only infinitesi-
mally from being tangential *.

On the rectangle y;2; as base not only an w-tube but also
an /-tube may be constructed, since the /- as well as the
w-lines are at right angles to yz. The same figure 2 with
OO’ representing the /-axis gives

Oly mS 1 On 081; ak

Oy mm Ol’ aes h

and equation (80) reduces to

a1
Ol’

co)

s( AB ee Oe hd x) x,--C2 6)

so that the flux theorem applies to the /-tube also.

Any infinitely thin tube constructed of either y- or 2-lines
will at the origin be perpendicular to the plane xl. Take
the cross-section at the origin to be rectangular, formed of
infinitely short z- and /-lines, 7; and 1,. We find as before

Bye _ 13% yw _12h
Of. Fie Ol lL, oy’
Bee _ 19m — 01_1 dh
Ox 4 02’ Gio. Oa

* In the general field the x-lines are twisted in the yz plane (v. infra)
and a closed surface everywhere perpendicular to them cannot be
uniquely constructed; 2. ¢., if we go from the origin always’ perpen-
dicular to the «-lines distances, first y, and then z,, we do not arrive at
the same point of the final z-line as will be reached by going the same
distances in the opposite order, mathematically the conditions of uncon-
ditional integrability will not be satisfied for the y; and z, displacements.
For the purpose of reckoning the flux this feature of the tubes is
immaterial, all the surfaces formed by joining up the points obtained by
displacements in any order will over an infinitesimal region only differ
from each other in area by second order quantities. The curvatures of
the «-lines, and the second order displacements which they have under-
oP at O', Y and Z out of the hyperplane zyz, are also of negligible
eitect.

714 Prof. 8. R. Milner on
With these substitutions (8c) and (8d) become

S( VEF+HS wh) = 0, ae

2. ( YEE HE. al) = 0. . a

It thus appears that a theorem expressing the constancy
of the flux over the cross-section of the tube is derivable for
each one of the four electromagnetic tubes which can be
constructed with any point of Te field as origin. The cross-
section over which the flux is reckoned is determined by the
particular hyperplane which contains the infinitesimal portion
of the tube concerned. The z- and /-tubes lie initially in the
hyperplanes «xyz, /ye respectively, and have the same cross-
section y;2,; the y- and ¢-tubes lie in yal, zwl, and have the
same cross-section yl, The same quantity H?+ H?
appears in each case as the function whose flux is constant ;
it will be convenient to represent it by a single symbol R.
In terms‘of the invariants of the field as usually expressed
we have by (6)

R= VB? = { ?— 7)? 4 ARP oe
5 (e a +4(eh)?}#. 5. (10)

In the special case when e and h are Sere at right
angles to each other (eh) = 0 and

R = Ve? —h?.

This is a result which has already been given in Prof. Whit-
taker’s paper. The Lorentz transformation also shows that
when e and h are perpendicular, H = 0 and R = H, hence
when they are viewed at any point in. the appropriate hyper-
plane the electromagnetic w#-lines are lines of pure electric
force, they may however differ from ordinary electrostatic
lines by their whole lengths not being containable in a single
hyperpiane.

§ 4. A Theorem, complementary to the preceding, relating
to the Twist of each Tube.

The expression of the constancy of the flux of R for the
four tubes only accounts for half the information derivable
from the eight electromagnetic equations. The second four
of the equations in (8) are concerned with what we may call

\ 2

Electromagnetic Lines and Tubes. 715

the “internal composition” of the vector R. An electro-
static tube in three-dimensional space is characterized by
only two quantities at each point: its direction, and the
electric force e which is inversely proportional to its
sectional area at the point. An electromagnetic tube in
hyperspace has not two but three such characteristic
quantities. Not only will the same tube in two different
places differ in direction, and in the magnitude of the vector
function of position R, which latter is inversely proportional
to the sectional area of the tube, but R= V7 EH?4 HH? will
differ in the relative proportions of the collinear EK and H
of which it is composed. The composition of R is adequately
expressed by a parameter « such that

hie H artis ey Sigh og a)

bate Ce lac
sina measures at any point the relative amount of magnetic
force in R which cannot possibly be transformed away. We
have further from (6) -

ae 2EHH 2 (eh)
sin 2a >= — — ——.
EK? + H? R?

Thus also sin 24 gives a direct measure at each point of the
departure of the field from orthogonality of e and h, as

expressed by the value relative to R? of the scalar-product
) I

invariant 2(eh).
Differentiating (11) we have

EdH—Hdb = Reda,

the substitution of which in (8 a'-d') gives

Be 4 (Be oe) = 0, |

Ow Oy F
siti(S 2+ 52)=0 |

‘ : - (12)
Ox . (0921 O08 x2 a. f Remy oe ray
yaa si) =0,

a2 : Oly obs) hac |

Be (42 . Bese) _ 0, |

These equations disclose that the variation of in each of
the four coordinate directions is determined by the extent to

a eR

716 Prot. S. R. Milner on

which twisting occurs in a corresponding tube. In a twisted
tube the bounding lines will not be parallel to the tube axis
but will tend to run in spiral curves around it. Fig. 2
(p. 712) shows the effect of a twist on the wx-tube. Let

oo — be the angles Y’O'Y,, Z'O'Z, through
which O'Y’, O'Z' have been rotated from OY, OZ in the

plane ye, while Y'YA, Z’ZB are the angles nse, oe
through which the #-lines through Y and Z have been
rotated in the planes zz, wy respectively from the z-line
through O. We see at once by expressing AY’, BZ’ in

terms of these that

OO zx ie Oz, Oy O dz; uM
Oy =r Ox >) 5. 5 aie ‘ : . (13)

When Y'O'Y,, Z'O'Z, are not equal there is distortion as
well as twist; the two may be separated in the usual way

by writing

byz= — bey = — 3 pzy' + bzy'')s Nom (14)

Vyz= Vi —3( dey’ oe bzy'')

measures the pure twist, or the rate at which the

eos

Mes yi, is undergoing rotation in its own plane in
the direction from y to zas we proceed along the «-tube,

Oye
Ou

while expresses the rate at which it undergoes dis-

tortion. These considerations are applicable to any tube.
For the z- and J-tubes erected on the same base W219
Obyze Ody
Ot7e el
z-tubes, based on 4l,, are twisted in the al plane ; let
Odzi Ode
10702
(13) and (14), and corresponding formule for the other
tubes

represent the respective twists. The y- and

be their respective twists. We then have from

00:2 OFyt _ r Odyz 0621 OO xz ee ¢ Oda
Oy ar So ys =+2 Ais ae Al =+42 Bei
Obzx 4 OFay _ _ 9 Obye EU ie OF ye _ _ 4 OGut
OY Z OL’ Of) Ol Oye

Electromagnetic Lines and Tubes. 717
and equations (12) become
222i Beg Bh |
Ow ol OY Oe (15)
Peas Ad as om)
per sche Om _ 4 95 Oba?
fo}! Ow Oz OY

These equations, one for each tube, are complementary to
the corresponding equations expressing the flux theorems,
and show that the tubes give a complete representation of
the electromagnetic field. Hquations (9) show that the
tubes, by the variation of their cross-sections, determine
the variations of R, equations (15) show that by their
twists, they determine the variations of a, along each of
the four coordinate directions. It should be noted, however,
that while the variation of R given by each tube—aza, l, y, -—
is that along the length of the tube, the variation of « given
by the twist is along a direction perpendicular to the tube,
but specially associated with it,—l, a, z, y.

§5. An Electromagnetic Five-Vector characteristic of

the Field.

The meaning of these results can be made clearer by
observing that the geometrical construction by which the
electromagnetic lines passing through a given point have
been obtained still leaves their directions in hyperspace
subject to a certain amount of arbitrariness. They are only
uniquely fixed when the arbitrary initial hyperplane is given
(cf. § 3). In fact the final directions of the lines can be got
from an arbitrary initial set of axes by four successive rota-
tions in the planes of xy, wz, yl, and zl, and rotations in the
planes zl and yz are not required to produce collinearity of
e and h. They may be made of course, but they do not
affect the collinearity. Thus in a hyperplane in which col-
linearity is observed at a given point, let the observer suppose
that he is in motion with a velocity v along the a-axis,
zt. e. along the direction of EH and H. The Lorentz transfor-
mation shows that this will affect in no way the magnitudes
of E and H at the point; it follows, therefore, that the
observer has no means of ascertaining his velocity in this
direction, or of concluding that he has none, by observation
of the field at this single point. We have therefore no right
to assume that this velocity is zero, but in a general theory
should write it as an arbitrarily given quantity. The corre-
sponding transformation is equivalent to rotating the axes
through an arbitrary angle in the plane of w/, and thus to

718 Prof. S. R. Milner on

laying down in hyperspace a/- and J'-lines.in a different
direction from before, although still in the unaltered al
plane. Similar considerations apply to the y and z axes;
indeed, when collinearity of e and h has been obtained it is
evident that there is nothing to distinguish their actual
directions in the fixed plane yz. |

It thus appears that what is uniquely fixed in hyperspace
is not the directions of the individual axes, but the orienta-
tions at the point of the two coordinate planes al and yz, in
which planes the axes themselves may be drawn arbitrarily.
The entity that we have really to deal with in a four-dimen-
sional theory of the electromagnetic field is of the type which
is best represented not by lines but by surfaces; in other
words, it is not of the four-vector, but of the six-vector type,
being a function of position the properties of which at any
point are associated with two absclutely orthogonal planes
(vl and yz) which cut each other only at that point *. The
six-vector in question is of a restricted type, characterized by
the equality of its two parts; it consists of R= Vv H?+ H?
associated with the yz plane by ‘‘ acting” in any direction per-
pendicular to it, 2. e., along any line in the plane al, combined
with the equal (but imaginary) quantity 7R similarly asso-
ciated with the al plane. Although-a six-vector, since its
two parts are equal, it. only requires five independent
quantities to specify it: the four quantities required to fix
the orientations of the planes, along with the magnitude
of R. We will eall it consequently a “ five-vector,”
thereby distinguishing it from the electromagnetic six-
vector (h~ ze). The six quantities required to specify com-
pletely the field at a point are known when a is given:in
addition. . RES

§ 6, The Construction of Unique Sets of Tubes.

_ These considerations enable us to explain the special sym-
metry which the tubes show in the planes al and yz, and at
the same time to derive a set of tubes which are really
uniquely laid down in hyperspace. Having obtained from
any initial hyperplane the axes oriented so that lies along
the collinear E and H, let the axes be further rotated through
the angle 0,7 in the plane wl, This will not affect E, H, or
the axes y, 2. Starting from the origin along the new direc-
tion w’ of w, we can construct an a’-line just as before by
- * The plane «/ comprises all points of hyperspace for which y=0,
z=0; yz all points for which «=0, 7=0; the two planes thus cut only
at the origin.

Electromagnetic Lines and Tubes. 719

infinitesimal rotations in the planes wy, a'z, yl’, zl! alone.
a ;
For the corresponding w'-tube we shall have

fe sa
era V KE? + H? ; Y121)

= (cos Cis 2 + sin 0.7. =) ( / B+ H? . 21)

= iy (oO aeeb).

Hence the flux theorem applies to a tube starting from the
origin in any direction in the plane of xl, the properties of
the w- and /-tubes are in fact symmetrical in this plane.
Since @,7 is arbitrary, we may choose it so as to satisfy any
stated condition, for example, the condition that ;

24 = cos 0.192 + sin a =
It is clear that in this way an 2’-line may be drawn such
that there is no change in the composition of R along it, and
it is now also uniquely laid down in hyperspace. When
& =0, (15) shows that Oo = 0, which means that there
is no twist round the /' direction perpendicular to z#'. Hence
the “twist of the al plane,” 7. e¢. the twist round its axis of
any infinitely thin tube drawn in it, is a maximum round the
particular '-line which does not vary in composition along
its length, while the l’-line at right angles to z' is charac-
terized by no twist round it, and along it a maximum rate
of change of composition. In fact, in the al plane (and all
these conclusions apply also to the plane yz) the vectors
representing the twist and the gradient of « are in mutually
perpendicular directions *.

0.

* Some sort of a visualization of the effects of twist can be got by
picturing an z-line as being something like a ribbon instead of being the
same all round like an ordinary line. We can suppose it shows E on
the face and H on the edge of the ribbon. The Lorentz transformation
enables one to alter the view point, and with untwisted ribbons to
change the h-aspect of the field completely into e; when they are
twisted however it is impossible to find any view point from which the
e- or the h-aspect alone may be seen; they must both show simul-
taneously. This agrees with what has been deduced: when e and hare
perpendicular, one of them may be transformed away, when they are not
perpendicular the lines are twisted and neither can be transformed away
completely. It also enables us to visualize how twisted tubes might
produce space-time variations in the ratio of H to E. I[t is, however,
only a crude analogy and must not be pushed too far. Asa fact the
composition of the lines changes in a direction perpendicular to the axis
of twist, and not along it as the analogy would suggest.

720 Mr. A. Bramley on Radiation.

Summary.

When viewed in a suitable hyperplane, 2. e., when suitably
transformed, the electric and magnetic forces at any point
of the general electromagnetic field can be made to coincide
in direction. This direction determines an electromagnetic
line, continuous through hyperspace, from a set of which an
electromagnetic tube can be constructed. Four such tubes,
mutually perpendicular, can be constructed containing any
point, and each is characterized by the constancy of the flux
of the quantity

R={(e?—h’)?+4(eh)?}s.
over its cross section. A complementary theorem for each
tube relating to the twist of its generating lines determines
the internal constitution of R as expressed by the ratio of
the magnetic to the electric force present in the compound
vector. R is shown to be a “ five-vector,’”’ 7. e., a six-vector
with its two parts equal; it is a function of position asso-
ciated with two absolutely orthogonal planes uniquely fixed
at each point of the four-dimensional field.

The University, Sheffield,
29th June, 1922.

LXIf. Radiation. By AntHurR BRAMLEY.
To the Editors of the Philosophical Magazine.

GENTLEMEN,
[ the following discussion we shall make use of the idea

that energy possesses mass, a principle which has been
so fruitful in explaining the behaviour of light-rays in a
gravitationai field and which is a natural consequence of the
electromagnetic theory.

If energy possesses mass, then the fundamental laws of
mechanics ought to apply to it. Following this idea, we
shall attempt to show how the laws of radiation are related
to the fundamental principles of mechanics and electro-

dynamics.
The values of the potentials are :
1 0°
Vier a ee 5?

1 9?U V

Dicks taf

U c2 0! Pp e”
10U

Mr. A. Bramley on Radiation. 721

We shall suppose that the charged system is rotating in
the XY plane around the origin as centre with uniform
circular motion. Transforming to coordinates moving with
the charge we have «=rcos ot, y=rsin at, or calling ot =0
we have

in oe ne at oe
But =o Sy,
oe BG
ERLE La eB,
Baw let 6,6) I= 8

since the terms involving the velocities are negligible in
comparison with the other terms.

Now 0 7.0, ey = 0 (0419?) = 3 (1++2)o,

dt 00°dt 00 dt/ 00
Ba (14 Bars 2, (222 4.2%)
If 280 +108 =i.
or = eae
which is very probably the case for small oscillations
then 0 +192 mo kt

But the time during which the electron is oscillating 1s so
great that kt-'< <a,
pip
or 52 =? 567
Phil. Mag. 8.6. Vol. 44. No. 262. Oct. 1922. 3A

122 Mr. A. Bramley on Radiation.
Also yrU= — Bp,
prdbdrdz= pyrd0drdz,
d6=d0, V1—P?,
dt=dt, V1—p",
or ; pPi=p V1 — 5 ;
Ale Vidi1=—p\=—- V/1—? .p.
But @ satisfies |
Vie ape
o;= V1—? o @.
Assuming that ~ <1, and using Boussinesq’s definition

of an harmonic,
go 07 AN Oe,
Or 4/1? Or”

0d V1 608 is oer
Moreover, Uj ee, =. 0 — 0:
Now fet as
/ IB? OFZ
1 09,
De ee
fl — 2? OF
ee is fee
Ko= C Ot (Ug)— oe 00”
me) 1.0¢
ee!) > See
ee
oa C2 eile Cc += )88,
a ae
C2 V1 B? 1 Ge 00 e
Further
_10U:_2Uy__pd¢__ =8 24
r 00 fol) or a7 ee or”
Hy= Us _ BU y

a: St i ln NE GIR Lie ee ee

Mr. A. Bramley on /tadiation. 723
Bey Oe in OP
ei 08 eee eae
pee eee OF

a/ 1—f? Z ro / 1 — R? 00
pane Or Tah, Ose
Pee a areal AC RAL OPS

Now for the contractile electron the potential ¢, is
symmetrical in the distorted space of the fixed system and

H;=

é e . ?
equal to —— where 7, is the distance from the centre of
Arr,
the sphere in this system.

P(xyz)

O

Rk

Let P(ayz) be a point in space in the XY plane, 7) the
projection of 7; on that plane, and 6; the angle POE where
E is the position of the electron whose orbital radius is R.

Then we have

rere ters
a rere 22,
roe =r? + KR? —2rR cos 6,
rear?+R?+27?—2rR cos 6).
Therefore | = (7° + R42? —2rR cos 6,)'”,
Sure : 1

dr a/7?-+ RP a?—DrR 03 By

Whence we have

which gives q,

Sills caro Wamuate Dokl Lito oh Le:

Oz (9? + R?—2rR cos 6, +27)?" dar”
Odi _ —r+Reos9, Ou

Or (7° + R?—2rR cos 0, 4+ 2”)9?" 4a’
Od1 _ —rhsin 8, e

Oy int (7? + R2—2rR cos 0, + 2? i? * Aa’
ee

Go

724 Mr. A. Bramley on Radiation.

Then :

Tis CO gia <BR RCo

R Aor ‘ /1— ° (7? + R? 27 —IrR cos 6,)3/ 9
mee ¢

ar. /1— 8? (et Ree 2 Ricose ye

B es é 7); iL 1
0 4’ ce? * A/T — RB? (0? + R?427— 27K cos 0,)1?

1+? e R sin 0,

- 1—B? 4° (0? + R?4+ 2?—27R cos 8, )??
ands) El, = ()

es B r—Reos 6,
dor” / 1 — 8? (7? + R24 2?— 27K cos 6,)9? ”
——< ere es
8 Aa’ a) feet GS he 2 casin)
Caw 1 it

where r,;=(7?+ R?+ 2?—2rR cos 0,)!*= distance from the
electron’s centre to the point in question. .

The calculated values of H and E along the axis of
revolution agree with those found by other means except for
the terms involving the accelerations.

It will be observed that this part of the force varies in-
versely as the square of the distance of the point from the
moving charge, and is therefore inappreciable at great
distances. |

Turning to the part of the intensities which involves the
accelerations we have two components,

i= 2 1
0 seers V1—82 (0? + R? +2?—2rR cos 6,)¥2’
i 1 ile
Hy= e ub

“der we VW TOR? + RE+ 2 — BPR c08 0,’

Thus we see that the part of the electromagnetic field which
depends on the acceleration of the particle is specified by
two vectors, the electric and magnetic intensities. These
are mutually perpendicular but not equal in magnitude,
except for the special case that w=c. There is another
important difference between the part of the field which
depends on the acceleration and that which does not; in the

Mr. A. Bramley on Radiation. 725

latter the intensities are both inversely proportional to the
square of the distance, while in the former they are iny ersely
as the first power, so that at great distances from the moving
charge the part of the field which depends on the accole-
ration will become very great in comparison with the other
part.

The energy of the field is :

=1)7?+ H? + 2( (HH,) +2(BE,) +H, + Hy’},

where the terms with suffixes depend on the acceleration,
We shall consider the latter part only.
Since the energy per unit volume is 4(H,’ eae we have

for the volume density +Hs(1 ~ e nove lhe dale 2

The stream of energy passing any point per unit of area
is equal to c[E. H], the direction of the stream being along
the perpendicular to the plane of the orbit.

Now
“2
((E. ep UU it ih

len “uc 1— 8?’ (r? + R?+ 22 — 2rR cos 4)

in the direction of the axis of Z.

This flow of energy is zero when the electron is stationary
and has a maximum value when @ +1, attaining an infinite
value as the motion of the electron equals that of light.
This shows that the high speed electrons are the most
efficient radiators of energy. This energy also varies in-
versely as the square of the distance, as in the case of all
radiant energy.

We are now in a position to calculate the force acting on
an element of volume due to the radiation emitted from that
particle.

Bw = [vil,
1
Fr=E, + —[H.ue—Hz. ug],
= 6.
es
Be=He+~ fe Oe ug—Hg EO, lis

(ew vie Goer}

726 Mr. A. Bramley on Radiation.
By=Ey+ — (He. —H,.u}

é —U i 1

ee /1—2? (7? + R24 27 — 2rR cos 6,)¥? ae

where IF represents the force acting on unit charge and Ug is
the tangential velocity.

The force acting on an element of volume dr’ at any point
P is therefore

i700;
ee Se aaa =
Am” ce? 4/1 — 8? (r+ R24 2?—2rR cos 61) ¥? ’

Lae Mee ens
Agr ; Cc i V1 — 2? ; (ee + R2 ao 25 == Dn GOs Oia r

F,=
F,=

IE we suppose that an electron is composed of a perfectly
conducting sphere surrounded by an Boon charged

shell of uniform surface-density, then the {( [E.H],ds over
the entire inner surface of the shell is equal to zero. Thus
we see that no energy is radiated inwardly.

Suppose we take any spherical element of volume dz, then
the energy radiated from it, if it is of uniform density, is

c\\(E. H_|,ds over any surface enclosing the element

e ue 1

Sa uc? 1— RB?’

where e is the elementary charge on this volume element.

Thus we see that each element of the electronic shell
radiates the same amount of energy.

We shall now make use of the idea of electromagnetic
mass in dealing with radiation. If electromagnetic energy
possesses mass, then there ought to be an equilibrium estab-
lished between the mutual force of attraction and the radiation
forces. Thus the energy will condense around the electron
until this equilibrium value is attained, when it will be
emitted in quanta. The force acting on the element of
volume due to the radiated energy is per unit volume

Mr. A. Bramley on Radiation. 727
But the amount of ensrgy radiated is
ee ies Ee

‘, the force acting on unit energy density

according to the third law.

In this caleulation we have taken account of the con-
densation of the energy.

The force per unit of mass

£2 tog 8-H}

2 (ert. dp dr=e 2 log [E. Hi.

But this mass is similar to that due to a charged particle,
so that the force of attraction is not the ordinary Newtonian
force of gravitation but rather the electrostatic force, there-

fore multiplying by 10% +41

Now

we have 4A7rypa da=—, ca ale log [E. H]. ac,

10% dé
where p is the density and da the thickness of the shell.
But M=47ra’p da= =

where E is the total energy condensed.

4-1 u a?

Ce
cues e WOeaa ee PP

But if we examine the as. for the intensities, we see
that the frequency 27v=a

CAL 27a?
so that Be 10 a

h=6:57 x 10=" app.,

taking the radius of the electron =1°'5 x 10-1 [ Lorentz’s
value |.

728 Mr. A. Bramley on Radiation.

Now, if the electron is revolving around in a circular
orbit, a suggestion of which we have just availed ourselves,
then the radiation ought to experience a centrifugal
force.

The equation of equilibrium will be

102 @? 2
Ampada.y. Qo
: 2
or B= aS (wr )a.@
41 ¢
ae (w1 ) Qara )

But wr? =const. according to the theory of central forces,
since the radiation force acts along the axis of Z, whereas
the plane of revolution is the XY plane.

a A eho 3/2
h= Tom &6 x 10-8 Bi") Vv r}a

= DE oUt ey 10).

Both of these values are very nearly coincident with the
experimental values of h, for our knowledge of a is very
limited.

There is also another remarkable coincidence in these
values. They show that the momentum due to radiation is
identical with that due to the centrifugal force; another
example of the Principle of Equivalence.

It is hardly necessary to add that this equation H=hy has
been made the starting-point of the quantum theory of

stationary states. ‘Aniun Beste
2167 Kincaid Street,
Eugene, Ore., U.S.A.
August 20, 1921.

Pez: ..:]

LXIII. The Effective Capacity of a Pancake Coil.
By tx, BREE,
Purpose.

es has been shown in a previous paper + that the effective
capacity of a coil may be computed as

Cy =| 4 ("ee a) ys | da, ra Cl)

vis an arbitrary parameter along the wire ;
L is the inductance of the coil ;

M(@)dz is the mutual inductance of the section between
x and «+dz to the rest of the coil ;

where

di
= a(xv)dzx is the charge on the element dw, 2 being the
current through the coil terminal ;

2% 1, & are the values of wx at the coil terminals, the
value x, corresponding to the ungrounded ter-
minal of the coil.

The conditions which were assumed in deriving this
formula are :—

(1) The constant Cp exists.

(2) The product of the frequency used into the con-
ductivity is so high that the wire of the coil may
be considered as a perfect conductor: 7.¢., the
electric intensity is practically perpendicular to
the surface of the wire at any instant.

(3) The dimensions of the coil are sufficiently small to
make it legitimate to neglect the phase differences
introduced into the retarded potentials by currents
and charges in different portions of the coil.

(4) The formula still applies if Cy is not a constant in
general, but is constant within a range beginning
at very low frequencies.

It is the purpose of this paper to apply this formula to the
case of a pancake coil.

* Communicated by the Director of the Bureau of Standards,
Washington.

t See “The Distributed Capacity of Inductance Coils,” by G. Breit,
Phys. Rey. xviii. p. 649 (1921).

~

730 | Mr. G. Breit on the Effective

Description of Pancake Coil.
By a pancake coil is meant a coil whose wires are all
wound in one plane in a spiral, as shown on the figure
(see fig. 1). It will be supposed that the number of turns

Fig, 1.--Pancake coil.

in the coil is large, that the turns are close together, and
that the thickness of the insulation is negligible.

Thus the coil may be replaced by a disk on whose surface
the potential varies in the same manner as it does in the
coil.

Notation.

The radius of the pancake will be denoted by a.

Points in space will be referred to by cylindrical co-
ordinates |

Gee

with centre O at the centre of the coil, and with axis

perpendicular to the plane of the coil.

Fig. 2.--Cross section of pancake coil by plane through diameter.

sme

—

The meaning of these symbols is shown on fig. 2.

Capacity of a Pancake Coil. to4.
Simplifying Assumption as to Potential Listribution.

An arithmetical computation of the e.m.f. induced in
various parts of the coil for a coil with a finite number of
turns revealed the fact that the e.m.f. induced between
a point on the surface of the coil and the centre is approxi-
mately proportional to the square of the distance of that
point from the centre. ‘The computation above mentioned
consisted in calculating the e.m.f. induced between the
centre and a number of points at various distances from
the centre for the case of a coil having a finite number
of equally spaced turns. Maxwell’s formula in elliptic
integrals was used, and numerical results were tabulated.
These were then plotted, and the graph revealed the
approximate relation stated.

The relation is frankly approximate, but is believed to be
accurate enough for the calculation of the coil capacity.
The computation which follows takes this for its starting-
point.

General Plan of Attack:

The first step will be to compute the distribution of charge
on the wires of the coil which will satisfy the law assumed
for the potential distribution. Then the quantity M(z) will
be determined from the same law. The two expressions
will next be substituted in (1), and hence ©, will be
obtained.

This will be done for three cases—namely that of the
coil when ungrounded, and also when grounded—either at
the centre or else at the outer edge. |

The first part of the work consists, then, in the solution
of an electrostatic problem—namely that of finding the
charge distribution. The second part is ordinary inte-
gration.

Solution of the Hlectrostatic Problem.

It is convenient to transform the cylindrica] co-ordinates

(7, 2, @)
to elliptical co-ordinates
(wu, v, 0)
by the formula
r+ jz =acosh (u+jv), <5 alee (2)
where poe ai,

or its equivalents r = acosk u cos v,
z = asinhwsin v.

732 Mr. G. Breit on the Ejfective

The surface w=constant gives a spheroid of revolution
whose equation is
92 wa

iy ese (a
as is seen by eliminating v from (3); and the surface
v= constant gives a hyperboloid of revolution whose
equation 1s

y? 2 1
Fors ons °° |
as is seen by eliminating u from (3).

The two sets of surfaces represented by (4) and (5) are
orthogonal because (2) is a conformal transformation.
Also the planes 9=constant are perpendicular to both (4)
and (5). Thus the co-ordinates (u, v, @) are orthogonal.

It is readily shown that the Laplacian in these co-
ordinates is

Liar OV ko @ oV
cosh wu =A eek a z COs VOU (cos v=)

cosh? u—cos?» 92V

' cosh? u cos? v ° 0G?

=

In particular, if V is independent * of 0,

SWS RGiles O80) OV 1S OV
V2 = See (cosh u >) + ee (cos ae ) (6)

* ‘This expression may be derived by remembering that if
Ti, Uo, V3

are three orthogonal co-ordinates of such a kind that the differentials of
length corresponding to the three differentials

dij, dz, dads

are
alg O25
fre On a
then . Vv
27 — O hy OV fe) ie 3
OS UR se (;. hs Sa) “O22 Ur hs ae.

wn (2)

See W. E. Byerly, ‘Fourier Series and Spherical Harmonics,’ p. 289,
equation (6).

Capacity of a Pancake Coil. 733
It is advantageous to transform this by
= gin t.
eke oe. Pa (0)
y= 7 sinh;w,
which reduces (6) to

pee anya yO Mei Oe OV
vV= 24a Be +2 {a ge (8)

Now the electrostatic problem to be solved is that of
finding for V a solution which together with its first
derivatives is finite and continuous, which is independent
of 6, which satisfies the Laplacian

ete a wis Sais ut (O)

which vanishes at infinity at least to the first order and
which at the disk becomes

V = Vo—L
But the equation of the disk is

i ee (Ns

r di

ei: (10)

in which case (3) reduces to
T= 0. COS\U,
Hence, using (7), equation (10) becomes

V = V.-Ld =p) %.

Now the expression

[anPn(v) + Bnldn(v) | [anPr() + bnQn(m) J,

where P,,, Q, are Legendre functions of the first and second
kind respectively, when substituted in (9) satisfies (9) in
virtue of (8). If, then, one should be able to find such

values of an, Bry Gn, 6,, and such values of n that
V = > [ onPr(v) + BrQy(¥) | [ anP, (we) =F brQn (u) |

should vanish at infinity to the first order and should
degenerate into (11) when v approaches zero along the axis
of pure imaginaries, then, in virtue of the uniqueness of the
solution of (9) for given boundary conditions, the summation
written gives the value of V.

If the summation written is an infinite series it also
gives V, provided it is universally convergent as to pw
and v.

(11)

734 Mr. G. Breit on the Effective

Further, for a given vy, V may be represented by a series
of the form ~
> AnPn(u);
n=0

the summation being taken over all positive integral values
of n, because V obviously satisfies the conditions which make
such an expansion legitimate. ‘The coefficient A, is inde-
pendent of w but, for different vaines of v, varies and is thus
a function of vy. It must be clearly of the form

iSuy = atnPn(v) ne B,QiV),

for otherwise (9) would not be satisfied. Here n is a
positive integer. The function P,(v) is therefore a finite
polynominal *, viz.

t. See en =)
eee or
na —in=2)\ 3)
pW
and (),(v) is an infinite series when | v/>1, viz.

han ie LODE rere 1 (n+1)(n+2) 1
Qa) = eB be. nee Bie 2IOn+3) ven

(n+ V(n+2)m4+3mt4) Vo
2.4(2n+3)(2n+5) Sito ee oe

But points at infinite distance from the origin are given
by real, positive, infinitely large values of u, and con-
sequently in accordance with (7) by infinitely large values
of v on the positive half of the axis of pure imaginaries.
Such values can be denoted as usual by +j«. It is clear
that if n>0, the expression for P,(v) becomes infinite for
v= +) because it is a sum of terms of the same sign,
and each term becomes infinite. Hence, if n<0; «,=0.

Further, there is symmetry about the plane z=0. Hence
by (3) and (7) only even values of n can be taken. Thus the
most genera] possible expression for V is

V = 2 don Pon(H) Qon(v). c : : (1 2)

The coefficients A, as, must now be determined in such a
way that

ro) y aa
© aan Pon(H) Qealj 0) = Vo-LUL— yp). . (13)
n=()

1 Mise La es

Pe

* See W. E. Byerly, ‘Fourier Series and Spherical Harmonics,’
p. 145, equations (9) and (10).

Capacity of a Pancake Coil. Top

where the symbol Q»,(j.0) stands for the limit of
Qon(j).5) as 6 approaches zero taking only real and
positive values.
Since now
1—p? = 3[ Po(w)—P2(w)],
all do, but ap, a vanish, and 40» ad) are determined by the
relations

la
ay Qo (7 - 9) = Vo—3L<, | (
14

ee
Ao Q.()- 0) — 2L ate |
Thus
li
Vo—3L5
dt es Q.(j «sinh w)
V= sinh w) +2 Lo z le ;
Qj .0) Qo ) Q(z. 0) CH)
} (15)
The surface density of charge in coulombs is obtained as
epueigeh 25) ey See
~ 8989 4 An’
where K_ is the dielectric constant of the medium, and a

is the directional derivative of V with respect to the normal
drawn away trom the surface. The same may be written as

ee
Am On’ ;
where ga Coo)
i yee
Now at the disk the normal is parallel to OZ. Hence
oT =(S)_,if 20 ang ee -) if 2<0,
On Oz 2z—0 7 foe z=0

_ Hence by (3)

and by (15), (16)
pu J Qo (J - 9) 0)
~ seein YZ) G50 oy
di 7 Q,' tJ 0)
+3L5, oe yy Pl) }

036 Mr. G. Breit on the Effective

But it may easily be shown that *

psa) 20) ne
Qo(j .0) ce
and
j Qe! (7 - 0) ae)
CGO) ar
Hence

oe oa ee LS )+3 ah Pa(u) \

This solves the electrostatic problem proposed.

Computation of the Function M(2).

In order to find the function M(z), a choice must be

made of the variable x Here r will be chosen as this
variable.

It was assumed that the e.m.f. varies as 7”.

This means
that

r y2
{Moy qr= 15:

or, differentiating,

ip
Mr) 2L5. . .
* These formulas can be derived from using the following facts :—
(2) Qa(z) = Flog 25,
Pg—]
I
Ole gag 7% Ont DP, WAY)

(ec) The recurrence formulas
(2n+1)zP,,(z) = (wn F1)P,, 41 (2)+nP,,_1(2).

Using (c) in (0), the expression for Q,,, in terms of Q. can be derived
by writing the identity

Then from (a) expressions for Qo,, are obtained. On differentiating

these expressions and passing the limit in the result of the differentiation
as well as the original, the result follows at once.

Capacity of a Pancake Coil. 737
Computation of the Function a(x).
As stated in the introduction, the differential of charge is

2 a(x). da.

The independent variable here chosen is 7. As? varies
from 0 to a, and as @ varies from 0 to 27, the whole coil is
traversed by the point (7, 0). The differential of area
is 27rdr, and the differential of charge is then 4aordr,
where o is given by (17) becau-e expression (17) gives the
surface density only on one side of the coil.

By (3), on the coil » becomes acosv; so that

na
—— /1—cos?v = /1 =

Substituting this in (3), and expressing the fact that
di

4rordr = (5) a(r)dr,

it is found that

A= Vea) (ay 3) SPV)? on
(\(a)=

x

It now remains to substitute (18), (19) into (1). If the
ungrounded condenser terminal is connected to the centre,
x, is to be taken as 0 and 2 is to be taken as a. IE,
however, it is connected to the periphery, x, is to be taken

as aand a, as 0.
~The first of these gives

Bee P20: 20)

and the second

C: _ 4a (2 J a (21)
jy =Biss eS a |. ek a Zu
7 [9 s Lo

If, now, the coil should be used with the centre grounded,
and the ungrounded terminal of the condenser should be
connected to the periphery, formula (21) applies, and in
that formula V)>=0. ‘This gives

SKa 3 ;
peewee ASAE [a TS ie ra Oy -
Sean es (22)
This is the effective capacity if the centre is grounded.

Phil. Mag. 8S. 6. Vol. 44. No. 262. Oct. 1922. aB

738 Mr. G. Breit on the Effective
Again, if the periphery is grounded and the centre is not,

di

formula (20) applies, and in that formula Vo=L

et is
seen from (10) by setting V=0 when r=a. This gives
| 14a
n= SS 2:
Co loa 2

for the capacity with periphery grounded.

Finally, if the coil is insulated and the condenser is
unshielded, as much current enters the coil as leaves it ; so
that (20) and (21) must give the same value for Cp.

Multiplying (20) by 2 and adding to (21), it is found
that

_ 32Ka

G= Fee. 6 er

if the coil is ungrounded.
It is worth mentioning that if Cy be eliminated from (20)
and (21), it is found that ue which, in virtue of the
Le

1
{ P2(u)dp = 0,
0)

shows in a different way that the coil is insulated.
Expressing the results in micromicrotarads, the capacity
1s

identity

when grounded at centre.....;... 0°567 Ka ueuf,

when grounded at periphery ... 0°330 Ka puyf,

when insulated. =) 4)) ieee 0-252 Ka ppl.
Now, according to the results cf a previous calculation *,

the effective capacity of a pancake coil of small depth when
insulated is 0°437 Ka.

Thus, so far as the effective capacity is concerned, there
is an advantage in using pancake coils of large depth as
compared to pancake coils of small depth.

EHaperimental Verification.

The formulas (22), (23) have been verified experimentally
on a coil which is shown in fig. 3. This coil is not cireular

* See G. Breit, 2. c.

Capacity of a Pancake Coil. fel

but hexagonal. The quantity a is therefore not quite
certain.

In the computations it was taken as the mean of the radii
of the inscribed and escribed circles, which are 26:5 ems.
and 29°5 ems. respectively. Hence the mean is 28 cms.
The dielectric being air, the capacity with centre grounded
should be 16 mwuf, and if grounded at the periphery it
should be 9 wuf. The values as measured are 16 put and

9 upl.

Fig. 3._-Photograph of pancake coil.

Measurements were not made, more accurately than
1 pf, on account of the difficulties connected with such
measurements.

The capacity of the same coil was also measured without
the copper foil, leaving only the copper braid. No change
was detected in the capacity.

It also appeared that the copper rods used in fastening
the braid could affect the capacity. A row of them was
soldered to the braid, but no detectable change in capacity
was noticed.

3B 2

740 Prof. Hackett on Relativity-Contraction in a Rotating

Conclusion.

The effective capacity of a pancake coil has heen calcu-
lated, and the calculations have been verified experimentally
in two cases.

The results of the onleulation are that the capacity of
the coil

when grounded at centre is ......... 0°567 Ka wpt,
when grounded at periphery is ... 0°330 Ka ppf,
when insulated is ............ .. ila ee 0°252 Ka pp,

where a is the radius of the coil and K is the dielectric
constant of the medium.

Washington, D.C..
Jan. 14, 1922.

LXIV. The Relativity-Contraction in a Rotating Shaft moving
with Uniform Speed along its Avis. By VWurrx HW.
Hackert, M.A., Ph.D., Professor of Physwes, ‘College
of Science for Ireland, Dublin *

§ L. Introduction and Summary.
SOLUTION is offered in this paper of the problem of

the relativity-contraction in a rotating shaft moving
with uniform velocity along its own axis. The standpoint
adopted is that of the fixed ether and the FitzGerald-Lorentz
contraction combined with the restricted principle of rela-
tivity. ‘The validity of Huclidean geometry is assumed
throughout the paper.

A hypothetical modification of Fizeau’s method for
measuring the velocity of light is considered—a rotating
shaft carrying two disks with apertures which correspond
to the toothed wheel in Fizeau’s experiment. It follows
readily that when a rotating shaft is moving with uniform
velocity along its own axis, to a stationary observer, looking
in the direction of motion, it appears twisted in the opposite
sense to the rotation. This effect has been pointed out by
R. W. Wood +, and he has discussed the experiment, but not
in a sufficiently precise way to serve as a basis for the
subsequent discussion in this paper.

The arrangement may act asaclock. It measures time

* Communicated by the Author. Read at the meeting of the British

Association, September 1921.
+ Wood, ‘Physical Optics,’ 2nd edit. p. 690.

Shaft moving with Uniform Speed along its Avis. 741

on the same principle as the ideal clock consisting of a beam
of light reflected between two mirrors, with the addition that
a disk fixed on the shaft at any cross-section and rotating
with it can, owing to the twist, indicate the local time there.

In the latter p: os of the paper, the contraction in the shaft
due to the motion of translation and the twist is considered
as a strain-displacement. One of the principal axes of the
strain is assumed to be the direction of resultant velocity
7? 4 wu. The principal contraction in the latter direction is
found to be W1— (v?-+u2)/c?.

This result holds for a shaft of any ferm, since the twist
and longitudinal contraction do not depend on the form of
the oath Passing from the case of a solid circular cylinder
to the limiting case of a disk rotating without any motion of
translation, the reasoning in this paper gives the circum-
ferential contraction as equal to that usually accepted for a

rotating ring, viz. W1—vw?/c?, where wu is the velocity at the
rim, Tt follows that the conichion in the radius is vee
the same magnitude.

§ 2. The Velocity of Light and a Rotating Shaft.

Stationary System.—A rotating shaft can serve in theory
for the determination of the velocity of light by the following
modification of Fizeau’s experiment. Two similar disks are
mounted on the shaft in planes normal to the axis separated
by a distance 1. Hach disk is perforated by a number of
equidistant apertures lying on a circle concentric with the
shaft. In the subsequent discussion we ignore ordinary
elastic strains, or, in other words, assume that the elastic
constants are infinite.

The axis of the shaft is taken as the axis of z. The disks
are similarly placed so that the apertures in each disk pass
simultaneously through the plane of («, z) as the shaft turns.
We need only consider light rays travelling in this plane
parallel to the axis of the shaft so that they can pass through
an aperture in each disk for suitable speeds of rotation.
Let the period of the lowest of these speeds be T. For this
speed, light travelling through an aperture in one disk to-
wards the other will pass through another aperture there
which has just been brought into position by a rotation of
the shaft through an angle ¢, where ¢=angle between two
successive apertures in either disk. We have then

eat. ee. - at)

:

742 “Prof. Hackett on Relativity-Contraction in a Rotating

Moving System.—On the theory of restricted relativity, if
the apparatus is transferred to a system 8’ moving with
speed v along the axis of z and parallel to the axis of the
shaft, the properties of the system remain unaltered. For
the moving observer there are, therefore, definite speeds of
rotation for which

(1) light rays parallel to the axis of the shaft can
pass through the apertures in the disks in either
direction,

(2) the measured speed of light in this hypothetical
experiment is c.

To an observer in the stationary system, this is impossible

unless compensations take place. He knows

(3) the velocity of the moving system,
(4) the distance between the two disks modified by
motion to 1V1—v?/e?.

Reasoning on these data, he concludes that the experiment
can only succeed if the forward end of the shaft when in
rotation is twisted, with respect to the rear end, in the
opposite sense to the rotation through an angle, say @; and
this twist must be such as to compensate for the different
light-times between the disks in and opposite to the diree-
tion of motion. During the light-time for the former
direction the shaft turns through an angle 6+, while the
light has the relative velocity c—v. In the other direction,
the shaft running at the same speed turns through an angle
@—O, but the ight has the relative velocity e+v. At this
speed, condition (1) is satisfied, since light, emitted through
an aperture in one disk towards the other, reaches it just as
an aperture is passing across the path of the ray in the plane
of (,2). The period of rotation T’ and the twist required
to satisfy this condition can be determined by the fixed
observer using his own units from the equations :—

(c—v) (6 +0) T/20 =1 V1 —v?/e? = (c +0) (6—8) T'/27,
which give |
CHGS Og
b.¢. (1—v/e?) Waa = 1 103 /e?.

Shaft moving with Uniform Speed along its Aais. 743
Using (1) we get

Eo So] re.)

lv 20 luw
= VS es = SSS (3)
CV1L—we Th ae V1
For the moving observer, however, there is no twist in the
shaft. His units of time and length have altered, so that
equation (1) holds giving the speed of light as e.

§ 3. Lelativity Clocks.

This combination of rotating shaft and disks may be
regarded as a set of relativity clocks regulated by the
property that the shaft must rotate with the slowest speed
for which it can transmit light through the apertures in
either direction. The fixed observer considers that such
clocks in the moving system run slow according to (2).
It will be shown below that there is an automatic synchroni-
zation of the clocks. This is produced by the twist in
the mechanical coupling, and satisfies Hinstein’s test for
synchronism.

In the simplest form, each disk can serve as a local clock.
To give the same value toc, the same method of fixing the
unit of time must be adopted in all systems. It is con-
venient, here, to take as the unit the time of describing one
radian. Using this unit, the angular position in radians
of a special aperture with respect to (#, z) gives the time
directly, and from (2)

o = V1—v?/c? = angular velocity of S’ shaft in S-units. (4)

To the moving observer there is no twist in the shaft.
If he arranges that the timing aperture shall lie for every
disk in the plane of (x, z) when the shaft is not rotating, he
will conclude that in rotation the timing apertures pass
through this plane “‘ simultaneously ”’.

It is easily seen that Hinstein’s test for synchronism 1s
satisfied. The first disk may be taken as the origin, and the
second at a =/S/-units since 8’ ignores the contraction.
Let all the apertures be numbered in the opposite sense to
the rotation 0, 1, 2, 3, ete., bsginning from the timing aper-
ture, and let mirrors be fitted into the apertures in the second
disk. For the speed w, a ray leaving the first disk by No. 0
aperture is reflected at the second disk by No. 1 mirror at 2’
and returns through No. 2 aperture. Thus the time of

744 Prof. Hackett on Relativity-Contraction in a Rotating

arrival at vw’ = mean of departure and arrival times at the
origin.

According to the reckoning of the fixed observer, however,
the passage of the timing aperture at 2’ through ‘the plane
of (xz) is from (3) later then at the origin by w’va/e? W1—v?/e?
owing to the twist in the shaft. This is the local time eftect

and leads to the Lorentz time-transformation adopting the »

usual conventions.

Counting time from the instant when the origins in each
system coincide and the timing aperture at the origin in 8’
passes through the plane of (z,z), we have at every point
along the shaft

rotation of the 8’ shaft
its angular velocity in S-units.

fin S=time =

The rotation of the S’ shaft is got by adding the twist
to the angular distance of the timing aperture from the plane
of (a, 2). The latter is ¢’ in the units we have adopted, and
from (3) and (4) we have 0 = w’v/c?, giving

WAR IEG EA 5: 1 — P
¢ in S-units = W=ae @ +0) — ie Wie ee 5 Cae tale Ne
(5)
§ 4. The Strain in the Rotating Shaft.

The discussion in §2 has shown that to a fixed observer
a rotating shaft with a motion of translation in the direction
of its own axis is in a state of strain. For convenience the
term contraction is used here for the ratio of new length to
original length. We shall now proceed to find the principal
contractions for this state of strain. For the sake of clear-
_ness, the assumptions involved in this discussion are set
forth below.

We have as data the following deductions from the appli-
cation of the principle of Restricted Relativity to the systems
considered in §2:— |

(I.) The twist in the shaft is independent of its radius,
whether it is hollow or solid, and is given by (8).

CII.) The FitzGerald-Lorentz contraction of the distance
between two planes perpendicular to the axis is not altered
by the rotation, otherwise the relations deduced in (2), (3),
and (5) in agreement with the ordinary theory could not
exist.

The contractions of the relativity theory are independent

Shaft moving with Uniform Speed along its Avis. 745

of the physical characteristics of the body, and consequently,
as already stated (§ 2), the fixed observer eliminates from
his consideration any distortion due to centrifugal forces by
the following condition :—

Ordinary elastic strains are ignored or, in other words, it
is assumed that the elastic constants are infinite.

He interprets his observations in terms of Euclidean
geometry, and so he makes the following geometrical
assumptions :—

(A) Each cross-section of the shaft remains a Huclidean
plane, so that its radius alters in the same ratio as
its circumference.

Assumption (A) is the only possible one from the Euclidean
standpoint. Its justification lies in the simple form in which
the principal contractions appear below.

Using these deductions and the foregoing assumption, it
will be seen below that we can derive expressions for the
contractions along the principal axes if we know their posi-
tion. At this stage a further assumption must be made,
more speculative than the preceding which arises directly
from the Kuclidean point of view of the fixed observer.
According to (A) one of the principal axes of strain lies
along the radius, 7. e. along the direction of the centripetal
acceleration. The other two must then lie in a plane normal

to the radius, and we assume that

(B) one of these lies along the direction of resultant
velocity.

This last assumption may be justified by analogy with the
Wiedemann effect. To a fixed observer a rotating shaft in
the form of a thin tube moving along its own axis will be in
a state of strain similar to that of a steel tube placed ina
coaxial spiral magnetic field. In this instance, if hysteresis
be eliminated, as it can be by special experimental methods,
one of the principal contractions must be along the ee aiaad
magnetic field. The formula for the twist in the tube,
deduced on this assumption, has been confirmed by experi-
ment. In one case the tube is twisted by a spiral magnetic
field, and in the other by its spiral motion. The physic: al
analogy is so close that it seems to the author to justify the
fixed observer in applying the same analysis to the twist in
each case and in making assumption (B). about the direction
of the principal axis.

We shall now examine the state of strain in the shaft from
the point of view detailed above. In considering the analogy

746 Prof. Hackett on Relativity-Contraction in a Rotating

of the Wiedemann effect the shaft was taken as a thin tube;
this restriction is not necessary for the general mathematical
treatment, as can be seen by reference to deduction (1.),
though it may be helpful to think of the shaft as a thin tube
in the following discussion.

We consider that the new co-ordinates 2’, y’, z' of any point,
as interpreted by the fixed observer, are given in terms of
the original co-ordinates as in the ordinary strain theory.
The axis of the shaft is taken as the axis of z. The fixed
observer infers a rotation 0’=7z according to (3), and a
FitzGerald-Lorentz contraction parallel to ¢ according to
deduction (II.). In accordance with assumption (A), each
radius in any plane parallel to (wz, y) is assumed to be con-
tracted in a ratio which depends only on 7, and the circum-
ference alters in the same ratio.

As the rotation @’ cannot be assumed generally to be
small, the steps of deducing the strain-components are
given below :—

w' = excos 0’ —ey sin 0,
y' = ex sin 0 + ey cos 6.

We get for the relative displacement &', m’, & around
&, Y, g, of a point whose undisplaced co-ordinates are w+&,
yt, 246:

Toe roy Oz! | !

gia £0 + 9 E— try
oy’ Oy

— - u e
n= a ae 4) a + Cre

But part of these relative displacements 1s a pure rota-
tion 6’ around an axis through a’, y’, 2’ parallel to the axis of
the cylinder and arising from the general rotation. We shall
obtain the strain components &', 7’, ¢ by combining the total
effect &,', n', &/ with a rotation —0’, where

£'— £/ cos 0'+7n// sin 6,
n' = —&/ sin 0'+/' cos 0.
Whence

a¢er)

ie Cle
pi EY 4 ee tray),

1= po + + Er(ee).

Shaft moving with Uniform Speed along its Aais. 747

When we refer to a point «=r, y= 0, we get

g'= E(e+r S$) + a chalga GO)

CO — ent+err€, esc mee BACT ies Bier ter) fe (7)

The radial strain at any point is determined by equa-
tion (6). To find the principal axes of strain in the plane
of (y, z) we need only consider equations (7) and (8), which
may be written

FCT ASE s ghee IE tee te ost eel as CD

Cee Rape te es eye yo ee | ee CO.)
where Se eNews at Nee ure, Ga)

f= V1—v/e by deduction (II.). (2)

The next step is to express e, /, s in terms of the position
of the principal axes and their contractions. Since the strain
is not pure, the principal axes will be rotated from their
initial positions, and both their final and initial positions will
have to be considered.

Fig. 1.

\ Final
| ‘ JY position
\
Y
/attia/
pos/tion

We assume this part of tle strain is produced by contrac-
tions along and perpendicular to axes Y and Z, which in
consequence of the strain have been rotated into positions y¥
and z, as shown in the diagram.

448 Prof. Hackett on Relativity-Contraction in a Rotating
We have
y'= py,
2 = 9K.
The co-ordinates y, z and axes Y, Z introduced here have
no reference to the co-ordinates and axes used to deduce

equations (6), (7), (8).
Returning to 7, € axes, we have

n' = y' cos6—z' sin 6,
C= y' sind +2' cos 6,
Y= ncosa+€sin a,

Li — sin «+ (cos 23.
which give

n' = (pcos « cos 6+ q sin & sin 8)
+ €(p sin « cos 6—g cos « sin 6),
¢’= (pcos a sin 6—g sin a cos 4)
+ €(p sin asin 6+ q cos & cos 6).
Comparing with (9) and (10), we get
e= pcosacosdot+qsinasiné, . . . (13)

s=psinacosé—qcosesinéd, . . . (14)
f=psinasind+qceosacosd, . . . (15)
O=pcosasind—q sin xcos 0. :. . yay

We get from (14), (15)
fsin 64+ scos6 = psin«,

f cos é—ssin 6 = 9 cosa,

giving :
= (f+s cot 5) a Saye acai (17)
g = (f—s tan yee gt ah ele
from (16)
ee deine aE ee 12. OUT a gre

a eae!

Shaft moving with Uniform Speed along its Aats. 749

Thus
pp? f+scotd ;
g? f—stand’ - (20)

ae SEEN ° e ° ° . ° ° “ (21)

oe 9
ee bere eich 3 OD

At this point assumption (B) is introduced, 2. e. one of the
principal axes in the final position lies along the direction of
resultant velocity, and we write

an

efor OE a Se le OO

In the expression for the twist obtained in § 2 we have to
note that 0’ is in the opposite sense to @, since translation is.
in the positive direction along z, and we introduce a negative
sign, writing

0 /l = —ar/? V1—v?/2 = 7;
hence
Serr wc’ 4/ Lose? 8. (2A)
and from (12)

Inserting these values in (20), we find
pr Pe ale Cae al yo aa teed ge (2G)
giving

Geant — Uw A/a? 0? tle. eam, (27)
and from (23) and (27)

sin a = sin 6 /1—7/P? — w/e? |W 1—w se. . (28).

Inserting these values in (19), (21), and (22), we find
the values of e, p,and qg. The contractions are, most con-
veniently, stated for the surface of a cylindrical shaft having
a uniform motion of translation v and a rotational speed wu at
the periphery, where both velocities are measured by the
fixed observer.

Symbol. Direction, Contraction.

p Resultant velocity. V 1—v/? —v?/c?.

e Circumferential. VIP —w/e2 [| f1—v?/c.
Longitudinal. Vv 1—v"/c*,
q Normal to acceleration I

and resultant velocity.

750 Prof. Hackett on Relativity- Contraction in a Rotating

The way in which the form for the simple longitudinal
contraction is maintained in this more complex motion sup-
ports the assumptions which we have made, especially
assumption (B)}, giving in (23) the direction of one of the
principal contractions.

These results hold generally for any solid shaft in the
same state of motion, since, as stated in deductions (I.)
and (II.), §4, the twist and longitudinal contraction are
independent of the form of the shaft.

§ 5. The Contraction in a Rotating Disk.

The expressions deduced in the last section hold for all
values of » and wu. They should hold in the limiting case for
which v=0 when the shaft is rotating around an axis fixed
relative to the observer. In this way we derive a solution
of the problem of the rotating disk which enters so frequently
into discussions of the restricted and general principles of
relativity. Writing v=0, we find

circumferential contraction = /1—uw?/c?.

Before discussing this result, it may be well to state the
solutions which have been previously given of this problem.
Following Ehrenfest, it has been frequently stated * that if a
measuring rod is applied tangentially to the edge of a disk
in rotation in its own plane about its centre, the rod is
shortened in the direction of motion, but will not experience
a shortening if it is applied to the disk in the direction of the
radius. The result was originally put forward as a speculative
inference from the restricted principle of relativity. It raised
the difficulty that the ratio of the circumference to the dia-
meter is no longer constant, but this has since been met by
the statement that the disk is no longer a Euclidean plane.

On the other hand, Lorentz finds that both radius and
circumference contract in the ratio of 1 to 1—v?/8c° from an
investigation based on the general principle of relativity.

This problem is a special case of the “ general question as
to how far the dimensions of a solid body will be changed —
when its parts have unequal velocities, when, for example,
it has a rotation about a fixed axis. It is clear that in such
a case the different parts of the body will by their interaction
hinder each other in the tendency to contract to the amount
determined by /1—v[c?”f.

* Einstein, ‘Theory of Relativity,’ p. 81; Jeans, Proc. Roy. Soe.

vol. 97. A, p. 68 (1920).
t Lorentz, ‘Nature, February 17th, 1921, p. 79.

Shaft moving with Uniform Speed along its Avis. 751

The statement just made does not seem to apply to the
problem treated in this paper. For the contraction due to
the helical motion of the shaft is given in terms of thie
resultant velocity by the usual formula; and according to
deduction (I.) in §4 the state of strain in the periphery of a
solid shaft is in no way different from that of a thin tube of
the same external radius and in the same state of motion.
In the limiting case no distinction can, therefore, be drawn
between the strain in the rim of a rotating disk and a
rotating ring. ‘The radius of each must contract in the same
ratio as the circumference, viz. in the ratio 1 to V 1—w?/c?.

We have then rg=r V1—o?r,?/c?, where 7, = radius
when the angular velocity is m ; this gives

dy dy) (Ul — wr? cP.

Summarizing these results, we have then :—
The radius and the circumference of a solid disk rotating
with constant speed about an axis at right angles to its plane

contract in the ratio of 1 to W1—v?/c?, where wu is the
velocity at the rim.

A measuring rod laid along the radius contracts in the
ratio of 1 to (1—1w,?/c?)??, where w, is the velocity at that
position in the disk.

The simplicity of the assumptions made and the analysis
given in this paper give support to the view that the above
conclusion is correct, within the limitations of the Euclidean
outlook adopted. It takes a middle course between the
results stated by Hinstein and Jeans and the solution given
by Lorentz. It is conceivable that a solution may, however,
be found beyond the limits of Euclidean geometry which may
include all points of view. ee

§ 6. Note on the Wiedemann Effect.

A vertical iron wire carrying a current twists in a vertical
magnetic field. This is recognized as an effect of magneto-
striction due to the resultant magnetic field in the wire.
The effect is simplified if a steel tube is used in which a
spiral, or more accurately a helical, magnetic field acts whose
axis coincides with the axis of the tube. This is produced
by combining a longitudinal field with a circular fieid due
to a current flowing in a wire passing along the axis of the
tube,

cee

752 Dr. T. J. Baker on Breath Figures.
The analysis given above immediately applies; we get
from (11), (13), (14), and (19)
O'/l = 7 = sfer = sin 26/r . p?— q?/2pq.
In the case of magneto-striction we can write p=1+«¢
and g=1-+e,, where e, and e, are small. The difference

between the position of the principal axis before and after
the strain can be neglected so that 6= «a, and we get

6’ = sin 2a (€,—€) l/r. F
This is the formula given by Knott and verified experi-
mentally by the author *.

The author desires to express his obligation to Prof.
W. MeF. Orr, F.R.S., for his interest in and criticism of this

paper.

LXV. Breath Figures. By T. J. Baker, D.Sc. (Lond.)f.

F one breathes upon a sheet of glass which has been
cleaned with soap and water and polished with clean
linen, water-vapour condenses uniformly on the glass in such
i. manner that the surface as seen by reflected light appears
dull and rather white. If the tip of a small blowpipe-flame
is caused to traverse the surface of such a plate and the plate —
is then breathed upon as soon as it is cold, a whitish con-
densation appears on those parts which the flame has not
touched, whilst the track of the flame is marked by a form |
of condensation which, owing to its transparency, appears |
black by contrast with the neighbouring parts.

This and certain allied phenomena were described by
Aitken ¢ in 1893, and several letters discussing the subject
appeared in the pages of ‘ Nature’ § during the period 1911
to 1913, but no general agreement as to the cause was
reached. |

A lens shows that the white portion of the deposit consists
of lens-shaped drops which are isolated from each other,
whilst the black condensation consists of a continuous film of

water.

* Knott, Trans. Roy. Soc. Edin. vol. xxxv. p. 388; Hackett, Proc.
Roy. Dub. Soc. vol. xv. (n.s.) p. 416.

+ Communicated by the Author.

{ Aitken, Proc. Roy. Soc. Edin. p, 94, 1893.

§ ‘Nature,’ May 25, June 15, July 6,1911, Dec. 19, 1912, Feb. 6719138.
See also vol. vii. of Lord Rayieigh’s ‘ Collected Scientific Papers.’

Dr. T. J. Baker on Breath Figures. 133

The late Lord Rayleigh held the view that the part of the
elass swept by the flame had been rendered cleaner than the
neighbouring portion, whilst Aitken urged that the track of
the flame had been rendered dusty by solid particles deposited
from the flame, and that these particles aided condensation of
moisture. He pointed out that by scraping with a match-
stalk across the flame track a dusty deposit could be rubbed up,
and he considered that this contamination of the surface is
responsible for the breath figure. It is true that the track
which an ordinary blowpipe-flame has followed can be
detected by the eye before any moisture has been deposited
on the plate, but a flame of carbon monoxide leaves no such
deposit, and Aitken’s explanation seems inadequate because
this flame yields an excellent breath figure. Lord Rayleigh
showed that if the outside of a test-tube were heated to
redness the “black” or transparent condensation could be
obtained on the corresponding part of the interior of the
tube. This was at first contested by Aitken, who maintained
that the flame, or the hot gases from it, must strike the glass
directly to produce the result ; but further experiments led
him to accept Lord Rayleigh’s statement, and he then sug-
gested that a chemical change in the glass itself might account
for the effect.

Quincke found that when a drop of strong sulphuric acid
is warmed on a glass plate, which is afterwards washed and
dried, the ‘“ black”? condensation may be obtained on the
part which has been exposed to the acid. Craig suggested
that this might be due to the soaking in of the acid, thus
forming a hygroscopic film; and as coal-gas always contains
sulphur compounds, he contended that a coal-gas flame
‘playing on glass might deposit enough suiphuric acid to act
in this way. Butit was known that a flame of pure hydrogen
burning in air also gives breath figures, and Craig therefore
suggested the possibility that some nitric acid might be formed
by the flame, and that this acid might determine the pro-
duction of a breath figure. If this is true, it follows that
hydrogen burning in pure oxygen should fail to be effective.
This summary fairly represents the main features of our
knowledge of the subject up to 1913. From that date
onwards the author has intermittently carried on the inquiry,
with the result that other relevant phenomena have been
discovered, and a partial explanation of the results can be

offered.
Phil. Mag. 8.6. Vol. 44. No. 262. Oct. 1922. 30

Peo”

T54 Dr. T. J. Baker on Breath Figures.

Heperimental,

With the exception of certain cases specifically mentioned,
the surfaces used were first washed with soap and water,
then rinsed with tap-water, dried with a clean cloth, and —
finallyrubbed vigorously witha clean linen handkerchief until
the white condensation produced by breathing lghtly upon
them showed uniformity. ‘The surfaces so prepared will be
described for convenience as ‘ cleaned” surfaces. It must
also be understood that the term ‘‘ breath figure’ connotes
that condition of a surface which reveals itself by the black
form of condensation. A breath figure is not visible until
moisture condenses on the surface.

At the outset it seemed probable that a complex substance
like glass was not best suited for the purpose, and some pre-
liminary experiments were made with other materials. In
general, it was found that the chemical composition of the
substance is not important, because breath figures were easily
obtained on porcelain, rock-crystal, mica, Iceland spar,
platinum, nickel, silver, brass, and mercury. Aitken’s sug-
gestion that chemical change might explain the effect cannot
be maintained in face of the fact that the figures can be
obtained on rock-crystal and platinum. Again, since the
burning of coal-gas in a blowpipe-flame might lead to the
deposition on a cold surface of such substances as carbon,
sulphur compounds, and tarry matter, it was decided to try
the flames of hydrogen and carbon monoxide. Hydrogen
prepared from hydrochloric acid and magnesium was passed
through soda-lime to arrest any acid spray, and was burnt at
aplatinum jet. This flame produced excellent breath figures.
Burning carbon monoxide from sodium formate and sulphuric
acid gave equally good figures. From these experiments
we may conclude that breath figures do not require for
their production the deposition of any solid matter from
the flame. Further, since no water is produced when carbon
monoxide is burnt, the effect cannot be attributed to depo-
sition of moisture.

Lord Rayleigh has pointed out that the pattern of a breath
figure may be recorded permanently by the chemical! depo-
sition of silver on the glass plate. ‘The track of the flame is
distinctly marked by a difference in the appearance of the
deposit, and this difference is most marked near the margins
of the track, i. e. where the hottest part of the flame
impinges on the glass. This was confirmed both when the
flames of hydrogen and coal-gas were employed.

Dr. T. J. Baker on Breath Figures. 7155

Injluence of the temperature of the flame.

Aitken mentions that the flame of burning alcohol does not
produce satisfactory breath figures. The author made a
small spirit-lamp with a test-tube as reservoir and a tuft of
elass-wool supported in a glass tube as wick. This was
supplied with *‘ industrial spirit,’ and was regulated to give
a flame about + inch high. No breath figures could be
obtained, probably because such a small flame is not very
hot. It was found that the jarger flame of an ordinary
spirit-lamp gave a breath figure, but the effect was distinctly
weak.

Some methylated ether was purified by standing over
caustic potash and potassium permanganate for several days
and then distilling. The portion coming over at 34°°5 was
collected and used in the lamp described above. No breath
fizures were obtained. It may be noted that an ether flame
always leaves a small deposit of soot on the plate, and the
non- production of a breath figure in these circumstances
seems to negative Aitken’s suggestion that dust is an 1m-
portant factor.
~ Ifa mouth-blowpipe is used in conjunction with an alcohol
or with an ether flame good figures are easily obtained.

Again, if a glass plate i is drawn rapidly across the extreme
tip of a well- shaped blowpipe-flame, the breath figure shows
a perfectly uniform “black” track ; but if the. plate cuts
across the flame near to the red acing cone, then the breath
figure shows two clear “ black” lines which correspond with
the hot exterior of the flame, whilst the space between them
exhibits more or less of the white condensation corresponding
to the relatively cool interior of the flame.

These facts seem to indicate that the condition of the
surface necessary for the production of these figures is only
attained after it has been exposed to a flame whose temperature
is above a certain minimum.

Sir J. J. Thomson (‘ Conduction of Electricity through
Gases,’ p. 194) says “ionized gas is produced by flames ‘of
coal-gis whether luminous or not, by the oxy-hydrogen
flame, by the alcohol flame of a spirit- lamp, by a flame of

Sarapnte oxide: it is not, however, produced in very low
temperature flames such as the pale lambent flame of
ether.”

The parallelism between the ionizing effects.of the flames
mentioned in the foregoing abstract and the facts just stated
in connexion with the production of breath figures suggests

302

756 Dr. 'T. J. Baker on Breath Figures. —

the possibility that ions derived from the flame may be the
cause of these figures; but the extraordinary permanence of
the effects and the fact that actual contact with the flame
gases is not essential render the hypothesis untenable.

Permanence of breath figures.

The peculiar condition of the surface of glass which causes
it to reveal the flame-track when it is breathed upon is
singularly permanent, and persists for many months. As
stated by Lord Rayleigh and Mr. Aitken, a breath figure may
be removed by rubbing with soap and water, but the author
has occasionally experienced great difficulty in getting rid of
the last traces of the effect. Rubbing with a dry cloth
weakens the figures, but does not destroy them.

Huperiments with chemically cleaned glass.

Up to this point the glass plates had been cleaned as
described on page 754, and it now appeared necessary to
examine the behaviour of glass which had been subjected to
chemical cleansing processes such as are employed preparatory
to silvering. After the final washing in distilled water the
plates were supported on glass reds in a desiccator, and were
left there until dry. |

When these chemically clean plates were breathed upon
the condensation was almost entirely of the “‘ black” kind—or,
in other words, the glass was covered with a continuous film
of water. When a flame was made to traverse a chemically
clean plate and was afterwards breathed upon, no breath
figure, or at most a very imperfect one, appeared, thus sug-
gesting that a film of some contaminating material must be
present upon the glass before it is exposed to the action of
the flame.

Glass plates which have been cleaned as described on
page 754 are certainly covered with a contaminating film, and
in this connexion the work of W. B. Hardy andJ. K. Hardy
(Phil. Mag. July 1919) is significant. These investigators
found that truly clean glass surfaces will not slide over each
other, but seize owing to cohesion. A very small amount of
contamination lowers the resistance to relative motion, and
sliding becomes possible. The author has applied this method
to test the condition of the surfaces of glass plates used in
obtaining breath figures, and the following details are repre
sentative of the results obtained.

A chemically clean watch-glass was placed on an equally
clean sheet of plate glass as in Hardy’s experiments, and it
was found that a horizontal pull of 8 grams was needed to

Dr. T. J. Baker on Breath Figures. 7

start motion. This was not true sliding, but consisted of
je: ky movements accompanied by a gritty sound resembling
that produced by a diamond when drawn across glass. One
half of the glass surface was then rubbed with clean linen
and the watch-glass was placed upon it, and it vas found
that a pull of 4 grams was sufficient to cause steady sliding
without noise. Evidently the linen had contaminated the
glass.

A blowpipe-flame (coal-gas) was now swept across that
half of the sheet which had been rubbed with linen, and
once across the unrubbed and therefore chemically clean
half. Each of these flame tracks was tested with the watch-
glass, and a pull of 5°5 grams was required in each instance
to cause movement, and this motion was not smooth.

It appears reasonable to suppose that in one case the flame
had removed the film of contamination left by the linen, and
that in the other it had slightly contaminated the half which
was chemically clean.

In another experiment the flame of pure carbon monoxide
was used instead of a coal-gas flame, in order to eliminate
the possibility of contaminating the glass by the products of
combustion of coal-gas. Steady sliding of the watch-glass
occurred on the linen-rubbed part with a pull of 4 grams ;
but when the watch-glass was placed on the track of the
CO flame a pull of rather more than 7 grams was required,
and the movement was of the type associated with a clean
surface. This pull of 7 grams is very close to the value
(8 grams) required on chemically clean glass. If a thin
clean glass rod with a rounded end is drawn gently across a
glass sheet which has been rubbed with clean linen it slides
freely and noiselessly, but when it encounters a flame track
the increased friction is easily felt, and a faint squeak may

be heard.

7

Lc

Chemical deposition of silver on a flame track.

If a film of silver is deposited chemically on a sheet of
“cleaned” glass across which a flame of carbon monoxide
has been swept, it is seen that the mirror is whiter and freer
from pin-holes on the flamed part than it is elsewhere. This
points to the greater cleanliness of the flamed portion.

A test-tube was washed out with soapy water followed by
tap-water, and was then thoroughly dried by rubbing the
inside with clean linen. A narrow belt of the tube near its
middle was then heated externally in a small Bunsen flame
to a temperature which was much below its softening-point.

(ers _ Dr. T. J. Baker on Breath Figures.

When the tube was cold a silvering solution was introduced,
and the quality of the mirror obtained was better on the zone
which had been heated than elsewhere. ‘The borders of this
belt were badly silvered, and it seems probable that the con-
taminating film which covered the interior of the tube had
been driven from the heated belt to the cooler parts on each
side of it, where the increased contamination would affect
the deposition of silver adversely.

If the tube has been heated until softening is imminent,
the silver deposits on this part in a manner which suggests
that the original smoothness of the glass surface has been
partially destroyed.

The point which was at issue between Lord Rayleigh and
Aitken appears therefore to be decided in favour of the
former, who believed that the passage of a flame across the
glass cleanses it, and thus favours the condensation of moisture
in the form of a continuous film instead of droplets.

Nitric acid not a cause of breath figures.

It remains to consider the suggestion of Craig, viz. that
some nitric acid might be formed in a flame and be deposited
on the glass, thus determining the formation of a breath
figure. If hydrogen were burnt in pure oxygen this possi-
bility would be excluded. A jet of hydrogen was ignited
electrically in a large tube through whicha stream of oxygen
was passing, and by a simple device a small glass sheet was
passed through the flame. The glass showed an excellent
breath figure when breathed upon. The oxygen used was
prepared in one case from sodium peroxide and water, and
in another experiment from potassium permanganate, but
absorption with pyrogailol showed that it contained rather
more than 1 per cent. of (presumably) nitrogen. The result
is not conclusive, because a small amount of nitric acid might
have been formed ; but it seems unlikely that the reduction
of the nitrogen from 79 per cent. (in air) to about 1 per cent.
should have been without influence on the strength of ,the
breath figure if nitric acid plays any part in the process ;
and it is difficult to see how the presence of a minute quantity
of acid could account for the great increase in friction
described in the preceding section.

Transference of breath figures to a second plate.

In that which follows it will be convenient to refer to
a plate which has been traversed by a flame as a “flamed ”
plate.

Dr. T. J. Baker on Breath Figures. 759

A flamed plate which had been used for certain expe-
riments was by chance placed with its flamed surface down-
wards on another glass plate which had not been used. Next
day the plates were separated, and it was observed that both
plates gave breath figures, one being an exact copy of the
other. Moreover, the original showed no diminution of
intensity. This accidental observation was many times
confirmed, and it became clear that closer investigation was
necessary.

The transference of the effect from one plate to another
lying upon it suggested that some volatile material was
concerned i in the process, and that the escape of this material
might be assisted by reduction of pressure or by increase of
temperature. A flamed glass was therefore put face to face
with a “cleaned ” sheet of glass and the two plates, clipped
together, were placed in the receiver of an air-pump, and the
pressure was reduced to a few cms. of mercury. A few
minutes later the plates were withdrawn, separated, and
breathed upon, when a perfect copy of the original was
obtained on the “cleaned ” plate, whilst the flamed plate still
retained its power of producing a breath figure with unim-
paired intensity.

Thus transference occurs as effectively in a few minutes
under reduced pressure as it would in the course of hours
under atmospheric pressure.

Hxperiments were then made with plates separated about
+ mm., and a clear transfer was obtained in about 15 minutes.
The separation was increased to about 2 mm., and again a
transfer occurred, but much diminished in intensity. With
a separation of 1 cm. it was not possible to detect with any
certainty that transfer had occurred.

The flames of hydrogen and carbon monoxide also pro-
duced transferable figures ; and, since neither of these gases
yields solid products on burning, it appears unlikely that the
volatile material causing the transfer can have been provided
by the flame. It was also found that the transferred figure
resembles the original in offering marked resistance to the
steady sliding of a watch-glass across the glass plate, although
the effect, as might be expected, is weaker. Whatever may
be the explanation of the phenomenon of transfer, this fact
indicates that the transferred figure represents a portion of the
glass which has been partially cleared of the contaminating
film.

At this point it was thought desirable to discover whether
the peculiar properties of a flamed plate are modified by
breathing upon it. One half of a flame track was covered up

760 Dr. T. J. Baker on Breath Figures.

wand the other half was breathed upon. As soon as the
deposited moisture had evaporated the protecting cover was
removed, and the usual procedure for obtaining a transfer
im vacuo was followed. Not the slightest difference in the
two halves of the transfer could be detected. It is therefore
permissible, and sometimes convenient, to test the flamed
plate by breathing upon it before using it to obtain a
transfer.

Length of time during which a flamed plate retains its
power of giving a transfer.

A flamed plate was kept in a warm room for 60 hours,
and at the end of this time it was left zn vacuo for 24 hours
in contact with a “cleaned” plate, and on this a good
transfer was obtained. Another flamed plate after 9 days’
exposure to the air behaved similarly, but the transfer was
fainter; and ina third instance a plate produced a transfer
18 days after it had been flamed. The loss of the volatile
material is plainly very slow under ordinary conditions of
temperature and pressure.

Condition of the flamed plate after zt has been exposed
to reduced pressure.

A flamed plate was left in vacuo for 24 hours. It was
then removed and clipped face to face with a “ cleaned ”’
plate, and the two were kept in vacuo for 20 hours. No
transfer occurred, but the original flamed plate gave a
breath figure as good as though it had not been exposed to
low pressure. Ina second trial the time allowed for transfer
in vacuo was extended to 48 hours, but no trace of transference
could be detected.

These facts seem to confirm the hypothesis that transference
is due to the escape of material from the flamed track on the
plate, and that the whole of this escapes under reduced
pressure in the course of a day. At the same time it is
clear that the surface of the glass which has lost this matter
is still in an abnormal condition, and whatever this condition
may be it is one which persists for many weeks.

Secondary transfer.

A transfer was obtained in the usual manner. ‘The plate A
on which this transfer had been effected was then clipped in
contact with a “cleaned” plate B, and both were placed
in vacuo for 10 minutes. At the end of this period they
were separated, and plate B was examined. No transfer

Dr. T. J. Baker on Breath Figures. 76]

could be detected. The plates were again putin contact and
left in vacuo for 2 days, and then it was found thata transfer
from A to B had occurred. Plate A still retained its power
of producing a breath figure.

These secondary transfers afford further confirmation of
the view that the substance which modifies the glass surface
is volatile, and it is worthy of notice that the glass on which
the first transfer was obtained still retained its property of
yielding a breath figure.

Permanent record of a transferred figure.

When silver is deposited chemically ona glass sheet which
has received a transfer, the pattern is recorded precisely in
the same manner as the figure on the original flamed plate.

Transfer produced by heating the flamed plate.

A flamed plate and a “cleaned” plate were held face to
face by clips, and the back of the flamed plate was heated
by a Bunsen flame until it was uncomfortably hot to the
touch. When cold the ‘cleaned ”’ plate showed an excellent
transfer, and the flamed plate itself still gave a perfect
breath figure. The original flamed plate was now placed in
contact with anothsr ‘‘ cleaned” plate, and the process was
repeated. This resulted in a very clear transfer, but rather
fainter than the first, indicating that not all the volatile
matter had been expelled by the first heating.

Experiments were then made with the plates slightly
separated, and transfers were obtained even when the
distance between them was fully 2 mm., but the outlines
of the figures were less distinct, probably owing to diffusion
of the volatile matter during its passage across the inter-
vening space.

Temperature required to expel the volatile matter.

The volatile matter is rapidly dissipated at 100° C., for it
was found impossible to obtain a transfer froma flamed plate
which had been. heated in a steam-oven for # hour. In this
connexion Lord Rayleigh’s observation that a breath figure
may be obtained on the inside of a test-tube by heating it
externally to redness may be recalled. Such a figure should
be incapable of transference because the high temperature
of the walls of the tube would have expelled any volatile
matter which may have been there. To test the point a flat
sheet of fused silica was held so that the tip of a small
blowpipe-flame impinged on the middle of one face until a

762 Dr. T. J. Baker on Breath Figures.

red-hot spot was visible. When quite cold the face of the
plate remote from the flamed face was placed in contact with
a “cleaned” sheet of glass, and both were subjected to
reduced pressure. No transfer was visible even after the
lapse of 46 hours. A similar negative result is obtained if
the flamed face of the silica plate is used.

Hlectric breath figures.

- The tracks of electric sparks which have traversed the
surface of a “‘ cleaned ” plate of glass are rendered visible by
“black” condensation when the plate is breathed upon.
Figures so obtained are transferable, and in all their pro-
perties resemble those produced by flames.

Electric discharges in air produce ozone, but the present
Lord ne has shown that this gas does not yield breath
figures on glass.

Nitric acid is also formed, but the author has found that
good figures can be obtained on glass which has been tra-
versed by sparks in an atmosphere of hydrogen. It is
therefore unlikely that the figures are caused either by
ozone or nitric acid. By passinga large number of sparks
between two platinum sheets which rested upon a sheet of
“cleaned” glass a well-defined area was obtained, which
could be tested for friction by Hardy’s method. To cause
steady motion on the unsparked portion of the glass a pull of
4] orams on the watch-glass was sufficient, whilst on the
sparked area a force of 74 to 8 grams weight was necessary
to start motion. These results are almost identical with
those obtained on the track left by the flame of carbon
monoxide (p. 757), and it is highly probable that sparks
remove the contaminating film from the glass surface either
by their heating or disruptive effects.

It is well known that when a sheet of cleaned glass is
placed upon an insulated metal plate which is connected to
one pole of an induction-coil, and a coin, connected to the
other pole, is laid on the glass, the passage of a discharge for
a few seconds will produce the conditions for the developme:t
of a breath figure on the glass. The ‘ black ” condensation
corresponds to those parts of the coin which are in relief and, |
in addition, the tracks of any sparks which may have traversed
the glass surface from the edge of the coin will also be
rendered evident as wavy lines of “ black ” condensation.

It may be assumed that discharges from the under surface
of the coin to the glass beneath it occur most abundantl
from those parts which are in highest relief, so that the film

|
|
E

— ~~

~~ ee, ee ae Ie eee eh

Dr. T. J. Baker on Breath Figures. 763

of contamination is. removed from those parts of the glass
more rapidly than from the neighbouring areas, and a
breath figure revealing the design of the coin may be
developed. If the time during which the discharge takes
place is too prolonged, the whole of the film beneath the
coin is removed, all details disappear, and only a disk of
“ black ” condensation is obtained when the glass is breathed
upon.

Hlectrical conductivity of breath figures.

The author has found that the passage of a flame or of a
stream of electric sparks across the surface of ‘‘ cleaned”
glass greatly reduces the insulating property.

Two ebonite rods were capped with small pads of tin-foil,
one of which was earthed, and the other was connected by a
wire to a charged electroscope. By pressing the two pads
simultaneously on the surface of the glass to be examined
any leakage across the intervening portion of the glass is
easily detected.

Flaine tracks produced by burning coal-gas, hydrogen,
and carbon monoxide all show considerable conductivity.
Coal-gas appears to be most effective, and this is not un-
likely, because sulphurous acid is one of the products of its
combustion.

If a flamed plate is heated in a steam oven for about

- 30 minutes the conductivity of the flame track is found to

have been reduced very greatly, and a similar result is
obtained after a flamed plate has been left in vacuo for a few
hours. ‘Transfers are also found to possess a certain small
conductivity.

It is a matter of indifference whether the electroscope has
been charged positively or negatively.

The reduction in conductivity brought about by heating to
100°, or by exposure to reduced pressure, suggests that water
derived from the burning of the coal-gas or hydrogen may.
account for the effect, but it is not clear why the flame of
carbon monoxide or electric sparks should produce con-
ductivity.

Discussion of results.

It is probable that a breath figure produced by a flame or
by electric sparks is to be attributed in part to the burning
off or volatilization of the thin film of contamination which

‘Is present on a surface which has been rubbed with “‘ clean”

linen. The track of the flame or spark then presents an

764 Des, a: Baker on Breath Figures.

uncontaminated surface on which moisture condenses in the
form of a continuous transparent film.

It would be expected that the flame-cleaned track would
speedily become contaminated again and cease to function,
but the extraordinary persistence of the property associated
with the production of a breath figure (p. 756) indicates that
other factors have to be considered. What these factors are
cannot be asserted with confidence, but it is not improbable
that the structure of the surface layer of the glass itself
suffers a change during its momentary exposure to a high
temperature, and it is also possible that some of the decom-
position products of the contaminating film are occluded by
the glass along the flame track.

If a chemically cleaned sheet of glass is traversed by a flame
of carbon monoxide, and, when quite cold, is immersed in a
silvering solution, it is found that the silver begins to deposit
first along the flame track. Since no film of contamination
previously existed on the glass, it would appear that the
difference in the rate of deposition of silver is due to a
physical change in the surface of the glass.

It is more difficult to offer an explanation of the trans-
ference of a breath figure from a flamed plate to a ‘‘cleaned”
plate, but since the process is hastened by reduction of pres-
sure and by rise in temperature, and occurs even when the
plates are not in contact, it is clear that some gasecus material
passes from one to the other. Also, it has to be borne in |
mind that the transferred figure is an area from which
the contaminating film has been more or less removed
(ped ag)).

We may imagine that the contaminating film ona “‘ cleaned”
plate tends to prevent the ready escape of gas-molecules
which have been occluded by the glass surface, but that
where this impediment has been removed by the passage of
a flame, or by sparks, a violent outrush occurs when the
temperature is raised or the pressure is reduced, and these
molecules on striking the opposed surface of the “cleaned ”
plate break up and scatter that portion of the contaminating
film on which they impinge and thus expose a relatively
clean surface on which moisture will condense in the
“black” form. This suggestion may be extended to explain
a second transfer from the first.

The molecules which are active in producing this result
probably arise from the occluded products of decomposition
of the contaminating film, for it has been found that no
transfer, or at most a very faint one, can be obtained from a
chemically clean plate which has been flamed.

a a ae a a ll

Repulsive Eject wpon Poles of Electric Are. 765

The results of the experiments on the electrical conduc-
tivity of flame tracks and of their transfers point to water
molecules being one of the active substances, but the experi-
ments with carbon monoxide and with sparks show that the
effects caunot be attributed to water molecules alone.

The author has obtained some evidence that the transfer
in vacuo of a breath figure to the sensitive surface of a photo-
eraphie plate is capable of development, but the necessary
conditions are as yet uncertain and require further investi-
gation.

King Edward’s School,

Birmingham,
Feb. 21, 1922.

LXVI Repulsive Effect upon the Poles ee the Hlectric Arc.
By A. SELLERIO *

1. FN December 1919 Prof. W. G. Duffield published

the results of a careful series of experiments
carried out in conjunction with Messrs. Burnham and
Davis, on the same subject as the present paper f.

As similar experiments made by me are not mentioned
there, it seems that my Notet of 1916 is unknown to
the authors. It may be useful to put together both the
results concerning this interesting subject§: but before
doing so, I must observe that in “evaluating my readings
I did not take account of the electromagnetic force V
due tv the earth’s magnetic field, and of the electro-
dynamic action Ii between the fixed and the movable parts
of the circuit. In recommencing my experiments some
years ago, I remarked that V, EK were on the contrary not
to be neglected, and in the meantime I was notified of
Prof. Duffield’ Ss work, in which the different sources of error
are accurately separated. His investigation enables me
to estimate the corrections concerning my results, without
further trials.

* Communieated by the Author.

+ Phil. Trans. Roy. Soc. of London, A, vol. ccxx. p. 109 (1919). A
further note on metallic and composite arcs is recorded in Science Abs.
1920 (Roy. Soc. Proc. xevil. p..326 (1920)). -

{ A. Sellerio, “ Effetto di repulsione nell’ arco elettrico,’ MNwovo
Cimento, xi. p. 67 (1916). This paper and the first noticed of
Prof. Dutheld will be denoted by the initials 8. and D.

§ Dutheld’s research was the subject of a note by Mr. Ratner, Phil.
Mag. xl. p. 511 (1920), and Prof. Tyndall, zbedem, p. 780.

766 Dr. A. Sellerio on the Repulsive Effect upon

It may be well to give an outline of my apparatus : a sort
of torsion balance, shown in plan in fig. 1 (from the Nuovo
At O were connected to the horizontal rod a suspension
fibre normal to the plan hanging down from a graduated

Fig. 1.

Sehermo

se im) -|

b !
Ne m te 0 6 :

ue u
a Contrappeso

Microscopio

torsion head, and a little iron style dipping into a mercury
trough. Therefore the current flowed through the mercury
either in the direction OMBA, or in the opposite one.
Further details are referred to loc. cit.

When the arc is started, the arm OM tends to recoil, and
to hold it stationary it is necessary to give to the fibre a
certain torsion «, corresponding to a force

R= 0°036 « dyne.
In the force F there are to be distinguished :—

The true repulsion rising within the are gap ;

Some disturbance occasioned from the heat, as air
convection currents, ete. ;

The influence V of the earth’s magnetic field ;

The electro-dynamic action E between OMB and the
- fixed circuit.

2. The earth’s action V can be easily calculated, as
follows :—

As the arm OMB is free to turn about a vertical axis, the
recorded forces are only the horizontal one, and consequently
the earth’s field is acting upon OMB only with its horizontal

the Poles of the Electric Are. 767

component, H,. When B is +, the current direction is
OMB, thus according with Fleming’s law, the repulsion
is apparently increased.

The resultant H,.OM.z2 of the forces acting on OM is
applied to the centre of OM, hence on transferring it to the
end M it is to be reduced to

In a similar way, the couple due to the forces acting on

BM being (H,. BM .2z) a, we may replace it with a force
<M
CH, . BM ..2) 2 OM

applied to the extremity of the arm. As this force is much
less than the preceding, when the length of the carbon rod
BM during a set of experiments is reduced by burning, no
appreciable error is caused.

Adding the two forces together, we have

OM. BM?,

AV: = Te i a tt Beso yyt
or, by setting the current I in amp.,
yk OB
= Sie OM . . . e . . (1)

This simple calculation of V may bring a remarkable
economy in carrying out further experiments. Of course, it
would be better to compensate both the actions V and E by
some magnet or circuit conveniently disposed.

In my apparatus OM = 15 cm., MB =~4 em., H, = 0°37
(Palermo), then

: Ree LN ees Se")

Much more laborious and doubtful would it be to caleu-
late H, 7.e. the coefficient K of the law

eI ee ee, (2)

whenever the geometric data of the circuit were known.
It is then preferable to estimate it experimentally, as Prof.
Duffield did. He found (p. 124) with 8 amp. E=75 degrees
= 1°8 dyne, thus

K = 0028;

H=O08T oe. . .. @)

768 Dr. A. Sellerio on the Repulsive Effect upon

As my arrangement was a very similar one, I can adopt
without committing (especially with small currents) a con-
siderable error, the same value of H, and according to
(1')—(2), the correction (V + E) becomes

—(:2961—O0°038 7 ......... for the anode, \

9/1
3
O22 I61 0:03 7 es. for the cathode. ae
3. In my experiments almost all the readings were taken
by keeping the arc length L constant and varying the
current. But, having observed that the carbon quality has
afar greater influence than L on the results *, it would be
useless to relate here the individual series of measurements.
In Table I. I have therefore recorded only the mean
values of P upon anode and upon cathode for are length
L=1~4mm.,, by a given current, as they result from the
whole of my readings after the corrections (3’).

TasBue I.
ANODE. CATHODE,
Current = Deflexion =F pee Deflexion F De
a: oe. dyne. dyne. a. dyne. dyne.
Ou ee: 36 1:30 0-14
SOT tee 62 2°23 0:57
Dapreae es 88 316 0°93
Os AAPA 114 4:10 1:24
Tot crane de 140 5:04 1-49
Se eae a l66 5°90 IG
OR SeE Mens 204- 7:35 2°26
LO Seeks 240 8°65 2°69
fal eres hee: 290 10:4 3°49 16 0°58 0:29
ND one ee 336 12-1 4:28 48 1:73 0:96
east eeeee 380 13:7 4°77 80 2°87 1:66
VAY ieecsa ae 436 15°7 5°68 112 4:08 2°31
1S ear Arnie 480 17-3 6:07 144 Seis) 2-87
ING eee Arte 540 19°4 6:97 176 6°35 3°38
Ihab eee 600 21:6 788 210 7°55 Bon
SH eebiacit a Sic Si 260 9°35 4°95
VO Ree aes, ee ee ae 318 11°50: 6°35
Pht vole cae ce ab 376 13°60 7°42

4. Comparing Table I. with Duffield’s results, we shall
see, in spite of several numerical discrepancies, a good agree-
ment in the general behaviour of the observed effect. So
far as concerns the mean value (4(P,+P_), which is

* When the carbon rods are very close together (L=~0), the repul-
sions P, and P_ become evidently greater.

i
j

the Poles of the Electric Are. 769

independent of the earth’s magnetic field, we shall find
also a numeral concordance. For instance, by extrapolating
for 11 amp. :—

From Dutteld’s Table VII. (Burnham),
L = 3°5 mm., anode P = 1°55, cathode P= 1°75,
4(P,+P_) =1°65 dyne ;
From D.’s Table VII. (Davis),

mean value L = 2°5 mm., anode P == 2°02, cath. P = 1°78,
£(P.-— P_) = 1°90 dyne;
From Table I. (above),
mean value L = 2°5 mm., anode P = 3°49, cath. P = 0°29
ae ees) = 1°88 dyne.

There is, on the contrary, a remarkable difference between
the separated values P, and P_, which may be due to the
difficulty in eliminating V *. The readings from Table VII.,
which are almost unaffected by V and H, show, according to
my results, P, >P_, 7. e. a greater effect upon the anode.

5. In order to find out how other circumstances may
influence the pressure P, I have tried some expsriments
with cored carbons, finding an increased effect upon the
negative pole and a reduced one upon the positive. This
behaviour is to be attributed to the metallic salts of the core
(S., p. 77).

I have also noticed that the readings for ascending and for
descending current are often a little different, as happens,
for instance, in P. D. measurements, for both the shape and
the matter of the carbon (occluding gases, metallic salts,
grain, &c.) are altered by burning.

The diameter of the electrodes has no great influence in
the present research, of course only while it remains large
relatively to the crater size. For, putting a carbon rod
12 mm.in diameter against a similar one of 3 mm., when
the latter is acting as anode, the arc hums and the repulsion
becomes greater (S., fig. 8).

Further remarks, made also with thick carbon rods either ~

* In Duffield’s experiments, setting approximately H,=0°47 (Ingland)
and taking OM=OB=11 cm. from his hgs. 1 and 17, formula (1) would
give V=0'261. Instead of 0:26, w2 get from fig. 8, 0:14, and from
p- 124, 0°48.

Phil Mag. 8. 6. Vol. 44. No. 262. Oct. 1922. 3D

770 Dr. A. Sellerio on the Repulsive Effect upon

by increasing the current strength over the hissing point, or
by shortening the length L, &c., have generally shown that
when the are is not quite steady and silent, the forces acting
upon the poles become greater.

6. In order to test whether the observed effect P is due to
any disturbance C produced by heat, as convection currents
of hot air, an arc was struck between two fixed carbon rods
A, B (fig. 2, from the Nuovo Cim.). No repulsion was
exerted upon a movable carbon B’, although it was so
near to B' as to become white hot. On the contrary, the
influence of the heat is to cause the poles to approach each

Fig. 2.

M O

other ; the deflexion due to C was 10 to 20 degrees, or 0°36
to 0°72 dyne (S., p. 70). Then if we momentarily denote
with O the observed effect, the true repulsion is to be set,
P=O+C.
= This disturbance is not entirely avoided by enclosing the
apparatus in a box, to prevent air currents, as stated by
Prof. Duffield. He tried to determine the value of C, and,
having found approximately 0°44 dyne at 8 amp., proceeded
to draw a correction curve C. However, “It is unfortunate
that information is very difficult to obtain in the crucial part
of the curve, where the current is small,” as he says, thus
we cannot yet decide with certainty whether the true repul-
sion P is present witn the smallest currents, or whether it
starts up only at a current minimum.
This interesting question is included in a general one,

the Poles of the Electric Arc. foie

ae. either the repulsion P is a gradual and uniform effect,
or not. Fig. 3 shows a typical specimen of my readings.
As the corrections Vee C, whatever their exact values
may be, do surely vary in the same way on increasing the
Fig. 3.
600|%
=f
560 -

Pi carbonce+ 12mm )

Negat iyo

720
° Wied normate

80 x >> atbi lante

e } x x Amp.

x

Bl eae ay Dai Oe Be ei Oh AB RO

current, i. é. they as well as their derivatives are both con-
tinuous functions of I, the ‘‘ Knees” remarked on graph 3
must remain after allowing for the corrections V, H, ©,
and consequently they are to be attributed to the true
repulsion P.

If the explanation of P as a recoil effect by the departure
of carbon ions or molecules is accepted, the different traits of
the curves « or P are easily to be explained (S., p. 86) by
assuming that when the current exceeds a certain value
greater particles are also expelled from the craters.

aD2

772 Dr. A. Sellerio on the Repulsive Effect upon
7. The next point to be considered, as I pointed out loc. ct.,
is the specific pressure p = = that is, the pressure upon the

unit surface of the crater, A being the whole area.

Taking from Duffield’s fig. 22 the values P (as unaffected
by E, V, and corrected for C), and, on the other hand, from
my fig. 10 the values A fora circular crater normal to the
carbon rod BM, we obtain

TABLE II,

1, Pa 4. 6. 8. 10. ann
2U0 DC \ P O10. “044° 090 4-430 Osea
Cathode ene eo Z ‘ yne
AMOCOIN ue one, 0:017 0:088 0'16 0:24 0°31 em.”
Oathode A ............ 0:005 0027 0050 0073 0096 >>
ANNO DOD ea Stara 5°83 50 56 59 68 dyne/em.?
Cathode p 2. s:-.- 20 16°5 18 19°5 22 >>

This table shows that the unit force does not vary much.
with the current strength.

It must not be forgotten that, owing to the uncertainty in
the data P, A, the above values are only recorded in order
to give a rough estimate of the specific pressure p. To
pursue accurately this inquiry, coherent values of P and A
for the same carbons and the same arc length are required.

Besides the mechanical pressure, recent manometric obser-
vations with drilled carbons have shown for current strength
under 20 amp. * a hydrostatic pressure up to 30 dyne/em.?,
an order of. magnitude not far away from that of the
mechanical pressure p.

In a research f on the electric are Hgt/C~ between Hg as
anode and a thin carbon rod as eathode, I have calculated a
pressure of 6500 dyne/cm.’? upon the positive pole Hg. In
fact, it is known that in a mercury are a cavity of 1 mm. or
more in depth has been often observed }, corresponding to a
pressure range over 1300 dyne/em.’, with currents of a few
amperes. With a carbon-carbon arc the specific pressure p
is, as shown, a hundredfold smaller.

* H. E.G, Beer and A. M. Tyndall, Phil. Mag. xlii. p. 956 (1921).

t A. Sellerio, ‘‘Contributo allo studio quantitativo dell’ arco eletrico
fra mercurio e carbone,” Nuovo Cim. xxiii. jan —febr. 1922.

{ Stark u. Reich. Phys. Zeit. iv. p. 823 (1902). See also, Stark u.
Cassuto, Phys. Zeit. v. p. 269 (1904).

| ie ew oS he

the Poles of the Hlectrie Are, aie

ON THE NATURE OF P,

8. For the purpose of explaining the observed mechanical
pressure, many hypotheses have been suggested *, starting
from different points of view on the arc mechanism, and a
conscientious discussion would carry us beyond the limits of
the present paper.

Without any assumption, it may be observed that of
course in the arc a loss of matter by each electrode anda
transport from anode to cathode occur, thus—on the anode
at least—a recoiling effect must be occasioned. Whatever
the nature of the forces propelling the carbon particles
may be, the recoil due to evaporation can be estimated as
follows :—

Let N, p, v be respectively the number of particles leaving
the crater in 1 second, their mass, and their velocity of pro-

pulsion; then the considered recoiling effect is given by

Nee ene eo aoa (Al)

By pw, v mean values are to be understood, for it is quite
improbable that all the particles possess the same mass and
are projected with uniform speed, therefore the right-hand
side of (4) is a substitute for an expression Ynpv, with
N=2n.

In calculating the repulsion P as a recoil effect, by a
formula similar to (4), Prof. Duffield assumes for v the

velocity of agitation of carbon atoms at the temperature
4000° C. of boiling, that is,

2 4273 :
18°39. 10+ Ve \ 973 at 2°97 1G? em. /Sec.,

18°39.10* being the molecular velocity of H, at 0°.
It seems to me that this assumption is hardly defensible, as
will be best shown by means of the following analogy. If
we keep a compressed gas in a bulb, by opening a tap the gas
escapes, Impressing a reaction upon the bulb, as in tur-
bines. The velocity responsible for this recoil is not at all
the molecular velocity of agitation (a function of the gas
temperature), it is the velocity v with which the gas departs
from the bulb (a function of the pressure difference in and
out of the vessel). The value v will be obviously less than
the molecular speed ; thus the pressure P does not reach to

the high range estimated by Duffield.

* Besides the works already noticed, see a recent paper of Prof. A. M.
Tyndall, ‘On the Forces acting upon the Poles of the Electric Arc,”
Phil. Mag. xlii. p. 972 (Dec. 1921).

174 Dr, A. Sellerio on the Repulsive Effect upon

In my paper on the are Hgt/C~, above mentioned,
assuming for the vapours issuing from the anode the
boiling temperature 357° C.=630 abs., I have found a value
V =1000 em./sec., whilst at the same temperature the atomic
speed would be

v2 7 630
. 4 eee ee OR
Iksseayg) = Iu cole 973 7 28,000 cem./sec.
Even when the assumed temperature at the Hg crater is
really higher, both the considered speeds remain too different
from each other.

9. With a C/C arc, if the carbon consumption did occur
only by evaporation in the crater, we should have, denoting
with d the mean vapour density, a loss of mass per second,

m = Avd.

As, however, in an ordinary are the carbon out of the
crater is consumed also by burning in the air, we must
write

m= Ard,
whence
m

With reference to some of Duffield’s researches”, we can
take for the anode with 1=8amp. and L=3mm.,
m = 135.10-° g./sec., the area A being 0°24 cm.’, as shown
in Table II. The density of the carbon vapours at 4000° C.,
being taken as 0°00009, the abs. density of Hs: ataO nee
becomes

12 0-00009 ie
273

then we have from (5)

” = 650 cm./sec.,

a velocity range far removed from the atomic speed 2°97 . 10°
mentioned above.

We will calculate in a simple manner what should be the
average velocity v, so as to give account of the observed
effect P.

* ‘Consumption of Carbon in the Electric Arc,” Roy. Soc. Proc. A.
vol. xcii. p. 122 (1915). Above data are taken from a “ Note upon the
Alternating-Current Carbon Arc,” by Prof. Duffield and Mary D. Walker,
Phil. Mag. vol. xl. p. 781 (1920).

the Poles of the Electric Arc. 175

| If we momentarily assume as known the net carbon con-
sumption m, t.e. the loss occurring by evaporation at the
crater surface in 1 second, we can write instead of (5), the
equation

m = Avd.

It follows from (4), with m = Nu,

ee een
or for unit surface
peat eee ce ad
f ay Oe Pier
p :
a= et Tae ede

Taking p from Table II., we get a mean value
V = 400 cm./sec.

in a good agreement with the foregoing upper limit 650.

