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THE

LONDON, EDINBURGH, anp DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

CONDUCTED BY

SIR OLIVER JOSEPH-LODGE, D.Sc., LL.D., F.R.S.
“SIR JOSEPH JOHN THOMSON, O.M., M.A., Sc.D., LL.D., F.R.S.
| JOHN JOLY, Midi: D:So., F:R.S., F.G:S.

GEORGE CAREY FOSTER, B.A., LL.D., F.B.S. 1

AND /

WILLIAM FRANCIS, F.L.S. /

a re Ae a ANCIENT

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

VOL. XXIX.—SIXTH SERIES.

JANUARY—JUNE 1915.
\BRAR eS
se

JUN 15 1915

LONDON:
TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET.
SOLD BY SIMPKIN, MARSHALL, HAMILTON, KENT, AND €O., LD.

SMITH AND SON, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN ;—
AND YEUVE J. BOYVEAU, PARIS,

“‘Meditationis est perscrutarl occulta; contemplationis est admirari
perspicua .... Admiratio generat questionem, questio lnvestigationem,
investigatio inventionem.”—Hugo de S. Victore.

“Cur spirent venti, cur terra dehiscat,

Cur mare turgescat, pelago eur tantus amayror,

Cur caput obscura Phcebus ferrugine condat,

Quid toties diros cogat flagrare cometas,

Quid pariat nubes, veniant cur fulmina ceelo,

Quo micet igne Iris, superos quis conciat orbes

Tam vario motu.”

J. B. Pinelli ad Mazonium.

CONTENTS OF VOL. XXIX.

(SIXTH SERIES).

NUMBER CLXIX.—JANUARY 1915.

Pa @
Prof, E. Taylor Jones on the most Effective Adjustment of :
PmAcbion—-COlLyw (Plates yin eM cece Sa a dao alanis 1
Mr. C. V. Raman on Motion in a Periodic Field of Force.
Fellas TES chek ete oi Coils att RIAD RR ed Lea i Mg eg ae 0A 15
Messrs. W. J. Jones and J. R. Partington on the Ideal
eermremivinias Ol Gases it): Mee ee Oe le eh 28
Messrs. W. J. Jones and J. R. Partington on a Theory of
SENS APMUCA DION |) Jue leo howe RL ic Lk a Be
Dr. H. Stanley Allen on the Magnetic Field of an Atom in
Relation to Theories of Spectral Series ................ A()

Prof. G. A. Schott on the Motion of the Lorentz Electron. 49
Dr. 1. J. Schwatt: Note on the separation of a Fraction into
paren PRIRACHIONS AY sia Neen Ee Pal Nuke ile ghntge Sly aera hood L 6
Dr. I. J. Schwatt: Note on the Expansion of a Function... 6
Mr. R. V. Southwell on the Collapse of Tubes by External

Peeeemen MIO ty arene oapaeicn sg tidine ama aki ahead Nees ude dU 67
Dr. A.D. Fokker: A Summary of Einstein and Grossmann’s
Mite mey Or OiravitaAtloniyy sy maski We ols “dir aca he dane ele ZG
Mr. A. EH. Young on the Form of a Suspended Wire or Tape
ieimMmons the iMitect Ob Stutmessy, 2 oo ee Uy 96
Dr. F. A. Lindemann on the Theory of the Metallic
TMC Mmeme vaste) cate ams RN SPURNS GER Ca a cl aithrg) Saver (gh) eae Pag
Dr. H. Stanley Allen on the Series Spectrum of Hydrogen
amdgine Strichare ot the Atom) Ss... seis fae clea ewok 140
Prof. A. W. Porter on the Variation of the Triple-Point of a
Substance with Hydrostatic Pressure ................ 143
Mr. J. Rice on the Form of a Liquid Drop suspended in
another Liquid, whose density is variable.............. 149
Dr. Emil Paulson on the Spectrum of Palladium.......... 154

Mr. G. H. Livens on Lorentz’s Theory of Long Wave
LAI UIRHECOLITGANS Nila Ne RUE an aPC NN IA COVER, YORCARMRREC TR Tc RLS
Mr. G. H. Livens on the Electron Theory of Metallie Con-
CHIEN STENT CT al aN NA le GOR Raa Be ee Naa oe 73
Mr. F. Lloyd Hopwood on the Plastic Bending of Metals .. 184
Dr. 8. A. Shorter on the Shape of small Drops of Liguid .. 190

LSS FT TT SE NT

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

Page
Mr. N. P. K. J. O'N. McCleland: A Study of the Absorp-
tion Spectra of Organic Substances in the Light of the
Mlectron Theory” chic. st eee oe ee ele > 192
Notices respecting New Books :—
Dr: J... Kippax’s The Call of the Stars. >.>... - = sees 206

Proceedings of the Geological Society :—
Dr. A. Dunlop on a Raised Beach on the Southern Coast
Ob Jersey “ey lacie dave a eds wel yee 207
Prof. 8. J. Shand on Tachylyte Veins and Assimilation
Phenomena in the Granite of Parijs (Orange Free
SRHLE)) a sose oe ieinlwshcialee eo aie 207
Intelligence and Miscellaneous Articles :—
Pressure of Radiation on a Receding Reflector: Corri-
geuda, by Sir-Ji: Larmor’ .... .. Jab ose ae 208

NUMBER CLXX.—FEBRUARY.

Lord Rayleigh on some Problems concerning the Mutual
Influence of Resonators exposed to Primary Plane Waves. 209
Prot. E. C. C. Baly on Light Absorption and Fluorescence.—
Bart iio d., Hatin feisee elles alee db 16 OE rr 223
Prot. H. Nagaoka and Mr. T. Takamine: Anomalous Zeeman
Effect in Satellites of the Violet Line (4859) of Mercury.
@alates MME TV.) Scone oblate rr 241
Mr. H. P. Walmsley and Dr. W. Makower on the poe
Pay of the Recoil Stream from Radium A. (Plate V

Ee PE CMLL eke Slopelienia) oe Slates SO 253
Mr. oy. Tunstall and Dr. W. Makower on the Velocity of
the « Particles from Radium A. (Plate V. fig. 2.)...... 259
Mr. W. Morris Jones: Frictional Electricity on Insulators
amo Miebalishiiie . Bedi Saye tl Sods On ee ee 261
Lord Rayleizh on the Widening of Spectrum Lines ...... 274
Mr. E. J. Evans on the Spectra of Helium and Hydrogen.
Migiiety 01). i ease te eee mn mmerbar rec: here Po. 284

Profs. P. Ehrenfest and H. Kamerlingh Onnes ona Simplified
Deduction of the Formula from the Theory of Com binations
which Planck uses as the Basis of his Radiation Theory |. 28m

Mr. P. G. Nutting on the Visibility of Radiation ....... 301

Dr. A. C. Crehore on the Gyroscopic Theory of Atoms and
Maleenles .... eae ek. oa Cee 310

Dr. N. Bohr on the Series Spectrum of Hy drogen and the
Structuresof- the Atomic). Bae ee Ge he BOO Lae: ake eee 332

Notices respecting New Books :—

Dr. L. Silberstein’s The Theory of Relativity ......., 335

CONTENTS OF VOL. XXIX.——SIXTH SERIES, v

NUMBER CLXXI.—MARCH.

Page
Mr. A. Fleck on the Condensation of Thorium and Radium '
15 UE SULTS DGS alah Mk a ea a AR an Pa ae AE PE BOT
Mr. F. Lloyd Hopwood on a Qualitative Method of Investi-
eae PU NCE MIONIC EMISSION 02 cacccc so i5 2 os ele ww/nie) «5° 362
Dr. Norman Campbell on the Ionization of Metals by Cathode
EM oa Sots Ber has Oo. G ok oes 8.0) 5 pot sdaneba a6 369
Mr. G. H. Livens on the Law of Partition of Energy and
2 GUN GOMINT DUI CTT Aes Ss ae ae Pee Pare nn EN 383
Dr. Genevieve V. Morrow on Displacements in certain
Speeual Wines\ol Zinc and Vitanium 2... 0.3... on ee 394
Prot. W. H. Bragg on the Relation between certain X-ray
Wave-leneths and their Absorption Coefficients ........ 407
Mr. H. Pealing on Condensation Nuclei produced by the
meron ot Nighton Lodine: Vapour) 2.5.2.2... 4. 4 ore 413

Prof. 8. Kinoshita and Mr. H. Ikeuti on the Tracks of the
a Particles in Sensitive Photographic Films. (Plate VII.) 420
Mr. G. H. Livens on the Electron Theory of Metallic Con-
Seer OMe Pes Mat he el «ee oid ai cial'e Ss cieheis ule 8 Wastes 425
Notices respecting New Books :-—-
Mr. Heyward Scudder’s The Electrical Conductivity and
Jonization Constants of Organic Compounds ...... 432

NUMBER CLXXII.—APRIL.

Lord mace OniseAHoltam MOTs Pah acai aeons, sik ay Oars ete 433
Mr. G. B. Jeffery on the Equations of Motion of a Vi iscous

Sea ene Mey MN ENSEN Ueno vaiayere < sm ag-a a clsere a3 Sto te whats 445
Mr. G. B. Jeffery on the Two-Dimensional Steady Motion

NTSC OMS EL UTS Pratt Nene 9 lei led eye)ie, ahay. che een els wa a Septet 455
Prof. 1). N. Mallik on the Theory of Dispersion .......... 465
Mr. J. H. J. Poole on the Average Thorium Content of the

aE AMEE ORO TOUS ne Nee © 3.014 Sachaerdend «Gey vis als ais gate dare eee 483
Dr. J. H. Vincent and Mr. C. W. Jude on the Duplex

Emomomocrapine (Colabe VIWa ie ow. eae sa cae aeies 490
Prof. J. A. Pollock on the Nature of the Large Ions in the

SRNR serosc es epi Pree aS Mladen wicks aig’e’al'a. vig nehgiaerees ol4
Mr. W. Ellis Williams on the Motion of a Sphere in a

ere ermeru nner aie Woke ic a cine a ena ejeraun’s'«, wl gaapers als 526
Dr. A. O. Rankine on the Relative Dimensions of

TWO S US SRLS SNE ERE SRNR Cee OU CR OS TS aon

Prof. Louis Vessot King on the Precision Measurement of
Air Velocity by means of the Linear Hot-Wire Anemo-
LNG PLOIRUROR ON Fl elsrae Oxia Siweis cs so vee Gee Ce) GOO

v1 CONTENTS OF VOL. XXIX.—SIXTH SERIES.

Page
Mr. S. Butterworth on the Coefficients of Self and Mutual
induction of Coaxial Cols) 2 2. 578
Mr. T. Carlton Sutton on the van der Waals Formula (and
the latent Meat ot Vaporization) <2 .)... >...) eee 593
Dr. Allan Ferguson on the Boiling-Points and Critical
Temperatures of Homologous Compounds ............ 599
The Harl of Berkeley and Mr. E.G. J. Hartley on a New
Form of Sulphuric-Acid Drying-Vessel ................ 609

The Earl of Berkeley and Mr. D. E. Thomas on a Sensitive
Method for Examining some Optical Qualities of Glass

Plates eos oe ue a Glas Bee BA ce ce er 613
Prof. O. W. Richardson and Prof. F. J. Rogers on the Photo-
electric iiects Ti ik a a 618
Mr. Fernando Santord on the Contact Difference of Potential
ou Distilled Metals... wos steeds cee nents me 623
Notices respecting New Books :—
Bulletin of the Bureantor Standards .) 4.2.02 ee eee 62-4

NUMBER CLXXIII.—MAY.
Dr. C. V. Burton on the Scattering and Regular Reflexion

of light by Gas Molecules Part). 2...) een 625
Prof. J. A. Pollock on a New Type of lon in the Air .....- 636
Miss M. O. Saltmarsh on the Brightness of Intermittent

Dara ton ee eee lis tian es ce) et, ot oale et eee 646
Dr. L. Vegard: Remarks regarding the Series Spectrum of

Hydrogen and the Constitution of the Atom .......... 651
Mr. G. H. Livens on the Electron Theory of the Optical

Properties of Metals;-—Part Ile)... 52 5.2 2 See eee 655
Messrs. Arthur Holmes and Robert W. Lawson: Lead

and the End Product of Thorium (Part IT.)............ 673
Dr J. de Waltonon Mipples! 2.2... 23). ee oe 688
Dr. Frank L. Hitchcock on the Operator V in Combination

with Homogeneous unchions’ | ..... 2. 42. sae ee 700
Dr. L. Silberstein on Radiation from an Electric Source, and

line Specivay= The Hydrogen Series 72.2)... ee 709
Dr. H. Stanley Allen on an Atomic Model with a Magnetic

CORB ate arsmite AUT Valea sehet ya defo lehiin dae so ie 714
Mr. R. W. Varder on the Absorption of Homogeneous

I ECANS fo GS stil APR MN Aled TES il Sb Se Ch 425

Mr. T. Carlton Sutton: Photo-electric Constant and Atomic
EEE eee Bebere arei ard San UCNS, AAP ae eR AON WA ee COMERS (34.

eM See fed

CONTENTS OF VOL. XXIX.—-SIXTH SERIES,

NUMBER CLXXIV.—JUNE.

Prof. 11. Lamb and Miss L. Swain on a Tidal Problem
Mr. W. Barlow and Prof. W. J. Pope on Topic Parameters

snd Mlornhorropie Relations) |)... .ieiaie at ae es
Dr. A. C. Crehore on the Construction of Cubic Crystals
Mmoateencoretical Atoms. (Plate mt.) 12. elon ee oa
Dr. W. Marshall Watts on the Principal Series in the
Pecan om unemaullkali Metalg Ise e i bi sos.

Dr. Norman Campbell on Ionization by Positive Rays ...
Dr. Wiliam Wilson on the Quantum-Theory of Radiation
manos SCOURGE isos Oise Gal Gla allone sida Ae eid's\\e oy alee)
Mr. K. K. Smith on Negative Thermionic Currents trom
PRM SHO Meh eran le rae Ny mate km HG Ta Ee ucla hie Suds alegre aed ee
Dr. J. J. Schwatt on the Higher Derivative of a Function,
the variable of which is a Function of an independent
PELLBIOUS: SEAN RS BAG Mca ne) tM kc a iat Te
Mr. G. B. Jeffery on Self-Intersecting Lines of Force and
MIC MNDIA Le SUTTAUCES) hac 8 cy ol a janepehs 3/4506 oi'e 5 So) ayyes «) aah as

Prof. hk. R. Sahni on the Photographic Action of a, 6, and y

ae ipecr Mme (Ep avbe SelM (aye arty cusboesieye city Wetec ep ois icools ale She ates
Proceedinys of the Geological Society :—

Mr. Charles Irving Gardiner on the Silurian Inlier of

isis (vlommoubishire yin ye ie ya eset sce

Mr. Samuel Rennie Haselhurst : Some Observations

on Cone-in-Cone Structure and their Relation to its

BUONO NSU ean ealiaae Wee ace i Ba Ra A

836

841

PLATES.

I. Illustrative of Prof. E. Taylor Jones’s Paper on the most
Hffective Adjustment of an Induction-coil.
II. Illustrative of Mr. C. V. Raman’s Paper on Motion in a
Periodic Field of Force.
Ill. & LV. Illustrative of Prof. H. Nagaoka and Mr. T. Takamine’s
Paper on Anomalous Zeeman Effect in Satellites of the
Violet Line (4359) of Mercury.
V. Fig. 1. Iustrative of Mr. H. P. Walmsley and Dr. W.
Makower’s Paper on the Magnetic Detiexion of the Recoil
Stream from Radium A.
Fig. 2. Illustrative of Mr. N. Tunstall and Dr. W. Makower’s
Paper on the Velocity of the a Particles from Radium A.
VI. Mlustrative of Mr. E. J. Evans’s Paper on the Spectra of
Helium and Hydrogen.
VII. Illustrative of Prof. S. Kinoshita and Mr. H. Ikeuti’s Paper
on the ‘l'racks of the a Particles in Sensitive Photographic
Films.
VIII. Dlustrative of Dr. J. H. Vincent and Mr. C. W. Jude’s Paper
on the Duplex Harmonograph.
IX. Illustrative of Mr. W. Ellis Williams’s Paper on the Motion
of a Sphere in a Viscous Fluid.
X. Illustrative of Prof. Louis Vessot King’s Paper on the
Precision Measurement of Air Velocity by means of the
Linear Hot-Wire Anemometer.
XI. Illustrative of Dr. A. C. Crehore’s Paper on the Construction
of Cubic Crystals with Theoretical Atoms.
XII. Llustrative of Prof R. R. Sahni’s Paper on the Photographic
Action of a, 6, and y Rays.

ERRATA,

Page 104, lines 15 and 16, for a read —_

als
Page 105, lines 6, 15, and 20, for T read T,.
Page 109, line 1, for cos* read cos *W.

‘ Yo LOW af 5
Page 114, line 13, fo 1758 read 7168"
Page 115, in Fig. 3, for é read &.

wl cos £\ 4 wi cos £\2
Page 118, line 3, for ( 1— 7 ) read (1 =) :

Page 122, line 21, after “ standard ” omit the comma.

Page 127, line 6, for “reciprocal” read simultaneous.
Page 689, line 8, for “12—p'/p” read “1—2p'/p,”

Page 698, line 17, for “x= ” vead “x= 2)”

THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE.

AND

I. On the most Hfective Adjustment of an Induction-coil.
By ®. Taytor Jones, D.Se., Professor of Physics in the
University College of North Wales, Bangor ™.

[Plate I.]

a a recent paper t it was shown that the secondary
potential and spark-length developed by an induction-

coil, when a given current is interrupted in the primary

circuit, are greatest under the following conditions :—

(1) The ratio n/n, of the two frequencies of elec-
trical oscillation of the system has one of the values
“D1 piel Oe ne Pi

(2) L,Cy;=(1—#) LC, f.

The first is the condition that maxima of the two potential
waves in the secondary circuit should occur simultaneously,
the second that the sum of the amplitudes of the two waves
should be a maximum for a given value of k.

It was further stated that the most effective of the ratios
specitied in (1) is m/nj=3, and that in this case the
conditions are satisfied by the adjustment &£ =°756,
LC, ="429 L,Co.

* Communicated by the Author.

+ Phil. Mag. xxvii. pp. 580-586, April 1914,

+ L,, L, are the self-inductances, U,, C, the capacities, and & is the
coupling coefficient of the primary and secondary circuits,

Phil. Mag. 8. 6. Vol. 29. No. 169. Jan. 1915. B

2 Prof. E. Taylor Jones on the mosl

The physical meaning of the above results may be further
considered in the light of what goes on in the primary
circuit after the interruption of the current. An expression
for the potential-difference V, of the plates of the primary
condenser, at any time ¢ after the interruption of the
current %, has been given by Dibbern*. If the resistances
of the circuits are neglected, Dibbern’s expression becomes

QTrignyNe” ( 1

y(n? — 2,7) \4r? ng?

i — —1,C,) sin 2rnyt

QTrigny No 1
Cy (2? — 4”) \42r?n,?

—14G,) sin 2arngl.

If n/n, has one of the values 3, 7, 11,...., then at the
time t=1/4n,, sin2a7njt=1, and sin2a7nt=—1. Conse-
quently the value of V, becomes at this instant

to if Ny No

2aCy Ng ; yy
This may be expressed in terms of the ratio L,C,/L,(;.
Calling this m, and writing a for —— we have
nyng=2ay/m(1— I’),
ny —m, = V/ 2a(m-+ 1)—day/m(1— &*) F-
igh L—Bm*LyCray/m(1— FF

Hence V,= a
: : IC, / 2a(m +1) —4ar/m(1—k?)

Inserting the value of a and condition (2), i.e. m= BS :
we find that this expression vanishes. oe

It appears, therefore, that if conditions (1) and (2) are
satisfied, the amplitudes of the two potential waves in the
primary circuit are equal, and that at the instant in question
the potentials are at their maxima, but in opposite phase.
The primary condenser is therefore uncharged, while at the
same moment the secondary potential is at its maximum f,
the two waves in the secondary circuit being then at their
maxima and in the same phase. Further, since dV,/dt=0,
and dV,/dt=0, there is no current in either circuit. The
whole of the energy therefore exists at this moment in the

* E. Dibbern, Ann. de Physik, xl. 4, p. 988 (1918); Inaug. Diss.,
Kiel.
7 G.-¢. p. o8¥. eG, OC L.

Lijective Adjustment of an Induction-coil, 13
electrostatic form in the secondary circuit, and the secondary
potential must in these circumstances have its greatest
possible value for a given initial energy-supply $17)’.

It was shown in the previous paper to which reference
has been made * that the value of this maximum secondary
potential, still on the assumption that the resistances are
negligible, is

ea flee fis f
a TAG Lh ua

The energy equation is therefore
OD) ae Typ
41% =4C0, Vom han F °

Lig

The factor Ly,/L.; arises from the manner in which the
capacity O, of the secondary circuit is defined. It is the
charge on one-half of the secondary coil (and the bodies
connected to its terminal), divided by the difference of
potential of the terminals. The charge on this portion of
the secondary circuit is C.V2,, but as some of this charge is
at a lower potential than that of the terminals, the energy
must be less than 34C,V3,,. The correcting factor is Ly./Ls).

There are many values of & and L,C,/L.C. which satisfy
the above conditions. The first four, probably the only ones
having any practical importance, are given in Table I.

TABLE I,
No/ 1. k pe ey Ae
3 756 49
7 914 164 |
11 950 098
15 ‘965 070

If we define the efficiency of an induction-coil as the ratio
of the maximum electrostatic energy in the secondary to the
electrokinetic energy in the primary circuit just before the
interruption of the current (2. e., Lyi”), then the efticiency

4 > Prof. EH. Taylor Jones on the most

is unity in each of the adjustments specified in Table I. if the
resistances of the circuits are negligible. But it is easy
to see that of these adjustments the first (2. ¢, nf/m=3,
L,C,="429 L,C,) is the one which gives the longest spark
for a given primary current. For let us suppose that an
induction-coil is so constructed, and the primary condenser
so chosen, that 4=°914 and L,C,='164L,C,. This is the
second adjustment of Table I., and it allows the whole of
the primary energy }L,%? to be converted at a certain
moment after the interruption into electrostatic energy in
the secondary circuit. Now by inserting coils of suitable
self-inductance, say J, in the primary circuit (coils which do
not act inductively on the induction-coil) we can reduce the
coupling coefficient to°756 ; and if the primary condenser is
chosen so that (1h, +/,)C,; =°429 L.C., we then have the first
adjustment, with n/n;=3. The system has again unit
efficiency, but the maximum electrostatic energy in the
secondary is now $(L,+/,)i?, and is greater than the
former value in the ratio of L,+, to L,. The capacity C,
of the secondary coil being unaltered, the secondary
potential is increased in the ratio VL,+/, to “L,. The
3/1 adjustment is thus more effective (though not more
efficient) than the others because it allows a greater quantity
of initial energy to be converted into electrostatic energy
in the secondary circuit, in which therefore the charge and
potential developed are greater than in the other cases.

There are other ways in which the coupling coefficient
could be reduced (e. g., from *914 to *756), but none of
them are so effective as the plan of introducing external
inductance into the primary circuit. For instance*,
external inductance may be introduced in the secondary
circuit. This increases C2, and to some extent L,., without
altering L, and L,;, and therefore, by (36), lowers the
secondary potential. Again, k may be reduced by re-
moving a part of the iron core. This diminishes Ly,
without causing any increase in Lp;/L., or any change
in C,, and therefore, by (3b), reduces V2,. The reduction
of the coupling may also be effected by withdrawing the
primary coil with the core to a suitable distance along
the axis of the secondary. This process reduces Lys, L,,,
and L,, without changing L,, and should cause no change
in the secondary potential except such as may be due toa
slight diminution of C, or of the ratio Ly;/Lyp.

It should be clearly understood that these reductions
of the coupling are supposed to bring the system from

* In each of these supposed modifications the primary capacity is to
be adjusted so that L,C;=(1—4°)L.C,.

Lffective Adjustment of an Induction-coil, J
one state of unit efficiency to another, e. g. from the
second to the first adjustment of Table I., for it is only
in these states that equations (3a), (36) hold. If, on the
other hand, & lies between two of the unit-efficiency values,
the spark-length for a given current may be increased
by withdrawing the primary coil and core to a suitable
distance along the axis of the secondary. For example,
the coil with which I have experimented has a coupling
coefficient of °876, and the longest spark occurs when the
primary capacity is about °06 microfarad*. A marked
increase is, however, produced in the spark-length by with-
drawing the primary coil and core to a distance of 36°5 cm.
from its nearly symmetrical position in the secondary, and
increasing the capacity to ‘15 microfarad.

We may conclude that any induction-coil in which the
coupling is greater than ‘756 can be improved in spark-
length by connecting external series inductance in the
primary circuit so as to reduce the coupling to this value.
An induction-coil should not be constructed so that the
coupling is less than °756, for there is no convenient way
of increasing the coupling to this value. Nor should the
coupling coefficient of the primary and secondary coz/s have
precisely this value, though if no external coils are used
thisis the best arrangement. Better effects are obtained by
coupling the coils more closely than this, and adding series
inductance to the primary circuit so as to reduce & to this
value. Tie ratio l,/C, is thereby increased, and it is upon
this quantity, when the adjustments (1) and (2) are effected,
that the maximum secondary potential for a given primary
current chiefly depends.

In fig. 1 (A) are shown curves representing the twe
potential waves in the primary circuit of an induction-coil
in which the above adjustment has been effected, and in
which the damping of the oscillations is negligible. The
amplitude of each wave is taken as 5000 volts, and the
periods are ‘0024 sec. and:0008 sec. These values are chosen
so as to correspond approximately (for ij=10 amperes) with
those found in an actual case, as described below. Fig. 1 (B)
shows the result of superposing the two curves of (A). The
equation to curve (B), the ordinate of which represents the
potential of the primary condenser ¢ seconds after the inter-
ruption, is

V,=—5000 sin 150,000 ¢— 5000 sin 450,000 ¢,
the angles being given in degrees.

* It was explained in the previous paper (/. ¢. pp. 584, 585) how the
most effective primary capacity in any such case can be calculated.

& b>
an ©
ra ol
Ze
a 2 fe .
Beda ozens 3any amas) tebe ey 3 i e ee eateess a
Fae 7 CE aoe oe
eee rd 8.50 ; oe a
See Bae a a
Su Ser HH

eo a8

ett a

10n.

potential waves in the

Co
=

Prof. E. Taylor Jones on the most

2 ce
ae

i
-aad asunssGnan

Primary potential.

The curves in fig. 1 cover one half-
wave ; their continuation in the second h

a repetition with the ordinates reversed

a)
oy
AS
=
; 2
: d
- He H a a ae a rae rm
Fy SoHE HGH a os
scree fatET ER 8
fa)
oe ae Se,
HHeereliE a ea ial |

ae Hee ltl era ©

ae a %

™m

-§

alee
Eilat ee 9

en as 600,000 and 200,000 volts

.2(B) the effect of their superposit

oO
5

°
3

ircui

‘, 2(A) shows the correspondin
fi

ig
secondary ¢

The amplitudes here are ta

Effective Adjustment of an Induction-coil. f

respectively. It will be seen that the two positive maxima
coincide at °0006 sec. after the interruption, giving the
maximum potential of 800,000 volts at the secondary
terminals. At this moment the potential of the primary
condenser is zero. Fig. 2(C) shows (for comparison with
the oscillograph curves described below) the curve obtained
by squaring the ordinates of fig. 2 (B).

The above theoretical conclusions apply strictly only to
the case of an ideal induction-coil, in which the resistances
of the circuits and other causes of damping of the oscilla-
tions are negligible. Some applications of the theory were,
however, given, in the former paper referred to *, in which
it was shown that the results of the theory agree closely
with experiment in so far as they concern the conditions
under which the greatest spark-length is given by an
induction-coil. The value of the secondary ‘potential is of
course greatly modified by the damping.

Another experiment will now be described, from the
results of which some further information may be gathered
as to the working of an induction-coil when in its most
effective adjustment.

The coil experimented upon was the one employed in the
previous experiments, viz.: an 18-inch coil in which the
coefficient of coupling of the primary and secondary coils
(the secondary terminals being connected to the oscillo-
graph and to a variable spark-gap) is °876. The primary
circuit was fed by a number of storage-cells, and included
an amperemeter, a rheostat, and a slow interruptor which
broke the current about once per second. The condenser
was connected directly across the interruptor.

The experiment consisted in increasing the self-inductance
of the primary circuit by the addition of air-core coils, and
varying the capacity of the condenser, until the longest
spark was produced by a given primary current. The curve
of secondary potential was then photographed, and the con-
stants of the circuits determined. The best effect was found
to be obtained with four extra coils in the primary circuit,
and with a condenser of ‘2 microfarad capacity. It was not
very easy to decide with great exactness upon the best self-
inductance and capacity, owing to the fact that the spark.
length varies slowly with these quantities in the neighbour-
hood of the maximum. A difference of °025 mfd. either
way in C, caused no appreciable change in the spark-length.
From the practical point of view this slow variation near the
maximum is in itself no disadvantage, since very exact

* TI. ce, pp. 582-586.

8 Prof. E. Taylor Jones on the most

adjustment is not necessary in order to produce a spark-
length practically equal to the maximum.

The constants of the circuits were measured by methods
which were fully described in the previous paper, and found
to be as follows :— |

f
— ° 6 a
Lo 19°57:. 108 c.6.s.,
pes SO nar
reCS a ; i
ke = *7A8,

Le; = 20°4 henries,

LAY

aes 680,

Re — Q—9

Mia 825.

| Re the resistance of the primary circuit for steady
currents, was 8 ohms. |
Hence |
6, = 28, Cy = 1-738. lO Wee:
6.= 4R,C, = 4°709 .10~°
Go = IRC, === 8 ° L070

_ The quantities R,/L,, R,/L,, were determined from the
decay factors of the circuits when oscillating separately.
The effective resistances Ry, Rj, are much greater than the
steady-current values ; they depend also upon losses due io
leakage, hysteresis, and absorption. A considerable part of
Ry is. probably due to leakage through the electrometer. The
decay factors are not constant; the values of R,/L,, R./L,
given above prevail during oscillations of fairly large ampli-
tude in the primary and secondary circuits respectively.
The calculated frequencies are :

ny = A393,

Nig == L208,
giving ts) iy ==) 2) Oe
Also L,C,/LeC, = 448,
while 1—k? = *440.

The differences between these values and those of the
ideal case (n/n; =3, k='756, L,C,/L,C,=°429) are within
the limits of experimental error, and may be due to the
above-mentioned difficulty of adjusting L; and ©, accurately

Effective Adjustment of an Induction-coil.1 9

to give the best effect. If L, and C, were reduced, by
amounts which would cause no appreciable change in the
spark-length, the differences would disappear.

From the above data the secondary potential can now be
calculated *, and the result, expressed in volts for ip=10
amperes, 1S

V.=601300 . e-7! sin (149000 ¢— 1°95)°
— 206900 . e~ 4% sin (435000¢— 5°7)°.

From this expression values of V,., the difference of
potential at the secondary terminals ¢ seconds after the
interruption of the primary current, were calculated for

Fig. 3.

Bo tee Be

etd

SoA AseGE ae

Secondary potential.

various values of ¢ up to °003 second, a time covering rather

more than one whole period of the longer wave. The results
are exhibited in fig. 3, in which (A) shows the two oscillations

* The general expression for V, is given in the previous paper, Z. ¢.
pp. 565, 674. The statement in the footnote on p. 574 requires correction,
When the condenser is connected directly across the interruptor, so that
the battery (e.m.f. 2) is included in the primary oscillating circuit, we
have initially V,=0, but finally V,=—E, The results as given on
pp. 974, 575, apply to this case provided V, is taken as H+-the potential-
difference of the primary condenser.

10 Prof. EK. Taylor Jones on the most

of the secondary circuit separately, and (B) the result
of their superposition. It will be seen that two positive
maxima in the oscillations agree at about t=‘0096 sec.,
giving rise to the maximum secondary potential (596900
volts) at this time. Fig. 4 shows the square of the secondary

Fig. 4.
4

H fe

Square of secondary potential.

potential plotted against the time. This curve shows clearly
the peaked maxima and the flattened zeroes which are
characteristic of the 3/1 ratio. |

In Plate I. figs. 6 and 7 are shown photographic
records of the secondary potential wave obtained with the
oscillograph. In these curves the ordinates are proportional
to the square of the secondary potential, owing to the idio-
static connexion of the instrument, and these curves are
therefore directly comparable with fig. 4. The currents
interrupted when these photographs were taken were 1-5
and 2:0 amperes respectively *. It will be seen that the
greatest ordinates of the curves are proportional to the
squares of the currents. In period f, rate of decay, and form
the calculated curve of fig. 4 agrees well with the photo-
graphs. The photographic curves are perhaps rather more

* The greatest spark-length for 2 amperes, between spark-balls 2 cm.
in diameter, was 18:2 cm. The external inductance being removed, and
the primary coil being still in its symmetrical position within the
secondary, the greatest spark-length at 2 amperes (C,=0'6 mfd.) was
12°38 cm. No sparks passed when the photographs shown were taken.

+ The time curve shown on the photographs has a period of 1/768 sec.

Effective Adjustment of an Induction-Coil. 11

peaked at the summits, and more flattened at the zeroes,
than the calculated curve, which indicates that the caleu-
lated damping factor of the shorter wave is rather too great
in comparison with that of the slower oscillation. This may
possibly be due to the existence of an appreciable difference
(arising from various causes) between the values of the
effective resistances of the circuits when oscillating sepa-
rately, and their values when the circuits are oscillating as a
coupled system.

Turning now to the primary circuit, Dibbern’s formula
allows the potential wave in the primary condenser to be
calculated *. Using the values given above for the con-
stanis of the circuits, and taking 2, the primary current
interrupted, as 10 amperes, Dibbern’s expression becomes in.
the present case

Vit = —5020 e~ 788 sin (1490008 + 8-02)°
— 4940 e—1440f sin (4350007—9°07)°, 2. (4)

where V, is the primary potential in volts.

The amplitudes of the two oscillations in the primary
circuit are thus nearly equal, as required by condition (2).

The two oscillations represented by (4) are shown in
fio. 5 (A), the result of their superposition in fig. 5 (B). It
will be seen that the negative potentialt of the primary con-
denser reaches a maximum of 6800 volts at about ‘00025 sec.,
and a minimum of 2250 volts at about :00065 sec. after the
interruption. Thus, at the moment at which the secondary
potential reaches its greatest value the primary condenser,
instead of being uncharged as would be the case in an ideal
induction-coil, is still charged to about 2250 volts, and this
is due almost entirely to the difference in the damping factors
of the two oscillations.

The effective resistances of the circuits therefore act in
two ways in reducing the efficiency of the arrangement.
First, they give rise to dissipation of energy and consequent
decay of the amplitudes of both oscillations. Second, ow ing
to the difference between the damping factors of the two

oscillations, there is some energy stored in the primary con-
denser at the moment when the secondary potential is at its
maximum.

* 'The existence of the two oscillations in the primary circuit after the
interruption is well shown by a current oscillogram taken by Wertheim-
Salomonson, Physik. Zeitschr. xi. p. 589, fig. L (1910).
+ As explained in the footnote on p. 9 abov e, V, here means E+ the
potential of the condenser. E is here 80 volts.
t The potential of the primary condenser is taken negative when it
opposes the battery E.M.F.

596900

ible, and if the system

, 18

10 amperes

O
(=)

to
,

ily calculated in the present case. The
Ta

y potential, for

If the resistances were neg]

Prof. E. Taylor Jones on the most

lency is eas

The effic
maximum secondar

a2
volts.

Pre of i Hn

tees ie

Tate qe
ee
i Eat
i oe i ae

ee eae sadiiiel
HEHE ETRE EE ee POT e ue ee ceed
ae ne L Lo ae ABET
Hite [Efeeetl ad mee fe fay GE ee Cae er et tT

PERSE lL fis i i LEE sits ini aa i
Hes ae te ee a see i oe
Ere EE
ear
ae Bie fal ALE o i
Sie eafdeeet fit eee ie + :
Ee et ea eta ELE ja Hi
ate Beara ea
a
HH HH HEE Le teerrttT seats gecessates
HRA HHE eet SL
ET a

Pees

is HH
cea ie
H cna ort :

id eee Ha and

Urry)

Lo
5 are

Os
ap)
If we knew the

Io’, 2. €. since

imum electro-

— ae
LO “en

9
L,
The max

2
les rather over

5969
1987
- 2250?
ed in the primary condenser,

ystem is

les.
secondary circuit is therefore *559 x 12°75,

i) jou

known.

efficiency adjustment, the
Ij. the capacity could be at once

a
ent of maximum potential.

available to provide for the
secondary is (.V». and may be

9 18S

y then stored in the prim

) jou
uaa
‘inal J

co)

are

i
le is stor

acity

les
scharge at the mom
ge then on the

ed when the ea

potential would be, by (8a), 798700

Primary potential.

ally in the s

The energ
jou

jou
7

in the first unit-
ey origin

‘atio Lo

?
a

rilf a

oe
"255 henry, 12°7

71 joules.
ser (capacity -2 mfd

Consequently the efficiency is
9° joule.

Consequently of the or
ated, h

The. char

The ener
ealeulat

L, is about
issip

and the remainin g

static energy in the
secondary di

value of the

maximum secondary
volts.

or about

conden

about °

were exactly

‘Effective Adjustment of an Induction-coil. 13;

determined from the known values of 2”, L,, Ly, and ‘Sil Oe
since .

ta, ti
fig ei?)
With regard to the ratio Liy;/Ly., this differs from unity

C [2 ° L,C,. . F) a . (5)

because the current in the secondary coil during the oscilla-.

tions is not uniformly distributed, but is greatest at the

central winding and nearly zero at the ends. [If all the

windings of the secondary had equal inductive effects on
the primary, when reckoned per unit current, and if the

current in a turn of the secondary at a distance z from the.

| ee Tz :
central winding were proportional to cos—~, where h is.

h
the length of the secondary coil, it is easily seen that Lo,/Li»

would be equal to 7/2. In the actual case, however, the in-

ductive effect of the secondary windings (per unit current)
diminishes from the centre towards each end. This was

tested by ballistic galvanometer experiments in which the

mutual inductance of the primary and a single turn of wire,

wound on the secondary (or primary) in various positions,.

was compared with its value for the central position. From
the results of these measurements it was found that this
mutual inductance could be represented approximately by
the expression a—bz?—cz'*. Consequently Li, is propor-
tional to

+h/2 é
(a—bz? —cz’) cos = dz,
—h/2 :
while LL, is proportional to
+h/2
(a—bz*—cz"*)dz,
=/2

since the current in the primary coil is uniformly distributed.
The value of L;/Ly. 1s thereby reduced from /2, and
becomes in the present case *95 7/2.

Another correction is necessary if, as in the present
experiments, the secondary terminals are connected to a
capacity which is not negligible in comparison with that of
the coil. In this case the secondary current is not quite
zero at the ends of the coil, but should be represented as pro-

portional to COST; where h’ is greater than hk. The value
L

* The dimensions of the coils are :—primary, length 90 cm., mean
diameter 6°9 cm.; secondary, length (between the terminals) 51 cm.,
diameter (outside) 24 cm.

YD ea se ae eee ena on Ee

- eS .-

14 Most Lffective Adjustment of an Induction-coil.

of h' may be estimated if we know the ratio of the externa!
capacity C, to the total capacity ©, If C. is smali in
comparison with C, the approximate value of the latter
(obtained from equation (5) by neglecting the present

correction) may be used here. If the current in the secon-
Ate Tz ?
dary windings varies as cos yr the charge per unit length
e e e TZ e 1 Y
will be proportional to sin are Hence the ratio of C to C,
/} ~
is equal to the ratio of

wh’ a ae eed me
| sin Wn dz to ( sin yr
vJh2 e 0
C, awh
dt. €. C = COs op! °

This determines h’/h, and we then have

+h/2
lite 1 te TZ
ioe a Fin
tls ji) —h/2 h

In the present experiments C, is the capacity of the
electrometer and the spark-gap terminals, and this is about
one-sixth of the total secondary capacity C3, the value of

which is already known approximately. Hence cose = -
! U
and ~ = 1°12, from which 2 =1°10 = The effect of this
2

1
correction is therefore further to reduce L,,/L,, by about
i0 per cent.
Taking both corrections into account we haye approximately

Lig;/Liys = "8507/2 = IS;
from which by (5)
C,='000052 microfarad.

The charge of the secondary circuit at the moment of
maximum potential is therefore 5:2 x 5-969 x 10‘ c.a.s., or
31.10-* coulomb. I£ the whole of this charge escaped from
the terminals, in the form of a spark or other discharge, the
discharge current would be, at n interruptions per second,
nC.V>. Further experiments are, however, required in order
to decide whether this complete discharge takes place, or
whether some of the electricity does not return through the
secondary coil and continue to oscillate in it.

On Motion in a Periodic Field of Force. 1

The conditions for maximum potential are the same when
a rapid interruptor is employed as for the slow break used in
the above experiments. They also hold whether the primary
current is supplied by a small storage-battery, a 100-volt
battery, or the 200-volt mains. Plate I. fig. 8 shows the
course of the secondary potential at two successive ‘ breaks ”
effected by a motor mercury interruptor. ‘The effect of the
small potential at the “make” ig also noticeable. On
this occasion the 100-volt battery was used to supply the
primary current, the total (steady-current) resistance in the
primary circuit being 11 ohms. The greatest spark-length
was 18°4 cm., and the mean primary current, as indicated
by an amperemeter in the primary circuit, was ‘3 ampere.
The current immediately before interruption would, however,
be over 2 amperes.

Bangor, November 1914.

II. On Motion in a Periodic Field of Force.
By Oiy EvAMeaN set AL*

[Plate IJ.]

Vibrations maintained by a Periodic Field of Force.

‘| experimental study of the motion of a dynamical

system in a periodic field of force leads to results of
quite exceptional interest. One aspect of the problem, i. e.
the oscillatory motion of the system about a position of equi-
librium in the field, has some affinities to the case of vibra-
tions maintained by a variable spring which I have dealt
with in my previously published work, but the two classes
of investigations lead to results which differ from one another,
yet are related in a mostremarkable way. By experimenting
on stretched strings subjected to a variable tension, I showed
that a normal variation of spring will enable the oscillations
of the system to be maintained, when the frequency of these
oscillations is sufficiently nearly equal to 3 of, or 2 times, or
8 times, or 4 times, &c. the frequency of variation of the
spring, these ratios forming an ascending series f. By ex-
periments on the vibrations of a body about a position of
equilibrium in a periodic field of force (to be described
below), I have shown that the frequency of the oscillations
maintained may be equal to, or half of, or one-third, or one-
fourth, &c. of the frequency of the field; in other words, it

* Communicated by the Author.

mi ae “om Mag. Oct. 1912, “The Maintenance of Forced Oscillations
of a New Type.’

16 Mr. C. V. Raman on Motion tn a

may be any one of a descending series of sub-multiples of
the frequency of the field. It appears, in fact, that we have
here an entirely new class of resonance-vibrations. It will
be noticed that if the two series referred to above are both
written in the same order of descending magnitudes of
frequency, thus,

G99). As OB 2 ae
22 129 Dorp? 27
Me: AC ea oll
12 29 3° 49°5> 6?

the last two terms of the first series, and the first two of the
second series coincide, and these two are to some extent
typical of the rest. For, as I have shown in a previous
publication *, the 1st, the 3rd, the 5th, and the odd types
generally in the first series bear a family resemblance to
each other, giving symmetrical vibration curves. The 2nd,
the 4th, and the other even types similarly resemble each
other in giving markedly asymmetrical vibration curves.
Since the first term in the ascending series is the 2nd in the
descending series, we may expect that the 2nd, 4th, 6th, &c.
in the latter would give analogous types of motion, and that
similarly the Ist, 3rd, 5th, &e. would show resemblances
amongst each other. These points will be dealt with more
fully as we proceed. |

The vibrations studied which form the subject of this
section were those of the armature-wheel of a synchronous
motor of the attracted-iron type, about a position of equi-
librium in the magnetic field produced by an intermittent
current circulating in the coils of an electromagnet. The
phonic wheel or synchronous motor devised by La Cour and
Lord Rayleigh is, as is well known, of great service in
acoustical investigations. In my own work on vibrations
and their maintenance, it has been of considerable assistance..
Apart, however, from the various uses of the instrument in
different branches of Physics and in Applied Electricity, it
possesses much intrinsic interest of its own as an excellent
illustration of the dynamics of a system moving in a periodic
field of force, and the present paper deals almost entirely
with experiments carried out by its aid and with its applica-
tions to the study of vibrations.

The instrument used by me was supplied by Messrs. Pye
& Co., of Cambridge, and has given entire satisfaction. The
motor consists of a wheel of soft iron mounted on an axis
with ball-bearings between the two poles of an electromagnet.
placed diameirically with respect to it. The wheel has thirty

* See Physical Review, Dec. 1912. ‘

Periodic Field of Force. 17

teeth, and when a direct current is passed through the electro-
magnet, sets itself rigidly at rest with a pair of teeth at the
ends of a diameter opposite the two poles of the electro-
magnet. The equilibrium under such conditions is of course
thoroughly stable, and, in fact, the wheel possesses a fairly
high frequency of free angular oscillation for displacements
from this position of rest, and any motion set up by such
displacement rapidly dies out, apparently on account of
Foucault currents induced in the iron by the motion. This,
in general, is also true when an intermittent current supplied
by a fork-interrupter is used to excite the electromagnet,
except however in certain cases, when it is observed that the
equilibrium becomes unstable of its own accord and the
wheel settles down into a state of steady vigorous vibration
about the line of equilibrium: or that an oscillation of
sufficient amplitude once started maintains itself for an
indefinitely long period.

An optical method can be conveniently used to study the
frequency and the phase of the oscillations of the armature-
wheel maintained in the manner described above. A narrow
pencil of light is used, which first suffers reflexion at the
surface of a small mirror attached normally to one of the prongs
of the fork-interrupter furnishing the intermittent current,
and then falls upon a second similar mirror attached to the
axle of the armature-wheel parallel to its axis of rotation. The
apparatus is so arranged that the angular deflexions produced
by the oscillations of the fork and the wheel are at right
angles to each other, and the pencil of light which falls upon
a distant screen, or which is focussed on the ground-glass of
the photographic camera, is seen to describe a Lissajous
figure from which the frequency, and the phase-relations
petween the oscillations of the fork-interrupter and of the
armature-wheel, can be readily ascertained. It is then
observed that the period of the vibration of the armature-
wheel is equal to, or twice, or thrice, or four times, &c. the
period of the fork: in other words, the frequency is equal to
or 4 of or 4 or + or } or ¢ that of the fork.

In making the experiments, the motor-wheel is relieved of
the large stroboscopic disk that is usually mounted upon it,
and in working down the series, the adjustment of frequency
is secured by suitably loading the wheel. The fine adjust-
ment for resonance is effected by altering the current passing
through the interrupter with the aid of a rheostat, and if
necessary by regulating the contact-maker on the fork. Any
oscillation of the wheel, when started, dies away except in
the cases referred to above ; in other words, no frequencies

Phil. Mag. 8. 6. Vol. 29. No. 169. Jan. 1915. C

, OO —_—

- »

Pe £8 2 s- ae ee.

18 Mr. C. V. Raman on Motion in a

intermediate between those of the series are maintained.
In obtaining the 1st case, in which the oscillations of the
wheel and the fork are in unison, it is generally found
necessary to increase the “ spring” of the wheel by passing
a steady direct current through the electromagnet of the
motor from a cell connected in parallel, in addition to
the intermittent current flowing in the same direction from
the interrupter circuit.

Tt is found that the Lissajous figures for the Ist, 3rd, and
oth cases are distinctly asymmetrical in character, the 3rd
being markedly so. The 2nd, 4th, and 6th types are quite
symmetrical. Jhis, it will be remembered, was what was
anticipated above, and in fact the Ist, 3rd, and Sth types
differ rather markedly in their behaviour from the 2nd, 4th,
and 6th types. These latter are maintained with the greatest
ease, while the former, particularly the 5th, are not altogether
so readily maintained. In fact it is found advantageous, in
order to maintain the 5th type steadily, to load the wheel
somewhat unsymmetrically and to put it a little out of level,
in order to allow the oscillations to take place about an axis
slightly displaced from the line joining the poles of the
electromagnet.

It is noticed also that the lower frequencies of vibration
have much larger amplitudes. This, | would attribute princi-
pally to the greatly reduced damping at the lower frequencies
owing to the slower motion, the larger masses, and the weaker
magnetic fields employed.

We are now in a position to consider the mathematical
theory of this class of maintained vibrations. To test the
correctness of my theoretical work, I have prepared a series
of photographs of the simultaneous vibration-curves of the
fork and of the armature-wheel, which are reproduced as
figs. 1 to 6, Pl. Il. These curves were obtained by the
usual method of recording the vibrations optically on a
moving photographic plate, it being so arranged that the
directions of movement of the two representative spots of
light on the plate«lie in the same straight line. The upper
curve in each case shows the maintained vibration of the
armature-wheel. The lower represents that of the fork-
interrupter. The frequency ve the former, it will be seen,
is + or $ or 4 or ¢ or 2 or 4xthat of the latter. The
precise features of the vibration-cunve noticed in eachi eee
will be referred to below, in connexion with the mathematical
discussion.

The equation of motion of a system with one degree of
freedom moving in a periodic field of force, and subject also

Periodic Field of Force. 19

to the usually assumed type of viscous resistance, may be
written in the following form,

OU a COUN =0) 52 i. Cd)

where F(U) gives the distribution of the field, /(¢) its
variability with respect to time, and 2a is a constant. If we
are dealing with oscillations about a position that would be
one of stable equilibrium if the field were constant, F(U)
may as an approximation be put equal to U. We then have

OWE on = On) 0

In the experiments described above, the periodicity of f(t)
is the same as that of the intermittence of the exciting
current. If an alternating current had been used, the fre-
quency of f(¢) would have been double that of the alternations.
In any case we may write

af (t)=a, sin nt + dg sin 2nt + a3 sin 38nt+ Ke.

+ by) +0, cos nt+b,cos 2nt+53 cos dnt+ Ke. . (3)

Since U is shown to be periodic by experiment, we may write
U=A, sin pt + Asin 2pt+ Az; sin 3pt+ Ke.

+ B+ B, cos pt+ Bz cos 2pt+ B; cos 3pt+ Ke. (4)

As a typical example of the even types of maintenance, we
may take the cases in which n=4p. We have

af (t) =a, sin 4pt + a, sin 8pt+a3 sin 12pt+ ec.
+ bo +, cos 4pt + be cos 8pt +b; cos 12pt+ ke. (5)

In this case, and also in the case of the second, sixth, and in
fact in all the even types of maintenance, we find that the
quantities A,, A,, A,, &c., and Bo, By, By, &., do not enter
into the equations containing A, and B,. We therefore
write them all equal to zero. The significance of this is that
with the even types of vibration maintained by a periodic
field of force, the even harmonics are all absent from the main-
tained motion. This result is fuily verified by a reference to
the vibration-curves of the 2nd, 4th, and 6th types shown in
figs. 2, 4, and 6, Pl. IJ. It will be seen that the vibratory
motion of the armature-wheel has that type of symmetry so
familiar in alternating current curves, in which all the even
harmonies are absent. In other words, the image of one-
halt of the curve above the zero axis, as seen by reflexion in
a mirror placed parallel to this axis, is exactly similar to the
other half below it.
C 2

20 Mr. C. V. Raman on Motion in a

Substituting now the odd terms alone left on the right-
hand side of (4), for U in equation (2), we have the following
series of equations :—

—(by— p*)A, + kp B, = —b,A; + 0,B,+0,A;— a,B;—&e.+ &e.

— (by— p”) By — kp A, =a, A; + 0,B;+a,A; + 0,B;+ &e. + Ke.
—(byp— 9p?) A34+ 3kpB; = —b,A,+ 4, B,—b.A; + a2B; + &e. — Ke.
— (b) — 9p?) Bs —3kpA3=a,A,+6,B, +a,A;+6,B; + &e. + Ke. (6)

and so on.

Hvidently, the possibility of this being a consistent set of
convergent equations depends upon the suitability of the
values assigned to the constants hk, p, bo, ay, b,, &e.

It is not possible here to enter into a complete discussion
of the solution of these equations. One point is, however,
noteworthy. From the first two of the set of equations given
above, it will be seen that such of the harmonics in the steady
motion of the system as are present serve as the vehicles for
the supply of the energy requisite tor the maintenance of the
fundamental part of the motion. Paradoxically enough, the
frequency of none of these harmonics is the same as that of
the field.

We now proceed to consider the odd types of vibration,
2. €. the Ist, the 3rd, &c. Taking the 3rd as a typical case,
we put n=3p and get

af (t) =a, sin 3pt+ az sin 6pt +a; sin 9pt+ Ke.
+ bo +b; cos 3pt + by cos 6pt +b; cos Ipt+ Ke. . (7)

Substituting (4) and (7) in equation (2) and equating the
coefficients of sine and cosine terms of various periodicities
to zero, we find that the quantities As, Ag, Ay, &c. and Bo,
B3, Bg, By, &e., do not enter into the equations containing
A,and B,. We therefore write them all equal to zero. The
significance of this is that the maintained motion contains no
harmonies the frequency of which is the same as, or any
multiple of the frequency of the periodic field of force. This
remarkable result is verified by a reference to fig. 3, Pl. II.
from which it is seen, that the vibration curve is roughly
similar to that of the motion of a trisection point of a string
bowed near the end, the 3rd component, the 6th, the 9th, &e.,
being absent at the point of observation.

Périodic field of Force. 21

We then obtain the following set of relations by substi-
tution :

— (bb —p?) A, + kp By = —b, Ag+ 4,B,+0,A,—a,B,— &e.

— (bo —p?) By —kp Ay =a, Ag +0, Bo+ aAy+b,By+ &e.

— (b)— 4p”) A, + 2kpB,= —b, A; + 4,B,4+ 6,A;—a,B;+ Ke.

— (bo —4p?) Bo— 2hpAg=a,A, +0,;B,+a,A;+0,B;+ &e. (8)

and so on.

It must be remembered that these relations are all only
approximate, as (U) in general contains powers of U higher
than the first which we have neglected, and which no doubt
must be taken into account in framing a more complete
theory. The general remarks made above with reference to
equation (6) apply here also.

The exact character of the vibratory motion maintained by
the periodic field of force in any case, depends upon the form
of the functions F(U) and f(t) which determine respectively
the disposition of the field and its variability with respect to
time. One very simple and important form of /(¢) is that
in which the field is of an impulsive character, in other
words is of great strength for a very short interval of time
comprised in its period of variation, and during the rest of
the period is zero or nearly zero. Such a type of variation
is not merely a mathematical possibility. In actual experi-
ment, when a fork-interrupter is used to render the current
passing through the electromagnet intermittent, the magneti-
zation of the latter subsists only during the small fraction of
the period during which the current flows and at other times
is practically zero. When the current is flowing the accele-
ration is considerable: at other times, the acceleration is
nearly zero, and the velocity practically constant. These
features are distinctly shown in all the vibration-curves
(except those of the first type) reproduced in Pl. II., the
sudden bends in the curves corresponding roughly to the
extreme outward swings of the fork, 7. e. to the instants when
the magnetizing current was a maximum. It seems possible
that a simpler mathematical treatment than that given above
might be sufficient to discuss the phenomena of the main-
tenance of vibrations by a periodic field of force when the
periodicity of the field is of the “impulsive” type ; in other
words, when the dynamical system is subject to periodic
impulsive “ springs,”’ one, two, three or more of which occur
at regular intervals during each complete period of the
vibration of the system.

These experiments on vibrations maintained by a periodic

A 4-

22 Mr. C. V. Raman on Motion in a

‘field of force are very well suited for lecture demonstration,
as the hissajous figures obtained by the method described
above can be projected on the screen on a large scale, and
form a most convincing demonstration of the tact that the
frequency of the maintained motion is an exact sub-multiple

of the frequency of the exciting current.

On Synchronous Rotation under Simple Excitation.

It is well known that with an intermittent current passing
through its electromagnet, the synchronous motor can main-
tain itself in “ uniform” rotation, when for every period of
the current, one tooth in the armature-wheel passes each
pole of the sheet oui In other words, the number of
teeth passing per second is the same as the frequency of the
intermittent current. From a dynamical point of view it is
of interest, therefore, to investigate whether the motor could
run itself successfully at any speeds other than the “syn-
chronous”’ speed. Some preliminary trials with the motor
unassisted by any independent driving proved very en-
couraging. The phonic wheel I have is mounted on ball-
bearings, “and rans very lightly when the large stroboscopic
disk usually kept fixed upon it is taken off, ad there is no
current passing through the motor. When a continuous or
intermittent current is flowing through the motor, the latter
does not, however, run very lightly, ‘being subject to very
large electromagnetic damping, apparently due to Foucault
currents in the iron. In the preliminary trials, however, I
found that, using the intermittent unidirectional current
from an interrupter-fork of frequency 60, the motor could
run successfully of itself at half the synchronous speed, 2. e.
with 30 teeth passing per second. It of course ran very
well at the usual synchronous speed, 2. e. with 60 teeth
yassing per second. By increasing the speed, it was found
that the motor could also run well of itself at double the
synchronous speed, 2. e. with 120 teeth passing per second.
Using an interrupter-fork of low frequency (23:5 per second)
the motor, it was found, could also run of itself at trvple the
synchronous speed. No certain indication was, however,
obtained of the intermediate speeds, 7. ec. 1$ and 24 times
respectively the synchronous speed.

To test these points, therefore, independent driving was
provided. This was very satisfactorily obtained by fixing a
small vertical water-wheel to the end of the axis of the
motor and directing a jet of water against it. The water-
wheel was boxed in to prevent any splashing of water on the
observer. By regulating the tap leading up to the jet, the

Periodic kield of Force. 23.

velocity of the latter could be adjusted. The speed of the
phonic wheel was ascertained by an optical method, 2. e. by
observing the rim of the wheel as seen reflected in a mirror
attached to the prong of the interrupter-fork. When the
motor “bites,” the pattern seen becomes stationary and
remains so for long intervals of time or even indefinitely,
and the speed of the wheel can be inferred at once from the
nature of the pattern seen.

It was found in these trials that the motor could “ bite ”
and run at the following speeds. (frequency of interrupter
60 per sec.)

(a) 3 the synchronous speed : SOUS pattern of rim of
moving wheel seen as a single sine-curve: wave-
iength 4 the interval between teeth. Number of
teeth passing electromagnet per second = 30.

(0) Synchronous speed : stationary pattern of rim of
moving wheel seen as a sine-curve, wave-length =
interval between teeth. Number of teeth passing
electromagnet per second = 60.

(c) 15 times the synchronous speed: stationary pattern
of rim of wheel seen as three interlacing waves.
Number of teeth passing electromagnet per second
= SNe

(d@) 2 times the synchronous speed: stationsry pattern
seen as two interlacing curves. Number of teeth
passing per second=120.

(/) 3 times the synchronous speed : stationary pattern
seen as three interlacing curves. Very s satistactory
running. Number of teeth per second=180,

(y) 4 times the synchronous speed: stationary pattern
seen as 4 interlacing curves. Number of teeth per
second = 240.

(h) 5 times the synchronous speed: stationary pattern
seen as 9 interlacing curves. Number of teeth per

second= 300.

The outstanding fact of observation is that while speeds
which are equal to the “synchronous ”’ speed or any integral
multiple of it are readily maintained, only the first two or
three members of the other series (7. ¢. having ratios 4, 14, &e.
to the synchronous speed) can be obtained, and the ‘ grip”
of the wheel by the periodic magnetic forces, 7. e. the
stability of the motion, is hardly so great as in the integral

24 Mr. C. V. Raman on Motion in a

series. This fact may be explained in the following general
manner.

We may assume, to begin with, that the independent
driving is less powerful than that required to overcome re-
sistances, so that the wheel is a little behind the correct
phases. In the case of the integral series, one or two or
more teeth pass for every intermittence of the current, the
wheel being in the same relative position, whatever this may
be, to the electromagnet, at each phase of maximum magne-
tization of the latter. This is not, however, the case with
the fractional speeds. It is only at every alternate phase of
maximum magnetization that the wheel assumes the same
position (whatever this may be) relative to the electromagnet.
At the intermediate phases, it is displaced through a distance
approximately equal to half the interval between the teeth.
Whereas with the integral series, every phase of maximum
magnetization assists the rotation, in the fractional series the
wheel is alternately assisted and retarded by the successive
phases of maximum magnetization, and it is the net effect of
assistance that we perceive, this being of course comparatively
small.

As the synchronous, half-synchronous, and double-syn-
chronous speeds can all be re eadily maintained without inde-
pendent driving, they can be very effectively exhibited as
lecture experiments by lantern projection in the following

way. The synchronous motor (which is quite small and
light when the stroboscopic disk is removed) is placed on
the horizontal stage of the lantern and the rim of the wheel
is focussed on the screen. In front of the projection prism,
where the image of the source of light is formed, is placed
the fork-interrupter with the necessary device for intermittent
illumination fitted to its prongs. When these are set into
vibration and the synchronous motor is set in rotation, the
‘“pattern ” corresponding to the maintained speed becomes
visible on the screen, and the effect of reversing the direction

of rotation can also be demonstrated.

We now proceed to discuss the mathematical theory of
the maintenance of uniform rotation in each of these cases.
The first step is obviously to show that with the assumed
velocity of rotation, the attractive forces acting on the disk
communicate sufficient energy to it to balance the loss due
to frictional forces. ‘Taking the line joining the poles as the
axis of w, the position of the wheel at any instant may be
defined by the angle @ which a diameter of the wheel passing
through a given pair of teeth makes with the axis of reference.
Tf n is the number of teeth in the wheel, the couple acting

Periodic Field of Force. 25

on the latter for any given field strength at the poles is
obviously a periodic function of n@ which vanishes when

2
a= xe and also when ge emt 2)

7d 7v
integer.
We therefore write

, Where v is any

Couple= Field strength x [ajsinn@ + asin 2nd + agsin 3nd + Kc. }
= Vield strength x f(7@) say,

where the terms a, dy, a3, &c. rapidly diminish in amplitude.
It will be seen that the cosine terms are absent. Since the
field strength is periodic, we may write the expression for
the couple acting on the wheel thus

Couple = Lf (8) [b, sin (pt +e.) +0 sin (2pt+e.)+ Ke. |
= Lf (n@) F(t), say.

The work done by the couple in any number of revolutions

= (Ly (8) JMG) ic

It is obvious that this integral after any number of eee
revolutions is zero, except in any of the following cases
when it has a finite value proportional io and increasing
with ¢; 2. e. when

nd =pt or 2pt or 3pt or 4pt and so on,
or when
2nO = pt or 2pt or 3pt or 4pt and so on,

or when
3n0= pt or 2pt and so on.

It is therefore a necessary but not, of course, always a
sufficient condition for uniform rotation to-be possible that
one or more of the above relations should be satisfied. The
first series corresponds to the synchronous speed and
multiples of the synchronous speed. ‘These have been ob-
served experimentally by me up to the fifth at least. The
second series includes the above and also the half-synechronous
speed and odd multiples of the same. These latter have also
been observed by me up to the fifth odd multiple. Since ay,
is much smaller than a,, the relative feebleness of the main-
tenance of the half-speeds observed in experiment will readily
be understood.

The third series has not so far been noticed in experiment.

26 Mr. C. V. Raman on Motion in a

It is obvious that the maintaining forces in it should be
excessively feeble compared with the first or the second.
Perhaps experiments with interrupter-forks of higher fre-
quencies and independent driving of the motor may succeed
in showing the existence of controlled rotation-speeds at
these ratios.

Combinational Rotation-speeds under Double Hacitation.

When the electromagnet of the synchronous motor is
excited simultaneously by the intermittent currents from two
separate interrupter-forks having different frequencies, main-
tenance of uniform rotation is possible not only at the various
speeds related to the synchronous speeds due to either of the
intermittent currents acting by itself, but also at speeds
related jointly to the frequencies of the two currents.

The preliminary experiments on this point were made
without the assistance of any independent driving of the
motor, and it was found at once that differential rotation of
the motor was easily maintained, the number of teeth passing
per second being equal to the difference of the frequencies
of the two interrupter-forks.

When the “ differentially ” revolving wheel was examined
by reflexion in mirrors attached to the prongs of the two
interrupter-forks, it was found that the patterns seen in
neither of them were stationary. They were found to be
moving steadily forward or backward with a definite speed,
with occasional slight to and fro oscillations superposed
thereon. This continuous rotation of the patterns seen was
obviously due to the fact that the frequencies of the forks
and their difference did not bear any simple arithmetical
ratios to each other, and it enabled a rotation-speed main-
tained by joint action to be distinguished by mere inspection
from one maintained by either of the two currents separately.
Using this optical method, and assisting the rotation of
the motor with independent driving by a water-motor, various
other combinational speeds were found to be maintained.
Of these, the most powerfully and steadily maintained was
the simple summational rotation. The summationals and
differentials of the second series, 7. e. those in which the
half-frequencies of the fork enter, were also noticed. The
rotation-speeds were determined by actual counting and a
stop-watch.

The mathematical theory of these combinational speeds is
very similar to that given for the case of excitation by one
periodic current. For the field strength in this case is also

Periodic Field of Force. OH

a periodic function of the time, and the function F(¢) which
expresses its value at any instant may be expanded in the
following form

F(j)=a > bsin [ (rp, + sp.)t+ BE],

where p,/2m7 and p./2m7 are the frequencies of the two
interrupters, and 7, s are any two positive integers. Using
the same notation as before, we find that in any complete
number of revolutions, a finite amount of energy proportional
to the time is communicated to the wheel only in any one
of the following sets of cases :

nO =(rpytspo)t,

or 200 = (rp. + sps)t,
or ONO (ip istsy)e.

and so on.
The cases actually observed in which rotation is maintained
fall within the first two of the sets given above.

Summary and Conclusions.

The vibrations of a dynamical system maintained by a
periodic field of force have been investigated experimentally
and theoretically, and itis shown that they form a new class
of resonance-vibrations, in which the frequency of the main-
tained motion is any sub-multiple of the frequency of the
exciting force. The possible speeds of synchronous rotation
of a motor of the attracted-iron type under simple and double
excitation are also investigated. The experiments and ob-
servations described in the present paper were carried out
in the Physical Laboratory of the Indian Association for the
Cultivation of Science, Calcutta, where further work on
the dynamics of vibration is now in progress. One very
interesting case which has been worked out is that of the
Combinational vibrations of a system maintained by subject-
ing it simultaneously to two simiple harmonic forces varying
its. spring. This is experimentally realized by attaching a
stretched string at its two extremities to the prongs of two
tuning-forks of different periods, the directions of motion of
which are parallel to the string. If M and N are the
frequencies of the forks, it is found that the string is set
into vigorous transverse oscillation if its tension is so
emusic that the natural frequency is nearly equal to

3(Mm+Nn), where m and n are integers. Further details

of this i investigation will be published 1 in due course.

III. On the Ideal Refractivities of Gases. By Wi.ttaM
JAcoB JONES and James Rippick Parrineron, Assistant
Lecturers in Chemistry, Manchester University *.

‘| ae formula deduced by Lorentz + and by Lorenz i

Boat pat
wr+2° ad

where w denotes the refractive index of a given substance
for a given wave-length, and d its density, expresses very
accurately the observed relations over a very wide range of
densities. When yw is nearly equal to unity, as is the case
with gases, (uw +1)/(u?+2) approximates to 2/3, and formula
(1) degenerates into the formula of Gladstone and Dale § :

7 ]
te z= (a constant) =

iam )
on 4

w—l
a

We shall designate the excess of the refractive index of a
gas over unity, i. e. w—1, its refractivity. The refractivity
is therefore the value of & for unit density.

In their calculations of refractivities from the experi-
mental results, investigators have, with the exception of
C. Cuthbertson and E. P. Metcalfe ||, and L. Stuckert 4,
failed to realise the necessity for taking into account the
deviations of the gases employed from the ideal state.

Now the relation between temperature, pressure, and
volume of a gas is at pressures less than 5 atmospheres very
accurately expressed (at least in the case of permanent gases,
and with very close approximation in the case of less perfect
gases) by the characteristic equation of D. Berthelot™*:

=k (a constant): 2 3, 21) eae

(p+ pa) OD =RT, ———
where a ae be Ue
he SS
o4 Pe 4

T., Pey and vr, being the critical constants. Jf we consider a

* Communicated by the Authors.

+ Lorentz, Wied. Ann. ix. p. 641 (1880).

1 Lorenz, ib¢d. xi. p. 70 (1880).

§ Gladstone and Dale, Phil. Trans. exlviii. p. 887 (1858).

| C. Cuthbertson and E. P. Metcalfe, Proc. Roy. Soc. A. Ixxx. p. 406
1908).
© L. Stuckert, Zettschr. fiir Elektrochem. xvi. p. 87 (1910).

** DT). Berthelot, Wém. du Bureau. internat. des poids et mesures, xiii.
{1907).

On the Ideal Refractivities of Gases. 29

gram-molecular mass, M, of a gas enclosed in a volume 2,
we have
eM

(0)

d (4)

Equations (2), (3), and (4) then enable us to express the

variation of w for a given gas for a definite wave-length,

with temperature and pressure ; for from (2) and (4) we
have

Mi:

—
p—l

and by substituting the value of v from (5) in (3) we further
have :

OPS aa ht, WALA NI)

Mk : ;
Diate TM2h2 aa —)=RT. Ba Wheat (6)
(SNe |

It the value of mw for a given wave-length for a gas is
known at a standard temperature and pressure, and, in
addition, the critical constants of the gas, then all the mag-
nitudes in (6) are known with the exception of k. The
value of k& is then determined by solution of (6). If this
value of & is then substituted in (6) we obtain a cubic equa-
tion in (w—1), the solution of which gives the value of pu at
any desired temperature and pressure. This equation cannot,
however, hold strictly up to the critical point, where the
applicability of Berthelot’s equation fails *.

It is easily seen from the theoretical considerations, on
which the Lorentz-Lorenz equation is based, that the problem
is complicated by the deviations of real gases from Avogadro’s
hypothesis. D. Berthelot has shown that these deviations
lead to the introduction of a correcting term, which, for a
given temperature and pressure, can be calculated from his
characteristic equation. This equation then assumes the
form

r 9 Lee F

Dev the ES [1+ qo: 7-71-67) |; sata RE)

where R denotes the value of the gas-constant for the par-
ticular choice of the units of p and of v; 7 and 7 denote the
ratios p/p. and T,/T respectively, where p-. and T. denote
the critical pressure and critical temperature respectively.

* Cf. Nernst, Theoretische Chenve, 7th edit. p. 241 (1918).
+ Berthelot, loc. cit. p. 52; Nernst, loc. cit.

30 Messrs. W. J. Jones and J. R. Partington on the

If we denote the expression

hse 7 .7( 1-677). .

128°
by @ we have from (7) and (5)

fe =6 kor. |

For a low pressure po, the factor @ obviously approaches
unity, and, if 4) denote the refractive index of the gas at
the pressure py), we have

fork ae | ae (10)
fo —1
Dividing (9) by (10) we obtain
2 wy—l
i = eee (11)

Doge
Tf for a second gas, for which the value of

! ! QE)
The wet 3(l—ba)
at the same standard pressure, p, and temperature, Tt, is
denoted by ¢', the values of the refractive indices are pw’ and

fy at p and py respectively, then

Digits zal aa 2
Fy aoe
Whence dividing (12) by (11) we eliminate p/yo, and obtain

eet: =f :
a On ee . .

The experimental determinations of the refractive indices
of gases, at normal temperature and pressure, have usually
been made by interference methods involving the counting
of the number of bands which cross the field of view on
subjecting the gas to small changes of pressure. Strictly
speaking “each of these determinations would require cor-
rection for the changes of pressure from the normal value.
Since, however, these pressure-changes were inconsiderable,
it is quite legitimate to assume that the values of the re-
fractive indices, as given by the several investigators, repre-
-sent the values at normal temperature and pressure. These

Ideal Refractivities of Gases. dl

values, however, still require correction for the deviation
from the ideal state shown at normal temperature and pressure
by the gas. We would nevertheless urge that in future
investigations greater attention be paid to the deviations of
the gases from the ideal state within the previously referred

to pressure-changes. It is obvious from equation (13) that
for any number of gases [1], [2], [3],... &e.

Mo—1 : ofo— 1 : su—l:.....

= ,(;4—1) 3 Polo —1) : b:(34—1) :
ee

where (;40—1), (o@—1l). ... &e. Perse the ideal re-
fractivities, corrected for the deviations of the gases from
Avogadro’s hypothesis, expressed in terms of the refrac-
tivities of the gases at normal temperature and pressure and
of the critical constants of the several gases.

In Table I. are given the mean uncorrected values of the
refractivities at normal temperature and pressure for the
several wave-lengths indicated, duly weighted after con-
sideration of the original memoirs, the values of ¢, and the
corrected ideal refractivities (f4)—1).

TABLE LI.
| lane

GC: Wave- length | at 0° C. a eal ay

as. eles @. {\Refractivity

in mm. X 10°. 760 mm. He. 1) 107.1

| (u—1) 10". Bo—1) ‘|
ivdrogen’!.: 0.004.055. 436 | 1412 1:00051| 1413
vy ds SRL RS 486 | 1406 100051) 1407
vy i) SEA RU a | 578 1391 100051 1392
«AYN (ON A a 589 1392 1:00051 1393
Che AC ae Sa 656 1387 100051 1388
ApUiie > To GORI A a e 43 2965 0:99964 2964
ay AUS tes aE le 486 2948 0-99964- 2947
vy AUIS RN CEN aD 578 2927 0.99964 2926
Me 9 Cai scree datos 589 2926 0:99964 2925
Mh Ae Sond OL, 656 2916 0:99964 2915
Oxygen hye ae Aa A 436 2747 0:99927 2745
Mr tie usin aUe As 486 2739 0:99927 2733
POM RN shel escent shia 578 2706 0:99927 2704
sa) Sie LL ERE aE 589 2711 0:99927 2709
BAR Rica ar ian 656 2697 0:99927 2695
Nitrogen age IN LUMA 436 3020 0:99972 3019
3 En et ae 486 3012 0:99972|; 3011
Se i RAR OO 578 2976 099972; 2975
Fe A OSE SN a 589 2976 0:99972|} 2975
PMMA cod 656 2982 | 0:99972 | 2981
PASE OUP auins vast ew tanie si 436 2851 | 0999382 ~8+t9
PURE Soe uit 8's 486 2838 099932 | 2836
RAMEN SaRMaH leh wc, 578 2803 099932} 2801
CS | Oa | 644 | 2796 | 0-99982| 2794

32 Messrs. W. J. Jones and J. R. Partington on the

Refractivity deat

Gas Wave-length | at 0° C. and Refractivit

: in mm. X 10°.) 760 mm. Hg. Bs : a "10.
(u—1) 107. (H)—1) 10°.

Kerygotoml «eb. ss.orsseeenn 486 4318 0:99744| 4307
PPD tub aos aoscemeinn 578 4276 0:99744| 4265
st cen ek ee mene 671 4253 0°99744| 4242
PMENOM |i Jereete. cee aaeee 486 7130 0-99266| 7078
sis Wile etetiraleoe nce natin 578 7030 0:99266| 6978
spam Meee de ocopeatarouounae 671 7007 0:99266| 6956
Carbon monoxide......... 436 3420 0:99959 3419
sd a Cee 589 3350 0:99959| 3349
Carbon dioxide ......... 436 4575 0:99308| 4544
ee he Sone 589 4498 0:99308| 4467
sah eatin apts 671 4470 0°99308| 4439
Sulphur dioxide ......... 436 6960 0-98040 6824

TTY Vi nee anise 546 6820 0:98040| 6687 .
SUR aes neen 589 6760 098040} 6628
STR pune eect 671 6610 098040; 6481
Hydrogen sulphide...... 486 nit 0-98913 6498
or fulbhe Ln pense 546 tak 0°98913| 6428
J coll Cee west 579 a 0:98913| 6400
PORN Ube) Att Trae a 671 Ae 098913] 6351
Nitrous oxide? .e5-4-¢- +a 589 5160 0°99267 5122
Iisrichoxide 5. yaaa. 589 2950 0:99892| 2947
Atm OMIA @-ceeseeee seen eee 589 3790 098891 3748
Gy aNOreny creas scores sees 436 8710 097999} 8536
LRM cpt eran Sere 671 8430 0:97999| 8261
Metlaie. yo.c.steeecacecec 436 4500 0°99845} 4492
oe «10 Bsa neencetast es 578 4420) 099843} 4413
Sad c/ Bjorn saa Gee 589 4410 0:99843| 4403
Persil cate ncita dey Lane 656 4400 0:99843| 4393
WBibhame! on sadt aeadeenaee 436 7820 098924] 7736
Pee et. wo Nvach accesories 546 7690 0:98924| 7607
BEN SAS cas cee we ee 671 7630 0-98924| 7548
Bitlsylene: sci 6 bla 436 7430 0:99222| Tari
Meee eee AR 671 7170 0-99222| 7113
IAICeGYAEING eins eateesee eee 436 6190 0-99179 6139
i A ee ARR 671 5980 0°99179| 5930

Chiovinene sinc ssa 589 7730 098417} 7608 -
Hydrogen chloride ...... 589 4470 0°99272| 4437

In Table II. the dispersions of a number of gases have
been expressed in terms of the ideal refractions for wave-
length 589.

TasBe II.

ire len athe eee ae | 436 486 | 656 671

Gas. |
EDV GTOC CH ie... .c2t keen see +20 +14 —5 ==
CAHN rs eae REED. Be ate +39 +22 —10 —
Oyen sak ee +36 +24 —14 —
INTO MEM) eee Lb oles | +44 +36 — ae
Sulphur dioxide ......... | +196 — |. alee

Meiitame sarc n 8 teen: | +89 — --10 |

Ideal Refractivities of Gases. 33

In Table III. are given the values of the dielectric con7
stant ¢ for several gases at normal temperature and pressure,
together with the values of Wh e which on the electromagnetic
theory of light should, for insulating media with very small
magnetic susceptibility, when corrections for dispersion are
taken into account, be equal to the refractive indices of the
said gases. The measurements of the dielectric constants
have all been executed with frequencies exeeeding 10°, so
that they may be regarded as referring directly to waves
of infinite length. Although strictly speaking a small
correction for the deviations from the gas laws should be
applied to the values of the dielectric constants, which
refer to normal temperature and pressure, its magnitude
does not, in We, exceed the limits of experimental errors.

For the purpose of comparison the values of the refractive
indices should strictly speaking be extrapolated to infinite
wave-length. Since in all cases we are dealing with normal
dispersion, this procedure would lead to smaller values of
the refractive indices. As a matter of fact, however, the
higher values of the refractive indices, corresponding to the
shorter wave-lengths, are in better agreement with the ob-
served values of »/e than are the extrapolated values.
Assuming the correctness of the usual dispersion formule,
and the theoretical relation between the refractive index and
the dielectric constant, this result would seem to indicate that
the determinations of the latter are affected by some constant
error. It would seem more probable, in view of the better
agreement among the values of the refractive indices given
by different, observers, than among the values of the dielectric
constants, that the origin of the discrepancy is to be sought
in the latter rather than in the former.

TABLE ITT.

~ Phil. Mag. 8. 6. Vol. 29. No. 169. Jan. 1915. D

34 On the Ideal Refractivities of Gases.

A question of great chemical interest is that of the in-
fluence of the constitution of the molecule on the refractivity.
This has already been studied by Briihl*, who from a con-
sideration of the uncorrected values arrived at the conclusion
that: ‘...die Molekularrefraktion der Gase ist keine rein
additive, sondern eine unter gewissen Umstiinden deutlich
konstitutive Higenscheft derselben.” A similar view was
later expressed by C. and M. Cuthbertson f. Such a con-
clusion could not, however, be final until account had been
taken of the reduction to the ideal state. In Table LV. are
given the values of the refractivities of a few gases calculated
from those of their simpler constituents, and the observed
refractivities of the same gases, all reduced to the ideal
state.

TABLE LV.
Gas. | (uy—1) 107 cale. (#5 —1) 10° obs.

CLO CHAE Cl) eee: | 4550 | 4437 |
NET, OCNEL SED) let | 3577 | 3748 |
CO, \\(COA-O)s: eho: 3029 | 4467 |
NMOL (NON) coe 5656 5122

(C,H,-+6H) ...... 5216 |

GsH, (C,H,+4H) ...... 9965 7736

Some additive relation underlying the whole series is
immediately apparent. In the cases where the elements of
the constituent compound gases (e.g. CO) are united by
different linkages from those in the more complex gases
(e. g. CO), the divergence from the additive relation, how-
ever, becomes considerable. Thus, if the refractivity of
methane be calculated from that of the saturated hydro-
carbon ethane, it is in much better agreement with the
observed value than when it is calculated from that of the
unsaturated hydrocarbon acetylene.

The results collected in Table IV. show that, even with
the ideal refractivities, the conclusion of Briihl still holds
good. }

* J. W. Brihl, Zectschr. physik. Chem. vii. p. 1 (1891).

+ C. and M.Cuthbertson, Proc. Roy. Soc. A. ]xxxiii.p.171(1909). These
authors applied a correction in the cases of sulphur dioxide and hydrogen

sulphide by multiplying the refractivity by the ratios of the theoretical
to the observed densities.

Nee
Or

A Theory of Supersaturation.

Summary.

The refractivities of several gases have been referred to
the ideal state by means of the characteristic equation of
D. Berthelot. ‘he corrected values are called the. “ ideal
refractivities.” |

A comparison has been instituted between the ideal
refractivities and the dielectric constants of the gases.

The conclusion of Briihl that refractivity is highly additive
in character has been confirmed in the case of the ideal
refractivities.

IV. A Theory of Supersaturation. By W.J. Jones, I.Se.,
and J. R. Parrineron, M.Sc., Assistant Lecturers in
Chemistry, Manchester University *.

4 ees solubility s of a given solid substance in a given

solvent depends on the temperature, the pressure, and
the size of the solid particles in contact with the saturated
solution. If we consider spherical particles of radius r, we
have

se Cay tania ia tne /12. 0) WU aaN

where '' is the temperature and P the pressure.

The problem of the surface energy between a solid and
its saturated solution was first systematically treated by
J. W. Gibbst, and from a different standpoint by J. J.
Thomson {. The more recent investigations of Ostwald,
Hulett, and Freundlich have been considered in two papers
by W. J. Jones §, in which the theory is applied to various
special cases.

The importance of the theory of surface energy in the study
of supersaturation does not, however, appear to have been
realized. The results of the investigations referred to may
be summarized in the formula

RD oo $9 Ae 20 ( wa 1 ) (2)

M | SEN, Pp a) 4)" 3 ; ; ; a
where R is the gas-constant, T the absolute temperature,
M the molecular weight of the dissolved substance, o the
energy per unit area of the surface of separation of the solid
and the solution, p the density of the solid, s, and sy the

* Communicated by the Authors.
+ ‘Scientific Papers,’ New York, i. p. 310.
{ ‘ Applications of Dynamics to Physics and Chemistry,’ p. 251.
§ Zeitschr. physik. Chem. \xxxii. p. 448 (1912); Ann, Phys. (4) xh.
p. 441 (1913).
D 2

36 Messrs. W. J. Jones and J. R. Partington on a

concentrations of the solute when spherical particles of radii
7, and 7, respectively are in contact with the saturated
solutions. If 7, is infinite, the corresponding value of s, is
equal to the ordinary “ solubility,” which may be called the
‘normal solubility,” and denoted by s,. The normal solubility
is therefore the concentration of a solution in equilibrium
with a plane surface of the solid solute. The difference
between the solubility of spherical particles of radius r and.
the normal solubility is therefore given by the equation
RT ei Be,

M loge — or. : C 4 - - > (3)

Hence we find the value of ¢s for any value of r in terms
Of sa:

For log. s. we can substitute the expression *
ay T i

BoSe + const.,

where X=A,+2’'T is the heat absorbed when a gram-molecule
of the solid dissolves in the nearly saturated solution ; X, and
«' are constants. Thence

2Mo Niantic
Apr pn Oe Be sho (oe ar

GON OS I
log sA= RT?

Over a small range of temperature, p is a linear function
of temperature; a closer approximation is given by the
equation

1 dp
p al

The value of p is practically independent of r.

As a first approximation o is also a linear function of
temperature +, and hence we may assume that the equation

1 do

oat.
represents a closer approximation: o is also practically inde-
pendent of .

We have now to consider how the radius of the particles
in equilibrium with a given supersaturated solution (7. e., a
solution which contains more solute than corresponds with
the normal solubility) alters with the temperature, when the

* Hardman and Partington, Trans. Chem. Soe. xcix. p. 1769 (1911).
+ Frankenheim, Lehre von der Kohiision, 1836.

=@ (a constant):. . . . 2a

= 8 (a coustant)i..-.. . eee

Theory of Supersaturation. a7

amount of dissolved substance remains constant. If we
consider a supersaturated solution of given concentration at
a temperature T, then equation (3) shows that there is a
particle of definite radius + which will be in equilibrium with
this solution. The size of this particle we will call the
equilibrium-size. Towards particles of greater radius the
solution behaves as supersaturated, 7. e. the introduction of
such particles into the solution would bring about crys-
tallization. ‘Towards particles of less radius the solution
behaves as unsaturated, i.e. if such particles are introduced
into the solution they dissolve. These results have been
confirmed experimentally by Ostwald and by Hulett (doc. cit.).
Now let the temperature be raised to (T+6T). The equi-
librium size will also be changed, and the alteration may be
calculated as follows. Let (*+6r) be the new equilibrium
radius, and (o + 6c), (op +6p) the corresponding values of the
surface energy and density. Since the concentration remains
practically constant,

Geo) an we SAN He SiON Pnsiig pated bareNontwer ne tas (7)
We have at the temperature T+ oT,

i 2.M.(o+0c) x, il
oT -Lo" yng +. loge (T+61)+%
Sa pon=A- »LR@ Lop +6L)(*r+ér) RL+6T) BR Se ( ) il (8)

The exponents in equations (4) and (8) are therefore equal;

: T+6T . :
and if we expand log 7 in a series, and assume oT to be

infinitesimally small, we have:

Ga NoaOah ihe Seat do _dp__ dr _ a
et ws aM e) te Srna

It is, however, known by experiment that « is approxi-
mately equal to 8*, 2. @.,

PER ANEEIS gid ES UNS egy
ne
or eee cre MR ENN ee pL (11)
Co
Thence we find from equation (9):
my Lee lr
at (— 3 tay +hy.r) — 2 =O, > eas: (EA)
where ;
ey ey) : 7) A i ol ie
Pot Ty oa a

* Partington, ‘Thermodynamics,’ p. 453.

88 Messrs. W. J. Jones and J. R. oan on a

In this equation 2’ is small compared w ith > T a> and thence

ky may be neglected in comparison with &, Equation (12)
then reads :
aan ee
dr kv®—r
By integration of (13) we obtain the equation on which
the theory of supersaturation now proposed is based :

kor

~o'. . e

If is positive, z.e. when heat is absorbed on dissolution
of a solid in a nearly saturated solution, then the expression

kr—

<1. If, is diminished, the corresponding value of T

kr
isalso diminished. If, on the contr ary Vis negative (7. e. heat
; kp —
is evolved on dissolution), then i a 1, and in this case T

increases when 7 is diminished.

Ordinary supersaturated solutions belong to the first type.
An example of the second type (\<0) is furnished by a
normally saturated solution of gypsum.

The constant C in equation (14) denotes the temperature
at which the given concentration is equal to the concentration
of the normally saturated solution, since when r=, then
(beaGe

The spontaneous crystallization of supersaturated solutions
of the first type, when they are cooled below a certain tem-
perature, remains to be explained. In the sense of the
theory described above, the size of the particle required to
induce crystallization is smaller the lower the temperature.
It may therefore be assumed that at a sufficiently low tempe-

rature the necessary size is only a relatively small multiple of
the molecular size, an assumption which becomes more pro-
bable from the following considerations. In order that a
particle of this size may be produced in the solution itself,
it is obviously necessary that several molecules shall simul-
taneously collide inside a small element of volume. This
leads directly to the hypothesis of de Coppet*, according
to which the formation of crystals is dependent on a
‘** favourable collision” of the molecules concerned, and such
“favourable collisions ” occur all the more frequently the
further the solution is removed from its state of normal
saturation. An estimate of the size of particle required can

* Ann. chim. phys. (5) vi. p. 278 (1875).

Theory of Supersaturation. 39

be obtained from the results of Ostwald " according to
which a quantity of solid solute less than 10~° gram was no
longer capable of bringing about the crystallization of a
supersaturated solution of sodium chlorate prepared from
107 parts of salt and 100 parts of water. This corresponds
with a radius of about 10, and this must therefore be less
than the equilibrium size in equation (3). At lower tempe-
ratures very much smaller particles would, however, be
active; and it is conceivable that molecular groups having
radii from 0-Oly to O-lw could be formed by multiple
molecular collisions in the solution.

A diminution of active radius would result in solutions of
the second type from a rise of temperature. Such solutions
should therefore crystallize spontaneously when heated above
a certain temperature, which would be higher the less
the supersaturation. There appear to be no quantitative
experiments in this field.

In the case of ordinary supersaturated solutions, the
active size of particle will be formed from a smaller number
of molecules the lower the temperature. The favourable
collisions will therefore change with falling temperature
from higher to lower orders, and the probabilities of such
collisions will then increase enormously. De Coppet seems
to have regarded the diminished molecular velocity resulting
from the lowering of temperature as the chief cause of the
spontaneous crystallization of supersaturated solutions. In
the sense of the theory now proposed, the main cause of that
phenomenonis rather the variation of the equilibrium size
with temperature, z.e. the shift of the probability of the
formation of particles of active size from smaller to larger
values with fall of temperature.

A rougn calculation shows that the equation (14) gives
results of at least the right order of magnitude, which is all
that can at present be expected. If we consider a solution
of Glauber’s salt normally saturated at 27° C., then

Te 6 —2 (a4 24 — 300.

Also 2M =284=300 approx. In the cases which have been
quantitatively examined +, o is of the order 10° erg. We can
further assume that p=38, and A»=500 cal. =25 x 10° ere.

Then

h= sw = 25x 10* cm.71.

* Lehrb. allgem. Chem. 2 Aufl, ii. p. 754.
t+ Cf. Jones, loc. cit.

40 Dr. H. Stanley Allen on the

If we assume that the active size is s=10-! cm., then
A
T=300( =) — 284°,
Ws)

i.€., at this temperature, or with a supercooling of 16° C.,
the solution would crystallize spontaneously.

The theory now described therefore leads to the following
properties of supersaturated solutions:—

(1) It is possible for a solution to contain more solute than
corresponds with equilibrium in contact with a plane surface
of solid solute (2. e. large erystals of the latter).

(2) Such a “supersaturated ” solution can be in. equi-
librium with particles of solid of a definite size. Smaller
particles dissolve in the solution, larger particles bring about
its crystallization.

(3) According as the solid dissolves in its nearly saturated
solution with absorption or evolution of heat, the size of
“‘active”’ particles of solid, z.e. such as induce crystallization
in a given solution, decreases with fall of temperature, or rise
of temperature respectively.

(4) The solution may, at a suiiciently low or high tempe-
rature, respectively, crystallize spontaneously.

V. The Magnetic Field of an Atom in Relation to Theories
of Spectral Series. By H. Stantey Aten, J/.A., D.Sc.*

N the course of a discussion on the structure of the atom
it was pointed out by the present writer, that it may
be necessary to take into account not only the electrostatie,
but also the magnetic forces in the neighbourhood of the
atom. It was suggested} that the atom ‘might be regarded
as a central core, carrying an electric charge and producing
a magnetic field Soni: to that due to an elementary magnet,
the core being surrounded by electrons in orbital motion.
Such a magnetic core might arise from a spherical volume
distribution of electricity rotating about a diameter with a
specified angular velocity.

One of the most important questions to he considered in
connexion with any atomic model is the possibility of ex-
plaining the lines in the spectra of the elements, and in
particular the relations between the frequencies of the lines

* Communicated by the Author.

+ ‘Nature,’ vol. xcui. p. 680 (1914).

{ ‘Nature,’ vol. xcii. p. 718 (1914), Discussion on the Structure of
the Atom, Royal Society, p. 17, March 19, 1914.

Magnetic Field of an Atom. Al

in spectral series. Nicholson™ has been successful in calcu-
lating the frequencies of the lines in the nebular and coronal
spectra by employing Rutherford’s model involving only
electrostatic forces. In these cases, however, only a ‘simple
nucleus is dealt with. The theory put forward by Bohr is
confessedly not dependent on the usual dynamical laws,
although it involves the calculation by ordinary mechanics
of the steady motion of the electron in the electrostatic field
of the positive nucleus. All the relations that have been
obtained between the lines in a spectrum involve the
frequency of the vibration. Lord Rayleigh{t has pointed
out that in the case of vibrations under electric or elastic
forces it is the square of the frequency that is involved. If,
however, the vibrations take place under the action of mag-
netic forces, the acceleration, instead of being proportional
to the displacement, is proportional to the velocity of the
moving electrified particle, and relations involving the
frequency of the vibration may be obtained. A theory
based on this consideration has been put forward by Ritz§.
He assumes the existence of molecular magnets, and sup-
poses that the electron is describing a circular orbit in a
fixed plane perpendicular to the axis of the magnet. The
elementary magnets are the same for all elements. To get
the different lines of a series he supposes that a number of
the elementary magnets are placed end to end, so that the
magnetic field is due to two poles whose distance apart is
always some multiple of the length of the elementary magnet.
It is a characteristic feature of the theory of Ritz that every
spectral line is brought about by the difference of two
actions.

Hinpirical Formule for Spectral Series
It will be convenient to summarize here the empirical
formule that have been suggested to represent the distribu-
tion of the lines in a spectral series.
It N denote the wave number (2. e. the number of waves
in 1 em.) Balmer’s series for hydrogen may be written

N=N, (j a 9)

WM?

where Ng is Rydberg’s “universal” constant (usually taken
as 109675) and m is a positive integer, 3, 4,5 ....

* Nicholson, Monthly Notices R. A. S., 1912-1914.
+ Bohr, Phil, Mag. xxvi. pp. 1, 476 (1918).

i Rayleigh, Phil. ” Mag. xliv. p. 096 (1897).

§ Ritz, Ann. der Physik, xxv. p. 660 (1908).

| See Baly’s ‘Spectroscopy,’ Chapter xvii., 1912.

42 Dr. H. Stanley Allen on the

If v denote the frequency of vibration, y=Nc, where ¢ is
the velocity of light, and the formula may be written

ie
age i- aa)

where vp has the vaiue 3°29x 10” sec.-1. Bohr’s theory
identifies vp with 27?me*/h?, the numerical agreement between
the two quantities being remarkable.

In the case of elements other than hydrogen *, more
complicated formule have been proposed. The typical
Rydberg series is of the form

1 1
an, ee
°° (1+ po)? Ont pe

where py and yw are fractions. This may conveniently be
written
ee at eh iL
N=Nol np DF}:
Rydberg states that the true formula should be given by

“ai
Mm => fh

commenting on this Hicks makes the following significant
statement :—

writing for D some function of m+ yp, say m+ ut+

ae a slight

“Tf a series is represented by D=m+p+ =
alteration will represent it equally well by putting D a
continued fraction, viz.

D=m+p+ — — at

in other words,

ING = — 2 ) m+ py?
Me eral

or N=A—B§ s/(m? + 2am+b)—(m+a) }*,

which looks quite different, and points to the frequencies
depending on the reots of a quadratic” f.

* Curtis (Proc. Roy. Soc. vol. xe. p. 605, 1914) finds that the results
for hydroger. may be represented by a modified Rydberg formula.

+ Hicks, Phil. Trans. vol. cex. p. 85 (1910). The notation of the
original has been slightly modified.

Magnetic Field of an Atom. 43;

Ritz* has obtained remarkable agreement between the

observed and the calculated results by taking
(m+ pu)?

Several investigators have made use of the form suggested

by Moggendorf and Hicks, in which
D=m+pta/m.

The value of the expression N,/D? for integral values of
m has been called a sequence. Four sequences exist, and it
has been shown by Hicks, van Lohuizen J, and others that the
majority of the lines in spectral series can be determined by
the difference between two sequences.

D=mt+p+B/m? or D=m+pt+

The Magnetic Lield of the Atom in the Quantum
Theory of Spectral Series.

In a letter to ‘ Nature’ (vol. xcii. p. 630, 1914) I have
drawn attention to the important work of Professor Carl
Std@rmer on the path of an electron in the magnetic field of
an elementary magnet. He has investigated the motion of
an electron when it is subject to the action of a central force
varying inversely as the square of the distance from the
centre of the magnet. Such a case would arise if the atom
consisted of a magnetic core, electrically charged and sur-
rounded by one or more electrons. Stormer finds certain
remarkable periodic trajectories in the form of a circle whose
plane is perpendicular to the axis of the magnet, and whose
centre is at some point on that axis. If this point coincide
with the centre of the magnet we obtain circular orbits in
the equatorial plane of the magnet. Further there are other
trajectories which never get outside closed toroidal spaces,
in the case of stability, or which approach asymptotically
the circle in question in the case of instability.

Let the magnetic moment of the core, considered as an
elementary magnet, be M and its positive charge be H,
electrostatic units being employed throughout. The equation
of motion of an electron (charge e, mass m) moving, with
angular velocity w,in a circular orbit of radius r in the
equatorial plane is

marore= Mea eile! io Moves ht CL)
It must be noticed that there are two possible directions
* Ritz, Phys. Zettschr. vol. iv. p. 406 (1908); vol. ix. pp. 244, 521

(1908). _
+ van Lohuizen, Science Abstracts, vol. xvi. no. 179 (1915).

44 Dr. H. Stanley Allen on the

of motion of the electron in its circular path, in the one the
mechanical force due to the movement in the magnetic field
is directed towards the centre, in the other away from it.
These may both be included in the formula by supposing
that M may be either positive or negative, the positive sign
being taken when the mechanical force is directed towards
the centre.

This equation is not in itself sufficient to specify the
motion completely. ‘There is but one equation of motion,
the radial one, while there are two independent variables,
the speed and the radius of the orbit”? (Schott, ‘ Electro-
magnetic Radiation’). In order to obtain a second equation
we assume that the angular momentum can be expressed in
the form

me ==Thyl Dams sy Ve rr
where 2 is Planck’s constant, and 7 is a coefiicient whose
value is not for the present specified.
On eliminating + by means of equations (1) and (2) we
obtain a quadratic equation for w, which may be written

(Mo + E)?= Aa,

or M’o?—w( A—2ME)+H?=0,.. . 2
The
milestones Satie
Same”
Remembering that M may be either positive or negative,

we see that there are in general four possible values for w and
four corresponding frequencies v, since @= 27».
If we divide equation (1) by the square of equation (2)

ee ee Ae :
we obtain the value of 3 in the form

or PR when the effect of the magnetic field is
neglected,
I Aagnlte (4)
r — 2) . . . . e ° °

As in Bohr’s theory we consider next the work that is
required to move the electron from its orbit to a position of
rest at infinity. Denoting this quantity by W, we find,
assuming the mass of the core large and the each of othee
electrons negligible,

W = eS marta Joe)!

Maanetic Field of an Atom. 45

In Bohr’s theory W is capable of being expressed by the
equation ;

te hw
AV oe
; Ao?
where o is an integer.
Let us now put
a Mew
wen a
U=W+ ca (6)
K 1 |
= — — 5mro? + ao :

Then by means of (1) we find that U reduces to

wae eae Ans
5 mo", which is the kinetic energy of the electron,

Making use of (2) we see that

the
Eee a hoe ee eek)

When there is no magnetic field present tT and o become
identical and U=W. In the presence ot the magnetic field
we have to take into account the small term Mew/r. We
proceed to find the approximate relation between 7 and o in
this case.

Combining (6) and (7) we find
rho _ cho, Mea
dg — Ag tp

__ cho Agr?mM Ke?

~ Agr Th? :
3 2
[Slane site egl aE ES) ehE NG (8) 8)

or, approximately,
i 1677°mM Ke?

a2)?

Solel Ria ens'y ialt yidi wants sti est Ragen mntee (9)
oO

167? mM Ke?
where §= ek

The value of W may be expressed in a form more
convenient for our purpose as follows :

_ che

W=—

= 2) (MotB... . - (0)

A6 Dr. H. Stanley Allen on the

Substituting the value of » derived from the quadratic
equation (3) we find

Ia2mebeo lt A \/ A \ Asare
se Nee oie ee)
i eae Fae = ear Tol CY

We now proceed to express the quantity inside the square
bracket, which may be denoted by w, as a continued fraction

a ie el
of the form a = pe ae
since ¢= i eu s , @ is the root of the quadratic equation

av + abu — me 0, and therefore

“ev Ge

So we find a =landb=-

ME
Hence SS, :
2a? me? K?a

W rere 7)" ‘le 1 Ho) 6 e (12)
Ti gatad ith
~ MET

or approximately
Wee 2a? me? Ko 1 13)
Th? fe yes (Lo

An examination of the numerical magnitude of the
quantities involved in the term ME/A shows that in general
the value of this term must be less than unity.

In Bohr’s theory monochromatic radiation is supposed to
be emitted during the passage from one steady state of
motion of the electron to another. How this change takes
place is left undetermined, but it is assumed that the amount
of energy radiated, that is the difference between the energy
in the first orbit and that in the second, is exactly one
quantum. That is

iv= OW = We-Wa ....” see
The frequency of the radiation is therefore given by
Qar?me? HE? ( ooky? = 21"
a eee T° —— a I, ence, (15)

or approximately

27? me? Ki?

2 aa: E fy “a _ MB 3 (16)

Magnetic Field of an Atom. AT

Substituting the value of 7 in terms of o previously
obtained, we find that the frequency is given by the approxi-
mate formula

2ar2me? KE? 1 1 an
Dee ine 5 EE Oe BYE Hast)
( | o2+ eA E a Sal
Ba oemM be’ 38 _ l6m'mM Be? _ 5

hh? 2 he

where

We may note that equation (17) is equivalent to

2a? me? H? f il i
V H3 r Te ao . (18)

The formula proposed by Ritz to represent spectral series
may be written

gene dial
»=%1 pa—pa ts : ° > 6 (1:9)

where D is of the form m+u+/m? or m+u4+/(m+ ph)",
and vp is the frequency corresponding to Rydberg’s constant.

When E =e, that is when the core carries a charge
equivalent to the loss of one electron, the factor outside the
bracket in equation (17) reduces to 2a?me*/h?, which Bohr
identifies with vy. ‘The bracket becomes identical in form
with that in the formula of Ritz if we take c=m+yp and
OS:

This implies that o, instead of being an exact integer (nz),
as in Bohr’s theory, is equal to an integer plus a certain
fractional quantity w, which depends on the element and on
the particular series considered. The presence of this
fractional part must be assumed ; it is not explained by the
action of magnetic forces.

We conclude that, on the assumptions stated, we can
account for the existence of four sequences in spectral series,
the denominator of each sequence being of the form proposed
by Ritz, and we can determine the lines in a spectral series
by the difference between two sequences.

When, however, we examine the numerical value of the
constant 8 as given by Ritz, we find that the values obtained
for the magnetic moment M by identifying Ritz’s constant
with the 6 of our analysis are many times too large to be
possible. Thus taking for illustration the case of lithium,
Ritz gives for 8 in the principal series the value 0°0257,
which would correspond with M=45x10-8 emu. The
magnetic moment of the magneton is 1°854x 1077) BE...

48 The Magnetic Field 0) an Atom.

Thus the core of the lithium atom would have to be equivalent
to about 2500 magnetons! Weare forced to the conclusion
that the magnetic field can be responsible for only a small
part of the term in question. The assumptions that we have
been compelled to make as to the constitution of the atom,
namely, that the magnetic field may be regarded as equi-
valent to that set up by an elementary magnet and that the
electrostatic field may be treated as varying inversely as the
square of the distance from the centre, involve so much
simplification that we can hardly expect the result to do
more than point the way towards the correct form for the
expression D in the denominator of a sequence. If, for
example, we treat the core of the atom as a positive nucleus
surrounded by a continuons ring of negative electricity,
analysis shows that the electrostatic field gives rise to a term
in the expression D of the same form as 6/c”. Thus in the
case of an element containing a large number of electrons, it
may be possible to obtain an approximate formula which
would agree with that proposed by Ritz (or perhaps that
proposed by Hicks), but in the case of elements like helium
and lithium, which contain only a few electrons, the diffi-
culties in the way of Bohr’s theory put forward by Nichol-
son * still remain serious if not insuperable.

The general conclusion that may be drawn from the
present work is that the magnetic forces set up by tle atom,
thongh they may play a part in controlling and perhaps
stabilising the motion of the electrons, are insufficient to
account for more than a small fraction of the effect that
would be necessary to give the observed distribution of lines
in spectral series.

Summary.

It is shown that a formula, similar to that of Ritz, repre-
senting the distribution of lines in spectral series can be
deduced from the assumptions following :—

(1) The core of an atom gives rise not only to an electro-
static field varying inversely as the square of the distance
from the centre, but also to a magnetic field such as would
be set up by an elementary magnet.

(2) The steady states of motion of an electron in the field
of the atom are determined by the ordinary laws of electro-
dynamics, combined with specified assumptions as to the
angular momentum and the energy of the electron.

(3) The energy of the radiation, as in Bohr’s theory, is

* Nicholson, Phil. Mag. vol. xxviii. p. 90 (1914).

Motion of the Lorentz Electron. 49

given out in quanta, which represent the differences between
the energies in two steady states of motion.

When, however, the numerical value of the appropriate
constants in the formula of Ritz is considered, it is found
that the magnetic forces set up by the atom are not in
themselves sufficient to account for more than a small
fraction of the effect that would be necessary to give the
observed distribution of lines in spectral series.

University of London, King’s College.

VI. On the Motion of the Lorentz Electron. By G. A.
Scuort, B.A., D.Sc., Professor of Applied Mathematics,
University College of Wales, Aberystwyth *.

ee a theoretical investigation of the origin of

X-rays I found it necessary to take into account the
effect on the motion of the electron of the reaction due to its
own radiation, and from this point of view examined some
simple cases of motion in order to gain a clear idea of the
result to be expected. The following communication in-
cludes these preliminary studies, but is also intended to serve
as an introduction to a more complete investigation to be

published later.
The Equations of Motion and Energy of the Electron.

1. The vector-equation of motion of the electron may be
written in the following form f

(CEES OL Ah aa a ena
where
mV
aa as) 2
G= AGE G2) >) ° e ° e e e ° ° . e ° e (2)
2 ee hse Dee ere
K 2e°v 2e?(vv)v e*(vv)V e (vv)? (3)

— 3c(c?—v") a d0(e?—v*)? * e(e?—v?)? " c(eP—v*)*"
G denotes the electromagnetic momentum of the electron in
the form due to Lorentz, K the reaction due to radiation,
2. e. the radiation pressure in the form due to Abraham f,
and F the external mechanical force. If we accept the
Principle of Relativity for accelerated as well as for uniform

* Communicated by the Author.

t Schott, ‘Electromagnetic Radiation,’ pp. 175, 176, 246 (quoted
below as E. R.).

t Abraham, Theore der Elektriziédt, ii. p. 128.
Phil. Mag. 8. 6. Vol. 29. No. 169. Jan. 1915. 0

90 Prof. G. A. Schott on the

motion, the expression (2) for the electromagnetic momentum
follows as a matter of course, but the expression (3) for the
radiation pressure requires a special hypothesis to justify its
introduction. It must, however, be borne in mind that the
deduction of the Lorentz momentum, as for instance by
Planck *, also implies the existence of a kinetic potential,
and that this has only been defined for reversible changes,
whilst accelerated motions of an electron involve radiation
and therefore are irreversible. If, on the other hand, we
adopt the usual equations of the Hlectron Theory of Larmor
and Lorentz together with the hypothesis that the electron
occupies a finite though small region of space, whether sur-
face or volume, then the terms on the left of (1) represent
inerely the first two terms of an infinite series. If a be a
length of the same order of magnitude as the linear dimen-
sions of the electron, and J a second length of the order of
the radii of curvature and of torsion of its path and of the
distance within which its speed is doubled, this series pro-
ceeds according to ascending powers of a/l, and converges
with sufficient rapidity only when a// is small compared with
1—?. When the acceleration of the electron becomes very
large, or its velocity nearly equal to that of light, the series
fails entirely ; indeed it is probable that under these con-
ditions the usual definition of the electromagnetic mass, im-
plied in (2), can no longer be upheld. For the rigid spher ical
electron of Abraham this has been proved definitely by
Sommerfeld t; he shows. that when the velocity of a uni-
formly accelerated electron is equal to that of light, the
largest term in the mechanical force on it due to its own
charge is proportional to the square root of the acceleration
when the latter is small. Unfortunately Sommerfeld’s
method cannot easily be extended to the case of the Lorentz
electron, so that it is impossible to be quite sure of what
happens here, but it does not seem likeiy that the result
would be very different. However that may be, it is clear
that the expiration (1), (2), and (3) must be used with
‘caution in cases where the velocity may be expected to
approach that of light, or in very strong electric or magnetic
fields, where the sete iae and curvature of the path of
the electron may reach large values. Thus we must be
eareful in using them for an ‘electron which approaches very
closely to the nucleus of Rutherford’s model atom, and in all
problems of a similar kind. May not the failure of the

-* Planck, Srtzungsberichte der Preussischen Akademie der Wissen-
schaften, 1907, p- 8.
ye aur Elektronentheorie, ” Gottinger Nachrichten, 1904, p. 411.

Motion of the Lorentz Electron. 51

ordinary mechanics and older electrodynamics so often
alluded to ky present-day investigators of theories of the
atom, be after all due to neglect of proper precautions and
to unjustifiable usage of confessedly imperfect analytical
expressions as much as to defects in the fundamental prin-
ciples of the electron theory ?

_ 2. The equation of energy may be derived from the equa-
tion of motion by multiplying it sealarly by the velocity v;
after a few simple algebraic transformations it is obtained in
the following form *:

eee am es iave th. ich 08 Weg)

where 4 1 3
T=2n4 q—ay 1} HU Alea M95)

2ce? (vv ) :

Q= ( Scie ated edt} Visa: QO)

2ce? Vv" (vv)?
R= { (@—2 pea sgt)

Here T denotes the kinetic energy of the electron and is
given by (5) in the usual form; (vF) gives the rate of
working of the mechanical force; the remaining terms in
(4) are derived from the radiation pressure. Of these R is
essentially positive and denotes the irreversible rate of loss
of energy due to radiation; the expression (7) is the well-
known one due to Liénard. On the other hand, Q repre-
sents a reversible rate of loss of energy; hence —@ must be
regarded as work stored in the electron in virtue of its
acceleration, so that we may speak of it as acceleration
energy. Its existence isa direct consequence of a mechanical
theory of the eether f.

3. In order to simplify the equations as much as possible
it is convenient to introduce a new system of units; we shall
choose the

new unit of length =2e/3e?n=1°83.107-*%cm.,

nm EIN pe — es ac — (Or le. Ln sece
“5 5 velocity:==c=3 .10 em./sec.

yi » force =de4m?/2e?=4'3 . 10° dyne,
a mi jeleroy —=<e— Oo . LOT ero.

The numerical values given in the last column have been
* B.A. pp. 176, 177.
PUR neo:
H 2

52 Prof. G. A. Schott on the

calculated for the electron with e=4°65.10-H.S.U. and
e/em=1:'77.10". When the new units are used we must
replace the factor m in (2), 2e?/3c in (3), c?m in (5), and
2ce”/3 in (6) and (7) by unity, the quantity @ in (2) and (5)
by v, and the velocity ¢ in the expression c?—v? in (3), (6),
and (7) by unity.

4. Introduction of a new time-variable.-—Using the new
units we put *

r=) Yaseen. <i
0

We shall use an accent to denote differentiation with
respect to the new time-variable 7, but for the sake of brevity
shall use the symbol w to denote the velocity relative to r.
Then we find in succession

v=w/(1-v?)= TC Faty Whence J/ (lv) = rae :
? w’ (ww )w

Sa eee
a w"’ (ww )w+3(ww')wi+w?w 4(ww’')?w
Egle ose Car (ees

Substituting these values in the expressions (2), (3), (5), (6),
and (7), we find

G=w, . . ° . ° ° : ° e ° ° ° . e (9)
Be A ea te ar wat ne Aa

a yearn 1l+w? J VW(1+w?)’ (©

Te=,/ (lw) ne
eee cna esis

a / (1+?) ae

!\Q
Rew? — (Ow), (13)

With these values the equation of motion (1) becomes

w —w'’+Rw=FV(l+w?). . . . (4)

Similarly the equation of energy (4) becomes
TT's RUS DSF)... ... ... oem

The curious similarity of form of the last two equations is
worthy of remark.
* E.R. p. 292.

Motion of the Lorentz Llectron.

Qt
os

Rectilinear Motion.

5. As an example of the use of the equations we have
obtained, we shall now consider the case where the electron
moves in a straight line under the action of an electrostatic
field in the same direction. We shall take the straight line
as the axis of x, so that w'=w’=2’. Then we find from

(13) and (14) respectively :

w'?
Saree is aa to 9 LOD
12
w!—w' + papi PV (+e). . iv oka aa

In order to reduce these equations to a simpler form we
write

= sinhy, whence v=$= tanhy, and T= cosh y—l1, (18)
(16) and (17) now give

eva teed an Sm NS! SGM ak EQN
Nie ge Om eva)... yah arti COON)

When F is known as a function of 7, (20) may be solved at
once in the form

T T
x= Far—e | BPe-td7+A+Be, . . (21)
0 0
where A and B are arbitrary constants to be determined
from the initial conditions. A third arbitrary constant will
be introduced when we determine w from the differential
equation e<=w=sinh x, but we may make this constant zero
bv choosing the origin of coordinates so that w vanishes
when ¢ and 7 vanish. |
6. Determination of the arbitrary constants A and B.—One
relation can be obtained at once between A and B, for we
ace at liberty to choose the origin of time so that v, and
therefore also x, vanishes when t=0. This condition with
(21) gives
A+B= . Et AA Ge ok GR)

Substituting for A in (21) we obtain

x=" Barer Fer dr + BEI). a2 (2S)
0)

a) ry

Bearing in mind our choice of the origins of space and time

54 Prof. G. A. Schott on the
and using (8) and (18), we find

ee
va| sinhiydr, . +». - -
0
Ar
=| cosh y drt. .
v¢)

We have now fully utilized the initial conditions so far
as they relate to the initial values of the coordinate and the
velocity of the electron, but there still remains an arbitrary
element—the arbitrary constant B in (23) to be determined.
Here we are brought face to face with one point of difference
between the ordinary mechanics of Newton and the electron
mechanics founded on the electron theory. Very slight con-
sideration shows that the presence of the third arbitrary
constant is due to the fact that the equation of motion of
the electron, (1), or (14), or (17), when regarded as a dif-
ferential equation for the coordinate, is of the third or der,
and that the differential coefficient of the third order arises
from the radiation terms. It is important to bear in mind
that these terms must be present whether we adopt the
Theory of Relativity for accelerated motions, or base our
mechanics on the hypothesis of the extended electron ; only
in the latter case every additional term of higher order which
we introduce into our equation of motion brings with it
another arbitrary constant. These additional arbitrary
elements, in so far as they must be determined by the initial
conditions, represent the effect on the motion of the electron
of its past history, a point which I have emphasized on
previous occasions *. Unfortunately, the past history is un-
known in many pr filers s, and therefore we are compelled to
make some additional hypothesis to overcome the difficulty.
We must choose it so as to preserve the continuity of the
electron mechanics with the ordinary mechanics, which we
know suffices in all cases where the velocity of the electron
is infinitely small compared with that of light: thus the
proper hypothesis suggests itself, namely, that in these cases
Newton’s Laws of Motion hold without alteration. Hence
we assume provisionally :

When the velocity of the electron is zero, its acceleration is
equal to the external mechanical force per unit mass.

This hypothesis has the advantage, as we shall see later,
that it leads to simple results which can be controlled by
experiment.

* Schott, Annalen der Physik, 1908, p. 63; HE. R. p. 155.

Motion of the Lorentz Electron. ,

- In order to apply the new hypothesis to our problem
we sat find the acceleration. From (18) together with
the expressions given in § 4 we obtain

t
° Ww

TG aaa De Ae pe MU aL GACY,

Again we find from (23)

v= {B-|" Berrie ber 5) 7) any a el WCCO

ae e O e
Hence applying our new hypothesis we obtain
i 7 f .
by, Say ER i kee a Ye RR BR 2a
where the suffix is used to denote initial values. Substituting

this value in (23) ... (25) we get finally

ne i
x= Fare ent dr ten) Me 29)
e })

= sinh at Bur—e'(" Be-7 dr+ Fy(e™—1) har, (30)
0 0

(Por

— { cosh { ( Farmer” Fe-td7 + Fy(e™—1) Madr, (31)
ey w0 20
We also find from (19) and (29)

R= f Fr —{"Rerar} em (32)
ae 0 q j e . ° . ° . . ® e On j

In order better to appreciate the import of our hypothesis
we shall now apply the solutions (29) ... (32) to the particular
case of a uniform force.

8. Hzample—Motion of a Lorentz electron in a uniform
electrostatic field parallel to the line of motion. I have already
treated this example elsewhere *, but without taking the
radiation pressure into account.

In the present problem IF is a constant, so that we may
omit the zero suffix as no longer necessary. Then we find

yas Pe coshy—1l, Fi=sinhy, R=F*.: (33)

Kliminating y between the second and third of these equa-
tions, we obtain precisely the same relation between & and ¢
as we do when we neglect radiation. This surprising result
is a direct consequence of the hypothesis of $6 ; in order to

* H.R. p. 181.

56: JProtiG, A. Schott on the

understand this better we must examine the energy relaticens
of the electron.

From (11) we obtain by means of (18) and (33)
T= coshy—I=Ba. 2... o> ee

This equation shows that the whole of the work done by
the external field is converted into kinetic energy of the
electron, just as if there had been no radiation at ali. None
of it is radiated.

Again, from (12) we find by means of (33) and (34)

QO= T= sinkive y.— lee ee

Thus we see that the energy radiated by the electron is
derived entirely from its acceleration energy ; there is as it
were an internal compensation amongst the different parts
of the radiation pressure, which causes its resultant effect to
vanish.

The total energy radiated is on the present hypothesis
only a very small fraction of the kinetic energy, unless the
external force be exceptionally large. From (33) ... (35) we
find by means of (18)

Rt Fsinhy 1+ Ve) ns
re —~ cosh y— if =F, /j-9 VAC eed een (36)

In applying this equation we must bear in mind that we
are using the new units of §3; hence when we return to
C.G.8. units we must replace F by 267 Hh /3e'm? = 2e?X /3e*m?,
where X is the electric force in H.8.U. From the value of
the new unit of force given in $3, viz. 4:3. 10° dyne, and
thatob ye, wiz. 465. 105 “10 B.S. U., we find that F in (36)
is equal to 1:08. 10~**, and that Re/'T is about 4:03.107™
when X is 30,000 volt/em. and v or B is 0°5.

9. In order to test the truth of the hypothesis of § 3 we
must examine what happens when it fails. Still confining
our attention to the case of an electron moving in a uniform
electric field along the line of motion, let us return to equa-
tions (18) and (23) ... (25), which are true quite independently
of the hypothesis in question. Bearing in mind that F as
well as B is a constant, we see that we may write instead

of (28)

B= BO ene... s.r

where 6 is another constant, 7. e. a quantity independent of
x or v, but generally a function of F. We may regard 6 as
a measure of the deviation of our hypothesis from the truth.

Motion of the Lorentz Electron. a
We now find instead of (33)

X=F {r+ 8(er—1)}, R=F{1 4 Ser}, . (38)

while we Have as before
Le Wee
v=S=tanhy, wv =| sinh y dz, ‘=| cosh y dr.
0 0

Changing the independent variable from 7 to x” we obtain

sya ‘sinh ROT) sinkivdy |
.T.), 14der ~ J, 1+d—74+y/F . . . ° ° ° e ° =a"

=n *cosh ydy _ * cosh ydy
ae EE Oct 9 L+6—T4+y/F? *

R= Wil + det P= {1 +5—7+y/FP,

t x xX |
{ Rat=F( {1+ der} cosh xdy=F {L+d—T+y/F} cosh ydy. |
0 0 7/9 }

The equations (38) and (39) show that the analytical
character of the solution is completely altered by the failure
of the hypothesis under consideration ; what change will be
produced in the numerical results depends on the magnitudes
of B, F, and 6. In estimating this change we must bear in
mind that what we measure by experiment is the increase of
velocity produced in a measured distance by a field of known
strength, and perhaps in certain cases the total energy
radiated in the process. Knowing 6 and therefore vy we
can calculate « and the energy radiated by means of (39):

(39)

but in order to measure 8 independently of the hypothesis to
be tested we must not use a deflexion method, either with an
electric or a magnetic field, because that would again involve
the hypothesis and require very troublesome calculations.
We must measure the kinetic energy, e. g. by a thermopile,
and thence calculate y and 8 by means of (18).

When the exponential term’ in (38) for y is negligible in
comparison with the first, we have the case already considered
in §8; for the sake of brevity we shall speak of it as the
Newtonian motion. On the other hand, when the exponential
term preponderates we have another extreme case, which we
shall call the exponential motion and shall now examine,

10. The exponential motion—We retain only the expo-
nential term in (38), and accordingly only the term y/f in
the expression 1+6—7+y/F, which occurs in (39). Then
the denominators in the integrals for x and ¢ vanish at the

58 Prof. G. A. Schott on the

lower limit, so that ¢ becomes infinite although 2 remains
finite. For this reason it is convenient to extend the in-
tegrals from a finite lower limit x, to the upper limit x,
the suffixes y and , being used to indicate initial and final
values respectively. Using the notation of the exponential
inteoral we find from (39)

X, SI i 1 5 . : 1
Pedy es ( a Ky = 5 {Bi (x1) — Bi(yy) — Bi(—yx,) + Bi(—y0)}

sh Maca |
_ (* cosh x TL day ate : :
oa | dx 5 { Bila) — Bile) + Bi(—x.) — Bil x0) } eo)
R=y%, | ty Rdt= x, sinh y,:— cosh y;— YX sinh yo+ cosh xo. . |
e ly |

These expressions involve neither I* nor 6, but only x and
x1, 80 that in this extreme case of the exponential motion
the result depends only on the initial and final velocities of
the electron, and not at all on the strength of the field or on
the precise value of 6. This fact of itself is sufficient to
prove that the exponential motion is not realisable experi-
mentally, at any rate not with the electric fields at our
command; a numerical example may make this clearer.

Let us take the case of an electron which has its speed
increased by an electric field of 27,700 volt /cm. (giving F
equal to 10-14) from 8)=0°01 to 8, =0°30, 2. e. from %)=0°01
tOlag — Oot:

With the help of tables of the exponential integral * and
of the hyperbolic functions we obtain the following results
ior the two limiting motions :—

Newtonian motion. Exponential motion.

Units of § 3. C.G.S. units. Units of $3. C.G.S. units.
t,—2L, 48.1012 0-88 0-302 55.100 a
it, ... 305.108 186.10-10 3-458 1 tines
J” Rae... 3:05.10-15 24 .10-21 0-049 39.1078

A comparison of the numbers in the last four columns of
this table shows conclusively the enormous difference between
the two limiting motions, and there can be no question that
the Newtonian motion is in far better agreement than the
exponential motion with what we know from experience.
Even if the hypothesis of § 6 be not exactly true, its deviation

* Dale, ‘Tables of Mathematical Functions,’ p. 85 and p. G4.

Motion of the Lorentz Electron. ao

trom the truth, as measured by the number 6, must be ex-
ceedingly small. In order to obtain some idea of its amount
we must study the general motion of §9 a little more
fully.

11. The limits of accuracy of the hypothesis —As we have
ulready remarked in $9, the theoretically best method of
testing the hypothesis in question depends upon a comparison
of the kinetic energy, T, acquired by the electron with the
work, Fw, done by the external field. We see from (18)
and (39) that T differs from Fa by a finite amount, the
difference being derived from the acceleration energy of the
electron. Suppose then that as a result of experiment we
find

i
T=cosh y-—1=(14+/)Fe=(14+/ F( sinh ydt, (41)
20

whege fis a number, which is probably a small fraction with
the same sign as 6. We must express 6 in terms of f by
means of (88), (39), and (41). Let us substitute for x in
(41) its expression in terms of 7 and 6 given by (88), expand
both sides of the equation in ascending powers of Fée™ by
means of Taylor’s theorem and integrate with respect to T.
Rearranging the terms according to powers of Fée™ we find

aL ih sinh 6)

FS (Tye? sinh F(¢ —6) —(1 +7) He? cosh Kir —8) + 1 +7)F (cosh Fé
1—t? Ohare.

+ i cosh 6)

a eget +7? )e* cosh F(r—6)—2(14+/) Fe? sinh (pO) a +7) ( 2 sinh I°6

2(4—F")
fee —/ cosh B(r—d)—1}—(1+/)icosh FO—1!}. . . . . . . 2 GY)

We must combine this equation with (38) so as to eliminate
7 and determine 6, but the calculation is so difficult that the
result will hardly repay the labour expended; hence we shall
content ourselves with finding limits for 6.

We first observe that the series on the left side of (42),
being derived from exponential series by integration, is
absolutely convergent for all values of Fée, and that the
coefiicients of all powers of F6 increase with 7 provided
that tanh F(r—6é) is greater than fF, a condition which is
satistied in actual experiments on account of the smallness of
IF’. Hence the first term on the left, which for such values
of 7+ has the sign of 6, is less than the right-hand member
when 6 is positive, and of course 7 also positive, but is

60 Prof. G. A. Schott on the

greater (numerically) when 6, and of course /, is negative.
Thas when 6 is positive, we can obtain an upper limit for its
value by omitting all positive terms in the factor of Fé and
all negative ones in the right-hand member of the equation.

In this way we find
Pée™{ tanh F(r—6)— (1+/)F} < /{1— sech F(7 —6)}.

This expression can be simplified very considerably without
raising the limit appreciably in any actual experiment. In
fact we see from (38) that F(7—6) is less than y or tanh 18,
whence we easily prove that sech F(r—6) is greater than
/ (1—8*), and tanh F(r—6) greater than @— Féet, so that

FSer{8—(1+/)F —Fber} < f{1— v (1—6°)}.

From this equation we find, again making use of (38), that
1—Q\V2F
ss < (go) 2128)
(B, ) 1+ 9

a {B—(14+/\F! sae (43)
ee) an

OF course, as we have stated above, (43) presupposes that

6 is positive.

12. Hitherto no experiments appear to have been
made in which both the kinetic energy and the work done
by the external field have been measured directly as our
investigation supposes, but in the course of some determina-
tions of e/m the fall of potential has been measured direetly,
while the speed of the electron has been determined, usually
by means of the deflexion produced by a known magnetic
field. The calculation of the speed, and hence of the kinetic
energy, from the magnetic deflexion involves an error due
to the radiation, presumably of the same order as f but un-
known, so that experiments of this kind cannot be expected
to supply us with an accurate value of 6. Nevertheless they
may be expected to give us some information as to its order
of magnitude.

One of the latest determinations of this kind has been
made by Hupka™* for velocities ranging from one quarter to
one half of the velocity of light and falls of potential from
4000 to 20,000 volt/em. measured to within about 1 in 400.
Assuming e|/m to be 1°77 . 10°, Hupka calculated the velocity
8 from the measured fall of potential by means of the Lorentz
formula (18) for the kinetic energy, of course neglecting
the effect of radiation which we wish to estimate. In his

* Hupka, Aun. der Phys. 1910 (1), p. 169.

provided that }

Motion of the Lorentz Electron. 61

experiments he measured the magnetic force required to
produce a prescribed radius of curvature in the path of
the electron, and compared their product with the ratio
8// (1—8?) to which it should be proportional for the
Lorentz electron. This proportionality was found to hold
throughout the whole range of the measurements to within
about 1 in 4000. It is obvious that this constancy of the
ratio of the two quantities to be compared could only be
possible either if the hypothesis were nearly true, or if in
the event of its failure the errors compensated each other
exactly. Of course it is extremely improbable that the effect
of radiation on the kinetic energy should balance its effect
on the magnetic deflexion so as to produce exact compensa-
tion, but in the absence of a complete theory of the magnetic
deflexion absolute certainty isimpossible. We may, however,
draw the conclusion that the number f/f, which measures the
difference between the kinetic energy and the work done by
the external field, is of the same order of magnitude as the
errors in Hupka’s experiments. By far the ‘ereatest error
is that in the determination of the fall of potential, given
above as 1 in 400; hence we conclude that fis about 1/400.

From six experiments with about equal falls of potential
we find that the fall of potential used by Hupka for a velocity
8=0°5 is nearly 20,000 volt/cm., which corresponds to
F=7'2.10-". The corresponding upper limit for / given
by (43) is 0-47, which is far beyond the error possible in
the experiments ; hence we may apply (43). On account of
the very small value of I’, the last factor of the right-hand
member of the first equation is alone effective in determining
the order of 6. Taking logarithms of both sides we find

io og io ae OE ee SAR)

13. Let us now consider the case where 6 is negative.
From (38) we see that F(7—6) is greater than y, so that
the whole investigation of §12 applies provided that the
sion ‘“‘less than” be replaced by “‘ greater than.” Thus (44)
cives'a lower limit for —6.

We may, however, obtain an upper limit for —é by a
different line of argument, based on the fact that according
to (38) x increases to a maximum as T increases, and
thereafter diminishes again. The maximum is given by
T= log (— 1/5) and is Soul to F {log (—1/8)—1—8}, and
there is a een maximum value of 8, which is
tanh F{log (—1/8)—1—6}{. Experiment shows no trace of

the existence of such a maximum, so that we may be sure

62 Motion of the Lorentz Electron.

that if it exists the velocities hitherto found for electrons lie
very much below it. If therefore we calculate the value
of (—1/6) from the highest value of 8 found for a given
value of I’, this will certainly give us an upper limit for —6é.
In this way we find

ii Bee aa
——OG (ae eo Sg! a )
O€ € leas (49)

With the same experimental data that we have used in § 13

we find
Logy(— 1/3) > 10%,

practically the same limit as in the former ease.

Hence we may assert as a result of Hupka’s experiments
that the deviation 6 of the hypothesis of §6 from the truth
amounts to less than one part in the ten-million-millionth
power of ten for a field of 20,000 volt/em. This is the same
thing as saying that for an electron moving with a velocity
small compar ed with that of light in an electric field of the
intensity stated, the acceler ation differs from the mechanical
force per unit mass bya fraction 6 at most, in excess or
defect.

It is possible that the deviation 6 may depend upon the
intensity of the electric field, but the experiments give no
certain information on this point. The probable error seems
to be rather smaller for a field of 5000 volt/cem. than for the
stronger field, but the number of determinations is too small
to afford a decisive result. Consequently it would be unsafe
to draw any definite conclusion frem the experiments re-
specting the dependence of 6 on the field-intensity. What-
ever this may be, it does not appear to be very considerable ;
hence it seems probable that our hypothesis may also be
applied to variable fields of intensities of the same order of
magnitude as those used in these experiments.

Since according to §6 the hypothesis is equivalent to
Newton’s Second Law of Motion for slow ly moving electrons,
we have verified this law to a degree of accuracy far beyond
that attained in astronomical investigations.

How far the law can be applied to electrons starting from
rest in very intense fields such as those inside and close to
the atom remains doubttul.

VII. Note on the separation of a Fraction into Partial
Fractions. By I. J. Scawart *

| eee following method for the separation of a fraction,
whose denominator is a power of a linear expression,
into partial fractions is simpler than the methods I have
given before f.

To separate into partial fractions

> eT

ag @ +a)?
Let e=y—a, then

s ea a a

NG =a) == iP Seis sual)

F(iy—-a)=y? > m SEE (eae (2)
a=0 B=0
Letting «+ 8=y, (2) becomes

Bg Oi Ma) (—Dr(1 TS) year (3)
a=0 y=a fy a J

Now since

therefore
g u n— a Ee
Ny —a) => ye > (— 1) em, es: eG.
y=0 a=0
We shall now distinguish between the two cases n < p and
n= p.

on <"p.
We may write (5) in the form

Que te w e (b)

* Communicated by the Author.
+ Quarterly Journal of Mathematics, No. 174, 1913; Archiv der
Mathematik und Phystk, xxii. 1914.

64 Separation of a Fraction into Partial Fractions.
Letting y=n—g, and y—a=h, we have

n=9
n Sem, a" @ ‘ h )
F(y—a) =s > h=0 p |

g=0 y? #
Therefore
Sy (—1) Mn—p 10 ie
F Re h=0 \.
() = 3 G
(1.) n= p.

Let n=p+q, then (5) becomes
ae pe ee ee eh a a \ ee
(y—a) = 3 cu aos x (- Dy scm ‘| nae Jar
or
q ; a4
F(y—a) = & yt-v > Wey wg *\ ars
ey 0 a=0 rags
S (— Lyem,(! ae *“\ars

Dalle
4D tet ee ea ee

y=qtl Yea

The last result may be changed to

q : oe
F(y—a) = > yt (—1yrarm, (2 gy i
y=0 a=0

qty
yey Qty—o
> ) m moma Pe po)

P. a ;
ae ee yr ? (9)
or
q | 2s
F(y—a) = oe yr-¥ |= s (— Lear, —.(?* il ; a
fhe")

spat y* ot,

Therefore

F(2) = 3 (e+ayry] & (— tearm, (P7977) |
Ye

Ss ee “Ltatng tye (7 i)
1 Vata

(11)

=e
I

On the Expansion of a Function. 65
q dey, G—
ey (etary = 3 (757 )atat+-?
y=0 B=0

=o 3 =) wbqt-Y-B,

B=0 y=0 \
We therefore obtain
B -
F(a) = s oP (23% )ar-re] > (=1a%m, (2592 al
B=0 y=0 B a=0 \ a
qty eh vith
5 EC tremn 7)
e = (w+a)?

University of Pennsylvania,
Philadelphia, Pa., U.S.A.

VIII. Note on the Expansion of a Function.
By I. J. Scawarr*.

te expand
log"(1+ 2) St Ai A ata eh eee (A)
in powers of w.
We have
log (1 +.) i if ees (oo 9
ie) = —— = LO I = @ Zz
a 9 l+uz 0 ap=0 ‘i ay=0 neni ( }
Now
log?(1+a) = 2 ~ log (142+) aoa ts S (—1)” ae Ss y (—Dseader
2) a l+wa Q a=0 Chae de 0
ta; weotatl
=) is > re —dix.
0 2 5G ) (ee ae ll
Letting aj+#,=8, then
“4
log? (1+a2)= iponpes (— Os
0 2=0 B=ay

25 8 er ae
i B=0 a ) (a+ 1)(@+2)°

Writing a for 8 and a, for a, we obtain
wut? ay y

Pea ee —0 atl (3)

lop?(1+2) = 2 s (—1)%-
ao=0
* Communicated by the Author,

Phil. Mag. 8. 6. Vol. 29. No. 169. Jan. 1915, Fk

66 On the Hupansion of a Function.

Again

log? Cu) = af Jog? {14+ w)> de

5 UR me coaiae hea! Lr Se
= BIN Si ee

ay 0 ag=—l
tw oD ae ajta;t2 ao ]
=3f & 2 (pve. 3 a
Jo ap=0 ap=0 Ay Si 2 oo = Diy al

Letting ay+a,= 8, we have

hte. al Sete See
02 7 =o) 2, (
=) ) 4 0 ap=0 B=a0 ao -- 9 a= 0 ay + a em

i B=0 ajg=0 # (a@g+ ZA) B+ 3) a,}=0 pest 1 ;

Writing a, for B, «, for x, % for «,, then
gx > o> 2 1»

= agt3 «a4 a
log (l+a)=3! > (-— ae x : > :
ag=0 a TO iS ar 2 a2=0 2 ar 1
ales eut8 (2 “B=1 1 |
= 3) 2 (= ie al SF pees - 4)
aj=0 a +3 \~e=1 ag=0/ 48+ 3—P

We now assume

Oy ynotrn n—l *B=1
log? (1l+2) =n! Gee, > o—_, » se

a ag N\g=1 ag=0 ag+n—B
Then
nN ' lx
log aril (1+ x) = ina) ( log (1+ x) oS

i ae \)! tr 3 aie gynorn (i &) re S (—l)rarde
Q ap=0 ay+n B=! ag=0 aa + he 0
ee agty+n /n-1 Fil if
=(n- hes > (ieee ( 0 ) le
\ Le ao=0 y=0 oF ) aot Nn \p-1 Poa) ae 2, ae ‘

Letting ao +y=6, then

log" L+0) = (nti 35S) ci ga an (0 fT s\ tS de

ag=0 sae ay ao+ 7 B=) a,=0 ag+n —B
& 8 é+n+1 me We fi
=(n+1)!5 > (—1)° (a )
( é= aire © (a +n)(S-+n+1)\gm 2, . 7 ae

“B

The Collapse of Tubes by Kuternal Pressure. 67
Writing 4 for 6, a for ao, a, for a), ete., we have

ip Gay nm “B—-1
8 (Son!

atl (] = 1) 1) ie a = c
foe"! * (1+ 2) = (+1) pest x (— ) ay tntl om tn\s 2 ag=0 ag+n+il—B
Or

44 - 1)! S 1)” yoru ti ( ul S J 1 ;
log" ay, Me ) atn+l i 2) agtn+1—B°> © (6)

which proves the correctness of the assumed result (4).

University of Pennsylvania,
Philadelphia, Pa., U.S.A

IX. On the Coilapse of Tubes by External Pressure.—111.
By R. V. Sourawe uy, M.A. » Mellow of Trinity College,
Cambridge™.

[> an interesting paper recently communicated to this
magazine tT, Mr. Cook has dealt with the resistance
offered to external pressure by short steel tubes—that is to
say, by tubes such that an appreciable part of their strength
was due to the sealing plugs which in the experiments were
employed to preserve their cylindrical form at the ends.
The subject is one which possesses practical as well as
theoretical interest, since his results, and those of other
workers on the same lines, are foundations upon which we
may endeavour to base scientific rules for the spacing of
“collapse rings” in boiler-flues.

I propose in the present paper to consider the bearing of
these results in regard to design. It appears to me that a
discussion of the problem trom this standpoint is becoming
urgently necessary, since the large amount of scientific interest
aroused by it in recent years { has resulted In a steadily
increasing number of formule, based either upon analy tical
investigations or upon isolated series of experiments, and
tending, by reason of their variety, merely to bewilder
anyone who has not made an extensive study of the subject.

In the first place, however, I desire to remove a slight
difficulty which may be encountered if Mr. Cook’s paper is

* Communicated by the Author.

+ July, 1914.

{ A full discussion of the problem, with a bibliography complete up to
the date of publication, was given in the Report to the British Associa-
tion Committee on Complex Stress Distribution (Birmingham, 1913),

F 2

68 Mr. R. V. Southwell on the

read in conjunction with my theoretical discussions of earlier
date*. This relates to the quantity termed the “ critical
length,” of which Mr. Cook’s definition differs slightly from
my own.

Both theory and experiment suggest that the length of
a tube sensibly affects its resistance to external pressure only
in the case of comparatively short tubes, and the earliest de-
finitions of the term “ critical length,’ given almost simul-
taneously by Profs. A. EH. H. Love f—as “the least length
for which collapse is possible under the critical pressure ””—
and A. P. Carman {—as a “minimum length, beyond which
the resistance of a tube to collapse is independent of the
length,”’—were in recognition of this fact. Prof. Carman
concluded further, from the early experiments of Fairbairn §
and from others which he had himself conducted, that “ the
collapsing pressure varies inversely as the length, for lengths
less than the critical length”’||. That is to say, the curve
suggested by him as expressing the experimental relation
between collapsing pressure and length, for a tube of given
thickness and diameter, consists of two discontinuous branches
—a straight line, representing constant collapsing pressure,
for all lengths above the critical length, and a rectangular
hyperbola intersecting this line at a point corresponding to
the critical length.

If these views are adopted, the critical length for any
definite size of tube may be determined from experiments,
by estimating (1) the straight line, parallel to the axis of
length, which best represents the collapsing pressure for
tubes of considerable length, and (2) the hyperbola which
agrees best with the results for the shorter tubes; their point
of intersection gives the required value. This is substan-
tially the procedure adopted by Mr. Cook, who finds that
within the range of his experiments the critical length L,
thus defined, is given satisfactorily by the formula

3
(a oe

being the thickness and d the diameter of the tube.

* Phil. Trans. Roy. Soc. A. vol. cexiil. pp. 187-244 (1918) ; Phil. Mae.
September 1913. .

+ ‘ Mathematical Theory of Elasticity,’ (2nd edition, 1906) p. 530.

{ University of Illinois Bulletin, vol. iii. No. 17 (June 1806),

§ Phil. Trans. Roy. Soc. vol. exlviii. p. 889 (1858).

|| Carman, loc. cit. p. 7.

Collapse of Tubes by External Pressure. 69
By calculation, 1 had previously obtained the formula

os Arad: ERR E)

« being some constant, depending upon the type of the end
constraints, of which I have not been able, except in certain
ideal cases, to obtain an exact value by analysis*; but, as I
have already stated, my definition of the critical length is
different from Mr. Cook’s. I had conciuded, as a result of
my analysis, that tubes of length such that the strengthening
effect of the ends is sensible, but small, will collapse under
a pressure given by

p= [ac +e al, SMe ali, Mea (a)

1 being the length of the tube, and « and 8 constants for
any given materialt. Clearly, as J is increased the col-
lapsing pressure given by this equation falls rapidly, and
becomes sensibly equal to Bt’/d’. Hence, adopting a slightly
modified form of Professor Love’s definition, I took the
eritical length to be “the least length for which collapse
is possible under [a pressure sensibly equal to] the critical
pressure.” L being thus defined, and 6 some small number
which we agree to regard as negligible, we have

ad i t?
or d* t?
“Ta = Pop:

whence equation (2) may be derived, « being equal to
Vv a] Bd.

No hyperbolic relation between collapsing pressure and
length occurs in the exact analytical treatment of our
problem. But the convenience of a relation of this form,
and its satisfactory agreement with experiment, suggest that
an hyperbola might with advantage be substituted for the
discontinuous curve which represents the exact theoretical
expression for the collapsing pressure. A curve of this
type is illustrated by the thick lines in fig. 3 (which may be
regarded as connecting pressure and length) of my paper in
the Philosophical Magazine for May 1913: it is composed

* Phil. Trans. Roy. Soc. A. vol. cexill. p. 227 (1913).
+ Cf. my equation (1), Phil, Mag. September 1913, p. 003,

70 | Mr. R. V. Southwell on the

of a series of intersecting ares, of which those shown in
the figure (as I pointed out in the same paper *) are very
appr oximately enveloped by a rectangular hyperbola.
The curves of which these arcs are fragments are those
members of the family represented by the earige
Gls eel 4
Paes | ey as KG -1)/p Me

in which k has positive integral values. & denotes the
number of lobes characterizing the cross-section of the tube
after collapse, and of the other quantities appearing in (4),
besides those which have already been defined—

E is Young’s modulus, and

iL e se . 72 .

— is Poisson’s ratio, for the material of the tube;
m

Z is a constant, depending upon the type of the end -
constraints.

Now it is easy to show that the curve represented by (4)
is touched by the rectangular hyperbola

BEA ae) ae ee
p= 58 ld k ee -—1)} d? ye

at a point given by
=! 4f oo m?—l d* 2
= AT 2 (pe a ieee a

and the occurrence of & in (5) shows that the family of
curves (4) is not exactly enveloped by any one hyperbola.
But the hyperbola touching that member of the family for
which £=3 gives values for the collapsing pressure which
are in satisfactory agreement with those obtainable from the
exact (discontinuous) curve, and which err on the side of
safety throughout the practically important range of lengths.
We may therefore take the equation to this hyperbola, viz.

m2 3
No ey) ge

as representing the saa pressure of short tubes, and
Prof. Bryan’s formula f

i ee te
32 = 2 T K a ° = . © ~ e ° ° (8)

* p. 698.
t Cf. Phil. Mag. September 1913, p. 603.
t Phil. Mag. September 1918, p. 504.

Collapse of Tubes by Eaternal Pressure. 71

for those cases in which it gives a higher value for 38 than
we should obtain from (7).

The theoretical value of the “critical length,’ as Prof.
Carman and Mr. Cook have defined the term, is then given
by the point of intersection of (7) and (8). We find

rrr ees

ye m?—1 a® ,
—— =o r= ce RI Te . . e 7 Q
L y) DO te (9)

which agrees in form with Mr. Cook’s equation (1). The
latter equation is therefore supported by my analysis,
although the “critical length” with which it deals is a
quantity differing from that which was considered by me in
my earlier papers.

The significance of the foregoing investigation lies in the
fact that the expression for the critical length given by (9)
is almost entirely independent of the material composing
the tube. In so far as it varies with the elastic constants,

4 i
Lo 4, 1—-_.,, Rian pe orev (iC)
and the value of the quantity on the right of (10) ranges
from 0°985 in the case of glass (for which 1/m=0-258) to
0-951 in the case of copper (for which 1/m=0°428)*. TJé is
highly probable, therefore, that Mr. Cook’s equation (1) has
an application much wider than the range of his experiments.

The point is, I think, worth investigation in future ex-
perimental work. If Mr. Cook’s equation should be found
to be thus generally applicable, it will introduce important
simplifications into the problem of design. For a knowledge
of the critical length and of a formula giving the minimum
collapsing pressure for a tube of given thickness, diameter,
and material is, as I shall now attempt to show, sufficient for
all practical purposes.

At the present time there is no generally accepted formula
for the collapsing pressure of long tubes. The results of
theory are naturally unreliable in practice, since the perfectly
elastic and homogeneous tube which it presupposes is an ideal
not practically realizable : those of experiment have the dis-
advantage of restricted scope and, further, give relations
between the collapsing pressure and the ratio of thickness to
diameter which are not expressible in any simple formula.
I have tried to show f that these experimental relations can

* The authorities for these figures are given by Love, op. et. p. 103;
the figures given above represent an extreme range of values.

+ Phil. Mag. September 1918.

2, Mr. R. V. Southwell on the

be explained ; but what is needed in practice is a simple and
comprehensive formula, involving constants which can be
determined for any given material from one or two of the
ordinary tests, without the employment of elaborate tube-testing
apparatus; and I do not know of any formula hitherto
published which satisfies these conditions.

The theoretical formula for long tubes has been given in
equation (8) of this paper. In form it has been found satis-
factory as a representation of experimental results for thin
tubes: the experimental constant given by Carman for steel
tubes is some 25 per cent. less than the theoretical vulue*,
but this reduction may be explained as due in part to un-
avoidable inaccuracies in his experimental tubes and in part
to his employment of rather too large a range of experiments
(his formula is known to give excessive values for the col-
lapsing pressures of his thicker tubes) in the determination
of the constant.

In endeavouring to explain the complete failure of the theo-
retical formula (8) to give the collapsing pressures of fairly
thick tubes, I have emphasized + the important part which
elastic breakdown plays in accelerating collapse. It would
seem, indeed, that we must not expect any iong tube to
withstand a pressure which is more than sufficient to impair
its elastic properties. ‘Thus, if y- is the stress corresponding
to the yield-point of a material in compression, we ought to
base our design upon the hypothesis that a tube of this
material will certainly collapse under a pressure given byf

Poy...

For tubes of less than a certain limiting thickness the
formula (8) gives a smaller value of the collapsing pressure
than this, collapse being possible, owing to the occurrence
of elastic instability, under a pressure which is not sufficient
to impair the elasticity of the tube, so long as it remains
circular. What we want, then, is an expression for #8 which
is practically equivalent to (8) in the case of very thin tubes
and which in no case exceeds the value given by (11).

* Cook (oc. cit. p. 52) estimates the reduction as 30 per cent., but I
think this figure is somewhat excessive; my estimate is based on the
figures K=30,500,000, 1/m=0°3.

+ Phil. Mag. September 1913.

{ Equation (11) tacitly assumes that the compressive stress is uni-
formly distributed over an axial section of the tube-wall. Though in-
accurate in the case of very thick or short tubes, this assumption is
substantially correct for all tubes of practical dimensions.

Collapse of Tubes by Eaternal Pressure. (ie

A simple expression fulfilling these requirements is

t 2 na
= LOO. ° e ° ° ° 2
2 a pe 12)

As the thickness is reduced, it approaches the limiting value
2K ¢*/d’, which is slightly less than that of Bryan’s formula
(8); and since the ratio d/ét cannot be less than unity, its
value is in all cases appreciably less than that which is given
by (11). Hence, provided that a cube is accurate in form
and of uniform material, it is clear that the pressure given
by this equation would be insufficient either to collapse it or
to impair its elasticity, so that the only modification required
for purposes of design is the insertion of a factor of safety
which shall make proper allowance for practical imperfec-
tions. Since the equation as it stands gives an estimate ot
the collapsing pressure which errs on the side of safety, L
believe that 2 would be amply sufficient as a factor of safety;
but this point can be investigated by a comparison with the
experimental results hitherto obtained.

he similarity of (12) with the Rankine-Gordon formula
for columns will be at once remarked. Dr. Lilly * was the
first to recommend the employment of an equation of this
form, but in place of y, and HE he suggests the insertion
of constants which are to be determined empirically from
the results of a complete series of tube-collapsing tests.
In general, when the comparison can be made, I imagine
that the difference between his formula and my own will be
negligible ; but the latter seems to me preferable, first, in
that it expressly provides against overstrain of the tube, and
secondly, owing to its greater scope: the appropriate factor
of safety may be determined from experiments on tubes of
any material, and when the corresponding formula for any
other material is required, it will be necessary only to change
the values of y. and EH, which are quantities determinable
from one simple test T.

It may be remarked here that in the case of the lap-welded
Bessemer steel tubes tested by Prof. Stewart f, the average
value of the stress at the yield-point was stated to be 37,000
pounds per square inch. If we assumea value of 30,000,000
pounds per square inch for the modulus of elasticity (which

* ‘Trans. Inst. Civ. Eng, Ireland, vol. xxxvi. pp. 188-164 (1910).

+ For practical purposes it will be sutticiently accurate to substitute ye,
the yield-point in tension, for y.; and an ordinary tensile test is therefore
all that is necessary.

{ Trans. American Soc. Mech. Eng., vol. xxvii. pp. 780-822 (1905-6).

74 Mr. R. V. Southwell on the

does not appear to have been determined), equation (12) will
give, as the collapsing pressure of these tubes,

74,000
1p = evs a ° ° . . ° ° (13)
1 * Bio
Dr. Lilly * has given the formula
80,000
BS ep ae Ci ae 14
a ae a
t * 10008

as very closely representing the results of this series of
experiments. Thus (13), which could be written down
without reference to any experiments besides that required
to find the yield-point and Young’s modulus, gives values
which closely agree with those actually obtained from
hydraulic tests, and which (as may be shown by drawing a line
to represent (13) in fig. 3 of Dr. Lilly’s paper) err almost
invariably on the side of safety.

It remains to discuss the bearing of Mr. Cook’s results
upon the problem of spacing ‘‘ collapse rings.” At a first
glance equation (1) may appear paradoxical: for if, as
has sometimes been assumed, collapse rings ought to be
spaced at distances equal to some multiple of the critical
length, the thinnest tubes will receive practically no re-
inforcement, whilst the thickest must be fitted with rings at
quite short intervals. This manifestly unsound result is due
to the inaccuracy of the assumption noticed above; for the
function of collapse rings is to strengthen a tube against
collapse by instability, and the ratio in which the resistance
is required to be increased (which, on the theory adopted by
Mr. Cook +, will be equal to the ratio of the critical length
to the distance between collapse rings) is obviously greater
in the case of the thinnest tubes.

At the same time, considerations of safety suggest that
collapse rings ought not to be placed too close together ;
for a point may be reached at which failure, if it occurs
at all, will involve rupture of the tube-wall close to the
rings t,—an occurrence which has far more serious con-
' sequences than the simple flattening of long tubes at collapse.

* Loe. cit. p, 145.
+ Namely, that the collapsing pressure is inversely as the length, for
tubes below the critical length. .

t In the author’s experience the walls of quite short tubes, which fail

at high pressures, were almost invariably sheared through at collapse.

Collapse of ‘lubes by Katernal Pressure. us

The best procedure would seem to be to arrange the spacing
of collapse rings in such a way that the resistance of thin
tubes is brought up to the value given by (il): they will thus
be enabled to withstand any pressure which is insufficient to
cause elastic breakdown, and a greater pressure than this is
in any case inadvisable.

If we take the collapsing pressure for a tube of critical
length as given by the equation

3
p=2h— 2 MRA Oe it

(which, as compared with (8), errs slightly on the side of
safety), and employ Mr. Cook’s equation (1) for the critical
length, the collapsing pressure for a tube in which the
collapse rings are spaced at distances s will be given by
the equation

= | ° ° ° ° ° (16)
Lee a a |
S a

Then if the resistance is brought up to the value given by

(11) we have |
t ae
OP eG ga (
a 346 7

or ats a ee am
ETT ae es eee rh

so that when collapse rings are to be employed we may use
(11) to determine the value of ¢/d, and fix the rings at
intervals given by (17).

Now the strengthening effects of collapse rings will be
mil if s>L: hence, by (1) and (17), their employment is
advantageous only when the dimensions given by (12) are

such that
t Yo oy
d SE s/t ° ° ° . . . ( i 8)

It may be remarked that this result will hold whatever be
the value of « in (2), provided only that the latter equation
is correct in form. Mr, Cook’s experiments suggest that it
is. ‘Taking for example the figures given by Prof. Stewart,
we find from (18) that collapse rings are of no advantage
on a tube of which the diameter is less than 28 times the
thickness.

eau x 2Ee,
s a?

76 Collapse of Tubes by External Pressure.

Substituting from (18) in (12), we find, as an alternative
form of the result, that collapse rings are useless when the
collapsing pressure exceeds a limit given by the equation

a a

This limit being of the order 1300 per square inch, the
equation suggests that collapse rings may be advantageously
employed in all boiler work. But questions other than that
of strength may dictate dimensions in practice, and hence
(18) is a more convenient form of the result.

Summary.

The paper consists of a review, written from the practical
standpoint, of recent theoretical and experimental work on
the subject of tube collapse. As a conclusion, the following
rules for design are suggested :—

1. Work in terms of a “collapsing pressure” 4, con-
nected with the (specified) working pressure p by the

equation
1D), - Dp, eee ee

where / is the factor of safety, for which (it is suggested) a
value as low as 2 will be sufficient.

2. When collapse rings are not to be used, fix the proportions
of the tube by means of the formula

3. When collapse rings are to be used, fix the proportions
of the tube by the formula

B=2" y. me

(making allowance for corrosion, &c., in cases where the
thickness suggested by this formula seems insufficient), and
use rigid collapse rings spaced at equal * intervals s, given

* T am indebted to Mr. Cook for suggesting the importance of egual
spacing, as a means of realizing the experimental conditions of “ en-
castred”’ ends when Adamson’s flanged joints are employed. Such joints
tend to keep the tube eicular, but their influence on the slope of the tube-
wall is asomewhat doubtful quantity. Myr. Cook points out, however,
that with equal spacing of the rings, each section of the flue will be kept
cylindrical at the ends, just as the ends of each span are virtually
“encastred” when a long continuous beam, uniformly loaded, is
supported by several equidistant piers.

Einstein and Grossmann’s Theory of Gravitation. 77

3
Sse Ay a Neate (LEY Bis

4. Collapse rings can be advantageously employed when,
and only when, the dimensions of a tube are such that

ee re agra:

In the equations (11), (12), (17) and (18), ¢ denotes the
thickness and d the diameter of the tube: these quantities,
and s, must be expressed in terms of the same units; y, is
the stress at the yield-point in compression (for practical
purposes the yield-point in tension may be substituted), and
Eis Young’s modulus for the material of the tube: these
quantities, and 48, must be expressed in terms of the same
units.

For additional security, it would perhaps be advisable
to substitute 1°5 for the factor 1°73, equation (17) having
been based on an estimate (15) for the collapsing pressure
of long tubes which is somewhat in excess of Carman’s.
Moreover, the figure 1°73 is based solely upon equation (1),
and, as Mr. Cook has remarked*, his tests cannot be
regarded as sufficient in number or covering a great enough
range of dimensions to confirm this equation definitely.

August 21, 1914.

by the equation

X. A Summary of Einstein and Grossmann’s Theory of
Gravitation. By Dr. A. D. FoxKer (Leiden) f.

i ERHAPS it might be useful to give a brief account

of the principal features of Hinstein and Grossmann’s
gravitation theory t, leaving aside as far as possible the
mathematical complications, but emphasizing the simple and
fundamental physical points.

Unfortunately, this theory will be of little direct importance
to experimental physics. Not because it fails to indicate
any experiments which could bring evidence for its validity
or non-validity, but because the foreseen effects probably are
far too small to be detected by present experimental methods.

* Loc..cit. p. 56.

+ Communicated by Prof. W. H. Brage, F.R.S.

{ Entwurf einer Verallgemeinerten Relativitatstheorte und einer Theorie
der Gravitation, Phys. Teil v. A. Hinstein, Math. Teil v. M. Grossmann,
1913 (Teubner). A. Einstein, Phys. Zevtschr. xiv. p. 1251 (1918).

78 Dr. A. D. Fokker: A Summary oj

The largest effect occurs in the case of the bending of light-
rays in a strong gravitational field. The strongest field
available is the sun’s. A light-ray passing near the sun’s
surface should suffer a bending through an angle of 0°83
seconds of are, so that the position of a star when the sun is
nearly touching it should be shifted away from the sun’s
centre. Perhaps the next eclipse may reveal such an effect*.
Another consequence of the theory is the influence of the
gravitation potential on the rate of action of physical pro-
cesses. or example, the vibrations in the atoms should be
siower at the sun’s surface than on the earth. As a matter of
fact, there has been observed a general shift of solar spectral
lines to the red side of the spectrum as compared with lines
from terrestrial sources T, but the solar conditions are so
complicated that no definite conclusion can yet be drawn
from this.

2. Notwithstanding the fact that the theory cannot give
much hope for new discoveries in experimental physics, it
cannot be said to be chiefly a mathematical speculation. For
throughout its development the lines followed are lines of
physical thought, and Hinstein’s intuition has only trusted
truly physical principles. The reason why these had to be
so few is that a century of experimenting has failed to bring
to light an appreciable influence of gravitation on other
phenomena.

Indeed, when the theory of relativity had concluded that
gravitation was to be propagated with the speed of light f,
it was difficult to look for an extension of Newton’s theory,
which henceforth had to be considered as a first approxima-
tion, without any new experimental indications as to the
direction in which this extension was to be sought. There
seemed to be too much freedom, the number of possible
assumptions seemed not to be restricted enough to point in
any definite direction.

Einstein considers of fundamental importance the fact
that all bodies fall with the same acceleration ; combined
with the assumption of the identity of gravitating and inertial
mass, it led him to work out the consequences of his ‘ Aequi-
valenz-Hypothese,’ which will be described further on.

Next he bases himself on the principle of the conservation

* War has made this impossible.

+ Compare EK. Freundlich, Phys. Ze:tschr. xv. p. 369 (1914).

{ For the consistency of this propagation with astronomical observa-
tions cp. H. A. Lorentz, Proc. R. Ac. Sc. Amsterdam, viii. p. 603
(1900) ; H. A. Lorentz, Phys. Zeitschr. x1. (1910).

Einstein and Grossmann’s Theory of Gravitation. 79

of energy, which seems to be a sufficiently trustworthy basis,
and admits that the principle of relativity shall be valid in a
particular case, that is, its formulee will be correct whenever
the gravitation potentials are constants independent of the
coordinates. To be complete, we state explicitly that it is
believed that the results of experiments in a laboratory are
not altered when the laboratory, as a whole,is taken to a
place where the gravitation potentials have other values.

Starting from these ideas, Hinstein has been able to give
the laws according to which matter is affected by a gravi-
tation field, and a gravitation field created by matter. It is
of special interest that. his theory is not constructed after
the known model of the electromagnetic theory (though we
find certain analogies to be present). He places the phe-
nomena of gravitation cn a higher plane, and accordingly
shows how the equations of the electromagnetic field are to
be altered when allowance must be made for the presence of
a gravitation field.

It is characteristic of his theory, that the field of gravita-
tion is not given by a single potential, but Dy a set of ten
potentials, functions of the coordinates *. These ten poten-
tials are the components of a symmetrical tensor. Further,
the important thing which is acted upon by gravitation and
which produces the field, formerly the ‘‘ mass”’ alone, is in
the present theory a tensor of stresses, momenta, currents of
energy, and energy. ‘This will be made clearer afterwardsf.

The Equivalence [Hypothesis.

3. The equivalence hypothesis briefly assumes the equi-
valence of a homogeneous field of gravitation and a uniform
acceleration of the : system of coordinates.

Consider the case, that an experimenter is working in a
room without any window, so that he cannot know anything
about things outside, and does not know whether he is in
relative motion against an outside world or not. Let the
only remarkable thing he notices be this, that all bodies fall
down to the floor when he lets them loose, all with the same
acceleration, in the same direction. Shall he be able to
state whether his room is in a homogeneous gravitation field
or whether there is no gravitation field at all, the cause of
falling down and apparent weight of his bodies being a
uniformly accelerated motion of his room thr ough space! ?

He cannot. He has no criterion. Of course, if all bodies
did not fall with the same acceleration, there would be no

* See § 11. t See § 12.

80 Dr. A. D. Fokker: A Summary of

reason to admit the possibility of his room being accelerated.
There would be no reason to suspect that there was anything
the matter with his room, affecting all bodies in the same
way. Again, there could be no reason for doubt, if he could
find an inertial mass which did not gravitate. Yet, as far
as the experiments of Hétvés have gone*, they seem to
confirm that bodies which are attracted by the earth with
the same force have equal inertial masses as measured by
the centrifugal forces excited by their motion in the earth’s
daily rotation.

It is clear, that when we consider the motions of bodies,
using the ordinary mechanics, no distinction can be made
between a homogeneous gravitation field and a uniformly
accelerated system of coordinates.

4. We may put the equivalence hypothesis in another
form. It we know the laws of motion in a field without
gravitation, we know the differential equations connecting
the time and the coordinates referred to a certain system of
coordinates. We are not obliged to describe the motion
with reference to this system. If we choose to do so, we
may describe the motion with reference to another system
which relatively to the first is uniformly accelerated. Of
course we shall have then to introduce alterations in our
equations.

The equivalence hypothesis states that the alterations to
be introduced are the same as those which we have to make
when there is a homogeneous gravitation field affecting the
motion.

It extends this statement beyond the region of mechanics,
It assumes that for all physical phenomena, when we give
the laws referred to an accelerated system of coordinates, the
differential equations will undergo the same variations from
what they were in the resting systems as they would suffer
if we produced a gravitation field.

5. Starting from this idea, it is easy to deduce in an
elementary manner some important consequences J.

For example, consider two sodium atoms, placed one above
the other at a certain distance h, and fixed on the Z-axis of
a system of coordinates that has a constant acceleration (y)
upwards. Let at some instant a signal, consisting of a train
of wavelets, be sent from the upper sodium atom to the

¥ B. Kotvos, Mathem. u. naturwissensch. Berichte aus Ungarn. vii. 1890.
Wiedemann, Berbldtter, xv. p. 688 (1891).
+ A. Einstein, Annalen der Physik, xxxy. p. 898 (1911).

Einstein and Grossmani’s Theory of Gravitation. 81

lower. To cover the distance h a certain time is required,
r e e .

say — asa first approximation, when ¢ denotes the velocity

of light. Then the velocity v of the upper atom at the
moment of sending the signal is less than the velocity

mtr of the lower atom, when the signal arrives there.

Thus the wavelets on arrival, according to Doppler’s principle,
Oe IN pe

will appear to have a wave-length which is (1 + a) times

shorter. According to the equivalence hypothesis, the same
would be observed if the sodium atoms were at rest in a
svstem where a homogeneous gravitation field existed having

atom, compared with another that is placed in a spot where
the potential i is less by an amount Ad, would seem to vibrate

a gradient —-=y. This means that a vibrating sodium

(1+=) times as fast as the latter. Thus, an observed

terrestrial sodium line ought to be shifted to the violet, when
compared with a solar sodium line.

Again, let a beam of light be sent at a certain moment by
a horizontal collimator at a point c=a, y=0, z=b of the
moving system towards the axis of Z. To cover the distance
a, a certain time is required. When the beam of light is

observed by a telescope fixed to the Z-axis, this telescope will
in the meanwhile have acquired an upward velocity vy+ ele

if v9 is the velocity of the collimator at the moment of send-
ing the beam. According to the common aberration theory,
the telescope will have to be directed a little upward to
observe the beam. In the accelerated system the light-rays
will apparently be curved lines. By hypothesis, the same is
to be the case in a homogeneous gravitation field. In order
to account for this apparent rotation of the wave fronts we
must, following Huyghens’s principle, conclude that the
velocity of light is greater in upper regions, where the
gravitation potential is higher.

Of course these considerations are only approximate, but
they bring out important conclusions of the theory.

The transformation from a resting into an accelerated
system of coordinates.
6. The equations which relate the coordinates and the
time, §, 7, ¢, and 7, of the resting system to the coordinates
Phul. Mag. 3. 6. Vol. 29. No. 169. Jan. 1915. G

82 Dr. A. D. Fokker: A Summary of

and time, 2, y, ¢, ¢, of the moving system must now be indi-
cated more exactly. In deducing them use is made of the
postulate, that the velocity of light at each point of the
moving system should be independent of the direction of
the light-beam, and therefore a function of the coordinates
only. In our case, where we shall suppose the system
(x, y, 2, t) accelerated along the axis of Z, the velocity is a
function of z only. lLorentz* gave the equations in the

exact form
E —dy 7—Y,
C=a\z—z), J. es

cT=)(z—Zo),

where c¢ is a constant, the velocity of light in the resting
system, and

a=i (e+e), b=h (e#—e-*).

The constant & is connected with the (variable) velocity of
light, c’, in the points of the moving system by the relation
Ic’! =k (z—2).

The approximate equations given by Einstein ¢ and valid
for very small values of t, so that t* may be neglected, are
easily deduced from these. They are

ey.

le— xu
E=a, =;
Ci Cis

C= (14 3 2)(e—m) = 2-2) +7

From the last equation we see
TE gg
Sty eee

that at t=O the starting acceleration of the different points

2
of the system is given by g)= ee . Speaking exactly, the
0

-acceleration is not the same for the different points of the
system. Nor is it the same throughout the time. A per-
fectly constant acceleration, by the way, would lead to a
contradiction with the old relativity theory, because it would
lead to an infinite velocity. We can see more distinctly

* H. A. Lorentz, Het Relativiteitsbeginsel. Drie voordrachten,

bewerkt door Dr. W. H. Keesom, 1913 (De Frven Loosjes, Haarlem).
+ A. Einstein, Annalen der Physik, xxxviii. pp. 359, 444 (1912).

Kinstein and Grossmann's Theory of Gravitation. 83

what the acceleration will be if, availing ourselves of the
relation a?—}?=1, we notice from (a) that

C—er?=(z—2)?, C=/(2—H)? +7,
and, therefore, for a point with a fixed value of z,

OE RSME NN
d/o ener.
so that the velocity of the moving system will never exceed
the velocity of light. From this follows for the acceleration

aa C(z—<2)* ;
I= Ga A ee

This gives for t=O the starting acceleration gp as found
above. The constant & has no particular meaning. It re-
lates the value of the velocity of light as measured in the
moving system to the velocity of light in the other. If go
be the acceleration of the origin z=0 at the time ¢=0, and
we want our system of coordinates defined in such a manner
that at this same point and time the velocity of light c’o9 be
the same as in the resting system, then we have to take

2
e G e

k=2. For we see easily that z9= — — , and if c= —kz,

G Joo

is to be equal to c, then we must have £=go9/c, as stated.
Lastly, we may notice that io the differential equations

d€=adz+be'dt, edt=ac'dt+bdz
correspond, by virtue of a?—b?=1, the reciprocal equations

dz=ad€—bear, cdt=acdt—b dl.

Jotion of a free particle.

7. Now, knowing the motion of a free point through a
space without gravitation (and such is the motion in our
resting system) to be in a straight line, Hinstein, by using
the relation of the coordinates of the two systems, could find
the equations of motion when referred to the moving axes.
He found it possible * to contract them into the form of
Hamilton’s principle :

}f \Hide ' —i{(\\s

* Ann. d. Phys. xxxviil. p. 458 (1912).
G2

84 Dr. A. D. Fokker: A Summary of
in which H' was to be put
H’=—mWvc?—v?,

v being the point’s velocity with components de ne ,a

: es dt ?dt “dt
m 1tS mass.

There is a striking resemblance between this form and
the form in which the equations of motion may be given in
a space without gravitation. It is known that according to
the principle of relativity in that case the function H must
be written

Dh ma dé\? dn\? dé 2
H=—my/e eG) Ge

and that Hamilton’s principle in that case is

é{\H at} =0.

If we define as the length of the four-dimensional line-
element determined by dé, dy, d¢, dr,

ds=/ @dP dE dp aa,
we may, omitting the constant tactor —m, put
§3\dst=0,

which means that the moving particle between two points
of its path traces the shortest line possible through the four-
dimensional space.

If we try to express the line-element ds in the differentials
of the new coordinates, we find, as a matter of fact, that

ds=V/ Cdr? —d&—dr? —d@=V/ ce? dP? —da? —dy?—d2.

Thus we see that 84 \ds .=0 expresses equally well our
equation 64 \ Hide ‘=(, and that the free particle, which in
the system of (a, y, Z, t) is a free falling particle, still moves
along the shortest line possible through four-dimensional
space, if ds again defines the length of the line-element
given by dx, dy,dz,dt. The length of an element ds appears
thus as a quantity not altered by a transformation of
coordinates.

The only quantity in the equation which is a function of
the coordinates, and therefore might and does play the role
of a gravitation potential, is the quantity c”, and we see that
it is nothing but one of the coefficients determining the
Jength of ds in terms of da, dy, dz, dt.

Einstein and Grossmann’s Theory of Gravitation. 85

It is very satisfactory that the equation of motion is now
put in a form which is not affected by our transformation of
coordinates. The statement that a free particle always takes
the shortest possible track between two points of four-
dimensional space is very simple, and reminds one of the
principle, put forward by Hertz in his Prinzipién der
Mechanik, that a free system jmoves along the straightest
line possible.

The Gravitational Potentials gyy.

§. The next step is to consider Hamilton’s principle in

the form
}} ds} ==).

to be valid still farther beyond the present case. It holds
in the old theories, and it holds after the special transforma-
tion of the coordinates which is equivalent to a certain fairly
homogeneous gravitation field. _We will now assume that
it will hold also after an arbitrary transformation of co-
ordinates, which will be equivalent to an arbitrary unhomo-
geneous field.
Now, when we execute an arbitrary transformation

Bess Oy Sy Uy)
N= Joa Ye 2.0),
C=f3(x, y, 2, t),
i fal Puy Zeb)
then, of course, substituting in ds? for the differentials dé, dy,

dt, dr their expressions in de, dy, dz, dt, the line-element
will be expressed by a form

ds? = 91,da? + Aqyoda dy + 2q3dadz + 2q,,dvx dt )
+ gosdy? + 2gogdy dz + 2qody de |
+ go3dz? + 2q3,d2dt

rw
+ gud, 4

in which there are ten coefficients g,,, in general all of them
functions of the coordinates. In the equation of motion
these ten functions will give the influence of the gravitation
field on the motion of a particle. 3

Thus it appears that in the theory there will be henceforth
a set of ten gravitation potentials.

In the particular case of the fairly homogeneous field we

86 Dr. A. D. Fokker: A Summary of

considered this set degenerated in a set of four:
19 eee a== —_ se ae! a
Gun = 922 = 933 = —1, Ju—C > (912= 913 = 914 = 923 = 91 = 934 = 9),

of which only gy=c’? was a variable function of the co-
ordinates. In the absence of a gravitation field even this
potential becomes a constant.

If we write the fundamental equation

s{\ds;—0, . . .) se
in the form of Hamilton’s principle,
}4 \ H'de} ee
by putting
H'’=—m oe
at
then we know that this equation is equivalent to
Ce ea
at 5) Of
It is through the ten potentials g,, that H’ depends on the
coordinates.

Laws of Conservation.

9. In order to show how phenomena are affected by gravi-
tation, and the gravitational field is created by matter, and
how the laws of conservation of energy and momentum are
preserved in the theory, we shall have to use tensors. To
introduce them it will be best to show how the laws of con-
servation can be expressed for a special case in electro-
dynamics, and in absence of a gravitational field.

Let d and h denote the electric and magnetic vectors in
free space. We may conceive stresses X,, X,, X,, &e.,
existing in the electromagnetic field, and also an electro-
magnetic momentum Iz, I,, I, per unit volume. If X,
denotes the pressure per unit area exerted on the field at
the positive Y side of a surface perpendicular to the axis of
X by the field at the negative side of this surface, and if
X, denotes the force per unit area in the direction of X
exerted through a surface perpendicular to the field at the
negative side of it on the field at the positive side, and °
if X denotes the same with regard to a surface perpen-
dicular to the Z-axis, then the X-component of the total
force upon an element dz dy dz is

_ (2%, BOX

oa ue a + So) dirdy dz,

Einstein and Grossmann’s Theory of Gravitation. 87

and this must be the increase of the momentum present in
the element, if the law of conservation of momentum is to
be fulfilled. Thus we see that this law is expressed by the
equation :

Ox 0x), (Ox or

+a tata

Ow Pe) y Oz ot
Similarly, if Sz, S,, S,, and E denote the currents of energy
in the field and the energy per unit volume, the law of
conservation of energy is expressed by the equation :

OS: Os, ON, OK as
Ou ay 0- uae

Now, introducing the symmetrical coordinates

US a0 HRS, eek
and writing Lo(co=1, 2, 3, 4, v=1, 2, 3, 4) for our stresses,
momenta, and energy, so that

Ly lie luis Liu x x 5G cL,
Tiny Ling Lng Ling ie NE) MG ioly
1 UN nese eke i Up Sy Ses iy eae en)

Helis) Lane = SUNT sin

C C ONG
we see that these equations have all the same form,

Olin Oly, OL); Olu
sem ()
TH Os one ae?

and that the laws of conservation of energy and momentum
ean be concentrated in one formula :

> Obie, ae 0
ih CORE
If we express the quantities L,, in terms of the components
of d and h according to the formule of Maxwell, Poynting,

and Abraham, we see that the set of the functions L,, form
a symmetrical square. Here they are:

+{@? ta 2d,” +h?— 2h,”}, ie d,d, ray h,h,, rey dd. a h,h., 2(d,h. rap d-h,),
—dd,—hh, 4{4?—24,?+h?—2h,2},—-d,d-—h,h., i(d-h.—4,h.),
—d,d,—h,h,, —d,dy—hzhy, 4{4?—24,?+h?—2h,°}, i(dchy—dyh,),

i(dyh.—d-h,), i(d-h,—d,h.), i(d.hy—ad,h,), —3}(d? +h’).

88 Dr. A. D. Fokker: A Summary of

When there are external forces acting upon the field
through the electrons, the laws of conservation are no
longer fulfilled for the field alone, the equations are to be
replaced by four other ones, contracted into

Olice
2 Oar

where I*, denotes the force from the charge upon the field,

1
FF, = —p(a or [v,h.—v.h,]),

and the work done upon the field -

— the

i= —p ; (vide + Vydy + V.4.),

these being equal and opposite to the force of the field upon
the charge

But I'6 being the force exerted by material agents upon
the field, there must be an equal loss of momentum and
energy of these agents per unit time. Denoting by: M,, the
stresses, momenta, currents, and energy in the matter, the
laws of conservation demand that

Thus, finally, the laws of conservation are now expressed
by
>» OCs == Mo)

Ou, =r

Tensors.

10. A set of sixteen quantities such as L,, is called a
tensor of the second order. It is a fundamental property
of a tensor that when a transformation of coordinates is
executed the components Ls, of the tensor transform them-
selves like dagdz,. For instance, when we transform our
coordinates in the following manner

L=Puly +Vote! +P13@3' HP ues’,
Lg = Prot + Pogty’ + Pasits 5 Pails
3 = Py3hy' + Pog’ + p33 ts + 3s’,
U4 = Prstey’ str Poss’ ate Psat’, Fie Psst, :

then a tensor as a ‘‘ geometrical quantity’? 1s mathematically

Einstein and Grossmanin’s Theory of Gravitation. 89

define] as a set of quantities L,, which transform themselves
like

We can convince ourselves of the fact that L,, is trans-
formed as dag dx, either by direct calculation or by remarking
that the stresses, &c. in the electromagnetic field have the
same dimensions as stresses, &e. in matter, 7. e.1n a (viscous)
gas. When the distribution law of velocities (&, , ¢) is
T(E, 0, ©) dé dy dé, so that in a space dS there are fd& dn df dS
molecules with the given velocity, then for X,, the amount
of momentum carried across unit of an area perpendicular
to the axis of X, we find

} m da dx
X=) d& dy de Jee dt di’

Similarly, for X,

v
Xo= (rae dy ee ee ey.
Jf =v dt dt
and so on.

We see the products dw dw, dx dy, that is dvzdx,, come in,
and entering into details we could prove that indeed the
components of a tensor of stresses, momenta, and energy
transform themselves as d.t¢ di,.

This property is of great use when we wish to write
equations in a form that is invariant against transformations.
Tt causes the four quantities

S52 @=L238

to transform themselves as dive, that is, they are the com-
ponents of a four-dimensional vector (which might be called
a tensor of the first order). ‘Therefore, when they are equated
to the components of a vector, such as the force (Ic), then
both members of the equations

5 Obey _ yp
Oy

are transformed in the same way, and the equation persists
in the same form:

90 Dr. A. D. Fokker: A Summary of
sonal

11. As long as we deal only with the linear orthog
transformations of the principle of relativity, the properties
of tensors are relatively simple. But our aim is to consider,
and we did already consider quite general transformations
vy = fi (y', &2', a3', vy), so that in the transformation formule
for the differentials

dx, = > py. da,
K
the coefficients p, are not such as to make the transformation

a linear orthogonal one. Therefore the coofficients 7. in.
the reciprocal equations

(ih ee JE, GP. da,
4
are not the same functions of the coordinates as p,., and we
must now distinguish between different kinds of tensors of

the second order, namely, covariant tensors which transform
themselves by the formula

Vv if
LoS ~ Pas PB» Te
a

contravariant tensors, for which

!
6 ua = Taw TT Bv 0,6,

a8
and mived tensors, which follow the rule
DD iy aoe thee Lee:

For instance, the quantities g,, which define our invariable
line element

Ose ?
ds’ = Sg duds,
pV
form a covariant tensor :
Choa Calor Winey) Cha
Jar 9Jo2 Yes Yaa
931 932 933 Ys (Gur = Jun)
Ya Yao G43 Yaa

On the contrary, if y,, is the minor of g,, in the determinant,
divided by the determinant itself, then the tensor of the y,,,

Via Vien Pisy wis
Yor Yoo (i) Yost uehyos
Ys1 Yao" a3. V34

Yar > 497 Yas) 1744.
is a contravariant tensor.

Einstein and Grossmann’s Theory of Gravitation. 91
ory O

Each of both tensors can be taken as a representation of the
gravitational field.

Action of Gravitation on Matter.

12. The tensor (Tey) which is closely connected with the
stresses, momenta, and energy is a mixed tensor.

Of course, when a gravitation field acts upon matter (let
us include an electromagnetic field in the term matter) it
cannot be expected that the laws of conservation of momentum
and energy will hold for matter in itself alone. Obviously
the gravitational field can impart energy and momentum to

oT, does not
B

02,

vanish now. Einstein gives for the influence of gravitation
on other phenomena the formula

OTey 15: Bou» '
2a, pate Te Try. ° e e ° (3)

BVT

the material system. In fact, the form >

We notice that the terms which on the right-hand side of
the equation determine to what extent a given field will
influence the physical phenomena, are precisely the com-
ponents of the tensor of stresses and energy. This was
alluded to in section 2.

As soon as matter and gravitation field are considered
together, then of course the laws of conservation must be
fulfilled. The existence must be supposed of a tensor of
stresses, momenta, and energy in the gravitational field
itself. Its components will be functions of the potentials
Juv OY Yuy and their derivatives, and, when this tensor is
denoted by t,,, we shall demand that the laws are expressed
by the equations

oe ine, Cte =.

Drigerential equations for the creation of « gravitational
jield by matter.

13. This demand, this application of the laws of conserva-
tion, has been a guide in investigating the form of the dit-
ferential equations by which the gravitational field is
determined. There must be ten of them, because we have
ten potentials. Of course they are to be expected to be
extensions of the known equation of Poisson:

Ad = kp,

where p is the density of the attracting mass, and ¢@ the

92 Dr. A. D. Fokker : A Summary of
potential. We expect that our differential equation will be
Gor = KT oy,

where Gs, denotes a tensor derived by differential operations
from our potentials, containing differential coefficients up to
those of the second degree. It has in special cases and with
certain simplifying neglections to become the same as Ad.

Now, when we put Try =Gor/K in the right-hand side of
the first equation of the previous section, we ought, as the
second equation indicates, to be able to show that the right-
hand side is identical with a sum of differential coefficients.

Indeed, Einstein has succeeded in doing this. He finds
that the identity exists if we put

1 = = O90 OY» if O9re OY
ea Re O9re OYre
yy J BTp Be, fe) vo 028 2 aTp Oop a8 Aa Ole 0. Lo
(4)
Here g denotes the determinant of the g,,, and 6,,, a quantity
which equals 0 for o #v and 1 for c=».
Otoy
Owy
identity are the differential coefficients of the tensor
Ouze Ore 1 S OJ» OYzp
eG i 5 ny ae )
w=! Hi ¥B Dao’ Og 2B BTp av lee ee OLa” C&z
(9)
so that the stresses, momenta, and energy of the gray itation
field are to be pken as given by this opncnn ey

A very important result is’ seen when we compare this
formula with the preceding. It then appears that

It appears that the differential coefficients in the

tov =

(Toy + tov) = x2
app Oa

OVur
(v= g YaoIon Sm.) (6)

i. e. the tensor of stresses, Ke. of the gravitation field enters
exactly in the same way into the differential equations
determining the potentials as the material tensor does. The
gravitational stresses, momenta, and energy evert the same
power in creating the field as the material ones do. This is
uite satisfactory. There is no reason why the energy ec.
of the field would behave otherwise than energy of- matter.

Einstein and Grossmann’s Theory of Gravitation. 93

As already said, the given differential equation fits in
with the formula
| ps oa bee (Toy + tov)
Oty

showing that the laws of conservation are fulfilled.

SP ORM ay das Sec cae Quie)

Approwimative simplifications.

14. The differential equations for gravitation appear to be
very complicated. However, there is a way of simplifying
the equations and getting successive approximations. It
has already been said that in the case of constant potentials,
let us now say in absence of a gravitation field, the funda-
mental tensor of the gu» becomes

—1 0 0 0
0 —1 0 0
0 0 —1l 0
0) 0 0 Co

and accordingly the tensor of the yp

—1 0 0 0
0 —1 0 0
0 0 —1 0
Gun OM OL Mei sey,

The first thing we can do is to assume that in the actual
case of our agline system the values of g Juv and yyy will diifer
only slightly from those given above by very small quantities
gpy and Vu » and that, ene as a first approximation,
we can mail in the differential equations those terms con-
taining products of two gi, or y,, or their derivatives.

a hen, besides, we absttact from the actual oxisting
motions, assuming that the velocities are so small as to

i He
make ~ and -, negligible, then the equation of Poisson
C G

for g;, is the only one retained, and we get Newton’s theory,
where gi, plays the role of the usual gravitation potential.
Considering the significance of gu=cC+%, in the form for
the line slenent te the conclusions may be drawn about the

dependency of the velocity of light and of the rate of action
of processes on the gravitation potenti: als which we mentioned

before (§ 5).

94 Dr. A. D. Fokker: Al Summary of
Relativity of Inertia.
15. When we take into account terms with ss then
€

equations of motion may be derived for a moving particle
which furnish us with some remarkable conclusions as to
the relativity of inertia.

It was pointed out long ago by Mach* that we cannot
speak of mass in an absolute sense. Just as we can only
speak of the motions, velocities, and accelerations of a body
relative to other existing bodies, so we only come to consider
the inertize of different bodies when we study their mutual
action. Mach concluded that we are not justified in thinking
of the mass as of something absolute belonging to a particle,
but that it may be due to some inducing influence of bodies
one upon the other.

If this is true we should expect first, that the inertia of a
particle is increased by heaping up other masses in its
neighbourhood, and secondly, that an accelerated mass
induces an acceleration of the same direction in other masses.
For if two masses are accelerated together (amidst other
bodies) their mutual acceleration is zero, and the resistance
against the mutual acceleration, which is the inertia that
they mutually induce one in the other, ought not to come
into play. So that when A and Bare accelerated together the
force required to give A this acceleration is less than it was
when B remained at rest. This is the same as saying that
the acceleration of B actually gives rise to an accelerating
force in A.

It is remarkable that these conclusions follow as con-
sequences from Linatein’s theory. Indeed, by bringing the
other masses nearer, the inertia of a particle is expected to
increase, and a sudden acceleration of neighbouring masses
would cause an acceleration of the particle.

Unhappily, the amount of the expected effects is so small
that there is scarcely any hope of discovering them.

Concluding Remarks.

16. In the absence of experimental evidence, certain points
may be laid stress upon which distinguish Hinstein’s theory
from other theories, and give it high intrinsic merits.

It is an advantage that the theory regards the velocity of
light no more as an absolute constant. There was something
unsatisfactory in the unexplained existence of a certain
constant critical velocity.

* “Prinzipien der Mechanik in ihrer historischen Entwicklung
dargestellt.’

Hinstein and Grossmann’s Theory of Gravitation. 95

Of fundamental importance is the conclusion of the theory
as to the relativity of inertia. Our classical conception of
inertia dates from Galilei, and can be said to be derived from
the observed behaviour of bodies acting upon one another.
But the underlying tacit assumption was that the bodies
would behave just the same if they were an isolated system,
and cut off from the remaining part of the universe. This
is corrected by the theory of LHinstein, which makes an
influence of this remaining part of the universe responsible
for the inertial properties of single systems.

Still more essentially in favour of the theory are the
following considerations, which really form the very nucleus
of all conceptions of relativity.

In order to describe physical phenomena we must construct
systems of coordinates, space-coordinates, and a time-co-
ordinate. With reference to these systems we can express
physical relations by certain equations. Now there are two
possibilities. Hither the equations exist only with reference
to certain specialized systems of coordinates, or they exist
independent of our choice of coordinates, and retain their
form after an arbitrary transformation of coordinates*. In
the first case the equation can be suspected to owe its
existence to a special artifice of choosing the coordinates,
and not to correspond to a real relation. In the second case
the equation can only owe its existence to a real relation
existing in the nature of things.

That the real relations in nature, and the equation ex-
pressing them, are to be independent of any choice of
coordinates whatever, is the principle of relativity in its
purest and most general form.

This principle was in the older theory of relativity limited
to those systems of coordinates connected by the linear ortho-
gonal transformations for which the Huclidean four-dimen-
sional element d7?=da?+ da,.?+dz,? + dx? was an invariant.
The Verallgemeinerte Relativitdtstheorie tries to apply the prin-
ciple in its full extent for all transformations which leave
the non-Huclidean general form ds?=Xqyy dv, dxy invariant.
It is for being able to express the laws in their covariant
forms that the complicate ‘absolute differential calculus”
with its tensors is worked out.

In fact, the fundamental equations (2) and (3) preserve
their form unchanged whatever transformation of coordinates
is executed. So do equations (5) and (6). The same cannot
be said of the equations (4) and (7). These preserve their
form only when the transformation is a linear one. The

* A. Einstein, Phys. Zettschr. xv. p. 176, Feb. 1914.

96 Mr. A. EH. Young on the Form of a Suspended

reason for this lack of generality is the condition that the
laws of conservation should be fulfilled. This limits the free
choice of coordinates and specializes the admissible systems.
Tt is, however, to be expected that to these differential equa-
tions there will correspond other general equaticns, which
preserve their forms for more general transformations.
Hinstein and Grossmann have found these equations in
March 1914, in the form of a variation principle analogous
to Hamilton’s principle. Indeed, Hamilton’s principle is
more general than the principle of conservation of energy.
Their results have been published in the Zeitschrift fir
Mathematik und Physik, |xii. p. 215 (May 1914).

The general covariancy of the equations is the great
achievement of Hinstein and Grossmann’s theory.

Leeds, July 1914.

————d

XI. On the Form of a Suspended Wire or Tape including
the Kffect of Stiffness. By ALFRED Ernest YOUNG,
Assoc. Mem. Inst. CE, Fellow of City Guilds Institute,
late Deputy Surveyor-General in the Federated Malay
States™.

N the Philosophical Magazine for July 1903 there

appeared a paper by Professor Richard C. Maclaurin

on “The Influence of Stiffness on the Form of a Suspended
Wire or Tape,” the opening paragraph of which states :—

‘‘ Some of the greatest improvements in modern surveying
are due to the substitution of a steel tape or wire for the old
surveyors chain. The newer instrument can, with proper
precautions, be made an exceedingly accurate measurer of
distances. So minute have been the corrections applied in
some recent surveys, that it it has been questioned whether
we may, with propriety, any longer regard the form in which
the ‘chain’ hangs as a catenary. It is true that the sur-
veyor’s tape is so thin as to be very flexible, but for some
purposes there are advantages in using a circular wire,
which is, of course, more rigid than the tape for the same
weight. It may be thought that, at any rate for the circular
wire, the hypothesis of perfect flexibility (on which the
investigation of the form of the catenary rests) may intro-
duce an error comparable with those for which corrections
are applied in the best modern surveys. The object of this
paper is to settle the matter by investigating the correction
that must be applied when the rigidity of the tape or wire is
taken into account.”’

* Communicated by the Author.

Wire or Tape including the Effect of Stifiness. 97

Professor Maclaurin then proceeds to form from first
principles the differential equation for the curve in which
a flexible wire hangs under its own weight and an applied
tension. This equation, which is of the fourth order, he
proceeds to solve by approximation, and eventually arrives
at a formula for the effect of stiffness on the ‘Sag Correction’
as deduced from the ordinary catenary formula. In an
example which would be considered extreme in Seay
practice, 7. e. a steel tape 10 chains long, } inch wide, #5 inch
thick, wholly suspended under an end tension of 14 lb., this
formula gives a stiffness correction of only 0°000000023 inch
—equivalent to 0°184 inch in a million miles—from which
Professor Maclaurin concludes that the neglect of the
stiffness of the chain need cause little anxiety to the
surveyor.

Before the writer had seen Professor Maclaurin’s paper,
his attention had been drawn to the effect of stiffness while
he was engaged in measuring with a steel tape a base line
in connexion with the Trigonometrical Survey of the
Federated Malay States, and he had worked out a formula
for this correction on the assumption that the sag was small,
as it generally is in practice. The tape was in fact regarded
as an elastic beam subjected to an end tension ie addition to
its own weight, and supported “clamped” or “free” at
the ends. On comparing his results with Bioteesn Mac-
laurin’s, they were found not to agree, and for many years
the writer was unable to account for the discr epancy, failing
to discover any flaw in Professor Maclaurin’s investigation
or his own. He at length handed over the question to
an assistant, Mr. D. T. Sawkins, B.A. (Cantab.), who was
employed for some time in the Malay States Survey Depart-
ment as a surveyor, and who succeeded in discovering the
cause of the discrepancy, which appears to be due to Pr ofessor
Maclaurin having neglected to use the complementary
function in the “approximate solution of his differential
equation. This complementary function comprises the
larger part of the correction, and though ‘the whole cor-
rection is so small as to be seldom or never appreciable in
practice, it is much larger than would appear from Professor
Maclaurin’s paper. The object of the present paper is to
investigate the correct formula for the stiffness correction,
to show where the error occurred in Professor Maclaurin’s
solution ; afterwards to give some results in the Bunemal

catenary formula and its variation when the elastic extension
of the tape is taken into account; and, finally, to give the
sag correction when a heavier tape is included in the series

Phil. Mag. S. 6. Vol. 29. No. 169. Jan. 1915. H

98 Mr. A, E. Young on the Form of a Suspended

in sag, which results the writer thinks may be of interest
and perhaps of utility to those engaged in making surveys
with the long steel tape.

That a steel tape is not perfectly flexible is easily seen by
the surveyor. If it were, a small piece held out in the
fingers would at once droop like a piece of thread, and if
a long tape were suspended in several sags or bays over
supports, it would drop away from the supports leaving
a sharp point or cusp thereat, but on the contrary, even
with the thinnest tape there is a distinct crest of curvature
concave downwards at each support, with points of contrary
flexure at some little distance on each side. The influence
of the rigidity is to decrease the sag in either case, but it
will easily be seen that the effect must be greater when the
ends are bent over supports than it is when the tape is
simply suspended at the ends with no bending moments at
or points of contrary flexure near them.

We will first take the case originally solved by the writer
of a long thin tape stretched by a large tension so that the
sag is quite small, and with its ends supported at the same
level so that the chord is horizontal. The tape is then
symmetrical about its centre, and it will be convenient to use
Cartesian coordinates, the axis of x being horizontal and
tangent to the tape at its lowest point and the axis of y
vertical through the same point.

(Fig. 1.) Let 7 be the length of the whole tape, w its

FIG. 1.

weight per unit of length, T, the horizontal component otf

the applied tension or, which is the same thing, the actual

tension at the lowest point, and M, the unknown bending

moment at the lowest point. Taking Mp as positive when it

tends to produce curvature convex to the axis of 2, the

bending moment at any point distant # from the origin is
Wk

M=Mo+Ty——-

Wire or Tape including the Effect of Stiffness. 99

Since the curvature is supposed to be small, we may put
a? dy M

d= BD where E is Young’s Modulus and I is the moment

of inertia of the cross section, which gives

Cy Mag Mile) ae

TE ae lee OI ens oe

The well-known solution of this differential ene iS

: Ns be Mio ance oe
y=Asinhy / >! “2+B coshy / Fe ae ste ort Roe ) (2)

where A and B are constants to be determined by the end

conditions. Since y=0 and gy 0 when «=0, we have

da
M wkKl
B=. °_ - and A=0, so that
T, T? an so tha

(M wHI Wa?
= (fo) (cosh / 5 L— 1) + op ae)

The value of M, will differ according as the ends are “ free”’
or constrained so that the tangent to the tape is horizontal
thereat. In the former case there will be no bending

=( when le and in the

moment at the ends or ay

Gite an 2
latter the slope will be zero at the ends or oY 0 when
(aie
on
Taking tirst the case of free ends, we have
d?y To /M,) wKI Et ete i
d= BLT, aye ) cosh maar
wHI 1
d i geeecres (at Se eee
an 0 Ih q 4 Ty, |
cosn EI : 9

Inserting this value in equation (3) we have

wk (cosh / ae aw ae?

pa ae REED top: 3

Ty? 2 | J as 0 ce
cosh EI

100 =Mr. A. E. Young on the Form of a Suspended

To find the sag correction we have with sufficient accuracy
s-2= 5) (%) dx, and in the present case for the sag

correction over the whole span we have

L
2
2(s—2) = { i (4) an
Ade

Putting for simplicity af, = =a, we have

dy _____wsinhax gee
ug aT, cosh is Lo
1 ; i
9
and (5. dx
2/0 a
1
a z 2 Gi 2x mee 4 sinh? e de
Ow tee 2 yey
0 a cosh 3 a? cosh 5
ae? =a? 22 coshaa”” Qsinhax sinh 2az 2
i —— f$ — _
Ty E a? cosh = a? cosh 4q? cosh? 2a? cosh? al
Sanne
Pe ay aie 1 ]
eee OAL ae 973 ee eae iy op. Ts
es Watt ah Aa? cosh? S

ROA. oe
247? is independent of
the stiffness, and is the ordinary sag correction on the
supposition that the curve in which the tape hangs is the
catenary or, which is the same thing to the order given, the
parabola or circle. The remaining terms give the correction
to this due to the stiffness. When dimensions are given in
inches the quantity a is of the order 5 in ordinary surveyors’
tapes ; in the example taken by Professor Maclaurin, a # in.
tape under 14 lb. tension, a=3'l. Taking / in the same
unit, it will be seen that both sinh“ and cosh S are sensibly

equal to 4e¢? and enormously large, so that tanh =1.

The first term of this expression

The third term can therefore be neglected in comparison

eat}

(9)

Wire or Tape including the Egect of Stiffness. 101

with the second, since the latter is at least J times (in inches)
larger, and the fourth term is quite inappreciable. The
correction for stiffness is in this case, therefore, given by

wl wlKI
aT? === Si e ° e e e (6)

which is to be subtracted from the catenary sag correction.
Professor Maclaurin’s expression for the correction is

whl tan 6(2— sec? ¢) eles eed
oT? nae eee | where d= sin7! oT,

Expanding this expression in terms of ¢, which we may put

m 473

equal to ae we get for the correction eT which 1s
T): In
the example taken By Professor Maclaurin, /= 10 dite =
7920 inches, width= in., thickness = Jy in., w=0-°000564 lb.
per inch, B= 3x10" Ib: per square inch, Pa 141b. From
which, since [=+4,6t? we deduce

equal to feel writer’s expression multiplied ter = (ir)

Bap La hive: paras ee nite
a=/ a =3 110, a?=9-67, 7, =0'3192,

wlKT _ _0°3192?
To F920 x 9°67
which though quite inappreciable is about 60 times as great

as Professor Maclaurin’s value. In working this example
To has been taken =T=14 lb., though, strictly speaking,
wl?

Lae ea

—(-00000131 inch,

or
Ty=1(1— (5 7 ) nearly =13'84 Ib.

The effect of the stiffness on the form of the tape is best
seen perhaps from the expression for the tangent of the
slope,

GaP ae sinh aw
—- = a Ay _— —______—— e
(ia Biles al

a cosh 3

The first part shows that the curve is a parabola with vertex
downwards, the second is the alteration to this caused by
the stiffness which tends to make the slope less, the effect

102s Mr. A. E. Young on the Form of a Suspended

increasing as we approach the ends, where it attains a
maximum value of
al
w tanh —
Bs ticall
aS = ell
aT ae) )

Since in the example taken a=3:11 the maximum effect in
this case is only 2'"7 in a slope of about 9°9'. The effect
can be also seen by reckoning « from the end instead of the

middle of the tape and writing a’ = Uo so that

2
sink (5 i!
a Tee ') wl Et yak
fe 7,42 a ae Pyro T, L2 ee =| , very nearly,
a cosh =-

putting tanh = =1. Since e*"=22 nearly this shows that

at only 1 inch from the end the effect of stiffness on the
slope is reduced to yy of its maximum value, 2. e., to about 31,
of a second of are.

We will now consider the case of ends clamped or con-
strained so that the tangent to the tape is horizontal thereat,
in which not only is the effect of stiffness greater but the
conditions approach more nearly those found in practice,
because, in using the long steel tape, the surveyor generally
supports it at several points forming several bays in sag, the
tangent to the tops at each point of support being horizontal,
and a bending moment with curvature concave to the axis of
we occurring thereat. Putting then =o where = in
equation (3), we get

a. ano NW aha! an
0=( Te = sinh 9 + oT,”

M,= “ i= see i
: 2 sinh S

Substituting this in (3) we get
_ (cosh az—1)

a?
Yaw py)

(7)

: al
2a sinh 3

Wire or Tape including the Effect of Stifiness. 103

I
hee dy 2
and | ( rs ) dx

~og
L ee
uw (2f , Iesinnas 8 i? sinh? ar d
= pall at — - = | az
ie . al. 5 2 al
0 sinh 5 4 sinh 3
3 = =
_w Fx lecoshar , Isinhar | sinh Qaz Px z
ewan MPM ene al
0 asinh zy a” sinh 5 1lb6asinh? = 8sinh? a
2 2 2; 2 Ve
na w? ee ale Li l I? (8)
= = a 9 ete e . ry ° ry . e
iy |i ue!

J ie aide
8a tanh 5 “16 sinh? a
a y
The third term can be neglected in comparison with the
second and the fourth is quite inappreciable, so that in this
ease the correction for stiffness is

__ Bu 8 (mls /T
8T 7a ‘a a) T) Le

This is 2 aay to times the correction for free ends, and in
Bie MET 3x 7920 x 311
the example considered is 3 = 9237 times as
great =0°012 inch in the 10 chains, a quantity which would
be worth considering in base-line measurement, though it
would not be likely of course that such a large bay as 10
chains in sag would occur in such an operation.
di

Writing c= ; —' in the expression for - , it reduces to

at (5 a =|
de a ToLota | Dew?

so that when w«’=0 at the support, the effect of the stiffness

is equal to a = 9 inthe example. To find the bending-
0

moment at the support we have

a
2 tanh -

w al w al wl 3
=> ,1- = = -—(1-— >) nearly, =— =~ practically
a” l a 2 w? 2a I Je
2

104 Mr. A. E. Young on the Form of a Suspended

The point of contrary flexure can be found oF solving for w
or w' from the equation

al cosha (5 —& “

d*,
I = or —]
dc? ; 9 GY al g
2 Sim —
2
whence
al al
log. 9 2-30 loge >
gS Pah wat aa =3°6 inches in the example.

In this short distance the effect of the stiffness on the slope
has decreased from 9°9' to about 0!:5, which shows how
rapidly this effect dies away.

If we make = in the expressions for y we obtain the
deflexion of the middle point of the tape below the ends, and
if we make T)=0 in these resulting expressions we should
obtain the deflexion of the middle point of a beam loaded
uniformly and supported at the ends either free or clamped.
In the first case we have

we? weoshar—l wi? w
= 1 sec 5)

aac BIg Waa aE NBG) Be
osh —
| 2
pel?) _ ae vl? dail? 3
=~ Snll= fe 5 Binge terms involving T, )
D> wit

~ Jn the second case, we have

. al Poe © ee!
M : I(cosh S ——— 1) wt i i tanh z
0

— —— = —=

| ‘ | aL 5 2a
8 2a sinh a ALO .

+ terms involving a and hi gher 1 powers)

co er ale t al 4
=ale== 4° «192

wlt
= Sea’ when T,)=0.
These results agree with those given in books on Applied
Mechanics.

Wire or Tape including the Effect of Stiffness. 105

In the preceding investigation we have assumed that the
tape is used on level ground, and also that the tension used
is so great that the curve in which the tape hangs is so flat
that even at the ends we may consider siny and tan w=
where y is the angle of slope. The greatest value of this

angle oe where J is the whole length of the tape, and we

have seen that with a tin. tape 10 chains long under the
rather low tension of 14 lb. this angle is only about 9° when
the whole 10 chains is in sag. In actual practice, however,
tapes are used on considerable slopes up to 45°, or even 60°,
with the horizontal. It might be assumed, and it will be
proved later, that the ordinary sag-correction—defined as
the difference between the curve and its chord—can be

obtained in these cases by writing weosy for w in the
: 2/3
wl Peevanly :
formula Ar? and the same remark applies to the correction

for stiffness because, even in the extreme case of constrained
ends, the angle through which the tape is bent is not the whole
angle y but the difference between the slope of the chord and
vy, and this angle is less the steeper the slope, being given
lwlcosw
Oe aa
for stiffness is concerned, therefore, the case of chord hori-
zontal is the most unfavourable that occurs in survey practice,
and the solution found can be used for slopes even steeper
than that in Professor Maclaurin’s example. But we will
now proceed to consider the general case dealt with by
Professor Maclaurin.

* (Fig. 2.) Professor Maclaurin starts with the following

So far as the correction

sufficiently closely by

S$
ig i

FIG. 2.

fundamental equations of equilibrium of a small element of
the tape:

dT GD tae |

oh LS —wsinw=0 CRE ae 24 8
dvr aU

rn 2 Se BOS AP SSO sm Ue 12

i me ae re 0 COS W= ( (12)
dl

— +U =a a eB a Fe
ds

106 Mr. A. E. Young on the Form of a Suspended

Moreover, L=EI/p= HI a , which with (13) gives

ay
ES U0... 2

In these equations T is the tension, U the shear, L the
bending-moment, s the length of the curve measured along
the are from some fixed point, the angle the tangent makes
with the horizontal, 1/p the curvature, w the weight of the
tape per unit length, H Young’s Modulus, and I the moment
of inertia of a cross-section of the tape about a line through
its centre of gravity perpendicular to the plane in which the
tape hangs.

Eliminating L, U, and T, the following is the differential
equation arrived at for the curve in which the tape hangs:

where the dot + denotes differentiations with respect to s.
When the tape is perfectly flexible the right-hand side of
(15) 1s zero, and we get _

cos vy +2 sinyw=0,

5 | ,
whence Me =—2 tani, d/o —ap— Coss

ue :
which of course represents the common catenary.
The flexibility of the surveyor’s tape being so nearly
perfect, Professor Maclaurin proceeds to solve (15) by
ae ike - Cos”
approximation, substituting y= ——,
side of (15), and finally arrives at the following for the
intrinsic equation of the curve :

in the right-hand

e 20 9
sEetanay+ =, sin cos? . +» Sees

From this, after further transformations, he finally derives
the formula for the stiffness correction previously quoted,
VIZ,

Eile (2—sec? ®”)
Awe? bag tall sae a) SI
I 2
l n RI) wl aes To :
where o=sin oT, 1 as

Wire or Tape including the Liffect of Stiffness. 107

The investigation and formule contain no stipulation as to
the end conditions.

Mr. Sawkins pointed out that multiplying each member of
equation (15) by cos, each is a perfect differential, which
Professor Maclaurin does not seem to have noticed.

Integrating them we have

WwW COS? Yr

: —FI( Yoon yt yisiny) +A, aa)

where A is a constant of integration. Putting ~y=0 in this
and equation (12), we see that A is the tension at the point
av=0 if there is such a point, 7. e., the horizontal component
ot the tension=T\.

Integrating again, we find

ee =O aii foals he as)
If we take the origin so that s=0 when y=0, then B=0;
so finally

ore ae Folgistmiyn i oot (LO
ds

which is the equation to be solved in the general case.

This equation might have been derived straight away
from statical considerations. It is simply the statement of
Professor Maclaurin’s (14) that the differential coefficient
of the bending-moment is equal to the shearing force which
is evidently equal to Ty sinyr—wscosy, where Ty is the
horizontal component of the tension or the tension at the
point ~=0 and s is measured from the same point. In
actual practice, when the tape is used on a slope there may
not actually be such a point, but in applying the formule
which follow, the tape is supposed to be continued to such
an imaginary point from which the distances s are supposed
to be measured. The actual length 7 of the tape is in such
cases either s.+s, or ss—s, according as the vertex of the
curve occurs in the tape or not. When s; is equal and
opposite to s,, we have the symmetrical case or case with
chord horizontal.

Tf in equation (19) we put cosy=1 and snyp=y, we

have

a?
BIE typ aus,
or
dup Th ws

Te. HE

108 Mr. A. E. Young on the Form of a Suspended

whence

Bidsrtiyee ; . aD : »_ to
a an a where a =F’

=A sinhas+ a since B=0 with s=0 for w=0.
0
And since

w= \cospds= Ves ao - is ds,

we have the sag correction
L

= As—x)= | “ds,
0
2 ype?
= ‘i i ae he sinh as + A? sinh? as | ds,
al see al
i we 2 ote (' cosh > is 2) ee ee t j
ee wae Bae a : a Aa 4. ;

In the case of constrained ends we have w=0 when

l
s=-; hence

2 vires wl
ARs eel
e r Si I ae
2T, sinh 5
and sag correction
ce n- B 30? Wg l B
=i 94 Tite gape aaa son
T, I 4 Sa tanh 5 16 5 sinh? aoe

the came as alr eady found by the Cartesian eee
If in equation (19) we put as a first approximation

[?
BIS Y =0,

ws
we find tan y= 7: . :
C ‘
Ws ee A'S
or w=tan 1 =tan™’- ;
To €
hence
dal) COA) d?ayp 2 sinycos*
Se ee re Sse Ge
ds Ge ns ds? ¢

: 3
sin COs :
me one Core =—wscosw+ Ty sin yp.
C

Wire or Tape including the Effect of Stiffness. 109

2EI .
Hence s=ctany+ ee? tt ar cos? yr,

which is precisely the value arrived at by Professor Mac-
laurin, but in deriving this we have neglected altogether the
complementary function of our differential equation. ‘The
fallacy here involved is so subtle that the writer wonders
if a parallel case has occurred in any other physical
investigation. The only somewhat similar case he has come
across is in the investigation of the Harth’s Precession
(see Routh’s ‘ Rigid Dynamies,’ vol. ii. p. 325, Hd. 1892),
but there it is shown that the original neglect of the
complementary function has not vitiated the result. He
will think himself fortunate if this paper should induce
investigators to give more careful attention to the question.
The effect of the stiffness can be seen more clearly by
dividing the angle y%& inio two parts, one of which ¢ is the
angle due to the ordinary catenary action, and the other
@ the correction to this due to the stiffness. We have,

then, w=¢+60,

where p=tan == talies0s:
: w 0

putting 6=-;-

toy

Hquation (19) becomes

ET Soe =—bscosp+sin v=sec ¢ sin @,
s

To
2
or a =a’ sec dsin 6 ;
but ‘

a? 2b BONE

~ =— (4b) ‘and ‘see b = (1+ 0?s?)2.
a0 ectailitte 23s ¢
oe aacl -~ b?s”)2 sin 0 + +88) Sos OCaer (20)

ds?

This or equation (19) must be solved for the complete
general case, but the writer has not been able to find the
general solution of either. In the surveyor’s tape @ will
always be a very small angle, even with steep slopes and a
low tension, so that we may put @ for sin@ and solve (20)
by approximation. Neglecting 06°s° in comparison with
unity we have

i Srlrure ab= 2b°s,
ds*

110 Mr. A. E, Young on the Form of a Suspended
913
whence 6=A sinhas+B coshas — Ae: :

and since @=0 when yw=0 and s=0, we baie B=0.

cos ds= { (cos g cos 8—sin ¢ sin 8) ds

= =) Cos sis— | (5 +@sin 6) ds nearly.

The first integral is the ordinary catenary formula, and
the second is the correction to this due to the rigidity.
If we put tan~*bs and sing@=bs as a first approximation

and neglect a in comparison with bs, we find A from the

equation y=bs+A sinhas, and then the sag correction due
to the rigidity 1s given by
i

2 2bis?
2( (Abs sinh as +4A? sinh? as— a) as

The first two terms will be found to give the expressions
already quoted, and the last gives
OE a eae El
6a7 7 Cae
which is the value found by Professor Maclaurin.

Through the kindness of Professor H. H. Turner, F.R.S.,
the writer is indebted to Professor A. R. Forsyth, F.R.S.,
for suggesting the following method of carrying the ap-
proximation further.

Write @=A sinha

2b? s
tv

and substitute this in equation (20) expanded to the first
term in 6?s*, or

2 @?
Ly = o7(1 4 4b?s) (2 ie =a) + 2b%s5(1— 207s?) ;
we have
dv fi 2
~~ — a@v= — a’b?As? sinh as— AS sinh? as — 5b°s°,

ds? 2 r

Wire or Tape including the Effect of Stefiness. 111

of which the particular integral only is required, as the
complementary function has already occurred in the ap-
proximate solution. The particular integral of the first
term is found to be

eat (La3s3 + 2as) cosh as— (}a’s* + §) sinh as} *,
of the second
A 3 aS 2
ae eee +3 sinh as—6as cosh as) 5

and of the third 5)5s8 i 30b%s

2 at

a
The full expression for as far as terms in s° is thus

3.3 b? 3
ar=bs ai +...+A | sinh us + yo { (ws + a) cosh as

RUS Le \e Ri;
—( + [)sinh as . |

3
-- (4 sinh 3as +3 sinh as — 6as cosh as)

9b?5 5b6°s* 306°s

a” a? a‘
It is remarkable that the coefficient of coshas in the
<a eta ie CHAS AE ORS 7S) 0 di ond
expression by A, viz., ie Baik is in ordinary cases

greater than unity and so greater than the coefficient of
sinhas in the first term. For instance, in the example

already quoted the greatest value of the coefficient for
s=5 chains=3960 inches, bs=0°1596, a=3'11 is

bs'as 0:1596? x 3:11 x 3960

IO Galas eu isy hor

This will give a smaller value for A when the end con-
ditions are put in, but the value of the term by A is again
increased by this factor, so that the expression for the
alteration in the sag correction due to the stiffness is not

altered in the most important term. This has been tested by
l

Ser,

2
1 y 2 " a . Rte
forming ly av’ds from the expression
ab?s* cosh as
Ww=bs+A (sinh CS ar an *)

* This particular integral was also found by Mr. Sawkins, to whom
the device of dividing w into ¢ and @ is due, by another method.

(21)

_- —_-

112. Mr. A. EB. Young on the Form of a Suspended

for both free and constrained ends. By continuing the
expansion of (1 +s)? it appears that the integral by A can
be written

Be) iawn 6.6
a [sinh Av as cosh as : s BS. 70's } ;

4 oi At. > 3 in

which shows that further terms rapidly decrease. The rest
of the factors in (21) appear to be all of a smaller order than
those of the first approximation.

This is as far as the writer has been able to carry the
investigation, and though it has not been found possible to
obtain a general solution of either equation (19) or (20) he
ventures to think that, so far as the surveyor’s tape is con-
cerned, he has shown that the approximate solution first
derived is sufficient, and proves that if only an adequate
tension is used and the lengths of the bays in sag are not
extreme, the effect of stiffness on the sag correction is quite
negligible.

The General Catenary Formula.

The ordinary catenary formula as applied to surveyors’
tapes has been worked out fairly completely both when the
tape is considered inextensible and when the elastic extension
due to the tension is taken into account. Anable and useful
paper on “ A System of Accurate Measurement by means of
long Steel Ribands ” was read before the Royal Society of
New South Wales in 1885 by Mr. G. H. Knibbs, C.M.G.,
formerly Professor of Surveying at Sydney University, and
now Commonwealth Statistician, in which formule are given
for the cases of chord both horizontal and inclined, but
applying more particularly to the method by which the sag
correction is eliminated by altering the tension, a method of
chaining which has, however, not been found very convenient
in practice, the more usual process being to preserve a
constant end tension and calculate corrections. Another
paper, on “The Measurement of Distances with Long Steel
Tapes,’ was read by Mr. C. W. Adams, Chief Surveyor,
Blenheim, New Zealand, before the Victorian Institute of
Surveyors in 1888. Jn this the series for the sag correction
in terms of the end tension is rigorously developed for a
number of terms, and tables are supplied for use with various

: wl : , :
ratios of 1 and for various slopes. These papers contain

practically all that is required by the surveyor engaged on
all except the most refined traverse work. The theory as
applied to Base-Line measurement and the effect of the
elastic extension are dealt with in more detail in Appendix

Wire or Tape including the Effect of Stiffness. 1138

No. 8 of the Report for 1892 of the United States Coast and
Geodetic Survey. And lastly, in 1912 in No. 1, New Series,
of the Professional Papers of the Ordnance Survey there
appears a Discussion on the Theory of Measurement by
Metal Tapes and Wires in Catenary, by Professor O.
Henrici, F.R.S., and Captain E. O. Henrici, R.E. The last
mentioned paper investigates thoroughly the effect of the
elastic extension when the tape has heen standardized in
catenary under tension, and shows that there will. then only
be a very small correction when the tape is used on a slope.
The sag formule in this paper apply also more particularly
to the case where the tape is standardized in catenary and
used in base-line measurement under the same conditions,
the slopes being obtained by measuring the difference of
level of the supports. It is desirable, however, to have
formule which contain only quantities observed in the field,
and in traverse work the surveyor generally uses a tape
which has been standardized under a certain tension on the
flat, and he usually applies the same tension in the field either
at the upper or lower end when working on slopes. He
observes on the vertical arc of his traversing theodolite the
actual angle of slope or inclination of the chord of his tape,
and the lengths of the bays in sag depend to a great extent
on the oround he is working over.

The following investigation of the catenary at any angle is
a development of that given in the above mentioned paper of
Mr. Knibbs:

Let s)s9, v2, and y,7. refer to two points on the catenary,
the subscript 2 referring to the higher, so that s,—s,=/ the
whole length of the tape, and let k=the length of the chord of
land € the angle it makes with the vertical. We have then

x av
Yo=c cosh — ; yi=ecosh— ;

¢ ¢

5 eae ee
Ss=cesinh—; s;=csinh—;

c c

Sa— S13; wo—a,=ksin€; y—y,=hk cos €.
vs v
Therefore kcosf=e (cosh — — cosh a)
t By

e lo e v
=e(sinb Boral a
6; C
€ ¢ ¢ 5 Us ome 1 hk: sin
2? —k? cos? C= 2¢? (cosh se ee 1) —— ptr *(co sh— Lee 1)
G

k* sin noe sin’ ¢
=k? sin? 4+ — [22 3608 oa SO

Phil. Mag. 8. 6. Vol. 29. No. 169. Jan. 1915. I

114. Mr. A. E. Young on the Form of a Suspended
Therefore

Pa pe4 sath kisin® ©) 1k? sin® &

29
Ve © 360c! +soa0et - (22)

We have thus obtained a development for / the length of
the tape in terms of & the chord, € the angle the chord makes

dk
with the vertical, and c=—° where T, is the unknown
W

horizontal component of the tension.
By reversing the series we obtain the following develop-
ment of k? in terms of /? :

_p E Sin lt (sin be sin® Sale bs
a 12) eae

Ssimt?e smi sm? € ) is :
-( 1728 ~ 864 * 20160/e7 °° | «MAD

And by extracting the square root we obtain after reducing

Ty Sin lye. asim osc “( 3 7 cos? €

a a= it eeu causa)

sin® 4 5 43 cos?g¢ | 11 cos*@\ ;
ois = aay on 1h J ae | fe

If we make {=90° so that the chord is horizontal we have

iB 3/4 518 1
— = = — : +5 es ——— . 2
“ 1] 42 + GAO 7 TIES LE | esinh™'g, - (25)

Tee

2
as it ought to be; and if we write for c? its value—}- = — —z

WU" Sas
where T isthe end tension, and then carry the correction due

2
to a to the next terms we find

4
wl? 11w*l*
PSs ii me

which is the series given in Mr. Adams’ paper quoted above
and which can be written approximately

w wl? a
r=t [1-3 — 3 (gap)

Returning to the general case we have to find an expression

for c in terms of ie end tension and the angle of slope.

Considering the portion of tape let V. and V, be the vertical

Wire or Tape including the Liffect of Stifiness. — 110

components of. the tension at the points 2 and 1, then we
now that V,—V,=w. Le Tyand 1, are the full tensions

and h=v,—y, the difference in height between the points,
we have

T2=we+V2; Te=we?t+V?; T,—T=wh-.
oe ier Nes
(T,+T,)wh=(V2+ Vj)wl ;
(2T,—wh)h=(2V.—wl)l ;

ml 2 72
Vv, SH al Te oes AGO)

[ 2]
and since w2c?=T,?— V,? we have
2 272
eaB(1_¥) [1 Y 8%). es
w HO) OUD: AT, 2 :

as a by Professor and Captain Henrici.