It the observed p is half due to recoil and half to impact
of particles moving towards the electrode, v becomes still
less, i.e. 280 cm./sec. The values 400 or 280 mean that if

400. 2
650°" ° per cent., or 65
waste occurs in the are gap by evaporation, the calculated
pressure agrees with the observed.

S i.e. 43 per cent. of the carbon |

10. Hitherto we have made no hypotheses on the electrical
nature of the particles issuing from the crater. It may be
pointed out that the repulsion P, owing to the intensive
evaporation due to heat, may be also produced from neutral
carbon particles carrying no current, a trivial phenomenon.

If we assume, on the contrary, that the particles possess
an electrical charge, and that the arc mechanism may be in
the first instance reduced to a stream of positive ions of
mean charge e moving from anode to cathode and carrying
a current portion, al, and to an inverse stream of negative
ious (or electrons) carrying the rest (1—a) .I, a system of
equations may be written *, from which it follows

€ al
- = U . . ° ° e ° . . g
vies (9)
* S., p. 80. To correct 6,W,=iN,4,v,2=0,2,.TE. According to
Thomson (‘ Conduction’. ... p. 426 (1903)) the ratios a, (1—a) should
be proportional to the mobilities of the positive and negative carriers.

776 Repulsive Effect upon Poles of Electric Arc.
As

O=axl,
we have
é vl . ; !
Se Bige (9)

and putting v= 400cm./sec., I= 8amp.=0°8 E.M.U.,
P= 1:43 dyne (Table II.), the ratio e/w for the positive

particles issuing from the anode results :
= 225 B.M.U.

Ue gly a ell? of the cheeened is Ge dp noe, : < 500.

The value e/w being 9580 for hydrogen atoms, becomes
9580

oe 740 for carbon atoms if carrying one elementary

charge. Then the range 255 (or 500) does not conflict with
the values theoretically admissible.

i may be observed that the meaning of values e/p less
than the theoretical is that carbon particles on starting from
the crater are not fully disintegrated and ionized.

Holding a different point of view, Prof. Tyndall has also
reached the conclusion (loc. cit.) that the observed pressure P
can be best accounted for as a recoil) by departure and by
impact of carbon ions. The electrons, probably, contribute
very little to the mechanical pressure P.

SUMMARY.

The experiments of Dufheld, Burnham, and Davis with
carbon-carbon are are generally in a good agreement with
‘mine. The main results are:

1. In the electric arc there is a repulsive effect upon the
poles, increasing with the current. The range of P is less
than 10 dyne with currents up to 20 amp. P does not vary
much with the arc-length L, except when L=~0, when P
becomes evidently greater.

2. The carbon quality hasa great influence on P. It seems
that metallic salts cause an increase in the pressure on the
cathode and diminish that on the anode (S.).

3. With uncored carbons the repulsion on the cathode
appears smaller than that on the anode. On the contrary,
the specific pressure p per unit crater surface is greater on

Path of an Electron in Neighbourhood of an Atom, 777

the cathode (S.). The range of p, and p_ is about
10 dyne/em.?.

4. It is yet not certain whether the law connecting P
with I is a linear one, and whether the pressure does arise
with every current strength (D.), or requires a current
minimum (S.). Probably with increasing I, the pressure
P does not increase by a uniform law, for the graphs show
some “' Knees”, which suggest different are stages (S.).

5. The value P calculated, in testing the recoil hypothesis,
by taking for the velocity of the carbon particles their atomic
speed, is too great (D.). Whilst, by taking the propulsion
velocity of carbon atoms starting from the positive crater,
the calculated repulsion is in a far better agreement with the
experimental results (S.).

6. The propulsive velocity of carbon particles has been
estimated as 280-400 em./sec. (8.).

In conclusion, I think we cannot yet say with full knowledge
whether the observed effect is intimately associated with the
electrical processes of the arc, or whether it simply accom-
panies in an ordinary way the evaporation of electrodes at
high temperature. Only after having established the former
view by further investigations, available information on the
nature of the are will be given by P measurements, im-
proving the theoretical construction whose foundations have
been established by Thomson, and Stark.

Istituto fisico della R. Universita,
Palermo, 22 March, 1922.

LXVIL. The Path of an Electron in the Neighbourhood of an
Atem. By Bevan B. Baxer, A, BSc, FLRSL.,
Lecturer in Mathematics in the University of Edinburgh *.

i: XPERIMENT has shown that when an electron

collides with an atom, thereby causing it to emit
radiation, the frequency v of the radiation is related to the
amount U of the kinetic energy of the electron absorbed by
the atom by the equation

Why.
where / denotes Planck’s constant of Action. Professor
Whittaker f has recently shown that, in order that all

* Communicated by the Author.
+ E. T. Whittaker, “On the Quantum Mechanism in the Atom,” Proc,
Roy. Soc. Edin. xli. pp. 180-142 (1922).

778 Mr. B. B. Baker on the Path o7 an

exchanges between the kinetic energy of the electron and
the radiant energy should conform to this relation, it is
necessary that the atom should contain a mechanism which
is such that an electron approaching an atom will induce in
the atom what may be called a ‘magnetic current” ; the
model which he has suggested to typify such a structure
consists of a number of elementary bar magnets lying in
a plane and rigidly connected like the spokes of a wheel,
so that they rotate together in the plane, each magnet
having one pole at the centre of the wheel and having its
direction at every instant radial from the centre ; it is, in
fact, such a structure as Sir Alfred Ewing has proposed to
explain induced magnetism *.

If we suppose such a magnetic wheel to be placed with its
centre at the origin and its plane in the plane of yz, and -
suppose an electron to be projected towards it along the axis.
of w, then the electron, by its motion, creates a magnetic field
which will cause the magnetic wheel to rotate, and the rota-
tion of the magnetic wheel will set up an electric field which
will retard the motion of the electron. Denoting the radius
of the magnetic wheel by a, the magnetic moment of one of
the elementary bar magnets by wa, the sum of the values of
pw for all the elementary bar magnets composing the wheel
by M, the moment of inertia of the wheel about its axis by A,
the charge on the electron and the mass of the electron by e
and m respectively, then Whittaker has shown that if ae

2¢

velocity of projection wt) of the electron is less than ——=
m

the collision between the electron and the wheel is in the
nature of an elastic impact, 7. ¢., the electron is stopped at a
certain point and forced to return along its path, the mag-
netic structure giving back to the electron the energy it had

: , 2eM
previously received from it; but that if a> a

m
electron 1s able to pass completely through and away from
the magnetic structure so as to be free from its influence,
and the magnetic structure is left in rotation. In this latter
case, the amount of energy U lost by the electron and
gained by the wheel, is given by the equation

2e?M?
U= uae |
and the absurbed energy appears in the atom as a magnetic

* Cf. Hwing, “On Models of Ferromagnetic Induction,” Proc. Roy.
Soc. Edin. xlu. pp. 97-128 (4922).

Electronin the Neighbourhood of an Atom. TT9

; 2eM
current, specified by the angular velocity O= a , so that

the absorbed energy and the angular velocity are connected
by the equation

ry U=eMO.

Whittaker has further shown that the disturbance in the
atom after the collision consists in the displacement of a
single electron, and that the radiation emitted by the elec-
tron in its oscillatory subsidence to its normal state must
satisfy the equation U=hv.

2. In his paper Whittaker has assumed that the electron
is projected towards the magnetic wheel * in a line per-
pendicular to its plane and directly towards its centre. If
we suppose the atoms to contain such structures as have
been described, we must suppose the magnetic wheels in the
substance to be bombarded to have ail possible orientations,
and the electrons to be projected from any direction. It is
therefore of interest and importance to discuss the general
ease, when the electron is projected in any direction and
passes in the neighbourhood of one of these magnetic
wheels.

We will suppose, as before, that the magnetic wheel has
a radius a, and that if wa is the magnetic moment of one of
the elementary magnets, the sum of the quantities w for
all the magnets forming the wheelis M. Further suppose
hat the plane of the wheel is the plane of yz and that the
wheel is free to rotate about its axis, which is the axis of a;
the wheel is therefore restricted to have only one degree of
freedom. Let the amount of rotation of the wheel at any
particular instant be specified by the angle y between the
axis of y and a definite fixed radius in the plane of the wheel,
the angle being considered positive when it is such as would
turn the axis of y towards the axis of z, the rectangular axes
of wyz torming a right-handed system. Let the moment of
inertia of the wheel about its axis be A, so that when the
wheel is rotating with angular velocity w its kinetic energy
is $Ay”. Let the mass of the electron be m, its charge e,
and let its position at any instant be specified by spherical
polar coordinates (7, 0, 6) connected with the rectangular

* Note.—Whioen referring here or elsewhere to a magnetic wheel, it is
to be understood that it is not suggested that an atom actually contains
a mechanism similar to that here described, but merely that the atom
behaves as if it contained such a structure.

780 Mr. B. B. Baker on the Path of an

coordinates (x, y, 2) by the relations
x=reos6, y=rsin@cosd, z=rsinOsin db;
the kinetic energy of the moving electron is therefore
lin(a? + 1°62 + 7? sin? 6°). te
We have further to determine the potential energy of the

system due to the mutual interaction between the electron
and the magnetic wheel. To do this the magnetic wheel,

when it is rotating with angular velocity ab, may be looked

May
upon as a magnetic current of strength —5 flowing i in a

circle of radius a. Now just as an teenie eee flowing
round a circuit may be replaced by an equivalent magnetic
shell bounded by the circuit, whose magnetic moment per
unit area is proportional to the current-strength, so we may
replace the magnetic current by an electric shell, bounded
by the circuit, such an electric shell being equivalent to a
charged condenser in electrostatics. For convenience we
shall. suppose the electric shell to have the form of a hemi- —
sphere of radius a bounded by the circumference of the
magnetic wheel, the charge per unit area on either plate of

the condenser being — The electric potential at any

point P due to the condenser is therefore Be .@, where w

is the solid angle subtended by the magnetic wheel at the
point P. The potential energy V of the system is thus
Mee

; it is independent of the coordinate ¢, and may be

eae in powers of 7 in the form:
when r<a,

oe
V=Mye | 1—"P,(cos @)+ 5 “;Px(cos #)—..

bel? og oy a 2n+1
Gry ok a (7) Poner(cosd)+...] 3

(1)

when >a,

ays?
VeMype [55 P, (cos) — 5-5 & 7 P3(cos 8) +..
2 1\ de oe In— oe
aa eee ee ee 1 (cos é)+...|.

Electron in the Neighbourhood of an Atom. 781

We shall write V=MweF (r, 0), where F(y, @) is a function

of x and @ alone, and F(*, (= =, where @ is the solid
angle subtended by the magnetic wheel at the point (7, 6)

The Lagrangian function L=T—V, where T and V are

the kinetic and potential energies of the system respectively,
is given by the equation

L=4AwW? + 4in(r? + 7°? +7? sin? O62) — Mew F(r, 0). (3)

The equations of motion of the system are therefore
. _ue(2F )
Aah —Me(S. ; +, CONES te tac?) oe Ne (4)
m(7*—r6? —r sin? Of?) + Moyo" <r a ee ee
m(7rO + 2r7rd — r? sin 8 cos 06”) + MAE) == 0) nee ao)

m(r? sin? 06 + 2r sin’Org + 27? sin 0cos0.66)=0. (7)
3. If we multiply equations (4), (5), (6), and (7) by bh, ;

8, and @ respectively, add, and integrate we obtain
Awe? + dm(7? + 6? + 7? sin” O¢?) = constant,

which is the equation of conservation of energy of the
system ; if we suppose that initially the wheel is at rest and
that thie. electron is projected from an infinite distance with
velocity wo, the equation takes the form

LAW? +lmr=Lmu?, . 2. . (8)

where v and are respectively the velocity of the electron
and the angular velocity of the wheel at any moment.

Moreover the coordinate @ is ignorable, the equation of
motion corresponding to this coordinate being (7), which on
integration gives

mrsin?@.g@=constant; . . .. (9)

this integral may be interpreted as the integral of angular
cae atid of the electron about the axis of the lige
Equation (9) shows that if ¢ is initially zero, it will remain
so always, 7. ¢., @ will have a constant value throughout the
motion, as we should expect from the symmetry of the
system. If, however initially, when r is infinite, @ is not
zero, and Hevetus necessarily sin@ is not zero, we see that

782 Mr. B. B. Baker on the Path of an

the constant on the right-hand side of equation (9) is infinite,

and since, from equation (8), @ cannot become infinite, the
value of r must remain infinite, or, in other words, the elec-
tron can never approach the neighbourhood of the magnetic
wheel. To investigate the cases of interest we must there-
fore suppose & to have a constant value which we can take
to be zero without loss of generality, so that the motion of
the electron is always in the plane of wy.
The equations of motion of the ae then reduce to

bape Me(Se r+ Se 7 6\= 0, ane
ee eee
Metre ae -OF

BE ee eee LZ

4, Hquation (10) may be integrated immediately, giving
Awr—MeF(r, 6)=constant, . . . (13)

Remembering that the wheel is initially at rest and replacing
F(r, @) by its value in terms of the solid angle subtended at

the electron by the magnetic wheel, the value of ar, when the
electron has reached any point P, is given by the equation

Aja 5. Ao, 9 a 2 eles

where Aw denotes the increase in the solid angle subtended
by the magnetic wheel at the electron, in its motion to P.

From equations (8) and (14) we see that when the elec-
tron moving with its initial velocity wu from a point at
infinity, comes into the neighbourhood of the magnetic
wheel, its velocity begins to diminish, whereas the wheel is
set into rotation ; the kinetic energy of the electron, in fact,
is being expended in setting the wheel into rotation.

It may happen that the velocity and direction of pro-
jection of the electron are such that in its path it does not
pass through the magnetic structure, 2. e., at no point of its

path do we have 0= = or v= and » <a simultaneously ;

then in that case, when the electron has completed its path
and passed again to an infinite distance, the total increment
in the solid angle i is zero and therefore, from equation (14),
the magnetic wheel will come finally to rest: from equation
(8) we see that the electron, in its later path, receives back

Electron in the Neighbourhood of an Atom. 783

from the magnetic wheel the kinetic energy it had previously
given up to it.

If, however, the direction and velocity of projection of the
electron are such as to allow it to pass through the magnetic
wheel, and, moreover, its energy is sufficient to allow it to
pass away to infinity without returning through the mag-
netic structure, then the total increment in the solid angle
will be 47, and the magnetic wheel will be left in rotation
with an angular velocity ©, given by the equation

If wu denotes the final velocity of the electron when it has
passed again out of the influence of the magnetic wheel, we
obtain from equation (8)

ee we. ac)

In this case the electron has given up to the magnetic
structure an amount of kinetic energy U given by

U=2A0",
or, using the value of 0 given by equation (15),
2M?e?
l= ac iT)

From equations (15) and (16) we see that, in order that this
should be possible, the initial velocity of projection of the

é
electron must be at least as great as ——, and, moreover,

/ Am

the direction of projection must be suitably adjusted.

5. The remaining possibility is that the electron should
penetrate the magnetic structure but should not have suffi-
cient energy to pass out of its influence. In this case the

greatest value of w that can be attained by the wheel is
given by

ay? = Avo
and therefore, from equation (14) the greatest value of Aw
is given by the equation
AE Eo ES et ae ean 63

Me

After attaining this value, the electron will return towards
the magnetic wheel and must pass through it again in the

784 Mr. B. B. Baker on the Path of an

opposite direction, passing away again to infinity on the
same side of the wheel from which it came originally, the
revious motion being exactly reversed. From equations
(8) and (14) it follows that, in the return path, the electron
receives back from the magnetic structure the kinetic energy
it had previously lent to it, and the magnetic wheel will
return finally to rest.

The value of Aw given by equation (18) is obtained on the
assumption that the electron gives up to the magnetic struc-
ture all its kinetic energy; that this is so in general can
be seen from the following considerations. When w has
reached its maximum polee: wr=0, and therefore from

equation (10)
oF, oF
or ae

From the expansions for F(7, @) in powers of r given from

O=0. es.

equations (1) and (2) it is apparent that a and a cannot

be simultaneously zero except when 7 is infinite ; we have,
: e
In fact : :

when r<a, |
=H * cos 6+ ys Ocos — 3cos0)— .. |
cee * sin 0— (1S cos?d—3) sin 0+ .. a
08 e A@ (20
when r>a. | )
2 4
o=- “e030 -+ 5 9; (5cos*—3 08 8) — ..., |
OL ey 1a 3 : a é
ere Zsind+* 4 (15 cos @— 3) sin 0 “a

=0( only when cos0=0, 7. ¢.,

oF
Or

and therefore, when ry

when 0= = or 0= 2 and oe =0 only when sin 9=0, 7. e.,

2
when 0=0 or 0=7.
In general, therefore, equation (19) will only be eto.
when 7=0 and 6=0 simultaneously, 7. e., when all the
kinetic energy of the electron has been given up to the
wheel.

00

Electron in the Neighbourhood of an Atom. 785

There may, however, be certain exceptional values of », 0,
7, 8 which will satisfy equation (19), even though the electron
continues in motion ; that even in these circumstances the
electron will pass through the magnetic structure on its
return path may be demonstrated thus.

Denote by a, and a the accelerations of the electron in
the directions of + increasing and 6 increasing respectively ;
then equations (11) and (12) may be written

Me ; oF

4,= ee je ° e ° . (21)
Me - oF

w= — ty 8. ° ° . . . (22)

Suppose, for definiteness, that the electron is projected from
a part of the plane of wy for which both wand y are positive ;

then w will be always positive and from equations (20),
(21), and (22) we obtain the results:
whenr>aand O0<6@< = aa 0 amd: ap. 0;
when r > a and ager a Eee Oranoseg > .U:
when r>aand w<O0< a,<Oand ag<0;

ta 2 Paes a, > O-and a’ < Os:

when r >a and

wheny<aand 0<6@< | ao. > O and ap< 0);

ite
whens Gand - ore ms p< 0 ardso, < 0;

whenr<aand 7w<O0<- a,< Oand a>0;

OT ;
when r<aand > <@0<273; a> QO and # > 0.
a

The radial acceleration is therefore directed away from the
centre of the magnetic wheel when « is positive, and towards
the centre when w is negative, and thus always tends to
retard the motion of the electron on its outward journey.
Moreover, when z is positive and r>a the curvature of the
path is towards the axis of y ; when « is positive and r<a
the curvature is towards the axis of #; when w is negative
and r<a the curvature is towards the axis of y; and when

Phil. Mag. 8. 6. Vol. 44. No. 262. Oct.1922. 3E

786 Path of an Electron in Newghbourhood of an Atom.

w is negative and r>a the curvature of the path is towards
the axis of a.

Now suppose that the electron has penetrated the mag-
netic structure and reached a point. for which 7>a (if it
only reaches a point for which r<a it will obviously return
through the magnetic structure); then in order that equa-
tion (19) should be satisfied we must have |

ple. CONE VO vail),
G0 Mer OU) Or. Metin.
therefore, when 5 <te 7, v and r@ must have the same

ome ;
sign and when 7<0< = r and 76 must have opposite

signs ; thus on the outward path, in both cases, the electron
must at such a point be moving towards the axis of #, and
so long as r>a its motion. will be such as to bring it to an
even more favourable position for passing through the
magnetic structure on the return journey.

6. It has therefore been shown that the electron can
permanently transfer an amount of energy to the magnetic
wheel only if its velocity and direction of projection are —
such that it penetrates the magnetic structure and passes
away out of the influence of the wheel without returning on
its path. In such a case the wheel is finally left in rotation

; 2M
with an angular velocity Q= = and the amount of

kinetic energy U transferred from the electron to the struc-
Ze ;
ture is given by Ue in order to attain this result

the initial velocity of the electron must be at least as great
2Me
/ Am
These results are precisely the same as those obtained by
Whittaker in the particular case which he considered, and
his further discussion of the way in which this absorbed
energy is converted by the atom into radiant energy and
the deduction of Planck’s relation connecting the energy
and the frequency of the emitted radiation may equally be
applied in this more general case,

as

r 737 |

LXVIII. On the Theory of Freezing Miatures. By ALFRED
W. Porter, D.Sc, F.AS., Funst.P., and REGINALD E.
Gipes, 6:Sc., A.lnst-P*

7s 1874, Professor Guthrie carried out extensive experi-

ments on freezing mixtures, and was the first to point
out the fallacy of a belief which has persisted till even the
present day. Guthrie said:—“In regard to freezing
mixtures, I confess to have been here very much misled by
the confident but rather erroneous statements of others, to
which I attached faith trebly blind,—blind, because no re-
corded experiments really support them, blinder still because
a little thought in the right direction must have shown their
fallacy, and blindest of all because the one experiment of my
own in this direction shows that the minimum temperature
of an ice-salt eryogen is reached, whether we take the ratios
three of salt to one of ice, or one of salt to two of ice, and so
points to the wideness of the margin of ratios which may
obtain between the weights of ice and the salt” f.

He showed that the same temperature, viz. the cryohydriec,
was reached for a wide range of proportions of the constitu-
ents, and that the initial temperature of the salt need not be
zero; in fact, in his extreme case the salt was initially at a
red heat.

The present object is to consider an equation representing
the heat changes which occur in a freezing mixture, and to
illustrate how well it bears out the truth of Guthrie’s
remarks.

For the sake of simplicity it is best to concentrate one’s
attention on a definite mixture, say that of salt and ice, and
to suppose that initially the constituents are all at 0° C. and
present in the following amounts: ice I gm., water—>zero,
salt S gm. The vanishingly small quantity of water is
introduced only to ensure that there will not be any discon-
tinuity under the conditions which accompany the reaction.
All possible cases can now be divided into three sections :---

(1) That in which the masses of ice and salt are such
that there are both free salt and free ice present
at the end of the change of temperature.

(2) That in which there is no salt remaining.
(3) That in which there is no ice remaining.

* Communicated by the Authors.
+ Proc. Phys. Soc. of London, vol. i. Jan. 18, 1875,

788 Prof. A. W. Porter and Mr. R. E. Gibbs on the
Case (1).

At the end of the change of temperature, let the mass of
solution be (M+m) om., hers

ie of water,
m= mass of salt in solution.

Since the solution is in equilibrium with the salt, it will be
saturated, and as it is in equilibrium with ice, it will be atthe
freezing-point ; hence the final temperature must be the
eryohydric. | | |

In determining the connexion between M and m at the
end of the change, the external work done can be neglected

owing to the very small change of volume at the moderate
_ pressure of one atmosphere obtaining during the experiment.
_ In these circumstances the heat change in a cycle can be taken

as zero with sufficient approximation ; ; or, in other words, the
particular path of transformation is immaterial so far as heat
changes are concerned. Representing the cryohydric
temperature by —r° C., the heat equation will be

7(1s;+Sss) =ML,+mL,,

where s; and s, are the specific heats of the ice and salt
respectively, and where

L, is the latent heat of fusion of ice at —r° C., and
L, is the latent heat of solution of salt at —7° C.

In writing this equation, all the salt and ice has been assumed
to cool down initially to - 7° C. and the transformation to
take place then at this low temperature.

Nothing would be gained by aiming at meticulous accuracy
in regard to numerical values. The general trend of results
can be illustrated by using constant and approximate values
for which the calculations can be made easily. As the
solubility of salt varies very little with temperature, one can
assume m/M to have a constant value 4. Assuming also the
following approximate values,

pe LI Oe ee

Li 00) L.=6 fat “35° C.) ae
boas
one obtains Maan 5 ae }
Olw

* In calculating Ly at —r the formula = 1—s has been em-

or
poset It would even be erroneous to employ the more usual equation
a - 7 =l- 8, because this gives strictly the latent heat under
equilibrium conditions, 2. ¢. under a pressure corresponding to a melting-

point of —7, whereas the pressure is approximately atmospheric
throughout.

Theory of Freezing Miatures. 789

| Q
Hence, if S is small, a ean be substituted for =
whence ma oe ks :

and therefore Me eee 5

This result is applicable, provided S¢ —

On the other hand, if I is small compared with 8, one obtains
epee Paths)
mt: and M= = ae

These results indicate that the cryohydric temperature will
be attained, provided
(i.) S€ay 1 +8)
and. an) fl +5).
This is in agreement with ordinary experience, though the

fact that they do not both totally disappear is usually
attributed to the substances not having been taken in the

Fig. 1.

Oy 40)" 26. SOaee BO \Go ap. = 80. 9p... 100

Percenk Salt
&—O Theoretical, P26 oe rich

pd bake Sate Exper/me nfal,

proper proportions. The above work, however, shows that
no such “proper” proportion exists; and, in fact, the
experimental results are of the kind one would expect
according to theory.

In fig. 1 this range is represented by the central horizontal
portion of the curve.

790 Prof. A. W. Porter and Mr. R. B. Gibbs on the

Case (2). No salt remaining.

As shown in the above equations, this means that in the
original mixture, S is less than 34 of the whole. ‘The final
temperature must be the freezing-point of the solution, as ice
is in equilibrium with it, but it will not be the oryohy rdric
temperature, as the solution is unsaturated (except in the
limiting case).

The * general equation can be adapted to this case by
writing m=S ;

7(Is;+S8s,) = ML, +81.

The law connecting the fr eezing- -point with the concentration,

63S

M+S_
The application of this law gives

63S
Was (Isi+Ss,)=ML, + SI.
This has first to be solved for M, and then 7 calculated from

the previous equation. The values of 7 for various values of
IT and S are shown in the following table :—

at least for dilute solutions, is T=

TABLE I.
8. I. M. Te
1 gm. 99 gm. 6°12 gm. 875° C.
us wey Sy, 12 ae.
3, Ohiaiss 10:0); EG ony
4. ome oe 16°5

In fig. 1 this range is represented by the left-hand sloping
portion. of the curve.

Case (3). No ice remaining.

This necessitates that the mass of ice taken is less than
#5 of the whole mixture. The final solution will be
saturated because it is in equilibrium with the excess salt.
On the other hand, it will not be at the freezing-point
(except in the limiting case), as it is not in contact with
ice. Again, the general equation can be adapted this time
by writing M=I. Thus

t(Isi+ Ss,) =ILy+mLs.

The final concentration is of course m/I, and will be approxi-
mately one-third. Hence

Ss
Ae + 5) =1(70 12 Ble
or approximately, as [ is small,

_ 3601
i [2s :

Theory of Freezing Mixtures. 791

In fig. | this range is represented by the right-hand sloping
portion of the curve, which is practically a straight line.

Series of experiments were carried out to test the
validity of the above work. Not very much importance
was attached to the absolute value of the eryohydric tem-
perature reached all along the central portion of the curve,
as this depended very largely on the purity of the materials
used. One set of results is recorded below :—

Weight of Salt. Weight of Ice. ae oe
=m. em. Salt Tas: Ts
‘80 27°6 2°82 8:0
98 32°2 2°95 12°5
1:20 344 3°37 15:0
LOr7 511 17°3 19°8
5:0 23°0 17'8 20°0
12°5 54:2 18°7 19:8
13-7 50°7 De Poa 19:8
15°6 44:3 26°0 20°8
20-0 45:0 30°8 20°5
21:2 47°0 31:2 20°5
28-0 59°1 32°2 198
36°0 363 49°8 ys
. 3882 17:2 68°9 19°9
45°8 4:1 91°8 12-0
46:1 3°1 93°8 70
377 2°3 94-0 30

The above results were obtained by mixing the ice and
salt in a small vacuum flask, great care being taken to
make the mixing as complete as possible : this was fairly easy
until the percentage of salt was high, say over 75 per cent. ;
but for high percentages it was probably imperfect at best.
The results agree with those expected from theory except
for high concentrations of salt ; and, even for such, they
are sufficiently close to substantiate the previous work.
Poor mixing and the thermal capacity of the vessel would
cause such a deviation.

In the practical case of the use of refrigerating mixtures,
the body to be cooled is always to be taken into account, and
it may produce considerable modification. If its thermal
capacity is 0, the equation would now read

3 ° Hi 2 +6 | =ML,,+mL;.
Ai esd J

792 On the Theory of Freezing Mixtures.
It is most convenient to write

fiat ye

A few curves (tig. 2) have been drawn for different values of

O 10 20 30 40 50 60 7FO 80 Go /00,
Ferce nfage of Salt,

K to show the effect produced. Possibly the most interesting
case is to find the value of K for which the cryohydric is
just reached. Assuming thatall the salt and ice is used and

that m/M=34, we have

21°6 E ab : +0 | =—I16m
or G=8:35 =2'1 (13s),
and hence iG alle:

therefore to cool a body whose thermal capacity is @ down to
—21:6° C., the quantities of materials to be used are 0/5°4 gm.
of salt and 6/2'8 gm. of ice. In practice it is always
necessary to take somewhat larger quantities to allow for the
formation of dew on the exposed surfaces. This is a fairly
serious factor, as 1 gm. of dew is equivalent in its heat
change to roughly 8 gm. of ice. It will be seen, therefore,
that if K<2:1, there exists a central horizontal portion of
the curve, whilst if K > 2-1, the two sloping lines intersect at
a vertex lying on the 25 per cent. ordinate,

THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

~
}- ™S

\
uxt SERIES. ]

fp

rT

t nT Oe o VEMBER 1922.

LXIX. The Emission of Electrons by X-Rays. By
G. Saearer, VM.A., 1851 Exhibition Scholar, Emmanuel
College, Cambridge *

HE general object of this investigation was to obtain

more definite information as to the emission of

electrons from matter under the influence of X-ray bom-
bardinent.

(1) Aistorical.—Very shortly after the discovery of
X-rays it was shown that all forms of matter emitted
electrons when bombarded by X-rays. Later work revealed
the fact that, in this emission, the electrons had all velocities
up to a certain maximum ; this maximum yelocity can be
determined by the quantum equations—

tmv? = Ve = hy,

' where e, m, v represent the charge, mass, and maximum
velocity of the electrons, v the frequency of the X-rays,
V the applied potential, and h Planck’s constant.
In addition to this general electron emission, there is a
special emission associated with the characteristic X-radiation
of the matter bombarded. Until recently there has been
some doubt as to the exact nature of this special emission.
Experiments by Barkla and the author f failed to reveal any
special distribution of velocity associated with these electrons.
* Communicated by Prof. Sir E Rutherford, F.R.S.
+ Barkla and Shearer, Phil. Mag. xxx. p. 746 (1915).
Phil. Mag. Ser. 6. Vol. 44. No. 263. Nov. 1922. 3 EF

794 Mr. G. Shearer on the

On the other hand, Robinson and Rawlinson * by the mag-
netic spectrum method showed that there were present
groups of electrons with special velocities. Kang Fuh Hu +
also obtained some evidence of the existence of these groups.
Simons { carried out experiments on the same lines as those
of Barkla and Shearer, and concluded that sub-groups of
electrons were present when the characteristic X-radiations
of the bombarded matter were excited. Recently the question
has been definitely settled by the experiments of De Broglie 9.
By the use of the magnetic spectrum method he has shown
that the energies of these groups correspond to h (v—p,),
h (v—vy,), &c., where v, vg, vy, &e., represent the frequencies
of the incident radiation and of the K, L, &c. radiations of
the matter bombarded. Similar results have recently been
obtained by Whiddington |.

While these experiments show that the electronic radiation
consists of a general emission corresponding to the “ white ”
radiation from the tube and groups of electrons whose
energies obey simple quantum relations, they tell us nothing
of the magnitude of the emission, nor of how this magnitude
depends on the type of matter from which the electrons are
ejected. :

Laub {J showed that the efficiency of an element asa source
of electrons increased with its atomic weight, but made no
attempt to obtain a law governing the variation. Moore**,
as a result of some experiments on the relative ionizations
produced in various gases, deduced the law that the number
of electrons emitted per atom is proportional to the fourth
power of the atomic number. These experiments were of a
somewhat indirect nature, and the range of atomic number
investigated was small.

The experiments described here were undertaken,
primarily, to throw light, if possible, on this question.

(2) EHeperimental Arrangements.—In order to avoid the
difficulties of interpretation introduced by the use of ioniza-
tion methods, it was decided to measure directly the number
of electrons emitted by observing the rate at which an
insulated radiator acquired a positive charge under the
influence of X-ray bombardment.

* Robinson and Rawlinson, Phil. Mag. xxviii. p. 277 (1914).
+ Kang Fuh Hu, Phys. Rev. xi. p. 505 (1918).
t Simons, Phil. Mag. xli. p. 120 (1921).
§ De Broglie, Journ. de Phys. (6) 11. p. 265 (1921).
|| Whiddington, Phil. Mag. June 1922.
q Laub, Ann. der Phys. (4) xxviii. p. 782 (1908).
** Moore, Proc, Roy. Soc. A. xix. p. 887 (1915).

_
j

oe ean sll

mission of Electrons by X-Rays. 795

Fig. 1 shows the final form of the apparatus used. Rays
from an X-ray tube enclosed in a thick lead box passed
through a small aperture in the box and entered the
cylindrical brass examination vessel A through a hemi-
spherical glass window W. Inside the vessel, in the path
of the rays, was suspended an insulated brass cube C, which
served to carry the materials to be examined. In order that
more than one substance might be investigated without dis-
mantling the apparatus, the cube was made capable of
rotation about a vertical axis by means of a ground-glass
joint G, which carried the whole of the insulated system.

By turning this joint through 90° at a time different
materials could be exposed to the action of the rays. The
beam of X-rays was such that the cross-section was never as
large as the area of the material under examination, so that
only the window and the matter on the face of the cube were
exposed to direct X-ray bombardment. An electromagnet M
was arranged near the window so that any electrons from
the window might be bent back into the walls of the vessel
and thus prevented from reaching the cube. ‘The inside of
the vessel was lined with filter paper to reduce the eftect of
the scattered and characteristic radiations from the cube.
Inside the vessel and insulated from it was placed a wide-
meshed wire cylinder which could be charged to any desired

potential.

ae 2

796 Mr. G. Shearer on the

The vessel was connected by wide-bore glass tubing toa
Gaede mercury pump. A charcoal-filled tube was attached
and cooled with liquid air. By these means the vessel was
kept at a very low pressure so as to make all ionization effects
negligible. As an additional precaution, hydrogen was used
as residual gas in some of the experiments.
he substances to be examined were mounted on the faces

of the cube, and were all of sufficient thickness to give the
maximum electron emission.

The source of X-rays was a Coolidge tube Hi a tungsten
anticathode. This was actuated by a Butt induction coil
and mercury interrupter. Owing to the smallness of the
effect to be measured, it was not possible to use a mono-
chromatic source of X-rays.

‘The rod carrying the cube was connected through earthed
shielding tubes to a string electrometer. The rate at which
this acquired a positive charge was taken as a measure of
the electron emission, and was determined for various sub->
stances. In order to correct for small variations in the
intensity of the rays during a set of observations, a standard-
izing ionization chamber was fitted. This was connected to
a Dolezalek electrometer of low sensitivity, a steady deflexion
method being used.

(3) General Results —The early results with this apparatus
showed that the electron emission was of a more complicated
nature than had been anticipated.

If the cube was allowed to charge up to a considerable
potential, it was found that the rate of charging up fell off
rapidly at first and only became steady after a potential of
from ten to twenty volts had been reached.. Such an effect
might have been due to ionization effects, but a simple
calculation showed that the magnitude of the effect was very
much larger than that due to the ionization of a gas ata
pressure of ‘01 mm., and the pressure in the vessel was
certainly less than ‘00L mm. It appeared, therefore, that
this effect was due to the presence of a large number of slow
electrons. When the cube attained a voltage of from ten to
twenty volts the electric field was of sufficient strength to
prevent the escape of such electrons.

The existence of these low-speed electrons was also
suggested by certain observations on the effect of the
magnetie field used to deflect the electrons from the window.
Tt was found that this field reduced the emission from the
cube although the stray field at the surface of the cube was
not more than a few Gauss.

Emission of Electrons by X-Rays. 797

In these experiments the surface of the cube was perpen-
dicular to the direction of propagation of the X-rays, and
therefore parallel to the direction of the electric vector in
the X-ray beam. It was possible, therefore, that there was
a large number of electrons whose initial direction was
nearly parallel to the surface of the cube. Such electrons,
even if their velocities were considerable, might be bent
back into the cube by electric or magnetic fields of relatively
small strength. This hypothesis was tested by comparing
the reducing effect of electric and magnetic fields.when the
angle of incidence was 90° and 45°. It was found that the
percentage reduction was the same in the two cases. Had
the effect been due to the bending back of electrons emitted
in directions approximately parallel to the face of the cube,
the reduction of the emission would have been larger in the
first case than in the second.

The conclusion reached was that, in addition to the high-
speed electron emission, there exists also an emission of
electrons of low speed. After considerations such as these
had led the writer to this conclusion, it was found that
similar effects had previously been observed by Campbell *
in his work on delta rays.

In what follows, first the properties of the high-speed
electrons and then those of the slow electrons will be
discussed.

A. [High-Speed Emission.

(4) Relative Hlectron Emission from Metals.—The effect
of the low-speed electrons was eliminated by charging the
inner wire cylinder to a voltage sufficient to prevent their
escape from the surface of the cube. The residual effect was
then that due to the high-speed electrons.

The metals investigated were Aluminium, Iron, Nickel,
Copper, Silver, Tin, Gold, Lead, and Bismuth. The X-ray
tube was operated under varying conditions, and_it was found
that the relative values for the various metals depended very
little on the conditions of the tube. This point. will be
discussed more fully later.

Table I. shows the results obtained, the value for the tin
being taken as 100.

TABLE I.

Rie ese sok, ode tin Petes Mine Oiurune, on, Au. Pb, Biz
Electron Emission ......... bie 40 58 2567)" 94° 100 184° 189 Tot
Atomic Number ............ ior be ae =e 4G D0" 79, «682 SS

Atomic Weight. “2222-002. i Oo oy ar 108, 119 197, 207) 208

* N. R. Campbell, Phil. Mag. xxiv. p. 783 (1912).

798 Mr. G. Shearer on the

It is clear that the efficiency of a metal as a source of
electrons increases with its atomic weight or number. Fig. 2
shows the results graphically. The relation between the
number of electrons escaping from a metal and its atomic
number is very nearly a linear one. The electron emission
may be expressed with considerable accuracy by an equation
of the form

n' = k(N—a),
where n/ is the number of electrons escaping from the metal,
N is the atomic number, and & and a are constants, the value
of a being approximately 10.

SO ae

50 y
|
ce Ge
@) 20 4.0) 60 50 -Qilida

Atomic Numeer.

PLeEcTRON EMISSION

Since N denotes the number of electrons in the atom, it
might appear that this result implies that the number of
electrons emitted from an atom is proportional to the
number of electrons in the atom with a small correction due
to the presence of the term a in the equation. Such an
interpretation is, however, not permissible, as what has been
measured in these experiments is the number of electrons
which succeed in escaping, and not the number liberated
from the atoms under bombardment.

In the present state of knowledge of the laws governing
the passage of electrons through matter, it is not possible
accurately to deduce the number of electrons liberated from
an atom from the observed number actually escaping from
the surface. An approximation may, however, be obtained
by ee certain assumptions which are prebably near the
truth.

|
_

Emission of Electrons by X-Rays. 799

In the first place, let it be assumed that the number of
electrons escaping from the radiator falls off exponentially
with the depth from which they come. If Ip is the initial
intensity of the X-rays, 8 the area bombarded, n the number
of electrons liberated per unit volume per unit time per
unit intensity, and jy, and py the absorption coefficients of the
X-rays and electrons in the radiator, then the number of
electrons from a layer dv at a depth a which actually
escapes 1s

dn'=n.8.1,e “t" "de.

The total number escaping is therefore

n! = nSIo/(m + pe).
Hence, since «4, is small compared with ps,
pouNel;
fey

nr

If, on the other hand, it is assumed that the number
escaping falls off exponentially with the distance traversed
by the electrons in the radiator, the following expression is
obtained :

n' = nSI,/4py *

where zy is the exponential coefficient of absorption for the
electrons.

Whichever of these two absorption laws is taken, it follows
that the number of electrons liberated per unit volume
is proportional to mw. times the number actually escaping.
From this the number liberated per atom can easily be
derived.

If A is the atomic weight of the radiator, m the mass of
the hydrogen atum, and p the density, the number of atoms

* The actual expression for 7’ is

5 (uy +12) ene
' ‘ = x = x
of i}:
0

w hich, on evaluation, gives

n'=inSI, [= pe ats Be log (1+) J.

If this is expanded in terms of ee yand 2 is neglected, it gives the result
quoted above. pe” p28

800 Mr. G. Shearer on the

per unit volume is p/Am. Hence the number of electrons
liberated per atom per unit time per unit intensity is

nAm
>]
p
e e . i
or, substituting for n in terms of n’,

k. Amn’ ;
p

where & is a constant depending on the exact form of the
absorption law chosen.

If itis assumed that Lenard’s law that “ is constant hoids
p

under the conditions obtaining in these experiments, the
result of these calculations is that the number of the electrons
liberated per atom per unit time per unit intensity is propor-
tional to the product of the number of electrons actually
escaping and the atomic weight of the substance from which
they are liberated.

It has been shown that the experimental results led to the
conclusion that n’ was proportional to (N—10). Hence
the number of electrons liberated per atom is proportional to

| A.(N—10).

In deducing this result several assumptions have been
made, some of which are only rough approximations to the
truth. Probably the most serious one is that Lenard’s
law—that p/p is constant—can be applied to this case.
Hven under the conditions of Lenard’s experiments, the
result was only an approximate one. A strong argument in
favour of its application to these experiments is that it causes
the density to disappear from the final correction to be
applied to the observed electron emissions. The values for
the electron emissions from the elements gold, lead, and
bismuth were found to be approximately equal. On the
other hand, the densities of these elements are 19°32, 11°37,
and 9°80 respectively. If the correction to be applied to
deduce the electron emission per atom were a function of the
density, all regularity would disappear from the results. If
the correction depends only on the atomic weight or atomic
number, no such difficulty presents itself.

When this result is compared with the only other result so
far obtained, a serious disagreement presents itself. Moore *
found that the electron emission per atom was proportional

* Toe. cit.

Emission of Electrons by X-Rays. 801

to the fourth power of the atomic number. This result was
obtained indirectly from observations on the relative ioniza-
tions of different gases. The heaviest element used was
chlorine, so that the range of Moore’s experiments falls
almost entirely outside that of these experiments. Had it
not been for the fact that Moore’s law fits in well with the
absorption law of Bragy and Peirce, we might have con-
eluded that both his law and that deduced here are both
approximations to a more general law, the former being an
approximation holding for elements of low atomic weight,
while the latter is a better approximation for the heavier
elements. Bragg and Peirce * have shown that the X-ray
energy absorbed per atom is proportional to the fourth power
of the atomic number of the absorbing element. The com-
bination of this result with that of Moore suggests that the
number of electrons emitted is proportional to the X-ray
energy absorbed, a result pointed out by Moore. The range
of elements used in the experiments of Bragg and Peirce was
well within that of these experiments although outside that
of Moore’s. On the other hand, if the above results are true,
no such simple law appears to hold. Even when allowance
is made for the approximate nature of the various assumptions
made in deducing the final result, it is extremely difficult to
see any way in which these experiments could possibly be
reconciled with a fourth-power law.

In connexion with this result, it is of interest to refer to a
result obtained by Kaye+ on the relative efficiency of
various metals as anticathodes in an X-ray tube. Kaye found
that the X-ray output of a tube increased linearly with the
atomic weight of the metal used as anticathode in the tube.
Later experiments by Duane and Shimizu t showed that
the proportionality was to the atomic number rather than
the atomic weight. In these experiments we are dealing
with the transformation of electron energy into X-ray energy,
while the problem that is the subject of this paper is the
inverse one—that of the transformation of X-ray. energy
into electronic energy. It has been shown that there is a
linear relation connecting the number of electrons escaping
from a metal bombarded by X-rays and the atomic number
of the metal. These results imply that, when the number
of electrons entering a metal is kept constant—as in Kaye’s
experiments,—the ee “ay energy emitted is a linear function

* Bragg and Peirce, Phil. Mae. xxviii. p. 626 (1914).
t Kaye, Phil. Trans. A. 209, p. 128 (1908).
¢ Duane & Shimizu, Phys. Rev. xiv. p. 525 (1919).

802 Mr. G. Shearer on the

of the atomic number, while, if the X-ray energy falling on
a metal is kept constant, the number of electrons emerging
from the metal is again a linear function of the atomic
number of the metal.

(5) Haperiments with Substances other than Metals.—
Certain experiments were carried out on the electron emission
from salts. Only a few salts were tried, and the dataare not
sufficiently extensive to warrant the deduction of definite
conclusions. The chief salts investigated were As,O;, KI,
and RbI. These were chosen in the hope that they would
throw light on the electron emission from the elements
Arsenic, Rubidium, and Iodine, and thus serve to fil! up
some of the gaps in the electron emission-atomic number
curve obtained from the investigation of the metals. The
values found for these salts, with tin taken as 100, were 47,
128, and 158 respectively. The values for KI and RbI
were both much higher than is to be expected if the effect is
an additive one and the values of the electron emission were
such as would result from the interpolation from fig. 2.
The atomic numbers of K, Rb, and I are 19, 37, and 53, so
that, if the effects are additive, it is to be expected that the
values for these iodides would be less than 100, whereas it
was found that they were considerably in excess of this
value. These salts were placed on the faces of the cube in
the form of a layer of small crystals, and it is possible that
the effective area under bombardment was thus considerably
greater than in the case of a sheet of metal. In spite of
this possibility, it seems difficult to account for the high
values obtaived for these salts. It is interesting to note that
the elements potassium and rubidium are both very active
from a photoelectric and thermionic point of view.

A few experiments were made on other salts of potassium.
It was shown that the electron emission increased with the
molecular weight of the salt. Thus, both the carbonate and
the sulphate gave very much smaller value than the iodide,
while the eftect from the sulphate was greater than that
from the carbonate.

(6) Special Electron Emission.—It has long been known
that when a characteristic radiation of an element is excited
there is an increase in the electron emission.

Although, in these experiments, no attempt was made to
use monochromatic radiations, still it was expected that it
would be easy to detect this special electron emission. In
order to test for its effect, observations on two metals—
especially copper and tin—were made under widely varying

r-.- —_—,.):C

Emission of Electrons by X-Rays. 803

conditions of the X-ray tube. The voltage of the tube was

varied and the ratio of the electron emissions from the two
metals was measured. Results of such measurements are
given in Table II. The observations have been taken in
groups, each group corresponding to a certain range of
potential on the tube, the potential being measured by the
equivalent spark-gap between points.

Tasue II.

Range of Spark-Gap. Cu/Sn,
Q= 5 cm, 0°57
a= 9 0°55
9-12 _,, 0°55

12> 12), 0°56

It is clear from this table that, under the experimental
conditions, the ratio of the electron emission from these
elements is practically independent of the potential used to
excite the tube. It was expected that there would be a dis-
continuity in the ratio at the point where the K-radiations
of tin became prominent. ‘lhis should occur at a potential
of about 50,000 volts. No such discontinuit ty appeared.

The reason probably lies in the fact that in these experi-
ments the tube was fitted with a tungsten anticathode. The
applied potential was never sufficient to excite the K-radiations
of tungsten to any extent, while the L-radiations would be
very largely. absorbed in the walls of the tube and in the
window of the examination vessel. Under these conditions,
the radiation used was what is generally termed “ white.”
Its quality varies with the potential used to excite the tube,
but not to avery marked extent. Ulrey* has shown that,
under conditions which are very similar to those obtaining
in these experiments, there isa maximum X-radiation at a
wayve-Jength which obeys the approximate law—

i
V? = constant,
(max.)

where V is the applied potential. Thus quite a large
difference in the potential produces a relatively small change
in the position of the wave-length to which corresponds the
maximum X-radiation.

It would appear, therefore, that the electron emission
associated with the characteristic radiations of the elements
bombarded is not of sufficient intensity as materially to affect

* C. T. Ulrey, Phys. Rev. xi. p. 401 (1918).

804 Mr. G. Shearer on the

these experiments; and these results are to be taken as repre-
senting what happens when the characteristic radiations are
not excited to any extent.

In view of De Broglie’s results, it is to be expected that
many of the electrons emitted in connexion with the
characteristic radiations will have relatively small velocities,

and will, therefore, have difficulty in escaping. from the
radiator. Only a ‘small fraction’ of those liberated will
escape and contribute to the effect measured in these experi-
ments.

(7) Selective Emission in the Direction of Electric Vector.—
On the classical electromagnetic theory of light it seems
probable that there will be a large preponderance of electrons
emitted in the direction of the electric vector in the X-ray
beam. In fact, the photographs obtuined by C. T. R. Wilson
seem to show that suchis the case. In order to find evidence
for such an effect, two adjacent sides of the cube were covered
with the same metal, and measurements of the electron
emission were made when the rays fell perpendicularly and
at an angle of 45° onthe metal. In the second case the area of
metal under bombardment was 1°4 times the area in the first
case, and for this reason an increase of 40 per cent. in the
emission is to be expected. When the rays fall perpen-
dicularly, electrons emitted in the direction of the electric
vector should have difficulty in escaping from the metal, but
when the angle of incidence is 45° this difficulty should not
be so marked. |

The results of such measurements made with lead as
radiator showed an increase of 35 per cent. when the angle
of incidence was 45°. This increase is rather more than
accounted for by the increase of area of the radiator, and
these observations show no evidence of any selective emission
in the direction of the electric vector. It is probable that
by the time that the electrons emerge from the metal their
direction of motion is very different from what it was initially,
owing to encounters with the atoms of the metal ; and experi-
ments such as these could not be expected to throw light
on the initial direction of motion of the electrons. Wilson’s
photographs were taken with a gas as source of electrons, and
in this case the initial direction of the electron is directly
observed.

(8) Variation of Hlectron Emission with Applied Voltage.—
A few experiments were carried out to see how the number
of electrons emitted per unit intensity from any one radiator

Emission of Electrons by X-Rays. S05

varied with the voltage applied to the tube. Such experi-
ments are rendered difficult by the necessity of obtaining
some means of measuring the intensity of the X-ray beam.
Asa rough measure of this, the ionization produced in the
standardizing vessel was used. The electron emission per
unit intensity was measured for various potentials ; ‘Table ETT.
gives a typical set of such observations.

Tei ELE,

Electron Emission per

Voltage. Unit Intensity. nv?2.
28000 85 14200
36000 81 15400
45000 ral 15000
65000 61 15500
78000 58 14800

This Table shows that the electron emission per unit
intensity diminishes as the applied potential increases.
Column 3 shows that the product of the number of electrons
emitted per unit intensity and the square root of the voltage
is approximately constant. Owing to the method adopted
for the measurement of intensity, too much reliance must not
be attached to this result, but it gives some indication of the
nature of the variation.

This suggests an intimate connexion between this result
and that obtained by Ulrey, to which reference has already
been made. Asan approximation the X-:adiation used in
these experiments may be considered as a monochromatic

radiation of wave-length Ds aay Ulrey’s result, combined

with that just given, leads to the result that the number of
electrons emitted per unit intensity is proportional to the
wave-length, or inversely proportional to the frequency of
the exciting radiation. As the size of the quantum is pro-
portional to the frequency, this may be interpreted as
meaning that the number of electrons emitted per unit
intensity is directly proportional to the number of quanta
involved.

B. Low-Velocity Emission.

(9) In studying the properties of these electrons, the
total number of electrons escaping from the radiator was
measured under the influence of varying, accelerating, and
retarding electric fields. or this purpose, the wide-meshed
wire cylinder described above was charged to positive and

806 Mr. G. Shearer on the

negative potentials, the walls of the vessel being connected
to earth. :

Under the influence of a retarding field the electron
emission diminished rapidly at first, then more slowly, finally
reaching a constant value. This stage was reached when a
negative potential of about 20 volts had been applied. On
the other hand, an accelerating field produced an increase in
the emission, and the potential necessary to ensure constancy
of the emission was considerably greater than in the case of
a retarding field.

Fig. 3.

ELECTRON EMISSION. —

120 -§0 —40 +120

Cc
0 +40 + 8¢

The curve in fig. 3 is typical of the effects of accelerating
and retarding fields on the total electron emission. Such
curves were obtained for various metals and for some salts.
The resalts of these observations was that there did not
appear to be any change in the velocity distribution of these
slow electrons with a change in the nature of the matter
from which they came. Exact quantitative measurements
of the properties of these electrons were rendered difficult by
the large effect which the state of the surtace had on the
emission. specially in the case of the salts examined,
fatigue effects were observed which were presumably of
similar origin to those observed in the photo-electric effect.
No special precautions were taken to obtain very clean
surfaces. ‘The state of the surface has very little effect on
the emission of high-speed electrons, whose properties were
the main object of this investigation.

Experiments made on the ratio of the number of low-
speed electrons to the total emission showed that this was
also independent of the material bombarded. ‘Table IV. shows
some of the results obtained in these measurements. The
numbers have been reduced so that they all show the same
electron emission with no field acting.

Emission of Electrons by X-Rays. 807
TaBueE LY.
Voltage. Ou, Sn. Pb.
0 100 100 100
+400 156 140 139
— 400 81 79 80

The results of the following experiment throw some light
on the source of these slow electrons. The wire cylinder
was removed, and measurements were made on the effect of
retarding fields. It was found that, under these conditions,
the diminution was much more marked, and in some cases
the brass cube even acquired a negative charge with a
sufficiently large retarding field. This implied that more
electrons were “being driven from the walls into the cube
than were coming from the cube owing to the direct action
of the X-rays. As precautions had been taken to prevent
the X-rays from falling directly on the walls of the vessel,
the only sources of electrons from the walls were the high-
speed electrons and the scattered and characteristic radiations
from the cube. These X-radiations would be of small
intensity compared with the direct X-ray beam, and their
effect would be still further reduced by the paper lining.
On the other hand, the walls were subject to direct bombard-
ment by allthe fast electrons, and it seems certain that it was
this electronic bombardment which liberated the low-speed
electrons. As the energy of the slow electrons is so small, it
is reasonable to assume that one high-speed electron can
produce a large number of low-speed electrons; and this
would account for the observed fact that the number of
electrons driven back by the action of the retarding field
was greater than the number of electrons liberated by the
X-ray beam. With the cylinder in position, the field
between the walls and the cylinder prevented the electrons
from the walls from reaching the central insulated system,
while, owing to the wide mesh of the cylinder, the number
liberated from it would be small.

The properties of these slow electrons are very similar to
hose of the delta rays produced by bombardment by alpha
rays.

These results indicate that the low-speed emission is of a
secondary nature, and owes its origin not to the direct action
of the X-rays, but to the high-speed electrons ejected by the
X-rays. The process appears to be exactly analogous to the
phenomenon of ionization ina gas. ‘lhe photographs obtained
by C. T. R. Wilson show that each high-speed electron is

808 Eimussion of Electrons by X-Rays.

capable of ejecting a large number of slow electrons from
the atoms with which it comes into collision. Exactly the
same process should occur during the passage of electrons
through a metal. In this case the number which will
succeed in escaping will be relatively small owing to their
low speed and consequent rapid absorption in the metal.
The fact that quite a small accelerating field considerably
increases the number of these electrons, shows that many of
them are unable to escape unless their energy is increased

by outside fields.
(10) Summary of Results.

(a) The electron emission from various metals and a few
salts under the influence of X-rays has been measured, and
an attempt has been made to deduce from these measurements
the relation between the number of electrons emitted per
atom per unit intensity per unit time and the atomic number
or atomic weight of the substance from which they are
liberated.

(6) It has been found that under these experimental
conditions the special electron emission associated with the
characteristic X-radiations of the substances bombarded is in
these experiments only a very small fraction of the total
electron emission.

(c). These experiments have yielded no evidence of any
selective emission in the direction of the electric vector in
the X-ray beam.

(d) It has been shown that the electron emission per
unit intensity diminishes as the penetrating power of the
radiation 1s increased.

(e) The properties of the low-speed - electrons which
accompany the high-speed emission have been investigated,
and this emission has been ascribed to a secondary effect due
to the action of the fast electrons.

This work was carried out at the Cavendish Laboratory,
_ Cambridge, and the author has great pleasure in acknow-
ledging the continual kindness and many helpful suggestions
he has received during its progress from Prof. Sir Hrnest

Ruthertord, F.R.S.

[ 809 ]

LXX. Impact Tonization by Low-Speed Positive H-Ions in
fIydrogen. By A.J. Saxton, M.Sc., Assistant Lecturer
wn Physics, The University of Sheffield *.

Introduction.

VHERE now exists a considerable amount of evidence on
the conditions necessary to ionize a “normal” atom
of a gas. In every case we are concerned with the energy
exchanges between the atom to be ionized and the source of
the ionizing energy.
We may formulate the conditions governing this inter-
change of energy as follows :—

(a) In order just to ionize a “normal” atomf of a par-
ticular gas or vapour, z.e. to detach completely one electron
from it so that this electron possesses no kinetic energy as
a result of the process (single ionization), always requires
the same total absorption of energy whatever the nature of
the ionizing source. Thus whether the atom is ionized by
electron impact or by the absorption of radiation, the total
ionization energy necessary to change it from the “ normal”’
unexcited state to the ionized state is always the same.
The long wave-length limit (threshold frequency) for the
photo-electric effect in the vapour thus corresponds to the
ionization potential for electron impact.

(b) “Single” ionization of an atom may be produced by
absorption of the ionization energy from one or more of the
following sources :—

1. Impact by an electron.

2. Impact by a positive ion.

3. Absorption of radiation (photo-electric ionization).
4. Impact by “ normal” atoms (thermal ionization).

(c) If only one of the above sources is concerned, e. g.
electron impact (about which we possess the most data), the
absorption of the energy may take place in a single process
or by stages ; in this case by successive electronic collisions.
If a partially ionized atom receives the extra amount of
energy necessary to ionize before radiating the portion it
has already received, ionization will result.

* Communicated by Prof. S. R. Milner, F.R.S.

{1 With polyatomic molecules energy may be required first to dis-
sociate the molecule.

Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3G

810 Mr. A. J. Saxton on Impact Lonization by

(d) More than one of the above sources may be concerned
in the ionization of an atom. Thus an atom may be partially
ionized by absorption of radiation, and then the process may
be completed by electron impact *.

2. Ionization by Positive Ion Impacts.

The method of studying the ionization produced by one of
these four methods alone must be by some form of discharge
through the gas at low pressures. The conditions of the
experiment must approximate to the following ideal condi-
tions. Only the source of ionization, e. g. positive ion impact
which is being studied, must contribute to the ionization of
the gas, other sources being eliminated as far as possible by
suitably designing the apparatus. We must also be able to
distinguish between ionization produced by a single process
and that produced by the cumulative effect of several colli-
sions.

Assuming the foregoing principles concerning the energy
exchanges in ionization, and given the ideal conditions stated
above, we may expect the following result :—That for a posi-
tive ion accelerated through an electric field and striking
an atom of the gas, in order just to ionize the atom its
kinetic energy must be equal to the ionization energy of the
atom. ‘Thus if it has fallen unimpeded through an acceler-
ating field of V, a singly-charged ion will have a kinetic

energy of i
Ve=tm’
2g e

This accelerating P.D. of V reduced to volts is the ioniza-
tion potential for positive ions. In a p rticular gas it should
have the same value as the ionization potential for electron
impact. : i

This ionization potential for positive ions should be inde-
pendent of the nature cf the ions, the effect depending only
on their kinetic energy. These principles will apply only to
the case where an atom is ionized by the impact of a single
positive ion (possessing one positive charge) which has been
accelerated unimpeded by a P.D. of V volts.

The case of positive ion impact may differ, however, from
the case of electron impact in the following manner :—
Though the kinetic energy of the accelerated positive ion is
the same as that of an electron accelerated through a P.D.
of the same value, the energy exchanges with the “ struck ”

* H. D. Smyth and K. T. Compton, Phys. Rev. xvi. p. 501 (1920).
(Iodine vapour.)

|
4
3
|

A

ee are et

Low-Speed Positive H-Ions in Hydrogen. 811

atom may not be so simple, the positive ion itself being a
complex system. ‘Thus the ionization potential might depend
on the state of excitation ot the positively-charged “ strik-
ing”? atom.

3. Lxvperimental evidence on Tonization by Positive ons.

To test the validity of these principles, we may examine
existing evidence on impact ionization by positive ions,
taking inio account to what extent the experimental condi-
tions approximate to the ideal conditions for the test.

(a) High-speed Positive Lons.—a-rays. Millikan * has
shown that ionization by a@-rays usually results in single
ionization, more rarely in double ionization. Double ioniza-
tion, i.e. detaching the second electron from the already
ionized atom, of course requires a greater amount of energy
than that necessary for single ionization. Assuming that in
the flight of an a-ray through hydrogen there is no loss of
energy in non-ionizing collisions, and that each pair of ions
produced levies « toll on the kinetic energy of the a-particle
of the same amount, and knowing the initial kinetic energy
and the total number of ions produced, we may calculate the
energy required t» ionize a single molecule. ‘This energy is
that acquired by a single charge moving through 35 volts Tf.
Since, however, much energy may be wasted in producing
partial ionization or in useless kinetic energy of the ejected
electron, we may regard this figure as a maximum value for
the ionization potential for a singly-charged positive ion.

Canal or Positive Rays.—By applying a cross-field of a
few volts behind a perforated cathode in a discharge-tube,
Stark { used a galvanometer to measure the ionization current
produced by the impact of the canal rays on the gas. In
nitrogen at a pressure of 0°134 mm. he obtained signs of
ionization when the cathode P.D. had a value of 500 volts.
These conditions are very different from the ideal conditions
necessary. At such high pressures the mean free path of
the positive ion would be very small, and therefore we do not
know to what extent when they produce ionization their
kinetic energy is comparable with that derived from the
total P.D. of 500 volts.

Moreover, in both these cases of ionization by high-speed
positive ions it is probable that the mechanism is quite

* Millikan, Phys. Rev. Dec. 1921, p. 446.
+ Rutherford, ‘ Radioactive Substances and their Radiations,’ p. 159.
t J. Stark, Annalen der Physik, 1906, p. 427. See also K. Glimme
and J. Koenigsberger, Zevts. fi Phystk, 6, iv. pp. 276-297 (1921).
3G 2

812 Mr. A. J. Saxton on Impact Lonization by

different from that operating with low-speed impact where
the striking particle does not penetrate the atom.

In the case where the kinetic energy of the positive ion
is great enough for it to penetrate the atom (a-ray), the
amount of ionization per cm. path increases as the speed of
the ion decreases. The amount of perturbation increases
with the time taken for the ion to cross the atom. Thus
a-rays produce most ionization near the end of their range.
Glasson’s * experiments on ionization by cathode rays indi-
cate a similar effect with high-speed electrons.

(6) Lonization by Collision.— Townsend | measured the
current between two metal plates with different field-strengths
between them, the negative plate being illuminated by ultra-
violet light. For small distances between the plates the
results could be readily explained on the assumption that
the photo-electrons emitted by the negative plate (and the
electrons they produced in the gas by ionization) when
accelerated through the field produced a-ions per cm. by
collision with the molecules of the gas. For distances
between the platesabove a certain value he obtained currents
which were larger than would be expected on the above

simple theory, and ascribed this increase to ionization by

positive ions which produced G-ions per cm.

For example t, in hydrogen at 8 mm. pressure with a
distance between the plates of -3 em. and a field of 700 volts
per cm. giving a P.D. of 210 volts between the plates, he
obtained an increase ascribed to the action of positive ions.
Thus positive ions falling through 210 volts in hydrogen at
8 mm. pressure ionize by impact. The M.F.P. of a positive
ion would be very small at this high pressure, so that it
could not obtain an unimpeded fall through more than a
fraction of a volt. This suggests that the ionization pro-
duced may be due to (a) either successive collision or (b) an
accelerated positive ion does not lose the whole of its kinetic
energy on every collision, and so may acquire a velocity
corresponding to a P.D. which is greater than that along
its M.F’. path.

(c) Cathode Fall in Discharge-tubes.—It is not yet clear
whether the positive ions accelerated through the cathode
fall of potential produce electrons (cathode rays) by impact
with the molecules of the gas or with the metal or occluded
gas of the cathode itself§. The values of the minimum

* J. L. Glasson, Phil. Mag. (6) xxii. p. 647 (1911).

+ J. S. Townsend, ‘ Electricity in Gases,’ Chapter IX.
t bed. p. 317.

§ Ratner, Phil. Mag., Dec. 1920, p. 785.

Low-Speed Positive H-Lons in Hydrogen. 813

eathode fall of potential are very similar to those obtained
for the minimum sparking potential in gases at the same
pressure. In hydrogen their values are between 200 and
300 volts. In both these cases ionization by collision of
positive ions becomes very important, but the corresponding
pressures are so high that in no case will the aecelerated
positive ion fall unimpeded through the total P.D. Thus
the essential condition for the test is not fulfilled in this
case.

(d) Positive Thermions from Glowing Filaments.—Stark *
measured the current between a glowing carbon filament as
anode and a metallic cathode 6 mm. apart in air at °22 mm.
pressure with different applied P.D.’s. After obtaining
saturation of the positive current for smaller values of the
applied P.D., he obtained an increase of the current at
300 volts which he ascribed to the positive ions from the
filament producing ionization in the gas. McClelland t
obtained a similar result using an incandescent anode in air
at -66 mm. pressure, when he found that an increase of
current took place at 240 volts. In both these cases the
pressure is so high that the P.D. between the two ends of
the M.F.P. is only a small fraction of a volt.

The only experiments in which the necessary conditions
have been at all fulfilled are the following three cases, in
which, however, the positive ions were those emitted by
glowing coated and uncoated filaments, so that their nature
was not known exactly. Pawlow t measured the ionization
produced by positive thermions from coated filaments, when
accelerated through small potentials, by Lenard’s method.
He obtained signs of ionization in hydrogen at as low as
10 volts, and found that the minimum potential at which
ionization could be detected varied with the supply of positive
ions, being smaller for a greater intensity of the source.
He also found that positive ions were much less efficient in
producing ionization than electrons accelerated through the
same voltages. Thefact that the minimum potential depends
upon the original number of positive ions suggests loniza-
tion by successive collision. Franck and Eva v. Bahr in
similar experiments with air and hydrogen obtained signs
of ionization in a gas at potentials below the ionization

* Stark, Annalen der Physik, 1906, p. 427.

+ McCielland, Phil. Mae. xxix. p. 362 (1915).

I Pawlow, Proc. Roy. Soc., July 1914, p. 398.

§ Franck and Eva y. Bahr, Verh. der Deuts. Phys. Gesell. Jan. 1914,
p. 57.

814 Mr. A. J. Saxton on Impact Ionization by

potentials for electrons in the gas. They concluded that
there was no sharp ionization potential for positive ions in
the gas, and also found that the minimum potential at which
ionization could be detected was lower the greater the
intensity of the source of positive ions. Horton and Davies*
made a thorough investigation of the ionizing properties of
the positive ions emitted from an incandescent tantalum
filament in helium. Their results indicated the production
of fresh ions by collision of positive ions accelerated through
20 volts. Further investigation led to the view that the
ionization produced was not due to the ionization of the gas
molecules by direct positive ion impact, but to the bombard-
ment of the walls of the ionization chamber by positive ions
releasing 6-rays. They conclude that the positive ions do
not produce ionization when accelerated through potentials
of 200 volts. Thus the only investigations satisfying the
necessary conditions are to some extent contradictory.

4. Description of Apparatus.

The following experiments were undertaken with the hope
of obtaining more definite information about the conditions
under which ionization is produced by low-speed positive
ions. Other workers have used the positive ions from glow-
ing filaments so that their nature was not definitely known.
In the present work the nature of the ionizing positive ions
was known with greater certainty. They were produced by
electron impact in hydrogen, and the speed of the colliding
electrous was great enough to dissociate the molecule on
ionization. Thus the positive ions formed would be H-
nuclei (protons). They were accelerated through hydregen
at very low pressure and their ionizing properties studied.

In figs. 1 and 2, F is a tungsten filament heated by a
battery of 8 volts supplying a current of 2 to 3 amps. and
insulated on paraffin-wax blocks. The filament leads are
sealed into a glass tube, the end of which fits as a ground-
glass stopper into the side tube B. With this arrangement,
when the filament burnt out, it could be replaced more
easily. The electrons from the glowing filament F are
accelerated towards the nickel electrode A by the P.D. of
V volts between the negative end of the filament and A.
If V is greater than the ionization potential for electrons in
hydrogen, ionization by electron impact occurs near to A, -
and the resulting positive ions are accelerated towards the -
filament by V. Some strike the filament, but some pass on

- ™ Horton and Davies, Proc. Roy. Soc., March 1919, p. 383.

Low-Speed Positive H-Jons in Hydrogen. 815

through the gauze G,. These positive ions may be further
accelerated by the P.D. of y volts between the gauzes,
whence they pass into the ionization chamber G,D, where
the ionization they produce. can be measured. The nickel
gauzes G, and G, are of fine mesh, and are fitted on frames
so that they do not touch the glass walls of the main tube.
This lessens the possibility of a current leak across the glass.

Fig. 1.

B
pA tee
F G, Gs
Fig. 2
[hh
Ey

One set of wires in each gauze is set vertical and the other
set horizontal so that the corresponding holes in the gauzes
are opposite. EH, and E, are side electrodes for using a
cross-field to measure the ionization.

Before fitting, the tube was thoroughly cleaned and dried.
After fitting to the supply tube, it was exhausted as much as
possible by an automatic mercury pump, the glass walls

816 Mr. A. J. Saxton on Impact Tonisaiion by

being heated to drive off occluded gases, and the filament
was made white-hot. The hydrogen was prepared by electro-
lysing a solution of baryta in distilled air-free water. The
prepared gas was then allowed to stand for several days over
P2205. <A gold-leaf tube was fitted to eliminate mercury
vapour from the ionization tube. Pressures were measured
by a sensitive McLeod gauge. To keep the pressure as
constant as possible and thus minimise the pressure-change
due to the “clean-up” effect of the filament burning in
hydrogen, bulbs of large capacity were fitted in the delivery
tube to the ionization apparatus. The ionization currents
were measured by a sensitive quadrant electrometer, and all
the batteries were insulated by paraffiin-wax.

Summary of Results —This apparatus showed some faults
of design which were afterwards remedied in a new apparatus.
It was found, however, that positive ions produced ioniza-
tion in the hydrogen when accelerated through 19 volts,
which was the minimum accelerating potential possible.
Consider first the production of the ionizing positive ions.
To produce these in quantity it was found that the arcing
potential V must be greater than 16 volts. The pressure of
the gas was tried at values between ‘005 mm. and ‘01 mm.
of mercury. These pressures were chosen because they
were about the lowest at which both the positive-ion current
and the resulting ionization current were measurable on the
electrometer. The pressure must be low enough so that the
M.F.P. of the positive ions produced near to A will be
greater than the distance AG,. Thus most of the positive
ions will not collide with the gas molecules before reaching
the ionization chamber G2D. On the other hand, if the
pressure is too low the M.F.P. of the electrons in the gax
will be much greater than FA, so that none of them will
collide with the molecules before reaching A. The positive-
ion current had values varying from 107° amp. upwards.
It increased with the arcing potential V, the pressure of the -
gas, and the filament current which controlled the supply
of the thermo-electrons from the filament. The supply of
positive ions from this low-voltage are in hydrogen for
larger values of V showed much unsteadiness, and thus it
was difficult to compare the ionization produced with different
accelerating potentials as the ionizing current itself was not
sufficiently constant. penal:

A more serious drawback was due to the fact that the
filament itself emitted positive ions. An attempt was made
to cut off this positive thermionic current by fitting a

Jonization Current.

Low-Speed Positive H-Lons in Hydrogen. 817

nickel-foil sereen partly round the filament, but this was found
unsatisfactory, apparently owing to the diffusion of the ions.
With the negative end of the filament connected directly to
the gauze G, so that e=0, it was found that many thermo-
electrons shot out into the space between the gauzes. To
prevent this, G; was made negative with respect to the nega-
tive end of the filament, a retarding P.D. of 2 volts being
applied. With w=3 volts as a minimum, this effect could
no longer be detected.

The ionization produced was measured by Lenard’s method,
using the gauge Gy as a collecting-plate instead of D to
shorten the distance the positive ions had to travel before
ionizing by collision. A retarding field of 50 volts for
positive ions was applied between G, and G2. The arcing
potential. V was kept constant, and « increased to increase
the total accelerating P.D. of w+v volts. The ionization
current plotted against the total accelerating voltage is
shown bya typical curve in fig. 3. The readings were taken

Total Accelerating Voltage (x+Wvolts,)

in rapid succession to avoid any alteration in the ionizing
current. Values of the accelerating voltage below 19 volts
were not possible, and readings to repeat were difficult to
obtain because of the variation in the ionizing current which
depended upon three variables, the temperature of the fila-
ment, the pressure of the gas, and also to some degree on
the value of 2, the ‘‘ drawing-out P.D.” A second method
was tried to detect ionization by applying a cross-field of

818 Mr. A. J. Saxton on Impact Lonization by

4 volts between the side electrodes EK, and Hy, and putting
both Gz and D to earth. EH, was maintained at a potential
of —4 volts, and E» connected to the electrometer. Since Hy».
was positive with respect to the electrode Ky, it should collect
the negative ions produced by collision. It was found,
however, that H, was always charged up positively by an
amount which increased with the accelerating P.D. of the
positive ions. The effect is probably due to the formation —
of a positive space-charge in the ionizing chamber. ‘This
' positive charge deflects the incoming positive ions to the:
side electrodes.

d. Description of Apparatus 2.

To obtain more definite results, two improvements on the
old apparatus were desirable. The positive-ion current from
the filament itself must be eliminated. It was also desirable
to work at much lower accelerating voltages to detect the
potential at which ionization sets in. A diagram of the
second apparatus is shown in fig. 4. The side electrodes.

Fig.

to}

4,

Fr

ssa Sos | \

i | |

V volts |Acc.P. D.i lonization E
ie open oe em iy aaa ibe OAS SS ee nase <S a tea al etapa TO
| ESS | yoo Fea eee

Drawing- out PD.|
|
A G

EK, and E, were omitted and the end electrode E increased
in area. The tungsten filament F was placed in a side tube
to cut off the positive ions given off by the filament itself.
The filament was surrounded by a nickel-foil screen, which
rested on the top of the cylinder A.

Electrons from F were accelerated by the arcing poten-
tial V into the nickel cylinder A through the gauze opening,
and there produced positive ions in the hydrogen by collision.
A P.D. of « volts is applied between A and G, to draw out
the positive ions, which, passing through the gauze Gj, are
then accelerated by a P.D. of y volts between G, and Go.
A retarding P.D. for positive ions of z volts is applied in
the ionization chamber between G» and E.

Low-Speed Positive H-Jons in Hydrogen. 819

The different currents in the apparatus were measured as
follows :—

(a) The filament was heated by an insulated 8-volt battery
with resistances. ‘The current which was usually about
3°d amps. was measured by an ammeter.

(6) The thermionic current between F and A, which was
always of the order of a few milliamps., was measured by a
Paul single-pivot galvanometer with shunt. The constancy
of this current rather than its absolute value was the reason
for measuring it. If this current had a constant value, it
indicated a constancy of conditions (pressure of gas and
filament temperature) in the apparatus. V was measured
between the negative end of the filament and the cylinder.
It had a minimum value of about 16 volts in order to produce
a satisfactory supply of positive ions.

(c) Thepositive-ion current leaving the cylinder for the

gauzes to produce ionization in G2H was measured by a

suspended magnet galvanometer, and was usually of the
order of a few microamps. The galvanometer was nee ee,
between the cylinder A and the insulated battery giving a
and y, and thus measured the total positive current ened
the cylinder. To compare the amount of.ionization produced
by a stream of positive ions accelerated through different.
voltages, it was essential that this current should keep con-
stant. The positive ions were drawn out of the cylinder by
the field 2, which penetrated inside the cylinder. This posi-
tive current varied with the original thermionic current, the
drawing-out P.D. 2, and the pressure of the gas.

It was found that for small values of w and large values
of V, especially with a heavy thermionic current, the current
leaving the cylinder was negative although the P.D. #
opposed anegative current. The filament vas screened from
the gauze G, by the nickel-foil screen so that no electrons
accelerated by V could shoot through directly against x

‘towards the gauzes. With smaller values of V the current

leaving the cylinder was small but positive, and was probably
a mixture of positive and negative currents with a pre-
ponderance of the former. The effect was probably due
to the formation of a negative space-charge (inside the
cylinder A) which repels incoming electrons to the gauzes
and to the ordinary diffusion of electrons. It was impossible
to be certain that the positive-ion current did not contain
some negative current against the field a unless the value
of # was greater than V. These positive and negative
currents could not be measured separately. The field due
to the space-charge could not be greater than the field

820 Mr. A. J. Saxton on Impact Ionization by

producing it. Thus, since V cannot be less than 16 volts
and 7 >V,«#>16, which sets aminimum value to the total
accelerating field of w+ y.

(d) The positive-ion current entering the ionization
chamber was measured for different values of « and y and
for different pressures (the retarding field being kept at
zero) by measuring the rate of charging up of the electrode E
connected to the electrometer.

For higher pressures (‘1 mm. He) the positive-ion current
reaching EH was only a small fraction—about one-tenth—of

the original positive-ion current leaving the cylinder.

As the pressure was diminished, a greater proportion
reached H, indicating that at higher pressures many of the —
ions were stopped by collision with the molecules. The
whole of the batteries, leads, and measuring instruments
were insulated on paraffin-wax blocks, and tests carried out
to ensure that none of the measurements of the different
currents were affected by spurious leaks.

Detection of Lonization.—The first method employed to
detect ionization by collision by the positive ions was to take
fixed values of x and y and to measure the charging up of H
as the retarding field z¢ was increased gradually. No pre-
cautions were necessary to distinguish between ionization
and radiation produced in Gok. "In the case of electron
impact the collecting electrode is negative with respect te
the gauze, and so it may charge up positively by releasing
electrons under the influence of the radiation, or it may
charge up positively as it collects the positive ions. In the
present case, since E is positive with respect to Go, radiation
will not release electrons from KE, which will collect the
negative ions produced on ionization. The ionization
currents were measured by the steady potential assumed by
the electrometer quadrants when connected to earth across
a high-resistance leak. The values of ¢, i.e. the P.D.
between G» and H, were corrected for this potential assumed
by E.

"The type of curve obtained plotting the current to E
against an increasing retarding field z is shown in fig. 5.
Great difficulty was experienced in keeping the positive
ionizing current sufficiently constant so as to render the
resale « comparable for different values of ¢.

For small values of z the current (potential assumed by HB)
was positive, but gradually diminished as z increased until
for a certain value of z it became negative, further increase
of z resulting in an increase of the negative current. This

a ee

Cee SL el
q

Total Current Zo EZ.

Low-Speed Positive H-lons in Hydrogen. 821

“cross-over” potential is shown in fig. 5 at 25 volts. The
effect of increasing the thermionic current and so the posi-
tive ionizing current is to shift the graph along to the right
so that the cross-over potential is increased. These facts
can be explained readily by the fact that the incoming
positive ions ionize by collision. The current to H is thus
made up of two currents—the positive ionizing current and
the negative current of the negative ions produced. When z

is sufficiently large, this negative electron current is sufficient

Fig. 5.

x = BO lv. y= Avy.
Total] Accelleratt

eS) O +5 10 15 20 ZS 30 35 40 45

Total retarding Field (Z volts)

to swamp the positive current retarded by z and by collision
with the gas molecules. In this manner definite evidence of
ionization by positive ions accelerated through potentials as low
as 18 volts has been obtained.

Other ionization tests were carried out with the retarding
potential z of constant value and greater than the minimum
value of «+y, the total accelerating BaD. fable. [.. wWlus—
trates the relation between the positive ionizing current
and the negative ionization current for different applied

Pye

822 Impact Ionization by Positive H-Lons in Hydrogen.

TaBueE I.
Total Current ee
| ace. P.D. +ive ionizing ae
(e+y). | leaving cylinder. Tare C,/C,.
(Volts). (Scale Divns.). | [10—-’° amp.].
One Cx
36 +50 9 56
mt +100 15 6:7
92 +60 10 6
+ +120 20 6
03 +145 23 6°3
“3 +290 60 4:8
Hydrogen ...... p= 02mm. Hg. V=18 volts. z=100 volts.

The positive ionizing current was varied by varying the
filament current. The results show that the amount of

donization is proportional to the ionizing current. The

former current collected under a field of 100 volts is probably
increased by the fact that the original electrons released

‘by the positive ions themselves produce some fresh ions by
-collision. By obtaining the ratio of the two currents we
can estimate a maximum value for the efficiency of the
positive ions as lonizers. ‘The ratio of the ionization current

to the ionizing current for the results in Table I. is about

1/130. Thus of at least 130 positive ions reaching ihe
ionization chamber G»2H, only one ionizes a molecule cf

hydrogen. Electrons accelerated through the same voltage

-and at the same pressure in hydrogen would produce about
‘one positive ion per ionizing electron*. Hence when

moving with these low velocities, the efficiency of an electron
is much greater than that of the positive ion in producing
ionization by collision. Most of the energy of the positive
‘ions is lost probably in non-ionizing collisions.

Summary.

An attempt is made to formulate the conditions governing

‘the exchange of energy when an atom is ionized by impact
-or by the absorption of radiation. This is applied to the

case of ionization by positive-ion impact with the conclusion
that the ionization potential for positive-ion impact in a gas

* J.B. Johnson, Phys. Rev. 1917, p. 609.

On the Partition of Energy. : 823

should be the same as that for electron impact. Results
of previous workers show wide disagreement in the value of
the minimum accelerating P.D. necessary for positive ions
to produce ionization.

Two types of apparatus are described, the positive H-ions
being produced first by electron impact i in hydrogen. The
peculiar difficulties of the experiment made it impossible to
test for ionization produced by positive ions accelerated
through less than 18 volts, and, as the ionizing current itself
depended upon three variable factors, it was more difficult
to obtain comparative observations. The results indicate
that positive H-ions produce ionization in hydrogen when
accelerated through P.D’s. as low as 18 volts, and that they
are much less efficient in producing ionization than electrons
under similar conditions.

In conclusion, I desire to thank Prof. Milner, F.R.S., who
proposed the problem, for his kindly interest and advice.

Sheffield University,
July 31st, 1922

————

LXXI. On the Partition of Energy.—Part II. Statistical
Principles and Thermodynamics. By C. G. Darwin,
F.RS., Fellow and Lecturer of Christ’s College, Cam-

- bridge, and R. H. Fow er, Fellow and Lecturer of
Trinity College, Cambridge *

§ 1. Introduction and Summary.
. a previous paper T we have-developed a method of

calculating the partition of energy in assemblies
containing simple types of quantized systems and free
molecules. In this method advantage is taken of the
multinomial theorem, which makes it possible to obtain
integrals expressing accurately the various average values
concerned in partition problems, and then the ‘integrals
are easily evaluated asymptotically. The “ fluctuations ” can
also be readily found, and from them it can be shown
that the possession of these mean values is a “normal”
property of the assembly in the sense used by Jeans f.
This method has the advantage of obtaining directly the

* Communicated by the Authors.

+ Phil. Mag. vol. xliv. p. 450 (1922).

¢ Jeans, ‘ Dynamical Theory of Gases,’ passim. Average and most
probable values are of course in practice the same.

824 Messrs. C. G. Darwin and R. H. Fowler on

average properties of the assembly, which are, on any
statistical theory, those which experiment determines,
instead of the most probable values, as is usually done.
It can also be carried out simply and rigorously without
the use of Stirling’s theorem, and thus provides satisfactory
proofs of all the usual partition laws, including Maxwell’s
Distribution Law.

In this discussion the partition laws were all obtainable
without any reference to thermodynamical ideas, in par-
ticular without any mention of entropy. This we claim
as an advantage. But a great deal of work has been
done on partition laws, in which the idea of entropy has
played a leading part; so that, for this if for no other
reason, it is fitting to examine its position in our pre-
sentation of statistical theory. But the power of our
method on the statistical side invites a somewhat more
general review of the fundamental connexion between
ciassical thermodynamics and _ statistical mechanics both
of classical dynamics and the quantum theory. In the
former work we were content with purely statistical results,
and identified the temperature scale simply by the perfect °
gas laws; here we attempt a more strictly logical deve-
lopment, and prove the Jaws of thermodynamics for
assemblies composed of systems of a fairly general type,
and, by linking on to Gibbs’ work, also for general systems -
which obey the laws of classical mechanics.

After summarizing our previous resuits in § 2, we pass
in § 3 to a comparison between the empirical temperature
in thermodynamics and the parameter which acts as
temperature in our previous work. In §§4, 5, 6, we
make a critical study of the usual presentation of entropy
in statistical theory. This is ordinarily introduced by
means of Boltzmann’s Hypothesis, which relates it to
probability, and, though no objection can be made to much
of the work based on this hypothesis, it appears to us
that the development is often marred by somewhat loose
reasoning. Though much that we here say is general
and not at all dependent on our special methods of
treatment, yet it has been far easier to examine the validity
of the arguments on account of the way in which it is
possible to combine assemblies together at will. In con-
sequence of this discussion we are led, in §7, to a pre-
sentation of entropy which is very closely related to that
of classical thermodynamics, which frees it from the com-
binatory complications with which it is normally associated
and brings it back to direct dependence on the partition

the Partition of Energy. 82

5
functions which form the basis of our method. In § 8
the definition is considerably simplified mathematically by
replacing the “entropy ” by the “characteristic function
as the basal thermodynamic quantity. In §9 we show
that for an assembly in a temperature bath our method
is exactly equivalent to Gibbs’ use of his ‘canonical
ensemble,’ and in § 10 we consider briefly the possibility
of inverting the argument so as to obtain information
about the elementary states from thermodynamic data.

99

§ 2. Summary of our previous paper.

It was shown in the previous paper how the partition
of energy could be evaluated for various types of assembly.
Those chiefly treated were quantized systems, for which the
energy was sole variable ; but it was also shown how to
apply the method in the case of a perfect gas, where both
energy and volume are variable. It is easy to see that the
method is applicable in considerably more general cases.
The partition of energy could be evaluated when any of the
types enumerated were mixed together, the essential point
of the method being the existence of a certain function, the
partition function, for each type of system. By means of
these functions all the rather tiresome combinatory ex-
pressions can be very easily dealt with so as to obtain mean
values, and also the fluctuations about those mean values.

The partition function of asystem—which with a different
notation is the “ Zustandsumme ” of Planck—is constructed
as follows. The possible states of the system may be divided
into cells; these cells are fixed and finite for quantized
systems, but for the systems of classical mechanics must
ultimately tend to zero in all their dimensions. Associated
with each cell is a weight factor, determined by the usual
statistical principles. The weight of any cell of a system
obeying Hamiltonian equations is proportional to its ex-
tension, The relative weights of the cells of a quantized
system are determined by Bohr’s Correspondence principle,
and the weights are all assigned definite magnitudes by
the convention that a simple quantized cell shall have unit
weight. For consistency in physical dimensions the cells
for Hamiltonian systems are divided by the appropriate
power of fh to give their weight. Associated with any cell
there is a definite energy, depending on the cell and on
certain external parameters 7, w2,...3 this last isa slight
extension of our previous paper, which must be made so as
to deal with questions of external work. Then, if p, is the

Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 23H

826 Messrs. C. G. Darwin and R. H. Fowler on

weight and e, the energy in the rth cell, we define as
the partition function

i = Spe oa

summed over all the cells. The partition function is thus a
function of 3, x, 2, ..... For systems of classical mechanics
the extensions of the cells, that is the p’s, all tend to zero,
and the sum is replaced by an integral.

As an example of the systems treated we may mention
first Planck’s line vibrator, which can take energy in
multiples of-e. This has the partition function

;
fis: (2:1)
More important is the free monatomic molecule. If it is
of mass m and is confined in a volume V, its partition
function is
(2am)??? V
hog 1/3)?”

(2-2)

The partition function of a diatomic molecule is simply
given by multiplying (2:2) by the partition function corre-
sponding to the quantized rotations (assuming that the
atoms cannot vibrate relatively to one another).

Now suppose that we have any number of types of
systems together in an assembly ; let there be, say, Ma
of type A with partition function fy. Then it was shown —
that the average energy among the systems of this type
was given by

= fo
Ky = Mas sy log fa, nuts Aa: (2°3)
where 3 is uniquely determined in terms of E the total
energy by
0
It is easy to show that 3 is always less than unity. The

average number of systems A in the rth cell is also easily
found, and is

Gea My nyo Sanifo. 0 Sl

In connexion with the relation of entropy to probability,

we must also recapitulate some of the work from which the

above results were derived. The statistical state of an

the Partition of Energy. §27.

assembly, say of two groups A and B, M and N in number
respectively, is specified by giving sets of numbers dp, a,
Megs io Oy. bj seo, where .a./is the number of :the “A's
which lie in the sth cell. Suppose that this cell has a
weight factor p, and that the sth cell for a B has a weight
factor gs: then the fundamental basis on which the whole of
statistical theory rests gives an expression,
“M1 om N! by By

as 2 : 96)
a —————— hi) 1 date enoe Go G1 LL Cte 2°6
reentia.f) awe meee Ie esta ka0)

for the number of ‘* weighted complexions ” corresponding
to that specification. If this number is divided by the
total of all the weighted complexions, which we call OC,
which correspond to any distribution of a’s and 0’s con-
sistent with the same total energy, then this ratio is the
proper measure of the probability of the specification, and
must be used in calculating the expectation of any quantity.
It was shown in the former paper that C and the associated
averages can be expressed as contour integrals (exact for
quantized systems) which lead asymptotically to the formulee

(2°3), (2°5).
§ 3. Temperature.

In considering the connexion between statistical theory
and the principles of thermodynamics, we must begin by
correlating the ideas of temperature in the two theories.
Throughout our former paper we have treated the para-
meter 3 as of the nature of the temperature, and it is
here of some importance to observe that 3 has precisely
those properties which must be postulated of the ‘‘ empirical
temperature ” when the foundations of thermodynamics are
rationally formulated *. The basal fact of thermodynamics
is that the state of two bodies in thermal contact is deter-
mined by a common parameter which is defined to be the
empirical temperature. The temperature scale is at this
stage entirely arbitrary, and any convenient body whatever
may be chosen for thermometer. Cn the statistical side
we have shown that when two assemblies can exchange
energy so that there is one total energy for the two
together, then their states are defined by a common para-
meter 3. The analogy is exact, and we are therefore
logically justified in identifying 3 with the empirical
temperature in precisely the thermodynamical sense.

* See, for instance, Max Born, Phys. Zeit, vol. xxii. pp. 218, 249, 282

(1921),
ale 2

828 Messrs. C. G. Darwin and R. H. Fowler on

Since we may take any body for our thermometer by
which to measure the temperature scale, we may if we
like at once define the absolute temperature of statistical
theory as proportional to the pressure in a body of perfect
gas at constant volume. Now if there are P molecules in
volume V, it follows from almost any theory that

lah ai
Be x7 3 Chins; Merete hee (0 |.

where @,in, 13 the mean kinetic energy of translation of a
molecule. In terms of 3 we have by (2°2) and (2:3)

ie e
log 1/3”

Y 3
Pein: — 2

and so we must have
log L/S’ SL /kT, S's os eee

where & 1s a universal constant.

This appeal to the properties of an ideal substance is,
however, not quite satisfactory. It is avoided in theimo-
dynamics, where the absolute temperature is defined in
connexion with the Second Law. Now we wish to show
how our theorems lead to the laws of thermodynamics, and
so we must not postulate a knowledge of absolute tempe-
rature, but must only consider it in connexion with entropy.
Our development of the Second Law will not of course
have the complete generality for classical systems of such
treatment as Gibbs’ (though we have later extended it
to his case), but still it will suffice to deal with assemblies
as general as those that have been used by most writers who
have deduced special thermodynamical conclusions from
statistical premises. |

§ 4. The usual presentation of Entropy.

Entropy is usually introduced into statistical theory by
means of Boltzmann’s Hypothesis relating it to probability.
This hypothesis is based in general on the fact that, on the
one hand, an assembly tends to get into ils most probable
state, while, on the other, its entropy tends to increase, and
so a functional relation between the two may be postulated.
The general line of argument is somewhat as follows *:—

We can assign the numerical value W, for the probability
of the state of any assembly. If we have two such assemblies

* Planck’s classical work on Radiation Theory is a representative
example of the use of the argument here quoted. |

the Partition of Energy. 829

which are quite independent, then by a fundamental principle
of probability, the joint probability is the product of the
separate probabilities ; that is,

WW; = Wei. ° . ° ° . ° (4:1)

Ou the other hand, the joint entropy is the sum of the
separate entropies, and so

Sit S. => eee . ° mo ae e ° (4°2)
Then to satisfy the functional relationship we must have

= OE Wy Sour sw ntny (AS)
k being a universal constant. Next, to evaluate W, a
definition is made of ‘thermodynamic probability ” as the
number of complexions corresponding to the specified state :
this is made a maximum subject to the condition of
constant energy, and the maximum of & log W is equated to

the entropy <r which is then shown by examples to be the

. eutropy of thermodynamics. Observe that there are two

separate processes involved: in the first the determination
of the maximum fixes the most probable state of the
assembly by itself. In the second the assembly is related
to the outside world by determining its entropy ; and then
the absolute temperature scale is introduced by the relation
0S8/oH=1/T.

Now there is much to be criticized in this argument. In
the first place there is a good deal of vagueness as to what
is happening. For the addition of entropies can only be
realized by some form of thermal contact *, and is then
only in general true when the temperatures are equal ; and
both these conditions require that the assemblies shall not
be independent, So it is only possible to give a meaning
to (4:2) by making (4:1) invalid. Again, without more
definition the probability of a state is quite ambiguous :
for example, we can speak of the probability of one par-
ticular system having, say, some definite amount of energy,
and for independent assemblies (4°1) will be true of this
type of probability, but it will have no relation whatever
to entropy. ‘This objection is supposed to be met by the
definition of ‘*thermodynamic probability” ; but that is a
large integer and not a fraction, as are all true probabilities,
and so (4*1) cannot be maintained simply as a theorem in
probability.

* There is perhaps an exceptional case, that of radiation, worked out

by Laue and cited by Planck (‘ Radiation Theory,’ ed. 3, p. 116); but a
general theorem must be generally true.

830 Messrs. C. G. Darwin and R. H: Fowler on

Now it is established that actually the “thermodynamic
probability ” does lead to the entropy, and so we must con-
sider how it is to be interpreted in terms of true probability.
It is clear that the “ thermodynamic. probability ” must be
divided by the total number of admissible complexions, and
that when we consider an assembly of given energy this
number is ©. In so far as we have to calculate the most
probable state, only ratios are concerned and the de-
nominator is immaterial, for we only have to deal with
the equation *

S'—S" = £ log WW 4 Wi) 3 ese

there is no difference between using “thermodynamic ”
and true probability. But when we attempt to determine
the value of the entropy itself by (4°3), we shall in all
cases find that when W is a true probability the maximum
value gives always S=O—a trivial determination of the
arbitrary constant in the entropy. This result may be
verified for any of the examples of the former paper. We
have merely to substitute values for the a’s and b’s of the
specifications, and make use of Stirling’s theorem in
the form

log (a!) = wloga—a ... 2)... eee

This is an approximation which is known by experience
to suffice in entropy calculations. The zero value of 8,
roughly speaking, expresses the fact that a ‘‘ normal”
distribution is so enormously more probable than any other
that by comparison it is certain, and so for it W=1.

It is thus clear that the straightforward process is useless,
and we must consider how it is to be modified so as to retain
the relation with true probability while giving the actual
value of the entropy—in effect we must find a way of
justifiably omitting the denominator C. As long as we
consider the whole assembly this is impossible, for C
depends on 3 and cannot be regarded as an ignorable
constant when changes of temperature are contemplated.
But if instead we consider the entropy of a group of
systems immersed in a temperature bath, it becomes
simple. Take, for example, a group of M A’s—systems
of the veneral quantized type described in § 2—and suppose
them immersed in a bath of a very much larger number
of B’s. We can now define the entropy of the A’s when
their specification is a, a,.... as k times the logarithm
of the probability of that specification. In calculating

* See Khrenfest and Trkal, Proc. Acad. Amsterdam, vol. xxiii. p. 162
(1921).

the Partition of Energy. 831

this probability we are indifferent about the distribution
among the B’s, so we sum the complexions involving all
values of the 0’s consistent with the selected values of
the as. Then

Pie By. aa}
M! ach N! bis Uh ‘
Rite! ay lowed eas wu, eae TT penne es Je

(46)
where the a’s may have any values (which do not involve a

greater total energy than that of the whole assembly), while
>, denotes summation over all different values ot the 8’s

such that
Lensls =k — > € ys

and of course, as always, ,a,=M, >,bs=N. Now, provided
that N is much larger than M, the factor

| at |
eatin Ks

will be practically independent of the a,’s and of the energy
of the group of A’s—that is to say, it may be taken as
constant and omitted from the calculation, and we are left
with the ‘‘thermodynamic probability ” as the only variable
art.

; It is only in this sense that a strict meaning can be
assigned to Boltzmann’s “Hypothesis; and it is of the
greatest interest that the conditions under which it has
meaning correspond exactly to the conditions of the
‘canonical ensemble” of Gibbs, as will be shown later.
But, even so, it is not a very convenient expression, for
we must always suppose that the assembly is a part of
some much larger one, whereas the expression for the
entropy is purely a function of the group and the tempe-
rature. It is therefore more convenient to abandon the use
of the principles of probability and to define the entropy
us k times the logarithm of the number of complexions
(weighted if necessary). We shail call this the kinetic
entropy. This number of complexions has the multiplicative
property (4°1), but now in virtue of its own combinatory
formula and not of an appeal to an inapplicable probability
theorem. The new definition does not appear to have the
same simplicity as the old, but that is only because in
the old the necessity for a detailed definition of what is
meant by probability was concealed. It would appear that

CO

832 Messrs. C. G. Darwin and R. H. Fowler on

some such argument as this is necessary to justify the
use of “thermodynamic probability,” the quantity used
with success by so many writers.

The argument of this section has really been dealing
entirely with the junction of assemblies which had the
same temperature ; it may be more conveniently visualised
as dealing with the separation of an assembly into parts
which are thereafter isolated from one another. Actually
of course our work must include the fact that entropy has
the property of increasing when assemblies at different
temperatures are joined. We have not yet had cause to
discuss this, as we have so far been mainly criticizing theories
which were developed by considering only assemblies of the
same temperature. | :

$5. Entropy as a non-fluctuating quantity.

The kinetic entropy as defined above is a fluctuating
quantity, whether we find it for the whole assembly or
for a part. On the other hand, the entropy of thermo-
dynamics is a function of the state of the assembly and
must be regarded as constant, and we must see how the
two may be best related. Now we cannot get away entirely
from the question of fluctuations, but we can conveniently
simplify the definition so as to dissociate them from the
entropy. Consider an assembly composed of A’s and B’s.
At every moment its state is specitied by the values of
CANA ERC ae OFS arn , and these numbers all fluctuate, and
with them the energy Ey, and the kinetic entropy Sa. But
if we want to treat of the entropy of the A’s as opposed to
that of the b’s, we must suppose the A’s to be suddenly
isolated. After the isolation they will have a certain
definite energy determined by the chance state at the
moment of isolation, and this energy will determine the
temperature and so the thermodynamic entropy. So, to
define a function representing the thermodynamic entropy,
it 1s most reasonable to choose some simple non-fluctuating
function of the state of the whole assembly; we can then
allow for the fluctuations in the entropy of its parts by
imagining them suddenly isolated, and calculating their
entropies from the energies they chance to have at the
moment of isolation on the same principle as was previously
done for the whole assembly. There are several suitable
definitions—for example, we can use the total number of
the complexions, or the average number, or the maximum

:
|
7

ee 2 a
‘

the Partition of Energy. 833

number, in each case attributing to the A’s the amount
of energy they had at the moment of isolation. Now
if these quantities are calculated, it will be found that,
to the approximation (4°5), they all have the same value.
This value is easiest to find for the maximum number.
It is unnecessary to take an assembly of systems of more
than one type, as we have seen ‘that the additive property
will hold. We have

S/k = log M!—%, log a,!+ 3,4, log py.

We must here make the unjustified application of
Stirling’s theorem to numbers some of which will un-
doubtedly be small; it should be possible to justify the
process, but we shall not do so. Then, making use of
(2:1), (2°5), we have

| | eae
S/k = M| log f—log $.9-S log /'], es ee
2M log fe Milos (1 /ayye es sale ee ABB)

since

i fo
= Ms 53 log f.

Equations (5:1) and (5:2) remain equally true for a
group of free molecules :to the same approximation. ‘This
formula for 8 is the direct consequence of Boltzmann’s
Hypothesis, and 8 has the necessary additive property for
combining the parts of the assembly. Moreover, it agrees
completely with the entropy of thermodynamics in all cases
where they can be compared: this agreement justifies our
use of (4:5) in these calculations. But it is indifferent
whether we define the entropy as the total, average, or
maximum number of complexions, and (4°5) is always
inexact ; it is therefore unsatisfactory to make the formal
definition of non-fluctuating entropy in any of these ways.

‘Now (5:1) and (5°2) give precisely the thermodynamic ex-

pressions in all comparable cases, and this suggests a direct
definition in terms of partition functions. We may thus
suppose that the combinatory processes are correctly looked
after by the partition functions, and may define the entropy
by either of the relations (5:1) or (5°2). Pending its formal
identification with the entropy of thermodynamics, we shall
describe it as the “ statistical entropy.”

834 Messrs. C. G. Darwin and R. H. Fowler on
§ 6. “ The increasing property” of Entropy.

We have now obtained a quantity S,,, the statistical
entropy, which is evidently related to the entropy of thermo-
dynamics Sy, but we must examine what right we have to.
make the identification complete. By its definition (5:1),.
Sst. has the additive property for separation, and we can
easily show that for Junction it has the property 8, + 8. < So).
which may be called the increasing property.

Consider the special case of two assemblies, and suppose
that in their junction only changes of temperature are con~
cerned—not of volume or any other parameter. We shall
also simplify by supposing that in each assembly there is.
only one type of system, different for the two. As we do.
not intend to base our final result on the present paragraph,
this wiJl be general enough. By definition, for the first.
assembly before junction the entropy is given by

Sst. /k = M' log f'(8')—E’ log 9’.

Now when the energy H’ is given, the temperature 3’ is
determined by (2°4), and this is equivalent exactly to the
condition that Ss’ should be a minimum for given HE’. So,.
if S$ has any value different from $',

Sst. /k <M’ log f'(3) —H’ log 8.

Similarly,
1m1 arly Set. /k< M" log f'""(3) — Hi” log 8,

if 3 is different from 3’, the temperature of the second.
assembly. It follows that unless 3’=$3''=3 we have

(Sst.’ + Sstl)/k< M' log 7’ (8) + M” log /'""(S) — Hi log 5,

where EH’ + H'’=H, the energy of the joint assembly. Now
with a ot: choise of 3 this is Sgjk, where Sgt, is the
entropy of the joint assembly after combination ; so we
have proved that (when vo volume or other such changes.
take place)

Sst Set. sts (E'+ bh” =E),

unless the initial temperatures are equal, in which case
! a ‘
Sst. + Sst. = Sst.

Thus statistical entropy has the increasing property.

It is often taken for granted that if we can find a function
of the state which has the increasing property, then that
function must be the entropy: this assertion tacitly underlies
Boltzmann’s hypothesis. But the identification of S., with
Sin., the entropy of thermodynamics, cannot be established in

the Partition of Energy. 835.

this way because the function L=Sg.+0E also has the
increasing property, where } is uny universal constant.
Now when we set out to define the absolute temperature
scale, we must start with the general function } which has
the increasing property, for we have as yet no right to

choose any particular value for b. If we attempt to define T

by the relation 9$/QH=1/T, we find

it eget VOSw
aes Um 0

+b = klog 1/5+6,

which can never determine absolutely the relation of ‘I’ to S

so long as 6} is undetermined.

This impasse is one aspect of the fact that in thermo-
dynamics the absolute temperature and the entropy are
introduced in the same chain of argument—the absolute
temperature as integrating factor and the entropy as the

resulting integral. Thus—and this is a point that has.

been overlooked by some writers—it is impossible to identily

the entropy by using assemblies in which temperature is

the only variable, for any function of the temperature
is then a possible integrating factor. There is only one way

of making the identification, and that is to evaluate dQ, the

element of heat, for an assembly of more than one variable
from our statistical principles, and to show that a certain
unique * function of the temperature 3 is an integrating
factor for it. The use of functions with the increasing
property can apparently never lead to precise results without
this appeal to dQ. We shall therefore abandon the whole of
the development of the preceding sections (4-7), including
the Boltzmann Hypothesis, and shall establish from first
principles that in fact the quantity dQ has a unique inte-
grating factor depending only on $3, and that this does lead
to Sst, for the entropy.

§ 7. The Entropy from first principles.
By the definition of dQ +, we have
GQ =a dara eww” (TL)

where E is the energy of the assembly, the «’s are certain
parameters defining the external fields, and the X’s the
associated forces. Let us suppose a generalized assembly

* Of course, an arbitrary constant multiplier excepted.

+ The “ heat” dQ taken in in any small change is defined in thermo-
dynamics to be the increase in internal energy plus the external work
done by the assembly. See e. g. Born, Joc. cit.

836 Messrs. OC. G. Darwin and R. H. Fowler on

composed of groups of systems ; let there be M, systems
which have a partition function /,.. This means that if the
possible conditions of one of these systems are that it should
be able to have energies ¢,1, €,2,..., and associated with

each of these states there is a weight factor p,,1, Pr, -
then

se 9

Ji — De Pr, to" ty

or the limit of this expression if, as for mechanical as
opposed to quantized systems, all the dimensions of the
cells must tend to zero. In order to allow for changes
of condition other than those of temperature, we must
suppose that each e,; is a function of the parameters
U1, 2, ...3; for example, in the case of free molecules in
a vessel the wall may be represented by a local field of
strong repulsive force, and then the potential of this
repulsive force must be contained in ¢,; With these data
we find at once by (2°3) that

: 0 ve
H=3,Masclogf, - - - + (TH)

| and the average number of the rth group of systems in their
tth cell is by (2°5)

yg = My iS 1 f(s ants @3,)-. 0
We also require to evaluate the external work done by the
assembly in any small displacement represented by small
changes in the parameters. Now the potential energy due
to the external bodies is contained in e¢,.;, and it will give
rise to reactions on the external bodies. If the positions of
the bodies are defined by the parameters 2, a, ..., the
reactions will be a set of generalized forces of amounts

ies t i ae ep
02, ae Ox, ae |

for each single system of group ¢ in the ¢th cell. The total
generalized force tending to alter. the parameter 2, will

thus be
ot rat anos ent)
Ox,

and its mean value will be

X, am Das ( - 3°, :)

= SMe pe 29%? (32 c,,.) fhe, Uy, Xo, ead

Ox,
4 a LE eNOS Milog f:(8; #), 45,
log 1/8 O21 Tirade ote) No) aah eae it

the Partition of Energy. ' 837
Then
dQ = FAD a XxX, day + De AX, +

= vl E lee ihe ao - ds

+ Mae — log f+ cas Py ae So ee i

S,M,d log flog 1/9. ao sou fr].
(7"4)
It follows that log (1/3)dQ is a perfect differential. We

can therefore at once define the absolute temperature scale,
so as to make dQ/T a perfect differential, by the equation

lod S Se d/eTP oop) OED

~ Jog 1/5 Ts

No function only of 3, except log 1/S, can be an integrating
factor of dQ, and Fhereford the absolute temperature so
defined is unique, apart from k, the constant undetermined
factor which it always an! Moreover, by definition
of Sth, dQ/T=dSin., and therefore, except for one arbitrary.
additive constant, |

Sa: = eM, (tog frtlog 1/9. 32. log ma

ds A whites Sal! ipahy (7°6):

where S,¢. is defined by (5:1). The identification of our
statistical entropy with the entropy of thermodynamics is.
complete, and the rest of thermodynamics follows in due
course, so long as the assemblies considered are of such:
types as to be representable by partition functions.

§ 8. The characteristic function of Planck.

We have presented the formal proof in its most familiar

form, but we can now make the presentation mathematically
much simpler. The expression (5°2) invites us to make
our fundamental definition not that of entropy but of
the ‘* characteristic function ” of Planck. This function VW,
which is closely allied to the “free energy,” is defined ii
thermodynamics by

eed pond pT

838 Messrs. C..G. Darwin and R. H. Fowler on
The characteristic function has the properties

ov

K= Por (3°2)
<= wT, (8°21)
: ey

= ae Cyne Me a ie (8°22)

Derinition.— The characteristic function for any part of
an assembly is k times the sum of the logarithms of the partition
functions of all the component systems of the part when the
argument of the partition functions is 3=e7 VAT,

With this definition we can show at once that the thermo-
dynamic processes are mathematically equivalent to those
we have been carrying out from the statistic: 1 point of view.
There is no need to repeat the work, as the mere change
from T to S$ exactly transforms (8°2), (8°21), (8:22) into
ATL), (7°6)) (7 3)ok V= kb, M loo f(3, #1; 253\..)

The characteristic function contains two arbitrary con-
stants, which occur in the form S)—E,/T. Of these, Ho is
seen to correspond to the arbitrary zero of the energy of the
systems, which appears in each exponent of the partition
function. ‘The constant So depends on the absolute values
adopted for the weight factors. We have made the con-
vention of taking this as unity for simple quantized systems;
but it is only a convention, and quite without effect on the
various average values, which are all that can ever be
observed. Indeed, the only conditions attaching to the
weight factors are precisely analogous to those attaching to
entropy in classical thermodynamics—a definite ratio is
required between the weights of states of systems which
can pass from one to the other (as in the dissociation of
molecules);—but as long as two systems are mutually not
convertible into one another, it makes absolutely no difference
what choice is made for their relative weights.

Many writers have attempted to give reality to the con-
vention that weieht has an absolute value, and from it have
defined absolute entropy. Such a definition cannot possibly
make any difference in any thermodynamic results ; but the
object was mainly to deal with the Nernst Heat Theorem,
and there it has been successful. It is, however, much more
rational to do without this somewhat mystical idea, and to
suppose that the theorem is a consequence of the equality of
weights of any allotropic forms in the states of lowest energy

the Partition of Energy. 839

that they may possess. The abandonment of absolute entropy
involves of course the acceptance of the paradox that the
entropy of 2 grammes of gas may not be twice that of
1 gramme ; but this paradox causes no real difficulty *.

§ 9. The “ canonical ensemble” of Gibbs, and its relation
to a temperature bath.

It is important to consider also the general question of
the truth of the Second Law as deduced statistically for
assemblies which are of some more complex type for which
the energy cannot be separated up in the way that has
hitherto been possible: any general proof must cover such
cases. Now Gibbs’ work deals with these generalized
assemblies, and he establishes with great simplicity the
necessary theorems, provided that he may start by postulating
the conditions of the * canonical ensemble.” But the idea
of “canonical” is not very easily defined, and it leaves
a slight feeling that there might be somewhere in it a
petitio principi. He later turns over to the “micro-
canonical”? conditions, but the calculus becomes rather
heavy. With our present method we can very quickly
show that Gibbs’ ‘canonical ensemble of phases” is, for
the purpose of averaging, equivalent to having our assembly
of systems in a temperature bath. i

Consider an assembly composed of a mechanical system
of n degrees of freedom, with coordinates Q;...Q, and
momenta P,...P,, together with a very large number M
of systems of any of the types we have treated. The
mechanical system exchanges energy with the others, but
for the greater part of its motion we may, as usual, think of
it undisturbed and in possession of a definite energy of its
own. For simplicity we may suppose the temperature to be
the only variable in the partition function 7 of the systems
of the bath, though thisis quite immaterial. Let the various
weight factors be po, pi, ... and energies €, €,... so that the
partition function is /($)=%,p,3°. For the mechanical
system we must take any element of phase dO(=dQ,.. OP.)
as having weight d0//h”, by the principles described in § 2 of
our former paper.

Now consider arrangements in which the mechanical
system is in dQ, while for the bath there are ap, ay, do, ...
systems respectively in states 0,1, 2,..... By the methods

* Ehrenfest and Trkal, loc. cit.

840 Messrs. C. G. Darwin and R. H. Fowler on

of our former paper the number of weighted complexions
will be |

: M! Per ee!)
PM nh (9°1)

Ag ! Ay ! ite
and we must have .
& a, = My: Drage = Ns? Sees

The probability that the mechanical system is in dQ is
measured by the total number of complexions for which
it is there, and so (9:1) must be summed over all values of
the a’s consistent with (9°2). Now if 3 is the solution of

d :
H—e= MS Clog f, “ey ee eae
this sum is

Cf (9) 349 dO
set 2M (s 5) bes ir

by virtue of § 6 of our former paper. Here $% is, strictly
speaking, the exact temperature of the bath at the moment
under consideration, and so will be liable to fluctuation
according to the value of e€; but, by virtue of the assumption
that the bath is very large, e wil] practically always be
insignificant in the solution of (9°3), and so S‘may be taken
us a constant. Then the probability that the mechanical
system is in the cell dQ is proportional to 3¢dQ, and all the
other factors are constant and may be omitted in taking
averages. Using (7°5) we thus obtain Gibbs’ expression
for the density-in-phase of the ‘“ canonical ensemble,”
namely

ds tna ley

This leads to the impossibility of perpetual motion and
all his work on the laws of thermodynamics.

§10. The Deduction of the Elementary States from

_ Thermodynamic Data.

An interesting result follows from the inversion of the
argument of $8. Suppose that we have an assembly of
unknown constitution in which the temperature is sole
variable. Then a knowledge of the specific heat determines |
the characteristic function, and thence the partition function.
Tf this can be expanded in terms of 3, we can determine
the energies and weights of the elementary states ; but the
matter is complicated by the fact that we cannot tell in

the Partition of Energy. 841

advance the size of the units of energy in which the
expansion is to be made.

The problem is exactly analogous to that solved by
Poincaré * in his deduction of the necessity for quanta
from the fact that Planck’s radiation formula agrees with
experiment. The machinery required has been examined
by one f of us in a recent paper. We give here a sketch
of how it may be applied to the present problem : reference
must be made to the original works for further detail. If
we write T=1/kT, we may suppose that the partition function
is known in terms of +t. The relation of the partition
function to the weights and energies from which it is
- generated may be put in the form of a Stieltjes’ integral :—

flr) = | emdw(e.

Here dw(e) represents the weight corresponding to e, and

it is indifferent whether we are concerned with quantized -

systems or mechanical ones with continuous distributions
of weight. The function w(e) can be determined by an
extension of the method of the Fourier integrals, which
(roughly speaking) leads to

1 a1
w(€) ete f(r) Jen

where « can have any positive value ch, a certain limit.
This is a complete solution, but it requires that / should
be known for complex values of 7, and in practice it would
be given in the form of a table, of course for real 7 only.
In general it would not be possible to find a simple analytic
expression to fit with the tabular values. This difficulty
can, however, be turned {, so that only the practical difficulty
of carrying outa large number of mechanical quadratures
would remain. For it is possible to associate with the real

function f(T) a complex function

H(q) = {saya

where g may have complex values, and then

1 (et J(q) dq
eS, oo Ga &
ole AHN Sat kg) 5
This is the formal solution of the problem; but it must be
doubtful whether it is reatly a practical method.
* Poincaré, Journal de Physique, ser. v. vol. ii. p. 5 (19
+ Fowler, Proc. Roy. Soc. A. vol. xcix. p. 462 (1921).
t Fowler, loc. cit. § 5.
Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. ae

842 Mr. J. H. Van Vleck on the normal Helium Atom

In this paper and its predecessor we have discussed the
relation of statistical theory to thermodynamics in detail
only for a rather limited class of systems, though it is
practically the class in which alone success has been attained
by anyone. It is at least doubtful how much further the
combinatory calculus can be pushed; as soon as the multi-
nomial theorem ceases to apply (as it would for imperfect
gases) great difficulties are encountered in our method, but
these difficulties are largely inherent in such problems.
In spite of these limitations, it would appear that the
potentialities of the method are by no means exhausted.

LXXII. The normal Helium Atom and its relation to the
Quantum Theory. By J. H. Van Vurcn, Jefferson
Physical Laboratory, Harvard University, Cambridge,
Mass. (U.S.A.).*

Pe I. of this paper is of a non-mathematical character,
and is concerned with the difficulties associated with
finding a_ satisfactory quantum theory model of normal
helium. After a résumé of existing models, a study is made
of the model suggested by Dr. EH. C. Kemble in which the
two electrons are arranged with axial symmetry, the one
symmetrical type whose energy has not yet been computed.
As the result of a rather laborious calculation, a value is

_ obtained for the ionization potential of this model which is

not in agreement with experiment, the discrepancy being
slightly greater than for the Bohr model.

Because of the failure of this model with axial symmetry,
it does not seem possible to devise a satisfactory symmetrical
model of helium based on the conventional quantum theory
of atomic structure, and the remainder of Part I. therefore ©
deals with the modifications in the ordinary conception of
the quantum theory or of the electron which may be
necessary in order to escape from this dilemma. Two
suggestions on reformulation of the quantum conditions made
by Langmuir are criticized, and a frankly empirical rule for
determining the stationary states is suggested which leads to
approximately the correct energy values for the helium
atom, the hydrogen molecule, and the positively charged

* Communicated by Prof. Lyman.

and its Relation to the Quantum Theory. 843

hydrogen ion. The difficulties confronting modification of
the law of force between negative elec trong as an alternative
method of explaining the dilemma of the helium atom are
also discussed.

Part II. assumes no knowledge of the quantum theory and
is an outline of the mathematical method used in finding the
orbits in the model of helium in which the two electrons are
arranged with axial symmetry. This method consists in
developing the perturbations as power series in a constant
of integration, and is readily adaptable to other problems
in the dynamics of atomic structure. A simple check on
the accuracy of solution is furnished by the theorem that
the average absolute values of the kinetic energy and half the
potential energy are equal.

Part III. deals with applications of quantum conditions to
the determination of the energy for the model of helium
possessing axialsymmetry. Various theories for determining
the stationary states prove to lead consistently to the same
values for the constants of integration.

The writer wishes to express his gratitude for the encourage-
ment and assistance given him by Dr. HE. C. Kemble in the

problems studied.

Parr L—Txe DILEMMA oF THE HELIUM Atom.
Résumé of Existing Models of normal Helium.

Probably the greatest success the quantum theory has yet
achieved is found in the Bohr atom, in which the electron is
allowed to move only in certain quantized non-radiating
orbits. However, the quantitative success of the Bohr
theory in explaining spectral lines and ionization potentials
has been confined to atoms containing only a single electron,
viz., the hydrogen atom and an abnormal helium atom which
has been robbed by ionization of one of its two electrons
normally present. In generalizing the theory to apply to
atoms with more than one electron, it is natural to begin with
the simplest possible case, namely, the normal helium atom,
which contains only two electrons.

Numerous attempts have been made to construct quantum
theory models of normal helium. In Bohr’s own model
(fig. 1) the two electrons revolve about the nucleus at the
extremities of a diameter *. In the Langmuir semicircular

* Phil. Mag. vol. xxvi. p. 492 (1913).
a 2

844 Mr. J. H. Van Vleck on the normal Helium Atom

model (fig. 2) the two electrons describe an oscillatory
motion in a plane, always being symmetrically situated with
respect to an axis passing through the nucleus*. In the
Landé ¢ aud the Franck and Reiche t models the electrons
have more complicated orbits of unequal size, which are
coplanar only in the case of the Landé model (fig. 3).
In the Langmuir double-circle model (fig. 4) the orbits are
circles lying in two parallel planes *. . ;

Pig wls Pio.

Ris ge Fig. 4.

The dimensions of these models must be determined by
applying the Sommerfeld-Wilson quantum conditions, which

state that
{ pdq=nh,

where fh is Planck’s constant, n is an integer, g is an appro-
priately chosen generalized coordinate, and p is its conjugate

rt
L A “ &

momentum Le . The integration is to be extended over a
Y

* Physical Review, vol. xvii. p. 839 (Mar. 1921).
+ Phys. Zeits. xx. p. 233 (1919) and xxi. p. 114 (1920).
{ Zetts. f. Phys. i. 2, p. 154 (1920) ; Phil. Mag. vol. xlui. p, 125.

La
.
-

and its Relation to the Quantum Theory. 845

complete cycle of values of g. Having thus determined the
dimensions, we can compute the energy and compare the

calculated value a that found experimentally from ioniza-
tion potentials. In no ease is the agreement satisfactory, so
that apparently none of these models can be correct if the
Sommerfeld quantum conditions are accepted. For a more
thorough exposition of the difficulties confronting these
models the reader is referred to a recent paper by E. C.
Kemble in the Philosophical Magazine *

Since none of these models can be regarded as thoroughly
satisfactory, it is natural to inquire whether there cannot be
some other possible model. In making this investigation we
must bear in mind that the extreme chemical stability of
helium indicates that the arrangement of its two electrons is
particularly simple and symmetr rical, for an electron revolving
in an orbit outside that of its mate would presumably be a
valence electron. Symmetry with respect to a point yields
the Bohr model, already mentioned. The two simplest cases
of symmetry with respect to a plane (figs. 2 and 4) have been
investigated by Langmuir, and yield impossible ionization
potentials of approximately the same size (—4°6 and —8°5
volts) t+. It is extremely doubtful whether other more
complicated, and therefore less probable, orbits symmetric
with respect to a plane would yield ionization potentials
differing very widely from those of these two simple limiting
cases I.

Study of Model with Awial Symmetry.

The only remaining type of simple symmetry which has
not been studied is that with respect to an axis. It
therefore seemed desirable to compute the ionization
potential of a model possessing this kind of symmetry,
which was suggested by Dr. E. C. Kemble §. Since the
two electrons I. and IT. move in three dimensions so as to
always be symmetrically located with respect to an axis,

* Phil, Mag. vol. xlii. p. 123 (July 1921).

+ Physical ‘Review, vol. xvii. p. 339.

{ Cf. identity of elliptical and circular energy lev ais 4 in the hydrogen
atom (relativity corrections neglected), In the two Langmuir models,
projection of motion on plane “of symmetry is a straight. line or circle.
The most general motion symmetric-with respect to a plane would pro-
ject into a sort of precessing ellipse, which may be regarded as
intermediate between the above two cases. The more oeneral mation
might involve impossible singularities, such as continual distortion of
shape of ellipse.

§ Loe. cit.

846 Mr. J. H. Van Vieck on the normal Helium Atom

their cylindrical coordinates are
Th, ©, IE. R; Z, @-o7,

the Z axis being that of symmetry. This model may be-
regarded as a sort of hybrid of the Bohr and Langmuir
models. The constant angular momentum, which the elec-
trons possess about the axis of symmetry, reminds one of
the Bohr model, and the projection of the motion on the
plane Z=0, which is normal to the axis of symmetry, is
a sort of precessing ellipse. The type of motion in the
RZ plane can be seen by returning to fig. 2. This motion
is of an oscillatory character, and is similar to that of the
Langmuir semicircular model, except for the effect of a
centrifugal force term introduced by the rotation of the
RZ plane about the axis of symmetry as the coordinate ¢
steadily increases. The motion may be approximately
described as the projection of a sine curve on a barrel-shaped
surface of revolution, the two electrons always being on
opposite sides of the barrel. 3

Rough preliminary calculations for this model indicated
approximate agreement of the computed and observed
ionization potentials, and it was therefore necessary to carry
through a more accurate solution of the dynamical problem,
which took almost six months. The orbits of the electrons
in the helium atom were determined as power series in a
constant of integration, a mathematical method often used by
astronomers in the three body problems of the solar system.
A more detailed description of the mathematics used in
solving the dynamical problem, as well as the method used
to check the accuracy of solution, is.given in Part II.

After solution of the dynamical problem, the constants of
integration were determined by ihe quantum conditions (see
Part III.), and the ionization potential was then computed,
which proved to be 20°7 volts for the removal of one electron
or 74:9 volts for the removal of both electrons. This does
not agree with the experimental value of 25:-44°25 volts”,
but the discrepancy is only slightly larger than for the Bohr.
model, which yields 28:8 volts, the closest agreement
obtained by any model based purely on the Sommerfeld
quantum conditions. :

The Dilemma of the Helium Atom.

As already mentioned, the extraordinary chemical stability
of helium indicates that the arrangement of its pair of
electrons is particularly simple and symmetrical, but all

* Franck and Knipping, Phys. Zeit. xx. p. 481 (1919).

and its Relation to the Quantum Theory. 847

models possessing this property now appear to have been
weighed in the balance and found wanting, as they lead to
impossible energy values. The conventional quantum theory
of atomic structure does not appear able to account for the
properties of even such a simple element as helium, and to
escape from this dilemma some radical modification in the
ordinary conceptions of the quantum theory or of the electron
may be necessary*. One such possibility is :

Reformulation of the Quantum Conditions.

Any reformulation of the quantum conditions which aims
to explain the anomaly of the helium atom by permitting
new energy values must yield results identical with those of
the ordinary Sommerfeld quantum integrals in the cases of
the hydrogen atom and a vibrating diatomic molecule, for in
these instances the Sommerfeld conditions are verified by a
mass of experimental evidence.

Two very interesting suggestions on reformulation of the
quantum conditions have been made by Langmuir tf. One of
these suggestions is that in a system with two negative
electrons the ordinary Sommerfeld integrals should be
replaced by the condition that the maximum angular
momentum of a single electron be set equal to = =e a)

: 2
Langmuir’s semicircular helium atom (fig. 2), this maximum

r) ° ~ .
value a would be achieved when each electron is at the
ZW.

~

middle of its path, so that we may regard the atom as
having two quantum units of angular momentum circulating
about the nucleus in opposite directions. This new condition
for determining the constants of integration yields the
correct ionization potential for the semicircular model of
helium +, and therefore merits serious consideration. How-
ever, it appears to be a contradiction, rather than generaliza-
tion, of the Sommerfeld quantum conditions, for in the

* The possibility of some asymmetrical model of normal helium
should perhaps not be entirely rejected, despite its apparent contra-
diction to the view of chemists on the symmetry of helium. In this
connexion it should be mentioned that the energies of the Landé
and the Franck and Reiche models are not computed directly, but are
obtained by extrapolation of spectral series terms. This extrapolation
pre-supposes the validity of the Landé theory of the helium spectrum.
Also the continuation of a curve can never be predicted with absolute
certainty by extrapolation.

+ Physical Review, vol. xvii. p. 339, vol. xviii. p. 104 (1921) ; also
Science, vol. lii. p. 434.

848 Mr. J. H. Van Vleck on the normal Helium Atom

hydrogen atom and the diatomic molecules of band spectrum
theory it is the total angular momentum of the entire system
(comprising two bodies), rather than the maximum angular
momentum of a single electron, which must be equal to
h
Qa’
condition to be equally applicable to his semicircular model
of the hydrogen molecule, for the latter is almost identical
with his semicircular helium atom, as the two electrons
oscillate back and forth about the centre of the line joiming
the two hydrogen nuclei. However, computations made by

Also one would expect Langmuir’s new quantum

the writer indicate that the maximum angular momentum of

fics. he
a single electron must be ‘I9D 9— s instead of exactly 5— ,ifthe

ionization potential of hydrogen is to have the proper value,
so that the scheme which works so well in helium does not
seem to yield correct results for a similar case in hydrogen.
Langmuir’s other suggestion consists essentially in
replacing the centrifugal force term found in the ordinary

dynamical Bohr theory of the hydrogen atom by a statical

force of equal magnitude, leading to a static atom. This
force may be accounted for by assuming that the electron is

2
a (= . The super-

an electrical doublet of strength
2em \2a

position of this new static force on the ordinary Coulomb |

force appears contradictory to the scattering experiments
of Rutherford on the validity of the inverse square law,
and to the dynamical orbits found in band spectrum and
specific heat theory. This static theory yields the correct
ionization potentials for the helium atom and the hydrogen

molecule *, but the strength of the electrical doublet

would have to be modified to depend on the mass of
the attracting nucleus, which is highly improbable, in order
to explain the observed shift between the lines of the Balmer
series of hydrogen and those of the Pickering series of

helium, a shift which the ordinary dynamical theory.

naturally accounts for as a correction for the motion of the
nucleus. A further objection to the electrical doublet
interpretation of the new static force is that the doublet
would presumably orient itself so as to be attracted rather
than repelled by the nucleus, giving a force of wrong sign
(centripetal rather than centrifugal). Other difficulties
confronting any static atom are explanation of the Stark and

* Bulletin of the National Research Council, no. 14, p. 347.

and its Relation to the Quantum Theory. 849

Zeeman effects, the selection principle, and the Sommerfeld
fine-structure.
In periodic motions the so-called action integral is

7

2 dt,

0

where T is the kinetic energy and 7 is the period of the
motion. In view of the fact that in periodic motion this
quantity is an adiabatic invariant, and that, according to the
relativity principle its value is independent of the particular
set of Galilean axes chosen as a reference system, one might
expect any form of quantum conditions to be expressible in
the form of a restriction on the value of the action integral.
If the ionization potential of the Langmuir semicircular
helium atom is to have the proper value, its action integral
must be equal to 1°578 A, while the corresponding value for the
semicircular hydrogen molecule is 1:°399h*. A very good
approximation to these valnes is obtained by assuming that
the action integral associated with one electron can have the

m

7 \ 2 : :
value (5). h, where mis an integer. m must be taken equal

to zero for the hydrogen atom and the K ring of X-rays,
while we shall set m=2 for the normal helium atom and
the positively-charged hydrogen ion (systems with three
bodies) and m=3 for the hydrogen molecule (a system
with four bodies). This yields 1:571h for the action
integral of the helium atom and 1°393h/ in the case of
the hydrogen molecule: Also for a model of the positively
charged hydrogen ion in which the electron revolves
about the centre of the line joining the two nuclei f, this
rule gives an ionization potential of 11°48 volts, which
agrees well with the experimental value of 11°5+°7 volts
found by Franck, Knipping, and Kriiger{. In the cases
of the helium atom and the hydrogen molecule, the agree-
ment is not quite as good as might be desired, hut it does not
appear impossible to explain the discrepancy as due to
experimental errors in measurement of the ionization
potential and in determination of the atomic constants e, h;m.

* These quantities are readily computed from data in Langmuit’s papers
(Science, vol. lii. p. 484; Physical Review, vol. xvii. p. 352). Compu-
tation for helium has also been made by E. C. Kemble (Science, vol. li.
p- 581). In calculations, ionization potentials of helium and hydrogen
atoms were taken as 25:4 and 13°55 volts, and heat of dissociation of
hydrogen as 84,000 calories/mol.

+ Cf. Sommerfeld, ‘ Atombau und Spektrallinien,’ 2nd ed. p. 514.

} Verh. d. D. Phys. Ges. xxi. p. 728 (1919).

850 Mr. J. H. Van Vleck on the normal Heliun Atom

The writer has not been able to find any theoretical basis for
these empirical rules, and they appear rather hard to reconcile
with the Bohr Analogy Principle, but it should be remembered
that the type of dynamical system to which they are applied
is somewhat different from that met with in the hydrogen
atom, where (with neglect of relativity correction) the action
integral can only be an integral multiple of h*. Also the
atomic models to which these empirical rules appear applic-
able are those whose physical properties seem to be in best
agreement with experiment. ‘This is especially true as
regards the semicircular modei of the hydrogen molecule,
while the case of the helium atem will be discussed on
later pages. The moment of inertia of the semicircular
model of the hydrogen moiecule agrees well with the value
found from band spectra, while its zero angular momentum
about the axis of symmetry is in accordance with specific
heat theory, the diamagnetism of hydrogen, and the behaviour
of the many-lined hydrogen spectrum in a magnetic fieldy.
No other model has been proposed which possesses these
properties.

Modification of Law of Force between Negative Electrons.

In the absence of any thoroughly satisfactory attempt at
reformulating the quantum conditions, we must consider as
an alternative the modification of the law of force between
negative electrons at atomic distances. Any alteration of
the law of force between negative electrons and positive
nucl-i (such, for instance, as would result from a highly
aspherical nucleus) would probably invalidate the Bohr
theory of the hydrogen atom and contradict the experimental
evidence of Rutherford on the validity of the inverse square
law, but his work yields no information on the forces
between two negative electronst: On the other hand,
A. H. Compton concludes that the spiral tracts of beta
particles indicate that the field of an electron does not have the

* Cf. remark by Bohr: “ Diese Storungen (7. e., mutual action between
electrons) geben namlich ftir die beiden Partikeln des Heliumatoms zur
Bewegungen Anlass, deren Charakter sich als iberaus verwickelt erweist,
und zwar so, dass die stationaren Zustainde nicht festgesetzt werden
konnen in direkter Anlehnung an die Methoden die fur bedingt
periodische Systeme entwickelt worden sind” (Zetts. fiir Physik, Band 2,
Heft 5, p. 465).

+t Of. Langmuir, Science, vol. lu. p. 434, Physical Review, vol. xvii.

. 339. :
‘ t In a very recent paper (Proc. Roy. Soc., Feb. 1922), Crowther and —
Schonland, however, conclude that some modification of law of force at
very small distances, either between two negative electrons or an

lectron and a nucleus, appears to be demanded by their experiments on
She scattering of beta varticles.

——

and its Relation to the Quantum Theory. 851
spherical symmetry demanded by the Coulomb law *. One
way to account for the magnitude of the observed dis-
symmetry is to assume that the electron is a magnetic doublet
of strength about 10°79 E.M.U.* (such as would be produced
by a rotating ring of electricity having one quantum unit
sof angular momentum). However, the strength of the
magnetic doublet would have to be about 5x 107" Daa Fea Ope
a value fifty times as large as that given above, in order
to have the desired effect on the ionization potential of helium.
The existence of a magnetic doublet of such a large size could
not be reconciled with observed molecular magnetic moments,
which are very much smaller, and would invalidate the
classical theory of the scattering of X-rays, as the magnetic
forces acting on this magnetic electron would be com-
parable with the electrical forces when the wave-length of
the impressed beam of light did not exceed the atomic
diameter. It is doubtful whether it is possible to attribute
to an electrical or magnetic origin departures from the
inverse square law of sufficient magnitude to explain the
anomalies of normal helium. Instead it would be necessary
to introduce a ‘‘ mystery force,” which is negligible except at
atomic distances, and which does not have a mechanism based
on the Maxwell field-equations. This new force should
explain the spiral tracks of beta particles as well as the
properties of helium. In support of the idea of introducing
this rather arbitrary ‘mystery force,’ it should be stated
that it is not improbable that such a bold hypothesis may be
necessary in order to explain the stability of atomic nuclei.
A cogent argument against modification of the law of force
between negative electrons is that it cannot account for the
absence of a satisfactory model of the positively charged
hydrogen ion, which contains only one electron. Also a
simple computation shows that a mystery force between elec-
trons depending only on their relative distance (consequently
developable in a series in inverse powers of the distance)
would invalidate the Sommerfeld-K ossel theory of X-rays T,

* A. H. Compton, Phil. Mag. vol. xli. p. 279, Feb. 1921.

+ This Ah as a consequence of the fact that centrifugal force
alld = 7s acting on an electron having one quantum unit
of angular momentum varies inversely as the cube of the radius as
the latter changes from element to element in the K ring. Therefore
an inverse cube mystery force comparable with the centrifugal force
in the helium atom, so as to give the required alteration in the
ionization potential, would also be comparable with the centrifugal
force in the K rings of elements of higher atomic number, and hence
would have an appreciable effect on the energy. With mystery forces

involving higher inverse powers, the effect would be even larger.

852 Mr. J. H. Van Vleck on the normal Helium Atom

and to retain the latter it would appear necessary to make.
some artificial and improbable assumption concerning the
character of the mystery force, such as having it depend on
the vetocity in such a way that it became negligible at the
high velocities found in the electrons of the K and L rings.

Conclusion.

As yet it appears possible to devise a satistactory symmetrical
model of the normal helium atom only with the aid of some
such radical innovation as reformulation of the quantum
conditions or modification of the law of force between
negative electrons. The probability of the latter alternative
is discounted by the success of the ordinary quantum theory
of X-rays. It is to be hoped that with one such bold
hypothesis we can a dee obtain the proper energy
values for models both of the helium atom and the hydrogen
molecule. The models of normai helium which are physically
most plausible seem to be the Langmuir semicircular one or
that with axial symmetry. The zero resultant angular
momentum of the Langmuir model is perhaps in best accord
with the observed diamagnetism of helium, and if the semi-
circular model of the hydrogen model is correct, one would
expect the normal helium atom to be of a similar type.
However, the type with axial syminetry has the advantage
of requiring smaller departures from the conventional
quantum conditions or less readjustment of the law of force
in order to obtain the correct ionization potential*. Some
very interesting experimental evidence on the structure of
the helium atom is given in a recent article by Millikan f.
Observations taken with his oil-drop apparatus indicate that
_when an alpha particle collides with a helium atom it hurls
out both electrons about 16 per cent. of the time, while
during the remaining 84 per cent. of the collisions it ejects
only one electron. ‘This, Millikan concludes, eliminates the
possibility of the Bohr model of normal helium, as the chances
of the incident alpha particle having just the right direction
to collide with the second electron after already striking
the first one would be exceedingly small in a model of this
character. Models such as those of Landé or Franck and
Reiche, in which the two electron orbits differ considerably
in size, are also rejected, as the innermost orbit exposes

* The modifications would have to produce a change of 30:0 volts in
the ionization potential for the semicircular model as compared with 4:7
volts for the type with axial symmetry.

+ Physical Review, vol. xviii. p. 456, Dec. 1921.

and its Relation to the Quantum Theory. 853

too small an area for possible collisions. Instead, Millikan
concludes that the correct model is one in which the orbits
are of equal size and inclined at an angle of 60° or 90°, so
that the two electrons might be in the same part of the atomic
volume about one-sixth of the time. The type in which the
two orbits are oriented at 90° does not appear to be allowed
by the quantum conditions, while that in which the inclination
is approximately 60° is that studied in detail in the present
article. However, it appears to the writer that in addition
to this model with axial symmetry the Langmuir semi-
circular model is also in accord with the experimental
evidence, although Millikan does not mention this possibility,
for an incident alpha particle would probably eject both
electrons when they are close tozether at the extremities of
their paths (see fig. 2).

Part Il.—So.utTion oF DYNAMICAL PROBLEM OF MODEL oF
HELIUM IN WHICH ELECTRONS ARE ARRANGED WITH
AXIAL SYMMETRY.

Introduction.

Besides its direct bearing on the study of the helium
atom, the determination of the orbits in a model of helium
in hich the electrons are arranged with axial symmetry is
of interest as a solution of a special case of the problem of
three bodies, and as an illustration of how the standard
methods of celestial mechanics may be employed to solve
the dynamical problem of sub-atomic physics. As no set of
coordinates was found which would separate the variables in
the Hamilton-Jacobi partial differential equation and thus
yield an exact solution in closed form, it was necessary to
haye recourse to methods of perturbations, similar to those
used by astronomers in lunar theory, ete. The method of
celestial mechanics which is particularly applicable to our
problem is that in which the perturbations are developed as
power series in a parameter *. The particular parameter
selected was a constant of integration depending on the
inclination of the two electron orbits relative to each other.

* The other standard astronomical method, one based on successive
approximations and mechanical intezrations, cannot readily be employed,
because the constants of integration are not known in advance but must
be determined by quantum | conditions after the solution is obtained.
This other method, however, could be and was used by Langmuir in his
semicircular helium atom, as the only arbitrary constants were scale of
model and origin of time, while two additional constants appear in the
present problem.

854 Mr. J. H. Van Vieck on the normal Helium Atom

The solution is thus obtained in the form of a family of orbits,
each member of which corresponds to a particular numerical
value of the constant of integration in which the power series
development was performed, and hence to a particular angle
between the plane of the two electron orbits.

Derivation of Equations of Motion.

If the Z axis be taken as that of symmetry, the cylindrical
coordinates of the two electrons I. and IJ. are

LR: Z; by 1 RA em (cf ios: Waele

Because of the very large mass of the nucleus, its motion
may be neglected, so that the total kinetic energy of the
system is that due to the two electrons, viz. : .

T=m[R?+ R26? + 27], Ate (JL)
while the potential energy is
—4e¢? e
\fe= VARIO aR? 2 (2)

where m is the mass and —e is the charge of an electron.

2
The term om represents the mutual Rorenizal energy of the

electrons, while the term
—4¢
VW R242
corresponds to the attraction exerted on the two electrons by

the positive nucleus, whose charge is + 2e. The Lagrangian
equation

(Ola oP eu,
dt sa) op "oe
gives us immediately the first integral

2mR2h =p, « So eal et) ten

where p is a constant equal to the resultant angular
momentum of the system. The corresponding Lagrangian

and its Relation to the Quantum Theory. 855

equations for R and Z are
e2 2e?R

mR ~—mR¢? = ip (R24 ZA 32 } (4)
: — 2°, ; :
mL= Gis Si OR a ae (9)

If we eliminate b by means of the relation (3), equation (4)
becomes

mi= Ri tae eazy LO

It will be found convenient to use certain reduced units
expressing the fact that tlie scale of the model is arbitrary.
Let

p? 1 é 2e*R
)

oat 2e sake cat Le Meek” Se }
or Ae OF ok Sake Ba Smet, 6 1)

where A is a constant depending on the size of the model.
The equations of motion then take the form
4

A if
PB =o == Ly —= (24+ O72 wa aa bac St ce iciaet tats (8)

Cre ae ae
Baers ote @)

Development of Solution as Perturbations from
Circular Orbits.
Let
n= V¥1—wcos? Bt, S=w? cos Bt,

where w is a positive constant less than unity. If we neglect
the term 37 in equation (8), which is due to tlie force of
repulsion between the negative electrons, r=, €=S will be
an exact solution of the dynamical problem provided the
constants p, A, and w satisfy the relation C=1—w. The two
electrons then revoive about the nucleus in the familiar
circular orbits found in the problem of two bodies. ‘Lhe
intersection of the planes of the two orbits is a nodal line
perpendicular to the axis of symmetry. .w? is the sine of the
angle between the plane of one of the orbits and the plane
normal to the axis of symmetry.

To obtain a solution of the complete equations of motion
let us write

r=ntp, . €=S+s

856 | Mr. J. H. Van Vleck on the normal Heliwn Atom |

p and s are then the perturbations from the circular motion
caused by the forces of repulsion between the two electrons.
Since p and s are small compared with 7, we can expand the
various terms in equations (8) and (9) as power series in p
and s, and with sufficient accuracy for our purposes neglect
terms of third or higher degree in p and s. When these
expansions are performed, (8) and (9) become

3 3(1— i
Be tel gta ont
il:
=f O-1+ w+ gnt3on'S +p" =
15 Le D3 5
~31|- | 12m eat os
tte 2a]
3
= Be s= 3057+ 3s8? + p? [58- Bog

=
+ps[3n—1on8"] +[ 58-28" ]. ge reer

= w) 3am af

In obtaining these equations simplifications have been
effected by combining coefficients of like powers of p and s,
and by using the identities

77+S?=1. o on == af), w=-8.

Development of Perturbations as Power Series in the
Constant w.

The differential equations of motion are perhaps most
readily solved by developing p and s as power series in the
constant of integration w. With this method the solution
can be built up step by step by equating to zero coefficients
of successive powers of w in equations (10) and (11). The
equations obtained by equating these coefficients to zero are
linear and prove to be readily integrable. In order to
obtain this series development it is first necessary to expand
the various powers of n= W1—weos? Bé which appear in
equations (10) and (11) as power series in w cos? Bé by
means of the binomial theorem. After these expansions are.

and its Relation to the Quantum Theory. 857
performed, (10) becomes to terms of the 7th order in w :
bo [1—1°5w cos? Bt +°375w? cos* Bt + :0625w’ cos* Be
+ *02344w* cos’ Bt +°01172w’ cos! Bt + °00682w* cos” Be |
+ p[°875+7:5w cos? Bt—4°125w? cos* Bt + 1°9375w? cos® Be
+°96094w* cos® Be + *78516w* cos! Bt+ °69830w* cos! Be]
+ p[ —3w— 1:5w? cos? Bt— 1°125w? cos* Bt —-9375w* cos® Be
—°82031w° cos’ Bt —:73824w® cos’ Be]
= [C—*8754+ w—'0625w cos? Bt—*015625w? cost Bt
—°007813w* cos’ Bt —:00488 22 cos® Be
— 00341 7w? cos’? Bt—-002563w' cos” Bt —*002016w’ cos#Be]
+ sw? cos Bt| 3—6w cos? Bt + 3w? cos* Be |

+4
Be

Be: [—3-+3w cos? Bé] + p?[ —4°5w cos? Be
—12w’* cost Bt + 4°5w? cos® Bt — 3w* cos® Bé— 3w’ cos” Be]
+ p?| 3w + 3w? cos? Bt + 3w? cost Bt + 3w* cos® Bt + 3w? cos* Be]
+ s?(1°5—10°5w cos? Bt + 16°5w? cost Bt —7:5w? cos® Be]
+ psw? cos Bt[ —3+ 19°5w— 23°625w? cos Be
+ 5°4375w" cos® Bé-+-"8672u* cos® Bt]. s. . « « (12)

Similarly, equation (11) becomes
2
BR?
— -0625w? cos* Bt —'03906w* cos® Bt —°02734w* cos Bt
—°02051w® cos!” Be}
+ 3ws cos? Bt + pw? cos Bt[ —6+7:5w cos? Be]
+ s?[4°5w!? cos Bt —7-5w3? cos? Be]
+ ps{3—15w cos? Bt] [1—*5w cos? Bt —*1250w? cos* Be
—‘0625w? cos’ Bt —-0390w* cos® Bt —°0273w? cos! Be]
AA MER Ge.)
Let us now assume that p, s, and C can be expanded as

Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3K

+s=3pw? cos Bt{1—“dw cos? Bt —*1250w? cos* Be

858 Mr. J. H. Van Vieck on the normal Helium Atom

power series in the constant of integration w, so that we may
write *
p=piw-+ pow? + p,we+.. .,
S==101/*| Ss) ++ Sth” Sg? =o). |,
C=C,4+ Cywt Cow? + Cow? +. . .,

where the coefficients P1s P25 P39 Sty So, 83, ete., are functions of
the time not involving w, and Co, 1, Og... are purely
numerical constants independent of w.

Determination of Zero and First Order Terms in w.

Since the equations (12) and (13) hold for all values of w,
the coefficients of the various powers of w in these equations
must each vanish separately. There are no terms of order
lower than w?? in equation (13), while if we equate to zero
the terms in equation (12) which do not involve w a e-,
coefficient of w°) we obtain Co=°875.

To determine the first-order term pi we equate to zero the
coefficient of the first-degree term in w in equation (12).
We thus obtain

This gives on integration
P1= Por + °010000 cos 2Bt-+ D, cos [4/F Pte) it
where D, and ¢« are arbitrary constants and where
49, = 96875 + C,.

It is necessary to set D,=0 in order to avoid introducing
‘* Poisson terms” in the higher order coefficients pp, So, ete.
These terms are those of a type in which the time enters
explicitly as well as through trigonometric functions

(e.g., ¢ cos v ZBe), and would probably lead to very large
perturbations, in which the distance of the electrons from the

5 |
cot (Ce — is a function of w, since the relation between the angular

momentum p and the scale A of the model depends on the relative
inclination of the two electron orbits. That the expansion of s involves
fractional powers of the form indicated above follows from the fact that
the expression for sB-?+s in (18) contains terms of the form
Dpw"t# cos?”t1Bz¢ (D a purely numerical constant and m an integer).
It is interesting to note that just such fractional power series in a
parameter are frequently met with in lunar theory.

and its Relation to the Quantum Theory. 859

nucleus might tend asymptotically to the value infinity *,
Even if this difficulty were non-existent, the introduction of

an extra frequency of vibration Ji = by setting D,540

would lead to more complicated motions, an additional
quantum number, and a negative energy content of smaller
absolute value, none of which are characteristic of the
normal state of the atom.

The values of po and C, are determined by equating to
zero the coefficient of w®? in equation (13). We thus obtain

a + s;=3[ po, cos Bt+°01 cos 2Bt cos Bt] =3po, cos Bt

+ °015[cos Bt + cos 3B],
whence for a periodic solution
Pou — ‘005, s,= E cos (Bt—e,) —°001875 cos 3Be.

The constant E may be set equal to zero, since we already
have an arbitrary term in cos Bt (viz., the term S=w*” cos Bt
corresponding to the unperturbed motion). The first terms
in the expansion of p, C, and s in terms of w are therefore

C=°875000—-973125w,
p=[— °005000 + -010900 cos 2Bélw,
se=(— 001STo cos 3BI)wi?. i Se 4)

Determination of Higher Order Terms in w.

Making use of the relations given in (14) we can equate
to zero the coefficients of w? and w®? in equations (12) and
(13) respectively, and so determine py, Cy, ands,. Knowing
p2 and s, we can determine third-order terms, and thus
continue the process to any desired degree in w. ‘The
calculation of the first-order terms given above in detail is
typical of the method used in computing the terms of higher
order. With the aid of the trigonometric reduction formulas
which express cos” Bt as a sum of linear trigonometric
terms, the type forms obtained by equating to zero the

* Cases, however, are sometimes found in dynamical theory where
the Poisson and secular terms (in which the time appears explicitly)
combine in such a way as to yield aconditionally periodic motion. Bohr
has shown that when the perturbing potential has axial symmetry, the
motion may be regarded as conditionally periodic if we consider only
first-order terms in the perturbing field (Quantum Theory of Line
Spectra, pp. 53-6), but this approximation is not sufficient in the present
problem owing to the large amount of mutual action between electrons

5K 2

860 Mr. J. H. Van Vleck on the normal Helium Atom

coefficients of w* and w"*/? in (12) and (13) respectively
may be written: | |

B22py + Son-C, = Fy + F, cos 2Bt+ F, cos 4Bt... 3
+F,, cos 2nBt,:.. 12. eee

B25, + S:—3Pn cos Bt= Gy cos Bt+G, cos3Bt...
+ Gy cos Qa+1) Bi... eae

where the F’s and G’s are purely numerical coefficients.
The periodic solution of equations (15) and (16) is

Pn = Pon + Pin COS 2 Bt + pon cos 4Bt. . .+ pun cos 2nBt,
Sn = Stn C08 3Bt+ Son cos SBE... +5y, cos (2n +1) Bt,
Cr=Fpor— Fo, |

where

—Pon= Pint 1G pin= ciate as (j£0)

Por = 2Pint-3%o Pn G—4) ’ )
yee G5 +3 (Pint Pim)
‘ 1—(27+1)

Pins P2ny -++Pan are thus determined by equating to zero
the coefficients of cos 2Bt, cos 4Bé, . .. cos 2nBt respectively
in equation (15), while sy, so, . ++ Snn are found by equating
to zero the coefficients of cos 3Bt, cos 5Bt,.. . cos (2n+1)Bé
respectively in (16). pon is determined by equating to zero
the coefficient of cos Bt in (16), thus avoiding the necessity
for introducing in the expression for s, the Poisson term
Gétcos Bt (G some constant), in which the time enters
explicitly.

Final Deternination of Orbits.
Following this method of attack, we obtain for the final
result :— : |
p= [| —'005000w — 0036430? — -002386w? —-001504u4
—°000887w° —-000386w' —-000024w'+ .. .1
+ [:010000w + -009592w? + °008435w? + 0074504
+ 006632w? + -005799w* + °005038w’' |cos 2 BE
- + [:002507w? + 00381 75w? + 003300w* + -00317 7m?
+ °002988w* | cos 4 Bé
+ [°0004134w3 + -0007303w* + -0009320w°
+ °0010386w®] cos 6BéE
+ [:0000811w! + -0001825w° + 0002760w® | cos 8 Be
+ {°0000169w? + °0000465w*] cos10Bé + *0000036* cos 12 Be,

and its Relation to the Quantum Theory. S61

s=w?| —:001875w —:001470w? — -001165w? — 000944
— ‘0007 66w? —:000607w*| cos 3 Be

+w?[ —'0000125w? —-0000140w? — 00001250! —-0000067w*
—°0000026w* | cos 5 Be

+ w?[ —-0000007w' — -0000011w* — -0000013w°
—°0000013w*] cos 7Bt + negligible terms in cos 9Bé, etc.,

C= —P _ —-375 —-973125w+-03207 20? + °021074w?

~~ &meA
+ °014987w!t + °011207w® + (008694w* + °006522w7
Me Og eee (17)
R=A|[Vv 1—w cos? Be +p|. Z=A[w'?cos Bé+'s],
t
o= me mca adn Ae (9)

aoe €.
2mk?2 z
0

This family of orbits is the most general solution in which
R and Z are periodic functions of the time. The power
series expansions of the perturbations p and s have been
carried far enongh to enable one to compute the energy
through terms of the seventh order in w*. ‘This solution
contains four arbitrary constants (p, w, an epoch angle for »
and one for R and Z) ft, while two other arbitrary constants
are eliminated by the requirement of periodicity for R and
Z. The mean period of @ differs from that of R and Z,
giving a preces-ion of the line of nodes formed by the inter-
section of the orbit with a plane normal to the axis of
symmetry. The Cartesian coordinates #=Rcos@ and
y=Ksin @ therefore contain trigonometric terms having
two distinct periods, and the motion is conditionally
periodic, the orbits not being reentrant (except when pro-
ay on the RZ plane). In carrying out the solution we

ave nowhere assumed the existence of solutions in which
R and Z are periodic functions of the time, but were
automatically led to solutions of this character on performing

* For proof of convergence of power series expansions in w, cf. Moulton,
* Periodic Orbits,’ pp. 15-19.

+ In performing the calculations, the epoch angle for R and Z has
been so chosen that R=Z=0 at t=0. This involvesno loss of generality,
as the origin of time is immaterial in computing the energy.

{ The mean period of ¢ is found by taking 27 times reciprocal of mean
angular velocity which is the constant term in the Fourier expansion of

ie
2mR?’

862 Mr. J. H. Van Vleck on the normal Helium Atom

the power series developmentin the parameter w. For w=0,
the perturbations p and s vanish, and C=‘875 ; this is an
exact solution, which is nothing else than the Bohr helium
atom, in which the two electrons move in the same plane.
As the constant of integration w is given increasing values,
the orbits intersect each other at greater angles, and the
perturbations become larger. ?

Evaluation of Energy.

After the orbits have been determined, the next step is to
compute the energy, so that the ionization potential may be

calculated. If we eliminate ¢ by means of the relation (3)
and use the reduced units of (7), the expressions for the
kinetic and potential energies given in equations (1) and (2)
become

_ 28 (ae) e]
tT=5 | B? teh

2
ieee
AL 2r yee

Next, making use of the relations r= +p, f=S+s, expand
the various powers of r and € as power series in the perturba-
tions p and s, neglecting terms of third and higher orders.
This gives .

2e? C= ata Ia eee fe eee :
nse. [1+ + iyo (2np + 288+ + p%)
Cli. 2p wae
n n 7]

ane 2e" 1 P p” : 21 of
V= late Ly ea +(28s+2np+p + $s”)

ae 3( pn” + 2nSps + $94) | 4

If w=0, so that the two electrons move in the same plane,
p=s=0 and C=°875; the above expressions then reduce to

so that we have the familiar expressions for the energy of
the Bohr helium atom. We thus see that practically all the
terms in the above expressions for the energy are pertur-
bative terms of small magnitude. .

and its Relation to the Quantum Theory. 863
The next step is to write p and s in the form

p=a_+ a, cos 2Bi +a, cos 4Bt + ag cos 6 Bt+ a, cos 8Bt
; +a; cos LOBt+a, cos 12Bt,
s=; cos 3Bt+b; cos; 5Bt+ b; cos 7Bt,

where the coefficients ao, a,... a, 3, b5, 67, are power
series in w given in equation (17). Also using the
binomial theorem we may expand the various powers of
n= V1—w cos? Bt as power series in w cos’ Be. If the series
occurring in products be multiplied together, T and V
will consist of terms which are products of powers of sines
and cosines of integral multiples of Bé. By means of the
addition formulas the products of powers of sines and cosines
may be reduced to sums of linear trigonometric terms, thus
giving T and V as Fourier series in the time, so that

T=T)(w)+T,(w) cos 2Bét+T,(w) cos 4Bt+...
V=V,(w) + Vi(w) cos 2Bé+ V.(w) cos 4Bi+ ....

It is only necessary to actually evaluate the constant term in
this final Fourier expansion, as the periodic terms will cancel
out when T and V are added together to obtain the total
enervy, which is constant. After reduction to power series
form in w, the constant parts Ty and Vo of the kinetic and
potential energies prove to be

T,= a [1-750 —-053750w—-028650w —-0190620°
~-013920w*—-010658w>—-008473w°—-006938w7], . (19)
Vo= — “< [1°750—-053750e —-028650w?—-019046%
—-0138884w! —-010613w> ~*008439w°—-006839w"], . (20)
while the total energy W is T,+ Vo. |

Check on Accuracy of Solution.

One of the standard methods of checking the accuracy
of computations in Astronomy is to compute the energy
and see if it remains constant. This method could be used
in our problem, but would involve the calculation of the
coefficients of the various periodic terms in the Fourier
expansions of T’ and V, which would be extremely laborious,
as over twenty pages of computations are required to
determine the constant term alone. A much easier method
of checking is furnished by the fact that in motion under the

864 Mr. J. A. Van Vleck on the normal Helium Aton

inverse square law the average absolute value of the potential
energy is twice the average kinetic energy *. Since the
average value is simply the constant part of the Fourier
expansion, and since a power series development is unique,-
the coefficients of like powers of w in the bracketed power
_ series in equations (19) and (20) must, therefore, be identical
if the computations are correct. There is absolute agreement
in the first three terms, while the small errors in the fifth
decimal place in subsequent terms are insignificant, and
due mostly to neglect of third and higher powers of
the perturbations.

Parr III.—APppPuLicaTIoN OF QUANTUM CONDITIONS
To Mopet with AXIAL SYMMETRY.

The same value for the energy is given consistently by
several different types of quantum conditions, viz., the value
_ obtained by choosing the constants of integration (p and w)
so as to satisfy the relations

Noh
= = g e e « e es e (21)
= h cs

where n, and ny are integers, ‘I’ is the average value of the
kinetic energy (equal to the negative of the total energy),
and v1, v, are the two intrinsic frequencies of vibration, given

by T

y= es the frequency of vibration of the coordinates 7 and z,
on q y
Vo = me , the frequency of rotation of the cyclic coordinate 0)

ne
(i. e., ae lites the mean angular velocity of the electrons
us

about the axis of symmetry). ..9:° 2 9) 9..¢5)) -) ae

Equation (21) states that the resultant angular momentum

* For proof of this relation, see Sommerfeld, ‘Atombau und
Spektrallinien,’ 2nd ed., p. 472. Sioa

+ For proof that the », and v, defined in (23) are the intrinsic
frequencies in the Fourier expansion of the Cartesian coordinates ~, y, 2,

see Bohr, ‘ The Quantum Theory of Line Spectra,’ p. 38.

and its Relation to the Quantum Theory. 865

7

p must be an integral multiple of In the actual numeri-

hr
oar”

cal determination of the orbits so as to satisfy (21) and (22),

T and v, were evaluated as the constant terms in Fourier
expansions. The quantum numbers 7 and ny were each
taken equal to unity to give the normal orbits, those of
lowest energy. Tie value of w was found by trial and
error to be *7216, giving an .energy of 74°9 volts, already
discussed. The remaining pages will be concerned with
showing that various theories devised for quantizing the
stationary states demand that equations (21) and (22) be
satisfied.

Sommerfeld Quantum Conditions.

From the standpoint of the Sommerfeld conditions (viz.,
that \ pidgi= nih) the result (21) is obtained by assuming
that the cyclic coordinate @ together with its conjugate
momentum p satisfy a quantum integral, so that

2Q0r

| pdd=nzh.

0

This is in agreement with Hpstein’s theory that when partial
separation of variables can be effected in the Hamilton-Jacobi
equation, the Sommerfeld conditions should be satisfied by
the phase integrals associated with the coordinates which

can be separated™ (i. e., \ pdqi=nih for the particular values
of i for which p; may be regarded as a function of g; only) ft.

. ; i ’
Also, as mentioned by Bohr {, the value 5 for the resultant
Us

angular momentum appears to be demanded by the con-
servation of angular momentum, independently of quantum
theory considerations.

For a conditionally periodic system with any number of

* Verh. d. D. Phys. Ges. vol. xix. p. 127.

7 It is interesting to note that Epstein’s conditions demand that the
resultant angular momentum of any three body system, and hence ofany
model of helium (not necessarily in the normal state) be equal to an

‘ ; h ;
integral muitiple of =~, for in this much more general case the resultant
aT

angular momentum can be proved conjugate to a cyclic coordinate of
period 27. (For proof, see Whittaker, ‘ Analytical Dynamics,’ p. 345,)

t ‘The Quantuin Theory of Line Spectra,’ p. 35 (Mémoires Dan, Acad.
1918),

866 Mr. J. H. Van Vleck on the normal Helium Atom

degrees of freedom *, which has two intrinsic frequencies of -
vibration vy; and v2, and in which separation of variables can be
effected, the Sommerfeld quantum conditions demand that
the average value of the kinetic energy be that given in
equation (22) f. The general type of motion in the par-
ticular dynamical system we are considering is presumably
not conditionally periodic, but, instead, the great majority of
orbits seem to be characterized by large perturbations, in which
the radius may tend steadily, though very slowly, to the value
zero or infinity. For this type of motion the Sommerfeld
quantum integrals have no meaning (except in case of the
cyclic coordinate ¢) and no technique appears to have been
devised for quantizing the general orbits in dynamical
systems of this character. However, the simple relation
given in (22), though not often mentioned in the literature,
is one which is satisfied in practically all cases in which
quantum theory dynamics have been applied successfully,
and consequently may itself be regarded as a quantum con-
dition of considerable generality. Therefore, when particular
classes of orbits can be found which are conditionally
periodic and characterized by two intrinsic frequencies of
vibration, one would expect this relation to be satisfied. This.
amounts to saying that, since orbits characterized by con- —
stantly increasing perturbations cannot occur in the normal
state, we need quantize only the families of orbits which are
conditionally periodic, which contain two intrinsic frequencies
and four arbitrary constants (two of which are epoch angles),
and which therefore resemble the general motion in a con-
ditionally periodic system with two degrees of freedom f.

* If the number of degrees of freedom exceeds two, the motion is
partially “ degenerate.”

t To prove this the case we observe that by Kuler’s therem on homo-
geneous quadratic functions }

pt NOR ts Gi le, uO
D} Ogi Vi=sPi N, Since Roe Ogi

Oa gees eh fb
The relation T= 5 (n,7,+n,v,) is obtained immediately by taking time

average and using the facts that \ p,4q,=n,h and that the frequency of
q, is either », or p,,

{ If the Poisson and secular terms in which the time appears explicitly
should prove to combine in such a way as to make the general motion
conditionally periodic, then, if separation of variables could be effected,
the Sommerfeld quantum conditions could be applied directly and the
general motion could be specified with the aid of three angle variables.
The relation (22) would then be obtained by equating to zero the quantum
number 7, associated with the third intrinsic frequency », not appearing
in the particular family of orbits studied in solving the dynamical problem.

and its Relation tu the Quanium Theory. S&T
Schwarzschild Angle Variables.

The “angle variables ” (Winkelkoordinaten) are intrinsic
coordinates which are 27t times the frequencies of vibration
of the system, and thus possess the characteristic properties
of being linear functions of the time such that alteration of
any one of them by an amount 27 leaves the configuration
of the dynamical system unaltered. The two angle variables
for the family of orbits given in equations (17) and (18) are
therefore 27p,t and 2av.t, where vy; and vy have the values
given in (23), The intrinsic momenta P; and P, conjugate
to the angle variables Q, and Q, are constants defined by
the canonical equations

dQ a oH é AQe Gs oH
ee OP, ae Pi
where H is the Hamiltonian function (7. e., the energy
regarded as a function of P; and P,). The general solution
of the above equations can be shown to be *
1

ee

= aa 2m[ R? == Z? \dt,

0
7T
1
= aI pdb=p.
0

Equations (21) and (22) follow immediately on setting P,

and P, equal to integral multiples of -- in accordance with

Schwarzschild’s quantum conditions, which demand that
27

frag, = nih °

0

Ehrenfest’s Adiabatic Hypothesis.

Ehrenfest’s adiabatic hypothesis states that motions
“allowed ” by the quantum theory are transformed into new
“allowed” motions as the character of the dynamical system
is altered by changing very slowly some parameter
appearing in the energy. We shall take this parameter a
proportional to the perturbative force of repulsion between

* For proof, use methods of Epstein (Ann. d. Phys. vol. li. p. 168).

868. Mr. J. H. Van Vleck on the normal Helium Atom

the two negative electrons, so that the total potential energy

V of the system is |
ar 4¢* OG! :

=— JRE + on (cf. equation (2)).

For a=0 there is no mutual action between negative electrons,

and then each electron describes a circle or ellipse character-

istic of a central force obeying the inverse square law.

Since the motion in this undisturbed system is periodic, the

ae integral must be an integral multiple of h, so that we
ave

T

2 ia (ny +n9)h, : . 5 co . (24)

0 .
where 7 is the period. Also we shall assume that the
resultant angular momentum about the axis of symmetry is

an integral multiple oe which apparently is demanded by

the conservation of angular momentum, and which is required
if Sommerfeld’s quantizing of space in polar coordinates is
accepted. Now let the parameter a be increased from the
fictitious value zero to the actual value unity. Since there
are no forces operative which have a moment about the axis
of symmetry, the resultant angular momentum retains its
=. and the axial symmetry is preserved.
Also if we assume that the motion always remains condition-
ally periodic as the perturbing field is thus gradually
increased, then, using an equation given by Ehrentest *, it
is readily shown that the average kinetic energy has the
value demanded by (22).

original value

Bohr’s Quantum Conditions.

By quantizing the perturbations in a manner analogous to
that of the Sommerfeld conditions for conditionally periodic
motions, Bohr has devised a general method for determining
tne “allowed ” motions whenever the perturbing potential
has axial symmetry f, although his treatment is intended
primarily for cases where the departures from the undisturbed
orbits are so small that only first powers in the perturbations
need be considered. These quantum conditions demand that

* Ann. d. Phys. vol. li. p. 348, equation (m).
+ ‘The Quantum Theory of Line Spectra,’ pp. 53-6.

and its Relation to the Quantum Theory. 869
the angular momentum about the axis of symmetry be an
ee h eee
integral multiple of ;—*, and that the motion be adiabati-

at

eally derivable from an unperturbed orbit for which the
action integral has the value given in (24). Bohr’s conditions
are therefore in agreement with the application of the
Khrenfest adiabatic hypothesis given above.
Jefferson Physical Laboratory,
Harvard University.
March 13, 1922.

Note-—Since this paper was written, an article by Bohr
has appeared in the Zertschrift fiir Physik, vol. ix. (1922),
p. 1, in which he conjectures that the Kemble model with
axial symmetry and with crossed orbits (studied in the
present paper) may be the correct solution of the normal
helinm atom. It is therefore to be regretted that calcula-
tion has given an ionization potential of 20°7 volts instead
of 25:4 demanded by experiment.

According to Bohr the normal helium atom is capable of
formation from a free electron and ionized helium atom by
continuous transition through a series of intermediate orbits.
The family of orbits in the present by varying the constants
of integration p and w furnish a means of transition from the
Kemble model to a stationary state of lower energy given by

h : :
w=0 and ea which gives the coplanar circular orbits

of the original Bohr helium model. According to Bohr
(p. 32) this mode] cannot be formed by a continuous
transition from the stationary states found in the orthohelium
model (coplanar orbits of unequal size), but the statement
which I have just made makes it appear capable of formation
by continuous transition from the stationary states of the
parhelium series. The instability which may result from
the possibility of degeneration into coplanar orbits of lower
energy makes it plausible that the normal state of the helium
atom may not be characterized by crossed orbits with axial
symmetry. |

* Bohr’s conditions demand that the resultant angular momentum of
. Noh mts
a single electron about the axis of symmetry be = , while in our con-

ditions this value was taken for the resultant angular momentum of both
electrons, a quantity twice as great. This, however, is probably not a
contradiction, as Bohr’s method was derived for systems with only one-
electron,

SE Seto

|

wi4
Cc |

be sn0 ]

LXXIII. The Use of a Triode Valve in registering Electrical
Contacts. By G. A. Tomiinson, B.Sc.*

fs eee electrode valve can be applied with advantage
to certain forms of apparatus in which use is made
of electrical contacts. A common case is that of a relay in
which it is usual to cause a feeble movement of one instrument
to make a contact and close the circuit of a second compa-
ratively powerful instrument supplied from an independent
source. An improvement in several respects can be made
if the first contact is placed in the grid circuit of a valve,
and the second instrument is connected in the anode circuit
and is operated by the anode current.
An arrangement used by the writer is shown in fig. 1.

—-------y5

eee

ANODE CURRENT.

FIG |

A small tongue of platinum C is moved by the first in-
strument between two platinum points p; and ps, and on
making contact either raises or lowers the potential of the
grid by about 4 volts by introducing the batteries b, or by.

The reaction on the anode current, which is illustrated by
the diagram of the valve characteristic, operates the in-
strument D. Thus the points P,; and P, show the anode
current when the tongue is in contact at p; and po, the dif-
ference being the range of current available, which is about —
3 milliamperes. ‘The actual values of the grid potential are
adjusted by the battery 6 to vary between —2 and +6 volts,
to obtain the best range of anode current.

In this particular case the relay is required only to detect
small angular movements of an instrument in either direction
from the zero position, and the instrument used in the anode
circuit is a pivoted moving-coil galvanometer.

* Communicated by the Director of the National Physical Laboratory.

Use of Triode Valve in registering Electrical Contacts. 871

The resistances 7, and vr, in the circuit enable a fine
adjustment of the points p; and p, to be made. ‘These
points can be advanced by micrometer screws until both
just touch the platinum tongue. This contact can be ob-
served by temporarily inserting the galvanometer G in the
circuit. The E.M.F.’s of the batteries b, and 6, act in the
saine direction in this local circuit, and with suitable re-
sistances 7 and 7, a small local current flows when p; and pz
are both in contact. The points can then be separated by
any desired amount. The resistances 7; and 7, also prevent
a short cireuit if an accidental contact is made from p, to po.

A relay of this type has several advantages. The current
to be transmitted through the contacts is very small, being
only the grid current of the valve. Variation in the re-
sistance of the contact within wide limits has no effect on the
action, owing to the great resistance already in the circuit
between the grid and the filament. The amplification of
mechanical power, which is the function of a relay, is pro-
vided for by the electrical amplifying properties of the valve.
A further advantage, that may be important in some cases,
is that this relay may be operated by much weaker forces
than could be used with an ordinary relay. In the latter
type the contact is placed directly in the circuit of the second
instrument, and a certain contact pressure 1s necessary to
ensure the passave of sufficient operating current. With
the valve relay an extremely light contact between clean
platinum surfaces is sufficient to charge the grid of the valve,
and very weak forces will therefore work the relay in a
satisfactory way. Thus it has been found that a contact
force of 0:000001 grm. is quite sufficient to charge the grid
and produce the required change in the anode current.
Since there is practically no current transmitted by the
contacts, there is no objectionable coherence of the surfaces,
and the movement of the contact tongue can be reduced to
a very small amount if desired. For example, the relay has
been operated with the travel of the tongue only about
1/1000 mm.

Certain modifications to meet different requirements may
be suggested. If the contact is for any reason intermittent,
a comparatively steady current may be obtained in the anode
circuit by connecting a suitable condenser across the grid
and filament. A high- resistance grid-leak may be used if
it is desirable for the anode current to assume its normal
value immediately the contact in the grid circuit is broken.

The writer has also made some experiments, using an
electrical contact in the way described for quite a different

872. Mr. EH. A. Milne on Radiative Hquilibrium :

purpose, namely as an indicator for precise measurement.
A compound lever with a magnification of about 600 was
arranged to be moved by a micrometer at the one end, and
earried the contact at the other end. Using this to measure
the thickness of a parallel slip-gauge with the lower face
resting on three steel balls and the contact on the upper face,
it was found that repetition of observations could easily
be obtained with variations not exceeding 0°5 x 106 inch.
These experiments indicate that the advantages obtained by
making contact in the grid circuit of a valve may eliminate
some of the difficulties hitherto experienced in this method
of measurement.

This method appears to have advantages in connexion with
the reception of feeble wireless signals with the aid ofa relay ;
and it is also proposed to try it, on account of its freedom
from sparking at the grid-circuit contact, in connexion with
the location of the height of the mercury surface in the
vacuum space of a standard barometer.

July 1922.

LXXIV. Radiative Equilibrium: the Insolation of an Atmo-
sphere. By Wi. A. Minne, M.A., Fellow of Trinity
College, Cambridge *.

§ 1. Iwrropvcrron.—The generally accepted theory of the
existence of the earth’s stratosphere was put forward in 1908
by Gold+. Gold showed that when radiation processes were
taken into uccount the continued existence of an adiabatic
gradient to indefinitely great heights was impossible ; for the
upper pertions of such an atmosphere, being very cold, would
radiate very little, but on the other hand, being backed by an
extensive cushion of warmer air besides the warm surface of
the earth, would be subjected to low-tem, erature radiation
of considerable intensity, and the consequent excess of absorp-
tion over emission would raise their temperature and so disturb
the adiabatic gradient. Such upper portions, however, could
not exchange heat with the rest of the atmosphere by con-
vection, for they would tend to rise, not fall. Consequently

* Communicated by the Author.

+ “The Isothermal Layer of the Atmosphere and Atmospheric
Radiation,” Proc. Roy. Soc. 82A. p. 43 (1909). A preliminary
announcement was made at the British Association meeting in 1908;
see ‘Nature,’ vol. Ixxvili. p. 551 (1908). See also Geophysical
Memoirs, No. 5 (Met. Office), vol. i. p. 65 (1913).

the Insolation of an Atmosphere. 873

their temperature would continue to increase until the extra
emission due to increased temperature balanced the absorption
and a new steady state set in—a state of radiative equilibrium.
The direct absorption of solar radiation is small and, though
important, does not affect the argument. (It is of interest to
mention that exactly the same course of argument shows that
even in the absence of convection a strictly isothermal atmo-
sphere is impossible ; for the outer portions would not be able
to absorb as much as they emitted, and so would cool, causing
convection. )

Gold embodied these ideas in analysis, in order to determine
the temperature and the height of the tropopause, and he
showed that the theory generally was adequate to account
for the observed values. His procedure, however, was in
part empirical. In the light of Schwarzschild’s * theory of
radiative equilibrium in a stellar atmosphere, an immediate
rough evaluation of the boundary temperature is possible ;
if I) is this temperature, then T)*=4T,*, where T, is the
effective temperature of the system (earth plus atmosphere)
as determined by the amount of energy radiated away into
space. This energy is equal to the mean value of the absorption
of solar radiation, assuming that the earth is on the average
neither losing nor accumulating energy. The value of Tj,
deduced by Abbot f from the solar constant and the earth’s
albedo, is about 254°, giving Tp =214°. The observed mean
value of the temperature of the str atosphere over the British
Isles is about 219°. Schwarzschild’s formula, To =1T,*, was
indeed obtained independently by Humphreys t in this con-
nexion, and applied to the stratosphere. Gold, however, did
not proceed in this way. Accepting the observed division of
the atmosphere into two shells—an inner one in convective
equilibrium with a known temperature gradient, and an outer
one at a uniform temperature,—he determined the height at
which the convective gradient should terminate, in order that
the atmosphere above this height should, as a whole, gain as
much heat by absorption as it lost by radiation ; the temper-
ature of the convective region at this height then gave the
temperature of the isothermal re'ion. It appeared that a
satisfactory balance was obtained if the point of division was
taken at a height given by p=+p,, where p is the pressure
at any height, p, the ground-pressure. It appeared further
that there was very nearly a balance of radiation in the upper

* Gott. Nach. 1906, p. 41.

+ Annals Astrophys. Obs. Smithson. Inst. ii. p, 174 (1908). ‘The

Sun’ (Appleton, New York, 1912), p. 323.
{ Astrophys. Journ. vol, xxix. p- 26 (1909).

Phil. Mag. Ser. 6. Vol. 44. No. 263. Nov..1922. a)

874. Mr. B. A. Milne on Radiative Equilibrium :

layer of the convective region extending from p=4p, to
p=ip,, from which Gold deduced that in this layer the
convection would be small.

It is the object of this paper to point out a certain difficulty
in Gold’s work, and to consider an idealized problem which
is suggested by it in the absorption of radiation by an atmo-
sphere subject to insolation.

Since the paper was first written, the author has become
aware of a paper by Emden* which anticipates portions of it.
Emden criticized Gold’s theory on certain points, and investi-
gated the general theory of the radiative equilibrium of an
atmosphere by a method similar to that of the present paper.
Where necessary, the paper has been recast to take account
of Emden’s work. 3

§ 2. Criticism of Gold’s solution.—One of the most in-
teresting points in Gold’s discussion is his isolation and
explicit formulation of the condition for a convective atmo-
sphere. In such an atmosphere, transfers of energy are
being effected both by radiation and by convection, and
across any plane there will be a net radiative flux and a
convective flux. Now convection can only transfer heat
upwards, not downwards. But assuming a steady state, the
upward convective flux plus the net radiative flux must be
equal to the downward solar flux. Hence the net radiative
flux (as due to the earth and atmosphere together) must be
less than the downward solar flux. But the downward solar
flux at any point cannot exceed its value at the boundary ;
and at the boundary the downward solar flux must be equal
to the outward flux due to the earth and atmosphere. Hence
another form of the condition is : the net outward flux across
any plane must be less than its value at the boundary. Again,
the upper Jayers must be gaining more heat by convection
from below than they are losing to layers above. Hence,
for a steady state, emission of radiation must exceed absorption
in the upper layers (for emission must equal absorption plus
net gain by ecnvection). Whenever these inequalities become
equalities, radiative equilibrium holds ; if they become re-
versed the state cannot be a steady one, for it would involve
convection downwards fF.

* “Uber Strahlungsgleichgewicht und atmosphiarische Strahlung :
ein Beitrag zur Theorie der oberen Inversion,” Sitz. d. K. Akad. Wiss.
zu Miinchen, 1915, pp. 55-142.

+ Gold’s conditions have been applied by the writer to stellar
atmospheres in a paper recently communicated to the Royal Society.

the Insolation of an Atmosphere. 875

Now Gold applied these conditions in various ways to show
under what circumstances a convective atmosphere can or
einnot exist: e.g., he showed that a convective atmosphere
cannot extend indefinitely, yet must extend above p= p1.
But he did not point oat that his final solution was incon-
sistent with these conditions. We shall show that although
on the assumptions made the layer (47, 0) is neither gaining
nor losing heat as a whole, yet its upper portions are emitting
more than they are absor bing, and its lower portions absor bing
more than they are emitting ; consequently the upper layers
must cool and sink, the lower ones warm and rise, convection
will occur, and the state of isothermal equilibrium must be
destroyed. Further, although the layer (4), +p ;) satishes
the conditions for convective > equilibrium as a whele, emission

exceeding absorption, in the upper portions absor ption exceeds
emission, so that a steady convective state in this region is
not possible ; ; the smallness of the excess of emission cver
absorption oe the whole layer, attributed by Gold to the
slightness of convection required, is merely the result of the
excess in the lower portions being balanced by the deficiency
in the upper ones.

Actually we can prove a more precise result than this,
under very general conditions. We shall show that the
excess of absorption over emission at the base of Gold’s
isothermal layer, per unit optical mass, is numerically equal
to the excess of emission over absorption at the top, whatever
the temperature distribution in the convective layer and
whatever the law connecting the coefficient of absorption
with height. To do this we shall employ the approximate
form of the equations of transfer of radiant energy. It may
be inentioned here that though Gold uses the exact formule
(involving #2 functions) which take full account of the
spherical divergence of the radiation, his results can be
obtained more simply to the same degree of precision by
using the approximate formule and by making free use of
the optical thickness and the net flux of radiation. The
quite small errors of the approximate formule are swallowed
up in the uncertainty of the numerical data that have finally
to be employed. The uncertainty arises in the final trans-
lation of the optical thicknesses into actual thicknesses; but,
as in other cases of radiative equilibrium, many of the results
hold in a form independent of the numerical values of the
absorption coefficients.

Let 7 be the optical thickness measured inwards from the

aél2

876 Mr. E. A. Milne on Radiative Equilibrium :

outer limit of the atmosphere ; if pis the density at height h,
k(h) the mass-absorption coefficient, then

ze) = ( k(h) pdh.
oh

Let I(r) be the intensity of radiation at 7 in a direction 0
with the outward vertical, where 0<@< 47; and let I'(7)
be the intensity at yw with the inward vertical, where
O<w< 4a. Assume the material is grey (i.e. has an _
absorption coefficient the same for all the wave-lengths that
are important—in this case the wave-lengths that are pre-
dominant in the low-temperature radiation considered).
Let B(r) be the intensity of black body radiation for the
temperature ruling at the point rt; and let 7K (7) be the net
upward flux of energy per unit area across a horizontal plane
at 7. Then

0 ee eee
AT
dl! :
cos, =B-I, 1 eer
30 ‘Ar
4E(s) = I(r) sin 6 cos 0 dd — I'(7) sin W cos Ww dy.
0 J0
Consider the expression
w(t’) —7F(7”), G Siar

Here wF(7') is the net amount of radiant energy entering
the lower boundary of the layer (7', 7’), wF(7”) the net
amount leaving the upper boundary. Hence the difference
1s the excess of absorption over emission for the whole layer
(7’, 7’). Thas F(7) behaves as an integral, whether or no
radiative equilibrium holds ; this is interesting, for in certain
forms of radiative equilibrium it appears naturally as an
integral * of (1) and (2) in the form F =const.

Let 7, be the value of 7 at the earth’s surface. Now
suppose with Gold that the complete atmosphere t=0
to T=T, consists of two shells—an outer one at a uniform
temperature from t=(0 to T=7,. (say), and an inner one in
convective equilibrium from tT=7, to T=7,. Then the outer
one will be in radiative equilibrium as a whole, provided

EG) (0) = 0," a ee pee
and this is the equation which determines 7».
* Monthly Notices, I1xxxi. p. 862 (1921).

the Insolation of an Atmosphere. 877

We now approximate. Setting t=27, t;=27,, etc., and

using ¢ as the variable specifying position, equations (1),
(2), (3) can be written approximately *

dl we fs
SPOT I—B, FT le B-I’, Coie (6)
Gh nares oe eS (7)

and the equation for ¢, is
EGG ONO yee es Gye (8)

since the incident radiation I'(0) is zero t. Solving (5)
and (6) with the assumption that the air near the ground
has the same temperature as the ground and that the earth
radiates like a black body, we find

ty :
1) =f B@ eae bene dts. kr CD)
t

rO= | Be@ede aD, ete aee ES aan © U9

These can be inserted in (8), and ft, determined as soon
as B(t) is known as a function of ¢ in the convective
region.

Now the excess of absorption over emission in a small
element of volume dv is

kpdv| J Ide + T'do'—47B |
= 2nkpdv| | Tsin @d0+( I’ sin ydyp—2B |
= 2rkpdv | 1(t) + I(t) —2B@) |

approximately. Denote the expression in square brackets
by H(t). Then for the values of the excess of absorption
over emission at the top and bottom of the isothermal region
we have respectively

E(0) = 1(0) —2B(0),
BG ie el Gy 2B) es, CL)
EO p= ral) 20). es C2)

or, using (8),

* For details, see e. g. Monthly Notices, lxxxi. p. 868 (1921).
+ Ignoring solar radiation. See below.

878 Mr. HE. A. Milne on Radiative Equilibrium :

We shall now prove that I(¢,)=2B(¢,). From (9) re=
membering that B is constant in (t,, 0), we have

t
1G) = a) "B(t) e—tdt + B(é,) e—(4—-4),
bo

KO} r= be B(t)e—*tdt + B(t,)(1 — e—#) + B(t,e-4,

whence

1(0)—e-®I1 (t,) = B(t,)(1—e-#).
. Further, from (10),
Ia.) = Bite) =e-4). 4. ee
Inserting in the equation for ¢t,, namely (8), we find
I(ts)(1—e-®) = 2B(t.)(1—e~),

which is the equality required. Making use of this, we have
from (11) and (12)

BG) = '@) = -HO). >... Ses

Now I is essentially positive. Hence there is an excess —
of absorption over emission at the base and a numerically
equal excess of emission over absorption at the top. This is
the result stated. The excess can only be zero if I'(é)
is zero, i. é. if ty 1s Zero.

It should be noticed that the departure from radiative
equilibrium at the base and at the top is very appreciable.
The ratio of the excess, 27kpdvli(t,), to the emission,
4orkp dv B(t,), has the value

aL net) 3). og he ae

if t= 1:0 this.is 0°32. and if ¢,=0°56 it 1s 0°22 -sanditecam
be shown from Gold’s data that these limits for t, correspond
to widely separated values of ¢,, the total absorbing power of
the atmosphere. Again, H(t) is a continuous function of ¢;
and hence, since it is positive when t=¢., it will be positive
in the upper parts of the convective atmosphere, violating
the condition for convection. As we approach the earth
it decreases, soon becoming negative, showing that in the
lower portions the condition is satisfied.

We have assumed the atmosphere “ grey ” as regards the
low-temperature radiation, and we have ignored the direct
absorption of solar radiation. But a variation of the co-
efficient of absorption with wave-length does not affect the
gist of the argument; a strictly isothermal upper atmo-
sphere would still be an impossibility unless its optical

the Insolation of an Atmosphere. 879

thickness were zero. As regards the solar radiation, Gold
made an allowance for this by choosing tr, so that the left-
hand side of (4) was slightly negative; but again the
argument is unaffected. It appears then that Gold’s
analysis, though doubtless giving the broad outlines of the
phenomenon, is inadequate in its details.

§ 3. Now the complete phenomenon must be very. com-
plex. Complications arise from the rotation of the earth,
the change of insolation with latitude, cloud-structure,
scattering, and the light from the sky, besides probably
the world-wide circulation of the air; and the suddenness
of the upper inversion has always been to some extent
a difficulty. Instead of attempting to take account of
the various influencing causes simultaneously, it would
appear to be more in accordance with scientific method
to construct a number of idealized models, to work out
the theoretical solution for each separately, and then to

_ examine the extent to which the earth’s atmosphere partakes

of their several characteristics.

§ 4. Lhe problem in principle.—As a contribution towards
this, it is proposed in this paper to consider the theory of the
radiative equilibrium of a mass of absorbing and radiating
material subject to insolation. The material is supposed to
be stratified in parallel planes, and to be subject at its outer
boundary to a parallel beam of incident radiation. The
latter will be supposed in the first instance to be normal to
the surface ; later we shall examine the effect of oblique
incidence. ‘The material will be taken in the first instance
to be grey ; but later we shall suppose that there may be one
coefficient of absorption for the incident radiation, another
coefficient for the low-temperature radiation emitted by the
material itself. Further, we shall assume the material to be
infinitely thick, and to be in radiative equilibrium throughout
its mass. ‘The assumption of infinite thickness involves little
or no loss of generality ; we could, if we liked, consider a
mass of finite thickness with an inner boundary consisting of
a black radiating surface, but since our results will only
involve the optical thickness, we need only suppose the ab-
sorption coefficient or the density to become suddenly very
large at an assigned depth in order to deduce the case of an
inner boundary ‘from the solution for an infinitely thick slab
of material.

The material being in a steady state must emit energy at
its outer boundary equal to the incident radiation. Across

880 Mr. E. A. Milne on Radiative Equilibrium :

any plane parallel to the surface there will be a net outward
flux of radiation derived from the material just balancing the
inward flux of the residual solar radiation. In the far interior
the latter will be greatly attenuated, and consequently the
outward flux there must be small too. We should expect,
therefore, that the temperature gradient in tlie far interior
would be small; and this proves to be the case. In fact,
not only is there a definite limiting temperature at the outer
boundary, as in the Schwarzschild case, but there is also a
defimte linuting temperature in the far interior. This is one
of the most interesting characteristics of the model we are
discussing.

Let 7 be the optical thickness measured from the outer
boundary to any point; I, I’ the outward and inward
intensities at any point at angles @ and w with the normal ;
B(t) the intensity of black radiation for the temperature at
the point t; 7S the intensity of the parallel beam of incident
solar radiation defined as the energy incident per second per
unit area normal to the beam. Here I and I’ are to refer
only to radiation derived from the material. It must be noted
that since we have assumed the solar radiation to constitute
a parallel beam, the definition of its intensity is necessarily
different from the standard definition for conically spreading
pencils *.

The residual solar intensity at any depth 7 is mSe-r. The
equations of transfer are

dl |
cos 0 7 = I—B, eee LC)

i/
cospS = BAT. ii eels

The amount of energy emitted by an element dv per second
is 477kp Bdv. That absorbed is

kp dv| \ Ido +\ I'do' J,
me Tkodv.
Hence the equation of radiative equilibrium is
("te sin 0 dé +" I'(7) sin dye +4Se-7 = 2B(r). (18)
The flux relation follows from (16), (17), (18), namely,
ie (7) sin 80s 0 d0—| “Tn tT) siny coswdw = 48e77. (19)

* See Planck, Wérmestrahlung, p. 15 (8rd edition).

together with

the Insolation of an Atmosphere. — - 88a

From (16) and (17), with the appropriate boundary con-
ditions,

I(t) — erowo (je ee? sec 6 dt, ere (20)

*r
I'(7) ee wow ier ae sec dt. . . (21)
0
Substitute these in (18); write t=7+ycos@ in the first

integral, t=T—ysecp in the second, and replace cosé

and cos ¥ by uw. We find
1 7%
y du B(t +yp)jeYdy
py 2 ako

1 T/p
+| iy. B(r—yp)e-Ydy +48e-7 = 2B(r).
0 0

We can now reverse the order of integration in the repeated
integrals*. Setting

O() =| Boyds, 0%) = Ben,

we find finally for the integral equation for the temperature
distribution,

; is TO(r+y)—C(t-y) _
C = 1Se-7 \ e
(7) Boe. ? Qy CNG

~~ CGED.
[OCD

If we invert the orders of integration before making the
substitutions for t, we obtain another form,

Be) =4( BEB e—r|) 4486 3)

dy

which is the standard form for integral equations f.
Solutions of these may be sought directly. For an

* For details, cf. Monthly Notices, Ixxxi. p. 865 (1921).

T In equation (23) Zi denotes the exponential-integral function.
The integral equation in the form (23) is substantially equivalent
to the integral equation obtained by L. V. King in the analogous
problem for scattering (Phil. Trans. 212A. p. 875, 1912); it bears
the same relation to King’s equation that the author’s integral
equation for the atmosphere of a star in radiative equilibrium
(M. N. lxxxi. p. 873, 1921) bears to Schwarzschild’s integral equation
for scattering in a stellar atmosphere (Berlin Sitz. 1914, p. 1183).
But the form (22) is more convenient when solutions are being sought
by successive approximation, and for other purposes.

882 Mr. EH. A. Milne on Radiative Equilibrium :

approximate solution, however, it is fe to employ the

approximate forms. of equations (16) o (19), obtained in
the usual way. These are
1 dl 1 al’
~ = = |— 2
Foe Gee B, a Fe = B-I’,’. .. .@2)325)
4+ 48e "= 2B). ae
Tol’ = 86) oe ee

From the two latter,
IT = B+i8e77,
I' = B—38e"",

Bat PO) —v: Consequently By = 38. Inserting this
approximate value of I in (24), we find 3

es RV ir
the Re ae
whence, using the value of B, already found,
| Bir) 2 8S(t—te- se Pe

It follows that there is a limiting temperature in the far
interior, given by B, =3S8. If'now T) is the boundary
temperature, T, the Paste temperature of the whole
mass viewed from the outside, T., the temperature in the
far interior, and o Stefan’s constant, we have

and thus
| Ts

|
—_
FS
fom)
He
|
bs|
-
leo

(29)

It is important to notice that T,, is different from T,,
contrary to what might have been anticipated ; also that
the relation between T, and T) is different from that in the
Schwarzschild case, where the net flux is the same at all
_ depths. Notice also TA=2T(".

These values and the general distribution of temperature -

given by (28) are only approximations. To test them,
let us re-employ (28) in (20) to obtain I and so check
the radiative equilibrium at the boundary and the net flux

ee eek ee ee en ee ee eee

the Insolation of an Atmosphere. $83

there. We find
4e-7
Tea) = a8 (1 . a)
whence
| ++ cos 0

ia dla 1+cos@

(30)

This gives the distribution of the emergent radiation—the
law of bolometric darkening. Inserting in (18), the total
absorption near the boundary is found to be proportional to
(2—3 log 2)S, the emission to 38S—z. e. 0°987 instead of
unity, an error of only 1:3 per cent. Again, from (19),
the net flux at the boundary is given as (3 log 2)7S instead
of wS—z. e. 1040 instead of unity, an error of 4'1 per cent.
The smallness of these discrepancies shows that (28) and the
values (29) are satisfactory approximations.

To obtain a better approximation, knowing now something
of the form of B from (28), we can assume

B(r) = a—be-7

and choose a and 6b so that the correct net flux is given
at the boundary and the condition of radiative equilibrium
is satisfied there. It is found that the condition of radiative
equilibrium in the far interior is then automatically satisfied,
save for terms which tend to zero. We tind

b
10, f= “1+ c080’
whence from (18) and (19)
a—b(2—leg 2) = $8,
4a—b(1—log 2) = 48.

These give

a= S/log 2 = 1:44278,

b = 48/log 2 = 0°72138;
whence B)>=0°72138, B, =1'4427S, and |

Se be eye be ee 72

Thus the values of T, and T, in terms of ‘7, come out about
1 per cent. smaller than on the previous approximation. The
relation T.4=2T,¢ still holds. ‘The change is so trifling that

we shall not attempt to obtain further approximations, which
can be sought by using the integral equation. We shall

884 Mr. E. A. Milne on Radiative Equilibrium :

content ourselves with observing that in the exact solution
the differential coefficient B'(r) has a singularity at r=0,
becoming infinite * like log rv. This is easily proved.

§ 5. Hatension to non-grey absorption.—Let us now suppose
that the material has a different coefficient of absorption for
the incident radiation, say equal to n times that for its own
low temperature radiation ; » will usually be a small fraction.
The inward solar intensity at 7 is now Se~”. Hence in the

flux equation, (19), e~7 must be replaced by e-”", and in
the equation of radiative equilibrium, (18), Se-7 must be
replaced by nSe-"7. Proceeding as before, we find that

I+ tn

nr

Bit) =8 fi (—dn)\enr"|

i
B, = S++2", B, = 38(1 +4),
Tot yt Tot SS ban 1 le

As n->0, T,->», To*>3T;', and the temperature dis-
tribution tends to

B(r) = S8($47).
The limiting case is, in fact, the Schwarzschild case for
a constant net flux 7F. Notice that TA= 2T)4/n.

~§ 6. Extension to oblique incident radiation. Next suppose
that the external radiation is incident at an angle « with the
normal. If we preserve the same intrinsic intensity, the
amount incident per unit area is now S cosa and the amount

crossing unit area at depth + 1s Sicosae—™ 5°¢*. |) Wemeamr yy
obtain the solution by putting S cosa for S and nseca
for n in the foregoing formule. We find

B(r) = 8 tS [cos «—(cos «—4n)e—"* 2], (28"")
B,, = Scosa(cosa+4n)/n, By, = 48(cosa+3n),
To: Ty4: Tyt = cosa($+n-1 cosa): cosa: $(cosa+4n)
= L4n-lcosa:1:4(1+inseca). . (29!)
Notice that T,4=2 cos aT)4/n.

* Cf. Monthly Notices, lxxxi. p. 3867 (1921).

the Insolation of an Atmosphere. 585

§7. These formule offer several points of interest. As
a increases from 0 to 47 and cos a—>0, |’) tends to a definite
non-zero limit, although T, tends to zero; Ty steadily de-
creases as a increases, the limit being given by oTy'=47n8 ;
T.. tends to zero. It appears, then, that for sufficiently
oblique incidence the boundary is warmer than the interior.
Consider now the temperature distribution given by (28").
When cose=3n, B(r) is constant and equal to gS or
S cos a> and the state is isothermal everywhere ; and when
cosa<tn, the temperature steadily decreases inwards in
the interior. In spite of this there is at each point a
net flux in the outward direction ; so that here we have
a case where the net flux is in the opposite direction to the
temperature gradient. This would seem to be a novelty
in the theory of radiative equilibrium. (It is easy to
assure one’s self that no contradiction with the second
law is involved.) These results are based only on the
approximate formule (28’) and (29''), but further investi-
gation confirms them. It is easy to see in a general way
how these curious temperature distributions arise. When
the solar radiation is nearly tangential, its effective intensity
is very weak, but owing to its obliquity it is entirely
absorbed in a thin layer close to the surface (provided
n is not zero). This layer is enabled to assume a definite
temperature, but no residual radiation penetrates to the
interior, which remains near the absolute zero. The out-
ward net flux is maintained at any point in virtue of the
outward radiation from the large amount of cold material
inside the point overpowering the inward radiation from
the small amount of warm material outside it. In the
limit when 2=47, the distribution of temperature is dis-
continuous; the temperature is zero everywhere, except
at points in the surface.

§ 8. Effect of rotation—These results can only be applied
to a thick spherical atmosphere on the assumption that the
solar energy incident on any one place is all re-radiated
from that same place. Making this assumption, let us
tentatively take into account the effects of rotation. We
will calculate the time mean of the temperatures in any
given latitude X on the assumption that the axis of rotation
is perpendicular to the ecliptic. If ¢ is the hour-angle of
the sun, its zenith distance « is given by cose=cos ¢ cos X.
Taking (29") as giving the “instantaneous” temperature
during the day and taking the latter as zero during

886 Mr. E. A. Milne on Radiative Equilibrium :

ihe night, and using bars to denote mean values, we
have

ow Le = is 4 (cos X+ jn)

oT 0 0 7 No 0 = ?
Oe a7 : oe

= {= B= =k ScosA cosddd =S8

gy ee > J _ tenia eee
TET 6 tka eee Pe T

(31)

When this averaging is taken into account, the approxi-
mately isothermal state (T,.=1)) is found to oceur for
cosk=n/¥ 2; for this value of X,

Tyi/T'= 4. +40 V2) = 1-055,
which is sufficiently near unity. The general run of the

change of temperature distribution with latitude has the
same features as before.

§ 9. Comparison with imden.—The formal problems dis-
cussed by Emden in the paper already mentioned and his
method of solution are very similar to those discussed above,
except that he takes the material to be bounded below by a
black surface. Emden considers the radiative equilibrium
of an atmosphere subject to external solar radiation in
two cases: (1) the case of “grey radiation,’ by which
he means the case in which the mean coefficient of
absorption for the solar radiation is equal to that for the
atinospheric radiation; this is the case n=I1 above; (2) the
case in which the radiation spectrum can be divided into two
ranges which have different mean coefficients of absorption,
the solar radiation being entirely confined to one of them;
this is practically our general case in which v is not unity,
In each case he considers the solar radiation to be “ gleich-
miissig verteilt,” i.e. not as being confined to a parallel
beam, but as uniformly distributed over the solid angle 27 ;
consequently he does not consider the variation of the state
of equilibrium with latitude. The two main results to
which he draws attention are: in case (1) the whole atmo-
sphere must be isothermal, at a temperature equal to the
“effective”? temperature T, calculated from the incident
radiation with allowance for the albedo (see p. 873 above) ;
in case (2) the state is not isothermal and the boundary
temperature Ty) is connected with the effective temperature

c

the Insolation of an Atmosphere. 887

T, by the relation
re eit vei) ha) | ay) ) eG eee) (82)

where /; and i, are the coefficients of absorption for the solar
and terrestrial radiations.

Both these results are in apparent contradiction with those
obtained in this paper. The source of the discrepancies is in
each case Emden’s assumption that the incident radiation
may be taken to be diffuse. The way this occurs is as
follows:—The mean coefficient of absorption for diffuse
radiation incident on a thin layer of material is approxi-
mately twice the coefficient of absorption for a parallel
beam incident normally, 7. e. twice the coefficient of ab-
sorption as ordinarily defined. This fact allows us to
approximate to the equations of transfer (equations (1)
and (2) above) by replacing them by the ‘equations of
linear flow”; in equations (5) and (6) we have explicitly
adopted a new optical thickness ¢ equai to twice the optical
thickness t obtained directly from the ordinary coefficient
of absorption ; in equations (24) and (25) we have retained
the optical thickness t and simply replaced the factors cos 0
and cosy by the value 3; the result is the same as if all
the diffuse radiation were supposed to be confined to beams
at an angle of incidence of 60° with the planes of stratifi-
eation. Emden approximates in the same way as we have
done, but since he takes the solar radiation to be diffuse
he is adopting for this also a coefficient of absorption twice
the value for a permanent beam. His results may therefore
be expected to agree with ours if in ours we put cosa=}3,
a=00°; and this in fact they do. But they lose part of
their significance. His result for case (1) is of course
true for diffuse radiation; indeed it is obvious thermo-
dynamically, without proof, that material exposed to iso-
tropic incident radiation will, if in radiative equilibrium,
take up a temperature equal to that of the radiation: the
case is practically that of a black body enclosure. But
our results show that if the incident radiation occurs as
a parallel beam—as, in fact, solar radiation does—-then
the isothermal state is merely the particular distribution
of temperature that happens to correspond to an angle
of incidence of 60°. Further, Emden’s result does not
suggest another of our results—that when n¥1 there also
exists an isothermal state of equilibrium: namely, for
cos a=4n for a fixed parallel beam, and for cosX=n/ V2
when rotation is taken into account. Hmden’s formula (32)

888 Mr. E. A. Milne on Radiative Equilibrium :

above should be compared with our formula for a fixed beam

‘Incident at @ (from (29’’)),

To = T,[3(.+4nseca)]?, . . , . (BB)
where n=hk,/k,; and with the rotational mean formula

fro1 )
CCIE, Ta aan sco an

Hmden’s formula differs but little from the latter when 7=0,
as is to be expected.

Emden does not obtain the integral equation for the
temperature distribution. For the sake of completeness
it seems worth putting on record the integral equation
for the general case involving n and «. It is deduced
in the same way as (22) :—

OMG esa)

Cir) daemon af - e~vdy
+| AGED esi, eo

§ 10. Liffect of an internal boundary.—We shall next
consider the case in which the material is bounded
internally by a black surface at t=7, insiead of extending
to infinity. It has already been mentioned that as the
formule only involve the optical thickness 7, we may deduce
the results for this case by supposing that immediately
beneath t=7, the density suddenly increases indefinitely.
The temperature distribution above 7, is unaltered. It
might at first be supposed that the black surface would
assume a temperature equal to T,,, but this is not so. For
the infinite density gradient we have postulated at the
level +, implies an infinite radiation gradient there, and
(unless we are prepared to accept the existence of an
infinite temperature gradient at the black surface) the
surface will take up a temperature intermediate between
T(7,), the temperature of the material in contact with the
surface, and |’,. This temperature, say ‘l,, is easily cal-.
culated. For since the surface must re-radiate all the
radiation falling on it, we shall have oT,4=7B,, where
B, is given by .

By = UGS cos ae "Roo aes,

e

the Insolation of an Atmosphere. 889
From the equations
eee eee cd ko LOY
Pesaro cone rrrese ii) i CO
eunad I’ = B—438e-"7 *(cos «+ 4n).
Hence

Bs: = B(x) +38(cose—injem sera, | | (37)
If cos a>4$n, which will usually be the case in applications,
B, is greater than B(7,). Thus the temperature of the
surface exceeds that of the material (say air} in contact
with it. Hence convection currents would be set up, and
the state of radiative equilibrium would be destroyed. his
is a simple way of demonstrating the impossibility of the
existence of a state of radiative equilibrium throughout
the entire atmosphere.