Now we have to a first approximation h=/cos¢, so we
may write

oa ee
(Pas =e 0? 6 ((1- a oe as \ Pee ee),
w

oT AT? 3
and if this be substituted in equation (24) we have
“1? sin? ¢
k=l) 1— ee a
| wl cos € elle sd)

at | (1- ie co

! % | - Toe ANS? J
n Bee Sly GUN nee ye se) ~...}.@0)

640° «4752

ao wicosf\? wl? |?
1 ; (1— oT ) co AT, 2 {

116 Mr. A. E. Young on the Form of a Suspended

We can now correct this expression for the assumption
that h=lcos € nearly, whereas h=kcos € exactly. Putting
X=l—k we have strictly

2 fie Q2 ieee S yA) IP.
pee wi COs a2) | ss e 2) cos ) Bee

Ww \ Palle

and expanding this to the first power of X we have

au ie wl cos See ae? 2r cos? €
C= sin? € 5 | (1— IE pan) (14 a

Taking for » the value derived from the second term of the
expansion in equation (30), and carrying the factor in A to
the numerator, we see that this correction can be included
in the term multiplied by cos? in the third term of the
expansion (30), which now becomes

wl sin? €

AG Ae wl eos GN? wee
y oe) Di) BE rae
Deal | (1 Bie ) — A'T.? |
wilt sin? ¢ 2 ae s)
ar ra | : wleose y wee 1 saan = 394.) oo (31)
LC metas

which is rigorously correct so far as it goes. Of course if
T, the lower tension were used we should ‘have to change the

: 4 wl cos
minus sign of aT 5 in the denominators into plus.

cans :
Tf we carry both the factors in — in the denominator of
¥ 2
the second term to the third we obtain the following :

wl? sin? €

pee yes wl cos €
2a (1—

(Bese |)

wl* sin? € ie cos” =)

es : (7930 — 128
2 ne af

(32)

from which it will be seen that the sag correction is given
with sufficient exactitude for all practical purposes by

He ena t epeat t ¢ (14 ae cos € eS where T is the full end tension, and

the upper or lower sign is used according as the tension is

Wire or Tape including the Lffect of Stiffness. = 117

applied to the upper or lower end of the tape. Itmust be
remembered that this gives the difference between the curve
and its chord, and if the chord is inclined we must multiply
by sin € again to reduce to the horizontal, but in practice it
will be better to do this after applying any further corrections
such as for temperature or stretch to k.

We have assumed that the tension observed is the full end
tension applied to the tape; and if the tape is used in one bay
and the spring balance or weight is applied directly to the
end without resting it on any other support, this will be the
tension recorded by the balance which will set itself to the
angle assumed by the tangent to the tape at its end. When,
however, as is generally the case in traverse work, the
tape is used supported at several points, part of the full end
tension is made up by the reaction of the adjacent support,
and the tension recorded by the balance or represented by
the weight will in this case be the component of the full
tension which is parallel to the chord, and which is with
chord horizontal simply To. In this last case formula (25)

Ny
with ¢ put = or gives the sag correction. When the chord

is inclined, however, let P, be the reaction at the upper end

normal to the chord and ‘I’, the component of the tension
parallel to the chord, then taking moments about the lower
support we have P.k=4wlksing, or P,=4wilsin €, and
resolving T’, and P, horizontally we have |

Tp =we='l', sin €—J wl sin € cos €;
so that

al ee Stns (1 wl cos zy

r= ===
Ww? i

Substituting this value of c? in equation (24) we have

ee gis We Pier sioney
- wil cos £\?
ort? 1— can )

wilh sin? ¢ (a 7 cos” )— ye 33)

“ay (. wl cos & 640 1152 eee |] (Oe
Cee )

wl? eos? €

and if we carry the term in me in the denominator
Ss |

118 Mr. A. E. Young on the Form of a Suspended
of the second to the numerator of the third term, we have

‘eles Sib ole 7G

ea ae see Ce
247 | ees
ant ty )
w'l* sin? g 3 5 san ]
“a saa ae 1153 ) 7
2 Te fie

If the component at the lower end is used we shall have
T,=T,—wlcosf, and the sign of the terms in the deno-
minators by cos € must be changed.

Also if the tape has several bays in sag it will be seen
that the value of '! decreases by wl cos £ from bay to bay;
so that in calculating the sag correction of them all (supposed
equal in length) we have, using the first term of the sag
correction from equation (33), a series of the form
Mesa C [ 1

2Tn

5 : |. Ga
: ee et) ea
ee af,

where I’, is the observed tension at the upmost support.
By expanding these factors and bringing them to the nume-
rator this expression can he put into the form

wT sin? cy fa n*wl COs Gen (8 - ae ene 7
ae ae is =),
= ny Se C {1 nicl COs G ze ey \ nearly, (36)
24D iE ile
If in the example already considered the tape had been
used on a slope of 30°, or £=60°, supported at every 2 chains
with a tension of 14 lb. applied to the upper end, we should
have total sag correction

7 5 x 0:000564? x 1584? x 0°866?

24x14
f, , 5x 000564 1584 | (5x 0000564 x ae
OF ae 14x2 | ( 14x2

=1:078 (1+ 0°160+0°025) =1:194 inches,

which shows that the third term can usually be neglected.

Wire or Tape including the Lffect of Stiffness. 119

Thus, so far as the surveyor engaged in traverse work
is concerned the sag correction on any slope is given by

nl sin? € oa i+ sik =| }

and the higher the tension the more nearly is this correct.

In any case rot doubt the next term within the curved brackets

a which should be increased to
DAs

zs
—4(si9s 3) if the slope is anywhere near 45°.

is very nearly —3

Effect of the Hastie Hatension.

The effect of the elastic extension of the tape is investigated
in the papers above quoted by including it in the equations
of equilibrium. As it is an effect of a smaller order than the
eatenary curve, it is justifiable to assume the tape to have
taken the catenary form unstretched and then to add the
effect of the stretch. The amount of stretch in any element

birdie! : :
of length ds is aa where T is the tension at that point, E is

Young’s modulus, and A is the area of the cross section. As
the curve is very nearly the catenary, we have T=T, cosh w,

iy Nek )
where w= —, v is measured from the vertex of the catenary
C v

T
and c=’, Wealso have ds=c cosh u du.
w
Tds
aT = we? cosh? u du;

and integrating, we have

? 2 S] 2 /
{ ee” | cosh UC ina a le 5): - BD

If the chord is horizontal we have for the stretch in the
whole tape

ae eta 20 +u) amd t= (Sim he oa! = =e
pe fies Z an == © = —_— = hare
HA 2 ie 2Ou 26 Fase

Expanding this we have

wad 7’ Sage. 9 Qy,5
Wwe Zul ZU
Stretch= (1 +u+ ——+ —-4.. )
. KA 3 ) af :

120 Mr. A. EK. Young on the Form of a Suspended

and since we have Pde lat Rae
Oe amy (38)
or BEA wl?
C= “(i= a) nearly 5
where T is i Les tension, this reduces to
wl? mel =|
Stretch= 5 =\i= ae )(Q2 +39) = mae — on). (39)
Now re is the amount by which the tape is stretched
when it lies flat, so that 1: ee is the amount by which it

contracts in length if it has been standardized flat under an
end tension T, nals is then used in catenary under the same
end tension, the contraction being due to a diminution in the
mean ienenon owing to the cur ears and haying nothing to
do with the sag correction.

Professor Maclaurin in his paper compares the stretch
correction with the stiffness correction, and finds the former
=1:757 inch; but he assumes that the tape has been
standardized under no tension, 7. e. he uses the expression

a If the tape in question had been standardized flat

under 14 lb. tension the shortening when used horizontally
in catenary owing to decrease of stretch would be
me 42 .
0000564 x 7920° x 8 x 60 ra eNON:
12x3x10'x14
which we see is just about the same as the value found for
the stiffness correction with constrained ends, viz. 0°0123 inch;
and as these two corrections are of opposite sign they mutually
destroy one another in this case.
Considering the stretch correction when the tape ison a
slope, we have
Up Toe — Zu, _ sinh 2a, Pets m)

Tds= —
“y Uy 2 2 2

= ze (cosh (5 + Uy) sinh (tt — Uy) — Mp = mJ
= Ney cosh? ~~ (us fue) -2 sinh Des sh (asus ae

2 YZ ~=

—sinh (t¥%—2w,) + ug— Uy "

; : ] —U} Uo tu
=s}? cosh ——— gosh — 2 cosh a Suh

» ‘ 2 “3

a =- Z -_

— eh)?
3.2 ae

Wire or Tape including the ffect of Stiffness. 121

Now
mh 4 al
Uy tu Ug— Uy To+T,
2 cosh 2=— cosh -— = cosh u,+ cosh uy = —_— >
2 2 Lo
and
Us + U Usg— Uy ; : }
2 cosh sinh 2——* = sinh wy — sinh wu, = —;
2 2 ‘ G
Stretch
r ‘ 71
mee es el eH)? ae (Ug— My)” | (40)
“Cae ae 6 COMI ate ee
a LosP ily one ai) a
2KA 12KA
but Yy—ay Using
Ly __ b
Vis Oh nearly
2 1 ¢ @ y) aN
and T,=T, —wh=T,—wil cos €, nearly;
. 2
also ¢ =sm oa , nearly;
Stretch

my a Ey, 4
=f peep meaty. GD
This of course reduces to the expression (39) when €=90°;
and it will be seen that the second term rapidly becomes of
more importance than the third. In the example considered
these two terms are equal for €=86° 50’. The correction
to be applied therefore to a measured distance when the
tape has been standardized on the flat under Ty, lb., and is
used on a slope inclined € to the vertical, is

wil? cos cy wl? sin* ).
2HA 1ZHAT

The first term is simply the stretch caused by the com-
ponent of the tape’s own weight, and if the tension at the
lower end is used its sign must be changed. When the tape
hangs in several bays the above expression must be multiplied
by the number of bays to obtain the full correction.

If the mean tension were kept constant as indicated in
the paper by Professor and Captain Henrici, the first term
of the correction would vanish. This would mean, however,
adjusting the tensions for every change of slope, which in

122. Mr. A. EB. Young on the Form of a Suspended

practice would be a rather tedious operation. If also the
tape were standardized in catenary, the third term would be
replaced by

wl? wl’ cos’ €

pecs ree EERE Sos ee «ee Se a
19mAT, 1 ~ SO) = Jomat,’

a very small quantity indeed, as pointed out and exemplified
in a different form in the paper just quoted.

Effect on the Sag Correction of using Tapes of
different Density.

In the standard traverse work of the Malay States
Survey Department, the tape used is of steel or invar wire of
rectangular section about 3 inch wide and -;), inch thick,
weighing about 5:1 oz. Av. per chain of 66 feet. This wire
can be obtained from Messrs. J. Chesterman & Co., Sheffield,
in continuous lengths of 1000 feet or more. It was customary
to obtain it in lengths of about 10 chains or 660 feet, and to
mark it off locally at every chain standard when flat at about
90° F. (the mean temperature of working) with 20 Ib. tension.
In order to obtain the odd linkage of his traverse lines the
surveyor was also supplied with a steel box tape 66 or 100 feet
long, with divisions etched on it to hundredths of links
standard, also at about 90° F. under 20 lb. tension. This
box tape could be attached to a loopat the end of the , tape
by a split hook; and it was arranged in marking off the latter
that the divisions on the box tape should read correctly.
Obviously it was desirable that the box tape should have the
same density as the 5), tape; but it was not possible to show
the etching on the former clearly with a less width than 4 inch.
The thickness was reduced somewhat to counteract this, but
could not be reduced with safety to give a less density than
7°3 oz. per chain. The question having arisen as to whether
this extra density at the end of the series could appreciably
affect the sag correction of the ,}, tape for which tables had
been calculated, the writer was led to investigate the question
mathematically; and as he has not seen the result published
anywhere, he appends it to this paper in case it may be of
use or interest to other surveyors.

In the case in point the denser tape is at one end of the
series, and this is the case the writer first solved; but he
afterwards worked out the more general case where the
denser tape may be anywhere in the series; and as this
includes the other the working out of it only will be given.

Wire or Tape including the Lffect of Stiffness. 123

(Fig. 4.) Let 1 be the whole length of the tapes in sag,
the chord being supposed horizontal, w be the weight per unit
of length of the lighter tape, w' the excess weight per unit of
length of the heavier over the lighter, r/ the fraction of

zed MOLY

“TAG aaa oe a
|

FIG. 4.

the whole length covered by the heavier, and ni, (1—7*)/
the distances from either end to the middle of the heavier
tape, R, and R, the vertical reactions, and T, the horizontal
component of the tension.

Taking moments about the R, support, we have

wl? ae ol

Bley te rl(1—r)l, or R= = +w'r(1—r)l.

We also have the following moment equations for the
several sectlons:—

>)
wae
Ty= Rye Tee from «=0 to e=nl—4$rl;

2 /
L WH UW
ry hye — ean 1 (e—nl+ rl)’, from e=nl—irl to v=nl+$rl;

9
wae? |
Ty=Rye— —- —w'rl(w-—nl), from x=nl+ gril to w=l.

Therefore
aay.
T. “ =R,—we, from «=0 to z=(n—Ir)l;
OL o
di 2 }
1a =R,—we—w'(a—nl+ 3rl), from e=(n—4$r)l to v= (n+ Sr);
di
A ae =R,—we—-w'rl, from v= (n+ dr)l to a=.
dau r
Therefore l rdy\? ew
Te ( z) dx=2T,? x sag correction =
-0 AX, ;
7 rr ar) v(m +3r)/
a : Rai) ; 9, ef i) ‘ :? .
| (Ry — we)?da—2u (R,-—we)(e—nlt+trddve+u- (a—nl+ dr) da
wv! (r— gryl v(n—tryl

r/
~2w'rl} (Ry—wa)de t+ wl di.

“(antiryl et Se

124 Mr. A. E. Young on the Form of a Suspended

The integrals are rather tiresome to evaluate, but many
terms cancel out, and the following fairly simple resulé is

obtained for the Sag Correction:

eed, 2 ! f ie ! ff Vous
tp? + 12 0w'r 3) n(l—n) —— | £1209? 5 n(1l—n)——1
nT, | Winer te ) 6 srs
r |
If we make n= 5, we have the case of heavier tape at one
end of the series, and the formula reduces to
Vi 9 1/99 2 19 9 >) me >)
2472 [ w? + Qww!(322— 2Qr*) + w'27(42— 372) ]. 2 (48
221
: (w+ w')?
If we make r=1, we get simply ae. as we should.
AT?

If in equation (42) we write w'rl= W the whole extra weight
of the heavier tape and then make the r’s within the brackets
vanish, we obtain the formula for the effect of a concentrated
weight W attached to the tape in any position, thus

ar (wl? + 12wlWn(l—n)+12Wn(l—n)], . (44)
which of course is greatest for n=4, or weight at centre of
tape. Putting w=0, we see the ettect of a concentrated
weight at the centre is 3 times as great as the effect of the
same weight uniformly distributed.

Returning to equation (43), we see that the error intro-
duced by using a slightiy heavier tape at the end of the
series is as a fraction of the whole length in sag approximately
Pww'r2

AT 9” )
whole chain of heavier tape is in use, so that when /=2 chains
-=J, /=3 chains r=4, and soon. Thus /r?=constant=1,
and as inthe case for which the investigation was made
w=5'1 oz., w'=2°2 oz., Ty=20 l|b., we have the above
DAK 22 11-2 Sgn: be

Tx20°x 162 ~ 409,600" say 1 in 40,000. The
average effect occurs when only half the heavier tape is in
use and is + the above ratio, or about 1 in 160,000. As an
accuracy of only about 1 in 30,000 was aimed at in the

The maximum value of this error occurs when the

fraction =

traverse work on which these tapes were used, this correction

was therefore negligible.

Formula (44) might be of use when for some purpose an
attachment is made to the tape such as a thermometer to
obtain the temperature at a certain point; and in this con-
nexion the writer worked out in a similar way the effect of a

(42)

hn Saal We see

Wire or Tape including the Effect of Stiffness. 125

number of equal concentrated loads each = W spaced
equal distances along the tape in sag. The result is Sag
Correction =

l (m?— 1)
Lawl? yey a W?2(m? — 1) | (AE
2aT? ae +2Ww aroun ( Wile (45)
where m is the number of spaces = number of loads +1.
If we make m=2, we obtain

mtr [w?+3Wwl+3W?],
the same as making n=4 in formula (44).
In order to test these formule the writer had some expe-
riments made which were carried out by Mr. W. A. Wallace,
a Surveyor ot the Lele ana Survey Department of the
Malay States. A 4g tape 7 chains long was stretched in sag
Ee 20 Ib. tension, and the movement of one end was
noted when a weight of 1 lb. was placed at successive
intervals of 1 chain along the tape, and also when 1 lb.
weights were placed at each chain. The tape weighed
5'1 oz. per chain of 66 feet. The following are the results

giving the differences between observed and calculated
effects :—

Position of Weight. | at : Sn nae |
f : | |

Weatrlchiainns...c.) acevo. | 0:357 link | 0:°346 link | +0°0185 link |
He Drelpimse: Wl. Poot) Osa OL “0056

co Ee ES RRR NA GO Wea Oxral Oe 0-692 ., | +0:018
MS EI PG GGda ee Nie 0-602! yeh 20028" yam
Ma ie Gat A Nossote wos | “00a ain
Bae Gichaite inl. SEA grat h. (oogon ee |

The differences are greater than the writer would have
expected, and may be due to differences of elasticity in the
tape which has of course been assumed to be homogeneous.
Mr. Wallace was not satisfied with the conditions under
which the tests were made. Allowance was made for change
of temperature, but he notes that there were passing clouds
which may have affected the temperature of the tape more
than the thermometers. There was also a slight breeze, and
it was intended to repeat the experiment under more favourable

126 Form oj a Suspended Wire or Tape.

conditions, but unfortunately this intention was not carried out.
The agreement, however, is close enough to show that no
blunder has been made in working out the theory.

The writer also made experiments to test the ordinary sag
formula while he was engaged in measuring in 1900 a base-
line with a g-inch tape of about the same thickness 31, inch
and 500 feet long. This was supported during the mea-
surement at intervals of 50 feet, and stretched by an end
tension of 20 1b. In the tests supports were placed also at
every 5 feet, and the movement of one end of the tape was
observed when it was supported at these and various other
intervals. The writer has not the actual figures at hand, but
the agreement, speaking from memory, was nearly perfect
between observation and calculation, the difference between a
change trom 5 to 50 feet intervals in the supports being
about +4, inch. Such a test includes of course the effect of
bending.

It was for long the writer’s desire to test the sag formule
on slopes of high magnitude, say 45° to 60°, with a good
length of tape in sag and with various tensions, in order to
see practically where the limits of applicability of the formule
lie, but unfortunately he was unable to carry out this intention
before retiring from the Malay States Survey. He gives the
following sketeh of how he intended to proceed in case any
other surveyor may have the desire and opportunity of making
the tests.

The tests should be made with as long a tape as possible;
and as wire can be obtained in continuous lengths of 1000 feet
and more, there is no difficulty in getting a good long con-
tinuous and presumably homogeneous tape. The chief diffi-
culty would be in finding a site, which would have to include
either a tower or precipitous hill or cliff, say 500 feet
above a plain where base-lines could be measured. Two or
three base-lines should be measured on this plain giving
slopes from 45° to 60° from their ends with a point on the
summit; and it would be well to have each in two sections,
to give acheck in the calculated distances to the latter. The
base-lines should be measured with the tape to be used on the
slopes, and the station on the summit carefully triangulated
into. Very careful vertical angles should be observed reci-
procally if possible, so that the direct chord distances from
base stations to summit could be calculated with precision,
refraction being eliminated as far as possible. These calcu-
lated distances could then be compared with those given by
the tape used on the slope from base to summit directly, and
with various tensions. As there should be no interference

On the Theory of the Metallic State. 127

from wind, and as the temperature should be as uniform as
possible, probably the best site could be found in some tropical
country with steep limestone cliffs sheltered from winds by
jungle; and it would be advisable to use an invar wire. The
actual form of the tape could be studied by attaching light
paper marks to it at known intervals to which reciprocal
vertical angles could be observed by theodolites mounted at
the upper and lower ends. The writer concludes by hoping
that some reader may have the will and opportunity to make
such tests.

University Observatory, Oxford.
June 1914.

XII. Note on the Theory of the Metallic State.
By F. A. Linpemann, Ph.D.*

< i outstanding physical properties of metals have

hitherto been attributed to the presence of a number of
so-called free electrons in the interstices between the atoms.
These electrons, which are supposed to behave like a perfect
gas, may be made to explain most electrical phenomena in
metals with fair accuracy.

The conduction of electricity is considered to be due to a
drift of electrons caused by the field. The electrical resis-
tance is accounted for by the collisions of the electrons with
the atoms, and Ohm’s law is explained by the large number
of collisions in unit time. The resistance may be expected
to change if an alternating current of such a high frequency
is induced that there are no longer a large number of
collisions during one phase. This has been observed to be
the case if currents of about 10" periods per second are
induced, as is done when infra-red light is reflected. From
this one may conclude that an electron must undergo at

2m
Net
where mis the mass, N the number per cm.°, e the charge,
and ¢ the time between the two collisions, a lower limit for
N may be determined. Putting in the known values one
finds about three times as many electrons as atoms. This is
approximately the same number as one finds from a con-
sideration of the dispersion of metals.

Various theories have been proposed to explain the fact
that the electrical resistance of a metal at different tem-
peratures is approximately proportional to the heat-content.

* Communicated by Prof, J. W. Nicholson. ,

least 10'4 collisions per second. As the resistance is
p

128 Dr. F. A. Lindemann on the

The latest and most elaborate assume the electrons to have
certain definite velocities which are independent of the
temperature”. Like Planck’s zero-point energy (Nullpunkts-
energie) this motion cannot be interchanged or observed.
The variation of the resistance with the temperature is attri-
buted to the change in the mean free path due to the change
in the amplitude of the atom’s oscillations. These theories
do not account for the supra-conductivity at temperatures
below 3° absolute.

Conduction of heat in metals is usually assumed to take
place analogously to in a gas, the electrons taking the place

Nmel

of the gas molecules. The formula found is )\ = 3%

/ being the mean free path, c the velocity, and y the specific
heat of 1 gram of electrons. Comparing this with the

: om Nei Wes
expression for the electrical conductivity ¢ = —— = ——
: 2m 2mc

» ney peer ; +s
one finds c= Gat This reduces to the Wiedemann-

Franz law if one assumes the atomic heats (and therefore
also the speed of the electrons) to be independent of the
metal. If one supposes the law of the equipartition of

energy to hold, y must be ion where M is the mass of an

5) mec

atom of hydrogen, m the mass of an electron, and a
3RT

must be Or where 2 is the number of atoms in a gram-

atom, and ‘I’ the absolute temperature. Thus - reduces to
oO

2] 2 2 2

a or ean as M =" This value is in very fair
agreement with the constant of the Wiedemann-Franz law,
and the variation of the constant with the absolute temperature
is in accordance with the facts.

It is unnecessary to go into the various theories that have
been put forward to explain the various secondary phenomena,
such as the Peltier effect, the Hall effect, the Thomson effect,
the Richardson effect, and so on. Hach has necessitated
secondary hypotheses, and none of them is very convincing.
It is sufficient to point out that the main points enumerated
above are in absolute contradiction with one another or with
the facts. The most obvious difficulty, of course, is the

* W. Wien, Berl. Bere 6. iv. 19138; W. H. Keesom, Phys. ZSaxame
p. 670 (1913).

Theory of the Metallic State. 129

question of the atomic heat of metals. Measurement shows
that there cannot be more than one free electron per hundred
atoms if the electron obeys the law of the equipartition of
energy, for the atomic heat of metals corresponds in every
respect to that of metalloids. As shown above, the con-
ductivity leads to the conclusion that there are more free
electrons than atoms. If one gives up the law of the equi-
partition of energy, which at first sight seems the simplest.
way out of the difficulty, one cannot explain the conduction
of heat and the Wiedemann-Franz constant. Thus the
electrical conductivity leads to a large number of free
electrons. Its temperature coefficient leads to no specific
heat. But the heat conduction cannot be explained except
by a normal specific heat. Again, the measured heat
capacities are incompatible with a large number of electrons
or with a normal specific heat.

The expression free electron, suggesting, and intending to
suggest, an electron normally not under the action of any
force, like an atom in a monatomic gas, might almost be
called a contradiction in terms. If one assumes that the
electrons are not attracted by the ions (and this assumption
is essential, for otherwise they would recombine with them),
the forces between the electrons themselves will prevent
their being free in the true sense. Indeed, the force pre-
venting one electron from moving between two others
at a distance of 3.107° cm., corresponding to about one
electron per atom, is so great that the equipartition energy

on could only shift it by about 1/20 of the distance apart.

These figures may be modified, of course, by assuming the
ions at a distance to attract the electrons, and some such
supposition must be made, as the electrons would otherwise
not remain in the metal at all. But the forces exerted by
the neighbouring electrons whose repulsion is not neutralized
are sufficient to prevent any similarity to a gas. The
hypothesis put forward in this paper is, that far from forming
a sort of perfect gas the electrons in a metal may be looked
upon as a perfect solid.

This conception would appear not to lead to any serious
contradictions, and even to supply an explanation for one or
two phenomena which the old theory hardly touches upon.

The following assumptions would appear to be necessary
to explain the facts :—

1. Though attracted, according to the inverse square law,
by ions at distances sensibly greater than the atom’s radius,
the electrons are repelled at distances less than 7, by a force

Phat. Mag. sv6. Vol. 29. No. 169. Jan. 1915. K.

150 Dr. F. A. Lindemann on the

equal to kf(r). This hypothesis, though not explicitly stated
in the ordinary electron theory, is accepted in its main

. e . é
outlines, for if an electron were attracted with a force WB?

the small kinetic energy attributed to it, sit T, could never

enable it to dissociate from the ion. The hypothesis seems
to be rendered fairly plausible by the experiments on the
reflexion of electrons by atoms described by Franck and
Hertz *. P |

2. The number of electrons per cm.’, N, the dielectric
constant of the ions D and the constant k of the repulsive
force kf(r) are such that @(N, k)~W(N, D). The functions
and will be defined later on. The equation probably
a

reduces to = const., or if D is constant, to N~,/k.

N
This hypothesis is introduced to account for Wiedemann-
Franz’s law.

We need only consider crystals for the time being, as all
metals consist of au agglomeration of, often microscopie,
erystals. A metal crystal would consist of two interleaved
space-lattices, one consisting of atoms or ions, one of
electrons.

To work out the exact mathematical consequences of this
conception will be a matter of great difficulty. This paper
will be confined to a general review of the various phenomena
and an outline of the way this hypothesis might explain
them.

Electric conduction —lf a metal crystal is brought into an
electric field the electron space-lattice will shift in respect to
the atoms until the attraction of the more distant ions
counterbalances the force exerted by the field. Ifa source
of electrons is brought into contact with the one end and if
they can flow out at the other, e. g. if the crystal is connected
to the two poles of a battery, the attraction of the more
distant ions will be counterbaianced by the repulsion of the
inflowing electrons, and the electron space-lattice will move
continuously through the atomic space-lattice. In other
words, the electron space-lattice or crystal may be said to
melt at the one end and fresh layers may be said to freeze on
at the other end when a current flows. If the distance to .
which the repulsive force of the ions extends, 79, is less than

= half the distance between the centres of the atoms, the

* Verh. d.d. Phys. Ges. xv. p. 929 (1913); Phys. ZS. xiv. p. 1115
(1918).

Theory of the Metallic State. Eat

electron space-lattice can move unimpeded tlirough the atom
space-lattice, as long as the atoms are at rest or as long as

p—2r :
9 This

would correspond to the supra-conductive state described by
Kamerlingh-Onnes as occurring in pure metals at tempera-
tures below about 4°*. If, however, the metal is not pure
this supra-conductive state can never be attained, for the
regularity of the original atomic space-lattice is destroyed by
the other atoms embedded in it, and the electron space-lattice
would always encounter a comparatively large resistance
which would be independent ot the temperature. This
corresponds to the formula W=W,+/(I) discovered by
Nernst t, in which the resistance W is equal to the resist-
ance (1) of the pure metal plus a constant W, depending
upon the impurities.

To return to the pure metal, as the temperature increases
the amplitude of the atomic vibrations increases and the
electron space-lattice can no longer pass without resistance.

Every electron will have to pass through the spheres of
repulsion on its path and will transfer the kinetic energy
gained from the electric field to the atoms. The mean
velocity of the electron space-lattice v is obviously pro-
portional to the current us the number is constant; the

mean velocity imparted to one electron by the field is 5—r,
m

where zis the time between two collisions with a repulsive

their vibrations do not extend to an amplitude

it ; ha
sphere. t= —, whilst the electron is in the immediate
v
neighbourhood of an atom if v’ is the atom’s frequency.
! ( B
During the rest of the time r= a: where d is the distance

during which the electron is further removed from the atom’s
centre than 77+ A, A being the atom’s amplitude. Now, as
we always have a very large number of electrons in any
observable current, there are always a large number otf
electrons within the distance 7) +A of an atom, consequently
the time during which the entire electron space-lattice can
move unimpeded is infinitesimal. It follows that Ohin’s law
holds good as the current a~v~a. Further, it may be
expected to hold for any current whose duration is of the

1
order —, or greater.
Vv

* Teiden Communications, 124C,
t Berl. Ber. ii. p. 23 (1911).
K 2

132 Dr. F. A. Lindemann on the

The calculation of the variation of the electrical resistance
with the temperature cannot be attempted here. It depends
obviously upon the amplitude of the atomic vibrations and
upon the force acting upon the electrons whilst they are within
the sphere of repulsion. As, according to Debye, there are
vibrations of almost every frequency less than v,,, and as all
their amplitudes may vary, it will certainly be very difficult
to take an average of all the probable forces acting upon the
electron space-lattice. We may take it, however, that there
exists a law of force £/(7) which entails a resistance propor-
tional to the square of the amplitude A, 2. e. proportional to the

E e e . e e
energy HE as A?=_, if is the quasi-elastic force holding
a

the atoms in position. As has been shown, @ is roughly
proportional to N, the number of electrons per em.**.
Therefore the resistance is a function of N and &, say
@(N,&)E. The dimensions would seem to lead to the

1/2
formula a a pik, p being the density of the electron
space-lattice. As p, N and & are independent of the tem-
perature, the resistance is thus in accord with the experi-
mental facts.

This proportionality of the resistance to the temperature
only holds good of course for pure metals. In alloys con-
sisting of metals which do not form mixed crystals, 2. e. which
consist of an agglomeration of pure crystals, the resistance
might be expected to be the sum of the resistance of the
components and the temperature coefficient would be normal.
In other alloys the homogeneity of the space-lattice would
be disturbed and the resistance would be larger. The
temperature coefficient would probably be smaller, for the
heat-motion might in some cases render the passage of the
electrons more easy, as the interspersed atoms which are in
the way might be moved into a more favourable position.
Somewhat similar phenomena may be expected in a liquid
metal, whose conductivity should be considerably less than
it is in the solid state.

Conduction of heat.—Debye has shown that a homo-
geneous space-lattice would have apparently infinite heat
conductivity f. This diminishes the less homogeneous the
space-lattice becomes. Debye’s theory explains Eucken’s
apparently paradoxical experimental results on heat con-

* Verh. d. d. Phys. Ges. xiii. 24. pp. 1107 & 1117 (1911).
+ ‘Vortrage wher die kinetische Theorie der Materie und Elektricitat ”
(Teubner), 1914.

Theory of the Metallic State. 133

ductivity of crystals at low temperatures *, namely, that the
reciprocal of the heat conductivity, the thermic resistance,
is approximately proportional to the temperature. Accord-
ing to Debye the heat is transported in the form of elastic
waves. These are scattered by inhomogeneity in the
elastic constants of the space-lattice, caused by variations
in density due to heat-motion. If a metal is composed of
two interleaved space-lattices, as assumed in this paper, its
measured heat conductivity will be the sum of the con-
ductivity of the atomic space-lattice and that of the eiectron
space-lattice. At ordinary temperatures the conductivity of
the atomic space-lattice may be neglected, as it will be
of the same order as that of a crystal. The conductivity of
the electron space-lattice will be comparatively very great,
for it corresponds to a crystal at a very low temperature.
Now the formula for the conduction of heat developed by
Debye is only valid for temperatures of the order T>8y,,
Ym being the limiting frequency. The electron space-lattice
will have a very high limiting frequency according to
‘ 9 \1/3 NU
Debye’s formula v,,= (=) olagia? on account of its small

mass and comparatively small compressibility. For N one

can put ag p being the proportion of atoms dissociated ;
is :
p, the density, is laa, m being the mass of an electron,
whilst the compressibility « depends upon N and the dielectric
constant. Now there is no reason why, if one atom expels
an electron, all the others should not do the same ; therefore
p is probably one, perhaps two or more. In this case « is
of the same order as it is for the solid +, though its exact
value depends upon the distance at which the attraction of
the ions becomes noticeable and upon the dielectric constant

ie : Saas Me
D of the material. Therefore v,, is of the order AV ~ Yims
M being the mass of the atom and yp, the limiting frequency
Me.
of the atomic space-lattice f. As a/ = is between 100 and
m

600, the electron space-lattice at 300° corresponds to the

* Ann. d. Phys. (4) xxxiv. p. 185 (1911).

+ According to Haber’s empirical formula, it should be exactly the
same (vide Verh. d. d. Phys. Ges, xiii, 24. p. 1117 (1911)).

} For the sake of simplicity only the compressibility has been taken

into account. In other words, the velocity of a transversal wave is
assumed proportional to the velocity of a longitudinal wave.

134. Dr. F. A. Lindemann on the

metal at 3° to 1/2° Abs. In this region we cannot apply
Debye’s formula for the conduction of heat. We can only
conclude from his reasoning that it must be large. Only an
exact theory could give some idea as to his “free path” J,
which he defined as the distance in which the energy of the

elastic waves is diminished to 4 part. He finds the heat
1
F e . De PUY e e e ‘ap Son Ge
conductivity \ = aa l, g being the velocity of sound Haas

y the specific heat. As has been or will be shown, p, «, and
y depend upon N and D, / can only depend upon the
number of layers of atoms per cm. ~v~"® or upon N and
probably T. Therefore X=/(N, D, T). Although we lack
an exact theory for the conductivity of a crystal at very low
temperatures, we can conclude trom the measured con-
ductivity of the diamond that it does not vary with the
temperature. Therefore \ reduces to >W(N, D), and this

W: uY ,
theory gives the law of Wiedemann-Franz, — = Ei const.,
oO

if A=W(N, D)~¢(N, k= 5. A consideration of the
; : 1

dimensions appears to lead H the equation A~ Nip"

= N/E?
course simply introduced to show that the observed pro-
portionality of electric and heat conductivities is not incon-
sistent with the electron space-lattice hypothesis. The
_ suggested theory does not pretend to predict this law as
the old theory does; but, on the other hand, it would not
seem to lead to the absolute contradictions for which the old

If this be true, = K. This assumption is of

Xr
— reduces to
(OF

theory is noted. If « is of the form ae

Fizpss" Thus supposing, for instance, D to be equal for all
metals, k~N? would lead to the law of Wiedemann-Franz.
As in the ease of the electrical conductivity, impurities should
produce inhomogeneity of the space-lattice and thereby
diminish the heat-conductivity.

Specific heat—At emphasized above, the question of tne
specific heat of the electrons has been the chief stumbling-
block of the old theory. The argument which leads to the
difficulty, namely, that as the electrons conduct heat so well
they must have a large heat-capacity, is sound only as long
as the electrons behave like a gas. If they form a solid,
on the other hand, the converse is nearer the truth.

There are analogies, as stated above, in the conduction of

Theory of the Metallic State. 135

heat by a diamond at 20° Abs. Its conductivity is almost
as great as that of copper, although its specific heat is
negligible. The specific heat of the electron space-lattice
may probably be calculated fairly accurately from Debye’s
formula, which has proved so successful in the case of solids *.
The atomic heat is

Bn \
120) om ede SU, i eee i Ns
ele (Brm)?\, e—1 BYm ue Um pei?
et —J

In our case, as shown above, the velocity of sound, pil PED

is very large t, en account of the small density, so that c,
1 D4 3

reduces to the form == R ae and the specific heat of

the electrons is well below the limits of measurement.

Taking one electron per atom and « equal to the compressi-

bility of silver, for instance, v,, would be
AG) 4°42) 5 1.012 = 1:93 10 or 6y,—= 94.000.

Thus the atomic heat at 300° would be 1°51.10~° cal. or
the specific heat y=0:266 cal.

This explains, too, why those phenomena which depend
upon the energy-content of the electrons are so minute.

Contact potential—The electrons in the metal will have
many points of similarity with a solution in spite of forming
a space-lattice. Their mutual repulsion must be counter-
balanced by the attraction of the more distant ions. We
thus have an analogous phenomenon to the internal pressure
in liquids in the theories of van der Waals and Reinganum,
or to the osmotie pressure in solutions. ;

If two metals are placed in contact, the electrons will flow
from the metal with higher internal pressure into that with
lower until the potential difference balances the difference in
pressure.

Other things being equal, the work necessary to remove
an electron will be inversely proportional to the eube root
of the atomic volume. Hence, in general, the metals with
large atomic volumes, such as the alkali metals, will become

* Ann. d. Phys. (4) xxxix. p. 789 (1912).

+ It is interesting to find that the velocity calculated by this formula

cm.
. 8 ‘ .
is of the order 10 Seo? which is about the value found for the trans-
mission of energy through a cable. It is difficult to see how these high
velocities can be explained on the old electron theory, for a wave can
never travel faster in a gas than the velocity of the molecules.

156 Dr. F. A. Lindemann on the

positively charged on being placed in contact with a metal
of smaller atomic volume. Similarly, their electrons will go
into solution more easily than those of other wetals, 2. e. they
are electropositive.

Thermoelectric effect.—The internal pressure of the electron
space-lattice will obviously depend upon the temperature
mainly on account of the thermal expansion of the metal.
But it will not necessarily change with the temperature
according to the same formula in different metals. Therefore
if a ring is formed of two different metals and the two
junctions kept at different temperatures, the difference of
the internal pressures at the one junction will not necessarily
be counterbalanced by the difference at the other junction.
In all cases in which the pressure differences are not equal
and opposite, a current will flow in the ring and continue
flowing as long as the temperatures of the two junctions are
kept constant.

Jn a metal in a state of strain, the relative positions of the
atoms and electrons would be slightly different from the
positions in the unstrained state. This would lead to changes
in the elastic coefficients of the electron space-lattice, and
thus probably to thermoelectric effects between parts under
different stresses.

Peltier Effect—The electron space-lattice in a metal has a
certain stability, and may consequently be expected to have
a certain latent heat of fusion. The passing of electricity
from one metal to another entails the melting of the electron-
crystal in the metal connected to the negative pole and the
solidifying in the other metal. The difference in the different
latent heats of fusion will be absorbed or liberated at ihe
junction according to whether the current flows from the
metal whose electron space-lattice has a large Jatent heat of
fusion to the one whose latent heat of fusion is smaller or
vice Versa.

_ Richardson Effect —Just as any other crystal, a erystal com-

posed of electrons must have a certain vapour pressure. As

in the case of solids, this niay be expressed by a formula of the
B

form Ae *T’, a similar formula to that found already by

Richardson.

Reflexion.—As observed above, the resistance of a metal
should be independent of the period of the current if this is
greater than vj, the period of the atoms. The resistance
opposed to the more rapid alternations of the currents
induced by reflexion of shorter waves than the ‘“ remaining
rays”? must depend upon the number of electrons involved.

Theory of the Metallic State. 137

In addition to this the electron space-lattice may be expected
to have a large number of proper frequencies, which will
modify the coefficient of reflexion. The proper frequencies
must actually become most numerous in the region in which
the deviations really commence. As will be shown, their

number should be any v? dv.

Putting in the values assumed above this is of the order
els?
ae

Thus for waves 1 mm. long there would still be 2 .10° fre-
quencies per tenth-metre. Planck’s infinite number of
resonators of different frequencies may thus have a physical
meaning, though in our case the number is confined to 3N,
and the frequencies are less than v,.

Photoelectric Effect—One would need special assumptions
to calculate the proper frequencies of the electron space-
lattice as Born and Karman did for atomic ou eee
Fortunately, however, we can use the method which Debye
proved was permissible as a first approximation for atomic
space-lattices, namely that used by Rayleigh in developing
the first radiation formula. According to this, the number
of nae frequencies in any interval dy is ‘Arr p?!e'/2v2 2dv
per cm.*,

dr, © being measured in Angstrom units, 1O—!° metre.

suede : 4 :
The factor p%/?ks3/2 18 ad being the velocity of sound in

the space-lattice, which, as shown above, is determined
by the atomic volume and the dielectric constant. Now if
light be allowed to fall on the metal, it may happen that a
sufficiently intense wave is induced in the space-lattice to
disrupt it and project an electron. ‘This is the more hkely
to happen the more proper frequencies there are in the
space-lattice in resonance with the incident light-wave.
For a given metal this number is proportional to v?. The
probability of a resonator getting the energy necessary
to free an electron fv is inversely proportional to».
Thus the photoelectric current should be proportional to
the frequency, which is confirmed by experiments. On the
other hand, as shown above, other ‘things being equal, the
number of proper frequencies of a given “colour is inv ersely
proportional to the third power of ‘the velocity of sound, or
roughly to the atomic volume. Thus the theory also accounts
for the observed fact that the photoelectric sensibility for red
light is greatest with the alkali metals whose atomic volume
1s oreatest. This point of view disposes at once of the

138 Dr. F. A. Lindemann on the

difficulty emphasized by Bragg, that the energy of photo-
electrons is often greater than the incident energy of the
light falling on one electron. A large part of the wave-front
ean act upon the electron space-lattice. The additional
assumption of zero-point energy would of course still further
simplify matters. Whether the selective photo-effect is due
to peculiarly numerous proper frequencies within a small
region, such as Born and Karman found in ordinary crystals,
must be decided by experiment. If, as seems more probable,
it is due to other intra-atomic electrons, a gas should show a

rep)
selective effect though it should not have a normal effect.

Hall [ffect. —The Hall effect would be explained exactly as
it is done in the ordinary theory. A magnetic field would
tend to make the electron space-lattice drift off at right
angles as it would do to cathode rays. To explain the
inverse Hall effect this theory needs the same hypothesis of
atomic magnetic fields us has been introduced to explain the
phenomenon on the old lines.

Thomson Hffect—As the electron space-lattice moves,
this being the electric current, the temperature inequalities
in it may be expected to move with it. The quantitative
relations involve, however, the question of the interchange
of energy between the atoms and the electrons, and how
elementary our knowledye of this question is, is proved by
the quantum theory.

Nernst-Ettingshausen Eeffect.—Though it is probable that a
magnetic field must deform a longitudinal elastic wave in an
electron space-lattice, thus producing a transverse electro-
motive force during heat conduction, the problem cannot be
attacked without more detailed knowledge of the interaction
of the atoms and electrons.

Hitherto only crystals have been considered, but in the
actual metals with which experiments have been made we
have to deal with an agglomeration of crystals. Obviously
this will not invalidate the conclusions drawn above, for to
all intents and purposes all the effects are additive. One
could only expect to find a difference if the crystals were of
the sme order of magnitude as the molecules, for only then
would the boundary effects become of the same order as the
volume effects. Perhaps the fact that cold drawing increases
the resistance of wires, whereas annealing diminishes it, may
be an indication that this is true ; for cold drawing obviously
breaks up the crystals which tend to join up again through
annealing. At first sight one might expect the specific
electric resistance of a metal crystal to be different in

Theory of the Metalle State. 139

different directions. The few measurements available seem
to confirm this view, which would not seem to be readily
derivable from the old theory. On the other hand, the
elastic properties also vary in different axes, and the
variation of the atomic amplitudes occasioned thereby might
compensate the change in the distance which the electron
space-lattice would have to pass through, and thus sometimes
mask the phenomenon. ‘This effect would of course not
be noticeable in an agglomeration of crystals oriented at
random, such as the specimens are which are used for
experiments. According to the proposed theory, an increase
of pressure would lead to an increase in the atomic frequency
and consequently to a decrease in the amplitude. This would
entail a decrease in the resistance, such as has been found
experimentally. It would seem difficult to explain this
phenomenon by the accepted theory. The thermodynamic
aspects of the space-lattice theory are particularly simple.
As the electrons form a crystal, Nernst’s theorem may
certainly be applied to them, and all the consequences already
deduced by this method hold good. The admissibility of
applying this theorem, as has been done, to electrons con-
sidered as a perfect gas is much more doubtful *.

It will be objected that the assumption of a force kf(7)
and of a number N and a dielectric constant D to satisfy
the condition ¢(N, k)~wW(N, D), are simply made to explain
the phenomena, without any regard for a priori probability.
On the other hand, once these assumptions are made, all the
essentially metallic phenomena may be explained without
any intrinsic contradictions, including some facts, such as the
electrical resistance of alloys and the photoelectric effect, on
which the accepted theory throws no light at all.

The accepted theory, besides leading to the absolute contra-
dictions touched upon in the introduction, entails special
hypotheses for many of the secondary phenomena. Its one
triumph, the derivation of the constant of Wiedemann-Franz’s
law, is based upon the theorem of the equipartition of energy,
whose applicability to electrons as they are supposed to exist,
is generally recognized as absolutely inadmissible.

Conclusions.

The free electrons in a metal may not be treated as a gas,
for a gas can only conduct heat well if its heat capacity is
large. Experiment proves that the free electrons conduct
heat well, but that their heat capacity is too small to be

* Is. Griineisen, Verh. d. d. Phys. Ges. xv. 6, p. 186 (1913),

140 Dr. H. Stanley Allen on the Series Spectrum of

measured. Theassumption of a large free path to compensate
a small number leads to contradictions with the optical pro-
perties of metals. It is suggested that the free electrons in a
metal form a space-lattice, which corresponds to a crystal at
a very low temperature, as the small mass of the electrons
leads toa high frequency. The point of view from which
this theory would explain the various metallic phenomena is
indicated. Although it explains the exceptional behaviour
of alloys and the general outlines of the photoelectric effect,
special assumptions are needed to arrive at the law of
Wiedemann-Franz. The contradictions entailed by the old
theory, more especially the one mentioned above, do not
arise.
Sidmouth, Dec. drd, 1914.

XIU. The Series Spectrum of Hydrogen and the Structure of
the Atom. By H. Stantey AuuEN, M.A., D.Se.*

HE series spectrum of hydrogen can be represented with
considerable accuracy by the formula of Balmer, which
may be written
n=N(q ee el
A
where n is the number of wave-lengths per centimetre, N is
the series constant, and m=3,4,5....

In a recent investigation by W. HE. Curtis}? the wave-
leneths of the first six lines of the series have been deter-
mined with an accuracy of 0:001 A.U. It was found that
Balmer’s formula was inexact. This was shown by the fact
that the values of N calculated by the above expression for
the different lines are not the same. (See Table I., which is
reproduced from the paper referred to.) Curtis found that
the results could be better represented by the Rydberg
formula,

ay “TN a ana el (2)
(2+pyP (m+n)

By putting p=0, p= +0°0;69, and N=109,679-22 the
formula gives a fit quite within the limits of experimental
error. Very little improvement is obtained by introducing
athird constant p. If this be done the values of the
constants are

p= + 0:0,8, L= + 0°0;70, N= 109:679°23:

* Communicated by the Author.
t W. E. Curtis, Proc. Roy. Soc. vol. xe. p. 605 (1914).

Flydrogen and the Structure of the Atom. 141

In a paper in the Philosophical Magazine on the magnetic
field of an atom in relation to theories of spectral series *, I
have shown that a formula of the type given by Ritz can be
deduced by the methods of Bohr’s theory if the magnetic
field of the atom be taken into account. Formula 17 of
that paper gives the frequency in the form

Qar?me?B? 1 1
ean stares eal pare a (3)

133 ale B 2
| o.+ =| out =|

—_ 167*mM Ke?

where B 13 eelg ord) Wenatchee ui(Cly

Seeing that Bohr’s theory of spectral series has achieved

its greatest success in dealing with the hydrogen spectrum,
it appeared that it would be of special interest to determine-

whether the inclusion of the effect of a magnetic field would
lead to results consistent with observation in this case.

In the case of hydrogen EH, the charge on the nucleus, is.

equal to e, the charge on the electron, and the factor outside
the bracket becomes equal to 27°me*/h®, which is equivalent

to Rydberg’s constant. It has been pointed out by Bobr ft

and Fowler { that a correcting factor must be introduced
involving the mass of the electron and that of the core.
In order to test the applicability of the formula to the

hydrogen series, we may put o,=2, o,=m (where-

ij—o.4,...), B in the first bracket = 0 and B in the
167?mMe?
ite i

It must be noted here that this implies a slight modifi-
cation of the scheme suggested in the previous paper. For
by assigning different values to B in the two brackets we
suppose that the magnetic moment of the core (M) has
different values in the two types of steady states of motion,
the emission taking place in the passage between these
types.

The inequality in the order of magnitude of p and p in
the formula of Curtis indicates that the two types of state
concerned are in some way different.

The formula for the wave-number may now be written

1 1
ren} ay B al Ug ee We catins

(m+—3) \
* Supra, p. 40.

m
+ Bohr, Phil. Mag. vol. xxvii. p. 509, March 1914.
{ Fowler, Bakerian Lecture, Royal Society, 1914.

second bracket =

142 On the Series Spectrum of Llydrogen.

The applicability of this formula might be tested by
finding values of the constants N and B which would
yield the closest fit for the wave-numbers observed. It
seemed preferable, however, to assume that B could be
ealeulated from formula (4) involving M the magnetic
moment of the core on the supposition that this contained
an integral number of magnetons. The magnetic moment
of the magneton was taken as 1°854x10771 E.M.U. or
0°618 x 10-*' E.8.U. It was found that the best results were
obtained by taking either 5 magnetons (B=5:24 x 10~°) or
6 magnetons (B=0'29 x 10~°) in the core. The results can
be tested by seeing whether constant values are obtained
for N when the observed values of » are substituted in
formula (5). The values of N calculated on the various
assumptions are collected together in Tables I. and IL.

TABLE I,
Values of N (Balmer’s Law).

| Line. | m. | m (observed). | N. | pb.
| | Pe ant a ee |
a 3 15,233°281 | 109,67962 + 0:03 |
8 | 4 ~ 20564880 — Sha + 0°02 |
y 5 93,082-644 26 + 0:02
é 6 24,373-165 | 24 + 0:03 |
¢ 7 25,181-458 | 24 + 0-04 |
z | 8 | 25,706-075 | 25 + 0-03 |
|
Tapin LL.

Values of N (suggested Laws).

Allen. Allen.
M=6 magnetons. | M=5 magnetons.

ree Curtis.
rae Wea p=0°0;69.

|
|
|
109,679°23 | 109,679°22 109,679°29

aes

Bei 4 ‘93 29 ‘30

y 5 2074 ‘24 24 |
8 6 22 | ‘D4 ‘25 |
eel. 7 23 25 25 |
2 ike 24 25 25 |
Mean ...... ‘DOS ve "26 |

Variation of Triple-Point with Hydrostatic Pressure. 143

An examination of the results in Table II. shows that
though the values obtained for N by the formula of Curtis
are slightly more consistent among themselves than those
obtained on the hypothesis of magnetic action. the diver-
gence in the latter case is not “too large to make this
ee ple untenable. The results with 5 magnetons are
slightly better than those with 6 magnetons.

In support of the view that the core contains 5 magnetons
we have the fact, first pointed out by Chalmers, that the
magnetic moment produced by an electron moving in a

ee orbit with angular momentum //27 is exactly

> magnetons.

The value of the series constant N would then be
109,679°26, and the convergence frequency of the series
would be 27419°815. These values differ but little from
those given by Curtis, and consequently the wave-lengths
ot the remaining lines of the series would be practically
identical with those given in Table V. of his paper.

On the other hand, the hypothesis that the core contains
6 magnetons appears to receive support from the observed
diamagnetic properties of hydrogen, and it may be pointed
out that the magnetic moment of the core in this type of
state would then be 3 times (corresponding to m=3) the
moment of a sphere rotating with the angular velocity
specified in earlier papers. This would give 109,679°25 as
the series constant, and 27419°813 as the convergence
frequency.

Thus it appears that it is possible to account for the
series spectrum of hydrogen on the lines of Behr’s theory
combined with the assumption that the core of the atom
can produce a magnetic field equivalent to that set up by
either 5 or 6 magnetons.

In conclusion, I desire to express my thanks to Pro-
fessors J. W. Nicholson and O. W. Richardson for their
advice and suggestions.

a of inoeaon King’s College.

XIV. Da i y. ariation of a eae of a Vee

with Hydrostatic Pressure. By AuFrED W. Porter,
OSCR RS.
VF GNHE fact, the truth of which is now well recognized, that
the saturation vapour-pressure of any liquid or solid
is a function of the hydrostatic pressure, carries with it
sundry consequences. One of these is that the triple-point
* Communicated by the Author.

144 Prof. A. W. Porter: Variation of Triple-Point

must no longer be considered as a fixed point but as a point
which also varies with change in the hydrostatic pressure.
That this must be the case can be seen almost immediately.
Recall that the triple-point is the temperature at which the
vapour-pressure curves of the solid and liquid meet ; so that,
if this temperature 1s exceeded, we can only have liquid and
its vapour in equilibrium together ; and if it is not reached
we can only have solid and its vapour. Now the position of
the triple-point for ice-water-steam, as usually determined,
is at ‘0074° C. Let the hydrostatic pressure, however, be
raised (for example) to one atmosphere: the melting-point
of ice becomes 0° C. If the triple-point is a fixed point we
shall, under these conditions, have a range of temperature
from 0° C. to *0074° ©. for any point of which it may be
said that hoar-frost cannot form because the substance is
above the freezing-point, and dew cannot form because it is
below the triple-point. It is clear that this dilemma can be
resolved cnly by postulating that the triple-point must have
moved to the freezing-point, viz. O° C., corresponding to the
hydrostatic pressure of one atmosphere.

This can be proved by a more detailed examination in
which proper account is taken of the variation of vapour-
pressure with hydrostatic pressure. We will first of all
prove an auxiliary theorem.

The slopes of the vapour-pressure curves
near the triple-point.
In determining these slopes it is customary to take
Clapeyron’s equation
dt
L i | C—O) =
for the three possible variations, and by neglecting v, v2, the

volumes of ice and water, in comparison with that of the
vapour, v3, to obtain the approximate equations:

dtr
Ly; — Tv, at 5)
Lg; = Tvs ae ’
Ly,= Tir, — 2%) prs ;

dt
whence, since at the triple-point

Tiy3 = Lyp + Lys

of a Substance with Hydrostatic Pressure. 145

we obtain
dti3 dts; OP Cor Un Apis

ae wor gs vn Pidtl<

This gives an approximate value for the difference of
slopes of the two vapour-pressure curves.

We will now show by a more precise treatment, in which
proper allowance is made tor the variation of vapour-pressure
with hydrostatic pressure, that this equation is not approxi-
mate merely, but exact when certain exact meanings are
AT 13 dt 23

le a

The hydrostatic pressure may be imagined to be applied
by means of a permanent gas placed in the vapour chamber.
It is much better, however, if we make use instead of a semi-
permeable membrane which separates the liquid and vapour

given to

Vig. 1.

Ligard Vapour Z

S-p.M

one from the other ; so that, if the membrane is permeable
to the vapour alone, any pressure whatever, p, may be
imposed on the liquid, and the vapour will then be able to
take up the vapour-pressure 7, which corresponds to this
hydrostatic pressure. With such an arrangement, shown
more in detail in fig. 1, a cycle of operations can be made,
each reversible in character, and the mechanical efficiency of

Git

this cycle may be given its universal value 7

In estimating the work done during the cycle we note

that this is given directly as the sum of all such values as
\ pdv for its various stages. Great simplification is brought
about, however, if use be made of the facts that

LD) iD)
{ padv=pove— pr1 -{ vdp always ;

1 1

and that for any cycle of operations the sum of all the values

of pove—pit; 18 necessarily zero, and therefore the work

done is given by the sum of all the values of —| vdp.
Imagine then a four-stage cycle performed, and estimate

Phil. Maa. 8S. 6. Vol. 29. No 169, Jan. 1915. L

146 ~=Prof. A. W. Porter : Variation of Triple- Point
the value of this latter quantity for each stage. We will call
ue separate values §,, So, 83, S..

i. Move the pistons A and B from left to right so that
one gram of liquid evaporates at constant pressure.

8, =0 because the pressures p and 77, keep constant.

ii, Raise the temperature an amount dT. This change
Ree the pressure of liquid and vapour by Fav a nd
Ae dT respectively.

[ Note. Part of the change in 77, will be consequent on the

change in temperature (p constant) and part on the change

of p itself. ]
If U and V were the initial volumes of the liquid and
vapour, and w, and v, their specific volumes, the volumes

during stage il. are uy. —u, and V+v, respectively.
We have therefore

$.=(U,—u,) OP at + (V+0) on aT.

or
iil, Push the vapour back into the liquid at the constant
temperature T + dT a

and p+ or dT respectively.
=0 because the pressures keep constant.

iv. Lower the temperature by the amount dT,
0 0
cy at
or roy
Now the heat given out at the higher temperature is L,
the latent heat of vaporization. Hence
S(pdv) aT
ae ke

hid (v. ate oii, i Le

m= —-U=] ay.

The cycle here employed can of course be exhibited
graphically. It is best to employ two diagrams—one for
the liquid and the other for the vapour.

of a Substance with Hydrostatic Pressure. 147

The cycles for the two will be traversed opposite ways
because when the total volume of the liquid diminishes

Fig. 2.
NX
PRUNE
N
Ea qaid Vapour

that of the vapour increases. The final equation can be
obtained almost by inspection. The sense in which the
cycles have been traversed happens to be that in which the
heat taken in at the higher temperature is —L. They could
equally well be taken with their directions both reversed,
the sign of the heat entry being reversed simultaneously.
The ordinary equation is obtained from this by taking p as
the pressure of the vapour alone, for it corresponds to the
case in which the liquid and vapour are in contact without
any intervening semi-permeable membrane. Both are then
at the same pressure, at any rate when capillary and other
special surface forces are absent.

foku

ot”

For this ordinary case of also equals and con-

sequently

This equation is of course exact for the special circum-
stances to whach it relates; it is not the general equation,
however.

Another special case is that in which the pressure p of the
liquid is kept constant throughout the cycle. This can be
done ; for the variations of p can be made independently of T,
In this special case we obtain

=m),

The meaning of L in all these expressions is absolutely the
same; but it is expressed in slightly different terms.

Let now a diagram be made in which it is these values of 7,
corresponding to different temperatures but to precisely the
same hydrostatic pressure, which are plotted against the
temperature. There will be two such curves, one for the

L 2

148 Variation of Triple-Point with Hydrostatic Pressure.

solid and one for the liquid. The difference of the slopes of
these curves at their point of intersection will be given
exactly by the equation

0713 O73 Ug Uy dp»

or ey A ae Hi

where the specific volumes of the liquid and solid are repre-
sented by u,and uw. Fig. 3 is such a diagram.

The locus of the triple-point.
On this diagram the vapour-pressures are shown for two
hydrostatic pressures differing by dp. Their point of inter-
section shifts from C to C’ owing to this change of pressure,

Fig. 3.

and the temperature of the triple-point shifts from 7 to 7’ or
through the amount - dp. Draw a vertical through C’

cutting the lower curve at A, and draw a horizontal through
C cutting this vertical at B. Then

AC'’=AB+BC'

(Fe )a= 35 USE), (St), ae)

as can be seen by inspection of the figure.

* This can be obtained at once from the general expression for the
total differential of the vapour-pressure in passing from one triple-point.

to another :
dr= a dp+ on 3
ey dp eal eat dr.

Form of Liquid Drop suspended in another Liquid. 149

Now make use of the relation for the slopes of the curves,
and we get

O73 7 Wu Oly on i dis
S ra oD Saari TENE aye ail

But (=) is the variation coefficient of 74; with negane
T
to p at constant temperature ; and this is known to be =;

similarly

so we obtain finally
de _ a
dp dps :
that is, the variation of 7, the temperature of the triple-

point, with hydrostatic presure is equal to the variation of
the melting-point with hydrostatic pressure.

It should be observed that = is not the slope of the line

CC’ on the 7, T diagram, but the slope of the corresponding
curve on a p, 7 diagram.

This question has been here discussed with special reference
to the case of ice-water-steam. But the results obtained are
of course general for all triple-points and can be extended
_to the case of multiple-points.

Dee. 5th, 1914.

XV. On the Form of a Liquid Drop suspended in another
LTaquid, whose density is variable. By J AMES Rick, I.A.,
Lecturer in Physics, Liverpool ees sity ™.

N the August number of the Phil. Mag., Lord Rayleigh
has considered the Equilibrium of Revolving Liquid
under Capillary Force.

The following paper offers a partial solution to a similar
problem, viz., to discover the form assumed by a liquid drop
suspended in another liquid, whose density varies with the
depth.

The investigation was suggested by an effect which is
observed in carrying out the well-known experiment of

* Communicated by the Author.

150 Mr. J. Rice on the Form of a

Plateau, in which a drop of oil is suspended in a mixture of
water and alcohol having the same density as the oil. If it
happens that the evaporation of the alcohol from the surface
of the mixture produces a density-gradient in the latter, then
the drop of oil flattens into an oval form.

It is assumed that the density of the surrounding liquid
varies continuously with the depth. |

The figure represents a vertical section of the drop (which
is of course a surface of revolution) through its highest
point. The axes are in the first instance tangent and normal
to the section at this point.

Let o represent the constant density of material of the
drop, and p the variable density of the surrounding liquid.

Then p=pi f(y)

where p; is the liquid-density at the level of O, and f(y) is

a function of y which approaches the value unity as y

approaches zero.
Let R be the radius of curvature at P, and R, that at O.
One easily obtains as the condition of equilibrium

Oo tg el?
Hs i 7o)= hore! a) pay,

where T represents surface-tension. If we assume this to be
uniform, it is easily shown that the places of maximum
surface-curvature lie in the level at which the densities of
the drop and the surrounding liquid are equal.

Liqud Drop suspended in another Liquid. 151

For at such a point

Chel Al i
int pa)=”

and therefore if T is constant

Without analysis this is evident from the following con-
sideration. A difference of pressure exists between points
at the same level, inside and outside the drop, and _ this
difference is proportional to the curvature of the drop at
this level. Such difference of pressure increases from the
top of the drop until the level of equal densities is reached,
since in this region the density of the drop exceeds that of
the surrounding liquid. Thereafter, the pressure-difference
decreases ; for below the level of equal densities the density
of the drop is the lesser.

Let us assume as a further restriction that

p =p(1 a a

where /. is a linear constant.
We readily obtain

9 as 2
legals o + KEP) y — FPL

RRO ke an yi
or putting p= nae h
Sane
and 2 5)
JP.

R RO Rk, 2he?

It is easily seen that y=b is the level of equal densities and
maximum curvature. In other words, 2) is the vertical
height of the drop.

Transfer the axis of # to CAX/ at this level, and the above
equation becomes

RABE Pinter: ar)

152 Mr. J. Rice on the Form of a

This equation shows that the curvatures at points equi-
distant from the level CX’, above or below, are equal. The
symmetry with respect to this level is an obvious consequence
of the assumed uniformity of the density gradient.

It is not difficult to show that an ellipse of small eccen-
tricity is a possible form of meridianal section.

For this to be so it is necessary according to the previous
equation that

ab, 6 26 , P—¥
TIT © Ge 2he *

where a= CA,
b=CO}
and b'=the semidiameter conjugate to UP.
If ¢ is the eccentric angle of P, this reduces to
l 2
2 2 Sica ay ! ae (_ COS" ®,
a ((1—e* cos’)? (1—e? cos? p)? J. 2he*
Pe)
Se Eee a AON Gog
G Pel cos* d + (| +3 )e cos @+ &e.,
Oates
= ieee dp.
If »_ vb _ 8x volume of drop
the? l6mrhe? i

and if also e were so small that e*, e°, &e. could be neglected
in comparison with e’, the above equation would be approxi-
mately satisfied.

It is clear that for a similar order of approximation, a
larger drop is possible, the greater the values of h and .

The method employed by Lord Rayleigh, in which the
differentials of the coordinates with respect to the are are
used, does not, unfortunately, lead to an equation so readily
integrable as the one preceding equation (3) of his paper.
Still some headway may be made, although the approxima-
tions, if pushed very far, would become excessively laborious.

i. g., the equation (1) becomes in these terms,

d (. )
ds gs), ihe ‘t > —1/?

EB eee tL ee )
dx Ry ek 2), ahh (2)

Liquid Drop suspended in another Liquid. 153

The difficulty of solution arises from the fact that the rigl
hand expression in (2) is a function of y and not of wv joe is
the case in Lord Rayleigh’s paper).
However, remembering that the curve is symmetrical with
regard to the axes, we may write
4

ae
[?

where « and / are to be found by approximation from (2).
£. g., the approximately elliptical form discovered above
can be found by using as the first approximation

rise, 2 2
PY = hd:

b?— 7? = av”,

where a (=1—e?) is a ratio slightly less than unity.
After substitution and an integration, (2) then leads to

J 3

Ayer ie ctl!
ds Thay ersive?’
no constant being required.
de y di
Also = eee

ds uals
and, therefore, since

we obtain

] a 2 ey b?—aax? >
(ee Oa ae sours \=1. . yetelaea

On expanding and equating the constant term to unity,

and the coefficient of 2? to zero, we find that @ is given by

a(l—a)= Fea,

ike (a VEN
a ne (L—e*)dhe? = the?’
as before.

It will also appear that with a sufficiently small value of
é” or 1—a, that the terms on the left of (3) which involve
w* and x are e negligible.

ve g-, the term in z* turns oat to be

ry 79
3 b ‘
64° Azc3
Its maximum value is
Pilte: 3
3a*l?/6417c' or AN

154 Dr. Emil Paulson on the

By applying the same method to the approximation con-
taining a fourth power of «, I find that an approximate
solution is given by

2 ast
Pe eS TN lace
b = 7. =au-+ pec?
where # is chosen to have a value slightly less than unity,
and 6 is then determined by

B/theP=a(l—«)+ (1l- a)? |.

The steps are laborious, and it is not necessary to reproduce
them. The solution is the oval of a quartic curve (which
also possesses imaginary parabolic branches). It is clearly
slightly less eccentric than the approximate elliptic solution.
It is nearer the truth, inasmuch as the steps leading to it
involve the neglect of terms in (1—«)* and higher powers,
whereas the elliptic solution neglected terms in (1—a)”.

XVI. On the Spectrum of Palladium.
By Dr. Emit Pautson *.

AYSER + was the first to show the existence of triplets

in the spectrum of Palladium. This was found by

him to he repeated 6 times completely and 3 times in-

completely in the whole spectrum. Designating the

wave-number of the first line in each triplet by A and

those of the two other lines by B and C respectively, the
wave-numbers of the triplet are given by the relations :

B=A+3967°90 A,=3967-90
CO=A+45159:09 A,=1191-19,

A, and A, being the differences of the components of the
triplets.

Afterwards, without being aware of the work of Kayser,
I ft discovered the pairs with the difference 1191, but did
not find the complete triplet ; [ also found many other pairs
with the differences 1628 and 403 respectively. It was, how-
ever, to be assumed that all these pairs and triplets could be

* Communicated by the Author.

+ H. Kayser, “‘ Die Spectren der Elemente der Platingruppe,” Adz. d.
Berl. Akad. 1897; Astrophys. J. vii. 1899.

+t E. Paulson, “ Beitrige zur Kenntnis der Linienspectren,” Diss.
Lund, 1914, pp. 88-54. ~

Spectrum of Palladium. 155

brought together in greater groups of lines in conformity
with what I have found in several other spectra *.

In fact a line has been omitted by Kayser, so that his
difference 3968 is the sum of two differences with the values
1628 and 2340. The first of these I have already given in
my thesis above referred to. Thus, the complete group con-
tains 4 lines. It seems, however, as if one line more should
be included. ‘This appears only three times, but it follows
from its intensity and general appearance, that it belongs to
the same group.

In the table below are given certain wave-numbers 7 of
lines in the Pd-spectrum, which are arranged in such a
way that lines belonging to two corresponding columns
A-D in all the rows (1-21) show the same differences.
These differences are given in the columns headed Aj, A,,
and A. In the last of these columns, A2, the difference
7757 bas also been inserted in brackets. The wave-
numbers of the corresponding lines (E) are placed in
column D. Lastly, the mean of each difference is given
at the foot of the table.

The wave-numbers of the lines in succeeding columns in
the table above are given by the relations :

B=A +1628°33
C= A +3968-00
D=A+45159-14

(=A +11725-45).

It will be noticed that, excepting the lines 3404 (10K),
3380 (Su), 3142 (6), 2763 (8R), and 2441 (6R), all strong
lines for the interval 4213 to the end of the spectrum are
embodied in the above mentioned system. Among the lines
with larger wave-lengths it gives, indeed, many with stronger
intensities which cannot be placed in the system.

We will next consider how these lines, without doubt
standing in a certain connexion to one another, are divided
in the electromagnetic field. The separation of the lines is
indeed very different in the same group as well as by passing
from one group to another and does not indicate any simple
relation between the lines, for instance that the groups
should be produced by the same electron.

* “On the Spectrum of Yttrium,” being printed in the Astrophys. J. ;
“ |vin System von Wellenzahlen im Scandiumspektrum,” Phys. Zetéschr.
1914, p. 892; “Gesetzmissigkeiten im Bau des Lanthanspektrums,”
Ann. d. Physik, xlv. p. 1203 (1914).

7 The measurements of the wave-lengths are taken from Kayser.

Dr. Emil Paulson on the

156

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Spectrum of Palladium. LoL

Certain regularities exist, however, and ought not to be
passed without mention. The lines in the groups 5 and 13.
are divided in the following way * :

/ A (GR) Tripl, +154 0 -153
ea eee
oe a © (QR) Tripl. +154 0 1°55
\ D (7B) Quadr, +219 +1:05 —1:05 —2-19
; A (9B) Tripl +150 0 ~153
aaa 134 B (6R) Quadr. +2°24 +0°84 —0°84 —2°25
| © GR) Trip, +153 0 -154
\ D (7R) Sep. unknown.

The first and the third lines in both these groups become:
triplets with identical values of a for all the lines. The-
second and fourth lines are divided into four constituents.

dn
each. If also here the values of jz are the same, cannot be
determined.

The lines in the groups 12 and 14 are divided in the
following manner :—

( A (4) Tripl. +164 0 - 1°63

| B (6R) Doubl. +0°72 -0°72
Group 12 + Sa aaa a ae

| C (4R) Sep. unknown.

\ E (10R) H

( A (5R) Tripl. +168 0 —171

| B (SR) Doubl. +0°65 —0°65
Group 14 ¢ Vrs bela ie ee

| © (8) Sep. unknown.

EF (10R) “

The separation is only known for the first two lines, which
became triplets or doublets, with probably identical values of
dn
Ni

In group 8 all the lines are divided into triplets. The

Both groups contain the line E instead of D.

dn
values of 52 are probably the same :-—
( A(6R) Tripl, +210 0 —2°12

eh 33 ae
Group 8
aoe } GORY.) E198 1,0: | 2:00
\ D (10R) ,, +190 0 -1:89

For the other groups there are not sufficient measurements
for comparison, although some other similarities could be
pointed out.

Lund, Oct. 1914.

* After J. E. Purvis, Proc. of the Cambr. Phil. Soe. vol. xiii. (1906) p. 326.

puss) (4

XVII. On Lorentz’s Theory of Long Wave Radiation.
By G. H. Livens *.

FYXHE Lorentz form of the theory of radiation, which

regards the radiation from a thin metallic plate as
arising from the motion of the electrons inside the plate,
will probably always remain as a deciding factor in the
general theory of this subject, since it involves no principles
which cannot certainly be regarded as well established by
independent theory and experiment. The final formula to
which this theory leads and the extent to which it depends
on the assumptions made must therefore be matters of the
first importance in the general theory. Basing mainly
on the two assumptions that the period of the radiation
considered is long compared with the interval of time
between two consecutive collisions of an electron with an
atom and that this latter interval is also long compared
with the time of duration of a collision, Lorentz derives
a formula, identical with the Rayleigh-Jeans fermula,
which is apparently correct in the long wave part of the
spectrum but fails hopelessly for obvious reasons in the
visible and ultraviolet regions. These two assumptions of
course naturally restrict the analysis to long waves, but it is
the expressed opinion of Prof. Lorentz f that the same
Rayleigh-Jeans formula would be obtained as the general
result for the other parts of the spectrum if only the
difficulties of the analysis could be overcome. The main
object of the present paper is the discussion of a partial
generalization of Lorentz’s analysis, in which one of the two
above-mentioned restrictions is removed. A formula, which
is concluded to be practically identical with the Rayleigh-
Jeans formula, and applicable to all parts of the spectrum, is
obtained on the single assumption that the duration of the
impact of an electron with an atom is always negligibly
small ; and the conclusion carries with it a partial confir-
mation and possible limitation of a certain well-known result
in the optical theory of metals.

The method to be followed is identical with that given by
Lorentz with the single exception that it will not be found
necessary to assume the relation between the period of
oscillation and time interval between successive collisions
which is implied in his theory. It is, however, necessary to
retain the assumption he makes regarding the smallness of
duration of a collision in order to avoid making arbitrary

* Communicated by the Author.
+ See “‘ Discussion on Radiation,’ B.A. Report, 1912.

On Lorent2’s Theory of Long Wave Radiation. 159

hypotheses regarding the dynamical character of the
collisions. The case to be analysed is, therefore, virtually
that in which the electrons and molecules are assumed to be
perfectly rigid elastic spheres, the molecules being, however,
of comparatively large mass so that their energy and motion
may be neglected.

I do not find it necessary to depart very widely from
Lorentz’s admirable exposition of his theory given in his
book ‘The Theory of EHlectrons’™, and I shall take the
liberty of quoting verbally in many cases from his work, to
which I must here acknowledge my great indebtedness.

We will, therefore, with Lorentz, ‘consider a thin metallic
plate in which a large number of free electrons are moving
about in a perfectly” irregular manner, consistent with the
general laws of the cons servation of their total energy and
momentum. We know that an electron can be the centre of
an emission of energy when its velocity is changing, thus,
as a result principally of the numerous collisions of the
electrons with the atoms, resulting in alterations of the
directions and magnitudes of the velocities of the electrons,
a part of the heat energy of the irregular motion of the
electrons will be radiated away from the metal. This radiant
energy, which is subsequently to be the subject of a detailed
examination, is, however, presumed to be so small compared
with the energy pi motion’ of the electrons that it can be
neglected in any dynamical considerations respecting those
motions extended over any finite time. To this extent the
analysis offered is only a first order approximation to the
actual state of affairs.

We know also, that as a result of the same collisions
between the electrons and atoms, part at least of any regular
or organized energy acquired by the electrons during ‘theit
free motion between the atoms can be dissipated into hea
energy of the irregular motion of the same electrons. Tn
this way it is possible for a metal to absorb a portion of the
energy from an incident beam of radiation, because the
electric force in the electromagnetic field associated with
the radiation will pull the electrons about during their other-
wise free motion between collisions, imparting kinetic
energy to them which will be dissipated by collision at the
end of each path into irregular heat-motion.

Now let w and w’ be two infinitely small parallel surface
elements, w being on the plate itself and w’ at a distance »
outside it on the normal to the plate through the centre
of w. Then of the whole radiation emitted by the metal

* The Theory of Electrons (Leipzig, 1909), Chapter IT.

160 Mr. G. H. Livens on Lorentz’s

plate, a certain portion will travel outwards through w
andw'. Suppose we decompose this radiation into rays of
different wave-lengths and each ray again into its plane-
polarized constituents in two planes at right angles through
the chosen normal to the plate (these two planes and the
plane of the plate being parallel to a system of properly
chosen rectangular coordinate planes in which s=0 is the
plane of the plate). Now consider in particular those of
the rays in this beam whose wave-length lies between the
two infinitely near limits % and X+dX and which are
polarized in the plane y=0; the amount of energy emitted
by the plate per unit time through both elements w and w’
so far as it belongs to these rays, must be directly propor-
tional to w, w', and dX and inversely proportional to 7”, and
it can therefore be represented by an expression of
the form
ww'dnr

9 e

i fe

The coefficient H is called the emissivity of the plate and
is a function not only of the positions of w, w’, and » but also
of the conditions and type of the metal composing the piate.

Let us now consider the opposite process. Suppose that
a plane-polarized beam such as that specified in the previous
paragraph is incident, through the small surface w’, on the
patch w of the metal plate: then we know that a certain
portion of the energy of this beam will be absorbed in the
metal and converted into heat-energy, instead of being
re-emitted as a portion of the reflected or transmitted beams.
The fraction expressing the proportion of the energy
absorbed is called the coefficient of absorption of the plate
under the conditions specified, and is denoted by A.

Starting from the thermodynamic principle that in a
system of bodies having all the same temperature, the
equilibrium is not disturbed by their mutual radiation,
Kirchhoff finds that the ratio

LO ie ueas
er Smt at)

between the emissivity and absorbing powers under the
same conditions is independent both of the direction of
polarization and the position and peculiar properties of
the metal plate. This ratio, a function merely of the
temperature T and wave-length 2, is now the chief object
of search in the general theory of radiation, determining
as it does the complete circumstances of the steady thermal
radiation from any body.

Theory of Long Wave Radiation. 161

But in the conceptions we have adopted, the calculation
of both E and A under the assumptions specified can be
directly accomplished. —-

If we consider that the thickness A of the metallic plate
is so small that the absorption may be considered as_pro-
portional to it, we shall find by an obvious calculation, after
Lorentz, that

A = 7A*,
c

¢ being the usual velocity constant and o the conductivity
of the metal.

Now the interpretation of o in terms of the electron
constants of the metal, although a matter of some difficulty,
is nevertheless fairly certain. If N denote the number
of free electrons per unit volume in the metal, each of
mass m and with a charge e, moving with velocities the
average square of which is w,,”, then we know that in all
applications involving steady or slowly varying currents the
conductivity o is given by

oo= sae
Y 37 Mum ’

wherein J» is a constant, a certain mean length of path, which
is determined by the formula

lin — — ==
nt R?’

in which n is the number of atoms per unit volume in the
metal and RK the sum of the radii of an atom and an
electron.

However, in applications involving more rapid alter-
nations in the current the above formula is found to be
insufficient and requires modification along lines already
laid down by various authors. According to Jeans { the
correct form to be used for alternating currents with a
fe Con.
requency 5 1s

ex)

o= SUAS 9
Aoy?17c?m

a
+ NA

* “The Theory of Electrons,’ p. 280 (note 33),
int The Theory of Electrons,’ chapter I., and p, 266 {note 29).
ft Phil. Mag. June 1909,

Phil. Mag. 8. 6. Vol. 29. No. 169. Jan. 1915. M

162 Mr. G. H. Livens on Lorentz’s

or, on substitution of the value of a), we get

Ne?1,,

mu
pes duh oun muti
377 adidas

l=
; BAU

a formula reducing to the Lorentz-Drude formula for large
values of 2.

We have therefore for the coefficient of absorption under
the conditions specified and for plane-polarized radiation of
wave-length 2X,

N. C7lim&

8 MU
A=a/ 5p Ly Bem
ie ON

Now let us consider the radiation from the plate, still
closely and often verbally following Lorentz. We need
only consider the radiation normally from the small volume
wA of the plate, as this is the only part of all the radiation
through w trom the whole plate that gets through w’. Now
according to a well-known formula of electrodynamics, a
single electron moving with a velocity v (a vector with
components vz, vy, vz) in the part of the plate under con-
sideration, will produce at the position of w’ an electro-
magnetic field in which the x-component of the electric
force 1s given by

Aner dt?
if we take the value of the differential coefficient at the

proper instant. But on account of the assumption as to
the thickness of the plate, this instant may be represented

for all the electrons in the portion wA by = if ¢ is the

time for which we wish to determine the state of things at
the distant surface w'. We may therefore write for the
«-component of the electric force in the total field at w!

i} dv,
ie — aoe [Bea 5 3° 0) Roane (1)
Rs

and then the flow of energy through w’ per unit of time
will be
CH,2w’

as far as this one component is concerned.

Theory of Long Wave Radiation. 163

Since the motion of the electrons between the metallic
atoms is highly irregular and of such a nature that it is
impossible to follow it in detail, we must rather content
ourselves with mean values of the variable quantities cal-
culated for a sufficiently long interval of time. We shall,
therefore, always consider only the mean values of our
quantities taken over the large time between the instants
t=0 and t=@. For example, the flow of. energy through
w’ is, on the average, equal to

1 <5
Cw'a H,di = cl,?w' ‘say.
0

Now whatever be the way in which H, changes from one
instant to the next, we can always expand it in a series by
the formula

GY a Sane
ee aac

s=1 ; ae
where s is a positive integer and
Za sae
a Al sin 7 Edt.
1 ° 5 5 oa) OOF
The frequency in the sth term of this series is g 80 that
the wave-length of the vibration represented in it is
28
s
If @ is very large the part of the spectrum corresponding to
the small interval of length dd between wave-lengths A and
2c0
A+dnxr will contain the large number e dX of spectral

lines represented by terms of this series.

Tf now we substitute the Fourier series for E, into the
expression for the mean energy flux through w’, we shall find
in the usual manner that it is equal to

OT al r& 2
cH,?w Seto GCOS see UE aah So. a st ad RR Be (2)
G= Il
To obtain the portion of this flux corresponding to wave-
ee between % and A-++ dA we have only to observe that
») .
2e
the 52 dX spectral lines, lying within that interval. mav be

considered to have equal intensities. In other words. the
value as may be regarded as equal for each of them. so that
they contribute to the sum & in (2) an amount

2cbaZdr

A M 2

164 Mr. G. H. Livens on Lorentz’s

Consequently the energy flux through w' belonging to the
interval of wave-lengths dd is given by
Céw' as"? 2a
rasa 6h?
and we now want to find ss
From the value x EH; given by equation (1) we see that

ara a908 r > { e's x 7 ce at},

where the square bracket round as vz serves to indicate the
value of this quantity at the time i— The sign =} now

refers again to a sum taken over all the electrons in the part
wA of the plate.
On integration by parts we find

sé i ge stt

or what is the same thing

f
se fae: $7
ds= FH7 2, =( v, COS —p (‘+ 4 dt.
c

Now each of the integrals on the left is made up of two
parts, arising respectively from the intervals between the
consecutive impacts of the electrons and from the intervals
during these impacts. If, as mentioned above, we can
suppose the duration of an encounter of an electron with an
atom to be very much smaller than the time between
two successive encounters of the same electron, we may
neglect altogether the part that corresponds to the collisions
and confine ourselves entirely to the part corresponding to
the free paths between the collisions. But while an electron
travels over one of these free paths, its velocity v; is constant.
Thus the part of the integrals in a; which corresponds to one
electron and to the time during which it traverses one of its
free paths is therefore

tt7 Sv] ,
; Gost 14 — at
ref) s 9 (+ 7) :

where ¢ is now the instant at which this free path is com-
menced and 7 the duration of the journey along it; but this
is equal to

Theory of Long Wave Radiation. 165

We now fix our attention on all the paths described by all
the electrons under consideration during the time @, and we
use the symbol § to denote a sum relating to all these paths.

We have then

Chense ACihe ve) Serres San
gga gat oF | ie +5):
We now want to determine the square of thesum 8. This
may be done rather easily because the product of two terms
of the sum whether they correspond to different free paths
of one and the same electron, or to two paths described by
different electrons, will give 0 if all taken together. Indeed
the velocities of two electrons are wholly independent of one
another, and the same may be said of the velocities of one
definite electron at two instants separated by at least one
encounter. Therefore positive and negative values of 1,
being distributed quite indiscriminately between the terms
of the series 8, positive and negative signs will be equally
probable for the products of two terms. We have therefore
only to calculate the sum of the squares of the terms in 8 or
simply
NG Ue a ST Ts
ae oi: sin? Jy co (tt "4 5). = Gres aCe)

Now since the ae motion of the electrons takes
place with the same intensity in all directions, we may
replace v,? by 4v7. Also in the immense number of terms
included in the sum (3) the quantities t and v are very
different, and in order to effect the summation we may
begin by considering only those terms for which the product

(« sin all has a certain value. In these terms which are

still very numerous, the angle gall (s+ 5 + 3} has values

0
that are distributed at random over an interval ranging
from 0 to sa. The square of the cosine may therefore be
replaced by its mean value 4, so that

oy v- * 9 STT
a= Sa pias tS (ae a a0 )

or if we introduce, after Lorentz, the length of the path /
instead of the time in it, this may be written

all
cia sin
‘ sre v0 |

(4)

166 Mr. G. H. Livens on Lorentz’s

The metallic atoms being considered as practically immoy-
able, the velocity of an electron will not be altered by a
collision. Let us, therefore, now fix our attention on a
certain group of electrons moving along their zigzag lines
with the definite velocity wu. Consider one of these electrons
and let us calculate the chance of its colliding with an atom
at rest ina unit of time. This chance is obviously equal to
the number of atoms in a cylinder of base 7R? and height wu,
R being as before the sum of the radii of an atom seal an
electron ; it is therefore equal to

nt Ru,

n being the number of atoms per cubic centimetre in the
metal.

But in unit time the electron under consideration travels
a distance uw, hence the chance of a collision of the electron
with an atom per unit length of its path is

nim Ru
C= — =nrk’,

Uu
and thus the mean free path of an electron is, as before,

me nik?

It is important to notice for future reference that J,, is
independent of uw. This is a consequence of the assumed
rigidity of the atoms.

Now during the time @ one of the electrons moving with
a velocity w describes a large number of paths, this number
being given by

l »)
and we now want to know how many of these paths are of
given length J.

For this, let f(l) be the probability that the electron shall
describe a path at least equal to /, then /(/+dl) is the prob-
ability that the electron has Reserved a path J and shall
describe a further distance dl, and this will necessarily be
the product of f(l) and another factor, this second factor
expressing the probability of no collision occurring within

* Lorentz does not make it clear that the JZ» introduced here is, in
fact, identical with that /» used in the formula for the conductivity ;
the expanded argument here given, however, proves directly what was
probably already known to him,

Theory of Long Wave Radiation. 167

the length di. This factor is known from the above to be
Cae (1- )

i+ dl)=(1= -)/0,

or what is the same thing

so that

so that pee!
the arbitrary constant of the integration being determined
by the condition that /(0)=1.

Thus the probability of the electron describing a free
path between / and /+dl is expressed as the product of the
probability that it has described a free path /, and that it
will collide in the next small distance dl, and is therefore

ely, odd =~ endl,
Thus of the total number of paths described by the electron
in the time @ the number whose length lies between / and
1+dl is
ud

2
m

endl,

so that the part of the sum in (4) contributed by these
paths is

sin sal . ( sitbm , §\\”
pon sin{ —— *,—
ud 20u ue PAC We )
5 e-Vmdl = ude—Y’m | —-__——_— dl.
is ST stl in
DAGHYS WL) 20u

On investigation of this expression from /=0 to =, we
find the part of the sum in (4) due to one electron, which is
therefore

ud Hey SiTl Ct a Ca 2u0l,,
aye) sins (= a-= je dl = ——— 3"
(5) : 20u (ss Sob al
0

20u is E27

Now the total number of electrons in the part of the
metallic plate under consideration is NwA, and by Maxwell’s

168 Mr. G. H. Livens on Lorentz’s

law, among these

tnNuda/ Le Y -9 udu

have velocities between u and u+du; the constant g is
related to the velocity u,, already introduced above by the
formula

q =
Thus the total value of the sum in (4) is given by

lnNudsy /L, . eee es

ma 7 2h
0 G2?

or, using z = qu’, by
Nw JL = we me
rape PS ae be Pm leg
WnG22 5.

This integral cannot be eee in definite terms, being
of the integral- -logarithmic type, but we can obtain various
good approximations to its value. In fact a direct use of
the first theorem of mean values in the integral calculus
soon shows that we have

© ge -*az if CPs im
s?ar?l a Sea lin *¢ ie
m ‘mM
0 & =() 0
il
im 0 lmeg
Lt
PANU
Z, denoting some mean value of z, which is ultimately, how-
yn)
; : S?9r 7b Oe
ever, a function of the one constant ( — in the

integral ; I find on trial that z) is such a function of this
constant that its value lies between 1 and 2, the values it
assumes for small and large values respectively of the con-
stant. If, therefore, we define uy by the relation

que” = ~05 ( ae
we shall know that w,?, ultimately a function of (32) :

must, however, lie between the limits

2 2 2
3 Um and Um, 5

Theory of Long Wave Radiation. 169

and then we shall have

a 0) Boe 14 s7e7L,,N umwAd
Cer D)
3a Sear lin?
BS An
60%c% a it 120?

and the expres sion for the partial energy flux joven the
element w’ thus takes the form

\/ 2 s*Ne*l nun, mG

5) 1 6 02077?? ; Tl
t v we
ee : ING) oe
But in virtue of the relation 7 = or this becomes
aA 2 Ne? als an Aaa.
3 ena Agrc?l,? ;
1 a Uy?

We therefore conclude that the emissivity of the plate 1s
given by

pe? See iNez) a WX
Mi aN 30 (1 ae Arr” al

Uy?

On combining the two expressions for E and A we find
that

hy 32a ln?
FO,T 87 EH _ 8m,” BU?
a, el Me Se 1+ Amel? ’
Up DS
which is exactly Lorentz’s result if
Woe kota?
ON ao
0 8
3c
Ole 2
Ug” ss g im p)

a value which certainly lies within the above limits possible
for w2, but which can only be said to be satisfied exactly for

one particular value of

If, therefore, the formula adopted for o is exact, our
analysis verifies that Kirchhoff’s law does not apply exac stly
in the case under investigation. OF eourse the diser epancy

is small except perhaps for extremely short waves, but it is
worth noticing.

170 Mr. G. H. Livens on Lorentz’s

It would, however, appear more probable that it is the
formula for o that is at fault* and not Kirchhoff’s more
general law, the truth of which can hardly be doubted. It is,
however, in any case interesting to notice that although the
formula for o adopted above may not be exact, the formula
necessitated by Kirchhoff’s law in combination with the
above analysis provides an interesting verification of its
general form.

In any case, however, we may conclude that for all prac-
tical purposes the complete radiation formula applicable all
along the spectrum is given by the usual Rayleigh-Jeans
formula

in Str MU”
is) Sa
as Lorentz predicts, a formula which is, however, only phy-
sically applicable in the extreme ultra-red part of the
spectrum.

But this general conclusion is utterly absurd both from
a mathematical and a physical point of view, and it therefore
appears that some fundamental error has been committed
either in the physical assumptions made or in the mathe-
matical analysis based on these assumptions.

It is very difficult, if not quite impossible, to indicate any
steps in the above analysis about whose mathematical rigour
any doubts can be raised, but it is worth noticing that the
final result obtained is not consistent with the preliminary
assumptions, inasmuch as the Fourier series initially assumed,
which can have no meaning unless it is convergent, ulti-
mately turns out to be divergent, so that the theory would
appear to lead to a result which is ultimately a contradiction
in terms, or, at least, apparently so. Some light is, how-
ever, thrown on this question by an examination of the
physical basis of the theory.

The one advantage possessed by the present form of theory
over Lorentz’s original form is that the number of physical
assumptions on which it is based is reduced from two to one,
so that it is now possible to determine the actual extent to
which the physical basis of the theory is responsible for the
result obtained. We have merely assumed that the duration
of every collision of an electron with an atom is vanishingly
small compared with the other periods involved in our
analysis, and as long as this assumption is justified our
result must be correct. But, as a matter of fact, in actual

* This probability is fully borne out by a more detailed investigation
of the question as to the proper expression for c.

Theory of Long Wave Radiation. 7a

practice this assumption is justified only to a comparatively
rough extent and only when all the other times involved in
the analysis are large compared with the usual intra-
molecnlar periods, so that the results obtained can only be
applicable in the extreme ultra-red region of the spectrum,
where it is of course known to apply. As soon as the en-
counters between the electrons and atoms are sufficiently
long compared with the period of the light discussed, the
effect of the collision will make itself felt in modifying the
radiation formula, a conclusion drawn some time previously
by J. J. Thomson.

It is just the assumption concerning the shortness of the
collisions which is the predominating factor in restricting
the general application of the Rayleigh-Jeans formula.

The fact that the general radiation formula which is to
be applicable all along the spectrum must ultimately centain
some general account of the actual collisions of the electrons
with the atoms can be illustrated in various ways. Let us
confine our attention to one of the free electrons in the
metal considered in the above analysis ; the average flux of
energy through w’ arising as a result of its motion is

ew! OKRA Le Cres: Od[ vr] ar
aries) di Jat = 94) t+ dLv,]-

Now if the collisions are all of short duration and A[v]
denotes the total change in [v] during a typical one of these
collisions of total duration At, then the energy radiated

through w’ during this collision is of total amount

ew! {Ale |}?
87r20°r78 ING Tes

which is inversely as At. Thus if, as in the above, we
assume the duration of all the collisions to be infinitely short,
the total amount of energy radiated away will be infinitely
large. This merely means of course that as soon as the time
of a collision becomes appreciable a closer investigation
will be necessary, involving necessarily some account of the
nature of the collision. |

Tt is now no longer surprising that a divergent series is
obtained in the expression for the total energy ; in fact the
shorter the collisions the farther up the spectrum does the
agreement between theory and practice hold, but then the
bigger is the total enerey.

These results are still further illustrated By they mesulls
of a general theory developed by Thomson * on molecular

* Phil. Mag. [5] xiv. p. 217 (1907).

Ie Lorentz’s Theory of Long Wave Radiation.

kinetic principles. The method followed by Thomson is
analogous to that of Lorentz, but it avoids the probability
considerations involved in that author’s theory. He views,
with Lorentz, the radiation as a result of the changes of
velocity produced in the collisions of the electrons against
the molecules, and he concludes that the manner in which
these changes take place must, as stated above, ultimately be
of influence on the final formula for the radiation. Assuming,
then, as possible arbitrary types of acceleration of an electron
during a collision functions of the time of the form

oF Ee
a? +

(ie), Ama (ue)
in which A and a are constants, he arrives at forms of
FK(, T) of the following types respectively,

Sle
(I.) F(, ) = MaKe? AS
OMUpy? — 228%
Sn ee ee
which give for the total energy radiated respectively

22 9 (*
Smit? ( See an Smu,? (” —s2ee
io) == e a lene e x? Aa;
0

(II.) Fa,T)=

3 NG 3a J,
and
(II.) Smrtyp? i —!22dp, _ 8p? fi pn Anee
: ; é = ee ue aX,
eG nr DO NG

where in each integral on the right we have written c=a/2.

Now in each of the two cases here illustrated the con-
stant a turns out to be approximately equal to the time of
duration of an encounter of an electron with an atom, so
that if we assume this time to be infinitely small both forms
of Thomson’s theory agree in giving

Smu ?
rt

FQ, T) =

as the complete radiation formula all along the spectrum, but
in both cases the total energy is infinite of the order (1/a’).
It would thus appear, both from these two examples and —
also from the more general case discussed above, that any
general theory which leads to the Rayleigh-Jeans formula
as the formula generally applicable all along the spectrum
must involve some assumption which essentially implies that
the total amount of energy radiated is infinite, so that it

Electron Theory of Metallic Conduction. 173

cannot represent any real physical example of a radiating
body of any known type. Physically this implies that the
general restrictions limiting in actual practice the validity
of the physical hypotheses on which the theory is based,
must also limit the applicability of the formula obtained from
the theory, an obvious remark which it appears, however,
necessary to insist upon, because it is the expressed opinion
of certain mathematical physicists that, for example, the
Rayleigh-Jeans formula obtained by Jorentz on certain
obviously restricted assumptions is of a general validity in
no way limited by the restrictions naturally imposed on
these assumptions. The results obtained by Thomson are
almost conclusive evidence that this contention is in no wise
justifiable, and the results of the above form of Lorentz’s
theory are also against such an opinion.

I hope to discuss, in further detail, in a future communi-
cation some of the points raised in the latter part of this
paper and not fully disposed of.

The University, Shetheld,
October 9, 1914.

Note added Dec. 2nd, 1914.—Since the above paper was
sent to press I have discovered that Prof. H. A. Wilson has
anticipated the main point of the above analysis although
he apparently failed to appreciate its bearing (Phil. Mag.
Nov. 1910). He, however, unfortunately includes it as a
small part of a paper, all the other results of which are
either incomplete or inaccurate, and I think it deserves
better and more elaborate treatment. Some advantage may
therefore be gained by amplifying the point as above.

XVIII. On the Electron Theory of Metallic Conduction.—I.
By G. H. Livens *.

Introduction.

Q)* of the greatest successes achieved by the so-called
theory of electrons has been in its application to the
explanation of the details of the conduction of electricity in
metals. Encouraged by the wonderful success of the earlier
and more tentative applications of the theory by Drude and
Riecke, numerous writers have endeavoured by the intro-
duction of statistical methods to develop the theory in still
greater detail. Many of the more fundamental results in

* Communicated by the Author.

TA. Mr. G. H. Livens on the

the theory have thereby received verification and justifi-
cation from several entirely independent investigations,
although it must be admitted that some of the results of
these investigations have not always been so happily coinci-
dent as one might desire. This remark applies particularly
to the formula for the electrical conductivity which expresses
it in terms of the electron constants of the metal. After
Drude’s initial attempt to deduce a formula for the conduc-
tivity, the problem appeared to be finally settled, at least as
far as its application in the theory of steady currents, by
the very general and elaborate investigations by Lorentz *.
The formule obtained by Lorentz, however, appear to require
modification and amplification in order to make them
applicable for very rapidly alternating fields, particularly
those associated with radiation. The application of the
theory of electrons to these extensions, initiated by Thom-
son t, was fully carried out and in great detail by Jeansf
and H. A. Wilson §, their method of procedure being, how-
ever, essentially different from that followed by Lorentz.
It appears, however, that the formula obtained for the con-
ductivity by Wilson, who alone carries the calculation right
through on the statistical basis, does not agree with Lorentz’s
formula in the limiting case, being in fact half as big
again.

It is maintained by Nicholson || that as far as the optical.
properties of metals are concerned Wilson’s formula 1s more
satisfactory than any other yet proposed. He aiso considers
that Wilson’s treatment is the most satisfactory yet pub-
lished and that from the theoretical standpoint it is
complete ! 4]

The object of the present communication is to prove that
a rigorous treatment of the problem along the lines laid
down by Wilson and Thomson leads to a formula differing
trom Wilson’s by a factor 2/3, which is Just what is required
to make it consistent with Lorentz’s result in the limiting
case. There is a discrepancy in Wilson’s treatment of the
problem which has considerable bearing on the final result
obtained.

The discussion of the actual bearing of the results of the
present discussion on the optical side of the question will
be reserved for a future communication, as it is merely
desired to show that the principles underlying the discussions

* Vide ‘The Theory of Electrons.’ t Phil. Mag. Aug. 1907.

t Phil. Mag. June and July, 1909. § Phil. Mag. Nov. 1910.

|| Phil. Mag. Aug. 1911.

4 Nicholson informs me that he has subsequently modified his views
as to the exactness of the formule under dispute.

Electron Theory of Metallic Conduction. 75

of Thomson and Wilson are consistent with Lorentz’s
general theory. It is perhaps necessary to add that Wilson
gives two independent deductions of his formula, one
following Jeans and the second following Thomson. It is
the second deduction with which I shall here concern myself,
the first, being more concerned with the optical side of the
question, will be discussed in the future communication.

With each of the aforementioned authors we shall consider
that the phenomenon of electrical conduction in the metal
arises entirely from the motions of a swarm of electrons
moving about in a perfectly irregular manner in the free
space between the atoms of the metal, which are presumed
to be of such comparatively large mass that their energy
and motion may be neglected. ‘The electrons and atoms
are presumed to be perfectly elastic spheres so that the
velocity of an electron is not altered by collision, the atom
being at rest.

The general principles underlying the determination of
the conductivity («) to be here reviewed, depend essen-
tially on the fact that the energy dissipated by a steady
current, driven by the electric force H, is presumed to be
the same as the energy acquired by the electrons on account
of the action on them of the electric force during their
otherwise free motion between the collisions, and which is
dissipated on the collision of the electron at the end of this
path. The rate of dissipation is known to be of’, and it
can be calculated by statistical considerations of the motions
of the electrons.

We shali first make the calculation on the assumption
that a steady field of constant strength E is in action
parallel to a fixed direction. ‘This is not precisely the
problem discussed by Thomson and Wilson, but the analysis
is much easier and has the additional advantage of bringing
out very clearly the correction which it is necessary to
introduce into the original analysis of these authors. The
extension to rapidly alternating fields will be given in a
subsequent paragraph.

Analysis for steady jields.

We choose a definite system of rectangular axes with the
w-axis parallel to the direction of the electric force. Referred
to these axes the velocity of the typical electron at time ¢
has components which we shall denote by (€,, 7,, §), so that
the resultant velocity is », where

pee aD 2 2
Pee i,

176 Mr. G. H. Livens on the

The equations of motion of the typical electron during its
free motion between two collisions while under the action of
the electric force is

dé: _ dnt _ aC,
m— =ell, ea” ae

where m denotes the inertia mass of the electron and e the ,
charge on it. Thus

E,= ey +&, 1,=7; C=,
(£, 7, £) being the velocity at the instant of beginning the
free motion from which the time is alsomeasured. The work
done by the electric field on this electron during the whole
of the time between two impacts, an interval of length 7, is

thus
: + /eEt
Bédt = cE (— Vat
\"- E. e ( a +e

B22? T?
= - 5 + eHér ;

and we now require the sum of the quantities of this type
corresponding to all the free paths of all the electrons per
unit of volume covered during the unit interval of time.
Denoting this sum by 8 we find that the total amount of
heat developed per unit time per unit volume is

é Ae

H=oite=s 5 + er).

Thomson and Wilson both proceed by making a statement *
which is equivalent in the present notation to saying that

S(eHEr) = 0,

their reason being presumably that since positive and nega-
tive values of & are equally probable there will be equal
positive and negative terms in the sum which will thus on
the whole be zero. ‘This statement, however, does not appear
to be quite correct, since, granted exactly identical conditions
for electrons with the component velocity &, the value of 7 is
less when £ is positive than when it is negative by an amount
of the order eHi7, so that there must in any case be a residue

* Jt is perhaps only fair to add that this statement, in the form I give
it, probably never occurred to either author, since in the particular case
they examine they have another plausible reason for neglecting the
corresponding term of the sum.

Electron Theory of Metallic Conduction. ean

in this term which is of the same order of magnitude as the
other term of the sum actually retained.

The summation will be best effected if, after Lorentz, we
interpret it in terms of the lengths of free paths and
velocities in them instead of the time of duration. From
the equations above we find that the projections of the
particular path there under consideration along the co-
ordinate axes are of lengths

eK 7?
ae g +é: l,=NT, = Gr.

so that the length of the path is practically
Pale LP le?
= i eli 3
| = pq? + Fe Gi
where we use
m= P+? + C

and neglect squares of the small order term involving H.
From this it is easy to see that to the same order of

magnitude
( Sac)
T= {— ons | |9
ip 2mr
so that the above expression for the work done by the

electric field on the electron in its free path is to the same
order of approximation

22/2
A nei —)

2mr? ? 2mr°
i eK? |? (1 ee it eHEél
~ Om 7? 7 yp ?

so that we have to evaluate

asf P (1 E) 4 oy
| 2m 7 ae r
Wherein it is to be remembered that all velocities are to be
taken at their initial value at the beginning cof an impact.
Consider first the contribution to H made by a single
electron which would in the absence of an electric field be
moving freely with a velocity 7, and which will therefore
move so that it resumes this value at the beginning of each
free path. This assumes with Thomson * that the whole

* Wilson does not state that this assumption underlies his analysis,
but without it his analysis is meaningless. It is, I believe, his failure to
realise the importance of a clear detinition of this point which is the
cause of the errors he makes in the analysis.

Phil. Mag. 8. 6. Vol. 29. No. 169. Jan. 1915. N

178 Mr. G. H. Livens on the

eifect impressed on an electron by the field during its motion
before a collision is obliterated by the collision. During a
unit of time this electron traverses a large number of free
paths, this number being given by

mee)
br

where 1,, is the mean length of a path. Of this number we
know that there are a number

ee
whose lengths lie between J and/+dl. These contribute to
the above sum an amount

e* S| P “Ulm l Lilm
Saad 1— a [2° Hee ME 2 dl.

Integrating this expression from 0 to ~ and noticing that
£ is not a function of J, we find the whole contribution to the
sum 8 due to one electron in the form

Te Ea Ana
It is now clear that the mean free path /,, which was merely
introduced in a general manner (much on the lines adopted
by Lorentz in his book ‘ The Theory of Hlectrons,’ page 282,
note 36) can be assumed to have its undisturbed value, which
it assumes in the absence of an electric field *.

This last expression must be summed over all the electrons
in a unit volume. If, as above, we assume that each colli-
sion destroys the etfects of the electric tield, then we may
assume that the distribution of the initial velocities among
the electrons is that expressed by Maxwell’s law ; in other
words the number of electrons per unit volume with their
velocities between the limits (& 7, €) and (€+df, »+dn,

£+db) is a
3
Na/ Tet ae dn dé,

wherein N is the total number of electrons per unit volume

* Some doubt may be expressed as to the general validity of the argu-
ment just repeated, but I think, on due consideration, it will be difficult
to replace it by any other. Besides, the argument used by Lorentz to
deduce the law of distribution of the lengths of path is probably inde-
pendent of the action of the field, if 2» is properly interpreted.

Electron Theory of Metallic Conduction. 179

and g is a constant which is connected with the mean square
of the velocities, viz. w,, by the relation

Bes:
Gh ane 9°

Sidi)

The contribution of this group of electrons to H is thus

2 pay 2\ p-or2
pe Nets L( -2)o dédndt

m T

ie Me
“i Na / ©, eL" EdE dn dF.

Integration of this expression over all positive and negative
values of the variables (&, 7, €) furnishes the complete value
for H. The integral due to the second part of the expres-
sion obviously vanishes, and so we are left with

eNeL /a ay a ae at edn dé

To evaluate this we may, as usual, put &* equal to 37? and

dé dn dé equal to 427r°dr, and then we find that

/ 2 2
Ha srt ay Ore-U" dr
TT Jo

3m
i (Be g )p°
3m 1

AN Ve : Ne*ln m9
QT) MU,

And since H =cE? we see that
Z Ne?2lin

G—y 3
Om | ntti.

which is precisely Lorentz’s result.

Analysis for rapidly varying fields.

The ideas of the preceding analysis are directly applicable
in the more general case of a rapidly alternating field such
as we find associated with radiation. We may, for such a
case, take the electric force E to be simple harmonic with a
period p: say

H= EK, cos( pt +e).
N 2

180 Mr. G. H. Livens on the

We have then the equations of motion of the typical
electron in the form

mé, = eH) cos( pt + €) y= t; =
so that now

pat eK {si : )
= ap sin(pt+e¢)—sin a +&,
t=; C= &

and also the projections of the typical free path along the
three coordinate axes are of lengths

eK
a= ET + mp
Ljy=NT, l.=€r,
and thus to the usual order of approximation we have the
length of the path given by
2&eH 7

! cos €—cos( pt +e)—pT sine \,

[2 eee ae ce { cos e—cos( pt +¢)—pT sine } :
and again
eHog 2 an a
T= [1 ape 1 cos e—cos(pt+e)—pT sine i ] :

In this case the work done by the electric field during the
particular free path under consideration is

" eH cos(pt +)E,dt
0
= eK cos(pt-te) { 9 sin(pt + e)—sin | + E | at
0

Me eg 7 oe
= oe | sin(ptT+e)—sinle

KE,
mi

= | sin(er +e) )—sine -

This nee is againi better interpreted in terms of the
length of free path and the velocity in it. On substitution
therefore of the value of t from above we find that to. _the
second order in EH? this expression is equal to

{ 2
Epic: al sin (& +e)— cine}

2 Le
x 25 cos (e ) COs € — cos (e +e) ~ Pr sin €

oe Hy Eee sin (Fe te — sin s.

Electron Theory of Metall Conduction. 181

And we must now sum this expression over all the free paths
of all the electrons per unit volume per unit time. We
may notice, however, that since the phase e of the electric
force at the beginning of the path may have any value what-
ever, we may at once replace all factors involving e by their
mean values. For example, the mean value of-

| 1
Gs) eos? (z + c) Sy eee

ss ) : ie
(1i.) cos (2 +6) cose 1s 3 008 ;

ii.) cost (+e) is

: l
(iv.) cos he +e] sine is —
Yh
and the others are all zero.
We have thus to sum expressions referring to each free

path of the type
2h 2 2
Os [2 eee 2 {cos 14 Pim Pe .
n° op 7

2mp? 2r r

Now, as before, let us confine our attention to one particular
electron which in the absence of any external field would
continue to move with velocity 7, and which therefore will
begin to describe each path with this velocity. In unit time
this electron will describe on the average the number r/lm of
free paths, and of this number there are

—

r
7 2
Ve

whose length lies between / and /+di/. The contribution of
these terms to the above sum is therefore

eK? 2 » pl & El A pl LenS
¢ »{ 2sin (1+ =| 3 sin Ee ate

On integration of this expression from 0 to 2 we find the
oe contribution by this particular electron. Noticing
that

oe im dl

9 «
Dil
Yoo ~ ToT
ae uot 2
[site tinane 2,
0 aie en pel?
. )

182 Mr. G. H. Livens on the
and that

ye ye

ne
( Peink onlin gj 7 a
Sie

we find that the contribution by the one electron to the total
sum is

pee.

Hy ln via
2rm ° (oe ;

The number of electrons per unit volume with initial velocities
in the limits between (&, 7, €) and (£+dé, n+dn, €+d£) is,

as before,
Qe >
ny /% e 2” dE dn dé,

so that the contribution of these electrons to the total energy
dissipated is

Ne? Ba Te -e— gt |
oO on py ee
1+ oy

and again integrating over all beta values of (&, 7, €) we
find the total value of H in this case to be

ea Ve i e72 par

0 14+f3 cn

Now the mean value of cH? is 3cH,’, so that on com-

parison we find that
__ 8m Ne’l,, jm ee
aa 7

| 27¢ )
or put s=qr’ and p= pe merce!

ee el
0 1 4 nS ¢ zs 407mg

ihe v2 Nel, VEN
e 37 MUm (0) pa OT elm?

aoe
Ras

Electron Theory of Metallic Conduction. 183

This is the same result as obtained by Wilson, reduced, how-
ever by the factor 2/3 which brings it into line with the
result obtained by Lorentz, and which must be applicable
to the present problem in the limiting case of very long
waves.

Conclusion.