§ 11. The “greenhouse” efect.—Inserting in (27) the
value of B(7,) from (28'’), we have

B, = 8 cos a[ (cos a+ 5n)— (cos a—3n)e-™71 82] In, (38)

Now if the black surface were exposed to the direct in-
solation 7S cose, without the intervention of an atmosphere,
it would take up a temperature T,’ given by

ol, */7 = B. = Scosa.
Hence
Me (cos a—4n) (1 —e7771°¢¢2)

pa = 1+

: (39)
Thus, when cos «>4n, the surface is maintained at a
temperature higher than it would be in the absence of
au atmosphere. The ratio T,*/T,’* increases as n decreases,
the limit as n—>0 being 1+7;. The case of diffuse incident
radiation is roughly given by putting cos#=4, and then the
condition is n<1, 7%. e. that the atmosphere or “ protecting
layer ’ must be more transparent to the incident radiation
than to the radiation returned. This is the radiation part of
the ‘“‘ greenhouse” or “ heat-trap” effect, which is some-
times the subject of fallacious statements ; 1t must of course
be distinguisbed from that part of the effect which is due to
the prevention of convection. .

§ 12. Extension to a partially convective atmosphere—We
will now generalize the problem a little further. Suppose
that we have a state of affairs in which the material above a

Phil. Mag. Ser. 6. Vol. 44. No. 263. Nov, 1922. 3 M

890 Mr. E. A. Milne on Radiative Equilibrium :

given level t=7, is in strict radiative equilibrium, that below
the given level merely in radiative equilibrium as a whole ;
below 7=7, the temperature distribution may be of any form
(with or without a lower bounding surface) subject only to
the condition that the whole system below 7, radiates out-
wards as much as it absorbs; in general, convection of heat
will be required in the region below 7=7; in order to maintain
a steady state. Then it is easily seen that the temperature

distribution above 7, is exactly the same as if the lower region —

were in radiative equilibrium in the strict sense ; for the
upward intensity at 72, namely I(7,), is the same in the two
cases. Hence the temperature distribution we have already
found applies to the region above t,. The importance of
this point from the point of view of applications to the
earth’s troposphere and stratosphere is evident *.

§ 13. Zhe boundary between troposphere and stratosphere.—
It is convenient to denote the regions below and above the
_level 7, in our ideal problem by the words “ troposphere”
and ‘stratosphere ”’ respectively, without implying any
reference to these actual regions in the earth’s atmosphere.
Then §12 shows that under the conditions there stated
a stratosphere cannot be isothermal unless its optical thick-
ness is zero or cosa=4n. If the optical thickness is not
zero and cos « >4n, the lower parts of the stratosphere must
be warmer than the upper. ‘This agrees with § 2, where it

was found that Gold’s stratosphere is warming up at the.

base.

We are now in a position to frame in a precise manner
the problem of where the division between troposphere and
stratosphere should occur, in the ideal case. Let us suppose
that there is a certain distribution of temperature which the
processes of convection tend to set up throughout the whole
atmosphere. Let the corresponding black body radiation-
function be expressed as a function of optical depth, say
B.(7). This temperature distribution together with the lower
boundary surface implies a definite upward intensity of
radiation at any point 7, say I,(7), which is determinate and
caleulable when B.(7) is given. Let tT. denote the optical
depth of the surface of separation between troposphere
and stratosphere which it is required to determine. Then

* The points which are the subject of §§10, 11, 12 are substantially
made by Emden, in the form appropriate to diffuse radiation. But
Kmden’s analysis is in parts a little complicated by his introducing
unnecessarily early into the investigation an empirical expression for
the water-vapour in the earth’s atmosphere as a function of height.

(the

the Insolation of an Atmosphere. Sor

below 7, the temperature is given by B,(r) ; above 7 it is
given by the function B(r) given by formula (28"). At 7,
the upward intensity of radiation is that appropriate to the
state of radiative equilibrium ; it is the value I(7,) deducible

from (26’’) and (27'’),
I(1s) = B(rs) +38 (cosa—4nye-rs we,

Flence v2 is the root of the equation

cee a Ugreise tin <irin she, nO)

Suppose this equation is solved. Jt by no means follows
that ‘
B.(t2) = B(t2) 3

i.e. 2¢ by no means follows that the temperature immediately
below the junction is continuous with that immediately above tt.
Further, evenif it happens that these temperatures are equal,
it does not follow that the condition for a convective atmo-
sphere is satisfied in the region immediately below 72. For
a physically possible distribution both these conditions must
be satisfied. Hence, in general, it is not possible to determine
a level rT. such that a prescribed temperature distribution exists
up to 7, and a radiative one above it.

The question must therefore be studied in the reverse order:
what conditions does the existence of a stratosphere of non-
zero optical thickness impose on the temperature distribution
in the upper troposphere? It would make the present paper
too long to take up the investigation here. But it appears to
be possible to show thatif the temperature is continuous at 79,
then in general (but not necessarily) the temperature gradient
is discontinuous there. This is, of course, what is observed.

In the earth’s stratosphere, on theother hand, the ob-
served absence of vertical gradients strongly suggests that
if it is in strict radiative equilibrium its optical thickness is
practically zero. For the particular relation (cosa=tn or
cos A=nj,/2) which is necessary for an isothermal strato-
sphere of non-zero optical thickness cannot be satisfied save
in very high latitudes; and even here (as we shall see) this
would be prevented by the additional radiation due to
world-wide convection. Further, we have seen in § 2 that
if the absorption of solar radiation is neglected, an isothermal
stratosphere would soon cease to be isothermal and would be
disturbed by convection currents. If now the optical thick-
ness of the stratosphere is practically zero, a state of radiative
equilibrium will probably extend a little way below the tropo-
pause, and the observed suddenness of the demarcation must

3M 2

892 Mr. KE. A. Milne on Radiative Hquilibrium :

be due to a sudden diminution of absorbing power. This,
again, would indicate the tropopause as the boundary of the
water-vapour atmosphere. ‘The contrary has, however, been
urged by Gold * on different grounds, and it is difficult to
deny the force of his arguments. The matter is obviously
one of considerable difficulty.

14. Applications.—This coneludes the discussion of the
idealized problem of which it is the main business of the
paper to give an account. The theory is capable of a
number cf applications to the earth’s atmosphere, but the
principal of these, at least in the case when the incident
radiation may on the average be taken to be diffuse, have
already been made by Emden. Perhaps the result most
directly useful is the correction to the Schwarzschild boundary
temperature due to the absorption of the incident radiation
there, given by formule (33) or (34), or Hmden’s form (32).
(The Schwarzschild temperature is given by putting n=0.)
From a discussion of the observational material, Emden finds
that n may be taken to be 5. With cosa=34 this gives an
increase of 1 per cent., making the calculated value of To
(see § 1) about 216°. If n is taken equal to ,1,, Ty becomes
219°, the observed value.

Another application made by Emden is to show that an
atmosphere entirely in radiative equilibrium would be an
impossibility, even in the absence of the warming effect due
to the earth’s surface (§ 10 above). Tor, taking into account
the water-vapour distribution, radiative equilibrium implies
at a sufficient depth temperature gradients in excess of the
critical gradient for stability ; so that convection currents
would be set up. Hmden finds that this would occur at
a height of about 3 km.

When Emden wrote, the variation of the temperature of
the stratosphere and height of the tropopause with latitude
was not fully appreciated. And in the light of this variation,
the small improvement in agreement, due to the introduction
of n, between the Schwarzschild temperature and the ob-
served mean temperature for S.E. England becomes largely
meaningless. What is astonishing, a priori, is that the two
temperatures should agree as well as they do. The agree-
ment can only mean that the actual temperature of the
stratosphere over S.H. Hngland must be very close to the
mean temperature over the earth. The agreement is partly
helped by the circumstance that the latitude of England is

* Geophysical Memoirs, No, 5, p. 129 (1913).

the Insolation of an Atmosphere. 893

close to 60°; and we have seen that the value «=60° plays a
special] part in the theory.

It is therefore interesting to inquire whether the theory
developed in the present paper has any bearing on the
question of the origin of the variation with latitude. The
facts to explain are that the temperature of the stratosphere
increases as the latitude increases, and that the height of
the tropopause decreases. Reference to recent books on
meteorology and the physics of the air shows that there
is no accepted detailed explanation.

Assuming that a stratosphere is optically very thin, it will
be very nearly isothermal and its temperature will be equal
to T). Formula (34) shows at once that the ratio of T, to T,
(the effective temperature of the insolation) increases as
» increases, provided » is not zero, and that the increase
becomes relatively large for high latitudes. As seen in $7,
this is due to the increased absorption of the solar radiation
in the more superficial layers. ‘This does not seem to have
been suggested before, and it may in fact be one of the con-
tributary causes of the increased temperature in higher
latitudes. But the first of formule (31) shows that the
actual value of Ty) decreases as X increases. All the theory
indicates is that Ty decreases much more slowly than 1, as
X increases, and so the absolute increase of T, as observed is
not accounted for. The tendency to increase would be helped
if it could be shown that n increased with X, i. e. if the ratio
of absorption of solar radiation to that of terrestrial radiation
increased with latitude. Some effect of this kind there must
be ; for carbon dioxide is more important as an absorber of
solar radiation than of terrestrial radiation, and owing to the
decreased humidity in high latitudes the ratio of carbon
dioxide to water vapour is there greater. But this, again,
would not appear to be sufficient.

It must now be reealled that we have assumed throughout
that any portion of the surface radiates away an amount of
energy equal to that incident on it. But we know that this
is not true for the earth’s surface. Heat is convected from
the equatorial regions toward the polar regions: otherwise
the change of surface temperature with latitude would be
much more severe than it is. Hence the equatorial regions
must radiate less than they receive, the polar regions more.
Now if vF is the additional net outward flux (positive or
negative), the radiative distribution of temperature is obtained
by adding the term F(4+7) to the right-hand of (28"), and
the boundary temperature is given by

olj/r =4F +48(cosat in), . . . (Al)

894. Mr. EH. A. Milne on Radiative Hquilibrium :

A similar formula holds when rotation is taken into account.
Since F' is positive in high latitudes and negative in low,
T,) should be greater in high latitudes and smaller in lone
than it would be in the absence of convection. Moreover,
for some particular latitude F will be zero, and here the value
of Ty will be the same as in the absence of world-wide con-
vection. The agreement between calculation and observation
for S.H. England thus implies that in this latitude the amount
radiated is about equal to the solar radiation incident.

But T, for high latitudes ewceeds that for low. Now
am(F +S cose) is the total outward radiation to space.
Hence (on the assumptions made) the total radiation to
space in high latitudes must exceed that in low, unless the
change of n is very considerable. This is a surprising
result, but not necessarily impossible; if the stratospheric
temperatures are really maintained to great heights under
the influence of radiation there seems little escape from
it. The difficulty is, of course, not new. Gold dealt with
it as follows *. Gold showed that if the absorbing power
of the atmosphere increases, then the theoretical height of
the tropopause increases. In the notation of § 2, 1t can be
deduced from (4) that tv, though increasing with 7, is
fairly insensitive to it. Roughly : speaking, then, on Gold’s
theory the isothermal state sets in at a fixed optical depth
below the outer boundary ; hence the more absorbing the
atmosphere the smaller is 2/7, the smaller is Pol Pry and
the greater is the height of the tropopause. The known
increased humidity over the equator, with consequent in-
creased absorbing power, would thus account for the observed
increased height of the tropopause; and the lower tempe-
rature of the stratosphere follows from the increased height
through which a convective gradient holds, even allowing
for the higher ground temperature. But the above difficulty
still remains, for the increased absorbing power implies a
decreased outward radiation.

Gold argued from the improbability of this that “ the
atmosphere is not a ‘grey’ body, but must have nearly
perfect transparency for some spectral region.” It is well
known that the coefficient of absorption varies considerably
from place to place in the spectrum, whereas we have
assumed it to possess but two values—one for solar radiation
and one for terrestrial. But it is very doubtful whether
this removes the difficulty. For there still has to be an
equilibrium of radiation. One might reason generally
that the transparency of the air in certain spectral regions

* Geophysical Memoirs, i. p. 128 (1918).

the Insolation of an Atmosphere. 895

would permit the escape of extra radiation which would
have no effect in controlling the temperature. But the
boundary temperature is not necessarily lower : it may be
higher or lower, according to the spectral regions in which
the air is not transparent. It has recently been shown by
the author * that if a thin layer of material is exposed
on one side to black radiation of effective temperature T,
and is in radiative equilibrium, then it will take up a
temperature Ty which is equal to 2~#T, (the Schwarzschild
value) when the material is grey, but which, whatever the
optical properties of the material, will satisfy the inequalities

Py hg ee:

and that Ty will approximate to 'T', if the material is trans-
parent save in the extreme ultra-violet, to $T, if transparent
save in the extreme infra-red. (Here “ ultra-violet”? and
“infra-red” must be interpreted as relative to the value
of YAmax. corresponding to T,.) If, for simplicity, we
neglect the absorption of solar radiation, the theorem can
be applied as it stands to the stratosphere. It shows that
the temperature will be less than for uniform absorption
only if the absorption occurs principally on the long wave-
length side of Amax—in this case about 10 pw. But water-
vapour is least opaque f to long wave-length radiation in
the region 7 w to 204, more particularly in the region
Sw to 12. There is, indeed, important carbon-dioxide
absorption { in the region 13 w to 16 yw, but this is usually
considered to be not large compared with the water-vapour
absorption. Without a detailed numerical investigation
it would be difficult to estimate the resultant effect ; but
if carbon-dioxide absorption is important near the equator,
it should be more important in higher latitudes, and this
would go against the argument. If a relation Tp=eT,
holds above the equator in virtue of selective radiative
equilibrium, it ought to hold too in higher latitudes ; for it
is difficult to see why the atmosphere at 20 km. above the
equator should be optically different from that in higher
latitudes in the direction of being relatively more absorptive
above the equator on the red side of Amax.. Moreover,
if we do modify the selective optical properties of the
atmosphere from equator to pole, then Gold’s explanation

* “he Temperature in the Outer Atmosphere of a Star,” Monthly
Notices R.A.S. Ixxii. p, 368 (1922). See also Fabry, Astrophys.
Journ. xlv. p. 269 (1917).

+ Abbot, Annals Astrophys. Obs. Smithson. Inst. ii. p. 167 (1908).

t See, for example, Humphreys, ‘ Physics of the Air, p. 88, 1920.

e

896 On Radiative Equilibrium.

of the change of height of the tropopause no longer elds,
at least without further examination.

In view of these considerations, it seems on the whole
tenable that the outward radiation from the equator is
less than from higher latitudes, and that the variation of
stratospheric temperature must be principally due to the
general circulation of the air in the convective region.
(The connexion between the variation of temperature of
the stratosphere and the observed uniformity of pressure
at 20 km. over the whole earth has been pointed out by
W. H. Dines*.) The higher upper-air temperatures of
high latitudes may still be helped by the increased direct
absorption there, in the way we have seen.

§ 15. ee Mae is shown that if the atmosphere is
divided into two shells—a lower one (the troposphere) in
convective equilibrium, and an isothermal one (the strato-
sphere),—then the stratosphere cannot be in strict radiative
equilibrium unless its optical thickness for low-temperature
radiation 1s zero, even if it is in radiative equilibrium as a
whoie. The only exception is when the lower region is also
in radiative equilibrium as a whole and when, in addition, a
special relation (cosa=4n, or cosX=n/,/2 when rotation
is allowed for) exists between the angle of incidence of the
solar radiation and the ratio of the coefficients of absorption
of solar and terrestrial radiation. The theory of atmospheres
in radiative equilibrium subject to insolation is discussed in
detail for various cases, including the dependence of the
temperature distribution on the angle of incidence of the
solar radiation. An integral equation for the temperature
is obtained. Comparison is made with HEmden’s work.
From an application of the results to the earth’s atmosphere
it is inferred that the variation of the temperature of the
stratosphere with latitude cannot be accounted for on
radiation principles unless the total radiation of the earth
to space is greater in high latitudes than in low latitudes.
This is probably the case, and the observed distribution
of stratospheric temperature is probably connected with
the general circulation of the air ; however, the increased
direct absorption of solar energy “in the upper levels in
high latitudes must have some effect.

It is intended to insist principally on the general theory,
and the applications are only made tentatively.

July 17, 1922.
* Geophysical Memoirs, No. 13, p. 71 (1919).
; 4

LXXV. On the Molecular Theory of Solution. I.
By 8. C. Braprorp, L).Sc.*

[* a previous paper +} a preliminary attempt was made to

consider the phenomena of solution from the point of
view of molecular energy and attractive forces. Recent
advances in atomic theory make it fairly certain that atoms and
molecules are surrounded by fields of force. Indeed it has long

been recognized that cohesion and surface tension are due to ~

molecular forces. But the part played by these forces in the
phenomena of solution has not been considered sufficiently.
We have to take into account the cohesion of the solvent, the
adhesion of solvent and solute, and the cohesion of the solute.
Whena solid is brought into contact with a liquid, the surface
tension of the solid, due to the unbalanced cohesive forces at
its surface, is reduced by the counter attraction of the liquid
particles for those of the solid. On this account an appreciable
number of solid particles may have sufficient kinetic energy
to overcome the diminished surface forces and escape into
the liquid. But any that come again within the range of
attraction of the solid surface will be reclaimed, so that
eventually a statistical equilibrium may be atiained when the
numbers of particles leaving and returning to the solid are
equal. This state, corresponding to the solubility of the
solid, is determined by the equation

s?

Ng=nge “, es gen ene a)

where vq and nz are the numbers of particles in unit volume
of liquid and solid respectively, « is the most probable speed
of the particles, and s is a velocity satisfying the condition
that the momentum normal to the surface, 4 ms’, of a particle
of the solute is just sufficient to carry it through the surface
layer. Similar reasoning applied to a cooling solution tf
shows that it will pass through a metastable stage, as the
diminishing kinetic energy allows the aggregation of the
particles, until they reach sucha size that the force of gravity
dominates their Brownian movement, the particles settle out
of solution, and the statistical equilibrium is re-established.
By the application of Perrin’s formula

ny
log —

* Communicated by the Author.
+ Phil. Mag. vol, xxxviii. pp. 696-705 (1919).
{ Biochem. J. vol. lv. pp. 553-555 (1921).

898 Dr. 8. ©. Bradford bn the

it was calculated that the radius of a gelatin particle just
large enough to settle would be about 0:06. This agreed
approximately with the value found experimentally by the

application of Stokes’ law. .
As a first approximation, the force between two particles
was taken as |
LS eee Mer mL,

ab

Yap being the distance between their centres. From this it
was deduced that the initial force to be overcome by a
particle of solid escaping into pure solvent is proportional to

Mis Mul M, M ‘it
faq 2 + ee GD)
2 ws iige
where the subscripts w and s refer to liquid and solid re-
spectively. The smaller this force, the greater the solubility.

In the normal case, when the molecular cohesion of the
solute is greater than that of the solvent, the last term
of (11.) will be the largest, and the force opposing solution
will be greater as the cohesion of the solute increases. That
is to say, the solubility of ordinary salts is smaller the greater
their cohesion. When the cohesion of the solute is less than
that of the solvent, as often happens with organic substances,
the middle term of (iii), representing the adhesion of solute
and solvent, will be greater than the last. In this case the
solubility will inerease with increasing adhesion of solute
_and solvent. And since the adhesion increases with tie
cohesion of the solute, the solubility of such substances is
greater, the larger their cohesion. Similarly the solubility |
of a given solute, in different solvents with less cohesion, will
increase with the cohesion of the solvent, while the solubility
of the same solute, in solvents with greater cohesion, will
diminish with increasing cohesion of the solvent.

Relative molecular cohesion may be estimated in a variety
of ways. ‘lraube * used the enhanced or diminished surface
tension of a solution as a measure of the adhesion of solute
and solvent. He considered solubility only from.the point
of view of adhesion, but was able to show that the solubilities
of organic liquids, which reduce the surface tension of water,
follow the order of the surface tensions of their solutions.
As, however, he neglected to take into account the mutual
cohesions of both solvent and solute as well as their kinetic

* Ber. deut. chem. Ges. vol. xlii. p. 86 (1906).

Molecular Theory of Solution. 899

energy, he failed to explain why the solutions of many other
substances, which increase the surface tension of water

decrease with increasing surface tension of their solutions.
When all the factors are considered, as above, it is found
that the solubilities of substances generally, whether solid,
liquid, or gaseous, can be accounted for, and that when
substances are arranged in the order of their solubilities s they
are in the order of every other property of solutions. It was
pointed out that exceptions may be expected when the

distances of the electric charges of the particles may be
influenced by the configurations of the unlike particle-, so
that the adhesive forces may differ from those calculated
from the respective cohesions according to formula (iii).
In this way, however, the theory gives a picture of the
processes of solution and crystallization, and indicates for
the first time the cause of the widely differing solubilities
and solvent powers of different substances.

On the assumption of molecular fields of force, 1t may
be easy to understand why such properties of solutions
as solubility, degree of hydration, heat of solution, molecular
volume and compressibility, should follow the alee of their
surface tensions. But, at first sight, it may be surprising
that the same’ should be true of the depression of the
freezing-point, elevation of the boiling-point, vapour pressure,
and electrical conductivity, with the suggestion that, were
sufficient determinations available, osmotic pressure would be
included. The inference can hardly be avoided that these
properties, also, are influenced by the molecular fields. As
with solubilities, Traube attempted, unsuccessfully, to account
for the coincidence by considering only the adhesion of solute
and solvent. That these properties are affected by the mole-
cular attractions is, however, a direct result of the present
theory.

A solution may be considered asa liquid in which a number
of its ultimate particles have been replaced by others
having the same average kinetic energy but exerting
different fields of force. The concentration of solvent will
be less and the internal pressure of the solution will differ
from that of the solvent. Kleeman* has shown that the
intrinsic pressure of a solution is given by the relation

oe Be =f 2 ie oe cae ;

where Py »2 and Pye are the attractions exerted, respectively,
across a plane by.the molecules of the kinds w and s on the
molecules of the same kind in a cylinder of unit cross-section

* ‘A Kinetic Theory of Gases and Liquids,’ p. 202 (1920).

900 Dr. S. C. Bradford on the

and infinite length standing on the plane, and Prys refers
to attractions between molecules of different kinds. If
a solution be separated from pure solvent by a membrane,
permeable only to the latter, the pressure of the solvent
particles on the membrane ‘due to their kinetic energy will be
less on the side of the solution, because there are fewer solvent
particles. Consequently less solvent will diffuse through the
membrane from the side of the solution than from the side of
the pure solvent. To neutralize this effect it will be necessary
to apply a pressure to the solution equal to that which would
have been exerted by the missing solvent particles. Adopting

eR Ls, tel RT ah
Porter’s equation *, this pressure is equal to pa In addition
v—

to this effect there is another due to the altered intrinsic
pressare of the solution. Ifthe solute particles have stronger
fields of force than those of the solvent particles, there will
be a greater attraction for solvent particles on the side of
the solution that will increase the diffusion of solvent into
the solution. To balance this an additional pressure must
be applied to the solution. On the other hand, should the
molecular field of the solute be less than that of the solvent,
this pressure will be negative. Thus osmotic pressure is the
sum of two effects, kinetic energy and molecular attraction,
and may be expressed by the equation

T= +$(0), «le

where the second member of the rigbt-hand side corresponds
to the action of the molecular fields. If this term be
less than the first, a solution (e.g. of salicin), having a
surface tension less than that of the pure solvent, may yet
show a positive osmotic pressure and not a negative one as
predicted by Traube.

These deductions were made at the time of writing the
previous paper. Simultaneously Wo. Ostwald and Mundler ft
were led independently, by different reasoning, to a similar
expression for osmotic pressure. They made the last term
of (iv) correspond to the adsorption law, putting the equation
in the form

I] =cRT+ ke”,

and were able to show that the few reliable determinations
of osmotic pressure available could be made to conform to

* Trans. Farad. Soc. vol. xiii. p. 123 (1917).
t+ Kollowd. Zeitschr. vol. xxiv. pp. 7-27 (1919).

Molecular Theory of Solution. 901

this equation. More recently Kleeman * deduced expressions
for osmotic pressure in terms of molecular motion, attraction
and volume, remarking that “it is evident... that osmotic
pressure must arise through these properties of matter and
the equations are therefore fundamental in character. They
are, however, of little use in practice ...since we have no
means yet of determining experimentally how the quan-
tities . .. vary.”

Treatment of the problem was omitted from the preliminary
paper in the hope of finding a more definite correlation. This
was not far toseek. On account of the meagre, and sometimes
doubtful, data available for osmotic pressure, cryoscopic deter-
minations were considered. From the present point of view
the depression of the freezing-point of a solution may he
regarded as the sum of two effects: a normal depression cor-
responding to van’t Hoft’s formula

0°02T?

De L

due to the presence of a solute with the same molecular field
as the solvent, together with an effect produced by the altered
internal pressure of the solution. The molecular depressions
of organic substances, having molecular fields not greatly
different from that of the solvent, will not deviate much.
from the normal. But aqueous solutions of many salts show
a marked increase in surface tension that indicates consi-
derable alteration in internal pressure. The freezing-point
depressions of such solutions should differ appreciably from
the normal. Since both surface tension and intrinsic pressure
are due to molecular attraction, we may take the one as pro-
portional to the other. And since the freezing-point of water
is depressed by increase of pressure, it follows that the
freezing-points of aqueous solutions of salts should be greater
than the normal by an amount which is proportional to
the increase of surface tension. It must not be forgotten,

however, that substances exhibiting greater molecular attrac-
tion than water may tend to aguregate in solution. | This
must happen at the higher concentrations. In this case the
normal depression would be reduced correspondingly to the
smaller number of solute particles. Moreover, the field of
force surrounding the aggregated particles would differ
from that round a simple particle and should be less on
account of greater concentration of the lines of force within

* Loe. cit.

902 Drs. C: Bralemclcn ie

the aggregate. These effects are the more likely with solutes
having large molecular fields. Or again, as pointed out
above, there may be a closer approximation of solute and
solvent particles, on account of a suitable relation between
the positions of the electric charges, with a corresponding
alteration in the external field around the particles. Thus it
is possible that a solute may reduce the surface tension of a
solvent although the molecular field of the solute may be the
greater. This effect is the more likely with solutes having
hydroxyl or acidic hydrogen groups. |

Such considerations do not increase the prospect of
finding an exact ratio between the increments of surface
tension and freezing-point depression. On this account the
constancy in this ratio shown by the few substances for
which there are data available is the more striking. In the
table below, the observed depressions of the freezing- -point are
taken from Landolt and Bérnstein’s Tabelien, 1919. But as
the determinations of surface tension were not made for the
same concentrations as the freezing-point observations, the
values given in the table were read from smooth curves
drawn through points corresponding to the published figures,
taking the surface tension of water as the zero point on each
curve. On account of the sparseness of the observations
there is a slight uncertainty in the values at the lower
concentrations. More extensive determinations are desirable
and are being undertaken.

The last column but one in the table shows tie ratio
between the increments in surface tension and depression of
freezing-point, the normal depression being taken as 1°°86
per gram-mole per litre. The chlorides of barium, calcium,
magnesium, and strontium behave like those of sodium. or
potassium. Copper and magnesium sulphates are the only
other substances for which both eryoscopic data, and surface-
tension determinations by the method of capillary rise, are
available. Surface tensions observed by other methods do
not correspond with the values obtained from the capillary
rise, nor are they proportional to the increments in the
depression of the freezing-point. The above-mentioned
sulphates give depressions of freezing-point at moderate
concentrations which are even less than the normal, so that
it is clear that some disturbing factor, such as agoregation,
comes into play.

With these two exceptions the eoneinners in the ratio is
remarkable, being in many cases about 0°64. It is interesting
to inquire the meaning of this figure. It has been assumed
that the surface tension of a liquid is proportional to its

Molecular Theory of Solution.

|
Grams |g. moles |

‘ De-
100 c.e.| litre.

Sodium Nitrate.
0°43 0:05 0°17°
Aa ie 0:20 0°67

Potassium Nitrate.
202.) 0°20 064°

2°53 0°25 0-77
5:06 0°50 1:47
T59 | o't5 211
10:12 | 1:00 2-66

Sodium Sulphate.
1-42 0-10 0°438°
284 | 020 081
4°27 0°30 116
T1l 0°50 1°84

Potassium Sulphate.

087 | 005 | 023°
fr ao: || 0-48
3-49 | 0:20 | 0-81
775 | 044 | 166

Sodium Carbonate.
: 0°21 ; 0-02 | 0-10°
053 005 | 0-23
1:06 | 0°10 0:44
eke b> O20 0°83
hat | 050 1°88

vi) OOL +.) G:05°

0-28 | 0:02 | 0-10

0:69 | 0:03 | 0-23
| 1:38 |-0:10 | 0-45
| 277 | 0:20 | 088

| Sodium Chloride.
| 063 | O11 | 0:38°
| 1°36 0:23 0°80
P Bibl 0-43 1°45
| 410 | 070 | 2-40.
Potassium Chloride.
1°56. | O21 0-71°
310 | 0°42 1:40
5 60 | O75 2°53
{

pS

Potassium Carbonate.

Observed| Normal

De-

pression.) pression,

0-02°
0-04
0:09
0-19
0°37

0-21°
0-43
0°80
1:30

0°39°
0-78
1°39

Increase.

Surface,
Tension

74°23
74:57

7452
7457
7486
75°04
75°22

74°40*
Te.
75°06
75°69

74°32
74°50
74:80
75°30

74:19
74°32
74:50
74:83
THxDT

T415
74:18
14:28
74°48
74°89

74°32
74:54
74°91
75°40

74°40
74:70
75°10

|
|

‘Increase.

0:13
0:47

0:42
0-47
0-76
0-94
1°12

0°30
0°64
0°96
1:59

0:22
0°40
0:70
1:20

Ratio.

0-61
0°62

0-64
| 0°65
| 0-73
0°75
O71

0°80
0°69
0°62
0°57

0-64
0-60
0:63
0:69

0:69
063
0°64
0°63

+ 0°65 | |

| 0-64 |

| O-75

| 0-79 |

068
0-64

0°85
0°88
0:82
0 86

0-80

| O90 |

| 1:16

903

eg
!
|

Surface |
| Tension |
| x 0°64.

|
|
|

* These figures were calculated from the two sets of determinations at different
temperatures.

904 Mr. R. A. Mallet on the Failure of

intrinsic pressure, and Walden * has calculated the value of —
the ratio from determinations made at the boiling-point. If
surface tension is really proportional to intrinsic pressure,
the same ratio should hold at ordinary temperatures, nor is
there anything in Walden’s calculations inconsistent with
this. Accordingly we may write

OII=75'36c,

dII being the increment of intrinsic pressure due to the non-
ageregated solute particles and 6c the corresponding in- |
crement of surface tension in dynes per centimetre. Since
1000 atmospheres increase of pressure depress the freezing- —
point of water 8°:5, the depression, 6D, corresponding to
the increase of internal pressure dII would be

60 X75°3 x 875
a a de
: 1000
= 0°64 60%. S00 0) io

The correspondence between the observed increases in the
depression and the calculated values of 0°646c, shown in
the last column of the table is remarkable. It appears,
therefore, that the observed increases in the depression
of the freezing-point of aqueous solutions of salts above
that due to a normal solute, are caused by the enhanced
intrinsic pressures resulting from the greater molecular
fields of the solutes. For non-aggregated salts the increased
depression is given by the equation (v.). Incidentally this
may be regarded as an experimental verification of Walden’s
relation.

The Science Museum,

South Kensington,
London, 8S. W. 7.

LXXVI. Onthe Failure of the Reciprocity Law
in Photography. By R. A. Matuet, B.A.+

VNHE failure of the photographic plate with a silver

bromide-gelatine emulsion to obey the ‘‘ Reciprocity
Law” of Bunsen and Roscoe was first observed by Abney f.
The first quantitative work on the subject was done by
Schwarzschild §, who proposed as an empirical “ Law of

* Zett. physik. Chem. vol. lxvi. p. 885 (1909).

+ Communicated by Prof. ‘T. R. Merton, F.R.S.
t Proc. Roy. Soe. liv. p. 143 (1893).

§ Astrophysical Journ. xi. p. 89 (1900).

the Reciprocity Law in Photography. 905

Blackening”’ the following relation :
See De

where 8 is the degree of blackening, I the intensity of the
incident light, T the time of exposure, and k& and p are
constants. The constant & is fully discussed by Plotnikow *,
and depends on several factors, mainly on the thickness and
composition of the film. The exponent p is known as the
Schwarzschild constant. Its value was determined by
Schwarzschild for Schleussner plates, which he exposed to
the light of a normal benzene lamp at various distances for
varying lengths of time, measuring the depth of blackening
by comparison with a scale of blackness standards made by
means of a Scheiner sensitometer. He found p=0 86.

Other workers in this field are Becker and Werner 1,
Leimbach {, Helmick § who used a Lemon spectrophoto-
meter, Stark || who used a Kénig-Martin spectrophotemeter,
and Plotnikow who used a Kruss polarization colorimeter.
Apart from the fact that all these workers have obtained their
results by methods depending on the comparison of light and
shade by eye, and that their numerical results vary some-
what, no one appears as yet to have suggested any physical
meaning for the constant p. It is, in fact, uncertain whether
it depends on the density, and whether it is constant for a
given make of plate under all conditions. Stark has shown
that it is subject to considerable variation over abnormal
ranges of exposure, and he and others have shown that it
varies somewhat with the wave-length of the light used.
Schwarzschild (doc. cit.) used both light and heavy blackening,
and found it to remain constant, but some further investigation
seemed desirable.

It was decided to use a method which did not depend on
visual comparison of varying shades of blackness, and to this
end a Goldberg wedge screen was introduced between the
plate and the source of light. This was carried out as
follows :—

A brass plate, having in it a slit about three inches high
and three millimetres wide, was let into a wooden hoard,
having at the back a recess of such size that a quarter-plate
wedge screen fitted in closely, in such a position that the
slit was opposite the middle of the wedge. Immediately
behind the wedge was placed the plate-holder, consisting of

* Lehrbuch d. Photochem. p. 667.

+ Zeit. f. wiss. Phot. x. p. 382 (1907).

t Zeit. f. wiss. Phot. x. p. 137 (1909).

§ Phys. Rev. xvii. 2, p. 142 (1921).

| Ann. d. Phys. xxxv. 3, p. 461 (1911).

Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3N

906 Mr. R. A. Mallet on the Failure of

an ordinary single metal dark slide, so arranged that it could
be slid across the slit and stopped in any of several positions
by means of a small spring loaded plunger. The wooden
‘board was fixed to the bench in an upright position, and a
well-fitting wooden shutter arranged to slide across the front
of the slit. This shutter was worked by hand. The source
of light wasa 36-watt 12-volt gas-filled motor headlamp bulb,
connected in parallel with a voltmeter to a 12-volt accumu-
lator battery. This battery was connected through a suitable
resistance to the town mains so as to form acharging circuit.
In the course of three months the voltmeter reading fell by |
0-2 volt, so that the drop during one experiment was
negligible. The lamp was enclosed in a light-tight box with
a square hole cut in the front, with the object of removing
irregularities due to stray reflexion from tke walls of the
room. The lght was used at a constant distance of about
one metre from the plate, at which distance even illumination
over the whole of the slit was obtained. All exposures were
timed by a stop-watch, the error in moving the shutter being
of the order of 1/10th of a second, which may be neglected,
in view of the length of the exposures used.

Jt is evident that with an arrangement of this kind, if a
plate be exposed behind the wedge, a line will be photo-
graphed on it, the length of which will be a function of
the time of exposure and of the intensity of the incident
light. Furthermore, if two lines are photographed side by
side and the time of exposure varied while the intensity is
kept constant, then

Tyee Typ = Tyee TY,

_ __ plt—th)
and hence p= log Ty lee 8?

where p is the wedge constant, T) and T, the times of
exposure, ¢, and ¢, the lengths of the lines.

Determination of the Wedge Constant. |

The wedge constant was determined for the light used by
exposing Paget “ Half-Tone”’ plates behind it to light of two
different intensities, the time of exposure being kept constant.
The intensities were in the ratio 1 : 25, the variation bein
obtained by altering the distance between the light and the
plate from one metre to five. Then, since Tp>=T,,

Ter = Tie 9h,
oe log I,—log I,

and hence
ty —ty

the Reciprocity Law in Photography. 907

The following are the values obtained :—

Exposure

(mins.), ¢) (mm.). ¢, (mm.). 0.
30 33°10 14:79 0:0754
"32°96 14°65 00764
30 34°92 16°34 0:0740
25 34°87 15°80 0:0733
34°72 15°51 0:0728
2°5 36°72 18°64 0:0773
39°66 16°48 0:0729
25 34:08 16°23 00783
33°98 15°24 0:0746
2°5 35°54 17-74 0:0725
2°5 46°07 27°34 00746

In the case of the last of the above results the plate used
was a Wratten ‘“ Instantaneous,” which is considerably faster
than the Paget. The mean of the above is 00747, with a
probable error of +0°0004.

The mean obtained is somewhat higher than the value
given by the manufacturers, Messrs. Ilford, in their calibration
ot the wedge used, but this is accounted for by the fact that
the wedge constant increases in the violet and ultra-violet.
Toy & Ghosh*, who investigated this point, found that the
value of p begins to rise at a wave-length of about 4500 A,
-at which point the density of their wedge was 1:25. The
density at 4000 A they found to be 1°6 and at 3000 A about 3:5.
A usual method of calibration is to focus the (small) filament
-of an electric lamp on various parts of the wedge, and to throw.
the spot of hght passing through on toa white screen. Light
froma similar source is passed through two Nicol prisms on
to an adjacent portion of the same screen, and the two spots
adjusted by means of the nicols to the same intensity.
Since the normal photographic range extends further into
‘the ultra-violet than the visual range, the value for p given
by this method will be lower than that obtained by the
photographic method.

Determination of the Schwarzschild Constant.

Three types of plate were used, so chosen as to give a large
‘variation of properties. They were the Paget *‘ Half-Tone,”’
the Wratten “ Instantaneous,” and the Imperial ‘‘ Eclipse.”
‘The Paget “‘ Half-Tone” is a process plate, about one-third
faster than wet collodion plates, and giving great contrast.
‘The Imperial “ Eclipse ” is one of the fastest plates obtainable,
-and the Wratten “ Instantaneous” is intermediate between
‘the two, being a medium slow landscape plate.

* Phil, Mae. Dec. 1920, p. 775.
3.N 2

908 Mr. R. A. Mallet on the Failure of

The approximate: relative speeds of these plates were
obtained by exposing a specimen of each behind the wedge
to light of the same intensity for the same length of time,
and measuring the lengths of the lines obtained as nearly as
possible without further treatment of the plates after develop-
ment. Then, at the end of each line,

TS)? cme’ =e

where S is the speed of the plate and e* is a constant.

Since p is not identical for each plate, it is necessary to
reduce all lines to a standard length, and take the ratio
of the actual time to the time for that length as proportional
to the speed. It ¢,, t,, etc. are the actual lengths of the lines,
and ¢, the standard length, then

_ loglh>p—k+>p log (ST)

t=
p
aon = wet eagles (ST).
: p
al Mf ee = log ks,
OS mr T, ae 0) ==
Si Oe perp i
zal = — ( Gilat + Te | to—t,).
08 y Di 2) es Ds (t2—to)
Hor dametiplatesy au press Mik O°0864, t==2i53
» Wratten plates ......... p/p=0'0884, t=38-0
», Lmperial plates ......... p/p =0°0884, ¢=46°7.

t) was taken as 35, which is approximately the mean value
of t. From these figures the relative speeds of the plates
are roughly :—Paget 1, Wratten 29, Imperial 165.

The Schwarzschild constant was determined by photo-
graphing lines on the plate with long and short exposures
alternately, the intensity of the light being kept constant
throughout, and the duration of the long exposures being
usually one hundred times that of the short. The actual
times of exposure are given in the table below. In the case
of the Wratten and Imperial plates, the intensity of the
incident light was reduced to a convenient extent by inter-
posing one or two neutral screens between the source of
light and the wedge, the screen being let into the front
of the box in which the light was enclosed at a distance
of about 5 cm. from the filament. This was necessary,
as, owing to the sensitivity of these plates, the exposures
would otherwise have had to be made too short to admit

the Reciprocity Law in Photography. 909

of accuracy in the timing. The plates were fully developed
in hydroquinone and caustic soda developer, and “cut”
by a few seconds’ immersion in potassium ferricyanide
to remove the slight chemical fog and to give a more
easily measurable end to the line. After this treatment,
the Paget, and in one or two instances the Wratten, plates
could be measured directly, but in other cases the plates had
to be printed onto Paget’s, and one, or sometimes two, further
transparencies made before it was possible to measure them.
The plates from which the wedge constant was determined
were treated in the same manner. All measurements were
made on a Hilger travelling microscope, and are accurate to
‘1 mm., and in some cases to °01_mm. :

As is well known, irregularities occur near the edges of a
plate. For this reason, only two long and two short lines
were made on each plate, and thus only the central portions
were used.

It will be seen that the results for. the same type of plate
differ appreciably, but this is to be accounted for by variations
in the thickness of the film on the plate, and the errors
arising in this manner are eliminated by taking the mean of
a considerable number of determinations on different plates.

Paget Plates. Wratten Plates, Tmperial Plates.
Exposure E ! E :
Gaamdsy. D. xposure. p. « :. Exposure. Pp.
32 & 15976 0-830 10 &1020 0-874 20 & 2000 0°848
i0 & 1000 0°840 0°845
32 & 15976 — 0°833
0-900 10&1000 0867 10&1000 0-868
10&1020. O861 | 0°880
60 & 6000 0-867
0879 10& 1008: > 0857 —- 10% 1000" © 0's 61
0°798 0:897
60 & 6000 0:856
0°873 10&1000 0:867 10&1000 0854
- 0°834 0-861
45 & 4500 0-792
0 817 10 & 1000 0°831 10 & 1000 0°765
0-840 0°830
. 45 & 4560 0°962
0-902 5 & 550 0-878 10&1000 0847
0°837
45 & 4500 0°861 10&1000 0-834
0-871 0815 10&1000 0°808
0°865 0°865
45 & 4500 0°847 _ 10&1000 ~ 0829
0°846

30 & 2000 0-869

30 & 2000 0°885
0°868

910 Mr. R. A. Mallet on the Failure of

From the foregoing are obtained the following mean
values of p :—

Paget “* Halt-Tone”’ plates v.40... 0:865+ 0:005
_ Wratten ‘ Instantaneous” plates. 0°846+0°005
Imperial ‘‘ Eclipse” plates ......... 0°3846 + 0-005

The probable error, which is certainly not due to errors in
measurement of the lengths of the lines, was calculated i in
the usual manner.

It is noteworthy that the mean values for the fairly slow
Wratten plates and for the very fast Imperial plates agree
exactly. Becker and Werner (loc. cit.) have stated that the
value of p tends to increase ae sensitivity, but this does
not appear to be confirmed by the present investigation.

The question then arose as to whether p was really a
constant for any one type of plate for any variations of
intensity and duration of exposure within the normal range
(great over-exposure, 2. e. solarization, being left out of the
question), or whether it was dependent on the intensity or on
the time of exposure, or on both. Hvidently, if p is constant
for all conditions of time and intensity, the gradation of two
lines on the plate of about equal length, but made under
differing conditions, will be the same.

To ascertain this, two plates were prepared, having on
them lines of approximately the same length, but in which the
intensity and the duration of exposure were so altered that
IT?=const. Six lines were made on the two plates. The
intensity of the incident light was as 1 in the case of three
lines to 6°25 in the other three, and the time of exposure
adjusted so anne IT’=8 in two cases and =12 in the other

four.

The plates used were Paget “‘ Half-Tone.” They were fully
developed but not “cut,” and the relative blackening at
different points of the lines was compared by means of a
photometer, involving the use of a photoelectric cell and a
string galvanometer which had been designed by Mr. G. M.
B. Dobson, to whom I am greatly indebted for the loan of
this instrument and for his assistance andadvice. Owing to
the limited range of the galvanometer it was possible to
measure the densities of the lines only over a range of 10
to 14 mm.

The distances from the ends of the lines were plotted
against the actual galvanometer readings, which have been
used as an arbitrary scale of blackness. It will be seen from
the figure that the gradations of the lines are identical,

—

the Reciprocity Law in Photography. 911

despite the alterations of intensity and duration of exposure.
The points represented by +A etc. refer to the different
plates measured, and the values plotted are over the same
range of density so as to show the similarity in the shape of
the curves. From this it is evident that the Schwarzschild
constant has a definite value for each type of plate, and
that this value remains constant over a range of density
extending from zero to approximately unity, since the density
of the wedge used is about 0°075 per mm. and the portion of
the lines considered in the curves given is about 14 mm.
long.

Fig. 1.

600

uw
°o
oO

Calvarometer Readings.
b
°
°

Millimetres.

It is further evident from a consideration of these curves
that the value of the Schwarzschild constant is the same at
the “threshold,” 2. e. at the point where the plate is just
darkened, as at greater densities. Owing to the “cutting ”
of the plates used in determining the constant, the value
given by them was not that at the ‘‘threshold,” but
the measurements made with the photvelectric cell extend
to the extreme ends of the lines, and the curves coincide

perfectly at this point also.

912 Messrs. Roberts, Smith, and Richardson on

It is evident that the Schwarzschild constant has a very
definite physical significance, but what this may be it is not
possible to say. From the values obtained with the three
types of plate it is evident that it is not greatly dependent
on either the size of the grain or on whatever factors ulti-
mately determine the sensitivity of a plate to light, but a
fuller knowledge of the composition of the silver halide-
gelatine emulsion and of its physical properties would seem
to be necessary before a complete explanation can be arrived at.

This investigation was undertaken at the suggestion of
Prof. T. R. Merton, F.R.S., to whom I am deeply indebted
both for the loan of almost all the apparatus used, and for
much valuable help in the course of the investigation.
Thanks are also due to Mr. C. H. Bosanquet for several
valuable suggestions.

Balliol College Laboratory,
Oxford.

LXXVII. Magnetic Rotatory Dispersion of certain Paramag-
netic Solutions. By R. W. Roperts, M.Sc., J. H. SMirs,
M.Sc., and 8. 8. Ricnarpson, D.Sc., A.R.C.Sc.*

NE of the anomalies met with in the examination of the
Faraday effect is that, whereas the salts of iron in
solution give rise to a negative rotation, those of the strongly
magnetic elements, nickel and cobalt, produce a rotation of
the plane of polarization in the opposite sense—that is, in the
direction of the rotation produced by the great majority
of compounds both inorganic and organic. In the case of
cobalt sulphate the rotation, though positive, is almost zero.
It is noteworthy, however, that the measurements on which
such statements are based have been made only with refer-
ence to the D line or at most a few lines in the visible
spectrum, and some years ago it occurred to one of us that
further information might be obtained by examining the
course of the rotatory dispersion in the ultraviolet. Hxperi-
ments were carried out in 1916 on CoSQ,, and the spectrum
photographs showed that the rotation of this salt becomes
strongly negative in the ultraviolet. The principal absorp-
tion-band lies on the borders of the visible and ultraviolet,
and the rotation, which has a small positive value up to the
edge of the band in the visible spectrum, becomes negative
where the spectrum reappears, and remains negative through-
‘out the ultraviolet up to the point where general absorption

* Communicated by Prof, L. R. Wilberforce, M.A.

Magnetic Rotatory Dispersion of Paramagnetic Solutions. 913

cuts off the spectrum completely, which occurs in the region
of 3000 A.U.

The result with cobalt indicated that a systematic ex-
amination of the the rotation produced by paramagnetic
substances in the ultraviolet would be of interest, and the
present communication refers to the results obtained with
the sulphates (fig. 1), chlorides (fig. 2), and acetates (fig. 3)
of nickel and cobalt, also ferrous sulphate, in aqueous
solution™.

hea ration IN RADIANS X10

Su LPHATES

-2 »
A x lofu 4)

The method employed has already been described in
connexion with the rotation in certain organic compoundsf.
The solution was contained in a short tube (1°135 em. long)
placed between the poles of the electromagnet. The rotation
obtained for water showed that a reversal of the current
(7 amperes) produced a change of magnetic potential 30380
Ci.-2auss.

In the accompanying graphs dispersion curves are given

* The results for nickel and cobalt sulphates were confirmed in this
Laboratory in 1918 by the Rev. W. D. Ross, M.Sc.
+ 8. S. Richardson, Phil. Mag. vol. xxxi. p, 282.

8s go 95 * 100 [ots “uo

914 Messrs. Roberts, Smith, and Richardson on
Fig. 2.

| Ve

CHiorI DES

\\

fROoTATION IN RADIANS x 1072

e,
es

5

RADIANS x /0 od
®
\
‘\

a

Ro TATION
G

fo} 1S 80 85 qo qs 100 (os ula

Magnetic Rotatory Dispersion of Paramagnetic Solutions. 915

for the solutions and for pure water*. When the water of
crystallization of the salt is taken into account, the mass of
water per c.c. is practically the same in the solution as it is
in the case of water alone; the difference in the ordinates
therefore represents the rotation produced by the salt alone.

As small differences of rotation were to be measured it
was necessary to work with fairly concentrated solutions,
and the graphs refer to the following concentrations :—

Mickel, Chlomdep os 30). tasaacs « ‘216 molar.
Cobalt BS ge ont Fe ee he Sp GED es,
Nickel Sulphate. gi< <% ozsejes ‘624 =,
(ne ae Se Pare eke Ee rr “B04. 55
Ferrous _,, 5.9 sells tr cn EU Se
Niekel Acetate (sid. 6.8. evs “664. *
Cobalt BM a a ae a act "20K - J.

The results indicate that the cobalt atom, like the iron
atom, is capable of producing a negative rotation, but of
smaller numerical value. ‘The disappearance of this on the
low-frequency side of the band may be attributable to a
preponderating positive effect of the (SO,), (C,H;0,),

Cl ions. This view is supported by a fact that Sel a
highly dispersive element (e.g., chlorine) is present the
residual negative effect in the ultraviolet is smaller, but the
point requires further investigation. In the case of nickel
the rotation remains positive pageae ee: the range mea-
sured, but does not undergo the large increase in the
ultraviolet which is observed with diamagnetic substances.
It is possible therefore that the nickel atom, whilst not
able to overbalance the positive effect of the other atoms in
the salt, still exerts sufficient depression nearly to neutralize
the dispersion. From this point of view, our results indicate
that in respect of paramagnetism the cobalt atom occupies
a position intermediate between those of iron and nickel.

_ The experiments are to be continued, and we hope to
publish results for other paramagneties shortly.

We beg to tender our thanks to Prof. Wilberforce for the
facilities and apparatus placed at our disposal and to
Dr. Smeath Thomas who kindly prepared the ferrous sulphate
solution.

The George Holt Physics Laboratory,

‘University of Liverpool.

* To obtain the rotation of the salts in radians per cm.-gauss the
ovaph-readings must be multiplied by 6°616x10-’.

——

is O16. 4

LXXVIII. Colour- Vision Theories in Relation to Colour-
Blindness. By F. W. Hpriper-Green, U.B.E., M_D.,
F.R.C.S., Special Examiner and Adviser to the Board of
Trade on Colour Vision and Hyesight *

ee importance of colour-blindness as a key to any
colour-vision theory does not seem to be sufficiently
recognized, though the fact was well known to Helmholtz f,
who showed that Hering’s theory explained the facts of
colour mixing quite as well'as his own, and stated: ‘‘ As far
as I see, there is no: other means of deciding on the
elementary colour sensations than the examination of the
colour-blind.” This cannot ‘be too widely known, because
any other method assumes that the three-sensation ‘theory i is
correct, and is useless when ‘this theory is denied. As has
been shown by Houstoun {, my non-elemental theory explains
the facts of colour mixing quite as well as either of the
above mentioned. Recently I have examined about 200 cases
of colour-blindness by colour-mixing methods as well as my
own, in order to ascertain certain facts of crucial importance
in deciding between an elemental and a non-elementual

_ theory.

Relation of Luminosity to Colour.

If the sensation of white were compounded of the addition
of three elementary processes and one of these processes
were subtracted, the position of the apex of the luminosity
curve would not be the game in the colour-blind as in the
normal, It is, however, well known that there are numerous
dichromics rhe have a luminosity curve aL to the
normal.

A case of colour-blindness regarded from the point of
view of a three-sensation theory may, for instance, be one-half
red-blind ; the composition of the theoretical white will then
be 4R+1G41V. As far as luminosity is concerned, this
white may be compared with light of various wave-lengths
by the colour-blind subject, just as normal white is by the
normal sighted. Now, as the apex of the luminosity curve
depends upon the point where the aggregate stimulation of
the three theoretical sensations is greatest in terms. of
luminosity, this apex will be displaced towards the point of
maximum stimulation of the other sensation, namely green,
the luminosity of the blue being so low as to be negligible.
Numerous cases can, however, be found in which the apex of

* Communicated by the Author.
+ Physiologische Optik, 2nd edition, p. 377.
t Phil. Mag. vol. xxxviil. p. 402 (1919).

Colour- Vision in Relation to Colour-Blindness. 917

the luminosity curve is at the same point as the normal. <A
striking case of this kind was examined revently—adichromic
with shor tening of the red end of the spectrum ; the apex of
his luminosily curve for the light of the Pointolite are was
at X 585up, which is the apex for the normal-sighted.

Explanation of the facts of Colour-Blindness.

The facts of colour-blindness are quite inconsistent with
any three-sensation theory. Supporters of such an elemental
theory have in many cases contented themselves with
describing a case of colour-blindness in the terms of the
theory Sahaut showing that the ascertained facts are con-
sistent with the theory. No one, for instance, has shown
how on an elemental theory 50 per cent. of dangerously
eolour-blind can get through the now obsolete wool test.
On the non-elemental theory “the explanation is easy enough:
the man has defective caste discrimination, but not sufficient
to prevent him matching wools in favourable circumstances,
particularly when colour names are not used. On an
elemental theory, why should the trichromic mark out about
half the number of monochromatic divisions in the spectrum,
designate yellow as red-green, and have an increased simul-
taneous colour contrast ? When there are three definite
colour sensations, how can colour-blindness be explained ?
The recent paper by Houstoun * should be read on this
point.

The Anomalous White Equation without Colour-Blindness.

Just as a man may make an anomalous Rayleigh equation
without any evidence of colour-blindness, so may a man
make an anomalous white equation w ichout being colour-
blind +. Asan example of this, a man was examined who
presented no sign of colour weakness. He passed my card
test, lantern test, and spectrometer with the ease and
accuracy of an absolutely normal-sighted person. His
luminosity curve was taken by the flicker method, and
corresponded with the normal. The wave-length of the
apex of the luminosity curve was at 585m, which is the
normal point. When, however, his white equation was
taken, he put only 8 scale divisions of green, instead of 134
or 14, which is normal, and the mixed light appeared red to
the normal-sighted. Animportant fact was noted—namel
that after fatigue with red of the region of A670up, the

* Proc. Roy. Soc. Edin. vol. xlii. pt. i. no. 7, p. 75 (1922).
t+ Proc. Roy. Soc., B. vol. Ixxxvi. p. 164 (1913).

918 Dr. F..W. Edridge-Greon on Colour- Vision

equation changed for him in the same way as for the
normal-sighted*. After fatigue with the red light, he required
only 4 scale divisions instead of 8. It is quite obvious that
this was not a case of partial red-blindness from the point of
view of the three-sensation theory, though he was not as
sensitive to the red end of the spectrum as the normal sighted.

The White Equation nad Colour- Blindness.

The colour-blind have been classified by some as red or
green-blind, in accordance with their white equations-—that
is, the amount of pure spectral red, green, and violet required
to match a simple white; those who put too much red in the
equation being classed as red-blind, and those who put too
much green in the equation being classed as green-blind.
There are, however, many who, whilst agreeing with the
normal equation, are quite satisfied when a considerable
additional amount of green or red is added to the equation.
This explains why in certain cases some have been described
as red-blind by one observer and green-blind by another.

A remarkable fact, which does not seem to have been
previously observed, is that many colour-blind persons who
strongly object to the normal match, but are satisfied with
an anomalous equation, will completely agree with the
normal equation when the comparison white light is increased
in intensity so that it is much too bright to a normal-sighted
person. This clearly. shows that the normal mixed white
produces the same effect as far as colour is concerned, but
has a more powerful effect as to luminosity. This is in
complete accordance with other observations, and is found in.
those cases in which there is abrupt and slight shortening
of the red end of the spectrum. If there be shortening of
the red end of the spectrum which does not affect 1670 wy,
and 4670 up has its normal light value, the mixed light will
be more luminous than the simple white in proportion to the
shortening. This portion of red light not producing any
effect has to be subtracted from the white light. These
facts are quite inconsistent with a hypothetical red sensation
which is affected by light of all wave-lengths. Another
illustration may make this point clear. A man with
shortening of the red end of the spectrum and normal
colour discrimination will put together as exactly alike a
pink and a blue or violet much darker. If, however, the
pink and blue be viewed by a normal-sighted person through
a blue-green glass which cuts off the red end of the spectrum,

* Proc. Roy. Soc., B. vel. xcii. p. 232 (1921).

Theories in Relation to Colour- Blindness. 919

both will appear identical in hue and colour. This proves
conclusively that the defect is not due to a diminution of a
hypothetical red sensation, because all the rays coming
through the blue-green glass are supposed to affect the red
sensation, and yet we have been able to correct the erroneous
match by the subtraction of red light. On the other hand,
there are colour-blind persons who, whilst disagreeing with
the normal white equation, agree with it when the com-
parison white is diminished in intensity.

A totally erroneous view of a case may be obtained
through methods based on the three-sensation theory. A
man may be examined and found to put too much green in
his white equation; he is therefore classified as partially
ereen-blind. Further examination shows that he can pass
the wool test, but fails to see a deep red light formed by
rays from the red end of the spectrum. The fact that he is
insensible to these rays explains the facts of his case, in-
eluding iis error in colour mixing. As certain red rays are
invisible to him, these have to be subtracted from white
light. If his white could be seen by a normal-sighted
person, it would appear greenish white. Therefore, if the
colours used in the white equation have their normal value,
he will put more green than normal in the equation, as he is
really matching a greenish white.

Hven the facts of colour mixing are far more satisfactorily |
explained by a non-elemental than by an elemental theory.
For instance, a considerable amount of one spectral colour
may be added to another without altering its appearance.

Houstoun has shown very clearly that from a mathematical
and physical point of view only one substance is necessary,
and that there is no evidence of more than one. All the
facts tend to show that the visual purple is the visual
substance, that the cones are the terminal perceptive visual
organs, and that the rods are not perceptive elements,
but are concerned with the formation and distribution of
the visual purple. Vision takes place by stimulation of the
cones through the photo-chemical decomposition of the
liquid surrounding them, which is sensitized by the visual
purple *. The ends of the cones being stimulated through
the photo-chemical decomposition of the visual purple by
light, a visual impulse is set up which is conveyed through
the optic-nerve fibres to the brain. The character of the
stimulus and impulse differs according to the wave-length
of the light causing it. In the impulse itself we have the

* ‘The Physiology of Vision,’ G. Bell & Sons, London, 1920, p. 134.

920. Mr. A. H. Davis on Natural

physiological basis of the sensation of light, and in the
quality of the impulse the physiological basis of the sensation,
of colour. But though the impulses vary according to the
wave-length of the light causing them, the retino-cerebral
apparatus is not able to distinguish between the character of,
adjacent stimuli, not being sufficiently developed for the
purpose. At most seven distinct colours are seen, whilst
others see, in proportion to the development of their colour-
perceiving centres, six, five, four, three, two, or none. This
causes colour- blindness, the person seeing only two or three
colours instead of the normal six, putting colours together
as alike which are seen by the nor mal-sighted to be different.
In the degree of colour-blindness just preceding total, only
the colours at the extremes of the spectrum are recognized
as different, the remainder of the spectrum appearing grey. |

LXXIX. Natural Convective Cooling in Fluids.
ay, Ae Avis. IVE Son

[From the National Physical Laboratory. ]

CoNTENTS.
Introduction.
I. THEORETICAL.

IJ. ExPERIMENTAL.

Formule.

Apparatus.

Experimental Observations.

Results. |
(a) Representation in the form H/k= F(c?’gd?a6/k’).
(6) Representation in the form H/k= F(d°0gae/kv).

Cooling Power of Fluids.

Introduction.

B* general reasoning from the principle of similitudef,

introducing certain plausible assumptions, it is possible
to obtain the following formula to represent the heat loss by
natural convection from similar bodies similarly immersed
in viscous fluids.

AL/kO=F(e?g Liad/k*)f(cv/k), . . . Cd)

* Communicated by the Author.
+ Davis, Phil..Mag. xl. p. 692 (1920).

Convective Cooling in Fluids. 921

where
h=heat loss per unit time per unit area of the body,
k=thermal conductivity of the fluid,
¢=capacity for heat of the fluid per unit volume,
v=kinematical viscosity of the fluid,
@=temperature excess of the body,
a=coefbcient of density reduction of the Huid per degree
rise of temperature,
g=acceleration due to gravity,
L=linear dimensions of the body.

For gases, cv/k is practically constant, and so experiments
with them cannot reveal the effects of this term. However,
the formula, restricted by its omission, has already been
shown for gases to be in good general agreement with
experimental results. In the investigation, early work on
miscellaneous small bodies has been considered *, and also
data for moderately heated large vertical surfaces from 2
inches to 9 feet in height}. Particular attention has been
given to the case of wires and long cylinders f, owing to the
wide range of size, temperature excess, gas nature and
pressure for which results were available.

Liquids, however, cover a wide range of values of cv/k,
and so experiments on the cooling of wires in liquids were
undertaken to study the effects of this term.

But, further, it was suspected that g and v might always
occur together in the equation in the form g/v, for any steady
velocity of the viscous streams will be determined by a
balance between the accelerating forces due to gravity and
the retardation due to viscosity. The formula would then
take the simple form

hL/kO=F(L8gaclhv). . 2... (2)

The present paper investigates the possibility of such a
simplification, and therefore consists of two main parts—one
theoretical, in which more formal consideration is given to
convection in a viscous fluid ; and the other experimental,
where convective cooling of wires is studied for a series of
fluids of different viscosity with a view to experimental
verification of the formule put forward.

* Davis, loc. cit.
+ Dept. of Scientific and Industrial Research. Food Investigation

Board Special Report, No. 9.
{ Davis, Phil. Mag. xliii. p. 8329 (1922).
Phil. Mag. 8S. 6. Vol. 44. No. 263. Nov. 1922, 30

Y22 Mr. A. H. Davis on Natural
Part |.—' THEORETICAL.

Convective cooling is taken to refer to the total heat
transfer from a hot body by the medium of a fluid
moving past the surface. Such cooling is said to be
“natural” or “free”? when the fluid is still, except for the
streams set up by the heat from the hot body itself, and is
said to be “forced” when the body is immersed in a fluid
_ stream, usually considered to be moving with such velocity
that the currents set up by the hot body itself are negligible.
The present paper is limited to natural convection.

In 1820, Fourier * stated the equation of heat conduction
in a moving fluid, and in 1881, Lorenz ft, upon certain
assumptions, gave a formula for heat loss by natural convection
for the special case of a vertical plane surface immersed in
an infinite viscous fluid. In 1901, Boussinesq t, dealing with
inviscid fluids, gave a general solution of the ‘problem of -
natural convection from heated solids in infinite fluid media.

The following investigation follows Boussinesq closely, but
introduces the modifications necessary in extending the
inquiry to viscous fluids.

Adopting the same mathematical symbols as those already
used, let us consider the natural convective cooling of a hot
body immersed in an infinite viscous medium and maintained at
a certain temperature, 0 degrees in excess of that of the liquid
at infinite distance, to which ali temperatures are referred.

Let p and v be respectively the density of the fluid and its
kinematical viscosity. For an element of the fluid at the
point wy 2, let 7, u, v, w, P be the temperature excess
(assumed steady, i.e. independent of time ‘¢’), the three
components of its velocity, and the non-hydrostatic part of
its pressure. For elements of the fluid at infinite distance
these quantities are all zero.

OU OU ie, Lor
Ow at Oy aie 02 =. p Ou rae u + vV/7u, |
10P 19 (3)

Oey ae ee

* Fourier, Mémoires de ? Acadénue, xii. p. 507 (1820).
+ Lorenz, Ann. der Physik, xiii. p. 582 (1881).
{ Boussinesq, Comptes Rendus, cxxxii. p. 1382 (1901),

Convective Cooling in Fluids. 923

w, v, and w’ being the accelerations of the fluid parallel to
Bee Hay

If k be the thermal conductivity of the fluid and ¢ its heat
capacity per unit volume, then 7’, the rate of change of tem-
perature for a given particle with respect to time, is given by

7 = —-V"r. sire ae (4)

Also, the derivative 7’, like the derivatives w’, v', w’ of the
velocities, is obtained by finding the increase in t when
v, Y; 2, increase by udt, vdt, and wdé ; in this way we have
the quadruple equation

O(u, v, wW, T O(u, v, w,T
(uv, wv’, +) = Ole 27) +v ( ae )
» OLY Vv, W, a
Sosa Desires OS
+ ee 6)

To the five differential equations (3) and (4) it is
necessary to add ths following seven boundary conditions,
in the first of which, /, m, n, denote the three direction-
cosines of the normal drawn from the interior of the fluid to
any element of surface of the body.

At the surface of the solid w=v=w=0 and r=8@,

At infinite distance (P, u, v, w, T)=0. (6)
In words, at the surface of the solid the fluid takes the
temperature @ of the solid, and the velocity is zero.

Following Boussinesq, let us endeavour to replace the
independent variables 2, y, z and the functions 7, v, v, w, P

’ by others, & 7, €,T, U, V, W, II, respectively proportional to
each of them, but whose ratios are chosen ina manner to

eliminate the parameters 0, ga, k/c, p, v.
Let us Sn the following substitutions :

A. i) haa _ (9gae We Gaye
= Ge = ky ) Y> c= kp ey
2\1/3 .2\ 1/3 2\ 1/8
TOI, _ ) U. — — ) V, w= ae ) W,
cy Cy ey /
(B22
ps (=== )n.

It is readily found that the substitutions eliminate the
parameters satisfactorily if cv/k is a constant and equal to
unity. For liquids ecv/k may have very large values
(glycerine 8000, etc., see later), but for gases, as indicated
by the Kinetic Theory, it is constant and approximately

3.0 2

924 Mr. A. H. Davis on Natural

equal to unity. For gases, therefore, the above substitu-
tions should be fairly satisfactory on this ground.

There is an alternative condition under which the
substitutions are satisfactory, even without cv/k being a
constant. The condition is that the accelerations wu’, v’, w’
of the particle shall be negligible compared with vV/7u, etc.,
which would appear to be justifiable for very viscous fluids.
It implies that on coming into the region of the hot body
a particle of the fluid almost immediately takes up its final
velocity and suffers but little subsequent acceleration.

Consequently, the above substitutions appear satisfactory,
and the differential equations (3) and (4) take the following
form.) (0° V \Vmay ibe retained if viet but otherwise
they must be omitted since wu’, v’, and w’ are neglected *).

0oU ov OW pe +2T oT or \
aft ay) OE a” Vasataa tom
Se US see) ow
Se Oe ten ae |
Ol gy (ON O eae
By 7 te tat * ae)
ol | ew ow Pee
a ee W'+ OF =F On” D) + 0’ |
where (U’, V'7,W,T) = po, ee al) |
+ VRC VW) 4 yy 90 MD) |
On

* IT am indebted to Mr. W. G. Bickley, M.Sc., for the following
notes :—
(a) If ey/k is not ae to unity, equations 8 are mathematically

ele if we write a ow for U’, etc. Evidently, if cv/k is large, the
term ie of is correspondingly small and may be omitted. Retaining it,

however, the solution of (8) becomes
T a P : ;
Q? Caan (ee a B = definite functions of

| (nas Bh

The experimental curve shows that the cccurrence of cy/k in these
functions is in such a manner that large variations in cy/k have
imperceptible effects.

(6) Equation (5), and the resulting one (8), would be more general if the
O(24,0,@,7)

cy

term were introduced on the left-hand side. This would

include unsteady motion, but would in no way affect the changes of
variables.

(8)

Convective Cooling in Fluids. 925

Also, let us put the equation of the solid in the form

A pee: ae (“ee )" | =o. es)

Thus, if the coefficient (@ gac/kv) changes, this amounts to
considering, instead of the actual solid, similar bodies having
linear dimensions inversely proportional to this coefficient.
Then the direction-cosines 1, m,n, of the normal will remain

the same at corresponding points, and the boundary con-
ditions become

At the surface U=V=W=0 and T=1,
At the distance 4/£+7?+ ¢? infinite (II, U, V, W,T)= a - (10)

The system of equations (8) and (10) determine (U, V, W,
T, II) as the functions of &7& and substituting in the
integrals for the eight new variables their equivalents as
given by (7), we have five relations of the form

me (u,', w) EP , 1

0’ aye ee ae = definite functions of |
Cy pP Cc ‘ ; (11)

\

Ogac\\? Ogac\' Ogac\?
ee 4 ce) 2 oa ey

The flux of heat furnished in unit time by unit area of
such a body, equal to that which the contiguous liquid layer
communicates to the interior the fluid, is given by

k Ge +m = 7 t” £7),

Introducing the new set we have then

=k(é gaciiv) (1 =~ +m oe +n g) vRELZ)
At corresponding points of the eae f (&, 0, )=0
limiting the bodies considered, the direction-cosines 1,m,n

and the derivatives sto 2) have the same values respec-
y) n;
tively ; so the ee. coefficient is a function of the shape
and orientation of the bodies only.
Thus the result may be stated in the following form.
For a family of similar bodies similarly oriented, and haying
linear dimensions L given by

Lice (6gae/kv)— 8, i.e. (L?@gac/kv) =const.,

926 Mr. A. H. Davis on Natural

the heat loss per unit area from corresponding points is
given by 3

NeackeOgacky) ?, see

and this will also be true if 2 represents mean heat loss for the
whole model. So for bodies of this shape and orientation
we have :

h=ki(Ggacikv)? fArdgaciky)) .:). eae
which may be written

hlyjkO=Ve0gacky).. - 2 ee

This equation is the simplified form of equation (1) it was
desired to establish, and it has been put to the test of
experiments in a later part of this paper. It is desirable to
notice here one point in connexion with it. :

For a series of fluids for which cv/k is constant, the
equation may readily be shown to agree with that obtained
by Boussinesq for inviscid fluids ; 2. e. Boussinesq’s grouping
of variables for invisicid fluids is satisfactory for viscid
fluids for which cv/k is constant. This equivalent grouping
is given by omitting (cv/k) from formula (1).

Part [I].—ExPpERIMENTAL.

Formule.

For long horizontal wires of diameter ‘d’ it may readily
be shown that formule (1) and (2) may be rewritten as
follows in terms of the heat loss H per unit length of wire
per degree temperature excess :

emileriaieiae (16)
H/k= F(d?8gac/kv).

When cv/k is a constant, the equations are identical in
form and, consequently, evidence for diatomic gases already
shown elsewhere to be in agreement with one of these
expressions is necessarily in agreement with the other.

If cv/k is not constant, the second equation is a special case
of the first. The experiments now to be described on the |
cooling of wires in liquids will indicate the form of the ev/k
term in (16) and also whether the simpler expression is
satisfactory. 7

Apparatus.

The method of experiment consisted in stretching a wire
horizontally at a convenient depth in a vessel full of the
liquid under examination, and measuring the electric energy

Convective Cooling in Fluids. 927

supply necessary to maintain a measured temperature
difference between the wire and the general body of the
liquid. The electric energy supplied was determined from
the measured current through the wire, and from its
resistance as obtained from a Wheatstone bridge. This
resistance also gave the temperature of the wire.

Kor each of the liquids the cooling of wires of two sizes
was studied, the diameters being 0°0083 em. and 0°0155 em.
respectively. Both were thought to be pure platinum, but

Fig. 1.

[011 anya

(©)

tests of the resistance at the temperature of melting ice and
that of steam gave a very low value for the temperature
coefficient of the finer wire.

The apparatus is shown diagrammatically in fig. 1. The
platinum wire TT, immersed in the liquid under examination,
formed part of the fourth arm of a Wheatstone bridge,
having equal ratioarms PP. By adjusting the slider 8 the

925 Mr. A. H. Davis on Natural

bridge was balanced with a small current (0°01 amp.), thus
allowing for slight temperature changes of the fluid. An
additional resistance dR now introduced into the third arm
destroyed the balance, which was then restored by increasing
the current through the bridge and thus heating the wire.
TT until its resistance had sufficiently increased. Correction
for the cooling at the ends of the test wire was automatically
effected by having in the other arm of the bridge a shorter
piece OC of the same wire* also immersed in the liquid
under examination. To be effective, this compensating wire
must be greater than a certain minimum length. Ayrton
and Kilgour f have given a calculation showing that for a
6-mil wire at about 300° C. temperature excess in still air,
the effect of the heavy leads extends about 1 cm. from each
end. For finer wires it would be less, and also it would be
less if the cooling medium were a liquid, for these carry off
more heat than air, so that the end effects become less
important. In the present experiments CC was never less
than 2 cm., and so the end effect should be entirely
eliminated.

The energy dissipated in the uncompensated length (/) of
TT was calculated from the current (2) through the wire
and its (hot) resistance R. The temperature excess (@)
of the wire above its surroundings is given by dR/R,a, where
Ro is the resistance of the wire at 0° C. and ‘«’ the temperature
coefficient of resistance. The heat loss H (in calories) per
em. length of wire per °C. temperature excess is given by

(ine
H —— 4°18] ern 0, e e e ° e (1 7)
where
0 =dR/Roa,

4°18 being the factor required to convert watts to calories
per second.

Details of the wires used are given in Table I.

The rectangular vessel containing the liquids under
examination had a height of 12 cm. and a base of 17x10
em., and the wires (3 cm. apart) were 3°5 cm. below the
surface of the liquid. This vessel required rather larger
quantities of some of the liquids than were available.. A
smaller one was found with carbon tetrachloride to give the
same result as the larger, and so it was generally used

* Callendar introduced this method of eliminating end effects in
resistance measurements, using it for many similar purposes. Proce.
Phys. Soc Lond. xxxiii, p. 187 (1921).

+ Ayrton and Kilgour, Phil. Trans. A, clxxxiii. p. 3871 (1892).

Convective Cooling in Fluids. 929
TABLE I.
Details of the Wires used.

Rie | Uncompensated
og gd’. | length. a. Ro.
; | | a.
| rate Ae roan ee at es
em. | cis. ohms.
“0083 | 5°62 10-* 5:95 ‘00183 2°127
‘0155 | 3866 x1074 | 6:5 003885 355

|

instead. It was 10 em. high witha base 16x6cm. It was
used quite full and with the wires 5 em. below the surface
of the liquid. In all cases the vessel stood upon an insulated
levelling table on a concrete slab on the floor, and no ripples
were observable on the liquid surface. It was completely
covered in by a draught-proof enclosure of cotton-wool over
a cardboard frame, for draughts might by local cooling set
up convection currents in the liquid, and might also cause
evaporation of the more volatile liquids. A’ thermometer
with 1/10° C. divisions checked the temperature of the fluid.

Table 1J. is a summary of the physical constants of the
liquids used. It was complied from published data in books

TasiE IT. +
Physical Constants of the Liquids used.

Liquid. 104%. | 1042. Ass Uo peo ase Ny cv/k. ac*/k? | Nes 95/ Ny.
Toluene ......... 3°42 | 10°99 0:866 0°40 0:0062 2a) wiles |
Carbon tetra-

chloride ...... 266 | 11:8 | 1°582 07198 | 0-0108 8:03 | 1630 "728
AMINE .......0. 4°] 85 | 1:023 | O°514 | 0:055 69°3 1390 465
live oil......... B02) 70) | O-91o | O47 (1-17 1402 842 363
Glycerine ...... 68 Boi 26M ay) 0 Os 9°3 7940 611 *42*

s=specific heat. n =viscosity. o=density. C=sp: v=n/p.
* My+10/ My:

of physical constants, and since pure liquids were used it is
suffiviently exact for present purposes where (as will be seen
later) we are concerned mainly with a small fractional

+ The majority of these values were obtained from ‘ Physical and
Chemical Constants, etc.,’ by G. W. C. Kaye and T. H. Laby.

930 Mr. A. H. Davis on Natural

power of the values given. Also, as only moderate heating
of the wires was involved, little account has been taken of
temperature changes of the physical properties of the liquids,
though the point is referred to later.

The ratio arms were usually 100 ohms eech, 1000 ohms
being occasionally used for the larger currents. Hquality of
the ratio arms was tested by reversing the connexions and
noticing whether the balance was thereby disturbed.

To eliminate disturbing resistance changes when the heating
current was passed, all connectiny wires were compensated
by similar wires in the opposite arms of the bridge, and the
resistances R and dR were constructed of manganin as also
was the slide wire 8. It was found on test, by removing
the platinum wires TT and CC and substituting heavy oil-
immersed manganin of the same resistance, that the balance
of the bridge obtained with a small current (0 01 amp.) was
undisturbed when a heavy current of 1 amp was passed, and
that therefore the. compensation was satisfactory. <A
terminal head on the heavy copper leads to the wires
facilitated short circuiting of the platinum wires when
subsequent occasional test of the apparatus was required.

It was possible that the heating current might set up
thermoelectric effects, but a test showed that these were
absent. The method was to switch off the heating current
after it had been adjusted to the value necessary to balance
the bridge, and no appreciable drift of the galvanometer was
then observed while the wires were cooling.

Thermoelectric effects might also have existed through
local inequalities of temperature caused otherwise than by
the heating current, and these would reveal themselves by
drift of the galvanometer on making the galvanometer
circuit while the battery circuit was still open. Some slight
effect was occasionally found, but it was only important
when getting tlhe original balance with the small current ;
and for this it was eliminated by the familiar device of
working with the galvanometer circuit always completed
and adjusting the bridge for no deflexion when the battery
was reversed,

A large storage battery was used at first, and it gave
troublesome galvanometer effects due to leakage, discovered
and traced by reversal of the battery and galvanometer
circuits. To eliminate these a separate 10-volt battery was
used, and this and all parts of the apparatus and leads were
insulated by standing upon ebonite disks, etc.

The increment resistance dR was obtained from a box
of manganin coils (resistance 0°01 to 1:0 ohm) provided

Convective Cooling in Fluids. 931

with mercury cups. A separate test showed that the resist-
ance of the coils was independent of the current. Values
of dR down to 0:001 ohm were obtained by shunting a
resistance of 1 ohm by 1000 ohms, ete. These values were cali-
brated directly on a Cambridge resistance bridge, and agreed
with calculation. Using a current of 0:01 ampere through
the bridge it was possible toobtain the initial balance correct
to 0:0001 ohm, or rather less, an amount associated with a
galvanometer deflexion of 2/5 mm. on reversal of battery.
Thus, with a posible error of setting of 0:0001 ohm, values
of dR=0:1 would be correct to 1/10 of one per cent., and
generally the error would be negligible, but with the
smallest values of dR used (0°001 ohm) the error might be
between 5 and 10 per cent., and averages were taken. With
the heating current, of course, the bridge was much more
sensitive. With the 3-mil wire and a current of $ amp. a
resistance change of 0'0001 ohm in the third arm would
cause a galvanometer deflexion of 1 cm. Incidentally it
was noticed that a change of | per cent. in this current
value caused a deflexion of 13 cm., and this was also
approximately the case with the larger wire.

As a further check on the apparatus, the cooling in gases
was studied by a volt-drop method instead of using the
Wheatstone bridge arrangement. Various experimenters
have used this method, measuring the current through a
given length of wire and the volt drop along it. To mini-
mize end effects, experimenters have usually used potential
leads of very fine wire, attached at sufficient distances from
the heavy current leads to eliminate the cooling effects of
these latter. This, however, does not eliminate whatever
cooling is due to the fine potential leads, and the volt-drop
method used in the present experiments is superior in this
respect. The method was to pass the current though TT
and CC in series, and to measure the respective potential
-falls by means of a potentiometer. The difference gave the
potential drop along the uncompensated length, the end
effects being eliminated completely. ‘he current | was
obtained by passing it also through a standard manganin
resistance and determining the volt drop for this also. The
resistance T'T— CC at the temperature of the air was obtained
from the potential drop when a very small current 0°01 amp.
was passing. The results obtained by this check method
were within 1 or 2 per cent. of those obtained by means
of the Wheatstone bridge. It is not so convenient a
method for the liquids, owing to the zero changes which ure
easily dealt with by the Wheatstone bridge method.

932 Mr. A. H. Davis on Natural

Heperimental Observations.

Certain general characteristic effects were noticed during
the experiments.

With slight heating of the wire the conditions ultimately
set up were usually quite steady, except perhaps for a slight
drift with the more viscous liquids due to the fact that the
prolonged heating necessary before the steady state was
reached gradually increased the temperature of the whole
mass of the fluid. However, with greater heating, unstead-
iness supervened, the galvanometer deflexion and also the
current changing spasmodically, as though the convection
current had become turbulent or reached some critical
condition.

Various measurements were made of the drift mentioned,
which was found to be of the order expected from the
known energy dissipation in the volume of liquid used.
Observations were taken while a heating current was
passing, sometimes of the decrease of current necessary as
time elapsed to keep the temperature of the wire constant,
and sometimes, with constant current, of the ultimate rate
of temperature rise of the wire. Observations were also
taken of the resistance of the cold wire just before the
heating current was passed, and again at such a time after
switching off that all local heating of the liquid had been
dissipated. In the case of glycerine this dissipation occupied
some minutes, and a true drift of 0-001 ohm during an
experiment appeared as great as 0:01 ohm when taken
too soon after switching off. Results were satisfactorily
consistent, and in general were equivalent to a change in
@ of the order of one quarter of one per cent. per minute.
The less viscous fluids attained the steady state too quickly
for this drift to have appreciable effect.

Another effect was noticed with the more viscous liquids.
For instance, with the 5-mil wire immersed in olive oil
slight heating currents gave a tendency for the galvano-
meter to drift in a certain direction as though the whole
mass were being gradually heated up. With greater
heating, however, an opposite drift was observed. This
was found to be due to oscillations having a period of a
winute or so, which slowly died out and left the usual
steady state. Thus, while on switching on a very slight
heating current the wire merely rose gradually in temper-
ature to its final value (2° C. excess), reaching this in about
three minutes, it was found that for rather greater heating
(5° C.) the temperature of the wire first overshot its final
value by about 1/8° C., reaching a maximum in about
one minute, and then fell gradually to its steady state. With

Convective Cooling in Fluids. 933

still greater heating (10° C.) two oscillations were noticed,
a maximum being reached in 4} minute followed by a
minimum # minute later, from which the temperature
then rose to its final value. ‘These oscillations resemble
those of a damped system, the three stages due to different
degrees of heating corresponding te three different degrees
of damping. The phenomena cannot be attributed to the
galyanometer, and must have their origin in the liquid
convection currents set up, for the galvanometer normally
reached its final deflexion in a few seconds, whereas the
observed variations occupied minutes.
Results.

The resulis obtained are given below in tables compiled
from smooth curves through experimental points, this making
it possible to give them much more concisely than would
otherwise have been the case.

It is necessary to state that, in general, results were
reproducible to within 5 per cent. With respect to repro-
ducibility, the glycerine series was least satisfactory, but
this may have been due to its extremely viscous nature and
to the very rapid manner in which its viscosity changes
with temperature.

In Table III. results are given for the various liquids,

Tasur Til.
Relation between 10*H and @.

[Heat loss (H) in calories per cm. length per ° C. excess temperature. |

Temperature excess (0)° C.

eer: Wi Ba ok ea Dae ee
Liquid Sse gdac/k. i i
Peg | DAeBeRO) | 20se 51. 50:
Gases* ...,.. 0:0083 | 474 10-° 0°82 | 0:83 | 0:86 | 0-91 | 1:03
00155 | 3:09x10~4 GO9s 10s" 1-101) 1-28
Toluene...... 0:0083 | 0-624 S693 104" | 1-1 |) 126
00155 | 4:06 98/107 |121 |186 |154 |182
Wiig e..053--- 0:0083 | 0:920 be) OS 77 | S4 1 92-1103
0:0155 | 5-98 Sole ile OS: 1) LET «| 129. |14:9
Aniline ...... 0:0083 | 0-782 75| 84 | 96 (108 |121 |146
00155 | 5°08 9-9/11:0 [123 |141 |16-2 |19-4
Olive oil ...| 09083 | 0-472 Brae vigor he: |. Glfieel) 7k 4p Brp
00155 | 3:08 GS 176 | 87 11-99 1120
Glycerine ...) 0°0083 | 0°348 96 | 94 | 95 | 103
00155 | 2:23 PSS UOT. | 11 9ie | Taos Lie

* For gases 9 was varied up to 200° C., and the values of 10*H obtained for

the two wires at 2=200°C, were 1:34 and 1°60 respectively,

(934 Mr. A. H. Davis on Natural

showing the relation between the linear emissivity H of the
wire in cals. per cm. length per degree temperature excess,
and the temperature excess 0.

(a) Representation in the form H/k= F(e’gd*ad/k?) f(ev/k).
In Table IV. these results are represented in the form
showing the relation to the equation for inviscid fluids,
and H/ké is given for various values of (c’gd*aO/k*), the value
of ev/k for the liquid being stated. The heat losses obtained

for air do not appear in this table as they fall under very
much smaller values of (gd?a@c?/k?).

TABLE IV.
Relation between H/k and gd?ac?@/k?.

| gdactOR?.
eet s Wi
| Liquid. | ev/k. ae |
: 2. | 5. | 10.| 20.| 50. | 100.) 200.

| | |
| Toluene...... 72 | 0-:0083 | 25) 26) 30| 3-2] 35| 41| 47
| 0-0155 | #91 32] 36] 41, 46
eel. ae 80 | 00083 | 23 26/29) 32) 3-5| 40
| 0-0155 | | 3:2) 36] 421 4-7] 52
Pailin oy, 69 | 0-:0083 | 19 22] 25) 28] 32) 381 43
| 0-0155 25| 2-7| 3:0] 35/40) 46
| Olive oil ...| 1400 | 0:0083 | 1-4] 1:6] 1-8} 2:0) 21 |
| 0-0155 1-7| 1-9| 21| 24128] 33
"Glycerine... | 7640 | 0-0083 | 1-4] 1-4] 15] 1-7 |

0-0155 1-4| 15] 1-7 2-0 | 24

It is clear from the table that for a given liquid the
relation between H/k and c?gd*a0/k? is practically independent
of the diameter of the wire.

There appears, however, to be a tendency for the finer
wire to give values of H/k a few per cent. higher than the
other, ant this is most probably due to the temperature
variation of the properties of the fluids for which no
allowance has been made. |

In considering the cooling of the wires in gases, where
much greater temperature excesses were involved, it was
found that approximate allowance of this kind eoureenta a
similar tendency*. It appears on approximate calculation
trom what data are available (mainly temperature coeffi-
cients of viscosity) that in the present experiments the

* Davis, Phil. Mag. xliii. p. 829 (1922).

Convective Cooling in Fluids. 935

correction to be expected is sensible, and is of the sign and
order of magnitude required to bring the results for the
two wires into even closer agreement. In fig. 2 single
eurves have been drawn representing mean values for each
of the liquids.

Although our own figures for gases fall on a very distant
region of the graph and so are not shown, a curve has been
given to represent gases. It was obtained from the paper
already mentioned, where cooling was studied for a wide
range of wire diameter and of temperature excess, for
various gases—oxygen, hydrogen, and air—over a thousand-
fold range of gas-pressure.

It is seen that the curves for the different fluids form a
family of more or less parallel lines, with ev/k as parameter.
The magnitude of the effect of cv/k is not great, for
although glycerine has a value more than 10,000 times that
for air, the value of H/é for a given ‘value of c?gd*a6/k? is
only reduced in the ratio 3:1. It is satisfactory that
toluene and carbon tetrachloride, with values of cy/. so
nearly the same, should be represented by one line. The
curve for diatomic gases (cv/k=0°74) may be shown to
agree with published results for cooling of wires in CO,
(cv/k=0°83), so that. here again it appears that cv/k
determines the position of the curve.

A logarithmic plot of the relation between H/k and cv/k
for various constant values (1, 10, and 100) of e?gd*®aé/k?
yields more or less linear relations; and it appears that within

936 Mr. A. H. Davis on Natural

the limits of our experiments the complete equation for the
cooling of wires in fluids is given by 7

H/k oc (e?gd?ad/k?)*(ev/k) ~Y, Meer eS)
where « varies from 1/10 to 1/6
and y varies from 1/10 to 1/8.

It is noticed that « and y have nearly equal values, and if
we put v=y, equation (6) becomes

Hikoc (@egacikp)".. ent

which is in agreement with the theoretical equation (16).
There appears to be no published evidence to show whether
this simplified grouping is of more general applicability, but
in this connexion it is interesting to note that Lorenz, in
calculating the convective cooling of a vertical plate in a
viscous gas, obtained a result reducible to this form. ‘he
assumptions he made led in his special case to the result
represented by e=y=1/4. ‘This index is rather higher than
those given above, but it is known from an extended curve
for gases that for larger cylinders 2 tends to increase to a

value of 1/3.

(b) Representation of results in the form H/k= F(d*0gac/kv).

We now proceed to show graphically the extent to which
the present experimental results agree with this theoretical
equation.

If satisfactory, results plotted with H/k as ordinate and
(d®?6gac/kv) as abscissa should yield one line independent of
the nature of the liquid used. Strictly it is necessary to
make allowance for the temperature change in the physical
properties of the liquids. Unfortunately, reliable data do
not seem to be available for some of the properties, but it
appears that the viscosity provides the most rapid change,
and its temperature coefficient is usually known.

Consequently, while in this paper temperature variations
in the other variables are neglected, an attempt will be made
to allow for them in the case of viscosity. As a first
approximation, the value taken for the viscosity in any
experiment is that appropriate to a temperature intermediate
between the temperature of the hot wire and that of the
main volume of cold liquid in which it isimmersed. If the
cooling wire is 50° C. hotter than the cold liquid, the viscosity
taken is that for a temperature 25° C. in excess of that of

Convective Cooling in Fluids. 937

the liquid. The factor nr+25/nr is given in Table II. and
it is seen that the temperature change is considerable,
particularly with glycerine.

From results obtained, and using the above correction for
temperature changes in viscosity, Table V. and fig. 3 have
been derived, and each liquid is represented on the graph by
four points. The extreme points of each set refer to the
extremities of the range of the experimental results. They
refer respectively to the cooling of the fine wire at 2°C.
excess and that of the other wire at 50° C. excess.

TaBLe V.
Relation between H/k and gd®ac/kv.

|
) 2° C. excess. 50° C. excess. |
| Fluid. Wire diam.
|
gd>a0c/kv, | H/k. | gda0c/kv.| H/&.
Sr 0-0083 0,120 | 1:41 | -0,235 | 1°66
0:0155 0,83 0153 | 1:99
2 0-0083 229 2:55 | 7-90 3°86
0-0155 1-49 3-42 | 51:3 562
Aniline ......... — 0:0083 0225 | 2:04 | 1:21 3:56
| 00155 1:47 261 | 7:9 4-74
Olive oil ...... 0:0083 0,67 | 1:34 | -047 2-18
| 0:0155 0,44 |1:73 | -30 3-07
Glycerine ......; 0°0083 0,87 | 118 | -0,21* | 1-51*
| 00155 0,56 | 1:44 | -0134* | 1:99*

|
{

* These refer to 9=2U° C
+ On the graph of fig. 3 are also plotted the following values for the wires
at 2U0° C. excess in air :—

Wire diam.... 0°0083 0-0155
gd3abe/ky ...... 5:1210-* 3:00 10-2
EE ie eee, 1:81 2-14

It is seen that the points all le very well on one curve, so
that equation (16) appears to represent the results of the
present experiments very satisfactorily. ‘The upward exten-
sion of the curve beyond the points plotted is the
representation on the present basis of the upper part of the
curve for gases already referred to.

Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3P

938 Mr. A. H. Davis on Natural

Fig. 3, -
200
i004
59
4. Gescs
QO CCl,
20- x Aniiine
Ss e Glycerine
r © Olive Oil

6 4. ; ; f | 10000 «575
‘OD 00001 O00! OOF oO 10 00 1000 10 10

The Cooling Power of Fluids.

It appears from the theoretical considerations and experi-
mental evidence put forward in this paper and previously,
that the convective cooling of similar bodies immersed in
fluids may be represented by the formula

hL/kO =F (cg L8a6/k*) f (cv/h),

and the form of the function /(cv//) is now indicated to be
such that we may write the simpler expression
AL/ké = FC LP Ogac/ky) .

It is to be expected that such a grouping of variables is of
more general applicability than for the special case of infinite
fluids, especially in so far as it can be obtained from the
Principle of Similitude. Where liquids by their natural
convective motion carry heat from a hot surface and yield it
to a cold one, as with the common case of a hot body
immersed in a fluid contained in a cooler vessel, one would
in general expect approximate agreement with the formula

Convective Cooling in Fluids. 939

if @ represented the temperature difference between the two
surfaces. A few experiments would soon test the applic-
ability in any doubtful case.

For any circumstances to which the formula applies, the
form of the function F may be obtained from an experiment
in which only one of the variables (say @) is altered. We
have some knowledge of its form in certain cases, lor
cooling of large bodies in free air or in enclosures, large
compared with the size of the cooling body, we may write

AL/kO cc (LP @yac/kv)",

where x is usually about 1/4, but tends to be as great as 1/3
for large bodies or very hot ones, and to be much smaller
for fine wires only slightly heated, as in the present
experiments. Measurements of the conductivity of fluids
haye been made by methods involving fine wires very
slightly heated along the axis of a small metal tube: the
validity of this method imphes that n=0 in this extreme
case.

Since for a given set of circumstances the value of n
depends upon the size of the model and upon its temperature
excess, it is obvious that the cooling power of a fluid depends
upon the circumstances in which it is to be used. Retaining
from the above formula only those quantities which relate to
the properties of the cooling fluid, we have

hc k(ac/kv)”,

and for the conditions so far studied n varies from 0 to 1/3,
so that the relative cooling power of a fluid may vary
between the limits

k and (ack?/v)*.

Thus it appears for these conditions that the conductivity
of the fluid is the preponderating physical property deter-
mining cooling power. ‘This result is probably fairly general,
for while it may be possible to devise experimental arrange-
ments in which & does not enter, suitable conditions for its
omission seem generally unlikely to be realized, for it
evidently enters vitally whenever the temperature of a
particle of fluid (and thus the vigour of its motion and the
amount of heat it absorbs or yields) depends upon the
thermal conductivity of the adjacent layers.

‘I'he effect of the physical properties, other than thermal
conductivity, appears to be such that high specific heat and
high coefficient of expansion have the same degree of
importance in facilitating cooling that high viscosity has in
restricting it.

ak 2

940 Mr. A. H. Davis on the Cooling Power

I desire to express my thanks to Dr. G. W. C. Kaye and
Dr. Ezer Griffiths for the kind and encouraging way in
which the facilities for the present work have been provided,
to Mr. W. G. Bickley, M.Sc., for critically reading a draft
of the theoretical part, and tu my wife for assistance with
the numerous calculations involved in the reduction of the
experimental observations.

May 1922.

LXXX. The Cooling Power of a Stream of Viscous Fluid.
By A. H. Davis, M.Sc. *

[From the National Physical Laboratory. ]

iB some previous papers} the author has studied the
phenomenon of convective cooling, both natural and
forced, from the point of view of similitude, and has shown
how excellently experimental data for gases agree with a
grouping of variables that Boussinesq { + deduced by hydro-
dynamical reasoning for inviscid fluids. The most recent §
of this series of papers considered for natural convective
cooling the necessary modifications of Boussinesq’s analysis
in dealing with the problems of viscous fluids, and the new
formula thus obtained was tested experimentally i in certain
conditions and found to be satisfactory.

The present note develops the theory of jorced convection
in the same way, studying the effect of introducing a
viscosity term into Boussinesq’s analysis for inviscid fluids.

The problem concerns the cooling of a hot body immersed
in an infinite fluid stream maintained at a certain tempera-
ture, 0 degrees in excess of that of the fluid at infinite
distance to which all temperatures are referred. The fluid
stream is rectilinear, and moving with uniform velocity »,,
at distances from the body sufficiently great. This velocity
is supposed to be sufficiently great for the natural convection
(gravity) currents set up by the hot body itself to be
negligible. We thus neglect the coefficient of expansion of
the liquid.

Let p and v be respectively the density of the fluid and its
kinematical viscosity. Ata given time ¢, and for an element
of the fluid at the point w, y, z, let 7, u, v, w, P be the

* Communicated by the Author.

+ Phil. Mag. xl. p, 691% x11, p. 899; xliu. p. 329:

if Boussinesq, Comptes Rendus, CXXXll. p. 1882; cxxxili. p. 257 (1901).
§ See p. 920,

of a Stream of Viscous Fluid. 941

temperature excess, the three components of its velocity, and
the non-hydrostatic part of its pressure. If /, m,n, are three
direction cosines of the general stream of velocity ‘‘v,,” we

have as boundary conditions :
At infinite distance from) wu, v, w=v,, (l,m, 2),
the origin (Ezy): (1)
At the surface of thesolid = (w, v, w) =0, id.

The hydrodynamical equations of continuity and of
motion are

Be oF ey 1OP wa yvtu, |

O« Oy Oz paw ' es
1aP 19P ; i
— = 492, = SE a HV |

p Oy p Oz a

u’, v’, and w’ being the accelerations of the fluid parallel to
the axes. |

Let & be the thermal conductivity of the fluid and ¢ its
heat capacity per unit volume, then 7’, the rate of change of
temper2zture for a given particle with respect to time, is

given by

POCO NAT ag ens, (3)

Also we have

U,V, W, T U,V, W, T
u',v’, ata 22? iy la eto)
Ox OY

O(u, U, UW, T) 0(u, U, W, T)

+w ye £ >i nc?)

The equations in P, u, v, w are everywhere quite separate
from those in 6, and hydrodynamieally the problem is the
same as that where 6=0, the motion of the fluid being
determined entirely by the given general stream and the
configuration of the immersed body. Everywhere u, v, and w
will be proportional to v,, and P proportional to pv.,,?.

Let us endeavour to replace the independent variables
t, «, y, z and the functions 7, u, v, w, P by others #,, &, n, ¢, T,
U, V, W, IL respectively proportional to them but whose
ratios are chosen to eliminate 6, k/c, p, v, v,,.

Let us consider the following substitutions :

ae eae, 2). (u,v, w)—v_ (CU, V, "o (5)
. 9)
sor, F=pv,7Ul, t= (v/v,,” ey:

9492 Mr. A. H. Davis on the Cooling Po ioe

It is readily found that the substitutions effect the
eliminations satisfactorily if cy/k=1. For gases cv/k is
approximately equal to unity, and the substitutions should
be satisfactory on this ground. Also they are satisfactory
if the accelerations wu’ v'w!' of the particle are negligible
compared with vV/7u, etc., as would appear justifiable for very
viscous fluids.

In either of these circumstances the equations take the
form (U' V' W’ being omitted unless cv/k=1):

0g ot:

aU 8¥ eW_ plot, on ae
de tay t ee 7 OF or tee

Ol So rg pO | OM oe |

ae +(Sa+ 3, 3 e) |

Ol an (Om = Com! |

on Ge Ses)

OL Ow: OW oa (6)
| sen tae + on 2 oe |
where . |
SF rr nari py HO Os Va Wl), 2 OC eee
(U,V,WT) =U o -

,woU.¥, WT), oU,V, WT. |

Let us put the equation of the solid in the form 3
FU ofP) (2 Y> 2) | =0. : y : ‘ s (7)

Thus, if v,,/v changes, this amounts to considering instead
of the actual solid, similar bodies having linear dimensions
inversely proportional to v,/v.. The direction cosines (,, mj, 7;
of the normal will remain the same at corresponding points,
and the boundary conditions become :

At the surface U=V=W= (0, ie !
At the distance /&+ 77+ C infinite U=l, V=m,W=n.

The system of equations (6) and (8) determine U V W II T
as functions of &, 7, €, and ¢,, and substituting in the integrals
for the new variables their equivalents, we obtain

yes TE OEMID SES :
ai ay 2 definite functions of ; (9)

KOO 2, ¥, 2) 3. 0, OM.

of a Stream of Viscous Fluid. 943

The flux of heat h furnished in unit time by unit area of
such a body, equal to that which the contiguous fluid layer
communicates to the interior of the fluid, is given by

Ow 02
Introducing the new variables, we have
ih ih ge
h= (ke, 6/0) (ao +m +m$p). ae CL)

_ At corresponding points of the surfaces /(€,7,€)=0
limiting the bodies considered, the direction cosines l;, m4, n,
have the same values, and at corresponding times t,;=const.

I

M(t, Som, $7 +n, 87). gta EL

the derivatives —————~— at corresponding points have also
O(E, m £) aT

the same values. Consequently, at corresponding times the
trinomial coefficient is a function of the shape and orientation
of the bodies only.

Thus, for a family of similar bodies similarly orientated,
having linear dimensions L given by

Recev/? eV Alijn=coust., “x. + (12)

the heat loss per unit area at corresponding tim es from
corresponding points is given by

PEMD IO) Wei Ae a) Vat vi 4 ( hOy)

and this will also be true if “ h” be the mean heat loss taken
at the given instant over the whole surface of the solid.
So for bodies of this shape and orientation we have

b= (he O/y) fife ott oe (4)
which may be written
hL wee cw 7e z
pF {ese |e, cide AP (ale)
When the conditions have become steady—that is, inde-
pendent of time ¢,—the formula reduces to

AL Oa Ay
g =F (=). AUR ae sta (16

And, further, if the conditions do not settle down to complete
steadiness, but settle down to periodic fluctuations, then these
fluctuations will be similar in form for corresponding cases,
and the average value of the heat loss will still be given

by (16).