It would thus appear that a rigorous treatment of the
problem of conduction based on the above-mentioned ideas
leads to a formula which is entirely consistent with the results
of the analysis on more general lines given by Lorentz.
The two methods of reasoning appear at first sight to be
independent. Lorentz bases his calculations on the existence
of an average steady state of motion, in which the law of
distribution of the velocities among the electrons, which
differs from Maxwell’s well known Jaw owing to the action
of the electric field, is definitely calculable by similar
statistical considerations of the effect of the collisions. On
the other hand, Thomson aad Wilson merely assume that the
whole etfect imparted by the field to the electron in its
motion along a free path is destroyed by the collision at the
end of the path, so that the electron starts off each path with
the velocity which it would have according to Maxwell’s
formula. which is presumed to hold in the absence of the ex-
ternal field. The two forms of the theory are thus mutually
consistent, and the fact that they lead to the same formula
for the conductivity is very strong evidence in favour of the
formula.

But the two views are probibly much the same in the end,
although that of Lorentz is probably the more general. In
fact it is impossible to imagine the existence of an average
steady state, as Lorentz imagines, unless some such action as
that implied in Thomson’s assumption is in play. Lorentz *
himself clearly appreciates the force of this remark and has
made various suggestions in explanation of it, so that it does
not appear necessary for me to go into further details. I
hope, however, to be able to return to it in a further com-
munication.

The University, Sheffield.
Oct. 1914.

* See Vortrdge tiber die Kinetisehe Theorie der Materie u. der Elek-
trizitdt (Leipzig, 1914), p. 187.

[Wee SS Aas,

XIX. On the Plastic Bending of Metals. By ¥. Luoyp
Hopwoop, B.Sc., A.R.C.Sc., Demonstrator of oa
St. Bartholomew's [ospital WMediea! College, London *

F a uniform straight rod, clamped at one end and not
otherwise supported, is allowed to sag from a horizontal
position under the influence of its own weight, one of three

things will happen:—

a. It will quickly take up an equilibrium position in which
the external forces are balanced by internal elastze
forces, or

b. Plastic yielding will occur and the sag will steadily
increase with time, the rod coming SLOWLY to rest
ultimately in an approximately vertical position, or

(If the ratio of the length of the rod to its diameter
exceeds a certain critical value)

c. The plastic yielding will take place so rapidly that
owing to its momentum the rod will bend until its
free end passes beiow the clamp and oscillates about
a point below the clamp. (See fig. 1.)

Fig. 1.

8
8
8
6
8
8
8
i
r)
9
8
a
e
§
)
8
&
®
8
o
]
1)
1]
B
B
8
&
B
B
L)
8
B
y
a
]
L)

The complete mathematical theory of Case a is worked out
in treatises on Elasticity.

Cases } and ¢ involving as they do finite flexure and plastic
yielding, under what is, in effect, a travelling load, are

* Communicated by Prof. A. W. Porter, F.R.S.

On the Plastic Bending of Metals. 185

incapable of exact mathematical treatment, and even an
approximate theory presents almost insuperable difficulties ~*.

The present writer has obtained some empirical results for
Case 6, which, in the absence of other information, seem to
him of sufficient interest to justify their publication.

The measurements here recorded were made on thin wires
of commercially pure lead, but similar results have been
obtained with thin wires of tin and cadmium. The lead
wires were specially drawn for the experiment and were
kept straight and not wound on bobbins. They were laid aside
for several months at ordinary temperatures fur annealing
purposes.

Observations were made as follows :—

A suitable length of wire, having one end fixed in a
horizontal clamp, was gently released by hand from the
horizontal position at a known instant. A number of whole-
plate photographs of the specimen, a vertical plumb-line,
and the recording timepiece, were taken at times extending
in some cases over several days. The photographs were
taken against a black background, and accurate measure-
ments of the coordinates of any point on the specimen were

eo)
Tenege ae

E
OA=/NITIAL POSITION
OB=FOS/TION O-9 MINUTES AFTER RELEASE

0¢= « 256
Oia ht J6a° vis ’ "
Oe= we) 4490 : “

obtained by laying the developed plate on squared paper and
Inspecting with a low-power microscope. Fig. 2 shows the
* Vide Math. and Physical Papers of Sir G. Stokes, vol. ii. p. 178.

186 Mr. F. Lloyd Hopwood on the

progressive yielding of the specimen referred to in Table I.
the curves being plotted from measurements made on the
photographs.

Taking the origin at the edge of the clamp and the hori-
zontal and vertical directions as the axes of a and y respec-
tively, it is found, on plotting the measurements from any
single photograph, that log x and log y are linear over a
considerable range, that is # and y are connected by a reiation
of the form

(1 Te

In fig. 3, log w and log y are plotted for the whole rod for
several photographs of the same specimen.

Fig. 3.

A = Photo. N° |.
Bes) cas
C= « 7
D= « 9
E= « x!

1-5 i Oe

It is interesting to note that the divergences from linearity
which are shown by points remote from the clamp in the

Lead Wire.

Plastic Bending of Metals.

earlier photographs are inclined in the opposite sense to those
for later photographs.

Table I. gives complete particulars of a set of eleven
photographs taken during the sagging of a single specimen,
while Table II. shows the agreement between the observed
values of # and those calculated from equation (A) for one
of these photographs (number seven).

TABLE I,

Length=11-05 cm.

Lead began tofsag July 21 at 12> 35™ 49°5° p.m.

187

Diameter ='023 cm.

| | Time |
Yabo a. | nm. | Temp.) Remarks,
| Minutes. |
ipsaimniyes O
First Photo taken July 21 at 12 36 44 90 | 9-44 |-638 |18°2 C.
Second ,, me , 12 38 16 2°43 | 9°01 |:636 |18:2
Third 5 9 ” ZS 21 oa iON Godlee
Fourth ‘,, a Th Disa Sisy Norisaea tava iter
Fifth - Af Rs 1 40 57 | 65:2 | 7-08 |°616 |18°2
Sixth A BA iy 2 46 36 131 | 6°50?|'612|18-1 |Faint plate.
Seventh ,, a Re 3 8 43 153 | 6°54 |°606 |18:1
Highth _,, if Me Arie 266 | 6:13 |:600 |18°2
Ninth - » duly 22 at 10 12 29) 1296 | 5-26 |-580|17-8
Tenth ‘ » duly 2at 225 3] 2990 | 4:81 |-571 116-9
Eleventh ,, » duly 24at 3 2518) 4490 | 4°59 |-566 118-2
|
TABLE IT
|
y- observed. caleulated. | observed. calculated.
OF. 0 0 | I iAce 36°35 36°41
ie 6:5 6-54 bs pa teeee 37°65 37°69
Bis 100 9°95 | LQ. 38°9 38:95
On 12°75 12°73 } 20.4. 40°05 40°18
4... 15°15 15:15 ibemeilve 41°3 41:39
ie 173 17°54 ees 43°5 42:57
6... 19-4 MPH Niece a. 43°7 43°73
(bee 21:3 21:27 24... 448 44°87
Br: 23°71 23°08 One. 45°95 46:00
tes: 248 24:77 W2Oa is 47°1 47°10
Oe 26°5 26°40 | mare 48°15 48°19
Tere 28:0 27:97 Woke es 51-4 51:47
ee 29:6 29°48 . || 35. 56°55 56°40
133.3 31:0 30°95 | 40... 61:45 61:16
a ee a 32'3 32°52 fearaOeat 708 70-01
[Srna Ke 33°75 33°75 1.60%... 80-2 78-19
LG eh, es sok 35:10 || *79°55} — *98-2 92°76 |
|

* Coordinates of extremity.

188 Mr. F. Lloyd Hopwood on the

The caleulated values of x given in the above table were
e 1 e oS
obtained from the equation

x= ODL y Oe.

Both the quantities a and n of equation (A) are functions
of the time which has elapsed since sagging began, a showing
a big percentage variation and n a comparatively small
variation, during a complete run.

The relation between a and time of sa
expressed by the equation *

a=e—Bloet.'. .. .) Renees

This is clearly shown in fig. 4.

g@ is accurately

Fig. 4. :

a & L0G. ib (72s)

The approximate values of « and 8 are
Oe eo ales ls
The result (B) is of some importance, for it is of the same

* Equation (B) obviously cannot hold from the instant sag commences.

Plastic Bending of Metals. 189

form as that obtained by P. Phillips*, connecting the ex-
tension with the time a metal wire is subject to a constant
pull, and by C. EH. Larard f, connecting the torque and time
when a metal is twisted to destruction at contant angular
velocity.

In the light of their experiments, we may assume that the
particular length of the specimen over which equation (A) is
true coincides roughly with that in which plastic deforma-
tion is taking place, the remaining portion of the rod not
being stressed beyond its elastic limit and acting merely as
a load.

Fig. 5 shows that n is a parabolic function of the time

Log. 72 « Log t

after an initial stage—taking about 15 minutes in present
case—is passed.

For all the specimens on which the writer has experimented
the value of n is in the neighbourhood of one half. This is
rather suggestive of the well known Parabolic Catenary,
in which the load on any element is proportional to the

* P, Phillips, “On the Slow Stretch in Rubber, Glass, and Metals
when subject to a Constant Pull,” Phil. Mag. ix. (1905),

+ C. HE. Larard, “On the Law of Plastic Flow,” Proc. Phys. Soc.
1913, 3

190 Dr. 8. A. Shorter on the

horizontal projection of the element. The slight variation
of n with time may be ascribed to the rotation of the spe-
cimen as a whole about a horizontal axis through the origin.

Summary.

The sagging of thin uniform rods in the form of canti-
levers under the influence of their own weight is divided
into three cases, and empirical results are given for one of
these cases. 3

The case considered is that in which plastic yielding occurs
and the momentum generated in the rod is insufficient to
cause it to swing past, and oscillate about, its final equi-
hibrium position.

It is shown that for the thin wires used (lead, tin, and
cadmium), if the origin be taken at the encastré end, the
equation to the curve they assume is of the form e=ay".

The significance of the terms a and n is discussed.

In conclusion it affords the writer much pleasure to
express his thanks to Professor A. W. Porter, F.R.S., for
his kindly interest in this paper.

XX. On the Shape of small Drops of Liquid.
To the Editors of the Philosophical Magazine.

GENTLEMEN,—
N the November number of the Philosophical Magazine
Mr. J. Rice puts forward a theory to account for the
peculiar shape (a biconcave disk with rounded edges) of the
red corpuscles of blood and of suspended particles in a
lecithin emulsion. Now experiment shows that small par-
ticles of liquid suspended in another liquid always assume
a spherical shape—as is to be expected from considerations
of surface-tension. Hence the most natural explanation of
the above shape is that the effect of surface-tension is
modified by some other factor or factors. Mr. Rice, however,
assumes that the peculiar shape is due to surface-tension, and
attributes the deviation from sphericity to the smallness of
the particle and consequent variability of the surface-tension.
Assuming that the surface-tension T at any point is a
function of the thickness of the disk at that point, Mr. Rice

Shape of small Drops of Liquid. iL)

calculates that the shape of particle for which the value of

the integral
{ {24 S,

taken over the surface § of the particle, is a minimum.

The fallacy of the above method lies in the assumption
that the above integral represents the *‘ surface-energy.” At
the beginning of his mathematical treatment of the subject,
Mr. Rice writes: “‘ Now let T be the surface-tension or surface-
energy per unit area of the interface ”’—a sentence in which
the fundamental error is tacitly introduced. This identifi-
cation of surface-tension with surface-energy per unit area is
allowable only so long as the surface-tension is constant.
The surface-tension is the rate of increase of the free energy
with the area of the surface. If the area 8 of a film is
increased by external forces from 8, to S, the work done is

aie
{ Td8,
Si
which is equal to the change of free energy of the film.
If the film is so thick that T is independent of S, the above
expression becomes

isc:

Hence in this case the variation of the quantity TS is equal
to the variation of the free energy, so that T may be called
the “surface-energy per unit area of surface”*. When the
film is so thin that ‘I'is a function of the thickness (7. e. of the
area), it is as incorrect to call the surface-tension the “ surface-
energy per unit area of surface’ as to call the pressure of a
gas the “negative volume-energy per unit volume,” or the
tension of a spring the “elastic energy per unit extension.”

The fact that a blood corpuscle is not spherical simply
shows that it cannot be regarded as a drop of liquid sus-
pended in another liquid. The occurrence of factors modifying
simple surface-tension effects is quite common even outside
the sphere of biology. A familiar instance is the formation
of a soap layer at the interface between an alkaline solution
and a vegetable or animal oil. This makes possible the
formation of long thin cylinders of solution in the oil, which
do not break up into drops till exceedingly thin—a_ process
impossible in ordinary cases.

Yours faithfully,
The University, Leeds. S. A. SHORTER.
26th Nov. 1914,

* Or more precisely “free surface-energy......

Bye aie

XXI. A Study of the Absorption Spectra of Organic Sub-
stances in the Light of the Electron Theory. By N. P.
K. J. OPN. McCieiann, B.A., Pembroke College, Cam-
bridge *.

[N.B.—The small numbers refer to the table of references given at
the end. ]

HE large number of experimental results now available
has made it possible to trace a qualitative, and quite
empirical relation between the constitutions of organic
substances and their absorption spectra. Thus, it has been
found that certain groupings of atoms are likely to give
rise to bands in particular regions of the spectrum, but all
theories which have hitherto been proposed to explain this
fact appear to have postulated a characteristic type of vibra-
tion within each kind of group. Thus in the isorropesis
theory of Baly a vibration between the forms

—CH,. C==-0 > —CH=C—OH
| |

was suggested in the ketones, and between

C=0 C—O
etl |

C=O C—O

in the diketones, and so on. The only generally accepted
law was that selective absorption in the ordinarily examined
region f originates in the unsaturated condition of atoms,
and that the accumulation of these unsaturated valencies
shifts the absorption towards the red end of the spectrum.

In this paper a theory is proposed which, starting from
the atom, builds up the spectrum in a perfectly general way
and makes it possible to predict the positions of absorption
bands with reasonable accuracy from the constitution, given
certain fundamental constants. At present, these have only
been obtained in a few cases owing to the lack of necessary
data, and the author is unable to obtain these for want of a
spectroscope.

* Communicated by Sir J. J. Thomson, O.M., F.R.S.

The author being under orders for active service has been obliged to
leave some points in this paper unfinished, for example an extended
table of numerical results might have been given. It is heped that
these deficiencies may be made up in the future.

+ Until recently spectroscopes in general use covered the region
4 600—230 (up). Stark and his collaborators have lately published
results down to 180. (See below.)

Absorption Spectra of Organic Substances. 193

The model of the atom from which the results are obtained
is that in which the valency electrons are supposed to move
in circular orbits round a central nucleus; when the valencies
become saturated, the corresponding electrons are supposed
to be withdrawn out of their original orbits, perhaps into one
of greater radius (about the line joining the nuclei) the
periods in which are much greater than in the original. On
this hypothesis, it is clear that when an atom is unsaturated
as to one primary valency like the carbons in ethylene, there
will remain one electron only in the outer orbit. If, now,
a disturbing force such as light passes through the system,
the electrons will vibrate about their orbits, and, from what
has been said above, it appears that the periods ‘of primary
electrons of some of the commoner atoms happen to lie in or
near the region examined in the ordinary instruments. The
case when there is one such seat of disturbance has been fully
dealt with by Drude! and others ; here cases where several
such “ vibration centres ” occur in the molecule will be dealt
with.

Since an electron moving in an orbit is a current, the
vibrations of one will affect those of another in accordance
with the laws of electrodynamics (see Appendix); the
problem is therefore reduced to compounding vibrations by
means of the mutual induction method.

It is not of course claimed that the whole of the mathe-
matical treatment is original: it is, however, claimed that
most of the cases worked out are new, and that this method
has not been applied to the problem before.

There are, however, certain sources of uncertainty : in the
first place ihe work is so complicated that only the simplest
cases, 2. ¢. those in which there is a certain amount of sym-
metry, are profitably discussed ; we assume, therefore, that
slight changes not involving the introduction of a fresh
vibration centre will not make any difference to the type of
spectrum.

Again, there are on our theory several rings of electrons
in a molecule, and these will all have some influence on
one another. But while every electron in the molecule
vibrates in a complex manner compounded of all possible
periods, the most marked periods will be those which
most closely approach the natural period of the electron in
question. The effect of the other rings may explain the fine
lines into which broad bands occasionally split under favour-
able conditions. In connexion with this it must be observed
that the convenient abbreviation that “the band at (say)
300 is due to oxygen in the carbonyl group” does not

Phil. Mag. 8. 6. Vol. 29. No. 169. Jan. 1915. 0

194 Mr. N. McCleland on the Absorption Spectra of

mean that it is due to vibrations of the electrons of free
oxygen valencies alone, though these may contribute the
greater part of the effective absorbing power. Further, the
introduction of fresh vibration centres is bound to affect the
induction constants of those already present by altering the
relative positions of the orbits. These difficulties do not
greatly disturb the qualitative results, but introduce an
uncertainty into the values of the constants which can only
be overcome by accumulation of experimental results, espe-
cially in regions of the spectrum which have recently been
made available by Schumann.

There are two kinds of motion possible for the electrons,
mamely, in the plane of the orbit, and perpendicular to it.
There are also two kinds of absorption: the one is observed
in comparatively dilute solutions, very thin films or short
columns of vapour, the other ? through considerable thick-
nesses of pure substance. It is likely that these two kinds
correspond to the two kinds of vibration. There are reasons
for thinking that the first kind, which alone is dealt with here,
depends on the vibrations in the plane.

In the numerical part of the paper, since the data cover
only a small part of the spectrum it is impossible to obtain
sufficient equations to give all the unknowns. Weare there-
fore obliged to make some assumptions as will appear.

Notation and Units.

l represents coefficient of self-induction, or apparent mass,
m and sometimes /, g, @ of mutual induction. ,r the friction
of damping. c the stability. He'?’is the disturbing periodic
force of the light.

The unit of time is so chosen that when the wave-length

is In be, p= “ SENS

lis assumed to be the same for all electrons and is taken
as dl

m and ¢ vary for different atoms, the former also depending
to some extent on the constitution of the molecule.

For example, the mutual induction between two atoms is
increased when there is a mass of unsaturation outside them,
e.g. the mutual induction between the N= in MeN=N—Me
is less than that in the case of C,H; -N=N—C,H;. This is
readily explained as due to the repulsion of the unsaturated
electrons in the phenyl groups forcing those of the nitrogens
closer together.

Organic Substances and the Hlectron Theory. 1s,

Bands are spoken of as on the near or remote side of one
another with reference to the red end of the spectrum.

The data refer as far as possible to alcoholic solutions,
these being most abundant.

I. One vibration centre only.

This is the case developed by Drude }.
The fundamental equation of motion may be put in the
form

Uae panda Gn == We RO) aca) a) id)
If we assume «= Xe”? for steady motion, we find
X(y?—rip—c)=—E . . . Gi)

giving X. The development of this is well known ; in par-
ticular it appears that the frequency of free vibration,
corresponding to that of the ray most strongly absorbed, and
so to the head of the band, is given by

iE EO e any ayers \CLEL }
and also that the sharpness of the band depends on the
smallness of =.
The only substances belonging to this type that have been
examined are the aliphatic iodides **. The I vibration
centre gives rise to a band at about » 252.

In what follows, the cases where there are several vibration
centres will be treated as follows :—

First, the ‘frequency ” equation (corresponding to (iii.) )
will be investigated to find the positions of the heads of the
bands, then in a few cases the “characteristic” equation

(corresponding to (ii.)) will be examined to determine their
nature.

II. Ywo vibration centres.
The equations of motion are
Lae =f myy +70 +¢e,7= He?! Vv
mi + loy + roy + coy = Het, J

assuming the relation

(iv.)

Mjg=Mg,=M.

QO 2

196 Mr. N. McCleland on the Absorption Spectra of

The frequency equation is

(1,9? —c,) (lop? —c.) =m*p*, . . . (v.)

and, as is well known, the roots of this, z.e. the frequencies,
lie outside those of the simple vibrations of each centre, 7. ¢.,
the bands are forced apart.

The characteristic equation for the x vibration centre is

X [ (lip? — rp — €1) (lop? — retp —cg) —m*p* |
= —H[(l,—m)p? —reup—cg|. . (vi)

This is much too complicated to deal with conveniently,
but in the symmetrical case we easily find the frequencies

p= “_ and the characteristic equation becomes
l+m
X| ((+m)p?—rip—cj=—HE. . . . (ii)
When p? =- “we see the actual value of X is ey
l+m pr
and! at p= “we find the actual value is Bene

l—m
7p

where tan 9 = 5 From which it appears that the two
bands in such spectra mmay be locked on as derived from the
vibrations governed by

(l+m)a+raé+ca=Hert, (l—m)#+>7r cosec ne +ca= He.

Of these, the former represents a band on the near side of,
and sharper than that of the single vibration centre, while
the latter represents a diffuse band in the more remote
regions.

The former band will therefore tend to break up into
lines.

It must, however, be noticed that if m is small, the bands
will probably coalesce to form a single wide band.

We assume that in asymmetrical cases, provided the
asymmetry is not too marked, a similar kind of result will
be obtained as to the relative sharpness of the bands; the
experimental evidence in favour of this is abundant.

Substances belonging to the above type are fairly
numerous: the ketones *1%1 and ethylene® derivatives
may be quoted, The two bands demanded by theory
are found, but the more remote band is beyond }200.

Organic Substances and the Electron Theory. 197

Ill. Two similar vibration centres with a third the
frequency of which is great. The coefficients of
induction are as follows :

eq are 9 ly

Ii is easy to see that the frequencies are given by

lp? —c, mp”, TP” =0
pe Ep.) ope
IP, gp pte

and ¢, is by supposition large compared with f, g, l,.
The equation reduces to

(ce — Lp?) { (lp? —c)? — m?p*} + PAC? + 9°) Lp? —¢) — 2fgmp*} = 0.

= r 7 On al T T Fal 2 hyat
It is easy to see that if z > (es two values of p? are

: G C pag
approximately ea ( 1— cy (i Gi : : ) )

The bands are therefore both shifted towards the visible
region by the introduction of the new vibration centre.
The effect is also seen to be equivalent to a diminution of c.
Therefore, in examining the effect of say a methyl group,
we will not look on it as a separate vibration centre, but will
diminish the value of ¢ for the atom to which it is attached.

IV. Four centres placed partly symmetrically.

# and y correspond, so do & and », so that (a, &) is a
group of the kind considered above, and (y, 7) is an identical

group :

e.g. E ®
HC=0O

[og

HC=0O

IES:

198 Mr. N. McCleland on the Absorption Spectra of

The coefficients used are given in the following table :—

ey Mee Wee

TMNT m ih g
y | m l g 7
sel g iL om
ig Se ee

Proceeding as before, we find that the frequency equation
breaks up into the two equations

[ (1+ m)p? —e] (+m) p? —q] =(f+9)2p* . (iii)
[U—m)p?—e]| (h — mi) p* al =O pt. (ix.)

and the characteristic equations reduce to

X[ { (l+ m)p? — rip — ch {(L, +m) p? — rep — 4} — (f+ 9)?p*]
= —H[l+m)p—(7+9)p?—rup—aq] (x.)

and a similar equation for &.

It can be seen that equations (viil.) and (ix.) correspond to
equations (v.), and (x.) to (vii.), the difference being in the
induction coefficients only. Now (v.) and (vi.) are the
equations of the (wx, &) (or y, 7) group independently (allowing
for the change of notation).

It appears from the above that the four bands of the
system may be looked on as derived from the two bands of
either group by displacement. Thisin the case of the bands
given by (viil.) is toward the regions of greater wave-lengths,
and the bands produced by the displacement in this direction
wili tend to be sharpest. The direction of displacement of
the bands given by (ix.) depends on the relative values of
the various coefficients. It appears, then, that the groups
affect one another in the same general way as simple
oscillation centres.

Thus, for example, a group which gives two bands A, B
(fig. 1) will give rise to four bands, C, D, H, F, when
associated with a similar group, and the appearance of the
curve will be as in fig. 2.

If mis small, C and E may coalesce to form a single broad
band.

We have seen that the band C is likely to be sharp, while
F may be very diffuse. ‘This, it is suggested, is the so-
called general absorption, which is really part of a diffuse
band in the most refrangible regions ®*.

Organic Substances and the Electron Theory. 199

The asymmetrical case of the above is undeveloped, but
it is fair to assume an interpretation similar to the above,
viz., the thrusting apart of the bands due to the con-
stituent groups.

Bigs Fig. 2.

There are many substances of the above type, e. g.,
diketones 1°, diolefines ®, and unsaturated ketones 7°, but
in no case has the investigation covered the whole region
in which bands should occur. The results obtained in the
regions examined do, however, correspond accurately with
this theory.

The importance of §§ III.—-IV. lies in the fact that we can-
not in most cases deal with isolated centres in practical work,
since even when they can exist their bands are usually out-
side the ordinary regions. We have therefore to investigate
the influence of groups and calculate from them the constants
for the free centres. Also, as stated above, even the simple
bands in reality are made up of fine lines ; it is of importance
to show cause why these should move together. The above
reasoning, while by no means conclusive, suggests that this
will be the case and no exceptions have been met with in

practice.

V. The Benzene System.

This consists of six identical vibration centres arranged in
accordance with the laws of isomerism, in a ring or
otherwise.

The coefficients of induction are :

self-induction / ;

mutual induction between ortho carbons i,
meta as M»
para A Re 78

200 Mr. N. McCleland on the Absorption Spectra of
The equations of motion are:
Liaty +, %2 + Mls + Mpl'g + Mndis +m dg + re, + cv, = He”, Ke.

The frequency equation is

rae Mp» Mnf. OCo | eae
|
Nes) Dee lp? —c |
| &e (xi.)
and from this the squares of the frequencies are :
¢ ¢
(a = 3 ae (B) SSS,
(+ m,+2m,+m, [+ 1% — Mn+ My
< (& c
(y) (8) —_—__———-

L+ m,— Mo +My, ; l—m,—2m,—M,»,
The second and third of these represent coincident roots of
the equation. This cannot mean more than the coincidence
of two bands, 7. e. a very deep band.

Since m, will clearly be greater than mm or m,, the above
are in order of magnitude.

The four bands will be called the «, B, y, 6 bands
respectively.

Characteristic equation.
This degenerates into
X[(l+m,+2m, a Mm) p> —rip—c|=—H. -. (xi)

From this we see that the least refrangible band may be
derived from the equation

(l+mp+2m,+m,,)pietre+cv= He? (a)
and the others from
La +7 cosee n+ce= He,
where

the
L=(l+m—m,—m,), tan 47=—

pm, + 3M + 2m) for 8 band.

pam, + 3m») aes

=(/—m,— ma, SUL) —

q a

= [9 2 1) ) oe Re
( m+ 2m my ) PA Am, + 2p)

Organic Substances and the Klectron Theory. 201

Since, as has been said, m, is larger than m,, or m,, it
follows that there is a diminution of the induction coefficient
and a reduction of tan, 7.e. an increase of the friction
coefficient in the series. The bands therefore gradually be-
come more diffuse on passing towards the extreme ultraviolet,
while the first may be expected to be exceptionally sharp.
This it in fact is, for it breaks up even in solution, giving
the seven well-known benzene bands described by Hartley,
4,5,11 *

Only the «4 and 87% series of bands have been actually
observed ; the constants have been calculated from these,
and it was then found that the y and 6 series are out of
range of all the present spectroscopes.

Non-symmetrical ring systems (toluene, pyridine, &c.).—
The complete investigation of these cases need not be given.
They are, in general, too complex for treatment, but in any

particular case can be worked out numerically when the

requisite data are available. It appears that the bands will
be of the same general character as the benzene system, but
the double bands 8 and y will separate into their com-
ponents f. These, however, will probably not be sufficiently
far apart to be distinct, so the total effect will be a single
broad band.

Also the band corresponding to a will become less sharp

as the departure from symmetry becomes more and more
pronounced.

VI. A benzene nucleus with an independent oscillation
centre outside it.

We must assume that the coefficients of the benzene
system are unaltered by the presence of the new centre.

The coefficients for this are denoted by A, w, fo, fin Mp:

This case also is too complex for general treatment. No
reasons can be given for assuming that the benzene group in
general acts like the single centre of § II. or the simple group

of § IV.

* Tt may be stated as a general principle that the “nearest” band is
the one most likely to break up into small bands, and this is most likely
to happen when the frequency and characteristic equation break up into
factors.

In addition, the substance in question must be fairly volatile, since
bands always tend to coalesce as the temperature increases.

+ In the monosubstituted derivatives, and in pyridine, it appears that
one component each of the @ and y bands is altogether unchanged. A
similar result is obtained when an independent oscillation centre is
introduced into the molecule, and also in the case of quinone, diphenyl,
diphenylmethane, &c.

202. Mr. N. McCleland on the Absorption Spectra of
If, however, the new centre is such and so situated that
@ is small, w,, wm, “yp negligible, the frequency equation
reduces to
(—hp?)A—aw*p?A'=0, . . . (xu)
where, introducing the numerical values of the constants,
A =(1—6°25p?)(1— 4:17 p?)?(1 — 2°59p?)?(1 — 1897),
A’ =3°60p?(1—5:14p?)(1— 3:1 4p?) (1 —3:00p?)(1 — 4:1 7p?)(1 — 2-5 9p”).
Hence the 8 and y bands should appear and

(A) If the new centre is on the near side of p?= 3-14
i.e. X 227, two bands will be seen in the normal
region (e.g. anilines *14"*?),

(B) If it is on the far side of 227, only one band
(e.g. phenols * ?°).

We can also work out the cases of two centres attached to
one ring in the same way. The para case is the only one
which factorizes, and hence we expect (p. 201 note) para
derivatives to show narrow bands more readily than the
ortho and meta. This is found correct.

VII. A “ group” attached to a benzene ring, e.g. C,H; . CHO,
benzaldehyde.

If w and y are the inductions between the atoms of the
group and the carbon to which they are attached, and
the approximation holds, the equation is
At (hip? — 4) (lap? —e) — m4_"p*} + A'p*{ w*(lap? — ¢2)

+? (hp? —¢) —2apmyp?}=0. (xiv.)
There will be two bands on the near side of 7227 and

sometimes a third very near 227, e.g. benzaldehyde “,
styrol, methylazobenzene ”.

VUIL Two similar groups each united to “a oxime
C,H;N=N— (Oplalee
If @ is the mutual coefficient of the two nearest carbons of
the rings, the frequency equation breaks up into
ta-(hzm)p }(A+0a))=+p(otwp)da’. . (xv.)
It follows that if the group between the phenyls gives rise
toa band on the near side of 1250, at least two bands appear,

otherwise only one, e. g. azobenzene”, hydrazobenzene "*,
and stilbene.

Organic Substances and the Electron Theory. 203

IX.. Two rings united by one or more methylene groups (the
effect is apparently just felt through two).

The frequency equation is

aU Ne Utrera tc ove UNV Is)
e.g. diphenyl methane **, dibenzyl.

When the rings are united directly, as in diphenyl, the
approximation required for the above does not hold. It will
be noticed that the B and y bands will be found in the above
eases (VI.), (VII.); (VIL.), and (1X:).

It is easy to see that (xv.) is related to (xvi.) in the same
way as (xili.) to the benzene equation, A=0 (xi.).

The following values of the constants may be quoted as
giving numerical results agreeing closely with the observed
values :—

Values of ¢. 5;
Carbon, —CH, -297, —CHMe :278, —CMey, °252.
—CMeHt -247, —CHt, -241.
Oxygen, hydroxylic, 60.
ketonic, 22.0),

[In the lower aldehydes the stability constant is much
smaller, as can be seen from the position of the bands.
This is especially marked in formaldehyde, and is in accord
with the chemical nature of these substances. |

Nitrogen, —NH, -17 (assuming /=1, 7. ¢. 1 electron free).

—NMe, oleae
=N -12 in azo derivatives.

Todine, Lake:

Values of m.

C to C, +13 if adjacent.

_ *02 if separated by one carbon atom.

C to O in ketones, -48.

C to OH (in benzene derivatives), 1:49 (¢).

(‘to =N azo. ,, i -57 if adjacent.
07 if separated by one
—C to N (amido), 9543. atom.

=—Nto =N, °38.
(in azo derivatives),
I to I att. to same carbon, °38.
to adjacent carbons, “O4.
In benzene m, =*275> assuming m,, and m, almost equal.
m,, =o There are only two equations to
m, —()56 ealeulate the three unknowns.

204 Mr. N. McCleland on the Absorption Spectra of

The value of mis increased in ratio 1:1°6 if the atoms
are between two unsaturated carbons or if they are between
two phenyl! groups.

Using the above figures satisfactory results are obtained,
the difference between calculated and observed positions of
the bands being small except in very complex substances.
In these cases, however, the numbers and approximate
positions of the bands calculated and observed are in
agreement.

In conclusion the author wishes to thank Prof. Sir J. J.
Thomson for his interest in this paper, and Mr. G. Birtwistle
for much valuable advice and criticism.

August, 1914.
References.

. Drude, Theory of Optics, Sect. IT. ch. V.
. Russell & Lapraik, Trans. Chem. Soc. xxxix. p. 168 (1881).
Brit. Assoc. Report on Dynamic Isomerism, iu.
. Hartley, Phil. Trans. 1, clxx. p. 257 (1879).
Phil. Trans. A, ecvill. p. 475 (1908).
Star, Steubing, Enklaar, & Lipp, Jahrb. Radioaktiv. Electronikh,
x. p. 189 (1918).

D> OT C9 LD be

7. Stark & Levy, zbed. p. 179.

8. Hartley, Trans. Chem. Soc. xlvii. p. 685 (1885).

9: ms ,, lili. p. 641 (1888).
10. Hartley, Dobbie & Lauder, re » Ixxxi. p. 929 (1902).

11. Baly & Collie, A m . Ixxxvil. p. 1832 (1905).
12. Baly & Ewbank, $5 ” . lxxxvil. p. 1855 (1905).
13. Stewart & Baly, s se », Lxxxix. p. 492 (1906).
14. Baker & Balvy, m 7) xe p. 1122 (180m:

15. Purvis, ss ‘ » xevil. p. 1546 (1910).
HG) tae ee ’ » xcix. pp. 811, 1699 (1911).
ie ee “5 - .» xeix. p. 2318 (19ED):
18. Purvis & McCleland, _.. 5 ,» el. p. 1514 (1912).

19. 5, Hi 55 - >» ci. p: 1810 (tobe
2). 5 ¥ B . cil. p. 483 (1918).

rH Veal is : ap » ceili. p. 1088 (1918).
22. Purvis, - + » ceili, p. 1630 (1918).

23. Baly & Tuck, xciil. p. 1913 (1908).

24. Ley & Engelhardt, Zeit. Pha Ys. Chem. Ixxiv. Pp: t (1900).
25. Hantzsch & Lipschitz, Ber. xlv. p. 3011 (1912).

APPENDIX.—I am indebted for the following proof of the
existence of the mutual induction term to Mr. Birtwistle,
of Pembroke College.

The field due to a system of electrons moving in small
closed orbits about mean points 247121, 22Yo22, &e. has been
expressed (so far as the part affecting radiation is concerned)

Organic Substances and the Electron Theory. 205
by Prof. H. M. Macdonald (‘ Electric Waves,’ p. 176) in the

form

Xx pe eee lols eis aes, &e. (A)
i a

a on Gey Or em
PS Ne SIM Biv ER A ood

where 7, is the distance of the field point xyz from a,y,2),

&m¢,, the position of electron 1 at time ¢,

”
Ey/my'C1', 3) 29 a 7c

e being the velocity of light,

r ps O e,&,' fo
and Hest as age te PSR:

ihe summation being taken for all the electrons.
The electrokinetic energy is

All (F/+ +)dr,

a) 0H _O0G
Tey) ae?

and since from (B) and (C)

where

&e. (C), and4ae?f=X (D);

nN fe) “1b
Jp esc

4 Y; Sy
the energy is, using (A) and (D),

1 0 yak’ of A613 OW. | .
so WLS Ty se _ £5 1 Aa | + (n) + (0) fdr,

the integral being taken through all space.

Tf the velocities are small enough compared w ith ec, the
expression for the energy reduces to the followi ing (the dashes
being now omitted) :

yagi 0 OW
- (WS Ee Sat Ot Ou = eC?

206 Notices respecting New Books.

or to

~ a We (et (220i) +

which is of ie form
Lye?é,? + kee + Mype1¢0£ + a thaaes

the coefficients L,, My, &c. being independent of ¢.

This justifies the introduction of “mutual induction ”
terms into the equations of motion, and the analogy to
currents.

If cis not much larger than the velocities, then L,, My,
&c. are not truly constant.

XXII. Notices respecting New Books.

The Call of the Stars: a popular introduction to a knowledge of the
Starry Skies. By Joun R. Kippax, M.D., LU.B., Author of
Comets and Meteors, Churchyard Literature, &c. 418+ xviii
pp-, 9 in.x54 in. 54 illustrations. G. P. Putnam’s Sons,
New York and London: Knickerbocker Press. 10s. 6d. net.

(| ED author aims at “a concise and accurate story of the starry

heavens, together with the legendary lore that time and fancy
have associated with them.” The backward gaze indicated in the
second object prepares us to find that the most modern work
receives very little attention: for instance, the studies of stellar
movement, which have engrossed so much attention of late, are
dismissed with a few words in the opening chapter; and very little
is said about modern solar research. Since the author has col-
lected his information chiefly from other text-books, this feature
might perhaps be ascribed to the fact that he must necessarily be
behind them in date; but itis a pity that he did not consult, for
instance, Darwin’s “‘ Tides,” which has been a classic for many
years now. Apart from this defect, the book is a well-written
and interesting account of the main facts; and it can do no
possible harm to be reminded, alongside the modern account of
Saturn, of the mythology and worship of the associated deity in
times past. The illustrations are excellent; and there is a good
collection of poetic references. ‘There are a few slips, as when the
‘* isothermal layer” of our atmosphere is called a ‘‘ reversing layer ”
p- 312; but on the whole the information is remarkably accurate
and sound ; and the index has been made with care.

My 20K tal

XXIII. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from vol. xxviii. p. 842.]
November 18th, 1914.—Dr. A. Smith Woodward, F.R.S., President,
in the Chair.

ee following communications were read :—
1. ‘On a Raised Beach on the Southern Coast of Jersey.” By
Andrew Dunlop, M.D., F.G.S.
Last June Mr. E. F. Guiton drew attention to a raised beach
recently exposed on the southern coast of the island. It is on the
eastern slope of the ridge between Le Hocq and Pontac, and the
section, facing northwards, shows the following succession of beds

from above downwards :—
Thickness in feet inches.

(1) Earthy loam, with a layer of rubble ..........:............. 4, 0
Cypsrit brownish-red Clay: 1. 0k vecactee. hoes eee dco eo i! 0)
(3) Yellow loamy clay, containing waterworn pebbles and

Puovea Daley O ahaa venasys\ cup vaee ee eae ime Remy Helse, det paadn kn Oe ee 3 4
(ANRC OALSeHOLO WIN SAME ote cers rtp ae Se Ne Meee teenie Ns 3 6
(5) Waterworn pebbles, closely packed in a matrix of coarse

GLO WAIISAD GN omy esc Mase iatene suet AN Neat econ coals as 4 6

The rock beneath is fine red granite. The section is terminated
at its western end by sloping rock, and there, between the rock and
the lower beds, is a layer of stiff brownish-yellow clay about
2 feet thick, which is continued for a short distance under the
bed of pebbles.

The base of the section is about 50 feet above mean sea-level.

The pebbles in both the upper and the lower beds are mostly of
the fine red granite of the locality, but there are some of diabase
and of quartzite, as well as a few of flint. Flint is, of course,
foreign to the island, but there are many flint-pebbles on the
recent beaches, especially on the north-eastern coast. Flint
pebbles have also been found in at least two low-level raised
beaches, and flint-pebbles and fragments have been noticed in the
yellow clay. Pebbles and fragments of Devonian shale have also
been found in what appears to be a remaining fragment of a raised
beach on the south-western coast.

Col. Warton recently pointed out a raised beach, not previously
noticed, in the railway-cutting near the Eastern Railway-station,
This is also on the south side of the island, not far from the coast.
Its base is about 55 feet above mean sea-level, and it is covered by
a thick bed of yellow loamy clay. ]

2. ‘On Tachylyte Vems and Assimilation Phenomena in the
Granite of Parijs (Orange Free State).’ By Prof. 8. James Shand,
Disc, PhDs EGS:

The district described is the neighbourhood of Parijs Township,
which is situated on the Vaal River and hes upon the northern
portion of the Vredefort granite-mass.

The so-called ‘granite’ near Parijs is a red and grey streaky gneiss,

208 Intelligence and Miscellaneous Articles.

often traversed, both parallel to and across the foliation, by veins
of red pegmatite: these are of a later period of consolidation than
the rest of the rock. The author concludes, from field-evidence,
that the grey facies of the gneiss results from assimilation of the
country-rock by an ascending magma ; while the red facies represents
the residual portion of the same magma.

The special interest of the district, however, lies not so much in
the granite, as in a system of tachylytic veins which everywhere
intersects the granitic rocks. These veins range from a fraction of
an inch to 2 feet in thickness, but in the thicker veins there are
numerous inclusions of the country-rock. They are irregular in
form, thickness, and direction, and are due to the intrusion of a
basic magma which underlay the district. The author describes the
microscopic characters of these tachylytes, and comments on their
general glassy and cryptocrystalline nature, which he does not regard
as a result of chilling, but suspects is dependent upon the viscosity
of the basic magma.

He brings forward evidence to prove that the position occupied
by the tachylyte is independent of tectonic features, but follows
directly from solution and corrosion of the granitic rocks by the
basic magma.

XXIV. Intelligence and Miscellaneous Articles.

PRESSURE OF RADIATION ON A RECEDING REFLECTOR :
CORRIGENDA. BY SIR J. LARMOR,

a correct statement (Phil, Mag. Nov. 1914, pp. 706-707) is

that the ethereal momentum inside and the mechanical -
forces sustained by the bodies inside are the result of the stress
transmitted across the boundary : the sign of the extra tangential
traction must therefore be changed.

Also equation (A) makes 6’ of opposite sign to @: to adapt it to
(B) the sign of its first term must thus be changed.

Equation (B) means that for oblique total reflexion, as for
direct, the force intensity is altered in amplitude inversely as the
wave-length, and in no other respect. Thus the phenomena can
still be formulated under asimple scheme which expresses that the
radiation carries momentum with it, and on total reflexion the
amplitude of the fundamental displacement in the ether is
conserved*. Then perhaps the validity may be asserted beyond
the first power of v/a: but for an ordinary partial reflector, the
propagation is in part into matter whose molecular structure is
naturally too complex for such a scheme.

* IT observe now that in a clear and concise investigation in the
Phil. Mag. for December, conducted in terms of momentum alone in
Prof. Poynting’s manner, and thus applicable only to propagation in free
space, Mr. T. Harris finds that the resultant thrust on the perfect receding
reflector is along the normal. This fact either may be regarded as a
simple geometrical consequence of the principle above stated, or else
may be formulated as an alternative principle. |

K. T. Jos. Phil. Mag. Ser. 6, Vol. 29, PL. I.

Primary current 1°5 ampere

IEIG\5 “Ws

Primary current 2 amperes.

Fra. 8,

Two successive breaks by motor interruptor.

))

RaMAn.

Phil. Mag. Ser. 6, Vol. 29, Pl. I.

, £ = =AN
‘, ? P
t , Yin ; as

r F ‘ ‘ : - s
, ; ¥ :
F r t as ’ . Cc

i : : £ AZT

f= : 5
’ * 7 , @ i aa fo
a) Z ’ > 5
+ + J bd a hole oa

THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

XXV. Some Problems concerning the Mutual Influence of
Resonators exposed to Primary Plane Waves. By Lord
Rayueicnu, O.M., F.R.S.*

ECENT investigations, especially the beautiful work of
Wood on “Radiation of Gas Molecules excited by
Light” f, have raised questions as to the behaviour of a cloud
of resonators under the influence of plane waves of their
own period. Such questions are indeed of fundamental
importance. Until they are answered we can hardly ap-
proach the consideration of absorption, viz. the conversion
of radiant into thermal energy. The first action is upon
the molecule. We may ask whether this can involve on
the average an increase of translatory energy. It does not
seem likely. If not, the transformation into thermal energy
must await collisions.

The difficulties in the way of answering the questions
which naturally arise are formidable. In the first place we
do not understand what kind of vibration is assumed by the
molecule. But it seems desirable that a beginning should
be made ; and for this purpose I here consider the case of
the simple aerial resonator vibrating symmetrically. The
results cannot be regarded as even roughly applicable in
a quantitative sense to radiation, inasmuch as this type.is

* Communicated by the Author.
+ A convenient summary of many of the more important results is
given in the Guthrie Lecture, Proc. Phys. Sce. vol. xxvi. p. 185 (1914).

Phil. Mag. 8. 6. Vol. 29. No. 170. Feb. 1915. P

210 Lord Rayleigh on the Mutual Influence of

inadmissible for transverse vibrations. Nevertheless they
may afford suggestions.

The action of a simple resonator under the influence of
suitably tuned primary aerial waves was considered in
‘Theory of Sound,’ §319 (1878). The primary waves were
supposed to issue from a simple source at a finite distance ¢
from the resonator. With suppression of the time-{factor,
and at a distance r from their source, they are represented *

by the potential
e—tkr

d=, -. . .

in which k=27/d, and X is the wave-length ; and it appeared
that the potential of the secondary waves diverging from the

resonator is
e —tke e —thr"'

Lo re

1 fp

so that
dire’? Mod?ap = 4a/ kc? : 0. Dees)

The left-hand member of (3) may be considered to represent
the energy dispersed. At the distance of the resonator

Mod? = 1/c?.

Tf we inquire what area 8 of primary wave-front propa-
gates the same energy as is dispersed by the resonator, we
a
have

S/e? == Aar/k?c?,
or S = 4r/P i Vlas). oe >

Equation (£) applies of course to plane primary wayes,
and is then a particular case of a more general theorem
established by Lamb T.

It will be convenient for our present purpose to start
de novo with plane primary waves, still supposing that the
resonator is simple, so that we are concerned only with
symmetrical terms, of zero order in spherical harmonics.

Taking the place of the resonator as origin and the direction
of propagation as initial line, we may represent the primary
potential by

hb = etroos? = 1 +ikr cos 0—$k*7" cos? O+ . Me

* A slight change of notation is introduced.

+ Camb. Trans. vol. xviii. p. 848 (1899) ; Proc. Math. Soc. vol. xxxii.
p. 11 (1900). The resonator is no longer limited to be simple. See also
Rayleigh, Phil. Mag. vol. i. p. 97 (1902); Scientific Papers, vol, v. p.8.

Resonators. exposed to Primary Plane Waves. Ala

The potential of the symmetrical waves issuing from the
resonator may be taken to be

Since the resonator is supposed to be an ideal resonator,
concentrated in a point, 7 is to be treated as infinitesimal
in considering the conditions to be there satisfied. The first
of these is that no work shall be done at the resonator, and
it requires that total pressure and total radial velocity shall
be in quadrature. The total pressure is proportional to
d(d+r)/dt, or to ae and the total radial velocity is
d(p+y)/dr. Thus (6+) and d(p+w)/dr must be in the
same (or opposite) phases, in other words their ratzo must
be real. Now, with sufficient approximation,

ae —ikr a
7

EUR ao os a) ne CRIN LG)

Ptp=1+ (iy), “Se _ - ,

e
so that
Ay eR sheer ae Data HEP)
ah
If we write
NCE lO = Nera Wed enti (S)

then
ON Aseria ISM) ee Te ATLL ea evhale 469)

So far « is arbitrary, since we have used no other condition
than that no work is being done at the resonator. For
instance, (9) applies when the source of disturbance is
merely the presence at the origin of a small quantity of gas
of varied character. The peculiar action of a resonator is
to make A a maximum, so that sine=+1, say —1. Then

==) Maat) Ce AO Son eae eee (10)
and
ier
SPH eae en ak (11)
As in (3),
Ame Nod? ap) Aire [2 = Afonso) Soto CEB)

and the whole energy dispersed corresponds to an area of
primary wave-front equal to 2/7.
The condition of resonance implies a definite relation
between (6+) and d(P+y)/dr. If we introduce the
P 2

212 Lord Rayleigh on the Alutual Influence of
value of a from (10), we see that this is

(Oe Vege eee
doi b)idr 2=t? ~ (i)
and this is the relation which must hold at a resonator so

tuned as to respond to the primary waves, when isolated
from all other influences.

The above calculation relates to the case of a single re-
sonator. [or many purposes, especially in Optics, it would
be desirable to understand the operation of a company of
resonators. A strict investigation of this question requires
us to consider each resonator as under the influence, not
only of the primary waves, but also of the secondary waves
dispersed by its neighbours, and in this many difficulties are
encountered. If, however, the resonators are not too near
one another, or too numerous, they may be supposed to act
independently. From (11) it will be seen that the standard
of distance is the wave-length.

The action of a number (7) of similar and irregularly
situated centres of secondary disturbance has been con-
sidered in various papers on the light from the sky *. The
phase of the disturbance from a single centre as it reaches a
distant point, depends of course upon this distance and upon
the situation of the centre along the primary rays. If all
the circumstances are accurately prescribed, we can calcu-
late the aggregate effect at a distant point, and the resultant
intensity may be anything between 0 and that corresponding
to complete agreement of phase among all the components.
But such a calculation would have little significance for our
present purpose. Owing to various departures from ideal
simplicity, e.g. want of homogeneity in the primary vibra-
tions, movement of the disturbing centres, the impossibility
of observing what takes place at a mathematical point, we
are in effect only concerned with the average, and the average
intensity is times that due to a single centre. j

In the application to a cloud of acoustic resonators the
restriction was necessary that the resonators must not be
close compared with 2X; otherwise they would react upon
one another too much. ‘This restriction may appear to
exclude the case of the light from the sky, regarded as due
mainly to the molecules of air ; but these molecules are not
resonators—at any rate as regards visible radiations. We
can most easily argue about an otherwise uniform medium

* Compare also “‘Wave Theory of Light,” Ene. Brit. xxiv. (1888), § 4;
Scientific Papers, vol. ili. pp. 53, 54,

Resonators exposed to Primary Plane Waves. 213

disturbed by numerous small obstacles composed of a medium
of different quality. There is then no difficulty in supposing
the obstacles so small that their mutual reaction may be
neglected, even although the average distance of immediate
neighbours is much less than a wave-length. When the
obstacles are small enough, the whole energy dispersed may
be trifling, but it is well to observe that there must be some.
No medium can be fully transparent in all directions to
plane waves which is not itself quite uniform. Partial
exceptions may occur, e.g. when the want of uniformity is
a stratification in plane strata. The dispersal then becomes
a regular reflexion, and this may vanish in certain cases,
even though the changes of quality are sudden (black in
Newton’s rings) *, But such transparency is limited to
certain directions of propagation.

To return to resonators: when they may be close to-
gether, we have to consider their mutual reaction. For
simplicity we will suppose that they all lie on the same
primary wave-front, so that as before in the neighbourhood
of each resonator we may take

Gea bdo ON tea kn ad CLA)

Further, we suppose that all the resonators are similarly
situated as regards their neighbours, e. g., that they le
at the angular points of a regular polygon. The waves
diverging from each have then the same expression, and
altogether

a kr) = p— thr ’

where #1, 7,... are the Lone of the point where v is

measured from the various resonators, and a is a coefficient

to be determined. The whole potential is 6+, and it

suffices to consider the state of things at the first resonator.
With sufficient approximation

a ‘ cmon :

dtp = 14" (Lair) tas ap es + (16)

R being the distance of any other resonator from the first,

while (as before)
MU hag Me Maks ecu orsligele Oy)

We have now to distinguish two cases. In the first.
which is the more important, the tuning of the resonators is

* See Proce. Roy. Soe. vol. 86.4, p. 207 (1912).

214 Lord Rayleigh on the Mutual Influence of

such that each singly would respond as much as possible
to the primary waves. The ratio of (16) to (17) must then,
as we have seen, be equal to —7,, when 7, is indefinitely
diminished. Accordingly

(18)

which, of course, includes (10). If we write a = Ae,
then

ijl?

cos eee sinkR
SS
[ ais +[1 a aye

A? = (19)

The other case arises when the resonators are so tuned
that the aggregate responds as much as possible to the
primary waves. We may then proceed as in the investi-
gation for a single resonator. In order that no work may
be done at the disturbing centres, (6+) and d(@+w)/dr
must be in the same phase, and this requires that

—1ikR
on Lae so See real,
Cami R

or . + = real tik+: pes

(20)

The condition of maximum resonance is that the real part
in (20) shall vanish, so that

3 Sin AR’
ia thd i+ a } .. ee
Lk

14 ee

or A= (22)

The present value of A? is greater than that in (19), as
was of course to be expected. In either case the disturb-
ance is given by (15) with the value of a determined ys
(18), or (2i).

The simplest example is when there are only two re-
sonators and the sign of summation may be omitted in (18).
In order to reckon the energy dispersed, we may proceed
by either of two methods. “In the first we consider the
value of yw and its modulus at a great distance r from
the resonators. It is evident that Ww is symmetrical with

Resonators exposed to Primary Plane Waves. 215

respect to the line R joining the resonators, and if @ be the
angle between r and R, 7-72 = Ros @.
Thus

r . Mod? ar = A? {2+2 cos (KR cos @)} ;

and on integration over angular space,

Tw
Qar® i} Mod? w. sin 0d0 = 87A? Legare

sin k&R (2!

Introducing the value of A’ from (19), we have finally

srk-*(1 4 sane
1 sinkR ~

Qa" (* Mod? y. sin dé =
Jo

lf we suppose that /R is large, but still so that R is small
compared with r, (24) reduces to 87k~? or 2A*/a. The
energy dispersed is then the double of that which would be
dispersed by each resonator acting alone ; otherwise the
mutual reaction complicates the expression.

The greatest interference naturally occurs when £R is
small. (24) then becomes 2k?R?.2A?/a, or 167R?, in
agreement with ‘Theory of Sound, § 321. The whole
energy dispersed is then much /ess than if there were only
one resonator. ;

It is of interest to trace the influence of distance more
closely. If we put kR=2am, so that R=m), we may
write (24)

Ss = fee ° Jal, 2 e e e ° ° (25)
where § is the area of primary wave-front which carries the
same energy as is dispersed by the two resonators and

an 2am -+sin (277m) i
arm (27m) ~* + 2 sin (2a7m)° ays ( 6)

If 2m is an integer, the sine vanishes and

—o = D7
H 1+ (am)-?? Sir WW AN aR ae EN (27)

not differing much from unity even when 2m=1; and
whenever 2m is great, F approaches unity.

216 Lord Rayleigh on the Mutual Influence of

The following table gives the values of F for values of 2m
not greater than 2 :—

\{
| 2m. iB: | 2. ial 21. PB,
area || aa ee eee)
0-05 00459 | 0:70 0-7042 1-40 1-266
| 0-10 01514 | 0:80 0°7588 1:50 1:269
| 020 03582 | 0-90 0:8256 || 1-60 1-226
0:30 04836 =| 1-00 0-9080 1:70 1-159
0:40 05583 =| 1-10 1-006 1:80 1:088
0:50 06110 | 1:20 151135) ||| 1260 1:026
0:60 | 0:6569 1:30 1:208 | 2-00 0-975

In the case of two resonators the integration in (23)
presents no difficulty ; but when there are a larger number,
it is preferable to calculate the emission of energy in the
dispersed waves from the work which would have to be done
to generate them at the resonators (in the absence of
primary waves)—a method which entails no integration.
We continue to suppose that all the resonators are
similarly situated, so that it suffices to consider the work
done at one of them—say the first. From (15)

Da Nal = lee re eae dy _ a
eae i lage at | : dr op

The pressure 1s proportional to ap, and the part of it
which is in the same phase as dyy/dr is proportional to

af aas sinkR ee
Accordingly the work done at each source is proportional to
afi4 pee (28)

Hence altogether by (19) the energy dispersed by
n resonators is that carried by an area 8 of primary wave-
front, where

= sin kR
S= Lace: (29)
Th * [3 COS an +|1 hess sin KR D0 a e. =
™ ER

the constant factor being determined most simply by a

Resonators exposed to Primary Plane Waves. 217

comparison with the case of a single resonator, for which
n=land the &’s vanish. We fall back on (24) by merely
putting »=2, and dropping the signs of summation, as there
is then only one R. )

If the tuning is such as to make the effect of the aggregate
of resonators a maximum, the cosines in (29) are to be
dropped, and we have
g wr? 1

Ta oss sin KR”
eR

(30)

As an example of (29), we may take 4 resonators at the
angular points of a square whose side is b. There are
then 3 R’s to be included in the summation, of which two
are equal to } and one to J,/2, so that (28) becomes

Ne fa siD eo: sin (kb 2) 7 |
bat pen iacmnyzann ire renin el)

A similar result may be arrived at from the value of rat
an infinite distance, by use of the definite integral *

ACen Oct G ean oe (32)

(9) Lv

As an example where the company of resonators extends
to infinity, we may suppose that there is a row of them,
equally spaced at distance R. By (18)

une putkR = p-2ikR — g-BIAR

ee ee G3

a AR 2R SS oe | (33)
The series may be summed. If we write

} I —=XAr 2 7-312 z
Sy eS Crees —_ ay — Se ee tar Oram Re (72)

where / is real and less than unity, we have

GO ein he
Te Sia
and
i Ms Qn
>= aay loan Gua les 2) crim oe (35)

no constant of integration being required, since

>= —h-Nog(—h) when «= 0.

* Enc. Brit. 1. c. equation (48); Scientific Papers, ii. p. 98.

218 Lord Rayleigh on the Mutual Influence oj
If now we put h=1

= —log (2 sin5) +3i(e—m) +2inm, JS ie

if a Die batoy ois 8 ; e
la ihe log (2 sin oD) +d. (kR—) + 2inm | A oun
If kR = 2m, or R = mA, where m is an integer, the

logarithm becomes infinite and a tends to vanish *.

When RB is very small, a is also very small, tending to

a@=R=2 log (kR). '. “eae

The longitudinal density of the now approximately linear
source may be considered to be a/R, and this tends to vanish.
The multiplication of resonators ultimately annuls the effect
at a distance. It must be remembered that the tuning of
each resonator is supposed to be as for itself alone.

In connexion with this we may consider for a moment the
problem in two dimensions of a linear resonator parallel to
the primary waves, which responds symmetrically. As
before, we may take at the resonator

@=1, ddldr=
As regards yf, the potential of the waves diverging in two
dimensions, we must use different forms when 7 is small
(compared with X) and when 7 is large f. When 7 is small

4 4 ad:
igh setae a shit Se a sei a
one? C wy ‘
+o ti doe ti De

and when 7 is large,

= — (57) et HE! oct nel ae? } 4
ie mS) ; {! 1. 8ibr) 1.2. (Shree oe

By the same argument as for a point resonator we find, as
the condition that no work is done at r= 0, that the imaginary
part of 1/a is —t7/2. For maximum resonance

=m. 6. . .
so that at a distance y approximates to

ee ape
— — e —i(kr— 7). gt le Ae 42
T/? fo
* Phil. Mag. vol. xiv. p. 60 (1907) ; Scientific Papers, vol. v. p. 409.
+ ‘Theory of Sound,’ § 341.

Resonators exposed to Primary Plane Waves. 209

Thus oN
| Qar . Mod’ = —, Re DWE pore: CLO.)

which expresses the width of primary wave-front carrying
the same energy as is dispersed by the linear resonator
tuned to maximum resonance.
A subject which naturally presents itself for treatment is |
the effect of a distribution of point resonators over the whole
plane of the primary wave-front. Such a distribution may
be either regular or haphazard. A regular distribution, e. g.
in square order, has the advantage that all the resonators
are similarly situated. The whole energy dispersed is then |
expressed by (29), but the interpretation presents difficulties
in general. But even this would not cover all that it is |
desirable to know. Unless the side of the square (b) is |
smaller than A, the waves directly reflected back are accom-
panied by lateral “spectra” whose directions may be very
various. When b<A, it seems that these are got rid of. ;
For then not only the infinite lines for ming sides of the
squares which may be drawn through the points, but a for-
tiort lines drawn obliquely, such as those forming the .
diagonals, are too close to give spectra. The whole of the
effect is then represented by the specular reflexion.
In some respects a haphazard distribution forms a more
practical problem, especially in connexion with resonating
vapours. But a precise calculation of the averages then
involved is probably not easy.
If we suppose that the scale (0) of the regular structure is
very small compared with 2X, we can proceed further in the
calculation of the regularly reflected wave. Let Q be one
of the resonators, O the point in the plane of the resonators
opposite to P, at which ar isi orequired:;) Ol ==, OO= a.
PQ=r. Then if m be the number of resonators per unit
area,

mH enthr
p= 2nma| y dy — 4

v0
or since ydy=rdr,
L Mie’)
= 2rma | eran.
ee

The integral, as written, is not convergent; but as in the
theory of diffraction we may omit the integral at the upper
limit, if we exclude the case of a nearly cireular boundary.

220 Lord Rayleigh on the Mutual Influence of

Thus 2a
and Hl Agr? A? ,
Mod? = ape . see sii

The value of A? is given by (19). We find, with the same
limitation as above,

> staph == 2arm coskRdR=0,
R ane
> sue = 2am sin KR dR=2am/k.
R Ne
Thus A?=1/(k + 2am/k)?
and Modes oe” (46)

(kK? + 2am)?’

When the structure is very fine compared with X, /? in the
denominator may be omitted, and then Mod? y=1, that is
the regular reflexion becomes total.

The above caleulation is applicable in strictness only to
resonators arranged in regular order and very closely dis-
tributed. It seems not unlikely that a similar result, viz. a
nearly total specular reflexion, would ensue even when there
are only a few resonators to the square wave-length, and
these are in motion, after the manner of gaseous molecules ;
but this requires further examination.

In the foregoing investigation we have been dealing
solely with forced vibrations, executed in synchronism with
primary waves incident upon the resonators, and it has not
been necessary to enter into details respecting the consti-
tution of the resonators. All that is required is a suitable
adjustment to one another of the virtual mass and spring.
But it is also of interest to consider free vibrations. These
are of necessity subject to damping, owing to the communi-
cation of energy to the medium, forthwith propagated away;
and their persistence depends upon the nature of the reso-
nator as regards mass and spring, and not merely upon the
ratio of these quantities.

Taking first the case of a single resonator, regarded as
bounded at the surface of a small sphere, we have to establish
the connexion between the motion of this surface and the
aerial pressure operative upon it as the result of vibration.
We suppose that the vibrations have such a high degree of

Resonators exposed to Primary Plane Waves. 221

persistence that we may calculate the pressure as if they
were permanent. Thus if ~ be the velocity-potential, we
have as before with sufficient approximation

Giro ae ide
v/a = 9 pod ee oe

, a dr

so that, if p be the radial displacement of the spherical
surface, dp/dt= —a/r”, and

dips — ole) Gyan ao a EO)

Again, if o be the density of the fluid and 6p the variable
part of the pressure,

dp = — adw/dt = or(1—vkr)d?p/dt?, . . (48)

which gives the pressure in terms of the displacement p at
the surface of a sphere of small radius 7. Under the circum-
stances contemplated we may use (48) although the vibra-
tion slowly dies down according to the law of e’, where »
is not wholly real.

Tf M denotes the “‘ mass” and p the coefficient of resti-
tution applicable to p, the equation of motion is

agpynn

dp eee) vik
M ie + pp + dara? (1 —tkr) TP =()) ule rese(Lon

or if we introduce e”” and write M’ for M+47rar*,

n?(— M+ 4arako* £1) +u=0. SONS C50)

Approximately,

n= /(p/M’).{1 +2. 2rokrt/M'} ;
and if we write n=p+iq,

Ba) (He ao. 2maki/ MSY SP)
If T be the time in which vibrations die down in the ratio
mae l= 1/9.

If there be a second precisely similar vibrator at a distance
R from the first, we have for the potential

2
ip CDs
I= |; e kk — ° e ° ° e 52
ve R hr (92)
and for the pressure due to it at the surface of the first
vibrator

Oo r= oO)
——— = > ° ’ e : ° dO )
kh dt* .

222 Mutual Influence of Resonators.

The equation of motion for p, is accordingly

d*p ie de Piet eae
wo? a ae a
= qe + pp, t4mor* ) (1-tkr) i. ee 0;
and that for ps differs only by the interchange of p; and pp.
Assuming: that both p, and ps are as functions of the time
proportional to e*™, we get to determine n

n2{M'—4Anor® .ikr$—p= +n?. 4rar'R-1e-*®,

or approximately

n=a/ #1 es 7 (ikRte#®) f - (54)

If, as before, we take n=p+zq,

( 2arar* os

p=a/ nt (1+ — cos MR), - (55)
LY ee
I=P Rue |

We may observe that the reaction of the neighbour does not
disturb the frequency if coskR=0, or the damping if
sinkR=0. Whea £R is small, the damping in one altern-
ative disappears. The two vibrators then execute their
movements in opposite phases and nothing is propagated to
a distance.

The importance of the disturbance of frequency in (55)
cannot be estimated without regard to the damping. The
question is whether the two vibr ations get out of step while
they still remain considerable, Let us suppose that there is
a relative gain or Joss of half a period while the vibration
dies down in the ratio of e: 1, viz. in the time denoted
previously by T, so that

ki sin KR). |... 2s

()1— po) T = ow.

Calling the undisturbed values of p and g respectively P and
Q), and supposing k£R to be small, we have

a SEer

ay Q RM’

in which Q/P=2aek7*/M'. According to this standard the
disturbance of frequency becomes important only when

kR <1/z, or R less thanA/7?. It has been assumed through-
out that 7 is much less than R.

Terling Place, Witham.

= Ty,

XXVI. Light Absorption and Fluorescence.—Part Il. By
eC. (CL Bany, WWSc. BE,S., Professor of _ Inorganic
Chemistry in the University of Liverpool rE

Z. a previous paper ¢ the existence of constant differences

between the frequencies of absorption bands and fluores-
cence hands exhibited by many organic compounds was dealt
with, and I showed that it was possible to account for these
on the basis of the energy quantum theory. Several com-
pounds are known each of which shows three absorption
bands and at least one fluorescence band, and in these cases
there exists a constant difference between the frequencies of
the centres of these bands. I pointed out that the absorbed
energy must be emitted again at some frequency which is
characteristic of the molecular system, which frequency may
be either in the visible spectrum, when we have fluorescence,
or in the infra-red. Then on the ener gy quantum theory a
single quantum absorbed at the higher frequency must be
given out asa whole number of quanta at the lower frequency.
If, therefore, v, be the frequency of the characteristic vibra-
oun in the caval red, the energy can only be absorbed at the
frequencies 1, vo, v3. Kec., Taber V1, Vo, V3, GC. are successive
multiples of p;. Obviously, t therefore, there must be a
constant difference (v,) between the frequencies V1, Vo, V3, SC,
and this difference must equal the frequency of the infra-red
vibration.

Since that paper was published a number of compounds
have been accurately examined, and the above relation is
found to hold good. These resu!ts will it is hoped be pub-
lished in a separate communication. I omitted in the previous
paper to point out that the existence of these constant
trequency differences has also been noted by v. Kowalski
in a paper dealing with the phosphorescence of a number of
organic compounds Ihe

Two years ago a paper was published by Bjerrum §, in
which he dealt with the short-wave infra-red absorption
bands of certain compounds. He points out that if v be the
characteristic frequency of the atoms in a given molecule,
then, if v, be the frequency of rotation of the molecules,
three bands should be shown in the neighbourhood of the
band v. The frequencies of these will We V—V,p, Vo V+D,
respectively. Since the central vibration is pure it will

* Communicated by the Author.

+ Phil. Mag. xxvii. p. 682 (1914).
t Phys. Zevt. xii. 1911, p. 956.

§ Nernst, Festschrift, 1912, p. 90.

224 Prof. E. C. C. Baly on

evidence itself only as a narrow absorption line, and will
probably escape detection owing to the comparatively large
width of slit necessary in infra-red work. ‘The result will be
that the band wil] appear to bedouble, each portion being broad
since v; is the average rotational frequency of the molecules.
Bjerrum further pointed out that on the energy quantum
theory the rotational frequencies must have well defined

: hn :
values given by the formula v,= 373]? where I isithe moment
ri a7

of inertia, h the Planck constant, and n a whole number.

As a result of this, an absorption band in the short-wave
region of the infra-red should consist of a series of maxima
symmetrically distributed about a central line of frequency v.
Hach pair of maxima will correspond to a definite rotational
frequency of the molecules. Now Friulein von Bahr has
made very accurate measurements of the absorption band of
water vapour at 6°26, and found clear evidence of these
pairs of maxima*. From these she calculated the wave-
lengths corresponding to the rotational frequencies, and
showed an excellent agreement with the absorption bands
as observed by Rubens and v. Wartenberg. Eucken f has
shown that a still better agreement is obtained on the basis
of there being two degrees of freedom possessed by the
water molecule. The experimental evidence therefore most
strongly supports Bjerrum’s theory.

Now it is well known that the absorption bands of many
organic compounds in the ultra-violet can be resolved into
groups of fine lines, each group very frequently possessing
a well marked head. It has occurred to me, if the bands in
the short-wave infra-red region are due to atomic vibrations
and those in the ultra-violet are due to electronic vibrations,
that it should be possible to combine the two in the same
way as has been done by Bjerrum. That is to say, if v be
the characteristic vibration frequency in the ultra-violet,
then we should find pairs of absorption lines with frequencies
equal to v+v,, where v, stands for the frequencies of the
short-wave intra-red bands. It is evident that the applica-
tion of the theory can be tested on ultra-violet absorption
bands with far greater accuracy than in the case of the
infra-red bands for two reasons. Firstly, the measurement
of the ultra-violet bands is much more accurate : and,
secondly, we know far more of the absorption bands in the
short-wave region of the infra-red than of those in the long-
wave region.

* Phil. Mag. xxviii. p. 71 (2914).
t Deutsch. Phys. Ges., Verh. xv. p. 1159 (1913).

Light Absorption and Fluorescence. 225

Some preliminary support for the idea is gained from the
fact that benzene exhibits nine absorption maxima which are
obviously arranged symmetrically around a central position.
This is also the case with several other compounds.

Again it follows that exactly the same structure should be
found in the case of the fluorescence bands of the same sub-
stances. Finally, from what was stated in my previous
paper, the central line of the fluorescence and absorption
maxima should be consecutive multiples of a whole number
which equals the frequency of one of the infra-red bands.

I have calculated the values of the ultra-violet absorption
lines of one or two compounds, and find that they form in
each case a series of pairs symmetrically distributed about a
central line, and that the frequency of every infra-red band in
the short-wave region has a corresponding line or pair of lines
forming part of the structure of the ultra-violet absorption
band. In short, the application of the Bjerrum conception
to both ultra-violet and fluorescence bands is completely
successful, and moreover the conclusions in my previous
paper are entirely confirmed. This may be seen from the
following.

Benzene.

Hartley * investigated the ultra-violet absorption band of
benzene and published measurements of the wave-lengths
of the component lines, which amount in number to about
90. He showed that these fine bands or lines seem to form
ten groups, each with a well marked head. He gave the
wave-leneths of these heads as 2670, 2630, 2590, 2523,
2466, 2411, 2360, 2335, 2326, and 2279 Angstréms. The
values of 1/AX for these, expressed in four figures, are 3745,
3802, 3861, 3963, 4055, 4148, 4237, 4282, 4299, and 4388,
respectively. Now it is obvious that the first four and the
last five of these can be symmetrically distributed about the
fifth with frequency (1/) of 4055. The following table
shows the arrangement of the ten bands and the values of
the frequency differences.

In the third column are given the frequency differences
between each line and the central line, and in the fourth
column the means between the two values where such
exist. The values of 1/y, or the wave-lengths of the infra-
red bands are given in the fifth column, while in the last
column are to be found the wave-lengths of the absorption

* Phil, Trans. ecvili. A. p. 475 (1908),

Pilul. Mag. 8. 6. Vol. 29, No, 170. eb. 1915. ()

26 Prof. E. C. C. Baly on

bands as measured by Coblentz*. Four of the calculated
bands were observed by Coblentz, and as can be seen the
agreement is exceedingly good.

TABLE I.— Benzene.

1-5. in |

Infra-red bands.
| Angstroms. 1/r. | Vr. Mean pr.
| Calc. Obs.
2670 | 3745 | 210 | 393] 3:25
2630 | 3802 253 249. || 408 |
2590 | 3861 | 194 515
i 2a2darn sl 3963 C2 Wha 92:0) | Oe 10°78
| 2466 | 4055 Ona
be oan) 1" atas 93 | 925 || 1081 | 1078
We eeeO | aoa |g | 550 | Sl
| 2385 | 4282 | 227 | 4-40 | 4-40
earn a) |, 4999 | 244 290] 408 |
ipigzar Sid) ¢4888, i. (888° | | 300 |

On the other hand, there are altogether sixteen absorption
bands shown by benzene between 3:0 w and 15 p, which is
the limit reached by Coblentz; and the question at once
arises as to whether each of these gives rise to a correspond-
ing line or pair of lines in the ultra-violet band when com-
pounded with the central vibration 1/.=4055. The complete
list is given in Table II., and as can be seen every single
infra-red band gives a pair of ultra-violet lines, except the
band at 3°23 uw, which only gives one line on the red side of
the centre.

Certain of the calculated infra-red bands have not been
observed by Coblentz, and they are included in the table
because the corresponding ultra-violet pairs were given by
Hartley as the heads of the ultra-violet band groups, and
therefore the presumption would be in favour of these being
important lines. Of course it is perfectly possible that the
corresponding infra-red bands were missed by Coblentz
owing to their being very narrow, but there is also another
possible explanation. Jt must be remembered that only 36
out of the 200 benzene absorption lines are accounted for in
the above table. It would seem likely that that in addition
to the combining of the short-wave infra-red frequencies with
the central frequency 4055, there is also coupled with these
the rotation vibration of the molecules. If this were so
then the probable result would be the existence of several

* Publications of the Carnegie Institution, Washington, No. 35 (1905).

Light Absorption and Fluorescence. 227

series of absorption lines, each series starting from the
central line. There would be pairs of arithmetical series,

TaBLeE I1.—Benzene absorption (Hartley).

Peo Ne in | Infra-red bands,
Angstroms, I/X. Vz. Mean vr.
1 Cale. | Obs
2687 3722 333 333 | 3:00p
.| A 2670 3745 310 | 3-23 3°25 ps
| A 2630 3802 253 248-5 4:02
| 2612 3828 227 298 4:38 4:40
2600 3847 208 208 4-80 490 |
A 2590 8861 194 194 | B15 |
2582 . | 3873 182 182 5:50 5:41
Po as ir 3895 160 160 6:23 6:20
| 2560 3907 148 148 6°75 6°75
|. 9553 3917 138 138 | 725 7:22,
2546 3928 127 127 | 7-85 7:80
2538 3940 115 115 8-67 8°67 |
2529 3954 101 101°5 9:85 9-78
2527 3958 97 97 1030 | 1030 |
A 2523 3963 92 92-5 | 1081 | 10-78
2519 3970 85 85 (| 11°80 11:80
2516 3974 81 Ste OBO 1945
| 9514 3978 77 Ti eos) 3111) 2-95
A 2466 4055 7) |
2420 4132 nn API 11295 | 12:95
9A17 4136 1 81 | 12°30 | 12°45
2416 4140 85 85 | 1180 | 11°80
| A241] 4148 93 Os OR ine
Pe 8409 4152 97 97 10°30 10:30
| 2406 4157 102 101-5.) 9°85 9-75
2398 4170 115 115 _ 8-67 867
9391 418%) 1D 127 | 7°85 780 |
2385 2493 138 138 | 7°25 725 |
2379 4203 148 148 NOG ion i Gre
2373 4215 160 160 | 623 | 6:20
A 2360 4237 182 182 5°50 5:50
2354 4249 194 194 | B15
2346 4263 208 908 || 4:80 4-90
ih) 9384 4284 229 SL ae aes 4-40
|; A 2326 4299 244 2485 | 4-02
|} A 2279 4388 333 333 3-00

\ |

one on each side of the central line, and the result would be
the appearance of heads which mark the regions where
several lines of the different series happen to tall together.
On this explanation the heads will have no especial signiti-
cance in the present connexion, and so will not necessarily
be expected to correspond to bands in the short-wave infra-
red region. It is possible that in this combination of the
three vibrations electronic, atomic, and molecular (rotational),
the explanation is to be found of the structure of all band
Os

228 Prof. E. C. C. Baly on

spectra. This, however, cannot be entered into here. It is
sufficient to say that it does not seem probable that there is
any especial significance in the band heads in the absorption
spectrum, and further that the whole of the remainder of
the absorption bands of benzene beyond those given in the
table may arise from a combination of the rotational fre-
quencies of the molecules with the frequencies given in the
table. These rotational frequencies certainly are of the
right order of magnitude.

Turning now to the frequency of the central line 4055, it
follows, if the deductions in the previous paper are sound,
that this must be a whole multiple of the frequency of one
of the infra-red bands. It is almost exactly 10 x 405*, which
corresponds to a wave-length of 2°47 uw, which corresponds
to the band observed by Coblentz at 2°49. Now the next
multiple of 405 is 9x 405=3645, and this should form the
central line of the fluorescence bands of benzene. The
fluorescence of an alcoholic solution of benzene has accurately
been measured by Dickson t, who found 6 bands with fre-
quencies of 3436, 3537, 3631, 3733, 3795, and 3848. Now
in order to compare these with the absorption measurements
they must be corrected for the effect of the solvent which
tends to shift the maxima towards the red. The values must
therefore be increased by a few units. If 14 be added to
the third frequency we have 3645, which should be the
central line of the system. It may be assumed, therefore,
that 14 should be added to all the values, and this is done in
the third column of Table III. In the fourth column are
given the frequency differences from the central line 3645,
and in brackets are the mean values for the corresponding
intervals found in Table IT.

TaBLE ITI.— Benzene fluorescence (Dickson).

| eae 1/d. A414. | ya
2910 | 8436 | 8450 | 195 (194)
2827 WaraoTA Lasol 4 i 94 (92)
2754 Pe asGslepiel: Beta. | 0
2679 |, 8783. | 3747 | 102 Gon)
| 2635 pep Sig5ey Juames09 | 164 (160)
| 2599 ie) BS48) 1) 4 (S862 5 4. | 217208)
} |

Ldieeaneric.t. hue 3
The agreement is remarkably good.

* The reason why this number is selected will be dealt with in a
further communication.
+ Zeit. wiss. Phot. x. p. 166 (1912).

Light Absorption and Fluorescence. 229

Tt will be seen that in the cases of the compounds noted
below there are included many pairs of absorption lines
which correspond to infra-red bands that have not been
observed. In the present state of our knowledge it is im-
possible to say whether these infra-red bands really do exist,
or whether the absorption line pairs are due to the combina-
tion of the molecular rotational frequencies.

When the substitution products of benzene are considered
some difficulty tends to arise, owing to the fact that the
molecule becomes unsymmetrical. There is no doubt that
the simplicity of the structure of the absorption spectrum of
benzene is due to the symmetry of the molecule. The absorp-
tion spectrum of an alcoholic solution of benzene shows clear
evidence of the symmetry. In toluene and its homologues
much of this symmetry disappears, and therefore it cannot
be expected that the absorption of the vapours of these sub-
stances should be so symmetrical. On the other hand, Miss
Ewbank and I * showed that in the case of the disubstituted
derivatives of benzene the para isomer is always the most
symmetrical of the three, and this is well shown in the case
of the three xylenes. As will be seen below, p-xylene shows
remarkable evidence of symmetry, while in toluene, o-xylene,
and m-xylene, the want of symmetry is evidenced by there
being fewer absorption lines on the ultra-violet side of the
central line.

Toluene.

The ultra-violet absorption of toluene vapour has been
examined by Hartley +, Grebe, and others. Hartley stated
that the absorption lines resolved themselves into groups
with heads at X=2670, 2633, 2605, 2587, 2530, 2471, and
2419. Grebe arranged the lines into a number of series
with constant differences. The most accurate measurements
of the absorption of toluene vapour have been published by
Cremer §, who also arranged the lines in series of constant
frequency differences.

In Table IV. are given Cremer’s measurements of the
toluene absorption lines, arranged in reference to the line at
X= 2471 taken as centre.

Again, in this case there is an exceedingly good agreement
between the calculated and observed values of the infra-red
bands, the latter being those given by Coblentz, whose paper
is quoted as authority throughout. The lettering of the

* Trans. Chem. Soc. lxxxvii. p. 1847 (1905).
t+ Phil. Trans. ceviii. A. p. 475 (1908).

{ Zeit. wiss. Phot, iii. p. 376 (1905).

§ Zeit. wiss. Phot. x. p. p. 349 (1912).

230 Prof. BE. C. C. Baly on

bands in the table has the following meaning :—The bands
marked A are those which Cremer found to be the strongest,
those marked B are those which according to Cremer form
the heads of the series, while those marked H are the bands
which Hartley considered to be the strongest.

TasLe LV.—Toluene absorption (Cremer).

|
5. wan | Infra-red bands,
Angstroms. 1/X. Vr. Mean vz. |
| Cale. Cbs.
ABH 2667 | 3749 | 298 || 336n |. 3:34p
2647 || Sis 269 271) 4) 3:69
AH 2635 | 3795 252 252 || 3:97 4.00
A 2630 | 3802 245 || 4:08 |
A2615 | 3824 293 294 || 4-46
ABH 2603 | 3842 205 | 4:88
A2600 | 3846 201 201 =|: «4:98
2595 | 3853 194 | 515 510
2589 3862 185 1865 || 5:38 5°35
AH 2585 3869 178 | 5:62 551
A 2580 | 3877 170 || 5°88 5:80 |
2572 | 3888 159 | 6-29 6:20
AB2567 | 3895 152 6:58 { oe
2565 | 3899 146 | 6:85 6:86
2560 | 3906 We | 7-09 7-25
2554 | 3915 132 13L |, 7°68 7:70
A 2550 | 3922 125 || 8:00 8-10
2541 | 3936 111 h CO |
9539 | 3939 | 108 107-5: || 9:35 “Ir O97
AB 2536 3948 104 | 9-61 9-73
H2529 3954 93 | 10°75 10-60
9524 | 3961 86 86 —||- 14-60 11-15
2522 | 3965 82 | 12:20 12-08
2517 | 3974 73 || 13°70 13-78
B2507 | 3988 59 61 16:39
2477 | 4037 10 105 || 95-24
H2471 | 4047 0 |
A 2464 | 4058 11 105 || 95:24
| 2433 | 4110 63 61 |): 16:39
BH 2420 | 4133 86 86 ||: 11:60 1115 94
B2407 | 4154 107 107°5 || 9:30 9:27
B 2394 | 4177 130 131 || 763 7-70
2359 | 4235 188 186°5 5:38 5°35
9354 | 4248 201 201 4-98
2341 | 4272 225 994 || 4:46
| 2326 | 4299 252 D52N a S07 ZEOE
| 9315 | 4320 273 271 | 3°69
| |

Only two of Coblentz’s toluene bands are not accounted
for, namely, those at X=8'40 and 8°53 4. Now Cremer in
fae. series of lines notes that certain members are absent, and
as 180% these missing lines should have a wave -length of

2545 Angstrims "and a frequency of 1/A=3929. This
gives a value of », = 118, which corresponds to a band in the

Light Absorption and Fluorescence. 231

infra-red at X=8'48 py, an almost exact mean of the two
bands observed at 8:40 and 8°53 yw.

It is interesting to note that the central line at 1/A= AOLT
is very nearly the same as that of benzene (4055). Now
4047=10x405 very nearly, and therefore 9 x 405=3645
should again be the central line of the fluorescence maxima.
If Dickson’s values for the fluorescence of toluene be taken,
it is clear that the maximum at 3650 must be the centre of
the system. The maxima can be arranged as in Table V., the
figures in brackets again showing the corresponding values
of vz obtained in the absorption band. The accuracy of
determination of the wave-lengths of fluorescence maxima is
not very great, and if the correction for solvent be applied
here there appears to be an error of about 9 units in the
value of 1/A for the central line. This is well within the
limit of experimental errer.

TasLe V.—Toluene fluorescence (Dickson).

} |
eet 1X. ee
| 2886 3465 185 (186-5)

2809 3561 89 (86)
2740 3650 0

2676 3737 87 (86)
2646 3779 129 (131)
2622 3814 164 (159)

Again a very good agreement is shown between the
values obtained from the absorption and fluorescence spectra.

The want of symmetry of the toluene molecule is well
shown by the fact that out of the 19 infra-red bands only 5
give pairs of absorption lines, the remainder being evidenced
only by lines on the red side of the centre.

p-Aylene.

Although the vapour absorption spectrum of p-xylene was
investigated by Hartley, more accurate measurements have
been published by Mies*, who showed that the fine lines
ean very readily be arranged in series. He observed in the
spectrum a certain number of str ong lines which he denoted
by A, and also a number of slightly less strong lines which
he denoted by B. The A lines “and the B lines form two
series with constant frequency differences. Then, further,
he found other lines, the frequencies of which differ by
definite amounts from the frequencies of the members of the

* Zeit. wiss. Phot. vii. p. 357 (1909).

232 Prof, E. C. C. Baly on

A and B series. He thus established four series which
he denoted by A, C, B, D, the C and the D series being
connected with the A and the B series respectively. In
Table VI. are given the absorption lines of p-xylene arranged
in reference to the line 1/A=3869 as centre, and the letters
refer to Mies’ classification.

Taste VI.—p-Xylene absorption (Mies).

BALI Infra-red bands.
Angstréms. WE el Vy. Mean vz.
| Cale. Obs.
2814 3554 | 315 316 317 | 325 p
2800 3571 | 298 298 3°36 3°38
A 2785 3591 27 277-5 3°60 |
C2771 3609 | 260 | 258 || 388 |
B 2757 Sage Weeder a 24 415
D 2744 S614 | 225 || 223) | |) 450 |
A 2722 3673 | 196 | 1965 || 5-09 !
9717 3680 | 189 | | 5:28 530.28
C 2709 Seon i irs | 17k «|| «2 |
B 2695 3710 | 159 | 160 || 6-25 ees
2691 SnIG: de tos. | | 6:55 6°55 |
2685 Srode | 145)" 6:90 690 |
D 2682 3728 14a 1405 712
2680 3731 | 138 | 7:25 7-25
A 2664 3758 | 116 117 8:55 8-60
2661 ATs. ao olit 11L 9-04 9:05
2657 3764 105 104 9-60 9°62
C 2650 3773 96 97. ‘|| 10°31 10°20
2643 S785.01 | /B4 84 ~—||:11-90 11-90
B 2637 3792 | 77 78 || 1282 | 1258 |
D 2624 Bei) Wl) Lae 59 ||: 16-95 |
A 2611 3830 | 39 3 | 25°64
C 2598 3850 | 19 195 || 51-28
B 2584 3869 0 |
D 2571 3889 | 20 | 195 || 51-28
A 2558 3908 | 39 39 25°64
C 2545 3929 | 60 59 16-95
B 2533 3948 | 79 i | 12:82 1258 |
9530 3953 84 84 11:90 11:90 |
D 2521 3967 | 98 97 10°31 10-20
2517 3973 | 104 104 ||: «9-60 9-62
2512 3980 | 111 111 9-04 9:05
A 2508 3987 | 118 117 8:55 8:60
9 | p= . 7°25
© 2494 4009 | 140 1405 712 { es
B 2483 4028 | 161 160 6-25 { eae
D 2471 4048 | 178 178 5°62
A 2460 4066 | 197 196°5 5:09 |
2447 A087 | 218° | 299 4:50 |
B 2434 4109 | 240 | 241 415 |
9495 4124 | 255 258 3:88 |
A 2412 4146 | 277 277°5 3-60 |
C 2400 4167 | 298 298 336 | 3:38
B 2389 4186 317 316 3:17 | 3:25

Light Absorption and Fluorescence. 233

Again, every infra-red band is accounted for, and also the
greater symmetry of the molecule is shown by the fact that
out of the 14 infra-red bands 11 give rise to pairs of
absorption lines.

The frequency of the central line 3869=15 x 258 almost
exactly, and 14x 258=3612 should be the central line ct
the fluorescence. Only 4 fluorescence bands were observed
by Dickson, and the frequencies of these should arrange
themselves symmetrically round this as centre. The four
frequencies when corrected for solvents are 3504, 3584, 3665,
and 3744, and these obviously can be arranged round 3624
as centre as shown in Table VII.

Taste VII.—p-Xylene fluorescence (Dickson).

Rasa 1/r. 1/412. Ve.
2865 STO | aa 7 Ten
| 2801 | 8872 3584 40 (39)
2739 PASSES A Bay 4] (89)
2681 3732 3744 120 (117)

Now, 3624=14 x 258°8, and the fundamental interval is
therefore very near that of the absorption band system (258).
The small number of the fluorescence bands makes it im-
possible to arrive at greater accuracy.

The absorption lines and fluorescence maxima can also be
arranged in the same way for o-xylene and m-xylene, and
they are shown in Tables VIII., [X., X., and XI. In these
tables certain lines are marked A or F. Those marked A
are the lines which Hartley considered to be the heads of
the band groups, while those marked F give the same values
of v, as appear in the fluorescence spectrum. The wave-
lengths of the lines are taken from Mies’ paper *.

The frequency of the central line 3909=13x 300-7. The
next multiple is 12 x 300°7=8608, which may be taken as
the centre of the fluorescence bands. In Table IX. the
frequencies of the fluorescence bands are increased by 13
units to correct for the effect of the solvent.

* Zeit. wiss. Phot. viii. p. 287 (1910).

234 Prof. E. C. C. Baly on

TaBLe VIII. o-Xylene absorption (Mies).

2 Ato: | | | Infra-red bands. |
Angstroms, | 1/R. Vr. Mean rz. |
| | Cale. | Obs. |
MTs 4, S601 308... | 325y | 3825p |
2770 | 3610 299 | 338° | 388 |
F 2730 3663 246 | 4:07
F 2699 3705 204 | 4:90 |
2691 3716 193 | 518 B24 |
| A2683 | 3727 182 | 5:49 ciel
HA 2668" 1) 3748 161 | 6-21 6:20 |
2666 | 3751 158 | 633 630 |
2659 | 3761 | 148 | 74 |] Gaeeae
2654 Bes Aalto, ade To oe
D650 |. 3774 is 4 | 7-41 |
A 2547 3778 IGE. -4 | 7-63
A2633° | 3798 | 111 9-01 802m
2628 | 3805 | 104 | 964) dean
2624 | 3811 98 97 || 1031 | 10204
2620 | 3817 92 91:5 | 10-91 :
2607 | 3836 7 von eles 1360
A 2601 3845 GhL “ss 15°63 |
A 2585 3869 | 40 25-00 |
9572) |) |) 3888 21 | 47-62 |
A2558 | 3909 0 | |
W527 8) (8957 AB || 20°83
SCS Oe Ms St yank yt Qala Mt 13:33 | 1360 |
2500 4000 91 91:5 || 10-91
2497 | 4005 96 97. ‘|| 10°31 10:20 |
2474 4042 141 | 141 | 7-09 7-25 |

TaBLE TX. o-Xylene fluorescence (Dickson).

Ain |
Angstroms. NPs Di te nea

a es oe |
3135 3190 | $2038 405 |
3038 3292 3305 303 (299)
| 2986 3349 3362 246 (246) |
2896 3453 3466 | 142,147) ae
2798 3574 Boek. oes 21 (21) |
2713 3686 | 3699 91 (91s |
2680 3731 3744 141 (141) |
| 2636 3794 3807 199 (204) |
2603 3842 3855 247 (246) |

It is to be noted that the fundamental frequency 301
appears both in the absorption and fluorescence spectra.

Light Absorption and Fluorescence. 235

TABLE X.—m-Xylene absorption (Mies).

| > Nin | | _ Infra-red bands.
Angstroms. | I/A. Vr. | Mean v,. |
Caley, |" ‘Obs,
2802 | 3369 295 | PSO SE ei |
2721 3675 189 5:29 5°25
Ceres 2 Ota BOS? 182 181 5°52
2703 | 38699 165 163°5 6°12 6°20
2694 | 3712 152 6°58 677
E2OSi ew aioe 142 142 7-04
2684 | 3726 138 7°25 725
2668 _ 38748 116 | 862 8°70
2663 | 3755 109 | 17 oF
2658 _ 8762 102 105 || 9°70 9°68
| 2655 — 38767 97 || 10°30 | 10:20
2648 | 3776 88 875 11-43 | 11:42
2640 | 3788 76 745 || 13°42 13-00
F 2601 _ 3645 19 20 |, 80°00
2588 | 38864 0 |
F 2574 3885 | 2) 20 | 50:00
2.540 3937 73 74:5 13-42)° 1 13500
2531 3951 | 87 875 11°43 11-42
2520 | 3968 104 105 9°70 9:70
F 2496 4006 142 142 704 |
2484 | 4026 162 163°5 6:12 C20
FB 2473 4044 180 FSi 5°92

The central line 3864 is almost exactly 13x 297, and
12 x 297 =3564, which should be the centre of the fluores-
cence. Dickson only finds three maxima of fluorescence,
and they all lie on the ultra-violet side of 3564. If this be
correct, then allowing 14 units for the effect of the solvent
the fluorescence bands may be arranged as in Table XI.

He ene I/X. | L/A+14. | ie
Deo) Sy) hseeg) | Sans | ~~ 19 (20)
2715 3683 8697) 183149)
2685 | 3724 =| ~~ 3738 | 174 (181)

It is, however, manifestly impossible to draw any definite
conclusion from only three maxima.

There can be little doubt from the above results that the
conception of combining the frequencies of the short-wave
infra-red bands with those of central lines of absorption and
fluorescence bands is justified. In the five substances dealt

236 Prof. E. C. C. Baly on

with, every single infra-red band observed by Coblentz, with
two possible exceptions, is represented by either one or two
lines in the ultra-violet absorption band, and further, the
agreement between the calculated and observed values of the
infra-red bands is remarkable.

The general conception can be put to a very severe test in
the following way. Dickson found in the fluorescence
spectrum of naphthalene 14 well-defined maxima which are
very regularly arranged. In fact their frequencies may be
expressed by the yeneral formula

1/A=3326 —47°12 xn,

waterermds (Bie 2 13. He finds small differences
between the observed values and those calculated from the
formula, especially in the case of the band with the smallest
frequency. It would seem, therefore, that in making any
calculations from the frequencies, it would be preferable to
use the values obtained from the formula. Now the absorp-
tion spectrum of naphthalene in the infra-red region has not
been observed, and the only fact known about it is that
Coblentz found a band at X=3'25u for a solution of the
compound in carbon tetrachloride. It is not possible, there-
fore, to check the values of frequency differences against
infra-red measurements. Since the fluorescence bands are
very symmetrically arranged, it is possible accurately to
calculate the frequency differences from the central line.
This central frequency must be a multiple of the funda-
mental frequency, and the next higher multiple should form
the centre of the ultra-violet absorption band. From this
new central frequency, by making use of the frequency
differences found in the fluorescence spectrum, it should be
possible to calculate the frequencies of the lines in the ultra-
violet absorption band.

In Table XII. are given the frequencies of the fluorescence
maxima of naphthalene as corrected by Dickson, and arranged
symmetrically with respect to the mean frequency 1/A=3020,
together with the frequency differences.

The calculated values of the infra-red bands are given so
that when this region is investigated, the observed values
may be compared.

Now the central frequency 3020=302 x 10, and as there-
fore the fundamental frequency of naphthalene is 302,
the central frequency of the absorption band must be
302 x 11=3322. In order to calculate the frequencies of
the absorption lines, we thus use 3322+v;, the values of
vx being those given in Table XII.

Light Absorption and Fluorescence. 237

TasLeE XII.—Naphthalene fluorescence (Dickson).

Calculated

1/A. | a | Infra-red Sens

| Q714 306 | 327 |
| 2761 259 3°86
| 2808 212 4°72
| 2855 | 165 6-06
| 2902 118 8:48

2949 70° | 142

2996 23-5 | 42-6
| (3020) | | |
| 3043 | 23°5 | 42:6 |
| 3090 70°5 | 14-2

3138 118 8:48

3185 165 6:06
| 3932 212 4°79
3279 259 3° 86
| 3396 306 320
: ee ck iy et op tet he

The absorption of naphthalene vapour has been investigated
by Purvis, who finds that the bands shown by the alcoholic
solution are not resolved into fine lines. It is necessary,
therefore, to make use of the solution spectrum of naphtha-
lene for the present comparison. This spectrum has been
measured by several observers*, and Mr. IF. C. Guthrie, in my
laboratory, has kindly repeated the observations, using the
new Hilger ultra-violet spectrophotometer, the accuracy of
which far exceeds that of the old method of qualitative
measurement.

In Table XIII. the first column shows the values of vp,
obtained from the fluorescence bandsand given in Table XI].
The second column contains the calculated frequencies of
the absorption bands, while the corresponding wave-lengths
appear inthe third column. In the fourth column are given
Mr. Guthrie’s measurements.

The agreement between calculated and observed values is
exceedingly g good in view of the fact that there are no infra-
red imedsurement: against which the frequency differences
(v,) can be checked. Certain of the calculated absorption
bands do not appear in the solution spectrum, and the broad
band at ~A=2965 does not seem to divide. One more absorp-
tion band has been observed at X=2670 beyond those that
have their counterpart in the fluorescence spectrum. — It

* Hartley, Trans. Chem. Soc. xxxix. p. 153 (1881); xlvil. p. 685
(1885). Baly and Tuck, Trans. Chem. Soe. xciil. p. 1902 (19C8). Purvis,
Trans. Chem. Soe. ci. p. 1515 (1912).

238 Prof. E. C. C. Baly oa

Tasie XIIJ.—Naphthalene absorption.

ie) |
Wave-lengths in Angstroms.

| Pa 1) UX.
Caic. Obs.
306 3016 3311
259 3063 3265
212 3110 3215 3218
165 3157 3168 Bibs |
| 118 3204 3121 3118
71 3251 3076
23 | 3299 3031 3025
0 (3322 on
| 23 3345 2990 ee
| 71 3393 5035 | oaks
| 118 3440 2907
| 165 | 3487 2868 2867
212 | 3536 2828 2830
| 259 i S581 2793 | 2798
306 bos 2757 2759
| 2670

may be claimed that the above calculation completes the
evidence and confirms the theory here put forward of the
complex structure of ultra-violet absorption bands.

Two interesting points may be noted, one of which is the
agreement of the only observed infra-red band of naphthalene
with the calculated value, and the other is the fundamental
frequency of 302. This frequency is practically the same as |
that of o-xylene, which is 301. Naphthalene can be looked
upon as containing ortho substituted benzene rings, and it
would seem therefore as if 301 or 302 might prove to be
the fundamental frequency of ortho disubstituted benzene
compounds.

There is no doubt that it should also be possible on the
present theory to explain the phosphorescence spectra as
observed by v. Kowalski* and by Goldsteint with certain
organic compounds. Both these authors investigated the
phosphorescence of the solid substance at very low tempera-
tures, and we have no knowledge of the shift of the bands
under these conditions as compared with the vapour. _ It is
impossible, therefore, to calculate the trequencies trom these
authors’ measurements in the way adopted above.

The following substances may be selected, namely, benzene
and p-xylene, both of which were investigated by v. Kowalski,

* Phys. Zeit. xii. p. 956 (1911).
+ Phys. Zeit. xii. p. 614 (1911) ; Deutsch, Phys. Ges., Verh. xiv. pp. 38
& 493 (1912).

Light Absorption and Fluorescence. 239

and the latter also by Goldstein. The fundamental frequencies
ot these compounds are 405 and 258 respectively. The
simplest method of calculation is to find whether the phos-
phorescence maxima can be arranged in each case with
reference to a multiple of the fundamental frequency, due
regard being paid to the fact that the phosphorescence
maxima are certain to be moved towards the red. The
accuracy of measurement reached by v. Kowalski and
Goldstein is only about 25 Angstroms, and therefore the
frequencies are only expressed in three figures. But in
spite of this, it is clear that the same relation holds good
here also.

Thus in benzene the frequencies of the phosphorescence
maxima can be arranged symmetrically with respect to
1/AN= 242, as shown in Table XIV., together with the cor-
responding values of v, (in brackets) found in the absorption
band.

TasLeE XI V.—Benzene phosphorescence
(v. Kowalski).

1/N. Vz.
230 12 (12°7)
| 233 9 (9:2)
239 o
242 QO
249 Tard)
252 10 (10:2)
260 18 (18°4)
263 21 (208)
270 28 |
74 32 (318) |
280 38 |
| 284 42 (40:5) |
289 a7 |
295 53

It may be pointed out that the frequency difference of 47
corresponds very nearly with the infra-red band at 2-18
(1/A= 46).

‘ -We thus have for the fundamental frequency of benzene
the following values :—

Phosphorescence . . 6x 403°3
Einorescence sy. O%*« 405
Abrorption e). 2). 10405

240 Light Absorption and Fluorescence.

In the case of p-xylene the following values are obtained

(Table XV.).
TABLE XV.—p-Xylene.
Phosphorescence Cathodoluminescence

(v. Kowalski). (Goldstein).
1/d. Vr Mean vz. eee | Vr.
234 92 22 (22-2) 176 | 80(29:8)
239 17 175(19and 16) | = 180 26 (26)
243 13 13°5 (13'8) 184 PP OL TD
249 7 8 (7°8) |) 188 18 (19) |
253 3 2°5 192 14 (14) |
(256) 0 | 196 10(10°5)
258 2 25 | 200 6 (59) |
265 9 Se Ga) | 205 0
270 14 13°5 (13°8) 208 2 (1:95)
274 18 173 (I6and19) | 212 6 (5-9)

282 26 | Central line 205°6.

Central line 256.

It is possible that there is an error in v. Kowalski’s | |

measurement of the last maximum on his list.
have for the fundamental frequency of p-xylene the following

values :—

Cathodoluminescence. 8 x 257
Phosphorescence . . 10x 256
Fluorescence . . . 14x258
Absouption.. 2) .9: 1as<zos

It is manifestly impossible to expect much accuracy from
the values of the phosphorescence maxima, seeing that the
measurements themselves of these are far from accurate, but
they certainly seem to give considerable support to the
theory put forward in this paper, a theory which would
appear to be proved correct from the absorption and

fluorescence measurements.

Inorganic Chemistry Laboratory,
The University, Liverpool.

We thus

Fae Wiel

XXVII. Anomalous Zeeman Lifect in Satellites of the Violet
Line (4359) of Mercury. By H. Nacaoxa, Professor of
Physics, and 'T, TAKAMINE, Graduate in Physics, Imperial
University, Tokyo”.

[Plates III. & IV.]

rq \HE changes wrought by magnetic fields on the satellites

of spectrum lines are generally different from those
observed in the principal line, both as regards the modes of
separation and in the distribution of intensities among the
different components. The effect is in these respects mostly
anomalous, if we follow the course of satellites to strong
fields, although some regularity usually observed in simple
lines may still be traced in fields of a few thousand gauss.
The present paper is a continuation of the experiments fT
made on the satellites of the green and violet lines 5461
and 44047 respectively, to those of 14359. ‘The study of
this line is specially interesting as it 1s accompanied by more
than ten satellites, whose displacements in magnetic fields are
so diverse, that different cases of anomalies already noticed
in the satellites of two lines above mentioned are also found
in those of A 4359.

The method of observation was exactly the same as that
already described in our former paper, so that it will be
unnecessary to enter into its details. Briefly speaking, the
position of satellites with respect to the principal line in
magnetic fields was determined mostly from photograms
taken with an echelon grating, and the course of the curves
giving the transition of the satellites was traced from those
obtained in heterogeneous fields. Some doubtful cases were
also examined in the photograms of interference points,
obtained either by combining the echelon grating with the
Lummer-Gehrcke plate or with the Fabry-Perot inter-
ferometer. Owing to the great complexity in the distri-
bution of the satellites, it was only after various examinations
of 180 photograms that the displacements of the satellites
were finally settled.

Typical photograms showing the lines in uniform and
heterogeneous fields are shown in PI. IIT.

Fig. 1 a and fig. 16 show comparisons of the lines \ 5461

* Communicated by the Authors.
+ Nagaoka and Takamine, Phil. Mag. xxvii. p. 333 (1914).

Pal. Mag. 8. 6. Vol. 29. No. 170. Feb. 1915. R

949 Prot. Nagaoka and Mr. T. Takamine : Anomalous

and 14359 in a field of 15,500 gauss. Figs. 2a, 2b, 2¢ are
enlarged photograms of the p- and s-components taken in
fields of 6300, 14,600, and 20,300 gauss respectively. The
general appearance of the lines in heterogeneous fields is
shown in fig. 3a and fig. 36, of which the latter is more
enlarged than the former. Faint lines are not distinctly
seen in photographic reproductions, so that these figures
only serve to illustrate the complex appearance of the satellites
with reference to the principal lines.

Principal Line.—It is generally assumed that the so-called
principal line is simple and has finite breadth ; it was on
this supposition that we have measured the Zeeman effect
of satellites belonging to the lines \ 5461 and 74047. This
is by no means always the case. Janicki™ showed that the
principal line of 75461 has fine structure and can be re-
solved into five lines. We recently found that the principal
lines of 1 4359 and A 4047 are of similar nature, and cannot
be treated asa single broad line. This fact will doubtless
have an important bearing on the Zeeman effect. It will
therefore not be out of place to!give a brief sketch of the
result, reserving the detailed description of the resolution
for another place.

By combining the echelon grating with a Lummer-Gehreke
plate, both of resolving power slightly exceeding 4 x 10°,
we { have shown that there are two satellites +17 and —23
m.A.U. very near the principal line 14359. The interval
between —23 and the principal is, however, so vague, that
the presence of a cluster of lines seemed to us very probable.
By combining an echelon grating or a Lummer-Gehrcke
plate with a sliding Fabry-Perot interferometer, and making
the air-plate about 5 cm. thick, thereby utilizing a resolving
power of about 2x 10°, we found that the assumed principal]
line is composed of a group of three strong lines, the con.
secutive distance between them being about 6m.A.U. The
middle line of the triplet may be considered as the principal.
When the analysing power of the interferometer is not
sufficient to resolve the principal line, the mean point appears
displaced by about 3 m.A.U. towards the side of longer
wave-length, from that given in our investigation con-
cerning the constitution of mercury lines. The reason
why Janicki did not observe the fine structure of the violet
lines must be ascribed to the low resolving power of his
instruments.

* Janicki, dan. d. Phys, xxxix, p. 489 (1912).
+ Nagaoka and Takamiv4 Proc. Phys. Soc. xxv. p. 1 (1912).

Zeeman Effect in Satelittes of Violet Line of Mercury. 243

We take this opportunity of filling in the omissions which
we have made in our former communication on the Zeeman
effect of X 5461 and X 4047. The principal lines have fine
structure, as given in fig. 4, fig. 5, fig. 6, which indicate the
position as well as the intensity of different lines forming
a triplet of the second subordinate series of mercury. The
new lines are marked with asterisks.

4. A: 0461

= DARW S wo N
S x Sr BS ko Nn 0) G
«s SS NY WANN 7) 7) t

ke SE

o )
: B88 2 Besse s $ iS
He KE
IRCA Gy, A : 4047
fo)

The result for 5461 is almost identical with that of
Janicki. He considers the principal line as a doublet of
equal intensity, but according to our experiments, the one
towards the violet appears stronger than those towards
the red.

244 Prof. Nagaoka and Mr. T. Takamine: Anomalous

In order to avoid the ambiguity in the position of the
principal line, the position of satellites is sometimes referred
to that lying farthest towards the violet. But we believe
that finer structures will no more be forthcoming, and in
discussing the Zeeman effect especially, it is convenient
to refer the position of satellites to that of the principal
line.

As the examination of this fine group of lines is only
possible so long as the light is quiet and steady, the Zeeman
effect of each of the lines forming a cluster is almost im-
possible to follow in the present stage of our experiments.
Owing to this circumstance, what we consider as the magnetic
separation of the principal line refers to that of the cluster.
Whether the structure remains unaltered in magnetic fields.
or not, seems to be an important problem, but special
means and arrangements will be required to decide the
question.

According to Runge and Paschen, the principal line
X 4359 is divided into a regular sextet, following the rule
a 5)
Eee), Oo i
To test the deviation, we compared the separation with that
of the green line, which almost exactly obeys the rule,

a, +2a. This is only approximately fulfilled.

2 :
0, + = +a, +54, +2a. These two lines were photo—

graphed side by side in the same field (Pl. III. fig. 1 a and
fig. 16). By increasing the field gradually, the following
results were obtained :—

| H in gauss. | +0A\{P+,). +06A,(P+,). | +6A,(P+,). | — | _
Nhat) he ee een Seen. TAL pie il 1
| 13900 | G2m.AU, 178 mh.U, wimA’v. 288 | 3:90
| 18200 Si; PAO EE 324° ,, +] 296° 9) 2eme
| 21000 Bt) BY Ga | 36) | |e
|’ 99800 |102 ,, |300 , | 400 , | 2090 | gap
bereatog®. A Od 0) deaG oc aie 09 >, 290 | 3°94

The branches of the sextet are designated by the letters.
Poy, Ps, P.. and, Fj, F_s, Ps, of which Page

vibrate parallel (p), and the rest perpendicular (s) to the

Zeeman Fiffect in Satellites of Violet Line of Mercury. 245

P : rn
magnetic field. If Runge’s rule be strictly obeyed, a Nas
OAs : 1
and >, =! The agreement is not therefore so close as

with the green line. Whether this discrepancy is to be
attributed to the fine structure of the assumed principal
line or not is an important question. When we recollect
that the green line has similar structure and still obeys
Runge’s rule, we cannot at once answer the question
positively.

Satellites.—The displacements of the satellites were mea-
sured by taking the branches of the sextet as reference lines.
Unfortunately the finite breadth of these branches makes
the line somewhat vague, so that micrometric measurements
were not entirely free from the error of pointing, especially
in low fields.

As already noticed in our former paper, the p-components
are simpler than the s-components. The distribution of linesin
the photograms ef 4359 is very intricate, and the tracing of
points lying on the curve representing the displacements
presents extreme difficulty, but we believe that the principal
feature of the mode of separation has been deciphered,
except for faint satellites or for those lying very near the
quartet P.2, Piz, of the principal lines, which are far
superior in intensity and obscure the traces of satellites.

The general feature of the p-components is shown in
fig. 7, Pl. EV., and of the s-components in fig. 8, PI. IV. It
will be noticed at a glance that the Zeeman effect of the
satellites is almost without exception anomalous. As was
already shown with the lines 1.5461 and X 4047, the anomaly
appears either as a dissymmetry in the intensity of the
displaced lines, or as deviations from the law of linear propor-
tionality of the amount of displacement to the magnetic
field. These two features are also characteristic of the
satellites of the violet line \ 4359.

The observations of the displacements of different satellites
are given in the following tables. The lines in which the
displacement 6X is proportional to the field H are first
tabulated. When this condition is not satisfied, the displace-
ment is generally represented by an hyperbolic curve, which
is given by

H?=adv+4+ 0dr’.

Sometimes b=0, and the curve becomes parabolic.

246 Prof. Nagaoka and Mr. T. Takamine :

I.—SimMpeLE DISPLACEMENTS.

(—160)

s-component.

\
|

(—107)

Anomalous

p- component.

edge, echt trnenes

Dw woreeee
KD bD = 7 bO OO bo CO

: side. | | N
aN
toate a. FX 102. a |
[na am ied eee 10800 | 0 0
| 1300 18 | (1:882)| 11300 | 0 2
| 2000 | B21 | 1-05 | Se ee 0
| 4300 43 1:00 | =
5100 46 0:90
9900 96 0:97
10200 101 0:99. |
10800 106 0-98
11400 isso ORS)
(—18)
p-component. | $-components.
a Ee EEE ee
—side. + side (outer). — side (inner) + side
| OA Le) | Ov
He. | Od. | H* 1Oe i He Ao Ne | H* LOPES il mone ox re HH. | Oae
2300| —17| —0-'74 | 1800} —53| (2°95 ?) | 1800) —27|(—2-087)| 2800} 34
2900) —20 —0°69 | 3300) —84| 2°55 4100; —78) —1:90 | 3300) 37
3300) —382 —0-97 | 5100)— —126 2°47 5100} —96} —1°88 | 6500! 77
4200} —37 —0°88 | 10200/—255; 2:50 |10800/—203) —1:88 | 7400) 83
7600) —47 —0°62 10800 —265) 2°45 |12300/—234| —1:90 |10200) 119
12800/—106 —0°83 11000|—270 2°45 |14500)—274| —1°89 |14200) 171!
11400|—279, 2°45 |16700/—307/) —1:84 |16400) 201
14300}—354, 2°48 |19200)—356) —1:86 18000) 220
pias —398, 2°42 |20600)—379| —1:84 23000) 279
(420)

—side.

—_—_—— —_

2200
7400
14500
16000

- $-component.

247

07 Mercury.

f Violet Line

ites O

Zeeman Effect in Sateil

FS SS ee ON
— side. + side.
OX Or
H. ale 1026 Ee al:
On WX 10. H On WX 10
9900; —47 | —0-48 | 1800 9 0°50
16600} —82 | —049 2300; 10 0:44
24500 |}—115 | —0:47 3000; 13 0°43
30000 | —149 | —0:50 | 3300) 15 0 45
4200' 20 0:48
== = Sel GO et 0:47
| 5700] 25 0:44.
6300; 28 0:44
6700; 28 0:42
7600) 32 0:42
9900| 43 0:43
12700| 55 0°43
14600} 66 0:45
16600| 76 0:46
20800) 94 0°45
21400; 97 0:45
23400 | 106 0:45
24500} 112 0:46
380000 | 137 0:46

(+ 46)

p-components.

— side (outer).

is OX.
4300} —79
7000 | —130
7300 | — 186
14200 | —258
18550 | — 343

$-components.

— side (inner).

Ov. i x 102.
—388 | —1115
—62 | —1:24
—74 | —1:27
—78 | —1:20
—83 | —1'18

—124 | —1:21
—175 | —1:22
—206 | —1:23
—224 | —1:23
—255 | —1:24
—287 | ~1:25

+ side (inner).

OX.

ooo i FF ————— ——————

+ side (outer).

ON 1
ah 2A ele
m1 10”,
(1°85?) 1800
1-48 | 3300
1:41 4300
1:33 =| 5100
1°36 5700
1°36 10200
10800
= ~ | 18200
19000
20600
213800

248 Prof. Nagaoka and Mr. T. Takamine : Anomalous
a eee i |
(+ 108 ) (+185)
Liga tree | »- components.
| ye SSS en : —~
H Pe Ly, 102 igre side. -- side.
| #. 19 | re
z > j
| H. OX. 2x 10°] HH. | oA.) ce
PSH; LE, 0 ae
129001 2° I 0 | |
| 3300); 2 , 9 i" 2900 | —20| 069 | 2900! 18! 0-64
SN Se | 5100 | —34| 067 16600) 88] 053
| 6700 | 0 te 0 7600 | —52| 068 | 21400) 112| 052
bo i | | 3 9900 | —66| 0-67 | 23400 124} 053
2700 | —85 | 0°67 |
| 13200 96 | 0°68 | |
IJ.—CompLEX DISPLACEMENTS.
(— 160) | (=107)

- ———~| :
— side. + side. | — side. — side.
| | | |
H. | 6Xobs. | 6Acale H. | Aobs. | H | OXobs. | dXcale H. | dXobs.
3 aaa Nae
| } | |. (a
9 Cas We eas | Penile a 4
900] —4 ° 2 al 11 | 1800) 7 3=| | as00) 4
3300] —9|—8 ¥ | 2900) 23 | 1800; —6; —5s8| | , 5
5700) —23 |~23 , | 5100) 33 || 2300/ -9| -8S/ |, 0]
© | | a i a
6300] —29 |-28 & | 5600| 33 | 3300) -17| -162 | | anol_yea
ol } Sal 2 =
6700| —32 |-32 X | 6300| 37 | 4300) -28| —255)| |
= | S |
6700} 40 | 9800, —93 | —91 1
| | |
9900 45 | 11400 113 oe |
| | | =
12700! 32 | 14300)/—152 |~153 x |
re =
| 16700 |—185 |—185 —
|

p-components,

ais

s-components.

|
|
|

S-component.

PN cale.

— 164

(H? x 10-5= —0:3636-+0-00175A2.)

Zeeman LHffect in Satellites of Violet Line of Mercury. 249

aot)
p-components. 1 s-components.
am aS FS iliac meanest —-
— side. + side. | — side. + side.
et | SXobe. H. OXobs. | tal OXobs. Xcale. | H. OXobs.
Te tas | | \ SNe
PCO a et aes 0ON Wig | 200) Wr 2 he | 1800) |. OI
23000 | +2/ 4200} 17 |) 1800 | —4|) —4 2 | 4300! 62 |
29000 | —5| 5700) 17 || 2200 | —5| -6 8& | 5600 | 103 |
S
33000 | —8 | 6300| 14 || 2800 | -10} —10 + | 7000 | 128
L4=)
49000 | —23 | 6700| 13 || 4000 | —22| —22 & | |
! Oo
7500| 8 || 5600 | -40| -39 | |
6500 | 52] —52 £
| bes
10800 |—128 }—125 a
| | = |
| | 12300 158 |-155 ~ | |
(+ 108 ) (+185)
s-components. s-components.
(eS eee ost caoN
— side. + side. + side.
— - : aie Wan eee |
H. €Xobs. | H. | ONobs. Joh | OXebs. OXeale. |
Ae OURS OY Aira Raa Aeon ae
| | |
1000. | —8 | 1300 | 18 | 1300 | 5 Sapa
htc! sais neeteCOn | 9 2y ats aS ate
1800.) 22,0 ees0an|) oe 2300 Leite elses = |
5100 | -82 4300 76 2600 hl) i) LO hay
7300 | —128 | | 3300), hip suinid ae ome, Px

5600 | 49 48

250 sero Nagaoka and Mr. T. Takamine : Anomalous

The simplest type of separation is observed in the satellite
whose position in non-magnetic field is +46. The p-com-
ponents consist of two branches, which are parallel to the
principal lines P_,; and P,;. One of the branches towards
the red is much more pronounced than that on the violet
side. The positive branch is especially bright, as itis always
distinctly to be observed by the side of the principal line.
The s-components consist of a quartet, parallel to the
principals Ps, Ps, but the intensity is widely different
according to the components. Those parallel to P,3; and P_;
are brighter than those parallel to P,, and P_;, while with
the principal, P.. are a little brighter than P:3; thus for
one branch of the quartet of this satellite, the order of
intensity is inverted. The linear relation of the displace-
ment with the field strength seems to be nearly fulfilled up
to very high fields.

The p-components of the satellite +185 are displaced
parallel to the principal, but the lines are weak and can no
‘more be followed beyond 23 kilogauss for the +branch,
while the —branch is obliterated by the principal line P,;.
The s-components of this satellite show anomalous behaviour
similar to the type already noticed with the p-component of
the satellite —242 (—239 according to our new measure-
ment) in the green line of mercury. Another anomaly
lies in the smallness of the amount of separation for
s-components in comparison with p-compenents.

The satellite —18 has a negative p-component which is
displaced wider than P_,; the positive branch is probably
eclipsed by Pj, so that it does not appear well defined on
the photograms. A singular anomaly is presented by the
_s-components. The separation takes place proportional to
the field strength in all the observed branches, but the
amount is less for the positive than for the negative ; one of
the latter branches is parallel to P_3, while the other is
farther down; there is thus greater dissymmetry in the
effect. One reason for the preponderance of the negative
displacement is probably to be found in the fusion of the
negative branches of the satellites —160, —110,and —94
with those of —18.

In the s-components of +108, two faint branches are to
be traced ; by combining them with the undisturbed line in
the p-component, we notice that the separation is of a triplet
type similar to the principals of the violet line 4047.

The magnetic force produces anomalous effect on p-
components of the satellites —160 and —94: the negative
branch is represented by a parabolic curve, so that the

Zeeman Liffect in Satellites of Violet Line of Mercury. 251

displacement is nearly proportional to the square of the
field ; it gradually fades away and disappears in fields of a
few thousand gauss. The + branch is curved for both lines,
the concavity being turned towards the —side; they both
approach the negative branch of —18, to which the lines are
finally immersed at H=9000 gauss for —94, and at
H=13000 gauss for —160. This singular phenomenon of
the ultimate fusion of lines was already observed in the
satellite —242 of the green line. Such behaviour has
already been observed by Wali-Mohammad * on the satellites
of cadmium and bismuth lines, and is probably characteristic
of the Zeeman effect of the satellites. It is questionable, if
we have to consider these two satellites as belonging to the
principal line, or to —18, in which they are finally immersed.
The positive branches of the s-components of these satellites
fade away in tolerably weak fields, but the negative branch
takes an hyperbolic course, and approaches —18, almost
asymptotically. Two of the satellites, —160 and —4, are
immersed in the upper, and —107 in the lower negative
branch of —18. This behaviour is similar to those already
observed for —160 and —94 in the p-components, and for
—242 in the green line of mercury. ‘The difference in the
behaviour is that, in the p-components, the fusion of lines
takes place with a component whose displacement is not
proportional to the square of the field in the initial stage of
separation, while with the s-component the contrary is the case.

The anomaly above~.described is mostly confined to
satellites which do not lie in the immediate neighbourhood
of the principal line.

We have noticed that the farthest satellite —242 in the
green line shared this characteristic to a remarkable degree,
while for 4359, the satellites lying at -160, —107, —94, and
+185 from the principal, show a similar mode of anomalous
displacements. The satellites in the vicinity of the principal
line seem to be much affected by it, and are generally
displaced proportional to the field, the course of the curve
running in most cases parallel to the branches of the
prneipal line.

The above fact is also borne out by the experiments of
Gmelin +, and Lunelund {, on the outermost satellite +224
of the yellow line 25790 of mercury, and of Wali-
Mohammad §, on cadmium and bismuth lines. Among the

* Wali-Mohammad, Diss. Gittingen (1912); Ann. der Phys. Xxxix.
p. 225 (1912).

+ Gmelin, Diss. Tiibingen, p. 41 (1909).

{t Lunelund, Ann. d. Phys. xxxiv. p. 505 (1911).

§ Wali-Mohammad, Joc. ett.

252 Zeeman Effect in Satellites of Violet Line of Mercury.

last-mentioned lines, whose displacements are proportionat
to the square of the field strength in the initial state, may be
mentioned +58 of the cadmium line \ 4800 for p- and
s-components, and —103 of the bismuth line 14722 for
s-components.

The comparison of the present experiment with the results
of previous investigators is of little importance, since the
anomalies here discussed were not noticed in most cases.
Gehreke * and v. Baeyer were the first to investigate the
Zeeman effect of a satellite of 74359 in weak fields ; the
same satellite was afterwards investigated by Lunelund, but
as the upper limit of the field did not exceed 3500 gauss,
the measurements did not bring out the princial features
discussed in the present paper. ‘The lines on beth sides of
Piz, which appear in strong fields according ©» Wendt f,
are probably the branches of the satellites —18 and
+46; the correspondence with our result is not exact.

In his theory of the Zeeman effect, Voigt f has shown that
for a simple coupling with an electron, the change of wave-
length for the p-component of a satellite takes place at first
proportionally to the square of the field, and then according
to an hyperbolic law. ‘The behaviour of some of the satellites
of the line here investigated corresponds to this theory as
already noticed. In addition to this, we meet with several
similar cases in the s-components; this evidently calls for
a new discussion of a system with another kind of coupling,
It may, however, be objected that the study of the mercury
lines 0. 5461, X 4359, and A 4047 is limited to a triplet of the
second subordinate series, so that other kinds of displace-
ments may appear with other lines and with lines of other
elements. But the results of different experimenters on
satellites of other spectrum lines show that the types of
anomalies are not numerous, so that we may arrive ata
satisfactory explanation of the satellites and their connexion
with the principal line, by following the reason based on the
mutual action of vibrating electrons. By the study of
the three lines above-mentioned, we can obtain abundant
experimental data tor testing the theory of the coupling of
electrons in explaining the different behaviours of the
satellites in magnetic fields.

Physical Institute, University of Tokyo,

July 28th, 1914.
fon and v. Baeyer, Ver. d. Deutsch. Phys. Ges. viii. p. 899
t Wendt, Ann. d. Phys, xxxvil. p. 535 (1912).
T Voigt, Ann. d. Phys. xiii. p. 815 (1913).

Be 258 is

XXVIII. The Magnetic Deflexion of the Recoil Stream from
Radium A. By H. P. Watmstny, M.Sc, and W.
Maxower, M.A., D.Sc.*

| [Plate V. fig.1.]

THXHE deflexions suffered in an electric and a magnetic

field by the recoil stream of radium B produced by

the disintegration of radium A have been studied by Russ
and Makowerf and by Makower and Evanst. It was shown
that the atomic mass of radium B has approximately the
value 214 predicted by radioactive theory, and that each
particle in the recoil stream carries one positive charge. It
is of importance to repeat these experiments with greater
accuracy in order to determine the atomic mass of radium B,
and it is also of interest to find out how nearly the velocity
of the particles in the recoil stream is in agreement with
that to be predicted from a simple application of the law of
the conservation of momentum. lxperiments on the mag-
netic deflexion of the recoil stream have been completed and
will be described, but the investigation of the electric
deflexion has unfortunately had to be postponed.

When an atom of mass M disintegrates and emits an
a particle of mass m with a velocity v, the atom recoils with
a velocity V given by the equation

(NI cr Ne OES ul a PN laa a

If the recoil stream and the « rays pass through a magnetic
field of strength H, the radii of curvature p, and p, are
(M—m)V ; _me

PRMER ane He?
carry only one charge, whereas the # rays carry two. It,
therefore, the recoil stream and the « rays pass through the
same magnetic field, the radius of curvature of the former

Fig. 1.

respectively since the recoil particles

Nine Caen ae

should be exactly double that of the latter. Using a line

source O,a slit S, and a screen P placed perpendicular to the

line OS (fig. 1), the particles passing through the magnetic
* Communicated by the Authors.

t+ Russ and Makower, Phil. Mag. Noy. 1910.
t Makower and Evans, Phil. Mag. Noy. 1910.

254 Mr. Walmsley and Dr. Makower on Magnetic

field will be bent into circles, and will strike the screen in lines
parallel to the source and slit but displaced from the point P.
Tf the distance OS=J, and the distance SP=/,, then if the
displacements d, and d, suffered by the « particles and recoil
stream respectively are measured, the radii of curvature are
given by the equations

iL
Pa pre Lie4+d2\ 1 (h+h)?+d
4d,” | | (2)
and elu : zi) : 3 ’
P; aaa Ad,? } ls aig d, f Ci aE ly) == d,. {
whence pa dy dlp aie, Ghee Ore 5 ee.
A = a IL=5 le cle Gikhy approximately.

If, therefore, the quantities d. and d, are measured, the
radius of curvature of the recoil stream in the magnetic
field can be compared with that of the @ rays and the value
of Mee for the recoil stream found without the neces-
sity for an absolute determination of the value of the magnetic
field or for measuring the quantities /; and /, with great
accuracy, since these quantities occur only in small correcting
terms.

The method depends upon the possibility of detecting the
positions at which the a2 rays and recoil stream strike the
screen P. This can be done easily in the case of « particles
by using a photographic plate as the screen. Various plates
were tried to see whether the recoil stream gives a develop-
able image, but the tests, though not exhaustive, failed. It
was found, however, that under suitable conditions, if the
radium B was allowed to decay on the photographic plate
in situ,a photographic image was obtained by the subsequent
emission of a rays from the radium C formed from it.

The apparatus consisted of a stout brass box, the internal
dimensions of which were 18 em. x 3°7 cm. x 0°9 em. (fig. 2).
The box contained the slit S and the photographic plate P.
It was closed at one end by means of a piece of plate glass A
ground on to a broad flange and at the other by a closed
ground joint B. To this was attached on the end of a brass
rod Ca plate which carried the active wire. The apparatus
was evacuated through the vertical tube D, which was pro-
vided with another joint by means of which it could be
detached from the rest of the apparatus when necessary. The
slit S was horizontal and was 0'l mm. wide. It was mounted

Deflexion of Recoil Stream from Radium A. 259

on a frame which could be adjusted in any position within
the box; the photographic plate was also mounted on an
adjustable carrier.

Vaan

|
|
M

The box was placed between the pole-pieces M of a large
electromagnet. The distance between the pole-pieces was
1 cm. and their face area 16 cm.x5 cm. A current of
11 amperes gave a uniform field of about 15,000 gauss over
the whole area, so that throughout their paths the @ rays
and recoil stream were in a uniformly strong magnetic field.
The whole of the glass apparatus near to the box was painted
black. The active source used consisted of radium A
collected on a platinum wire 0°4 mm. in diameter. ‘This
fitted into a groove cut in the plate C. The groove was
adjusted until it was horizontal and parallel to the slit S,
and reference marks were then placed in white paint on
the two parts of the joint B for simplicity in subsequent
adjustments.

In order to insure a high efficiency of recoil, the wire was
made active with radium A in an apparatus similar to that
described by Wertenstein*, in which the wire could be
removed from the emanation without drawing it through
mercury. From 50 to 100 millicuries of emanation were
used, and the wire was exposed to the emanation for six
minutes. At the end of the exposure the wire was dropped
for a few seconds into a glass tube maintained at 400° C. to
remove emanation. The wire was then fixed in position in
the groove cut in the plate C (fig. 2) by means of a speck of
soft wax. The stopper B was then replaced and rotated
until the wire was horizontal and the magnetic field was
excited. These operations usually occupied 2 to 3 minutes.
The box was rapidly evacuated, first by means of a Fleuss

* Théses presentées a la Faculté des Sciences, Paris, 1915, p. 90.

256 Mr. Walmsley and Dr. Makower on Magnetic

oil-pump and then by the use of charcoal cooled in liquid
air. Pressures were read ona McLeod gauge. The pressure
gradually decreased during the experiment, but was usually
of the order of 1/1000 mm. after the first three minutes.
Three minutes after introducing the stopper B the field
was reversed, and the photographic plate subjected to a
further radiation for 9 to 12 minutes. At the end of
the exposure the plate was removed and wrapped in black
paper for a period of about three hours to allow the active
deposit to decay. It was then developed in the usual way.
The plate used was a piece of Ilford Process plate 3°3 em.
x 08 cm., which was sufficiently large to prevent the film
becoming detached during development. ‘The activity of
the wire was also measured. Its y-ray activity at the
maximum usually corresponded to about one milligram of
radium C,

The best arrangement of wire, slit and plate is not easy
to foresee. From equation (3) it is evident that the accuracy

obtained for the ratio ? depends upon the accuracy with

which the distances d, and d, can be measured. These
quantities increase in the first place with the strength of the
magnetic field, which was made as intense as practicable
without over-heating the magnet. The accuracy with which

the ratio °* can be measured depends also on the width of

:
the images relative to their distanceapart. The width of the
image can be reduced to a certain extent by employing a
narrow slit and a narrow source, but the amount of active
matter which reached the plate is rapidly reduced by the
use of fine slits, and it is impossible to collect enough
radium A ona very fine wire. In practice we obtained very
satisfactory results by using as source a platinum wire of
diameter 0-4 mm. anda slit 0-1 mm. wide. The distance /,
was kept permanently 3:0 cm., the slit carrier being held in
position by metal stops. When /, was 1°84 cm. we obtained
a dense image easily measurable, but the maximum value of
d, obtained was only 2mm. With /,=2°85 cm. the image
was just measurable in the case of the recoil lines, and
d. increased to 3°7 mm.; but with 1,=3°5 em. using the
same amount of active material, the recoil image though just
visible was quite unmeasurable. In all cases the e-ray
images were remarkably sharp and distinct. This rapid
reduction in the intensity of the recoil images is rather
striking, and explains many failures in previous experiments.

In fig. 1, Pl. V., is given a reproduction of one of the

Deflexion of Recoil Stream from Radium A. 257

_ photographs taken which is not without interest. The
a-ray lines show structure, the image being denser at the
edges than at the centre. This is clearly brought out when
the line is examined under a low-power microscope. From
the manner in which the recoil images were obtained, it was
not to be expected that they would show any characteristic
structure. They were generally diffuse, and even in the
case of the strongest lines there was not a vast difference
between their density and that of the background of the
photographic plate, which was always slightly fogged by the
8 and y ray radiation from the active deposit. In one case
it was noticed that the centre corresponding to the recoil
lines was displaced relatively to that corresponding to the
a-ray lines, 7. e. one recoil line appeared to be nearer its
a-ray line than the other. This was apparently due to a
difference in the efficiency of recoil trom different parts of
the circumference of the wire which displaced the position
of maximum density within the lines. The ratio of the dis-
tance between the «-ray lines to that between the recoil lines
is, however, unaltered by this dissymmetry. A second fairly
strong line was often found near to the e-ray line obtained
after reversing the field. This was due to the « particles
from the radium C which had grown on the wire during the
experiment. In many cases this radium C line was stronger
than the recoil line, and occasionally a faint radium C line
was also visible near the initial radium A line. A direct
comparison of the velocities of the a particles from radium A
and radium C can be made by this method*, and thus the
empirical relation between the range and velocity of «& par-
ticles given by Geiger can be tested for this case f.

In measuring the photographic plates, the distances between
the centres of the lines were determined. Hach photographic
plate was mounted in a frame of black paper so as to expose
only the essential part of the photograph and mounted in a
clamp at a distance of about a metre from a travelling tele-
scope of low magnifying power. To illuminate the plate an
electric lamp was placed behind a sheet of opal glass and
connected in series with an adjustable resistance. It was
found that by adjusting the intensity of illumination of the
plate, the contrast between the clear portions of the plate
and the faint recoil lines could be best brought out at a
certain intensity of illumination. Although the recoil
lines were considerably broader than the fiducial line in the
eyepiece of the telescope, yet when this was set on the image
it was practically impossible to see the recoil line on account

* Tunstall and Makower, znfra, p. 259.
+ Geiger, Proc. Roy. Soc. A. Ixxxiii. p. 505 (1910),
Pil. Mag. S. 0. Volo2do No. 1710) feb. 1915. S

258 Magnetic Deflexion of Recoil Stream from Radium A.

of its faintness. This difficulty was overcome by arranging
that the fiducial line should cover only one half of the
length of the image. Some bias in making the settings
was inevitable, and it seemed impossible to increase the
accuracy of the readings beyond a certain limit by multi-
plying observations. This was due to a tendency to adjust
the fiducial line on marks and dust spots which though faint
occurred on the plates. We eliminated this as far as possible
by inverting the photographic plate at intervals, thereby
changing its appearance in this respect, and by using the
independent measurements of three observers. Throughout
we have assumed that the densest part of the image in the
case of the recoil lines was the centre. The magnification used
was controlled by the intensity of the recoil lines; the greater
their density, the greater was the magnification that could be
employed. Although the recoil lines were distinctly visible
to the naked eye on each plate taken, yet only about half the
plates could be used for making accurate measurements.

The measurements were made in sets of ten and the plate
vas always readjusted between each set, so as to use dif-
fereut parts of the divided scale of the travelling telescope.
About six sets were taken on each plate. The final values
of the ratio d,/d, for each plate given in the table are the
means of all the observations. The vernier readings were
taken to the nearest hundredth of a millimetre, but the
ineans were worked out to the next significant figure. The
results of the measurements on several plates are given in the
following table, an inspection of which shows that the radius
of curvature of the recoil stream in a magnetic field is
double that of the « rays with a high degree of accuracy.

TABLE.
| Pa, I 7 } |
| No. of Approximate L ds. | dy | Pa |
plate. field in gauss. | in cm. | incm. | a oe orl |
Rar aan or CE —
Prierea STO ae a5 1-84 0-4934 | 0-4959 |
ee | 1450085 2-85 04977 | 05017 |
Se eee | 15000 3:5 184 04981 | 0:5008
ATR aosed | 14600 35 | 285 04967 0:5006
Phpicats 15000 35 | 184 | 05006 | 0:5055 |
| Mean ... 0:5009 |

| | |

We are greatiy indebted to Mr. N. Tunstall for his
assistance in measuring the plates. With his valuable help
we have been able to increase the accuracy of the experi-

ment by using the results of three independent observers.

XXIX. The Velocity of the « Particles from Radium A.
By. N. Tunstatut and W. MaKxower *,

[Plate V. fig. 2.]
‘| De velocity of the « particles from radium C_ has

recently been very accurately determined by Ruther-
ford and Robinson+. The value given is 1°922 x 10° cm.
per second, and from this number the velocities of the
a particles from other radioactive substances are calculated
by Geiger’s formula

VP he te

which gives the relation between the velocity v of an
a particle and its range Rin air. The quantity & is a
constant.

A direct comparison of the velocity of the & particles from
radium A with that of the particles from radium C, does not
appear as yet to have been made; and as the apparatus
used to measure the magnetic deflexion of the recoil stream
from radium A{ was also suitable for this purpose, ex~-
periments were made to determine the ratio of the velocities
of the a particles from radium A and radium Cas accurately
as possible.

The apparatus used was the same as that used by
Walmsley and Makower§. The wire O (fig. 1, p. 253) was
made active by exposure for eight minutes to radium emana-
tion and quickly mounted in the apparatus, which was then
evacuated as rapidly as possible. The « rays passing
through the slit fell on the photographic plate, which
was subsequently developed. A field of about 14,000 gauss
was applied between the pole-pieces of the magnet. The
a rays from the radium A on the wire thus fell on the
photographic plate, producing an image of the slit. After
four minutes the field was reversed for fifteen minutes,
and a second image due to the « rays from radium A was
obtained. During the interval which had elapsed since the
exposure of the wire to the emanation, sufficient radium C
had grown on the wire to produce simultaneously on the
photographic plate a second and slightly less deflected
image. By this time all the radium A on the wire had
decayed, and the field was again reversed for ten minutes
so as to obtain an image from the « rays from radium C on

* Communicated by the Authors.
+ Rutherford and Robinson, Phil. Mag. Oct. 1914, p. 522.
t Walmsley and Makower, Phil. Mag. supra, p. 253.
§ Loe. ett.
$2

260 Velocity of the « Particles from Radium A.

the same side of the centre of the plate as the first image of
radium A, ‘The plate was then removed and developed, and
showed two pairs of lines due respectively to the & rays from
radium Aand radium ©. The appearance of the plate can be
seen from fig. 2(P1. V.). The plates were examined by means
of atravelling telescope, and the distances, d, and dc, between
the two lines due to radium A and those due to radium C were
measured in the way described for determining the deflexion
of the recoil stream*. From these measurements the ratio
of the radius of curvature of the « rays from radium A to
that for the rays from radium C could easily be calculated.
The results obtained are given in Table I., in which the
numbers recorded for each plate represent the mean of
thirty observations. In each experiment the distance from
the wire to the slit was 3°6 cm. In the case of plate 1
the distance from the slit to the photographic plate was
3°60 cm., and for each of the other plates this distance
was 2°85 cm. The small variations in the deflexions on
plates 2, 3, and 4 were due to differences in the strengths of
the magnetic fields used.

Taking the velocity of the « particles from radium C as
1-922 x 10° centimetres per second, this gives for the velocity
of the « rays from radium A the value 1°690 x 10° centimetres
per second, which is in close agreement with the value
1°693 x 10° centimetres per second calculated by Geiger’s
formula from the known ranges of the « particles.

* Loe. cit.

ee2enen!

XXX. Frictional Electricity on Insulators and Metals. By
W. Morris Jonzs, B.Sc., Research Student of the University
of Wales *.

ia a recent paper Morris Owen + describes a series of

experiments, in which he obtained absolute measure-
ments of the charges produced on solid bodies by known
amounts of frictional work. In particular, he rubbed
ebonite and glass with slate and copper under various con-
ditions as to pressure, and found that with a sufficient
amount of frictional work the charges reached a constant
maximum, and that this maximum was independent of the
pressure applied during the rubbing, but was reached with a
smaller quantity of work the greater the pressure.

As there seems to be but little known about frictional
electricity, the investigations described below were under-
taken with the object of discovering new facts which might
be of assistance in forming a quantitative theory. In these
experiments, after the method adopted by Owen, measure-
ments were made of the charge produced on a surface by
friction and of the frictional work spent in generating it.
Apart from any theory of the phenomenon, this seems
the most convenient way of stating the results of the
experiments.

Apparatus.

The apparatus employed was essentially the same as that
used by Owen, but there were several improvements. The
rubbing apparatus consisted of a slate wheel round the rim
of which rubbers of various soft materials were placed. The
wheel, of moderate moment of inertia and °8 cm. thick,
turned in bearings fixed in a wooden framework, and could
be set into rotation by a known weight falling through a
measured height. The weight was attached to a cord passing
over a light pulley in the ceiling and wound round a wooden
drum fixed on the wheel. A vertical pole graduated wp-
wards in decimetres gave the distance through which the
weight fell. In later stages of the work the wheel was
driven by an electric motor of variable speed.

The specimens rubbed were usually disks about 1:2 em.
in diameter, and were of various insulating and metallic
materials. They were mounted with an insulating cement on

* Communicated by Prof. E. Taylor Jones.
t Phil. Mae. xvii. p. 457 (1909).

262 Mr. W. Morris Jones: Frictional

small T-shaped strips of ebonite, which could be firmly fixed
to an ebonite holder, fig. 1. The holder was made of two
pieces of ebonite connected across by a block of well insu-
lating amber, and could be fitted into a bayonet-socket

1071.

®
Sas ANE eR SN

ot ORK
—~ — — =\— — —-WoopEN FAAMEWDI i

‘
Lever AaM

j
ay
tg

Esomire--- ---—-] \

forming one arm of a lever bent at right angles. The lever
turned about an axle fixed in the wooden framework sup-
porting the wheel, so that, when the holder was in position,
this arm of the lever was vertical with the specimen pressing
horizontally against the rubber, the thrust between the
specimen and the rubber being produced by a lead weight
placed in any suitable position on the other lever-arm. A
rub was effected by releasing the wheel, which was then set
rotating by the falling weight, the specimen being separated
from the rubber just before the wheel came to rest. Know-
ing the time of fall of the weight, the kinetic energy lost on
impact of the weight with the floor could be taken into
account in the calculation giving the work spent against the
friction of the specimen on the rubber. The amount of
energy lost at the bearings of the wheel was small and could
be neglected.

Immediately after a specimen was rubbed, the holder
supporting it was quickly withdrawn from the socket and

Electricity on Insulators and Metals. 263

suspended by a string over a deep metal jar into which the
specimen could be lowered. The jar was placed inside, and
insulated by quartz supports from, a larger earth-connected
jar, which served to increase the capacity and to shield the
inner jar from outside influences. The inner jar was connected
by a fine wire passing through sulphur plugs in the outer, to
the upper plate of a parallel plate condenser, andalso to one
of the terminals of a quadrant-electrometer, the lower plate
of the condenser and the other terminal of the electrometer
being permanently earthed. The deflexions of the needle
were observed by a telescope and illuminated scale. The zero
of the scale was obtained by earthing the terminal of the
electrometer by means of a suitable mercury-cup key operated
by a string. The whole measuring system was enclosed in a
wooden box with a window for the electrometer, the inside of
the box being covered with tinfoil and earthed. The box
served to- shield the whole apparatus from external influences
and by means of drying agents inside it kept the apparatus
perfectly dry. The needle of the electrometer was suspended
by a platinum wire ‘01 mm. in diameter, and throughout
the experiments was maintained at a potential of 100 volts
by a small battery of Weston cells.

In measuring the charge on a specimen after a rub, the
zero of the electrometer was first observed, the earth con-
nexion removed, and the specimen lowered into the jar and
the deflexion noted.

The insulation of the apparatus was frequently tested, and
the holder cleaned with a warm cloth, so that the specimen
should lose no appreciable charge during its removal from
the lever to the jar. Provided the holder were clean and
dry, no appreciable change in deflexion was noticed, even
when the charged specimen was replaced in the socket and
then again removed and lowered into the jar. (Good insula-
tion of the specimen was thus secured by the use of amber
as a connecting piece in the holder, and the upper piece of
ebonite could be freely handled when being placed into or
withdrawn from the socket without any leakage of charge
from the specimen.

Before each rub the specimen, holder, and rubber were
completely discharged by exposure to radiations from radium.

Preliminary experiments showed that it was necessary to
reduce considerably the seusitiveness of the electrometer
owing to the large values of the charges produced on some
of the specimens. This was done by shortening the sus-
pending wire, so that the needle could still be maintained
throughout the experiments at-a potential much higher than

264 Mr. W. Morris Jones: Frictional

that of the insulated quadrants, without unduly increasing
the defiexion.

The scale was calibrated by the difference of potential due
to a known current passing through a known variable re-
sistance. ‘The capacity of the measuring system was deter-
mined by the method of mixtures, using a standard condenser.
Knowing the capacity of the electrometer and accessories,
and also knowing the potential of the insulated quadrants
corresponding to any deflexion of the needle, the charge
producing that deflexion was obtained.

The Insulators.

In rubbing the specimens, care was taken to bring the
middle part of the specimen evenly into contact with the rim
of the rubbing wheel. If this was not done, the deflexion
obtained with insulators was always too small owing to the
smaller area of contact. This precaution also prevented the
sliding contact from breaking into a series of impacts. It
was also found necessary to give the insulators a rest after
each rub, for if an insulator was rubbed a number of times
in succession the deflexions rose to the maximuin, though the
work done was the same for each rub and less than that
required to give the maximum for a first rub. Owen also
observed this effect, and found that he had to give his
specimens a rest of about three hours after each rub with
slate or copper. In the experiments described here, the rest
needed was far shorter, this being apparently due to the fact
that the rubbers used were softer. To get the rubber as soft
as possible a few layers of the material were wound round
the wheel.

The insulators rubbed were fused quartz disks, quartz
crystal, glass, polished ebonite, amber, sealing-wax, and
crystals of fluorspar, Iceland spar, and heavy spar. ‘The
rubbers were bands of flannel, silk, and chamois leather
wound round the rim of the wheel, the axle of which was
earthed, and the normal thrust between specimens and
rubbers was in all cases 178 grams weight.

The following tables give the charges in electrostatic
units produced by various quantities of frictional work re-
presented in joules, when four typical specimens were rubbed
with flannel, silk, and leather.

Electricity on Insulators and Metals. 265

Tas.eE I.
Fused Quartz.

Positive electricity on the quartz when rubbed with
flannel, silk, and chamois leather.

Diameter of specimen 1°2 cm.

Frictional work Charge in E.8.U. when rubbed with
in joules. Flanuel. | Silk. | Teather.
"4885 8827 4-094 2°518
‘9771 9:447 5°038 3'149
1:4655 10:077 5°983 3779
1:9540 10-392 6928 4-094
24425 10°581 7558 4409
3°4195 10:800 8:187 5°393
4°3965 10:917 8817 5°668
5°8620 10-862 9°132 6:298
78160 10°916 9°447 6:928
9°7710 10:958 9°636 7118
12°2125 10:946 9°747 1492
14-6550 10°950 9°761 7557

‘TABLE II.
Glass.

Positive electricity on glass when rubbed with flannel,
-silk, and chamois leather.

Diameter of specimen 1°2 em.

Frictional work Charge in H.8.U. when rubbed with
in joules. Flannel. | Silk. | ‘Leather.
"4885 2-519) | 1:260 | 315
“O771 3°464 | 3°169 | "629
14655 4-094 ATOMS) i 819
19540 | 4°724 6298 | “945
2°4425 5241 7085 1:070
3°4195 6241 T7557 | 1-260
4-3965 6613 S075 STS |
58620 6928 9-888 | 1-732
78160 7873 10°328 | 1-886
9°7710 8819 10°643 2-078
12:2125 8820 10°738 2°204
14:6550 8818 10°725 2:212

266 Mr. W. Morris Jones: Frictional

TaBLeE ITT.
Ebonite.

Negative electricity on ebonite when rubbed with
flannel, silk, and chamois leather.

Diameter of specimen 1°2 cm.

| Frictional work | Charge in E.S.U. when rubbed with

in joules. | Flannel. Silk. Leather.

4885 5983 2-834 1-575

‘9771 7:358 4-409 2519

1°4655 8:187 5°668 3401

1:9540 | 8:502 6:298 3621

| 24495 | 8-943 6-613 3968

| 3°4195 | 9°762 7090 4-031

| 4°3965 | 10:077 7-206 4-220

| 58620 | 10549 7558 4:245

| 78160 10°833 TT 4:219

| 9°7710 | 10-991 7841 4-220

12°2125 11:085 7868 4346

146550 11:022 7872 4-250

Tarim:
Amber.

Negative electricity on amber when rubbed with
flannel, silk, and chamois leather.

Diameter of specimen 1:2 cm.

Frictional work Change in E.S.U. when rubbed with |
in joules. Flannel. | Silk. Leather. |
ee Sap e ee SUS BPS |
‘4885 6928 3:905 9204 4
9771 8439 4-724 2-992
1-4655 8-502 5:479 3°464
1:9540 8:817 5-983 3179
2°4425 9°132 6:298 3°887
3°4195 9:321 6°487 4-094
4°3965 9:193 6613 4-283
5°8620 9:257 6°718 4314
78160 9:233 6-739 4-345
9°7710 9:238 6°723 4°392
12°2125 9:269 6°738 4-440
14°6550 9:269 6°770 “409

Electricity on Insulators and Metals. 267

In figs. 2 and 3 the charges produced by various amounts
of work are plotted as curves. They show, as observed by

Quartz & FlrawveLc cy

y Quaatiz & Sith t+

D / Ly |

! : o, Quaate & LEATHER.) @ ___—o
| A De aa
tf =
/
g
/

ee ere
we
es

C Harce IN ELECTROSTATIC UNITS
aS
SS
~
SS ;
SS
SS

(3) 50 100 150

Fricrionkt WORK Iu IAILLIONS OF EGS

(i ———$————= a
| Esowrre & Flawweni—)
© ’ :

GLass & SILK A+)

AMBER & FlavveLc—:

, ‘ : a :
mie vA GLASS & FLANWELW+)

o {
_ Be EoowiTe ¢ Siex. i=) so

AMGER & Sinn. im 5
|

Ghass & LEaTHER. +}

CHARGE iN ELECTROSTATIC UMTS

oO 50 109 450

CRICTIONSL YYORK IN MILLIONS OF EROS

Péclet * and Owen, that with a sufficient amount of w ork,
the charges reach a constant maximum. Results have been
* Ann, de Chimie et de Physique, lvii. p. 3387 (1834).
+ Loc. cit.

268 Mr. W. Morris Jones: Frictional

obtained for the other materials rubbed and have been plotted
as curves, which show a similar form.

Table V. gives the values in electrostatic units of the
maximum charges produced on the insulators by friction
with the rubbers. It is interesting to note that the trans-
parent insulators were positively charged when rubbed with
the three materials, and that the opaque substances all
became negatively charged.

The maximum charges produced by friction with flannel,
silk, and chamois leather are much greater than those ob-
tained by Owen, whose specimens, however, were rubbed
with hard materials, viz. slate and copper. This is probably
due to the different nature and hardness of the rubbing
surfaces. The soft rubbers used in the present experiments
allowed a larger area of the specimen to come into contact
with the rubber. It is also possible that a hard rubbing
surface cuts up or wears away the surface of the specimen.

Metallic Specimens.

In the experiments with the metals, the rubber had to be
a good insulator and thoroughly dry, otherwise the charge
generated leaked away through the rubber to earth as fast
as it was produced. For this reason the metallic specimens
were rubbed only with silk bands. The insulation of the
silk was frequently tested by bringing the cap of a charged
electroscope into contact with the band when on the wheel,
and observing the rate at which the leaves collapsed. Owing
to the hygroscopic nature of the silk, the material had to be
kept warm and dry by placing a stove underneath the
wheel. ‘The insulation was further improved by having a
band of indiarubber between the silk and the slate.

In rubbing the metals, reversals of the sign of electrifi-
cation were frequent, and appeared to be due to several

Electricity on Insulators and Metals. 1 269

causes. If the surface of the metal had become tarnished
by oxide, the charge generated was low and in some cases
reversed in sign. This reversal was noticed in the case of
zinc and iron. If the surfaces of these metals were rubbed
when tarnished, the charge was positive for moderate amounts
of rubbing and negative for greater amounts, whereas the
surfaces when previously cleaned with fine ‘emery- paper
always showed a negative charge. Apparently the rubbing
gradually wore off the film of oxide and ultimately produced
a charge similar to that obtained on the pure metal. A
reversal of the sign of electrification also appeared when the
silk rubber had been used for some time, and was probably
due to the silk having become smeared with oxide. A fresh
silk band always gave an electrification of the proper sign
when the cleaned specimens were rubbed.

When the metals were all cleaned with very fine emery-
paper, all the specimens except thallium, lead, and bismuth
(which have the highest atomic weight) gave a negative
charge at the first rub. Continued rubbing, however, pro-
duced a polish on the metals, and then it was found that the
negative charges on aluminium, iron, copper, zinc, and
antimony became less and less and changed to positive,
though the other metals, on continued rubbing, did not show
this effect.

Owing to these reversals and to the difficulty of preventing
a roughened surface becoming smoothed out and polished by
the friction, the specimens were, as far as possible, finally
tested all with their surfaces at about the same degree of
polish and with fresh silk bands.

The metals rubbed were lithium, boron, aluminium, cal-
clum, iron, copper, zinc, silver, tin, antimony, platinum, gold,
thallium, lead, and bismuth. The metals which could be
obtained in the form of thin disks 1:2 cm. in diameter were
soldered on brass cylinders 3 mm. thick, and second specimens
were in some cases cemented on ebonite cylinders of the
same thickness. Metals such as iron were made up to the
same dimensions as the brass cylinders. All the above
metals were rubbed for a maximum charge, and measure-
ments of the charges produced by various amounts of work
were also taken for a few typical specimens. In the case of
the metals that readily tarnish in air,a few rubs were rapidly
given immediately : after cleaning their surfaces. The boron,
supplied amorphous, had to be made up into a thick paste
and allowed to set hard before it could be tested. Some
of the specimens employed were of the ordinary commercial
grade of purity, those of the rarer metals being refined.

270 Mr. W. Morris Jones: frictional

Fig. 4 shows the curves for some of the metals. In shape
they are, as might be expected, very similar to those of the
insulators. The maximum charge is, however, greater on
some of the metals than on any of the insulators.

Fig. 4.

Tin ow Brass CYLINDER 3am Lowe

Stak & Cooper cH

8 AL

Siw & SicvEeRi-y

2 2
=
=>
)
S
S
S
3 , Sick & agin Disc owEBONITE (=
“ |
Kt
Bll |
= a
= ait a] |
w i i H
= |
| i
wu | |
rey | }
= I {
5 | |
} |
|
1 |
| - }
! =6
| i Fresxio
O 50 100 7sc

With the metals, capacity plays an important part in the
value of the maximum charge generated. Three tin disks
1:2 cm. in diameter were mounted, one on ebonite and the
other two on brass cylinders 3 mm. and 1 ecm. thick. On
being rubbed, the specimen mounted on the longer metal
cylinder gave the greatest maximum charge, and the disk on
ebonite gave the least, while the specimen on the short metal
cylinder gave a charge of intermediate value. The three
results in this case are shown in fig. 4. Apparently, there-
fore, the charge generated on a metallic specimen by a
given amount of work increases with the capacity of the
specimen. |

The following classification of the metals according to the
sign of the charge, shows that the metals examined can be
divided into three groups :—

CHARGE IN ELECTROSTATI2 UNITS.

Electrieity on Insulators and Metals. mtn

Au\ Negatively charged
et | when rubbed

Sn { with silk. Sb) Positively charged if
Ag! Zn L polished, negatively
Cu > charged if rough,
Fe | when rubbed with

Bi ) Positively charged
Ps | when rubbed Al J silk.
Tl J with silk.

The sign and magnitude of the charge produced by the
friction of the metals do not appear to be a well-defined
physical property of the metals. A possible relation between
the maximum charges for different elements would thus be
difficult to obtain, since, except with metals of high atomic
weight, the sign of the charge seems to be largely deter-
mined by the physical state of the surface, and not merely
by the nature of the materialrubbed. Iu fig. 5 the maximum

Fig. 5.

Atomic WEicHT.

charge obtained by the friction of the surfaces of the metal,
polished when possible, is plotted against the atomic weight.
An examination of the curve shows that there appears to be
here some evidence of a periodicity in property with increase
of atomic weight, and that, like some other physical pro-
perties of the elements, this property possibly conforms to
the Periodic Law.

212 Mr. W. Morris Jones: Frictional
Additional Experiments.

Some of the insulators and metals were rubbed in strong
magnetic and electric fields, but no change in the maximum
charge produced was noticed. The influence of temperature
on the maximum charge was also investigated, and in this
case algo no effect was observed. In some experiments the
rubbing wheel was driven by a motor, and the influence of
pressure between rubber and specimen and that of velocity
of the rubbing surface were investigated. From the results
obtained, it appears that the maximum charge is independent
of the pressure and of the velocity*.

General Conclusions.

It has often been supposed that frictional electricity is of
the nature of contact electricity, the frictional work being
expended in bringing the surfaces into closer contactt. It
should, however, be noted that in Owen’s experiments, mere
contact without rubbing did not in any case produce the
slightest evidence of charge on his specimens, and the same
result was found in the course of the present experiments.
Frictional electricity appears, therefore, to be an effect of
a different order from that of contact electricity, and it is
worth while considering whether the facts cannot be accounted
for on some other hypothesis.

Let us assume that the rubbing friction has the effect of
removing electrons from either the rubber or the specimen
at a rate proportional to the rate of working. Then if e is
the total quantity of electricity liberated by a quantity of
work w, we may write

de dw
7 sa aaa ° (1)
where a is a factor depending on the nature of the materials
and also upon the normal thrust between the bodies; a is a
constant during any one experiment.

We now assume that during a rub, leakage of electricity
takes place at a rate proportional to the total charge present,
(, and to the rate at which fresh surface of the rubber is
coming into contact with the surface of the specimen, the
total leakage being e’. The velocity of the rubber is pro-
portional to the rate of working, so that this law of leakage

* This result was stated by Péclet.
+ Helmholtz, Wissenschaftliche Abhandlungen, Erster Band, p. 860.

Electricity on Insulators and Metals. 973

may be written
!
ab =s/p : APs . Q). . ake 3 Cea) (2)

where 6 is also a constant. ‘The total charge existing on the
specimen will now be

Pe ere iM a itil ak) (SY

From these equations we find

dQ
a—bQ

which, on being integrated, gives

=) LE EG eT are

=dw,

the constant of integration being determined by the fact
that Q and w vanish together.
On this theory the frictional electricity reaches a constant

maximum value, ie when the generation of charge is com-
)

pensated by the leakage.

The constants, a/b and 6, of (4) can be determined for any
particular case from two points on tle experimental curve,
and ( can hence be calculated for any value of w. Asa
rule, the theoretical curves so obtained show fair agreement
with the experimental curves, the chief difference being that
the calculated curve is generally rather steeper in the middle
part of the curve. Curves calculated in this way are shown
in broken lines in fig. 2, for quartz when rubbed by flannel,
silk, and leather.

In order to account for the result that the maximum
charge is independent of the pressure, it is necessary to
suppose that a and 6 contain as factors the same function of
the pressure. In the case of the metallic specimens, where
Q=CV, if © is the capacity and V the potential of the
specimen, the leakage in equation (2) should be assumed to
be proportional to V; in other words, the coefficient 6 is
inversely proportional to the capacity. It follows by equa-
tion (4) that, in the case of two specimens of the same
material but of different capacities, the maximum charge
should be greater for the specimen of greater capacity, and
that the slope of the rising portion of the curve should be
steeper for this specimen. Less work should be necessary,
therefore, to produce any given charge in the case of the

Phil. Mag. 8. 6. Vol. 29. No. 170. Feb. 1915. T

274 Lord Rayleigh on the

specimen of greater capacity. These results agree with the
facts as indicated by the curves of fig. 4.

It seems probable, therefore, that a satisfactory theory of
frictional electricity can be formed on the lines indicated,
though in the form given above the theory is doubtless far
from perfect.

In conclusion, I desire to express my best thanks to
Professor E. Taylor Jones for much valuable help and advice
during the course of the work.

Physics Laboratory,

University College of N. Wales, Bangor,
October 1914.

XXXI. On the Widening of Spectrum Lines.
By Lord Rayueies, O.M., PL RS.*

N ODERN improvements in optical methods lend ad-

ditional interest to an examination of the causes
which interfere with the absolute homogeneity of spectrum
lines. So far as we know these may be considered under
five heads, and it appears probable that the list is ex-
haustive :—

(i.) The translatory motion of the radiating particles in
the line of sight, operating in accordance with Doppler’s
principle.

Gi.) A possible effect of the rotation of the particles.

Gi.) Disturbance depending on collision with other par-
ticles either of the same or of another kind.

Gv.) Gradual dying down of the luminous vibrations as
energy is radiated away.

(v.) Complications arising from the multiplicity of sources
in the line of sight. Thus if the light from a flame be
observed through a similar one, the increase of illumination
near the centre of the spectrum line is not so great as towards
the edges, in accordance with the principles laid down by
Stewart and Kirchhoff ; and the line is effectively widened. ©
It will be seen that this cause of widening cannot act alone,
but merely aggravates the effect of other causes.

here is reason to think that in many cases, especially
when vapours in a highly rarefied condition are excited
electrically, the first cause is the more important. It was
first considered by Lippich t and somewhat later inde-
pendently by myself {. Subsequently, in reply to Ebert,

* Communicated by the Author.

+ Pogg. Ann. t. cxxxix. p. 465 (1870).

} ‘Nature,’ vol. vill. p. 474 (1878); Scientific Papers, vol. i. p. 183.

Widening of Spectrum Lines. 279

who claimed to have discovered that the high interference
actually observed was inconsistent with Doppler’s principle
and the theory of gases, I gave a more complete calculation *
taking into account the variable velocity of the molecules
as defined by Maxwell’s law, from which it appeared
that there was really no disagreement with observation.
Michelson compared these theoretical results with those of
his important observations upon light from vacuum-tubes
and found an agreement which was thought sufficient,
although there remained some points of uncertainty.

The same ground was traversed by Schénrock 7, who made
the notable remark that while the agreement was good for
the monatomic gases it failed for diatomic hydrogen, oxygen,
and nitrogen; and he put forward the suggestion that in
these cases the chemical atom, rather than the usual molecule,
was to be regarded as the carrier of the emission-centres.
By this substitution, entailing an increase of velocity in the
ratio ,/2:1, the agreement was much improved.

While I do not doubt that Schénrock’s comparison is sub-
stantially correct, I think that his presentation of the theory
is confused and unnecessarily complicated by the introduction
(in two senses) of the “* width of the spectrum line,” a quantity
not usually susceptible of direct observation. Unless I mis-
understand, what he calls the observed width is a quantity
not itself observed at all but deduced from the visibility of
interference bands by arguments which already assume
Doppler’s principie and the theory of gases. I do not see
what is gained by introducing this quantity. Given the
nature of the radiating gas and its temperature, we can
calculate from known data the distribution of light in the
bands corresponding to any given retardation, “and from
photometric experience we can “form a pretty g good judgment
as to the maximum retardation at which they ‘should still be
visible. This theoretical result can then be compared witha
purely experimental one, and an agreement will confirm
the principles on which the calculation was founded. I
think it desirable to include here a sketch of this treatment
of the question on the lines followed in 1889, but with a few
slight changes of notation.

The phenomenon of interference in its simplest form occurs
when two equal trains of waves are superposed, both trains
having the same frequency and one being retarded relatively

* “On the limits to interference when light is radiated from moving
molecules,’ Phil. Mag. vol. xxvii. p. 298 (1889); Scientific Papers,
vol. iil. p. 258.

t Ann. der Physik, xx. p. 995 (1906).

276 Lord Rayleigh on the

to the other by a linear retardation X *. ‘Then if A denote the
wave-length, the aggregate may be represented
cos nt + cos (nt—27X/X)=2 cos (wX]/r). cos (nt —wX/r). (1)
The intensity is given by i

I=4 cos? (wX/A)=2{1+ cos(Q7X]rA)}. . . (2)
If we regard X as gradually increasing from zero, I is
periodic, the maxima (4) occurring when X is a multiple of
» and the minima (0) when X is an odd multiple of $d. If
bands are visible corresponding to various values of X, the
darkest places are absolutely devoid of light, and this remains.
true however great X may be, that is however high the
order of interference.

The above conclusion requires that the light (duplicated by
reflexion or otherwise) should have an absolutely definite
frequency, ?.¢., should be absolutely homogeneous. Such light
is not at our disposal; and a defect of homogeneity will
usually entail a limit to interference, as X increases. We
are now to consider the particular defect arising in accord-
ance with Doppler’s principle from the motion of the radiating
particles in the line of sight. Maxwell showed that for
gases in temperature equilibrium the number of molecules
whose velocities resolved in three rectangular directions lie
within the range d&dndf must be proportional to

ene aT ae dnd.
If & be the direction of the line of sight, the component
velocities 7, € are without influence in the present problem.
All that we require to know is that the number of molecules
for which the component & lies between & and &+dé is
proportional to

en de. EU

The relation of 6 to the mean (resultant) velocity v is

)

MMM
JB) S
It was in terms of v that my (1889) results were expressed,
but it was pointed out that v needs to be distinguished from

the velocity of mean square with which the pressure is more
directly connected. If this be called 1’,

=a / (52): og)
so that vo Gee MP:

* In the paper of 1889 the retardation was denoted by 24.

Widening of Spectrum Tines. 277

Again, the relation between the original wave-length «A and
the actua] wave-length X, as disturbed by the motion, is

I I NON ay

= tee LO RSM rant g Lek a A)
¢ denoting the velocity of light. The intensity of the light
in the interference bands, so far as dependent upon the
molecules moving with velocity &, is by (2)

d=2{ 14 cos “(145 “)he pede, He hk (Co)

and this is now to be integrated with respect to & between
the limits +<. The bracket in (8) is

il Male ee are oe sin Tes
MAR | Nic Ne ING.

The third term, being uneven in &, contributes nothing.
The remaining integrals are included in the well-known

formula
+n
5 TT 12/92
{ e cos Qrar)\de= a Cree ts

|G)

I= Ye [ 1+ cos Ex p(— Zan) 1@)

The intensity I, at the darkest part of the bands is found by
making X an odd muitiple of 4, and I, the maximum
brightness by making X a multiple of 2.

“Ehats Ex i HO \ pa eal

P 6? BN?) 1,+4,
where V denotes the “ visibility ” according to Michelson’s
definition. Wquation (10) is the result arrived at in my
former paper, and 8 can be expressed in terms of either the
mean velocity v, or preferably of the velocity of mean
square v *.

The next question is what is the smallest value of V for
which the bands are recognizable. Relying on photometric
experience, I estimated that a relative difference of 5 per
cent. between I, and I, would be about the limit in the case
of high interference bands, and I took V=:025. Shortly
afterwards T I made special experiments upon bands well

Thus

SVL lel «ce NT)

* See also Proc. Roy. Soe. vol. Ixxvi, A. p. 440 (1905); Scientific
Papers, vol. v. p. 261.
Tt Phil. Mag. vol. xxvii. p. 484 (1889); Scientific Papers, vol. iil.

p: 277.

278 Lord Rayleigh on the

under control, obtained by means of double refraction, and I
found that in this very favourable case the bands were still
just distinctly seen when the relative difference between I,
and I, was reduced to 4 per cent. It would seem then
that the estimate V=‘025 can hardly be improved upon.
On this basis (10) gives in terms ot v

ay
= =e o/(loget0) = "6907, . - » (1

as before. In terms of v’ by (6)

Xx 4/30 € Ae
BE pee o,40) =-(49—.. 2 eee
A a /2.v' v/ (loge 40) eS (29

As an example of (12), let us apply it to hydrogen
molecules at 0° C. Here v' = 183910? cm./sec.*, and
esac l0. i bhus

X/A = 1:222 x 10°... ee

This is for the hydrogen molecule. For the hydrogen
atom (13) must be divided by 4/2. Thus for absolute
temperature T and for radiating centres whose mass is
m times that of the hydrogen atom, we have

XK _ 1-222 ./ (273) x ate = 1-427 x 108 \/ (tr):
A ee T fh

. ie ee
In Buisson and Fabry’s corresponding formula, which
appears to be derived from Schonrock, 1:427 is replaced
by the appreciably different number 1°22.

The above value of X is the retardation corresponding to
the limit of visibility, taken to be represented by V=:025.
Ia Schonrock’s calculation the retardation X,, corresponding
to V='5, is considered. In (12), ,/(log,40) would then
be replaced by ,/(log, 2), and instead of (14) we should

have
eee Ae ee NAG
A = 6186 x 10 7) ee (15)

But I do not understand how V="5 could be recognized in
practice with any precision.

Although it is not needed in connexion with high
interference, we can of course calculate the width of a

* It seems to be often forgotten that the first published calculation of
molecular velocities was that of Joule (Manchester Memoirs, Oct. 1848 ;
Phil. Mag. ser. 4, vol. xiv. p. 211.)

Widening of Spectrum Lines. 279

spectrum line according to any conventional definition.
Mathematically speaking, the width is infinite; but if we
disregard the outer parts where the intensity is less than
one-half the maximum, the limiting value ot & by (3) is
given by

Seria MOC MENA anh 8) AUS ERG)

and the corresponding value of » by

Nom cnn (loee2)
TT aC al (17)

Thus, if 6A denote the half-width of the line according to
the above definiticn,

ONWAGOUS 1) seu ei e Fic Je
Rm ean = 837 x10 aoe i Gs)
T denoting absolute temperature and m the mass of the
particles in terms of that of the bydrogen atom, in
agreement with Schonrock.

In the application to particular cases the question at once
arises as to what we are to understand by T and m. In
dealing with a flame it is natural to take the temperature of
the flame as ordinarily understood, but when we pass to
the rare vapour of a vacuum-tube electrically excited the
matter is not so simple. Michelson assumed from the
beginning that the temperature with which we are con-
cerned is that of the tube itself or not much higher. This
view is amply confirmed by the beautiful experiments of
Buisson and Fabry *, who observed the limit of inter-
ference when tubes containing helium, neon, and krypton
were cooled in liquid air. Under these conditions bands
which had already disappeared at room temperature again
became distinct, and the ratios of maximum retardations
in the. two cases (1:66, 1°60, 1°58) were not much less than
the theoretical 1:73 calculated on the supposition that the
temperature of the gas is that of the tube. The highest
value of X/A, in their notation N, hitherto observed is
950,000, obtained from krypton in liquid air. With all
three gases the agreement at room temperature between
the observed and calculated values of N is extremely good,
but as already remarked their theoretical numbers are a
little lower than mine (14). We may say not only that
the observed effects are accounted for almost completely
by Doppler’s principle and the theory of gases, but that

* Journ, de Physique, t. i. p. 442 (1912).

280 Lord Rayleigh on the

the temperature of the emitting gas is not much higher than
that of the containing tube.

As regards m, no question arises for the inert monatomic
gases. In the case of hydrogen Buisson and Fabry follow
Schénreck in taking the atom rather than the molecule as
the moving source, so that m=1; and further they find that
this value suits not only the lines of the first spectrum of
hydrogen but equally those of the second spectrum whose
origin has sometimes been attributed to impurities or
aggregations.

In the case of sodium, employed in a vacuum-tube,
Schénrock found a fair agreement with the observations
of Michelson, on the assumption that the atom is in
guestion. It may be worth while to make an estimate for
the D lines from soda in a Bunsen flame. Here m=23,
and we may perhaps take T at 2500. These data give
in (14) as the maximum number of bands

Rg =| 157,000.

The number of bands actually seen is very dependent
upon the amount of soda present. By reducing this Fizeau
was able to count 50,000 bands, and it would seem that this
number cannot be much increased*, so that observation
falls very distinctly behind calculationt. With a large
supply of soda the number of bands may drop to two or
three thousand, or even further.

The second of the possible causes of loss of homogeneity
enumerated above, viz. rotation of the emitting centres, was
briefly discussed many years ago in a letter to Michelsontf,
where it appeared that according to the views then widely
held this cause should be more potent than (i.). The trans-
verse vibrations emitted from a luminous source cannot be
uniform in all directions, and the effect perceived in a fixed

* “ Tnterference Bands and their Applications,” Nature, vol. xlviii.
p. 212 (1893) ; Scientific Papers, vol. iv. p.59. The parallel plate was a
layer of water superposed upon mercury. An enhanced illumination may
be obtained by substituting nitro-benzol for water, and the reflexions from
the mercury and oil may be balanced by staining the latter with aniline
blue. But a thin layer of nitro-benzol takes a surprisingly long time to
become level.

+ Smithells (Phil. Mag. xxxvil. p. 245, 1894) argues with much
force that the actually operative parts of the flame may be at a much
higher temperature (if the word may be admitted) than is usually
supposed, but it would need an almost impossible allowance to meet
the discrepancy. The chemical questions involved are very obscure.
The coloration with soda appears to require the presence of oxygen
(Mitcherlich, Smithells).

t Phil. Mag. vol. xxxiv. p. 407 (1892) ; Scientific Papers, vol. iv. p. 15.

Widening of Spectrum Lines. 281

direction from a rotating source cannot in general be simple
harmonic. In illustration it may suffice to mention the case
of a bell vibrating in four segments and rotating about the
axis of symmetry. The sound received by a stationary
observer is intermittent and therefore not homogeneous.
On the principle of equipartition of energy between trans-
latory and rotatory motions, and from the circumstance that
the dimensions of molecules are much less than optical wave-
lengths, it followed that the loss of homogeneity from (ii.)
was much greater than from (G.). JI had in view diatomic
molecules—for at that time mercury vapour was the only
known exception ; and the specific heats at ordinary tempe-
ratures showed that two of the possible three rotations
actually occurred in accordance with equipartition of energy.
It is now abundantly clear that the widening of spectrum
lines at present under consideration does not in fact occur ;
and the difficulty that might be felt is largely met when we
accept Schonrock’s supposition that the radiating centres are
in all cases monatomic. Still there are questions remaining
behind. Do the atoms rotate, and if not why not? I
suppose that the quantum theory would help here, but it
may be noticed that the question is not merely of acquiring
rotation. A permanent rotation, not susceptible of alteration,
should apparently make itself felt. These are problems re-
lating to the constitution of the atom and the nature of
radiation, which I do not venture further to touch upon.
The third cause of widening is-the disturbance of free
vibration due to encounters with other bodies. That some-
thing of this kind is to be expected has long been recognized,
and it would seem that the widening of the D lines when
more than a very little soda is present in a Bunsen flame can
hardly be accounted for otherwise. The simplest supposition
open to us is that an entirely fresh start is made at each
collision, so that we have to deal with a series of regular
vibrations limited at both ends. The problem thus arising
has been treated by Godfrey * and by Schonrock ft. The
Fourier analysis of the limited train of waves of length +
gives for the intensity of various parts of the spectrum ‘line

A aeaSuINE TRIE et mites cuir) «| ayy cap eta rea

where & is the reciprocal of the wave-length, measured from
the centre of the line. In the application to radiating vapours,
integrations are required with respect to *.

* Phil. Trans. A. vol. excy. p. 346 (1899). See also Proc. Roy. Soc.
vol. Ixxvi, A. p. 440 (1905) ; Scientific Papers, vol. v. p. 257.
+ Ann. der Physik, vol. xxii. p. 209 (1907).

282 Lord Rayleigh on the

Calculations of this kind serve as illustrations ; but it is
not to be supposed that they can represent the facts at all
completely. There must surely be encounters of a milder
kind where the free vibrations are influenced but yet not in
such a degree that the vibrations after the encounter have
no relation to the previous ones. And ia the case of flames
there is another question to be faced: Is there no distinction
in kind between encounters first of two sodium atoms and
secondly of one sodium atom and an atom say of nitrogen?
The behaviour of soda flames shows that there is. Otherwise
it seems impossible to explain the great effect of relatively
very small additions of soda in presence of large quantities of
other gases. The phenomena suggest that the failure of the
least coloured flames to give so high an interference as is
calculated from Doppler’s principle may be due to encounters
with other gases, but that the rapid falling off when the
supply of soda is increased is due to something special. This
might be of a quasi-chemical character, e.g. temporary asso-
clations of atoms ; or again to vibrators in close proximity
putting one another out of tune. In illustration of such
effects a calculation has been given in the previous paper *.
It is in accordance with this view that, as Gouy found, the
emission of light tends to increase as the square root of
the amount of soda present.

We come now to cause (iy.). Although it is certain that
this cause must operate, we are not able at the present time
to point to any experimental verification of its influence.
As a theoretical illustration ‘“‘ we may consider the analysis
by Fourier’s theorem of a vibration in which the amplitude
follows an exponential law, rising from zero to a maximum

and afterwards falling again to zero. It is easily proved
that.

9.9 i ri 9; 5) { o, °
Pa cos dweos ux{e Che ee ae
e 0

24/1
(20)

in which the second member expresses an aggregate of
trains of waves, each individual train being absolutely
homogeneous. If @ be small in comparison with 7, as will
happen when the amplitude on the left varies but slowly,
e— urns may be neglected, and e~('-"*4@ js sensible only
when wu is very nearly equal to >” +.

An analogous problem, in which the vibration is repre-

* Phil. Mag. supra, p. 209.

fea: Mag. vol. xxxiv. p. 407 (1892); Scientific Papers, vol. iv.
p. 16.

Widening of Spectrum Lines. 283

sented by e-“sin bt, has been treated by Garbasso*. I
presume that the form quoted relates to positive values of ¢
and that for negative values of ¢ it is to be replaced by
zero. But I am not able to confirm Garbasso’s formula +.

As regards the fifth cause of (additional) widening
enumerated at the beginning of this paper, the case is
somewhat similar to that of the fourth. It must certainly
operate, and yet it does not appear to be important in
practice. In such rather rough observations as I have made,
it seems to make no great difference whether two surfaces of
a Bunsen soda flame (front and back) are in action or ovly
one. If the supply of soda to each be insufficient to cause
dilatation, the multiplication of flames in line (3 or 4) has no
important effect either upon the brightness or the width of
the lines. Actual measures, in which no high accuracy is
needed, would here be of service.

The observations referred to led me many years ago to
make a very rough comparison between the light actually
obtained from a nearly undilated soda line and that of the
corresponding part of the spectrum from a black body at the
same temperature as the flame. I quote it here rather as a
suggestion to be developed than as having much value in
itself. Doubtless, better data are now available.

How does the intrinsic brightness of a just undilated soda
flame compare with the total brightness of a black body at
the temperature of the flame? As a source of light Violle’s
standard, viz., one sq. cm. of just melting platinum, is equal
to about 20 candles. The candle presents about 2 sq. cm.
of area, so that the radiating platinum is about 40 as
bright. Now platinum is not a black body and the Bunsen
flame is a good deal hotter than the melting metal. I esti-
mated (and perhaps under estimated) that a factor of 5 might
therefore be introduced, making the black body at flame
temperature 200 as bright as the candle.

To compare with a candle a soda flame of which the D-
lines were just beginning to dilate, I reflected the former
nearly perpendicularly from a single glass surface. The
soda flame seemed about half as bright. At this rate the

: Geers ‘ 1 1 1 :
intrinsic brightness of the flame was 5 X 35 = 50 OF that of

. = “or 1 © Ale
the candle, and. accordingly 10,000 of that of the black

body.

* Ann. der Physik, vol. xx. p. 848 (1906).
} Possibly the sign of uw is supposed to change when ¢ passes through
zero. But even then what are perhaps misprints would need correction.

284 Mr. E. J. Evans on the

The black body gives a continuous spectrum. What would
its brightness be when cut down to the narrow regions
occupied by the D-lines? According to Abney’s measures
the brightness of that part of sunlight which lies between

the D’s would be about =+, of the whole. We may perhaps

250
; i
estimate the region actually covered by the soda lines as 55
of this. At this rate we should get

1 i 1

25a m2 501 62507
as the fraction of the whole radiation of the black body
which has the wave-lengths of the soda lines. The actual
brightness of a soda flame is thus of the same order of
magnitude as that calculated for a black body when its
spectrum is cut down to that of the flame, and we may infer
that the light of a powerful soda flame is due much more to
the widening of the spectrum lines than to an increased
brightness of their central parts.
Terling Place, Witham,
Dec. 18

XXXII. The Spectra of Helium and Hydrogen.
iBy B. J. Evans, B.Sc., AR.C.Se.*

[Plate VI. ]
INTRODUCTION.

TRXHE investigation of the series spectra of hydrogen and
helium has acquired considerable importance in con-
sequence of recent theories on the structure of the atom, and
the application of the quantum hypothesis to radiation
problems.
It is known that some of the series spectra attributed to
hydrogen can be represented with great accuracy by the
Balmer-Rydberg-Ritz formula

hel: zai
p= K & — =) By tte - : 3 (1)

where vy represents the frequency, K Rydberg’s universal
constant, and m; and nz are whole numbers. If in equation
(1) n,=2, and ny takes the successive values 3,4,5..... ;
the ordinary hydrogen series in the visible spectrum is

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

Spectra of Helium and Hydrogen. 285

obtained. The formula for this series was discovered by
Balmer, and the series is usually referred to as the Balmer
series. If n,=3 a series of lines in the ultra-red is obtained,
and two members of this series have been observed by
Paschen. Also the series calculated by putting n,=1 has
been recently observed by Lyman” in the extreme ultra-
violet. In addition to the lines represented by formula (1),
some other series of lines have generally been ascribed to
hydrogen. In 1896 Pickering f discovered in the spectrum
of the star & Puppis a series of lines, which are closely
represented by the formula

aie. 1

y=K4 — ant Elis aaaore Malet aC aeh lean

On account of the numerical relationships existing between
this series and the Balmer series, Pickering attributed the
lines to hydrogen, and later Rydberg f, from analogy with
the spectra of the alkali metals, considered the Balmer and
Pickering series to be the diffuse and sharp series of hydrogen.
He further concluded that the complete hydrogen spectrum
should contain another series of lines, corresponding to the
principal series in the spectra of the alkali metals, and given
by the formula

Si ee a }
i ee ea) a KO

Rydberg’s conclusions were apparently strongly supported
by the observation of a strong line in the spectra of certain
stars and nebulee at the place calculated for the first line in
formula (3).

A few years ago, Fowler § obtained the two series of lines
represented by equations (2) and (3) by passing a condensed
discharge from a 10-inch coil through mixtures of helium
and hydrogen contained in an ordinary Pliicker tube.
Further, he observed in the spectrum of the mixed gases a
third series related to the series represented by equation (3)
in the same way as the Pickering series is related to the
Balmer series, and which was approximately represented by
the formula |

ed 1 1 A)
ues Geet | e . ° : (4)

* Lyman, ‘ Nature,’ xciii. p, 241.

+ Pickering, Astro-Phys. Journ. vol. iv. p. 369 (1896) ; vol. v. p. 92
(1897).

t Rydberg, Astro-Phys, Journ, vol. vii. p. 233 (1899).

§ Fowler, Monthly Notices R. A.S., Dee. 1912.

286 Mr. E. J. Evans on the

Fowler, in the absence of strict experimental proof,
considered that Rydberg’s theoretical investigations justified
the conclusion that the three series of lines were due to
hydrogen, and the series represented by (4) was called the
second principal series of hydrogen. It is interesting to
note that with the observation of series (4) the supposed
analogy between the spectra of hydrogen and the alkali
metals breaks down, as the spectra of the latter are known
to have only one principal series.

Recently the problem of the origin of the series in question
has been considered from the theoretical standpoint by
Dr. Bohr*, who has arrived at very interesting results.
Taking as basis Sir E. Rutherford’s t atom-model, he has
deduced with the aid of Planck’s quantum hypothesis the
following formula for the spectrum emitted by an atomic
system consisting of a central positive nucleus ani an electron.
moving round it :

Memes Mona 1 i)
Ea eee

where e and m are the charge and mass of the electron, E
and M the charge and mass of the nucleus, and A is Planck’s
constant. For a hydrogen atom according to Rutherford’s
theory, E = —e, and the formula can be written

Q77e*m it il
mee m { 2-ap >!
a a)

where the bracket multiplying h®? is very nearly equal to
unity on account of the great mass of the nucleus compared
with that of the electron. It was shown bv Bohr that the
above expression, on putting in the values of the constants,
agreed within the limits of experimental error with
formula (1).

It will be seen that formula (6) does not include the lines
observed by Pickering and Fowler, and given by formule
(2), (8), and (4). However, if in equation (5) H=—2e,
which according to Rutherford represents the helium atom,
the following formula is obtained for the frequency of the
lines:

2Qar°e4m | iL as Be

eee |, SES dh
3 dg iuia\ Ny \*
m1 + 3 i (9 (2
* N. Bohr, Phil. Mag. xxvi. pp. 1, 476, 857 (1913) ; xxvii. p. 506

(1914).
+ Sir E. Rutherford, Phil. Mag. xxi. p. 669 (1911).

Spectra of Helium and Hydrogen. 287

As the factor outside the bracket differs from that in
formula (6) only by a small correction due to the difference
in the masses of the hydrogen and helium nuclei, it will be
seen that ae (7) approximately represents the series of
lines given by (2), (3), and (4). Bohr therefore suggested
that the lines in question ae not due to hydrogen but to
helium. This conclusion was not disproved by previous
observations and experiments, for helium was always known
to be present when the lines appeared. Bohr also pointed
out that the reason why the lines considered are not observed
in ordinary helium tubes may be that in such tubes the
ionization of the gas is not as complete as in the star
£ Puppis or in Fowler’ experiments with the condensed
discharge. According to the theory, the presence of helium
atoms which have lost both electrons is necessary for the
appearance of these lines in a spectrum-tube.

Preliminary experiments by the author™ gave strong
support to Bohr’s conclusions. A helium spectrum was
obtained showing the first line (A 4686) of the series
represented by equation (3) very brightly, but no trace of
the ordinary hydrogen lines of the Balmer series. Also
this line couid not be observed in the spectrum obtained by
passing a strong discharge through mixtures of hydrogen
and neon, and of hydrogen and argon. Later Stark fT
observed the 4686 line in a helium tube in which the
hydrogen lines did not appear. Further evidence pointing

oO
to the same conclusion has been given by Raut, who has

made some interesting experiments on the voltage necessary
for the production of spectrum lines.

In a discussion in ‘ Nature,’ Fowler § pointed out that the
lines observed by him were not accurately represented by
formule (2), (3), and (4), but Bohr subsequently showed
that the deviations from the values given by the formule
could be accounted for within the limits of experimental
error by taking into account the correction due to the mass
of the central nucleus. This influence of the mass of the
nucleus was not considered in Bohr’s original paper.
Recently Fowler ] has published a very important paper on
series lines in spark spectra. He concludes from analogy
with spark spectra of the enhanced type, that the lines
represented by equations (2), (3), and (4) are enhanced lines

* Evans, ‘ Nature,’ xcii. p. 5

t Stark, Verh. d. Deutsch. Phys. Ges. xvi. p. 468 (1914).

t Rau, Sitzungsb. d. Phys.-Med. Ges. Wiirzburg, 1914.

§ Fowler, ‘Nature,’ xcii. p. 95. | Bohr, ‘ Nature,’ xcii. p. 231.
§ Fowler, Phil. Trans, A. vol. cexiv. p: 225.

288 Mr. E. J. Evans on the

of helium. Stark also came to the same conclusion with
respect to the line 4686.

In the above mentioned discussion in ‘Nature’ Bohr
showed that if his view was correct, the helium spectrum
should contain another series of lines, which are represented

by the formula
1 1
y= ml Ta\9 ——— WG . . . . ° (8
Leas *)

where K’ is the constant outside the bracket in formula (7).
This formula is obtained from (7) by putting ny=4 and
Rao, LO 2. s. The lines of this series should appear
near the hydrogen lines of the Balmer series, and their
wave-lengths were calculated to be 6560°4, 4859°5, 4338-9,
10) eee These lines and the Pickering lines can be
represented by one formula

Klar (ay | a

and the intensity of all the lines should decrease regularly
with increasing values of n. The lines would be difficult
to detect, especially if hydrogen was present in any quantity,
as it is known from Fowler’s experiments that the Pickering
lines are faint even when photographed with instruments of
small dispersion. The presence of the lines given by formula
(8) would greatly strengthen the experimental evidence in
favour of Bohr’s theory, but their absence would immediately
show that the theory was incorrect. The investigation of
the lines in question, and the determination of more accurate
values for the wave-lengths of the lines of the Pickering
series, formed the main object of the present research.

THE EXPERIMENTAL ARRANGEMENT.

The method which was chiefly employed for the production
of the spectrum lines was very similar to that previously
described by Fowler. A 20-inch coil with a condenser across
its terminals was connected to a helium spectrum-tube in
series with which was placed an adjustable spark-gap. The
spectrum produced was examined visually with a Hilger
direct-reading instrument, and the wayve-lengths could be
read off with an accuracy of 1 or 2A.U. For the purpose of
photographing the lines, four different instruments were
employed. In the first series of experiments the lines were
photographed by means of a small quartz spectrograph giving

Spectra of Helium and Hydrogen. 239
a dispersion of 250 A.U. per mm. in the red, 130 A.U.

per mm. in the green, and 76 A.U. per mm. in the violet.
Later the spectrum was photographed with a concave grating
of 1 metre radius ruled with 14,000 lines to the inch. By
means of this instrument it was found possible to obtain the
first members of series (3) and (4), but no trace of any of
the lines represented by equations (2) and (8) even after
exposures of 40 hours. A few members of the series
represented by equation (2) were obtained in the first place
by employing a spectroscope consisting of two glass prisms,
which gave a dispersion of 150 A.U. per mm. in the red,
54 AU. ber mm. in the green, and 30 A.U. per mm. in the
violet. finally these two prisms were replaced by others
giving greater dispersion and capable of resolving 2.A.U. at
6500. For the sake of comparison the dispersions given
by this prism spectroscope in the red, green, and violet were
75, 27, and 15 A.U. per mm. respectively. The time of
exposure varied from 20 minutes to 30 hours, depending
upon the dispersion of the apparatus and the particular lines
studied. The first member of series (3) at. 4686 was easily
photographed, but the series represented by formula (9)
required a very long exposure, and even then only a few
members of the series could be detected with certainty. All
the photographs were taken with Wratten and Wainwright’s

panchromatic plates and films.

The spectrum-tubes employed were of various shapes and
dimensions. The diameter of the capillary was varied from
8 to 3°5 mm., and its length from 3 to 8 cm. When the
lines represented by equation (9) were studied, a spectrum-
tube with a can any of ‘8 mm. was chosen, the spark-gap
was made 1°2 cm., and the pressure of the helium was
adjusted to about -5 mm. of mercury. Photographs of the
4686 line were taken with spectrum-tubes having capillaries
of different bore, and containing helium at pressur es ranging
from °36 to 2 mm.

Since the hydrogen lines of the Balmer series are in
ordinary circumstances much stronger than the odd members
of series (9), it was necessary, especially with the relatively
small dispersion employed, to obtain a helium spectrum
giving little or no trace of the hydrogen lines. In the
preliminary experiments spectrum-tubes with aluminium
electrodes were employed, but it was found impossible to
remove the hydrogen from the electrodes in a reasonable
time. Later, thin platinum electrodes of -4 mm. diameter
were employ ed, and a helium spectrum showing no trace of
hydrogen was obtained. In this connexion it is interesting

Pijul. Mag. 8.6. Vol. 29. No. 170. Feb. 1915. U

290 Mr. E. J. Evans on the

to note that it is comparatively easy to obtain a helium
spectrum showing no trace of the hydrogen lines of the

Balmer series if an ordinary discharge is sent through a
spectrum tube containing purified helium. With the con-
denser discharge, however, the hydrogen lines usually appear
in the bulbs, although they are often not present in the
capillary. ‘This was also observed by Curtis * during a
research on the band spectrum of helium.

The arrangement of the apparatus and the method of con-
ducting the experiments are made clear by reference to the

accompanying diagram.

In the diagram, E represents the spectrum tube and A,
C, F, and G bulbs containing carefully prepared coconut
charcoal. The tap B was placed at a distance of 1 metre
from the spectrum-tube, and the discharge scarcely ever
reached it. A Topler pumpand a tube of P.O; were attached
to H, and the whole apparatus was evacuated to a pressure
of 1/100th of a mm. of mercury. The charcoal buibs C, F,
and G were then heated for several hours until no more gas
was given off. Also the bulbs and the capillary of the
spectrum-tube were heated until the softening point of glass
was reached. The gases given off during the heating process
were absorbed by the charcoal bulb A, which was immersed
in liquid air. The tap B was then closed and the gas
absorbed by the charcoal in A removed by heating.
The bulb A was again immersed in liguid air, and the helium
containing a small quantity of impurities was then introduced

through the side-tube K and allowed to remain in that part
of the apparatus for over an hour. The charcoal bulbs C, F,
and G were now immersed in liquid air, and the tap B was
opened so that helium could enter the spectrum-tube. The
pressure of the gas was then adjusted to ‘3 mm. by means of
the pump and McLeod gauge, and the ordinary discharge
(without condenser) passed through the tube. The current

* Curtis, Proc. Roy. Soc. A. vol. Ixxxix. p. 146 (1913).

Spectra of Helium and Hydrogen. 291

passing through the primary of the induction-coil was
adjusted until the cathode was almost white-hot, and it was
kept at this temperature for nearly four hours. By reversing
the direction of the current passing through the tube, the
other electrode was treated in a similar manner. ‘This
process was repeated until no hydrogen lines were visible
in the Hilger direct-reading spectroscope. The condenser
discharge was then passed through the tube for 30 minutes,
and both the bulbs and capillary were examined for the
presence of the hydrogen lines. If the lines eventually
appeared, the process of heating the electrodes was continued
until the spectrum showed no hydrogen lines after the
passage of a heavy condenser discharge for one hour. During
these experiments the platinum electrodes spluttered to the
sides of the spectrum-tube and consequently became much
finer. This was also made evident by the bending of the
electrodes to touch the sides of the bulbs when a heavy
condenser discharge was passed through the tube. In later
experiments, the diameters of the bulbs were so adjusted
that the platinum electrodes would not touch the sides even
when perpendicular to their original directions. In the pre-
liminary experiments, all the photographs and observations
were obtained with the spectrum-tube in the vertical position,
but later, especially when investigating the series repre-
sented by equation (9), the tube was usually placed in the
end-on position with the capillary pointing towards the
spectroscope.

The spectrum obtained by passing the ordinary discharge

without condenser and spark-gap through a tube containing
helium at a pressure of *25 mm. was also studied. Under
these conditions a stream of charged particles, which were
deflected by a magnet in the same direction as cathode rays,
passed down the tube. The results of experiments carried
out in this way will be described later.

For the determination of wave-lengths, the lines of the
ordinary helium spectrum were employed as standards ex-
cept in the red and yellow, where they are too far apart. In
these regions, the impurity lines of mercury, sodium, silicon,
and oxygen were found useful. Also in some cases the
wave-lengths were determined with the aid of a copper or
barium comparison spectrum. The mercury lines only
appeared on one of the plates, but the sodium, silicon, and
oxygen lines were always present on photographs taken when
a heavy condenser discharge was passed through the tube.
These lines are due to the decomposition of the glass of the
capillary by the discharge.

U 2

bo
pe)
bo

Mr. E. J. Evans on the

EXPERIMENTAL RESULTS.
The Series given by Formula (3) and (4),

The preliminary experiments * showed that it was possible
to obtain photographs showing the 4686 line very strongly,
and no trace of the ordinary hydrogen lines of the Balmer
series. As the photographs were taken with a quartz spectro-
graph of small dispersion, the wave-length could only be
determined with an accuracy of about 1 A.U. The line was
therefore photographed with the prism spectroscope giving
the greatest dispersion, and the wave-length determined with
a greater degree of accuracy. The mean wave-length as
determined from these photographs was 4686-00 (+-05),
which agreed well with the value 4685-98 (+:01) determined
by Fowler from a photograph taken with a 10-foot concave
grating.

It was also possible to obtain this line, employing the same
spectroscope, by passing the ordinary discharge without
condenser and spark-gap through an end-on tube containing
helium at a pressure of °25mm. The line is shown in photo-
graph J. (PI. VI.), where it appears fairly strong and broad
in the capillary. The wave-length of the line was measured
and found to be 4685°94. The photograph does not show
the presence of the hydrogen lines at 6563, 4861, .... but
only the ordinary helium lines and the 4686 line.

These experiments show that it is possible to obtain the
first member of series (3) from a helium tube giving no trace
of the hydrogen lines of the Balmer series.

The Series given by Formula (9).

(A) Experiments with tubes containing aluminium elec-

trodes.

The second member of this series was observed visually
for the first time in the spectrum of an end-on tube with a
narrow capillary. ‘'wo members of the series are shown in
photograph II., which was taken with the two-prism spectro-
scope of low dispersion. ‘The wave-lengths of these lines
were determined with an accuracy of 0-2 A.U., and were
found to agree within the limits of experimental error with
the values calculated for the second and fourth members of
series (9). The two values obtained were 5411°63 and
4541-93 respectively. The wave-lengths of the lines as de-
termined by Fowler were 5410°5(+1°0) and 4541:3(+0°25).
In addition to the ordinary helium lines, shorter or capillary

* Evans, loc. cit.

Spectra of Helium and Ilydrogen. 29a

lines due to oxygen, silicon, and sulphur made their appear-
ance in the spectrum. The photograph also shows the
presence of the H. (6563), He (4861) .... lines of hydrogen
and the mercury line at 5460.) Thess lines str etch across
the capillary and have the same length as the ordinary
helium lines.. The 5411 and the 4542 lines also pass through
the capillary spectrum, but their lengths are only about
one half that of the ordinary helium iines. The line at 5411
was also found to be more intense and sharper than the 4542
line. The sixth member of series (9) could not be detected
on the photographic plate. A thorough examination of the
plate near the Hz and Hg lines did not reveal the presence of
any faint lines at 6560°4 cad 4859°5. Rough measurements,
however, showed that the widths of the H, and Hg lines were
approximately 8*l and 5 A.U. respectively. Also the line at
5411 was found to be narrow compared with the hydrogen
lines. Since the even members of series (9) were so faint
compared with the hydrogen lines, it was decided to repeat
the experiments with a prism spectr oscope of nearly double
the dispersion and tubes provided with thin platinum elec-
trodes.

(B) Eaperments with spectrum tubes provided with
platinum electrodes.

A condenser discharge was sent through an end-on tube
containing helium at a pressure of ‘4 mm., and three photo-
graphs were taken with the two-prism spectroscope of
greatest dispersion after exposures of 5,12, and 26 hours
respectively. In the last case the direction of the discharge
was reversed at the end of 13 hours. On all the plates” a
faint line appeared at 6560, which is the calculated position
for the first line of series (9). This line was shorter than
the ordinary helium lines, and the oxygen lines at 6455 and
6157, but longer than the silicon capillary lines at 6371 and
6347.

The plates taken with exposures of 12 and 26 hours showed
in addition to the 6560 line an ill-defined capillary line at
6564. The nature of the line made aecurate measurements
impossible, and the origin of the line is unknown unless it is
due to a trace of hydrogen. On the same plate there also
appeared a line at 5411. which was of the same general
character as the 6560 line but of less intensity. The third
member of the series at 4859°5 could not be detected on the
plates taken with exposures of 5 and 12 hours, and the only
lines present in this region were the oxygen capillary lines
at 4872, 4865, 4861, and 4857. The plate taken with an

294 Mr. E. J. Evans on the

exposure of 26 hours was also examined in the same region,
and the wave-lengths of two very faint lines, which extended
beyond the capillary, were measured. One of these lines,
which had a wave-length of 4861:7 A.U., was longer than
the 6560 line. The other line was fainter than the 4861
iine, and its measured wave-length was 4859°5. The fourth
member of the series which should appear at 4542 could not
be detected on any of the plates. All the lines between
7065 and 4713 on the plate taken with an exposure of 12
hours were measured, and with the exception of capillary
lines at 6721°5, 6641°1, 5739°3, 4829°3, 4813°7, all the others.
could be attributed to helium, oxygen, silicon, sodium, and
chlorine. These experiments show that when a condenser
discharge was passed through helium, two lines were obtained
whose wave-lengths agreed within the limits of experimental
error with the values calculated for the first and second
members of series (9), and whose intensities fell off with
diminishing wave-length. Alsoa very faint line was obtained
whose wave-length was very approximately the same as that
calculated for the third member of the series represented by
equation (9). However, in view of the faintness of the line
and the appearance of another stronger line of unknown
origin close to it, the question whether the line is the third
member of the series cannot be regarded as settled.

The experiments described above show that very long ex-
posures are required to bring out the lines of the series given
by equation (9), for even with exposures of 26 hours only
two lines could be detected with certainty. It was therefore
considered desirable to obtain photographs with the other
prism spectroscope, which gave a lower dispersion. Two
photographs were taken, one with an exposure of 21 hours
(photograph III.), and the other with an exposure of 31
hours. In the latter case, the discharge was reversed in
direction at the end of half the exposure. The wave-lengths
of the first two lines were measured for each plate, and found
to be 6560°69, 5411-95, and 6560°10, 5411:71 respectively.
A third line at 4860°5 was also obtained, but its wave-length
was 1 A.U. greater than the value calculated theoretically
for the third member of the series. Also, the line had too
great an intensity if it was to be regarded as the third
member of the series in which 6560 and 5411 were the first
and second members respectively. It is interesting to
point out that the wave-length (4860°5) is approximately
the mean of the two values 4861°7 and 4859°5 obtained for
the two faint lines photographed with the spectroscope of
greater dispersion. The appearance of the line also suggested

Spectra of Helium and Hydrogen. 295

that it might possibly be a double line. A faint fourth line
of the right order of intensity in the blue region of the
spectrum was also measured, and its wave-length (4542°2) is
found to agree within the limits of experimental error with
the calculated value (4541°80) for the fourth member of the
series.

When the above exposures were concluded, the helium was
pumped out of the spectrum-tube and replaced by oxygen.
A condenser discharge was then sent through the tube for
4 hours, and a photograph taken. The lines on this plate
between 6800 and 6157 were measured, and no evidence
was found of the presence of a line at 6560. A line at
6563°5, probably due to hydrogen, was however obtained.
The two lines at 6721 and 6641, and impurity lines due to
nitrogen and silicon, were also present.

EXPERIMENTS WITH THE ORDINARY DISCHARGE.

In these experiments, the ordinary discharge was passed
through an end-on tube containing helium at a pressure of
-25 mm. ‘The same spectroscope was employed as in the
experiments described in the last section, but the slit was
widened in consequence of the comparatively small intensity
of the light. The exposure was varied from 9 to 12 hours
and a copper comparison spectrum was sometimes utilized
for the determination of wave-lengths. The discharge was
also reversed in direction when half the time of exposure
had elapsed. A series of lines with wave-lengths approxi-
mately the same as those calculated from equation (9)
appeared in the capillary in addition to the 4686 line. The
wave-length measurements in this case were not as accurate
as before, because of the greater breadth of the lines. The
mean values of the wave-lengths ootained for the first four
lines are 6560°71, 5411°44, 4860°5, 4542-44. Tor one of the
photographs the measurements were extended still further
towards the violet, and lines which are possible members of
series (9) were obtained at 4339°97 and 4199-9. All the
lines on this photograph (photograph IV.) between 6678
and 3819 were measured, and with the exception of the
above lines and two others at 4358°7 and 4226°9, they were
all due to helium and oxygen. Inall the photographs taken
with the ordinary discharge the third line at 4860°5 was
stronger than it should be if it was the third member of
series (9).

The accompanying table gives the wave-lengths of the
lines which have been determined from photographs taken

296 Spectra of Helium and Hydrogen.

with the condenser discharge and also the ordinary discharge,
together with the values calculated from Bohr’s theory.
The values given for the first four lines were determined from
photographic plates taken with the condenser discharge,
whilst the last two values were deduced from photographs
taken with the ordinary discharge. The dispersion in the
particular region of the spectrum. and the probable error are
also included in the table.

Milica | “aimee” | Tare | DA
656038 | 6560-43 05 | 75 AU per mm.
5411-74 | 5411-67 0-2 | 45 AU per mm.
4859-54 ? |
4541-80 are 93 0-2 | 30 AU per mm.
4338°88 4339-97 0-4 | 36 AU per mm.
4200°03 | 4199°95 | Or4 | 00 AU per mm.

PSOE Saree ramen oe ee ae 2 eee

* Two or three oxygen lines in the neighbourhood of this line may affect
the observed value.

SUMMARY.

The experiments have shown :—
(A) That it is possible to observe the first member ( eee)
of the series

1 i ;
y=109750 1\? as nN 2 5
(15) (5)

which includes the lines given by formule (3) and (4), and

also a line which agrees in wave-length with the first member
(6560°4) of the series

Drea ai a9
y= 109750 oF (a b
D

which inciudes the Pickering lines and the lines at 6560°4..
predicted by Bohr, in a helium tube giving no trace of the
hydrogen lines.

( B) That the experimental values for the wave-lengths
of the Pickering lines agree with the theoretical values
calculated by Bohr within the limits of experimental error.

These results strongly point to the conclusion that the

Simplitied Deduction of Planck's Formula. 297

series spectrum of hydrogen consists only of lines which are
represented by the formula
“baie Gil 1B
P= 097104 (5 a s
ny No”
and that the series spectrum of helium in addition to the
ordinary helium lines consists of all the lines which are
represented by the formula

EES (en (ete eal

The author wishes to thank Sir Ernest Rutherford for
bringing the subject of the present investigation to his
notice and for valuable suggestions and encouragement
<taring the course of the research.

The University of Manchester,
Dec. 1914.

XXXII. Simplijied Deduction ofthe Formula from the Theory
of Combinations which Planck uses as the Basis of his
Radiation Theory. By P. Worenrest and H. KAMERLINGH
ONNES*.

W E refer to the expression

(N—1+P)! (A)
Ey GN Re RSA aM eRe”
which gives the number of ways in which N monochromatic
resonators Ry, Ro,... Ry may be distributed over the various
degrees of energy, determined by the series of multiples
0, «, 2e... of the unit energy e, when the resonators to-
gether must each time contain the given multiple Pe. Two
methods of distribution will be called identical, and only
then, when the first resonator in the one distribution is at
the same grade of energy as the same resonator in the
second, and similarly the second, third,.... and the Nth
resonator are each at the same energy-grades in the two
distributions.

‘Taking a special example, we shall introduce a symbol for
the distribution. Let N=4,andP=7. One of the possible
distributions is the following : resonator Ry has reached the

Nees
C>=

* Communicated by the Authors.

298 Profs. P. Ehrenfest and H. K. Onnes : Simplified

energy-grade 4e (R, contains the energy 4e), R, the grade
2e, R; the grade Oe (contains no energy), R, the grade e.
Our symbol will, read from left to right, indicate the energy
of R,, R., R3, Ry, in the distribution chosen, and particularly
express that the total energy is 7e. Tor this case the
symbol will be :—

TeEe044004 1

or also more simply :-—

| ceeeeOeeOQO:= |
With general values of N and P the symbol will contain
P times the sign « and (N—1) times the sign O*. The
question now is, how many different symbols for the dis-

tribution may be formed in the manner indicated above
from the given number of eandO? The answer is

(N—1+P)!
PIN-DT |

Proof : first considering the (N—1+P) elements e€... ,
O...O as so many distinguishable entities, they may be
arranged in

(N=1+P)! ...°. 2 =e
different manners between the ends [f |]. Next note, that

each time
(N—1)!P! ~ “(ap

of the combinations thus obtained give the same symbol for
the distribution (and give the same energy-grade to each
resonator), viz. all those combinations which are formed
from each other by the permutation of the P elements ef or
the (N—1) elementsO. Thenumber of the diferent symbols
for the distribution and that of the distributions themselves
required is thus obtained by dividing (2) by (3) q.e.d.

* We were led to the introduction of the (N—1) partitions between
the N resonators in trying to find an explanation of the form (N—1)!
in the denominator of (A). Planck proves that the number of dis-—
tributions mnst be equal to the number of all “combinations with
repetitions of N elements of class P,” and for the proof, that this number
is given by the expression (A), he refers to the train of reasoning
followed in treatises on combinations for this particular case. In these
treatises the expression (A) is arrived at by the aid of the device of
“ transition from 2 to +1,” and this method taken as a whole does
not give an insight into the origin of the final expression.

+ See Appendix.

Deduction of Planck's Formula. 299

APPENDIX.

The contrast between Planck’s hypothesis of the energy-grades
and Einstein's hypothesis of energy-quanta.

The permutation of the elements ¢ is a purely formal
device, just as the permutation of the elements O is. More
than once the analogous, equaliy formal device used by
Planck, viz. distribution of P energy-elements over N reson-
ators, has by a misunderstanding been given a _ physical
interpretation, which is absolutely in conflict with Planck’s
radiation-formula and would lead to Wien’s radiation-
formula.

As a matter of fact, Planck’s energy-elements were in
that case almost entirely identified with Hinstein’s light-
quanta, and accordingly it was said that the difference
between Planck and Hinstein consists herein, that the latter
assumes the existence of mutually independent energy-
quanta also in empty space, the former only in the interior
of matter, in the resonators. The confusion which underlies
this view has been more than once pointed out*. Hinstein
really considers P similar quanta, existing independently of
each other. He discusses, for instance, the case that they
distribute themselves irreversibly from a space of N, cm.’
over a larger space of N, cm.*, and he finds, using Boltz-
mann’s entropy-formula, S=klog W, that this produces
a gain of entropy } :

YN
S-Sy=hlog (x?) 5 - LOM AU REDCE:
At]

i.e. the same increase as in the analogous irreversible distri-
bution of P similar independent gas-molecules, for the number
of ways in which P quanta may be distributed, first over
N,, then over N, cells in space, are to each other in the ratio

Nai Ne ie h ond, 6 oui)

If with Planck the object were to distribute P mutually
independent elements ¢ over N resonators, in passing from
N, to N, resonators the number of possible distributions
would in this case also increase in the ratio (8) and corre-
spondingly the entropy according to equation («). We know,

* P. EKhrenfest, Ann. d. Phys. vol. xxxvi. p. 91 (1911); G. Krutkow,
Physik. Zschr. vol. xv. pp. 133, 363 (1914).
+ A. Einstein, Ann, d. Phys, vol. xvii. p. 182 (1905).

300 Simplified Deduction of Planck's Formula,
however, that Planck obtains the totally different formula

(N;—1+P)! (N.—-1+P)! (y)

Q.-1) Pt GDP! | ae
(which only corresponds approximately with (@) for very
large values of P) and a corresponding law of dependence
of the entropy ‘on N. This can be simply explained as
follows: Planck does not deal with really mutually free
quanta e; the resolution of the multiples of e in separate
elements e. which is essential in his method, and the intro-
duction of these separate elements have to be taken cum
grano salis ; it is simply a formal device entirely analogous
to our permutation of the elements ¢ or O. The real object
which is counted remains the number of all the different
distributions of N resonators over the energy-grades 0, €,
2e..- with a given total energy Pe. If, for instance, P=3,
and N=2, Einstein has to distinguish 2?=8 ways in which
the three (similar) light-quanta A, B, C can be distributed
over the space-cells 1, 2.

en Bea oo

LA eesti Acti aay

18 ae i GO a ee

PE Ty con ee aT
TN ae SO ae
OR ba tes Pitas a

WL eon alae
Venly 9 2 1
WARDEN Roe non neo

Planck, on the other hand, must count the three cases
II., III., and V. as a single one, for all three express that
resonator R, is at the grade 2e, R, at e; similarly, he has to
reckon the cases IV., VI., and VII. as one; R, has here e
and R, 2e. Adding the two remaining cases I. (R, contains
de, R, Oe) and I. (R, has Ge, Ry 3e), one actually obtains

(N—14P)! @—1+83)!
(SNPs GA ISt a

different distributions of the resonators R,, R, over the
ener gy-grades.

On the Visibility of Radiation. 301

We may summarize the above as follows :—Hinstein’s
hypothesis leads necessarily to formula («) for the entropy
and thus necessarily to Wien’s radiation-formula, not
Planck’s. Planck’s formal device (distribution of P energy-
elements ¢€ over N resonators) cannot he interpreted im the
sense of Hinstein’s light-quanta.

XXXIV. The Visibility of Radiation.
By P. G. Nurrine*.

HE quantitative relation between light and radiation has
long been sought by many investigators. Herschel,
exploring the spectrum with a thermometer, found that the
radiation continued beyond what was visible. The invisible
ultra-violet portions of spectra were long ago explored by
photography. Langley, twenty-five years ago, explored
the infra-red solar spectrum with his fine wire bolometer,
and in the visible spectrum measured the amounts of energy
of various wave-lengths required for reading print. Pfliiger t
and Konig and Dieterici § determined the relative amounts
of energy required to just produce a luminous sensation
in different parts of the spectrum. Konig || continued his
investigations from the threshold of vision up to an intensity
of about 500 metre candles.

About ten years ago it was clearly recognized that in
order to define light in terms of the radiation which excites
it, an intermediate function, the visibility of radiation, must
be formulated and its constants determined for the av erage
normal eye. Goldhammer 4, in 1905, partly reduced some
of K6nig’s data and expressed visibility as a function similar
in form to that giving the spectral energy of a perfect
radiator. Hertzsprung Hej in 1906, took a rough average of
all available threshold data and formulated visibility : as a
logarithmic hyperbola. The author tf, independently of
Goldhammer and Hertzsprung, reduced the data of Langley,
Pfliiger, and Konig, and in 1907 published this, a function

* Communicated by the Author.

+ S. P. Langley, Am. Journ. Sci, xxxvi. p. 359 (1888).

{ A. Pfltiger, Ann. Ph. ix. p. 185 (1902).

§ Konig and ‘Dieterici, Zs. Psy. Phys. Sinn. iv. p. 241 (1893).

| TA. Konig, Ges. Abhandlungen.

q D. A. Goldhammer, Ann. Ph, xvi. p. 621 (1905).

4% bs Hertzsprung, Z. Wess. Phot. iv. p. 48 (1906).

. G. Nutting, Phys, Rev. xxiv. p. 202 (1907); Bull. Bu. Stds. v.

Dp. be (1908).

302 Mr. P. G. Nutting on the

representing it, and the related Purkinje effect, and made a
rough determination of its principal constant, the maximum
ratio of the candle to the watt.

Recently Ives* has applied the flicker photometer to the
determination of visibility with excellent results, and has
published data for eighteen different subjects in the region
from ‘48 to ‘64u. I have here to present a similar set of
data for twenty-one subjects with extensions of the visibility

Fig. 1.—Diagram of Visibility Apparatus.

curves farther into the red and violet, and the results of a
number of direct determinations of the maximum ratio of
the candle to the watt.

* H. E. Ives. Phil. Mag. Dec. 1912. See also Thiirmel, Anz. PA.
xxx. p. 1154 (1910).

Visibility of Radiation. 303

The method of determining the visibility curves was
similar to that used by Ives. A wave-length spectroscope
was fitted with a Whitman disk flicker photometer, so that
the pure spectral hue and a white surface illuminated by a
standard lamp were viewed alternately (fig. 1).

Instead of a glow lamp as a source we used one of the
acetylene standard lamps designed by Dr. Mees, and tested
and described recently by Mr. Lloyd Jones*. This is

essentially a cylindrical flame from a + foot Bray tip sur-

rounded by a metal chimney in eh is a re-entrant
window screening out all but a horizontal section about
5mm. high. This source is extremely constant in intensity
as well as in quality.

Intensities were varied by means of a pair of Nicol prisms
before the slit, the slit remaining of constant width, and
therefore the spectrum of constant purity

The observing pupil was 0:57 x 2°57 mm. throughout, the
standard intensity 350 me. or the equivalent of 241 me.
through a pupil of 1 sq. mm. ; test curves run at twice, 4, 4
and +; this ulumination showed that it was safely outside
the range of the Purkinje effect.

The energies representing equal luminosities were deter-
mined by placing at the ocular a Rubens bismuth-silver
thermopile connected to a Paschen galvanometer, both
made by Dr. Coblentz. This gave the spectral energy
distribution of acetylene in the spectr um actually observed.
As a further check, the dispersion curve of the spectroscope
was determined, and the spectral energy computed from the
bolometric data of Coblentz and Stewart on acetylene ;
the two determinations agreed throughout.

The visibility data obtained are summarised in the followin g
tables. Three independent curves were run by each subject
on different days. In combining the curves, ordinates were
weighted according to height by reducing to equal areas
(equal total light in a constant ener ey spectr um). In fig, 2
are plotted individual mean visibilities together with ae
mean of all 21 subjects.

The average visibility curve (fig. 3) for the 21 subjects
agrees well with that of previous determinations. It is
shghtly more contracted than that obtained by Ives, the
greatest difference from Ives’ mean being on the left (blue)
side of the curve near the maximum. The mean wave-
length of maximum visibility is °555 as against *°553 obtained

‘by Ives.

* L, A. Jones, Trans. I. E.S. ix. p. 716 (1914).

Subject. |A 49 5

ee SS a a
_————— | |

L. A.J. ...| 192 -274, 715) S88 1-006) 1-042) 1-026 -9:
F.A.E. ...| 248) 350) 662] 819! 910} 967) 976) 9:
J.B. Hi. ...| 234 367| 705, 863) 952) 982) 977)
re ae 196| 287) 610] 787) 913) 970) 975
J.G.C. ...| 304| 440) 751| 855| 917} 947) 940
IH. L. H. ...| 237} 358| 686] 811] 911) 963) 965
iV. G. uM 146 250) 618| 794 915) 970| 986
L_E.J. _..| 223) 330] 612| 766 876| 929) 946
L. W.E. ...| 221 311} 679} 841) 960) 1-010} 1-016
L. M.F. ...| 253) 870} 688| 838, 934) 985) 986
M.B.S. ...| 185) 293| 650) 841) 972) 1-036) 1-031
|W. R.F. z) 399) 507 796| 919) 986)1-:000| 968
C. W.F....| 238} 313] 701| 854) 944) 988) 991)
Ney H. ...| 167) 255) 626) 834, 963) 1-023) 1:022
eit ip 263, 367] 661) 804 924] 975| 967
NETS. ..:) 290 369 654| 884) 1-021) 1-072) 1049
C.E.S. E 923, 305| 589) 760, 904) 986) 995
A. MeD, ...| 138) 245] 626| 794} 918} 982) 990
io ee 190 300] 658| 846, 972) 1-035) 1-032
K. i. ......| 245| 372| 697) 851) 932] 972) 979
P.G.N. ...| 182 273] 714) 878) 996) 1-055) 1-038
| |
Average | 227330) 671|"825| -944) -995) -998
|
Fig. 2.—Individual Visibility Curves compared with Mean.

On the Visibility of R idiation. 305

In fig. 3 are included the new dati for the violet and
extreme red given in tie following table (p. 306) for five
subject~. Th-se data were obtained by means of te mercury
lines 406, 436, 492, 546, and 578, together with helium lines
439, 447, 492, 52, 588, and 563 Cnet against the acety-
lene spectrum for enery.

Fig. 3.—Visibility Curves.

" Tos :
i EHEC HEE ne chs
ECE EE
SReerauetiare =
- Ene si
AE
He ,

4 /|
Lp |

eee ee

Goro): Konig’s data reduced by Nutting, 1907.

(—— ). Ives’ mean of 18 subjects.
( —— ). Author’s mean of 21 subjects.
(-—~-). Curve calculated by formula 1.

The theoretical formula whose values are given in the
table and plotted in tig. 3, is of the form

Wee Wes ise Coates apy acu cerel cele, crm ata (1)

in which R=Amax/A, and a=181. The curve ec computed for
the constants Am=0°595 and a=181 agrees very well with
the data of the new mean experimentally determined curve
between wave-lengths -48 and -65 4. The departure from
the actual curve in the extreme red and violet is of slight
consequence in computing the luminosity of sources on
account of the relatively low visibility of radiation in those
regions.

Plial. Maq. 8. 6. Vol. 29. No. 170. Feb. 1915. xX

Mr. P. G. Nutting on the

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Visibility of Radiation. 307

Now the spectral energy of a normal radiator at a tempe-
rature T is well represented in the visible spectrum by the

Wien-Paschen function
aC WN em uk ge ee)

Hence the light emitted by such a radiator will be given by
the integral of EVdd from 0 to ~. Call this integral L,

then *
L=A(> +1), . Gee rete ate)
in which
A=C,Vn dt Din ta—1)(ady) "21, and B=Cy/ahn.