944 Dr. F. H. Newman on a

It is desirable to recall two alternative conditions which
have been introduced into the proof of the formula. They
are that cv/k shall be equal to unity or else very large. If
these conditions are not satisfied, it may be shown that the
formula becomes

hl 7 oes ey
ig =! ("s ale re
where cv/k is expected to be of little importance in the two
extreme cases mentioned.

Whether the simpler formula is true for gases on the one
hand and for very viscous fluids on the other, and whether —
cv/k is important for intermediate circumstances, is a matter
for experimental investigation. From data already available
it is known that the simpler form is fairly satisfactory for
gases, being indistinguishable in this case from the grouping
of variables deduced by Boussinesq for inviscid fluids. This
is shown graphically in a curve between AL/k@ and v, L/2,
given elsewhere *, for the cooling of wires in a stream of air
(H/k and vl/vy in the notation of the actual graph).
Corresponding data for liquids are not yet available, but an —
isolated result has been given by Worthington and Malone Tt
for the cooling of a wire in water (cv/k=7 ; v=0:01006),
and this yields a result, H/k=21°6, vl/v=38, which is in
satisfactory accord with the curve mentioned for air. In the
analogous case of natural convection, cv/k has been shown to
be of little importance for a wide range of viscous fluids.

July 1922.

LXXXI. A Sodium-Potassium Vapour Are Lamp. By ¥.H.
Newman, D.Se., A.R.C.S., Head of the Physics Depart-
ment, Unversity College, Exeter }.

[Plate V.]

ETALLIC arcs operated in vacua give very intense
radiation, and the lines in the resulting spectra are
very narrow, whereas with a substance placed between the
poles of a carbon arc, working under ordinary conditions,
broad lines are obtained, which often show much reversal,
the centres of the lines being comparatively faint. This is
the case when the sodium D lines are excited, and a bunsen
flame, to which salt has been added, is not a suitable source
of sodium radiation. As the amount of salt is increased, the

* Phil. Mag. xli. p. 899 (1921).

Tt Journ. Frank. Inst. clxxxiv. p. 115 (1917).
t Communicated by the Author.

Sodium-Potassium Vapour Arc Lamp. 945

lines are broadened and show much reversal. Modern prac-
tical and research needs require intense radiation and a mono-
chromatic source. ‘The chief line must be sufficiently removed
from its immediate neighbours, so that if a relatively wide
slit is used, other wave-lengths in the immediate vicinity of
the one desired are excluded. The quartz-mercury vapour
lamp provides such a source; it is easy to construct and
work, and does not require continuous pumping to keep it
exhausted while running. A sodium vapour are lamp,
working on the same principle, has been designed and
constructed by Rayleigh*, but it is more difficult to work
than the mercury lamp. Iron electrodes are unsuitable,
since they fuse and drop off after the lamp has been in use
for an hour or two. Tungsten, which seems to withstand

the action of sodium vapour, is used instead of iron. This

Pigst.

lamp requires an applied potential difference of 200 volts
when working, although the actual drop of potential across
the arc is very much less. The author f has used, previously,
a sodium vapour electric discharge-tube which gives intense
sodium radiation, but requires continuous heating while the
electric discharge is passing. The sodium-potassium vapour
arc lamp described in the present work needs no applied
heat, can be worked with a small applied potential difference,
and requires no attention while it is running.

The form of lamp is shown in fig. 1. It is made of quartz,
the bulbs A, B being about 3 cm. in diameter, and Joined by
a piece of quartz tubing C of internal bore 5 mm. and length

* Hon. R. J. Strutt, Proc. Roy. Soc., A. xevi. (1919).
+ Proc. Phys. Soc. xxxiii. pt. 11. (1921).

0 OQLSQSLS ee eS ee ee

946 Dr. F. H. Newman on a

15 mm. ‘The electrodes are iron rods 4 mm. in diameter,
and are sealed with sealing-wax. ‘The current used must be
such that these rods never become so hot that the wax is
melted or softened. The liquid alloy of sodium and potas-
sium—two parts by weight of sodium and one of potassium—
is run into the bulb A, and the exit tube D then connected
to a glass tap, and the whole exhausted. The lamp can then
be disconnected from the pump. An electric discharge is
passed through the lamp, the alloy being made the cathode.
Under the action of the discharge the oxide on the surface
of the alloy disintegrates, and the surface becomes quite
clean. Tilting the lamp, some of the alloy flows into the
other bulb B. Using this method of introducing the alloy,
the part C remains quite free from the alloy. With direct
current the lamp works with a minimum applied potential
difference of 30 volts, although when once the arc is struck,
the fall of potential is only 10 volts with a current of
1°5 amps. ‘The arc is struck either by tilting the lamp in
the same way that the mercury lamp is started, or one ter-
minal is connected to a small induction coil and a momentary
discharge passed. No external heat is required, as that pro-
duced by the current is sufficient to vaporize the sodium and
potassium. As the temperature rises the current decreases,
and the potential difference across the terminals becomes
greater. With currents smaller than 2°5 amps. the tempera-
ture of the quartz at C is never such that a piece of paper
held at this part is charred, and the wax seals do not soften,
however long the lamp is working. There is no “browning”
of the silica. As the applied potential difference is increased,
the radiation becomes brighter, and greater luminosity can
be obtained by warming the part C with a small gas flame.
This part of the apparatus is, of course, hotter than the other
parts, owing to the high current density. The lamp works
satisfactorily at any potential between 30 and 200 volts, and
the current can be regulated by a resistance in series. If
the current rises above 2°5 amps, there is the characteristic
‘browning ”’ of the silica. The lamp does not require con-
tinuous pumping while it is working. The sodium-potassium
alloy absorbs all gases, particularly nitrogen and hydrogen,
while the current is passing. In this manner a very good
vacuum is maintained, however long the lamp is in operation.
This fact, and the low voltage at which the arc is struck, are
two important improvements on the other forms of sodium
vapour lamp. The present form will not work satisfactorily
with alternating currents.

With the lamp it was found that the potassium lines were

Sodium-Potassium Vapour Are Lamp. 947

very faint compared with the sodium lines under all condi-
tions, and they became weaker as the temperature increased.
The relative brightness of the sodium and potassium lines
differed in various parts of the lamp, and the subordinate
series lines of sodium varied in intensity compared with the
D lines, although the latter were always the brightest. At
the cathode the potassium lines were very weak, while the
subordinate series lines of sodium were strong (PI. V. fig. 1.)
At C the radiation was very intense (PI. V. fig. 11.), while at
the anode the potassium lines were brighter than they were
at other parts of the lamp (PI. V. fig. 11.). The spectrograms
were photographed witli a constant deviation type of spectro-
meter, Wratten panchromatic plates being used. ‘The sub-
ordinate series lines of sodium becaine faint when the current
was reduced, and when the lamp was heated externally
(Pl. V. fig. 1v.). The electric discharge gave a radiation
consisting almost entirely of the D lines (Pl. V. fig. v.).

The intensity of the spectrum lines emitted by a mixture
of vapours when subject to electrical stimulus depends on the
ionization and resonance potentials of the various vapours,
and also on the partial vapour pressures. On the Bohr
theory the spectrum lines have their origin in the movements
of an electron within the atom when it moves from one
temporary orbit to another. In the case of sodium the
innermost orbit is that represented bythe limit of the
principal series —that is, by the term 1°58. The second orbit
is represented by the term 2p, and the frequency of the
resonance line is that of the first principal line 15 S— 2p.
The theoretical value of the resonance potential of sodium
vapour is 2°10 volts, and electrons of this energy produce
the D lines. Electrons having a velocity corresponding to
about 5°13 volts are able to ionize sodium vapour and cause
it to emit all the lines, including those of the subordinate
series. The resonance potential of potassium vapour is 1°60
volts, while the ionization potential is 4°33 volts. Ina mixture
of sodium and potassium vapours, as the accelerating potential
is increased, the 7699, 7665 doublet of potassium should
appear first, then the 5896, 5890 doublet of sodium, followed
by the subordinate series of potassium and sodium respectively.
The doublet 7699, 7665 being near the limit of the visible
spectrum would be faint, and so it is to be expected that the
D lines will, under all conditions, be the brightest lines in
the spectra.

When the current density is increased, the subordinate
series lines increase in luminosity. Sodium vapour having
only one resonance potential, the elevation in energy of the

948 Mr. J. J. Manley on the

electrons colliding inelastically must be produced by succes-
sive impacts or by absorption of radiation of suitable
frequency. The electron normally in the 1°5 S orbit of the
sodium atom may be forced into the 2» orbit by direct
impact. It is possible, however, that the ejection may be
brought about by absorption of radiation of frequency
15S—2p. Before it is able to return to the 1°58 orbit and
emit this radiation, collision occurs with a second electron,
and the electron within the atom is ejected to an orbit still
farther removed from the innermost stable orbit It then
returns to the 1:58 orbit in two stages, the first step causing
the emission of a line in the subordinate series, the second
step giving the D lines. The chances of this type of
collision occurring will increase as the density ot the
free electrons becomes greater. ‘This also explains why the
subordinate series lines are very faint compared with the D
lines when an electric discharge is sent through the lamp.
for the current density in this case is comparatively small.

LXXXII. The Protection of Brass Weights. By J.J.
Maney, W/.A., Research Fellow, Magdalen College, Oxford”.

IXTEEN years ago, I applied to a set of brass weights
; a method introduced by Faraday for protecting iron
from rust ; and asthe experiment has been highly successful,
other workers may find the plan, or a modification of it
described below, helpful.

The weights, some of them badly corroded, were first
lightly tooled and then suitably polished ; next they were
heated in a semi-luminous gas flame until they were nearly
red hot, and then suddenly plunged into boiled linseed oil,
wherein they were left to cool. The weights having been
removed from the oil, were washed with turpentins and then
polished with old linen; subsequently they were adjusted,
standardized, and set apart for the use of students beginning
their course of (Juantitative Chemistry. Now, although the
weights have been in regular use throughout the 16 years
that have since elapsed, and have been subjected to what
may be rightly termed the severest test of laboratory con-
ditions and usage, numerous re-standardizations, the most
recent of which was carried out a few weeks ago, have con-
clusively shown that Faraday’s method for the protection of
iron is also remarkably effective when applied to brass, and

~ * Communicated by the Author.

Protection of Brass Weights. 949

as a result I find that the original relative values of the
whole set of weights are still retained ; no re-adjustment
has been required. These observations and conclusions are
also borne out by the fact that the uniform and somewhat
pleasing bronze-like tint acquired by the weights during the
treatment is still to a large extent almost unchanged.

To obtain indubitable evidence as to the intrinsic value of
the method described above, the plan was lately tried with
other weights but without success. And here we may observe
that owing largely to the admixture. of one or more
sulphur compounds, the quality of the present day coal-gas
is very different from that which formerly obtained, and
experimental work showed that to this fact must be attributed

Fig. 1.

my non-success with the second set of weights. The difh-
culty which thus so unexpectedly arose, has now been over-
come by a method which for convenience and effectiveness
leaves but little to be desired. The new plan is as follows :—

A ‘“‘vitreosil””? crucible having a capacity of 50 c.c. is
loosely charged with asbestos fibre to the depth of about
half-an-inch, as shown in the figure. A lid J, also of vitreosil,
is inverted and placed upon the fibre; on this lid are
arranged triangular-wise three pointed fragments of porcelain
or fused quartz for supporting the weight w. The weight
having been cleaned and polished and evenly covered with
linseed oil, which is applied with the finger and thumb, is
stood within the crucible, as indicated in the figure; the

950 The Protection of Brass Weights.

crucible is then covered with its lid, and the whole heated
with a Bunsen flame. (It is convenient to adjust the flame
so that it extends about half-way up the crucible.) During
the heating the weight is frequently viewed, and when it has
assumed a golden tint the process is complete ; the flame is
then removed, and the closed crucible allowed to cool.
Finally, the weight is rubbed with an old silk handkerchief
and then adjusted. Treated thus, the weight presents the
appearance of well-polished and lacquered brass. The
protecting film being tough, is not readily defaced, and,
if necessary, the polishing may without risk be prolonged :

but when the initial operations are correctly performed,
nothing beyond a light and brief rubbing is required.

In dealing with a number of weights, | great economy both
in time and labour may be -effected by substituting for the
crucible a vitreosil muffle. The muffle is fitted with a rect-
angular tray, which carries the bits of porcelain, placed in
groups of 3, for supporting the various weights. The
weights having been arranged, the tray with its charge is
introduced into the muffle, which is then closed, suitably
heated, and subsequently allowed to cool; the weights are
then ready for polishing and adjusting. It may be observed
that as the supporting fragments are pointed, they but
barely engage the surfaces in contact with them ; hence
the finished protective film is practically complete.

Some experiments were made to determine the average
mass of the protecting fil ; the results obtained with two
weights, the one of 50 and the other of 20 erms., may be
cited. Thecleaned and polished weights were first accurately
weighed, then protected and finished as already described ;
finaliy they were re-weighed. The larger weight had in-
creased by 0020 grm. and the smaller by ‘0006 prm. As
the respective areas of the two weights were approxi-
mately 17 and 12 cm., we find that the average mass
of 1 sq. cm. of film was ‘00012 grm. in the one case and
"00005 grm. in the other. It was found that the smaller
value most nearly represented the weight of a normal film ;
the other and larger value was exceptional.

Daubeny Laboratory,
Magdalen College,
Oxford.

Peron ho

Al

LXXXIII. Note on the Analysis of Damped Vibrations.
By H..8. Rowe *.

HE two primary difficulties in the analysis of damped
vibrations are the nature of the friction and the position
of the zero. In most cases it is sufficient to assume that the
friction is a combination of so-called solid friction—a
constant, and of fluid friction, proportional to the velocity.
In the ordinary view it is inconceivable that these two kinds
of friction can coexist at the same time and interface, for
the conditions supposed to produce these two kinds of friction
are essentially different ; dry and wet, or molecular film and
measurable film. In practical cases, however, the two kinds
of friction can coexist in a system as, for example, where a
body slides or turns on dry surfaces and is damped by fluid
friction. Thus the equation of motion may be taken as

make ee PaO fg ws (LL)
where the signs of F and of « are the same ; put
2=X>+ EF /e’*,

and the solution of equation (I.) is

gig Ane 2" cosin'ts 2 (AL)
where : n= Vn? —k?/4m?
and ae
write ies
and ka/2mn'=r; so that e*=6,

where 6 is the logarithmic decrement for half periods.

Assume that a datum line is drawn at a distance E trom
the true time axis, and let R; be the reading from this datum
corresponding to the ith half swing ; then

a ee. (LY
By — HS Age, ee C2)
ip AN So ee we (BD
ieee gS). SY
dg AAD Ory: eth is) @ Seen

* Communicated by the Author.

952 On the Analysis of Damped Vibrations.

Subtracting (1) from (2), and

Risky = Ay +0)... Si
Subtracting (4) from (2), and

RyRy = Apo(l = 0"). eee
Subtracting (3) from (5), and :

Ry— Ry = Ayd?(1—8"). se SD
Dividing (8) by (7), and

Ree R, SO... Cae ee nn
Substituting in (6), we have the first amplitude :
Ay = (R,—Ry)/1+6

_ (Ri—Ro)(Ri— Ry)
a

Dividing (7) by (6) and adding (2) and (3), we have

ie Lo Bie
oR (Bei ea
which gives the position of true zero.

The solid friction term § follows from (1), and since ¢?
is easily found statically, F the solid friction of the system
per unit mass follows by division.

Where the system is dead-beat, the foregoing method does
not apply, and one way of solution is then by tuning of the
system by adding mass or increasing c? or both so that
sufficient equations are determined tor elimination.

The curve, of which the vibration in II. is a projection, is
an equiangular spiral with alternating origins distant
2K /c? apart, and it may be traced in either of two ways
according to circumstances. In the first place an arithmetic
spiral (see Phil. Mag., July 1922, p. 284) may be drawn
and the radii vectores shortened logarithmically, or an equi-
angular spiral may be drawn and portions taken out each sub-
tending a, and such that the initial radius vector of one
por tion is 2F/c? less than the final radius vector of the
preceding portion. Clearly the parts run smoothly together
on account of the equiangular property of the spiral.
Here it may be added that since the evolute of the arith-.
metic spiral is a straight line 2F/c? long, the curve can
be drawn mechanically by coiling a fine thread round two
pins 2F'/c? apart. ‘The same curve is described by the hand

of the housewife in winding up a card of “ mending.”

Liffect of Variable Head in Viscosity Determinations. 953

The outstanding difference between solid friction and fluid
friction in damped vibrations is that in the one the dissipation
per cycle is proportional to the amplitude, whereas in fluid
friction the dissipation varies as the square of the amplitnde.
Since the energy of the motion varies as the amplitude
squared; the dissipation per unit time with fluid friction is a
constant fraction of the energy of the system, whereas with
solid friction the dissipation as a fraction of total energy is
inversely as the amplitude. Hence in solid friction the
rapid damping of small vibrations until finally the dead
region of width 2F'/c? is reached. Clearly in the two systems
the envelopes of the two vibration curves may be tangential,
in which event solid friction may be mistaken for fluid
friction. As the foregoing analysis indicates, five half
vibrations or two complete periods suffice not only to safe-
euard against this possibility, but also to apportion the
relative magnitudes of the two sets of frictional forces.

It seems possible that some such analysis as is here
outlined may be a useful instrument of investigation in
connexion with friction and lubrication, affording at least
some sort of criterion in so-called border-line cases.

LXXXIV. The Full Effect of the Variable Head in Viscosity
Determinations. By Frank M. Lipstone*,

late the publication in this Magazine of my paper on
h the Measurement of Absolute Viscosity (February 1922),
it has been pointed out to me by Mr. W. H. Herschel, of the
American Bureau of Standards, and by Dr. Guy Barr and
Mr. L. F. G. Simmonds, of the National Physical Laboratory,
that the logarithmic head correction in the ‘“‘ viscous ” term
of the equation is, strictly speaking, incomplete, inasmuch as
it is based on the assumption that the head varies directly
with the velocity. Barr also makes a necessary correction in
the final kinetic energy term of the approximate equation,
which should read as in equation (2) below.

As no attempt appears to have been made to finda general
equation embracing all these corrections, it is here proposed
to try to find the exact expression, however laborious and
cumbersome, in order to ascertain to what extent the results
obtained by means of the ordinary formule deviate from the
true value. The premises of the whole argument are in-
cluded in the generally accepted equation,

__aregpth Vp
SEO GEND Salt”
* Communicated by the Author.
Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3Q

954 Mr. F. M. Lidstone on the Full Effect of the

the coefficient of the K.E. term being taken as unity. This
equation is true provided the head h is constant ; but unless
some compensating mechanical contrivance is used, such as
that adopted by Hyde (Proce. Roy. Soc. A, xevil. 1920), this
condition is never absolutely realized in practice, since no
matter how the pressure is applied or maintained, as soon as
flow starts, there is a change in the hydrostatic head and the
total head becomes at once a variable.

We have then, by making dt depend upon dh, first to
integrate the whole expression with respect to A over the
interval H—F. :

A
Putting SEE ie Man) Ris Te

Shas Sarl
eA Le ie Ah(H—F )dt nd Bdh .
dh ~ (H—-F)dt’
. I~ PD 3° ae
(8 areata VEBy/ 1+ yop
SAMHOR) = AC

The plus sign being obviously the only one permissible, we
; .

oe ' pel
| get, writing C for LAB?
Hd /B HV/et
dt= onl — $$ “de,
{ 2A(H — In 2 / A(H—F) ps Ey

which, after a little manipulation, gives finally

7 =
TI'Jp Get — F)

= \ i ar H =e
8Vi{log,.v ta vo) ty / 1+ G-a/ 4G}
(1)

Now, since C contains 7, to evaluate 7 from this expression
_would lead to a number of very unwieldly power series.
However, it will be seen that C must be large in comparison
with H or F'; hence a small change in C will not greatly.
affect the result. As a first approximation, then, we can take
C as equal to

mw rg?(H— EF)?
4V*(log. H/T)?"
Calling this Cj, we get a value for 7 which we will call 7.

ny 16? CG ‘
gp? Or Uo.

We can now get a closer value for CU, namely

Variable Head in Viscosity Determinations. 955

Re-calculating with this value we get m2, which gives us

from which 7; is calculated, and so on. The first value
for C or C, being too large, 7; will be too small. Simi-
larly, 7. will be too large and 73 again too small. Thus
7123 .... form a series in which the even terms and the odd
terms respectively converge asymptotically to the true
value for 7. The rapid convergence of the series con-
siderably shortens what would otherwise be a_ rather
laborious calculation ; three or four terms are sufficient to
fix the final value.

As an example, we take one of the standard runnings of
water at 20°C., and select in particular that one as set
forth in detail in Archbutt and Deeley’s ‘ Lubrication and
Lubricants, 3rd edition, p. 157. This example is chosen
as there is a considerable fall in the head, although, on the
other hand, the kinetic energy term is small. We quote
the data of the experiment in full :—

H= 23:56 | g = 980°51
F= 11:60 | p= 099826
r= 00309 V= 4:00756
t = 136-0 I= 21-991!

From these figures we get as the first value for C or
C, = 890°52 giving 7, = °0100609,
feom which C,= 897°24 ..,° 7, = °0100692
s Cz = 880°69 F455 gy = 70100680
5 ©, = 88049" % 5, oy, = 0100682,
from which we can write down the final value of 7 as
-01006818. The value of 7 calculated from the same data,
using the equation
arrget( Be Vie CH? — *) log, H/F 9)
"= 3Vilog, H/F Wnu(—F)(H+F)’ °° &
is 0100721.

The difference between this and the correctly calculated
value ‘01006818 is not very striking, but when one con-
siders that here the kinetic energy term is small (only
‘5 per cent. of the total) this is not surprising. In order
adequately to demonstrate the shortcomings of the ordinary
formule, it would be necessary to make a running under a
considerably increased initial head, in order that the K.E.
correction should become appreciable, and to continue the
running until the final head was relatively small.

3 Q 2

| 956 |

LXXXV. Quantum Theory of Photographic Exposure.
(Second Paper.) By Vi. SUGBERSTHIN and Ay) ieebe
TRIVELLI™.

N the present paper an account will be given of some
additional experimental tests of the light-quantum
theory of photographic exposure proposed in the first paper
of the same title t, and some further theoretical formule will
be deduced from the fundamental one given in that paper.
Virst of all, however, due mention must be made of certain
very valuable experimental investigations, since published by
Svedberg, which seem again to corroborate the theory, also
of a paper by Svedberg and Andersson published somewhat.
earlier, but not brought to our notice until the first paper had
been dispatched for publication.

1. Concerning “The Effect of Light,’ Svedberg and
Andersson’s paper (Phot. Journal, August 1921, p. 325),
dealing under that head with only a very few size-classes of
grains (each class, moreover, of a very considerable breadth),
contains only the qualitative though definite conclusion that
‘for equal exposure the percentage of developable grains is
always greater in the class of larger grains.” The quanti-
tative, viz. exponential dependence of this percentage upon
the size (area) of the grains, is obtained and well verified
experimentally in the case of bombardment by erays,
Kinoshita’s experiments of 1910 having made it very
probable that each silver halide grain hit by an a-particle
is made developable. The latter being granted and the
discrete nature of a-rays being a palpably established fact,
the validity of the exponential formula, in our symbols
k=N (1—e-”*), had to follow as a necessary consequence.
Its verification is properly a verification of Kinoshita’s
statement, and by having thus tested it experimentally
Svedberg and Andersson have certainly done an important
piece of work, especially as Kinoshita’s result was contested
by St. Meyer and v. Schweidler. In the next section |
analogous experiments with @-rays are described, but the
results thus far obtained are not conclusive apart from
enabling the authors to state that one or two -particles
striking a grain do not as a rule make it developable.
Finally, returning once more to the eftect of light (p. 332),

* Communication No. 149 from the Research Laboratory of the.

Eastman Kodak Company. Communicated by the Authors,
+ L. Silberstein, Phil. Mag, July 1922, p. 257.

Quantum Theory of Photographic Exposure. 957

the authors remark only briefly that an analogous concep-
tion might also assist in the interpretation of the mode of its
action ; but add that if the quantum hypothesis be assumed,
‘“‘ the difficulty arises that the real blackening curve has not
the exponential form prescribed by this hypothesis if we
suppose each halide grain to be made developable when
struck by a single light quantum.” ‘They seem to forget
that the simple exponential formula yielded by a quantum ©
theory relates to the case of equal grains, which is not that
of real emulsions, and that in order to obtain the blackening
curve (say density J) plotted against the logarithm of
exposure) that elementary formula has to be integrated over
the range of sizes, which presupposes the knowledge of the
frequency curve of the emulsion, and the somewhat intricate
question of the relation between the photographic “ density ”
and the total of blackened aréas has to the treated*. ‘he
latter question, simple though it be for one-layered coatings,
becomes particularly mnie in the usual case of many
layers of grains. It is for this reason that the best way of
testing a similar theory consists in microphotogr aphic counts
and planimetric measurements of the individual grains. At
any rate, Svedberg and Andersson propose to turn to another
more complicated assumption f which, they expect, “ will
actually predict a blackening curve of S- shape.” They
propose to discuss this possibility on another occasion.

The second of the papers alluded to, due to Professor
Svedberg himself (Phot. Journal, April 1922 , p- 186), has a
more direct bearing upon our subject, and may turn out to
supplement our-own tests by offering, as it were, an inter-
mediate link in the conjectured mechanism of the action of
impinging quanta or light darts. In this paper Svedberg
proposes to explain the behaviour of the grains noted in his
preceding paper by a single hypothesis, and to test the latter
directly. His hypothesis is that the product of the light
action on the halide grain consists of ‘‘ small centres distri-
buted through the grain or through the light-affected part of
the grain according to the laws of chance,” and that a grain
will become developed if it contains one or more such centres.
If v be the average number of centres per grain, the per-
centage probability that a grain will contain at least one
centre (and will therefore be developable) is P=100 (1—e7).

* Concrete examples of such a kind will be treated in the third paper
on our subject.

+ Namely, that a certain minimum number of quanta must strike the
grain within a certain maximum part of its area in or der to “build up a
silver nucleus large enough to act as a reduction centre.”

958 Dr. L. Silberstein and Mr. Trivelli on the

Now, it would be enough to assume that these centres are
produced by discrete light-quanta impinging upon the grain,
and the formula P=100 (1—e~™) would follow at once.
(For, if n be the number of light-quanta per unit area, and a
the area of a grain,v=na.) But Svedberg does not make this
assumption *, and devotes instead the remainder of his paper
to testing directly the above formula for the occurrence of at
least one centre and the corresponding chance formula for
the percentage number of grains having no centres, of those
having one or two or three centres, etc., having succeeded in
making these centres or, in Svedberg’s own words, ‘‘ the
nuclei corresponding to the developable centres,” visible and
accessible to measurement. For details of these elegant
experiments the reader must be referred to the original
paper. Here it will be enough to say that the recorded
“dots” or visible traces of those centres were found distr?-
buted very much in accordance with the probability formule,
namely, in one experiment with light and one with X-rays.
Only two size-classes of grains were treated in each of these
experiments, and with regard to the dependence upon
exposure Professor Svedberg (p. 192) has thus far only
roughly stated that the percentage number of developable
grains “increases approximately exponentially as function
of exposure,” at least for normal and for over-exposures in
the case of light (and probably for all exposures in the case
of X-rays) though not for under-exposure to light. The
paper is concluded by the remark that to account for the
deviation from the exponential formula in the case of low
light-exposure, we should probably haye to adopt the
quantum point of view, and that in the case of light (a
quantum of visible light containing 5000 times less energy
than an X-ray quantum) “ several quanta would have to be
absorbed very near one another to forma developable centre ””
Such a view, however, can easily be shown to be untenable.
At any rate, Professor Svedberg proposes to test it by experi-
mental investigations which are planned in this direction.

* In the discussion which followed upon the reading of Svedberg’s
paper, Prof. T. M. Lowry mentioned such an assumption of a ‘‘ bombard-
ment by light corpuscles” as the simplest interpretation of Svedberg’s
photographs (of the “‘ centres”). Other speakers, however, were rather
hostile to such a view, and Mr. B. V. Storr considered it even equally
conceivable that the “ centres” distributed haphazardly might be present
before the light action, but such a state of things would have hardly
escaped Svedberg’s notice. At any rate, Professor Svedberg will no
doubt meet Mr. Storr’s objection by appropriate control experiments.
Control experiments of such a kind, viz. desensitizing experiments, are
now being made by Sheppard and Wightman.

Quantum Theory of Photographic Exposure. 959

The existence of the aforesaid “centres”? as seats of
incipient development, around which the developer's action
gradually spreads, has been known for some years *, and has
been observed, among others, by Trivelli. But the important
discovery that these centres are haphazardly distributed is
entirely due to Professor Svedberg. If his results are
ultimately confirmed by further experiments, especially for a
series of different exposures, it will be possible to consider
these centres as an intermediate link in the theory proposed
in our first paper (the centres marking the spots where the
grains were hit by the light-darts). Im the meantime,
however, our further tests have to be conducted by con-
sidering the lasé link of the chain, 7. ¢., by counting the
grains of each size-class affected and ultimately developed.

2. Before passing on to the description of our further
experimental results, a few words must be said in defence of
the property attributed in our first paper to clumps (aggre-

gates) of grains which apart from some single grains con-
stituted our chief material, An explanation seems the more
necessary, as another recent paper by Svedberg + contains
results apparently clashing with what we believe to be the
behaviour of clumps with respect to light. The property
assumed by us, as the expression of experimental facts, was
that a clump, 2. e., an aggregate of grains in contact with one
another, behaves as a photographic unit, by which is meant
that if any one of its component grains is made developable,
the whole clump will be reduced by a sufficiently long develop-
ment. We have since been able to test this behaviour in a
variety of ways.

On the other hand, Svedberg concludes from his experi-
ments that there is no transference of reducibility (develop-
ability) from one grain to another ‘‘in direct contact’ with
it. (See especially p. 184, loc. cit.)

This apparent discrepancy seem to be due to the circum-
stance that Professor Svedberg worked with an emulsion (a
single kind only) consisting of rather small and almost
spherical ¢ grains, whereas our “material, and especially the so-
called W-12-C experimental emulsion, with which all the
work in question is being done, consists predominantly of
large and very thin, flat polygonal plates or tablets which are
in mutual contact either along a whole edge or, still more
intimately, are partly piled upon or overlapping each other.

* Of. M. B. Hodgson, Journ. Franklin Inst., November 1917.

+ On “The Reducibility of the Individual Halide Grains,” Phot,
Journal, 1922, pp. 183-186.

960 Dr. L. Silberstein and Mr. Trivelli on the

The fine spherical grains of Svedberg could have only at
the utmost a point contact, and this might not have been
intimate enough. It is even credible that in view of the
Brownian motion of these minute bodies there was actually
no permanent contact between them, as becomes very likely
from Svedberg’s remark on page 185, that ‘“‘even over such
a small distance as 1 micron no noticeable transport of silver
ions takes place.”

At any rate, we have found in our case the property of
clumps as units well verified. Without attempting to
reproduce in this place all our available evidence *, we may
support and illustrate the said principle by the following
data. Fig. 1 represents the frequency curve and, in the

Janey, alo
00 800 wi2C
a 3154 GRAINS MEASURED.
5.71 - 1(0°GRAIRS PER SQ.CH. PLATE=H.
X= 1.04 p7CALC. X=i.01p708s.
175 700 o=SiZE—FREGUENCY PER 1000 GRAINS, OBS.

A=S!ZE—FREQUENCY PER 1000 GRAINS,CALC.
Y=120e@-081%-0.8d?

uiis0 600 B=SIZE-AREA PER S0.cM. PLATE.
Y= -0.8Kx-082 = ———
s ome C= T0060

cM. P

id

un
GRAINS.

uo

re)

i2)

PER SQ
i000

=
ow
w&
8

A-FREQUENCY PER
‘
7

wn
KS)

3
<
@

B-AREA IN {2 x 10-8

Ww
wu
<<
9

inset, a microgram of a sample of grains of the aforesaid
W-12-C emulsion. This emulsion was spread over the glass
plate in a single layer so as to obtain the maximum number
of grains per unit area with the least possible overlapping.
The emulsion, after the coating, was kept in its liquid state
long enough to enable the majority of the grains to settle
with their flat faces on the surface of the glass. Under
these circumstances they, and especially the larger grains,
form numerous clumps of from 2 up to 33 grains, as will be

* Discussed in a paper just sent to Phot. Journ. by Trivelli, Righter,
and Sheppard. [This paper has since been published in Phot. Journal
for September 1922, p. 407. |]

Quantum Theory of Photographic Exposure. 961

manifest from Fig. 2, curve marked VV. After exposure and
development the clumping of the survived grains was
determined all over again and is represented by the curve
marked N—K; the curve marked K is the difference of
these two curves and represents the clumps affected by light.

Fig. 2.
Wi2C CLUMPING CURVES

N=ORIGINAL CLUMPS PER SQ.CM. ONELAYER PLATE.
K= AFFECTED CLUMPS PER SQ.CM. ORELAYER PLATE.

K,=Ny ((— @-0.9938)
N—-K=REMAINING CLUMPS PER SQ.CM.

La K.= Wreacs°sn

= 8000

Km =~ @-0.4538
Na

@ MaNUMBER OF GRAINS IRA CLUMP,

w
°o
°
°

CLUMP-—FREQUENCY PER SQ.CM X 107? ON ONE LAYER PLATE
N
8
°o

This would suffice perhaps by itself to show that our clumps
behave as photographic units. But additional evidence is
afforded by figs. 3a and 30, in which all the individuals were
carefully blackened by hand on a microgram originally
enlarged 10,000 times ; the former of these figures refers to
the original unexposed one-layer grains, and the latter to

962 De. dh. Shenae and Mr. Trivelli on the

the grains surviving after exposure and development. A
glance will show that the majority of clumps, and especially
the larger ones, are removed entirely. Of such pairs of
samples as figs. 3a and 36, about forty were made, and the
behaviour was always typically the same.’ Further and more
direct experimental tests of the adopted clump principle are
now in progress, notwithstanding that we have but little
doubt about its correctness, always, of course, in relation to
the material which we are using. And we feel sure that the
same principle can be firmly relied upon in what follows.

3. Let us recall that the theoretical values of the per-
centage number y=100k/N of clumps affected, as given in
the fourth column of the table in our first paper, were
calculated by means of formula (12),

log (1-4) = —nal1— /clals er

with the values of the parameters

=

ie BY 2
eer . ae
a= 0091 Ww:

the meaning of all the symbols being as before. The agree-
ment of these values with the observed ones, ranging over
33 classes of grains and clumps, was excellent, thus proving,
at any rate, the essential correctness of the formula as far as
the dependence on size (area) a goes.

To test it with regard to the exposure or n, portions of the
same plate were subjected to the action of the same lght
source, ceteris paribus, for one-half, and for one-quarter of.
the time of the original exposure. The same method of
evaluating WV and k being adopted as before, the results
tabulated below under yop5. were obtained. Now, without
even taking the trouble of retouching the values of the
parameters in adaptation to the new observations, o was
taken as in (12a) and n equal to one-half and to one-quarter
of its original value, respectively. Since the exposure is, at
any rate, proportional to n, our formula with these n-values
should represent the two new sets of observations. The
following table gives in the first row the number of grains in
a clump *, and in the second row the average area a of each
class of clumps, in square microns, as before; the third and

* Starting from 2, since with these weaker exposures reliable counts
of single grains affected could not be secured. .

Quantum Theory of Photographic Exposure. 963

the fifth rows contain the percentage numbers of clumps
affected calculated by (12) with c=0°097 and

"572
Gr) Gules ~ =0°286 per pu?

: 2
and (IT.) we Ae

=('143,

respectively, and the fourth and the last rows those observed.
The last but one column refers to clumps of 12 and 32 grains,
and the fact that almost all of these have been affected
(Yovs. =100) gives an additional score of verifications of the
theory (though in the case of (II.) the observed “100” sets
in somewhat too soon).

Grains in Clump 2 3 + 5 6 7 8 9 10 1 itera
ae 173 303 488 62 74 86 98 Il 12 13 ...>25
(Veale, 282 442 643 743 81-0 859 89-7 924 942 955... 998
‘Wenn, 21 57 638 745 875 96 97 97 965 100 ... 100

Beas foe, 40 749s) bb 626 Gre 2735759 TES... 95-7
a 13 876 423 53 66 825 865 (?) 894 100 ... 100

The agreement, although in general not so close as in the
previous case, is certainly satisfactory and in three or four
instances even remarkably good. Notice especially the case
of four-grain clumps which show perfect agreement in all
three exposures, the calculated and observed values in the
original exposure (cf. first paper) having been 87:3 as against
87:1, and now 64°3 and 41:2 as against 63°8 and 42°3.
Almost the same is true of the five-grained clumps. But in-
general the agreement is good enough throughout the array
of clumps *.

4. Notwithstanding the good agreement and the consistency
of these three sets of results with regard to the values of n
and o, some critical remarks must now be made about the
meaning of the latter parameter. It will be remembered
from the first paper that o or mp? was originally introduced
as the (average) “cross-section ” of the light darts, and p as
their equivalent semi-diameter, and the mathematical réle of
this finite diameter was fixed by assuming that a grain is
made developable only when it is “fully ” struck by a light

* The outstanding discrepancies being attributed mainiy to the
uncertainty of the (average) sizes a of the clumps and perhaps also to
disregarding the effect of tue finite range of a within each class of clumps,
How this finite breadth of the classes can be taken into account will be
shown presently.

464 Dr. L. Silberstein and Mr. Trivelli on the

dart. This gave as the efficient area of a grain, instead of

a=Tr"*,
a =al1—p/r|?.

Now, exactly the same formula would arise if we assumed
that, no matter what the thickness of the light darts (and.
whether it is finite at all), a grain is made developable only
when the aavs of the dart hits it in a point not too near the
edge of the target (grain), thus excluding from the total area
a boundary zone of a certain breadth p. Such a condition is
not altogether fantastic, and one might support it by
imagining that if the grain is hit too near its edge, an
electron is still ejected and a “centre” of reducibility is
produced at the spot, but the wave of development, stopping
dead at the edge, has not such a good chance to spread over
the whole grain as when the centre is well within the target.
If so, then the empirical principle that a grain is either not
affected at all or is made developable entirely would require
a qualification, viz., the exclusion of that boundary zone.
This alternative, therefore, should and can still be tested. If
it is supported by experiment, the original interpretation
given to p or o can be abandoned, since it certainly is not
very satisfactory. Not that there is anything incredible in
the light darts having a finite thickness and a cross section
such as one-tenth pu”; so far as we know, they may be trains
of waves of even much larger transversal dimensions. But
the unsatisfactory point about this interpretation is that it is
hard to imagine why the grain to be affected at all, 2. e. to
have a photo-electron ejected, has to be hit by the whole of
that cross section. For, if so, then, unless some light darts
have a diameter of the order of 107° cm., no such things as
simple atoms or molecules could ever have their electrons
ejected by light*. Yet, a grain, as a crystal lattice, may,
after all, behave as a single molecule, at least in the present
connexion, and the original réle attributed to the cross-
section of the light darts, though repugnant, may still turn
out to be a useful working hypothesis. To ensure the
possibility of being fully hit and therefore affected, even to
the smallest available silver halide grains, it would be enough
to treat o in our formula as the average taken over a
sufficiently ample interval of sections down to very small
ones. It would be premature to enter into quantitative
details of the consequences of such an assumption. But it
seems proper to mention even at this stage that an assumption

* Whereas the photo-electric effect has been obtained with gaseous
substances, though not beyond every doubt.

Quantum Theory of Photographic Exposure. 965

of this kind can well be tested experimentally. In fact, if
that assumption be correct, then the light traversing two

or more equal photographic plates piled upon each “other
should contain, successively, a larger percentage of the
coarser light darts, so that the formule of type (12) repre-
senting the number of affected grains or clumps of various
sizes should have not only a decreasing n, but also a succes-
sively increasing average value o of the cross-sections of the
darts, a comparatively larger proportion of the more slender
darts being absorbed each time. In short, we should have a
kind of sifting effect. Such experiments which, to be at all
convincing, require obviously a much higher degree of
accuracy in counts and area measurements, are now in
preparation. Their results will be published in a subsequent
aper. In the meantime, the parameter o may and profit-
ably will be retained as a small but desirable correction of
the exponential formula without, however, being given either
of the alternative interpretations.

It may be well to add here also a few remarks about n, the
chief parameter in the fundamental formula. This was
originally defined as the number of light-quanta or darts
thrown upon the photographic plate per unit of its area.
Now, apart from the generally small correction term con-
taining o, the parameter n appears in the formula only
through the product

p=na,
where a is the area of the grain. Thus, essentially only the

value of this product (a pure number) can be determined
from microphotographic experiments. Suppose now that
the sizes of all grains of the given emulsion were reduced in
the same ratio, converting every a into ea; then, the same
experimental value of p would indicate a number of light

darts : times larger. Now, such would exactly be the

position if for every grain not the whole but only a fraction
e of the area were vulnerable, 2. e. deprived of an electron on
being hit by a light dart. The grain may be sensitive only
in spots scattered over its area, “and each perhaps of very
Gonats dimensions.. Provided that all theso spots occupy a
jived fraction e of the total area of the grain, the micro-

hotographic counts and measurements could not inform us
about the value of this fraction unless the exposure given to
the plate is known in absolute energy measure. ‘Thus, for
instance, if, as was tacitly assumed, e=1, the number of light
daris in the set (1.) of observations just described would lead

966 Dr Le siterctenaad Mr. Tavelloae

to n=0'286 per pw? or about 29 million darts per square
centimetre of the plate; but if, say, only one-thousandth of the
area of each grain were vulnerable, we should conclude that
- 29 milliards of darts were thrown upon each cm.’ of the plate.
But it would be idle to speculate upon this subject and, as
far as we can see, the only way of deciding whether that
suggestion Is correct or not and of deter mining the value of |
the fraction e¢ is to measure the exposure energy in absolute
units *. Now, in none of our experiments thus far reported ~
was the energy value of the exposure even roughly estimated,
not to say measured. But in order to decide this important
question, preparations for measurements of this kind are now
in progress in this laboratory, and their results will be
published in due time.

5. Liffect of finite breadth of size-classes of targets.—The
short name “target ” will now be used for either a single
grain or a clump of grains in suthcient contact to act asa
photographic unit.

In the three sets of observations hitherto reported, the
targets were classified according to the number of grains
contained in them (from 1 to a3 and for each class the
average size (area) was used as a in the theoretical formula,
without taking account of the finite breadth of any such
class, 7. e. of the interval, a, to a, say, over which its indi-
viduals ranged. It was ‘not possible with the said classi-
fication to secure reliable estimates of this breadth, which,
however, for some classes might have been considensnle
(perhaps of the order 1y?), and ‘at any rate varied from class
to class. It is likely that some of the outstanding dis-
crepancies are due to these neglected tactors and especially
to the latter.

To eliminate this source of error, and at the same time to
avoid the laborious planimetrization of targets within very
narrow limits, we propose henceforth to divide the whole
material of targets into deliveruiely broad classes, all of equal
breadth, say 2a.

ie then, the average size of any of these classes of targets
is Tee as he variable ain our formula, a correction has to
be made for the finite value of 2a. This correction can easily
be found.

Disregarding for the moment the o-term, the number of
targets of a class of breadth 2«=a,—a, affected by n darts, is

* Although even then the final result would be made doubtful by the
uncertainty “whether the total light energy (as required by Einstein) or
only a fraction of it is conveyed in discrete quantum parcels.

Quantum Theory of Photographic Exposure. 967
by the fundamental formula (7), first paper,

k=\ (a) [1—e-™|da,

where f(a)da is the number of targets of size a to a+da
originally present. Now, if 2 is of the order of 1/2 or even
ly’, we can take f(a) =const. within the integration interval
with sufficient accuracy for all of our experimental emulsions.
Thus, denoting by V the original number of targets in the
whole class, so that

ie) OND
k ena e Raz
ve shall h — =l|—
we shall have N DAs :
or, writing simply a for the average @=1(a,+a,), and there-
fore, dg=a+a, a,=a—4, .
i NO. op na
k —|—e-™. ae a ae

N 2ne

Remembering that 3(e"*—e~"*)= sinh (ne), writing for
brevity
N
v=log

Pen
and replacing a in the chief term by
ae
we have ultimately the required formula
sinh (ne) ;

Na

(13)

v=na'— log

(14)

Notice that the correction term depends only on nz, that is
to say, for X=const., on the product of the exposure and the

class breadth. If this product is a fraction, such as one-half —

or even two-thirds f, we can write, up to (n«)‘,
em es (CLAY

If, as explained, all the contemplated targets of the
emulsion are divided into classes of equal lreadth 2a, the

* This is accurate enough provided o/a is small. In the correction
term the semi-breadth a requires practically no amendment.

+ If a=1p” and the exposure is as in the previous concrete cases, the
value of this product does not exceed 0°6.

968 Quantum Theory of Photographic Exposure.

correction term in (14) is, for a given exposure, constant
throughout the array of classes, and v plotted against a
should give a straight line. If o were non-existent or
negligible, we should have a straight. line for v plotted
against a, itself.

The aforesaid classification of targets and the corresponding
formula (14) will be used for analysing all the experiments
now in progress. For the present, we are able to quote only
one such set of results condensed in the following table.

The targets (grains and clumps alike) were all divided into
five classes of equal breadth 2a=0°60y?, ranging from 0°20
to 0°80, from 0°80 to 1°40, etc., as shown in the first column,
which gives the average sizes @ in square microns. ‘The
third column gives the observed number of targets surviving
for every NV targets originally present, each of these data
being an average of counts on four different domains of the
plate. The fourth column contains the percentage number

y=100 5
calculated by (14), to wit, with

of grains affected, as observed, and the fifth, as

n— 0°25) per 27,

o=0°0081p7.

a. N. N—k. y obs. ycale. A.
0:50 190°3 173°3 8:9 9°0 —O0:1
1-10 140-0 103°5 26°0 (20°6) (+5°4)
1-70 62°6 43°0 313 311 +02
2°30 314 18°7 40°4 40°3 +01
2°90 19°8 10°3 480 48°4 —0-4

The agreement is, apart from the second class, bracketed
as an “outlaw,” almost perfect. The ‘‘ cross-section” of
the darts, or what o may stand for, is agreeably about ten
times smaller, 7. e. the diameter three times smaller than that
previously obtained with the same light source. ‘This is not
to say that the reality and réle of o is herewith settled. Yet
it is interesting that without o, that is to say, with a’ in (14)
replaced by a itself, no choice of n yields such a close agree-
ment. As to the correction term due to the finite class
breadth, it may be mentioned that in the present case it
amounts (as a subtrahendum from v) only 10 0°030.

Rochester, N.Y.,
June 23, 1922.

[ 969 J

LAXXVI. The Discharge of Air through Small Orijices, and
the Entrainment of Air by the Issuing Jet. By J.8.G.
Tuomas, D.Sc. (London and Wales), A.R.C.Sc., A.LC.,
Senior Physicist, South Metropolitan Gas Company,
London*.

[Plate VI.]

INTRODUCTION.

4 Naas present paper details some of the results obtained

in a preliminary investigation of the conditions deter-
mining the entrainment of air by jets of various gases. As
there exists considerable uncertainty as to the representation
by means of a formula of results for the discharge of gases
through fine orifices T, it has been considered desirable to
include also a short discussion of results, under this head,
obtained during the work.

In the case of the entrainment of air by a jet of gas either
lighter or heavier than air, the volume of air entrained per
unit volume of gas in the jet is ditferent according as the jet
is directed in an upward or downward direction. With a
jet of air, such an effect, if present at all, is small. As the
great majority of the practical applications of air-entrainment
by gas jets issuing from small orifices with which the author
is concerned refer to downwardly directed jets, it was decided
to confine the preliminary experiments with air to such
direction.

HXPERIMENTAL.

(a) The Discharge Tube-—The flow systems employed in
the present investigations are shown in fig. 1, Air was
delivered under constant pressure to the brass tube A, the
interior surface of which was carefully smoothed, and issued
as a jet from the orifice in the disk B, situate at the lower
end of the tube. Particulars of the preparation and mount-
ing of the disks and orifices are given later. The tube A was
provided with a water jacket through which tempered water
flowed, so that throughout the whole series of experiments
the temperature of the water in the jacket was maintained
constant at 13°C., with a possible variation of ‘5° C., the
temperature being taken by thermometers placed at the inlet

* Communicated by the Author.

+ See, e.g., Buckingham & Edwards, Sci. Papers, Bureau of
Standards, vol. xv. pp. 574 et seg. (1919-20); Walker, Phil. Mag.
vol, xliii. p. 589 (1922),

Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3R

970 Dr. Thomas on Discharge of Air through Small

and outlet to the jacket. The pressure of the air in the tube
A at a distance of about 1°5 inches above the orifice was
determined by means of the tube C, which was closed at its
lower end and provided with a number of small circular
openings on its cylindrical surface near its lower end. The
tube C was connected with a water manometer, which was
read to 1/50 mm. by means of a cathetometer microscope.
Calculation showed that the deficit of pressure due to motion
of the air in A was ‘negligible. Air delivered from the
orifice B passed downwards through the tube D, of vertical
length about 10 inches, and bent as shown. ‘The lower end
of D was shielded by a large open box, so that disturbances

RG

in the air of the laboratory were very largely prevented from
affecting the jet*. No Venturi tube, to increase the air-
entrainment, was inserted in Din the present series of experi-
ments, as it was found that the same tube was not equally
suitable for use with all the orifices emploved.

(b) The Air Induction Tube.-—-Air set in motion by the
action of the jet issuing from the orifice flowed towards the
jet through the flow tube EH, in which was placed a hot-wire
anemometer of the Morris type, constituted of two fine

* It may be remarked that without such protection, the effect of any
atmospheric disturbance upon the indication of the hot-wire anemometer
employed is under certain conditions considerably magnified by the
operation of the jet. A combination of jet and hot wire, such as

described herein, would appear to be an exceedingly sensitive device for
indicating small atmospheric movements.

Orijices, and Entrainment of Air by the Issuing Jet. 971

platinum wires. The end of this tube away from the jet
opened into a large vessel containing water, which served to
saturate the air with water vapour, a condition corresponding
to that of the air employed in calibrating the instrument, and
to shield the anemometer from outside disturbances. Slightly
different calibration curves were employed corresponding to
different atmospheric temperatures. The velocity with which
the air was set into motion in the tube E by the jet depended
upon the length of tube employed. This was chosen of the
minimum length consistent with securing steady readings of

the anemometer, and was equal to 54 inches.

The jacketted tube A was inserted into the tube D to such
a depth that the velocity of air-stream passing over the
anemometer wires was a maximum for the pressure employed.
A gas-tight joint was then made between the tubes A and D
bv means of the gland G screwing down on a rubber ring.
If for any cause it became necessary to remove the tube A,
it could be readjusted to its former position by reading the
position of the pointer P with reference to a scale marked on
the tube. The pointer S moving over a circular scale served
to indicate the azimuthal position of the orifice disk, and was
more especially used in connexion with subsequent experi-
ments with multiple-orifice disks.

(ec) Measurement of the Discharge.—The method employed
for determining the rate of efflux of air was to determine the
time taken for a definite measured volume of air to flow
through the orifice. The device employed for this purpose
is shown in fig. 2, andisa slight modification of the apparatus
employed by Coste*. The measured volume of air is
contained between two marks shown on the vessel A. Water
is delivered to the vessel through the jet B from a Marriotte
bottle or overflow constant-head device, the rate of flow of
water being controlled by means of the cock C. The cock
D is fully open during the flow of water through B, and
closed while the vessel A is subsequently emptied through
the cock H, the cock © being meanwhile unaltered. In this
manner the rate of flow of water through B is very conveni-
ently adjusted to the series of increasing rates of flow
employed in the present sequence of experiments.

Vessels A of various sizes were employed in the course of
the present experiments, the smallest having a volume
between marks of 8382 c.c. and the largest a volume of
7081 ¢e.c. The lower stem of the vessel A extended for a
eonsiderable distance below the lower mark, so that the

* J. Soc. Chem. Ind. vol. xxx. p. 258 (1911),
dR 2

972 Dr. Thomas on Discharge of Air through Small

conditions of flow became steady before the time of the
_ water surface crossing the lower mark was taken. In some
of the larger vessels a small bulb was blown on the lower
limb for the same purpose.

The pressure in the vessel A was measured by a water
manometer connected to F, and the temperature was
measured by a thermometer hanging near A. Air was
delivered to the orifice through the tube G, and passed over

calcium chloride and then through a small plug of glass
wool. Throughout, the volume of air delivered was corrected
for pressure, temperature, and humidity. The tube H served
for drawing air into A, the cock K being meanwhile closed.
The device maintained a very constant pressure at the
orifice, any variation occurring being somewhat less than of -
the order of 0°5 per cent. of the total pressure. ;
(d) Preparation of the Orifices.—Considerable attention
was given to the preparation of the orifices. Throughout,

a

Orijices, and Entrainment of Air by the Issuing Jet. 973

tlle endeavour was made to make the orifices as circular and
smooth as possible. In the case of the orifices numbered 1-
10 in the sequel, these were made in a stiff copper-nickel
alloy (88 per cent. Ni, 12 per cent. Cu), 0°0229 cm. thick,
and except for the smaller ones were made by means of a
machine designed by Dr. Charles Carpenter for the bulk
manufacture of single or multiple-orificed injectors for use
in gas-burners. By this machine, a disk about 6 mm. in
diameter is cut froma sheet of metal, aud simultaneously a
hole or holes punched in the disk by the passage of a flat-
ended accurately cylindrical needle or needles through the
disk, which is held meanwhile between blocks, the lower
carrying the needle or needles, which after passing through
the disk enter accurately bored holes in the upper block.
The machine is operated after the manner of an embossing
press.

Orifices in disks prepared in this manner possessed a very
smooth interior surface, and there was little burr on their
outer surfaces. This was readily ground away by rubbing
on an oiled stone, and the inner surface finally burnished by
means of a cylindrical needle.

The two smaller orifices employed were drilled in the same
material, great care being taken to see that they were as
accurately circular, and their interior surfaces as smooth as
possible.

Fig. 3 (Pl. VI.) shows microphotographs (linear magni-
fication about 37) of representative orifices of the series, They
indicate very slight departure from the cireular form. Actual
measurement showed that the greatest and least radii of any
disk agreed to within 1 per cent. except in the case of No.8,
where a maximum difference of about 2 per cent. occurred.
A microscopic examination of the interior of the orifices
showed that there was present very little roughness in
the finished orifices.

The prepared disk was carefully soldered on to a cap which
screwed on to the lower end of the tube B (fig. 1), this cap
being then itself soldered on to the tube. It was previously
ascertained that all joints in the various parts of the apparatus
were gas-tight. Tested under 10 inches of water pressure at
various times during the course of the experiments, the leak
in the discharge tube when the orificed disk was replaced by
a blank of the same thickness, and in the anemometer tube,
was found not to exceed 0°001 cubic feet per hour.

974, Dr. Thomas on Discharge of Air through Small

RESULTS AND DISCUSSION.

Table I indicates the general nature of the observations
and the calculations based thereon.

(a) Discharge.—Fig. 4 shows how the discharge through
the various oritices of the series depends upon the pressure
and diameter of the orifice. The results shown in this figure
all refer to orifices in disks’ of thickness 0°0229 cm.

f "DIAMETER OF ORFICE (CM)
| |NCl Gy4621, N°6 A 0870
| IN°2 ©1868, N°777-060

N°3 X20, N°8 G-0442
N°4 4049, N°9 D 0335
25 © 0942, NO V 0256

LOG OF DISCHARGE IN CUBIC FEET PER HOUR.

LOG OF PRESSURE IN CMS OF WATER

Particulars of the diameters of the various disks are given
in the diagram and are tabulated in Table II. below. In the
figure, the logarithms of the discharge are plotted as ordinate
against the logarithms of the corresponding pressures
as abscissee, all volumes being reduced to 0° C. and 760 mm.
pressure, dry.

It is clearly seen from fig. 4 that the experimental results
may be represented algebraically with considerable accuracy
by a linear relation between the logarithms of the pressures
and discharges respectively. The lines drawn in fig. 4 are
the “ best fitting ” straight lines which can be drawn through

g Jet.

SUN

of Air by the Ts

‘ainment o

ces, and Entrai

tf

Or

TABLE I,

Orifice No. 5. Thickness of Disk, 0:0229 cm.

Diameter of Orifice, 0:0942 em.

=. b- .) {fine ana oo Se ee eee

| Corrected
as discharge | Vol. of air
Temperature. Tal Tinie f through Anemometer. | entrained | Vol. of air
Pressure (0° C.) A Ee dischatee orifice. : by jet. entrained
a _Barometer.| marks on of ont Deflexi Nat ; by rere
Sere! | | (inches) bottle | measured | P®*20" | Galyano- ; emi wetness eT [eer ee
(cms. of | | anitaigtedel Moclanne dry, meter | Balance | Bridge | right—left., measured | issuing
water) | Bottle, | Water | ( - : (secs.) "| measured | 974 | tesistance.| current. (mm, ) at 0° C. | through
| "| jacket. | rade peo at 0° C, (chms) (ohms (amp.) and orifice.
and 760 mm.)
| | 760 mm.)
0°320 | a7 3), 12%) 30°45 2361 661 0435 10 1014 1:000 (2 1°56 3°59
0502 | 12:5 | 13-0 542 0533 10 7 sj 110 1:98 371
0°850 130 | 130 | All 0701 10 A o 258 2°81 4:01
1348 | 13:3 13:0 | 325 0:886 10 % ” 394 3°80 4:29
1°664 | 15°5 13:0 | 295 0974 10 ‘A 7" 466 4:46 4:58
P80: a1 13° 13°5 275 1:043 6 5 ” 314 5°30 5°08
265 | 130 | 13:5 239 1199 6 7 A 379 6°42 5°35}
JOD. \auleo | 130 201 1-428 6 %9 ” 448 8:99 6:28
£88. i isp |; 125 |. 30:46 175 1624 6 ” ” 303 =| «11:20 6°90
608 | 137 | 125 160°2 1771 2 1028 1:200 335 13°58 7:67
9°58 | ae 128°2 2°22] 2 1025 5 382 18:46 8°31
1}*44 | 12°5 | 4416 2188 2°432 2 ” ” 400 20°64 8:49
15°10 | ha 10:0) | 194°5 2°746 2 ai | 5
182 | 145 | 130 | 1776 3°015 2 sia hoe. as
204 | 1 130: | 168°2 3'184 2 3; )
200) 445), 130 | 30°44 153°0 3518 2 i | ‘5 |
|

976 Dr. Thomas on Discharge of Air through Small
the several points. Hach has been drawn through the

ey)

“‘centre of gravity” — , = of the respective observations,
1

at an inclination @ to the axis of logarithms of pressure given
by
tan ed =), ee)
2(a—%)?

where

Ra eae = a0,

«x and y being the logarithmic ee experimentally
determined and m the number ofobservations.

It follows that the dependence of the discharge Q (measured
at 0° C. and 760 mm. pressure) upon the excess pressure eé
can, within the limits of pressure employed in the present
series of experiments, be represented by a relation of the

form
QaNe. | a

The respective values of A’ and «@ are set out later in
Table II.

It is cf interest here to consider the relation of this
equation to that deduced for the relation of discharge to
pressure on the assumption that the discharge occurs under
adiabatic conditions.

Lamb * gives for the mass discharge under these con-
ditions the formula

ytl
2 \12 n\y )te, ie
nas’ = (7) nly GS) Sy ay

where p, and po are the respective pressures outside and
inside the vessel from which the discharge occurs. pp and ¢%
are respectively the density of the gas and the velocity .of
sound inside the vessel, S’ is the area of the vena contracta,
y the ratio of the <pecitic heats, and gq, the velocity outside
the vessel.

Writing p=pite, where e¢ is the excess pressure inside
the discharge vessel, this expression becomes after some
little algebraic reduction, assuming the expansion to take

place under adiabatic conditions, and the value of a to

be small, 1

2. Ae? y—le €. Dy + 32a
pis’ = $'(-) CoPo ( a ee, (1- a ie )] : (iil. )

* ‘ Hydrodynamics, 1906, p, 28.

Orifices, and Entrainment of Air by the Issuing Jet. 977

which may be further transformed ; and we obtain, finally,

. | 1 ¢\?
S=S' (2ep))'7{1————) . . . Civ)
MpiS = S'(2¢€p1) ( Dey Dy (iv.)
Now it is well known that experimental results are not
correctly represented by the assumption of the existence
of adiabatic conditions during the discharge of the gas
through an orifice in a thin plate*. ‘The procedure
followed by Buckingham and Hdwards is to modify the
adiabatic relation by the introduction of corrections, taking
into account effects due to viscosity, heat conduction, and
turbulence, such disturbing effects being regarded as re-
latively small. We propose to follow a somewhat similar
procedure. The adiabatic relation (iv.) failed to represent
the experimental results obtained in the present series within
the limits of experimental error. The expression S'(2ep,)!/?
is commonly employed for calculating the approximate mass
discharge through orifices in thin plates. The expression

1/2

= =| _ in which * is small, is a small correcting
ay Pi Pr

factor. By a suitable slight modification of this factor,

a formula may be obtained which represents the present

results within the limits of experimental errors.

We take T
! ‘ ne
aR te S'(2ep:"(1- ere

where the value of « is to be determined from the experi-
mental results. We have from (v.),

d logy, M me ang RE:

d logy € is 2 2p
assuming — to be small and §' to be independent of

the excess pressure. Identifying the left-hand side of this
equation with the “ best”? value of @ in (i.) as determined
from the experimental results, we have a=h—T The
1
appropriate value of « is to be determined from a con-
sideration of the ‘best fitting ” value of ¢ in this relation.
The observations being approximately uniformly spaced
through the range of excess pressures employed, and the

* See,e.g., Buckingham & Edwards, Sci. Papers, Bureau of Standards,
vol. xv. p. 599 (1919-20).

+ M being the mass discharge, is evidently proportional to the volume
discharge measured at 0° C. and 760 mm, pressure,

978 Dr. Thomas on Discharge of Air through Small

relation between a and ¢ being linear, it follows that
the best value of ¢ that can be employed in this relation
is approximately half the maximum excess pressure. Taking
e=12°5 cm. of water, we find «=166(0°500—2). :

From (v.), we have the volume discharge V,' measured
at 0°C. and 760 mm. pressure given by

Vv td S! (Zep)? Hee
Po P

where py is the density of air at 0° C. and 760 mm.,

pi representing the density under atmospheric conditions

during the experiments. 8’ was calculated by means of

51

fe) ?

LOGARITHM OF A’

-|-3 -f-0; -07
LOGARITHM OF DIAMETER OF ORIFICE

this formula, employing the value of the discharge corre-
sponding to an excess pressure of 12°5 cm. of water, and
the coefficient of contraction of the jet was calculated there-
from. The results are set out in Table II. herewith.

The relation of A’ in the empirical formula A’e* for
the discharge given in the fourth column of Table II.
to the diameter d of the orifice is shown in fig. 4a, in
which the logarithms of A! and of d are plotted as

979

g Jet.

‘

SSULN

y the I.

b

wr

t of A

Orifices, and Entrainmen

bo

or

eS) G09) Sy eS>

10

Diameter.
(em.)

01621
01368
01201
0°1049
00942
0:0870
00607
0°0442
0°0335
0:0256

002064

0:01469

0°01132

0008644
0006971
0005944
(002894
0:001534
0000881
0000514

)
|
|

TaBueE II.
Thickness of Disks, 0°0229.

—SS

Empirical Forinula
for discharge
in ¢.c, per sec.

Corresponding

best value of « Winlineroninl

measured at 0° tS : (sq. em.)
and 760 mm. S’(2ep,)? fre 2
A’e™, Po. ( pia

NG Gre) 36) 0:01327
|

11:79 6482 3:0 | 0:0097 1

DiB6) wer qo 27 Ee 000772
‘ Pe

7-12 6486 | 2:3 = 0:00588

5:94 ¢0'480 3:3 0:00488

AGS oy 25 J 000407

DBA eee = 0:8 0002113

epee! Om ans Br) ‘! 0-001101

O786e0°= Re | 0:000628

Ogee. ~106 | 0:000320

Coefficient of
Contraction.
S'
3°

0-679
0-701
0-684 J

0 730
0°718
0713
0622

980 Dr. Thomas on Discharge of Air through Small

ordinates and abscissee respectively. The relation between
the logarithms in the case of the larger orifices is clearly
linear, the points corresponding to orifices nos. 1-7 lying
very accurately on the best-fitting line given by

Xe) O00) eee

For orifices nos. 8-10, the value of the index increases
as the diameter of the orifice diminishes. This result,
together with the very approximate constancy of the
index a for disks nos. 1-6 shown in column 4 of the
table, indicates that these disks only of the series employed
can be regarded as thin disks. Disks 8, 9, and 10 are
to be regarded rather as orifices in thick disks, disk no. 7
affording a transition from one class to the other. The
results indicate that an orificed disk is to be regarded as
thin if the diameter of the orifice is not less than about
three times the thickness of the plate. In the recent paper
of Buckingham and Edwards * on the efflux of gases through
small orifices, the diameters of the orifices employed were
considerably smaller than any employed in the present
work, and moreover, were in no case greater than 1°7 times
the thickness of the disk, being in three cases out of four
very much less than this. Such disks would, in the light
of the present work, be characterized as thick disks.

The average value of the contraction coefficient in the
present work was found to be 0°674. For orifices of
diameter d cm. in then disks (nos. 1-6) of thickness 0°0229 cm.,
the discharge of air is given by the formule

Vol = BOQ dre e0eee0008 aoe
Se Nae 1/2
= 0674s CO (1-2-9 2 , eae

the symbols having the significance given in the text.

Of these, formula (vi.) is of a type which has some
physical justification. In addition, (vil.) is correct from
the point of view of physical dimensions.

For purposes of comparison between orifices in thin and
thick ‘* disks,” experiments were in like manner carried out
with very carefully prepared short channels bored in brass.
Particulars of the various channels are given in Table III.,
together with empirical formule, the detailed results being
plotted in fig. 5 and the empirical formule deduced as
already explained.

* Scientific Papers, Bureau of Standards, vol. xv. p. 584 (1919-20).

LOG OF DISCHARGE IN CUBIC FEET PER HOUR

Orifices, and Entrainment of Air by the Issuing Jet.

TABLE III.

ISL

Length.
(em.)

channel,
(cm.,)

0:0780
0:0612
0:0544
0:0471

0:0421 \

0°0421
00424
0:0420

0:0442

\

|

|
]

|
|

Empirical Formula

| for discharge (A’e®
Diameter of | ee)

(measured in
c.c. per sec. at 0° C,
and 76 mm.)
at excess pressure
€ cms, of water,

3:39 0919
2-09 "880
1°65 go 085
1:05 @264
0-783 ¢0°628
0:509 6721
0:479 60783
1:19 0509

1:32 60007

982. Dr. Thomas on Discharge of Air through Small
For the channels 1 A-5 A of constant length 0:2789 cm.,

the value of A’ in the empirical formula for the discharge A’ e
is given in terms of the diameter d by the linear relation
A'=73d — 2°33. The gradual increase of the index @ in the
case of channels 1 A-5 A as the diameter of the channei
decreases is seen from fig. 5. It will be noticed that at a
pressure a little greater than the maximum employed in the
construction of fig. 5, the straight lines shown corresponding
to the orifices 11 and 5A of the same diameter intersect
(actually the pressure was found to be 33°2 cm. of water).
For pressures greater than this, the discharge through the
longer channel is actually greater than that through the
shorter channel of the same diameter. A similar phenomenon
is represented by the points P and Q, in which the straight
lines corresponding to channels 8 and 11 in disks of thick-
ness ('0229 cut the line corresponding to channel 4a, of
slightly larger diameter and of about 12 times the length.

The phenomenon is clearly attributable to the difference in
the form of the issuing jet in the respective cases of discharge
through a channel in a thin or thick plate, the existence of
the vena contracta in the former case reducing the effective
area of the discharge and tending to counterbalance the effect
of the greater length of the channel in the latter. Attention
has been directed to the existence of a critical length of
channel, such that the discharge through an orifice of given
size Is a Maximum, In a recent publication of the Burean of

Standards, Washington *.

(b) Air Entrainment.—In fig. 6, the respective total
volumes of air (reduced to 0° C. and 760 mm. pressure, dry)
entrained by the issuing jet in the case of each of the jets
nos. 1-10, are plotted as ordinates against the respective
pressures as abscissee. The several curves are numbered
according to the number of the corresponding orifice. Curves
1-7 represent the normal manner in which the volume of
air entrained by a jet of air issuing from a given orifice
increases as the pressure at the orifice is increased. As
the pressure is increased, an initial approximately linear
increase of the total volume entrained is followed by a
subsequent increase at a continuously decreasing rate, the
curve becoming concave to the axis of pressure. ‘This latter
is to be anticipated, as the viscous and frictional drag upon
the stream of entrained air increases as the velocity of the
stream increases.

The curves corresponding to orifices nos. 8, 9, and 10,

* Technologic Papers No. 193, p. 17 (Sept. 6, 1921).

Orifices, and Entrainment of Air by the Issuing Jet. 983

which, as has been shown, must be considered essentially
orifices in thick disks, exhibit points of inflexion at R, Q,
and P respectively corresponding to pressures of 3, 5, and
12 em. of water. During the course of the experiments
with orifices 9 and 10, anomalous behaviour of the jet
in the region of these respective pressures was very readily
detected, as the galvanometer reading became very unsteady
unless the jet was protected very carefully from outside
disturbances. Steady deflexions could be obtained by.
reducing outside atmospheric disturbances to a minimum.

24

(CUB FT. PER HR.)

VOLUME OF AIR ENTRAINED BY JET

10 15
PRESSURE (CMS. OF WATER)

For values of the pressure at the orifice, below the critical
value, the galvanometer deflexions were exceedingly steady.
The galvanometer deflexion was unsteady until the value of
the pressure was increased to a definite value, above the
eritical value, and thereafter the deflexions were again
extremely steady. The phenomenon indicates the existence
of a small range of pressures over which the issuing jet
is essentially unstable, the efficiency of the jet as regards
air entrainment being very considerably affected by slight
disturbances in the surrounding atmosphere.

The nature of the instability of the jet is most clearly

984 Dr. Thomas on Discharge of Air through Small

seen from a consideration of fig. 7, in which the volumes of
air entrained per unit volume of air in the issuing jet are
plotted as ordinates against the pressures as abscisse, in the
case of orifices nos. 1-10. Curves 1 to 8 inclusive indicate
the normal behaviour of the jet as regards air-entrainment.
It is seen that a diminution in the size of the orifice is
accompanied in each of these cases by a greater air-entrain-
ment per unit volume of air in the jet. This point is of

Ib

wn a .

VOLUMES OF AIR ENTRAINED PER UNIT VOLUME OF AIR ISSUING FROM JET.

56.

i=)

PRESSURE (CMS. OF WATER).

importance in the design of gas-burners operating on the
Bunsen principle, as it is to be anticipated that a greater
degree of primary aeration of the issuing Jet of gas can be
effected by diminishing the size of the ejector orifice, and
employing a multiple-orificed ejector in place of an ejector
provided with a single orifice of greater area. In the
practical application of this result, however, the several

Orijices, and Entrainment of Air by the Issuing Jet. 985

orifices must be so disposed that the air-entraining power
of each individual jet is not reduced by mutual interference
of the jets.

It will be clear from the figure that there is a limit of size
below which the orifice cannot be reduced without the intro-
duction of disturbing factors adversely affecting the greater
degree of air-entrainment normally accompanying reduction
of the orifice. This is first seen in the region of F in curve 9,
where the degree of air-entrainment is somewhat less than
its anticipated value, as shown by the broken line. From
curves 9 and 10 it is seen that over the range of pressures
up to about 16 cm. of water, the degree of air-entrainment
effected by the jet issuing from the small orifice (no. 10) is
very much less than that effected by the jet issuing from the
larger orifice (no. 9).

A similar phenomenon to that described has been observed
in the case of coal-gas issuing from a fine orifice. In such
a case a flame burning at an orifice under conditions corre-
sponding to those shown at Fin fig. 7 would be very sensitive
to slight changes of pressure, the degree of air-entrainment
varying considerably with a slight increase of pressure. The
phenomenon probably explains, at least in part, the existence
of low-pressure sensitive flames, to which attention has been
recently directed, and which were studied in considerable
detail by Chichester Bell *.

With a view to ascertaining the cause of the apparently
anomalous bel aviour of jets issuing from orifices 9 and 10,
similar experiments on air-entrainment were performed with
air jets issuing from the comparatively much longer channels
nos. 1 A-5 A, 5B, and 5C, particulars of which have been
given in fig. 5 and Table III. The curves corresponding to
those in fig. 6, in which the total volumes of air-entrainment
were plotted as ordinate against the pressures as abscissa
showed well-marked points of inflexion in the cases of
channels 3 A,4A,5A,5 B, and 5 C, these points corresponding
to pressures respectively equal to 2°5, 5:0, 7°5, 11:0, and
13 cm. of water.

The curves for channels nos. 1 A-5 A showing the degree
of air-entrainment per unit volume of air in the jet are
shown in fig. 8. The curves generally resemble those given
in fig. 7. One distinction is of importance. A comparison
of curves 3A and 4A with curves 9 and 10 in fig. 7 indicates

* See, e. g., ‘ Nature,’ vol. cviii. p. 532 (1921). Phil. Trans. Part 2,
pp. 383-422 (1886).
Phul. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 38

386 Dr. Thomas on Discharge of Air through Small

that the effect referred to is not so pronounced in the case of
jets issuing from the longer channels. The same is seen from
curves 4A and 5A in fig. 8.

The existence of turbulence in the jet naturally suggests
itself as the cause of the phenomenon referred to. It is clear
that as the effect occurs only over a limited range of pressures,
and is not present at the higher pressurés employed in the
present series of experiments, and as moreover the mean

VOLUMES OF AIR ENTRAINED PER UNIT VOLUME OF AIR ISSUING FROM JET

fo) Shan lo esis ee 25
PRESSURE (CMS OF WATER).

velocity in the jet for this range of pressures is considerably,
below the critical velocity at which turbulent flow occurs,
the turbulence referred to is impressed upon the jet on
entering the orifice. It is therefore to be anticipated that
the effect would be less pronounced in the case of long
channels than in the case of small channels, as in the former
case the unstable turbulence initiaily impressed upon the
stream would die away to a greater extent than in the
latter case. This, as has just been stated, was found to be

Orijices, and Entrainment of Air by the Issuing Jet. 987

the case and is clearly brought out in fig. 9, which gives
the results obtained with channels all of the same diameter
but of different lengths. The effect is much less marked
in the case of the longer channel 5C than in the case of
the shorter channel 5 B or 5A.

The absence of the effect at the lower pressures is
attributable to motion of the gas at such low pressures being

IS - :
is ORIFICE ssc

| (cm)
3 5A 0:2789 0 :

NY

VOLUMES OF AIR ENTRAINED PER UNIT VOLUME OF AIR ISSUING FROM VET

PRESSURE (CMS OF WATER)

in general so slow that little initial unstable turbulence is
impressed upon the stream, and any such turbulence, if
produced, is damped out before emergence of the jet from
the orifice. At the higher range pressures at which the
effect is absent, the passage of the air into the channel
approximates to stream-line motion, the air within the
reservoir immediately adjacent to the disk being practically
stagnant, so that little turbulence is produced by a sudden

882

988 Dr. J. R. Partington on the

change of direction of motion of the air entering the channel
or orifice.

The work detailed above was carried out in the Physical
Laboratory of the South Metropolitan Gas Company ; and
the author desires to express his sincere thanks to Dr. Charles
Carpenter, C.B.H., for the provision of facilities for carrying
out the investigations, and for his continued interest in the
work.

709 Old Kent Road, S.E. 15.
14 Aug., 1922.

LXXXVII. Zhe Chemical Oonstants of some Diatomic Gases.
By J. R. Partineton, D.Sc.*

ile fee Theorem of Nernst supplements the two Laws of

- # Classical Thermodynamics by assigning to the
constant of integration, I, of the Reaction Isochore of Van’t
Hoff (1) a value otherwise determinable only by experiment
with the particular system of materials under consideration :

dlog K/dU=Q,/RI? .. . eel
loo, K= —QURI+L, . > ee
According to Nernst, l=2ni, . 1. 9) es

i. e. the integration constant is represented as the algebraic
sum of the products of the numbers of interacting molecules
(n) by the chemical constants (1) of the various pure gaseous
materials taking part in the reaction.

2. The value of 7 is the constant of integration of the

Clapeyron-Clausius equation, simplified by the usual approxi-
mations

Agq=RIP.dlog pd, . <2 ees)
where Xp is the latent heat of vaporization per gram molecule
at the temperature T. The value of A, as a function of
temperature is given by Kirchhoft’s equation

My=ry+ | (Coal ee | (SS

where C, is the molecular heat of the vapour at constant
pressure, and ¢ that of the condensed phase. Thence

r etal
loger=— Rt RI, al (C,—c)dT +7. . (6}

* Communicated by the Author.

Chemical Constants of some Diatomic Gases. 989

According to Langen*, the value of C, may be split into a
translational term independent of T and equal to 5/2.R for
a diatomic gas, and a rotational term, C,,, dependent on T’.
Equation (6) is then written

r 5 LW Geecer bot Coe Saag play
log. p= — RT + slog. T + zl. | (Cp-—c)dT +2, (6a)
log p (atm.) = — a +2°5 log T

Um tai he Fare
art al (Cp —e)dT+C. . (68)

The constants i, C then supply the terms to (3), and
permit the integration of (1). The values of © have been
calculated from a consideration of the experimental data by
Langen for several gases. Values of C had been calculated
by Nernst f by a somewhat arbitrary method, and although
his results provide a satisfactory approximation for particular
problems, they are not in accordance with modern theories
of specific heats, as was pointed out by the author in 1913 f.

3. It will be noted that the equations so far given do not
provide a complete solution of the problem of predicting,
from purely thermal magnitudes (heats of formation and
specific heats) and universal constants, the behaviour of
materials when placed together under specified conditions in
the absence of passive resistances §. The final step was taken
in the case of monatomic substances by Sackur and by
Tetrode ||, who were able to calculate the value of 7 in terms
of universal constants. If the temperature is reduced to
such an extent that the atomic heat of the condensed phase
becomes negligible in comparison with unity, then (6)
becomes

Xo

logep= — pn +2'5 loge T +2, pi COE)

since C,, is zero for a monatomic substance, and the value

* A. Langen, Z. Elektrochem. xxv. p. 25 (1919). .

+ ‘Recent Applications of Thermodynamics to Chemistry,’ 1907.
Theoretische Chenne, 8-10 Aufi., 1921, p. 799.

{ ‘ Thermodynamics,’ 1913, p. 496.

§ It may be that the influence of passive resistances could be included
by introducing a ‘‘heat of activation,” in the sense understood by Perrin,
Trautz, W. C. M. Lewis, and others, in connexion with dA, but this
problem is not considered in the present communication. —_- ”

|| Sackur, Ann. d. Physik, xl. p. 67 (1913). Tetrode, bed. xxxyiii.
», 434, xxxix. p. 255 (1912). Stern, Z. Elektrochem. xxv. p. 66 (1919).

990 Dr. J. R. Partington on the

of 7 is then given, according to these authors, by

3/2 75/2
i=log.@™™),* oe 4

where m is the mass of the atom, kis Boltzmann’s constant
(R/No, where N, is Avogadro’s constant), and / is Planck’s
constant. With numerical values (see § 7 below), and p in
_atm., this gives

J=—1-58941-5lopM, . . Ae ie

where M is the atomic weight, referred to the same standards
as No. :

4. The object of the present communication is the extension
of this line of investigation to a hypothetical diatomic
molecule which, 1t is believed, represents with some approxi-
mation the structure of a particular group of gases*. A
general solution would obviously enable us to predict the
results of all types of gaseous reactions without recourse to
experiment, and would provide a long-sought solution to a
fundamental problem of chemical affinity.

The method of calculation adopted is that of generalized
statistical mechanics f. An isolated system, possessing an
energy e, and composed of a large number of molecules
which exert no forces on one another, is assumed to be
definable in terms of a set of generalized coordinates
G1) Ya) -»-») and a corresponding set of generalized momenta
1, Pa --. related by the first canonical equation of Hamilton,

Oh = 06/071 . : Me Peo OS (8)
According to the Quantum Theory, |

H=((.. dq, dgo...dp, dpe ...,

which is independent of time and of the particular choice of
coordinates, has a definite value for each element of the
generalized space (Llementargebiet). In the case of an ideal
gas, the mean energy € coincides with the energy € in any
point of the element, and

G

Sec al) ¢ Bay

* Partington, Trans. Faraday Soc. 1922.

| J. W. Gibbs, ‘Elementary Principles of Statistical Mechanics.’
Planck, Warmestrahlung, 4 Aufl. 1921. Jeans, ‘Dynamical Theory of
Gases,’

Chemical Constants of some Diatomic Gases. 991

5. In the first case we suppose the gas molecule to consist
of two identical atoms rigidly ‘attached to each other ata
fixed distance. In addition to the coordinates of the centre
of gravity we require two angles, @ and ¢, defining the
direction of the molecular axis. Rotation about this axis is,
as usual, ignored. We then have .

N=2®3 H=Y3 B=2Z3 U=F3 B=:

Pi=ML 3 Pp=MY 3 pz=MzZ; py=MK7O; ps =mK?’ sin’ Od,
where K is the radius of gyration.

The energy of this molecule is given by

a pop +e) 4 js (62+ sin? 062) +e, . (10)

e= 5 (i

where € is the energy of i molecule at rest in the
generalized space. Thus

aoe 5K4 (it cal eens
Bema” \\ Mena nld: dda dbdde
(1)
The limits of the multiple integral are the boundaries of
the element of volume for a, y, and z; the angles 0 and 27

for @; the angles 0 and w/2 for ¢, and all the velocities
from —2% to +0. Hence, if V is the total volume,

eae Ke
Pipe a sca a
Le a
The free energy, w, of the system is then given by *

Ap — —kTN { log. Le - i aaa log. Dash (12)

where N is the iotal number of molecules in the volume V,
say N=N,, the number of molecules in one gram-molecule.
In the above case

Ene
(QarmkT)*?e **,

= KING log (QrmkT)? + Noe. . (13)

For equilibrium between the vapour and the condensed
phase

Pl Apeap( Vee Vi ee 8) et CQ

in which dashed symbols refer to the condensate. Substi-

tuting in (14) from (13), and neglecting small terms, we find
2Qar KK? Ve 5/2 Chic ap" INg erg f
k log. 873 —(2arkmT)* Sr ane ie.

* Planck, loc. cit. p. 210.

992 Dr. J. R. Partington on the

But e9—wW'/No=rAy’, the latent heat of vaporization per
molecule at T=0; hence, with the substitutions pV =N kT
= 5 oe ; and Ay>=NoAy, we find

K?(Qark)"”
CSP am

where H=A’ in the peueeaieed space of five dimensions *.

log. p= — ah 5 loge’ Uae 2 9 Oke M+ log,

6. For a diatomic gas of the type considered in § 5,
C,=7/2, and hence

log p(atm.) = mee ee e717 +C, log T+2°5 log M

+2log K+12°730. . (15a)

(hee 137 x 10-% 5 k= 655 X 10-7 = No = Onn ame
iatms—1013250 abe tonite. Gee Millikan, Phil. Mag. July
TEA

The equation representing the vapour-pressure of a
diatomic substance at such low temperatures that the energy
of the condensate is negligible in comparison with that of
the vapour (which will generally occur before the gas begins
to lose its diatomic character, except in the case of molecules
of very small mass and diameter, such as hydrogen) is

los p= we + Oplog, T +7,

or log p(atm.) = be71t t+ Cy log T+ C.

res i
By comparison of (15) with these we find

ei.

(AN?2)° 2. 7G
or C=2°5 log M+ 2 log K+ 12-730.

7. In the case considered, K?=7?, where 7” is the radius of
the molecule. For oxygen, r=1'8x10-? cm.t, M=32,
hence Co,=1°001. The four values given by Langen (loc.
cit.) range from 0-539 to 1:021, the mean being 0°829. In
the case of nitrogen, r=1:9X 1078 em., M=28, hence
Cy,=0°904. Langen gives only one value for nitrogen,
—0-05, from which one can perhaps only conclude that it is
somewhat less than the value for oxygen. The case of

i=2°5 log. M+2 log. K + log, ——{—

* The various methods of quantizing rotations are kept in mind.
+ Jeans, ‘ Dynamical Theory of Gases,’ 2nd edit. p. 341 ; all values of
7 from this.

Chemical Constants of some Diatomic Gases. 993

hydrogen is probably not satisfactorily covered by formula
(16), since the value of Cy becomes appreciably reduced
within a region of temperature before the value of c for the
condensate becomes negligibly small. Hydrogen, therefore,
should behave in a manner intermediate between that of a
diatomic gas (equation 16) and that of a monatomic gas
(equation 7). Nernst * has applied (7) to hydrogen, and
after the application of small corrections, has found
Cy, = — 1°23; whereas Langen, by formula (6), finds —3°767.
Hguation (16) gives Cy,=—2°255 (M=2°016; r=1°34
x60 cm.).

8. In the case of gas molecules composed of two different
atoms rigidly bound together, the calculation is similar,
except that the angle ¢ is now, since the molecule is no
longer symmetrical, to be taken between the limits 0 and 7.
In the cases to be considered it is still a sufficient approxi-
mation to take K?=7r*. This case is, therefore, covered by
the addition of log2 to (16). In the case of carbon
monoxide, r=1°90 x 107°, M=28; hence Coeg=1°205, whilst
Langen gives —0°04. [for nitric oxide, »=1°86 x 107%,
M = 30; hence Ono = 1:268, whilst Langen finds 0 92.
Perhaps all that can be definitely said of Langen’s values
for these gases is that they are somewhat larger than the
vaiue for nitrogen, and it is noteworthy that Nernst’s
empirical values for the compound gases are larger than
those for the elementary gases : 3°5 for CO and 3°5 for NO,
as compared with 2°8 for O, and 2°6 for Nf.

It is believed that the above method of calculation gives
results which are in all cases of the right order, and that
the values deduced by other methods are still so divergent
that a more searching comparison is not at present possible.
It is hoped that the method will shortly be extended to gases
with more complex molecules, in which internal motions also
occur. If these are considered as small vibrations, their
energy can be represented as the sum of squared terms in
the coordinates, and the above method can be applied to
them without difficulty.

East London College,
University of London.

t Grundlagen des Neuen Wéarmesatzes, 1918, p. 150. There are a
few misprints in this section, e.g. in (120) —0°5T should be —0‘d/nT,
and (2mm)? should be (24m)*’.

* Theoretische Chemie, p. 799 (1921).

904. |

LXXXVIII. The Motion of Electrons in Carbon Diowide.
By M. F. SxinKER, Rhodes Scholar, Exeter College, Oxford *.

1 some recent publications of the Philosophical Magazine,

Prof. J. S. Townsend and Mr. V. A. Bailey t describe
their experiments on the motion of electrons in hydrogen,
nitrogen, oxygen, and argon.

In this paper I wish to give the results of similar experi-
ments with carbon dioxide and to compare the results.

The apparatus used had the same dimensions and was
similar to the one described in the above papers. The
electrode E,, in fig. 1{f, however, was not exactly under
the slit in B, but was 0°6 millimetre to the right. In order
to find the velocity of agitation u it is necessary to find the
normal distribution-curve when the centre of the stream is
0°6 millimetre from the centre of the electrode H,. In this
case R, the ratio of the current received by the central
electrode to the total current, is given by the curve in fig. 2,
Z being the electric force in volts per centimetre. |

The curve differs slightly from the curve which corre-
sponds to the case in which the centre of the stream
coincides with the centre of the electrode Hp.

* Communicated by Prof. J. S. Townsend, F.R.S.

+ Phil. Mag. vol. xlii. Dec. 1921, and vol. xliii. March 1922.
{ Fig. 1, vol. xli. p. 875.

Motion of Electrons in Carbon Mowide. 995

The calculation of this curve will be explained in a future
paper by Prof. J. 8. Townsend and Mr. V. A. Bailey.

In order to find the velocity of the electrons in the direc-
tion of the electric force, two different magnetic forces may
be used. With this eccentricity of 0°6 millimetre the stream
may be deflected 1:9 millimetres to the left or 3:1 millimetres
tothe right. In these experiments all deflexions were to the
right, as the determinations with the larger deflexions are
the more accurate.

The results of the experiment are given in Table I., where

p is the pressure in millimetres of mercury,

k the factor by which the kinetic energy of the electron
exceeds the kinetic energy of a molecule of a gas at
is’ ©...

W the velocity of the electrons in the direction of the
electric force in centimetres per second.

TABLE I.
p Z Z/p k. WwW x107°.
20°23 416 0:206 1:19 _
9°82 2:08 0:222 1:283 1:18
9°82 4:16 0:444 1:29 2°41
5:06 2:08 0°411 1-277 2°45
9°82 8:33 0°888 — 4°91
506i 4°16 0:822 1°36 4°55
2°49 2°08 0°835 1:36 4°67
5°06 8°33 1:647 ~ 9:42
2°49 4°16 ays 1°72 981
1:26 2:08 1:66 1°64 9°47
5:06 16°67 3°30 -— 22°4
2°49 8°33 3°32 2°88 23'8
1°26 4°16 3°29 2°79 23°6
62 2°08 3°32 2°89 24°5
2°49 166 6°64 221 82°4
1:26 8°33 6°59 21-1 82°4
63 416 6°64 23°1 81°4
2°49 go'oo 13°4 60°6 118
1°26 16°67 132 60°1 124
1:26 33°33 26°4 81°3 142
63 16°67 26°6 91 150
63 33°33 53°2 147 202

ee OO — ree

996 : Mr. M. F. Skinker on the
§7.The values of W and £ are plotted against 2 in figs. 3-6,

together with the curves for hydrogen and nitrogen.
Fig. 3.

loa

cal

mapa)

997

Carbon Dioxide.

Motion of Electrons in

998 Motion of Electrons in Carbon Dioxide.

In the following table, u is the velocity of agitation of
electrons in centimetres per second, / the mean free path
of the electron in centimetres, p the pressure of the gas in
millimetres of mercury, and ® the proportion of energy of
the electron lost in collision with a molecule.

The formule connecting u, /, and X with the quantities

k and W being :—
u=1:15x10'x Wk,

We oe
1 Paste
WX = 2°46 x ae
TABLE II.
Zipaterce Neg: wxi07® uxi0s®. mxied. Meco.
50 139 19°5- 185°7 3°67 506
4() 117:5 17°75 1248 3:64 > | Age
30 96 159 1128 4:15 487
20 75 13°8 99:5 4°76 472
10 47 10°8 73:9 591 ~—- 460
6-5 20°7 Hise a 524 4:36 543
5:0 9 5:0 34:5 2:39 516
4 4:8 3:2 25:2 1:40 397
3 2:3 2-0 17°5 ‘809 321
9 18 2 Fs 15°4 ‘630 144
1 15 “5D 141 538 37:4
O°5 1:3 ‘25 13°1 454 8:95
0:25 12 12 12°6 ‘419 2°34

In order to determine whether or not there were any ions |
in the stream, the magnetic force was increased, to see if the
stream were completely deflected off E, and H,*. This was
found to be possible when using a magnetic force which was
comparatively small and which would not have been sufficient
to deflect ions from the plates. Also the quantity & and the
velocity W were found to remain constant with different

Z
values of Z and p when — was constant ; these results show
that there could not have been any permanent ions formed
in the gas.
* Fig. 1, vol. xlii. p. 875.

A Wide Angle Lens for Cloud Recording. 999

With values of : greater than 30 the loss of energy in a

collision is comparatively large, so that the velocity of agita-
tion is less than seven times W, and in these cases the formula
for W in terms of / and u is not so accurate as in the cases

r

where a is less than 30 and wu comparatively large.
p

"G

a 7,
Table II. shows for the higher values of 2 that the mean

free path increases with decrease of u, but for the lower
values it decreases with decrease of velocity of agitation.
In the other gases the mean free path increases for the
smaller values of the velocity of agitation.

The values of X% show that with this gas there is a
remarkable increase in the loss of energy of an electron in
a collision for comparatively small increases in the velocity
of agitation from the values 13 x 10’ centimetres to 15 x 107
centimetres per second.

Electrical Laboratory, Oxford,
July 1922.

LXXXIX. A Wide Angle Lens for Cioud Recording. By
W.N. Bonn, W.Sc.(Lond.), A.R.C.S., AJdnst.P., Lecturer
in Physics, University College, Reading*.

[Plate VIL]

HIS paper consists of a short description of a lens that

might be used for obtaining a photographic record of

the clouds visible at a meteorological station at definite

times, or for similar purposes, such as recording lightning
flushes.

The special feature of the lens is that its field of view
embraces a complete hemisphere; so that if the lens be
arranged to face vertically upwards, all the clouds visible
at the station at any one time can be recorded photo-
graphically on a single flat plate or film. The resultant
photograph (see Pl. VII.) is circular, any clouds at the
zenith being reproduced in the centre of the circle, and any
near the horizon appearing near the edge of the circle.
Such apparatus might, of course, be used at two stations
simultaneously to obtain the altitudes of the clouds.

* Communieated by the Author.

1000 Mr. W. N. Bond on a

The lens in its original form consists of a glass hemi-
sphere L (fig. 1) of radius vr. The light is incident on the
plane face, which is covered by a thin screen S, except for
a small circular aperture at the centre. It will be seen
that a ray incident in the plane of the outer face of the
lens will be refracted at the critical angle. Furthermore,
all refracted rays will arrive almost normally at the hemi-
spherical face of the lens, and will subsequently converge
to form an image, which, for objects at infinity, will lie on
part of a sphere III, concentric with the hemispherical face
of the lens and of radius ry/(#—1), where yp is the refrac-
tive index of the lens. The emergent cone of rays will in

the aperture in the screen S is small enough, the whole
image will be sufficiently in focus on a flat plate PP placed
at a distance from the plane face of the lens equal to the
mean distance of the various portions of the true image III
from this plane face.

The screen 8 should be covered on the outer side by a
plane plate of glass G, the whole being cemented together.
This arrangement avoids the finite thickness of the screen 8,
preventing some or all of the light incident at fairly oblique
angles from entering the lens. )

It will be seen that the photographic plate should be
placed at a distance h from the plane face of the lens of
about 2°57 (i.e. rather less than pr/(u—1)). It is easy

Wide Angle Lens for Cloud Recording. 1001

to show that a cloud at an angle @ from the zenith will
appear on the plate PP at a distance w from the centre
of the image, given by

w=hsin 0/ Vu?—sin? 0.

The lens is, of course, not corrected for chromatic aber-
ration, but this might be largely removed by employing a
monochromatic filter, which could be closen so as to facilitate
the photographing of clouds.

The finite size of the image of a distant point source,
due to the plate PP not coinciding with the true image ITT,
ean be shown to be very approximately d/7, where d is the
diameter of the aperture in the screen 8. If the hole in
the screen be of diameter d=7r/20, the aperture of the lens
is roughly f. 50; and the finite size of the image mentioned
above results in a blurring at the zones which are most out
of focus, which is equivalent to an uncertainty in 6 of
about +°.

The illumination of PP becomes less for large values of 0,
but the effect does not sem sufficiently pronounced to be
objectionable in cloud photography, and need not be
considered in detail.

The angle of the field of view of the lens could be
increased yet further by replacing the outer plane plate G
by a plano-convex lens, having its curved face outwards.
This arrangement might be used if it were desired that the
photograph should show the horizon and slightly below it.
An advantage of this arrangement would be that the zone
for which the image is most cramped together would not
be at between 80° and 90° from the zenith, but say from
95° to 105°, thus enabling the record of clouds near the
horizon to approach more nearly the clearness of that of
clouds at the zenith than would be the case if a simple
hemispherical lens were used.

It may be mentioned that the lens gives views similar to
those seen by a fish in water. The apparatus has, however,
probably been reduced to the simplest form advisable, though
a less perfect image could be obtained by placing the photo-
graphic plate in contact with one side of a thick parallel
plate of glass, the other side being covered by a thin screen
pierced by a small hole.

Finally, it will be seen that when using the simple hemi-
spherical lens, or the thick plate of glass just described, only
two constants (viz. h and u) need be known accurately, to
enable a complete network of degrees to be constructed for
use in interpreting the photographs.

Plul. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3T

1002" |

XC. A Problem in Viscosity: The thickness of liquid
jilms formed on solid surfaces under dynamic conditions.
By The Research Staff of the General Electric Company
Litd., London. (Work conducted by F. 8. GoUCcHER and
ie Warpoy+ ;

|Plate VIIL.]
Summary.

(ake problem of determining the thickness of the liquid

layer coating a solid body drawn out of a liquid is
discussed theoretically and practically. It is shown that if
the solid is a flat slab of infinite width, the forces deter-
mining the thickness are those of gravity (g) and viscosity
(n), and that the relation between thickness ‘¢), density (p),

209
| py

Tf the solid is a fine wire of radius 7, surface tension is
dominant and gravity negligible. If y is the surface tension,

the relation must be of the form : =/(®). It is found

and velocity of drawing v% is ? =

empirically that ee is of the form 4°8 in ¢.g.s. units.

These results apply to suspensions if (1) the diameter of
the suspended particles is not greater than ¢t, (2) the effect
of the particles in increasing 7 is taken into account.

There is no evidence of any special cohesion between
solids and liquids wetted by them ocher than that which
prevents slipping at the interface.

In many important industrial processes solid surfaces
are coated with a layer of liquid by drawing them out of a
bath of the liquid. The enamelling of wires or tubes is such
a process, and so in all essentials is painting with a brush.
We are aware of no theory or even complete experimental
investigation directed to determine how the thickness of the
liquid layer produced in such circumstances varies with the
properties of the liquid, the solid surface, the velocity of
drawing, and other possible factors. The experiments
described in this paper show that the matter is surprisingly
simple.

* Communicated by the Director.

A Problem in Viscosity. 1003
I. Theory.

The factors which must be effective appear at once from
fig. 1, which shows a longitudinal section of the solid A
drawn vertically with velocity v, out of the liquid B.
Observation shows at once that at least in some circum-
stances, the thickness of the liquid layer C is constant for a
distance above B very much greater than t.
Consequently it is reasonable to assume that in C the

stream lines are vertical ; and if they are vertical, continuity
requires that the velocity v along any one stream line is
constant. This constant velocity must represent a balance

Fig. 1.

(63)

>

ROSSER RCN
SSSA SANS

SAN

RM SUMO

MMO

between the forces acting on any element of the layer, and
of these forces one must be that due to viscosity of the
liquid and another that due to gravity. If the solid is a
plain slab of infivite breadth perpendicular to the diagram,
it is difficult to see what other force can act; it is possible
that, if it is sufficiently small, some special force of cohesion
between liquid and solid is effective, but we shall’ see that
the facts can be explained adequately without introducing
such a force. If, on the other hand, A is a cylinder, surface
tension may be effective ; for, owing to the curvature of the

ae D2

1004 Research Staff of the G. E. C., London, on

outer surface of UC, this tension will produce an increase of
pressure in C which will not be balanced by any corre-
sponding pressure over the ends of the layer. Accordingly,
the liquid in the layer will be squeezed out of it at the lower
end, and, possibly, at the upper.

The balance of these forces must be such that the outer
layer of C is at rest relatively to the liquid B in order that
the continuity of the liquid surface may be preserved.

This last condition may appear puzzling; for if the solid °
is continually moving upwards carrying the liquid C with it,
it would seem that the outer layer of this liquid must be
moving upwards. What really happens is that the inner
layer, next to the solid, moves upward with the velocity of the
solid ; the other layers nove upwards with a velocity con-
tinually decreasing outwards, the difference between the
velocities of different layers providing viscous forces neces-
sary to counteract gravity or surface tension. If the layer
at distance « from the solid moves upward with velocity v,
it will require a time //v after the drawing starts before a
layer of thickness w is found at a height / above the liquid.
Strictly speaking, it will require an infinite time before the
layer of full thickness, corresponding to v= 0, appears at a
finite height above the surface. “But a consideration of the
numerical values in the equation about to be deduced will
show that the time required fora layer of thickness differing |
inappreciably from « to form at a distance of severai centi-
metres above the surface amounts only to a few seconds.
Accordingly, if we wait a few seconds between starting the
drawing and taking observations the thickness of the liquid
on the surface will be practically equal to that corresponding
oO 0

With these considerations in mind the complete solution
of the case of the infinite slab is easy.

If x be taken horizontal and z vertical, with the origin on
the surface of the slab ; and if p is the density, 7 the viscosity
of the liquid, v its velocity relative to B, we have

d dv |
ae (1 <) py fey . A ho) (1)
with the boundary condition that at «=t, - =O)
a
Hence v= a — tw) +05 52.) 7 ee)

at w=0 if there is no slip v=vp, the velocity with which the

‘

a Problem in Viscosity. 1005

slab is drawn upwards; ¢ must adjust itself so that this
condition is fulfilled, 7. e.

ae
tan/ 2 i LS ae)

It is not easy to measure ¢ accurately while the slab is
moving, and of course, if it is stopped, the conditions are
changed at once. In practice the liquid layer usually sets,
owing to cooling, evaporation, etc., at some little distance
about the liquid surface. In our experiments we have used
a liquid with a melting-point above room temperature, so
that it freezes on the slab a little distance above the surface
of the bath; we can then measure the thickness ¢’ of the
solid film. If in these conditions the assumptions we have
made so far are legitimate, we have from the equation of
continuity

t
y= | ede, B pateseeupenn ey ec! «) CAs)

0

2v0n
= FSU am AC) ee, Se eee (0D)
doe (5)
But the assumptions cannot be accurately true, for since
the thickness of the layer decreases as the liquid cools and
a3 its viscosity increases, the stream lines cannot be vertical
or of constant velocity. We shall consider the effect of this
failure of the assumption in III.
If the surface were a circular cylinder of radius 7) we
should have instead of (1)

d dv
a7.) = pg ME Risto). hee CO}

which, using the same assumptions as before, gives in

place of (2)
ye — 2 ”
vant 4! 9 : — (tr)? log k. Sea)

t
When za is small (7) gives

which gives i

Cole

g :
was e as in (3).