L has the maximum value
é y= \2
i TIN ieee
n+a—-1

T= anlar eas
(Go 1 Nap
or about 6530 if we take n=5, a=181, and C,=14500.

The remaining visibility constant V,, must be determined
experimentally. It is the ratio of the candle (or lumen as
preferred) to the watt at the wave-length of maximum Visi-
bility. The simpler method is to measure in metre candles
as light and in watts as energy some given monochromatic
illumination, preferably of a wave-length near that of
maximum visibility. The first determinations of Wm were
made by this method by Dr. Drysdalef and the writert
seven years ago. We obtained 16:7 and 135 cand./watt
respectively, values of the right order of magnitude but
much too low on account of stray radiation. More recently,
Fabry and Buisson§ have made a determination by this
method, and obtained the value 55 cand./watt using the
green mercury line 5461 from a powerful mercury are.

The other method for determining Vm is indirect but less
subject to large systematic errors, and it gives, under certain
conditions, a direct relation between the international candle
and the watt. A source of light is used having a continuous
spectrum and whose spectral energy distribution is known.
With radiometer and photometer, the radiation at a given

* P. G. Nutting, B.S. Bull. v. p. 305 (1908) ; vi. p. 387 (1909);
“ Applied Optics,’ p. 158 (1912).

*« C. V. Drysdale, Proc. Roy. Soc. Ixxx. p. 19 (1907).

t P. G. Nutting, Elec. World, June 26, 1908.

§ Fabry and Buisson, Compt. Rend. cliii. p. 254 (1911).

Dee

at a temperature

308 Mr. P. G. Nutting on the

distance in a given direction is determined in metre candles
and in watts per square centimetre ; from these total candles
per watt is found. Then by graphical integration of the
spectral energy and spectral luminosity (energy times visi-
bility) curves, V,, is readily calculated.

For example, suppose that the spectral energy curve has
an area A, while the measured energy is W metre-watts then

i) KavA=A= W.

Call the area of the spectral luminosity curve B, and suppose
the illumination is © metre candles, when and where the
energy is W metre-watts. Then
Mh
7 =—hav— Vo b— Ce

Ym
eee CA c :

By division, V,= WRB: hence knowing C/W by direct
determination and A/B by graphical integration, the funda-
mental constant V,, may be readily determined. The pre-
cision attainable by this method depends upon the uncer-
tainties in the three quantities used, (1) relative visibility
V/V, (2) the specific quality C/W, and (3) the spectrak
energy H(A) of the source used in relative watts per unit
difference in wave-length.

I have recently tried the monochromatic method with
mereury green light and the total spectrum method with
various sources.

Filtered mercury light gave a very low value (4°86 metre-.
watts to 12°7 metre candles or V,=2-6 C/W) in spite of
every precaution to screen out the stray radiation and correct
for the remainder.

Mercury light dispersed with a high intensity spectroscope
gave better results, but the uncertainty is still large owing”
to (1) the photometric comparison of pure green with white,
and (2) the removal of the thermopile (or other radiometer)
from the spectroscope to face the euergy standard. We
obtained

220 me. 6°54 mw. 38°6 C/W_ J. H.
21D Re 38°0 N partly rested.
320 A 48-9 N rested.

In measuring the brightness, merely looking at anything
illuminated with mercury licht will greatly depress the eye
sensibility to Hg green, while the fatigue : aused by adjusting
the mercury Ja ip persists for perhaps an hour. Even using
a specially designed lamp and every precaution against stray —
radiation, L regard the final result as uncertain by 10 per cent.

With the total spectrum method the sources used were

- Visibility of Radiation. 309

acetylene, pentane, Hefner, Nernst (two efficiencies), tung-
sten and carbon (three efficiencies). The acetylene source
was a Mees standard burner (see above), the pentane and
Hefner were primary standard lamps, the Nernst was of
Westin house make, 1 03x13 mm. filament, the tungsten
an old type evacuated Jemp, and the carbon of the “ gem”’
type. The quality determinations (means of three to six)
are as follows :-— |

Observations. | Standard quality.| Mean horizontal.
| j e

M-W. M-C.| W/C. C/W. | W/em2/C atl m.

EBEDTICR 2,080. 5e80in 6°84 B71) 7:84 1276 6:24 x 10—5 watt
RENtANG ..caccehoecsee 14:2 1625! 875 1144 | 697 ‘
Acetylene ............ 2°45 BIL NST *598 1:23 , |

Nernst ‘SO amp. ...; 21°96 12°42 1:75 ‘D70 1529 a
a3 (Of) bal tee 332) 640 2-06 484 1:64 a
Tungsten 1:20 W/C.| 5°48 4:94 ITU 903 0°884 im

Warbon’4:0)..7h.5.2080 eee Oe 4:64 ‘216
LF) 3
ln we (Eee ee eee 1606 4:58 351 "286

|
ee 1420 410 | 412 243 |

In calculating the illumination constant Vm, these quality
determinations and the above visibility curve (V/Vm) were
used. The required spectral energy curves available were,
however, found to be inadequate. Hither the visible portion
is not known with sufficient precision or else the conditions
under which the whole curve was taken is not specified with
sufficient detail. After these curves have been freshly deter-
mined, for the sources whose luminous quality has been
determined, the constant V,,, should be determinable to perhaps
two or three per cent.

In the case of acetylene, spectral energy determinations
by Coblentz (B.S. Bull. vii. pp. 291-3) enable us to evaluate
Vm to about 5 per cent. uncertainty. We find for relative
integrals of energy and luminosity A/B=626/5:66=110°6.
For C/W we obtained 0°598, hence V,,=66°2 candles per
watt. The uncertainty arises from the uncertainty in the
“saturation ’’ of the infra-red part of the radiation for the
thickness of flame used. I hope soon to have a number of
more precise values of Vin.

I am greatly indebted to friends in the Photographic and
Chemical divisions as well as to colleagues in the Physics
division, who so cheerfully served as.subjects in obtaining
visibility data. JI am particularly indebted to Mr. Felix
Illiott, who recorded and reduced nearly all of the thousands
of observations.

Research Laboratory,

Eastman Kodak Co.
Rochester, N.¥., July 1914.

esata +9

XXXV. she Gyroscopic Theory of Atoms and Molecules *.
By ALBERT C. CREHORE f.

i obtaining the equilibrium positions of two atoms as
they unite to form a molecule, it was shown in a former
paper f that the distance between the atoms is great compared
with the radius of the orbits of their electrons, probably
more than a thousand times as great. The order of magni-
tude of the radii of the orbits of the electrons in the atoms
was there estimated as 107-!¥ em.§ This result follows from
the better known molecular dimensions and the calculated
ratio between molecular dimensions and the radii of the
orbits. Following this conception, certain modifications as
to the distribution of the electrons within the positive
electricity are suggested, not only to introduce the quanta
of Planck but also to account for both the X-ray spectra
recently published by Moseley and luminous spectra.

The present tendency among atomic theorists is to favour
with Rutherford an atom with a central positive nucleus
having electrons circulating in orbits large compared with
the radius of the nucleus. One reason for favouring this
atom is that it explains the observed scattering of the alpha
particles in their passage through matter. Another reason
is that it accounts for the large mass of an atom, as compared
with that of an electron, by the small radius of the positive
electricity, the mass of the hydrogen atom being about
1900 times that of the electron. The theory involves the

* Since this paper was communicated the work of calculating the
forces between any two atoms of the nature described in the text has
been in progress. The integral equations have been obtained when the
axes of the atoms lie in one plane, either parallel to each other in the
same or in upposite directions, or perpendicular to each other. These
equations enable us to specify the directions of the axes of the atoms in
a cubic crystal such as rock-salt or potassium chloride, and to show that
the whole crystal is a very stable arrangement. The distances between
the atoms agree with those calculated by Prof. Bragg within the limits
of error. The experimental work on crystals seems to be a confirmation
of the theory advanced in this paper, because the same fundamental
values here given of the size of the positive electricity and the speed of
revolution of the electrons in the atom have enabled us to construct
theoretically a crystal such as rock-salt, and to obtain the same distances
as found experimentally.

The derivation of the forces between two atoms and the proof that
these atoms may form a stable crystalline structure of dimensions
agreeing with the experimental determinations, forms the subject of a
future communication.

+ Communicated by the Author.

t A. C. Crehore, Phil. Mag. July 1918, p. 25.

§ Loe. cit. p. 56.

The Gyroscopic Theory of Atoms and Molecules. 311

principle that when an electrical charge is in motion, it
possess ‘$ an apparent mass that increases with the velocity,
and implies that all mass is due to this electromagnetic
origin. For high velocities, one-tenth or more of that of
light, experimental measurements show an agreemeut with
this theory. The suggestion has been made that perhaps
this electromagnetic mass constitutes the whole of the mass,
and we do not, therefore, need to assume anything else.
It is probable that the velocities of the electrons in the atoms
are of an order of 1/700 that of light, and to apply the
above result to them extends the law far beyond the range
of experimental evidence. The theory is. moreover, usually
applied to the positive electricity in the atom which is
supposed to be relatively at rest, a condition as far removed
ax possible from the experimental observation. !t requires
that as neutral atoms grow heavier with the addition of
electrons, the positive nucleus must grow rapidly larger in
radius to show increasing mass, the radius being directly
proportional to the number of electrons or to the mass, unless
some supposition is introduced which prevents the positive
electricity from amalgamating as it were, and preserves the
individual positive electrons so that their radii are not
changed, and the mass is increased by the additional numbers
of them, If this is the case, tt is difficult to understand the
nature of the forces tht hold them together, and I am not
aware that any definite hypothesis on this point has ever
been suggested. That some such thing must be the case
seems to be required by the experimental fact that when an
atom breaks up in radioactive transformations, atoms of
helium are viven off, each contai: ing an exact multiple of
the smallest positive charge. On the whele, this conception
that the principal mass is due to that substance which fills
the smallest volume, and that the radius varies directly with
the mass or artmlberr of electrons, is very unnatural. This
makes the volume eight times as much for double the mass.

On the theory here advanced, the mass of the atom is
shown to be proportional to the volume of the positive
electricity. This volume is first determined by independent
means, and when compared with the approximate volume of
the electron, the ratio is the same as the ratio between the
masses of the atom and the mass of electrons composing it,
about 1900. On this theory the mass per unit volume
everywhere, whether of positive or of negative electricity, at
slow velocities is the same and is cons stant, approximately
equal to 10” grams per cubic centimetre, as is shown in
a subsequent section.

312 Dr. A. C. Crehore on the

The distance 10-* cm. is sometimes referred to as of the
order of atomic dimensions. This seems unfortunate, be-
cause it implies that this is the approximate size of atoms.
This distance, obtained from the kinetic theory of gases,
means that the atoms in collision approach each other to
within about this average distance before rebounding, thus
behaving as though they are of this size. On the present
conception of atoms, it is the magnetic and electric fields
accompanying and surrounding ahent which determine this
distance of rebound, and in one sense this is the effective
size of the atom. !n another and more rational sense, the
size of the atom is determined by the size of the orbits of
the electrons composing it, and in this theory of the atom
these are of the order of 10712 em., from one to ten thousand
times smaller than the so-called frome dimensions.

It will aid in the discussion to poini out the characteristic
features of this corpuscular-ring gyroscopic theory of the
atom, not only because the small absolute dimensions of the
atom have altered the whole case, but because the iscendency
of the central nucleus theory has deterred many from serious
consideration of another form of atom. The small magnitude
of the electron orbits alters the case again, because the
electron, with a radius of the order of 107% em, is not of
a negligible size compared with the distances between
electrons in the same atom. LHven if the law of repulsion
between them is the same as for electric charges as we know
them in the gross, there is no reason for applying the inverse
square law of repul-ion to the electrons witlin the atom,
which at such distances does not hold for two charged
spheres. The law of repulsion may not be different for
single electrons and for aggregates of them, but in dis-
carding the inverse square law the equilibrium figures
originally calculated by Thomson using this law are greatly
modified.

In this theory, the volume of the positive electricity is
supposed to increase by a fixed amount with the addition of
each electron. This fixed increment in volume is the volume
of the smallest positive portion, or unit of electricity, having
a charge equal to and a volume larger than the electron.
It may be called the positive electron. The volume of this
elementary positive unit may be found by dividing the
volume of the positive electricity of any neutral atom by the
nuber of electrons it contains. There is just the same
difficulty here connected with the positive electricity as there
is in the central nucleus atom referred to in regard to the
amalgamation of the different positive electrons, and no

Gyroscopic Theory of Atoms and Molecules. 313

hypothesis is now introduced to permit of the positive
electricity again separating into definite fixed units when the
atom breaks up. It seems necessary to do this sometime.
Using the dimensions of the atom obtained in the previous
paper*, the volume of the positive electron comes out
2:7 x 10-8 cu.cm., corresponding to a radius of °86 x 107”
em. if taken as spherical.

The Number of Electrons per Gram of any Substance
Constant.

On any atomic theory, the number of electrons per atom
is approximately proportional to, and Rutherford makes it
one half of, the atomic weight. It seems as if there is
reason, from a comparison with the periodic table of the
elements, to take the number of electrons as about equal to
the atomic weight, but proportionality alone is sufficient to
show that the number of electrons per gram for all substances
is nearly constant. If the electrons per atom are equal to
the approximate atomic weight, then the number per gram
must be about 6 x 10”, the so-called gram-molecule constant,
which thereby assumes a very definite physical meaning.

lf the number of electrons per gram is constant, then it
follows on this theory that the volume filled by all the atoms
in a gram is also constant, since the vast majority of atoms
are neutral, and each electron is accompanied with positive
electricity having a fixed volume. It is possible, then, to
find the total volume of all the atoms in a gram, because it
is the same as the total volume of positive electricity in all
the atoms, the electrons being conta‘ned within the volume
of the positive electricity. It is found by multiplying the
constant 6 x 1023 by the volume of the unit positive elec-
tricity 2°7 x 10~** cu.cm., which gives 1°62 x 10-” cu.cm.
Disregarding the fraction ond taking the order of magnitude
as 10-2 cu. em., this constant expresses the volume of all the
atoms Ina cen of any kind of substance. If, for example,
a gram of water which ordinarily fills one cubic centimetre
could be compressed until its atoms are brought into con-
tact, or until all interspace between atoms is eliminated, it
would fill only 107? instead of one cubic centimetre. Or
again, if 10” grams of water, normally filling a space of
10” cu.cm., which is the volume of a cube one hundred
metres on a side, is similarly compressed until there is no
interspace between its atoms, it would only fill one cubie
centimetre.

* Loe. cit. p. O6.

314 Dr. A. C. Crehore on the

The spaces between the atoms on this theory are vast
compared with the total volume of the atoms themselves,
but, as before stated, this gives a somewhat false conceptiou
ot this atom. The electric and magnetic fields surrounding
each and extending to relatively great distances in a sense
determine the size of the atom. But, on the other hand, the
volume filled by the atoms themseves as above considered is
important. If we regard the ether as a continuum with
a uniform density everywhere, and consider that the positive
and the negative electricity are really portions of it differ-
entiated from the rest merely by possessing different energy
characteristics in some way not now specitied, the volume
filled by the atoms in a gram givesa definite meaning to the
term ether density. The density should be the reciprocal
of the volume of the atoms per gram, namely 10%, We
have just seen that, as far as density is concerned, we can
make 10” grams of matter out of one cubic centimetre of the
eether, assuming the density of the positive electricity to be
the same as the rest of the ether. An analogous case would
be to think of a large portion of the cosmos filled with -tars
and compare its density regarded as matter with that of the
Earth. The filmy structure of the cosmos compared with
the Earth would be analogous to that of ordinary matter
compared with the ether. ‘This conception is not new, but
the very close coincidence of the volume, 10~¥ cu. em., of
all the atoms in one gram as calculated from this theory with
the well-known reciprocal of the number 10”, which those
who have taken this view of the ether have accepted as the
approximate value of its density, is very significant.

We shall take 10’ as the value of the ether density, and
reversing the process indicated, determine from it the size of
the unit positive electricity or positive electron taken as
asphere. It comes out *735x107' cm. radius instead of
*86 as above.

On Mass.

If there are 6 x 10” electrons in every gram of any sub-
stance, and the single electron atom is that of hydrogen,

then the mass of the hydrogen atom is — = "166 >
gram. Taking the massof the electron as *878 x 10-7 gram,
the ratio of the masses of the hydrogen atom to that of the
electron is 1900. The radius of the positive electron as
above determined is *735x 107” cm., and a sphere having
a radius of *593x10~% cm. has a volume 1/1900 of the
positive electron. ‘Taking this latter figure as the size of

Gyroscopic Theory of Atoms and Molecules. d15

the electron, it then appears that the mass of every atom is
proportional to the volume of the electricity in that atom,
and that both the positive and the negative electricity have
the same mass per unit of volume, that is the mass is
10” grams per cubic centimetre, the value of the zether
density.

Beta Particles from Radioactive Substances.

A certain difficulty bas arisen in the central nucleus
atomic theory in explaining the beta particles from radio-
active matter. It seems certain* that the beta particles
in radioactive transformations cannot come from a ring
whose radius is comparable with 107% cm. For this reason
those who advocate the Rutherford atom with electron orbits
of a size calculated by Bohr 1, are forced to restrict the source
of the beta particles to the electrons which emanate from
the very inside rings or the nucleus itself. On the author’s
theory this difficulty vanishes, and any electron may give
rise to the beta particle as far as the size of the orbit is
concerned, because all orbits in neutral atoms are sufficiently
small, being of the order of 10-! em. radius.

The positions of the electrons outside of the nucleus in
the Rutherford atom have been estimated by Bohr by making
use of the conception of quanta, and Planck’s universal
constant “/.” Whatever the explanation of quanta may be,
it is now very generally admitted that they have a real
physical existence, and no atomic theory can ignore the fact.
It is just as possible on this new theory of the atom as
with the central nucleus atom to recognize the quantum of
Planck.

In this paper some of the older features of the corpuscular
ring theory are retained, but the whole is modified in certain
important particulars. The retained features are, first, that
the electrons are within the mass of the positive electricity
and are confined approximately to one plane, the enormous
frequency of orbital revolution being sufficient cause to
restrict them to this pline. Second, “the positions in this
plane are chiefly determined, as in the old theory, by the
electrostatic and magnetic forces, but with these differences,
that the forces of ‘Yepulsion no longer obey the inverse
square law on account of the near approach of the electrons
to each other, and that the existence of quanta also modifies
the equilibrium positions, somewhat changing the radii of
the rings in a manner to be described.

* J. W. Nicholson, Phil. Mag. April 1914, p. 544.
1 N. Bohr, Phil. Mag. September 1918, p. 488.

316 Dr. A. ©. Crehore on the

From these premises alone certain important conclusions,
which are in harmony with experimental observations,
follow almost axiomatically. In neutral atoms, as electrons
are added forming increasingly heavier elements, and as the
positive electricity increases in volume by a fixed amount
for each electron, retaining an approximately spherical shape,
it results that the diameter of the rings lying in a plane
must increase at a greater rate than the diameter of the
corresponding positive electricity, since the latter increases
nearly as the cube root of the number of electrons and the
former at a greater rate, somewhere between the first power
and the square root of the same number. There must, then,
come a time when, as electrons are added, the diameter of
the outside ring equals that of the positive electricity. No
more electrons than this can be accommodated within the
positive electricity, and there is, therefore, a superior limit
to atomic weights. As nearly as can be estimated, this
point is reached for that number of electrons which corre-
sponds to the element uranium, the heaviest of the elements,
and it is not surprising that there are no heavier atoms
found. In these heavier atoms the outside ring is most
unstable, and comparatively slight forces may drive an
electron outside of the positive electricity, where the law of
force suddenly changes allowing it to escape. It seems
natural on this view to expect to find self-radioactive
elements at the latter end of the periodic table, and in fact
we come to uranium, thorium, radium, radium emanation,
etc., all self-radioactive elements. The degree of radio-
activity should not follow precisely this order, as will be
evident when the irregularities introduced by the equilibrium
positions are taken into account.

Scattering of the Alpha Particles.

this form of atom as well as by the central nucleus atom.
A study of the electric and inagnetic fields surrounding a
single atom of this kind shows that it is possible for an alpha
particle to pass straight through the atomic field without de-
viation or to be reflected at any possible angle. The deviation
depends upon the relative directions of the axes of rotation of
the atom and of the approaching atom, and upon the direction
of the line of approach. The forces may be either attractive
or repulsive according to circumstances. While no quanti-
tative proof is here given of the precise law of scattering, it
must be conceded that the known laws of scattering supply

The scattering of the alpha particles may be explained by

Gyroscopic Theory of Atoms and Molecules. 317

no argument against this atomic conception. The immense
difficulties of obtaining the desired positive proof will be
evident to anyone who attempts it. The introduction of the
magnetic forces with the conception of polarity in addition
to the electrostatic forces is the source of the difficulty.

A-ray Spectra.

It seems as if the discovery of Prof. Laue* of a method
of measuring the frequencies of the X-rays by ihe use of
crystals is destined to give experimental measurements
which will eventually provide a complete working atomic
theory. Ihave endeavoured to account on this theory of
the atom for the series of X-ray spectra as determined b
Moseley f, and have worked out a distribution of the elec-
trons within the positive electricity to account for his two
principal series, the Kz and the Lz. Moseley has given the

general equation
Ya AON OO iia Ways aul eu hia eae CLO)

us representing approximately the experimental measure-
ments, where v is the frequency of the X-rays, A and } are
constants for a given series, and N is an integer, being a
series of ordinals increasing by unity from element to
element. ‘This equation gives a straight line by taking v? as

abscissa and N as ordinate. and may be written
IN Oba EO en it sae a (2)

where A is replaced by 1/a:.

The manner of finding the possible positions of the elec-
trons to give both the observed X-ray and the light spectra
in this form of atom may be something like the following.
It is first assumed in common with other theories that there
is no radiation of energy when the orbits of the electrons
are circular and the motion is in the ‘steady state at a fixed
anvular velocity. It is the disturbance of this state only
which gives rise to the radiation or absorption of energy.
It is also assumed that the angular moment of momentum
of each electron in every atom is the same and is constant,
that is ‘

mor, =k;",a constant,, . . . . (3)
where m is the mass, @ the angular velocity in the orbit, and
r, the radius of the orbit of the ring of n electrons. If is

* Laue, Friedrich, and Knipping, Miinck. Ber. pp. 303-322 (1912).
+ H. G. J. Moseley, Phil. Mag. Dec. 1918, p. 1024; April 1914,
p. 708.

318 Dr. A. C. Crehore on the

the frequency of revolution of each electron in its orbit,
then w= 27s.

Any disturbance of the uniform state of motion due to
outside causes will in general disturb the planes of the
orbits of the electrons as well as their eccentricities, and
thereby give rise to motions similar to those of gyroscopes
but more complex. Itis suggested that the resulting rapid
nutations, both natural and forced, cause the high fre-
quencies of the X-rays, and that the slow precessional
motions cause the light spectra. The difficulties of any
complete investigation compel me to resort to the analogy
of the gyroscope, the investigation of which is known. In
the case of a simple gyroscope with a rigid wheel, acted
upon by an external moment of force M, the frequency of
the resulting nutations is given by the expression

where @ is the angular velocity of the wheel, and C its
moment of inertia about the principal axis, and A its
moment of inertia about an axis in the plane of the wheel.
If we now conceive of the mass of a single electron as
uniformly distributed like a ring throughout its entire orbit,
then the moment of inertia C about the principal axis is
double A, the moment about an axis in the plane of the
ring. Hence, C/A =2, and

VE28% now + 3 en

That is to say, the frequency of nutation of the electron in
the single electron atom is twice its frequency of revolution.
Moreover, this frequency is independent of the external dis-
turbing force, though the energy is not, and is dependent
only upon the constitution of the atom itself. This shows
that the order of magnitude of X-ray frequencies, if due to
this cause, is not different from the trequency of revolution
of the electron in its orbit.

When the number of electrons in a ring is more than one,
the analo,y to the rigid wheel can hardly be used to give
even an approximation to the principal nutation frequencies.
We shall make the assumption that the frequency of the
nutations occurring in a ring of n electrons is proportional
to the number of electrons, 2, in the ring and is equal to

pte S'S

Gyroscopic Theory of Atoms and Molecules. 319
Comparing this with the equation (3) we derive the relation

VP n?

== aCOMsuaMh. ervey s «5. Ke)
Or PHA mee DN in ene 0) a RO)

where x replaces n2/7,, and is proportional to the square root
of the frequency of the X-rays. The corpuscular ring theory
gives the approximate values for both n, the electrons per
ring, and r,, the radius ef the ring, and hence values of «
proportional to the square root of the frequencies. Charting
these values points are obtained, as we pass across a series
of elements in the periodic table, which have some semblance
to the straight line series of Moseley. However, it is neces-
sary to restrict this to the Ke series which applies to the
lighter elements, as the number of electrons corresponding
to Zr, the first of the Le series, is too large to handle. The
best line to represent the Ka series when projected back to
the ordinal axis intersects at about three units instead of
unity as Moseley takes it, making b=3 instead of 6=1 in
equation (1). Rydberg * makes this constant exactly 3
in his revision of the Moseiey ordinals, and has in doing
this added 2 to each ordinal in the series of elements,
makin: N for aluminium 15 instead of 13. In deriving
the spectra in this paper, Rydberg’s interpretation of the
Moseley ordinals is used.

The next process is to abandon the approximate values of
the radii r, obtained from considerations of equilibrium and
proceed to calculate them on the basis of Moseley’s observa-
tions, assuming the points to he exactly upon his Ka line or
series. In so doing, we are at liberty to distribute the elec-
trons in rings in any manner, but that particular arangement
has been choxen which is demanded by the periodic system.
The resultin » arrangement, therefore, contains an explanation
of both the periodic system of the elements and the X-ray
series of Moseley.

In fig. 1, the line I represents the Ke series, the marks
upon it being the spectrum lines experimentally obtained,
but it begins at N=3, and the abscisse are proportional
but not equal to the square root of the frequency, being

equal to #= “The first point on the line at N=15 and

#=1°875 x 10” is found by taking Al to be the configuration
27=3, 9, 15, there being 27 electrons total with an outside

* J. R. Rydberg, Phil. Mag. July 1914, p. 147.

320 Dr. A. C, Crehore on the

ring of 15. Taking the volume of the unit of positive
electricity in absolute dimensions as above obtained, a fair
approximation to the absolute radius of the outside ring can
be made. Having thus found two points on the line repre-
senting this series, its position is determined. It is to be

igs
aia
N

HOU Ptp 7
a5, 10 RINGS /

TAyy {
a YETM,
/{] MER DS ue TMy if I]
el EUr |

Se a eke
NOpa_. G : ,
bt] NEL Ap Ape B

(ee 2g, A Se
X=W/k

noticed that all the spectrum lines in this series are not
caused by the outside ring of electrons. Al, Si, and P are
attributed to the outside ring; S, Cl, and A to the second
ring ; K, Ca, Sc, and Ti to the third ring; V, Cr, Mn, Fe,
and Co to the fourth ring ; Ni, Cu, Zn, Ga, Ge, As, Se, and
Br to the fifth ring; Kr, Rb, Sr, Y, Zr, Nb. Mo, and

possibly a few more to the sixth ring. Ina similar manner

Gyroscopic Theory of Atoms and Molecules. 321

the La series, which is represented by the line IT in the
figure, begins at Zr with a line due to the outside ring, and
for a few of the elements in this region, which have lines in
each series in Moseley’s table, the one is due to the first and
the other to the sixth ring of electrons.

If this mode of explaining the observed X-ray spectra is
correct, it is conjectured that several other series of lines
between the Ke and the Laexist. Ofcourse a determination
of the spectra on this theory depends entirely upon the accurate
determination of the frequencies of nutation of the electrons
in their orbits—a most difficult problem—but the simple
assumption made above that v=f,n points to a distribution
of the electrons in good agreement with the corpuscular ring
theory. There is no doubt that the complete solution of
these frequencies of nutation is more complex than the
assumption made; and even if this is one solution, it is not
surprising that Moseley has observed several other series of
varying intensity lying close to these principal series.

A way of approximating to the absolute value of the
frequencies is by the use of Planck’s constant. The energy
required to separate to a great distance an electron from the
unit of positive electricity may be taken as the minimum
energy change that ever takes place. If the electron starts
from the surface of the sphere of radius b=:735 x 10-” em.
as above determined, it is shown to be ¢e?/2b. LEquating this
to hs, where s is the frequency of revolution and fh is
Planck’s constant, we obtain

e? x
hs = 5 =15°5 x 10 eC et Te a ReMi es
where e477 10712, and) h=6:5 x 1072". _ Hence for the
single electron atom, which on this theory is taken to
represent hydrogen, the frequency of revolution of the
electron is*

and p= 2s=4:770x109. JS © ro

On the equilibrium theory, when there are two elec-
trons, they place themselves halfway from centre to circum-
ference of the positive sphere, and with only one electron
it would go to the centre, making the radius in the first
case *46 x ~!2 em., and in the second case zero. The suppo-
sition that the angular moment of momentum is constant for
all electrons and not zero, indicates that the true value

* The letter » is reserved for the nutation frequencies and s is
introduced for frequencies of revolution,

Phil, Mag. 8. 6. Vol. 29. No. 170, Feb. 1915. e

322 Dr. A. C. Grehore on the

of r; lies probably between these limits. The value
7,=°285x10-™ cm. has been chosen by an inspection of
the curve of radii for the atoms having 2, 3, 4, and 5 elec-
trons, it being assumed that these radii are the least affected
by the equal moment of momentum hypothesis. It must be
regarded as a rough approximation at best, but the results
that follow from it seem to be in close accord with the dis-
tances between atoms as experimentally determined.

Since the publication of the experimental equilibrium
figures obtained by electrostatically charged spheres* sus-
pended as pendulums in which the maximum number of
electrons was 20, an experimental series obtained with
magnets floating on water up to 75 magnets has been
published f. It is shown that the groups repeat themselves
periodically around the central groupings shown in fig. 2,
which accounts well for the pericdic system of the elements.

Fig. 2
9 © oe © ° ©0 6%e
e ° eo 6 oO SP) | ee ®6 08 @ Se
Se eo © © oo

Fig. 3 represents the atoms of the first nine series of the
periodic table according to these groupings. The outside
circle represents the boundary of the positive electricity on
the assumption of spherical shape, and the numerous small
black dots represent in magnitude the relative size of the
electrons. The number of electrons per ring is indicated by
the numbers in the lower left corner of each square, the
Rydberg ordinals in the lower right corner, and the symbol
for the element in the upper left corner. The radii of the
rings where electrons are shown are derived from fig. 1, and
this arrangement gives on this theory the X-ray spectra of
Moseley. Where certain rings are omitted, as, for example,
in chlorine, their radii are not obtainable from fig. 1. In
chlorine the ring of 6 and the single electron are omitted
because the lines lie to the right of the Ka series.

LTaght Spectra.

One of the pertinent criticisms that may be made against
any atomic theory is that it has not accounted for the
observed luminous spectra of the elements. Although. Bohr
has in a brilliant manner given an explanation of some of

* Loc. cit. Plate TTI.
+ E. R. Lyon, Phys. Rev., March 1914, p. 232.

323

Molecules.

Atoms and

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Theor

Gyroscopic

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324 Dr. A. ©. Crehore on the

the series of spectral lines, notably those of H and He, yetit
may fairly be said that luminous spectra have not been ex-
plained by any atomic theory. Nicholson * has shown ina
seemingly conclusive manner that these spectra are not
really accounted for on Bohr’s hypothesis. It is not sur-
prising that it is difficult to account for light spectra, but it
should not be inferred that there is not safficient basis in
this theory for a possible explanation. The range of X-ray
frequencies is from 1,000 to 10,000 or more times greater
than the frequencies of light, and it becomes necessary to
find periods of a comparatively low order of magnitude.
An estimate of the values of these slower or luminous
frequencies in the theoretical atom containing a single elec-
tron and representing hydrogen is given below, but it is
evident that there must be many periods of a low order in
more complex atoms that are not easily specified. Hach
atom must in fact be considered as a little gyroscope having
several rings or wheels of very great but different periods
of rotation. Ifthe axis of rotation is disturbed by any cause
certain nutations and precessions will necessarily occur.

In the theory of the ordinary gyroscope having a rigid
wheel there are two classes of periods due to the action of
external forces, the frequencies of the one being independent
of the external forces, and of the other dependent upon
them. If the gas giving the spectrum consists of individual
atoms, as may be the case with He, then these periods caused
by external forces may be of the first class and have constant
values whatever the outside disturbing force may be, they
being determined solely by the properties of the atom itself.
If the gas consists almost entirely of molecules of more than
one atom, then additional periods of a fixed value should be
obtained because of the constant effect of the atoms upon one.
another. The gyroscopic theory of these atoms is much more -
intricate than that of a rigid wheel. There is a flexible
rather than a rigid connexion between the electrons, and it
is necessary to have a more definite knowledge of the cause
which gives rise to the existence of quanta, or which holds.
each electron to a constant angular moment of momentum, if
this is a fact. .

Our theory has enabled us to make a complete specification
of the hydrogen atom consisting of one electron within the-
positive electricity, determining the volume of the positive
electricity, the radius of the orbit, and the frequency of
revolution of the electron, as well as the relative positions of

* J. W. Nicholson, Phil. Mag. July 1914, p. 90.

Gyroscopic Theory of Atoms and Molecules. 325
the two hydrogen atoms which form the diatomic molecule,
including their distance apart, and the angle of latitude that
the line joining their centres makes with the planes of their
orbits. Although it is the average value of the forces when
taken over a long time that determines the stable position of
equilibrium of the two atoms forming the molecule, yet the
instantaneous values of the forces on the individual parts
ot the atom varying during one revolution give rise to the
precessional motion which causes the light spectra. We
will now determine in absolute dimensions the distances
referred to, and then proceed to calculate for this simple
molecule the precessional period or frequency which gives
rise to the light spectrum of hydrogen.

The Hydrogen Molecule.

The mechanical force that two hydrogen atoms, having a
single electron each, exert upon one another when their axes
are parallel and in the same direction, may be derived from
equations (42) * and (44) of the former paper. ‘They show
that two such atoms come to stable equilibrium with each
other when their axes are in the same straight line, the
revolution of the electrons being in the same direction, the
phase angle between them being zero, so that the line joining
the electrons is always parallel to the line joining the centres
of their orbits. When the distance between the centres of
the atoms is

ya = oie ABO Cinna ey eee (bls)
where c is the velocity of light and s the frequency of orbital
revolution of the electron obtained from Planck’s constant
in (10), then it has been found that the atoms are in stable
equilibrium with each other.

It is to be noticed that the distance between the two
hydrogen atoms is very small compared, for instance, with
the distance between a sodium and a chlorine atom in rock-
salt, which is 2°814x10-8em.; 81:2 times smaller, and yet
the same values of the fundamental constants s and 7, serve
to show that we get an equilibrium distance of this larger
order when the sodium and chlorine atoms are used. This

(y YA
Vio

* In giving the coefficients Be,2, B4,2, &e., middle of page 70, Phil.
Mag. July 1918, a column of B’s was omitted. The first column
should read Boo=5;
gt 1S 15
B10,2 = q Bs,2; Biro= Bio2; Bisgja= % 312, 2.

o _
o Ss é
- Ba, 2 =5Bz Bé,2= 6 Ba9; Bsa=
o~ onl
5)

326 Dr. A. C. Crehore on the

peculiarity of the hydrogen atom will later serve to explain
some of the compounds into which hydrogen enters.

A small displacement in any direction whatever from this
position of equilibrium as origin gives rise to a restoring
force directed toward this origin. For small displacements
the restoring force is proportional to the displacement, giving
rise to harmonic vibrations about this point or origin. The
restoring force per unit of distance and per unit of mass along
the line joining centres of the atoms is

4 eB y
= Toe (12)
where a is the radius of the orbit of the electron and e its
charge, and £ the velocity of the electron divided by that of
light. Ifwe equate this force to the mass times the accelera-
tion per unit of distance, we find the frequency of oscillation
in the direction of the line joining centres to be

eB?

3 TES Fi
347 m?zaz

Vi (13)
where m denotes the mass. |
The restoring force per unit of distance and mass along
any line perpendicular to the line joining centres is
2 02°

25 a b>]

(14)

exactly one half the force in (12) along the line of centres.
The corresponding frequency of oscillation along the per-
pendicular line is, therefore,

My =o se

If we consider that the mass which is subjected to this
force is that of the single electron in the atom, we obtain in
numerical values taking e=4°77x 107; a=:285x10-”:
m="878 x10-% ; B=2msale ; s=2°385 x10"; c=3 x 10%,

Me= 112 x 195 }
My= *79x 108 J”

The wave-lengths of light corresponding to these fre-

quencies are
Aa=2680 x 10-8 em. along, i:
Ap=3800 x 10-S cm. perpendicular! ~

(16)

(17)

Gyroscopic Theory of Atoms and Molecules. 327

If we consider that the mass which is subjected to this
force is that of the positive electron in the atom, we obtain
in numerical values taking m=1°66x10~** and other
quantities as above,

N= '026 x 10% mi
Np='018 X 10% J

The wave-lengths of light corresponding to these vibrations
are

(18)

Na LFO0Ox LOrs ah (19)
Ayv— 165000 x 107% cm.

The light spectrum of hydrogen is not to be attributed to
these two simple vibrations of the electron alone (16) and
(17), but rather to the disturbances to which they give rise
in the motion of the single electron in its orbit.

It is remarkable that this calculated value of the frequency
of the electron perpendicular to the line joining centres
comes so close to the experimental value of the fundamental
constant in Balmer’s series of hydrogen lines. The wave-
lengths in Balmer’s series * of hydrogen lines are given by
the equation

9
m?
m2z—4

X= 3647-20 x 10-8

(20)

m being a series of integers.
. 9 . e
If we express Balmer’s series in terms of the frequency
instead of the wave-length, (20) may be written

sole (2 AA MING Sag MuANGARLAL 5.

m”
where

Bib tvis aso ties 15 SEUNG se ch ey 15
a 3647-90 21073 = 823 x10”, and a= —4b= —3°292 x 10”.
It has been shown that this law of Balmer’s can be
derived f from considerations of ordinary dynamics provided
there isa proper sort of gyroscopic connexion between the
twoatoms. Let wand v be scalar functions of the time and
of the position of points in the two atoms respectively at an

* J. S. Ames, Phil. Mag. vol. xxx. p. 55 (1890).

+ E. T. Whittaker, Proc. Roy. Soc. ser. A. vol. Ixxxv. No. A 578,
June 9, 1911, p. 262. The importance of this demonstration warrants
repeating in full, as given in the original paper, since we have shown
that the values are obtainable from the hydrogen atom.

328 Dr. A. C. Crehore on the

angle 6 such that the potential energy of the one atom is

represented by
Sau? + 4b (Sy.

2 ory
Jani + 10( 55)
where a and 6 are constants. ‘Then if the gyroscopic con-
0’v

nexion between the atoms gives rise to a term =, 08. aioe

and of the other by

the kinetic potential, the equations of motion are

tu Oy
au Oe — S462 =?
3,
i, OR O°u

Aez | DLae 7

If, now, wu and v are assumed to be simple periodic functions
given by the equations
u= A sin (nt +m@),

v=B cos (nt +m),

where m denotes an integer and n gives the frequency, the
solution of these equations gives

(a+m*b)A=mnB,
(a+mb)B=m’nA;

whence eliminating A and B, we obtain Balmer’s equation
for the frequency as in (21) above,

a
tn=b+ me

‘The low-frequency vibrations and the long wave-lengths
due to the vibration of the positive electron in (20) and (21)
may be considered as keat rather than light. When the
energy of these vibrations becomes onesies the whole atom
may depart so far from the equilibrium position that it will
not return to it and the substance becomes volatilized by
excessive heating.

Photo-Electriec Phenomena.

There are certain experimental facts connected with the
photo-electric effect which are not easily explained on any
atomic theory. It seems as if this theory contains elements
which will eventually lead to a more complete understanding

Gyroscopic Theory of Atoms and Molecules. 329

of these phenomena. When ordinary light falls upon a body
under the proper conditions electrons are emitted. The
velocity of each electron emitted is independent of the
intensity, but depends directly upon the frequency of the
light. The number of emitted electrons depends upon the
light energy or intensity, but the velocity cf the individual
electron does not. ‘The slightest change in the frequency of
the light produces a corresponding change in the emitted
velocity, and the velocity is a continuous function of the
frequency. No form of atom which is only capable of reso-
nance at particular fixed frequencies peculiar te itself would be
capable of such response to external forces. The gyroscopic
nature of the atom, however, renders it capable of responding
to the frequency of the impressed force. An analogous case
is to be found in the precessional motion of the earth due to
the comparatively slow revolution of the sun or the moon
in an orbit inclined to the plane of the equator, so that the
gyroscopic couple acting upon the earth varies with the
position of the sun or moon. It is well known that this
produces a periodic motion of the earth’s axis corresponding
to twice the frequency of the orbital revolution of the sun
or moon. ‘There is, similarly, produced in each atom upon
which the light falls a frequency double that of the light.

“Let fig. 4 represent the single electron atom

Fig. 4. upon which light is falling in the direction
SA indicated. If we imagine that the light pro-
f duces a pressure upon the electron, perhaps

in just the way that it produces a pressure
upon any small particles, then this pressure
will vary harmonically with the time corre-
sponding with the frequency of the light. This
is a very low frequency compared with that
of the electron in its orbit, and it will produce
a precession of the pole of the orbit having twice the
frequency of the impressed force, the light pressure.

The energy so received from ‘the light may accumulate
until an electron escapes. This is likely to happen always at
a critical velocity which is fixed by the character of the atom
rather than the intensity of the light and so not vary with
the light intensity. The energy, however, is abstracted from
the light. A greater light energy merely brings an in-
creasing number of electrons up to the critical point where
they quit the atom. A calculation of the manner in which
the electron may be ejected, especially in a complex atom,
is not undertaken at present, involved as it is with the
intricate precessional motions of the electrons. We are

330 Dr. A. C. Crehore on the

fortunate to be able to show with some degree of probability
that this form of atom contains the potentiality, which may
some time be more fully realized, of explaining these experi-
mental observations.

Synopsis.

1. The number of electrons per gram of any substance
is approximately constant. This follows from any atomic
theory which makes the number of electrons per atom pro-
portional to the atomic weight. If the electrons per atom
are approximately equal to the atomic weight, as in this
theory, then the well-known gram-molecule constant 6 x 10**
is equal to the number of electrons per gram of any
substance. Since, in neutral atoms, a positive electron
having a fixed volume accompanies each electron, it follows.
that the volume of all the positive electricity in a gram is
also constant. This volume is found by multiplying: the
volume of the unit positive electron by the number, 6 x 10°%,
of electrons per gram. Ina previous paper the volume and
radius of the positive electron were determined in absoluie
dimensions, 2°7x107®° cu.em., or ‘86x 107! em. radius.
Multiplying this volume by 6 x 10” gives 1:62x 10~ cu.em.
as the volume of all the atoms in a gram of any substance,
the electrons being within the positive electricity. The
coincidence of this volume per gram with the reciprocal of
the quantity 10%, which those who have taken this view
of the ether consider represents its density, is very sig-
nificant. Working in the reverse way from the ether density
as a basis, the radius of the positive electron comes out
73510712 em. Taking the radius of the electron as
"993 x 107% cm., the volume of the positive electron is 1900
times that of the electron. The mass of the atom is accounted
for by the larger volume of the positive electricity, the ratio
of the volume of the positive to the negative electricity being
the ratio of the masses of the hydrogen atom to the electron.
On this view the mass of positive or of negative electricity
is the same for equal volumes, equal to 10 grams per cu.cm.,
the ether density, and the mass of any piece of matter may
be calculated in grams by multiplying the volume filled by
all its atoms by 10?.

2. Beta particles in radioactive transformations may come
from any electron in the atom, since the order of magnitude
of the electron orbits is 10-" cm. In the central nucleus
theory they cannot come from the outside rings, and must be
restricted to the inner electrons or the nucleus itself.

3. The theory shows that a limit of atomie weights should

Gyroscopic Theory of Atoms and Molecules. dal

be reached when the number of electrons is so great that the
outside ring, which grows at a greater rate, equals in
diameter the positive electricity, w ‘hich point should occur
somewhere near the element uranium. Self-radioactivity is
attributed to the fact that when neur this limit the outside
ring of electrons is most unstable; and comparatively slight
forces may drive an electron outside the positive electricity,
where the law of force changes allowing it to escape, thus
breaking up the figure and requiring readjustment. The
self-radioactive elements should, according to this, occur at
the latter end of the periodic system, as they do.

4, There is assumed to be no radiation of energy from an
atom when the electrons describe circular orbits in the steady
state. A disturbance of this state may give rise to rapid
nutations of the electrons both natural and forced, accounting
for the X-rays. In the single electron or hydrogen atom
the natural nutation frequency is twice the frequency of
orbital revolution. ‘Che orbital frequency is determined from
Planck’s constant together with the size of the hydrogen
atom to be s=2°385x10, and the characteristic X-ray
frequency for ee should, therefore, be twice this
value.

5. When there is more than one electron in the same
orbit the natural nutation frequencies are not easily obtained
from analogy with gyroscopic equations, and an assump-
tion is made that these frequencies are (v=ksn) propor-
tional to the number of electrons per ring. Upon this
assumption, together with the grouping of electrons in rings
to represent the periodic table of the elemenis, a tentative
distribution of electrons is given, which would account for
the Ke and La series of X-ray spectra of Moseley.

6. The comparatively low frequencies of light are attri-
buted to the precessional frequencies of the electrons in their
orbits. A calculation of the simplest case, that of the single
electron atom, is made to determine the frequency that the
one atom in the hydrogen diatomic molecule causes in the
other atom. The hydrogen molecule is first determined
definitely, including the distance between the two atoms and
the angle that their two axes make with the line joining
centres. This distance is 1220 times the radius of the orbit
of the electron in the atom, equal to 347 x 107° em., and the
angleis 0°. The frequency of vibration of the electron in a
direction perpendicular to the line joining centres agrees
well with the principal constant in the equation expressing
Balmer’s hydrogen series. It has been shown by Whittaker
that if there are certain terms in the differential equations

332 Dr. N. Bohr on the Series Spectrum of

representing the gyroscopic connexion between the two atoms,
Balmer’s law may be accounted for, and the demonstration
is considered of sufficient importance to repeat in full.

The frequency of vibration of the positive electron in the
atom is 43°55 times slower than that of the negative electron,
giving long waves which may be considered as_ heat
radiation.

7. This atom is capable of response to any frequency of
light, because there are precessional frequencies produced in
the atom proportional to those of the impressed force. No
form of atom capable of resonance at fixed frequencies only
can possibly account for the experimental facts connected
with the photo-electric effect. It is suggested that light
pressure upon the electron, when the light falls at an angle
with the plane of the orbit, is responsible for inducing a
precessional variation of double the light frequency.

XXXVI. On the Series Spectrum of Hydrogen and the
Structure of the Atom.

To the Editors of the Philosophical Magazine.
GENTLEMEN,—

1° the January number of this magazine Dr. H. Stanley

Allen has published two interesting papers in which he
considers the effect on the series spectrum of an element if
the central nucleus of the Rutherford atom has, besides its
electric charge, the properties also of a small magnet. In
the first paper, it is shown that a nuclear magnet under
certain assumptions might give rise to a number of different
series of lines, instead of the single series of lines to be
expected if the nucleus consists simply of a point charge.
It is shown, however, that a magnetic field of the order of
magnitude which may be assumed to occur in the actual
atoms will be much too small to account for the different
series of lines which have been observed in the spectra of the
elements. In the second paper, the formule deduced in
the first are applied to the hydrogen spectrum, and it is
attempted by the help of the hypothesis of a nuclear magnet
to explain the very small deviations from the Balmer law
which have been observed by Mr. Curtis in his recent
accurate measurements of the wave-length of the hydrogen
lines. The moment of the nuclear magnet is found to be
approximately equal to that of 5 magnetons. The import-
ance of this result, if correct, is easily seen ; but it would
appear that some of the deductions made by Dr. Allen are
difficult to justify.

Hydrogen and the Structure of the Atom. 333

The application of the quantum theory in the calculation
of the effect of a magnetic field affords a very intricate
problem, since there are several possible ways of applying
the theory and each of them leads to different results.
The only guide on this question seems to be experiments on
the Zeeman effect. In the first place, it might be argued as
a serious objection against the method of calculation applied
by Dr. Allen, that an analogous calculation in the case of a
homogeneous magnetic field does not give results in agree-
ment with measurements of the Zeeman effect. I shall not,
however, try here to discuss this difficult and unsolved
problem *, but will only consider the way in which the
formule obtained in the first paper are applied in the second
paper to the hydrogen spectrum. In this application new
assumptions are involved, one of which seems hardly con-
sistent with the main principles of the theory. According
to Dr. Allen’s calculations, the presence of a nuclear magnet
leads to a splitting up of the lines in components situated
symmetrically with respect to the original lines, at any rate
if the square of the magnetic force is neglected. This
result, in itself, will not explain Mr. Curtis’s observation,
which consists in a small systematical deviation of the
“centre of gravity ” of the hydrogen lines from the position
calculated by the Balmer lawf. In comparing the theory
with experiments, Dr. Allen now uses only one of the two
components calculated. or this, apparently, no explanation
is offered ; it might, however, be justified by assuming that
the nucleus, on account of its smaJl moment of inertia, will
always take a position such that its magnetic axes will
coincide with the direction of the magnetic force due to
the rotating electron. In order to obtain an expression for
the frequency of the same type as the empirical formul:e
by which Mr. Curtis has represented his results, Dr. Allen
next assumes that the correction in one of the terms of his
formula can be neglected. This assumption amounts to the
neglect of the correction due to the nuclear magnet in one of
the “stationary states ” of the atom. It seems very difficult
to see how this assumption can be justified; for if the nucleus
is assumed to be a small magnet, it would appear necessary
to have the same magnetic properties for all the states of the
atom. According to the theory, these states ditfer only in
the size of the orbit of the rotating electron. If the correction

* In the special case of a homogeneous magnetic field the problem in
question is considered in some detail by K. Herzfeld (Phys. Zettschr. xv.
p. 193, 1914) and by the present writer (Phil. Mag. xxvii. p. 506, 1914).

+ W. E. Curtis, Proc, Roy. Soc., A. xc. p, 614 (1914),

334 Series Spectrum of Hydrogen and Structure of Atom.

is not neglected, Dr. Allen's formula will show a deviation
from the Balmer law which is much larger than that observed
by Mr. Curtis, and which has the opposite sign. It therefore
seems to me that the interesting suggestion of the nuclear
magnet and the calculation of its moment can hardly be
considered as supported by the experiments of Mr. Curtis.

I should like here to draw attention to an effect of
another kind, which involves a correction in the theoretical
formule for the hydrogen spectrum, z. ¢. the variation of
the mass of the electron with velocity. Jt seems necessary
to take this into account even if other effects may be in-
volved at the same time. Assuming that the orbit of the
electron is circular, and proceeding in exacily the same way
as that followed in the deduction of the Balmer formula on
the quantum theory, but replacing the expressions for the
energy and the momentum of the electron by those deduced
on the theory of relativity, we obtain the following formula
for the hydrogen spectrum :

‘3 a Maange & cee i )I
ia h3(m+M)\ n,? — n-? CN ae ke bi

where e and m are the charge and the mass of the electron,
M the mass of the nucleus, / Planck’s constant, and ¢ the
velocity of light. In the formula, terms are neglected which
involve higher power than the second of the ratio between
the velocity of the electron and the velocity of light. The
correction due to the last factor in this formula has the same
sign as the deviations from the Balmer law observed by
Mr. Curtis*. However, it accounts only for 4 of the
deviations observed.

In connexion with this discussion it may be remarked that
it seems hardly justifiable to compare the measurement of
Mr. Curtis with any theoretical formula unless the observed
doubling of the hydrogen lines is taken into account. Con-
sidering that the distance between the components is much
greater than the deviations from the Balmer law and that
the components are of unequal intensity, it is difficult to
know, in the absence of a theoretical explanation of the
doubling, the interpretation to be placed on measurements
of the “centre of gravity” of the lines. In a previous
paper I suggested that possibly the lines were not true
doublets, but that the doubling observed was produced by
the electric field in the discharge. As Mr. Curtis points
out, this suggestion does not seem consistent with the

* In the diagram in Mr. Curtis’s paper (loc. cit. p. 615) the curve

corresponding to an expression of the above type is inadvertently drawn
with its curvature downwards instead of upwards.

Notices respecting New Books. 339

observed ratio of the distance between the components
of H, and Hg, It seems also difficult to reconcile with the
observed unequal intensity of the components. It may
be mentioned here that there is perhaps another way of
explaining the observed doubling without introducing new
assumptions as to a complicated internal structure of the
hydrogen nucleus. For small velocities of the electron,
the calculation gives the same result whether the orbits
are assumed to be circular or not ; but taking the variation
of the mass into account, it can be shown that for higher
velocities the orbits will not be stationary unless they are
circular. In other cases the orbit will rotate round an axis
through the nucleus and perpendicular to the plane of the
orbit, in much the same way as if the atom were placed in a
magnetic field. It might therefore be supposed that we
would obtain a doubling of the lines if the orbits are not
circular. The frequency of this rotation of the orbit will
depend on the degree of excentricity. For very small
alterations from the circular orbit the ratio between the
frequency of rotation of the orbit and the frequency of

9) 9

aed

revolution of the electron is given by SOR which for
i Ne

n=2 is of the same order of magnitude as the doubling
of the hydrogen lines observed. In view, however, of the
great number of new assumptions involved in such ealeula-
tions, it seems to be of very little use to consider this
question in detail until more accurate measurements of
the distance between the components and especially of its
variation for the different hydrogen lines have been made.
I hope in a later paper to deal more fully with some of
the problems briefly considered here, and to discuss in some
detail the main principles involved in the application of the
quantum theory to the problem of series spectra and the
structure of the atom.
: Yours faithfully,
University, Manchester, N. Bour.
January 12th, 1915.

XXXVI. Notices respecting New Books.

The Theory of Relatwity. L. Stuperstern, Ph.D. Pp. viii + 295.
Macmillan & Co. Ltd.: London, 1914. Price 10s. net.

THXHIS book is founded on a course of lectures delivered at

University College, London; but the exposition has been
made more systematical and has been largely extended so as to
include all the most important aspects of the subject. The most

336 Notices respecting New Books.

important feature of the book is the attempt to make the subject
as real as possible by an examination of the experimental data
which are the main foundation of it. In most of the expositions
of the theory of relativity this experimental foundation is almost
lost sight of in the array of mathematical equations which, in
some quarters at any rate, seem to be the only vital thing. We
still meet with people who imagine that the whole of mechanics
is a series of deductions from Newton’s Law of Motion ; and that
in the exposition of mechanics no appeal is necessary to experiment.
The same might have been said of the pre-Newtonian principles
which experiment has shown to be erroneous. The same fascina-
tion for general pricciples led Einstein himself to forsake earlier
methods and to enunciate two general principles from which all
deductions were to be made. Dr. Silberstein is careful to sketch
the historical development of the subject from Maxwell, through
Hertz, Heaviside and Lorentz, to the final enunciation of his
fundamental principles by Einstein. In this sketch he is equally
careful to explain the part which the appeal to experiment has
taken. The result is that this presentation of the theory has
a real look about it, in contradistinction to those expositions which
are carried away by the fascination of the universalisation of the
application to moving bodies of a set of equations which were
primarily put forward to apply only to bodies at rest. The fact
is that without the result of Michelson and Morley’s experiment
there would be no justification for the theory atall. It is because it
gives the most direct explanation of their null result and is at the
same time not at variance with any other experimental fact, that
the theory may claim serious consideration. So much does the
reviewer feel this to be true that he would go further, and declare
that it will only be when further experimental data of a crucial
kind are obtained that the theory will run much chance of becoming
definitely accepted as scientific knowledge.

Meanwhile it is necessary that the consequences of the theory
should be examined in detail; for by doing this, information may
be gained of the kind of way in which further appeal to experiment
may be made. It is with this object in view that we would
recommend the present volume to be studied.

With regard to the mathematical methods adopted, we may
point out that most use is made of quaternionic formule; but
these are fully explained: so that no one need be deterred on this
account. Some use is also made of the matrix method of repre-
sentation employed by Minkowski.

It must not be thought that Dr. Silberstein is merely an
expositor of other people's work. The whole book savours of
originality, and no one who wishes to be abreast of this revolu-
tionary subject can afford to leave the book on one side.

It is beautifully printed and appears to be very free from typo-
graphical errors.

: ”
ia) oe

pee

vee
a

NaGaoka & TAKAMINE. | Anom a lou Ss

nes laa? Fie. 1b

: 5461 d: 4859
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(s) (p)

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Effect.

Heterogeneous Field.
H : 8700 to 17000.

P3 P3 Pro fee P-3 P.3

Phil. Mag. Ser. 6, Vol. 29, Pl. IIT.

iG, BO,
Pst Pt

Heterogeneous Field. (p)
H: 9500 to 15000,

NAGAOKA & TAKAMINE,

INES Ve

Hin kilogauss.

Fre. 8.

—-—> H in kilogauss.

~500 .
2

Phil. Mag. Ser. 6, Vol. 29, Pl. IV.

4359

OX in tne hel oe

-4C0 -300 -200
30

p-components.
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Phil. Mag. Ser. 6. Vol. 29, PI. V.

]. Mag. Ser. 6. Vol. 29, Pl. VI.

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LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND
JOURNAL OF SCIENCE: * ” e
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a ey os
[SIXTH SERIES.] we Pay,

MARCH 1915.

XXXVIII. The Condensation of Thorium and Radium
Emanations. By ALEXANDER Fieck, B.Sc.*

LARGE number of experiments have been made on
the temperature of condensation and other properties
of these emanations. As a result of their first experiments,
Rutherford and Soddy f found that they were condensed at
the same temperature, but more precise measurements made
them conclude that while radium emanation condensed at
—155° C., thorium emanation, on the other hand, was con-
densed over a range of temperature varying from —120° C.
to —155°C. The well-known experimental facts concerning
the chemical non-separability of certain groups of radio-
elements from one another and from certain common
elements have received a theoretical explanation by the
generalization of Russell, Fajans and Soddy, governing the
evolution of the radio-elements through the Periodic Table.
The elements which appear to be chemically identical occupy
the same place in the Periodic Table and are termed
“isotopic”? elements. The three radioactive emanations are
isotopic but, as they belong to the family of chemically inert
gases, they are, of course, chemically indistinguishable.
Professor Soddy suggested that it would be of interest to
examine whether the thorium and radium emanations could
be separated from one another by condensation, or whether,
as suggested by the work of Sir J. J. Thomson and Ashton
* Communicated by Professor F. Soddy, F.R.S
+ Rutherorg and Soddy, Phil. Mag. [6] Vv. p. 561 (1903).
Phil. Mag. 5. 6. Vol. 29. No. 171. March 1915. Z

338 Mr. A. Fleck on the Condensation of

on neon and metaneon, they would have the same temperature
of condensation *.

In a paper communicated to Section A of the British
Association meeting in Birmingham, September 1913,
Ashton brought forward evidence to show that atmospheric
neon consisted of two elements of different atomic weight.
These two gases could not be separated from one another by
means of fractional condensation, but the separation could
be effected by diffusion. The atomic weight of the new gas
was found by the positive ray method to be 22.

Experiments at Atmospheric Pressure.
Pp i}

The difference found by Rutherford and Soddy in the
behaviour of the two emanations when subjected to low
temperatures is definite enough, but it has to be remembered
that the concentration of radium emanation required to.
produce a given ionization effect must be approximately
6000 times the concentration of thorium emanation to produce
the same effect. The only possible way therefore to test
whether the two emanations have exactly the same conden-
sation point, is to have them thoroughly mixed before being
cooled to the low temperature. The methods used in the
first series of experiments were the same in principle as
those used by Rutherford and Soddy, in which the emanation
was mixed with air at atmospheric pressure. The apparatus
finally used is shown in figurel. An 80 feet gas-cylinder, A,
was used which was in direct connexion with a glass T-piece, B,
the vertical limb of which dipped below the water in the glass
jar. There were two marks on the stem of the T-piece, M,
and M,, and the level of the water in the jar was kept at M,.
whilst the level of the water in the tube was adjusted, by
altering the pressure of the escaping gas, to M,. Immedi-
ately beyond the T-piece was a short length of capillary
tube, C, so that by maintaining a constant pressure at B,
the amount of gas passing through C was directly pro-
portional to the time. The capillary tube was followed by
a tube containing phosphorus pentoxide, and then by a
piece of wide tube containing the source of thorium
emanation. This was a strong preparation of radio-thorium
which had been separated from a mixture of radium and
mesothorium. Gas coming through the apparatus was.
therefore mixed with a constant quantity of thorium
emanation after the equilibrium amount of emanation had
been removed. Beyond the active preparation was a three-
way tap, one limb of which went directly to the outer

* Soddy, ‘Chemistry of the Radio-Elements,’ pt. ii. p, 36.

Thorium and Radium Emanations. 339

atmosphere, while the other was connected toa copper spiral
made of tube of an internal diameter of 2 millimetres: The
end of the spiral was also connected to the outer atmosphere.

Fig. 1.
Pi
on ev
' Atmosphere
i 7
2k = AGS
Outer elii dig SS | i
t aed qi WY
Na rPentane GY
aie icl | < ] 7
iy —™ \vyvvv ZA

In the hollow of the spiral there was a vertical stirrer
driven by an electric motor, and the whole was enclosed by
a wide test-tube which contained pentane, cooled externally
by liquid air. The temperature was measured by a
previously calibrated iron-eureka couple, the hot junction
being kept at the temperature of melting ice.

Thorium Emanation alone.—The first experiments that were
made were with thorium emanation alone. Air was pumped
into A with the help of the liquid air machine until the
pressure was equivalent to that of 10 atmospheres. A constant
air-stream of 60 ¢.c. per minute was maintained across the
radio-thorium, the gas being sent direct to the atmosphere,
while the temperature of the pentane was adjusted to some
constant value. When the temperature had been fixed, the
air containing thorium emanation was then diverted through
the copper spiral and allowed to flow for 40 minutes. The
gas-stream was then stopped, the coil removed from the bath
and its y activity measured until the thorium active deposit
had attained its maximum after, approximately, three and a
half hours. The experiment was repeated, using a fresh
coil, for an immersion at liquid air temperature, and, again,
with the coil maintained at room temperature for the

7,2

340 Mr. A. Fleck on the Condensation of

standard time. The small activity obtained by this last
measurement was deducted from all measurements taken at
low temperatures. The fraction f of the emanation con-
densed is therefore given by the expression
a A,—Ar
aa LL.
ign aes As (1)

where Ay, Ay, and A, are respectively the activities obtained
for immersions of a coil at some low temperature T, liquid
air, and room temperatures. ‘I'he curve obtained by plotting
the fraction condensed against the temperature is shown
(figure 2, Curve A), and from it, it is evident that the

Fig. 2.

cmanalion condensed.
? S 2 >
> [a ~]

Fraction o
i

Curve A. Thorium emanation alone.
Curves B and C. Mixed emanations: B, radium; C, thorium.

thorium emanation is gradually condensed until about
—154° C., at which temperature it becomes completely
condensed.

Mixed Thorium and Radium Emanations.— The object
of this preliminary experiment having been attained, the
experiment with the mixed radium and thorium emanations
was proceeded with. The iron cylinder was exhausted and
14 millicuries of radium emanation introduced, after which
the pressure was again raised to 10 atmospheres. The
cylinder was allowed to stand over night to ensure complete
diffusion of the emanation, and the experiment carried out
the following day. In this experiment, the current of air
carrylng emanation was 100 c.c. per minute, and the time of
exposure was again 40 minutes at the given temperature.
After the exposure, the temperature of the coil was lowered
to that of liquid air and kept there for 34 hours, 7. e. until

Thorium and Radium Emanations. 341

the radium active deposit had come into equilibrium with
the condensed emanation and the thorium active deposit had
reached its maximum activity. The coil was then warmed,
and the radium emanation in it displaced by a stream of
fresh air. y-ray measurements were immediately commenced
and continued for at least three hours, after which the
activity obtained is solely due to thorium active deposit.
By multiplying this quantity by the factor 1°119 the value
of the maximum activity of the thorium active deposit is
obtained. When this latter quantity is subtracted from the
initial activity, the remainder gives the effect due to radium
emanation alone. The experiment is repeated, of course,
for liquid air and room temperatures, and the fraction of
the emanation condensed, both in the case of radium as well
as of thorium, is obtained by substituting in expression (1).
The experiments for the points shown in the curves were
carried out in the course of a few hours, so that the radium
emanation had not materially decayed while they were in
progress.

Curve B (figure 2) is that obtained for the condensation
of radium emanation, and curve C that for thorium
emanation.

It is seen that the two curves are quite distinct, thorium
emanation being apparently more easily condensed than the
radium emanation. It has to be noticed, however, that the
form of the two curves is the same, while Rutherford and
Soddy found that when tested separately their shapes were
different. It has also to be observed that the speed of the
gas-stream influences the position of the condensation curve,
in the case of thorium the curve is displaced to the right by
increasing the amount of gas passing through the coil per
minute. It is thus evident that the exact temperature at
which the emanations condense depends to a large extent on
the physical conditions prevailing during the experiment.

There must also be considered the effect of the vapour
pressures of the condensed emanations. It is well known
that even at the temperature of liquid air, radium emanation
has an appreciable vapour-pressure, and we can assume that
thorium emanation will likewise have that property. In
the case of radium emanation the vapour phase will be
continually swept away, while in the case of thorium the
condensed emanation quickly changes into the active deposit
which remains. Assuming for the moment that the two
emanations have the same condensing point, there would be
found relatively more thorium active deposit than radium
active deposit. Consequently the thorium emanation would

342 Mr. A. Fleck on the Condensation of

appear to be more easily condensed than the radium. The
above results are therefore not a proof that the two
emanations have dissimilar properties of condensation.

Experiments at Low Pressures with Radium Emanation.

It was then decided to experiment on the emanations in
llosed tubes to avoid sweeping away the vapour phase. The
main idea of the method was to have the emanations
enclosed in a tube at very low pressure and to assume that
a quantity of gas would diffuse throughout the tube instan-
taneously. This assumption was proved to be correct by
experiments that will be described later.

As no work on the condensation of the emanations at low
pressures has previously been done, it was thought advisable
to work, in the first place, with the single emanations.

Apparatus—For this experiment, the apparatus and
position of electroscopes is shown in figure 3. <A lead

Fig. 3.
Section
throug h

gs 3s

cylinder, M, 44 cm. long and 11 cm. diameter, with a hole
2 em. diameter bored along its axis was used, and it had on
top three lead bars of length equal to that of. the lead
cylinder and each of cross-sectional area of 25 sq.cm. On
the right-hand side was a large sensitive y-ray electroscope,
A, while on the left-hand side were (1) the very sensitive
B-ray electroscope, B, described by Soddy* in a recent
paper, and (2) the small y-ray electroscope, C. The electro-
scopes were arranged to be slightly above the axis of the
lead cylinder, so that if a radioactive tube was in its centre,
no rays except those from any portion of the tube purposely
exposed could reach an electroscope without passing through
considerable thicknesses of lead. The object of this arrange-
ment was that experiments could be started with a large
quantity of radium emanation, using the small y-ray electro-
scope for measurements, and that when the emanation
decayed to such an extent that the effect on this electro-
scope was too small, measurements could be continued on
the large y and finally on the B-ray electroscope.

Two thin-walled glass tubes, 60 cm. long and 1°5 em.

* Soddy, Phil. Mag. [6] xxvii. p. 215 (1914).

Thorium and Radium Emanations. 343

internal diameter, were made, and into one was introduced,
with the help of liquid air to get rid of uncondensed gases
in the usual way, 14 millicuries of emanation, while in the
other 0°07 millicurie was placed. It was estimated that
the glass walls of the tube cut down the § rays to 31 per
cent. of the total. The tubes were sealed at a capillary
provided for that purpose, when only one bubble was obtained
by each stroke of the Topler pump; the sealed end will be
referred to as the “liquid air end,” while the round end
will be spoken of as the “ experimental end.”

Method of Experiment.—Whenever the tubes were not in
use, the liquid air end always stood in a depth of at least
5 cm. of liquid air, so that any time after three hours one
could assume that there was no activity at the experimental
end. When a determination was to be made, the tubes were
withdrawn from the liquid air and the experimental ends
quickly placed 5 cm. deep in a bath of pentane which had
been previously cooled to some desired low temperature.
After an exposure of 40 minutes, during which the tem-
perature was kept constant, they were withdrawn and placed
alternately under the electroscope for about 20 minutes,
while measurements of activity were made, in a standard
position in which all the tube, with the exception of 10 cm.
of the experimental end, was screened by the heavy lead
cylinder. The liquid-air ends were then replaced for at
least three hours in liquid air before another experiment at
a different temperature was made. lExperiments for 40
minutes immersion of the experimental ends at liquid-air
and room temperatures were also made. ‘The weak tube
was measured on the §-ray electroscope and the strong tube
initially on the small y-ray electroscope.

A series of such determinations for these two tubes at
different temperatures were simultaneously made, and then
the strong tube was laid aside for 10 days. Another set of
experiments were made with it alone, measurements being
made this time on the large y-ray electroscope, after which it
was laid aside for the same period, and then redetermined.
This time the 8-ray electroscope was used for measurements.

It was proved in the first place that all the emanation was
condensed by liquid air by removing the liquid-air end from
liquid air and immediately measuring the activity of the
experimental end, when it was found that after extrapolation
backwards, the activity at the instant of withdrawal was
only 0:3 d.p.m., while after 40 minutes at the temperature

of the room the activity was 18°8 d.p.m.

The activities obtained for a low temperature experiment

344 Mr. A. Fleck on the Condensation of

usually started three or four minutes after the removal of
the tube from the pentane bath, and were extrapolated back
to the instant of withdrawal. Some convenient instant was
chosen, and by noting the interval of time between the time
of experiment and the chosen instant, and by assuming the
radium emanation to decay exponentially with a period of
average life of 5°55 days, the quantity obtained above by
extrapolation was multiplied by a factor to give the activity
that would have been obtained had the experiment been
carried out at the chosen instant. The effect of the decay of
the emanation was thus eliminated. After this correction
had been made the fraction of the emanation condensed is
obtained by substituting in expression (1).

Results. — The curves obtained are shown in figure 4.
Curve A is for the weak tube, curve B is the first curve
for the strong tube, and curves © and D the second and
third curves respectively for the same tube. Dealing first

Fig. 4.
1:0

{ emanation condensed
o ° > °
ro ab o o2

Fraction o

curve A(weak) ee. Be I“ strong) oN,
curve C(2"4sTron cuyve D 3°stron oo
60° -%0° - 100° 8) aa “140° 160° ra) 3

Radium emanation in exhausted straight tubes.

with the curves obtained from the strong tube, it is observed
that each curve shows a marked maximum point at or near
—161° C., and that as the concentration becomes less the
condensation curve is moved to the left. Thus curve D is
roughly parallel to curve B, but about 8° C. to the left.

Thorium and Radium Emanations. 345:

Curve C is intermediate in position. The curve for the
weak tube approximates more closely to the first strong tube
curve, and it is to be noted that although it exhibits no very
marked maximum, there are three points that consistently
indicate its presence. The last set of readings that was
made with the strong tube was prolonged into comparatively
high temperatures, and it is at once seen that there is a
second maximum at about 72°5 C., and that there is a long
portion of the curve between 115° C. and 143° ©. during
which the fraction of the emanation condensed remains
constant.

The results of this experiment show (1) that different
tubes, in general, have different condensation curves, (2)
that as the concentration of the emanation diminishes it
becomes easier to condense, and (3) that there exist two
maximum points, one at —161° ©. and the other at
—72°5 C,

Luminosity Experiments.

The existence of at least one maximum point was then
studied in another way. A good many years ago, Rutherford
noticed that if a tube containing radium emanation was
partially immersed in liquid air, the greatest intensity of
the luminosity occurred just above the surface of the liquid
air, and that as the level of the liquid air was raised the
ring of greatest intensity was raised as well. This expe-
riment was repeated by supporting a tube, containing 14
millicuries of emanation with a very small quantity of
other gases, on a cork floating in lquid air. The ring
of brightest intensity was obtained, but it was found that, no
matter how long the tube was kept in liquid air, all the
luminosity could not be obtained on this ring. There was
still a large fraction scattered over that part of the tube
beneath the level of the liquid air. It was proved that this
ring was not due to a collection of ice nor due to a particular
state of the glass, because the ring persisted after the whole
tube had been warmed to room temperature. To test whether
the effect was due to some electrical condition of the glass
which would cause an accumulation of the active deposit in
this particular region, even although there was no concentra-
tion of the emanation, a glass tube, silvered on the inside, was
filled with the above mentioned quantity of emanation. If
the effect had been due to electrical causes, the silver would
have dissipated any charge that had accumulated in one
region.

The ring of intensified luminosity was still observed. The

346 Mr. A. Fleck on the Condensation of

only conclusion that can be deduced is that the vapour pressure
of the emanation is lower at some temperature slightly higher
than that of liquid air, than it is within certain limits of
temperature intermediate between these two. That this low
value of the vapour pressure is not lower than the value at
liquid air temperature is shown by the fact that the emana-
tion, no matter how long it 1s exposed to very low temper-
atures, will not collect completely in the region beyond the
liguid air.

These observations, therefore, form further evidence of
the existence of a maximum, probably the one at —161° C.,
found in the quantitative measurements with the straight
tube.

Experiments by another method in a two-limb tube.—F urther
experiments were made with radium emanation in a two-limb
tube of the form shown in fig. 5. The greatest length of the

tube was 70 cm., while the short limbs were each 18 cm.
long and were 15 cm. distant from one another. As before,
10 em. of the long tube projected beyond the lead eylinder
(the same as used previously) and the activity of the un-
screened portion could be measured on either the small y or
on the f electroscopes, which occupied the same positions as
previously described.

The end of limb A was always immersed in liquid air
except when an experiment at some low temperature was in
progress, while limb B was enclosed in the larger gas-tight
tube D, through which a current of air, freed from CO, by
soda-lime and from moisture by sulphuric acid, could be
drawn by a water-pump. The taps T, and T, regulated the
gas stream, and the end of the thermocouple was at the same
level as the end of limb B.

The tube was charged with 14 millicuries of emanation.
‘The advantage of this tube was that it was unnecessary to
remove it from its position under the electroscope to make a
determination.

Thorium and Radium Emanations. 347

The tube was kept at room temperature until the active
deposit had come into equilibrium, and the y activity of the
exposed end was then determined as accurately as possible by
repeated measurements. Limb A was suddenly plunged into
liquid air, and y-ray measurements were continuously made
until the activity became very small, when alternate y and @
measurements were made for a time, after which the latter
alone were continued. The @-ray measurements continued
to decrease until the activity due to the tube was 6°9 d.p.m.
Calculated from the factor obtained when alternate readings
were made, it was found that the:tube had an initial 6-activity
of 7550 d.p.m., so that taking the ratio of these two values,
there is 0'0915 per cent. of the emanation uncondensed in a
good vacuum at liquid-air temperature.

The determination of the condensation curve for the
emanation in this tube was then proceeded with. The method
was to regulate approximately the temperature registered on
the voltmeter by surrounding the wide tube with a cylin-
drical flask of liquid air,a stream of air being kept con-
tinuously going through the tube. When the temperature
was satisfactorily adjusted and the activity of the experi-
mental end had been determined, the liquid-air vessel keeping
the emanation condensed in limb A was quickly removed and
continuous readings were made with the electroscope for 10
to 15 minutes. The temperature was kept constant by altering
the volume of air going through the wide tube. Limb A was
replaced in liquid air exactly at the end of five minutes, and
the increase of activity which had resulted during this time
was read off from the curve and then a correction made, as
previously explained, to eliminate the effect of the decay of
emanation.

In these experiments, what was actually observed was the
quantity of emanation volatilized, in distinction to the earlier
ones which gave the amount of emanation condensed. The
fraction of emanation volatilized is, therefore, B,/Bp where
3B, is the increase, after correcting for decay, obtained by
exposing limb A to room temperature for the standard time
when limb B is at the temperature T, and where B, is the
increase of activity similarly obtained when both limbs are
exposed to room temperature. The fraction of emanation
condensed / is therefore given by

348 Mr. A. Fleck on the Condensation of

Results —Fig. 6 shows the curves obtained; all values
were worked out by the same methods, but the points marked

Fig. 6.

ame |

Fraction of AS uae condensed

: O O=G B x
-60 -80° ~100° -120° -14.0° -/60° -1$0°
Radium emanation in exhausted two-limb tube (fig. 5).

with circles were obtained between March 20th and 26th and
those with crosses between the 26th and the 28th of the same
month. It will be seen that they lie on two curves, one of
which crosses the other.

With reference to curve A, it will be noticed that here
again there is evidence of a maximum point about —80° C.,
but that there is no trace of the maximum previously
obtained at —161° C. In view of the comparatively large
variation of the points between —100° C. and —150° C., it is
not wise to attach any value to the apparent change of slope
at —160° C. It will be noticed also that the curve in the
neighbourhood of the maximum is much less steep in this

a than in the case of the maximum at —72° C. in curve

D (fig. 4). This and also the absence of the maximum at
—161° C. may be due to the fact that in this experiment
there is always a slow gradient of temperature owing to the
upward current of air in the wide tube, whereas in the
previous experiments, where pentane was ‘used, the change
from the low temperature to that of the room was very quick.

The second curve was obtained in the course of attempts
to add more points and to verify those already obtained.

Thorium and Radium Emanations. 349

The surprising result was found that the points were lying on
a totally new curve, that the emanation did not start to be
condensed until after —150° C., and that then it became
completely condensed very quickly. At the conclusion of
the experiment the vacuum of the tube had not appreciably
deteriorated.

The only suggestion that can be made to explain the two
curves is that some quantity of gas had been liberated from
the walls of the tube which had a great effect on the con-
densation of the emanaticn. Whatever the cause, the results
show the great difficulty attaching to experiments of this
character.

’ Huperiments at Low Pressure with Thorium Emanation.

Apparatus.—For the thorium experiments the same pre-
paration of radio-thorium which had been used for the work
at atmospheric pressure was put into a small cylindrical brass
drum which was closed at the ends by filter-paper fixed on
with sealing-wax. Tor the first experiments a straight tube
was taken, fig. 7, at the bottom of which a small test-tube

Fig. 7.
eae game
P,0, Fray et ee Rar Experim ental
End : adio-Uhoyrum. Te: end.

filled with phosphorus pentoxide was placed, and a depression
was made in the glass so that this tube could not slide away
from itsend. The radio-thorium drum was then introduced
and made to take up a position mid-way along the tube,
where it was fixed by sealing-wax. There was also a glass-
wool plug placed between the drum and the sealed end close
to the former. The total length of the tube was 60 cm. and
the diameter 1°5 cm.

The tube was thoroughly evacuated by attachment over
night to a charcoal bulb cooled in liquid air, after which it
was sealed. The two ends of the tube will be distinguished
as the “P.O; end” and the ‘“‘experimental end.” In this
experiment the large y electroscope, A, in the same position
as shown in fig. 3, was used to measure the activities of the
experimental end. For this purpose the tube was placed in
the lead cylinder with 10 cm. of that end projecting to
the right.

The P,O; end was placed in liquid air and allowed to stand

350 Mr. A. Fleck on the Condensation of

there for three days, and then the activity of the experi-
mental end was determined. This quantity was found to be
be 2°90 d.p.m., and it must be treated as an additional
“natural leak’? and always deducted from any subsequent
measurements.

Test for instantaneous diffusion—An experiment to de-
termine whether the assumption as to instantaneous diffusion
was justified was carried out with this tube.

The P.O; end was allowed to stand in liquid air for three
days, and then the activity (1,) of 10 cm. of the P.O; end
was measured. This gave the activity due to the equilibrium
amount of active deposit that could be collected at this end.
The experimental end was then immersed for the same time
in liquid air and the activity (Iy) of the equilibrium amount
of active deposit that: could be collected at this end found.
The values of these quantities were respectively 332°6 d.p.m.
and 319-1 d.p.m.

Let k be the fraction of the emanation present in one half
of the tube crossing the glass-wool plug per second,
to the other half ;

Q, the total amount of emanation set free from the
drum ;
nthe number of molecules that cross the glass-wool
plug per second from the P.O; end to the experi-
mental end ;
and m the number that cross in the reverse direction.

For the P,O; end in liquid air, 1,=Q, and for the experi-
mental end in liquid air Aly=n—m; but in this case m=0,
and therefore Al,=n:

but n=k(Q—Iy)=Xly,
1 (k4+2X)= (AE?
Ty/Ip=h/(k +2).

Since I, is nearly equal to Ly, it follows that £ must be very —
large compared with A; 2.e. that the fraction of emanation
diffusing through the glass-wool plug is very large compared
with the fraction decaying per unit time. The assumption
as to instantaneous diffusion in the case of this thorium
emanation tube is therefore justified, and it follows that in
the case of radium emanation, where there is no glass-wool
plug, instantaneous diffusion also takes place.

Method of Experiment.—The method of making a determi-
nation of the fraction of emanation condensed at any
particular temperature was as follows. The P.O; end had

Thorium and Radium Emanations. 351

been standing for at least four hours in liquid air, so that it
could be assumed that any activity more than the irreducible
quantity mentioned above was decaying exponentially with
the period of thorium B. The tube was withdrawn from
liquid air, and during the ensuing 40 minutes the activity, L,
in excess of the irreducible activity at the beginning of that
interval was determined. At the end of that period of time
the experimental end was placed 5 cm. deep in a pentane
bath previously cooled to some low temperature. After a
40 minutes’ exposure at this temperature, the experimental
end was withdrawn and the P.O; end replaced in liquid air
for another definite time, 33 hours, when the tube was again
withdrawn from liquid air and the activity, N, of the experi-
mental end determined as quickly as possible. Then the
P,O; end was again placed in liquid air for at least four .
hours before another determination at some other temperature
was made.

When the measurement N is being made, the quantity L will
have decayed to Lx e—’, where 2 is the radioactive constant
of thorium B and ¢ is the time that has elapsed between
making the measurements L and N. In determining the
initial leak prior to the exposure, the emanation is uniformly
distributed throughout the tube and a certain quantity of
active deposit will be obtained on the experimental end. The
activity from this active deposit is exceedingly small while
that measurement is being made but will not be, by any
means, negligible when the final measurement is taken. If,
however, the time during which the initial measurement is
made is always constant, the final measurement will be the
sum of three quantities, the quantity Le, a variable
quantity due to the 40 minutes’ exposure at some low tempe-
rature, and a constant quantity due to the uniform distribu-
tion of the emanation during the initial measurement. The
activity, D, due to the 40 minutes’ exposure is therefore
given by

10) an ean ei 5)

where Q is a constant activity due to active deposit collected
during the uniform distribution while the initial measure-
ment is being made. The fraction of emanation condensed
at any temperature T is obtained by finding the values of D
for that temperature, room and liquid air temperatures, and
substituting the values Dy,-D,, and D, for Ay, A,, and A, in
expression (1). The numerator and the denominator of this
expression therefore consist of the difference of two values
for the expression (2), and therefore the quantity Q does not

352 Mr. A. Fleck on the Condensation of

require to be known. It is only necessary to keep the time
employed in making the initial measurements always constant.

Results.—The curve obtained is shown in curve A, fig. 8.
It exhibits no points that require detailed reference, but it
will be observed that the shape of the curve is similar to that
obtained by Rutherford and Soddy for thorium emanation
mixed with air at atmospheric pressure. The greater part
of the emanation is condensed between —165° C. and

—170° C.

Fig. 8,

bon condensed.
° ° °
= ~~ oo

Fraction of emanation
¢

13) ie
O : 2
OD @ - @
-60° -$0° - 100° -120° -/4.0° “/60° “/K0°

Curve A. Thorium emanation in exhausted straight tube (fig. 7).
Curve B. Thorium emanation in exhausted bent tube (fig. 9).

As the net activity for the amount of emanation condensed
at liquid air temperature was only 11 d.p.m., the curve was
not prolonged to high temperatures.

Haperiments at Low Pressure with Mixed Thorium and
Radium Emanations.

Apparatus and Method of Experiment.—Experiments were
now started on the mixed emanations in a sealed tube at
low pressure. A new type of tube, as shown in fig. 9, was
constructed of glass permeable to 8 rays and was filled with
0:07 millicurie of emanation. ‘The thorium preparation was
slightly stronger than that used in the previous experiment,
and was, as before, in a filter-paper drum which was fixed

Thorium and Radium Emanations. 353

in the bulb of the tube by sealing-wax. rays only were
used and the electroscope was in the position relative to the
lead cylinder shown in fig. 3.

The short limb of the tube is the P,O; end, and 10 cm.
of the long limb is the experimental end.

P,O; end from liquid air.
The P.O; end was, as before, immersed for three days in
liquid air, and the irreducible activity of the experimental
end determined, and this quantity is always treated as a
natural leak. As in the thorium straight tube, the P.O;
end must always have been four hours at least in liquid air
before a determination at some temperature is carried out.

The following is the method of carrying out such a deter-
mination. The activity, L, of the experimental end was
measured. The tube was then removed from the lead
cylinder and the experimental end placed as quickly as
possible 5 cm. deep in a pentane bath previously cooled to
some low temperature T. After 40 minutes of exposure
at this constant temperature, the tube was replaced in the
lead cylinder and the P,O; end immediately immersed in
liquid air. The activity of the experimental end is deter-
mined during the 15 minutes following this replacement, and
then the final measurement is made after the P,O; has been
immersed in liquid air for 34 hours.

The curve obtained from the measurements made during
the time immediately following the exposure is extrapolated
back to the instant of withdrawal of the tube from the pen-
tane bath. Let the quantity so obtained be M, and let the
final activity be N. M is made up of three factors, (a) the
activity contributed by L, 2. e. Lxe-°7), since ¢ for the time
of exposure = 40 minutes; (>) the activity contributed by
thorium active deposit obtained during the 40 minutes’ expo-
sure ; and (c) the active deposit from the radium emanation
condensed during that exposure. (c) when found can be
taken as a measure of the radium emanation condensed. It
is better, however, for the sake of accuracy to allow (6) to
increase to its maximum and to measure it after 34 hours.
If Eis the maximum activity obtained from the thorium

Phil. Maa. 8. 6. Vol. 29. No. 171. March 1915. 2A

354 Mr. A. Fleck on the Condensation of

emanation condensed, then E= N — Le-®, since ¢ = 5 hours.
When the experiment has been carried out for room and
liquid-air temperatures, the fraction of the emanation con-
densed is found by substituting the values of H,, H,, and
EE, in the expression (1). The value of (c) cannot yet be
determined.

A whole series of such determinations were made, and
then the tube was laid aside until no trace of the radium
emanation could be detected. A large number of deter-
minations as described above were then made for liquid-air
temperature alone, and the average value of the ratio

M—Lle-°”
N— Le-5

was found. This was determined as 0°326.

This factor was then used to multiply all values of
{N—Le-*} obtained during the experiments with the
mixed emanations, and for each temperature in that set of
experiments the value of

0°326{N — Le} + Le-°A

was subtracted from the corresponding value of M. The
result gives a measure of the radium emanation condensed
at a particular temperature at a certain time. The value so
obtained was corrected for the progressive decay of the
radium emanation as before to the value, G, which would
have been obtained had the experiment been carried out at
the chosen instant. The value of G is determined for room
and liquid-air temperatures and, as before, the fraction con-
densed is obtained by substituting Gy, G,, and G, in the
expression (1).

Example-——An example, for which the detailed curves
obtained are shown in fig. 10, will make the calculation
clearer. Curve A shows the curve obtained from the
measurements taken over the whole six hours during which
the experiment was carried out, while curve B is obtained
by plotting on a larger scale the measurements made imme-
diately after the exposure at the low temperature. This
latter curve was used to obtain the value of M.

Temperature of experiment, —159°5 C.
Time of experiment, 1 day after chosen instant.
Initial activity — 26°0 d.p.m
Trreducible activity = 4°§ d.p.m.

*. Value of L = 21-2 d.p.m.
Activity after exposure = 80°5 d.p.m.

*, Value of M

Thorium and Radium Emanations. 355

Activity after 34 hours = 44°8 d.p.m.
*, Value of N = 40°0 d.p.m.
Lx e707 aa 20°35 d.p.m
Lx e7° == 15°30 d.p.m.
*, Hp=N — Le- = 24°70 d.p.m.
, ol M—0°326(N — Le) — Le
=75°7—8:05—20°35 = 47°30 d.p.m.
Fig. 10.
Time in mnules ___, . curve B
20 4 i) 12 6 20

ts—>

y in arbilrary tint

Activil

C=]

O Time in hours __ 4 frayve A 6

Example of observations used to plot curves shown in Fig. 12,

Since the experiment was performed one day after the
chosen time this last quantity has to be multiplied by 1:2 to
eliminate the effect of decay. This gives the value of Gy to
be 56°6. It was found. in other determinations that H, and
H,, were respectively 6°3 and 57°5 and that G, and G,, were
| 10: 9and101°8. The fraction of thorium scariaion conden
at —159%5 C. is therefore

24-7 —6°3 :
tar eee
and of radium emanation at the same temperature,
56°6 —10°9 zit
TSE eae Ne ati

Variation of the fraction of the emanations condensed at
liquid-air temperature with lapse of time.—Another compli-
cation arose, however, in this experiment from a cause

Se AD

356 Mr. A. Fleck on the Condensation 07

similar to that already discussed (page 348) and shown in
fig. 6, namely, the alteration in the behaviour of the tube
in successive measurements. During the course of the
experiments three exposures of the tube were made at liquid-
air temperature, and it was noticed that the quantities of
emanation obtained in the experimental end (after all cor-
rections had been made) became smaller as time progressed.
When these quantities were plotted against the time, it was
found that in both cases the points lay on a straight line as
shown in fig. 11. Itis further observed that the same per-
centage decrease is obtained in both cases. This is an im-
portant observation, and if it could be confirmed, it would
be strong indirect evidence that these emanations are non-
separable by condensation. This introduces an uncertainty
but, from the fact that the points in fig. 11 lie on straight
lines, it was considered justifiable to correct for this by
reading off the values for E, and G, at intermediate times
from the curve (fig. 11).

Rios.
110
x
| =
100 2
S =
2 a
"40 o=
= :
=
= ce
= ¢o 30 u
3 S
< =
4
fF "Jo

‘oints for Radium are marked O. Thorium

Curves showing progressive change of the proportions of emanations condensed
at liquid-air temperature in the same tube with lapse of time.

Results.—The different fractions obtained at various tem-
peratures are shown in the accompanying table, and also
numbers to indicate the order in which the experiments were
made. Itis at once seen that at any temperature the frac-
tion of radium emanation condensed is greater than the
corresponding fraction for thorium emanation. Viewed as a
whole, this is the only result that can be arrived at, that
radium emanation appears more easily condensed than
thorium emanation. When more fully examined, however,
the points separate themselves into two lots each of which
lies on acurve. These curves have been separated and are

357

Thorium and Radium Emanations.

Temperature.

Order in
which exper.
was made,

198 Oo, cuspanecs
SAO 2 Seameanes
—152°6 ........
DoD Siatens
N92 wesnmenes

—161°°5 .........|

coast | Se ee ee
= 1649... ec ees
SMOG arcana ces
SGT OO ne enakue

1610 DO eccdiess..
= OOS eerenwes es
DO vawtactos setae
HN BOO cc deev els

Numerator
of
expression(1).

Ol
2°3
18
2°3
16:5
19
28'5
CU
21:3
38'7
15°7

SESE SSO |

Fraction
condensed,

TABLE.

Tiortum EMANATION. Rapium EMANATION.
pra = (oe eens aetna
Denominator Denominator
of expres- Fraction |Numeratorof| of expres-
sion (1). Read) condensed. |expression(1).| sion (1) from

from fig. 11. fig. 11.
468 0:0024 10:1 83:4
540 0:0426 91 96:0
45°05 0:0599 11:9 79'S
54:25 0:042 18°55 96°5
51:2 0°359 45°70 90°9
43°6 0:0435 76 768
52°6 0541 70°85 93:0
42:0 0183 31°8 740
46°3 0:46 54°1 81:9
55:9 0:693 97°6 99:2
40:0 0343 47°5 70:1
44°9 0:462 55°7 79:0
51:0 0-779 94:3 90:0
43:2 0839 64°6 761

‘Set of curves

| to which
points belong.

POPP Her eee ee

358 Mr. A. Fleck on the Condensation of

shown in fig. 12, and are to a certain extent similar tu the
pair of curves shown in fig. 6. There is, however, this
difference, that consecutive points in the former experiment

Fig. 12.

1-0 :
{ %
08 |

on condensed
= CJ
a
x

a
e 5] |

—*

(
|

Faction of emanals
S
<4 A

SSS SS . — - a sail

-120' a) 160 “180 ~/50 -4O -19Q"
Points for radium ave shown thurs 9—9—o
Pe eee onlin he) m.. 2c eee Geena

Curves obtained for mixed radium and thorium emanations in
exhausted tube (fig. 9).

lie on the same curve, but in this latter consecutive points
may lie on either. The curves of set A are roughly parallel
to but from 7 to 10 degrees lower than those of set B.

Repetition of this experiment.

A second experiment which was made was mainly concerned
with an endeavour to repeat the previous observations with
regard to the proportional decrease in the quantity of emanation
condensed as time went on. The 8-ray electroscope remained
in its former position, but the small y-ray one was placed
close to the P.O; end, so that the total amount of emanation
condensed in this end could also be determined.

Another quantity of radium emanation was sealed in the
tube, and the determination of the quantity of emanation
condensed by liquid air was made every third day. Only
one experiment was made for a higher temperature, and it
confirmed the previous result that radium emanation was
apparently more easily condensed than the thorium emanation.

Thorium and Radium Emanations. 309

However, it was found in this case that the quantities of

thorium and radium emanations condensed on different dates

remained appreciably constant, and that therefore the pro-

portional diminution effect was not an invariable property
of the mixed emanations.

Thorium Emanation alone.

An opportunity was taken after all the radium emanation
had decayed to determine the condensation curve for thorium
emanation alone in this tube.

The experiments were carried out as explained on page 353,
and the values tor E and the fraction of emanation con-
densed as stated on page 354.

The curve obtained is shown in fig. 8, curve B, and it is
seen that the greater part of the emanation is condensed from
— 162° C. to —180° C.

Theory of the Experiment.

It is difficult to compare the preceding experiments, where
the fraction of the emanation condensed at a given tempe-
rature has been studied, with the condensation of an ordinary
gas where the vapour-pressure of the gas rather than the
fraction condensing isa function of the temperature. It may,
however, be pointed out that even were the radium and thorium
emanations physically identical as regards volatility, a differ-
ence in the direction found, namely, that the shorter lived
emanation will appear more volatile, is to be expected under
certain circumstances. In the case of thorium emanation
there is a steady supply of, say, # atoms per second to the
gaseous phase and a steady disappearance of the same number
per second, Aw from the condensed phase, and (l—A)# from
the gaseous phase, where A is the fraction condensed. Equi-
librium is established when Aw more atoms enter the con-
densed phase per second than leave it. In the case of radium
emanation, the numbers entering and leaving the condensed
phase are equal. In the case of mixed emanations, indis-
tinguishable in volatility, the ratio of the concentration of
thorium to radium emanation must be higher in the gaseous
than in the condensed phase, since the ratio of thorium to
radium emanation condensing is higher than the ratio of
thorium to radium emanation volatilizing. Hence the thorium
emanation must appear more volatile. Whether or not this
effect is large will involve the absolute time which, on the
average, a molecule of emanation spends in the gaseous
phase before entering the condensed phase. If this is com-
parable with the period of average life of the atom, the effect
will be marked. |

360 Mr. A. Fleck on the Condensation of

Let the fraction of the condensed emanatien entering the
gaseous phase per second be g, and of gaseous emanation
entering the condensed phase per second be p, so that 19
and 1/p represent the average lives of the emanation molecule
in the condensed and gaseous phases respectively. Let the
fraction of the total emanation condensed be A.

For the radium emanation

gAyp=p(l—Ag)
Ap
or pl = GA.)

For the thorium emanation
gAy+AA = p(l—Ay)
ay Comer
A+q 1—Ay
Consider now a case where the radium emanation would
be completely condensed and where the thorium emanation
once condensed does not volatilize again, 7.e. letg =0. The
thorium emanation is not completely condensed unless A is
negligibly small compared with p. The ratio of the condensed
to uncondensed portion is that of p to X, or of 1/A to 1/p, that
is the ratio of the period of average life of the thorium ema-
naion to its period of average life in the gaseous phase before
condensing. If 1/A=1/p, that is, if, on an average, the
emanation spends 78 seconds after formation before con-
densing, only one half will be condensed. The results
obtained may therefore be due to the rapid disintegration of
the emanation and the apparent separation effected by con-
densation may be a time separation cf isotopes differing in

period, which is familiar enough, rather than due to a true
difference of volatility.

General Remarks.

It is observed in these experiments that the radium
emanation condensation curves have only the slightest
resemblance to each other, and the same remark applies to
the thorium emanation curves. Any quantity of emanation
in an exhausted sealed tube has at least one, and may have
two, condensation curves peculiar to itself. The natural
conclusion to be derived is that the residual gases play an
important part in determining the condensation curve of
emanation in any particular tube. This is perhaps not to be

Thorium and Radium Emanations. 361

wondered at, as their concentration may be many times as
large as the concentration of the emanation.

In none of the experiments with thorium emanation nor
with mixed emanations has any trace of the maxima pre-
viously observed been found, and their presence in the earlier
experiments, therefore, seems to be due to special circum-
stances. The fact that they were not found when P.O; was
enclosed in the tube, suggests that the presence or absence
of water vapour may have a deciding influence. They might,
of course, be due to the formation of a molecular complex,
but at present there is no evidence on which a decision as to
their cause can be based.

It is worth noting that these experiments involve, for the
first time, the consideration of the time which an atom
spends in the gaseous phase in equilibribrium with a liquid.
In the case of thorium emanation, a determination of this
magnitude might be based on methods similar to those here
employed, if the true condensation curve could be deter-
mined.

ISUMMARY.

(1) When thorium and radium emanations are mixed with
air at atmospheric pressure, thorium emanation appears to be
condensed about 5° C. above the radium emanation.

(2) This apparent difference is probably due to the radium
emanation in the gaseous phase over the condensed phase
being swept away by the air current.

(3) As the concentration of the emanation in a highly
exhausted tube diminishes the emanation becomes more easily
condensed.

(4) In certain circumstances the condensation curve of
radium emanation exhibits two maxima, one about —75° C.
and the other about —161°C. It is suggested that the
existence of this property may be dependent on the presence
of water vapour.

(5) The existence of at least one maximum in the con-
densation curve is contirmed by studying a glass tube,
containing a large quantity of radium emanation, floating
vertically in liquid air. A ring of bright luminosity occurs
just above the surface of the liquid air, and this ring is not
due to a condition of the glass nor to a concentration of the
active deposit away from the emanation.

(6) Internal changes are liable to take place inside the
tube which will completely alter the condensation curve
obtained.

(7) At liquid-air temperature in a highly exhausted tube

362 Mr. F. Lloyd Hopwood on a Qualitative

0°0915 per cent. of radium emanation remained uncondensed
in the one determination that was made.

(8) The condensation of thorium emanation in a highly
exhausted tube was also studied. |

(9) When the two emanations are mixed in such a tube
the radium emanation appears to be more easily condensed.

(10) This may not be due to a real difference in the con-
densation points but is probably caused by the rapid disinte-
gration of the thorium emanation.

(11) In one experiment indirect evidence was obtained
which seems to point to thorium and radium emanations
being non-separable by condensation.

I desire to take this opportunity of expressing my thanks
to Professor F. Soddy, F.R.S., for the strong thorium pre-
parations and quantities of radium emanation, for the
preparation of the tubes employed, and for the section entitled
‘“‘Theory of the Experiment ” in this paper for which he is
responsible. J am indebted to him also for his interest in
the experiments throughout their entire course.

I have to thank Miss Hitchins, B.Sc., also for the capable
assistance which she gave me in carrying out the manipula-
tions and lengthy measurements involved.

Physical Chemistry Department,

Glasgow University.
December, 1914.

XXXIX. On a Qualitative Method of Investigating Thermionic
Emission. By F. Luoyp Horwoop, B.Sc., A.R.C.Se.*

HE present writer discovered some time ago f, that the
movement of charged bodies in the neighbourhood of
a glowing carbon-filament lamp produced, under certain
circumstances, a displacement of the loops of the filament.
Similar observations were recorded by Eve t at about the
same time. The results of a further study of the phenomena
and their application as a basis of a qualitative method of
investigating the emission of electrified particles from in-
candescent bodies, are set forth in the present paper.

Experiments with Carbon Filaments. |

When an electrified rod, charged with electricity of either
sign, is brought near an unlighted ordinary 200-volt carbon-
filament lamp, the loops diverge in a similar manner to the

* Communicated by Prof. A. W. Porter, F.R.S.
7 ‘Nature,’ March 1914. t bid.

Method of Investigating Thermionic Emission. 3.68:

leaves of an electroscope, the divergence disappearing on the
removal of the charged rod.

If a negatively charged rod is brought near the lamp when
the filament is glowing, the loops diverge, the divergence
again disappearing on the removal of the rod, or if the
filament touches the glass walls of the lamp. |

When, however, a positively charged rod is brought towards.
the glowing filament, no movement of the loops occurs. On.
rapidly removing the positively charged rod from the neigh-
bourhood of the glowing filament the loops diverge. If the
divergence is sufficiently great for the loops to touch the
glass walls of the lamp, they immediately spring back to:
their original position, but if they do not tcuch the walls,
they will remain in the displaced position, in some cases for
several minutes. When in the displaced position produced
in this way, the loops respond very readily to the movement
of any bodies whether charged or uncharged, in their
neighbourhood—“ twiddling”’ the fingers near the lamp sets.
them into violent vibration. Placing the bulb momentarily
in contact with the metal cap of an electroscope gives a
negative charge to the leaves.

Should the bulb be touched by the hand or a Bunsen flame
be rapidly passed over it, the displacement of the loops, their
response to the movement of an uncharged body, and the
ability of the bulb to charge an electroscope on contact, all
disappear.

The above effects may be explained as follows :—

It is well known that when a carbon filament is raised to
incandescence in the neighbourhood of a cold conductor
maintained at a positive potential, both being enclosed in an
evacuated vessel, a continuous stream of electrons passes
from the hot filament to the cold conductor. The emission
under these circumstances has been studied in great detail
by Richardson* and others. If the cold conductor is not
maintained at a positive potential by external means, the
emission from the hot filament proceeds until a certain
limiting difference of potential between the filament and cold
conductor is attained. This limiting potential depends on
the geometrical configurations of the cold conductor and the
hot filament, and its value has been obtained in certain
cases, both theoretically and practically, by W. Schottky f.
In the case of an ordinary carbon-filament lamp, this limiting
potential difference is quickly established between filament
and walls of bulb when the filament is glowing, and an

* Richardson, Phil. Trans. 1903.
t Deutsch. Phys. Gessell.. Verh. xvi. 10, May 1914,

364 Mr. F. Lloyd Hopwood on a Qualitative

equilibrium state is attained. On bringing up a negatively
charged rod, no further electronic emission can take place
and the loops diverge in a similar manner and for the same
cause as the leaves of an electroscope. When a positively
charged rod is brought up, the conditions are made more
favourable to the electron emission, and the negative charge
induced on the filament by the rod escapes from the filament
to the walls of the lamp, giving a negative charge to the
inner surface of the bulb on the side nearest the rod. Thus,
owing to the escape of the induced charge, the loops are
unaffected by the approach of a positively electrified rod.
Before the removal of the rod, the filament will be in
equilibrium under the joint action of the charges on the rod
and on the inner surface of the bulb. On the removal of the
positively charged rod, the negative charge on the walls
induces a positive charge on the filament and produces the
observed displacement of the loops.

Touching the bulb with the hand or playing a Bunsen
flame on it, neutralizes the negative charge on the inner
surface by permitting the accumulation of a positive charge
on the outer surface of the bulb. The charging of the
electroscope is obviously due to the same cause.

The response to the motion of the hand will be due partly
to a redistribution of the charge on the walls and partly toa
change in the potential difference between filament and
walls.

Null Eject at High Temperatures *.

When various carbon filaments were heated above their
ordinary temperatures by running the lamps above their
normal voltage, they were found to be insensitive to the
motion of charged bodies in their neighbourhood, although
they were very sensitive to such motions at lower tem-
peratures.

It was sometimes found that filaments of high candle-
power lamps were also insensitive when run at their normal
voltage, but sensitive at lower voltages (temperatures).

It was thought that the null effect in these cases might be
due to one of the following causes :—

* In a preliminary note on this effect which appeared in ‘The
Electrician’ of July 31, 1914, a different explanation from that given in
the present paper was advanced, but in the light of further experiments
has had to be abandoned.

Method of Investigating Thermionc Emission. 365

a. An anomalous increase in the flexural rigidity of the
filament owing to some change in structure due to
a molecular rearrangement such as is known to
occur in iron and silica at high temperatures ;

b. The residual gas in the lamps becoming ionized to
such an extent by the hot filaments that it acts as
a protective conducting-sheath towards external
electrostatic fields ;

c. The emission of both positive and negative ions in
sufficient quantities to neutralize an induced charge
of either sign on the filament.

Eixperiments were carried out to test these hypotheses.

Some difficulty was experienced in devising an experiment
to test the mechanical properties of a carbon filament at high
temperatures, but finally the lamp shown in fig. 1 was used
for this purpose. The filament was subjected to stress by

Fig. 1.
(2)

Fig. 2.

placing it in a fairly strong magnetic field while the heating
current passed through it.

No evidence of any increase of rigidity with rise of
temperature was obtained, the yield of the filament in-
creasing with rise of temperature for all the temperatures
used.

To test hypotheses (6) and (c), an incandescent lamp with
two similar unanchored “hairpin” filaments was used (see
fig. 2). On subjecting the residual gas to the ionizing action

366 Nir. 1 Lloyd Hopwood on a Qualitative

of the beta rays from 20 milligrammes of pure radium
bromide, and to both hard and soft X-rays from a powerful
X-ray tube, it was found that, when due precautions were
taken to prevent the discharge of the charged rods by the
atmospheric ionization produced by the radiations, any
ionization of the residual gas in the lamp had no appreciable
influence on the displacement of the filament.

Galvanometric measurements of the thermionic current
between one filament when raised to incaudescence and the
other (cold) filament, when the cold filament was maintained
first at a positive potential and then at a negative potential,
showed a comparatively large negative emission, but gave no
certain indication of a positive emission from the hot filament.
Air was then readmitted into the lamp and pumped out
again by means of a Topler pump. It was noticed that
when the filaments were first raised to a dull red heat after
the re-evacuation of the lamp, a positively charged rod
produced a divergence on approach and a negatively charged
rod a displacement of the loops on removal, thus showing
that at the low temperature at which this took place, the
filament emitted positive ions only.

This effect was only temporary and disappeared on con-
‘tinued heating.

The experiments of Richardson, Wilson, and others *
show, however, that in similar cases a small permanent
leak of positive ions still persists after the larger temporary
one has disappeared, and that this leak (which can only
be measured by an electrometer method) is very much
greater at high than at low temperatures, although at the
higher temperatures it is much smaller than the negative
emission.

Now in the present experiments the inducing charges,
though at high potential, are very small, so we may conclude
for the reasons given above that the null effect at high
temperatures is due to the emission of sufficient positive and
negative lons to remove the induced charges, while the
positive emission at lower temperatures is too small to
dissipate the charges induced by the negative rod sufficiently
quickly to prevent the motion of the filaments. |

Experiments made with glowing platinum filaments in the
open air tend to confirm this conclusion, for at a red heat

* Vide J. J. Thomson, ‘Conduction of Electricity through Gases,’
2nd. ed. p. 214; and Schottky, Phys. Zevtschr. July 1914.
+ + Glass rods rubbed with silk, and ebonite rods rubbed with flannel,
avere used throughout.

Method of Investigating Thernuionie Emission. 367

they were found to be attracted by a positively electrified but
not by a negatively electrified rod, while at temperatures
near the melting-point of platinum neither rod attracts the
filament. Itis of course well known that platinum in air
emits chiefly positive ions at low temperatures and an excess
of negative ions at high temperatures.

Experiments with Metal Filaments.

The knowledge that a glowing filament is attracted by a
rod charged with electricity of the same sign as the ions
emitted by the filament, and that the removal of a rod charged
with electricity of opposite sign causes a movement of the
filament provided there is a solid dielectric between them on
which to receive the ions, has been used by the author as the
basis of a qualitative method of investigating thermionic
emission.

As the results obtained from experiments on the alloy
nichrome are typical of those obtained from the metals and
alloys mentioned below, a detailed account for this case only
will be given.

A long filament (40 to 50 cm.) of No. 40 8. W.G. nichrome
wire was attached by silver soider to the leading-in wires
of a metal-filament lamp from which the glass envelope and
original filament had been removed (see fig. 3).

This method of suspension was found to be most convenient

Fie. 3 and possesses the advantage that it is readily

svi connected to the ordinary lighting circuit by
means of the usual bayonet-type holders.

At a dull red-heat the filament was readily
d attracted by a red charged with electricity of

either sign, showing that there was no appreci-
| able thermionic emission of either sign at this
| temperature.

by At a bright red-heat (current 1°5 amps.) the
| filament was attracted by a positively charged
: rod, but not by a negatively charged one. A
sheet of glass was then interposed between the
filament and the charged rods. This had no influence on
the action of the positively electrified rod, but the filament
was attracted towards the glass screen when a negative ‘rod

was removed from its neighbourhood.

For the purpose of investigating the emission in air at
pressures less than the atmospheric pressure, and in other

868 Qualitative Method of Investigating Thermionic Emission.

gases, the filament was mounted in the apparatus shown in
fig. 4. The use of the ground-glass stopper facilitated the
replacement of one filament by another, while

the side tubes permitted the introduction of Fig. 4.
different gases and the regulation of the

pressure by attachment toa pump. The hot

nichrome showed the attraction produced by

a positively charged rod and the kick due to

the removal of a negatively charged rod in

air at atmospheric pressure and at pressures

down to less than one centimetre of mercury.

It gave similar indications in hydrogen
and CO,.

We conclude that nichrome at a bright red-
heat in air, hydrogen, and CO, emits a large
number of positive ions, and that ils negative
emission is insignificant.

Of the following metals and alloys all were
tested in air and a few in hydrogen and CO,. In every ease
they gave a positive emission.

The substances examined were :—

Tron, nickel, copper, nichrome (an alloy of nickel and
chromium), brass, phosphor-bronze, silicon-bronze, platinoid,
eureka, tinned copper, and “ galvanized ”’ iron.

So far as the writer is aware, the emissions from most of
the alloys mentioned above have not previously been in-
vestigated.

Advantages of the Method.

The method of investigation outlined above is very suitable
for a rapid qualitative examination of the emissions from a
large number of substances in air, and other gases, at various
pressures. It requires no special manipulative skill, and use
is made of only such apparatus as is generally found in a
Physics Laboratory. It provides some striking lecture
experiments. Incidentally, it was noticed that incandescent
filaments mounted as above (fig. 3) are very susceptible to
the approach of a magnet, and the laws governing the
mechanical action of a magnet on current-bearing conductors
may be readily demonstrated to large audiences by their
use.

Summary.

Experiments are described illustrating the difference in
behaviour of incandescent carbon filaments under the influence
ef small positive and negative charges. Explanations are

T he Ionization of Metals by Cathode Rays. 369

given of the observed effects and also of the null effect
obtained at very high temperatures. A method of investi-
gating qualitatively the thermionic emission from various
bodies is developed and applied to the case of the emissions
from wires of various alloys in air and other gases. A
number of these had not previously been investigated.

In conclusion the author desires to express his indebted-
ness to Professor A. W. Porter, F.R.S., and to Professor
QO. W. Richardson, F.R.S., for their interest in the above
experiments, and to thank the Staff of the Hlectrical Depart-
ment of St. Bartholomew’s Hospital for placing their powerful
X-ray installation at his disposal, and for the use of their
standardized amount of radium bromide.

XL. The Ionization of Metals by Cathode Rays.
By Norman CAMPBELL, Sc.D.

11. 1 the Philosophical Magazine for August 1914 (p. 286),

some experiments on the ionization of platinum by
cathode rays were described. It appeared that considerable
changes in the amount of ionization produced might be
effected by heating the metal or making it the cathode of a
discharge through oxygen. In order to throw light on the
precise nature of these changes it appeared desirable to
extend the observations to other metals and to cathode rays
of higher speed. Such an extension is made in the observa-
tions now to be described.

2. A slight alteration was made in the essential part of
the apparatus shown in fig. 1 of the last paper. Fig. 1 of
this paper shows the new arrangement. A and B are
nickel-plated brass cylinders. In A the cathode rays are
produced from the Wehnelt cathode W and caused to enter
B with a speed V, by keeping A at a potential V, higher
than W. The rays fall on the plate P, the ionization of
which is under investigation ; the speed, V, with which they
strike P is varied by varying the potential V, between A
and P. The electrons leaving P fall on B; the potential
between B and P will be called U, and will be counted
positive when P is positive.

Two quantities have to be measured : 2, the total current
carried by the rays falling on P, and i, the current received
by P when none of the rays reflected from P or the electrons
emitted from P by its ionization fall back on P. R, the
reflexion coefficient, in terms of which all the results are

* Communicated by the Author.

Phil. Mag. 8. 6. Vol. 29. No. 171, March 1915. 2B

370 Dr. Norman Campbell on the

conveniently expressed, is (2,—22)/i;. If there is only
reflexion and no ionization or only ionization and no
reflexion, then R is a measure of the reflexion or the
ionization respectively.

Fig. 1.
A Ww ie Te)
ong :
P

tg can easily be measured by making U negative ; it was
found that 7, was saturated when U= —20 volts ; this value
was sufficient to prevent any of the rays leaving P from
striking it again. To measure 2, accurately is not quite so
easy. The most obvious way to do so is to connect B to P’
and measure the total current received by the whole of the
lower part of the apparatus. But it was found that; in order
that the rays striking P should be homogeneous in speed,
and all have the speed V=V,—V, it was necessary that
V, should be not less than 40 volts, and in a direction to
retard the rays entering B. (Thatis to say, besides the rays
of speed V, entering B, there are some others of a speed
between 0 and 40 volts, which probably represent the result
of the ionization of A by the rays falling on it.) But if
the tube projecting from A into B is at a potential con-
siderably higher than P or B, many of the rays leaving P
will be attracted to the tube, instead of to B, and the current

Ionization of Metals by Cathode Rays. 371

received by B and P together will be less than the whole
current entering B. Another way of measuring 2, is to
prevent any of the rays leaving P by making U sufficiently
large and positive ; it is obvious that if B is kept connected
to W none of the rays from W either before or after
striking P will strike B*; all that enter B must come
to rest finally either on P or on the tube, and very few
will strike the tube after striking P because the tube is
surrounded by metal which repels the rays. This method of
measuring 7. is much more satisfactory than the other, and
was always adopted ; but it must be noted that, unless the
rays entering B have all exactly the same velocity the
number striking P will not be the same when 7 and when i,
are being measured. For the potential of B determines the
potential at the mouth of the tube by which the rays enter ;
when 2, is being measured the rays are entering against a
retarding potential V.—20; when 2, is being measured the
retarding potential is V, and is always greater. But since
it was found that the total current measured in this way did
not vary by as much as 10 per cent. when the speed of the
rays was varied (by means of V,) from 400 volts to 2 volts,
there is probably little uncertainty on this account, not
enough at any rate to affect the conclusions it is proposed
to draw.

The Changes in the Ionization.

3. It appeared that any change in P which reduced the
value of the ionization for one speed of the incident rays
reduced it for all speeds, so that in considering the nature of
such changes we can speak of “the ionization” without
reference to the speed of the incident rays. The precise
variation of the ionization with the speed will be considered
later.

Experiments were made with four metals: platinum, nickel,
copper, and aluminium. In the cases of platinum and
copper the changes were made both by heating the metal by
an electric current and by making it the electrode of a
discharge. In the earlier experiments on platinum and
nickel the apparatus shown in fig. 1 was enclosed in a vessel
constructed entirely of glass and protected from vapours
by a U-tube cooled in liquid air. In the later experiments
the apparatus was exposed to the vapours of sealing-wax,
tap-grease, and mercury. No difference between the two
arrangements could be found with platinum and nickel ; it

* For reasons which it is needless to set forth in detail, B had to ke
at a slightly lower potential than W when V was small.

2B2

372 Dr. Norman Campbell on the

is assumed that none would have been found with copper or
aluminium.

The changes produced by making the metal the cathode
of an electric discharge were of precisely the same nature
as those produced by heating in both the metals on which
the effect of both procedures was tried (platinum and
nickel). But the changes could be effected much more
rapidly by means of the discharge. The lowest ionization
that has been observed (in copper) was produced from state
A by the passage of a discharge for only 5 minutes; to
produce state B from state A by heat always required about
24 hours, and even after heating for 300 hours (when an
ionization as low as that mentioned had not yet been
attained) decrease was still in progress. But whereas
further heating always produced a decrease, the passage of
a discharge sometimes produced a slight increase. The
extent to which the ionization could be reduced appeared to
depend mainly on the pressure of the gas through which the
discharge passed; the discharge appeared to be most
efficient at a pressure of about 2 mm. of mercury, and its
efficiency appeared to be closely connected with the amount
of “‘ sputtering” which occurred. Moreover, it did not seem
to matter whether the plate was made the anode or the
cathode of the discharge, but on this point I cannot be
certain ; for though a valve-tube was included in the circuit,
the form of the electrodes between which the discharge
passed (B and P) was such that, at the pressure at which
the discharge was most efficient, P acted as the cathode
much more readily than B. It may he that P was acting
sometimes as cathode even when the valve-tube was
arranged so that it should act as anode. The discharge
acted in the same manner whether the gas through which it
passed was air, oxygen, hydrogen, or petrol vapour.

To illustrate the nature of the changes one series of
observations may be given in some detail. The metal was
copper and the value of R, taken as a measure of the
ionization, was that corresponding to incident rays with a
speed of 280 volts. When the metal was first inserted,
after the surface had been polished by emery and oil, and
heated: to drive off the oil, R was 2°44. The passage of a
discharge in 2mm. of air for 5 minutes changed R to 0°89.
Bombardment with cathode rays from W in a very high
vacuum (current carried by rays about 10-* amp. ; speed
150 volts) for 2 hours increased R to 1°42 ; further treat-
ment of the same kind for 22 hours increased R to 1°623 ;
further treatment for 52 hours left R unchanged at 1-621.

Tonization of Metals by Cathode Rays. a73d

The discharge in petrol vapour for 10 min. reduced R to
1:115: a further 15 min. increased R to 1:216; a further
15 min. reduced it to 1:187. Leaving the plate standing in
a mixture of petrol vapour and air for 72 hours inereased Kt
to 1:209. A discharge in oxygen changed R to 1369; a
further discharge to 1:097 and so on. Then the copper was
taken out and heated in a bunsen flame ; R became 2°39 ;
finally the surface was polished as at the start, and RK returned
to 2°45.

These experiments and many others of the same kind lead
to the following conclusions with regard to platinum, copper,
and nickel. They are very similar to those announced
previously. There are two states of the metal, A and B,
which can always be reproduced. A is that produced. by
polishing the surface with emery and (though with, less
certainty) by heating it in a bunsen flame. In state A the
lonization is nearly the same for all three metals; I found

Rir2a0. Ni 2°26: Cw 2:44.

State B is that produced by reducing the ionization i
means of the discharge, and then restoring it as far as
possible by means of the bombardment of cathode rays in a
high vacuum. The ionization in state B varies notably
between the different metals; I found

Pt 1:98; Ni 2-22; Cu 1-62.
The ionization can be reduced below that corresponding to

state B by heat or by the discharge. The lowest values I
have found are | | 3
Pt 1:203; Ni 1°201; Cu 0°888.

But I have not been able to find a method of reproducing
these values certainly, and it is not, therefore, by any means
sure that these are the lowest values which can be produced
or that there is a real difference between the three metals. It
will be noted that since in order to measure R it is necessary
to bombard the metal with cathode rays, it is quite im-
possible to measure accurately the lowest values of R.
Long-continued standing of the metal, whether in the most
complete vacuum that can be produced or in any mixture of
gases and vapours, produces a slight increase in R ; but the
variation in R which I have been able to produce by this
means is never so great as that which exists between the
actions of different discharges which are as similar as it is

possible to make them.
In aluminium the ionization in state A is much the same
as in the other metals; R=2°60. But by no action of the

374 Dr. Norman Campbell on the

discharge have I been able to reduce R lower than 2°31; I
am not at all sure that this value corresponds to state B,
because further action of the discharge increased R again
to 2°47. The lack of alteration in aluminium by the
discharge is doubtless connected with the absence of
sputtering.

It will be observed that these conclusions agree in all but
one respect with those advanced in the first paper. It has
not been found, however, that state A can be regained from
state B by bombardment with cathode rays in a high
vacuum. It was thought that this result had been obtained
because, in platinum and with the low speeds of the incident
rays then employed, the ionization in state B does not differ
very much from that in state A. Further experiments have
shown that it does differ certainly, and that the only certain
way to reproduce state A is to take the metal out of the
tube and polish its surface. Long-continued standing in
mixtures of gases or vapours has little or no effect.

4, There seems to be a simple and plausible explanation
of these changes. We may suppose that the state A is that
in which the metal is covered by a layer of gas, and that the
state B is that in which this layer is removed. It is, of
course, known that heat or the action of the discharge
removes the layers of condensed gas adhering to the surface,
and there are several lines of evidence which seem to show
that the layer once removed is not easily restored. This
idea would account for (1) the similarity of the values of R
for different metals in state A ; (2) the methods of changing
from state A to state B; (3) the difficulty of restoring
state A, though perhaps the difficulty is rather greater than
would have been anticipated. The reduction of the ioniza-
tion below that corresponding to state B would then be due
to some other change in the metal ; it might represent either
a roughening of the surface by the sputtering and a
consequent entanglement of the electrons liberated, or it
might represent the effect of double-layers such as Seeliger
has shown to exist. ‘The second alternative is supported by
the fact that the change is reversed by the action of cathode
rays which Seeliger has shown to restore the double-layer
removed by the discharge. It is difficult to suppose that the
mere bombardment with cathode rays would have much
effect in changing the molecular structure of the surface *.

* The surface was examined with a microscope in states A, B, and in
states giving less ionization than B. When the changes had been
effected by the discharge the appearance of the surface was always

different from that in state A, but no difference could be seen between
the surface in state B and in the state giving less ionization than B.

Tonization of Metals by Cathode Rays. 375

It appears possible to test these ideas by quantitative
measurements. If the change from state A to state B
consists in the removal of the gas and the exposure of the
metal, we should certainly expect in state B signs of an
ionization potential different from that in state A. If,
again, the changes beyond state B represent an increasing
difficulty of emergence of the liberated electrons, then we
should expect the ionization to be decreased in the same
proportion, whatever the speed of the incident rays, for the
speed of the liberated electrons is known not to depend upon
the speed of the primary rays. Unfortunately, the quanti-
tative experiments which have been made do not seem to
support either of these ideas.

Variation of the Ionization with Speed of the Rays.

5. Fig. 2 gives some of the results which were obtained
in a series of experiments with copper, no changes in the

5 70 Id 20
apparatus being made during the series except those effected

by the discharge. These: results are typical of all those
obtained, except in a few particulars, which will be noted

376 Dr. Norman Campbeil on the

explicitly. The continuous curves give the relation between
R and V’, the apparent speed of the rays, from 0 to 400
volts (upper row of figures on the abscissa), the dotted curves
show the part corresponding to the smaller values of. V’ on a
larger scale (the lower row of values on the abscissa).
Curves A,B,B’ refer respectively to the copper in state A,
in state B, and in the state which gave the smallest
ionization. 400 volts was the greatest speed obtainable with
the source of steady potential obtainable ; some ‘other
observations were made with speeds up to 30,000 volts, the
potential V, being obtained. by an induction-coil, and V,
being always 40 volts. These observations at higher speed
will be noticed later. : nea ne

An examination of the dotted curves would seem to
show that the result announced in the first: paper, that the
ionization potential remained unchanged by the treatment to
which the metal is subjected, is incorrect. In curve A, the
minimum of R occurs at 12 volts, in B at 9 volts, in B’ at
6 volts. And certainly the changes of this nature observed
with copper are greater than those observed in the earlier
or later experiments with platinum or in those with nickel.
But since copper shows all the changes which are being
investigated more markedly than the other metals, certain
considerations have to be taken into account before con-
cluding that this change in the position of the minimum
really indicates a change in the ionization potential.

(1) The exact position of the minimum varies somewhat
in different series. For state A values between 10 and
12 volts have been observed; in state B values between
8 and 10 volts; in state B’ values between 5 and 9 volts.
It is clear, then, that the position of the minimum is not an
exact indication of the ionization potential, and consequently
that the progressive change of position of the minimum as
the ionization is decreased may be capable of some other
interpretation.

(2) The most probable of such interpretations would be
based on a difference between V’ and the true speed of the
rays. V’ is the difference of potential between W and P
imposed by the battery; if there were.at the surfaces of the
metal other differences of potential, V’ would not be the
true speed of the rays striking P ; and, if these differences
of potential varied, the same value of V’ might correspond
to different values of the speed. As was explained in the
first paper, an attempt to allow for such differences of:
potential was made by observing the value of », the
potential between W and P when the current from P was:

Lonization of Metals. by Cathode Rays. 317

just zero; it appeared that the true speed of the rays should
be V'+v. But Iam byno means sure that the introduction
of v really does give the true speed of the rays; v was very
nearly constant (0°4 to 0:7 volt) throughout all the experi-
ments ; it varied very much less than would be expected
from the known volta differences of potential between the
metals employed. Moreover, the variation of v is certainly
very much less than would be expected from the large values
found by Seeliger for the potential differences in the
“ double-layers” which?are produced at the surface of metals
when cathode rays fall on them, and are removed by making
the metal the cathode of a discharge. For nickel, Seeliger
found potential differences as large as 20 volts. No change
in v of this order of magnitude, and indeed no consistent
variation of v with the state of the metal has heen detected.
Throughout the experiments no indication of the presence of
such double-layers has been found; perhaps all the
ionization investigated takes place at the outer surface of
the layer; but even in that case a variation in v ought to be
produced by its presence. I cannot suggest any reason why
V'+v should not give accurately the speed of the rays
maximum falling on P.

(3) There is uncertainty arising from the difficulty of
measuring 7,, which has been noted, but this source of error
does not seem serious. An examination of the figures shows
that unless 2, actually increases as Vz, increases and V'
decreases (such a change is almost inconceivable), an error
in measuring 7, could not account for the differences in the
positions of the minimum. If 7, falls more rapidly with
V’ than has been supposed, the difference in the position of
the minimum would be increased and not decreased. Such
an error may have been caused by the assumption, implied
in the method of measuring 2, adopted, that all the rays
entering B fall on P,and that there is no appreciable scatter-
ing even at the lowest speeds. (It must be remembered
that owing to the method of varying V’ adopted, these low.
speeds would only be found close to P.)

A lack of homogeneity in the rays could hardly account for
the variation of the minimum, for it is likely that such hetero-
geneity as existed would be the same in all experiments.

(4) The position of the minimum is determined, not only
by the variation of the ionization with V’ but also by the
variation of the reflecting power. It is to be observed that
the variation of the reflecting power at values of V’ less
than that required to produce ionization, does not agree with
that found in the earlier experiments or with that found by

378 Dr. Norman Campbell on the

Gehrts. These other experiments indicated a maximum of
reflexion at about 5 volts, which was especially marked in
Gehrts’ work. I am unable to explain this discrepancy.
Gehrts’ method of measuring 7; was more satisfactory than
my own, but in order to produce a maximum in my curves
it would have to be supposed that 2, had a maximum between
10 and 2 volts. On the other hand, the rays which I used in
these experiments were probably more homogeneous than
those employed by Gehrts or those employed in my earlier
work.

Accordingly it does not appear to be at all certain that
there is any change of the ionization potential with the
treatment of the metal. If there is a change it consists of
a reduction of the ionization potential as the ionization is
decreased. And this result might seem in accordance with
the idea that in state B the surface of the metal is exposed,
whereas in state A it is covered with hydrogen. Tor it is to
be expected that the ionization potential of metals should be
less than that of hydrogen; the ionization potential of
gaseous mercury is below 5 volts. On the other hand, if
this explanation is adopted, it is not easy to see why the
ionization potential should be reduced still further in passing
from B to B’.

Careful comparative experiments have been made along
the whole curve up to 400 volts; there is no indication
whatever of any other kink in the curve suggesting the
presence of an ionization potential greater than 12 volts.

6. Now let us turn to the part of the curve corresponding
to greater values ot V’. It is clear at once from the com-
parison of the curves B and B’ that the expectation that, in
the change from one of these states to the other, the
ionization should be changed at all values of V' in the same
proportion is not fulfilled. For whereas the maximum
value of R occurs in B at about the same position as in A,
in B' it occurs (if at all) at some much higher value. (The
steady potential available was not great enough to determine
the position of the maximum for B’, but it appeared to occur
between 400 and 9000 volts.) The following table gives
the ratios of R (measuring the ionization) for different
values of V’ in the three states. For each value of V' the
value of R in state B is put equal to 1; the figures corre-
sponding to states A and B’ are the ratios of the ionizations
in these states to that in state B.

A ke ei 40 80 200 =300 = 400 9000 26000
Hwetate A). o00..-3.. ioaee tao -, WAb) 1:52-1:62 161 1-66
i Ge age 5) eee 00-8 £00 TOO 1-00) £60 1:00 1-06

20 Ain 220 ee 036 . 038 049 O56 067 0-79 0-87

Ionization of Metals by Cathode Rays. d79

It will be observed that the numbers corresponding to
states A and B’ are by no means constant, and that there
is a marked difference between the trend of the numbers in
the two rows. In state B’ the ionization becomes more
nearly equal to that in state B as V' increases ; in state A it
becomes less nearly equal to that in state B, though the
ratio seems to tend to constancy (but not to equality) at the
higher values of V’. ‘These results seem to confirm com-
pletely the view that there is a marked difference between
the state A on the one hand and the states B and B’ on the
other, such as we might expect if in the state A the surface
consists of a gas, and in the states B and B’ of the metal ;
but 1 am unable to offer any explanation of the variation of
the ratio of the ionizations in states B and B/ with the speed
of the incident rays; the change does not seem compatible
with the view that the lesser ionization in state B’ is merely
due to a greater opposition to the emergence of the
electrons.

Some experiments were made in a completely different
apparatus in which ionization at the surface of a nickel
plate was produced, not by cathode rays, but by X-rays, or
rather by the secondary cathode rays excited by X-rays.
The experiments were difficult because of the smallness of
the effect measured, but it was quite clear that by the passing
of a discharge (and change from A to B or B’) the ioniza-
tion at the surface was considerably decreased. The speed
of the secondary cathode rays in these experiments, deduced
from the absorbability of the X-rays, was about 50,000 volts.
Accordingly, the difference between states A and B persists
up to this speed.

7. Experiments were also made on ihe speed of the
electrons liberated at the surface by ihe incident cathode
rays ; they consisted of observations of the current flowing
from P when the difference of potential between P and B
was varied. If U is this potential (counted + when P is +),
then the current is saturated if U is sufficiently large and
negative or sufficiently large and positive; in the latter case
none of the electrons liberated at P leave P, in the former
case all of them leave P. Let 7, be the current from P
corresponding to the potential U. Then it is easy to see
that, if there is no reflexion of the rays at the surface
bic —ly

is the fraction of the rays liberated at P

tia —t_x
with a speed greater than U. For reasons which will
appear presently, 7, was identified with (459, although the
current was not completely saturated at this potential ; it

380

was completely saturated when U=—20. The presence of

Dr. Norman Campbell on the.

reflexion makes the absolute values obtained unreliable (it

is due to this reflexion that the current is not saturated
when U=0, and that some of the rays appear to havea
negative velocity), but it should not influence greatly
the comparison of curves taken after different treatments
es

The following table shows the results for nickel in states
A, B,B’. V’ in all cases was 400, but it appeared, as was
expected, that so long as V’ was considerably greater than

20, the form of the curve was independent of V’. The

results with other metals were similar in all respects.

Ure +20 +15 +10 +8 46 44 42 41 0 —2 -10
pistateA) ... O ‘021 -065 °090 -121 -215 -358 “483 ‘“708" Bis ee
P(; 8). 0. 021 7075 -110 159 +250 417 “SiO yeieemmeees ‘991
p(s B) ..,. 0038 -O87 “121 -186 -295 °500 ‘68! “Si Saaeoe

These figures seem to indicate that, so far as there is any
change in the speed of the rays, the speed increases as the
ionization decreases—a conclusion which would be recon-
cilable with the view that the lower ionization is caused by a
difficulty in the emergence of the electrons, so that only
those with the greater speed emerge. But it must be
remembered that, since the discharge is passed between P
and B there may be a change in the reflecting power of B
when the ionization is decreased ; a decrease in the reflect-
ing power of B with a decrease of ionization might also
account for the apparent change in speed, for the number
of rays leaving P and failing to return would be increased.
The results which have been given show that the reflecting
power of P for slow electrons is diminished greatly by the
passage of a discharge, and since this change appears to be
independent of the direction in which the discharge passes,
it is probable that the reflecting power of B is actually
diminished by the treatment that causes the decrease in
ionization, and that the apparent change in the speed of the
electrons is to be traced to this cause. ‘

8. Some observations were made to determine whether the
power of the metal to reflect incident rays with speeds
greater than 40 volts was altered during the change from
state A to B or B’; no such change is to be expected, for
such reflexion probably takes place, not at the actual surface
of the metal, but after the rays have passed throngh the
surface. It is not, however, easy to determine the reflecting
power for rays which are capable of producing ionization in

—20
1-000
1-000
1-000

i
i)
if)
p
}
f
|!
ie
if ‘i
i
eg
mY
ait
Vi
(hy
:

Bees
2 Si Serer
aoe Sta 8 _

Ionization of Metals by Cathode Rays. 381

such an apparatus as was used. The electrons emitted in
the process of ionization have practically all speeds less
than 20 volts; the reflected electrons, when V' is greater
than 40, have almost all speeds greater than 20 volts.
Accordingly, it might be thought that the current carried by
the reflected electrons would be measured by the change in
the current flowing to P, produced by changing the potential
difference between B and P from 20 to V' volts. But it
must be remembered that when the reflected electrons fall
on B they cause ionization there, and the electrons so
emitted will be driven to P, so that by increasing the
potential between B and P and preventing the reflected
electrons from reaching B we may decrease and not increase
the negative current received by P. And experiment shows
that this effect is of importance, for at the higher values of
V' too, the negative current received by P when U=20 is
actually greater than i, that received when U=V’'. The
following figures give the ratio ig/iy for various values of
V' in states A, B, B’ corresponding to the three curves of
fig. 2; if there were no reflexion the ratio would be 1;
the occurrence of values greater than 1 indicates the effect
of ionization at the surface of B :—

40 80 120°" 160) 5 200° | 240») 280: | 320

7: ie Bee - 0858 0°903 0:°933 0-975 1:°082 1:054 1:109 1:170
i
aa wk 0855 0°894 0913 0953 0976 1:010 1:039 1-067
ly,
BES! cscs: 0-921 O91S 0:920 0919 0:920 0-945 0-956 0-997

At first sight these figures might seem to show that the
reflexion is actually less in state B’ than in state A; but
the difference between the figures can also be accounted for
by supposing that it is not the reflexion, but the ionization
produced by the reflected rays in B, which is the cause of
the change.

It appears, then, that it is very difficult to interpret
certainly the quantitative results in a manner to throw new
light on the changes investigated. They seem to confirm
the view that the state B does not differ from A merely as
B’ differs from B ; the change which produces B from B'
is not a mere partial reversal of the change which produced
B’ from A. But they do not seem to decide without
ambiguity whether the explanation of the changes proposed
in §3 1s tenable. It is not certain whether there is a change
in the ionization potential, and it appears that the decrease
of the ionization between states B and B’ must be due to

382 The Ionization of Metals by Cathode Rays.

some cause more complicated than an opposition to the
emergence of the electrons liberated, for this cause should
produce the same proportional decrease whatever the speed
of the incident rays. But I have no further suggestion to
make as to the nature of this change.

Summary.

The experiments described in a recent paper on “The
Tonization of Platinum by Cathode Rays” have been extended
to other metals and to higher speeds of the incident rays.

It is shown that the changes in the ionization which were
found to take place on heating the platinum can be produced
in that metal and in copper and nickel by making the metal
one electrode of an electric discharge in air, oxygen,
hydrogen, or petrol vapour at about 2mm. pressure. The
changes were greatest in copper; in aluminium hardly any
change could be produced. The changes seem connected
with “ spluttering.”

It appears that, in respect of this ionization, the surface
of the metal can be in the following states :—(1) State A,
which is that of the metal after it has been polished, (2) a
series of states B’ which are produced by the discharge.
The ionization in the states B’ is smaller for all speeds of
the incident rays than in state A. Of the states B’, one,
called B, is distinguished (1) because the ionization in that
state is greater (for all speeds of the incident rays) than in
any other of the states B’, (2) because it can always be
reproduced from any of the states B’ by subjecting the
surface to cathode ray bombardment. State A cannot (as
was stated previously) be reproduced from state B by such
treatment.

It is uncertain whether the ionization potential in states
B’ is different from that in state A; if it is different, it is
lower. It is also uncertain whether the speed of the
electrons liberated varies with the state of the surface; it
does not vary with the metal (in state A) or with the speed
of the incident rays.

The ionization produced by rays of any speed in a metal
in the various states B’ becomes more nearly equal and less
nearly equal to the ionization by those rays in the metal in
state A as the speed of the rays is increased.

It is suggested that the difference between state A and
the states B’ lies in the fact that in the former the metal is,
and in the latter it is not, covered with a layer of gas. If
this explanation is correct, the ionization potential of a

Partition of Energy and Newtonian Mechanics. 383

metal may be less, but is certainly not greater, than that
of hydrogen (11 volts). But on this view no explanation,
consistent with the measurements, can be offered of the
difference between the various states B’. It appears certain
that these differences cannot be due to the presence of the
“* double-layers,” the existence of which has been proved by
Seeliger ; if such double-layers were formed at all in the
conditions of these experiments, it seems that the ionization
must take place at the outer surface of them.

Leeds, December, 1913.

XLI. On the Law of Partition of Energy and Newtonian
Mechanies. By G. H. Livens®.

ARIOUS attempts have been made, notably by Jeans +
and Lorentz ft, to prove that the only possible law of
steady thermal radiation deducible from ordinary Newtonian
mechanical principles is that which corresponds to equi-
partition of energy among the various oscillations, a law
which is, however, totally in disagreement with the actual
state of affairs as experimentally determined §. Ultimately
these proofs reduce to the fact that equi-partition of energy
among the various large number of coordinates of any

dynamical system represents the only possible average par-

tition which can reasonably be expected in any steady
state.

It is therefore concluded that the Planck formula of radia-
tion necessitated by experience is inconsistent with our
ordinary mechanical principles, and therefore necessitates
an essential modification in our usual stock of dynamical
notions. Jeans even goes so far as to prove that the only

* Communicated by the Author.

Tt Phil. Mag. Dec. 1910.

t Vide A. Eucken, ‘Die Theorie der Strahlung und der Quanta’
(Halle, 1914).

§ It is to be insisted that, although the equi-partition law apparently
provides the right formula for long wave radiation, its application even
in this part of the spectrum is open to certain objections. In pure
thermal radi:tion, which presumably furnishes a continuous spectrum
which would defy all attempts at resolution, the total number of dif-
ferent wave-lengths in any small range of extent ddA even at the long
wave-length end of the spectrum, is infinite, or at least must be assumed
to be so in order to secure continuity of the spectrum. There would
therefore be an infinite number of different oscillations which on the
average secure the same finite amount of energy according to the equi-
partition law. There would, therefore, be an infinite amount of energy
associated with the small range dA in any part of the spectrum.

384 Mr. G. H. Livens on the Law of

conceivable mechanical scheme which can lead to Planck’s
law is one in which elements of energy ofa definite size play
an important and essential part.

In a previous paper on the statistical theory of radiation,
I have attempted to show that the application of certain
tentative ideas in the statistical theories of both Planck and
Jeans enable us to arrive at Planck’s formula, but without
any very unnatural assumptions. The ideas there tenta-
tively introduced, which may appear rather crude and un-
certain, involve a modification, not of our ordinary mechani-
cal notions but merely of the one additional fundamental
principle * that all dynamical coordinates which enter into
the usual expression for the energy of any system are
necessarily equally probable as receptacles of energy ; this
modification + applies most particularly to those coordinates,
infinite in number, introduced by the use of Fourier’s series
and usually expressed by the coefficients in these series.

It may well be asked, why is it that these coordinates are
not equally probable, seeing that they all enter into the
energy function in precisely the same way ? But it may be
equally well asked whether the mode of appearance in the
energy function, or more generally in the dynamical theory,
is sufficient criterion for the probability of the coordinate as
a general receptacle of energy in any statistical problem.
Besides the assumption that all the coordinates are alike in
this one respect is nothing if not impertinent, seeing that it
implies a good deal more knowledge of the higher order
coefficients in the Fourier series than any mathematical
theory would allow. And after all the coefficients in the
Fourier series are at least differentiated from each other by
their places in the series, and I see no reason to suppose that
their differences end at this.

If we are prepared to adopt such a modification of the
theoretical bases of the statistical method in mechanies, it
can be shown that the form of the theory which is to agree
with experience is at least not inconsistent with our usual
Newtonian mechanical notions.

We assume quite generally that the state of any dynamical
system is determined at any instant by its state at the
previous instant, and that this state can be specified by the
value of certain definite generalized coordinates. The motion

* [t is very important to recognize that this principle underlies all
deductions of the equi-partition law, and is additional to the mechanical
principles involved.
~ ‘+ A similar modification is implied in Planck’s theory, but a dyna-
mical reason, involving a discrete atomic structure for the energy, is
assigned for it.

Partition of Energy and Newtonian Mechanics. 385

of the system between the two instants is presumed to be
governed by the ordinary Newtonian system of mechanical
laws, so that a general set of coordinates sufficient for our
purposes would be provided in any set of generalized

Lagrangian coordinates of the system p;, ps, --+ P,, and the
momenta corresponding to these, say 91, 2, -+-9,- Lhe
equations of motion can be taken in any of the usual general
forms.

If we construct a 2n-dimensional space a single point
in this space, namely the point whose coordinates are
Pir Pa» +++ Pid Ws Yn +++ Ip, Will represent the state of the
system at any instant, and the general equations of motion
are the equations to the paths or trajectories traced out in
this space by the representative points as they follow out
the ditferent possible motions of the system. It is obvious
that through every point in the generalized space there is
one and only one trajectory, and that as a point moves along
a trajectory and so follows the motion of a system, its
velocity at any point depends only on the coordinates of the
point and not on the time.

In the usual manner we can therefore imagine every
region of the generalized space which represents a physically
possible state of the system to be filled with so many repre-
sentative points, that the whole collection of points may be
regarded as forming a continuous fluid. The general equa-
tions of motion then assert that this fuid moves along fixed
stream-lines and that the velocity at any point is constant.
Moreover, we know from Liouville’s theorem that if we
follow the motion of all the points from the inside of any
elementary parallelopiped

dpy, -dpr- + - IP Ey + Eay v2 dd,

at time fo, they will be found at time ¢ in the corresponding

parallelopiped
dp o dps S05 dp ° dq, ° dqz eee dq.

and the volumes of these parallelopipeds are the same. The
same is also true of the projections of the points on the
elementary area (dp dq,) parallel to one of the coordinate
planes defined by a generalized geometrical coordinate and
its corresponding momentum, the area remaining constant.
The density of the fluid, or the density of the aggregation
of the representative points thus remains constant through-
out all time, so also does its density parallel to any (p,g,)
plane. The initial distribution of the density is entirely at

Pint. Mag. 8. 6. Vol. 29. No. 171. March 1915. 2C

386 Mr. G. H. Livens on the Law of

our disposal and may be chosen as simple as we like. The
simplest and most convenient thing to do is to make it
uniform throughout the whole of the space with which we
are concerned.

This mass of fluid moving in the generalized space now
provides a basis for the introduction of the calculus of
probabilities ; but, as Jeans says, great care must be exer-
cised in settling the basis for the calculation of the prob-
ability. Of course for our analysis to be legitimate we are
not compelled to choose any one particular basis for the
calculation of the probabilities. We may select any basis
we please, and then the analysis will be legitimate if we
retain the same basis throughout the whole investigation.

In the present instance we agree to state that the prob-
ability of the motion in any one type of coordinate, say p,,
being in any state A, is, on some definite scale, measured by
F,(ay), where f, is some, at present undetermined, function of
its argument a,, which is the area, measured in definite units,
of the projection on the (p,9,) plane of the volume occupied
by points representing systems in which the motion in the
p, coordinate has the characteristic A,: the relation between
the probabilities for different types of coordinate are deter-
mined as soon as we know the form of the function 7, for
each of them.

If we are dealing with a system, or part of a system,
comprising an entirely large number of one particular type
of coordinate for which the function 7 is the same (say m,
coordinates of type p,), then we shall agree to say that the
probability of this system being in a state A, is

W,=F,(V,),

F, being some definite function of V,, which is the volume
occupied by the representative points for the system
characterized by the state A,. It is, however, important to
notice that if the characterization of the state A, is general
and bears no reference to, or preference for any special
members of the coordinate system, the only really possible
functional forms for the functions 7, and F, are such that

JA2) = C Ur, F (2) a C2%,

where a, is a constant, the same for all the coordinates of
the specified type. This follows at once from the fact that
in these circumstances the coordinates share equally, one
with the other, the responsibilities implied in the specification
of A,, the probability that any coordinate is in the conditions

Partition of Energy and Newtonian Mechanics. 387

in which it can exist in the systems characteristic of the
state A, being the same for them all.

The procedure here adopted is more general than that
adopted by Jeans inasmuch as it is not assumed that equal
volumes in partial spaces corresponding to different types of
coordinate are equally probable with each other.

Now consider a few generalities regarding a more compli-
cated system comprising a large number of separate types
of coordinates. Let A, Ag,...... be characteristics of different
parts of the system, such that the coordinates involved in
the specification of any one characteristic are not involved in
any of the others and are in addition all of one specified
types) et Wy, Wo, ..:..: be the respective probabilities,
calculated on the predetermined scale, that in any random
choice of a system fromall those possible the respective parts
shall possess the characteristics A,, Ag, ...... A complete
system obtained at a single random choice may possess two
or more of these characteristics simultaneously, and the
probability that it possesses them all is of the form

A eV Woe WW
If we now put
= clo,

then we know that 8 is proportional to Boltzmann’s measure
of the entropy of the system with the specified character-
istics, probabilities now being measured on the _ basis
provided by the generalized space as described above.

Now let H,’, E,’,... be the energies of those parts of the
system with which the properties A,, Ao, ... are associated
and let E be the total energy given by

E = Ey + 5, Tevae acters
The total entropy S is then given by
S => Aloo Wi.

Now the characteristics A, Ag,...... may be chosen so as
to determine the partition of energy. ‘To be precise let any
characteristic property A, be satisfied if the corresponding
energy HE,’ les between BE, —te,. and H,+4e,. Let it be
assumed as a property of the system that if left to itself it
will assume a steady state in which the energy is divided in
a definite manner, namely one in which 4H,’ becomes om
to the corresponding Hj, at least to within the small range €;
Then W must be equal to unity for these values of Hy’, en.’

and this is its maximum value. It follows that S is also
2 (12
ad ) tl

e@eeerne

388 Mr. G. H. Livens on the Law of

a maximum when each H,’ is equal to the corresponding E,,
subject to the sum being equal to HE. The analytical
condition for this is, in the usual way, that all the E,”’s are
given by the system of equations

poy OF
Se se
combined with the equation
Ey,’ + 15 Sip eels «6 =i
if now we proceed on the basis suggested above and put
W,=V,>

we find that

so that

Now suppose that the first part of the system is a perfect
gas. Its energy will be then simply the kinetic energy
B= lg’,
the sum & here extending to say m, terms, all representing
identical types of coordinates. In this case V, is the volume

of the generalized space in which ¥/q’ lies between H,— 4e,
and H,+4e,, which is known to be of the form

Vi= CE, ? "€1,
wherein C, is a definite constant. Thus
SONG 2 amy
ae oF, a EK, :
which since m, is very large is practically the same as

ae Cok Aime Diy
Noe, | ely

so that

OS. akm

Oey 2h, *
But if @ denote the absolute temperature of the system in
the steady state and R the usual absolute constant of gas

Partition of Energy and Newtonian Mechanics. 389

theory, E, is known to be equal to 4m,R@. Thus we have

Oro Re

or if we use «,4=R then we see that
OS OS a. ae
Gham Obs 4. ee

which is the general result expressed by the second law of
thermodynamics.

The present considerations do not therefore affect the trath
of the second law of thermodynamics, but this does not
appear to Justify us in the conclusion that the theorem of
equi-partition is also true, as the following analysis shows.

Suppose that any other part of the energy, say E,’, can
also be expressed in the same form as Hy’, viz. |

the summation now extending to m, terms. The value of
V, can then be calculated in the same way as V,, and now

we find
Os ; _ #2 h: a mh

ey Ue CCA
so that

ae MoRO
E,= — —.—>
a1 2)
a resultant which is not consistent with the theorem of the
equi-partition of energy since the average energy in this

: : : ae
particular type of coordinate is now only — ° and not
a

0 ?
2

“yas in the case of the perfect gas, which forms part of the

same system.

It would thus appear that if there is any reason to suppose
that the various coordinates, and in particular the coordinates
of the Fourier series, are to be differentiated from one another
on the lines suggested above, then equi-partition of energy is
hardly to be expected ; but any such differentiation in type,
although not actually contained in our usual stock of
dynamical ideas, is at least as consistent with these ideas as
the usual assumption made regarding this point, so that
there is every reason for adopting it as a useful, if arbitrary,
additional hypothesis to replace the one already in use.

Applications in Radiation —In attempting to apply the
statistical ohineaples of the preceding paragraph to the

390 Mr. G. H. Livens on the Law of

fundamental problem in radiation, we should again fail to
obtain anything like the definite radiation formula proposed
by Planck, but the result obtained- has now the additional
advantage of being extremely indefinite. This indefiniteness
is, however, not very surprising seeing that we know so
little about the dynamics of the system concerned in radia-
tion phenomena, and are therefore quite at a loss to determine
anything about the constants «, tentatively introduced in
the above analysis, or even the total number m, of the co-
ordinates of any particular type. We do know, however,
that for instance the possible vibrations, each of which pre-
sumably corresponds to a degree of freedom, of the type
specified by the fact that the radiation from it has a wave-
length lying in the infinitely small range between > and
A+ dX, are infinite in number, but such knowledge is, under
the present circumstances, worse than useless.

We are not, however, prevented from obtaining further
information on this subject because there are still two
methods of attack open to us. The first, or Planck’s method,
has been fully discussed in a previous paper, and the con-
clusion to be drawn from it is identical with that drawn in
the previous paragraph, unless it is preferred to retain in the
analysis the hypotheses of a finite limiting ratio between
the elements of energy and extent of the elementary cells
which form the bases for the application of the probability
calculus, in which case it is possible to obtain Planck’s
formula. The suggestion that Planck’s formula essentiaily
involves an assumption of this kind and nothing else, is due
to Larmor*, but I am not yet aware of any plausible
physical reason for it f; some such assumption is, however,
necessitated by the requirements of definiteness in the
ultimate formula, and it is not inconsistent with any of our
usual stock of ideas, so that for the present, at least, some-
thing will be gained by retaining it.

There is yet another method of attack still open and this
is the converse one followed by Jeans, but, contrary to the
conclusion drawn by Jeans from his work, I cannot agree
that anything very definite can be got out of it. The method
exactly reverses the argument of Planck and starts with the
assumption that his formula is correct. Let us therefore
assume that in any given system in thermal equilibrium the

* Proc. Roy. Soc. vol. Ixxxiii. 1909.

+ I should like to take this opportunity of applying a reservation to
certain remarks bearing on this question which were made in my previous

aper. On due consideration of the various possibilities I think it will

be difficult to avoid Larmor’s suggestion, even if we cannot find a
good reason for it.

Partition of nergy and Newtonian Mechanics. 391

components of the radiation with wave-lengths differing only
infinitesimally from 2» arise from the large number N of
vibrators. ‘Their total energy EK must, according to Planck’s
law, be given by

Ne
K= Reha <3
eke — |
where «= es h being Planck’s constant. Eliminating the
temperature between this and the equation
a8_1
ola
we get
Suk N
soa. be (1+)

which gives on integration
S=k | (N+ ) log (N+ ya log =
€ Gi APNE €

Thus on the usual bases of probabilities

log W= [ (a+ 2) log (N=) ue log =|

: wy é 5 5c 3
whence using P= — and with Stirling’s approximation,
(Ss

fi !
(ieee ¢

Thus on the basis of our previous measure of probability,
we see that the space occupied by points representing the
system with coordinates corresponding to these vibrations
with their total energy between (H—4e, H+e) has a
volume

v,= [oS $e ie

If we made ¢ iniinitely small, as we should generally be

entitled to do, this formula reduces to
N

v= Cn,

unless of course) Ne is comparable with E, in which case no
such simplification is possible.

In any case, however, it appears to be quite impossible,
owing to our lack of knowledge regarding both a, and N, to

392 Mr. G. H. Livens on the Law of

draw any conclusions regarding the volume of this space as
compared with that discussed above, or the size of the element
of energy which is used in the usual method of deducing
this formula, or even the number of vibrations involved
(which, however, on Planck’s method, must have a finite
limit). The equality of the spaces in the two cases would,
however, imply some such relation as that discussed in our
previous paper and mentioned above, between the element
of energy and extent of cells.

Tn conclusion it might be useful to illustrate the restrictions
and limitations of Planck’s Theory by the alternative deduction
given by Jeans.

Other things being equal, if a vibration can have energies
0, e, 2e, ..., then the ratio of the probabilities of these events,
as in the usual gas theory calculations, are

5 pe 3 ae. 5
l:eé 2 @ Cs OCR IMEC ary ci)

where, however, according to the calculations given in the

earlier part of this paper, g is not equal to but to

mele
2RT

=a «# being the value of the probability constant

corresponding to these vibrations.
If out of the N vibrations under consideration M have
zero energy, then the number which have energy € is Me 72°

the number having energy 2e is Me~*“ and-so on. Thus
ag = cee
N =M(lL—e eee eg ane 2)
M

ij— eo 24

If E is the total energy of the N vibrations
it tiete Se ee a
Mier 2 bic Ne

ar (1—e ry pug e

which is Planck’s law if

hails a he
an Ree ~ RN
and
ve es

Bui now e can be taken to be zero if @ is sufficiently small
and N sufiiciently big.

Partition of Energy and Newtonian Mechanics. 393

It is perhaps worth while here emphasizing again the

Ki essential difference between Planck’s formula and many
im interpretations of it. Although it is quite obvious that the
i only certain point about Planck’s law is that the formula

CE ar
Pe é bd re
¢ er? — |

expresses the energy per unit volume to be associated with
the component of the radiation with wave-length between
rand A+dA, many authors interpret the theory in a manner
that implies that the energy of a perfectly monochromatic
constituent must be finite. Such a statement can, however,
Y hardly be true, when we consider that ultimately an infinite
nt number of such constituents are to be associated with any
t small range in the spectrum.

l Conclusions.—In any case the general conclusion must be
i that Planck’s law does not require or necessitate anything in
h the form of definite multiples of a fixed unit of energy, nor
(i is it in this or in any other respect in contradiction with a
bi general interpretation of the ordinary laws of Newtomian
dynamics. It is not suggested that this formula does not
} involve anything but whale ean be derived from ordinary
dynamical principles ; but it is insisted that any statistical
| considerations regarding dynamical problems do, in fact,
a involve an additional hypothesis over and above these
| provided in our usual dynamical schemes, and that therefore
a a modified form of this hypothesis cannot be said to be
inconsistent with dynamical principles, since it has in reality
;

°c

nothing whatever to do with these principles.
The modification thus introduced into the theory involves
merely a revision of the principles of the calculus of
: probabilities as applied in such problems. After all it is the
method of application of this calculus which is most probably
the vulnerable point in any statistical theory, so that it is
hardly surprising that the new phenomena of radiation force
us into new paths in this direction. While it is possible
thus to shift the responsibility for the particular form of the
theory necessitated by experience from the definite dynamical
principles to the indefinite statistical ones, it would appear
that no conclusions regarding the general applicability or
otherwise of these pr inciples can be drawn from the theory.
The only impression left by the foregoing discussion is
rather one of indefiniteness. There appear to be so many
indefinite constants in the theory, that it is difficult to draw
any definite conclusions respecting any of the quantities

394 Dr. Genevieve V. Morrow on Displacements in

involved, such as those drawn by Planck and Jeans ;
although it appears that certain relations must exist between
these constants if Planck’s formula is to be obtained.
Investigation seems to be necessary to attempt to fathom
these relations between the constants of the theory, or are
we to accept them as unfathomable properties of natural
phenomena ?

The University, Sheffield,
November 4, 1914.

Note added Jan. 16th, 1915.—The main contention of the
present paper is capable of explanation in terms of a well
known difficulty in the ordinary statistical kinetic theories
connected with the “ continuity of the path”’ of a dynamical
system. It is in fact definitely denied that a dynamical
system which involves in its essential constitution a perfectly
irregular mass of vibrating elastic matter (or ether) when
started from any phase will traverse every other phase geo-
metrically consistent with the energy condition. In fact,
motions of the system in which more than a limited finite
number of the higher vibration coordinates possess an amount
of energy comparable with that of a dynamical coordinate
of ordinary type (translation coordinate of a gas molecule,
for example) are impossible both mathematically and
physically.

XU. Displacements in certain Spectral Lines of Zine and
Titanium. By GENEVIEVE V. Morrow, Ph.D., A.R.C.Se.L,
pee Scholar)*.

ITHIN recent years many observers claim to have
found displacements in the lines of the are and

spark spectra of various elements, whilst others afirm that
there is no alteration in the wave-lengths, and that the
differences found have been caused by inaccuracy or by
the methods used in measurement. Professors Exner and
Haschek t have found displacements in certain lines of zine
and titanium by their method of measurement—that of pro-
jection on a divided screen. It seemed of interest to obtain,
by means of the same apparatus which they had used, the
are and spark spectra of the same two elements, but under
various conditions and methods of production, and to measure

* Communicated by the Author.
+ Exner and Haschek, Svtzwngsber. der Wien. Akad. exv. II. A. (1906) .

certain Spectral Lines of Zine and Titanium. 395

some of the lines by means of a micrometer. The present
work has been undertaken to ascertain if under these con-
ditions there is any displacement in the lines.

Rowland observed that the position of the iron lines in
the sun’s spectrum was not always constant, and the subject
was further studied by Jewell*, who found marked dif-
ferences between the positions of the metallic lines in the
are and the sun. Humphreys and Mohler 7 noticed that
increase of pressure in the arc caused metallic lines to be
displaced towards the red, but that variation in the strength
of the current did not affect the position of the reversals of
lines. Mohlert found corresponding results on lowering the
pressure, but Duffield§ noticed that reversed as well as bright
lines were displaced towards the red, under increase of pres-
sure in the are. Jewell || found that the stronger reversed
lines were those whose displacement was greatest, and also
that an increase in the are in the quantity of material pro-
ducing the line always displaced it towards the red, but an
increase in the quantity of other material did not change the
position of the line to the same extent, if at all. Kent com-
pared the positions of the are and spark lines with each other,
and found that the part of the spark near the terminals gave
lines the wave-lengths of which were greater than those of
the are, whilst those produced at the centre, where there is
very little pressure, were not displaced, or only to a very
slight extent. He suggests this as a reason for the fact that
Eder and Valenta have observed no real displacements when
comparing are and spark spectra.

The question of the displacement of spectral lines is dis-
cussed by Exner and Haschek in their book ‘ Die Spektren
der Hlemente bei normalen Druck,’ and their results show
that the more intense a line is, the more strongly is it
displaced towards the red. They found displacements of
considerable dimensions in both are and spark spectra, and
in the bright as well as in the reversed lines. In another
publication ** they examined the spectra of the elements
potassium, tin, and zine under various conditions, and obtained
displacements in the lines which in some cases were more
than 0-1 Angstrém unit, far beyond the region of error in

* Jewell, Astrophys. Journ. iii. (1896).
+ Humphreys and Mohler, Astrophys. Journ. iii. (1896).
{ Mohler, Astrophys. Journ. iv. (1896).
§ Dufhield, Astrophys. Journ. xxvi. (1907).
|| Jewell, Astrophys. Journ. iii. (1896).
{] Kent, Astrophys. Journ. xvii. (1903) ; xxii. (1905).
** Hixner and Haschek, Sitzungsber. der Wien. Akad. exy. I. A. (1906).

396 Dr. Genevieve V. Morrow on Displacements in

the measurements. The displacements increased with the
increase of the density of the glowiag vapour. By a com-
parison of the are and spark spectrum of the same element,
they found that the wave-lengths in the spark were often
less than those in the are, but on the other hand the maximum
displacement in the spark was the larger, which the authors
attributed to the greater difference of density between the
central and outer layers of the spark.

Amongst those who have found no displacement in spectral
lines are Eder and Valenta®*. They compared directly on
the photographic plate the eight lines of zine, in which Exner.
and Haschek had found displacements between are and spark,
but they could not observe any alteration in the position of
the lines with respect to each other. They are of the opinion
that an apparent displacement may be caused in a line by
unsymmetrical broadening, but that the position of maximum
intensity remains unaltered, and that the so-called displace-
ments obtained by observers are caused by the measurements
not being made through the most intense part of the line.
Kayser ¢ also has found no displacements in spectral lines,
and suggests that the results of Hxner and Haschek and
those of Kent have been caused either by errors in measure-
ment or by bad adjustment of the apparatus.

The spectral apparatus which was employed in the eae
investigation was the same as that used by Exnerand Haschek
in their experiments. In order to avoid the possibility of
errors which might be caused by the projection method of
measurement used by them, and of which Kayser did not
approve, the plates were measured by means of a micro-
meter. This micrometer was constructed by Perreaux and
permitted of measurements extending 30 cm. horizontally.
It was of great importance to know if the thread of the
screw was ‘sufficiently fine and constant to permit of its
being used in the present investigation, so the pitch was at
first ascertained by the usual method. ‘The distance travelled
by the microscope along the screw was measured by focussing
the cross-hairs on a divided millimetre scale of plated brass
supplied by the Société Genevoise. This scaie had been
tested in Paris and found correct. Several sets of measure-
ments were made and curves drawn to ascertain if the screw
possessed periodic errors. The latter were not found, and
the mean difference between the measurements of the pitch
was ‘0005 mm. Turther measurements for finding the value

* Hder and Valenta, Sitzungsber. der Vien. Akad. exii. I, A. (1908).
+ Kayser, Zettschr. fiir wiss. Photographie, iii. (1905).

certain Spectral Lines of Zinc and Titanium. 39%

of the pitch were made in both directions along each centi-

metre of the screw, in order to avoid personal errors in the
veadings. The values found in seven sets of measurements
varied between 0°4985 mm. and 0°4995 mm., so that
0-499 mm. was considered as being correct.

For the present work it was necessary to know the value
of one turn of the screw in Angstrom units. To ascertain
this Professors Exner and Haschek kindly lent me a photo-
graphic plate of the spectrum of palladium, which had very
sharply defined lines. The iron spectrum was on this plate
as a standard, and the wave-lengths were taken from Row-
land’s tables. The palladium lines as well as 32 standards

were measured, and the value of one turn of the screw in

Angstrom units was found from the standards. A curve was
drawn taking the mean wave-length of each pair of standards
as abscissa, and the corresponding factor for the screw as
ordinate. This gave 1°3868 as a mean value for the factor.

To ascertain how accurate the readings had been, the wave-

coo)

lengths of the measured palladium lines were calculated.
The differences between those found from this one measure-

ment and those usually accepted as being correct did not
exceed afew hundredths of, an A.U. ‘To test the accuracy
of the readings further, the standards were measured in
both directions, and the mean error for each reading was
found to be 0:02038 A.U., which agrees very well with that
obtained above. From five measurements in each direction
for the same two lines, the mean error in one reading was

found to be 0°0225 A.U.

From the above results it was evident that the micrometer

was sufficiently accurate to detect a displacement of
0:04 A.U., which is the smallest recorded by Exner and

Haschek.

The spectral apparatus consisted of a Rowland concave
grating whose radius of curvature was 4°56 m.; it had
20 900. lines to the inch and 70,000 over the w AGE divided

Soantnce’ The mounting of slit, grating, and camera was

that of the well known arrangement of Rowland. The

camera and gr ating are fir mly bound together by a beam the

length of which is equal to the radivs of curvature of the
orating, and they move on carriages along two rails which
are set at right angles to each other , and. above the junction
of which ihe slit 1s doing:

The electrodes were held in clips which were so arranged
that during the passage of the current they could be moved
sideways, or their distance apart could be altered. For the
production of the are a direct current from 110 volts was

398 Dr. Genevieve V. Morrow on Displacements in

used, the strength being varied between 4 and 18 amperes
by mears of resistances. The alternating current which was
used for the spark was furnished by a primary current of
25 amps. at 110 volts, which was transformed to a potential
of 10,000 volts. In parallel with the spark was a Franklin
condenser of 750 m. capacity. The distance between the
electrodes in both arc and spark was from 4 to 5 mm.; and
an enlarged image of the source of light was thrown on the
slit by means of a Schumann condenser, which consisted of
two crossed cylindrical quartz lenses—one being vertical and
the other horizontal.

For obtaining the photographs Schleussner’s orthochromatic
Viridin plates were used. They were 30 cm. long and about
4 cm. wide and were placed in a curved position in the
camera, so that they were on the circumference of a circle
of 2:28 m. diameter. In this way the lines of the spectrum
along the whole length of the plate were equally sharp. On
each. plate either the arc or the spark spectrum of the sub-
stance examined was photographed, and immediately below
it and slightly overlapping it was taken the photograph
of the are spectrum of iron to be used as a standard. A
vertical screen with a horizontal opening was placed before
the camera so that only part of the plate was exposed at first,
and it was then lowered the necessary distance before the
iron was photographed, precautions having been taken that
no possible displacement of the camera could occur between
the two exposures. There were always common lines in
both spectra due to some impurity, which could be compared
in order to see if any accidental displacement had occurred,
but in no case was such observed. The time of exposure for
the arc spectra varied between 2 and 5 seconds, and in the
spark spectra it reached a maximum of 5 minutes. The
plates were developed with hydroquinone.

The wave-lengths of the standard iron lines were taken
from Rowland’s tables, and were so chosen that each line of
the substance investigated could be referred to four well
defined standard lines, that is to two on either side of it.
Each plate was measured ten times,—five times forwards
and five times backwards, and the mean of these measure-
ments taken to calculate the wave-length. Two values were
obtained for the wave-length of each line, one from each
ee of standards near it, and the mean of these two values

as taken as being correct. The difference between these
ines was in most cases only a few thousandths of an Angstrém
unit, but in a few instances it reached 0°025 A.U., that is

certain Spectral Lines 07 Zinc and Titanium. 399;

the mean error in measurement. For each line always the
same standard iron lines were used.

The tables which are given below contain the results of
the measurements of seven lines of the zine spectra and
thirteen lines of the titanium spectra. The first column
contains the wave-lengths obtained in the arc by using pure
metal electrodes and a current of 4 amperes. In the re-
maining columns the decimal of the wave-length is given
which was obtained under the conditions stated at the head
of the column. The numbers in brackets (; represent the
intensities of the lines valued from 1 to 100, the greater
number corresponding to the greater brightness. R means
that the line is reversed, and n that it is not sharp. The
wave-lengths obtained by Hxner and Haschek are given in
the last two columns of each table for comparison. Their
scale of intensity is from 1] to 1000, uv means the line is
reversed, and + that it is not sharp.

Zine.

Tables I. and II. contain the measurements obtained for
the zinc lines.

The zine which was used for the electrodes was the
chemically pure zinc of commerce. For the spark the
electrodes were about 2°5 cm. long, and 0°5 sq. cm. cross-
sectional area with chisel-shaped edges, which were placed
parallel to each other. The spark was vertical and parallel
to the slit. Many photographs were taken of the spark
spectra in the region of the lines chosen for examination, the
time of exposure of the plate being varied from 2 seconds to
3 minutes. Inthismanner some plates were obtained having
strong broad lines and others with sharply defined lines.
Photographs of the arc spectra of zine were also obtained,
but some difficulty was experienced, owing to the lower elec-
trode melting and falling away. This was avoided by melting
some zinc into a small carbon cup and using this as the
lower electrode. In order to ascertain the effect on the wave-
lengths of the lines caused by alteration of the strength of
the current in the arc, photographs of the spectra were taken
when the current had various strengths between 4 and 18
amperes, the time and exposure being kept constant. In
order to obtain lines of various intensities photographs were
taken also when the time of exposure was varied. It was
difficult to obtain weak but sharp zine lines owing to the
short exposure necessary, consequently part of the Schumann
condenser was cut off by a screen, and thus only some of the
light was allowed access to the slit.

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Displacements in Spectral Lines of Zinc and Titanium. 401

In order to find the effect, if any, on the wave-lengths of
the lines caused by the presence of impurities in the arc and
spark, brass electrodes were used, which of course caused
copper as well as zinc to be present in the glowing vapour.
The electrodes had approximately the same dimensions as
those of zinc. The are and spark spectra were obtained
under the same conditions as those of pure zinc, with
the exception that in the case of brass of course there
was no difficulty with regard to the melting of the electrode.
The time of exposure was varied between 2 seconds and
> minutes.

Table I. shows that by using a current of 4 amperes the
are spectrum of zine gives wave-lengths for three of the
chosen lines XX 3345°6, 3345°1, 3282°4, which correspond
fairly well with those given by Exner and Haschek when
the lines have about the same intensity; but the wave-
lengths for the lines 13303 and 3302 are not so great
as theirs, and even the greatest values obtained for these
lines in the present work are not so large as those of the
strongest lines of Exner and Haschek. ‘The wave-lengths
here given for these two particular lines when 4 amperes
current is used correspond more nearly with the values
AX 3303'03 and A 3302°67 given by Eder and Valenta. An
increase in the strength of the current to 17 or 18 amperes
seems to have no effect on the wave-lengths of the lines,
within the limits of error, although the intensity of the
lines is greatly increased, and they are all reversed. In the
spark spectrum obtained from 2 seconds’ exposure all these
lines except 1.3282 appear to experience a displacement
when compared with the previous arc measurements, and in
each case the displacement is towards the less refrangible
end of the spectrum. When 380 seconds’ exposure was used
all the lines showed a displacement towards the red when
compared with the previous arc measurements.

In the arc spectrum of brass obtained by using 9 amperes
current, the zinc lines show no displacement when compared
with those of pure zinc from 4 amperes current, with the
exception of A 3282, the wave-length of which is smaller than
with pure zinc. Witha current of 14 amperes all the lines
are reversed and the lines 13302 and 2X 3282 are displaced
towards the red, and with 18 amperes there is a considerable
increase in the wave-lengths of all the lines. The fact that
these reversed lines experience a displacement is in agree-
ment with the results of Duffleld, Jewell, and Exner and
Haschek. The spark spectrum of brass obtained by giving

Phil. Mag. 8. 6. Vol. 29. No. 171. March 1915. 2D

402 Dr. Genevieve V. Morrow on Displacements in

2 seconds’ exposures gives lines which show a great displace-
ment towards the red when compared with those of the are
spectrum of brass when a weak current is used. With an
exposure of 30 seconds the displacement is increased in all
the lines, which is still the case on exposing for 2 minutes;
but in the latter case the line 13282 has not such a large
wave-length as with an exposure of 30 seconds.

Table II. contains the measurements of the zinc lines
4810°7 and 7 4722°3.

There is no displacement observed in the are spectra with
increase of current when pure zinc electrodes are used,
which was also the case with the other zine lines previously
examined. Both lines are reversed with a current of 4
amperes, and although the lines are nebulous when 12 and
18 amperes are used, still the reversed part is in every case
quite sharp and easy to measure. Eder and Valenta give
2X 4810°71 and 2 4722°26 as the wave-lengths of these lines,
which are considerably smaller than those given by Exner
and Haschek. Inthe spark spectra there is a displacement in
each line towards the red when compared with the are spectra;
the differences between the measured wave-lengths for each
line for the two different exposures lie within the limits of
error in measurement. The limit which has been taken
abeve for displacements is reached for one line by an ex-
posure of 30 seconds, for the other by 3 minutes’ exposure.
Both lines are unsymmetrically broadened towards the red,
the darkest part of the line in each case being almost on the
edge of the violet side, whilst the line itself gradually shades
off to the red. The increase in the time of exposure pro-
duced no alteration in the measured wave-lengths.

The wave-lengths obtained for these two blue lines of zine
when the are spectrum of brass was photographed, showed
practically no deviation from the wave-lengths given by pure
zinc. Even when the current is increased to 18 amperes
there is no change in the wave-length, which is contrary to
the results obtained from the five lines described above. In
the table it will be noticed that there is a great difference
between the intensity of the lines when 17 and 18 amperes
current were used, which is accounted for by the fact that
there was a screen before the condenser lens in the one case
which cut off part of the light from the slit. From this it
can be seen that the change in intensity causes no difference
in the wave-lengths obtained. On using a current of 18
amperes the lines were very broad but the reversals were quite
sharp. The spark spectra obtained from brass electrodes

certain Spectral Lines of Zine and Titanium. 403

give lines the wave-lengths of which are much smaller
than those given by the arc spectra of brass or of pure
zine.

It may be seen that in no instance in the spark spectra of
either pure zine or brass do the lines 14810 and 24722
attain the maximum values given by Exner and Haschek,
that is 2 4810°85 and X4722°50 respectively, although the
lines show a displacement in the spark spectra of pure zinc
compared with those of the arc.

Titanium.

The following table contains the measurements of the
titanium lines.

To obtain the are and spark spectra of tilanium, pieces of
the metal about the size of a pea were placed in the electrode
clips and the current passed between points which had been
set opposite to each other. The spark was in each case
vertical and parallel to the slit. Many photographs of the
spectra were taken, the time of exposure in the case of the
spark being varied between 2 seconds and 1 minute, and in the
case of the arc the strength of the current was varied between
4and 18 amperes. In order to study the influence on the
wave-leneths of the titanium lines due to the presence of
other elements in the are and spark, titanium potassium
fluoride was used. This salt was dissolved in hot nitric
acid which deposited a gelatinous precipitate on cooling, but
this went partly into solution on shaking. Carbon elec-
trodes were used, and some of this solution of the titanium
salt was placed on the lower one. Then photographs were
taken of the spark and are spectra, varying the time of ex-
posure and the strength of the current as with the metallic
electrodes. As there were no titanium lines obtained by this
means in the are spectrum when a current of 4 amperes was
used, a paste of the salt was made with water, and some of
this was placed on the lower electrode, and by this method
some lines of weak intensity were produced. On increasing
the strength of the current more lines appeared on the plate,
but with 18 amperes the salt, which seemed to be driven
away from the region of the spark, melted and ran down
the side of the electrode, so that no titanium lines appeared
on the photograph. The thirteen titanium lines which
were chosen for the investigation lie between A 4000 and
r 3300.

Dr. Genevieve V. Morrow on ei. mn

404

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‘TI WIV ye,

certain Spectral Lines of Zine and Titanium. 405

The first column in Table ITI. gives the wave-lengths of the
thirteen titanium lines in the are spectrum obtained by using
a current of 4 amperes and metallic electrodes. As may
‘be seen the intensities of these lines are small, but they
are well defined and sharp. The lines X 3958°3, \ 3948°8,
»r 3904:9, A 3641°4, X 3372°9, A3341°9 correspond fairly well
with the wave-lengths given by Exner and Haschek when
the lines have about the same intensity, but the other lines
show a smaller wave-length than theirs, this being especially
the case with 23913°6, 1 3900°6, 1 3685°3, and 2 3383°9.
The wayve-lengths given here for 2 3900°6 and 23685°3
correspond with the measurements of Eder and Valenta which
are \ 3900°68 and 2 3685-30, those of Exner and Haschek
being 23900°72 and 23685:37 respectively. The other
lines 2 3913 and 23383 are not given at all by Eder and
Valenta. When the strength of the current is increased to
9 amperes, most of the lines show an inclination towards the
side of greater wave-length when compared with those ob-
tained from a current of 4 amperes, but a decided displace-
ment is seen in the lines 13958, 03913, % 3904, A 3383,
X 3372, and A 3341, and in each case towards the red end of
the spectrum. On further increasing the current to 18
amperes, all the lines are displaced more than 0:04 A.U. from
the previous measurements with a current of 4 amperes,
except A 3948 and > 3349, although the wave-lengths of each
of these is greater than with 4 amperes. The lines \ 3372,
® 8349, and 1.3341 are reversed.

In the spark spectrum obtained by an exposure of three
seconds, nine of the chosen titanium lines appear on the
plate, and all show a displacement towards the red when
compared with the corresponding lines obtained from the
are using a current of 4 amperes, although the intensities
are almost the same in each case. The line 3913 was
difficult to measure owing to the proximity of a line in the
iron spectrum, but the alteration in the relative positions of
the two lines can be seen even on examination of the plates
with the naked eye. When the time of exposure is increased
to 1 minute all the thirteen lines are seen on the plate, and
all show a displacement of more than 0-04 A.U. when com-
pared with the are spectrum from a current of 4 amperes.
The lines 7 38383, 13372, and X 3341 are not sharp except
at the ends, which part was measured in each case. The
line X 3349 is broad, and owing to the proximity of another
titanium line equally broad, was difficult to measure.

Only seven of the chosen lines appear in the are spectrum
of titanium potassium fluoride when a current of 4 amperes

406 Dsplacements in Spectral Lines of Zinc and Titanium.

is used. The lines 13958, 1 3948, A 3741 show a displace-
ment towards the red when compared with the are spectrum
of the pure metal obtained with the same current. None of
the other lines give smaller measurements than those obtained
from the pure metal under the same conditions. On in-
creasing the current to 114 amperes all the lines appear
except 23913, 23900, and 73641. The lines 23958,
A 3904, 13741, A 3685, A 3383, and 73372 all show a dis-
placement of more than 0°04 A.U. towards the red when
compared with the arc measurements of the pure metal with
a current of 4 amperes. The line 13948 is increased in
wave-length by 0°034 A.U.

In the spark spectrum obtained from titanium potassium
fluoride all the lines appear when the plate is exposed for
3 minutes, and most of them have greater wave-lengths than
the corresponding arc lines of the pure metal obtained from
acurrent of 4 amperes. There is a distinct displacement
towards the red in the lines 13948 and 13913. If the
exposure is increased to 5 minutes the lines experience very
little alteration in wave-length, the differences being within
the limits of error in measurement. But when the results
of the spark measurements obtained from the titanium salt
and those from the pure metal are compared, it is noticed
that in some cases the pure metal gives lines the wave-lengths
of which are greater than those of the salt. This is the ease
in the lines > 3900, A 3685, > 3641, A 3505, A 3383, dr 3372,
r 3349, and 73341. The same effect is observed with the
zinc lines X 4810 and A 4722.

Conelusion.

The foregoing measurements prove without doubt that the
wave-lengths of the lines in the spectra of the metallic
elements are not constant, but that they experience displace-
ments towards the red under certain conditions.

The wave-lengths of lines in the spark spectrum of the
pure metal are in general greater than those in the are
spectrum, but the difference alters for the various metals
and for the different lines of the metals.

In the are spectrum of the pure metal the lines are dis-
placed by increasing the current, if the point at which the
element vaporises be high enough for the increase in current
to produce an increase in the density of the vapour.

The presence in the are or spark of atoms of another
element appears to have no influence on the wave-lengths of
the lines of the substance being examined, the wave-lengths
obtained depending on the partial density of the substance
itself.

Relation between X-ray Wave-lengths and Absorption. 407

The measurements show clearly in many cases that through
over-exposure of the plate errors may be produced in the
results, since on increasing the time of exposure, and con-
sequently increasing the precipitate of silver on the plate, an
apparent alteration can be found in the wave-length of lines
which are unsymmetrically broadened.

The present research has been carried out at the Physical
Institute of the University of Vienna. I take this opportunity
of expressing my best thanks to Hofrat Professor Hxner and
Professor Haschek, at whose suygestion the work was under-
taken, for their active interest and helpful criticism during
its progress.

May 28, 1914.

XLII. The Relation between certain X-ray Wave-lengths and
their Absorption Coefficients. By W. H. Brace, D.Se.,
FLRS., Cavendish Professor of Physics in the University
of Leeds *

a the figure is shown the X-ray spectrum of rhodium,

in the second order, given by the (111) planes of

calcite. The abscissee denote the glancing angle, that is

to say the angle between the incident X ray and the crystal
Fig, 1.

An °e eeeee,

planes. ‘The ordinates represent the readings of the electro-

scope attached to the X-ray spectrometer. In determining

this spectrnm, readings were taken every two minutes of

arc, and every minute in the important regions. Each dot

in the figure represents a separate measurement, but only
* Communicated by the Author.

408 Prof. W. H. Bragg on the Relation between certain

a few of the readings taken in the regions between the
“lines” and on either side of them are shown in the figure.
It will be observed that the spectrum is practically a pure
line-spectrum. In fact, no general or “ white” radiation
can be found—-a condition which is no doubt due to a
particular state of the bulb. The small readings of the
figure, which lie between the lines, are due to various minor
causes, including scattering of X rays. If a diamond is
employed, they are proportionally far less. With the
spectrometer-slits opened to two or three millimetres so as
to obtain the maximum effect, the electroscope-leaf moved
260 divisions in five seconds when the glancing angle
(between incident rays and diamond) was 8° 30’ and the
reflected portion included the strong « line. When the
angle was half this, the reading was not more than one
division in the same time.

Calcite is an accurately built crystal, resembling diamond
in this particular. It is a far better crystal for accurate
work than rocksalt, which may be compared to a badly ruled
diffraction-grating giving false images or “ ghosts.” The
rhodium anticathode was placed so that the rays left it ata
grazing angle, and the source of rays was therefore, in effect,
a line parallel to the slits. ‘The pencil of rays was limited
just before incidence on the crystal by a slit 0°2 mm. wide.
With this combination of circumstances, the lines of the
spectrum are well separated from each other.

The spectrum contains four lines. The two of longest
wave-length constitute the doublet which has already been
observed and examined *. In combination they compose
the strong line which Moseley | has observed to be given
by a great number of substances, constituting in fact the
principal part of the K series of characteristic radiations.
The term “doublet” is not really justified, for there is no
reason to suppose the two constituents stand in any special
relation to each other. They may be spoken of separately
as @, and a, which terminology will be in touch with
Moseley’s.

The line near to 11° in the spectrum is the other well-
known constituent of the characteristic radiations of the
K series; it is known asthe 8 line. There is also a fourth
line which has not been noted before, so far as I am aware.
Jt is marked y in the figure.

Examination has also been made of the rays from bulbs
having anticathodes of palladium and of silver. It was not

* Nature, March 12,1914; Phil. Mag. May 1914.
+ Phil. Mag. April 1914.

X-ray Wave-lengths and their Absorption Coefficients. 409

possible to obtain such clear-cut results as in the case of
rhodium, because the bulbs were of an old pattern and the
anticathodes were surrounded by raised rims. The rays
leaving at a grazing angle could not be used, and there was
no longer a “ line-source.” The images were therefore
blurred, more especially in the case of palladium.

Nevertheless the figure makes it clear that the spectra of
all three metals are of exactly the same kind. Diamond was
used instead of calcite in order to effect the separation of
a, from a by the use of the high resolving powers of the
third order spectrum. In order to separate @ from ¥ it was
found best to use the first order spectrum ; the distance
between @ and y is twice as great as that between a, and a,
and the two can just be resolved in that order. In the third
y is too weak. It is certainly remarkable that the spectra of
these three substances should resemble each other so closely.
It will be of much interest to know how far the resemblance
extends to the spectra of other substances.

The angle of reflexion of a, in the first order of calcite is
6° 11'. The structure of calcite has been given by W. L.
Bragg (Proc. Roy. Soc. Ixxxix. p. 486). Assuming the
density of the crystal to be 2°71 and the mass of the H atom
to be 1°64 x 10~*4, it can be calculated that the spacing of
the (111) planes is 2845 ALU.

This gives
| N= xe 84a esim oe

=0°613 A.U.,

agreeing well with the value 0°614 A.U. previously found
by the use of diamond *.

The values of all the wave-lengths in the three spectra
may be readily calculated from the results illustrated by the
figure. A value for the line of Pd, 0°576 A.U. was given
in a paper published in the Royal Society Proceedings,
November 1913; the crystal used was rocksalt. Moseley ’ is
using potassium ferrocyanide calibrated by reference to
rocksalt, found 0:°584 for Pd and 0:560 for Ag. Recently
Malmer ¢ gives for the a lines of Pd and Ag 0°590 and
0°564, and for the 8 lines of the same substances 0°522
and 0499. He also used rocksalt. As far as the particular
crystal has influence, diamond and calcite are to be preferred
to rocksalt, for the reasons already given. The following
table gives the wave-lengths in nestrém units calculated
from the experiments described in this paper. The values

* Phil. Mag. May 1914. + Phil. Mag. April 1914.
t Phil. Mag. December 1914.

410 Prof. W. H. Bragg on the Relation between certain

for the y line of Pd and Ag are not so reliable as the
others.

0°503 | 0°554

These measurements were undertaken in the attempt to
throw some further light on the relation between wave-
length and absorption in certain cases. Barkla has shown
that in general the X rays characteristic of any substance
are strongly absorbed by substances of lesser atomic weight
as compared with substances of greater atomic weight. The
phenomenon may be more definitely expressed in terms of
wave-length. An instance is given in the table on p. 627 ~
of the December (1914) number of the Philosophical
Magazine. The 8 ray of silver (0°495 A.U., see the table
above) is very strongly absorbed by Pd as compared with
Sn or even Ag itself. Some figures given in the table
quoted are reproduced here (Table II.) in slightly altered
form, and will make the point clear.

Tani LI).

l
Log. of atomic absorption

coefficient x 1072.
Log. of
wave-length
X ray. | in ALU. Pd. | Ag.
|

jee sag a a | 16946 2064 | 1-260
1122 ah STS ae ae 17126 1-301 1:342
BY sateen 1-7364 1-350 1-394
As ah ale us Gel os | 1-7459 Lore 1-403
Bd aie. eee |- W657. | 1°452 1-477
Bh ait ie env | 17882 | 1-498 | 1°545

These are plotted in fig. 2. All the points representing
the way in which the absorption of the different rays by

X-ray Wave-lengths and their Absorption Coefficients. A1t

silver depends on the wave-length lie on a straight line
within errors of experiment. This was to be expected more
or less, because Owen has pointed out that the absorption
coefficient by a given absorber varies inversely as the fifth

Fig. 2.

ro)
S)

(o-)
o

1-60

Log (atomic absorprion coerticrent) + 22

1-65 T:70 1-75 180
Log. (ware lergth (7 AV)

power of the atomic weight of the atom emitting the radiation,
and we know that the latter, roughly, is inversely propor-
tional to the square root of the wave-length which it emits.
Consequently the absorption coefficient by a given absorber is
proportional to (wave-length) , and logarithmic plotting
should give a straight line. The slope of the lines in the
figure givesan index 3 much more nearly than 5/2 ; but the
range of wave-lengths in the table is too narrow for an exact
deduction. Measurements of the absorption coefficients in
terms of wave-length are being made over a much wider
range, and will be useful in making a correct determination
of the index.

If we consider the silver curve, we see that the absorption
of the silver rays is not remarkable in any way ; the points
for the silver wave-lengths lie on the straight line passing
through the points for the wave-lengths of palladium and
rhodium. Now it is certain that the high absorption which

412 Relation between X-ray Wave-lengths and Absorption.

occurs when a characteristic radiation traverses a substance
of less atomic weight is accompanied by, and to some extent
dependent upon, the production of secondary radiation.
Zine rays are highly absorbed by nickel, and at the same
time zinc rays excite nickel rays. But nickel rays cannot
excite zinc rays, and the absorption of nickel rays by zine is
relatively small.

When, therefore, we find that silver absorbs its own rays
on no higher a scale than it absorbs those of Rh and Fd,
which are of less atomic weight (the atomic weights of
Ag, Pd, and Rh are 108, 107, and 103, their atomic num-
bers 47, 46, and 45), we conclude that none of the rays
emitted by these three substances can excite any of the
silver rays. This is the case although many of the waves
are shorter than one or more of the characteristic silver
waves.

If we examine the palladium curve, we find all things the
saine, except that the short silver wave 0°495 is highly
absorbed by Pd, and no doubt excites Pd rays. Why should
it be able to do so? It cannot be merely because it is
shorter than some of the Pd waves, because the Pd wave
0°516 cannot excite Ag waves of greater length.

The most probable condition would seem to be that the
exciting wave must be shorter than all the characteristic
waves of the substance in which it excites those waves.
It will be observed that the 6 ray of Ag is just shorter than
the y ray of Pd. Further examination of parallel cases
will be necessary, of course, before this statement can be
generalized. It seems likely, however, that certain pecu-
jiarities in Barkla’s table of absorption coefficients can be
explained by its aid. For example, Barkla states that the
mass-absorption coefficients of Ni for the rays emitted by
Ni, Cu, Zn are respectively 56°3, 62°7, 265. We should
explain this on the ground that neither nickel ray can excite
the characteristic rays of nickel, because of course neither
is shorter than the shortest; of the copper rays, one, the
weaker, can excite the nickel spectrum and is_ highly
absorbed, but the other, the longer and stronger, cannot
do so and is not specially absorbed. The net result is that
the copper rays, though shorter on the whole than the nickel,
are nevertheless the more highly absorbed as a whole. Both
the zine rays can excite the nickel rays because both are
shorter than the shortest nickel ray ; the zinc rays are very
highly absorbed in consequence. It may be that the charac-
teristic rays of a substance form a system which can only be
excited as a whole.

(ea

XLIV. On Condensation Nucler produced by the Action of
Light on Iodine Vapour. By Haroupd Pearine, M.Sc.,
Lecturer in Physics at South African College, Cape Town*.

| aie object of this research was to continue the investi-
gation I carried out in collaboration with Professor
Gwilym Owen, D.Sc. (an account of which we published in
the Phil. Mag. for April 1911) ; and to test and examine the
objections which Ramsauer made to the explanation we gave
of our experiments.

From experiments we made with a Wilson’s expansion
apparatus it was shown :—-

(1) That when light fell on a mixture of moist air or
oxygen and iodine vapour contained in a freshly cleaned
glass vessel, nuclei are produced possessing the following
properties :

The nuclei are very unstable, disappearing in a few
seconds in the dark, and carry no electricity and need oxygen
and moisture for their production. The light required for
their production need not be very intense, nor of a high
degree of refrangibility. They usually reach their maximum
size in less than one second, and as a rule they were not large
enough to be caught by an expansion of less than 18°5 cm.

(2) No nuclei are produced after the iodine has been in the
apparatus for some days, but if the apparatus was washed
with nitric acid and finally with distilled water. nuclei
reappeared.

(3) Glass-wool possesses the peculiar property of facili-
tating the formation of nuclei, the number produced when
iodine-laden air is admitted into the apparatus through a
plug of glass-wool being much greater than the number
obtained on placing iodine directly in the cloud-chamber.
This property becomes less and less marked as the wool
becomes more and more saturated with iodine. We were
unable to decide whether this action of the glass-wool was
due to some impurity on its surface. In the case of the
nuclei produced without the aid of the glass-wool, we were
of the opinion that they were not produced by an impurity
on the surface of the glass, but were produced by some
chemical action between the iodine, water-vapour, and
oxygen. To explain the disappearance of the effect we
supposed that the chemical action was reversible, and that as
soon as chemical equilibrium was established no more nuclei
were produced.

The most conclusive evidence we put forward for this

* Communicated by the Author.

A414 Mr. H. Pealing on Condensation Nuclei

explanation was that the effect reappeared every time tlie
apparatus was carefully cleaned. When, however, the glass-
wool (which had not been previously used) was cleaned, as.
far as possible in the same way, it gave an effect which was
much less than it gave in the unwashed condition, Another
important flaw in the evidence was the fact that when once
the glass-wool had lost its property of producing nuclei, all
attempts to bring back its property of producing nuclei
failed. Obviously, as long as this was the case, the effects
in the two cases being so similar except in actual amount,
the evidence in favour of the explanation we gave could not
be regarded as conclusive. For this reason the present
investigation was carried out. Neurly all the experiments
were with glass-wool.

Haperiments with Glass- Wool.

First a word about experimental details.

(a) The apparatus used was very similar to the Wilson’s
apparatus used in the former experiments.

(b) The apparatus was not cleaned with extreme care as
it was shown that the accidental impurities which glass gains
when left exposed to the air have no influence on the effect
investigated. The whole apparatus was thoroughly washed
with a strong solution of soap and water and then carefully
rinsed out with ordinary tap-water. Sometimes strong
nitric acid was used before rinsing with the tap-water.

(c) The size of the nuclei was estimated by observing the
pressure-fall in the Wilson’s expansion apparatus necessary
to bring them down. As a rule a pressure-drop of 18§°5 em.
was necessary to catch the majority of nuclei, but very often
a much lower pressure brought a large number of the nuclei
down. The number of nuclei was estimated in a rough
manner by finding out the density of the condensation cloud
produced when an expansion was made, the expansion
chamber of the apparatus being illuminated by the light
focussed from a Nernst lamp. |

(d) In order to eliminate causes other than the glass-wool
of producing the nuclei, the apparatus was left standing with
iodine in it until the effects produced on expanding’ the air
contained were the same as those obtained before the intro-
duction of the iodine. In other words, no experiments were
made with the glass-wool until the apparatus gave, when
saturated with iodine, the ordinary Wilson effects.

(e) The effect produced by the glass-wool was observed by
passing iodine-laden air through a plug of it (about 25 em.
Jong) direct into the apparatus. :

produced by Action of Light on Lodine Vapour. 415

The table gives the results obtained when the plug was
freshly set up.

yy, B.

Wilson apparatus containing moist
dust-free air freshly saturated
with iodine vapour.

Wilson apparatus containing moist
dust-free air.

Result

Press. Fall. | (Ordinary Wilson Effect ) || Press. Fall. Result.

15°4 em. Nothing. 91 cm. Nothing.

15:8, Few drops. 166 ,, Thin shower.

Kio: Thin shower. 8) Good shower.

WOT Good shower. NS so he Coloured cloud.
aS ae Slightly tinted shower.

Ik 55 Coloured cloud.

D.
C. Same as C, but iodine-laden air
Same as B. one day later. admitted through freshly set
up glass wool plug.

Press, Fall. | Result. Press. Fall. Result.

15‘7cm. | Few drops. 157cm. | Few drops.

HGEEe 5, Few drops. ies | Good shower.

vag Fair shower. l8:3ae | Dense colcured cloud.
LSS, Heavy shower.

A comparison of C and D of the table shows that the glass-
wool is instrumental in producing a very large number of
nuclei, but it soon ceased to be effective, as after one day the
effect produced for an expansion of 18°7 cm. was a very
heavy shower. Its power was tempurarily restored when air
was drawn through the glass-wool after it had bubbled
through water. The power of the glass-wool to produce
nuclei was brought back and intensified in the following
manner. When the glass-wool had ceased to produce nuclei
and was strongly discoloured with iodine throughout its
length, a small quantity of tap-water was sucked throug h it.
The amount of water used was just sufficient to saturate half
the wool. When air was drawn through, the wool became
wet throughout its entire length and the air which got
through gomened an enormous number of nuelei. An
expansion of 18°7 cm. produced a dense fog, but after one

416 Mr. H. Pealing on Condensation Nuclet

day the effect diminished to a slightly tinted cloud. When
the glass-wool was dried again by drawing dust-free air
through it for a few hours, the effects obtained were again
largely increased. These effects were very persistent, and
experiments continued for more than a fortnight (during
which dust-free air was drawn through the glass-wool for
days) failed to remove the effect. By eliminating to a large
extent the iodine, the effects were reduced but not got rid of,
no doubt because of the difficulty of entirely getting rid of
the iodine by simply drawing air through. The presence
of fresh iodine quickly increased the effect again. The
effect did not disappear when dry dust-free air was used in
place of the ordinary air of the laboratory. After the giass-
wool plug had been treated in this manner, it was left
standing for a fortnight. The effect was found to have
almost disappeared, but it reappeared when a few cubic centi-
metres of distilled water was drawn through, but the effect
this time was not so persistent and disappeared almost
entirely in the course of a few days. A similar amount of
tresh distilled water brought back the effect, and this time
it was very persistent. These experiments were made in
Cape Town. Through the kindness of Professor Wilberforce
T was able to repeat some of them at Liverpool University,
using another apparatus and a different kind of glass-wool.
The glass was obtained in the spun condition and was cut
into lengths and the fibres separated just previous to the
making of the plug used in the experiments, the result being
somewhat heterogeneous glass-wool. A more important
variation from the Cape Town experiments was in the water
used to renew the effect in the glass-wool. This was of
special purity, and was supplied to me through the courtesy
of Mr. Powell of Liverpool University. The water was of
the degree of purity required for accurate determinations
of the resistivity of solutions. In this case also the effect
was renewed in the glass-wool when a few cubic centimetres
of this specially purified water was drawn through the plug.

Discussion of Results.

These experiments show that the purest water obtainable
renews the effect in any kind of glass-wool. The conclusion
drawn is that the effect does not depend in any way on the
impurities which water contains. Other reasons supporting
this conclusion are given further on. The fact that the effect
can be renewed seems to prove that the effect is not caused
by any impurity on the glass-wool. Owen and Pealing have
shown that the effect obtained by the interaction of the

produced by Action of Light on Iodine Vapour. 417

iodine and the Wilson’s apparatus was renewed by rinsing
the apparatus with distilled water, and the properties of the
nuclei produced were the same, except possibly in size, as
those produced by the glass-wool. Since the effect produced
in the glass-wool is renewed by drawing distilled water over
its surface, the conclusion seems irresistible that the nuclei
produced in the two cases are due to the same chemical
action.

The question arises, what is the nature of this chemical
action? In the communication referred to above we gave a
discussion of this. Three possible kinds of chemical action
were considered :—

1. That the chemical action was one between the iodine
and some impurity on the surface of the glass.

2. That the chemical action was a surface action between
the glass and the iodine or one of its chemical compounds.

3. That it was a chemical action between iodine and
water-vapour and oxygen caused by a catalytic action of
the glass.

We considered that the weight of evidence entitled us to
reject the impurity explanation in the case of the Wilson’s
apparatus alone, and this conclusion I now extend to the
glass-wool. We considered that the second explanation was
the correct one for the Wilson apparatus, and that the third
explanation might explain the results when using glass-wool.

Ramsauer objects to this view and puts forward the
following evidence in favour of the first explanation *.

He says :—“‘ In explanation of the effects there described
I should like to call attention to our experiments ¢, where we
have shown that glass-wool and every glass surface that has
not been strongly heated continuously gives off adsorbed
minor constituents of the air, which on the production of ozone
by ultra-violet light always lead to the formation of nuclei ;
now, according to Mr. Owen himself, ozone is formed in his
experiments. The regeneration of the effect on washing the
walls with distilled water is explained by the fact that the
water must contain dissolved small quantities of the vapours
and gases mentioned before { (for instance NH3), and as it
trickles down gives these up to the glass walls through
adsorption, ‘These observations led us to construct asbestos

* Phil, Mag. May 1912.

+ P. Lenard and‘C. Ramsauer, “‘ Ueber die Wirkungen sehr kurzwel-
ligen ultravioletten Lichtes auf Gase und iiber eine Sehr reiche Quelie
dieses Lichtes,” Heedelberger Akademie, Five parts: I. Aug. 2, 1910; II.
Nov. 5, 1910; III. Dec. 20, 1910; TV. June 9, 1911; V. Sue. 4, 1911.

t CO,, NH,, organic vapours, ete.

Phil. Mag. 8. 6. Vol. 29. No. 171. March 1915. 2H

418 Mr. H. Pealing on Condensation Nucle

filters of combustion-glass, which could always be cleansed by
being heated to redness, while glass-wool or cotton-wool
filters, which are generally used finally to free a current of
purified air from dust, have just the opposite effect, and
charge the air again with all the impurities which have been
previously removed, thus vitiating the results.”

In reply to these objections, [ would point out that his
conclusions rest upon the assumptions that ozone is produced
by the interaction of iodine and water, viz.

HeOee = Ae TA PG!
Hie, > Hi+O,.

So far as lam aware no proof exists that this chemical
action goes on under the conditions of our experiments.
(Ramsauer’s reference to our experiments establishing that
fact is a mistake.) What evidence there was, negatived that
conclusion. If such an action goes on, then the water should
become acid, but it was found that it was of the same degree
of acidity after several days’ exposure to the iodine as it was
before its introduction to the apparatus. But Ramsauer’s
explanation seems to break down much more completely
when we consider the reappearance of the effect on washing
the apparatus. Hxperiments showed that the introduction
of water into the apparatus up to about 100 cc. in amount,
had no appreciable result on the effect so long as the surface
of the glass was not rinsed in the process. Hence, if
Ramsauer’s explanation is correct, the amount of water
necessary to introduce sufficient impurity to bring back the
effect must be of the order of many kilograms. ‘That is, the
apparatus must be rinsed with several kilograms of water
before the glass walls are saturated with the minor impurities
he mentions, and the iodine effect would continue until all
these impurities were exhausted. It may be urged that only
a small part of the impurity contained in that amount of
water is absorbed by the glass. It is very unlikely that this
is the case with the water contained in the apparatus (of the
amount of 200 grms. approximately). Now in a case like
this the effect is very much smaller on the second day after
the introduction of the iodine, and by the fourth day has
entirely disappeared. ‘That is, on Ramsauer’s theory the
impurity necessary to keep the effect going at an appreciable
rate for two days is contained in an amount of water which
cannot be more than 200 grams. Now when we come to
consider the case of the yvlass-wool, we find that when so

produced by Action of Light on Iodine Vapour. 419

small an amount of water as 8 grams or less is used to clean
the glass-wool, nuclei are produced which exceed very much
in number the maximum effect obtained when using the
glass walls of the apparatus (although it had, previous to the
cleansing, produced practically no nuclei, no matter how
much air was drawn through it), and this action continued
for more than a fortnight. Hence in this case, if we adopt
Ramsauer’s theory, 8 grams of water would hold enough
impurity to keep the effect in action at a somewhat
accelerated rate for fourteen days. It is easy to see that we
adopt a very low estimate when we say that 8 grams of
water in the one case produces at least seven times the effect
of 200 grams in the other.

When the glass-wool is freshly used and presumably
saturated with impurities, the effect has practically disap-
peared in two days. To explain the continuance of the effect
for a fortnight on a subsequent occasion, other circumstances
being the same, we should have to assume that either all the
impurity was not exhausted in the first case, or else we get
a considerable supersaturation in the second. Both these
explanations seem untenable. The experiments with the
glass-wool show that the effect obtained does not depend
upon the amount of water used, but on the amount and
nature of glass surface exposed. This is particularly brought
out in the experiments with the specially purified water.
All the experiments support the third explanation that the
nuclei are produced in a chemical action between iodine and
oxygen and probably water-vapour aided by a catalytic
property of glass. The increased action produced by the
glass-wool is explained by its greater catalytic property.
The disappearance of the effect is explained by the disap-
pearance of the catalytic property, caused probably by the
deposition of iodine or one of its products upon the glass
surface. If this explanation is correct, then the action of
the glass walls of the apparatus must be catalytic also.

The second explanation, that the action ceased because
chemical equilibrium was established, is not essential and
there are many objections to this theory. All the experi-
mental facts are consonant with the third explanation, but I
am unable definitely to state that it is the correct one.

In conclusion I desire to express my thanks to Professor
Beattie for his very kind interest in the experiments and for
placing the resources of the South African College Laboratory
at my disposal.

2 EK 2

[ 420 |

XLV. The Tracks of the « Particles in Sensitive Photo-
graphic Films. By 8. Kinosuita, Assistant Professor of
Physics, and H. Ixsvutr, Research Student, Imperial
University, Tokyo *.

[Plate VII.]

N 1910 one of the writers J showed that each « particle
produces a detectable effect on a photographic film, 7. e.
whenever an @ particle strikes a grain of silver halide in the
sensitive film, that grain is subsequently capable of develop-
ment. It was also shown that this is the case throughout
the whole range of the « particles. These conclusions were
afterwards confirmed by the experiment of Reinganum f.
On a microscopic examination of a photographic plate, to
which the « rays had been tangentially projected, he observed
that the path of each « particle appeared as a trail of silver
grains. Reinganum drew attention to the fact that some of
the trails showed the effect of scattermg. Experiments on
this subject were later made by Michl§ and Mayer |} in
more detail. Recently, Walmsley and Makower{ succeeded
in taking some microphotographs on which the deflected
paths of the a particles were beautifully demonstrated. We
have also been engaged in the study of the same problem
and now allow ourselves to give a brief account of the
results.

In investigating the photographic traces of the e rays it
was considered effective to work with a possibly small source
of the rays. For, if a point-like source be established and
placed on a photographic plate, the expelled & particles will
jeave on it a set of radial traces, which can be followed with
greater ease and certainty.

What we have utilized for the source of the « rays wasa
sewing-needle, carrying at its pointed end a minute quantity
of the active deposit of radium. This could be easily
prepared by lightly rubbing the point on a metal piece which
had previously been exposed to a few millicuries of radium
emanation. After the active needle had been in contact with
a photographic plate fora short time, the plate was developed
in the usual way, when a fine spot became visible to the

* Communicated by Prof. H. Nagaoka.
+ S: Kinoshita, Proc. Roy. Soc. A. Ixxxiii. p. 482 (1910).
- ¢ Reinganum, Phys. Zevts. xii. p. 1076 (1911).
§ Michl, Wien. Ber. cxxi. 2 a, p. 1431 (1912).
| Mayer, Ann. d. Phys. xli. p. 931 (1918).
§] Walmsley and Makower, Proc. Phys. Soc. xxvi. p. 261 (1914).

Tracks of the a Particles in Sensitive Films. 421

naked eye. Asa matter of fact, it was not essential to keep
the needle in contact with the plate throughout the time of
exposure. A momentary touch seemed to be sufficient, if a
proper interval of time was afterwards allowed to elapse
before development, indicating that a part of the active deposit
was detached from the needle and left behind on the plate at
the point of contact. In some cases, a metal piece coated
with the active deposit was, while being held above the plate,
knocked with a finger, when similar results were obtained.
In these cases, some dust particles adhering to the metal
piece must have been set free by the shock and have settled
down on the plate.

On examining the plate under a microscope, the said spot
is seen to consist of a multitude of the radial trails of silver
grains around a circular dark nucleus, to which reference
will soon be made. A closer examination shows that these
trails of grains are, in general, to be divided into two sets.
Those constituting the first set emerge at the rim of the
nucleus and end very nearly at the circumference of a circle
drawn outside the nucleus and thus present themselves as a
halo. The second set of the trails, on the other hand, spread
out around the nucleus over a wide region with no sharply
defined boundary. By focussing the microscope, it can be
ascertained that the silver grains constituting the latter set

of trails are all found in the upper ‘most layer of the sensitive

film.

It was, however, not always the case that both of these
sets are equally conspicuous. In some cases, one of them
was particularly pronounced while the other was hardly
visible.

The nucleus above mentioned is undoubtedly the cavity in
the gelatin film produced by the point of the needle when it
was brought into contact, and has nothing to do with the
a particles. Its size is various in different photographs,
depending on the circumstances under which the needle was
held ; the greater the pressure applied to the needle, the
larger the size of the nucleus. It may be added here that
the pointed end of the needle had the shape of a truncated
cone terminating in a flat section, which, as measured under
a microscope, had a diameter of about *01 mm. When the
needle had been held with such a care that it hardly touched
the plate, the photograph showed no dark nucleus. The
same fact was often experienced when an active dust particle
was the source of the x rays. Figs. 2 and 7 (Pl. VII.)
are the examples.

In the photographs, in which a large number of trails

A22 Prof. S. Kinoshita and Mr. H. Ikeuti on the

are radiating from a centre, each trail does not appear
isolated from the others along the whole length, but, in the
vicinity of the centre, comes very close to or overlaps the
neighbouring ones. This effect gives rise to another circular
dark area in the middle, though it has no well-defined
boundary. The radius of the dark area varies of course as
the number of the trails. As this dark area covers the dark
nucleus above spoken of, it sometimes makes the boundary
of the latter indistinct.

The radius of the halo varies slightly in different plates
from one another, but the length of each constituent radial
trail or the difference of the radii of a halo and the cireular
nucleus inside it, is the same for all the haloes and equal to
about ‘054 mm. in the case of Ilford’s Process Plates. It is
most probable that this represents the range of the « particles
from radium OC in the substance, as, in these cases, we used
active deposit in which radium A had practically decayed
away. The silver grains constituting this set of trails are not
restricted within the uppermost layer of the film, but found
also in layers within some depth beneath the surface. From
these facts, it is evident that the halo is produced by a
similar process as the pleochroic halo seen in the mineral
such as biotite, and investigated in detail by Joly. In
the present case, however, a spherical halo cannot be
expected, as the film on the plate employed was equivalent
to only about 2 cm. of air in stopping the @ rays, as
calculated from its weight 0030 gramme per square centi-
metre *. In order to show the haloes at different stages
of formation, we have reproduced some of the micro-
photographs in figs. 1-6 (PI. VII.). These were taken
with a microphotographic apparatus by Zeiss, the magnifi-
cation being 380 diameters. Fig. 1 shows a halo in which
about 80 tracks are to ke seen, this being of course the
number of tracks on the plane focussed in reproducing and
forming only a small fraction of the total. The tracks which
do not radiate from the centre are due to other sources lying
outside the halo. <A fairly developed halo is shown in fig. 6,
which, on our estimation, contains about 200 tracks. The
spots covering the greater part of the figure are due to dirt
on the plate. Figs. 2, 3, 4 and 5 are haloes at intermediate
stages of formation. On the plate containing the halo in
fio. 2, there are, besides, several extended sources of radiation,
whose effect is visible on one side of this figure. We have
counted the number of grains in each trail and found it to be
about 16 on the average.

* Kinoshita, /. ¢. p. 437.

Tracks of the « Particles in Sensitive Hilms. 423

Attempts were also made to obtain a halo which is due to
the a rays from radium A as well as radium C. For this
purpose a metal piece was exposed to radium emanation for
a few seconds, and the active deposit quickly detached from
it was used as the source of the « rays. Closely examining
the haloes thus obtained, it was found that among the trails
of grains there are some which have a considerably shorter
length than the others and seem to be those produced by
the a rays from radium A. But we have not yet been able
to get one clearly shown as a corona, asin the case of the
pleochroic halo.

We have already drawn attention to the fact that the
silver grains, which constitute the set of the trails spreading
over a wide region, have their seats in the uppermost layer
of the film. Fig.7 is an example, of which the magnification
is the same as that of the haloes. This fact suggests the
view that these trails of grains are produced by the
a particles projected tangentially to the surface of the film
from the part of the active deposit just above the surface.
At first sight, some of the trails seem to be much longer than
those constituting the haloes. Caretrul inspection, however,
shows that this is only apparent and that each of them
consists of two or more elementary trails, having the average
range of °054, following one another.

So far, we have presumably treated the problem as the
results a the « ray effect, and no account was taken for
the B.rays which are emitted as well from the active deposit
of radium. It has long been known that the @ rays possess
the property of acting on a photographic plate; but owing
to difficulties involved in the experiment, very little is known
about the effect of an individual 8 particle. In the
experiment of Kinoshita already referred to, the photo-
graphic action of 8 rays was found to be about one-third
or one-quarter of that of « rays in the case of a thinly
coated Wratten’s Ordinary Plate. In this calculation it

was assumed that active deposit from radium emanation, in
which radium A had already decayed away, emits about the
same number of « and @ particles. Since it is now known
that the active deposit emits, under this cireumstance, about
twice as many § as & par ticles, the ratio of the photographie
action of a 8 particle to that of an « particle reduces to only
one-sixth or one-eighth. It therefore seemed likely that

a silver halide grain bombarded by a 8 particle is -not
Bede i capable of development. If this be the case, the
track of a 6 particle would not be so closely filled up with
the developed grains as in the ease of an « particle. It

424 Tracks of the « Particles in Sensitive Films.

is quite possible that the developed grains follow one another
with too wide intervals to present themselves as a track of a
8 particle, much more so when the lability of scattering of
the rays is considered. It will be of interest to recall that,
in the well-known experiment of C. T. R. Wilson, a
considerable difference was observed between the « and
8 particles in forming a track in air. We are inclined to
believe that the grains spreading out over a wide region
in a far less definite manner are to be attributed to the
8 particles.

We shall next describe some experiments made in magnetic
fields. A photographic plate was brought into contact with
an active needle in the way already stated and quickly intro-
duced between the poles of a powerful electromagnet, the
plate being held perpendicular to the lines of magnetic force.
When the plate seemed to have been exposed to a sufficient
number of « particles, it was withdrawn and developed
immediately. Working, however, in a field up to nearly
ten thousand gauss, no ‘indication of curving of the track
was observed. Taking the velocity of the « particle as
2x 10° cm. per sec., it can be shown that the particle
should, in a field of 10,000 gauss, describe a path for which
the radius of curvature is as great as 40 centimetres. Such
a slight curvature would not easily be recognized, as the
length of the path under examination is so minute, that it
is only about ‘054mm.

Evidences have already been given by the previously cited
investigators that many of the « particles suffer sudden
deflexions on the passage through the emulsion film. On
the microphotographs taken by Walmsley and Makower,
this effect is seen exceedingly well. This phenomenon can
be observed on our photographs too. Fi fig. 8 (Ply Vidg
is the microphotograph of a plate in which the a particles
passed from left to right, the magnification being in this
case 1,210 diameters. “It can be seen that, while one of the
a particles passed straight on, another suffered a sudden
deflexion of about 15° after traversing some distance nearly
parallel to the former. It may be remarked, that we have
examined a great number of sets of the radial tracks of the
a particles, but so far we have not been able to find any
which can he said with certainty to have suffered the deflexion
of an angle so large as 90°. The number of the tracks
showing such a large deflexion seems to be one in several
thousands at most.

In the halo with a thrust-mark inside, such as that in
fig. 5, many of the radial tracks are seen to be curved in

Electron Theory of Metallic Conduction. 425

a wave-form. ‘This is possibly due to ununiform contraction
of the gelatin film in the vicinity of the pierced point as
it dried from the wet state in development process, and
therefore cannot be interpreted as the effect of deflexions.

In a series of experiments, a photographic plate was placed
in contact with a flat piece of glass coated. with the active
deposit of radium in the usual way, and thus exposed to the
@ rays coming out of the source of large area. On
examining the plate under a microscope, a considerable
proportion of the tracks seemed to have suffered large
deflexions. Moreover, in these cases, it seemed that there
were as many tracks showing large deflexions as those
showing small deflexions. Consequently, further investi-
gations will be necessary to settle the question, whether these
tracks are actually due to single « particles or to two
particles passing through the film in different directions.

In conclusion, we wish to thank Professor Nagaoka for
his kind interest in this experiment. We must also thank
Professor 'Tawara of the Metallurgical Department for his
kindness in allowing us to use the microphotographic
apparatus.

September, 1914.

XLVI. On the Electron Theory of Metallic Conduction.—Il.
By G. H. Livens*.

| hh modern theory of electrical conduction in metals is

based on the assumption that every metallic body con-
tains a large number of electrons moving about quite freely
in the space between the atoms. Both electrons and atoms
are presumed to be perfectly elastic spherest, the latter being
of such comparatively large mass that their energy and
motion may be neglected. In the absence of an external
field it is presumed that the electrons are moving with the
velocities assigned to them by Maxwell’s law, aecording to
which there are

Na/ Lo eredédnd
V2 Edndt
electrons per unit volume with the components of velocity

* Communicated by the Author.
+ It is easy to generalize the procedure here suggested to the more
general type of theory which disposes of this arbitrary assumption.

426 Mr. G. H. Livens on the

between (&, », €) and (£+dé, »+dn, €+df). In this ex-
pression N represents the total number of free electrons per
unit volume and

fae ay + 6 ;

gis a constant connected with the mean value w,,? of the
square of the resultant velocity of each electron by the
relation 2

ae
ib 2m”

If a uniform field of strength E is brought into play
parallel to the z-axis of coordinates, this velocity distribu-
tion will be immediately altered. Hach electron will
acquire momentum parallel to the same axis at a rate eH
(e being the charge on it), but this gain will be held in check
by a perpetual transfer of momentum between each electron
and all the molecules by which it is influenced at any
instant. Hxactly how this transfer takes place we do not
yet know, but some such interaction between the electrons
and the atoms during the encounters over and above the
ordinary quasi-elastic reactions seems to be necessary in
order to ensure the maintenance of a steady state. We can,
however, make good progress in the theory without assuming
very much about such interactions, and two alternative
methods of attack have been suggested.

Lorentz assumes that a new steady state of motion of the
electrons is attained, once the steady current is well estab-
lished, and by statistical considerations regarding the effect
of collisions and the electric force on the distribution he
finds that in such a steady state the new velocity distribution
may be approximately expressed by saying that

Gf: 2geblae\ 3 |
wn [fi + UBlat),wagandt
E(t 2 )er dé dnd

is the number of electrons per unit volume with velocity
components between (£, 7, €) and (£+dé,n+dn, €+d€) : in
this expression m represents the mass of the typical electron
and J, the mean free path. ‘This of course means that there

are at any instant more electrons on the average moving ©

* It is here assumed that there are no thermal effects in action.
cea s theory is, however, sufficiently general to include these actions,
but they modify the distribution of velocities here quoted. The present
mode of analysis is easily extended to such cases.

Electron Theory of Metallic Conduction. 427

with any definite velocity in the positive direction of the
x-axis than in the negative direction. By calculating the
transfer of electricity resulting from such a distribution
Lorentz is led to his well-known formula for the electrical
conductivity.

On the other hand Drude, Riecke, Thomson, Wilson, and
others take rather a different view of the matter. According
to these authors the whole effect imparted by the electric
force on the electron during its free-path motion is oblite-
rated by the collision at the end of the path, so that each
free path is started with the velocity the electron would
have had throughout it, in the absence of any field of force :
the distribution of the initial velocities in the paths is thus
that specified by Maxwell’s law given above.

The object of the present paper is to show that these two
views, which at first sight appear to be rather contradictory,
are in reality probably the same, or at least that they are
consistent with one another. Although on a priomt grounds
one would certainly prefer to accept Lorentz’s view of the
situation, yet detailed consideration of the matter rather
inclines one towards the perhaps less general but certainly
more direct methods of Drude and Thomson. I will not,
however, presume to dogmatize on the relative merits of the
two forms of the theory.

We shall consider the problem of conduction from the
point of view that the initial velocities at the beginning of
the free paths are distributed according to Maxwell’s law,
each impact removing all effects imparted to the electron by
the electric field previous to it. The peculiarity of this
assumption is that it does not give the law of distribution of
the velocities of the electrons at any particular instant, but
rather the law of distribution of the initial velocities at the
beginning of the free paths being pursued at that instant, the
instant of beginning the free paths being, however, different
for the different electrons. ‘The actual law for the distribu-
tion of the velocities at any particular instant can, however,
easily be deduced and in the following manner.