> t . : . .
If " is not small we may expect surface tension forces to
Y I

0

be appreciable. The calculation is then more difficult and
we can give no complete solution. The flow would appear
to depend on the exact form of the meniscus at the surface
of the liquid. Buta dimensional argument gives us some
information.

1006 ~— Research Staff of the G. E. C., London on

If both gravitational and surface tension forces are appre-
ciable, we must have |

bg yee ov ;
flee r, Y ? ae) — C, e ° ° . (8)
where 7, are lengths and © a no-dimensional magnitude
characteristic of the system. We shall see later that con-
ditions can be found in which the gravitational forces are
inappreciable compared with those due to surface tension.
If we assume that this condition is fulfilled and assume
further that there is only one independent length 7,, viz. 7,

then (8) reduces to
ee hl G
- J ah ( ) A ea rat Cat Sele! (s)

It should be pointed out, however, that the last assumption
is precarious. For when surface tension is effective, 2) the
height of the meniscus over which the pressure due to
surface tension varies from ¥(7)-t) to zero is likely to be as
important as 79. (8) can be valid over the whole range only
if 2 is proportional to 7. But if it is true, we should
expect the importance of the surface tension relative to the

gravitational terms to be measured by 7 so that the

gravitational terms may be neglected when this expression
is large.

The dimensional argument cannot, of course, prove that
the distribution of the liquid as an even layer is stable.
The analogy of a water jet strongly suggests that it will not
be stable. If it is not stable, the conditions may be entirely
altered by freezing the layer as soon as it is formed. This
possibility will be considered later; but if they are not
greatly altered, the effect of freezing will merely be, as
before, to introduce a constant factor. The form of (9) will
not be changed.

Another ease of some interest may be mentioned briefly,
although we have not investigated it experimentally. If a
eylindrical tube is drawn out of the liquid, the gravitational
and surface tension forces will act in the same direction in
the layer on the outside of the tube and will act in opposite
directions in the layer on the inside. If the dimensions of
the tube are such that the two sets of forces are comparable,
the layer will therefore be thicker on the inside. It is
interesting to speculate what will happen inside the tube
when the surface tension forces are great compared with the
gravitational. It is easy to see that in such conditions the

. ae ee ae
td < Fd

a Problem in Viscosity. 1007

liquid must fill the entire cross-section of the tube, and will
not begin to fall out of the tube till the column becomes so
high that the gravitational forces become appreciable and
the conditions supposed are violated.

II. Apparatus.

In order to test these theoretical considerations arrange-
ments were made to draw metal strips or wires at known
speeds out of a stable liquid, the viscosity of which could be

Fig. 2.

NS

Ss
LZ LIL LC

varied by change of temperature and which would solidify
at ordinary temperatures, so that the thickness of the layer
could be measured. .
The apparatus was designed so that the viscosity of liquid
and the surface tension could be measured under the condi-
tions prevailing when the wire or plate was drawn from it.

1008 Research Staff of the G. EH. C., London, on

The essential features were the viscosimeter, the surface
tension apparatus, and the drawing device.

(a) Viscosimeter.—The viscosity was measured directly by
means of the torsional force exerted on a flat disk by another
disk parallel and near to it when both are immersed in the
liquid and when the second disk is made to rotate at a fixed
speed. If the geometrical arrangement is unaltered the
torsional force will be proportional to the viscosity of the
liquid and to the velocity.

A is a metal vessel containing the liquid and surrounded
by a thermostat B containing glycerine. Through the
bottom of this vessel passes a spindle C to which a pulley is
fixed driven by a small motor.

To the top of © is attached the rotating disk E in the
centre of which is a jewelled bearing. Above this is the
fixed disk separated from it and supported by a hardened
steel point resting in the jewelled bearing. A light steel
spindle F passing loosely through the metal cover serves to
keep the fixed disk parallel to E and to connect it with a
torsion measuring device consisting of a spiral steel spring H
and an aluminium pointer G which moves over a graduated
scale on the metal cover.

Vanes D are also attached to the spindle C which serve to
stir the liquid by causing it to rise through vertical channels
in the metal cylinder K and to fall over the top of K back
into the bath. The bottom of K is bevelled as shown, so-
that if the liquid contains suspensions these will fall in
between the vanes and thus be kept from settling.

_ The thermostat can be heated by means of the gas ring
burner placed below it.

(a) The viscosimeter was calibrated by means of solutions
of known viscosity. A 60 per cent. sucrose solution was
found exceedingly useful] for this purpose, as it gives a wide
range of viscosities with temperature change, and the
viscosities at various temperatures have been accurately
determined by the Bureau of Standards *.

It was found that the scale deflexions for a given speed
of the rotating disk were directly proportional to the viscosity,
and also that for a given viscosity the scale deflexions were
directly proportional to the speed of the rotating disk overa
wide range.

(0) Surface Tension Apparatus——The surface tension
apparatus was measured by Wilhelmy’s method, viz. by
means of the force required to break the film drawn out of

* It. C. Bingham and R. F. Jackson, Bureau of Standards Bulletin,
“No, 14) p59 917):

a Problem in Viscosity. 1009

the liquid by a given length of straight fine wire. If / is the
length of the film, f the force required to break it, then

y= a Fig. 3 is a perspective view of this apparatus,

which was mounted above the viscosimeter.

A platinum wire frame of the form shown served to pull
the film from the liquid, the horns remaining beneath the
surface of the liquid until the film broke. This was attached
to one end of a light aluminium pointer supported in a
metal frame by a torsion wire as shown. ‘This acted as a
balance for measuring the force required to break the film.

Fig. 3.

Wire Frame

\2)
% a

The metal frame could be rotated about an axis through A,
thereby raising the platinum wire frame from the surface of
the liquid until the film broke. The reading of the pointer
could be noted at the moment of breaking and the equivalent
force determined by means of a small scale-pan and weights
attached to the pointer in place of the wire frame.

The restoring force of the torsion wire could be supple-
mented by means of a rider attached to the aluminium
pointer, so that for a wide range of surface ‘tensions the
reading at the break-point could be brought on the
scale.

1010 Research Staff of the G. E. C., London, on

(c) Drawing Device.—The fine wires (tungsten or constan-
tan) used as cylinders were cleaned superficially by heating
in a reducing atmosphere. They were then wound on a
bobbin P (see fig. 2) above the surface of the liquid, and
drawn thence round an idle pulley L below the surface of
the liquid and finally round the winding bobbin N, which
was rotated at a regulated speed. Asan approximation to an
infinite slab a copper strip about 1 inch wide and 0°05 em.
thick was used. It was thought that if such strip were
drawn through the liquid from a bobbin outside it, the liquid
might be cooled appreciably when the drawing was rapid.
Accordingly the strip was originally wound on a bobbin,
wholly immersed in the liquid, and drawn thence direct to
the winding bobbin. 7

(d) Estimation of thickness—The thickness of the layer
on the fine wires was determined by weighing a known
length of the coated wires on a torsion balance (designed in
these Laboratories) capable of estimating a few milligrams.
with an error of 0:0001 milligram. The coating was then
dissolved off and the bare wire weighed. The layer on the
strip is uniform only at some distance from the edges ;
accordingly a known area was cut from the central portion
and weighed before and after removal of the layer. The
density of the solid layer was determined on a sample of
convenient size. :

(e) Liquids used for coating.—Molten waxes were first
used as the liquid for coating the solid surfaces. It was
found that filtered beeswax and carnauba wax were most
suitable.

A range of viscosity from 1 to 100 centipoises was.
obtainable within a temperature range from 50° to 140° C.
In some experiments the viscosity was increased by adding
fine insoluble powders to form suspensions. Waxes were
expected to he particularly advantageous, because their
surface tension is low and therefore not likely to be changed
by grease and other impurities of low surface tension. But
it was found in the course of the work that aqueous solutions
could be used with convenience ; for by the process of drawing
surface impurities were removed very rapidly, so that after a
short length had been drawn the surface tension was con-
stant and normal. Some of the observations recorded were
made with aqueous solutions of sodium silicate, which were
dried on the wire by passing it through a small electric
furnace a short distance above the liquid.

a Problem in Viscosity. 1011
III. Eeperimental Results.

The results of the experiments on flat strips are shown in
fio. 4, where ¢ {estimated from ¢' by (5)} is plotted against

3 7 ae from 0°122-0°320 c.g.s. units, v) from

2-46-7°85 cm. per sec., p from 0°8-1°2. ~The straight line is:

calculated from (3).

It was pointed out in I. that the assumptions used in the
derivation of equation (5) could not be accurately true.
Owing to the cooling of the wax, as soon as it leaves the
liquid surface its viscosity is increasing, so that it is to be
expected that the effective viscosity will be somewhat larger

Fig. 4.

Ee ee
hs oly, MR
DEUS SESE e IIR aes
eee Cy
BZe

paleg
pele ade ead pt ala Lb Tey
ng a See ig at aad
ia) 2 a a al SE ee eee |
ae Le

Vr

than that measured. The observed thickness of film should
therefore be thicker than that calculated on the basis of the
measured value, as is actually found.

But the difference is comparatively small; and when it is
remembered that (3) involves no empirical constant, but is
calculated from measured values only, no doubt will be felt
that the theory offered is completely confirmed.

The coating on the edges of the strip was much thinner
than on the central portion ; ; it was therefore to be expected
that surface-tension effects \ ould be prcminent in fine wires.
Preliminary measurements on the thickness of the coating
on wires of diameter less than 0°02 mm. showed that the
coatings were much thinner than were demanded by the
theory which takes only gravitation into account (eqn. (7)).

1012 Research Staff of the G. EH. C., London, on

Moreover, it made no difference to the thickness of the

coating whether the. wire was drawn vertically from the
liquid or horizontally through the top of the surface formed
by a rotating wheel within the liquid. Surface tension
must therefore be the dominant force. This result is in

in the

accordance with eqn. (9), for the least value of _ ;
observations was 300. PI

However, the possible effect of instability due to surface
tension must be considered. It was clear immediately that
the layer originally formed was unstable, for the solidified
coating on the wire was not even but was broken up into
blobs. These blobs were beautifully symmetrical and evenly
distributed when viewed under a microscope. Photomicro-
graphs (1), (2), (3) are shown in fig. 5 (PI. VIII). But
formation of the blobs evidently occurs after the wire has left
the liquid surface ; for it does not depend on the velocity of
drawing or the viscosity within fairly wide limits. No. (1)
was obtained at one-half the speed of No. (2) and at twice
the viscosity ; in these two cases there is approximately the
same thickness of coating. No. 3 was obtained at a higher
speed than No. 2, but at the same viscosity. Here the
coating is obviously thicker.

That the blobbing took place after the liquid coating had
been withdrawn from the liquid was further demonstrated
by drawing a wire through glycerine; a very thick film
was obtained which remained smooth and even for some
inches above the surface of the liquid before it could be seen
to break up.

A systematic study of the film thickness as a function of
viscosity and surface tension and velocity of drawing as well
as the radius of the wire was undertaken.

The following were the limits of the variables :—

Radius of wire r=0:00075-0-01 em.
Velocity of drawing v=5°5—-66'0 cm./sec.
Viscosity 7 =1-100 centipoises.

Surface tension y= 36-62 dyne/em.
Density p=0°8-1'4 grm /cm’.

The limits of v and 7 that could be investigated were set,
on the one hand by the impossibility of measuring very thin
layers, on the other by the thickness of the film; if ¢ were
greater than 79 the blobs which formed fell off the wire.

‘The results are shown in fig. 6 hy plotting ( ~) against

(7)

a Problem in Viscosity. 1013.

If the theory given is correct, the points should lie on a
smooth curve. They do actually lie on such a curve within
experimental error, and the curve is seen to be very nearly
a straight line. All our measurements on wires can simply
be expr ressed by the formula

eee ade py (10)

the constant 4°8 above being empirical.

It appears, then, that the two extreme cases considered
theoretically, namely , all gravity and all surface-tension, are
easily realized. Indeed, we have not realized an intermediate
ease, although it would probably be possible to obtain

ira
SoeSee00ee06 460

AN!
a

it with cy ‘linders of greater radius. Further, it appears
that there is no special force of ‘ cohesion’ between solid and
liquid surfaces which can produce a film even as thick as
the thinnest we have examined (0:00007 cm.). If v were-
infinitely small, 7. e., if the surface were allowed to drain for
an infinite tame: there is no evidence that the film would
not be completely removed, or at least reduced to molecular
thickness. The only action between solid and liquid im-
portant in these experiments is that which prevents slip at
the interface. It follows that the coating should be in-
dependent of the surface of the solid so lone as the liquid
wets it at all. As far as our experience goes ; the conclusion

1014 Prof. 8. Timoshenko on the Distribution of

is correct ; e.g., a tungsten wire coated with graphite (from
the process of wire-drawing) gave the same results as a
-clean wire.

In all the experiments described so far the liquids used
have been true liquids. But in such processes as enamelling
or painting the liquids are usually suspensions. The varia-
tion of the viscosity of suspensions with their solid contents
cand with the size of the suspended particles has been
investigated by several authors*. We have repeated some
of this work on liquids in which we were particularly
interested, and have confirmed many of their results. But
the question arises whether the viscosity of a suspension
measured by shearing it between parallel plates is the same
‘as that which determines the amount of liquid adhering to
a solid drawn out of it. We have made many observations
in this matter. It appears, as might be expected, that the
two viscosities are the same so long as the diameter of the
suspended particles is not larger than the thickness cf the
liquid layer drawn out. If the diameter exceeds that
thickness the liquid behaves in drawing as if it had a
viscosity much less than that measured by shearing. But a
-consideration of fig. 1 shows that such large particles cannot
be expected to enter the layer of liquid on the solid
surface ; they are squeezed out from it. Accordingly ©
the failure of formula (10) for these large particles is
simply due to the fact that the liquid which is being drawn
is that from which the large particles have been removed
-and of which the viscosity is correspondingly lower.

XCI. On the Distribution of Stresses in a Circular Ring
compressed by Two Forces acting along a Diameter. By
S. TIMOSHENKO fF.

Cee. the problem as a two-dimensional one, we
‘/ can obtain a solution in the case represented in fig. 1
‘by combining the known solutions of the problem of com-
pression of a disk f (fig. 2) and that of a ring § (fig. 3).

If we take the normal and the tangential tensions acting
-on the inner rim of the ring (fig. 3) as equal and opposite to
the tensions acting on the cylindrical surface of the radius
r in a disk (fig. 2), the stress-distribution in the case of

* KE. C. Bingham, Bur. Stand. Bull. no. 278 (1916). %. Humphrey
-and KX. Hatschek, Phys. Soc. Proc. xxviil. p. 274 (1916).

+ Communicated by Prof. HE. G. Coker, F. RS. .

{ See A. E. H. Love, ‘Treatise on the Theory of Elasticity,’ p. 215,
1920.

§ A. Timve, Z. f. Math. u. Phys. liu. p. 848 (1905).

Stresses in a Circular Ring. 1015
fig. 1 will be obtained by summing the stresses corresponding
to figs. 2 and 3.

Fig. 1. Fig. 2.

£

<—t—
eS ee

Fig. 4 shows the normal stresses on the vertical and
horizontal cross-sections of a ring when R=2r calculated by
the above method. The dotted lines on the same figure
represent the results of elementary solutions obtained by
using (1) the hypothesis of plane cross-sections or (2) the
hypothesis of plane distribution of normal stresses.

The stress-distribution in a compressed disk (fig. 2) is
obtained by superposing a tension *

pe
= (Ae on) eee |

* We suppose the thickness of the ring equal to unity,

1016 Prof, §. Timoshenko on the Distribution of
equal at all points, and two simple radial distributions :
__ 2P cos dy 2P cos ds

i On Tepe

#8

EF
Ag S|
A

AADs| | |
BRAC Ee
NZ

WL)
Teese
HE
ee Bee

If R=2r, the corresponding normal and tangential stresses
at the points of the cylindrical surface of the PEE r ae be
approximately represented by the following series *

pe oH (0°506 + 1:008 cos 20 + 0°443 cos 40
+ 0°158 cos 68 + 0°0467 cos 86 + 0:0083 cos 108),
= ap (3)
10 = =e (0°749 sin 20+ 0°374 sin 40

+ 0°141 sin 60 +0°0460 sin 80 + 0:0133 sin 108).
Distributing on the inner rim of the ring (fig. 3) the tensions

* The calculations have been made by the Runge approximate method.
See Theorie und Praxis der Revhen, p. 158. rom the calculations
made we may conclude that the error in stresses will not be higher than
4 per cent, if we take the first six terms of the series only.

Stresses in a Circular Ring. 1017

which are equal and opposite to the tractions (3), and using
the solution of the ease of a circular ring, the following
expressions are obtained for the stresses at any point :—

2P As 7 R? + p? 90 eo
R| — 0506, ae (22686-3246,

aa oR, t R?

5, Rt 2 ies
+0-4832— Jeos 20 +( 0°3691 a —0°6783 F

ajeg RS RA p!
+0 0368 — —0-0599 =, ) 0s 40 +(0-06504 on
p p Rl

6 8 6
—0-10026 F. + 0-0041319 S —0-009524%

RS

eos 68

5 8 10 (4)
+(0-008758-%, —_0-01225 8, =

fp + 000080795,
RS) ;
~0-0010888 =) ena (0-0007880 F, |
| pr? sete pce,
—0-001037-£-, + 000002960 =

RI0
—0°00008475 =) cos 100 |,

Peer Re Ree
ae ee . Ra a SES ae —(*- : .
rae [0 506", —a 268 — 04832
Ry -— 285" AAR ray tron root |
+-2°752 = cos 20 +(—0-3691 = +0 22615,
4
R¢4
(5)
0
0 :
3U

6 8 6
+0:05013 Ff —0-0041319~ 4 0-01904**,
p p

, pe ie Ri
+ (—0-008758 pe + 0°007352 £ —0-00040795 a

5 4
—0-0368—% +0'1798— cos AO + (-0-06504
) cos 60

R8 p®
a. 00018146" 603 80. { —0-0007880 =
a ae Sie RY
+0-0006911 frp —0-00002960
R09
+0:0001265 = cos 108 |,

Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922.

1018 Prof. 8. Timoshenko on the Distribution of

oe zeal alc 268—3'162 &, —0-4839 e

R?
+1°376~; )sin 20+ (0: 36918, =i) —
RS

—0:03680 ° +0: 1198) sin AO + (0° 065046,
Re

—0-07520 F - —0-0041319~ e+ 001428 = sin 68

i (0 008758-2, —0 009802 BK: ele

Ré hh.

RS
+0:0014517 #) sin 80+ (0 0007880 4,

—0: 0008638 67, i) 00002960

+ 0:0001054 a sin 106 | k

where p denotes the distance from the centre.

Superposing the stresses (4), (5), (6) on the stresses (1)
and (2) above, we obtain the stress-distribution in the com-
pressed ring of fig. 1. The normal stresses on the vertical
and horizontal cross-sections of the ring have the following
values :—

me I? R? + p”
o ie R?
8 p'

—0°6133 5; -—0-0915 ? = _—0 01146 £,—0- 001045,

ser 4233 +0: 02728, +- 0°003043 = + 0: 000 Ey

(68)_-0=— , 5-0-5065 nes + 22685-95525

+ oono200 go a

a dee Ue ag
ae aoe (R?+p2)? 0°506 — Pe R—P mm

te: be —0°7433 € +0°1090 ie =i); 013048,
R! RE Rs

6 OOO SM te fete OG eee 005221 —,
R10 p°

ite ae a 8

(60), — 2-268

Stressesin a Circular Ring. 1019

The results of these calculations are represented on fig. 4.
As a measure of the accuracy of these calculations, we have

R
( (40), xdp=—0-s022P, . be pert al.)
ey ree

so that the error is smaller than } per cent.
If the corresponding calculations are made for the vertical
cross-section, we obtain

R P
(0), do= 9p 01996, fe es laiiewera

This must be compared with the result

i 2 cos g@ sin ddd= yi ;

0 a

corresponding to a’ simple radial distribution (fig. 5). We
have again an error smaller than 4 per cent.

The method outlined here may be applied to the more
general case of a circular ring subjected to any forces acting
on its external rim. It is oniy necessary to use the corre-
sponding solution for a disk *. |

* A. E. H. Love, ‘ Treatise on the Theory of [lasticity,’ p. 216.

3U 2

S(1020 ).

XCIL. On a Revised Equation of State. By ALFRED }
W, Porvur, 05Sc.. HRS: Ff nst.eo

pee See eran equation of gases in the
reduced form is 1 Gao | xP: | 2(1- ph which

can be modified by putting y” instead of y in the exponential
term. This equation is very fairly satisfactory, when
n=3/2, in the region of low pressures; but it breaks down
for pressures above the critical value.

Berthelot also developed an .equation in which the
respective terms are based directly on experiment instead of
theory. It is explicitly applicable to low pressures only, and
is very much used for that region. This equalion 1s

16 B2¢ |
ay Bad

It will be observed that it does not pass thr eee ‘hee critical
point («=@S=y=1). Onfig, 1 are shown experimental
values of «8 plotted against « for isopentane, and on this
same figure values calculated from Berthelot’s equation are
represented by a.dotted-curve. _ Values-from van der Waals’
equation are indicated by small squares. |

The chief fault of Dieterici’s equation is that it makes the
critical volume only twice the least volume of the liquid,
whereas experiment shows that it is in most cases very
nearly four times. To get over this, Dieterici treats the
volume of the molecules themselves as being a function of
the pressure.

The first object of this paper is to point out that there is a
way of testing the equation which shows that this last-stated
modification (even if it should be necessary) cannot be the
only change required, and that it is no use making it until
other changes are made. :

If the equation be written

a=yl'(8) exp.(3).

where F(@) is a function of the volume alone, the value of

on

* Communicated by the Author.

Ona Revised Equation of State. 1021

y Ou Lb n eee fink
“—~ —] becomes — —~ whatever the function (8) may be.
a OY y"B ri ( ) Y

Calling this quantity / (it is connected with the internal

eile lg ie
Cae
REECE
Che

Rta ee id 2 iss |
SN
pe hey ie ol
JS) oe

ey enn A
ey Ede Eee

of 2.
O © Experimental. + + Berthelot.
C] () Vander Waals. —— Porter.

latent heat of expansion of the gas per unit change of
volume), the value of §/ should, according to the Dieterici

1022 Prof. A. W. Porter on a

equation, be a constant. In fig. 2 are shown two curves of
Gl plotted against the density (1/v) for isopentane de-
termined from the experimental values. Curve A is the
critical isotherm and B is the isotherm fer 503° abs. It
will be seen from this figure that instead of being constant
it undergoes very considerable variation. Near the critical
point the calculated points fluctuate, but a smooth curve
drawn amongst them shows that an equation of at /east the
second degree is required to represent them.

Fig. 2.
Ae
Es Ss) Geez ae
iS ee ae a es
—|
Sel ees
al en
arene =
=a
; Isopentane.
(Oe ) ” plotted against Density.
( aT ie p P 8 y,
Curve A, Critical [sothermal, T = 461 abs. C.
», 3B, Isothermai, T = 503 abs. C.

If the necessary extra terms are introduced, it becomes
possible to bring the equation for the critical isotherm into
the form

The three conditions for the critical point are satisfied if
N=33, G—iol. b="261,.. c= —"Zare
It is noteworthy that the negative value of ¢ is required not

only from the data of the critical point, but also by the
isotherm on fig. 2.

Revised Kquation of State. 1023

The values of a, 6, and ¢ will be functions of the tempera-
ture.

1 O(a —l a DIT ROC
At 2=0,8=«a, this becomes zero (and because a does
not do so, so also does 0 (a@8)/d«) when
Gis.
If we write 0 a 4
n

this occurs when
y” =4a,= 6°04 = (2°455)?.

Now the e#@ against « curve for nitrogen starts out
horizontally when y=2°54; hence n=2 nearly. Inserting
these values, which are obtained solely by making the
equation suit the critical state, it is interestiny to see how
nearly the equation becomes Berthelot’s equation when the
pressure is small. It can be written

vi .
eee | (1-+ = aa

Caw ye
5°28 ar aye. |
0 — ) = for large values of ~.
Ce 5 bac ia i

In Berthelot’s equation the numbers are 95°33 and 3°95
respectively.

If 4 be written 6,/y”, the value of mis found from the
curves of isopentane to be a high one—about 12 to 15. It

can be obtained also by considering the value of ge at the
eritical point. We have in general
y Oa nay 4. mb, 6
“OY BR VG 4 MER,
or at the critical point the right side is
1+na,+mb,+9¢,
or 14+3+°267m—°237q.

There are not data enough to find m and q with certainty.
But since this critical slope is for all substances nearly
equal to 7, it follows that m must be at least 10, which
agrees with what we find from the experimental curves.

1024 Prof. V. Karapetoff on General Equations

The critical isotherm calculated from the values of the
constants obtained above is shown in fig. 1 as a continuous
curve. How nearly it fits the measurements for isopentane
down to about 4 the critical volume is seen by examining
the circles which represent experimental points.

The following values given by experiment and also by
various equations for the case of 8 =4 are useful for
comparison to show the success and defect of the revised
equation.

Experiment. | Van der Waals. —_—Diieterici. | Porter.
eas ILS) 4 | Infinity | 156
a38=1:07 © 2 | “Infinity 9) ee

XCIII. General Equations of a Balanced Aliernating-Current
Bridge. By Vuapimirn Karapetorr, Professor of Electrical
Engineering, Cornell University, Ithaca, N.Y.”

N the last few years the use of the Wheatstone bridge
for the measurement and comparison of inductances
and capacities has cousiderably increased, partly due to
developments in the art of electrical communication, and
partly because of improvements in the sources of high-
frequency sinusoidal currents. Old classical arrangements
of alternating-current bridges have been further developed
and new arrangements evolved t. This seems, therefore, to
be an opportune time to deduce a general equation of the
a.c. bridge which would comprise the various actual bridge
arrangements as specific cases. Such a general formula
gives a bird’s-eye view of the existing bridge connexions
and will. enable new bridge arrangements to be devised
without deducing fundamental equations in each case or
constructing vector diagrams. It is hoped that the general
formula (D) given below will serve these two purposes.
Some time ago Dr. Poole showed { that the currents
and voltages in the usual arrangements of an a.c. bridge,
when balanced, can be represented by comparatively simple
vector diagrams from which the relationship between the

* Communicated by the Author.

t See D. I. Cone, “ Bridge Methods for Alternating-Current Measure-
ments,” Trans. Amer. Inst. El. Engrs. vol. xxxix. p. 1748 (1920).

{ H. H. Poole, “On the Use of Vector Methods in the Derivation of
the Formule used in Inductance and Capacity Measurements,” Phil.
Mag. vol. xl. p. 798 (1920).

of a Balanced Alternating-Current Bridge. 1025

desired quantities can be readily deduced. While his
results will be very valuable to the practical users of the
bridge, the other side of the problem, that is, a generali-
zation of the theory, may prove to be of interest to
investigators of new possibilities of bridge connexions.

Fig. 1 represents general connexions of an a.c. bridge,
with an impedance in each branch, and a mutual inductance
in each of the lower branches. The upper left-hand branch
consists of two paths in parallel, and the galvanometer is
connected at an intermediate point, A,’, of one of the paths.

Z2> —— hee

Golveanometer

or

Telepwone

A.C. Source

This is the arrangement used in the so-called Anderson
bridge, and is included in the general scheme because of its
further possibilities. The bridge is supposed to be balanced
on alternating current, that is, the galvanometer current is
supposed to be equal to zero.

The current in the lower branches is denoted by Ij, that in
the upper branches by I,.. Im the divided branch 2 the
current through the lower path is denoted by I, so that the
current through the other path is I,—I. The line current is
I,+1, The impedances Z in the two left-hand branches are
denoted with the subscripts 1 and 2, to agree with the
sketches in Dr. Poole’s article. The right-hand quantities
are provided with the subscripts 3 and 4, although Dr. Poole
uses again the subscripts 1 and 2, except in his fig. 5, where
the subscript 4 is introduced in the same place as in this
article.

1026 Prof. V. Karapetoff on Greneral Hquations

Assuming the currents and the impedances to be expressed
as complex quantities, we have the following three funda-
mental equations of the voltage drop in the parts of the |
brid ge :—

PAH e) XSi, 6 oe
1% leZg= liZe= Gy) Xa
(GoYA=10 ee)

Kliminating the currents from these equations, we obtain
the following general relationship among the impedances of a
balanced bridge :—

(Z3— Xone) [(Ze aR Xin) 4 ve aXxmi( Le a ii) Ne (Z, ae
[ (Zy+ Xms) (1+ Ya(Zo+Zs)) + YaLeZs5] . - (D)

In this equation the admittance Y, is used in place of the
impedance Za; the relationship between the two is Z,Y,=1.

While in any special case eq. (D) may be applied directly,
there are some typical special cases for which it is more
convenient to write simplified formsof eq. (D). Hight such
cases are considered below.

(1) No mutual inductances and a single-path branch 2.
This means that X»y=Xm3=0, and Y,=0; we then have a
simple bridge consisting of four impedances. Hq. (D)
becomes

Lls=Lyly... 5
(2) No mutual inductance and a single-shunt branch 2
Here again X,,=X,3;=0 and Z.=0. The points A, and
A,’ coincide, and Z, is in parallel with Z,. Hq. (D) becomes
LL = LL, (1 + Y ily) ° . c ° (2)

(3) No mutual inductance and a double shunt in branch 2
as shown in fig. 1. In this case the only simplification 1 is

that X,,1= Xn3=0, and eq. (D) gives
Ashig=Ly[Zi(1 + Ya(Z2+ Ze)) + YahAeZs].. - (3)
(4) Mutual inductance in branch 3 only and no shunt in
branch 2. We have X,,,;=0 and Y,=0; eq. (D) becomes
(Z3—Xing)Zg=Z,(Zy+Xmg)- - + - (4)

(5) Mutual inductance in branch 3 only and a single shunt
in branch 2. In this ease X»4=Z,=0 and Z, is in parallel
with Z,. Eq. (D) gives |

. (Z;— Xing) Dy Ly [ (Zi, at Xs) al mm YaZe) | oinitye (5)

of a Balanced Alternating- Current Bridge. 1027

(6) Mutual inductance in branch 1 only and no shunt in

branch 2. We have X»3=0 and Y,=0; eq. (D) becomes
Z3 (Ze 50 nl = (Z,— Xin) Za - eRe aie « (6)

(7) Mutual inductance in branch 1 only and a single shunt
in branch 2. In this case Xp3=Z,=0 and Za, is in -parallel

with Z,. Hq. (D) gives
Zs[ (Z_+ Xm) + YaXmZe] =(Z,—Xm)(1+ Yo%e)Z,. (7)

(8) Mutual inductance in both branches, but the branch 2 is
not shunted. In this case Y,=0, and eq. (D) becomes

(Z3— Xmg)(Zo a Xm) a (Z, — Xm) (Ly + ona) ° (8)

In the following table the special applications of the bridge
are those discussed by Dr. Poole in the article mentioned
above, and the references are to the figures in his article. All
these applications are covered by the foregoing eight special
eases of formula D, and the ease number is stated for each
application. In some applications the four resistances of the
bridge have first to be balanced with direct current, in other
cases it is not feasible. This is indicated by “‘ yes” or “‘ no”
in one of the columns.

It will now be shown how readily the familiar formulee for
the measurements shown in the table can be derived from
the formule (1) to (8), all of which are specific cases of eq.
(D). ‘The item numbers below refer to the items in the
table, and in each case an equation is used as indicated in
the table. All the results check with those given by
Dr. Poole.

Item No. 1:
(73+ jols)ry=7;(",+joL,).
Separating the real and the imaginary parts, gives
pe ere (!.)
eee ee ee | ED)
These equations combined give
ee ePalh a. a's) oe Like hie ee)

Eq. (9) shows that the bridge must be balanced on direct
current as well as on alternating current.

SS] ce

Prof. V. Karapetoff on General Equations

1028

tN)

frpod- ey

ery ol ey

eQem/C—

ry ol+ 4

Erol Fu

®rol+ oy

“Q

“Q0URTe | “ON “SLL
JUBLING JOE S$, e[00g

-tu09 aq OF SeTyIJURN’)

; ‘209,
Aouenbo.a yy soul 3
“Ayrovdeo ‘|-10q809F
| pue souvjonpul jenny | Aone) D
(0 <2) Ayrovdeo
pus soURJONPUL-Jpeg “Wosrepuy 9
| (9=94) £yoedvo |
pure souvjonpul-jjeg | Uos1epuy 6)
‘soiqioedvo OMT | “Aqneg oq zs
: "20URy
-ONPUI-JfPS PUL TVNIN, *[[OMxvy]AL ‘2
"‘ge0UBINPUT [BUINU OMT, | “[[OMXBIAL G
| ‘SOUBJONPUI-JJOS OMT, | “TOMXLT "Tl
‘parnsvent co peard *payeursito| “ony

moma AG | ULa}]

of a Balanced Alternating-Current Bridge. 1029
Item No. 2 (see Note at the end) :
(73 +joL3;—)jo Ms) joM,=( +joL,—joM,)joMs;,
from which
L.M,=L,M; ° ° . . < . . (12)
and r35M,=7,M, ; - : : . 5 ° : (13)
These equations, combined, give |
M,/M,=L,/L,=7,/7, a tete Lek” eet vie (14)

The bridge cannot be balanced on direct current, but the
ratio of the resistances must be equal to that of the self-
inductances, before two mutual inductances can be compared.

Tiem No. 3:
(73 -+j@L3—joMs;) r=, (74 +-JoL,+joMs).
Consequently 1309 =T3i"4 AEs i chs eed eras 9)
ro(iz3 — Ms) =7,(Ly+M;3) ... . . (16)
so that
1 /%2=1'3/74= (3 — M3)/Ly+M3). . . (17)
Item No. 4: z :
—jry/@C3= —77,/oCy
oe Me ee vats ais tsp ais se wured ee |) CLS)
tiem Nos 82 |
(73 +jols)r2=7y"4(1+772@Ca),
from which
To Tae NE Ak a ene ye a CLO)
= FOr I OM et A (20)
Item No. 6.

r,(7r3+joL3) =r,[7,(1 +joCa(rs +15) +jpoCar or |.
Kquating the real and the imaginary parts, we get

LoV3 SP 14 : : - - “ ° ~ ° - ; (21)
% Lis = 11Cq [ ral + 1p) + Tots | a - : (22)

or, combining the two,
Tg=Cobriret re(ti 73) }- (28)

When 7;=0, this expression becomes identical with eq. (20).
Item No. 7:
(7'3—j/@C3) joM,=(7, +joL,—joM,)7.

LOGOS a rote Vs. Karapetoff on General Equations
Separating the real and the imaginary parts, we get
: MO 3 erty sc Ae ee
and 73M, = (L,;— M,)74. ohne Sepa ate 5 (25)
Eq. (25) may be also written in the form
L,/M, — 1 + (1°3/7°4) e e . . e (25 a)
Item No. 8. This frequency bridge was described by -
Mr. Cone, zbid. p. 1749. Eq. (2) gives :—
rsrg= 147) —3/(@C,) | 1 +joCore). . . = (26)
Separating the real and the imaginary parts, we get
1309/7 = Vy Se Cyr|C;, ° ‘ 6 5 0 (27)
1/(@U) =o. 2) 2 eee,
The last equation gives

w?C C9979 = 1 SVS (29)

from which the unknown frequency may be computed. The
following special case is of practical interest. In eq. (27)
put r=, and re=2r;; then C)=2C,, and eq. (29) becomes

aC, =aCoro= Y paraen s ietes . (30)

As is mentioned above, the general equation (D), or any
of its particular forms, (1) to (8), may be used for the
derivation of new forms of the bridge. ‘lake for example
the simplest case, that of eq. (1). It may be written in the
form

(13-1 ps) a +22) = (v7, +941) G4 je). . eee

Separating the real and the imaginary parts, we get
13% g— Uglg=M— Vey. . . . . (82)
U3Po + P3lg=&\rg+ V4 5 5 < 5 e (33)

We may put, if we so choose, ryr.=r,7,, that is, require
the bridge to be first balanced with direct current. Then
eq. (32) becomes a3%,.=%,%,, and these two conditions,
together with eq. (33), may be used to investigate various
possible bridge combinations with resistances and induc-
tances. ‘Ihe condition r3ry=r)r, may be dropped and eqs.
(32) and (33) used for an investigation of various bridge
arrangements containing resistance, inductance, and capacity.
One or two of the 2s may be put equal to zero, with a
resulting simplification in the algebraic relationships. Ina
similar manner, eqs. (2) to (8) may be resolved into their

of a Balanced Alternating-Current Bridge. 1031

component parts and the possibilities of various bridge con-
nexions and measurements analysed, using only elementary
algebraic transformations.

When one or more branches of a bridge contain parallel
paths, quicker results may be obtained by using admittances
in place of impedances. Let, for example, the branches 1
and 4 contain ohmic resistances only, let branch 3 contain
an impedance r,+j@L;3, and let branch 2 consist of a
capacitive susceptance jwC, in parallel with a resistance ry.
We then have, according to eq. (1),

1 foes
(73 +)@]3)/ ie + jC Jann ee eae)
rs +j@L3=ryr/t2+jorynC,. . . .. (35)

Equating the real and the imaginary parts, eqs. (19) and
(20) are obtained.

or

Fig. 2.
Az

Gel/vanometer

Vote.—Fig. 2 shows the diagram of connexions of
Maxwell’s mutual inductance bridge (item 2 in the table).
At first sight it does not seem possible that fig. 2, with
its two separate circuits, could be a particular case of
fig. 1 which has one circuit only. In order to explain
the transformation of fig. 1 into fig. 2, intermediate
diagrams of connexions are shown in figs. 3 and 4. In
fig. 3, the impedances Z, and Z, are assumed to be very small,
otherwise the connexions are the same as in fig. 1 for the
case of a single path in branch 2. In other words, the points
O, B, and A, are electrically close to each other. In fie. 4
the limiting assumption is made that Z,=Z,=0, and the
three points are brought together. Under these conditions

1032 Equations of Balanced Alternating-Current Bridge.

no current can flow from the primary into the secondary
circuit by conduction, since the two circuits have only one
point incommon. The secondary current is produced only
by induction, and the two circuits may just as well be entirely
separate. In this manner fig. 2 is obtained from fig. 4.

Fig. 3.

2> A> 24

Galvanometer

AL

Galvanometer

This also explains the reason for which Z, and Z, in the table
are marked equal to zero. ‘Thus, the fundamental formula
(D) is also applicable to bridges in which the secondary
current is produced entirely by mutual induction, and the
primary source of current has no metallic connexion with
the bridge itself.

FE goaa. ]

XCIV. The Motion of Electrons in Argon and in Hydrogen.
By J. 8. Townsenn, M.A., Ft.S., Wykeham Professor
of Physics, Oxford, and V. A. Battuy, W.A., The Queen’s
College, Oxford *.

i, HE experiments on the motion of electrons in argon

which we have already published show such re-
markable differences between this gas and nitrogen or
hydrogen, that we considered it desirable to make further
experiments with argon which had been very completely
purified, and to extend the determinations of the velocities
over larger ranges of pressures and forces.

For this purpose it was necessary to construct an apparatus
suitable for measuring the velocity in the direction of the
electric force, and also the velocity of agitation when the
electrons move in a widely diverging stream after passing
through a narrow slit in a metal sheet.

In order to obtain accurate results it is necessary in all
cases that the gas should be free from impurities which tend
to form ions. With gases like argon, where the electron
loses a very small proportion of its energy in colliding with
a molecule, the gas should be free not only trom impurities
that tend to form ions but also from gases like nitrogen and
hydrogen, as the loss of energy of an electron in a collision
with a molecule of one of these gases, although small, is
large compared with the loss of energy in a collision with
argon. Also the effect of such impurities in argon is
accentuated by the fact that the probability of a collision
between an electron and a molecule is much greater in the
other gases than in argon.

It was found that impurities get into the gases from the
materials such as ebonite or elastic cement generally used
in the construction of apparatus for measuring velocities, so
that in nitrogen or hydrogen the results obtained after the
gas had been in the apparatus for a few days were slightly
different from those obtained immediately after the gas had
been admitted. In the case of argon the effect of these
impurities was noticeable after the gas. had been in the
apparatus for one day.

2. In order to eliminate the impurities emanating from
the apparatus, glass was used instead of ebonite to insulate
and fix in position the various electrodes and guard-rings, and
the connexions were made through glass capillary tubes
instead of ebonite plugs. The capillary tubes were slightly

* Communicated by the Authors.

Phil. Mag. 8. 6. Vol. 44. No. 263, Nov. 1922. 3X

i

=r

1034 Prof. Townsend and Mr. Bailey on the

tapered and ground to fit into metal sockets in the outer case
of the instrument, the wax used for sealing being applied
only on the outside of the joint. A great improvement was
thus obtained, and two instruments of different dimensions
were constructed, one with a slit 2 centimetres from the
receiving electrodes suitable for measur ing velocities in
gases like argon where the lateral diffusion of a stream of
electrons is very wide, and the other similar to that which
had been previously used with the slit 4 centimetres from
the receiving electrodes. When tested with hydrogen no
change was observed in the velocities after the gas had been
in the apparatus for several days, and with pure argon the
changes in two or three days were extremely small.

In the instrument with the slit 2 centimetres from the
receiving electrodes the guard-rings and the electrodes were
fixed in the positions shown in fig. 1. The electrons are set
free from the copper plate P by ultraviolet light admitted
through a quartz plate sealed in the cover of the instrument,
and the stream of electrons that passes through the gauze G
and the slit S is received by the electrodes H,, Hy, and Es.
These electrodes were mounted on two strips of plate glass
fixed to the guard-ring Rj,, so that the upper surfaces of the
electrodes were in the same plane with the upper surface of
the ring. The ring R, was 7:8 centimetres internal diameter
and 11°6 centimetres external diameter, and was at zero
potential. The ring Ry, of the same size as R,, was insulated
and fixed at a distance of one centimetre from R,. The
slit S was 2 millimetres wide and 1°5 centimetres long in a
sheet of silver foil stretched inside the brass ring A, and
fixed at a distance of 2 centimetres from the receiving
electrodes. The gauze of silver wire G was at a distance of
3 centimetres and the plate P at a distance of 6 centimetres
from the receiving electrodes. A uniform electric field was
obtained by maintaining the ring Ry, the plate A, and the gauze
G at potentials V, 2V, and 3V proportional to their distances
from the receiving electrodes EH. In most of the experiments
the plate P was maintained at the potential 6V, and the
electric force from this plate to the gauze was the same as
the force in the lower part of the field. The object of the
gauze was to ensure that the electrons should have attained
the steady state of motion corresponding to the force Z in the
lower part of the field before passing through the slit. This
condition may be obtained without the gauze by fixing the
plate P at the potential 6V, and for experiments with gases
at low pressures this gauze is unnecessary. But with large
pressures above 20 or 30 millimetres the currents become

Motion of Electrons in Argon and in Aydrogen. 1035

very small when the electric force is small and the plate P
is at the potential 6V. The current is increased by
increasing the pontential of the plate P, and with the gauze
at the potential 3V the electrons pass through a distance of
one centimetre under the force Z before reathing the slit.
With the gases at the higher pressures the number of
collisions of each electron with molecules of the gas in this
distance is very large, and the motion of the electron acquires
the steady state corresponding to the force Z before passing
through the slit.

Fig. 1
E

A A

SEES <

2) EY Le (ey ee
R, E, See ae R,

In the instrument with the slit 4 centimetres from the
receiving electrodes there are three guard-rings between Ry
and the ring A with the slit, as shown in the diagram,
page 875, Phil. Mag. Dec. 1921. The dimensions of the
electrodes and the guard-rings were the same in both
instruments.

In order to avoid errors which might arise from contact
potentials at the surfaces of the rings R or the electrodes E,
the metal surfaces were all electroplated with silver which
is less liable to become oxidized than brass. The variations
in contact potential which may arise owing to oxidation

aX 2

1036 Prof. Townsend and Mr. Bailey on the

would be too small to have an appreciable effect in most of
the experiments as the electric forces were so large, but it
was considered advisable to reduce as far as. possible an
error that might affect the experiments with the smaller
forces. :

_ In the original apparatus the electrodes were of unsilvered
brass, and when the experiments with hydrogen were repeated
with the silvered electrodes almost exactly the same results
were obtained.

3. The position of the slit in both instruments was
adjusted to bring the centre of the stream slightly to one
side of the centre of the electrode E,. This arrangement
makes the instruments very adaptable for the measurement
of the velocities in the direction of the electric force.

For this purpose the stream is deflected by a transverse
magnetic force H which is adjusted to bring the centre of
the stream to coincide with one of the gaps between H, and
the electrodes EK, and E;. The electrode EH, was a flat strip
4-5 millimetres wide and each of the: gaps ‘5 millimetre
wide, so that the distance between the centres of the two
gaps was 5 millimetres. Let this distance be 2a, 6b the
distance of the centre of the stream from the centre of
the electrode H, when H=0, H, the magnetic force required
to deflect the centre of the stream through the distance a+),
so that the current received by EH, is equal to that received

by E, and E;, H; the magneticforce in the opposite direction

which deflects the centre through the distance (a—b), the
current received by Hi; being then equal to that received by
KE, and H,. The velocity W in the direction of the electric
force Z is given by the equations .

H,W _ ath
7, ae C 5) e e e « e e Gi
lel Wie |
7 Foo

where ¢ is the distance from the slit to the electrodes EK.
Thus 6 is determined by the relation H,/H;=(¢+6)/(a—0).

The magnetic field which was uniform in the space
between the slit and the electrodes EK was produced by a
current in two large coils fixed in position on either side of
the apparatus. With the larger gas pressures the velocities
W are comparatively small, and it was convenient to deflect
the stream through the shorter distance a—b as the coils
became overheated when currents of the order of 15 amperes

Motion of Electrons in Argon and in Hydrogen. 1037

were flowing through them during the time required to
make the observations. The distance b was found with the
gas at one of the lower pressures when the velocities were
large, and comparatively small currents were required to
obtain either the deflexion (a+b) or (a—b). This method
was found quite satisfactory with nitrogen or hydrogen in
the second instrument where the slit was 4 centimetres
from the receiving electrodes, and the distance 6=°6 milli-
metre.

4. With the first instrument where the slit was 2 centi-
metres from the receiving electrodes, it would have been
necessary to double the magnetic forces in order to produce
similar deflexions. In this case the following method was
used to measure the velocities in argon at the higher
pressures. When the centre of the stream is at a distance b
to the right of the electrode H, (figs. 1 & 2), the current
received by H; is larger than that received by H,. By
means of a suitable magnetic force H, the centre of the
stream may be deflected through the distance 6 and thus
brought to the centre of Hg. The two electrodes F K, and EB,
then receive equal charges. The value of H, was found oF
measuring these two charges with the central electrode
maintained at zero potential, and adjusting the magnetic
force to the point at which the charges are equal.

The velocity W is then given by the equation

Se ae ca cca tee

C

In order to find 6, the stream is deflected in the opposite
direction through the distance (a—b), which is attained
when the current received by E, and E, is equal to that
received by H3, the required magnetic force H; being
ee by equation (2). Thus b is given by the relation

H,/H;=6/(a—b), and was found to be *87 millimetre.

In argon at the higher pressures the velocities W were
found by this method, ‘and the currents necessary to produce
the magnetic fields H, were from 10 to 15 amperes.

These results were tested by finding the velocities of
electrons in hydrogen with both instruments. The hydrogen
was admitted through palladium tubes sealed in the apparatus,
and the experiments were made with different forces Z and
pressures p, the ratio of the force to the pressure being
varied from the value Z/p='2 to Z/p=40. There was a
close agreement between the results obtained with the two
instruments.

The velocities obtained for the different values of the ratio

en

1038 - Prof. Townsend and Mr. Bailey on the ’

Z/p were on an average less by about 2 per cent. than those
obtained in the previous experiments * with hydrogen.

_ 5. The velocity of agitation u of the electrons is deduced
from measurements of the ratio of the charge received by
the central electrode H, to the sum of the charges received
by the electrodes H,, Hy, and Ez.

The theory of the method has already been explained fF in
detail, and may be expressed briefly as follows :—When a
stream of electrons moves under an electric force, the
number per cubie centimetre at any point is a function of
the quantity eZ/mu* when the steady state corresponding to
the force Z is attained, e being the atomic charge, Z the
electric force and mu?/2 the energy of agitation of the
electrons. If MO*[2 be the energy of agitation of a mole-
cule of a gas at 15°C., the velocity of agitation wu of an
electron would be 1: 15X10’ cm. per sec. if its energy of
agitation were equal to MQ2/2. When moving under an
electric force the energy of agitation of the electron is much
ereater than this quantity, and if mu2=kMQ? the quantity
eZ/mu? becomes eZ/kMQ?. This ratio may be written
NeZ/kNMQ?, where N is the number of molecules per cubic

centimetre of a gas at 760 mm. pressure and 15°C. ; and

since the quantities Ne and NM©? are known accurately,

the number of electrons per cubic centimetre at any point of
the stream is a function of Z/k and known constants.

The ratio R=n./(ny+ng+n3) of the charge nz received
by the central electrode H, to the sum of the charges 1, ng, nz
received by the three electrodes E,, E,, EH; may therefore be
expressed in terms of the ratio Z/k, and the values of R
corresponding to definite values of Z/k may be computed.
The value of R for any value of Z/k may be represented by

_ means of a curve, the form of the curve depending on the

sizes of the receiving electrodes and the size and Pe of
the slit.

It was necessary therefore to calculate the values of R
for different values of Z/k when the centre of the stream
fell to one side of the centre of the electrode HK, with the
slit two and four centimetres from the electrodes EK. As it
is difficult to construct the apparatus so that the displace-
ment 6 of the centre of the stream from the centre of K, is
some exact fraction of a millimetre, the points on four curves
were calculated which are given in fig. 3.

The curves 1 give R for receiving electrodes of the dimen-
sions shown in figs. 1 and 2, with the slit two centimetres

* Phil. Mag. Dec. 1921.
+ J. S. Townsend, Proc. Roy. Soc. A, lxxxi. p. 464 (1908).

Motion of Electrons in Argon and in Hydrogen. 1039

from the plane of the electrodes, the upper curve corre-
“sponding to the case where the centre of the stream falls
on the centre of H, (6=0) and the lower curve where the
centre of the stream is one millimetre from the centre of
H,(6='1). The curves 2 give the value of R under the
same conditions, except that the slit is four centimetres from
the receiving electrodes. In each case the curves for b=0
and b="1 are close together, and the correct ratio Rh for any
intermediate value of } is easily estimated.

Fig. 3,

G-2 O-4 60:6 ory, 1-0 /-2 [4 1-6

The method adopted for calculating the ratios R corre-
sponding to definite values of Z/k was similar to that used
by Mackie * to find the points on the curve 2 (b=0).

In the course of the calculations we redetermined the
points on this curve and obtained numbers almost exactly

the same as those given by Mackie.
~The calculated values of R, from which the curves were
drawn, are given in Table I.

TABLE I.
: m
| R.
Ze
k c=2 | ca—4
| 6=0 | b='l. b6=0 b=
05 196 195 Ven | he
1 231 230 ‘1607 1602 |
2 298 295 | ‘2077 2066
3 353 348 | 2495 2476
4 397 389 te,
5 435 “425 | 314 “310
1 Ser abt: “425 “415
It. 15 ie ald 506 -490
ae #8 aN | 570 548
| 25 eet ay ‘619 590

* J. H. Mackie, Proc. Roy. Soc. A, xc. p. 69 (1914).

1040 Prof. Townsend and Mr. Bailey on the

Jf the ratio R=n,'Gy + 12+ nz) be determined experi-
mentally when the stream is moving under an electric
Force Z, the value of-Z/k corresponding to R is given by the
curves (fig. 3), and the factor & is thus found. The velocity
of agitation wu of the electrons is then given by the formuia

u=1:15x10’* kk. None of the experiments were made
with the ratio R less than *24, as greater accuracy is obtained
with the larger ratios.

6. The accuracy of the normal distribution curves was
tested by measuring the velocities of agitation of the
electrons in hydrogen with each instrument. ‘The experi-
ments were made over tke same range of forces and
pressures as the test experiments on the velocities in the
direction of the electric force. The values of & obtained
with the two instruments were in very close agreement, and
on an average they did not differ by more than 2 per cent.
from the values of & found in the previous experiments.

It may be mentioned that in the previous experiments
the hydrogen was prepared by the electrolysis of barium
hydrate, and passed over hot copper into a drying-flask, from
which it was admitted through a tap into the apparatus,
In the test experiments with the new instruments the
hydrogen was admitted through a palladium tube without
bringing the gas into contact with any chemicals from which
an impurity might have been given off. There was no leak
in either instrument which could be detected by means of a
McLeod gauge, even when the apparatus was exhausted to
1/100th of a millimetre, and observations of the pressure
were made at intervals during a fortnight.

The results obtained with hydrogen may therefore be
taken as being well established.

7. The argon used in these experiments was obtained from
a cylinder supplied by the British Oxygen Company. The
gas contained about 10 per cent. of nitrogen, which was
removed by Rayleigh’s method. It was admitted to a vessel .
containing a solution of caustic potash, and oxygen added in
excess of the amount required to combine with the nitrogen.
Two platinum electrodes were sealed into tubes leading into
the vessel, and a discharge was passed between the electrodes
for several hours. ‘The change of pressure in the gas due to
the combination of the oxygen and nitrogen was noted, and
after sparking for about fifteen hours the pressure was found
to remain constant. The residual traces of nitrogen were
removed by continuing the sparking for several hours.
The gas was then passed slowly over hot copper-foil and
into a drying vessel containing phosphorus pentoxide.

Motion of Electrons in Argon and in Hydrogen. 1041

Two quantities of argon were thus prepared, one having
had the traces of nitrogen removed by sparking for 120 hours
and the other for 70 hours. These specimens of argon will
be referred to as the first and second respectively.

The velocities « and W were determined with both
specimens over large ranges of electric forces and pressures.
With the smaller pressures from 2 to 30 millimetres, where
Z/p is large there was not much difference between the two
specimens, but with the larg¢r pressures from 30 to 150
millimetres, where the range of the ratio Z/p was trom
‘1 to °8, there was a considerable difference. With these
values of the ratio Z/p the velocities of agitation were greater,
and the velocities in the direction of the electric force were
smaller, in the first specimen than in the second. These
results indicate the presence of a small trace of impurity in
the second specimen. From our previous experiments we
found that the loss of energy of an electron in a collision
with a molecule of argon is much less than in a collision with
a molecule of nitrogen or of any other impurity that the
gas might be likely to contain. Small pee. of impurities
have therefore the effect of reducing the velocity of agitation
of the electrons, with the result that the velocities in the
direction of the electric force are increased.

8. The following table gives examples of the experiments
made with the first specimen of argon. The pressures p of
the gas are given in millimetres of mercury, the electric
force Z in volts per centimetre, and the velocity in the
direction of the electric foree W in ecm. per sec. The
quantity k is the factor by which the energy of agitation
of an electron exceeds the energy of a molecule of a gas
= a tO

The velocities W and the factors & for electrons moving
in argon and in hydrogen may be compared by the curves
in figs. 4, 5, and 6.

The curves (fig. 4) give the velocities W in argon and in
hydrogen corresponding to the lower values of the ratio Z/p
from ‘l to 2. The velocities corresponding to the larger
values of Z/p are given in fig. 5 for argon, hydrogen, and
a mixture of hydrogen and argon in the proportion by
pressure of one of hy drogen to 24 of argon. In the ratio
Z/p for the mixture, p is the partial pressure of the
hydrogen.

The values of & are given in fig. 6. There are two curves
for each gas, the lower curves a giving k& for the smaller
values of Z/p from °06 to 1:6, as indicated by the scale at the
foot of the diagram, and the upper curves II for the larger

1042

Prof. Townsend and Mr. Bailey on the
Tasue II.
a Z. Zp. i. Wx10-
150 16° 0-112 96
100 10°5 0-105 95
“5 00 aes? “6168 | ae
M0) 386 | oees, | ono
80 16:8 0-21 126
ae ae Aer ee ee
50 21 0-42 172
ee pe ae
ee ee ey a
ee EE oe 9:19 340) |
is wee 63... | 920 | aoa
Te Ol aoe aos OS ee
OO Ge OT Gos 5g eh es sce) = a
oe ee ee a
On eee Coils em me
= eae 125) locum eee

Motion of Electrons in Argon and in Hydrogen. 1043:

Fig. 6.

‘Z/p scale forcurves If

2 & é is /0 12 /4& 16

Z/p scale for curves I

———

1044 — Prof. Townsend and Mr. Bailey on the

values of Z/p up to 16, as indicated by the scale at the top of
the diagram. Taking p as unity, the curves for argon show
that as the force increases, £ increases rapidly and attains
the value 340 when Z is 1°6 volts per centimetre, and after
a diminution to 310 at 5 volts per centimetre, & rises again
to 325 at 9 volts per centimetre and remains constant at —
that value for the larger forces.

J. The mean free path | of an electron may be obtained
from the formula for the velocity W : |

Wie coals oo ws a or

mu

This formula for the velocity of the electrons is obtained
from Langevin’s more general formula for ions or electrons
when the velocities of agitation are distributed about the
mean velocity u according to Maxwell’s law, wu being the
square root of the mean square of the velocities of agitation.

Jt is difficult to determine the distribution in the case of
electrons moving under an electric force, and according to
Pidduck’s * calculations the factor *92 is more correct than
815, but the exact value of the numerical factor is uncertain,
as the meanfree path depends on the velocity of the electron.
The general conclusions obtained from the experiments as to
the relative lengths of the free paths in different gases or
the variations of the free paths with the velocity do not
depend on the value attributed to the numerical factor in
the formula, and as the value °815 has already been used
in previous calculations, itis desirable to retain it for purposes
of comparison.

The effect of a collision on the velocity of an electron may
be shown by calculating the coefficient of elasticity f by
Pidduck’s formula. This method was adopted in the earlier
researches on the motion of electrons in air +, and in those
on oxygen, hydrogen, and nitrogen which were published
recently {. .

It is simpler, however, to give the proportion of the
energy of an electron which is lost in a collision, as this
quantity is found directly from the experimental results.
The loss of energy of an electron in a collision may be
estimated approximately from elementary. considerations.

* ¥. B. Pidduck, Proc, ond: Math. Soc. vol. xv. pp, 87-12
(1915-16).

+ J. S. Townsend and A. T. Tizard, Proc. Roy. Soc. A, Ixxxvili.
p. 336 (1913).

t Phil. Mag, Dec. 1921.

Pw

Motion of Electrons in Argon and in Hydrogen. 1045

When moving along its free paths between collisions the
mean velocity of an electron in the direction of the electric
force is W ; and since all directions of motion are equally
probable after a collision, the mean velocity in the direction
of the force is zero after a collision and 2W before a collision.
The loss of energy in a collision is therefore 2mW’. When
variations in the mean free paths and the velocities are taken
into consideration, it is found that the fraction A of its mean
energy of agitation mu?/2 which an electron loses in a collision
is given approximately by the formula

V 72
A= 2°46 Paes e . . . . e (5)

The following table gives the mean velocity of agitation w,
and the velocity in the direction of the electric force in
argon for different values of the ratio Z/p and the values of
{and 2 obtained from the above formule. Since / and x
depend directly on the energy of agitation which is pro-
portional to k, the. values of Z/p are chosen to correspond to
definite values of &. The values of J are for the gas at
one millimetre pressure. |

TABLE III.

| {

hiss 5s | Z/p. | WxX10-5. | ux10-7. ix10, | Ax 108

| 190 | a sa ee 115 20 1-79

| 120 | -195 3:9 12°6 14-7 164 |
| 140 | ‘75 | ° 34 13°6 118 154. «|
| 160 355 | 3:6 145 10°3 152 |
| 180 | 440 3°85 15-4 9-44. 154 |
| 200 525 4-15 163 9-02 Tee
| 240 | 71 4°85 17°8 8-52 ea,
| 980 | -95 6-0 19:3 Boe = See
| 820 | 1:25 rar 20°6 888 | 3:45
mers 40 90-2 113 9:7

| 394 110 65 20°7 9-42 24-3
Piste 115 82 20°7 7:92 386 |

i

10. The large values of k obtained in argon are due to
the fact that the loss of energy of an electron in a collision
with a molecule is extremely small, as shown by the figures
in the last column. This loss is* very much less than in
hydrogen or nitrogen. When moving with a velocity of
agitation 12°6 x 10’ cm. per sec., the fraction of its energ
lost by an electron in a collision with a molecule is 1'6 x 10~°
in argon, 5x 10-? in nitrogen, and 4x 107? in hydrogen.

1046 Prof. Townsend and Mr. Bailey on the

The increase of A with the electric foree when the mean
velocity of agitation remains approximately constant at about
20 x 10’ cm. per sec. is clearly due to a large loss of energy
in collisions with velocities greater than the mean, and a
change in the distribution about the mean as Z and W
increase.

As an illustration of what would take place under this
condition, it may be supposed that when the velocity of an
electron exceeds a value A, its velocity is reduced to B when
it collides with a molecule, and while its velocity of agitation
is again increased from B to A, under the action of the
electric force, the electron makes several] collisions with
molecules in which there is very little loss of energy.
The distance z that the electron travels in the direction of
the electric force Z while the velocity of agitation rises from
B to A is z=m(A?—B?)/2eZ, and the total number N of
collisions with molecules while travelling the distance =z is
approximately uz/IW. Hence N is inversely proportional
to the product ZW. Hach collision in which there is a large
loss of energy m(A?—B?)/2 is therefore followed by a
‘Jarge number N in which the loss is negligible, so that the
average loss 1s inversely proportional to N and therefore
directly proportional to ZW. Thus, although the inean
velocity of agitation remains constant, the mean loss of
energy in a collision increases with ZW. In this case the
velocities of agitation are distributed near the mean value wu
when Z and W are small, but as Z and W increase, the
number of electrons with velocities near the mean diminishes
and the number near the limits increases. |

Another example of the effect of a change of distribution
of the velocities of agitation about the mean, occurs when
electrons move in pure hydrogen and in a mixture of argon
and hydrogen. In hydrogen the loss of energy per collision
is much greater for the larger velocities of agitation than
for the smaller. Thus an effect which increases the number
of electrons with velocities near the mean will reduce the
average loss of energy per collision. With a constant
force Z the velocity W in pure hydrogen is in many cases
reduced by about 20 per cent. by adding argon to the
hydrogen, while the mean velocity w of agitation remains
unchanged. The loss of energy in the collisions with the
argon may be neglected, so that in these cases the average
loss of energy in collisions with molecules of hydrogen is
proportional to ZW when the electrons are moving in pure
hydrogen, and to ZW x°8 when the electrons are moving in
the mixture, the reduction being due to a change in the

Motion of Electrons in Argon and in Hydrogen. 1047

distribution of the velocities of agitation about the mean.
In pure argon the velocity corresponding to k=340 may be
taken asa lower limit to the velocity at which a large loss
of energy occurs in a collision. ‘This velocity is the velocity
due to a potential fall of 12°6 volts, and is a lower limit to
the ionization potential.

The increase in Xx 10° from 1°54 to 1:79 in pure argon
when ux 107? changes from 13°6 to 11°5 may be due toa
small quantity of impurity remaining in the gas. It will be
noticed that the mean free path / changes from *118 cm. to
*20 em. with this change in wu, so that the effect of an
impurity would increase as wu diminishes, since the propertion
of the total number of collisions in which there is a
considerable loss of energy increases.

11. The mean free paths of the electrons are much longer
in argon than in nitrogen or hydrogen. When moving
with a velocity of 12°6 x “107 em. per sec., the values of / are
*147 cm. in argon, ‘029 cm. in nitrogen, and -035 cm. in
hydrogen, the gases being at one millimetre pressure. If
the molecules were elastic spheres of the radius ¢ which is
obtained from the viscosity of argon, the mean free path of
the electron in argon at a millimetre pressure would be
*0286 cm.

With the range of velocities of agitation given in the
table, the free path / in argon increases rapidly as the velocity
diminishes, and much longer free paths would evidently be
obtained if experiments were made with higher pressures
and smaller forces. With the amount of pure argon at our
disposal we were unable to make reliable experiments with
values of Z/p less than °105, which gave k=95.

The free paths given in the tables for the velocities
11°5 x10’ cm. per sec. and 12°6x 10’ cm. per sec. are
probably too large, as may be seen by considering the effect
of a large increase of / for a comparatively small reduction
in w, on ‘the relation connecting W with Z,u, and Jl. If the

Zel

formula W= x 0°815 be taken as giving accurate values

of / corresponding to the mean velocity of agitation « when
a large change in wu produces a small change in J, the
numerical factor must be increased when a small reduction
in u produces a large increase in 7. The correction depends
on the distribution of the velocities of agitation about the
mean velocity u, and the rate of change of the mean free
path with the velocity. When these two factors are taken

into consideration, it 1s found that in the case of argon,

1048 Prof. Townsend and Mr. Bailey on the

where the velocities u are about 12x10’ cm. per sec., the
mean free paths obtained by the above formula may be
20 or 30 per cent. above their correct values, More accurate
determinations of .the mean free paths in argon for these
velocities of agitation may be deduced from the mean free
paths in a mixture of argon and hydrogen; and it is of
interest to compare the values of / obtained by the two
methods. |

12. The simplest method of finding the mean free paths
in argon when the velocity of agitation is less than
11'5 x 10" cm. per second, is to find the mean free paths in
a mixture of hydrogen and argon and also in pure hydrogen,
and to calculate thé mean free paths in pure argon from the
two sets of measurements. The velocity of agitation is
controlled mainly by the hydrogen ; and as there is so little
loss of energy in the collisions with molecules of argon, the
principal direct effect of the argon is to reduce the mean

free paths of the electrons, and therefore to reduce the

velocity in the direction of the electric force. In order to
produce any measurable effect on the velocities of the
electrons in hydrogen, it is necessary to add a large quantity
of argon to it. In some previous experiments” it was found
that when the partial pressure of the argon is four times that
of the hydrogen, the velocities in the mixture were not more
than 10 per cent. lower than the velocities under the same
forces in the hydrogen alone.

These observations show directly that the mean free path
in argon for certain velocities of agitation of the electron
must be of the order of fifty times the mean free paths in
hydrogen at the same pressure. As no accurate conclusions
could be deduced from experiments where the velccities
differed by only a few per cent., the experiments were
repeated, using much larger quantities of argon.

The velocities W in the direction of the electric force for
a mixture containing argon at a partial pressure twenty-four
times that of the hydrogen are given in fig. 5, the values
of Z/p being the ratio of the electric force to the partial
pressure p of the hydrogen. Thus, taking p=1, the curves
show that with a force of two volts per centimetre the
velocity of the electrons in pure hydrogen at a millimetre
pressure is reduced from 16 x 10° to 11-7 x 10° cm. per second
by adding argon to bring the total pressure up to 25 milli-
metres, The mean velocity of agitation is only reduced by
1 or 2 per cent. by the argon, so that under these conditions

* Phil, Mag. June 1922,

Motion of Electrons in Argon and in Hydrogen. 1049

the number of collisions of an electron with molecules of
argon at 24 millimetres pressure is less than the number with
molecules of hydrogen at one millimetre pressure.

At the-higher forces the difference between the two gases
is less marked. |

The effect of adding 4 per cent. of hydrogen to pure
argon may also be seen from the curves. Taking the case
where Z is 16 and the pressure of argon 24 millimetres, the
velocity W in pure argon is 46X10? cm. per second (as
shown by the point on the curve for argon corresponding to
Z/p="666). When hydrogen at one millimetre pressure is
added, the velocity is 21 x 10° em. per second (corresponding
to Z/p=16 on the curve for the mixture). The addition of
the hydrogen causes the velocity of agitation to be reduced
from 17°4X 107 to 8-22 x 107 em. per second, and this change
in w would not be sufficient to account for the inerease in W
if the mean free path in the argon were unaltered by the
change in the velocity of agitation.

13. The following table gives the mean velocity of
agitation u, and the velocity W in the direction of the
electric force in a mixture containing 96 per cent. of argon
and 4 per cent. of hydrogen by pressure, for different values
of the ratio Z/p, Z being the electric force in volts per
centimetre and p the partial pressure of the hydrogen.

TABLE LV.
|

| & | *Z/p: |Wx10-%., wxl0-7. | Imx10% | &x10% | JgXx 102.
_ 2s Ae ee ic) oa aie
| |

a 6:35 2:3 2°05 2 ea |
ee he 75 7-95 2:82 2:09 339 138
mee tO Y:] 3:26 2°08 2:06 154
fama Te 1:28 10-0 3°64 1°99 3°88 161
| 13 1-72 111 4-15 1:87 2:66 154
16 2:95 12:2 46 1-75 2-49 138

20 3:25 13-6 B15 151 2-29 110
| 30 | 655 16-7 6:3 112 1:99 61:5
| 40 | 108 | 19-0 7-28 ‘89 1:95 | 39°5
Peels} 156; =| 20-9 8:14 76 200 | 29°5
Beg 260, |.) 23° 9-62 ‘61 2-30 20:0
1100 | 424 | 25:4 115 -48 2-98 136
140 | 64:8 | 26°5 13°6 39 4-00 10-4

The mean free paths corresponding to the velocity wu given
in the last three columns of the table are: l,, for a mixture
containing hydrogen at one millimetre pressure and argon at

Phil. Mag. 8. 6. Vol. 44. No: 263. Nov. 1922. 3 Y

1050 Prof. Townsend and Mr. Bailey on the

24 millimetres pressure, / for pure hydrogen at one millimetre
pressure, and J, for pure argon at one millimetre pressure.
The free path J, is obtained from [,, and U, by means of the
formula:

oe aa

The free paths J, and J,, are shown by the curves in fig. 7.

Vig. 7.

wma.
meee a
Bae’ dee

lia:

14. The free paths in pure argon at a millimetre pressure
are shown by the curves (a) and (6) (fig. 8). Curve’(a) for
the lower velocities of agitation u, gives the free paths
obtained from the measurements of velocities of electrons in
a mixture of hydrogen and argon, and curve (0) the free
paths obtained from the velocities in pure argon. The
free paths tor the range of velocities from 11°5x 10" to
13-5 x 107 were found by both methods, and the curves tend
to coincide with the larger velocities. Exact concordance
can be expected only at points where the variation in the
free path with the velocity is small, or where the velocities
of agitation are very near the mean velocity w. For
velocities between 4107 and 14x10’ there is a large
increase in the free path as uw diminishes; and as explained
in section 11, the free path calculated by formula (4) from

Motion of Electrons in Argon and in Hydrogen. 1051

measurements of W and wv in pure argon is larger than the
true value corresponding to the mean velocity w.

The free path of an electron in argon at a millimetre
pressure has a maximum value of 1°6 cm. for the velocity
3°75 x 107 em. per second, which corresponds to a potential
fall of :39 volt. As the value of J is an average for a
number of different velocities having a mean value w, it 1s
probable that the mean free path for electrons all moving
with the velocity 3°75x107 cm. per sec. is greater than
1°6 centimetres. .

The free paths in hydrogen at a millimetre pressure are
indicated by the lower curve in fig. 8 for purposes of
comparison.

In hydrogen at a millimetre pressure the mean free path /
of an electron has a minimum value of ‘0195 cm. when the
velocity wis 7X 107 em. per sec. As w diminishes / increases,
and when u=13x10', 1=-044. In nitrogen a minimum
-alue of J equal to *026 occurs when u=9 x 10’, and / increases
to ‘057 when w=2°5 x 10%.

It appears from the curves showing / in terms of u that
the free paths in hydrogen and nitrogen would continue to
increase with further reduction in the velocity, and it is
probable that in these gases | attains a maximum value

oY. 2

FFT REIS

1052 Does an Accelerated Electron radiate Energy 2

for certain velocities smaller than that corresponding to
2 volt.
The large increases of the free paths of electrons as the
velocity diminishes are the most remarkable of the definite
results obtained from these experiments. There can be no
doubt that these conclusions about the mean free paths, and

the estimates of the loss of energy of the electrons in

collisions with molecules, are substantially correct, notwith-
standing the possible experimental errors or any uncertainty
as to the exact values of the numerical coefficients in the
formule that have been used.

XOV. Does an Accelerated Electron necessarily radiate Energy

on the Classical Theory?

To the Editors of the Philosophical Magazine.
GENTLEMEN,—

Y the kindness of Professor Born I have learnt that the
absence of radiation from the system of two oppositely
charged point electrons of Lorentz mass accelerated by a
uniform electric field, which I proved in a paper with this
title in your March 1921 number (p. 405), also follows from
a general theory which he worked out so long ago as 1999.
Professor Born’s paper (Ann. d. Phys. xxx. p. 1, 1909) forms
a discussion of the theory of rigidity and of the motion of a
“rigid” electron, on the basis of the principle of relativity,
and one of his conclusions is given in the following words :—-
‘“‘ Bemerkenswerth ist, dass ein Hlektron bei einer Hyper-
belbewegung, so gross auch ihre Beschleunigung sein mag,
keine eigentliche Strahlung veranlasst, sondern sein Feld
mit sich fiihrt, was bis jetzt nur fiir gleichformig bewegte
Elektronen bekannt war. Die Strahlung und der Widerstand
der Strahlung treten erst bei Abweichungen von der Hyper-
belbewegung auf.”. |
This remarkable result of the early days of relativity
seems to be but little known in this country, may I therefore.
be permitted to direct attention to it here? By “ Hyper-
belbewegung”’ is meant the motion of a particle whose world-
line in the four-dimensional universe is hyperbolic, or, which
comes to the same thing, the graph of which on an a, ¢
diagram forms an hyperbola. It is the equivalent in the
relativity theory of uniform acceleration in Newtonian
dynamics.

Yours faithfully,
September 30th, 1922. S. R. Mitner.

iP 10530 '|

XCVI. Simple Model to illustrate Elastic Hysteresis.

To the Editors of the Philosophical Magazine.
GENTLEMEN,-—

EGARDING the very interesting paper by Mr. 8. Lees, ~
M.A., St. John’s College, Cambridge, ‘‘ On a Simple
Model to illustrate Elastic Hysteresis ?’ in your current
issue for September, 1922, may I be permitted to refer to
an earlier publication by your present correspondent in the
pages of the ‘ Physical Review’ (Ithaca, N.Y., U.S.A.) for
June last.
Before the Physical Society there had been described the
model illustrated below, which involved both spring and

la AAA A A OD Af A 6
ola memos eae oo eo a ee

Ae, e, TV VU / / ev, , VV VW

& AS era AR a an Aen f'n

Pa See ee ee
¥, 7 ) U U UV U UJ
=a

—

cm
:
t |
is
dip
rims
3 ig 8
tin:
dt
dt}
d\_§
aoe
d{_}
‘ims
H+
a8:

solid frictional ‘constraints, similar to those employed by
Mr. Lees.

The model enabled the writer to explain not only mechani-
cal hysteresis, but electrical and magnetic hysteretic systems,

1054 Simple Model to illustrate Elastic Hysteresis.

involving formulze of the type
B=p.H+1.

The hysteresis was shown to be due, quite naturally, to
the I-Component.
319 Dorset Avenue, T am, Gentlemen,

Chevey Chase, Md., U.S.A. Yours etc.,
September 23,1922. A. PRuss.

> Lo the Editors of the Philosophical Magazine.
GENTLEMEN,—

From the sketch given by Mr. Press, it is quite clear that
his mode! will produce effects analogous to those of the
model described in my paper (Phil. Mag. September 1922).
Until to-day, I had not seen the short abstract of Mr. Press’s
paper in the July (1922) number of the ‘ Physical Review.”
I hope his full paper will be available before long.

As evidence of the curious coincidence of the papers, I
note that Mr. Press’s paper was read to the American
Physical Society on April 21st last. My own MS. was in
the hands of the publishers about April 5th last (the interval
between this date and publication I am not responsible for).

Referring to my fig. 3, since the model is symmetrical
- about OO,, only one-half of the model need be considered in
actual practice.

I should like to emphasize a result arising out of my
discussion, namely the fact that with either Mr. Press’s or
my own suggested model, an alteration by overstrain of the
point of departure of the stress-strain curve in tension from
the straight-line law, is accompanied by an alteration of the
corresponding point of departure in compression. This
result, although of considerable technical importance, does
not appear to have received much theoretical attention.

51 Chesterton Road, Yours etc.,

Cambridge. S. LEzs.
October 6th, 1922. |

[- 1055 ]

XCVII. Notices respecting New Books.

Cours de physique générale. Par H. Ounivirr. (Paris: Librairie
Scientifique. J. Hermann, 1921.)

Pus tome premier is devoted to a discussion of units, gravi-
tation, electro-magnetic, in the C.G.8. system, and a new
M.T:S. system (metre-tonne-second) recommended here.

What is the matter with Physics Training for Engineers ?
This is the question being asked to-day. The answer is sure to
be: [tis the C.G.S. source of Arrogance.

These niggling microscopic units are thrown aside by the young
engineer as soon as he is tree from the tyranny of the lecture and
examination room.

They are described by Halsey in his ‘ Handbook for Draftsmen
as a ‘Monument of scientific zeal with ignorance of practical
requirements. The object of Weights and Measures is to Weigh
and Measure, not merely to make calculations.”

The second is acclaimed as the unit of time, because it keeps g
down to an easily remembered number for calculations. But the
engineer prefers the minute to record revolutions, as an interval
that can be checked with accuracy on a stop-watch. And in the
astronomical units of Relativity the unit of time is nearly 1000
years.

But when it came to the choice of such diminutive units as the
centimetre and gramme, in preference to the commercial metre
and kilogramme, the decision was made for the supposed con-
venience of making the density of water unity, so that density and
specific gravity would be the same number, and a name would be
saved.

On the M.K.S. (metre-kilogramme system) the density of
water is 1000 (kg/m*), with the advantage of keeping the allow-
ance for buoyancy of the air in view, as a correction of the last
figure, say a deduction of 1°25; and soa Table of Density would,
if absolute, require the same deduction of 1°25 to give apparent
density in air. But this correction is out of sight in the C.G.S.
system, and we never hear it mentioned, although an accurate
measurement must be carried out by a human being in an atmo-
sphere where he can breathe, and not in vacuo.

The same theoretical pedantry has influenced our author in his
selection of the M.T.S. system, with the same view of keeping
specific gravity and density the same figure, to the same decimal,
but usually ignoring the decimal when air buoyancy makes itselt
felt ; paramount in the balloon.

A writer on Hydrostatics is equally loose when he tells us to
neglect the pressure of the atmosphere. He should observe the

?

1056 =~—— Notices respecting New Books.

distinction between pressure, as gauge or absolute, as he would be
compelled on changing from the non-condensing locomotive to
the condensing marine engine, or in any thermodynamical caleu-
lations. In the high pressures of Internal Ballistics of Artillery,
the difference may be disregarded as unimportant. But it is
strange to read in an elementary text-book of this neglect, when
the author is employing his favourite absolute dynamical units in
Hydrostatics, and speaks of an atmospheric pressure of about
seventy thousand poundals on the square foot as something of
triflmg account. .

On the M.T.S. system with g=9°81, m/s® the absolute unit of
force would be about the heft of 10-2 kilogrammes, and an atmo-
sphere of one kg/em*, or 10t/m* would be expressed by 981 in
M.K.S. units. . .

Our artillerist reckons his pressure in the ton/inch’, of about
150 atmospheres. <A normal pressure of 20 tons/inch* would be
3000 atmospheres, or 3 million M.T.S. units; and here Halsey
would begin to protest. The C.T.S. system (centimetre-tonne—
second) could reduce this to 300 units, say 35t/em? in the
gravitation unit the artillerist would employ in a measurement
of Force, and so on in a convenient scale for record and cal-
culation. |

It is when we come to Hlectro-Magnetic measurement we
find the powers of 10 require such careful attention, and a
system must be selected of universal acceptance in broadcasting
the theory.

A Comprehensive Treatise on Inorganic and Theoretical Che-
mistry. By Dr.'J. W. Metior. Vol I. pp. xvi4+1065.
Vol. IL. pp. viii+ 894. (London: Longmans, Green & Co.
1922.) £3 3s. net each vol. :

Dr. Mettor has undertaken a very heavy task in attempting to
write single-handed a Comprehensive Treatise on so vast a subject
as is now covered by the title ‘Inorganic and Theoretical
Chemistry.” The two volumes of the Treatise which have been
issued furnish abundant evidence of the special qualifications
which Dr. Mellor has brought to his task, and of the skill and
industry with which he has marshalled the data with which the
volumes are so well stored. But whilst the competence. and skill
of the author are beyond dispute, the magnitude of the work
which he has undertaken carries with it certain obvious dis-
advantages. Thus, in order to cope with the difficulty of bringing
even a single volume to completion, instead of spending his whole
life in keeping his information and indexes up to date, the author
has been obliged in certain instances to adopt rather drastic
methods of treatment, by resolutely closing down some sections
of the book in which new information is being gathered so quickly

Notices respecting New Books. 1057

that perpetual re-writing would be required in order to assimilate
it completely. These sections of the book happen to include
those which will be perused with the greatest interest by readers
ot the Philosophical Magazine, in which Magazine so much of
this newer knowledge has been published. ‘Thus, on page 104 an
estimate of the Avogadro Constant is given which dates from
1899, although later determinations are quoted on pp. 639 and
753. Again, the whole of the sections on the Atomic Theory
are based on the conceptions of 20 years ago, before the
complications arising from the discovery of the radio elements and
of isotopes had arisen, and before the simplifying factors intro-
duced by the experimental determining of atomic numbers had
appeared or made their influence felt. Strictly speaking, this
omission applies only to the text, since the list of International
Atomic Weights on page 199 and the Periodic Table on page 256
have been revised to include Atomic Numbers, as well as Atomic
Weights; but these are not referred to in the index nor explained
in the text, although atomic numbers are mentioned on page 255
and isotopes on page 266 as subjects for discussion in a future
volume. In view of the masterly way in which he has dealt with
other subjects, there can be little doubt that these later discoveries
will be adequately described and discussed in the volume which
deals with the radioactive elements; but there are many para-
graphs, such as the one on page 200, where the inquiry is made
as to whether the atomic weights are whole numbers, which would
certainly have been written otherwise if the facts in reference
to isotopes had been known when the text was first drafted.
Again, a considerable part of the author’s very able discussion of the
classification of the elements is tedious to the point of positive
irritation to a reader who is accustomed to see all these problenis
simplified, even if not completely solved, with the help of atomic
numbers. The fact that certain of these sections are already out
of date, if not actually incorrect, is part of the penalty which
must be exacted from anyone who insists on being the author
rather than the editor of a ‘‘ Comprehensive Treatise ” on a rapidly
growing science with two thousand years of history behind it. It
is necessary toadd that the chapter on Crystals and Crystallization
bears no evidence of being rendered obsolete or of requiring to be
re-written in the light of modern work on the analysis of crystal
structure by X-rays, of which a satisfactory summary is given,
including references as recent as 1920. Moreover, the author's
reputation, both as a mathematician and as a physical chemist, is
a sufficient guarantee of the adequate treatment of subjects such
as veaction velocities and energetics, where there is no risk of
obsolescence arising from rapid new developments, so that
the problems can be reviewed and the sections written up at
leisure.

The chemical, as contrasted with the physical, portion of the
work calls for nothing but admiration; and whilst the Treatise

——

1058 Notices respecting New Books.

has been based mainly on the requirements of chemists, physicists
will also find it of pre-eminent value as a work of reference, to
which they can turn for information on all the chemical topics
which are covered by the title.

Science in the Service of Man: Electricity. By Sypnuy G.
StaRLine. (Longmans Green & Co., 1922; price 10s. 6d. net.)

THE object of this book is to give the general reader an account
of the present stage of electrical knowledge.

After a brief historical sketch the industrial applications are
dealt with, such as the electro-magnet, dynamo, electro-motor,
telegraph, telephone and alternating current transformer.

Later chapters deal with electrolysis, discharge of electricity
through a gas, X-rays, radioactivity, the electro-magnet theory
and wireless telegraphy.

The treatment throughout is entirely non-mathematical. A
‘ book suitable as a school prize for a boy who delights in making
things for himself, as a welcome change from the schoolmaster’s.
favourites, Scott’s Poetical Works, or Macaulay’s History of
England.

Some dismal X-ray photographs of surgical interest cast a
gloom; otherwise much of the apparatus illustrated is simple
enough for a boy to make for himself,

La théorie Einsteinienne de la Gravitation. Essai de vulgari-
sation de la théorie. Par Gustave Mine. (Paris: J. Hermann,
1922.)

Tuts is a translation, by J. Rossignol, from the German in the
Deutsche Rundschau, and it is addressed in book form of 100:
pages to a public not supposed to be acquainted with higher
mathematics, but none the less capable of appreciating the
precision and clarity of the Relativity Theory. 7

It will serve as an introduction to the more extended treatment
of Eddington’s ‘ Relativity,’ and it is an eloquent presentation in
popular language of the new ideas that arise in a discussion in.
general company at the present day.

Wave-lengths in the Arc Spectra of Yttrium, Lanthanum, and
Cerium and the preparation of pure rare Harth Elements.
Bureau of Standards, Government Printing Office, Washmeton..
Scientific Papers. No. 421.

THIs paper is a continuation of the work already undertaken on.
tiie mapping of the red and intra-red spectra of the chemical
elements. The results for about 35 elements have so far been
published, and here the results of the study of the arc spectra of

Notices respecting New Books. 1059

yttrium, lanthanum, and cerium are given in detail, about 175
lines for Y, 400 for La, and 1700 for Ce. Measurements from a
number of Prof. Eder’s spectrograms are included in these tables.
The second part of the paper describes the preparation of rare
earth elements in the cerium and yttrium groups. The publication
is ready tor distribution, and those interested may obtain a copy
by addressing a request to the Bureau until the free stock is
exhausted.

Lhe Journal of Scientific Instruments: A monthly publication
dealing with their principles, construction and use. Produced
by the Institute of Physics with the cooperation of the National
Physical Laboratory. ee Number. [Institute of
Physics, 10 Essex St., Strand. W.C. 2.]

THis preliminary number of a Sabot Journal of Scientific
Instruments is due to the recognition of the fact that there is no
journal in the English language which covers the ground described
in its title. Incidental descriptions of apparatus undoubtedly
appear in researches published in other periodicals; but there 1s
no room in such cases to give more than casual accounts ; and
moreover the accounts that are given appeal only to the few that
are interested in the main subject of the paper while instruments
are usually of value for researches of quite different character
from those for which they were originally developed. It is
intended in the proposed journal to give measured drawings of
instruments as well as a scientific examination of their design.

This is a sanple number and it has been distributed broadcast
amongst scientific and industrial people. The possibility of this
free distribution was made possible by a grant from the Treasury,
through the Department of Scientific and Industrial Research.

It is not confined to any one branch of science. There is no
department in which instruments are not used. The present
number shows that it is intended to cater for them all. Physio-
logy, Astronomy, Ordnance Survey, Optics, Aeronautics, Engin-
eering, Laboratory Arts, are all pepEreenied in the present
number.

The publication of a journal of this kind is a work of national
importance. During the war great advances were made in many
directions largely due to the scientific aid that was given to makers
in the design and construction of instruments intended for novel
purposes. This journal will tend to perpetuate such cooperation ;
and should serve as a continuous stimulus to the manufacturer.

The preliminary number can only be succeeded by others if the
project receives sufficient promise of support. We are asked by
The Institute of Physics to mention that a great many persons
who have received a copy of the preliminary number of the pro-
posed Journal and who may wish to support it have not yet filled

1060 Geological Society.

in the form inserted in the Journal. It is hoped that all who
intend to subscribe will inform the Institute without delay so that
an estimate may be formed of the support which may be relied
upon.

. XCVIII. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
(Continued from p. 288. ]

February 17th, 1922.—Mr. R. D. Oldham, F.R.S., President,
in the Chair.

TIVE President delivered his Anniversary Address, the subject

of which was the Cause and Character of Harthquakes,
using the word in its original sense of the disturbance which
can be felt and, when severe, causes damage, as apart from
that which gives rise to distant records, only obtainable by
special instruments. ‘he character is sufficiently established as
a form. of elastic wave-motion, of extreme complexity; this
is present in all cases, and may be distinguished as the orchesis
of the earthquake. In addition there is, in some cases, a
molar, permanent, displacement of the solid rock, which forms
the mochleusis. Further it has been shown, definitely in one
case and inferentially in others, that, where mochleusis is present,
the disturbance of the surface-rocks, to which the earthquake
proper can be referred, is only the secondary result of a more
deep-seated disturbance, which has been distinguished as the
bathyseism. The origin of the elastic wave-motion must be
a sudden disturbance of some sort; the depth of origin can, in
many cases, be shown to be very moderate, not more than about
10 miles, and in this outer portion of the Harth’s crust the only
sudden disturbance conceivable is fracture, due to strain in excess
of the power of resistance. In certain cases such fracture,
accompanied or not by displacement, has been recognized at
the surface; and measurements of the displacements show that
a state of strain must have existed before actual rupture took
place, but give no indication of the rate of growth of the strain.
The commonly-accepted notion that the growth must be very slow
appears to depend on the assumption that the strain is due to the
same causes as those that have produced the folding and faulting
of the surface-rocks, and also on the assumption that tectonie, as
other geological processes, must necessarily be slow. The problem
can only be attacked through the variation in the frequency of
earthquakes ; the precautions needed in applying this method
are indicated, and, when applied, leave only one existing record
available, the Italian one. A discussion of this shows that the
rate of growth of strain is, at slowest, such that the breaking-

Geological Society. 1061

point will, on the average, be reached in, at most, a year, and, at
the quickest, may be of such rapidity as to be analogous to a
separate explosion for each earthquake. The possibility of so
rapid a growth of strain being due to tectonic processes, as
ordinarily understood, is considered and rejected, so that the
changes by which the strain is produced must be referred to the
material below the crust. Recent researches on the change of
bulk, resulting from a change in the mineral aggregation of the
same material, are referred to, as indicating one means by which
the required effect may be brought about; and, without restricting
the possibilities of other unknown processes, the results are
summarized as indicating that the cause of the great majority of
earthquakes is a rapid growth of strain, and that the production
of this strain must be referred to changes which take place in the
material underlying the outer crust of solid rock, which is directly
accessible to geological observation.

March 22nd.—Prof. A. C. Seward, Se.D., F.R.S., President,
in the Chair.

Sir Cuartes Joun Hormes, Director of the National Gallery,
proceeded to deliver a lecture on ‘Leonardo da Vinci asa
Geologist.’ The Lecturer began by referring to the growth in
recent years of Leonardo’s reputation as a man of science. This
rapid growth led recently to a reaction, and it was. now not
infrequently stated that Leonardo’s scientific discoveries were in
the nature of fortunate guess-work, and were neither proved nor
accompanied by experimental research. In view of this attitude,
the Lecturer felt that he could not present any statement of
Leonardo’s discoveries to a scientific body, such as the Geological
Society, except in the form of extracts from Leonardo’s own
writings, which would enable them to judge for themselves
whether his scientific reputation was firmly founded or not.

Reading extracts from the translations made by Mr. McCurdy
and Dr. Richter, the Lecturer pointed out how Leonardo was
really the first to have a large and accurate conception of the
causes underlying the physical configuration of the Earth. His
studies of aqueous erosion, of the formation of alluvial plains, of
the process of fossilization, and of the nature of stratification, led
him to a logical conviction of the immensity of geological time,
and were so far in advance of the dogmatic thought of his age,.
that they exposed Leonardo to the Ghar ge of Pein: There can
be no doubt whatever, that if he inal not confided these dis-
coveries to the almost undecipherable script of his note-books, and
kept them hidden there, he would have been one of the first and
most notable of the martyrs of science.

Caution thus compelled him to work in isolation, and to keep
his results concealed: he had no scientific instruments, no corre-
spondents to furnish him with observations on geological conditions
elsewhere ; yet his grasp of the physical history of the portions of

1062 Intelligence and Miscellaneous Articles.

Italy which he had personally visited, was so sound, so firmly based
on experiment and research, and so entirely in accordance with
modern knowledge, that he must be considered the one great
geological predecessor of Lyell.

Since publication of his discoveries was impossible, Leonardo
left a record of them in his paintings, as in the background of the
‘Monna Lisa,’ the ‘Madonna & St. Anne,’ and in a less degree in
our own ‘ Madonna of the Rocks’ in the National Gallery. Here
we find pictures of the primeval world as he imagined it, when
seas and lakes ran up to the foot of the mountains, to be slowly
displaced and silted up by the detritus which the rain carried down
from the summits. From this reconstruction the pictures derive
that sense of action, apart from place or time, which has fascinated
generations who could not understand Lecnardo’s meaning as we
-can understand it now.

XOIX. Intelligence and Miscellaneous Articles.

THE BUCKLING OF DEEP BEAMS.
To the Editors of the Philosophical Magazine.
GENTLEMEN,—

T wisn to thank Professor Timoshenko for his letter in the May
number of the Philosophical Magazine, in which he gives
references to earlier work on the above subject. I had discovered
‘some time ago that other people had preceded me in this investi-
gation, but I had not taken the trouble to look up their papers
as my own were already published. Since reading Professor
‘Timoshenko’s letter, I have, however, examined Mr. Michell’s
paper in the Philosophical Magazine for September 1899, and
was astonished to find how closely my own first paper (Phil. Mag.,
‘Oct. 1918) resembled his. We have solved the same problems in
much the same way, and agree perfectly except on one question,
the one numbered IV. in Michell’s paper, and Case 5 in mine.
‘Here I venture to say that he is wrong, for his solution makes
the torque zero at the ends, which is obviously not true for the
actual problem. My second paper carries the subject a little
further, but, of course, it is the first step that counts, and
Professor Timoshenko does not tell us whether Michell or Prandtl
anade the first step.

Yours faithfully,

College of Technology, JOHN PRESCOTT

Manchester.
May 21st, 1922.

Intelligence and Miscellaneous Articles. 1063
ON DAMPED VIBRATIONS.

T’o the Editors of the Philosophical Magazine.
GENTLEMEN,—

The note in Phil. Mag. July 1922, p. 284, entitled “ Note on
Damped Vibrations” gives a somewhat inadequate account of
a problem treated in detail by Routh, ‘ Dynamics of a Particle’
(1898 ed.) p. 65.

As a dynamical problem, the chief point of interest seems to
have been overlooked by the author of the above note, namely, the
discontinuity rendered possible by the assumption of constant
friction (Routh, J. ¢.).

Some further results are easily obtained in this direction ; thus
if the particle is placed at rest at distance w, from the origin it
will execute semi-vibrations (about alternative centres), where
n is the least integer determined by _x,| —2nIF*/c* < F/c’; if the
particle is projected with any velocity, | v,| may be taken to refer
to the first position of rest attained.

This excludes an easily determinable portion of the series of
seml-circles considered in the note.

The extended problem, including friction proportional to the
velocity, is discussed by J. Andrade, Comptes Rendus, 5 Jan. 1920.

Artillery College, Woolwich. C. EK. Wrient.
July 18, 1922.

THE MAGNETIC PROPERTIES OF THE
HYDROGEN-PALLADIUM SYSTEM.

To the Editors of the Philosophical Magazine.
GENTLEMEN ,-—

In a Paper communicated to the Royal Society (Proc. A, vol.
101, p. 264, 1922), I described a number of experiments on the
measurement of the specific magnetic susceptibility of samples of
palladium-black which had been charged with different amounts
of hydrogen. The results showed that the susceptibility
decreased rapidly with increase of hydrogen content.

My attention has just recently been directed by Mr. H. F.
Biggs to a research carried out by him and described in the
Philosophical Magazine, xxx. p. 131 (1916). The latter experi-
ments were made on a sample of palladiwm-foil, the susceptibility
of which was found to decrease very considerably with increase of
hydrogen content, my results for palladium-black thus agreeing
with his on erystalline palladium. Mr. Biggs’ work had entirely
escaped my notice, the only experiments on the magnetic pro-
perties of crystalline palladium of which I was hitherto aware
being those of Graham (Jour. Chem. Soe. vol. xxi. p. 430, 1869).
Graham’s deductions seemed so extraordinary, in the light of
what is now known about the magnetic property of free and

1064 Intelligence and Miscellaneous Articles.

combined hydrogen, that I thought 1t was worth while investi-
_ gating palladium-black, his experiments having been conducted
on crystalline palladium.

It appears that, from Mr. Biggs’ and my own experiments,
there is no evidence for the existence of a highly magnetic
hydrogen-palladium system, as qualitatively described by Graham,
nor for the existence of a magnetic “ hydrogenium,” whether the

palladium is in the crystalline. or amorphous “form.

ame Gentlemen,

29 Amherst Rank i
Withington, Manchester, Yours faithfully 4
July 20, 1922. A. HE: Oxuinry.

SHORT ELECTRIC WAVES OBTAINED BY VALVES.

To the es of the Philosophical Magazine.
GENTLEMEN,—

In the July number of the Philosophieal Macanae Messrs.
FE. W. B. Gill and J. H. Morrell have given a most interesting
account of experiments with thermionic valves used to maintain
electric oscillations of very high frequency.

I thought it might not be without interest to mention an
alternative method of obtaining intense oscillations of short wave-
length which depends for its success on the ordinary kind of
electrostatic coupling in the valve.

The experiments, carried out in 1917, were never published.

ilt--[1]1

The diagram illustrates the principle of the arrangement. A
is a small loop of wire, which, with the capacity of the valve,
forms a closed circuit of high natural frequency. B is a tunable
closed circuit joining grid and filament. When B is adjusted
properly vigorous oscillations occur in A which were used for
the usual telegraphic and telephonic purposes.

Yours faithfully,

The University, Leeds, ' RR. WHIDDINGTON.
July 31, 1922.

NEWMAN, Phil. Mag. Ser. 6, Vol. 44, Pl. V.

Cathode.

Tube C.
Llectric
Discharge.

‘THOMAS.

Phil. Mag. Ser. 6, Vol. 44, Pl. VI.

Lrg. 3;

Microphotographs of Orifices 1, 6, and 9.

SAWP SN Ses

aie erntig Feed

nate

Bonp.

Phil. Mag, fer 6, Vol. 44, Pl. VII.

Gaia C:
Phil. Mag. Ser. 6, Vol. 44, Pl. VIII

«x 100

THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SQLENCIE

DECEMBER 1922.

C. Atomic Systems based on Free Electrons, positive and
negative, and ther Stability. By R. Harcreaves, M/.A.*

Part I.—Sreapy Motion.

& the atomic scheme of which a planetary system is the
model, negative electrons have the position of planets,
and a positive charge is condensed at a central nucleus. It
is then postulated from experience, chemical, electrical, and
spectroscopic, that the nucleus shall oni an integral
number of standard charges. But if this postulate is
essential, it seems imperative to provide a structure in which
positive as well as negative electrons are discrete. The
scheme with multiple core takes no account of the mutual
repulsion of members of the core, and so ignores the primury
conception of separate existence attaching to an integral
number. An orbital motion, which the scheme provides, i is
however in all probability an essential feature of the atomic
ageregate.

“For. the present scheme the two types of electron
furnish the raw material ; they are assumed to be discrete,
to carry charges differing only in sign, and to possess
inertias differing widely in amount. It is found possible
to realize kinetic structures in which these constituents
are bound together by the electrical forces due to their
mutual action alone. There is no need to suppose the
normal laws of attraction and repulsion to be in abeyance,

* Communicated by the Author.
Phil. Mag. Ser. 6. Vol. 44. No. 264. Dec. 1922. 3 Z

1066 Mr. R. Hargreaves on Atomic Systems

no need for a vement as difficult to explain as the atom
itself ; an orbital motion is provided, and the materials
could not be fewer. The question how far these structures,
regarded as representations of the atom, meet the demands
from the various branches of Physics and Chemistry is
therefore a fundamental one, to which it is desirable to
obtain an answer as complete as possible.

§ 1. The characteristic feature of the scheme is a structure
or configuration consisting of two concentric rings, one of
positive the other of negative electrons in like number.
The charges are disposed at regular intervals on the circum-
ferences of their respective circles, while in angular position
elements of one type lie half-way between those of opposite
type (fig. 1). For this configuration we are seeking, not a

Fig. 1.

Arrangement of positive and negative electrons in concentric circles
for n=4. :

position of equilibrium but one of steady motion for each
element in its own circle. The symmetry of the structure
ensures a purely central force on each element; the main-
tenance of the symmetry demands a common angular
velocity, which requires an adjustment of the radii. The
condition to be fulfilled is

Moa, 2 moa, = FB 2i i ae

m being mass, a radius, F attraction to centre, with the sub-
scripts 1 and 2 attached respectively to positive and negative

based on Free Electrons. | 1067

elements. When account is taken of the fact that mg/m, is
of the order 1/1800, it is clear that if the electrical forces
F’, F, are on the same scale, it will be necessary to com-
pensate the greater mass of positive elements by a relatively
small radius. The attempt to do this fails on account of the
strong repulsive force called into play. If there is no great
inequality of radii then the force on the negative element
must be relatively small, that is we must seek a position
which is nearly one of equilibrium for negative but not for
positive elements. There is only one such position. In that
position the negative ring has a less radius than the positive,
with a difference small except for small values of the
number nx of elements in the ring, as will appear from
the following argument, certainly applicable when 2 is -
not small.

Fi

2.

N

Suppose, all charges on one circle as in fig. 2, and con-
sider the action on a negative electron N. The positive
pairs on either side give a diminishing series of contribu-
tions to a central attraction ; the repulsions due to negative
pairs form a second diminishing series. The first series is
the greater as representing nearer pairs. If N were now
placed in the middle of the chord P,P,, the most important
term of the first series would be out of action, and the
second series representing repulsion would be the greater.
Between these two positions of N is the position sought,
which for x great is found to show a difference in radii
‘about 2/3 of the sagitta of the are P,P}. Rough values for
the ratio of radii are 1°73 for n=2, 1:19 for n=4, 1:05
for n=8; the departure from unity already small and
pointing to the asymptotic law, variation as n-’,

a3Z 2

1068 Mr. R. Hargreaves on Atomic Systems

§ 2. We may for shortness use the term ion for “ positive
electron,’ leaving the unqualified word for the negative
electron: the mobility of the latter would suggest an inter-
change of terms if it were feasible.

The feature that the outer ring contains the more massive
ions contributes to stability through the protection afforded
to the mobile electrons. In part this is due to actual
screening, but the main defence lies in the fact that near
each electron are two ions, so that external action on the
former calls into play a strong counter-action from the
latter.

The angular velocity in the orbit is known when the central
force has been found, whether we are dealing with the double
ring or a single ring with core; but in the former case the
adjustment of radii requires a preliminary calculation. In
each case tabulation is needed when the number 7 in a ring
is small, and an asymptotic formula can be used when n is
not smali. For the double ring when the adjustment of
radii has been made, the angular velocity is given by

m0)'ay? = Ne?s a a ee
where eis the standard charge and N a number tabulated
below. An asymptotic formula (cf. § 13) is

N= "441 n+-424n1 +4... 3) 2

which is of service from n=10 upwards.
For a multiple core +ne,and a ring of n electrons the
formula is

mga? a? = INVER Se ee (iii. a)
where
N,="98n— “366460 logyg - 702182n5* 25 ia

is an asymptotic formula. The calculations required for N
are more serious than for N., but the asymptotic formula
simpler. |

The value of n for which N, becomes negative lies beyond
the range of n required for application to atoms. N,/n is at
first greater than N/n, is not much different for n about 30,
and then diminishes much more rapidly than N/n as n is
increased. As (i1.a) and (iii.a) have the inertias m, and my
respectively it is clear that the numerical relations connecting
w and a are very different, and must involve corresponding
differences of interpretation.

based on Free Electrons. 1069

§ 3. It is consistent with the hypothesis of discrete charges
to suppose a unit at the centre. The adjustment of radii for
steady motion is possible with a positive centre when n > 4,
not when n=2 or 3, for the mutual repulsion is not then
adequate to balance the central attraction, and so give a
position of equilibrium near which an electron in the ring
must lie. The inequality of radii is increased by the presence
of the central unit. If the latter is displaced axially while
the positive and negative rings are supposed to remain in
one plane the position is clearly stable for the central charge,
because the attractive elements are the nearer.

For a negative centre the adjustment of radii is possible
for all values of n; for n>4 the inequality of radii is
reduced, for n=2, 3, or 4 it is reversed and the negative
ring is outside. With the two rings in one plane the
position would, for axial displacement of the central unit,
be stable tor n=2, 3, or 4, unstable for other cases. A more
complete treatment in which account is taken of the relative
movement of the planes of the twe rings, shows that the
negative electron at the centre ts always axially unstable, the
positive always stable. Cf. § 33.

A positive structure is found for the cases n=2 or 3, some-
what unexpectedly, by taking two positive units on the axis,
one above the other below the plane of the rings, with a
separation wide enough to make the attraction of the pair on
an electron of the ring less than that of a unit at the centre.
This proves the beginning of a series which continues up to
the value »=6 and there stops. In the range from »=5
to n=8, containing important elements, B, CU, N, O, there
are two positions : one where the attraction of the pair on an
electron is less than that of a unit at the centre, the other
with closer axial units and an attraction greater than that of
a unit at the centre.

This completes the forms of the fundamental structures,
for which the symbols R,(0), R,(+), R.(+, +) may be
used—rings with vacant centre, positive centre, or, in a few
cases, two axial units. .

§ 4. As subsidiary to the main structure it is proposed to
consider the case of electrons describing nearly circular orbits
about the centre of R,(+), either inside the inner or outside
the outer ring. The central unit alone can maintain as many
as four electrons in one circular orbit, with only a small
residual attraction forfour. The central component of force
due to the ring fluctuates between the two signs, but its
mean value gives a reinforcement of the central attraction ;

———
ee

1070 Mr. R. Hargreaves on Atomic Systems

fluctuation and mean value being greatest for orbits near to
the rings. No truly circular orbit is possible, and the inner
and outer positions for satellites are in general alternative.

The cases of R,(+, +) from n=5 to n=8, in which the
closer position is taken, have the advantage of an increased
central attraction, for outer satellites certainly.

The presence of satellites alters in some measure the
character of the two rings, which may now be regarded as
a type of freely equilibrated nucleus controlling a limited
number of satellites.

As regards the number of satellites there is little doubt
that the case of one only is the most important on the
ground of freedom from instability ; if internal its relation
to the nucleus seems specially intimate. Outer satellites are
more exposed to attack but also more readily replaced than
inner satellites, when we consider the possible action of other
atoms or of free electrons.

$5. The polarizing action of external force is simplest
when the force is directed along the axis of the rings, and
then brings about a small separation of the planes of positive
and negative elements. A force in the plane of the rings
gives rise to displacements radial and tangential so nearly
balanced in opposite parts of a ring as to yield little resultant.
The bond between the rings is sufficiently strong, at any rate
when 7 is uot small, to limit the separation of planes to an
amount which does not sensibly modify the structure. For
satellites the resistance offered to axial forceis slight. Again
a central ion, though stable axially, suffers a displacement on
a sensibly greater scale than the separation of rings. If it
is detached by external force it will be subject to attraction
by the next neutral structure in its path, may be incor-
porated and so move forward by halting steps, functioning
as centre for a succession of atoms. The mode of conduc-
tion differs from that by free electrons much as electrolytic
from metallic conduction ; free electrons have large velocities
in random directions modified by external force, while ions
move with less velocity but follow more closely the action of
external force. There is the further distinctive feature
of incorporation at various stages. C7. § 34.

An external field of magnetic force in the direction of the
axis will give rise to central force in opposite senses on
the elements of the two rings. According to the sense of
the magnetic field or according to the direction of rotation
in the rings, the effect may be to increase or diminish the
difference of radii, thus altering the period.

based on Free Electrons. 1071

The external magnetic forces due to the separate rings are
not so nearly balanced as electrostatic forces, for there is a
difference in velocity which implies a difference in the
strengths of the equivalent continuous circuits, whereas for
electrostatic force only the difference in position exists. If
satellites are present there are unbalanced effects of much
greater amount, different also for internal and external
satellites ; they will be opposite in character to those of thie
double ring when the direction of revolution is the same, for
in the latter the balance is in favour of the positive. ‘To
these differences we must look for an explanation of
diamagnetic and paramagnetic properties.

In these two fundamental matters the presence of satel-
lites is seen to be influential, and in respect to spectroscopic
phenomena their importance is at once evident.

§ 6. It is proposed to bring forward material bearing on
(i.) atomic weights, (ii.) the gravitational constant.

In respect to (i.), contact is sought with the scheme of
atomic numbers suggested by van der Broek and developed
by Moseley and Bohr. As for the neutral ring there are
n ions of mass m, and n electrons of mass m., the primary
measure of mass is n(m +m). Since m,+mz, cannot difter
much from the mass of the hydrogen atom, a varying factor
somewhat greater than 2 is to be accounted for. Consider
first the point of view of inertia.

(a) The equation for internal motion Ne?=m,w?a;*> may
be written as ne*/a?= (nm,/N)o?a,= M,07a,, say; a form in
which simplicity is given to the member showing electrical
action by modifying the measure of mass. Thus if we treat
nM, as mass of the double ring, and seek to identify it with
actual atomic weight, we are supposing this modification of
mass to apply to external as well as to internal relations.
The modifying factor n/N by which we pass from m, to M,
ranges in value from 1:909 for n=2 to an asymptotic value
2°266, and proceeds with regularity but not at uniform rate.
The actual values of the quotient atomic weight + atomic
number are irregular, and for n large are in excess of the
limit just given: for nm not large the modification is on the
requisite scale. If the atomic number is taken to be
the number n defining the double ring, whether with or
without ‘centre, or again with or without satellites, there
will be various values of N comprised in one group. For
example, the case of R,(+) with 3 satellites would give a
sensible reduction of N and a correspondingly increased
value of the factor n/N. Irregularities would then he

1072 Mr. R. Hargreaves on Atomic Systems

significant in relation to the proportion of atoms with centre
and to the number of satellites carried.

The above seems the natural interpretation of atumic
number in relation to the present scheme: but if that
number represented a number of ions the grouping of cases
would be different.

(b) If we set up as an ideal the elimination of the gravita-
tional constant, so that a form m,?/r may appear in gravitation
as compared with e/r in electrical action, what value is sug-
gested for mass? A flat rate of transformation to M,, in heu
of the variable M, used above, would require yM,?=m,*; or
with y=6°67 x 107°, M,=3872m,. With Millikan’s ratio
of m,: mz this would imply M,=2-1m, or with the ratio
suggested below M,=2'13m;, values in close agreement
with the average in (a). This numerical relation seems
significant, but has no connexion with any special atomic
scheme, at any rate of an obvious character.

(c) In dealing with the above figures a point which
engaged my atiention was the closeness of ymy?/m,? to the
number $(-44127), the limit of N/2n. An attempted inter-

a SIN one?
pretation runs thus:—the coefficient in >;-{——)}, an
2n ay /
expression for internal kinetic energy, approaches in its

asymptotic form to equality with the coefficient in
y(m, + m.)?
Meo?

equality gives m,+m ,=1819m, or m;=1818m,. Millikan
gives for the unit of the atomic scale (mp say), €/mp=9690V ;
which with e/m,=1°767x10’x V gives mp=183lm,. An
assured connexion between m, and the mass of the hydrogen
atom seems to be wanting to give certainty to the value of.
the ratio m/7M».

The above value has been used wherever in the calcula-
tions the ratio occurs. Possibly my first impression in
respect to the number was too favourable.

§7. In concluding this sketch of the configuration 1t may
be pointed ont, that though by the addition of a central ion
and satellites some variety and complexity is given, yet the
margin of choice is strictly limited. In this respect the con-
trast with a multiple point-core is noticeable. A core +10
can maintain any number of electrons from 1 to 20 in orbital
motion in one circle, with a possible residue varying from
+9 to —10. The electrons may be distributed in several
circles with a further range of varieties. Whether this

274 2
x (“= ), a gravitational potential. The
So ME

based on Free Electrons. 1073

multiplicity of options is to be regarded as a valuable asset
or as a source of embarrassment may be left to the reader to
decide. The loss of such large freedom of choice is the price
paid for the nuclear cement provided. ‘The nucleus is of an
entirely different order, one of mass rather than of charge,
and it is endowed with orbital motion and possibilities of
internal oscillation. The whole structure—rings, central
ion, and satellites—does not admit of any but a quite small
residual charge. (C7. § 23.

We proceed to the mathematical theory on which this
general account is based, so far as it relates to steady motion.
A second part contains on investigation of the natural oscil-
lations and of the associated question of stability, considered
with reference to the present scheme and also that of a
multiple core.

Mathematical Theory.

§ 8. Let there be n charges e each with mass m, on a
circle of radius a,, and n charges —e each with mass mg, on
a cirele of radius ds, disposed as above.

For uniform motion in a circle we have

Ny@ Cie OO Bo...) a jaiity, CE)

When F, F, are expressed in terms of the radii, the problem
is to determine the ratio c=q, : a, so as to satisfy

B3ja,= Me] my xX Fy /a. Vp iroum aie oe ee (2)

With ratio known the forces can be expressed in terms of
one linear magnitude. Since m,.:m,= mw is of order 1 : 1800
a first approximation is got by determining w from F,=0,
and then finding » from

Cr Cite Oy) a ke A 9)

with the special value substituted in F,. The central repul-
sive force on one charge in a circle of n charges is given by

€7¢,/2a? ?

n—1l ‘ : O e 5 4

where (n= % Y cosec s/n. \ 4)
0

The attractive force on —e due to the n charges +e of the

circle a, 1s given by

— e?{ a.— a COs (Cise 1)m/n}
r=0 {Qy? + ay” — 2ayaq cos (2r-+1)a[n}??°

1074 Mr. R. Hargreaves on Atomic Systems

On introducing the ratio x equations (1) become
m,o7a,;>= & oF 4 de, t, MgW"ay> = € { f(a) —Se,} 5

(6 a)
(7)

in which = ieee
f(2) =

w cos (2r+1)a/n
aS + vw? — 2a cos (27 + 1)ar/n}3?"

If there is a poettine charge at the centre, formule (6 a)
are replaced by —

moa? = € = iG =) —te.—1 it, MW (lg ie) |
(6 b)
When the value of « satisfying f(«)=4c, is substituted

ta (7) we have the tabulated number N, while N, is

used for the corrected value when m,: m, is not neglected.
The kinetic energy of orbital motion is given by

. N, :
20 = a(mya,? + mas")? = ee (dae a Sa)

oA
for which it is generally sufficient to write
2T =nNe/a,. oer

The potential energy U, total energy of orbital motion T,
and total angular momentum H are then given by

H=T+U,. 2T+U =0, . 20 =o: oar
while
2TH? = m,N?n*et 2)

is a relation into which neither w nor a enters, written in
the form suited to the approximation (86). A symmetrical
form can be given to (8a) and (96) by using |

M=m 4m, Ma? = ma,?+ mea,”
te: 3M 5
and No N, = Mas: nna

-§9. As the numerical work of solution only appears in
the tabulated results, it may be of service to show the work
for the simplest case n=2. With a,=1, a;=w the equa-
tions are

p 26? e : Zane &
Moo” = rie DOWD) Rory see eet

(l+a7)3? 4? (1 a)8?% Lae

based on Free Electrons. 1075

and wis to be found from

Z 1 2 I
(1 + a?)3/? Biay) 450g ae te + 47) 3/2 Aw { *
The first approximation gives (1+a”)**=5 or # =/3,
and for a second we have

: -22#(1-4)-8(1-
(l+a*)? 4 me ae) 4 373)"
3

With 1
AOR eee BR H(I- 374)
=v (favs (ev)
c= /3( “)=v 97 (3V 3-1);
and

x? = 3o/3— = EV O84
Thus

2 ji Sa 1 e—l1 a 1
N,=2'| ay Lat i( -,)|= SEES ene
"aga 4ta a aye ga (1 3/3)
3/3—1 =
eI — GbV3-)) = 1:04904—-00019
| wit p= s lols ;
while #,='73205—-00034. In each case the correction
for w is quite small.

Table I. gives values of # and N for a vacant centre,
Table Il. values of z' and N’ for a positive centre, obtained
by direct solution of (6a, 6).

For the problem of two ions placed axially two variables
are needed: 2 as before, and y where 2y is the ratio of the
mutual distance of the ions to the radius of the negative
ring. - If we take the radius of this ring as 1, the nature of

the change in the equations may be shown by writing them
forn=4. They are

panerepies.. Fo omit Saal )
Ay? (PPR (1 4? 8?”
2—wV/2 2+ ar/2 2
(Lead)? Atattay 2) 4”

= 9 J 7 tT;
ek. > Se

(14 a?— 2/2 2 ( Lett x ny, 2/32 (0? +97 yee

1076 Mr. R. Hargreaves on Atomic Systems

The first equation is the condition for equilibrium of the
axial ions, the second that for the equilibrium of electrons |
in the ring (m# correction neglected), and the third gives
the characteristic number on which the angular velocity

depends.

§ 10. The position as regards a possible ambiguity in the
solution for « may be explained in connexion with the graph

(fig. 3) of /(v) here drawn for n=6. The curve is typical

Fig. 3,

| 2 3

Graphs of f(a) and f (*) , the latter dotted, for »=6.

of cases from n=4 upwards ; for n=2 or 3 the curve lies
above the asymptote. The graph shows a minimum value
n tor «=0, followed by a maximum and a rapid descent to a
minimum of small negative value, from which it rises to the
asymptotic zero value.

based on Free Electrons. 1077

Two solutions are possible : (1.) when the value assigned to
f(x) is greater than n and less than the maximum, and
(ii.) when the assigned value is negative and greater than
the second minimum. Case (i.) requiring ¢,>2n can only
occur if n>473, a number outside the range considered.
Case (ii.) does not occur with vacant centre ; but occurs
once, viz. with n=4 for a _ positive centre. Here
te, =°25+°707 = °957, and the assigned value of f(z) is
—043, lying between 0 and the minimum —:059, which
occurs near v=2°39. The second solution, on the up grade
of the curve, corresponds to a position of instability.

The comparative straightness of the graph near #=1
suggests a method of approximation to the value of f(z)
when n is great and «—1 small, which gives at once the
asymptotic value for n great and a serviceable approxima-
tion for moderate values of ». The solution of (6) by
numerical test needed for small values of n becomes
laborious between 10 and 20, and for prolonging tables
beyond this range the use of such a method as is proposed
is indispensable.

§ 11. With e—1=€ a small quantity and 0,=(2r+1)z/n,
the general term of f(x), viz.

(1—# cos 6,){1+ x? —22 cos 0,}-9/?,
may be written
{x(1—cos 0,.) —E}{2a(1 —cos 6,) + 2} -32
and expanded in terms of &, as
f(2) = 307 "?(2—2 cos 6,)~ 1? — Ex~2?(2— 2 cos 8.) 9?
— 3£2-37(2—2 cos 0,) 79? + 3Ea—52(2— 2 cos 0,) 5...
Thus if

n— n—1

1
>, = Scosec(2r+1)r/2n, 3, = 2, cosec* (2re-l\ar/2n ...;
T= r= ry
(11)
f(2) = $2°¥?2,—} fa, — 8, Pe 3+ Bd, .... (12a)
When the general term of f(2), viz.

x?(a— cos 0,){1+a?— 2 cos 6,}-*?

y)

1078 Mr. R. Hargreaves on Atomic Systems

is written as
x*(1—cos 0, +£){2x(1—cos 6,) + 2} -9?,
‘we find. |
1 ) ae 3&
(5) = 2e)?S, + dEa?23,—238 & 1253 — 64 Ds coe e (12 b)
For a vacant centre

N =f(=) —to,= f(«) +f (;)—— (Me) dea}

in which the bracket vanishes as the condition determining «.
Hence

N +e,=i(@ mM cee! > Lat ae (al? — ¢= 9/2)

or

N= Con — 2Cn + oe (33+ 21) alate shit toys (13 a)
on using > = 2 (¢2,— Cn).
The equation of condition /(#)=%c, by use of £ in (12 a)
i becomes

0= 4 (Con 2Cn) —_ #(1- *) (2s ale 2) =r eae s\eanne (14 a)

If this is multiplied by &/2 and added to (13 a), we have
S x 4 be
N = (em—2e,) (1+ alee 24 SF (180)
_Also, if (4a) 1s multiplied by 1+ a and &2;3 neglected,
1E (Sgt 4en—Con) = 4(C—2¢,) +3235/8..., (145)

- where on the right hand only the main term in the value of & ©

-is to be applied.
The values of & or e—1 and N are thus dependent on the

- sums of thrée series, which now require consideration *.

* A slightly different treatment may be Ese on Taylor's theorem
~avith the values

F (1) =4(c2n— en),
wr ) 8 1
—f' (I) =82s+2)= B | 2°T +801, (log eye ia)
A" )=—-3Zf'), and f= Be Tole

based on Free Electrons. 1079

§ 12. If we apply the trigonometrical expansion of the
cosecant *

rie Sera 1 2 (—1)’2ns
> cosec sir/n = & | ees ee
s=1 UL Se wie CS Te)
we have
or "xt n—l al i fea) aul Y va) Bt | Ma
— > cosee s/n = ¥ E + —— + > Rage by. > Bi ~1)" |,
"en ey N—S yay Ph+tS yairTn+tn—s
or oe leat SL)
7 n—1 co n—l (—1)"
— > cosec sr/n = > — iL
2n —- / r=0 ey m+s’ ( y b)

(15 a) being a general form, (15) only suited to the special
range of s.
The series is

ia OP 0 thet (Aig Hg dui)
Be 25a) an e n+1 ~"2n—1 An
1 1 1 1
esac tacit al (16)
But
i ee Ls. 1 1 1
a nit oe ee te (17 a)

and the series in the first bracket of (16) is the difference
between this formula as written for n and 2n, the next
bracket the difference as written for 2n and 3n, and
so on. 7

The first bracket is

loo~ 1 1 ii 1 1 ]
a eae are

* Tam indebted to Professor Proudman for the suggestion of this
method, and a rough sketch showing the term log 7/2.

1080 Mr. R. Hargreaves on Atomic Systems
and the process applied to (16) gives

q 1
a- & cosec s/n

PA

ee eee 2
To ee ae Ss, 3 Se a(t ate)

“ 1 iL ul 1
ter

dg ee em |
== rn —log =—— -_- = =
SE TS OS 2 se 7 S00 |
and so
BI Fea Mey cae chu |
m= 7 [lognty—legs) — a5 + a39008 7

ie r(18)
n T T
ED eescca trates ile mney NY
CZ ROD gee loge2+ A8n 23040n?-"’ |

T 49 ar?

n 8
Can Cn = fa {log.n+y+log—} oe Eo a 34560018 c

§13. The method is also applicable to find the sum of
cubes, but with a vast difference in the labour entailed.
The cube of (15a) has terms of types a’, 3a7b, and 6abe.
In the two latter it is necessary to separate each term into
partial fractions, and apply the approximate summations

Loa eal: 1 il
Tae ap Gee a
| (17 b)
ate Uae kt ae
13 D3 Cee Ss 3 Qn?"
From these with (17a) we have
n—1 1 Ly preci n—1 1 y 1 \
as ae C2 ee sai(7n +s)? nr(r +1)’ (17°)
7 c
hf amen aloes Renae On ...
Ge chis) 20 Lr ) ee t é

which are sufficient to give a second term of order n~? com-
pared with the main term. The separate summations are
very numerous, and it may be sufficient to add to this
account of the method the separate totals for the three

based on Free Electrons. 1081

groups, viz.
Ss

n?

for typea®, 28;—

” or) 3a’b, et 2 (log MESS 3)

oS.
morgen | OGUC 62 (logn-+y—los” ae

> cosec? sar/n = “, { n'S;+ 3nS. (log n+y—log 5 per tly 4 ie

(19 a)
The difference between this formula as written with n and
2n is
fink = { 2n8T, +01, (log ntytlog i. =), (19 b)
where f= 5 (27+1)-8, andso 7S3;=8T; ;
*=0
while oo g="/2.

In the case of ; it will be enough to write the main term
64n°T;/7°. This gives for (140) the form

En? | 2T,+3n 7T, Nice n+ y—log. = ae is) t
gee T55°

ead Sa eS

7 1 as

and in numbers
En® = 3°2521—13°1749n-? logy) n+ 5:0738n-?
N/n = °44127 +°42421n-?—1°45342n-* logign

+°36256n7*....}

|

Table III. has been caleulated from these formule.

§ 14. The formule of §11 may be applied when x is not
small to find the changes in w and N which result from small
additions to the central forces. If we describe these extra
forces as a central repulsion on m, of amount e’o,/a,”, and a

Phil. Mag. 8. 6. Vol. 44. No. 264. Dec. 1922. 4A

1082 Mr. R. Hargreaves on Atomic Systems

central attraction on m, of amount e?o,/a2”, it will correspond
to using o, and o; in place of the units occurring in the

equations (66). From (14a) and (13a, b) we find
ba =2 2ago,/N and oN = —o, +20). 2 ia

The correction fora positive centre is then got by writing
o,=0,=1, and gives

6” = 22&/N and (ON = @. 7). een

The correction for p, that is to take account of the ratio
M5 :m, instead of assuming it indefinitely small, is got by
writing og=—pNa~*, and is

Se =—2ueéa-?, 8N=—pENe-?. . (210)
§ 15. Iteis proposed to deal briefly with the potential of

charges distributed at equal intervals on a circle, with a view
to showing the mean ettect and the main fluctuation in the
force on a satellite. In the plane of a single ring radius a
we are concerned with a potential

n--1
Use = > {r? +a?—2ra cos (6 + 2pm/n\} 12,
p=0

where @ is the angle between the radius r of a satellite and
that to the nearest unit in the ring. The Legendre co-
efficients must be expanded in cosines of multiples of the
angle @+ 2p7/n, and note taken of the fact that

n—1

> cos m(+ 2p7/n)

p=0

vanishes unless m is a multiple of n, in which case it is
ncosm@. ‘The mean value of the attraction depends on the
term independent of ¢, which is

LQ) (574 7) lees =K(2),
aie +(5) a+ +(5 ) ate oy 22K (2),

according as r or ais the greater. K is the first complete
elliptic integral. Thus for the motion of each of s satellites
an approximate value of the effect of the double ring is

(22)

based on Free Llectrons. 1083

shown by

. 9 \
Ms ),,77° = e? | 1-46, =F a (ay? — 44”) .. | ;
On
=°¢ | 14+ a, | for n large,

or

, 3n&r?
ey La
=€E E 1 ws 9 =| 39 399 99

according as the satellites lie outside or inside the double
ring, a considerable departure of r:a from 1 being
presumed,

r
9077 = = | 1—4e.— > (a; ws) |
|
)

a” arr?
The coefficient of cosnd is n | an eae = ae Saad +... ],
.. (2n—1) |
where gee: és a rOe, and on this term depends the
— «6 ee — |
fluctuation.

For positions outside the plane we are concerned with

S| r+ a 2 Irar/1—p? cos (b+ 2pm/n)t-?

p—0

and it may be sufficient to write the value of the mean term

1 Gs. 3a4
see Cc 1) + (85! 30? +3)...

When a/r is widely different from 1, the field due toa
double ring is given by

ne(ay" —,”)

U.=— a (3u?—1) outside,

and by

U;= ne [-s ngs ar “(a ee ) (By —] | inside,

In a rough way one may say that the cone 3u7=1 divides
the lines of approach in which an electron would be repelled
and its path show reflexion, from those in which the path is
continued through the rings R,(0) with deviation.

4A 2

1084 Mr. R. Hargreaves on. Atomic Systems
: TaBLE I[.—Vacant Centre.

N. fae: (v—1)n?. N/a. 2. We (w—1)n?. N/a.
2 1°73205 2°9282 "52452 10. ~=1°031691 31691 44537
3 1:326485 29384 “48104 Jl = 1:026275 3°1793 “44467
4 1:°187775 3°0035 "46475 12 = 1:022137 3°1877 44424
5. Ll 2292 3°0555 -45676 15 =1:014265 32091 44355
6 1:0859 ~ 3°0924 "45225 16 =1:012538 3°2100 44990
7 1:063659 3°1193 *44935 18 1:0099278 32166 44962
8 1:04907 3°1413 “44755 20 1:0080602 3°2241 44968
9 1:038968 3°1564 44631 20 ~=——: 1003598 32382 44906

Tape I1.—Positive Centre.
nN. a’ ING | nN. a. ING
4 2:03384 2°27278 i 1:03784 4:9234}
5 1°30323 248557 12 1:030961 535605
6 L285 2°84013 15 1:018718 667414
a 111379 3°23398 16 1:016188 709815
8 1:08106 . 364543 18 1:0124821 797730
9 1:060784 406661 20 1:0099163 885814
10 =: 104729 4-49491 .

TasiEe I]1.—Values derived from formula.
(a —1)n?.
31710
3°1807
371886

> Dd bw bw tO
H Co bo

ho
Go -~Tt Oo

b> bs bo
eo)

31953

3°2009
3°2058
3°2100
3°2136
32167
3°2195
3°2219
3°2241
3°2260
3°2278
3°2295
3°2308
3°2320
3°2332
3°2342
3°2352
3'2361
3°2369
32377
323584
32390
3°2396

N/n.

"A4545
‘44473
‘44418

44375
44341

44314
44991
44273
‘44957
“44944
“44232
44293
“44214
‘44207
44201
‘44195
-44190
‘44185
‘44181
44177
“44174
44171
-44168
44166
44164
44162

2.

36
37
38
39
40
4]
42
43
44
45
46
47

100

(x— 1)n?.

3°2402
3°2407
3°2412
o 2417
3°2421
3°2425
32429
3°2432
3°2435
3°2438
32441
3°2448
3°2447
3 2449
3°2452

3°2462
3°2470
3°2476
3°2482
32486
3°2490
3 2493
3°2495
3°2498
3°2500

N/a.
-44160
"44158
44156
-44154
44153
44152
“44151
"44150
44149
44148
‘44147
44146
44145
44145
44144

44141
‘44139
44137
44136
44135
‘44134
44133
44132
44132
‘44131

Values in Tables I. and II. obtained by direct solution of the equations.

based on Free Electrons. 1085

TaBLeE 1V.—Some values of N,/n for a multiple core.

nN. en|2. Ne/n.
10 3 862 ‘614
20 9-935 503
30 16°834 "439
60 40:286 “329
90 66°254 "264

TABLE V.—Solutions for two axial ions.

2. y. o N. n/N.
2 4°588 1-803 986 2027 |
3 3°378 1:360 1-448 2072
+ 2°786 1-210 1°865 2°145
5 2356 1-138 2°331 2°145
6 2°033 1-099 2-877 2-130
< L757 1-075 3°158 2°217
8 1-493 1-060 3°595 2°225
a ‘Tat 1-082 3°720 27150
a 639 1°125 3°417 2-049
6* 556 1°208 3°165 1896
Fas “460 1552 3°233 1547

The figures starred belong to the closer position of axial ions.

Parr I].—Naruras OSCILLATIONS AND S'TABILITY.

§ 16. One main object in dealing with the natural oscil-
lations of a system is to ascertain whether it is intrinsically
stable or unstable. In the present case it is also important
to ascertain the periods, with a view to comparison with
spectroscopic results—electrical and optical.

For the two-ring scheme the number of variables in the
oscillation problem is 6n, of which the 2n referring to axial
oscillation stand separatefrom the rest. Variables attaching
to ions and electrons have very different coefficients of inertia,
but enter on like terms into the forces derived from potential
energy. As a consequence, the periods fall into two classes
(P,, P. say) specially related to ions and electrons respec-
tively; and if 27/p,@, 27/p ow are the periods, then m,p,?
and mop,” have values on the same scale dependent on n.
This allows us to halve the number of the equations required
to determine the separate types. Notwithstanding this
reduction, the problem is very laborious, and the amount of
work required for the case n=2 suggested the inquiry
whether it would be possible to base a solution on asymptotic
forms, and so general rather than individual, though
restricted in application to larger values of n.

1086 Mr. R. Hargreaves on Atomic Systems

If we use only the terms of highest order in n, reduction
to a standard form is in fact attainable. The validity of the
reduction is well assured for oscillations of type P., and as
here the motion of the mobile electron is primarily con-
cerned, it is to these it is proper to look for any tendency to
run away which instability would indicate.

§ 17. The results of the asymptotic treatment are that
oscillations of type P, are real for displacements radial or
axial, unreal (or the terms exponential) for tangential dis-
placements. The axial oscillations of type P, are also real,
but the approximations for motion in the plane are less
trustworthy : they point to radial stability and some degree
of tangential instability.

The individual solution for n=2 gives real periods for
oscillations of type P, close to the orbital periods, and real
periods for oscillations of type P, with axial and radial
displacements, but 4n exponential form appears in the
tangential displacements. Thus, in respect to oscillations of
type P., particular and general results agree in assigning
instability to tangential displacements, while giving stability
in other displacements.

$18. Apart from the analysis, simple considerations
appear to be applicable when the number n is sufficiently
great to give near neighbours a dominant influence. In the
double ring each electron has ions as next neighbours, and
in steady motion describes a small circle relative to each ;
if some disturbance should accentuate tle influence of one of
them, the relative orbit would tend to elliptic form. This
points to radial stability and tangential instability. On the
other hand, in the single ring (with core) each electron has
electrons as next neighbours, and if the influence of one is
exagoerated the tendency is to a hyperbolic form of orbit ;
which points to radial instability and tangential stability.

The second consideration is that a position of rest for a
negative charge in the straight line between two positive
charges is unstable. In the double ring it appears that
neither the inertia of orbital motion nor the departure from
alignment is adequate to overcome this tendency to in-
stability. The argument applied to the single ring (with
core) suggests tangential stability.

Again, in respect to axial motion the displacement in the
oscillation equation is associated with the inverse cube of
distance as coefficient. For next neighbours, therefore, we
are concerned with a factor varying as n%, while for the
action of the central core there is only variation as n.

based on Free Electrons. 1087

Hence in the single ring with core the repulsive action of
next neighbours me GAR ‘dominant for | arger values of n—
that is, ‘thie axial displacements unstable. The same argu-
ment suggests for the double ring a thoroughly stable
position in 1 respect to axial displacements.

$19. The individual solutions found for the single ring
with core comprise the cases of two electrons with core
+2 or +3, and three electrons with core +3 or +4. Axial
displacements are here oscillatory. Madial and tangential
displacements both show instability—for two electrons | by the
appearance of a pure exponential, for three electrons by a
complex form in which the real exponential has either sign.

The asymptotic solution gives axial instability ; “the
upproximations for plane motion are less trustworthy, but
point to tangential stability and some degree of radial

instability. The general conclusion for a single ring of n
electrons with point-core +ne is that radial instability
always exists, that axial instability accompanies increase
of n, while tangential instability disappears with increase

of n.

§ 20. There is a marked difference in the character of the
instability attaching to the two schemes. Consider first the
single ring with core. With n electrons and a core +n,
there is axial instability when n is sufficiently great. This
type of instability is taken to be fatal, and the question arises
whether relief can be found by distributing the electrons in
several rings. To obtain a first clue a less exact method of
calculation was followed, in which one electron is displaeed
from the plane containing other electrons and the core—that
is, the mutual displacements of these elements were ignored.
The condition obtained is probably less stringent than the
real conditions.

On this basis it appears that up ton=9a core +n will
give axial stability to n electrons; but for greater values
of n (number of electrons) stability dememde a rapidly
increasing excess of the core number. A tabulated result of
calculations made by this simplified method suggests that
relief can be found by distributing the electrons in a
succession of rings, which need not exceed but must reach
seven in dealing with the greatest value of n required.
These figures may be migited by more exact treatment,
cf.§ 32. The effect of distribution in several rings on radial
and tangential displacements has not been eases

§ 21. Consider now with respect to the double ring the
question of tangential instability, the only type of which

1088 Mr. R. Hargreaves on Atomic Systems

there is clear evidence in that scheme. With rings of
positive and negative elements P and N, if all the N’s are
displaced through an angle 2a/n, a new configuration of
steady motion is reached. At first this instability appeared
significant as providing the occasion for such a transition,
but on reflexion a wider view is suggested. The instability
is to be regarded as indicating that the whole motion consists
in the passage from one of these configurations to others in
succession.

_ Now, the equations set up for oscillation about the two-
ring position are not adequate to deal with this finite
transition—a problem of the motion of 2n bodies under
attractive and repulsive forces in less specialized positions.
The forecast I make of the motion (without calculation) is
that if N is displaced tangentially towards P, there will be
a gradual increase in the rate of approach, and N will be
carried a little beyond P and describe a loop about it before

Fig. 4.

N

proceeding to the next special position midway between two
P’s. Jn the loop phase or near the passage through the
apsidal position each N is under the almost exclusive control
of the next adjacent P. As a consequence, kinetic and
potential energies and acceleration are all on a greatly-
increased scale in this part of the motion,-much as in the
apsidal phase of motion in a hyperbola or elongated ellipse.
The distance between N and P is here of the orden ay — Ao,
Say p(d;—@,_) or pa€ at the apse. The acceleration at this
apse is then e?/m.p?a7—?, whereas in the steady motion it is
- Ne?/mj,a?; thus as Non, and Eon~* when n is great, the
first “ese ition will vary as n*/mz and the Posed as nfm,
a very wide disparity.

The system has been treated by mechanical rather than
electromagnetic methods, since electrostatic force has been
used nd a constant ene This treatment, however,
should give an approximate value of acceleration ; and since
radiation varies as the square of acceleration, it would
appear that in the brief space belonging to the loop phase,
radiation would be on a vastly greater scale than in the rest

based on Free Electrons. 1089

of the motion. Thus we have a succession of stages in each
o£ which occurs a pulse of radiation, with no appreciable
radiation between the pulses. How the interval of time
between the pulses may be related to the period found in
Part I. is not known.

If it were permissible to apply the formula, radiation
o X~*, to the comparison of the above radiation for ditterent
values of n, we should get Xx (pa), Xn~?; but the manner
in which pa may change with n is not known.

Now, in Bohr’s theory there-is what appears to be an
arbitrary assumption of unitary stages under circumstances
in which no step or stopping place occurs. The conditions
suggest rather an invariable orbit if radiation is ignored, or
a smooth spiral if it is taken into account. If results yielded
by Bohr’s theory are held to be in agreement with experi-
mental knowledge, there still remains an absence of
mechanism to explain its operation. If the above description
of the whole motion in the two-ring scheme is correct in
outline, a succession of stages is realized without abandoning
the conception of electromagnetic radiation.

§ 22. The above account of stability is, for the two-ring
system, based on oscillations of the special configuration.
When the wider view of the whole motion is taken, the
position in respect to tangential instability is certainly eased,
but the extent of the relief is uncertain. As positive and
negative elements still remain in close conjunction, the
position is not worsened in respect to axial and radial
stability. But it is natural to expect either an interruption
or a finite modification of the oscillations at regular
intervals.

A brief account will now be given of the actual periods
obtained in examining the question of stability. When the
atomic numbers are not small the periods for axial and radial
oscillation tend toagreement. The periods (quite numerous)
of type P, are comprised within a range represented approxi-
mately by a factor °873, and the shortest period has the
asymptotic value 27/16°72nm. The range is certainly very
near to that of the extreme columns of the K series. An
identification of this formula with the figures of the Ky)
series, or 27/14 oa with those of the « series, invoives a
“debermination ® of t, the radius pone the scale of the
configuration. It enh the number na,’ ‘eradually diminish
sath? increase of n; or if A, is atomic weicht, it gives
to A,a,° a fairly constant value 1:32 x 10~” with some rise
for lower values of n. For example, the use of the K (a)

1090 Mr. R. Hargreaves on Atomic Systems

column gives in the case of Barium (n=56) a=‘99 x 107°,
Wai Ir3ax dO:

The corresponding series of periods of type P, is not
limited to a narrow range; its shortest periods are about
50 times those of the shortest period above (vy) or about
43 times those of the other extreme (a). The table in
Sommerfeld’s Atombau gives no data in which the elements
of K and M series overlap, but the above factors are somewhat
in excess of values I get by comparison of M figures with
extrapolated values of the K series.

There is also a special ionic period, due to the axial
oscillation of a central ion, of much greater duration than
those enumerated above. When n is great, the factor con-
uecting it with the shortest period above is about 32n?/9.
The numerical value X}=3°73x 107° then corresponds to
388 x 107° in the K(a) series for Barium in the above
interpretation.

§ 23. Consider now the period of revolution of satellites.
When corrections due to the rings are ignored, the motion
of a single satellite is given by m.Q*r?=e?. If the period
27/Q is identified with A/V, the numerical connexion of r
. and A is given by (7x 10°)* x 1:405=(A x 10°)7.) Monge
external satellite, A will be of order 10 © if ris of order 107°.
Thus a period in the range of light corresponds with such a
value of » as appears in Chemistry in the roéle of radius
of activity. Also the influence of the rings is negligible
if a is of the order suggested in the last paragraph,
viz. a/r is then small ; but corrections can be applied by use
of (23).

For an internal satellite the same equation is (x 10"°)*
x 1:405=(A x 108)?. Identification with the L_ series
would here make rv of the order 107°, and the correction for
the rings would be negligible as v/a is small.

A value of the ratio 7/a near to 1 would involve a serious
perturbation in the mutual action of satellite and rings,
which may be the reason for the strong action of ultra-violet
light.

"rr the axial oscillation of satellites has a period 22/qQ,
then for s=1;,2,,3,1 find) g=1, 15155, 1-52; alsogjimadl
axial instability for s=4. For the cases of 2 and 3 satellites,
mQ7r?/e?="75 and 423 respectively. These cases show,
radial and tangential instability. Of the more complicated
orbits to which this instability points, the elliptic is the more
permanent form, liable, however, to a stronger perturbation
from the double ring than would attach to the circular orbit.

hased on Free Electrons. 1091

The quasi-hyperbolic type is transient, and points to inter-
change with the status of free electrons.

Setting aside the cases of R,(+, +), where stability has
not been examined, the most permanent forms of structure
are R,(0), Ra(+), and R,(+) with one satellite, two forms
neutral, one with a single positive residue. Less permanent
is the type with one negative residue (viz. two satellites), and
in a still less degree that with a double negative residue
(viz. three satellites).

§ 24. In respect to any atomic scheme there are two
crucial tests to be faced—the question of stability which
belongs to the domain of mathematics, and that of comparison
with assured facts in the domains of Physics and Chemistry.
As regards the problem of stability, examined with reference
to the present scheme and that of multiple core, aithough
the methods used are largely approximative, I believe the
results to be substantial in their bearings on both schemes
and in the main correct. The existence of instability in both
schemes, though of differing types, demands a discrimination
as to the fatal or admissible character of the instability, and
in the latter case as to the function which it may be held to
discharge. This again carries with it the suggestion that
the activity of groups of atoms and free electrons as displayed
in the world of phenomena may be dependent on some degree
of instability.

The application of the second test, with the task of inter-
preting an abstract mathematical theory, is one that calls
for a wide and intimate knowledge of experimental work,
‘and especially of results of recent research which seem to
probe the nature of the atom. To such knowledge I can lay
no claim ; and consequently I have felt much uncertainty in
interpreting the present scheme, and some hesitation in
criticizing what appear to be weak points in other theories.
For this there is no remedy but an appeal to readers who
may agree with my opinion that the multiple-core scheme
involves an essential irrationality, may be prepared to
consider the present alternative scheme, and can bring to
the matter a fuller knowledge of the relevant branches of
Physics.

Mathematical Theory.

§ 25. The first problem examined was that of oscillation

confined to radial displacements. The limitation is realized

by supposing this displacement the same for all elements
lying on one ring, and also supposing all elements to have

1092 Mr. R. Hargreaves on Atomic Systems

an equal angular velocity determined by the conservation of

angular momentum. The radial components then furnish

equations

; —2,, 2 2 Fa \ee Ip
mye "ry? (OP? 7, — 71) =7(X) — Fen, )
Mye~ 279?( 0779 — 19) = fo(X) — Fen 3 f

where to secure symmetry the notation of (6) is altered, viz.

fla)=Ka), file)=f( ‘) |

With # and 1 for original radii, p,; and p, for displacements
m=xt+p, and rg=1+p, make X—w«=p,+.xp,; and the
equation of angular momentum is

O(myr? + mgr”) = @ (1,2? + mg),
or O, = @(1—2x7!p;,— 2277p).
Tf then p,;= —4q’o’p,, the left-hand members of (24) are
Nu il +g? —Dalpi—4pepo}
and pN wv 2{1+ (9? +3) p.—4a7'p,—4 ux po}.

(24)

Further, we have
hn (X) aay Le =i x) —3en a (pi— xp Ww a)’ (a)
= N+ (pi- apo) fr’ (7) 3

and the equations then stand

(pi— pa)’ = # NAG —1) pi — 4 pe" p2}, |

temple See Nella Spey Se. pi Sere oe (25)
whereupon | .

PN.(@?— 1)? +.0%fe (?—1)—4pa"fe =0 3
represents to the first order in pw the result of elimination.
The small root is g?—1=4y2~?, and the ratio of displace-
ments associated with it is Pi= "ps to the first order. The
large root is g?—1l=—a*f,'(z)/wN,, and the ratio of dis-
placements Hof; (2)=2" 1 fo (a). Here p, and pp, are on
the same scale of magnitude; and, indeed, when «—1 is
small, 2.e. » not email wi aa fy! are opposite in sign
and nearly equal in magnitude, making p,+pp.=0 or
mp1 + Map2=0 a first approximation. The sign of f)'(«)
or J (vw) 1s npgaline and the oscillations real; a limiting
value g=16°72n when n is great, is found by using the

value of f’(1) given above ($ 12, footnote).
This root appears in more general equations as attaching

based on Free Electrons. 1093

to sums of displacements in one ring, and provides a useful
check on results obtained by approximative methods. The
periods differ widely, and the fact that displacements p; and
p, are of like order for one, while mp; and msp» are of like
order for the other, suggests a principle which may be
applied to separate the groups, and thus halve the number
of equations needed to determine the periods in each
group.

§ 26. To explain this method, let «2, be typical coordinates
respectively with positive and negative charges, and let U,
stand for the second-order term in the potential energy. In
dealing with the first group of periods we may omit inertia

. . 0U . .
terms with m, and write *=(), by use of which a new

022
form of U, is reached containing only variables #,, and
accordingly the group of equations with m, as inertia
coeflicient is formed. These equations give values in which
the ratio uw is neglected, and we can use w and N as tabulated
in lieu of Nu.

In the second group, wg? is finite when w is accounted
small, and so g is a large number; hence those inertia terms
which are linear in g may be ignored in finding a first
approximation. For the motion of positive charges, a,
appears on the inertia side with a multiplier m9?w”, on the
other side 2, and 2, have as a multiplier e?/a®, which is of order
mo”. Hence 2, is of order q?x, or 2, of order wry; and so
variables 2, may be ignored on the potential side of both sets.
of equations. ‘he first set then gives 2, in terms of x, with
a multiplier varying as qg~? or as mw, and the second set
determines a group of values of ug”.

The separation presumes that we are content with the
main terms in the sense that order is determined by the
small number mw, and the groups present g? of the one and
yg? of the other as numbers on the same scale. ,

§ 27. We now apply this to oscillations in the plane for
n=2. It is convenient to mark with an odd index
coordinates referring to positive charges mass m,, and an
even index for negative charges mass m2. If r,=a,+ and
6,-+wt attach to a displaced position, we can put a,=a,=1
and a,=a;= V3, ignoring the p» correction to the ratio
@,:d,. The potential energy is given by

U/e=D-1(1, 3) + D-*(2, 4) -D~*1, 2
—D-1(1, 4)—D-1(3, 2) —D-1(3, 4),

1094 Mr. R.. Hargreaves on Atomic Systems

and we are concerned with, the second order terms in p, 0,
VIZ. :

1 eae | |
U,/e= 39 [5 (p3—p1)(A4— 4.) =F V/3(p1— po) (0; —0;) |
pot px)? 3 2h ee 7
as Can 2 Be =P N3(pitps? ie a ay) [ —(p2” +p.)

+ 5(p:? +p”) +3y/3(p, + Ps) (p2 + pa)
ale V3(0;—0,)? 9 I ANe
mon ns Ts eed ee

+ (04—01)? + (G3— 05)? + (@,—83)") J. |

To deal with oscillations of type P, this form is reduced

OU,
Ope

Uaioe: ao [2p +en)*+3(0.— 0)"

)
|
|
] i (26)
|

by use of =()..., with the result

——=

= “(2(p1+p3)? +3(63 =O)? 7). ee

When each displacement varies as ¢?*’, the equations of
oscillations P, are

(p?—l)pi—2 V3 p81 = pi + ps, 3p7O1 +2 V3ppr.= —3(0,—6) t (28)
(0? —1)p3—2 V3p03=p, + p3, 3p703+2 V 3pp3 = + $(03—

leading to p?=0 and p?=—1, the last occurring three times.

If w were taken into account these roots would be unequal,

and we find for the type Pj, stable oscillations in the plane

with a period near to that of orbital revolution.
For the group P., coordinates of odd index are omitted

from U,, and
9
Us = 39 LOP2" + 5p4? + 8p204 — 70.” — 70, —40,.0,]. (29)
‘The equations of oscillation are then

: 2
‘Meyp"@" P= — r (Spo+4p,), map?w*A, = a (70. + 20,),

°

2
mapo"pr= — 7A Sey +4p2), msp*w°0,= + (10, + 20,),

based on Free Electrons. 1095
or
9 Ye?
mop?w" (Ps + Ps) = — ia Cera

2

mp? @" (po =p.) = — i= (p2—Pu);
ie (30)

mop*@? (8, + O4)= + 16 (A. + 44),

de?

ay ae F

mop?w” (O,—O,) = —~ (0.—0,).
16 |

7 : nae ee : ° : c

Terms linear in p having been omitted on the inertia side,
the above gives main terms in the value of p?. The first of
the real periods, numerical value p=71:2 /—1, will be
found to agree with the result of §25 for purely radial
oscillation.

For 6, and @, the values are exponential, and as the
exponent may have either sign, there is a clear case of
instability in respect to oscillations of tangential character
in special connexion with the motion of electrons.

For oscillations in a direction z perpendicular to the plane
of the orbit, we have

(23—23)” (22 —- 2%)? I |
mee. 8A Pal. \eatenzs MoE! |
2/¢ 16a,3 fea 2(a,’ + a,")*” : |

[ (21 — 22)? + (21 — 24)? + (23— 22)? + (23 — 24)? ], al}

a (2—23)” Le (Z9—&,)? ay

48/3 16
+5 [ (21— 29)? + (21-24)? + (23-22)? + (23—24)? |. J.
The order of equations being only half that for motion in

the plane, we may dispense with the special method ; the
types are then given by

m= EA (2; —<3) _ (22, —22—%) ]
24/3 8 |
pote Be €* (2 — 24) e*(22.— 21—23) f Ae
pe a 8 ee. T a

Subtracting forms for <, and 2s,

iL tS
m po? =e eon _ a) =—m.o’,

1096 Mr. R. Hargreaves on Atomic Systems

that is p?=—1 and subtracting forms for z, and 24, we
have p?=0. Addition leads to

{p’ +5 8(9+/3)}(a+ 23) = gis (94/3) (22424),
and |
{ up? + 3.(9 + 0/3) }(2 2g y= y (9 +4/3)(41+¢:) 5
which give
p=0 or pp=—35(9+/3),

that is p= +4738 / —1.

Axial displacement therefore gives rise to true oscillations

for both types P; and Py.

$28. The discovery of tangential instability prompted
inquiry as to the position in the system of multiple core.
It is convenient to use coordinates relative to the core. In
general, for a group comprising a mass m9 and other masses
m,, the relative kinetic energy is given by

pill ice = ZaMdiy! — (Qi Ay}? Ig + Ditty), en

Shen coordinates ae to m) are used. In the present
case the omission of the last term involves a modification of
periods of the order w. In seeking the main term in the
problem of two electrons and a core + 2e, we may therefore
write
2T,.=m(w+y+e7+y"),
and pass to polar coordinates 1+p and 6+ at.
Since U,/e?=1}(p+p')? +75 (@—0')? — 2p? — 2p”,

the equations of oscillation, ae account of mw? = 7e?/4,
are

(p?—1)p—2p0="8p—t(pte'), )

(p? —1)p'— 2p0'= "Pf p'—7(p + p'),
p+ 2pp= — 7; (0-8),

pee’ + 2pp'= (9-8), |

(34)

which give |
(p?—22)(p—p')=2p(@—0’), (p?+4)(0—0') = —2pip—p')

and

(p?—3)(p+e')=2pO+6), p?(O+6')=—2p(o +p’).

based on Free Electrons. 1097
The second pair gives p?=0 or p?+1=0. The first pair
gives p* + 6p?/7 — 23/49 =0,
p?=(—344V72)/7="3795 or —1-2367,

the first root implying instability which attaches to radial
and tangential coordinates. Axial oscillation is here stable.
With core +3¢ the quadratic,is altered to

p*+ LOp?/11—35/121=0.

It is not proposed to write out the work for three electrons
with cores +2e, +3e¢, +4e. The final equation is a quartic
of simple character.

By way of exploration of the source of instability, these
problems were also solved with the single modification of
repulsion to attraction in the mutual action of electrons,
masses and intensities of force as before, with the result
that complete stability was found.

§ 29. A search for further information in respect to R,,(0)
may be pursued in two ways—either by examination of
individual cases as n=4, 6, 8..., or by use of asymptotic
formulze. The tedious work which the former course would
entail seemed prohibitive, and in respect to the course
actually taken it must be understood that approximations are
based on the treatment of 1/n as a small quantity. The
treatment of axial displacement is simple, and yields

U./2=% (2—z,/)?/2D2—S (e—2,)?/2D3 . (354)

as expression for the terms containing any one <, D, then

being distance between this element and any repulsive.

element, or D,=2asinsm/n; and JD, distance between
the same element and any attractive element, or approxi-
mately D,=2a sin (2r -+1)2/2n.

The second order terms in potential energy which contain
displacements in the plane are more complicated, and here
approximation is needed. For two points whose distance in
steady motion is given by D?=a?+a’?—2aa’ cosw,-and for
which the increments of coordinates are (p, 0), (p’, 0’), we
find the terms of second order in D~! to be

—(p? + p” — 2pp' cos )/2D* + 3{ap + a'p' |
—(ap'+a'p) cos yp}?/2D", |
+ (0—6') sin v[ap' +a'p—3aa’{ap+a'p' al. (ooo)
— (ap'+a'p) cosw}/D?]/D*, |
—aa' (O—6')*[cosyr—3aa' sin y/D?]/2D*. J
Phil. Mag. 8. 6. Vol. 44. No. 264. Dee. 1922, 4B

1098 Mr. R. Hargreaves on Atomic Systems

In the case of attraction the terms containing (p, @) are
got by writing (2r+ 1)a/n for wy, (e,', 0, ) for (p', 8) and
summing for values of r from 0 to »—1, when such a sum
gives the terms in —U,/e? which involve (p, 0). In dealing
with repulsion a=a’', 2s7/n takes the place of w, (p,, Os)
replace (p’, 0’), and s ranges from 1 to n—1, when the sum
gives the terms in U,/e? which contain (p, @).

This exact expression is simplified when n is great, for
a~a' is of order n~* and D of order n7! for the near
neighbours whose action is most influential.

The terms of highest order which contain a given (p, @)
are then

Us/P=3 { (p—pr')?/2—a(6—8,')9}/D,?
FEL —(p—ps)"/2 + a(O—8,)7}]Dee (898)

Thus a form of the same type as (35a) is realized, but
only in virtue of the approximations used.
On the inertia side also a greater simplicity attaches to
the coordinate z than to p, 6; for if each has a factor e%,
pz occurs in the axial motion, (p’—1)p—2pad and pad + 2p
in the plane motion. No sufficient simplification is attainable
unless the terms linear in p can be neglected, and this
requires p to be great. Now p? varies as n?/N and ultimately
as n*, 2. e. its main term has a form n’y where y varies from
one root to another. The omission of linear terms is there-
fore permissible for large values of n so long as y does not
become small. This case of difficulty does not occur in
dealing with the type P,, but it appears in respect to type P,
and also in the work for a single ring withcore. Oscillations
P, have the additional advantage of showing an extra factor
m]mz in the value of p? which greatly improves the approxi-—

mation.

§ 30. Axial oscillations of type P, are represented by an
equation

m;p?w'2= —e* {> (z—2z,')/D,2—> (2—2,)/D3}

or pp?2= — oH { (e—2,’) cosec® (2r + 1)a/n
‘ — > (2—2z,;) cosec? str/n}. (37)
In accordance with § 26 we may omit <z,’, and the sum of
variables then shows a period given by
pp ?= —>,/8N=—n7T3/m? log 2 or p?=—(16:72n)?
in agreement with $25. For other periods the coefficients

based on Free Electrons. ' 1099

of z—z, enter into the calculations. These coefficients are
equal for s and n—s, and for near neighbours vary inversely
as the cube of the smaller of these numbers. We get
an approximate account of the mutual action if we take
only next neighbours for which s=1. For example, the
equation for z. with next neighbours <, and <z; stands

Mp, + a22—B (22-21 + 22 — 2) =0
or (pp? +a—2B8)22+ Blaitz23)=0, >. *. (88)
or say Y2g+ 2 +23=0,

in which
A= >, /ON—2n Tale N, (B=n? 8m Noe 2h. a9)

Equations (38) written for each variable constitute a
eyclic group, which may be treated by the method of deter-
minants or as an equation of finite differences. The latter
method gives z,=(—1)"(A sin ny + Beosny) where y=2 cosy.
The cyclic character is then expressed by the conditions
fn41=% and Z,42=2, which for n even require sin ny/2=0
or siny=0, for n odd cosny/2=0 or sny=0. The deter-
minant * itself is given by

eo eas ye (ERE ad oh (40)

For n even the solutions are given by |

ny (2, 4, ./0- 20): (41)

for n odd the solutions are given by
Rye hs. AEE):

The value y=2 or y=27 occurs only with n even;
y=-—z2 or y=7 is common to the two series and involves
21 =2,=23;=..., which was assumed as basis of the problem
in §25. All other roots occur in pairs which give equal
values to y or 2cosy; but these equal roots of the approxi-
mate equation will no doubt be replaced by closely adjacent
roots with more exact treatment. It is convenient to refer
to the values y= —2 and +2 as extreme values, though the
latter only occurs with n even, this extreme for n odd being
2 cos 7/n. ‘

x
>
-
II
=
hd
“
bel Ge

: It would be of interest to get solutions of the
> 9 | determinant with two other neighbours, in
0, 1, y, 1 | Which the first row of 4,’ would bey, 1, 4,0, &, 1;
where we could suppose k<1,

4B2

=
“
&
—
s

1100 Mr. R. Hargreaves on Atomic Systems
To the value y= —2 corresponds in (38)

pep’ +a—28=—28 or pp?+a=0,
the root attaching to the sum of variables. The other
extreme gives up?= —a+4£, a negative number since

a: 48=4T,=4-2072 ;

this corresponds to a period 27/14:6nw, the number 16°72
being reduced by a factor °873 or /3°2072/4-2072.

The reduced equations for p agree with those for z,
cf. (35, a, 6); in view of these reductions we find that the
periods for radial tend to differ from those of axial displace-
ments by amounts which diminish as n is increased. For
tangential displacements it appears from (3506) that we
obtain exponentials with exponents “2 times the values
found in the oscillations.

§ 31. For oscillations of type P,; we have to deal with
variables for both rings, and revert to the plan of separation
by odd and even indices. If we retain only the first term in
the repulsive series and the first in the attractive, then for
axial motion a specimen equation is

2,3
Mp wz; — 3 23 — 69 — Sy — A (22g— 4 —e5)..

Bin ipa Se —— Diy —ey =p A Bey ee
In accordance with § 26, we write
Aue Cause
“Oso See aoe O24 ereg

then taking only the first terms of these, viz. 22,—z2,—2,=0,
22,—23;—2;=0, we can clear (42) of variables with even

ae,
index and obtain
Np?x?
aS +2)—a—m=0. 2 2. (48)
As z, 5 refer to consecutive ions, the method used above
is applicable and gives to 8Np7s°/3n>+2 a series of values
ranging from —2 to +2, or to Np?m’/n® values from —3/2
to 0. The greatest numerical value is p/n=-331 WV —1, or
wave-length 50°5 times the least wave-length in P,. But in
treating P, all terms z—z,' were taken into account, here
only the first ; and a fairer factor of comparison is 49°2, got
by omission of factor T; in the first result.
For radial displacement the limitation imposed by neglecting

based on Free Electrons. LIOL

terms linear in p must be remembered in respect to the
smaller values of p corresponding to y+2 small; and even
at the other extreme the approximation is sensibly less trust-
worthy because the values of ce are only 1/50 of those for
the type PS

§ 32. For a central core +ne and a ring of n electrons
the equations of oscillation are

EAS TN rare ; p— Eee. np
Mop WZ =€ E pea oF | , Mp wp=e& ea a ;
s $s

My p? wad = — &*

D soy lw ee (4A)
the two latter being simplified as to the right-hand member
by supposing n great, and as to the left-hand by supposing
p great. A specimen equation for axial motion when only
next neighbours are taken into account is

87° N .p? pS Otes
( an is Ao —2)+2+25=0,

Sa Nps oar
rea
n

which gives to — a range of values from 0 to 4.
: n

Thus for higher values of n, p* is necessarily positive for

some part of the range with a transition taking place when

27°=n*. This gives to na value just under 8—a value whieh

hardly justifies “the approximation, and so leaves the point of
transition uncertain.

aNivpze eres ;

For the radial displacement, =") — 2 ranges from

0 to 4, and all values of p? are positive. In the tangential

displacement the form taken by the sequence equation is

== +2) 0,— geeial = 6:

Ag®N 1»?
and makes 4 range from —4 to 0.

Lastly, if the method of next neighbour is eae to the

case of n electrons in one ring with a different core number,
3
say Me, then ee == cee has values from 0 to 4, N’ being
n?

Me— Cn. Sn axial stability then demands 27m, > n°.
This points to a much more rapid increase of m, with increase
of n than the special method referred to above (§ 20). But
the oscillation method has various features of approximation,

1102 Mr. R. Hargreaves on Atomic Systems

and the real conditions probably lie between the limits
suggested by the two results; and in particular the case
m,=n=8 probably lies within the margin of axial stability.

§ 33. In treating oscillations for the double ring the centre
was taken to be vacant, and for n large it is clear that the
effect of a unit centre in altering periods of oscillation is
slight. Buta new period is necessarily introduced—that of
the central ion itself, which may be treated in conjunction
with the question of stability for central ion or electron
under axial displacement : a fundamental question to which
only a preliminary answer was given in Part I.

A reference to (35 a) shows that itis possible by summation
of equations of motion to isolate the two sums of z coordinates
for ions and for electrons. The periods thus given (and one
of them is that of the central charge) are such as would
follow from using the same coordinates z, and 2, for each
element of the separate rings, and this method is more
convenient for the purpose. Thus for a central ion with 2
‘as axial displacement, z, and z, coordinates relative to the
ion, equations of motion are :—

M2) =L—naz,+ nBZo, ) .
mM, (2) + 21) =Ltaz,—y (21-2), L se ae
Ms(2p + 2) = —L—Bzy +9 (a1—22)5 |
in which
a=e/a°, B=P/aX=aa, y=2bd=22°%ad,

n—-1
2h Ten 1+ v?—22 cos (2r+1)r/n}-3?,

and for n great d=n?°I3/77° approximately. 9

The external force is ene in a later apple
Eliminating Z, we have

myz,= (n+ Lac, —nB2.— (21 — 22), "|
Moe = — B&o+4(21—22),
or with z=—@’o*z and m’a2=Ne, +. . (47)
((?N +n +1—22°d) 2, =(n—2¢) a*20,
(wp? N — xv? — 22° )2.= — 20° be, fy
leading to
pg N?— Nai (26 +1) + (n+ 1)2°{26(2?—1)—-1}=0, (48a)

where the only approximation used is an omission of pw, and
N is the value proper to the case with centre. Where a

based on Free Electrons. 1103

positive centre is possible, that is for n=4, the last bracket
_in (48a) is positive, and the two values of qg? real and
positive. If the oscillation can be identified, this is a means
of determining p, which can be used with any value of
n=>A4 by calculation of ¢.

The asymptotic value is m,/m,=143°89?/n'q’? ; the roots
are Ng?=2¢/u = 2n°T3/u7? and Nq’*=3né in asymptotic form,
or with our previous value of w, g=16°72n and g/=4°712/n.
The number gq’ corresponds to the new ionic period, and q is
not altered, to this order in n, from the value previously
found for the sum of displacements.

For a negative unit at the centre the equations are changed

to —- msZyp=Nazy~—NB2Zo, my (29+ 21) = — 421 —Y(21— 22),
Mo(Zy + 22) = B2g + (<1 —22) 5
and elimination as carried out above leads to
pg? N? —pwa?N {22° +n—(n + 1) x*}

—nx*{2b(a? -—1)+1}=0. (486).

One of the roots is necessarily negative when #>1, and
on examination it appears that this is also true for n=2, 3,
or 4, cases for which v <1, that is 26(23—1)+1 is a positive
quantity. By the criterion of axial stability, therefore, the
admission of a positive unit centre and the rejection of a
negative unit are justified. :

§ 34. The equations (45) may also be applied to test the
cohesion of the system under external force in the direction
of the axis. Stability is a part of such a test, but we may
take a further step and inquire into the extent of internal
displacements when the various elements move with a
common acceleration f. Thus we write Z2=f, 2;=0=2,
and examine the magnitude of 2,2, for a given external

Maltiplying the second and third by n and adding all
equations, we get |
ZL={(n+1)m,+m.}f, or Z—m,f=nMf,
Z+mf=(n+1)M/ where M=m,+mz.
The three consistent equations are then
M f=az,—B2, nMf=y(21—25)—2A1, |
(n+1)Mf=y(a1— 22) — B22,
of which the solution is rai C49)
LV(@—B) + aB$a1=(y—nB) Mf,
{y(a—B) +aB}a=fy—(n+ lap Mf J

1104 Atomic Systems based on Free Electrons.
This gives
{y(S—«) —aB}(2.—-2,;)=f(n+ lje—nB} Mf=M fa

approximately for n great, and so

Mf=y(# -1)(a2—a) =38y(e2— 1),
or L=3nky(z.—%).

If now we write Z which is the force on one ion as é*/d?,
with the action of one atom on another at interatomic
distance in view, and apply numerical values, we get

Zo ee le L gs
Cer neenn 1 a

ae) ae

and a comparison with — oes ee shows that the relative
nN

displacement of the rings issmall compared with the differ- _
ence of radii. The value of z, or 2, is much greater than
their difference, viz. Mf=—(B—«)z, or Z= —3nEan, with
a numerical result 2,/a= —na?/9:75d?. On this feature that
the displacement of the central ion relative to either ring 1s
on a much greater scale than their mutual displacement,
is based the remark on electrolytic conduction in ¢ 5, Part L.

If Z is not a constant but a periodic force of period
27/Qw, the method used above will be found to give

€a—%2z| wQ*N?— Q?Na9(26+4 1) |
+ (n+ 1)a*{26(2®—1) +1}] = —2°(2¢6 —n9Z, | ;
or also >. (90)

Mgz@°N (Q?— q’) (Q?—g”) = —a3(26—n) Z,
mz" N (Q?’ — 9?) (Q?— 9”) = — (22° —n—1—-Q°N)Z |

with g and gq’ asin $33. When there is no central charge
the formule are

Moeqw?(Q?—q?)=Z, and myzy+moz7,=0.

The method could no doubt be applied to examine the
scale of displacement of satellites, but not without a sensible
complication of the equations.

In bringing this long task to a close I wish to acknowledge
the kindness of Sir Joseph Larmor in reading earlier sketches.
of this paper, and making various comments, criticisms and

Selective Reflexion of X 2536 by Mercury Vapour. 1105

suggestions, which have proved of great service in the
revision of the manuscript. The interest he took in this
question of the freely equilibrated nucleus and_ other
dynamical problems presented in the paper, provided a
valuable stimulus in the course of work which of necessity
comprised much tedious calculation. or this assistance
I am grateful, and feel all the more indebted since it
involved some encroachment on a very scanty leisure.

CI. Selective pa of % 2536 by ee
Vapour. By R. W Woop*.

| al earlier papers it was shown that there appear to be two
types of selective reflexion of radiation which is very
nearly in synchronism with the free period of the mercury
molecule at the 2536 absorption line. One type is due to
the abnormally low value of the refractive index of the vapour
on the short wave-length side of the line. The change of
refractive index at the boundary quartz-Hg vapour is
greater than the change for a boundary quartz-vacuum,
since the refractive index is less than unity, consequently
we have strong reflexion for radiations immediately adjacent
to the absorption line on the short wave-length side. The
high value of the refractive index of the vapour for
radiations of wave-length slightly greater than that of the
absorption line, makes the change of index at the boundary
small, consequently the reflexion for these radiations is
very feeble. This was shown by reflecting the light of
X= 2536 from a quartz are operated at high temperature

2536 broad and strongly reversed) from the inner surface
of a flat prismatic quartz plate which was sealed to a quartz
bulb containing mercury vapour at a pressure of several
atmospheres. ‘he reflected light was photographed with a
quartz spectrograph, and only the short wave- -length half of
the 2536 reversed line was found on the plate.

The experiment was also tried using the 2536 line from
the water-cooled quartz arc. In this case the reflecting
power of the quartz-mereury vapour surface was about four
times as great as the normal reflecting power of quartz in
this region of the spectrum. Since in “this case the light is
highly homogeneous, it was inferred that the high reflectivity
was the result of the absorbing power of the vapour, the case
being analogous to the selective reflexion of the aniline dyes.

It occurred to me recently, however, that it would be
desirable to repeat the experiment ‘using still more
homogeneous light, as a considerable portion of the 2536

* Communicated by the Author.

1106 Selective Reflexion of X 2536 by Mercury Vapour.

radiation, even in this case, can pass through a layer of
mercury vapour at room temperature, 10 em. in thickness.
We may thus interpret the result of the experiment as due.
to the selective reflexion of a certain very narrow range
of wave-lengths in the 2536 line, just as in the previous
case where the high temperature quartz mercury are was
used. I have accordingly repeated the experiment using a
a resonance lamp at room temperature as a source of
hight

“As I have shown in previous papers, the radiation in this.
cuse is almost completely stopped by a layer of mercury

vapour at room temperature a few millimetres in thickness.

The thick-walled bulb of fused quartz closed at one end by
a prismatic plate of the same substance, which was used in
the earlier work, was mounted in an pleciue furnace in close
proximity to a thermo-couple.

The resonance lamp was mounted in such a position that
its image, reflected from the inner surface of the prismatic
plate, was received by the lens of the quartz camera. This
adjustment was facilitated by attaching a small square of
white paper to the surface of the resonance lamp, in
coincidence with the area which radiated the 2536 mono-
chromatic light, when the lamp was illuminated by the
concentrated beam of 2536 light from a quartz monochro-
mator, the light coming originally from a water-cooled
quartz mercury are.

The paper square was illuminated by a concentrated beam
of white light, and by carefully adjusting the bulb in the
furnace, the image of the paper, reflected from the inner
surface of the plate, was seen in the camera. The paper
square was then removed, and two exposures made, with the
resonance lamp in operation, one with the bulb cold, the
other with the bulb at 400°, the photographic plate being
moved between the exposures. Just before each exposure
the prismatic plate of the bulb was super-heated, by brushing
it with a small pointed gas flame. This removed any
condensed droplets of mercury, which sometimes formed
on the inner surface of the plate.

The image reflected from the plate was much denser in the
case of the exposure with the bulb hot. A number of
exposures were now made giving longer times for the cases
in which the bulb was cold. These showed that the
reflecting power of the plate when backed by dense mercury
vapour (density corresponding to 400°) was between 3°95
and 4 times as great as the normal bole power of
quartz for the wave-length in question, a result which is im
agreement with the value found in the earlier work.

hoator 2

CII. Polarized Resonance Radiation of Mercury

Vapour... (By BR... W. Woop.
[Plate LX.]

[* my earlier papers on this subject I expressed the opinion

that the resonance radiation of mercury vapour showed no
traces of polarization, a somewhat surprising circumstance,
in view of the fact that the resonance spectra of sodium and
iodine are strongly polarized.

Lord Rayleigh published a short letter in ‘Nature’ several
years ago stating that strong polarization could be observed
in that portion of the excited vapour at some distance from
the window through which the stimulating radiation entered,
the percentage of polarization falling off as the window was
approached. This indicated that the polarization was pro-
duced only by radiation not quite in synchronism with the
molecule, of which we have a very marked example in the
case of the light scattered by air and other gases, in which
the wave-length of the radiation is very far removed from
that of the absorption bands of the gas.

During the past winter I have made a further examin-
ation of the subject, and have found that the radiation is
strongly polarized, but that the percentage of polarization
does not appear to depend upon the distance to which the
exciting radiation has penetrated. The mercury vapour
was contained in a highly exhausted quartz tube with
windows of the same substance ground and polished. The
exciting radiation was furnished by a quartz mercury are,
the lower half of the tube being covered with cotton, over
which a stream of water flowed continuously, this arrange-
ment being necessary to prevent the reversal of the 2536
line which excites the vapour. The radiation from a wide
slit, placed close to the arc, was passed in turn through a
large quartz lens, a quartz prism of about 40°, and a second
lens: the prism was cut parallel to the optic axis of the
quartz, consequently two polarized spectra were obtained,
and a polarized monochromatic radiation of wave-length 2536
could be obtained from a second slit suitably located. The
plate in whfch this slit was cut was coated with barium platino-
cyanide to facilitate its adjustment in the ultra-violet regions

* Communicated by the Author.

1108 Prof. R. W. Wood on Polarized

of the two spectra. Figure 1 shows the arrangement of the
apparatus.

A-is a quartz wedge of small angle cut parallel to the axis.
With monochromatic polarized light properly oriented, this
shows when viewed through an analyser a fringe system,
the visibility of which increases with the percentage of
polarization. B is a double-image prism of quartz, C the
camera with quartz lens, and D the exhausted tube
containing the mercury vapour, a cross-section of the
illuminated end of which is shown at E.

Rie ls

Photographs of the polarization fringes reproduced as
negatives on Plate IX. show clearly that they are distinct
quite up tothe wall of the tube, where the exciting radiation
enters, and that they are of equal visibility all along the
column of excited vapour. Photographs were also made
through the side of the tube, with the exciting radiation
passing down its axis, and similar results obtained. The
fringes are more distinct when the opposite end of the
quartz tube and the contained drop of mercury are cooled
with a bath of ice and salt. At room temperature there is
much secondary resonance radiation between the primary
beam and the observation window, and this undoubtedly
reduces the visibility of the fringes. I believe, however,

Resonance Radiation of Mercury Vapour. 1109

that the actual pressure of the mercury vapour also influences
the percentage of polarization, for it was found that it fell
rapidly if any other gas was admitted to the tube, helium
at 6 mms. pressure destroying practically all traces of
polarization. In cases where the vapour was excited by
polarized light (electric vector vertical as seen from the
camera) the visibility of the fringes indicated nearly
complete polarization. With unpolarized excitation the
visibility was somewhat less, though the fringes were still
very distinct.

These results show that in no ease is the polarization
complete, for if it were, the visibility of the fringes would
be the same with unpolarized as with polarized excitation.

The fact that the fringes are more distinct when the density
of the mercury vapour is lowered by refrigeration of one end
of the tube, indicates that molecular impacts probably have
something to do with the depolarization. Admixture of other
gases at low pressures reduces the fringe visibility, and
finally destroys all trace of polarization, though the
intensity of the resonance radiation may be quite unimpaired.

Air at lem. pressure practically destroys the polarization
while it is still pronounced in hydrogen at the same pressure.
Curiously enough, however, the intensity of the resonance
radiation of mereury in hydrogen is less than half of its
value in air. This isa marked exception to the daw which
was found by Wood and Franck, in the case of iodine
fluorescence, namely that gases diminished the intensity in
proportion to their electro-negative character, air being
much more destructive of the fluorescence than hydrogen.

The results obtained with air, hydrogen, argon, and helium
are given in the following table.

Intensity of Polarization

Resonance Rad. fringes.
Air O'Gi rig 9.56 ke ae 2°5 strong.
Hydrogen 0°65 mm. ...... Mi strong.
Are 4 tothe U cesede <acenes ; i nearly gone.
Hydrogen 4 mm. ......... 0:5 faint.
Ase Mentes?*. 0/55 0°5 gone entirely.
Hydrogen lem. ......... 0-2 a trace.
Aer ns, «2558.5 ees 3 a trace.
Argon, DMM: sec.220% tea: ae 10° gone.
AXON, 3 WIP tocot hares 5° faint.
Heluam 2 mig oe ad case 4: faint.

Helium: 6 mite... 22.0.2... 10° gone.

1110 Prot. R. W. Wood on Polarized

The decrease in the percentage of polarization caused by
the admixture of small quantities of other gases, and its low
value with pressures of mercury vapour much in excess of
the value which it has at room temperature, is of consider-
‘able theoretical importance, for if there is a brief interval
‘of time between the absorption of energy by the mercury
molecule and its re-emission, and the depolarizing factor
is a rotation of the molecule, it is probable that the magni-
tude of these quantities can be experimentally determined
‘by making a caretul and exact determination of the percent-
ages of polarization corresponding to different pressures and
temperatures. This matter is now under investigation.

In the case of the green fluorescent light emitted by
mercury vapour at higher densities, when stimulated by |
ultraviolet light in the spectral range well below the 2536
line, the time interval between absorption and emission is so
long that it can be shown experimentally by methods
which I described in the Proc. Roy. Soc. for 1921. The
vapour was distilled in an exhausted quartz tube in the
form of an inverted U, the rising column of vapour
remaining dark within the narrow beam of exciting rays,
but bursting into fluorescence at a distance of several
millimetres above this region, in the form of a_ pointed.
green flame, concave on its under side. The resonance
radiation, excited by a narrow pencil of rays of wave-length
2536, traversing the U-tube, was confined wholly within
the illuminated region, showing that the time interval
between absorption and emission, if it exists, is very much
‘briefer in this case. |
- The action of argon and helium in augmenting the
intensity of the resonance radiation is of considerable
interest. With argon at 0°5 mm. pressure the intensity of
the resonance radiation is noticeably greater than in a highly
exhausted tube. ‘The same is true with argon at 60 mms.
pressure, the intensity being more than doubled by the
ipresence of the gas.

Helium at 330 mms. increased the intensity to fully four
times its value in an exhausted tube. At first sight this
‘seems very surprising in view of the fact that in all cases
‘ previously observed, so far as I know, the introduction of a
foreign gas decreases the intensity of the fluorescence, for
example the introduction of helium or argon into fluorescing
iodine vapour. A little consideration, however, made it
‘seem probable that, in this case, the argon and helium

Resonance Radiation of Mercury Vapour. DB ile |

increased the spectral range of the frequencies to which the
mercury molecules could respond, in other words broadened
the 2536 absorption line of the mercury vapour, which was
thus able to divert as resonance radiation more energy from
the exciting beam. In vacuo, the mercury vapour abstracts
only the “core” of the very narrow 2536 line ; in helium
or argon practically the entire energy of the line may be
diverted. This was tested in the following way.

The 2536 light from the monochromator was passed
through a highly exhausted quartz tube containing a drop
of mercury. This removed the “ core” of the line, the light
emerging from the tube causing no resonance radiation in a
second exhausted mereury tube, though still nearly as intense
as the original beam. Helium was now admitted to the second
tube, and intense resonance radiation immediately observed,
proving that the mercury vapour in helium will resonate to

=
7

— —_
~~.
_— —_
~

~
—

ae OP Ow Ow wwe em www ww ee ewe

.
\
¢

@

the frequencies to the right and left of the core of the
exciting line, as indicated in fig. 2. Here we have the
intensity curve of the 2536 exciting line represented by an
unbroken line, the portion abstracted by mercury vapour in
vacuo by a dotted line, and the portion taken up by mercury
in argon or helium by the line of dashes. :

i ae,

CII. Electric Fields due to the Motion of Constant
Electromagnetic Systems. By 8. J. BARNETT ™.

§1. JN this article Maxwell’s equation for the electromotive
intensity, together with a theorem derived from it
in §600 of his Treatise, and hitherto but little used, will be
applied to the investigation of a number of simple but
fundamental fields. Some of these fields are well known,
some are new; all of them are worth considering from the
standpoint of Maxwell’s theorem.
Consider an electromagnetic system B which has a velocity
v relative to fixed axes C. Let H, H’, B, B’, w, W’, and
A, A’ denote the electromotive intensities, magnetic induc-
tions, electric potentials, and vector potentials observed at a
point P fixed in © by one at rest in C and by one moving
with the system B, respectively. The electromotive intensity
at P, or the force per unit charge upon an infinitesimal
charged body fixed at P, is, in Gaussian units,

Le a a 9g a ee lr:
to the fixed observer. To the moving observer it is

B= > (22) vy — 2 eB] eee

where

(7)=S+eva. ee

The electromotive intensity at P, if fixed to the moving
system B, is

IL 110A 1
eh Meg cen eo . (4)
to an observer in ©, and
pean e/a ;
P'=El+ ° (eB'}=—-=(2° )- vy Be ty

to an observer moving with.B.
Assuming a principle of relativity according to which

* From papers presented to the American Physical Society, Nov. 26,
1921, Feb. 25 and April 21, 1922. Communicated by the Author.

Fields due to Motion of Electromagnetic Systems. 1113
E=E’', B=B’ (so that also A= A’), Maxwell has shown that

the motion produces (to an observer fixed in C) an electric
field whose polar part is derivable from the potential

b=y—y = _ (AD). =v an ee nbr an)

From the assumptions made it is clear that the result is
only an approximation; but, as pointed out by Larmor *, who
has given another derivation of (6), the error is only of the
second order in v/c.

For a constant electromagnetic system,

(3) _ (*) =(), and therefore o° =—(vV)A.. (7)

If also the system is unelectrified (in B), as will be assumed
henceforth,

I
Vv'=0, and Vy=~V(Av). cused Mee ss (8)
In this case

B=+ (wV)A—*V(Av)=E'=" [Be]. . ()

The electromotive intensity of the field produced by the

motion is thus given completely by : x the vector product

of B and v.

§2. The vector potential at a point distant r from an
element of space (volume, surface, or length) dt in which
the current density (volume, surface, or line) is? is defined as

a=({ ne. a), *(10)
cr :

We have also the relation
curl A=B, or {caay={ (BAS sys. () «0 LCL)

where dl is the element of length of a closed curve bounding
the surface whose area is S. From one or both of these

equations we can always determine A, and hence ¢= (Ao).
If o denotes the electric density (volume, surface, or line)

* J. Larmor, Phil. Mag. xvii. p. 1, Jan. 1884.
Phil. Mag. 8. 6. Vol. 44. No. 264. Dec. 1922. 4 C

1114 _Mr. 8. J. Barnett on Electric Fields due to the

at any point of the system where the current density is 7, we
obtain from (6) and (10) the relation

‘an@iings eee iv
p=) = 2) Sar

Lr

or caeD.

C

This result was obtained from Clausius’s theory in 1880 by
EK. Budde*; the corresponding result with the correction
for the second-order term in v/c was obtained by Lorentz +
in 1895, and by Silberstein ft, from Minkowski equations,
a

From (12) it is easily shown that in the general case the
electric moment Q of the charges developed by the motion is
equal to 1/¢ multiplied by the vector product of the velocity
and the magnetic moment M ; that is,

Q== [eM], 2 OL: er

of which a number of special cases will be found below §.

§3. Two infinite plane parallel current sheets B and C with
equal and opposite currents, | per unit length, in motion
parallel to the stream-lines (fig. 1). From (10) it is clear

Fie. 1.

that A=0 over the central plane D parallel to the sheets,
and that elsewhere it has the direction of the current in the
nearer sheet. Tg obtain the vector potential at a point P

* EK. Budde, Ann. der Phys. x. p. 553 (1880).

+ H, A. Lorentz, ‘ Versuch,’ 1895, § 25.

t L. Silberstein, ‘The Theory of Relativity, 1914, p. 272.

§ Equations (6) and (12) have both very recently been derived for a
special case by W. F. G. Swann, who does not refer to the earlier work
by Maxwell and the others mentioned above. He has also obtained (13)-
for the special case of a doublet, but with the wrong sign. See Phys.
Rey. xv. p. 3865 (1920).

Motion of Constant Electromagnetic Systems. 1115

between the sheets and distant y from D in the direction of
the sheet B, in which v and I have the same direction, we |
may apply (11) to the area abed. We thus get Axbe=

Bxlexy=—= xbexy; so that
A= by— ey:

Similarly, we find for the region above B,

Between the plates :
1 Aart
o= = awe
Eo that — 2 Vy (Av) = — 85 = = [Bo]

is uniform and directed from B toward D. ‘The surfaces of
C and D have thus the charges + : and a5 per unit

area, in agreement with the general relation (12). Outside

the region between the sheets ¢6=+ a and —Vo=0.
Here OF = — (vV)A=0, so that the field is purely polar

and — En (ik) is the total electric intensity.
c

It is easy to see that if M denotes the magnetic moment
of the system, and Q its electric moment,

Q= (uM).

The charges developed by the motion, being themselves in
motion with velocity v, produce a magnetic field in the
direction of the original tield and with intensity

eae ‘3
ea lBN ld v=H-,
ae C
thus showing, in conformity with Larmor’s statement, that
the adoption of Maxwell’s method introduces an error of the
4C2

1116 Mr.S. J. Barnett on Electric Fite due to the

second order inv. We may proceed similarly in the other
cases which follow.

§4. Two infinite parallel wires B and C, with currents I
and —I, in motion parallel to ther lengths. For a point
distant 7; from B (whose current I is in the direction of v)
and 7, from C, we have

A=t((-— 9) ar
CNN a

ee ale ey ll

one ie 4 Caan
so that the electric field produced is that terminated by equal
and opposite charges on the wires, the charges per unit
length upon B and C being a and a Here also
dA OR. 6 al DE ea : reeees in th
wae 7 eee ore, as in the
last section,

(= : [uM |.

§5. Similar results are obtained for two circular cylindrical
coaxial conductors traversed by equal and opposite currents and
in motion parallel to the axis. |

§6. An infinite uniform cylindrical current sheet, with
current 1 per unit length of the aais, im motion normal to the

Fig. 2.

awis (fig. 2). Both inside and outside the cylinder the vector
potential is evidently constant in magnitude over each coaxial
cylinder and 1s tangential to the cylinders in planes normal

Motion of Constant Electromagnetic Systems. 1117

to the axis. Fora cylinder of radius » within the current
sheet, we have

2anrx Ap=a7r? B=?’ x amr
C
so that hi rB eel A
Z c
) oat,
and * (Av) = ae Aceh fe ae i
Thus |
2ar Ia
V(Av)=— SE =} [Br
C
is uniform is the sheet and is directed downward in the
figure.

The surface density of the charge which terminates the
uniform field with intensity — EY (An) is
c
é Zao |
ae Ane?

The charges per unit length terminating this field on the two
hemi-cylinders are

030 = , cos 8.

7; = x 2a= oe
eee th or 77) = 3 eo:
The electric moment Q, of these charges, positive on the
upper half and negative on the lower halt, is evidently
Ta7ly

ag [27 5p de= “Qe?

and is directed upward in the figure. The magnetic moment
of the system, per unit length, is

Tal

M=

Cc

directed into the paper in the figure. Thus
1
= OF [vM ].
Outside the sheet (7>a),
6 2
eek 2rla

cr

and o= : (Ao) = ala” = cos 7

Calle

1118 Mr.S. J. Barnett on Electric Fields due to the

This is the potential due to a line doublet situated on the
axis of the sheet with electric moment
. a mlat

= 2

C

per unit length and directed upward in the figure.
; 1
Th EE arig
us Q 5 [eM].

The outward radial component of the intensity at the
surface of the cylinder due to the doublet of moment Q is
2 7
eo cos = ae cos@. Thus the surface density of the part

of the charges on the cylinder terminating the outer field is

Co= cos 0=0;,

ane
2c?
and its moment Q,=Q,. Thus the total electric moment per
unit length of the cylinder is
1
Q=2Q\= i [vM ].

The total charge per unit length on the upper and lower
hemi-cylinders is

q=2%-
In addition to the electric intensity —- V(Av), there is
c

also the solenoidal intensity

Vea. 8
WH ote ax
which does not vanish. Inside the cylinder
] youl Monee 2arla
GMO Ae ON) Bieta ee

= ~V (Av) = 5, (eam

so that the total intensity is 2 [Bo].
Outside the cylinder

1 2nla? 0 /y Anrla?v ay
pee d Eyl a: SN ea eR PBL WORE LL Seema ete
F {WV )A bo= ne se epg
and
1 Bale? Oe \\) 2erlate (gy?
‘A {VA fy = — @ ox (2,) Pah, we ee

Motion of Constant Electromagnetic Systems. 1119

These are the component intensities due to an axial line
doublet with moment equal and opposite to that already
obtained. Thus

¥(eV)A— 2 (Av) =0= = [Bo].

ey el ayer
The field whose intensity is ~ (eV A is thus a circuital

field without divergence whose flux cancels that due to the
charges in the region outside the cylinder and doubles that
due to the charges inside.

§7. A spherical current sheet, with current I per unit length
of the diameter along the magnetic axis, in motion normal to
the magnetic avis. The magnetic field within the sphere is
known to be uniform, normal to the planes of the stream
lines, with induction

The vector potential is evidently tangential to circles centred
on the magnetic axis and normal to the magnetic axis.
Within the sphere, for a cylinder of radius 7,

Arl
i . Aly Arly
and co) = a (Av) = SoS Cos é= Dey “i gy.
1 Slee)
Thus _ ~ V (Av) = 55,5) has [ Bu |.

That part of the surface density due to the charges which ©
terminate this uniform field is evidently
lv

= @os 0.
3c?

O1=

The total charge (terminating the uniform field) on each
hemisphere is
ma*ly

a 2 — peas iE
Gi ee ey tga

and the moment of these charges is
1 4 ra*lv

= | o,.27asin@.ad@.acos 0=. .~— 5
“0 dar

1120 Mr.S. J. Barnett on Electric Fields due to the

The magnetic moment of the sphere is

so that, with proper attention to signs,
1
Qi= oy [vM J.

Outside the sphere the magnetic field is that of a point
doublet of moment M at the centre of the sphere. The
vector potential is thus

ee |
foots 8 Bia |
so that

= * (Ar) =— -(ptve] r) =— ave: [eM] ),

which is the potential of an electric point doublet with
moment

Q=* [eM]

at the centre of the sphere.

The outward radial component of the intensity at the
surface of the sphere due to the doublet of moment Q is
2
“cos 0. Thus the surface density of the part of the charges

terminating the outer field is

2 Iv
Oo== —— COS 0 = 2 = C0s 6 =a;
2 Darae Duce :

The total charges upon the upper and lower hemispheres
terminating the outer field are thus

and the moment of these charges is

Q,=2Q).

The total surface density upon the two hemispheres is thus

Iv
o=—a,+ Pag COS 6,

&

Motion of Constant Electromagnetic Systems. 1121

in conformity with (12). The total charges upon the hemi-
spheres are

tq= + (qi +9) = oe tae ‘
and the moment of the charges is

Qi + Q.=Q=- * [pM].
Within the sphere.

*(0V)A : See ier ——! LG is
which is equal to the iw — - A(Av).

Outside the sphere — — Ley(a is not Ror in even in
the same direction as the total intensity [Bo]. =(eV)A

is again the intensity of a solenoidal field Seria
divergence whose flux through the sphere doubles the

intensity inside and makes it everywhere equal to — [Bu].
C

§8. Two cylindrical coaxial magnetic poles with a radial
magnetic field between them, and in motion parallel to the

ae , oe pe

ae om wee .

Ss 2

awis (fig. 3). In this case the vector potential vanishes over
the central symmetrical plane of the field. The lines of
vector potential are circles, centred on the axis and normal
thereto. The magnetic flux across a cylinder of radius r and
length xw, one end in the symmetrical plane, is 27rvxB=

Cx aa
2rrA; so that A=Ba=— where C is a constant. To one
If ’

1122 Mr. S. J. Barnett on Electric Fields due to the

looking in the direction of wv the circles are right-handed
behind the central plane, left-handed in front. In this field

Q= * (A) =(), the electric field is circuital, without charges,
ahd p= ~ 94 = *(@y)A= = [Bo].
c i

The lines of electric intensity are left-handed circles centered
on the axis in planes normal. thereto.

§10. The introduction of conductors (with negligible
magnetic susceptibility) moving rigidly with the system B,
and uncharged in B, does not affect the electric field pro-
duced by the motion in any way at points not within the
material of the conductors. For it does not alter A’=A,
and therefore does not alter —V/(Av) or (vV)A. Thus, if in
the field of §3 a closed conducting box, moving with the
current sheets, were to surround the observer in C, the electric
intensity observed would remain unaltered. It is therefore
unnecessary to inquire whether the substances in B are
conducting or insulating. If they are magnetic, the vector
potentials of the magnetons will have to be taken account of.

Another way of looking at the matter is, of course, this :
The introduction of the conductors and the consequent
induction of electric charges upon them may greatly affect
the intensity at P due to the motion. But the effect of these
charges is entirely neutralized by the distribution produced

by the motional intensity * [eB] acting in and upon the

conductors themselves. Within the material of the con-
ductors forming a part of the system B, or rigidly moving
with B, the total electromotive intensity is

Jf) es - (vWV)A— : V(Av) += Lela — ; [Be] “ : [vB|=0.

§11. In the case of a conducting cylinder or sphere with
uniform intensity of magnetization I in motion normal to the
magnetic axis, all the formule, for points outside the system,
are exactly similar to those for the cylindrical and spherical
current sheet, but I has now the meaning attached to it here
instead of that of §§7 and 8. Within the material of the
system the electromotive intensity is zero.

If we apply to the magnetie elements, or magnetons, the
same method previously applied to the whole current sheet,
we evidently find that each magneton with component mag-
netic moment mw parallel to the magnetic axis becomes so

Motion of Constant Electromagnetic Systems. 1123
charged by the motion as to have the electric moment
1 ? : ’
—[vp]. Thus the sphere becomes electrically polarized, with
c

polarization, or electric moment per unit volume, given by
1 1
be Lee am [vl ],

where the summation extends over the unit volume. The
field of this polarization gives the polar part of the external
field, and, together with the internal part of the solenoidal

field, just balances the motional intensity : [vB] inside the

system ; or we may consider that the effect of the polarization
is neutralized by that of an equal and opposite polarization
due to the charges induced in the parts of the conductor
adjacent to the individual magnetons, and that the distribution
produced by the motional intensity gives the polar part of the
external field, which, together with the solenoidal part, just
balances the motional intensity *

§12. In the ease of two similar injinitely long magnets with
rectangular cross-sections placed parallel with opposite poles
facing one another symmetrically and in motion parallel to
their lengths and normal to the lines of induction of the magnetic
jield, the vector potential is evidently zero over the central
plane parallel to the motion and normal to the pole faces.
It is everywhere parallel to the velocity ; and its direction
is related to that of the central part of the magnetic field
exactly as in the case of $3. Its magnitude is independent
of the coordinate parallel to the length. Thus

OA

ot
and the total intensity outside the substance of the magnets,
viz.,

=—(vV)A=0;

E=— — *V(Av)= — A [Be],

lies in planes normal to the motion, or is two-dimensional
like the magnetic field. Within the magnets themselves the
total electromotive intensity is zero, and, as in § 11, there is

Si 1 ; mos
an electric polarization P=—{[vI] at points where the in-
c |
tensity of magnetization is I.
* See E. Budde, /. c. ante, and W. F. G. Swann, /. c. ante.

1124 Mr.S. J. Barnett on Electric Fields due to the

Experiments made by the author * in 1918 witha screened
condenser placed symmetrically between two much larger
parallel magnets in motion like those above, are consistent
with the view that the field is polar, as required by the theory.
It is immaterial whether the intensity is calculated from
| Bu], as was done by the author, or whether it is calculated
from —\V/(Av); or we may consider that the effect of the
polarization is exactly neutralized by that of the equal and
opposite polarization due to the charges induced on the parts
of the conductor adjacent to the individual magnetons, and
that the net electric field remaining is due to the electric

displacement produced by the motional intensity : asi.

§13. Maxwell’s theorem ¢= (Av) cannot in general be

applied immediately to the case of an electromagnetic system
forming a solid or surface of revolution about an axis of
symmetry and in steady rotation about this axis ft, although
it may be applied to each element of which the system is
composed and which has its own linear velocity and vector
potential. |

The electric field surrounding such a body, rotating either
in a neutral region or in an impressed magnetic field directed
along the axis of rotation or symmetrical about this axis, can
be determined at once from equations (1) and (4). In this

case = =0(, so that the field is purely polar, derived from

the potential x, which can be calculated from the motional
intensity : [vB], and is due to the charges produced in and
Cine

on the rotating electromagnetic system.

* §. J. Barnett, Phys. Rev. xii, p. 95 (1918); xv. p. 527 (1920) ; xix.
p. 280 (1922). |

+ In connexion with this experiment Swann, J. c. ante, has stated that
Maxwell’s equation (1) cannot be applied to the case of rectilmear motion
to show that the field is polar because (he states) i this case the vector
potential is not independent of the time. This is clearly an error. Several
examples of the contrary are given above, in addition to this particular
case. The theory given by Swann is quite unnecessarily complex.

{ The theorem was derived for the general case involving rotation, but
‘its application to the case of a symmetrical system in rotation about the
axis of symmetry involves the assumption that the tubes of induction
rotate with the system, which is inconsistent with Maxwell’s general
theory. See also § 16.

Motion of Constant Electromagnetic Systems. 1125

Jochmann* and LarmorTf have both worked out the
general form of the solution for the case in which the system
is a conducting solid of revolution and the magnetic induction
symmetrical about the axis of rotation, and have given the
details for the case in which the solid is a sphere and the
induction uniform.

The potential at a point distant r=(>a) from the centre
of the sphere and with co-latitude @ is

] a” 3 1 wBa?
9 Susi es aes ie ) :
% cr 63 on cr® ren a Sere
where @ is the angular velocity and C is a constant. When
the sphere remains insulated from the initiation of the motion,
w Ba?

C=— rare and only the middle term of wW remains.
When the pole is earthed, C=O. The sphere is uniformly
charged within, the volume density being — fa oe Sell

further details are given by Jochmann.

$14. If the rotating body is magnetic, it is electrically
polarized by the motion, the polarization at any point where

the intensity of magnetization is I being P= *[(or)I]. On

account of the conductivity of the body, however, an equal
and opposite polarization is produced by electric induction,
and no gross effect of the polarization appears. The ex-
istence of this polarization, but with the wrong sign, together
with its neutralization by conduction, has been pointed out
by Swann (J. ¢. ante).

§15. Experiments by the author §, by E. H. Kennard ||,
and by G. B. Pegram{[, in which a screened cylindrical
condenser was placed coaxially in the field of the rotating
system of §13, short-circuited, and tested for charge when
at rest and when rotating with the electromagnetic system,

* EK. Jochmann, Crelle’s Journal, xxxvi. p. 329 (1863) ; also Phil. Mag.
Xxvlil. p. 347 (1864).

+ J. Larmor, Phil. Mag. xvii. p. 1, Jan. 1884. See also, J. Larmor,
Roy. Soc. Phil. Trans, A. 186, p. 695 (1895).

{ Recently the above results for the sphere have been published by
Swann, /. c. ante, who does not refer to previous investigators.

§ S. J. Barnett, Phys. Rev. xxxv. p. 323 (1912).

|| E. H. Kennard, Phil. Mag. xxxiii. p. 179 (1917); see also, for a
related experiment, Phil. Mag. xxiii. p. 937 (1912).

q G. B. Pegram, Phys. Rey. x. p. 595 (1917).

1126 Mr.S. J. Barnett on Electric Frelds due to the

are all in accord with Jochmann and Larmor’s theory of
$13. That is, when the condenser remains at rest, the
charge is zero, because the field is polar, with external
charges ; when in motion, the charge is due to the intensity

; [(@r)B] in the conductors. The application of Maxwell’s

equations (1) and (4),as used by Larmor, to these experiments
was discussed in detail by Pegram in 1917. In many
experiments from Faraday to the present time the potential
difference between different circles of latitude on the
rotating body has been measured with a galvanometer or an
electrometer; recentiy the charge on the sphere of §13,
with axis earthed, and surrounded by a concentric sphere,
has been measured by Swann*. All the results are in
agreement with the theory of Jochmann, Maxwell, and
Larmor.

§16. In the case of a symmetrical electrical cireuit
rotating about its axis of symmetry, the rotation produces no
rotating effect on the tubes of induction of the magnetic
field,as indicated in § 13 and shown very simply by Pegram.
Applying this result to the Ampeéreian vortices of a magnet,
it was Pegram’s idea that when the magnet rotated, ‘each
vortex carried its lines of induction in ttemnale tiem but
that the lines of induction of the vortex did not share the
rotating motion of themagnet t. This idea has recently been
‘discussed in detail by Swann (1. c. ante), but it is not new.
It is only in this sense that the ‘moving line” theory of
unipolar induction is true.

We have already seen that so far as the net external result
is concerned, the effect of the.polarization due to the trans-
latory motion of the magnetons may be considered neutralized
by that of induced charges. The intensity e at any point P
of any plane through the axis of rotation due to the motion
of an element of the magnet with magnet moment m and
‘velocity v= |r] is

e=—V(av)+ (eV )a,

* W. F. G. Swann, Phys. Rev. xix. p. 38 (1922).

+ This is not discussed in detail by Pegram, bit was clearly in his
‘mind and is an immediate corollary of the case of the simple coil. After
Pegram’s paper was read at Cleveland, but in less complete form than
that of the printed article, I stated to him that in the case of a magnet
I thought the fundamental way to consider the matter was to assume
that each electron orbit carried its lines of induction with it. He
immediately assented, and remarked in addition that the rotatory part of
‘the motion was not to be considered.

Motion of Constant Electromagnetic Systems. 1127

while the intensity e’ at the same point due to the corre-
sponding element with (equal) moment m’ symmetrically
situated on the opposite side of the plane is evidently

e' = —Vav—(vV )a.

Thus at any point P the solenoidal parts of the fields’ due to
corresponding elements cut one another out in pairs, while
the polar parts of the intensities are equal and additive. The
polarization, due to the polar parts, being neutralized by in-
duction, the total gross effect of the motion of the vortices
vanishes.

§ 17. From the fundamental equations of electromagnetic
theory as developed by Cohn*, and later by Minkowskif ,
and still later by Einstein and Laub f{, a genera] expression
has been obtained for the polarization produced in an insulator
by its motion ina magnetic field. If K denotes the dielectric
constant of the medium, yw its permeability, I its intensity of
magnetization, B the magnetic induction, and v the velocity,
the formula for the polarization, in the approximate form

obtained by M. Abraham §, is

K 1
P= etn, a, [vB].

This polarization is easily shown to consist of two distinct
parts: one, P;, produced by the motional intensity 1 eB]
: c

acting on the moving part of the insulator; the other, P,,
due to the motion of the magnetons.

On the theory of Lorentz and Larmor the ether is at rest,
so that only the electrical fraction (K —1)/K of the insulator
is in motion. Hence

pat ee

Avr ¢

This result has been fully confirmed by experiments made on
air in 1901 by Blondlot||, on ebonite in 1904 by H. A.

* KE, Cohn, Ann. der Phys. vii. p. 29 (1902).

+ H. Minkowski, Gott. Nachr. Math. Phys. Kl. 1908, p. 53.

{ A. Einstein and J. Laub, Ann. der Phys. xxvi. p. 532 (1908).
§ M. Abraham, ‘ Theorie der Elektrizitaet,’ ii. § 38 (1908).

| R. Blondlot, C. R. exxxiii. p. 778 (1901).

1128 Prof. M. N. Saha on Temperature Lonization of

Wilson *, and on rosin, sulphur, and ebonite in the interval
1902-3 by the author f.
From Maxwell’s theorem

i i 1
Po= AlN = — ara el U 0
2= [el] = (1 1) (eB) |
Adding together the expressions for P, and P, we obtain
very simply the expression for P given above as hitherto
derived only on the basis of the Cohn-Minkowski equations ft.

The Carnegie Institution
of Washington,
Department of Terrestrial Magnetism,
April, 1922.

CIV. On the Temperature Ionization of Elements of the
Fligher Groups in the Periodic Classification. By Mucu
Nap Sawa, D.Se., FLLP., Guruprosad Singha Professor
of Physics, University of Calcutta, India §.

I.

HE theory of the temperature ionization of gases and its
application to problems of radiation and astrophysics

was given by the present writer in a number of papers
published during last year. In these papers the theory was
limited to the ionization of gas consisting of atoms of a

* H. A. Wilson, Roy. Soc. Phil. Trans. A. 1904, p. 121. Wilson con-
sidered his experiments to prove that the motional electric mtensity or
electromotive force was proportional to (K—1), which is not correct.
According to all theories, the motional intensity is independent of the

medium and equal to = (0B); while on the theory of Larmor and

Lorentz, the resulting polarization is proportional to (K—1), the result
supported by the experiments.

+ S. J. Barnett, Phys. Rev. xxvii. p. 425 (1908).

{ The permeability » differs from unity so slightly for all insulators
that it is impossible at present to distinguish experimentally between P
and P;. By embedding a large number of small steel spheres in wax,
however, M. Wilson and H. A. Wilson (Proc. Roy. Soc. A. Ixxxix.
p. 99 (1914)) formed a composite dielectric whose mean permeability, for
large volumes, was much greater than unity. On the assumption that
this procedure is justifiable, the results of experiments which they made
on the electric effect of moving the composite substance in a magnetic
field support the above equation for P. M. and H. A. Wilson concluded
that their results therefore supported the (HKinstein-Minkowski) principle
of relativity. As shown in $17, however, the result follows from
Maxwell’s theorem based on a much older, though less exact, relativity
principle.

§ Communicated by the Author.

Elements of the Higher Groups. 1129
single kind. Recently H. A. Milne and Henry Norris

Russell * have extended the theory to mixtures of elements.
By a comparison of the sun-spot and the solar spectrum,
Russell finds that the predictions of the theory with reference
to the relative intensity in the hotter and the cooler spectrum
of lines associated with ionized and non-ionized atoms are
found to be in general agreement with the facts. Russell
has also shown that the temperature of the sun and the sun-
spot comes out to be much more in accord with the figures
obtained from general intensity measurements when mixtures
of different elements are considered instead of one single
element.

But discrepancies have also been pointed out by Russell,
which suggests the need of some modification. The nature
of these discrepancies may be grasped from the following.

Let I, and I, be the ionization- potentials of elements A
and B. Then at a definite temperature and pressure the
ratio of the degreés of ionization of A and B is given by the
equation

I,—-I,

log ec = ‘log 5 Sh — log (725) = 5036

where 2,, x, are the fractions ionized, I,, I, are ret ak
in volts.

Tf I,=1,, x; should equal vy.

That this is not the case is shown from the fact that sodium
and barium have got practically the same ionization potential
(5-11 and 5°12 volts respectively), yet both in the sun and
in the sun-spot, barium is a good deal more ionized than
sodium. ‘The resonance line of Ba, X=5535°93, is absent or
very faint both in the solar and the spot spectrum, and itis
represented only by the enhanced lines (Bat, X=4934°07,
4554°04), which shows that barium is completely ionized
not only in the sun but also in the spot. The resonance
lines of sodium, A=5889°97, 5895-94, on the other hand, are
very prominent in the solar spectrum, and are greatly in-
teasified in the spot, which shows that in the sun a large
percentage of sodium is unionized, and in the spot the
percentage increases owing to a lowering of temperature.

What has been said of sodium and barium admits of a
widegoing generalization, viz. the alkaline earths are, as a
rule, “much more strongly ionized than their ionization
potential would indicate. The behaviour of the alkalies is

* Milne, The Observatory, Sept. 1921; Russell, The Astrophysical
Journal, March 1922.

Phil. Mag. 8. 6. Vol. 44. No. 264. Dec..1922. 4B)

1130 Prof. M. N. Saha on Temperature Ionization of

normal, and we shall see later on that, as a rule, elements
belonging to the higher groups are more easily ionized than
elements of the preceding group, and the successive steps of
ionization follow each other in rather quick succession.

First of all, let us consider the relative intensities of the
lines of alkalies and the alkaline earths in the sun and the
Spot spectra.

Alkalies. . | Alkaline Earths.

2) Intensity. | LP. Intensity.

_ Volts. Sun. Spot. | Volts. | Sun. Spot.
IND SeRoranse 511 30 60 Mia. eumes 765 30 oie
op, sain, ABO a Bnep Ubi! Wil Cacea a 6089) 20%) 525
Es i.e 4:16 = 1 SE | cckpisebren Baan 1 3
Osa heck 3°81 == ? Bante. eer 512 -—

* Russell, oc. cit. p. 130; the intensity given against Mg is that due to the
2p—3s line; the resonance line of Mg, \= 2852, is beyond the range.

The table shows that Ba is at least as highly ionized as
Rb, though I,—I,=-96 volt. Sr is only slightly less ionized
than Rb, both in the sun and the spot. Calcium is less
ionized than potassium. When we compare the intensity of
the lines of Ca and Na, we find that in the sun they are
almost equally ionized (calcium a bit more), but in the spot
the recombination between Cat and (e) is much less marked
than between Nat and e.

Prof. Russell suggests that if the effect of radiation could
be taken into account, the theory would be more improved,
and the discrepancies could be explained. (See Russell,
The Astro. Journ., May 1922.)

iil

It cannot be denied that the theory is always to be re-
garded as incomplete until the effect of the general field of
radiation can be taken into account. But it is doubtful if
this alone will explain all the discrepancies. Another factor,
the consideration of which is presented below, seems to play a
rather important part.

It is now well accepted that elements of the first group,
Li, Na, K, Rb, Cs, have only one electron in the outermost
ring, while metals of the second group, Mg, Ca, Ba, Sr, have
two electrons in the outer ring. Besides, these two electrons
are equally situated—in other words, whenever a Ca-atom is
subjected to the action of any physical agency tending to
tear off the electrons, it will act equally on both of the
valency electrons. In the case of the alkalies, it will act on

Elements of the Higher Groups. Tol

one electron* only. Let us take the case of electrolytic
solutions. Here the electrical forces act equally strongly from
all directions, and a calcium atom loses both electrons. The
inner electrons being more solidly fixed to the nucleus do
not get detached. A sodium atom, on.the other hand, loses
pader the same conditions only one electron, because tee is
only one electron which can be torn off by ‘the same ¢ agency
from a Na-atom.

This proves that in the normal case both the valency
electrons in Ca oceupy nearly identical positions in the
atomic system—they are contained in the same part of the
atomic volume, and are fixed to the system with forces which
are either identical or very nearly identical.

Let us now consider what will take place when a Na-atom
and a Ca-atom are subjected to the same ionizing agencies,

Coll pice
say bombarding electrons, light pulses, or thermal collisions.

For the sake of simplicity we consider the first case only.
We shoot at a Ca- and at a Na-atom with the same number
of electrons, which are possessed of such energy that they
can tear off just one outer electron when it hits at the right
place in the atom. Now, assuming the atomic volumes to be
the same and the I.P. to be the same, it is clear that the
number of successful hits on a Ca-atom will be twice as
great as the number of successful hits on a Na-atom, for in
the outer volume calcium has two electrons. while sodium
has only one. In other words, for the same strength of the
ionizing agent, Ca-gas will be, roughly speaking, twice as
highly ‘ionized as Na- -2as.

These considerations may be extended to all cases where
ionization takes place by encounter, either with a liyht
pulse or another atom.

In support of this view, an interesting observation by
Millikan { may be cited here. Helium has two electrons
which, according to Bohr and Lande f, are both in the same
part of the atomic volume. Millikan finds that when helium
gas is bombarded by e-particles, then, in one case out of
seven, both electrons are simultaneously carried off by the a-
particle. This could not take place if one electron was much
nearer the nucleus than the other, and was attached to the
nucleus with a greater force. If they are contained in the
same part of the atomic volume, then, according to the laws
of probability, in one case out of eight both electrons would

* The strength of the ionizing agent is assumed to be not so large as
to be able to tear off any one of the inner electrons.
+ Millikan and Wilkins, Phys. Rev., March 1922.
{ Zeitschrift fiir Physik, Bd. ix. p. 33.
4 D2

1132 Prof. M. N. Saha on Temperature Ionization of

come simultaneously in the same octant, and both would
be carried off by an «-particle which chanced to pass close
to them.

The alkaline earths resemble helium in so far that they have
two valency electrons in the same part of the atomic volume
in the outermost region of the atom”.

In solutions we have always Ca*tt-atoms and never Cat-
atoins, because the electric forces act equally strongly from
all directions. In the cases considered by us, Cat-atoms are
more probable than Cat t-atoms, because the ionizing agencies
act from one side only.

The cases of recombination of Cat and e, Na* and e may
be considered in the same light, and we find that in the case
of Cat and e, recombination is more difficult than between
Nat and e. If we consider an Nat-atom, we find that there
are no electrons in the outermost ring, and the positive lines
of force proceeding from the nucleus act equally strongly
within the 47 solid angles about the atom. From whichever
side the electron may. approach the Na*-atom, provided
other things (energy, distance) are of the right order, the
electron will be captured by Na*. Not soin the ease of Cat.
It has still got a valency electron in the outermost rings;
lines of force proceeding from the nucleus are strongly con-
centrated on it. In other words, to use the language of
Stark, there is a negative patch on one side. An electron
cannot be captured if it approaches Ca* from this side. It
can be captured only when it approaches the Ca*t-atom

within only a definite fraction = of the total solid angle

about the Ca-atom, where n isa number >1. We may call
‘n’ the “steric factor.”

These considerations show that for an atom like Ca, ioniza-
tion is easier, and recombination of Ca* and eis more difficult
than recombination of Nat and e. For trivalent and tetra-
valent elements like Sc and Si these considerations will apply
with even stronger force.

iif,

It isa more difficult task to take account of the above facts
in a statistical theory. . To Boltzmann we owe the idea that
when two atomic species A and B associate, every case of
approach of A and B does not result in a combination, but
only when A and B present to each other certain definite

* It may be pointed out that Langmuir places helium at the head of
the alkaline earths (see Loring, ‘ Atomic Theories’),

Elements of the Higher Groups. LASS

if : i
parts, —, -, of their respective surfaces, Boltzmann was
Na Io

of opinion that for the formation of diatomic molecules ‘2’
varies directly as the maximum valency of the element (viz.
2 for Ca, 7 for I, and so on) *.

In recent years the “steric factor” has been introduced
into thermodynamics by Stern+ in a new theory of the
dissociation of I, vapour.

Stern considered the case from the standpoint of both
thermodynamics and the kinetic theory, and came to the
conclusion that ‘n’ lay between 6 and 7 in case of combi-
nation of two I-atoms to form an I,-molecule, thus lending
colour to Boltzmann’s belief. The kinetic theory is not
very convincing, for the following reasons. According to
dynamical principles, two particles A and B approaching
each other from infinity cannot form a closed system until
and unless they lose a certain fraction of their energy,
presumably by radiation. Similarly, a molecule AB cannot
be dissociated into A and B if the system does not absorb
energy from the outside.

‘Thus a complete theory of ionization is incomplete without
a consideration of the mutual action between radiation and
matter, and we are beset with the same difficulties which
have confronted all investigators on the subject since the
days of Boltzinann.

Proceeding to the thermodynamical theory, the funda-
mental equation was derived from the equation

Sat oS. Sen U/'T, . F 6 ° ° (A)

where U=heat evolved, S., Ss, Sas were calculated from the
quantum theory involving certain assumptions. (Here ‘a’
is Ca™, b is Se,’ Say is Ca.) The above equations are derived
on the assumption that the steric factor n=1. Taking the
“steric factor’? into account, the probability that a and 0

. 1 '
would simultaneously present the definite portions, TTN awe of

their surfaces to each other is given by

N
w=(".=),
Na Te

where N = total number of particles of each species.

* Boltzmann, Gastheorie, Band ii. pp. 175-177. Jeans, ‘ The Dynamical
Theory of Gases,’ pages 209 to 217, 2nd edition.
t+ Stern, Ann. der Physik, vol. xliv.

1134 Prof. M. N. Saha on Temperature Ionization of
The diminution in entropy
o=kinW = — Rin(nanz),
so that instead of equation (A) we shall have
SatS8s—Sa—So= : 3
hence the equation of ionization takes the form, assuming
that only one species of atom is present,

log pee P= Slog T —6:°5+ log (mans).

The effective conization potential T-now becomes
U—2:3RT log (nan)
2°3RT log (rans)
23000
For the electron and the alkalies we can take n,=1. For
alkaline earths, if we follow Boltzmann, n,=2, but this
evidently does not suffice in the present case. It may be
pointed out that the present case is entirely different from
that considered by Boltzmann, for we are considering the
combination between an ionized atom and an electron,
whereas Boltzmann considered the combination of two atoms.

There is no reason why the steric factor should have the same
value in both cases.

or T,=I- volts.

LV.

On the basis of the above formula, let us consider the
effective I.P. of helium and the alkaline earths at different
temperatures. Taking n=2, 4, 6, 8 respectivelv, we have for

n=2, I.=I—:060m,
=4, I,=I—-119m, ( the temperature being
=o, l= Salsa. m-thousands.
=, = Laat) ane | :

Russell is inclined to take the temperature of the spot =
4000° K. and that of photospheric emission = 6000° K. If
this view be correct, the temperature of the spot is only
slightly above that of the arc. But we find that in the are
the Ba line A=5535 is quite strong, while it is entirely
absent from the spot. The discrepancy can, of course, be
explained by assuming that the temperature of the arc is not
uniform ; the absorption of X=5535 is due to the cooler |
mantle of gaseous barium next to the air. But more ex-
tended research is required to test this point.

Elements of the Higher Groups. 1135

Taking the temperature of the spot and the sun to be
5000° K. and 7000° K. respectively, and n=8, we find that
for Ba, I,=4°22 and 3°86 respectively, 7. e. in the sun
barium is ionized like Cs, in the spot like Rb. This very
nearly explains the complete ionization of Ba in the sun,
as well as in the spot.

For Ca, I.=5°18 and 4°82 respectively. Thus in the sun,
calcium is more ionized than sodium, while in the spot it ought
to be a bit less ionized than sodiim. ‘This satisfactorily ex-
plains the behaviour of the Ca-lines in the sun and spot

Tf we turn to the stellar spectra, we find that calcium ‘4g’
disappears from the B8-stage. By using the original equation
of ionization it was found that the temperature could not be
less than 13000° K. But, according to Wilsing and Scheiner’s
intensity measurements, the temperature is only slightly
above 10000°K. Taking n=8, the effective I.P. becomes
4°21 volts; the eee is complete at 10000° K*. Thus
the introduction of the “* steric factor ’’ seems to bring down
the temperature of different spectral classes in a line with
the temperatures obtained from intensity measurements.

In my original calculation of the ionization vot helium, the
LP. was taken to be 20-4 volts. This is now known Eo be
wrong, for numerous workers have established definitely that
real ionization begins at 25-4 volts. ‘The former figures are
therefore to be revised. Taking the steric factor = 8, the
effective I.P. at 25000° K. becomes 21 volts, and ionization
is 74 per cent. under concentrations corresponding to one
atmosphere pressure. If P=1071atm., the ionization be-
comes complete. Thus the temperature of the Oa stars given
in my former paper (Proc. Roy. Soc. Lond., May 1921,
p. 151), remains unaltered.

Manganese.—Let us next consider the element Manganese,
because the constitution of its series spectrum has been
recently elucidated by Catalan (Phil. Trans. vol. 223). He
finds the 1S term of Mn=59937, the I.P.=7°41 volts. The
resonance lines of Mn are the triplet X=4030°92, 4033-21,
4034-62, so that, as far as the variation in intensity of its lines
in stellar spectra is concerned, manganese isan ideal element.
According to Lockyer, they occur as faint lines in the spectra
of Ao-stars (intensity 1 on a scale 1-10, the Ca-line 4227 being
of intensity 2). Thus, in spite of the fact that the I.P. of Mn
is 1:23 volts higher than that of calcium, it, is more highly
ionized at T=10000° K. than calcium. Manganese has got

* Vide M. N. Saha, ‘‘ Elements in the Sun,” Phil. Mag., Dec. 1920.

1136 Prof. M. N. Saha on Temperature Ionization of

7 electrons in the outermost shell; its steric factor is there-
fore expected to be much higher than that of calcium,
which probably accounts for its comparatively high degree
of ionization. G;

According to Lockyer, Mnt is represented in stellar
spectra by AX=4344:19. It does not occur in King’s
furnace spectra *, though groups of lines due to Mn* oceur
about A= 3442-3497, 2914-2940. 4344°19 is certainly not a
resonance-line of Mn*. Probably it is of the same type as
He*4686. According to Lockyer, this line vanishes from
the stage €Tauri or 8 Persei. Thus in stellar ranges we have
not only Mn and Mnt, but Mn*t as well. None of the
lines of Mn** seems to be known.

Let us now treat some of the other elements in the order
in which they occur in the periodic tables. For elements of
the third group, e. g. Al, Sc, Y, La, neither the series
classification nor the variation in intensity of lines im stellar
spectra is satisfactorily known. We pass to the next group.

Group IV. Carbon, Silicon, Titanvum.

These elements are extremely interesting, because they
have got 4-valency electrons, and the steric factor is expected
to be unusually large not only for the neutral, but also for
the singly- and doubly-charged atoms. As a result, the
successive stages of ionization will follow each other in
rather quick succession. Unfortunately, the knowledge of
the spectra of these elements, as well as of the variation of
their intensity in stellar classes, is not so well known.

Carbon.—The spectrum of this important element is one of
the least known. The line X=4267 is supposed to be due
to C*; Lockyer, Baxandall, and Butler} treat the lines
4650°8, 4647°4 as specially enhanced: we can assume that
they are due to C**. One wonders what lines are to be
attributed to carbon itself. There are two strong lines,
N= 6583°0, 6577°5, which may be due to carbon. ‘The
variation in intensity of the lines of Ct and CTT in stellar
spectra is given below :— |

AL. BY. BG. Bd. BS. B22, Bi Bor “Oca Gr

Sh ADGT(G™ \uyee es Putte 8128S 5d eee ne
a An:
MT ee eer dee (te | a Ra eon tea) sire

+ Present, but intensity not exactly measured.

* King, Astro. Journal, vol. li. (1921).
+ Lockyer, Baxandall, and Butler, Proc. Roy. Soc. Lond. vol. lxxxii.
p. 852. Fowler, Report on Series in Line Spectra, p. 163.

Elements of the Higher Groups. 1137

The pair 4550°8 and 4647°6 are very prominent in the
spectra of Nove simultaneously with strong enhanced
nitrogen lines 4634°34, 4640°82. If the above considerations
be currect, carbon occurs in stellar spectra as Ct and Ct*,
and the two stages follow each other in rather quick suc-
cession. This is to be expected of an element having a
large “steric factor.”

Stlicon.—Lockyer ¢ classified the lines of silicon in four
groups, according to their mode of production, and has
shown that each group is represented with the greatest
intensity in stars at different stages of development. The
following is compiled from the Harvard Annals:—

Group I. Group II. Group III. Group IV.
Glues, f =8005(8i). “F988. “PoagitH. T5740
1 eee ? — — —
ee Renee te 12* Absent. “= —_
By SG Present. Present. — ==
1 Ree — Present, cee —
(it ee a 3 = —
RGeT ee. — 3 —- -=
ee ak.. _ 3 _- —
Pe sas — 2 1 —
B2. — 1 2 2
ee —_ — 4 5
it ee — — ape 15
ES la “= _ ? 12
Se es. — _- ? 6

* Intensity in the solar spectrum on Rowland’s scale.

It is not quite certain if lines belonging to Group III. and
Group IV. can be regarded as due to Sit? and Sit** re-
spectively. If Sommerfeld and Kossel’s spectral displace-
ment law { be true, the spectrum of Si** ought to have the
same constitution as the spectrum of neutral magnesium,
i.e. ought to consist of triplets and singlets. Group III.
and Group IV. may be both regarded as due to Sit,
Group ITI. belonging to combinations like 1S—mp,, Group IV.
to stronger combinations like 1S-mP. This is of course
only a suggestion. But there appears to be but little doubt
that in the stellar range available for us, silicon occurs not

+ The Sit-pair is present in the star y Cygni (F8A) and is as strong as in
a Cygni (A2F) (Lockyer, Month. Not. R. A.S. 1921). In FSA stars they are
weakened, and entirely absent from the solar spectrum. Additional lines of

Group IV. at \=5740°2, 4829°4, 4820°1, 4813°7 have been identified by
Fowler in the spectra of 8 Crucis type Bi, Monthly Notices, vol. Ixxvi, p. 196.

eS

t Sommerfeld, Atombau.

1135 Prof. M. N. Saha on Temperature Ioniz aon of

only as a neutral and a singly-ionized atom, but also at least
as a doubly-ionized element, and the stages follow each other
in rather quick succession.

Titantum.—The spectrum of Tiis very well studied, though
the series classification is not yet known. In their general
behaviour the lines of Ti, Tit, and Titt saeennole the corre=
sponding lines of Si, Sit, Sitt, but titanium becomes
ionized at a much earlier stage than Si, which is in ac-
cordance with the general rule that for elements belonging
to the same group, heavier elements have got the smaller
IP. ; Lines of Tit (XN=3759'47, 376147, 4570 eee
quite strong in the solar spectrum, and according to W. J.
S. Lockyer *, they are more intense in F5G stars (¢ Cassio-
peeize) than in A2F stars (a Cygni). The ionization of Ti
commences much earlier than that of 81.

Group V. Nitrogen, Phosphorus, Vanadium.

The enhanced lines in the spectrum of Nitrogen and their
occurrence in stellar spectra were first noted by Lockyer,
Baxandall, and. Butler tT. The subject has been subsequently
treated by Lunt, Fowler, and Wright.

According to these workers, the spectral lines of nitrogen
can be divided into 2 or 3 groups according to the
stimulus necessary for exciting them. The chief lines of
Group I. are X=3995°15, 4447-20, 4630°73, and the chiet
lines of Group II. are %¥ = 4097-48, 4103°54, 4640°82,
4634°34. There is, besides, another line at 4379°26 which
seems to belong to an enhanced group of still more pro-
nounced type. ‘Since lines of Group I. do not occur in the
low-temperature spectral classes like G, I’, and even at Aof,
I am inclined to think that they are due to N*, or belong to
some remote combination of the neutral nitrogen series In
the former case, Group II. would correspond to N**. They
first come out in the B2 classes (intensity 1), and gradually
increase in intensity as we go to the still hotter stars, as the
following shows :—

29 Canis Can.Maj. ¢ Orionis . ve ae
Majoris (Oe). (0e5). (Bl). 3 Centauri. y Orionis.

Nt (4097-45)... 18 6 4 2 1

* W.J.S. Lockyer, loc. cit.

t Lockyer and others, Proc. Roy. Soc. Lond. A. vol. clxxxii. p. 532.
Lunt, Monthly Notices, lxxx. p. 584 (1920). Fowler, Monthly Notices,
Ixxx. p. 693.

{ Of this last I am not quite sure.

Elements of the Higher Groups. 1139

The lines of N** are very prominent in the spectrum of
Novee in their later stages simultaneously with Het 4686.
and (** 4650°8, 4647°6.

Nitrogen, ae ing 5 electrons in the outer shell, would
have a large “steric factor” for not only the neutral atom,
but also for N* and Ntt, which probably accounts for the
quick succession of the different stages of ionization.

According to Lockyer ™*, Protovanadium or Vt, as repre-
sented by the line X=4053°9, is strongly developed in F5A
stars, but shows a weakening in Ao-stars, and disappears
somewhere about the B8-stage. In the stellar ranges we
have, therefore, V, V*, and V+ in rather quick succession.

It will be seen that the above discussion mainly centres
round the life-work of the late Sir Norman Lockyer. To him
is due not only the idea, but also extended and elaborate
studies of the enhanced and super-enhanced lines of elements,
and their application to the study of the ordered sequence in
stellar spectra. In this connexion, attention may be drawn
to the views which he presented in his ‘ Inorganic Evolution
of Klements.’ He tried toimpress the idea tliat the enhanced
lines are due to some proto-form or fractional pane of
chemical atoms. But in those days the atom was an “ elastic
solid sphere,” and his ideas did not find many adherents.
The real significance of his works is being realized in these
days of the Rutherford-Bohr theory of atomic constitution.

The present paper probably brings out the great importance
of these studies to problems of atomic physics. Lockyer’s
studies have been continued by American and English
workers, but the data hitherto available are not sufficient for
the purpose. The above discussion, though scrappy, prob-
ably points out the direction in which these studies ought to
be extended and amplified.

Note added during correction.—Nince the paper was written,
Russell has published a paper in the Astro. Journal, May
1922, where he has modified the view that Ba is absent from
the sun. The 1S-3P line of Ba (A=3071°59) has been
identified with a line of intensity 00, the resonance line
X = 5535 being probably masked by an iron-line at
A= 5535°68.

* erie! Roy. Soe. Lond. vol. lxiv. p. 896; Phil. Trans. “On the
Spectrum of a Cygnus,” 1903.

pie BLO)

CV. The Ionization of Abnormal Helium Atoms by Low-
Voltage Electronic Bombardment. By Frank Horton,
Sce.D., and ANN CATHERINE Daviss, D.Sc. *

ir a recent paper in the Astrophysical Journal t F. M.
Kannenstine describes experiments which show that
with alternating electromotive forces, an arc in helium can
be maintained, and even made to sirike, at about 5 volts,
provided the frequency of the alternations exceeds a certain
limiting value. Evidence of the maintenance of arcs in
belium at voltages below the first critical electron energy
for this gas (20:4 volts) has been given by other observers f,
and it might be expected that the limiting voltage for the
maintenance of the are in helium would be the difference
between the normal ionizing voltage (25:2 volts) and one
of the two critical electron energies for the production of
radiation (20:4 veltsand 21:2 volts), 2. e., the limiting voltage
might be expected to be either 4°8 volts or 4:0 volts. Kan-
nenstine’s: experiments seem to be the first in which a limit
approximating to either of these values has been reached,
and also the first in which the are has been made to’ strike
below 20:4 volts in helium. This striking of the are at
voltages below the resonance value in experiments with
alternating electromotive forces is not, however, so much at
variance with theory and with the results obtained by other
experimenters as it appears to be at first sight, for what
Kannenstine found was that after the cycle of voltages had
once been completed, the are struck at about 5 volts during
subsequent cycles if the frequency of the alternations
exceeded 220 per second. Hence the effect obtained was
not the striking of theare in normal helium, but the
striking of the are in helium containing some abnormal
atoms for the production of which a higher voltage had been
employed.

Kannenstine has pointed out that his results can be inter-
preted on the view that there is one of the states of the
helium atom, intermediate to the normal state and the singly
ionized state, in which the atom can exist for a perceptible
interval of time, which he gives as about ‘0024 sec. This

* Communicated by the Authors,

+ F. M. Kannenstine, Astro. Phys. Journ. vol. lv. p. 345 (1922).

{ K. T. Compton, E. G. Lilly, and P. S. Olmstead, Phys. Rev. vol. xvi.
p. 282 (1920). A. C. Davies, Proc. Roy. Soc., A., vol, 100. p. 599
(1922).

Lonization v7 Abnormal Helium Atoms. 1141

particular state of the helium atom is referred to by Kannen-
stine as the “* metastable ” state, and his results are consistent
with the view that the particular state of the helium atom
which is metastable is the one into which the atom is thrown
as the result of an encounter with an electron having 20°4
volts energy. It will be shown later, however, that the
results obtaned by Kannenstine can be explained without
the necessity of supposing one of the abnormal states of the
helium atom to be endowed with any vreater stability than
other abnormal states.

The suggestion that a metastable state of the helinm atom
exists was first made, on theoretical grounds, by Franck and
Reiche*, and the results of certain experiments made by
Franck and Knipping were interpreted by these experi-
menters as confirming the suggestion +. There is, however,
an essential difference between the term ‘ ‘ metastable” as
used by Kannenstine and as used by Franck, Reiche, and
Knipping ; for the former uses the term in the sense that the
helium atom remains in that particular condition for a small,
but measurable, interval of time, whereas Franck and his
collaborators use the term to denote inability of the helium
atom to revert from that particular state to the normal state
without the assistance of external agencies.

Evidence on this point is provided by the results of
certain experiments performed by the authors, and described
by them.in the Philosophical Magazine for November
1921. The conclusions to which these experiments led,
while agreeing with those of Franck and Knipping in some
respects, differed from them in regard to the production
of radiation in pure helium bombarded by 20:4 volts elec-
trons; for while Franck and Knipping concluded that
radiation is not produced at all in these circumstances, the
authors concluded that an emission of radiation undoubtedly
takes place, but to a smaller extent than at 21°2 volts, the
second critical electron energy in helium. The results
obtained by the authors in the experiments referred to,
indicated that the abnormal atoms produced by 20-4 volts
electronic-atomic encounters reverted to the normal con-
dition less readily than the abnormal atoms produced as a
result of 21°2 volts encounters.

In the paper already referred to, the authors describe
experiments in which they attempted to detect ionization

* J. Franck and O. Reiche, Zeits. f. Phys. vol. i. p. 154 (1920).
+ J. Franck and P. Knipping, Zets. £ Phys. vol. i. p. 820 (1920).

1142 Prof. Horton and Dr. Davies: Jonization of Abnormal

when helium atoms which were in an abnormal condition
were bombarded by electrons having energy in excess of
4°8 volts but less than 20-4 volts, and evidence is given
which shows that ionization can be produced.below 20:4
volts in this way. The attempts to demonstrate that such
ionization could be produced by electrons having only
4:8 volts energy, were, however, unsuccessful. More
recently further attempts to demonstrate this point satis-
factorily have been made by the authors, and although the
further experiments have, in some cases, only served to
bring to light the existence of complicating factors which
tend to frustrate the detection of the effect sought for, a
few of the experiments which were made under conditions
which varied over a wide range have yielded results which
can be considered as satisfactory evidence of the production
of ionization from abnormal atoms by bombardment with
electrons having only about 5 volts energy.

The apparatus used in the present investigation was the
same as that employed in the earlier experiments, and is
described in detail in our paper. Its design allows of the
production of abnormal atoms in the main tube by the
action of radiation which enters from an auxiliary side
tube. Jn this way abnormal atoms are obtained in the main
tube without necessitating the presence of electrons in that
tube. Radiation is produced in the side tube by bombarding
the helium with an electron stream and passes into the main
tube by the process of absorption and re-emission by other
helium atoms, as shown in our earlier paper. Hlectrons and
any positive ions produced in the side tube are prevented by
the arrangement of electric fields from entering the main ~
tube and so affecting the collecting electrode.

Having arranged for the presence of abnormal atoms in
the main tube, the effects of bombarding them by a stream
‘of electrons, the energy of which could be gradually in-
‘creased, were investigated with each of the arrangements of
electric fields designated i in our earlier paper as those required
for the obtaining of R, and (I-R), curves, 2. e , curves showing

the effects of radiation only, and curves in which the photo-
electric effect of radiation opposes the ionization current.
Hence, provided that in the absence of abnormal atoms the
variation in the measured current with variation of the
bombarding electron energy could be neglected, the pro-
duction of ionization by bombardment of abnormal atoms at
any given stage, would be indicated by a decrease of negative
current at that stage, in an R arrangement, and by either a

fTelium Atoms by Low- Voltaye Electronic Bombardment. 1143

decrease of negative current, or an increase of positive
current, if an Ge: —h) arrangement were employed.

The cases in which evidence was obtained of the production
of ionization of abnormal helium atoms by bombardment
with electrons having about 5 volts energy are illustrated by
the curve given below, in which the current measured by
the electr ce 1s plotted against the energy of the electrons
in the bombarding stream. In obtaining the results repre-
sented in this curve it was arranged to here e a large quantity
of resonance radiation coming vale the main fie of the
apparatus from the auxiliary side tube, and in order that an
intense stream of electrons should Gaines d the abnormal

CURRENT

ELECTRON ENERGY volts

helium atoms formed in the main tube by this radiation, the
filament supplying the bembarding electrons was heated to
the limit of satety. Pure helium gas was constantly
streaming through the apparatus during the experiment at
an average pressure of 0°12 mm. The arrangement of
electric fields was that required for an (I-R) curve. ‘The
curve shows that a positive current was measured which
began to increase when the electron energy exceeded about
4-5 volts. The detection of a positive current below 4°5 volts
is to be attributed to the ionization of 20:4 volts abnormal
atoms by 21 2 volts radiation. The measured current at this
stage could be made positive or negative by adjustment of
the electric field in the V, space*. One of the complicating
factors in the investigation was the recombination which
occurred between the bombarding electrons and the positive

* See Phil. Mag. vol. xlii. p. 746 (1921).

1144 Prof. Horton and Dr. Davies: Jonization of Abnormal

ions which were produced by the radiation, for as the energy
of the bombarding electrons was increased, the distance
through which they travelled also increased, and hence the
amount of recombination became greater, and fewer positive
ions were collected. This variation in the amount of re-
combination occurring tended to mask the effect of ioni-
zation by the bombardment of abnormal atoms, and was
responsible for the negative results obtained in some instances..
The downward slope of the curve given in the figure, before
4-5 volts is reached, illustrates the reduction in the positive
current due to increased recombination as the energy of the
electrons is increased. .

Some idea of the amount of recombination occurring
between the bombarding stream of electrons and positive
ions produced by the radiation which comes from the
auxiliary tube,is given by the following experiment :— With
the electric fields arranged for an (I—R) curve, and with V,
adjusted so that the measured current was of the negative.
sion, the current measured in a certain experiment was
3°94 x 10713 amp. when the energy of the bombarding elec-
trons was 3°5 volts. By means of a current sent through a
coil of many turns of wire wrapped round the main tube, a
magnetic field could be applied parallel to the axis of this tube
This magnetic field served to prevent the electron stream
emitted from the hot filament from spreading laterally by
concentrating it into the central part of the tube, thereby
diminishing the possibility of encounters between these
electrons and positive ions produced throughout the space by
the radiation. It was found that the effect of switching on
the magnetic field (all the electric fields remaining as before)
was to cause a positive current of 0°53x 10°~*% amp. to be
measured instead of the negative current of 3:94 x 10~% amp.
previously observed, the increase of positive current being
due to the smaller amount of recombination occurring.

Owing to the fact that when curves such as that given
in the figure were obtained the filament supplying the
bombarding electrons was extremely hot, it was desirable to
complete a series of observations as quickly as possible.
For this reason observations were taken at rather large
voltage intervals, and the genuineness of the observed effects
was tested by taking cbservations at decreasing values of the
voltage, as well as at increasing values, to see if the curve
retraced its course. The results do not make it possible to
decide with any great accuracy the minimum energy of the
bombarding electrons for which ionization is produced, but

Flelium Atoms by Low-Voltage Electronic Bombardment. 1145

they suftice to show that it les between 4 and 5 volts. The
results of these particular experiments do not, therefore,
indicate whether the abnormal atoms which are being ionized
are those resulting from 20°4 volts electronic-atomic
encounters, or those resulting from 21:2 volts encounters.

It seems possible to account for the beginning of ioniza-
tion between 4 and 5 volts without assuming that any one
abnormal state of the helium atem is more stable than any
other abnormal state. Although helium radiations corre-
sponding to various voltages are present in the side tube,
very little radiation corresponding to voltages other than
20°4 volts and 21°2 volts is likely to be passed into the main
tube for the following reasons :—

a. Only such radiations as correspond to transitions from
an abnormal to the normal state of the atom are capable
of being absorbed by normal atoms and subsequently re-
emitted.

8. Abnormal atoms which have absorbed radiation corre-
sponding to voltages higher than 21°2 can revert to the
normal state in several ways, the number of which increases
as the energy of the abnormal atoms increases. It is there-
fore improbable that more than a small proportion of the
abnormal atoms corresponding to higher voltages than 21:2
revert to the normal condition without passing through
intermediate stages, in which case the only portion of the
emitted energy which could be absorbed by normal helium
atoms would be that corresponding to the final transition.

Thus the fact that ionization of abnormal atoms did not
begin before about 4 volts in our experiments may have been
due to the absence of abnormal atoms other than those
corresponding to 20°4 volts and 21:2 volts, and the fact that
ionization did begin between 4 and 5 volts, although con-
sistent with the view that the 20:4 volts abnormal atom is
‘“‘metastable ”? in the sense used by Kannenstine, is not
sufficient to prove that this abnormal state is more “ meta-
stable ” than any other abnormal state.

When arcs are maintained below 20°4 volts in helium,
ionization of abnormal atoms only can he occurring. For
the maintenance of low-voltage arcs, therefore, it is of
importance that none of these abnormal atoms should be
lost. When recombination takes place the energy of the
radiation emitted during the final transition can be absorbed
by normal atoms, and the minimum energy which must be
supplied in order that every type of abnormal atom thus
produced may be ionized, is 4°8 volts. If the bombarding

Phil. Mag. 8. 6. Vol. 44. No. 264. Dec. 1922. 4H

1146 Tonization of Abnormal Helium Atoms.

electron-energy has a smaller value than this, some of the
abnormal atoms produced by absorption of radiation corre-
sponding to the final transitions at recombination cannot be
re-ionized, and hence the are can no longer be maintained.

In the experiments of Kannenstine, the re-striking of the
are below 20°4 volts must have been due to the fact that
the intervai which elapsed between the breaking of the are,
and the re-application of the energy necessary for the ioniza-
tion of abnormal atoms, was sufficiently short for there to be
still a considerable number of abnormal atoms present at the
end of the interval. _This does not necessarily imply that.
any particular abnormal atom remained in the abnormal
condition for the whole of this time, for the presence of
abnormal atoms at the end of the interval would be secured
if the radiation were absorbed and re-emitted by other atoms
several times during the interval. It does not follow,
therefore, that the period of ‘0024 second measured by
Kannenstine represents the “life” of individual abnormal
atoms.

In the event of radiation being absorbed and re-emitted
several times during the interval, there would be, as we have
already shown, a marked falling off in the number of ab-
normal atoms corresponding to voltages greater than 21:2,
so that the abnormal atoms present at the end of the interval
would be almost entirely of the types corresponding to
20°4 volts and 21-2 volts, requiring between 4 and 5 volts
energy for their ionization. It thus appears that the main-
taining of the are down to 4 or 5 volts and not below this
value, and the fact that it re-strikes when this voltage is
again established, can be accounted for without assuming
the metastability of any abnormal state of the helium
atom.

The authors desire to express their thanks to the Radio
Research Board of the Department of Scientific and In-
dustrial Research for the means of purchasing some of the
apparatus used in these experiments.

Royal Holloway College,
Englefield Green, Surrey.

Ar nia?’ j

CVI. The Ionizing Potential of Positive Ions.

To the Editors of the Philosophical Magazine.

GENTLEMEN,—

N a paper on impact ionization by low-speed positive

H ions in hydrogen, published in the November number

of the Philosophical Magazine, p- 806, Mr. A. J. Saxton

gives a general account of the results that have been
obtained on the subject of ionization by positive ions.

As an example of some of the experiments which I made,
Mr. Saxton refers to one where the ionization is produced
by electrons and positive ions moving in hydrogen at
8 millimetres pressure under a force of 700 volts per centi-
metre (‘ Hlectricity in Gases,’ p. 317), and states that ‘‘ the
M.F.P. of a positive ion would be very small at this high
pressure, so that it could not obtain an unimpeded fall
through more than a fraction of a volt,” and makes some
suggestions as to how the ions were generated in these
experiments. Mr. Saxton’s conclusions appear to be based
on the supposition that all free paths are nearly equal to the
mean free path and that the number of molecules ionized
by single collisions with positive ions is too small to be
taken into consideration.

The experiment in question was published in the Philo-
sophical Magazine, vi. p. 607, Nov. 1903, where I gave a
method of estimating approximately the ionizing potential
of positive ions from the determinations of the umber of
molecules ionized by a positive ion in moving through a
centimetre of the gas in the direction of the electric force.
The ionizing potential of positive ions in hydrogen was found
to be 20 or 30 volts. The voltage depends on the length of
the mean free path, and it was also stated in the original
paper that a somewhat lower voltage would be obtained if
the mean free path of a positive ion was the same as that
given by Meyer for a molecule of the gas. For hydrogen
at normal pressure and temperature the mean free path
given by Meyer is 17°8x10~* cm. (‘ Kinetic Theory of
Gases,’ p. 192).

The tree paths were not known very accurately at that
time, but as a result of recent investigations a more exact
formula connecting the viscosity of a gas and the mean free
path of a molecule has been obtained. It is therefore of
interest to calculate the ionizing potential of positive ions in

42

1148 The Ionizing Potential of Positive Ions.

hydrogen taking 11:25x 107° cm. as the mean free path
in the gas at normal pressure and temperature, which is the
mean free path of a molecule of hydrogen given by Jeans in
the last edition of his treatise on the Dynamical Theory of
(xases.

On this hypothesis, the number of collisions made by a
positive ion in traversing a centimetre of the gas at 8 mms.
pressure is 936, and it was found by the experiment that on
an average the number of molecules ionized by a positive ion
in moving through one centimetre is *059. For a first
approximation the ve locity of agitation of the ions may be
neglected in comparison with the velocity in the direction
of the electric force. Thus the ratio of the number of mole-
cules ionized to the total number of collisions by positive
ions is 1: 16,000.

If all the ionization by positive ions be attributed to
collisions which terminate free paths greater than «, then
e/’— 16,000 where /is the mean free path, so that r=9 68 xl
= ‘0103 em., and the potential fall along these paths would
be greater than 7:2 volts. The velocity corresponding to
this voltage is much greater than the velocity of agitation
of the molecules of the gas, so that the longer free paths of
the positive ions would exceed the value -0103 cm. by the’
factor 1'4, and the potential fall would also be greater in
the same proportion. Hence the ionizing potential of
positive hydrogen ions, as deduced from the above experi-
ment, must be at least 10 volts.

In order to make a more accurate estimate of the ionizing
potential it would be necessary to take into consideration
the initial velocities of the ions in estimating the number of
paths which are terminated by a velocity above a certain
value V,so that the above calculation only gives a lower
limit to the ionizing potential. If other experiments of the
series be considered where the gas is at pressures lower than
8 millimetres, it will be found that the ionizing potential of
the positive ions in hydrogen is above 15 volts.

Yours faithfully,
7th November, 1922. J. S. TOWNSEND.

fe snt49" j

CVII. The Propagation of a Fan-shaped Group of Waves in
@ Lispersing Medium. By G. Brerr, National Research
Fellow, U.S.A.*

B* a fan-shaped Group is meant a group of waves in
which the successive waves are plane, and have a
small constant inclination with respect to each other. It
may be produced by reflecting a plane wave-train in a
rotating mirror or by moving a source of light in the focal
plane of alens. A picture of such a group is given in fig. 1.

Fig. 1.

HUT 1,
Be,

The propagation of a fan-shaped group in a dispersing
medium has been discussed by Gibbs}. His calculation
shows that ata given point of the group (such as a maximum
of intensity) the orientation of the elementary waves is
constant in spite of the fact that each of the waves rotates
during its progress asa result of the dispersion. The im-
portant consequence of this result is that the image of a
fan-shaped group, which is first passed through a dispersing
medium and then through a lens, does not experience a dis-
placement on account of the dispersing action of the medium.

Having recalled this result of Gibbs, Prof. Ehrenfest
showed that the experiment recently proposed by Hinstein t
is not capable of settling Hinstein’s question, for the latter
incorrectly supposed that, according to the classical wave
theory, the image ofa fan-shaped group formed by a lens is
displaced on account of the dispersing action of a medium
which is putin front of the lens. [Another criticism leading
to the same result has been published by Raman§ ; and
Einstein also revised his views ||. ]

* Communicated by the Author.

t W. Gibbs, ‘Collected Works,’ vol. ii. p. 253; § Nature,’ vol. xxxiii.
p- 582, April 22, 1886.

{ A. Einstein, Sitz. Ber. d. Berliner Akad. p. 882 (1921).

§ C. N. Raman, ‘ Nature,’ p. 477, April 15, 1922.

|| A. Einstein, Sitz. Bsr. d. Berl. Akad. Feb. 2, 1922, pp. 18-22.

{2
|
|
i
bo
j
|

1150 Dr. G. Breit on the Propagation of a

In this connexion, it is of interest to point out that the
result of Gibbs can be understood without calculation in the
following manner :—

Statement of Fesult.

Consider the group drawn in fig. 1. The waves are
more crowded at A than at B. The medium is thus dis-
turbed at a higher frequency at A than at B, and the velocity
or propagation is therefore different at A and B. In spite of

this, as shown by Gibbs, the waves passing a point moving

with the group velocity—appertaining to the frequency of
the group at the point—pass that point always in the same
orientation.

Thus it is required to show that ata given point of the group
the orientation of the elementary waves is constant.

Proof in Special Case.
Consider the special group of fig. 2, obtained by super-
posing the sinusoidal waves of wave-lengths A,, A, drawn on
Fig. 2.

7 | Mi :
fig. 3. The minimum P is the point at which 24, 4 destroy

each other. Hence the lines L,M,, L,M, Joining points of
opposite phase of 2X4, A, on fig. 3 are vertical at P. If CD is

aanenand en ® -
Ee a

Fan-shaped Group of Waves ina Dispersing Medium. 1151

made equal to AB of fig. 1, and if in fig. 1 the difference in
on the top and on the bottom is just X,—A,, then these lines
represent the waves of fig. 1. ‘Thus at P the direction of
the elementary wave is always vertical, and therefore
constant. If the difference in X in fig. 1 is A'g2—NyAQ—Ay,
Cais ks
Mey NQ— DY
general case the argument may be stated as follows :—

the argument applies if we make In the

Proof in General Case.

The motion of a point of the group is such as to keep the
phase-difference between two nearly equal wave-lengths con-
stant to within quantities of the first orderin add. Therefore if
the angle between consecutive waves is so small that dispersion
effects may be treated as small quantities of the first order
in dd, then the motion of a point of the group may also be
said to be such as to keep constant the phase-difference
between the wave-lengths 2’;, X’, at the top and at the
bottom of an elementary wave.

Let now two points be considered both moving with the
group velocity corresponding, say, to the bottom, and both
situated in the surface of an elementary wave at a given
instant—one at the top and the other at the bottom of the
wave. The wave moves slower at the top than at the
bottom, but both top and bottom move faster than the two
points. Thus the two points are overtaken by waves coming
from behind, and each of these waves is turning during the
motion.

Since now the points move so as to keep the phase- difference
between the waye-lengths at the top and at the bottom con-
stant, and since the points have been once in the surface of a
wave (so that the constant phase-difference is zero) it is
apparent that if a wave reaches the top point it simulta-
neously reaches the bottom point, and thus the orientation of
a wave is unchanged if it is picked out by a point moving
with the group velocity *.

Briefly: both motion with group velocity and motion with

* It is essential to the argumeut here given that the words “ group
velocity ” should have a definite meaning. ‘I his implies that the Fourier
analysis of the group is confined to a sharp band on the scale of variations
in wave-length along the aperture. A closer examination shows that
this condition is fulfilled if the angle between the first and last waves
is large in comparison with the least angle distinguishable through
diffraction.

1152 Prof. E. K. Rideal on the Flow of

a constant orientation of the elementary wave are defined
by the same criterion, viz. zero phase-difference between two
adjacent wave-lengths. Thus it becomes obvious that one
implies the other.

To Prof. Lorentz and Prof. Ehrenfest the author is very
grateful for the discussion of the subject.

OVIIL. On the Flow of Inquids under Capillary Pressure.
By Eric Keicartey RipEAr”*.

Rees rate of penetration of liquids into capillary porous
materials, of importance not only in biochemical pro-
blems but also in the study of the phenomena of adsorption
by materials such as charcoal and substances constituting
the membranes of semi-permeable osmometers, has attracted
but little attention. Bell and Cameron (Journ. Phys. Chem.
x. p. 659 (1906)) showed that in the case of a few liquids
the rate of movement of a liquid moving through a hori-
zontal capillary was such that the relationship #?=kt (where
wv was the distance traversed in time ¢) held within the limits
of accuracy of the experimental method. Cude and Hulett
(J. A.C. 8S. xlii. p. 891 (1920)), in their study of the rate of
penetration of charcoal by water, obtained for the initial
period of penetration a similar relationship. Washburn
(Phys. Rev. xiii. p. 273 (1921)) has examined the problem
in more detail, and deduced for the conditions of horizontal

flow the equation a?= ye rt, where vy is the surface
tension, 7 the viscosity, 7 the capillary tube radius, and @ the
angle of wetting. The validity of a similar expression was
tested experimentally for liquids moving through capillaries
under the influence of their own capillary force as well as a
constant large external pressure. For all liquids which wet
the tube wall of the material, cos@ is evidently equal to unity.
The same value obtains for liquids which do not wet the tube,
since the angles of wetting noted in the literature are pro-
bably ficticious, and are due to observations on the alteration
in the radius of curvature at some point distant from the
point of contact with the tube wall, and not at the contact
point itself. Washburn has assumed that Poiseuille’s law
holds true during the flow after the initial period of turbu-
lence has ceased, and has calculated with the aid of this

* Communicated by the Author.

Liquids under Capillary Pressure. 1153

expression the rate of flow from the driving pressure made
up of three separate pressures, the unbalanced atmospheric
pressure, the hydrostatic pressure, and the capillary pressure. |
In the case of horizontal tubes the first two pressures are
eliminated. The effective total driving pressure, however,
varies with the length of the column, since the frictional
resistance to the flow increases with the length of the
column in the capillary tube. With this correction a some-
what different expression from that of Washburn is obtained
for the rate of penetration, which, however, reduces to the
form obtained by him on the neglect of terms which are
insignificant except for very small and very large values of
xz the distance of penetration. A simple derivation of the
relationship may be obtained in the following manner :—

Bigs I.

The forces acting on a column of liquid a cm. long ina
capillary tube 7 em. radius are:

(1) The surface tension forward, in magnitude 2a7y.
(2) A retarding force due to the viscosity of the liquid in
the tube.

According to Poiseuille’s law, neglecting the slip factors,
this retarding force may be expressed in the following form,
OD eo Til
dt 1 Byer

he. :
where = is the rate of flow; hence, solving for P the

pressure, we obtain

_ 8nx dv _ 8yx ree 80.

oar dt m7 :

The retarding force acting on the column is consequently
EB =r? P = 8rd.

The net force acting on the column thus varies not only with
the length of tube wetted, but also with the velocity of flow,
and is equal to

2rry— 8nu Tek.

1154 Prof. E. K. Rideal on the Flow of

The mass of the column in motion is mr?xd, where 6 is the
density of the liquid ; hence

Tr Ld dt = Wrry—B8yr Te,

a sil 2y 8a
~ bre 76”
ce e b
or fe gett 0 US at ea
a

On integrating this expression, we obtain

jla cd tb ei ee
BME ero a 2 * Daan? | Dy eat ee
Qn Se S2p5
or t= — «#’?——_- log ee Bs
yr . tory) aoa 512nF x? .

For small values of x this equation reduces to the form given
by Washburn for a liquid wetting the tube wall or

2 oe

x Dp
Experiments on the rate of penetration of liquids moving
through horizontal glass capillaries under their own capillary
pressure aloue without any external force, indicated that when
the liquid wets the tube the angle of wetting is zero, and
that the penetration coefficient is given by the expression

ue
Vv 2n
Haperimental.

A piece of capillary tube 1:2 metres long, of average
iuternal diameter 0°708 mm., chosen for uniformity of bore,
was mounted horizontally in a condenser tube maintained at
20° C. One end of the tube was fused into the side of a
wide boiling-tube containing the liquid to be tested. In each
experiment the boiling-tube was filled until the surface of
the liquid just covered the mouth of the capillary. The
liquid was drawn through the capillary by suction, and
forced back to within 15 cm. of the end by compressed air
so as to thoroughly wet the tube. This operation was per-
formed several times, the final removal of the liquid being
accomplished very slowly to ensure removal of all excess

Liquids under Capillary Pressure. 1155

liquid from the tube wall. The rate of flow was determined
by means of a stop-watch at increments of 10 cm. along the
tube. In the following curves (I.) are shown the distance
traversed in various times for a series of aliphatic alcohols ;
and from these data the derived curves (II.) were obtained,
indicating the relationship between wand Vt, which, ac-
cording to the theory outlined above, should be linear for
values of w within a relatively wide range.

Curves (1.).
B40}
780
720
660
600
540
; 480
:
S >
8 420 =
RS
SS 7 eae
& 380 i/
N Oh, Ree
& eS ~ xs
3 ¢
300 oy
wy »
200 ay A
\
180 S
no
Arce
120 meth
60 Water
re) 20 40 60 80 100

Distance of travel fp cms.

The values of the penetration coefficients obtained from
the slope of these derived curves are given, together with
those calculated from the data on the viscosity and surface
tension given in the tables of Landolt-Bérnstein according

as , where r = 0:0354 cm.
Qn

1156 Prof. E. K. Rideal on the Flow of
Curves (II.).

0 20 40 60 80 400
ie oe oo Qistance of travel 17 cms.
Substance. K eale. K. obs.
Tsobutyl alcool os ccnesseeesse= Sark) 3°70
Is@veOyonsh g5 Megonossosnossnocnt 4°10 4°20
Allyl BAMUin. = Steiracenae anaes 5°43 4°82
Ethyl Gk AMPA rcenaanre o 5°52 5°65
Methyl pate sh eaten ie 8:16 7:90
OMMMOTROCIRON, | cocassansonanipocesacs 8:73 8:60
MS OMZEME -Necgeanaatec ese ureter eee 8:90 9°90
ther’. aye ee eeeae Latico wane 11:38 10°95
Waiter sa sig Mane eescomecrsieic sins ners eal 11:40
Aeshone “Aer eee wack cara 11°70 12-70
Mthy Wacetate: accesories esc 9°60 10°20

It will be noted that the agreement is remarkably close in
those cases for which the values for the surface tensions and

a

Liquids under Capillary Pressure. 1157

viseosities at 20° C. have been determined with accuracy,
e g. water and chloroform. In the other cases, with the
possible exception of benzene, the variations in the values
of the determinations by various investigators for these
quantities is sufficieftly large to account for the dis-
crepancies.

A few experiments were made on the rate of penetration
of ethyl aleohol water mixtures, when the following values
for the penetration coefficients were obtained :—

Bibs ie K obs. K cale. static.
1 Fie NA ey Ree 11°40 11°31
SADT, a Seale ae ag teers 5°6 62
A EE ee nearer 5:1 49
Rie co Scyice e wietnas «sce 5:0 46
SOE: 25 okies ew canescee ett 5:4 5:07
LT) RECO eee ear 5°65 5°52

It will be noted that a minimum is obtained at ca. 50 per
cent. in agreement with the calculated * values ; but the
discrepancy between the calculated and the observed values
is considerably greater than the experimental error.
Data are lacking on the dynamic values of the surface
tensions of alcohol-water mixtures; but these figures, in-
cluding some experiments on the rate of flow of dilute soap
solutions, indicate that the surface film in the capillary tube
is being continually renewed during its progress through the
tube. The method is consequently applicable to the deter-
mination of dynamic surface tensions of mixtures which
frequently differ considerably from the stated values.

The penetration coefficient of a liquid is a physical con-
stant of importance in that it is related to similar constants
for gases and solids. Determination of the Maxwellian
period of molecular relaxation in gases (Jeans, ‘ Dynamical
Theory of Gases,’ p. 261; Boltzmann, Vorlesung tiber Gas
Theorie, pt. 1, p. 167) or the “‘sensibilité”? of Perrin (Ann.
de Phys. xi. p. 21 (1919)) indicates that the molecules, even
when acted upon by the mutually relatively feeble forces of
adhesion in the gaseous state, are highly damped, the re-
laxation period for nitrogen at 0° C. and 760 mm. being
1°66 10- sec. In the case of solids, the force fields or
adhesion forces are naturally much greater, causing the
molecular vibration to be even more highly damped. For

* Dunstan, J. C. S. Ixxxv. p. 824 (1904); Firth, J. C. S. xxxiii.
p. 268 (1920).

Se ee

1158 Flow of Liquids under Capillary Pressure.

metals, the period of relaxation (Langmuir, Phys. Rev. viii.
p- 171 (1916)) is in many cases identical with the period of

electronic vibration as determined by the ultra-violet radia- —

tion frequency (107 sec.), whilst for non-metals a close
approximation to the relaxation period is to be found in the
vibration frequency of the residual rays or natural infra-red

vibration frequency (10~}? sec.) (Rideal, Phil. Mag. xl. p. 462

(1920)). For liquids, it is to be anticipated that the period —

of molecular relaxation should lie between these values,
i.e. ca. 1071? sec., the more polar the medium the shorter
being its time of molecular relaxation. This period is given

by the expression =5 , where 7 is the viscosity of the fluid
and P the pressure.

In the case of liquids, P is identical with the internal
2 e
adhesional pressure P.= E (f(r)dr. Although f() cannot

be evaluated without a knowledge of the nature of the intra-
molecular forces, yet it is possible to obtain values for P

5 a
from various sources, such as the — term of van der Waals’
v

equation, from the latent heat of evaporation (Stefan, Wied.
Ann. xxix. p. 655 (1886)), from the coefficients of expan-
sion and compressibility, or from the surface tension of the
liquid. In this latter case

* = KS = K’ (penetration coefficient)’.

In the following tables are given the approximate values
of P, being the mean values of the determinations by the
varions methods, and the values of 7 calculated therefrom,
compared with the values obtained from the penetration
coefficient. [Ko being evaluated from the data for ether. ]

Substance. Be acca TO nee ey eo
Hbherre eile cake 1,590 131 11°38 (1°31)
Chiorotor mye. Pe-ne. 2,200 2-13 8-93 2:10

_Isobutyl alcohol ...... 1,900 14-7 3°76 12°2
Isopropy ly)? ete. 2,370 9°4 4°10. 10°1
A COLO MO me Saunt a mee 2,520 1°32 11:03 1-41

Whalen icc ntstcten 18,050 0-54 . 11:26 1:35

Balance Method of Measuring X-Rays. 1159

Better agreement is scarcely to be anticipated, since the
values of P for any liquid are not known to any degree of
accuracy ; P for benzene, for example, varying froin 1300
to 3810 atmospheres, according to the method employed for
its evaluation (Hildebrand, J. A. C. 8. xl. p. 1072 (1919)).
The parallelism between the two sets of determinations and
the decrease in the molecular relaxation period or increase
in molecular ‘ sensibilité” with increasing polarity is, how-
ever, clearly marked.

Summary.

The rate of penetration of a liquid into a fine capillary
under its own forces is shown to be expressed by the

Pape,
=— &

ee

capillaries the penetration coefiicient is

2
relationship ¢ = Flog a. For relatively large

ra
a

The experimental determination of the coefficient is shown
to agree with the calculated values. In the case of mixed
solvents the dynamic surface tensions and not the static
values are probably the governing factors.

The reciprocal of the penetration coefficient 1s proportional
to the square root of the period of molecular relaxation as
defined by Maxwell, and on analogy with reactions in the
solid state is probably important in reactions taking place in
liquid media.

The writer is indebted to Mr. R. L. Huntingdon for
assistance in the experiments detailed in this paper.

Chemical Department,
Cambridge University,
June 5th, 1922.

CIX. On a Balance Method of measuring X-Rays. By
Erotessor “S. Russ, 22Se., and Lh: H. Ciarx, M.Sc.,
Physics Department, Middlesex Hospital *.

NHE frequent and prolonged running of X-ray tubes
calls for some convenient method of recording. con-
tinuously the intensity of the X-radiation emitted during the
period of excitation. The balance method described below
indicates at any instant this intensity and is capable of
giving a continuous record of it. It 1s suitable for the
_measurement of the ionization produced by a powerful beam
of X-rays.

* Communicated by the Authors,

1160 Prof. 8. Russ and Mr. L. H. Clark on a

In principle, this balance method is similar to that devised
by Rutherford and described by Bronson (Phil. Mag. vol.
xl. p. 148, 1906). Hlectrical communication is made
between two lonization chambers by joining the insulated
electrode which each contains. Initially these electrodes are
earthed, but tne chambers are maintained throughout at a
constant potential difference by connecting them to the
terminals of a battery of cells. One chamber is exposed to a
beam of X-rays and the other simultaneously to the radiation
from a very small mass of radium. Under the action of the
two consequent ionization currents, the potential of the
insulated electrodes rises to a steady value when a balance is
set up. The potential of the electrode is indicated by
attaching to it a gold leaf, the deflexion of which is then
used to indicate the intensity of the X-radiation.

Vig. 1 represents a working model on the lines indicated.

Fig. 1.

B

Vessel A supports within it a small ionization chamber C,
measuring 0°'4x1:0x1°5 cubic cm., the two small opposite
ends of which are closed by windows of aluminium foil. This
chamber carries along its long axis an insulated brass
electrode, connected by a fine wire to the gold-Jeaf support
in vessel B. One wall of this vessel is made of very thin
aluminium leaf to allow entry of « radiation from a small
mass of radium.

Fig, 2
To Earth
i Radium >
ttl
LY
YlooAl
A B

Another form of the apparatus is shown in fig. 2. In this
case the ionization chamber C consists of a lead cylinder
1:2 em. long and 0°8 cm. in diameter fitted with an axial

<=> Rex

Balance Method of measuring X-Rays. 1161

electrode. Hach end of the chamber is closed by a very thin
mica window coated with aluminium leaf. A lead diaphragm
0°5 cm. in diameter carrying «a minute lead disk at its centre
allows an annulus of radiation to enter the chamber C, and
pursue a direct air path to the further window, without
impinging upon either the interior of the chamber or the
axial electrode.

Under working conditions 200 volts are applied to the
chamber C, the connecting tube and the vessel B are earthed,
the eeetral: electrode and ‘eold leaf being initially ganehed
then left insulated. The deflexion of the gold leaf is
observed through a tele-microscope or the image of the leaf
is projected on a screen.

rom a practical point of view it is important to know
how small a quantity of radium can serve to balance the
ionization current due to a powerful beam of X-rays. In
the two types of apparatus shown diagrammatically in figs. 1
and 2, 0:08 mgm. of radium bromide served to balance
currents set up in the given ionization chambers C, when the
latter were approximately 30 cm. from the anticathode of a
Coolidge tube. ‘The radium was spread over a circular
area of 2 cm. diameter, the preparation being covered by
a thin layer of mica, which allowed the arays to pass
through.

With the ionization chamber C set up at a definite distance
from the anticathode of a Coolidge tube, a series of obser-

vations was made to determine the way in which the
deflexion of the gold leaf varied with the intensity of the
X-radiation from the bulb. Keeping the spark-gap constant,
the filament current to the tube was varied over wide
limits and the intensity of the X-radiation was measured by
a distant electroscope. Simultaneously with these readings
the balance deflexion was observed.

The curve of fig. 3 was obtained with the apparatus shown
in fig. 2. It indicates a simple relationship between the
balance deflexion and the intensity of the X-radiation
entering the chamber C. The conditions of the experiment
were as follows:—180 volts on chamber C, distance of C
from anticathode 27 cm., alternative spark about 4 inches :
the distance of the radium from the aluminium window
in B, 2°2 cm.

The time taken for a state of balance to be set up was
found to be independent of the intensity of the X-ray beam.
If the leaf be started from its zero position it moves more
slowly to its position ef balance with a weak than with a strong
intensity, but the smaller actual displacement practically

Phil. Mag. 8. 6. Vol. 44. No. 264. Dec. 1922. 4F

1162 Prof. 8. Russ and Mr. L. H. Clark on a

compensates this so that the time for a balance does not vary
appreciably over a considerable range of intensity. ‘This is

Fie. 3,

cS

40

CH
So

BALANCE DEELECTION.
2S
oO

©

O 100 200 GOO ory [rain
INTENSITY.

well shown by the curves of fig. 4, which were obtained with
the model of fig. 1. The position of the gold leaf was read
at 3-second intervals from the moment the excitation of the
Coolidge bulb was begun. ‘The ultimate steady deflexions
were 21° 2, 29:2, and AT'S respectively, yet the time taken to
reach these values was approximately 30 seconds in each
case. )
Saturation of the ionized air in © is of course essential if
the method is to be used for the quantitative measurement
of the intensity of the beam of X-rays. With 200 volts on
3, it was found that saturation was obtained provided that
the radium was in such.a position that the leaf balanced at a
deflexion corresponding to a voltage on the leaf of about
120; this left 80 volts for the ionization current, If the
saan is removed to a greater distance the leaf deflects
more, but the corresponding rise in voltage now leaves
insufficient for saturation. One result of this is that a
spurious steadiness obtains in the balance ; when saturation

BALANCE -DEFLECTTON

Balance Method ef measuring X-Rays. 1163

occurs in CU oscillations are easily observed, vide fig. 4. The
oscillations under our experimental conditions were not due
to fluctuations in the radiating power of the radium, for, if
the ionization in C is produced by a radium source giving
beta and gamma rays, a very steady balance is obtained.
The oscillations which may be produced when an alpha ray

Fig. 4.
50
40
30
20 Soha
10 :
0 6 10 20 30 Z0

TIME [SSECINTERVALS J.

source is used has been studied by Geiger (Phil. Mag. vol. xv.
p- 539, 1908), who showed that they were due to actual
variation with time in the number of alpha particles emitted.
He showed that for similar ionization by beta radiation,
oscillations still occurred, though their significance was
minimized owing to the greater numbers involved. It might
be possible to detect oscillations in the X-ray case, con-
sidering their origin in the cathode stream, but under the
usual experimental conditions it seems certain that the lack

42

ee a

1164 Balance Method of measuring X-Rays.

of steadiness of the balance is mainly due to the instrumental
side of the high tension apparatus.

How far the ionization produced by X-rays in the air of a
vessel may serve as a reliable indicator of the intensity of the
former is for experiment to decide. With any change of
wave-length the issue becomes complicated by the selective
production of characteristic and scattered radiation at an
surface struck by the incident rays. The effect can hardly
completely be got rid of because of the necessity for windows
to the vessel. ‘The apparatus shown in fig. 2 was constructed
so as to reduce the above effect to a minimum.

Necessity for an International Unit of X-ray intensity.

The need has long been felt for a unit of X-ray intensity
which shall reeeive International sanction. Investigators who
wish to compare their results have of necessity recourse to
various indicators, over the performance of which there is,
bowever, not too much confidence. Attempts have from
time to time been made to express the output of X-ray tubes
in terms of the gamma rays from radium ; more, however,
with a view to comparing their relative output of energy
than to standardize X-ray intensity in terms of the constant
source of radiation which a sealed preparation of radium
affords.

The need mentioned above is also an urgent one in the
practice of medical radiology at the present time. The
radiologist who is dispenSing X-rays wields a highly compli-
cated collection of apparatus; he is provided with X-ray
tubes, the dimensions and output of which increase almost
yearly, but so far he has not been given the security he needs,
namely, a unit of intensity of the rays he is using. Itseems
that this can hardly be done without action of some sort by
physicists, and the suggestion is made here, as it has been
made elsewhere as well as by others, that steps be taken to
fix upon some International Unit ork -ray intensity which
will serve at one and the same time the interests and
requirements of physical and medical investigators.

f 1165 4}

CX. The Measurement of Light. By Joux W. T. Watsu,
M.A., M.Se., (Department of Photometry and Illumination,
National Physical Laboratory) *

a an article published under the above title in a recent

number of this Magazine t, Dr. Norman Campbell and
Mr. B. P. Dudding have criticised the logical foundation of
commonly accepted systems of definitions and in particular
the system adopted ‘for the definitions of the principal
photometric magnitudes. Their conclusions may, perhaps,
be briefly summarized as follows :—

(1) The fundamental photometric magnitude is illumina-
tion (I).

(2) Illumination is characteristic, not of the surface
illuminated, but of the circumstances in which that
surface is placed.

(3) It can be shown experimentally that illumination is
subject to the laws of addition 2f the conditions under
which the Purkinje effect operates be excluded.

(4) On the above bases the inverse square law can be
demonstrated by suitable experiment, and, this
done, the cosine law of illumination can be demon- —
strated.

(5) With the proven fact of the inverse square law
the definition of luminous intensity [candle-power]
in a given direction follows at once from that of
illumination (P=I?’?)

(6) By integrating the intensity in all directions a new
quantity F=(®do is obtained (@ js solid angle).
F is characteristic of the source alone, and is termed
the flux of light from the source.

(7) Brightness, B, is defined as the intensity of unit
projected area (B/S cos a) f(a), or normal brightness
is B/S where S is area.

Thus the system proposed by the authors gives the chief
photometric definitions in the order (1) illumination,
(2) luminous intensity, (3) luminous flux, and (4) brightness.
This order differs from that adopted by both the American
and the British National Nlumination Committees in their
systems of definitions proposed before the International
Commission on Illumination last year {. It is, therefore,
desirable to examine the reason for the difference with the
object of arriving, if possible, at the most natural order.

* Communicated by the Author.

+ Phil. Mag. Sept. 1922, p. 577.

t Commission Internationale de U Eclair age, Sme Session, Paris 1921.
Rapports (in the press).

1166 Mr. J. W.T. Walsh on the

In the American definitions * the first photometric magni-
tude defined is luminous flux, followed by luminous intensity,
illumination and brightness.

In the British definitions ¢ the order is luminous intensity,
luminous flux, illumination and brightness.

These two systems follow the respective customs of the
two countries as to the rating of illuminants. In Great
Britain a luminous source is rated in candles, while in
America the lumen is almost universally adopted.

The existing official definitions, then, adopt as their
starting point the magnitude in which the unit is realized in
practice, for both luminous flux aud luminous intensity are
characteristic of the source alone and are independent of
any other material body.

It is true that both luminous flux and luminous intensity
are impossible of measurement per se, and it is not until the
source illuminates some surface that either the flux or be
intensity can be measured. It would thus appear that,

‘deciding on our first defined magnitude, we have to oheae

between (a) the magiitude most intimately connected with
the material standard by which our unit is maintained, and
()) the magnitude which is directly measurable.

The authors of the paper referred to above apparently aim
at the second choice, but it is difficult to understand why,
after the irrefutable statement that ‘‘ light measurements are
based on judgments of equality of brightness of photometric
surfaces,” they go on without explanation to say tnat “the
fundamental photometric magnitude is illumination.”

In fact illumination appears to be the most abstruse of all
the photometric magnitudes and the one of which it is most
difficult: to form a mental concept. It is not easy, for
example, to decide whether illumination exists in the absence
of a surface.

Physical photometry being disregarded, it is clear that the
fundamental photometric magnitude from the point of view
of measurement is “brightness.”’ Illumination follows as
that which causes surfaces to have brightness. Luminous
intensity and luminous flux then follow as before.

In this connexion the proviso made by Dr. Campbell and
Mr. Dudding as to the avoidance of the Purkinje effect is
most interesting. On page 582 they say ‘ But the second
law of addition is not true in all circumstances ; it is not
true, for example, when the Purkinje effect is ‘apparent.
For, if R, and R, are red sources, B, and B, blue sources,
and if the illumination from R, is equal to that from By, and

* [llum. Eng. Soc. N.Y. Trans. xiii. (1918).
+ Uluminating Engineer, London, xv. (1922),

|

Measurement of Light. 1167

that from R, to that from B,, the illumination from RK, and
R, will not be always equal to that from B, and by.”
This statement is most surprising, especially if taken im
connexion with a previous statement that illumination is
“characteristic, not of the surface illuminated, but of the
circumstances in which it is placed. ?> These two statements

‘are not easily reconciled, for it is easily possible to Imagine

two pairs of "photometric surfaces, one pair of high reflexion
ratio and one of low reflexion ratio, so that in the former case
the addition law of illumination may be found to hold, while
in the latter case with the same illuminations it does not. Surely
if the statement first quoted be true, the second is refuted.

It seems, on the whole, more consistent to regard illumin-
ation as independent both of the nature of the surface
illuminated and of such essentially ocular phenomena as the
Purkinje effect. The latter can, and must, be reckoned with
when considering brightness, and the statement first quoted
above is quite intelligible if read as referring to the addition
of brightness, and if the words ‘“‘brightness due to” be substi-
tuted for ‘* illumination from.”

It seems, then, that we have to choose between a system of
definitions depending on (a) brightness, and (4) luminous
intensity or luminous flux, as the fundamental magnitude.

It seems to the writer at least a doubtful contention that
it is more logical to start with the magnitude actually
perceived and measured and to work back to the magnitude
in which the unit is maintained. In either case the physical
laws connecting the various magnitudes in the chain have to
be known and, in fact, they must be described, at any rate by
implication, in the definitions of the dependent magnitudes.

The gain, if any, in logical security seems to be more than
counterbalanced by a very marked loss of “ concreteness ”’
—never a pronounced characteristic of formal definitions.
The mind naturally finds it most easy to form a picture of
the inagnitude in which the unit is maintained, that being a
phenomenon having the closest association with a concrete
object. In the case of photometry it would seem that the
luminous intensity, or candle-power, of a source ina given
direction is far more readily understood as a basis of
definitions than is the brightness of a surface viewed in a
given direction. For the natural physical order is (a)
emission of luminous flux by a source owing to its luminous
intensity, (b) incidence of this flux at a surface, (c) brightness
of this surface due to the illumination and the power “of the
surface to reflect light. This is, then, the order in which the
mind expects the magnitudes to be defined, and it appears to

the writer the preferable order for that reason.

1168 The Measurement of Light.

The function of a system of definitions is, in the writer’s
opinion, so to describe a number of different quantities and
their relations to one another that a previous understanding
of any one of these quantities (regarded as the fundamental),
together with the definitions, enables all the remaining
quantities to be understood also. If this be granted, the
fundamental yuantity should be that most generally
understood. ‘The sequence of the remaining quantities is then
a matter of convenience.

In conclusion, it may not be out of place to point out
that Dr. Campbell and Mr. Dudding appear not to have
understood the “lambert” as a unit of brightness. It is
unfortunate that they were apparently only aware of the
brief statement in the Report of the Standards Committee of
the Optical Society of America* that “a lambert is the
brightness of a surface emitting one lumen per square
centimetre of projected area in the direction considered.”
They naturally remark that “ To Speak of a lumen emitted
in a direction is to talk nonsense.’

Although, of course, this description of the lambert is quite
unsound, it is unfortunate, to say the least, that the authors
did not refer to the official! definition of the lambert (by no
means a new unit). They would then have realized that
their difficulty arose, not from a miscalculation on the part
of the American Committee as they seem to infer, but from
an inaccuracy in the wording. “A lambert is the brightness
of a perfectly diffusing surface (i.e. one obeying the cosine
law of emission) emitting or reflecting one lumen per square
centimetre” T.

Summary.

(1) The fundamental photometric magnitude from the
point of view of visual measurement is brightness and not
illumination.

(2) The photometric unit is one of luminous intensity
(candle-power) or luminous flux.

(3) Of the two possible systems of definitions based on
these respective magnitudes, that based on the magnitudes
in which the unit is maintained seems preferable because
it follows the natural order of mental conception.

(4) The relation between the flux unit of brightness (the
lambert) and the intensity unit (the candle per square
centimetre) is pointed out.

* Opt. Soe. Am..J. iv. p. 230 (1920).

tT Report of Standards Committee of the Illum. Eng. Soc. N, Y.
Trans. xiii. (1918).

i LG" |
CXI. Notices respecting New Books.

The Cambridge Colloquium, 1916. Part J. By G. F. Evans.
New York, 1918. Published by the American Mathematical
Society.

TIXHIS volume by Professor Griffith C. Evans of the Rice

Institute consists of a course of lectures given before the

American Mathematical Society at its Highth Colloquinm, held

at Harvard University in 1916. The lectures dealt with the

theory of functionals and their applications, and also with various
other topics, including the theory of Integral Equations. The
second part of the present volume is to contain the lectures of

Professor Oswald Veblen of Princeton University on Analysis

Situs, which were delivered at the same Colloquium.

The present lectures select for discussion the general ideas of
Hadamard Stieljes, Borel, and Lebesgue in the theory of functions.
It will be of great value to students to have before them in this
book so clear an account of these modern developments. The
work of recent writers in particular on the Lebesgue integral and
on the very important development known as the Stieljes integral
is Summarized in excellent fashion. This volume and its com-
panion volumes should have an excellent effect in stimulating
further researches on topics which seem, indeed, to promise further
rapid extension.

A Treatise on the Integral Calculus: with applications, examples,
and problems. Vol. Il. By JosepH Epwarps. (Maemillan
& Co., 1922.)

We look on this work as our equivalent of Bertrand’s treatises.

The subject is developed in the good old-fashioned gentlemanly

style, and the reader is not tripped up perpetually by an appeal

te Rigour, Convergence, Epsilomology, and other impediments on
his road—‘ cherchant toujours la petite béte dans la démon-
stration.”

A summary of Elliptic Function theory finds a place, useful
as a Manual for the applications encountered everywhere in
Physical Science. Mention of these applications as they arise
may repel the mere mathematician, but will help a reader to a
grasp of general theory, as in the application of Compound
Representation to discontinuous fluid motion.

Historical reference too, the valuable feature of Bertrand’s
style, will add to the interest, as for instance in Mercator’s
projection, where it appeared long after Napier’s invention that
Edward Wright’s Table of Meridional Parts, 1599, is in reality a
series of logarithmic tangents and claimed as such by Wright in
his subsequent name of Nautical Logarithms.

The author does not see his way to the Continental abbreviation
of the hyperbolic function, to ch, sh, th,... in analogy with the
elliptic function en, sn, dn, tn,...

Numerous diagrams, drawn carefully with accuracy, make a

_ pleasing feature of the work.

¢

eae 1

INDEX to VOL. XLIV.

ee

ADSORPTION, (on ithe, of ions:
321; on the influence of the size
of colloid particles on the, of
electrolytes, 401; on a _ kinetic
theory of, 689.

Air, on the straggling of alpha
particles in, 42; on the discharge
of, through small orifices, 969.

Alpha particles, on the straggling
ot, by matter, 42; on the disin-
_ teoration of elements by, 417; ou
"the decrease of energy of, on
passing through matter, 680.

A lternating-current bridge, on the
general equations of an, 1024.

Aluminium, on the alpha particles
from, 420,

Amtonoff (Prof. G. N.) on the break-
ing stress of crystals of rock-salt,
62.

Arc, on the repulsive effect upon
the poles of the electric, 765.

lamp, on a sodium- -potassium
vapour, 944.

Argon, on the motion of electrons
in, 10335.

Atmosphere, on the insolation of an,
872. -

Atomic models, on a lecture-room
demonstration of, 395.

structure, on the spectra of

X-rays and the theory of, 546.

systems, on, based on free
electrons, 1065.

Atoms, on the binding of electrons
by, 193; on the distribution of
electrons around the nucleus in
the sodium and chlorine, 433 ;
on the paths of electrons in the
neighbourhood of, 777,

Bailey (V. A.) on the motion of
electrons in argon and in hydro-
gen, 1033.

Baker (B. B.) on the path of an
electron in the neighbourhood of
an atom, 777.

Baker (Dr. T. J.) on breath figures,
752.

Balmer series spectrum, on atomic
hydrogen and the, 538.

Barnett (5S. J.) on electric fields due
to the motion of constant eleetro-
magnetic systems, 1112.

Barton (Prof. IK. H.) on vibrational
responders under compound fure-
ing, 673.

Beams, on the buckling of deep,
1062. |

Belz {M. H.) on the heterodyne beat
method, 479.

Bond (W. N.) on a wide-angle lens
for cloud recording, 999.

Books, new :—Richardson’s Weather
Prediction by Numerical Process,
285; Robertson’s Basic Slags and
Rock Phosphates, 415; Ollivier’s
Cours de physique générale, 1055 :
Mellor’s Treatise on Inorganic
and Theoretical Chemistry, 1056 ;
Starline’s Science in the Service
of Man, 1058; Mie’s La Théorie
Einsteinienne ‘de la Gravitation,
1058; Wave-lengths in the Arc
Spectra of Yttrium, Lanthanum,
and Cerium, 1058; The Journal
of Scientific Instruments, 1059 ;
The Cambridge Colloquium, 1916,
1169; Kdwards’s Treatise on the
Integral Calculus, 1169.

Bosanquet (C. H.} on the distri-
bution of electrons in the sodium
and chlorine atoms, 433.

Bradford (Dr. S.C.) on the mole-
cular theory of solution, 897.

Brage (Prof. W. L.) on the distri- |
bution of electrons in the sodium
and chlorine atoms, 433.

Bramley (A.) on radiation, 720.

Brass weights, on the protection of,
948.

Breath figures, on, 762.

Breit (Dr. G.) on the ee
capacity of a pancake coil, 729;
on the propagation of a Ree.
group of waves, 1149.

Bromehead (C. E. N.) on the in-
fluence of geology on the history
of London, 286,

Broughall (L. St. C.) on the neon
spectrum, 204.

INDEX. A iia

Brown (W. G.) on the Faraday-
tube theory of electromagnetism,
594.

Browning (Dr. H. M.) on vibrational
responders under compound forc-
ing, 573.

Campbell (Dr. N.) on the elements
of geometry, 15; on the measure-
ment of chance, 67; on the mea-
surement of light, 577.

Canonical ensenible, on Gibbs’s, 839.

Capacity, on the effective, of a pan-
cake coil, 729.

Capillary pressure, on the flow of
liquids under, 1152.

Carbon bisulphide, on the ignition
of mixtures of, and air, 110; on
the molecular structure of, 292.

dioxide, on the motion of

electrons in, 994.

oxysulphbide, on the viscosity
and molecular structure of, 289,
292.

Carrington (H.) on Young's modu-
lus and Poisson’s ratio for spruce,
288.

Cavanagh (B. A. M.) on molecular
thermodynamics, 226, 610.

Centroids, on the calculation of, 247.

Chadwick (Dr. J.) on the disinte-
gration of elements by alpha par-
ticles, 417.

Chance, on the measurement:-of, 67.

Chemical constants of some diatomic
gases, on the, 988.

Chlorine atoms, on the distribution
of electrons in the, 433.

Clark (L. H.) on a balance method
of measuring X-rays, 1159.

Coagulation, on Smoluchowski’s
theory of, 305.

Coil, on the effective capacity of a
pancake, 729.

Colloid particles, on the influence
of the size of, upon the adsorption
of electrolytes, 401.

Colloidal suspensions, en the law
of distribution of particles in, 641.

Colour-vision theories in relation to
colour-blindness, on, 916.

Compressibilities of divalent metals
and of the diamond, on the, 657,

Compression, on the ignition of
gases by sudden, 79.

Conduction, on metallic, 6638.

Convective cooling in fluids, on
natural, 920,

Cooling, on natural convective, in
fluids, 920.

power, on the, of a stream of
viscous fluid, 940.

Cormack (P.) on the harmonic
analysis of motion transmitted by
Hooke’s joint, 156.

Coster (D.) on the spectra of X-rays
and the theory of atomic struc-
ture, 546.

Critical speeds of rotors, on the, 122.

Cylinder, on the unsteady motion
preduced in a rotating, of water
by a change in the angular velocity
of the boundary, |; on the rota-
tion of an elastic, 33.

Damped vibrations, on, 284, 951,
1065.

Damping coefficients, on the, of
oscillations in three coupled elec-
tric circuits, 3738.

Darwin (C. G.) on the partition of
energy, 450, 823."

Davies (Dr. A. C.) on the ionization
of abnormal helium atoms by
low-voltage slectronic bombard-
ment, 1140.

Davis (A. H.) on natural convective
cooling in fluids, 920; on the
cooling power of a stream of
viscous fluid, 940.

Diamond, on the compressibility of
the, 657.

Discharge tube, on the absorption
of hydrogen in the, 215.

Dispersion, on the magnetic rota-
tory, of paramagnetic solutions,

ame

Distribution law, on the Maxwell,
477.

Dudding (B. P.) on the measure-
ment of light, 577.

Earth, on fluid motion relative to a
rotating, 52.

Kdridge-Green (Dr. F. W.) on
colour-vision theories in relation
to colour-blindness, 916.

Kinstein’s theory, on the identical
relations in, 380.

Elastic bodies, on the rotation of,
30.

—— equilibrium, on, in an eolo-
tropic body, 501.

hysteresis, on a model to illus-

trate, 511, 1055, 1054,

stresses, on, in an_ isotropic

body, 274.

1172 INDEX.

Klectric arc, on the repulsive effect
upon the poles of the, 765.

charges, on the electrodynamic

potentials of moving, 376.

circuits, on the damping co-

efficients of the oscillations in

three-coupled, 373.

contacts, on the use of a triode

valve in registering, 870.

discharge tube, on the absorp-

tion of hydrogen in the, 215.

fields, on the Stark effect for

strong, 371; on, due to the motion
of constant electromagnetic sys-

tems, 1112.

waves, on short, obtained by
valves, 161, 1064.

Electrification at the boundary
between a liquid and a gas, on
the, 586.

“lectrodynamic potentials, on the,
of moving charges, 376.

Electrolytes, on the influence of the
size of colloid particles upon the
adsorption of, 401.

Hlectromaenetic lines and tubes, on,
705.

systems, on electric fields due
to the motion of constant, 1112.

Electromagnetism, on the Faraday-
tube theory of, 594.

Electron theory of solids, on the,
657. ;

Electrons, on the binding of, by
atoms, 193; on the electrodynamic
potentials of, 376; on the velocity
of, in gases, 384; on the distri-
bution of, round the nucleus in
the sodium and chlorine atoms,
433; on the path of, in the neigh-
bourhood of an atom, 777; on
the emission of, by X-rays, 793;
on the motion of, in carbon di-
oxide, 994; in argon and in hydro-
gen, 10383; on atomic systems
based on free, 1065.

Elements of the higher groups, on
the temperature ionization of,
1128.

Energy, on the partition of, 450,
823; on the partition of, in the
double pendulum, 382.

Entropy, note on, 832.

Equation of state, on a revised,
1020.

Ether and air, on the ignition of
mixtures of, 106.

Kixpansion, on the latent heat of,
590.

Fan-shaped group of waves, on the
propagation of a, 1149.

Faraday-tube theory of electro-
magnetism, on the, 594.

Films, on the thickness of, 1002.

Fluid motion relative to a rotating
earth, on, 52.

Fluids, on natural convective cooling
in, 920; on the cooling power of
a stream of viscous, 940.

Fluorine, on the alpha particles
from, 420.

Fowler (R. H.) on the partition of
energy, 450, 825.

Freezing mixtures, on the theory of,
787.

Gamboge, on the distribution of
particles in a _ suspension © of,
641.

Gamma-ray activity of radium
emanation, on the, 300.

Gases, on the straggling of alpha
particles in, 42; on the ignition
of, by sudden compression, 79 ;
on the velocity of electrons in,
384; on electrification at the
boundary between liquids and,

3865; on the chemical constants ~

of some diatomic, 988.

General Electric Company’s Re-
search Staff on the thickness of
liquid films on solid surfaces,
1002.

Geological Society, proceedings of
the, 286, 1060.

Geometry, on the elements of,
15.

Gibbs (R. E.) on the theory of
freezing mixtures, 787.

Gill (EK. W. B.) on short electric
waves obtained by valves, 161.
Gold foil, on the straggling of alpha

particles in, 49.

Gray (Prof. J.G.) on the calculation
of centroids, 247.

Green (Dr. G.) on fluid motion
relative to a rotating earth, 52.
Greenhill (Sir G.) on pseudo-regular

precession, 179.

Gwyther (R. F.) on elastic stresses
in an isotropic body, 274; on the
conditions for equilibrium under
surface traction in an eolotropic
body, 501.

Gyroscopic motion, note on, 179.

———e_

INDEX. 1173

Hackett (Prof. I’. E.) on the rela-
tivity contraction in a rotating
shaft, 740.

Hammick (D. L.) on latent heats
of vaporization and expansion,
590.

Hargreaves (R.) on atomic systems
based on free electrons, 1065.

Harmonic analysis of motion trans-
mitted by Hooke’s joint, on the,
156.

Harward (A. E.) on the identical
relations in Itinstein’s theory,
380.

Hedges (J. J.) on the distribution
of particles in colloidal suspen-
sions, 641.

Heli:nm, on the ionization of atoms
of, by low-voltage electronic bom-
bardment, 1140.

atom, on the, 842.

Henderson (G. H.) on the strazgling
of alpha particles by matter, 42;
on the decrease of energy of alpha
particles on passing through
matter, 680.

Henry (1). C.) on a kinetic theory
of adsorption, 689.

Heptane and air, on the ignition of
mixtures of, 101.

Heterodyne beat method, on the,
479.

Hicks (Prof. W, M.) on the quan-
tum-orbit theory of spectra, 546.
Holmes (Sir C. J.) on Leonardo da

Vinci as a geologist, 1061.

Hooke’s joint, on the harmonic
avalysis of motion transmitted by,
156.

Horton (Prof. I.) on the ionization
of abnormal helium atoms by low-
voltage electronic bombardment,
1140.

Hydrogen, on the absorption of, in
the electric discharge tube, 215;
on atomic, and the Balmer series
spectrnm, 5388 ; on impact ioniza-
tion by H-ions in, 809; on the
motion of electrons in, 1033.

particles, on the ranges of the,
423,

Identical relutions in Linstein’s
theory, on the, 380.

Ignition of gases by sudden com-
pression, on the, 79.

Insolation of an atmosphere, on the,

872.

lodine vapour, on the emission
spectrum of monatomic, 651.

Tonization, on impact, by low-speed
positive H-ions in hydrogen, 809 ;
on the temperature, of elemeuts of
the higher groups, 1128; on the,
of helium atoms by low-voltage
electronic bombardment, 1140.

Tons, on the adsorption of, 321; on
the ionizing potential of positive,
1147.

James (R. W.) on the distribution
of electrons in the sodium and
chlorine atoms, 433.

Jets, on the entrainment of air by,
969.

Kar (Prof. 8. C.) on the electro-
dynamic potentials of moving
charges, 376.

Karapetoff (Prof. V.) on the genera!
equations of a balanced alter-
nating-current bridge, 1024.

King (Prof. L. V.) on a lecture-
room demonstration of atomic
models, 395.

Landau - Ziemecki (St.) on the
emission spectrum of monatomic
iodine, 651.

Latent heats of vaporization and
expansion, on the, 590.

Lees (S.) on a simple model to
illustrate elastic hysteresis, 511,
1054.

Lens, on a wide-angle, for cloud
recording, 999,

Lidstone (I*. M.) on the variable
head in yiscosity determinations,
953.

Light, on the measurement of, 577,
1165,

Liquid films, on the thickness of,
1002.

Liquids, on the electrification at the
boundary between, and gases,
386; on the flow of, under
cepillary pressure, 1152.

McLeod (A. R.) on the unsteady
motion produced in a rotating
cylinder of water by a change in
angular velocity, 1.

McTaggart (Prof. H. A.) on the
electrification at the boundary
between a liquid and a gas, 386.

Magnetic properties of the hydrogen-
palladium system, on the, 1063.

rotatory dispersion of para-

magnetic solutions, on the, 912.

1174

Magnetic susceptibilities, on the
heterodyne -beat method in the
measurement of, 489. |

Mallet (Rh. A.) on the reciprocity
law in photography, 904,

Manley (J. J.) on the protection of
brass weights, 948.

Measurements, on the heterodyne

- beat method in physical, 479.

Mercury vapour, on the selective
reflexion of A 2536 by, 1105; on
the polarized resonance radiation
of 107:

Metals, on the compressibilities of
divalent, 657.

Milne (I. A.) on radiative equi-
librium, 872.

Milner (Prof. 8. KR.) on _ electro-
magnetic lines and tubes, 705 ;
on the radiation of energy by an
accelerated electron, 1052.

Molecular dimensions, on the, of
carbon oxysulphide, 289, 292; of
sulphur dioxide, 508.

—- theory of solution, on the,
897.

—— thermodynamics, on, 226, 610.

Molecules, on the distribution of,
in space, 46-4.

Morrell (J. H.) on short electric
waves obtained by valves, 161.
Mosharrafa (A. M.) on the Stark
effect for strong electric fields,

371,

Mukerjee (Prof. J. N.) on the
kinetics of coagulation, 3805; on
the adsorpiion of ions, 321.

Murray (I. D.) on the influence of
the size of collvid particles upon
the adsorption of electrolytes, 401.

Neon, on the spectrum of, 204.

Newman (Dr. I*. H.) on the absorp-
tion of hydrogen in the electric
discharge tube, 215; on asodium-
potassium vapour arc lamp, 944.

Nicholson “(Profs Ja W.)\%on) the
binding of electrons by atoms,
198.

Nitrogen, on the ranges of the H-
nuclei expelled from, 481.

Oldham (R. D.) on the cause and
character of earthquakes, 1060.

Oxley (A. E.) on the magnetic pro-
perties of the hydrogen-palladium
system, 1068.

Palladium-black, on the magnetic
properties of, 1063.

INDEX.

Pancake coil, on the effective

_ capacity of a, 729,

Papaconstantinou (Prof. B. C.}) on
kinetics of coagulation, 305, —

Partington (Prof. J. BR.) on the
chemical constants of some di-
atomic gases, 988,

Pendulum, on energy partition in
the double, 382.

Phosphorus, cn the alpha particles
from, 420.

Photographic emulsion, on the cha-
racteristic curve of a, 352, ,

exposure, on the quantum
theory of, 252, 275, 956. —-

Photography, on the failure of the
reciprocity law in, 904.

Photometry, notes on, 577,

Porter (Prof. A. W.) on the law
of distribution of particles in col-
loidal suspensions, 641; on theory
of freezing mixtures, 787; on a
revised equation of state, 1020.

Positive ions, on the ionizing
potential of, 1147,

Potentials, on the electrodynamic,
of moving charges, 376,

Precession, on pseudo-regular, 179.

Prescott (br. J.) on the buckling of
beams, 1062.

Press (A.) on a model to illustrate
elastic hysteresis, 1053.

Pye (D. R.) on the ignition of gases
by sudden compression, 79,

Quantum orbit theory of spectra,

_ on the, 346.

— theory, on molecular thermo-
dynamics and the, 226; on the,
of photographic exposure, 252,
257, 856; on the normal helium
atom and the, 842.

Radiation, on, 720.

Radiative equilibrium, on, 872.

Radium emanation, on the gamma-
ray activity of, 300.

Rankine (Prof, A. O.) on the mole-
cular structure of carbon oxy-
sulphide and carbon bisulphide,
292.

Reciprocity law in photography, on
the failure of the, 904.

Kteflexion, on the. selective, of
d 2536 by mercury vapour, 1105.
Relativity contraction in a rotating

shaft, on the, 740.

Resonance radiation of mercury

vapour, ou the polarized, 1107.

INDEX. 7 1175

Yesponders, on vibrational, under
compound forcing, 5738,

Richardson (Dr. 8. 8.) on the mag-
netic rotatory dispersion of certain
paramagnetic solutions, 912.

Rideal (Prof. E. IX.) on the tlow of
liquids under capillary pressure,
1182.

Righter (L.) on Silberstein’s theory
of photographic exposure, 252.
Ring, on the distribution of stresses

in a, 1014.

Roberts (R. W.} on the magnetic
rotatory dispersion of certain
paramagnetic solutions, 912.

Rock-salt, on the breaking stress of
crystals of, 62.

Nod, on the rotation of a thin, 32.

Rodgers (C.) on the vibration and
‘critical speeds of rotors, 122.
totation of elastic bodies, on the, 30.

Rotors, on the vibration and critical
speeds of, 122.

Rowell (H. S.) on damped vibra-
tions, 284, 951; on energy
partition in the double pendulum,
382.

Russ (Prof. $8.) on a balance method
of measuring X-rays, 1159.

Rutherford (Sir E.) on the dis-
integration of elements by alpha
particles, 417.

Saha (Prof. M. N.) on the temper-
ature ionization of elements of the
higher groups, 1128.

Saxton (A. J.) on impact ionization
by low-speed positive H-ions in
hydrogen, 809.

Sellerio (Dr. A.) on the repulsive
effect upon the poles of the
electric arc, 765.

Shaft, on the relativity contraction
in a rotating, 740.

Shearer (G.) on the emission of
electrons by X-rays, 795.

Silberstein (Dr. L.) on the quantum
theory of photographic exposure,
257, 956.

Skinker (M. F.) on the motion of
electrons in carbon dioxide, 994.
Slater (I*. P.) on the risé of gamma-
ray activity of radium emanation,

300.

‘Smith (C. J.) on the viscosity and
molecular dimensions of gaseous
carbon oxysulphide, 289; of
sulphur dioxide, 508,

Smith (J. 11.) on the magnetic
rotatory polarization of certain
paramagnetic solutions, 912.

Sodium atoms, on the distribution
of electrons in the, 433.

-potassium vapour are lamp,
on a, 944,

Soils, on the exchange of bases in,
343,

Solids, on the electron theory of,
G57,

Solutes, on the solvation of, 610,

Solutions, on the molecular theory
of, 897 ; on the magnetic rotatory
dispersion of paramagnetic, 912.

Spectra, on the quantum orbit theory
of, 346 ; on the, of X-rays and the
theory of atomic structure, 546,

Spectrum, on the neon, 204; on
atomic hydrogen and the Balmer
series-, 558; on the emission, of
iodine vapour, 651.

Spruce, on Youne’s modulus and
Poisson’s ratio for, 288.
Stark effect, on the, for

electric fields, 371,

State, on a revised equation of,
1020.

Straggling of alpha particles by
wnatter, on the, 42.

Stresses, on elastic, in an isotropic
body, 274; in an eolotropic body,
501; on the distribution of, in a
circular ring, 1014.

Sulphur dioxide, on the viscosity
and molecular dimensions of, 5C8.

Surface-tension, on the, of rock-
salt, 612.

Suspensions, on the law of dis-
tribution of particles in colloidal,
641,

Takagishi (E.) on the damping
coefficients of the oscillations in
three-coupled circuits, 373.

Temperature ionization of elements

. of the higher grou;s, on the, 1128.

Thermodynamics, on molecular, 226,
610; on statistical theory and, S09,

Thomas (Dr. J. 8. G.) on the dis-
charge of air through small
orifices and the entrainment of
air by the issuing jet, 969.

Thomson (Sir J. J.) on the electron
theory of solids, 657.

Timoshenko (Prof. S. P.} on the
distribution of stresses in a cir-
cular ring, 1014,

strong

1176

Tizard (fl. T.) on the ignition of
gases by sudden compression,
WS),

Tomlinson (G, A.) on,the use of a
triode valve in registering electrical
contacts, 870.

Townsend (Prof. J. 58.) on the
velocity of electrons in gases,
384; on the motion of electrons
in argon and in hydrogen, 1085 ;
on the ionizing potential of
positive ions, 1147.

Toy (F. C.) on the characteristic
eurve of a photographic emulsion,
302.

Triode valve, on the use of a, in
registering electrical contacts,
870.

Trivelli (A. P. H.) on Silberstein’s
quantum theory of photographic
exposure, 252, 950.

Valves, on short electric waves
obtained by, 161, 1064.

Van Vleck (J. H.) on the normal
helium atom, 842.

‘Vaporization, on the latent heat of,
590.

Vibrational responders under com-
pound forcing, on, 572.

Vibrations and critical speeds of
rotors, on the, 122; on damped,
284, 951, 1068.

Vibrators, on the distribution of
enerey among a set of Planck,
456.

Viscosity, on the, of carbon oxy-
sulphide, 289; of sulphur dioxide,

INDEX.

508; on the effect of the variable
head in determinations of, 953;
on a problem in, 1002.

Walsh (J. W. T.) on the measure-
ment of light, 1165.

Water, on the unsteady motion
produced in a cylinder of, by a
change in the angular velocity
of the boundary, 1.

Waves, on the propagation of a fan-
shaped group of, 1149.

Weights, on the protection of brass,
948

Wheatstone bridge, on the general
equations of an _ alternating -
current, 1024.

Whiddington (Prof. R.) on short
electric waves obtained by valves,
1064. :

Wide-angle lens, on a, for cloud
recording, 999.

Wood (Prof. R. W.) on atomic
hydrogen and the Balmer series
spectrum, 538; on the selective
reflexion of A 25386 by mercury
vapour, 1105; on the polarized
resonance radiation of mercury
vapour, 1107. .

Wright (C. E.) on damped vibra-
tions, 1063.

Wrinch (Dr. D.) on the rotation of
slightly elustic bodies, 30.

X-rays, on the spectra of, and the
theory of atomic structure, 546;
on the emission of electrons by,
793; on a balance method of
measuring, 1159.

END OF THE FORTY-FOURTH VOLUME.

Printed by Taytor and Francis, Red Lion Court, Fleet Street.

Woop. Phil. Mag. Ser, 6, Vol. 44, Pl. IX.

“

as

}
,
ie
,
rn
\

01202 5102

i. Lib