Consider the electrons and their motion at any instant in
their free paths : the number of the electrons per unit volume
which started their current free paths with their velocity
components between (&, 9, €) and (E+dE£, n-+dn, €+d€) is
given by Maxwell’s law, and is thus equal to

IN=Na/ Lo e-odedndt OC a CT. )

428 Mr. G. H. Livens on the
OF these dN electrons the number

.
dN= e ™mdr*® «| 67 (11 )
have been in motion since their last impact previous to the
instant ¢ for a time which is between 7 and 7+dr. Iw this
expression 7, represents the mean value of the various
values of 7 for this particular group of electrons T.

Now the velocity components of any one of the group of
electrons specified by (i1.) are at the instant ¢ given by

Cee BS,

Thus ali of them will have these components within the
limits (&, 7’, &'), (6° +d&, n'+dn’, +d’), and we may
put

de =Oe. dn =dn,. do =de:

If, therefore, we interpret (ii.) in terms of &', ’, &’, 7 we may
conclude that there is the number

sN'=8N= A/a eC ) oe dbl 'delde

of the electrons per unit volume w hose v elocity components

at the instant ¢ lie between (&’, 9’, ¢") and (& +dé', n +dn',

6’ + dé’) and for which the time 7 lies between 7 and t+dr.
If we use

19 ID 19 I
ey ee

and if, as is usually the case, we may neglect terms in-
volving E?, this expression reduces to
428 ne 1 2eErmé’ Tc

N q? 7 2 ae le :

—A / *~e dé'dn'dé'dz.
Tn 3 Edn dt di
Thus on integration over all the values of 7 we find that the
total number of electrons with velocity components in the
specified limits at the instant ¢ is given by

+ gq e-r"” ! ! !
N/E Spe dy dg 5

* Vide Lorentz, ‘ The Theory of Electrons,’ p. 308.
+ This value of r,, also appears to be the mean duration of a free path
in the more general case.

Electron Theory of Metallic Conduction. 429

which is practically the same as

Q 12 Qe tm ‘ sip
na/%, ROU (i+ wernt delay dt .

Now, noticing that 7,, only occurs in the small correction
term, we may replace it by the approximate value

Tn =P
so that we deduce the law of distribution of velocities at any
instant in the steady state and under the action of the
uniform electric force H in the form

2K
Wena / Fe a Ie a rene | agian de,

which is precisely Lorentz’s result *.

It thus appears that under the assumptions made by
Drude and Thomson the average steady distribution of
velocities is precisely that derived by Lorentz. I think also
it is elear that the argument may be reversed, or, in other
words, that the steady distribution caleulated on general
grounds by Lorentz necessarily implies that the distribution
of initial velocities is that specified by Maxwell’s law. The
two modes of formulation of the theory are consequently
ultimately identical, although they are apparently very
different in form.

The mode of deduction of Lorentz’s formula here suggested
can be extended so as to apply to the more general problem
with varying fields. As an example we may, for instance,
calculate the average distribution of velocities at any instant
when the perfectly irregular motion of the electrons is
modified by the application of a simple periodic electric
force, say)

Hi = Hy cos p(t +t’)
where ¢ is used for the time ¢ at which we evaluate the
distribution, and ¢’ is an auxiliary time variable which is
measured from the instant ¢ as origin. Under these circum-
stances we find, with the same notation as before,

Sah amie daa

Bie one O8P(E= 3 SING.

while n =n, oe.

* Tn this expression J is legitimately interpreted as the length of the
mean free path.

430 Mr. G. H. Livens on the

Thus the number of electrons with velocity components
between the limits (&', 7’, &') and (é+dé’, n' +dn’, Cae),
and for which the time t lies between t and t+dr, is

=— =p 4/2 3¢°dE'dn'dt'dr,

where « is used for the argument of the exponential function,
and is equal to

ae 7 \sin22) ye
x g ie. in aco p(t 5) sin S| ad) +5] a

which is practically the same as

I
“z=—¢ [Ee weeetns COS p (‘ — 5) sin | ——
pm 2 2 Te

Thus to the same order of approximation * we find that

Pens Ve. a 4eHé’ T)\ Nora ae .
ON pean a3e 2 [1+ Fn cos? (t= 5 Jain z |e mdé'dn!dt dr.

Integrating with respect to + from 0 to oo we find the
total number of the electrons with their velocity components
between (&'’’) and (&'+ dé’, n' +dn’, & +d&') is

3 9
wana /Zer[ fiat? 2eH€’ Dus cos pi + p” Te 5097 agg ae’;
ie oe

pm I+ ./ p's,

or again using

( ly

DL oy’
we ae
Cos pt + Pl, ” sin pt
aa 2el Ee’ yp ;
dN'=N a Se shee <n oF = dE'dn'de.
Lee

Following Lorentz we find that under these circumstances
the current of electricity at the instant ¢ is

l
Cos pt + een sin pt

2Nel Hy Jae Paine
oe me aban de

P
i =

* 7, €., neglecting all terms involving squares or higher powers of E.

Electron Theory of Metallic Conduction. 431
or again using
ee air? and déldyld¢! =4mr"dr!

so that the integral reduces to a single integral in 7! from 0
to «0, we get

_4nNel,, a) o Qa! 130 tlie °

U
-
12

On using v=q7” we Au

ANe?l Ben gene Ly Wq
oS is iD a Ba | Cag cont NG sin pt) de

3m T ee Piha |
v
or
2 sil ; Been (A cos pt + B sin pt),
37 mu
w herein
© ve edu 00 Vve-*dv
= i 5) B= ol q ( i °
A rr 1p Pl g) 5 Pad
If we use
2= tam"

we find that, again,

9 Ne’l_ A
je : Ky cos (pt —e) sece.

30 mu

The conductivity in such cases is defined usually by the
average rate of dissipation of heat, which in the present case
is the average value of

2 Nel, A
EJ =2 —— aa Ey? cos pt cos (pie) sec €

y Q Neil, A
lied ONL ye)
ry Q ’
o7 mMuU,,

which is simply

432 Notices respecting New Books.

whereas in terms of the conductivity it is known to be
4cE,” ; so that :

om? a
fa aie

a Nel. ¢, ee
oz 2 = = ( pr?
LS

OT MU,
“0

which is the result obtained on the more direct method
suggested by Thomson. [or very long waves, it reduces, as
it should, to Lorentz’s original formula.

A full discussion of the actual bearing of these results on
the electron theory of the optical properties of metals will
be reserved for a future communication which is in course
of preparation. The main point of the present paper seemed
worthy of special and separate attention in the more general
theory rather than in the restricted branch of it which is
concerned with the optical properties only.

The University, Sheffield.
Noy. 4th, 1914.

XLVI. Notices respecting New Books.

The Electrical Conductiwity and Ionization Constants of Organic
Compounds. By Hrywarp ScuppEer. Constable & Co., 1914.
12s. nett.

SS is a bibliography, extending to 568 pages, of the periodical

literature from 1889 to 1910, inclusive, including all important
work before 1889 and corrected to the beginning of 1913, and gives
numerical data for the ionization constants at all temperatures ati
which they have been measured and some numerical data of the
electrical conductivity. The compilation thus deals with all the
important work on the ionization constants aud the electrical
conductivity of organic compounds from the commencement up to
the year 1910, at “which date the Tables Annuelles International
de Constants et Données Numériques begin. The book con-
stitutes in its semewhat limited field a useful work of reference to
those engaged in the subject. But unless it should prove in the
future of value as a starting point in the comprehensive reexami-
nation of the whole question trom a theoretical standpoint, it may
be doubted whether such a great expenditure of time and labour
as the work has entailed has been profitably undertaken.

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XLVI. Lolian Tones.
By Lord Rayuries, O.41., F.RS.*

ie what has long been known as the Molian Harp, a
stretched string, such as a pianoforte wire or a violin
string, is caused to vibrate in one of its possible modes by
the impact of wind; and it was usually supposed that the
action was analogous to that of a violin bow, so that the
vibrations were executed in the plane containing the direc-
tion of the wind. A closer examination showed, however,
that this opinion was erroneous and that in fact the vibra-
tions are transverse to the wind. It is not essential to the
production of sound that the string should take part in
the vibration, and the general phenomenon, exemplified in
the whistling of wind among trees, has been investigated by
Strouhal{ under the name of Rerbungstine.

In Strouhal’s experiments a vertical wire or rod attached
to a suitable frame was caused to revolve with uniform
velocity about a parallel axis. The pitch of the eolian tone
generated by the relative motion of the wire and of the air
was found to be independent of the length and of the tension
of the wire, but to vary with the diameter (D) and with the
speed (V) of the motion. Within certain limits the relation

* Communicated by the Author. .

t+ Phil. Mag. vol. vil. p. 149 (1879); Scientific Papers, vol. i.
p- 418.

} Wied. Ann. vol. v. p. 216 (1878). .
Phil. Mag. 8. 6. Vol. 29. No, 172. April 1915. 2F

434 Lord Rayleigh on olian Tones.

between the frequency of vibration (N) and these data was
expressible by

N= 185 V/D,. - : ..eee
the centimetre and the second being units.

When the speed is such that the eolian tone coincides
with one of the proper tones of the wire, supported so as to
be capable of free independent vibration, the sound is greatly
reinforced, and with this advantage Strouhal found it pos-
sible to extend the range of his observations. Under the
more extreme conditions then practicable the observed pitch
deviated considerably from the value given by (1). He
further showed that with a given diameter and a given speed
a rise of temperature was attended by a fall in pitch.

If, as appears probable, the compressibility of the fluid
may be left out of account, we may regard N as a function
of the relative velocity V, D, and v the kinematic coefficient
of viscosity. In this case N is necessarily of the form

N=V/D..{0/VD),.+ - =e
where / represents an arbitrary function; and there is dyna-
mical similarity,if yoc VD. In observations upon air at one
temperature v is constant; and if D vary inversely as V,
ND/V should be constant, a result fairly in harmony with
the observations of Strouhal. Again, if the temperature
rises, vy increases, and in order to accord with observation,
we must suppose that the function / diminishes with in-
creasing argument.

‘An examination of the actual values in Strouhal’s experi-
ments shows that v/VD was always small; and we are thus
led to represent f by a few terms of MacLaurin’s series. If
we take

f(a@)=at be + ca’,
we get

ee ea 3
=a D D2 ¢ Vb: «| ay) Pees ( )

“If the third term in (3) may be neglected, the relation
between N and V is linear. This law was formulated by
Strouhal, and his diagrams show that the coefficient 6 is
negative, as is also required to express the observed effect of
arise of temperature. Further,

dN cv?

Daya yp

* In (1) V is the velocity of the wire relatively to the walls of the
laboratory.

Lord Rayleigh on Molian Tones. 435

so that D).dN/dV is very nearly constant, a result also given
by Strouhal on the basis of his measurements.

“On the whole it would appear that the phenomena are
satisfactorily represented by (2) or (3), but a dynamical
theory has yet to be given. It would be of interest to extend
the experiments to liquids.” *

Before the above paragraphs were written I had com-
menced a systematic deduction of the form of f from
Strouhal’s observations by plotting ND/V against VD.
Lately I have returned to the subject, and I find that nearly
all his results are fairly well represented by two terms of (3).
In ¢.G.s. measure

(5)

Although the agreement is fairly good, there are signs
that a change of wire introduces greater discrepancies than
a change in V—a circumstance which may possibly be
attributed to alterations in the character of the surface.
The simple form (2) assumes that the wires are smooth, or
else that the roughnesses are in proportion to D, so as to
secure geometrical similarity.

The completion of (5) from the theoretical point of view
requires the introduction of v. The temperature for the
experiments in which v would enter most was about 20° C.,
and for this temperature

ND ( VD)

et) 1306 gl0-* A
ar = 90190 = 1505 c.a.s.
The generalized form of (5) is accordingly
ND 20'1y

applicable now to any fluid when the appropriate value of v
is introduced. For water at 15° C., v=:0115, much less
than for air.

Strouhal’s observations have recently been discussed hy
Kriiger and Lauth+, who appear not to be acquainted with
my theory. Although they do not introduce viscosity, they
recognize that there is probably some cause for the observed
deviations from the simplest formula (1), other than the
complication arising from the circulation of the air set in

* ‘Theory of Sound,’ 2nd ed. vol. ii. § 872 (1896).
+ “Theorie der Hiebtine.” Ann. d. Physik, vol. xliv. p. 801 (1914).

yn ee

A436 Lord Rayleigh on olan Tones.

motion by the revolving parts of the apparatus. Un-
doubtedly this circulation marks a weak place in the method,
and it is one not easy to deal with. On this account the
numerical quantities in (6) may probably require some correc-
tion in order to express the true formula when V denotes the
velocity of the wire through otherwise undisturbed fluid.

We may find confirmation of the view that viscosity enters
into the question, much as in (6), from some observations of
Strouhal on the effect of temperature. Changes in v will
tell most when VD is small, and therefore I take Strouhal’s
table XX., where D=°:0179 cm. In this there appears

i = 11°, 1= 385, ING/Vi==67 105 Vi,
p= ole eco N/V = 6°48, Vo.

Introducing these into (6), we get

abe, 9B) 20-15-1957 eo
670-648 = (1- a )- 0p

or with sufficient approximation

2 DAN.
yy 8 eee = WING CLES,
Sinai eelt5 2074 i
We may now compare this with the known values of v for
the temperatures in question. We have

431 = 1853 x HOR p31 = 001161,

ne x 10%: on = 001243.
so that
Yo= "O96, v,="1420,

and Vo—v,='018.

The difference in the values of v at the two temperatures
thus accounts in (6) for the change of frequency both in
sign and in order of magnitude.

As regards dynamical explanation it was evident all along
that the origin of vibration was connected with the instability
of the vortex sheets which tend to form on the two sides of
the obstacle, and that, at any rate when a wire is maintained
in transverse vibration, the phenomenon must be unsym-
metrical. The alternate formation in water of detached
vortices on the two sides is clearly described by H. Bénard*.

* C. R. t. 147, p. 839 (1908).

Lord Rayleigh on Holian Tones. 437

“Pour une vitesse suffisante, au-dessous de laquelle il n’y a
pas de tourbillons (cette vitesse limite croit avec la viscosité
et decroit quand Vépaisseur transversale des obstacles aug-
mente), les tourbillons produts périodiquement se detachent
alternativement & droite et & gauche du remous @arriére qui
suit le solide; ils gagnent presque immédiatement leur emplace-
ment définitif, de sorte ywa Varriére de Vobstacle se forme
une double rangée alternée Wentonnoirs stationnaires, ceus
de droite deatrogyres, ceux de gauche lévogyres, séparés par
des mtervalles égaus.”

The symmetrical and unsymmetrical processions of vor-
tices were also figured by Mallock* from direct observation.

In a cameras theoretical investigationt Karman has
examined the question of the stability of such processions.
The fluid is supposed to be incompressible, to be devoid of
viscosity, and to move in two dimensions. The vortices are
concentrated in points and are disposed at equal intervals (/)
along two parallel lines distant h. Numerically the vortices
are all equal, but those on different lines have opposite
slons.

Apart from stability, steady motion is possible in two
arrangements (a) and (b), fig. 1, of which (a) is symmetrical.

Fig. 1.

h (b)

Karman shows that (aya is always unstable, whatever may be
the ratio of h to 1; and further that (d) is usually unstable
also. The single exception occurs when cosh (sh, l)= / 25 OF
it== 0283, With this ratio of h/l, (b) is stable for every
kind of displacement except one, for which there is neutrality.

* Proc. Roy. Soc. vol. Ixxxiv. A. p. 490 (1910).

+ Gottingen Nachrichten, 1912, Heft 5, S. 547; Karman and
Rubach, Physik. Zeitschrift, 1912, p. 49. I have veritied the more
important results.

438 Lord Rayleigh on Holian Tones.

The only procession which can possess a practical permanence
is thus defined.

The corresponding motion is expressed by the complex
potential (¢ potential, yr stream-function)

Meneame sin an( eye) Un |
eae enOe cinlec(caspeyiite oe

in which ¢ denotes the strength of a vortex, z=«#+v1y,
zy=iltih. The z-axis is drawn midway between the two
lines of vortices and the y-axis halves the distance between
neighbouring vortices with opposite rotation. Karman gives
a drawing of the stream-lines thus detined.

The constant velocity of the processions is given by

NEMA i! oP E
ee 17g . oe

This velocity is relative to the fluid at a distance.

The observers who have experimented upon water seem
all to have used obstacles not susceptible of vibration. For
many years I have had it in mind to repeat the ceolian harp
effect with water™, but only recently have brought the
matter to a test. The water was contained in a basin, about
36 cm. in diameter, which stood upon a sort of turn-table.
The upper part, however, was not properly a table, but was
formed of two horizontal beams crossing one another at
right angles, so that the whole apparatus resembled rather a
turn-stile, with four spokes. It had been intended to drive
from a small water-engine, but ultimately it was found that
all that was needed could more conveniently be done by
hand after a little practice. A metronome beat approximate
halt seconds, and the spokes (which projected beyond the
basin) were pushed gently by one or both hands until the
rotation was uniform with passage of one or two spokes in
correspondence with an assigned number of beats. It was
necessary to allow several minutes in order to make sure
that the water had attained its ultimate velocity. The axis
of rotation was indicated by a pointer affixed to a small

* From an old note-book. ‘“ Bath, Jan. 1884. I find in the baths
here that if the spread fingers be drawn pretty quickly through the
water (palm foremost was best), they are thrown into transverse vibra-
tion and strike one another. ‘This seems like eolian string..... The
blade of a flesh-brush about 13 inch broad seemed to vibrate transversely
in its own plane when moved through water broadways forward. It is
pretty certain that with proper apparatus these vibrations might be
developed and observed.”

Lord Rayleigh on Aolian Tones. 439

above the level of the water.

The pendulum (fig. 2), of which the lower part was im-
mersed, was supported on two points (A, B) so that the

stand resting on the bottom of the basin and rising slightly

possible vibrations were limited to one vertical plane. In
the usual arrangement the vibrations of the rod would be
radial, 7. e. transverse to the motion of the water, but it was
easy to turn the pendulum round when it was desired to
test whether a circumferential vibration could be maintained.
The rod C itself was of brass tube 8} mm. in diameter, and
to it was clamped a hollow cylinder of lead D. The time of
complete vibration (rt) was about half a second. When it
was desired to change the diameter of the immersed part,
the rod C was drawn up higher and prolonged below by an
additional piece—a change which did not much affect the
period 7. Jn all cases the length of the part immersed was
about 6 em.

Preliminary observations showed that in no case were
vibrations generated when the pendulum was so mounted
that the motion of the rod would be circumferential, viz. in

440 Lord Rayleigh on Holian Tones.

the direction of the stream, agreeably to what had been found
for the xolian harp. In what follows the vibrations, if any,
are radial, that is transverse to the stream.

In conducting a set of observations it was found con-
venient to begin with the highest speed, passing after a
sufficient time to the next lower, and so on, with the minimum
of intermission. Iwill take an example relating to the main
rod, whose diameter (D) is 84 mm., T=60/106 sec., beats
of metronome 62 in 30 sec. The speed is recorded by
the number of beats corresponding to the passage of two
spokes, and the vibration of the pendulum (after the lapse
of a sufficient time) is described as small, fair, good, and so
on. Thus on Dec. 21,1914:

2 spokes to 4 beats gave fair vibration,

agen ey Sina”: I Deveaeee £000) 50 cee
MUNA Env Re ae Oh) Taber OTe eae
s . e se s e a s Fé s s e J . s s good e J es . . s se
. a) 8S: @5 se: fa, .e@ = 5 e s . ’ . e . fair So, -@ ~-8)) 8. ane. s

from which we may conclude ‘that the maximum effect cor-
responds to 6 beats, or to a time (T) of revolution of the
turn-table equal to 2x 6x 30/62 sec. The distance (7) of
the rod from the axis of rotation was 116 mm., and the
speed of the water, supposed to move with the basin, is
2rr/T. The result of the observations may intelligibly be
expressed by the ratio of the distance travelled by the water
during one complete vibration of the pendulum to the
diameter of the latter, viz.

t.2ar/T 207x116 62

Wii Sox6x106 0 (on

Concordant numbers were obtained on other occasions.

In the above calculation the speed of the water is taken
as if it were rigidly connected with the basin, and must be
an over estimate. When the pendulum is away, the water
may be observed to move as a solid body after the rotation
has been continued for two or three minutes. Jor this
purpose the otherwise clean surface may be lightly dusted
over with sulphur. But when the pendulum is immersed,
the rotation is evidently hindered, and that not merely in
the neighbourhood of the pendulum itself. The difficulty
thence arising has already been referred to in connexion
with Strouhal’s experiments and it cannot easily be met in
its entirety. It may be mitigated by increasing r, or by
diminishing D. The latter remedy is easily applied up toa

Lord Rayleigh on .#Holian Tones. 441

certain point, and I have experimented with rods 5 mm. and
34 mm. in diameter. With a 2 mm. rod no vibration could
be observed. The final results were thus tabulated :—

Diameter ... $5 manten oO mima. 3°5 mm.
FeO O20 8°35 (5 7°38

from which it would appear that the disturbance is not very
serious. The difference between the ratios for the 5°0 nm.
and 3°5 mm. rods is hardly outside the limits of error ; and
the prospect of reducing the ratio much below 7 seemed
remote.

The instinct of an experimenter is to try to get rid of a
disturbance, even though only partially ; "but 1%’ 1s) often
equally instructive to increase it. The observations of
Dec. 21 were made with this object in view ; besides those
already given they included others in which the disturbance
due to fhe vibrating pene uae was augmented by the addi-
tion of a similar rod (85 mm.) immersed to the same depth
and situated symmetrically on the same diameter of the
basin. The anomalous effect would thus be doubled. The
record was as follows:—

2 spokes to 5 beats gave little or no vibration,

siete cd LSS nee APA Mia ieeh Uae ad tn ACR EE
Mae bts Dah ghana eiial ee Laie Rr Roa ee
i Ronee ane SU Og tiv tq ean less Laie VS bait

SNe ead GNC eae anes areas Uther "Ole MO yaieey ats c:

As the result of this and another day’s similar observations
it was concluded that the 5 beats with additional obstruction
corresponded with 6 beats without it. An approximate cor-
rection for the disturbance due to improper action of the
pendulum may thus be arrived at by decreasing the calculated
ratio in the proportion of 6: 5; thus

5 (8°35) =7°0

is the ratio to be expected in a uniform stream. It would
seem that this cannot be far from the mark, as representing
the travel at a distance from the pendulum in an otherwise
uniform stream during the time of one complete vibration
of the latter. Since the correction for the other diameters
will be decidedly less, the above number may be considered
to apply to all three diameters experimented on.

In order to compare with results obtained from air, we
must know the value of v/VD. For water at 15° C.
y=p='0115 o.as.; and for the 85 mm. pendulum
v/VD="0011. Thus from (6) it appears that ND/V should

449 Lord Rayleigh on Aolian Tones.

have nearly the full value, say °190. The reciprocal of this,
or 5:3, should agree with the ratio found above as 7:0; and
the discrepancy is larger than it should be.

Ax experiment to try whether a change of viscosity had
appreciable influence may be briefly mentioned. Observa-
tions were made upon water heated to about 60° C. and at
12°C. No difference of behaviour was detected. At 60°C.
p='0049, and at 12° C. w=-0124.

I have described the simple pendulum apparatus in some
detail, as apart from any question of measurements it de-
monstrates easily the general principle that the vibrations
are transverse to the stream, and when in good action it
exhibits very well the double row of vortices as witnessed
by dimples upon the surface of the water.

The discrepancy found between the number from water
(7°0) and that derived from Strouhal’s experiments on air
(5°3) raises the question whether the latter can be in error.
So far as I know, Strouhal’s work has not been repeated ;
but the error most to be feared, that arising from the circu-
lation of the air, acts in the wrong direction. In the hope
of further light I have remounted my apparatus of 1879.
The draught is obtained fromachimney. A structure of wood
and paper is fitted to the fireplace, which may prevent all
access of air to the chimney except through an elongated
horizontal aperture in the front (vertical) wall. The length
of the aperture is 26 inches (66 cm.), and the width 4 inches
(10°2 cm.) ; and along its middle a gut string is stretched
over bridges.

The draught is regulated mainly by the amount of fire.
It is well to have a margin, as it is easy to shunt a part
through an aperture at the top of the enclosure, which can
be closed partially or almost wholly by a superposed ecard.
An adjustment can sometimes be got by opening a door or
window. A piece of paper thrown on the fire increases the
draught considerably for about half a minute.

The string employed had a diameter of *95 mm., and it
could readily be made to vibrate (in 3 segments) in unison
with a fork of pitch 256. The octave, not difficult to mis-
take, was verified by a resonator brought up close to the
string. That the vibration is transverse to the wind is con-
firmed by the behaviour of the resonator, which goes out of
action when held symmetrically. The sound, as heard in
the open without assistance, was usually feeble, but became
loud when the ear was held close to the wooden frame.
The diticulty of the experiment is to determine the velocity
of the wind, where it acts upon the string. I have attempted

Lord Rayleigh on Molian Tones. 449,

to do this by a pendulum arrangement designed to determine
the wind by its action upon an elongated piece of mirror
(10-1 cm. x 1°6 cm.) held perpendicularly and just in front
of the string. The pendulum is supported on two points—
in this respect like the one used for the water experiments ;
the mirror is above, and there is a counter-weight below.
An arm projects horizontally forward on which a rider can
be placed. In commencing observations the wind is cut off
by a large card inserted across the aperture and just behind
the string. The pendulum then assumes a sighted position,
determined in the usual way by reflexion. When the wind
operates the mirror is carried with it, but is brought back to
the sighted position by use of a rider of mass equal to
“485 om.

Observations have been taken on several occasions, but it
will suffice to record one set whose result is about equal to
the average. The (horizontal) distance of the rider from
the axis of rotation was 62 mm., and the vertical distance of
the centre line of the mirror from the same axis is 77 mm.
The force of the wind upon the mirror was thus
62x °485+77 oms. weight. The mean pressure P is

62 x ‘485 x 981 = 93-7 dynes
Miele. PM em.” ©

The formula connecting the velocity of the wind V with the
pressure P may be written

P=CpV?,

where p is the density; but there is some uncertainty as to
the constancy of C. It appears that for large plates
C='62, but for a plate 2 inches square Stanton found
C='52. Taking the latter value*, we have
93. 3:
V2 23 fl 23 a

cod 2 —=

on introduction of the value of p appropriate to the circum-
stances of the experiment. Accordingly

V=192 cm./sec.

The frequency of vibration (r~') was nearly enough 206 ;
so that
ee eer 7-9
D2 OG 09D Ti.)
In comparing this with Strouhal, we must introduce the

* But I confess that I feel doubts as to the diminution of C with the
linear dimension.

444 Lord Rayleigh on Molian Tones.
appropriate value of VD, that is 19, into (5). Thus
are a:
silt Ws ied By
Whether judged from the experiments with water or from
those just detailed upon air, this (Strouhal’s) number would
seem to be too low; but the uncertainty in the value of C
above referred to precludes any very confident conclusion. It

is highly desirable that Strouhal’s number should be further
checked by some method justifying complete confidence.

When a wire or string exposed to wind does not itself
enter into vibration, the sound produced is uncertain and
difficult to estimate. No doubt the wind is often different
at different parts of the string, and even at the same part it
nay fluctuate rapidly. A remedy for the first named cause
of unsteadiness ts to listen through a tube, whose open end
is brought pretty close to the obstacle. This method is
specially advantageous if we take advantage of our know-
ledge respecting the mode of action, by using a tube drawn
out to a narrow bore (say 1 or 2 mm.) and placed so as to
face the processions of vortices behind the wire. In con-
nexion with the fire-place arrangement the drawn out glass
tube is conveniently bent round through 180° and continued
to the ear. by a rubber prolongation. In the wake of the
obstacle the sound is well heard, even at some distance
(50 mm.) behind; but little or nothing reaches the ear when
the aperture is in front or at the side, even though quite
close up, unless the wire is itself vibrating, But the special
arrangement fora draught, where the observer is on the high
pressure side, is not necessary; in a few minutes any one
may prepare a little apparatus competent to show the effect.
Fig. 3 almost explains itself. A is the drawn out glass tube;

Fig. 3.

B the loop of iron or brass wire (say 1 mm. in diameter),
attached to the tube with the aid of a cork C. The rubber
prolongation is not shown. Held in the crack of a slightly
opened door or window, the arrangement yields a sound
which is often pure and fairly steady.

r 445 |]

XLIX. The Equations of Motion of a Viscous Fluid. By
G. B. Jerrery, .A., B.Sc., Assistant in the Department
of Applied Mathematics, University College, London ”*.

| ae transformation of the equations of elasticity to curvi-

linear coordinates has been discussed by Lamé and
others, but the analogous equations for the motion of a
viscous fluid do not appear to have attracted the same
attention. The first section of this paper deals with the
transformation of the equations of motion. The terms in
the equations which express the resultant force on an element
of fluid are mutatis mutandis identical with the corresponding
terms in the equations for an elastic solid, but they are given
in a form which I believe to be new, and which lends itself
more readily to applications to particular systems of co-
ordinates. The terms which express the accelerations of the
fluid are different owing to the different assumptions which
underlie the two theories. The second section is devoted
to a discussion of the components of stress in curvilinear
coordinates. The theory is illustrated by applications to
cylindrical and spherical polar coordinates. In the remaining
two sections we discuss the special cases of axial and plane
motion respectively.

§ 1. Transformation of the Equation of Motion.

If u,v, w, X,Y, Z, are the components of velocity and
body force respectively, p the mean pressure, p the density,
and p the coefficient of viscosity, the equations of motion of
a viscous fluid are +

Py “PN 3e + 5H Se(Qe + byt Be) TAU

and two similar equations.

These three equations may be replaced by a single vector
equation. If v be the velocity and I the externally applied
force, then using the relation

curl.cur] v= grad. div. v—V"v f,

. Communicated by Prof. Karl Pearson, F.R.S,
+ Lamb, ‘Hydrodynamics,’ p. 538.
i In the usual yector notation
eurl. curl v=/yiv.v]]
=V(Vv)—v'v
= grad. div. v—V?v.

446 Mr. G. B. Jeffery on the
the vectorial equation of motion is

px =pK—erad. p+ 5 mera, div.v—ecurl.curlv*. (1)

The transformations of div. and curl follow at once from the
definition of these operators in terms of surface and line
integrals respectively.

If a, 8, y are orthogonal curvilinear coordinates, and if

a) s)* Gee

while hj, hg are defined by the same operation carried out
upon B, y respectively, so that elements of arc measured
along the normals to the coordinate surfaces at any point
are

da 68 by

hy? hy? hg’
then T

curl v=h, hz 2(2)- 2 i Ms
ista {ay (in) ~ dag) SL Sai.) 7 gaa) be

: v 0 /w :
div. v=dills | © (43), ) een) Ga eS

where u,v, w are the components of v along the normals to the
surfaces a, 8, y=const. respectively. In (3) put v=grad.¢d,
and we obtain

os hy O2) + he Od) hs Od
Vea shy { hohs 5°) +2, hshy =) OY ° (4 =) } (4)

By a second application of the operations implied in (2)
we have, denoting the direction of a component by a suffix,

aes { Ihe b\w
curlewty=hats[ 9 { "(2 () ~ 2a(;.))}

= == & (;;) i: = i.)) |.

* oh the corresponding elastic equation, Love, ‘Theory of Elasticity,’
. 138
t Love, p. 54.

Equations of Motion of a Viscous Fluid. AAT

This can be transformed into an expression which while it
appears more complex, is in reality simpler in application,
namely,

eurl, curl v=

0 5 hyv J ]
I, & (ai. v)—V7ut+ i, V 7h, — ae (V 7a) — ie so (W778) — we = sa)

Oe
+ 2he? oo x) =l5 )- yh, 2? o5(7. 2) + 2172 2(2)

2h $2 O(?).. ay MRR Nee IUGR ie etl a Ka = 76

This expression is simplified if it is possible to choose one
or more of the functions a, 8, y so that they are solutions of
V’?=0, or in the language of Lamé, so that they are thermo-
metric parameters.

It remains to find appropriate expressions for the com-

ponents of In Cartesian coordinates

Dv
De:
AA) — (2 tuo ts ve +w 2) (u,v, w),

and this may be written in vector form
Dv _ av
Dae +(v.V)v.
The component of this vector equation in the « direction is
Du LON Ow ae :
ne ae ea + Iu hoe 5A +hsw =) tCG)

Let > be any fixed direction in space whose direction
cosines are J, m,n, then the component of v in this direction

is
wm ER ea ob gM BD

+E (? OY mS a

448 Mr. G. B. Jeffery on the

Then
(3) =>
= 7, (13 +m ae je(tge tm §, +082)
+ aa(ge +m $2 +082)

ee ae Sa) +ma aE : 2.
)+

a (1%
“ae by ) }
(! a a Ct 52
2

G z
lee
ie
(

7 32)

Remembering that « = hw, B= har, y = haw, and using
the nine relations of the type *

lh Sing, 310s mcr

2 (ea 28 oF), 2S) = 2B 2)

and finally taking for the direction X the normal to the
surface a=const., we obtain

(57): ic ies [hs 08 (;,) je Is 2 (Eye
0 O oul:

ay ai pee Ce Was oe 2

oS da (;,.) : Iie (a

(7)

We can now write down the transformed equations of

motion.
Let v be the kinematic viscosity so that vy = p/p, and let

+ similar

0
terms in ByY-

ca Se aay
+0 6G. St) +m 2 (G52) +

hs O

ye ings, tO. yeey
= + ear’: + hav 5B + he Sy iat (8)

* Ibid. Note on moving axes, p. 539 ; or Lamé, Coordonnées Curvilignes,
Lod
p. 74.

Equations of Motion of a Viscous Fluid. 449

then the « component of (1) becomes

Su fol
Di + hau( Ia 25(; ‘er +h B= ° (z- -)w)— fila ° (;. ) — Inhss GA) Ww

ES pL aos eve 54 div. v) ) + Vu 7 Vly
+ ud (vray t! be SATB) +" 2 Ty

¢ SOL WO Oh, 0 (>

—2h,° Ae aes (i ye 2hyhe AD ae)

Ih? oe xl (j,) + 2k os 2. (F)]. OR ae

The corresponding equations in v, w, can at once be written

down from symmetry.

Application to cylindrical and polar coordinates.—It we

take cylindrical coordinates a, 0, z, we have

OO
Dt
Dv
Dt

Dw
Dt

lk = Ihe =e Le Mp ae
oT

Most of the terms in (9) vanish and we have *

~ =e Se Oe (div. v) +V’u—-—;— ee
UD Lop ; v 2° Ow)
+ =F,— S50 t? [gage liv) +V%- 54 = 96

y =F, —— oe +y AER (div. v) +V%0 |,

From (3)

OP Te MN) ou) Ow
SIONS (GIO) per on

and is zero if the fluid is incompressible, while from (8)

Dd Cn oD
Di ot Se CHOU: an O8
If 7, 0, @ are spherical polar coordinates,
1 il

h = i} he —=-—y; hs = ———
i a aes oi Aa aisiins

* Cf. Basset, ‘ Hydrodynamies,’ vol. ii. p. 244, where the equations are

given for an incompressible fluid.

Phil. Mag. 8. 6. Vol. 29. No. 172. April 1915. 2G

450 Mr. G. B. Jeffery on the

We have
1 Ow
rsin 8 dd

div. v = 5

i
2 u) + ang ey esin 8) +=
and Divo av 0 w Oo
EA. Dt at re r 00 e rsin 0 0d’

and the equations of motion are *

Du vt? _ 1 Op lao) 2
Cees or pee
BCLs 00. eee
Te "ie rod 7 sin 8 ad

De, Eo) aw cote. L Op 150i
DET oN ie rs =F, 0 » | 5g (div. ¥) + \77v

ous 2h eos 0 ow a aye
rsin?@ =” sin? 6 Od 38
Dw 4 ww il Op
a ce 66. pr sin 6 oa as sin 0 5

w aa 2 CE 2 ee
r? sin? @ * +2sin OG sin? 006]

(div. v)

§ 2. Transformation of the Stresses.

~a-_ e e e
If ns denotes the component in the direction n of the
stress across a surface whose normal is in the direction s
and if the Cartesian components of velocity are w’, v', w', we
have

eal Bu. s ae oa
D4
where 6 =div. v and A= — 3b

These are identical with the usual expressions for an
elastic solid except for the term p, which indicates a pressure
equal in all directions. The components of stress in an
elastic solid were transformed to curvilinear coordinates by
Lamé, and we can at once obtain the corresponding formulze
for a viscous fluid by inserting the uniform pressure p.

* Of. Basset, p. 246, where a slight misprint occurs in the third
equation.

Equations of Motion of a Viscous Fld. ADL
Changing the notation to that which we have employed we have*

ie 5 U Ww 1
“a= ee todas) onli) it “(10
\
!

oe “fz (haw) +5 he © (iar) | =f, |
J
and four similar equations.
For cylindrical coordinates we have, as before,
i
i=h,=V, l= ee
and equations (10) give f
aa =e
Cs 2
FT) a SOT al “Hor ay
As ow
= —ptr6+4 2u— er
—~ 1 ow . dv
ane ai elk
BE alee + oS
Galore on:
oe | AR ma IS or
- | Se CHGS, 06]:
In the case of spherical polar coordinates 7, 0, 6, we have
IL 1
lig = J = = [pen Mey
ita: ney 3 Yr sin @”

and equations (10) give $

P= pe

Or
@=—p+rd+2u(- a ot
fe: aed ) tke
do=—pt+ro+2 24( + - “cot 0+ ang sa):

ae 1 Ow dl Se
0b=n(; a0 Co nsmnid oa Tr ),

or= =1(;; 1 Ow , Ow _ =

rsin0dQd Orr
as Ov Lev
=. + 3873)

* Lamé, Coordonnées Curvilignes, p. 284.

+ Lamé, Lecons sur UElasticité, p. 184.

+ Elasticité, p- 199. Some differences of sign are introduced by the
fact that Lamé takes for the angle “ the : atitude instead of the co-lat ‘itude.

2G 2 4

452 Mr. G. B. Jeffery on the

Thus it will be seen that although the general formule
are somewhat complex, yet when we “apply them to particuiar

sets of coordinates they become in many cases comparatively
simple.

Aaial Motion.

The case in w ve we have symmetry of motion about an
axis can be discussed by means of the equations in eylin-
drical coordinates. We will suppose the fluid to be incom-
pressible and the external forces to be derivable from a
potential V. Then writing

y= —V—L
/ p y)
and putting go,=0 the equations in cylindrical coordinates
become
SO ao
Dt = = = +o Vu 5) “ (11)
Dv. w i a {
- | ee a . (12)
Dw _ ox oe 1
aria: +vV/*w, Bie
where
aie 3

The equation of continuity is

cS (wor) + < (wo) =0,

and hence

even if there is a velocity component ae. to the
meridian plane.
Equation (12) becomes

de Ldydv, 1oyor lov (, 2)
dio 0: da wdad: wd: =0(V

fet av = QO

and Pere fi.) rou

oh
GD

Equations of Motion of a Viscous Lluid. 453

and we have

oGh 8 O) _ o(: @—S)o. Oe ira)

Hliminate y between (11) and (13) and after some reduction
we obtain

S08 + Been) PZ LY=e(o0 -Z)ow
Lately

Then (14) and (15) are the equations for this type o£ motion.

One interesting result which flows from these equations is
that the only possible motion, which consists of pure circula-
tion about the axis without any accompanying motion in the
meridian plane, is that generated by the rotation of two
infinite coaxial circular cylinders about their common axis.
Put & = const. and (15) becomes

00
= (,
(ele
and hence from (1+), taking the case of steady motion,

OR gO),

ds & Te

the solution of which is

QO = Aw? + B,
or
v= Aw py
roy

which is the solution referred to. Hence if anv body of
revolution other than a cylinder be rotated about its axis in
a viscous fluid, the consequent motion of the fluid about the
axis must in all cases be accompanied by a certain motion
in the meridian plane. For the special case of a sphere
this was pointed out by Stokes*.

When the motion is entirely in the meridian plane, we
have ® = 0. Equation (14) is identically satisfied and (15)
becomes

CIN 6) en a 2 2 OV) ory
O(a, 2 a Oe — 3 5) OM:

* * Mathematical and Physical Papers, vol. i. p. 108.

AD54 Equations of Motion of a Viscous Flind.
which may be written

iG S 2
Dt wo

D e
where Di has the same meaning as before (p. 452).

These equations can readily be transformed to curvilinear
coordinates should occasion arise since, apart from transfor-
mations already discussed, we require only the form of the
operator ® and this can be obtained from the identity

€
P=V Ze

a Oa

§ 4. Lwo-dimensional Motion.

The equations for two-dimensional motion may be obtained
as a particular case of the general equations, but they are
more readily obtained from the Cartesian equations of
motion. Using the same notation as before and taking
the plane of wy as the plane of motion, we have

OUy Ow Ov OX :
ay, + ara: aS aioe +vV\/*u,

ov Oe On OX oe
Sl ey + Sane OF at UNV ats

ae ON Ome
Oy’ Ox
where vw is Harnshaw’s current function.

Substituting these values for wu, v and eliminating y we
have

and

Taking «, 8, conjugate functions of «, y, as orthogonal
curvilinear coordinates, we have from (4)

o={(@)-GVKE-S}

Oh, Vib) __ Or, Vb) . O(a, B)

rol Cun y) O(4, 8) O(a, 7)

= {G:) +) eee

Two-Dimensional Steady Motion of a Viscous Fluid. 455

so that the equation for yr becomes

Oy oy) + eK oe 0° 5) V4,

Bee” ) ean & oe
a de 2 Sar )

L. Phe Two-Dimensional Steady Motion of a Viscous
Fluid. By G. B. Jerrery, M.A., B.Sc, Assistant in the
Department of Applied Mathematics, University College,
London*.

ae object of this paper is to search for some ewact

solutions of the equations of motion of a viscous’ fluid. °
Much has been accomplished by assuming that the motion is
slow, and that the squares and products of the velocity compo-
nents may therefore be neglected. It has indeed been held
that this is the only useful proceeding, since the equations of
motion are themselves formed on the assumption of a linear
stress-strain relation, and this is probably only justifiable if
the motion is sufficiently slow. On the other hand, there is
very little evidence of the breakdown of the linear law in
the case of fluids, and in any case it is only possible to test
its validity by an investigation of solutions which do not
require the motion to be slow. It is, therefore, of some
importance to obtain some solutions which are free from this
limitation. In the present paper we confine our attention
to plane motion. Orthogonal curvilinear coordinates are
employed, and we discuss the possibility of so choosing them
that either the stream-lines or the lines of constant vorticity
are identical with one family of the coordinate curves. The
most important solutions obtained are those which correspond
to (1) the motion round a canal in the form of a circular arc,
(2) the motion between rotating circular cylinders with a
given normal flow over the suriaces, as in a centrifugal
pump, (3) the flow between two infinite planes inclined at
any angle.

If u, v be the components of velocity, p the mean pressure,
V the potential of the external forces, v the kinematic
viscosity, and p the density of the fluid, the equations of

* Communicated by Prof. Karl Pearson, F.R.S.

456 Mr. G. B. Jeffery on the Two-Dimensional

motion in two dimensions are

Ou, OMe OW alo. 7 OV: ;
el
09,208 Aor 2 lop ON ae
Si +us Uae Binh iy +vV7v

Eliminating the pressure from these equations we have

(1)

ro) D) Oy, WEA 4
ot (V Wr) + Due y) =vV/ Vv,
where ie ow a ov
Meron. 4) pow

ar being Earnshaw’s current function.

ake a system of orthogonal curvilinear coordinates de-
fined by conjugate functions 2, 8 of x, y. The equation for
wy may then be written

Oh V*h) bt oe ok, : ov
sas) = (set ae Se)
or if the motion is steady

BAVA) 82, BD? \ Gay i
=1( s+ 3g) . + ee

O(a, 8)

§ 1. Solutions for which the lines of constant vorticity
are a possible set of equipotential lines.

The coordinates can be so chosen that the curves #=const.
are identical with any given set of equipotential lines in free
space. Hence the characteristic property of this type of
solution is that it is possible to choose the system of co-
ordinates x, 8 so that Vv is a function of « only, say

Vea (4).

Substitute in (2) and we have

or fol ue d Ap i
Integrating with respect to 8

y=—vB (log 7’ (2))+F (2).

Steady Motion of a Viscous Fluid. ADT

Hence

vey= (52) + GE) f Leg logrten +P" fara,

Using a well-known property of conjugate functions this
may be written

) d(v+ zy) dl? 1 (a)

| ee | = Ba 5 (log f”
| d(a+ ip) | ge (a) Te 8 / (4)) = f(a)

so that we have to determine a function of «+78 such that
the square of its modulus is linear in 8. Mr. G. N. Watson,
to whom I submitted this problem, has supplied me with the
complete solution. If

| $(e+78) | °=AB+B,
where A, B are real functions of a, then
Pla +18) =K der),
where X is real but « may be complex. In this case
j= (0), lees | | Re
Applying this result to the problem in hand

day) _ ae K erat ip)
d(a+iB) ;

a ! if 9 9 g
f (log f"(#))=0, F"(a)= |e | 2 F(a),
Hence

f(a)= A caer
iN (@) = | K lee \\ (da) e a ew thate dy,

The constants «, merely determine the scales of measure-
ment in the cuehorene systems of coordinates, and we have
only two distinct solutions (1) k=1, A=9, (2) ee Vi

The first case gives

atiP=wvtiy,

and we have a solution in Cartesian coordinates

458 Mr. G. B. Jeffery on the Two-Dimensional
If a=0 this gives
w= —bvytAe*+Ba?+Cr, ©. . . Ce)

while if a is not zero, a shift of origin gives
>
p= —2vany+A\( em dey... CL)
(1.) and (II.) are the only distinct solutions for which the
lines of constant vorticity are a set of parallel straight lines.
In the second case, when e=1 and A=1

atiB= log (47+).
Hence, if 7, @ be polar coordinates

a= log 7, B=e:

and
p= —vO(2alogr+b) = rdr gui logr®+4(logr) +e”
= 7 T

This also leads to two distinct solutions according as @ 1s or
is not zero.
ia—o
p= — bv0+ Ar’t?+Br?+Clogr (b0 or i; (LIL.)
=2v0 + A(log r)?+Br?+Clogr (b=— 2),

while if a is not zero, a change in the scale of r gives
LC 2
dr ee
p= — 2vaé log rao {a patog re”
r F
eg ea

(III.), (IV.) are the only distinct solutions for which the
lines of constant vorticity are a set of concentric circles.
We have thus obtained all the solutions for which the lines
of constant vorticity are the equipotential lines in free space
of some possible distribution of matter.

(IV.)

§ 2. Solutions for which the stream-lines are possible
equipotential lines.

The characteristic property of this type of solution is that
it is possible to choose a set of curvilinear coordinates «, @
so that ~w=/f(a). It has not been found possible to solve
this case with the generality of the previous section. Sub-
stituting in (2) we have

Steady Motion of a Viscous Fluid. A59

or writing ae ey is Gay
Ay v

we see that «, B,/ are restricted by the condition that M
must satisfy the equation

Q°M 07M Bieta ou hfe Shan oM
& + Soe) + 20h" ( f (4) = (4) ~ 2

aap ey DEO, ete Ta),
This will be satisfied by any system a, @ whatever, if
f"=9, which corresponds to the otherwise obvious fact that

any solution of V/7W=0 is a solution of (2). Thus any
irrotational motion is a possible motion of a viscous fluid.

Suppose a+ ip = (a+ iy)”,

n—1

Wine? 2 82 \7

vf” (4)

then

equation (3) gives
2yv(n—1)(n —2)f""(a) + 4on(n— 1) af" (a) —2n(n—1) BP (a) fF" (@)
sun? f"(ar+ )=0.

If n=1, it is sufficient that /""’(«)=0, which leads to the
well-known solution for the motion between infinite parallel
planes :

v= Aa’ + Ba?+ Ce.

Otherwise, equating coefficients of powers of @B separately to

Zero,
f(a) =0, FAI @)=0, f"@)=0.

Hence /(«) is a linear function of a, which corresponds to
the case of irrotational motion. It appears, therefore, that
for no value of n other than unity is there any new solution
of this class.

Next consider polar coordinates 7, 0. Equation (2)
becomes

a(W, We afr) 4 4
aA, ) =—vrV/ vr, Seige te RGN twice eee (4)
and Sarg! its uy Meats) Lee
vi =o ( e TP oo)

First seek a solution of the form

v=f(r).

460 MrvGen: Jeffery on the Two-Dimensional
From (4) af. ia thee 7 YY) 20,
ae dr dr

and hence p=Ar loo r+ Bre+C log: 792)
Next seek a solution for which wis a function of @ only-
Let

y=0
then } 1

Ve 20

and | it

Vib= 5 40" +0").
Substituting in (4)
20'0” = 470" +0".
Integrating @? = 4)0/+10/" +a,
and ©” me

iG, = 2A" +y

+a0'+b,

Or ae ve OV be i ea de’ ‘i
ao /@/3 6vO’ 2 340! — x yy

This may be written

gv & / bv f dO’

2 Vv (0'—r)(0’ —p)(O!—Gv+r+p)’

where 2X, » are constants.
Write
©' =) sin? d +p cos’ d,

and the integral becomes

= a : ad
(2 0,)= =|" Ae aie
ie aa oH sin a)

Finally introduce new constants, h, m

P Nu » 6v—rA—2
pes A aa m=" se

by—A— 2p’ me Gy

and we have

ov = 0! = 2v(1—m?— km?) + 6vi?nsn*{m(@—))} (VI.)

where £, m, 6) are arbitrary constants.

Steady Motion of a Viscous Flaud. AGL

§ 3. Solutions which are independent of the degree
of viscosity.

If we consider the motion of a fluid between two con-
centric circular cylinders rotating with given angular
velocities, an increase in the viscosity of alte fluid would
necessitate an increase in the couples which maintain the
rotation of the cylinders, but otherwise the motion would be
unchanged. This will i true for any solution for which >
is independent of v, 2. e. the two sides of the equation (2)
vanish separately. Tee the equation in its Cartesian
form we have

Br, Var) _ a
aa OE VES ain Papal mig hl (0)

and Wale On tee al ta Wi eave OEY

It will be noted that equation (6) is the usual equation for
the two-dimensional motion of a viscous fluid on the assump-
tion that it is so slow that squares and products of the
velocities may be neglected. For an exact solution equation
(5) must also be satisfied, and hence

5) a
Vev=fb),
that is the ea is constant along each stream-line.
Substituting in (6),

ron {(S) +r} +rwrH=

Hence either (1) the resultant velocity is a function of w,
and is therefore constant along each stream-line, or (2) /’’(y)
and f(a) are zero. In the latter case

V7 = const.=4a, say
of which the general solution is

paar ty) +x,

where y is any solution of V°y=0. Hence any solid body
rotation superposed upon any irrotational motion is a possible
motion of a viscous fluid. The only other solutions of this
class are those for which the velocity and the vorticity are
constant along each stream-line.

§ 4. Discussion of solutions obtained.

Flow between two planes inclined at any angle.—We will
first consider solution VI. We may without loss of generality

462 Mr. G. B. Jeffery on the Two-Dimensional
take 0,=0, so that

= = 2v(1— m? — mk?) + 6vk?m?sn?(mé, k).
= 2v(1 + 2m? — m?k?) — 6ym2dn2(m, k).

The stream-lines are all straight lines passing through the

origin. If w be the velocity of the fluid yi

The constants m, & can be chosen so that the velocity is zero
when @=z, and hence, since uw is an even function of 6,
when 06=—a, while the values of W appropriate to these
two stream-lines differ by a given amount. Thus we have
a solution for the motion of a fluid between two fixed planes
inclined at an angle 2« due to a line source of given strength
Sens : i : ans Ol
along their line of intersection, or if we exclude the origin,
for the flow along a canal with converging banks. If @ is
the totel flux of fluid outwards from the origin

Q) = —4va(1+ 2m? — m?hk?) + 12umt dn®(ind, k) dé.
} 0
iG
ihe | dn®(z, k)\dz=H(C, k)
“0
where E denotes the elliptic integral of the second kind.
Hence
Q = — 4va(1 + 2m? —m’7h?) + 12vmE (ma, hk).
This relation, together with
3k? m?sn?(ma, kh) +1—m? — m7? =0,

is sufficient to determine m and k. (N.B.—a must be ex-
pressed in circular measure.)

If the angle between the planes is small, we may write
sn.2=2, and we have as an approximation when the planes

are nearly parallel
dw
ae Dy S| — mm? — 2/2 (1 — 2m 22
aia 2v{ 1 —m? — m7k?(1— 3m76")*.
If the angle between the planes is 2a we have

L—m?— nV? k*(1—3m?a7) =0,

or ree 1l—m
~ m?(L—3m"a?)’
and diy _ 6ym*(1—m*) (a2).

dg Saree

Steady Motion of a Viscous Fluid. 463

We see that to this approximation the velocity across any
cross section follows the same parabolic law as in the flow
between two parallel planes.

Solutions 1., II., and IV. lead to some interesting sets of
stream-lines. They cannot, however, be realized physically,
and they seem to be of little importance. We pass to the
consideration of solution ITI. ?

A centrifugal pump.—We have

= —bvO + Ar’t? + Br? + C log x,

and the components of velocity are given by

ie VON Py
U= Fs 36 = Fe

vist ONG 2) pbt+1 2 a C
FF Sp =A +2)1 + 2B) + Fi; e

From the equations of motion we can determine the mean

pressure. If there are no external forces
p= —AbpvBé + pf (r),

where f(7) can readily be determined if necessary. If we
include the whole of the space round the origin, p must be
smele valued, and hence B=0. In this case the solution
corresponds to the motion generated by the rotation of a
perforated cylinder, which, as it rotates, either sucks in or
ejects fluid uniformly over its surface. The fluid may either
flow away to infinity, or it may be absorbed by a coaxial
porous cylinder, which may be at rest or may be rotating
with any angular velocity. The total flux of fluid will
determine 6, while the angular velocities of the cylinders
will determine A and ©. Such an arrangement will be in fact
a centrifugal pump. When the fluid is viscous vanes are
not absolutely necessary, although, of course, they may
increase the efficiency of the machine. If there is no second
cylinder so that the fluid extends to infinity, then we must
have A=0 if the fluid is flowing outwards, for in that case
b>0. let the radius of the cylinder be a, and let it rotate
with angular velocity Q, aud eject a volume Q of fluid in
unit time; then

C=a20) b=(Q/2ryv,
and we find without difficulty,

1 ape 2 9
P=Po—- oe Ge a)? + Q?),

where po is the pressure at infinity. Thus we have the

464 Two-Dimensional Steady Motion of a Viscous Flind.

pressure head created by the pump when it rotates with a
given velocity and discharges a given volume of fluid per
unit time.

Flow under pressure along a circular canal.—Finally, we
will consider solution V.

w= Ar" log r+ Br? + Clogr.
If uw, v are the radial and transverse components of velocity

ae
ro0 i
pe Nae r+ (A+2B)r + Be
r is
“If A=0, this is the well-known solution for rotating con-
centric cylinders. Using the equations of motion in polar
coordinates we can without difficulty find the value of the

mean pressure p-
p=4vpA@ + pfi(r),

where
f(r) =2A (log r)?(Ar? + C) + 2 log 7(ABr? + AC 4+ 2BC)
ike Be dys
a. 3 (A+ 4B a — 9 72 =

The constants B, C may be chosen so that the velocity is
zero for any two values of 7, and we have the solution for
the flow of a viscous fluid round a canal bounded by two
concentric circular ares. The pressure will not be constant
across a radial cross-section, but will vary in a way which
is represented by f(r). It will be noted, however, that /(7)
contains only the squares of the coefficients A, B, C, and is
therefore of the order of the square of the velocity.

Suppose the canal is bounded by circular ares of radii
a, b, which subtend an angle @ at their common centre, and
let P be the pressure difference between corresponding points
on the two bounding cross-sections. Then

P=4pAa,

where pw is the coefficient of viscosity and is therefore equal
to vp. The condition that the velocity shall vanish when
r=a, b gives the following equations to determine the
constants B, C

{

A(r log 7? +7) +2Br+ s ——(). (=a Oe

On the Theory of Dispersion. 465

from which we have
A(a? log a?— 1? log ? +a?—1’)

ae 2(@ 8?)
ls Aa’b? (log a? —log 6?)
aay a*—b

If @ denotes the volume of fluid which flows through the
canal per unit depth in unit time, it is equal to the difference
of the value of y for r=a, 8.

Q=A(?? log b—a? log a) + B(L*—a”) + Clog b— log a)
poids: ae al I)
= gual ¥)— a—p(log5) |.

If a, } tend to infinity in such a way that a—l >d and

aa >! Q Dae

12’
which agrees with the known result for the flow between
parallel planes.

LI. Theory of Dispersion.
sige eaonts 1D Sie WU Menus GVO Tiled ted dae

ne: if is well-known that the electromagnetic theory, as

expressed by the equations of Maxwell and Hertz,
cannot account for aberration, dispersion, aud allied pheno-
mena. In analysing the reason for this, we note that the
theory is based on the following postulates :—

(1) The energy of the electromagnetic field is that of the
dielectric medium alone, arising from a certain
strained condition of the medium.

2) The conductors having static charges serve only to
limit the dielectric region so that no part of the
energy resides on them.

(3) The strained condition of a dielectric is due to electric
displacement or polarization f, g, h, subject to the

condition
OF, og + a ano Ung Mes BPE (9)
On Oy ) ez
This displacement is apparently held to involve
motion of the ether in the medium, subject to a pro-
perty akin to elasticity (due to inter-action of matter

* Communicated by the Author. First appeared as a Bulletin of the
“ Tndian Association for the Cultivation of Science,” Calcutta.

Phil. Mag. 8.6. Vol. 29. No. 172. April 1915. 2H

466 Prof. D. N. Mallik on the

and ether) defined by the so-called specific inductive
capacity of the medium, which thus appears as a
constant of the medium.

(4) Conduction as well as convection currents (as in the
case of an electrical discharge) involves a tranference
(‘‘a procession and not an arrangement ”’ as Faraday
put it) dependent on a certain property of the con-
ductor (called its conductivity). This transference
is that of electric charge, but what this charge is—
whether it is material or ethereal—is not further
specified, or is rather left entirely open™.

2. If we limit ourselves to two physical entities, matter
and ether, the electric charge whose motion constitutes
electric current must be regarded as a mode of manifestation
of the ether. If, on the other hand, we agree to regard a
unit of electric charge as an actual physical entity, distinct
from matter and ether (but related to them and partaking
the nature of both in a manner that will require further
investigation), we are able to give an account of the various
phenomena which are left unexplained on the above
postulates.

3. Weare led to this additional postulate of a unit of electric
charge as a physical entity, not merely on theoretical grounds
but as a result of direct experiment. For during electrolysis,
each monovalent atom is known to carry with it to the anode
a determined quantity of what has hitherto been called a
negative electric charge which can be measured and which
is independent of the nature of the transported atom. If we
assume that this also is the unit charge, which takes part in
electric conduction or convection, and if we call it an
“ electron,’ we have to conceive a monovalent material
atom showing no electrical properties as the result of com-
bination of a single electron with what may fittingly be
called one or more units of “ positive electricity.” This
would amount then to the statement that the electrical pro-
perties of bodies (as well as those of a dielectric medium)
are due to the presence of “electrons” associated with atoms
of matter, forming systems of various complexity, and such
a theory is found to be consistent with observed facts.

* Maxwell’s pronouncement on this point, indeed, clearly sets forth
the position (‘ Electricity and Magnetism,’ vol. i.) :

“Tt appears to me that while we derive great advantage from the
recognition of the many analogies between electric current and current
of material fluid, we must carefully avoid making any assumption not
warranted by experimental evidence, and that there is as yet no ex-
perimental evidence to show whether the electric current is really a

current of a material substance or a double current, or whether its
velocity is great or small as measured in feet per second.”

Theory of Dispersion. 467

. Maxwell’s remarks [Art. 1 (4) note] had reference to
1 negative results of the eae, whereby he proposed
to detect in a direct manner the inertia of an electric charge.
Experimental determination of the mass of an electron as
well as their velocities under certain conditions has supplied
the data sought by Maxwell on which to build up the further
development. of the electromagnetic theory.

5. We have thus justification for regarding the pheno-
menon of electric conduction in gases as due to “ ionization”
or generation of charged particles or “ions” which are
carriers of electricity.

6. An electric current would then, on this view, consist of
two parts—one due to electric displacement which we may
still regard as ethereal, and the other due to motion Bs
electrons; or if wis the total current, in the direction of

we shall have
(at OMe Mw ts aes ea)

where p is the volume density of (free) electrification whose
transfer or procession gives rise to convection or conduction

current, and #, its velocity in the direction of 2, while =

7. Now, when there is free electrification, we must noe

of Og oh

Serpe aa AN NEY eee Wee)
This must therefore be the further condition satisfied by
As 9, hs while in the case of conduction there must, in addi-
tion, be a viscous decay which has to be taken account of in
a suitable dissipation function.

8. When we proceed to interpret these equations, coupled
with the observed phenomena of metallic conduction and
dielectric polarization, in terms of the electron theory, we
are naturally led to the conclusion that :—

9. A dielectric medium must be conceived to have electrons
iterspersed in it, giving rise to a constrained (ethereal)
motion in the medium defined by the above relation ; this
also necessarily imposes a constraint on the motion of electrons.
On the other hand, motion of electrons in a conductor must be
free (subject however to certain dissipation of energy) so that
the static charge in them always resides on the surface.

10. A conception of this kind naturally suggests that the
dielectric property of a medium may be explained as arising
from the constrained motion imposed on it. We proceed to
show that it does, and that in this way we can explain
dispersion and allied phenomena,

2H 2

468 Prof. D. N. Mallik on the

11. On comparing the equations (1) and (3), we readily
see that these cannot be, obviously, satisfied at the same time
if f, 7, h are to have the same meaning in both.

12. Now if p=0, that is in /ree ether,

Oye 09 | oe

Qu 7 3 MMi (Oe
Thus, we may say that in free ether,
Oi 09). On
= == (();
= cha tees
that in a material medium,
oF at CHEN Oh _
Ou OY T 9:
while in all cases,
Ofo , AF . Oho
Sel oY (Dh Pe. — <= 0,5... 1S eee
Or non toe (4)

where fo, Jo, to are defined by the equation (4) and are equal
to f, g, h, when the medium is free ether,

13. In order to specify these quantities further, we observe
that from the equations

Ofo , Odo , Oho 9 4)

Ow oh OY zt OF :

and Oe 00) 4Oll am ay
Oe toy ac @)

we may write fp=ftA,&e,; «oe
where oe OB.) OCA,
—— . > Seo

Ou ar Oy a aoa =P (6)

14. This expression suggests that A, B, C are components.
of a quantity I which corresponds to the coefficient of mag-
netization on the usual theory of magnetism, so that just as
dM=magnetic moment of an elementary magnet =Idr
(where dv is an element of volume), similarly, we may take
dM=electric moment of an element of volume containing
electrical charges arising from the presence of electrons and

dMaldr,.’.. . 2 «ee

where, of course, [=(A, B, C).
15. Again, from (5), we get

Theory of Dispersion. 469
And ifswe agree that fy=u=total current (polarization
amarconvection), we concludethat A=px, 9... . . (9)

“, ¥, = being}the velocity of electrons, as before, and it is
easily scen that (7) and (9) are consistent, on the under-
standing that p(v—wx)=A (say) or e(a@—ay) = Adr,

e being the unvarying charge in vol. dr,
and #— )==the displacement of e in the direction of ..

Comparing the equations (3), (4), (5), and (6), we con-
elude that total polarization in any medium may be regarded
as made up of two parts—one involving ethereal (fg h) and
the other corpuscular (A B C) displacement.

Again, comparing (8) and (3), we observe that the total
current is to be regarded as similarly made up and that it is
the total current and total polarization that are subject to
the solenoidal condition.

From (6) since, in stationary media,

dA OB ac
OA -L OE + OU 4 ely ==\()),
: OL WOU, Oe et
we get the equation of continuity for electrons in motion,
ViZ.:

op Cpu
SIE CEG na
ot u Ox 5
Again, from (6), we can obviously derive

OW
where 2X is defined by AV’d+p=0.
This equation states that @ is the potential of a distribution
p, provided

where ky is the specific inductive capacity of the medium.
Since the medium here is the ethereal medium (whose pro-
perty is modified by the presence of electrons), /) is an
absolute constant defining the property of the free ethereal
medium.

Thus
ko O¢ rie uty ODus op
PN Cee Os USP ge 9 yeas Shou. )
A De ae &e:,) and: fp= + reaiae Kc (10)
so that Me Ak Coe) vfiL
f= tho 3 (;)

in the neighbourhood of an electric charge, e.

470 Prof. D. N. Mallik on the

16. On this understanding, the so-called electrostatic
energy of the ether will be W,, where

vi al (to th )\dr, —.) ee
0

while the potential energy of the material medium will be W,
where

Wa 7" ((ittae tinar, +

while the kinetic energy (T) of a material medium may be
taken to be

ott Tk Ava
tale +2? +y%)dr)) = i eer

17. From (10) we observe, what is a priori evident, that
the total force producing the Srentied condition of the eda
is that due to the free sthereal motion and that due to
electronic disturbance.

From the expression for W and T, (12) and (13), we get,

applying 5 | ( T—W)dr=0, where 6 is the operator of the

calculus of variation, and remembering
At fo = Oy = 5) e ° . e . (14)
fi=Vym .. . 2

where V?=
Hence, since |
fo=f +A'=f+pr, « «ae

and

VH=VitVARV/+ 7! Sy “6s > > lame

ney Aare
f+ & (pi) =V{v'/— oe). | ia
18. Also from Helmholtz’s principle we get
1 a
—_ a ( (10-4 mb-+ ne ais— ( (Xe + Yay +-Zd2), . (9)

where a, b, e=magnetic induction,
X, Y, Z=electric force.

Theory of Dispersion. ial
Hence we have, if

Meee (§ 21)
ko

mea Oh 0g Mea vsdl
a= Ani (5 =3) where og, a: AD),

(on the understanding that w=1), so that Vy is the velocity
of propagation in free ether.

19. From (14), (16), and (21) we get

wis dh Og
a= —4nV3 (50 —S! 1) VA NCD Th

in stationary media

) Oo8 | oY Open dey
=V.'| ee ) See _ SeL). (22)
0|Ve als, 4 Su BS i AE (22)
20. Let T,=kinetic energy of an electron (e),
T,=that of the ether,
V.=potential energy of the ether,
V,=potential energy due to extraneous forces.

Then since —f B h—yg)dr=momentum of the field in the
direction of x (J. J. Thomson, ‘ Recent Researches’),

or
— iy \(Bh— agar =5 ($3)

2, y, 2 defining the position of an electron.
Applying the Lagrangian equations of motion i the
system outs of the zther and the electron, viz.

ein or, eae) nt a(V,4V>) . =
oh

i On out Oz
we ve
Oly VON TOL mew:
alse) +3, = =" at i (Oho )dr Ow
a Oo” =0, obviously.
1 Vie
ae Has) ef ue gta oe
4 ’
== ALB Bel fiero
— OR th ht 2 a

Bu’

472 Prof. D. N. Mallik on the

since Meee a (2 3")

o=h+p2, ke. [§ 6].

Integrating by parts the first integral of (23) and
substituting

in the second integral and again integrating by parts, we
get
ye | Ca oT:
at\oe)* Oa —k Oe dy On Oa
en aa: OV, OV2
+4 pljy—8)are SS

4 , We
=F, \erde+( ain—s8)ar 0 ae (24)

which may be stated-in words as follows :-—

An electron in motion is subjected to a force which is the
sum of an electrostatic force and an electrodynamie force,
as the result of the action of the field. [ Poincaré, ‘ Elec-
tricité et optique.’ |

21. Again, if T, be the kinetic energy of free ether, we
have

Abe j (Fu+ Gv+Hw)dz,

where u=f+ A, &e.,
whiie W =the potential energy

9
= — { (f?+9?+h?\dr, where it should be remembered
0

ho= 7 V, being the velocity of light in free ether, and
f,g, h are connected by the equation of condition

One «Og. Oh

Of) oy 102°”

Introducing (in the manner of Larmor) an undetermined
multiplier : and writing |

We tte" + h?) sero [( (So +. a p Jar,

Theory of Dispersion. 473,

we have
éW'= io r+ . dr — 6[Sh 4 +) bp ir m.

since @ isnot to vary. Now integrating by parts into
surface (s) integrals and volume integr al, we have

OW’ =\{(Z ope oo i + )de+ | baper— {4 if + )ds.

Now from the Lagrangian equation (since /,g, h may he
taken as generalized coordinates)

AON lO VO Wi

eo ar
or se ee Th 6

where om is evidently the potential of distribution p.
That is (if the medium is at rest) the electrostatic force
due to the medium (say X)

uN aed
(the latter of which is due to p) on the understanding that
aT ieX,
Ko

22. It will he observed that k of the ordinary electro-
magnetic theory becomes fy. This amounts to the postulate
that there is no dielectric other than free sether.

Remembering that

ky O¢
a At Bw?
we conclude that
K= an Jo
hae

23. Returning to the equation (24), we observe that

OV : :
— —— =the force due to the system of electrons in volume

Ow
\d7 + the foree brought into play on account of the
displacement of electrons.

If we take a small sphere of the volume | dz, the system of

474. Prof. D. N. Mallik on the

electrons defined by o2 + g + = —p may be replaced

by a surface distribution I cos 0, per unit area of spherical
surface. And thus, as in the corresponding magnetic theory,

this part of the force in the direction of 2 will be a7 per
0

unit charge [ Maxwell, voi. ii. art. 399]. For the second
force we may, obviously, assume an elastic force in the
direction of w, due to displacement of electron, and this

will be
A 23
—p(v— 22) (say) ea . 4. et ei

while the frictional force will be of the form xA’. Hence,
for equilibrium

im i) Adt
i, pat

= Fl o(r+ 3 s)er+(p (Yy—zB). (28)
Putting
b= Mo faa=a, lerar=ep te
we have
OG ae (jy — 28), . « (29)

or for simplicity
ko
poA=C(f+4A)+ 7c (Gy—28), . . (80)

all the quantities having their mean values taken over a
small sphere, enclosing a ‘charge.

For motion, we have pz =A,

or \updt=A, - 2 . 2

where for A we take its mean value, as before.
Accordingly, since for a single electron of mass m, self-
inductance L, and charge e,
ten + Lie?) (a? + 24 2”), | (32)

we the

a 7 HA + (Le? +m) A= eelrts 3) ty 28)

a 7 e( 745 *) +e%(yb—@6). (83)

Theory of Dispersion. 475

Writing
Le? +m=nr >
4o0r Ar e? is |
a eee (34)
od j
ko
we get
Bea A = ayp, Se, wu ee) BD)

if the magnetic field is weak, and
TELS IN =voP from(@iS) iy fen eco)

From (35) we deduce
aes as Aan cs)
2 P= Pot COS (mt + €)

esto dhe 29 =0, or po is independent of coordinates, the

equations will be

A + po A=aof Ms MeN LOC B
ftA—VA2f=0

d7f

For a plane wave (z= const.), A?f/= 52

» &e.
and the solution is
PAs ew (5 —t)
S 3 NE
T= e'P ean i)
which yield

mal+e eas bp tel hs cde ate CR)
“TMi

20 cere
where n is the index of refraction and — = the periodic

0
time of vibrations of electrons unaffected by the field, while
> refers to the several groups of electrons that are set vibrating
on account of the impressed disturbance.

25. Returning to equation (35), and introducing a viscous
term (oA we get
A+ bo A+ po A=ag/. rere eN

476 Prof. D. N. Mallik on the

Now, assuming the solutions

A=A,e* (+ Saal
VO
_ 1B yi p he — jj
IE Ny Z V ’

Cp.

Po — tpbor pre’

we get

nve=1+>

pee that there is absorption in this case.
If po is very nearly equal to p, by cannot be neglected ;
this Aaa indicates that there is pina absorption, andes

these circumstances.

a }
If p is very small, then n?=2)?+ —, ~~ a and the real
part of x is po’ —ipho

U9” *

As p increases 1 diminishes. This explains anomalous dis-
persion.

26. It is not without interest to compare the above with
the various elastic solid theories that have been proposed for
the explanation of dispersion.

27. For this, Jet us recall the fact that in an elastic
medium there is, associated with an elastic displacement,
molecular rotation ; and if the properties of the medium are
to be capable of being expressed in terms of quantities that
enter into the statement of either theory, electric displace-
ment and magnetic force must correspond in some way with
the velocity of vibration and molecular rotation. Now in
the electrical theory we have two quantities defining the
property of the medium pw and &, as well as the quantities
ig, h (polarization) and H (magnetic force), while in the
theory of elasticity we have the constants p (density), n
(rigidity), and the quantities w (molecular rotation) and
Em € (displacement), and it will be necessar y to decide upon
a suitable mode of identifying these, s severally.

28. Now, on examining the expression for energy (kinetic
and potential) i in terms of these two sets of quantities, 1t 1s
easy to see that one such mode of establishing a concordance
between the two sets of phenomena is (as Larmor has done)
to identify electric displacement with molecular rotation
and magnetic force with sethereal velocity (in vibratory
moti one d

Theory of Dispersion. ATT
On this theory, the electrostatic energy

2a
k

eZ
will correspond to the energy of strain of an elastic medium,
since this is

(ge aE it hy )dr

a)
. e e 4.
_ 2n | (o/?+o/+o)dT + surface integrals if (i + .n JA=0,
o
so that on this theory we have

Qo fin? 6 2 : i Z
a =2nw,’, &c. (13), anc Renan ne (41)

provided we further postulate the identity of the kinetic
energy of a strained elastic medium, viz.:

Lf o(@ +P +e%)dr,

and electromagnetic energy

1 9 9
oe pu(a? + B?-b y” dz,
where o=density of the medium,

E=x£— 2, &c.=elastic displacements of the medium,

a, 8, y=magnetic force,

@,, @y, ®:=molecular rotation.

29. From (13) we derive, provided ’ = constant,
Oo

2a fy” a Wo»

and this leads to the conclusion that the resultant twist is.
made up of an ethereal motion in addition to an electronic
displacement, neither of which is however of the nature of a
pure rotation by itself.

30. We are now ina position to consider the equations
of motion that have been proposed™ to explain dispersion.
Boussinesq’s formula is

2 2
a?u Ue = (e+3

| dy SIDA) bas
Mae + ipa 3”) +nA?u,

Ow
where m is the mass, « the displacement of the ether, and
pz, U, those of “ matter,” and k=volume elasticity, &e.

* Glazebrook, B. A. Report, 1885.

478 Prof. D. N. Mallik on the

31. From these we may easily derive the corresponding
rotational equations, v1z.:

me, -+- ~0;,=nVy"’o:, -- = .. 9

where ;, 2, are the curls of u, v,w, U,V, W. Boussinesq’s

theory is thus seen to be capable of being interpreted as

being based on the postulate of twists, defining the disturbed

state of the medium.
For, if we write

on+ — Qv=f +A=fo,

: i
we get, putting pace

fee A= i) a
provided A2(O72—A)=0. . .- . Seen
co ko Op

Now yA

eer! 45 Sa

Further, the equation of equilibrium of a material medium
regarded as an elastic body would be W?Q72=0, &c., so that
(43) amounts to the statement that in forming the equation
of motion we must regard the material medium to be at
rest.

Again, U and wu are assumed by Boussinesq to be connected
by an equation of the form U=/(w), and in particular, for
dispersion, U is taken by Boussinesq

=Au+ i +Dy’u, where d, C, D are constants; (44)

whence on our notation,

O2=Xo,+ Dy’e., &c:, =) | See
or

A=f+ iDy fuer where 27=0,(1+2)+Q2z nies | (46)
C / : nu
If we admit that 7 is an harmenic function, the equation
can obviously be written in the form
An =ayf, - «+ ko eee

the constants being suitably adjusted.

Theory of Dispersion. A79

32. The equations of Helmholtz with the same notation as

in §30 are

mi=a’vyu + B?(U —u) : (48)
and wl =—6?(U— a)—aU —°U, j
whence Mor= #Y?o,+ B?(@z:— Oz)

PO Soo} | (8)

On transformation, for purposes of comparison with the
electron theory, 7. e. putting

DPA f EVAN i Quen ty'A, 2. . G0)
we get

m(rf tv) H=e—Af+vA) +B {(A—A)f+(V—vA}, (51)
and
u(r f+ v'A) =B?LO —A)f+ @'—v)A] —HAst+ vA)

7 Our Ry Be as eh (2)

The equation (52) is the same as the equation (48) pro-
vided \'=0, while from (51) we get (Gif X’=0) an equation
of the form bial:

ft A=ev7yt+ BF+ yA.

Remembering that fand A must vary as cos pt (say) we can
obviously adjust the constants and variables so as to put the
above equation in the form

fae heel ey

which is the second equation (38). [A and / differing in
value from the same quantities occurring in (51) and (52)
each by a constant factor. |

33. Kettler’s equations are of the form (Glazebrook’s

notation)

mit + nO U Saat

uCi+e"U= —2U — Bi, &e.
which yield as before

M@,+ uO, =27V"or

pCor + wQ,= — 20, — 620, &e.,

which are the same equations as (51) and (02), if we put
mor + wC'O.=f+A and wCo, + wO,=Af+ vA.

480 Prof. D. N. Mallik on the
On eliminating fand pucting Am—pe=0 these yield
ftA=v7 and CA" 4+/"=a2A+a2A.
And if 7 varies as cos pt, these equations are of the same

form as (38).
34. Inasmuch, however, as

BAC AB dC"
See aaos.) ah

and OG. OF.
oe Oy t Oz”

(A, B, C) and (7, g, h) cannot be interpreted as rotations,
each by itself, and to that extent these theories are less
gener al than the electron theory.

The main difference, however, consists really in the fact
that the elastic solid theories deal with ether and matter,
while the electron theory deals with eether and electron.

35. In attempting a comparison between the elastic solid
theory and the electron theory, we have identified the elec-
trostatic energy with the energy of (rotational) strain of the
elastic medium, and the electromagnetic energy with its
kinetic energy.

36. But the identification is not unique, as we do not know
which of the two expressions (in either system) is kinetic or
which potential, or in fact whether both the energies are not
(as they most likely are) kinetic. We may, therefore, if
we like*, regard electric force as identical with the rate of
elastic displacement and magnetic force with molecular
rotation.

37. On this scheme, the electromagnetic energy

iL 9 9 a
———— = = =<
8 if pa — B ny \dt

is to be identified with the energy of strain
a)

| 2n (@,?+ oy? + wo.) dT,

¢ + 5 n)A=0,

* Glazebrook’s Address as President of the Physical Section, B. A.
1893.

provided we have

Theory of Dispersion. 481

which yields the solenoidal condition for magnetic induction,
vies :

OG Gly tee

Bet dy t 327%
since O®: Ow, _ Oa: Bi)

Ou Oumar Tost ir

(provided np is constant).
38. When we proceed to identify the kinetic energy

io\ (u? + v? + w*)dr

of the elastic medium with the electrostatic energy, we
observe that if the elastic medium is ethereal, the equation of
condition should be

400 { (9? + Up? + Wo") dT = a (f?+9? +h?)dr,
E 0
while if the medium is a material medium we must have
2
to (Gr tot buthdraE | (atte thiyar

These yield the following results :—
9

—_ 1

where Vo is the velocity of light in the ether;

where V is the velocity of light in any medium (a, n or /, p).

Also
a(=)=§ oo 0% , OW = (ee og oN ee

di Pon Toes Won Ogi Ge) ae
where A is the dilatation of the medium.
Ou Of»
Also Ae - + = On qF AP ==);

i. e. the total displacement (ethereal and electronic) is sole-
noidal, while the volume density of electricity is proportional
to elie il expansion.

Pilals Mag iowGe NO a NOw kia. Apal 1915. 27

482 On the Theory of Dispersion.

39. The various equations appropriate to an elastic medium
are then found to have their analogies in the electric
theory.

Thus the equation

A , A be
(i | ins —2n = ae = =o&

yields the electrostatic equation for a material medium,
in the form

on properly choosing the signs.
40. The equation of motion of such a medium, viz. :

Savi
yields es Vee
or fr = Vf
Le fe Raveye( s+ Ala vey,

if Vy A=0,
while for the free ethereal medium we have
f=ViV's,
41. Again Boussinesq’s equation, viz.,

ToM +piU=nyé, yields,

ay, - 4, Aas
if we put oyw=f and py — a a SS
ee ee 9 9 »
f+ A=Voevi.
The other equation of condition of Boussinesq, viz.,

re VAN p
V=avau+C S + Dy?u,

5 = ° x Ip l= 9.7:
gives sunilarly Anji + DV

ret pitt A
=i +D Wir >)

and similarly for the equations of Helmholtz and Ketteler.

Average Thorium Content of the Earth's Crust. 483
42. A third method of identification will be to take
(@x, Wy, @.) a (a, B, Y)> (0°, Nos Co') =(f,9,h) and (ee 1’; C) = (fos Jos ho)

olving OZ Ol. 0
ae ou oy a,

as in the electronic theory, A=p, while all other equations
are practically the same as before.

43. Now electrical experiments lead to the conclusion that wu
is very nearly constant ina dielectric, but that & varies, while
optical experiments make the constancy of n and variability
of p probable, so that the second mode of representation
would appear at first sight to be more in accordance with
facts. It seems, however, that except in so far as the identi-
fication of the constants is concerned, the question of inter-
action of matter and ether is resolvable on any of these

hypotheses, for we may simply have os = == Vi where \.
is the velocity of propagation, although it may not be possible
to identify separately the various quantities, that enter in
these investigations, according to any of the schemes tabu-
lated above. On this understanding, if does not seem to be
possible to pronounce in favour of any of these in preference
to the others. In spite of the uncertainty that exists in this
respect, these various modes of representation are useful, as
we have so little knowledge of the intimate nature of electric

and magnetic quantities.

LIT. The Average Thorium Content of the Earth’s Crust.
By) els ds kOOLE)

HIS series of experiments was undertaken with the view

of supplementing a previous paper of Dr. Joly’s on the

‘“‘ Radioactivity of Terrestrial Surface Materials,” which

appeared in the Philosophical Magazine for October 1912.

The same composite rock mixtures as were used in the
previous series of experiments were also used in these.

The method of procedure and the design of the apparatus
have been fully described by him in the Philosophical
Magazine for May 1909. It may be briefly said that the
rock has first to be got into solution by fusing it with a
mixture of the alkaline carbonates; either a blowpipe ora
small electric furnace being used for this purpose, as a

* Communicated by Prof. J. Joly, F.R.S.
212

A484 Mr. J. H. J. Poole on the Average

temperature of about 1000° C. is requisite. The melt is
then treated with water and filtered. The filtrate is discarded
as it has been found to contain only a negligible quantity of
thorium. ‘The residue left on the filter-paper is dissolved in
distilled water containing 50 c.c. of pure HCl, and the solu-
tion made up to a convenient bulk with added distilled water.
This solution is set aside for some days to allow the thorium
disintegration products time to grow, as some of them are
lost in the process of getting the rock into solution. The
thorium content is determined by boiling off the thorium
emanation in a constant stream of air, which air then passes
through an ordinary gold-leaf electroscope. By determining
the rate of leak of the electroscope when the solution is
boiling and when it is not, we can estimate the amount of
thorium in the solution. The solution before being tested is
boiled in a separate vessel to drive off the radium emanation
if there is any present.

The electroscope is standardized with a thorianite solution
of known strength, as Dr. Joly has previously described.
One c.c. of this solution was either added to a known amount
of distilled water or to a rock solution whose relative activity
had already been determined. In either case, by observing
the new rate of leak we can easily calculate what quantity
of thorium corresponds to a gain of rate of leak of one scale-
division per hour.

The latter method was adopted to see whether rock solu-
tions had the power of concealing, as it were, the emanation
that they contained. If this were the case, one would expect
a higher constant for the electroscope to be obtained by the
second method. However, by both methods a practically
identical value was obtained. The whole method is of course
only a comparative one, the effect of any rock solution being
compared with that of the standard. For.this reason great
care has to be taken to keep the conditions under which the
standard solution and the rock solution are boiled as similar
as possible. In all cases some powdered steatite has to be
added to the solution to make it boil freely and to prevent
bumping. Many solutions were found to be very sensitive
to any change in the form of ebullition, and in these cases
it was often necessary to add some fresh steatite, as the
steatite after a certain time lost its power of producing free
ebullition.

The constant of the electroscope depends largely on the
rate of the draught of air through the boiling flask, and
thence through the condenser and electroscope. There is
one rate of flow which gives the lowest constant for the

Thorium Content of the Earth's Crust. 485
I. Acid Rocks.

|
Thorium grm. per .

2 Ra grm. é Ratio
Composite. a ejaes| grm. X 10°. Ra/Th x 107 |
SOS (1) @) |
eeiGranites 00 kl 7 | 204 | 2:00 | 1-33 |
23 Acid Intrusive & Volcanic SOP aol Se nD) a ke) pg
Ditto New Solution ............ nee ae 2°20 as | |
i

\
|

MicnulfoneG nocken 2 0a 10ne

II. Intermediate Rocks.

12 Syenites:
(1) 10 grms. in solution .........) 2°4 4-08 as 0-59 =|
(2) 6 grms. in solution ...... a: ‘ie ae 175 Neo |
ws) oerms: in solution..2..2...5.: aa: bh 1°68 a el
6 of the above Syenites ............ res wiki 4-2 0:57 |
EMINGIRIEGS: 2) .8. cv novazensesemmans: 16 0-99 ae 1-62 ;
MeMEe MYLES; 64. 0dseterncgaodteeseee| oO 1-79 set 1:68
PO MEOVpMVEIES .....523.. 5... Sewers ene 1°54 1-82 |
| Mean for 48 rocks=1°64 x 10-6
| me Basic Eine
im Basalts (chiefly Hebden 0:5 OSr elit ne: LOS |
PEGA UGOS) Seton -seenGissseoes ere y eee 13 0°50 Re 2-6
8 Diabases and Dolerites ......... em leCii el cia Beh i AOU |
14 Basalts and Melaphyres ...... I SAO EEO fc Sei Fa 2:38 |
WS'General Basalts .....:...062..0.05 | 14 0:63 063 | ee an
Ditto New Solution.............-. ao ce 0:56 sh i250 (|
Mean for 56 rocks=0°56 x 10-5
IV. Miscellaneous Rocks.
7 Vesuvian Lavas ...c....ccccseee- | 126 | 236 alias 534!
WSGRINCNSSES) (2 cvicwadesueaeccouheneenens [ PORT eRe ey | 2°42
| GeDeccanw Traps ...4.ccs-ssaseese sen: Peraecae alae uae 0:47
| Globigerina Ooze | | | |
| 56 per cent. soluble in HC! . | irO.Oone> | |
44 per cent. insoluble _,, “a | 0:28t
Allimvone Solution. .2...ccse-h | | 0:36
Krakatoa Ash : |
SOR sisi tsetse Re sseee HIPs lWeicultes. 0-8 |
Sol eligi pareac cane sare al ne shes 0-9
SONS MELT e erase yest cetec eps gy | ele es cs, 9h

V litniens per grm. of Globigerina Ooze.

Norr.—-(a) The results given under column (1) for the thorium content
were obtained while working with the higher constant, while
those in column (2) were obtained with the lower constant.

(6) Krakatoa Ash. In solution I. and solution II. 2 grms. of
KCl1O, were added to the fusion mixture. 100 ¢.c. of HCl were
used in solution II. instead of 50 e.c. Solution II. was made
in the usual way.

A486 Mr. J. H. J. Poole on the Average

instrument, for if the draught is too fast the emanation will
only be left in the electroscope for a very short time, and
hence it will only produce a smail effect; but if, on the othe
hand, the draught is too slow, most of the enamels will
have died out before the air reaches the electroscope at ail,
and so in this case also the effect will be small. Some of
the results given were obtained when the instrnment had a
higher constant than in the subs sequent experiments. The
higher constant had a Nis of 3°2x10-°, while thersau=
sequent lower one was 2°1x10-°. The change in constant
was due to the draught through the instrument being altered.
The results obtained with the constants are shown separately
in the table of results. It will be seen that whenever a
solution was retested with the lower constant, the new result
obtained agreed fairly well with the old. This is a satisfactory
feature of the experiments.

Dr. Joly has described, in his previous paper, how the
composite rock mixtures were originally made. Amounts
of the rock mixtures varying from five to fifteen grams were
used in the solutions. The smaller amounts were generally
only used in the repeat experiments owing to a certain
scarcity of material. One advantage, however, of varying
the amount of rock in the solution is that it would show if
any contamination of the solutions had occurred, as we would
expect a higher value for the thorium content of the weaker
solution if we assume that in each solution there was the
ee absolute amount of contamination approximately.

Anyway it od seem justifiable to assume that there could
Ati be any large amount of contamination if the two solutions
eave nearly the same result, as it appears highly a
that each solution should be contaminated in exact propor-
tion to the amount of rock it contained. This argument of
course does not apply to any contamination which the original
powder as a whole may have received. Ordinary incan-
descent gas-mantles are a very dangerous source of contami-
nation, and accordingly their presence in the laboratory
should be avoided.

The composite rock mixtures used may be roughly divided
into three classes, 7. e. acid, intermediate, and basic. Some
results obtained for some miscellaneous rocks are also given.
For the sake of comparison the radium content of these
rocks, as determined by Dr.. Joly by the electric furnace
method, are also given. It will be ‘seen from the results
that the thorium content decreases as we pass from acid to
basic rocks. In this respect the thorium content resembles
the radium content. However, there is no exact numerical

Thorium Content of the Earth's Crust. 487

proportion existing between the two, as will be seen from the
figures. ‘There is nevertheless a certain amount of evidence
that the ratio of radium to thorium also increases as we pass
from the acid to the basic rocks, and for certain rocks this
ratio seems to remain fairly constant. It is also remarkable
that the value of the ratio always les within such narrow
limits. In only three or four cases does it lie outside the
range 1°3 to 2°7xX107". This range is perhaps not narrow
enough to justify us in assuming “that there might be any
genetic connexion between uranium and thorium, but that
the ratio is as constant as it is, 18 certainly a remarkable fact.
The mixture of the twelve syenites gave rather confusing
and disconcerting results. Dr. Joly first determined its
activity using a solution containing 10 grms. of the mixture,
and obtained a result of 4:08x10-* grm. of thorium per
gram of rock. He thought that this result was rather high,
and suspected that the solution had been contaminated, so
two new solutions, one containing six grams and the other
five, were made up. Both these solutions gave a value for
the thorium content of approximately 1:7 x 107°. This result
would naturally lead us to suppose that the first solution had
been contaminated as first thought. Unfortunately for this
theory, however, when a solution containing six grams of a
mixture of only six of the original syenites was tested by the
present writer, it gave a result of 4:.2x10~°. It is hard to
explain how the two solutions that gave the high results
could both have been contaminated to pr oportional amounts
so that they would give nearly the same result. The dis-
cordance between the values might possibly be explained by
assuming that the thorium in some of the svenites was not
evenly distributed through the rock, so that the powder
would contain specks of highly active matter. In this case,
we could understand that in one case we might by some
chance get a good deal more of these particles than in another,
and so obtain a higher result. This theory is however
incomplete, as it fails to explain the fact that the activity
of the mixture seems to possess two fairly definite values,
namely, 1:7 and about 4:1 107%. It might be thought that
the second high value could be explained by the fact that
the mixture only contained six syenites. It seems, however,
that this cannot be done, as even if we assume that the six
syenites rejected in making up the solution contain no
thorium at all, still even in this extreme case we ought to
obtain a value for the thorium content of only double the
value for the twelve syenites. This value would be about
3°4x 10-°, which is appreciably lower than the actual result

488 Mr. J. H. J. Poole on the Average

obtained. It is of course highly improbable that the thorium
would be distributed in the way assumed. It is more likely
that both the mixture of twelve and of six syenites should
have very nearly the same thorium content. It will be seen
that the lower result agrees much better with the results
obtained for other rocks in the group, and that it also gives
a more normal value for the ratio of radium to thorium.
It would thus appear that the lower value is the more
probable one.

In order to see whether it was possible that the activity of
a solution might go on steadily increasing with time, the
following test was made. A solution containing ten grams
of the 23 acid voleanic and intrusive mixture was made up
on the 13th of April. This solution was tested on the 20th
of April, and again about a fortnight later on the 6th of
May. On both occasions it gave a result of 2°2x10~°.
This result would tend to show that the activity of a solution
becomes constant after the first few days, as one would expect
from the rates of decay of the various disintegration pro-
ducts involved. To further test the method, the manner of
making up the solutions was varied, and the melt from a
rock dissolved in its entirety directly in HCl, without being
first treated with water. However, the 18 basalts when
treated in this way gave nearly the same result as they had
given before by the usual method. It was then thought that
the limpidity of a solution might possibly affect the ease with
which it parts with its emanation. Some of the solutions
used in the above experiments were perfectly clear, but a
few of them were cloudy owing to the presence of a certain
amount of gelatinous silica. One might perhaps suppose
that the emanation in the solution might be partially en-
trapped in the silica, and so that a cloudy solution would
give a lower result than a clear solution. In connexion with
this conjecture three solutions each containing five grams of
Krakatoa ash were made. One of these solutions was made
in the ordinary way, and the resulting solution was rather
cloudy. In each of the other two cases two grams of KCIO;
were added to the fusion mixture. This addition seemed to
have a very good effect, as the melt on solution gave a per-
fectly limpid solution. In one solution also the amount of
HCl was increased to see if this would affect the result. It
will be seen, however, that the three solutions gave practically
the same value, indeed the differences are probably within
the limits of experimental error for the method. Of the two,
the cloudy solution gave the higher resuit. This may pos-
sibly be due to the fact that the presence of a small amount

Thorium Content of the Harth’s Crust. 489

of gelatinous silica may lead to better boiling of the solution.
The amount of HCl used, too, seems to have little effect on
the result, as the solutions made up with 50 ¢.c. and 100 ¢.c.
of acid respectively gave nearly identical results. Thus
from this series of trials it would appear that the exact
method of treating the rock has not much effect on the final
result, which is satisfactory.

It is rather difficult to arrive at a mean result for the
thorium content from the values obtained. We can either
simply take the mean of all the results obtained, or we can
attach to each result a weight proportional to the number of
rocks in the powder from which the result was obtained.
The means obtained by both methods are given. A slightly
higher result is obtained by the second method. This is due
to the fact that there is a larger number of acid rocks in the
powders than either intermediate or basic rocks. Of course
by neither method can we. hope to obtain the real mean
value of the thorium content of the earth’s crust. To do
this, we should know its average composition, 7. e. the per-
centage of acid, intermediate, and basic types in it. ‘The
amount of sedimentary rocks could probably be neglected
without making much difference in the result. Itis generally
thought that the composition of the lithosphere approximates
to that of a diorite or andesite, 7. e. it is intermediate in
chemical character. On this supposition the average thorium
content would be about 1:6 x107°. This value is of course
obtained only from surface materials, and we are not justified
in assuming that the thorium content is not different at some
distance below the surface. The mean for the acid rocks
alone is 2°08x 107° and for the basic rocks 0°56 x 107°.
These means are obtained by the second method. By the
same method we obtain a general mean for the experiments
of 1:50x107°. In these results the miscellaneous rocks are
neglected. The values are obtained from 86 acid, 48 inter-
mediate, and 56 basic rocks. If we take the simple arith-
metical means we obtain slightly different values, 2. e. acid
2°13, intermediate 1°50, and basic 0°51. The corresponding
general mean is 1°38 x 107°.

In conclusion I wish to express my sincere thanks to
Dr. Joly for his most kind aid and advice during the progress
ot the experiments.

Iveagh Geological Laboratory,
December 1914.

fr 490 |

LUT. The Duplez Harmonograph. By J. H. Vuxceyt,
M.A., D.Sc., ARCS. and C. W. Juni, BaSé.5 eae
Paddington Technical Institute*

{Plate VIII. ]

ConTENTS.
INTRODUCTION.
CONSTRUCTION OF THE APPARATUS.
Initial Amplitude and Phase.
Friction.
DESCRIPTION OF THE DUPLEX HARMONOGRAMS.
Unequal Frequencies in Simple Ratios; Friction Small.
Index of diagrams 1-18.
Frequencies ¢ as 2, 3, 4, 5.
Frequencies as fe 2, 3, 4,
Initial displacements all negative.
Various initial conditions.
Epicyclics, &e.
Index of diagrams 19-35.
Friction small.
Frequencies in simple ratio.
Frequencies slightly different.
Frequencies in simple ratio but unequally damped.
Three frequencies ; two opposite pendulums beating.
Index of diagrams 56-46.
The other frequency their mean.
The other frequency half their mean.
The other frequency twice their mean.
Four frequencies ; pairs of opposite pendulums beating.
Index of diagrams 47-54.
a. Mean frequencies and differences of frequencies of opposite
pendulums as two to one.
8. Mean frequencies equal and differences as in a.
y. Mean frequencies as two to one and differences equal.

INTRODUCTION,

rRNHE pendulum harmonograph has been developed in

various ways since its first introduction. By using a
vertically vi ibrating stereoscopic camera to photograph the
horizontal eae of the tracing point, pictures which,
when combined, show harmonic ena in three dimauainee
have been obtained. The well-known twin-elliptic penduium
in the forms given to it by Goold and Benham? draws the
resultant of four simple harmonic motions with the con-
dition that they consist of two pairs of nearly equal frequency.

* Communicated by the Authors.
+ “ Harmonic Vibrations,” by Goold, Benham, Kerr, and Wilberforce.
Newess and Co., London, n.d.

The Duplea Harmonograph. 491

if one receives the traces of an ordinary harmonograph on a
table at the top of an elliptic pendulum mounted on gimbals,
after Benham, the elliptic vibration can be combined with
the motion of the other pendulums. By the use of an elec-
trolytic method of tracing, we have already shown how the
sense and speed of the motion of the tracing-point can be
recorded on harmonograms*

Harmonographs in general have the defect that the
pictures, although entrancingly beautiful, require much skill
on the part of the experimenter, combined with good fortune,
for their production. In particular, the amplitudes and
phases of the constituent vibrations have usually depended
on the dexterity and luck of the operator. In the instrument
which we shall now describe little is left upon which the ex-
perimenter can exercise his skill. When suitably adjusted the
instrument will draw the same diagram with the regularity
of a printing machine. The amplitudes and phases of the
motion are under control, and the interference of the ex-
perimenter consists merely in placing the writing point on

o>
the prepared surface and removing it when the record has

been traced.

Tisley’s harmonograph draws the resultant of two simple
harmonic motions at right angles to each other. The Duplex
flarmonograph replaces each of these motions by the sum of

two simple harmonic motions in the same straight line.

CONSTRUCTION OF THE APPARATUS.

In fig. 1, A, B, C,and D are knife-edges of four pendulums
placed “at he corners of a square dan on the horizontal!
surface of a thick slate slab. A and C swing in the vertical
plane through AC and B and D in the vertical plane through
BD. Bach’ pendulum consists of a steel rod 113 em. long
and 2°2 cm. in diameter. Knife-edges are placed 22 cm.
from the top and swing in grooves cut in flat steel rings
placed over holes in the slate slab at the corners of the
)0 cm. square. A centimetre of needle is fixed vertically to
the top of each pendulum and passes through a hole in a
light wooden rod. The rods on A and B meet at right
angles at c in the centre of the square, where they are
pierced by a sewing-needle (forming a hinge) with its point
downwards. The point of this needle draws the curves on a

* ‘Knowledge,’ Jan. 1912,

492 Dr. J. H. Vincent and Mr. C. W. Jude on
table moved* by the pendulums Cand D. ‘The upper end of

this needle passes through a hole in a strut of thin metal

Fig. 1.

(not shown in the figures) rising from the lower of the two
rods Ac and Be. The rods from C and D are jointed at the

* Curves of the character dealt with in this paper could be drawn on
a fived table by the employment of other linkages, or by utilizing the
vibrations of coupled systems. For example, let the pendulums situated
at opposite corners of the square be similar in all respects, and let each
pair be suitably coupled by light springs. The pen is at the junction of
two light rods pivoted at the tops of two adjacent pendulums, its
position of rest being at the centre of the square. The motion of any
pendulum is the sum of two simple harmonic motions whose frequencies
depend upon the free frequency of the pendulum and the strength of tho
coupling. The motion of the pen will therefore be the resultant of two
motions each approximately parallel to a diagonal of the square, each of
these motions being made up of the two simple harmonic motions proper
to the coupled system, to one pendulum of which the pen is attached.
If p? and p? be the accelerations per unit displacement along the 2-dia-
gonal, due to gravity and relative displacement respectively, while

the Duplex Harmonograph. 493

point d which lies vertically beneath the needle-point when
all the pendulums are in their positions of rest. In the
figure, pendulum D is shown with its bob displaced positively
towards the centre of the square, while the other pendulums
are in their positions of rest. The rod from D bears a light
skeleton table which carries a standard lantern cover-glass
31 in. square, upon which the records are taken. In order
to prevent the rotation of the table and cunsequent distortion
of the picture, a line dd in the frame of the table forms one
side of a parallelogram whose other sides are the rod ws, a
steel rod Ea, 11 cm. long, fixed in the pendulum shaft, and
a light wooden rod w, borne on a needle-point at a and
pivoted with the table at 0.

Fig. 2.

™ The details of the construction of the table are shown in
fig. 2. This is a perspective sketch of the table as seen
from the side BC of the slate slab. The skeleton table (t)

q and o* refer to the y-diagonal, then the curves drawn will approxi-
mate to those having equations derived from

x=a cos[pt+a]+ ccos [ “p?+2p? .t+y]
y=b cos[gt+B]+dcos [ Vg?+2o07 .t+9)

by eliminating ¢ The method here sketched is not practical, since the
damping is large even when care is taken in designing the couplings.
The method also lacks the flexibility in adjustment, which is a valuable
feature in the duplex harmonograph described in the body of the paper.

494 Dr. J. H. Vincent and Mr. C. W. Jude on

is made of tinned sheet iron. The bases of the struts are
soldered together and joined to the centre of the table by a
large needle dh. This needle passes through holes at the
ends of the wooden rods w. and w; which proceed from the
tops of pendulums D and C respectively, and it also passes
through holes in the bases of their tin struts f, and fe.
Hxcessive vertical play of these rods and struts on the needle
is prevented by the use of glass beads g, which also serve to
diminish friction.

If the pendulums at the ends of one of the sides of the
square ABCD be clamped the others will draw an ordinary
harmonogram or Lissajous’ curve. In fig. 1 let x be the
displacement towards A of the tracing-point from its position
of rest, and y that towards B; the curves drawn by the
instrument will approximate to those having equations
derived from

a=acos | pt+a]+ecos |ri+y] ) (1)
y=b cos [gt +B] +d cos [st+6] j ate

by eliminating ¢. So that this apparatus is capable of
drawing a class of curves bearing a relation to Lissajous’
figures similar to that which epicyclics bear to circles.

Intt1AL AMPLITUDE AND PHASE.

A very important part of the apparatus is an iron bar of
square section which is fixed vertically under the middle of
the slate table with its flat surfaces at right angles to the
diagonals of the square. Any pendulum can be held at rest
with its bob displaced outwards (negatively) by means of a
light wooden rod, provided with a screw which rests on the
edge of a flat surface of the central iron bar. By altering
the length of this rod by means of the screw, the amplitade
can be accurately adjusted. A pendulum thus poised can be
released at the instant when either of the pendulums adjacent
to it has accomplished any desired part of its initial half-
swing, by the impact of the end of a rod of adjustable length
carried permanently by the releasing pendulum. ‘This re-
leasing rod can be either a plain straight rod for knocking
off the propping bar, or one provided with a cross piece at
the end to pull off this amplitude-bar.

When it is desired to release two or more of the pendulums
simultaneously, electromagnets are employed. By these
and similar means the initial amplitudes and relative phases

the Duplex Harmonograph. A95

of the pendulums are under complete control. Atter having
roughly adjusted the frequencies by timing with a stop-
watch, the accurate timing is performed under the guidance
of the simple harmonograms drawn by adjacent pairs. The
accuracy of the releases and the adjustment of the amplitudes
are similarly tested. The duplex harmonogram is then drawn
and can be identically repeated at pleasure.

FRICTION.

The beauty of many of the pictures drawn by previous
harmonographs is largely owing to the decrease of the am-
plitudes by friction and to slight defects in tuning. Our
pendulum-bobs were very heavy, and the friction of the
tracing-point was in some cases almost negligible. The
smallness of the friction is largely owing to the suspension
of the framed table, supporting the glass plate, from the
roof of the laboratory bya light thread. This device, though
theoretically inelegant, inasmuch as it introduces extra
restoring forces, has in practice no detrimental effect.

DESCRIPTION OF THE DUPLEX HARMONOGRAMS.

The pictures were in the first place white line drawings
on the smoked background. The plates show the drawings
reversed, light for dark, prepared for the press by a photo-
graphic process in which the seusitive surface was facing
in the same direction as the original trace. Figures 7

to 18 are reproduced full size, the others being reduced as
seven to eight.

UnequaL FREQUENCIES IN SIMPLE RATIOS;
FRICTION SMALL.

The frictional damping is not sufficient to separate the
successive paths of the tracing-point when the quantities
P> 7,7, and s in equation (1) are as small whole numbers.
So that, unless friction is purposely introduced, the record
has to be stopped when the complete path has been traced
once. A few diagrams taken under the conditions that p, g,
yr, and s are as simple whole numbers and that friction is
small will be described first.

496 Dr. J. H. Vincent and Mr. C. W. Jude on

Index of Diagrams 1 to 18.

| No | Frequencies | Pendulums Initial conditions
"| as | at rest. ‘amplitudes all a).
11) none \
2 A |
3)| +2; 3, 4,5 B
4 | C | |
Sal) | D_ ‘|Displacements all negative.
6 |) | none |
7 Rte |
8 | C | | All starting
9 | i) together.
10 | none
11 | B 7 All negative except A. |
124 D
13 | ple Seb none
14 A All negative except B. |
15 | C |
16 | | none A and B positive, C and D negative. )
| | (|D releases C, C releases B. \
La i none ; B releases A phen each has com- | Allcateee
| (| pleted a quarter of a vibration. f from nega-
1s | ‘ate | D ee releases A and C at tive position.
J wate

Frequencies as 2, 3, 4, 5.

The frequencies are in the ratios 2, 3, 4, and 5, while the
amplitudes are equal. All the bobs are started together
from their outside position. For fig. 1, in which all four
pendulums are swung, the equations to the trace become

2=acos [2pt+m7]+acos [4pt+r],
y=a cos [3pt +7] +a cos [dpt+7].

In figs. 2, 3, 4, and 5 the pendulums A, B, C, and D
respectively are at rest, so that the corresponding pairs of
equations are derived from those of fig. 1 by deleting the
appropriate term. In all five cases the record extends from
t=0tot=a/p. The length of the record in time is the same
in all the figures up to and including fig. 16.

bom |

the Duplex Harmonograph. AY
Frequencies as 1, 2, 3, 4.
Initial displacements all negative.

Figs. 6 to 9 form a similar group of diagrams to that
discussed in the previous paragraph, the trequencies being
now proportional to the numbers 1, 2, 3, and 4, while pen-
dulams b, C, and D are fixed in turn in each of the figures
7 to 9. The curves in each case represent one half the time
required to complete a cycle of operations, the appropriate
equations being derived from those of the previous paragraph
by substituting 1, 2, 3,and 4 for 2,3, 4, and 5. If pen-
dulum A is clamped the other three draw a diagram identical
with fig. 5 turned anti-clockwise through a right angle.

Various initial conditions.

The conditions are unaltered except that A is started from
its extreme position of positive displacement.
The equations to the trace in fig. 10 are therefore

v=acos pt +acos [dpt+a],
y=acos | 2pt+7] +acos [4pt+7].

Figs. 11 and 12 are drawn with pendulums B and D fixed
in turn. The equations corresponding to these cases are
found by deleting the first and second terms respectively in
the expression for y.

In figs. 13, 14, and 15 the initial displacement of B is
positive, while the other pendulums start with their bobs
in the outer position. The amplitudes are equal and the
pendulums are released simultaneously. The equations to
the curve shown in fig. 13 are

“e=acos[pt+7]+acos [apt+7],
y=acos 2pt 4-a cos [4pt+7].

The equations to the curves 14 and 15 are found by
omitting in turn the first and second terms in the expression
fora.

In fig. 16 pendulums A and B are started from the inside
position, while C and D commence from the outside. The
equations to the trace are thus

w=acospt +acos [3pt+7],
y=acos2pt+acos [4pt+7].
To draw fig. 17 we employ the methods described in the
paragraph headed ‘ Initial Amplitude and Phase” to control
the circumstances of release. The amplitudes and periods

Plat. Mag. Ss. 6. Vols20. No. 172. April 1915. 2. K

A9§ Dr. J. H. Vincent and Mr. C. W. Jude on

of the pendulums are unchanged. A, B, and © are sup-
ported in their negative positions by props adjusted to
give equal initial amplitudes. D is released from a cor-
responding position and is provided with a wooden arm to
remove the amplitude-rod supporting C when D is passing
through its position of rest; C in like manner reieases B,
which in turn liberates A. Thus the rods of all the pen-
dulums except A carry releasing rods attached to them,
which involves the re-adjustment of the periods. If we
measure time from the instant when the pendulum A is
released, the equations to the trace are

w=acos [pt+7] +acos[3p(t+7/6p+7/4p) +7],
y=acos [2p (t+ m/4p) +2] +4.c0s [Ap (t+ m/8p+ m/6p+-m/4p) +

the record enduring in this case from t=0 to t=2zr/p.

By employing a small wooden triangle pivoted near one
corner to change the direction of motion due to the impact
of a releasing rod through 90° (after the manner of a bell-
crank lever) it was arranged in fig. 18 that D should at the
middle of its swing release the opposite pendulum B. The
pendulum B in its mid-swing releases A and C simultaneously.
If we count time from this instant the equations to the trace
become

L== COs [pt+7] + @ COS [spi+m],
y=a cos [2p(t+7/4p) +7] +acos [4p(¢ + 7/4 + 7/8p) +a].

The time occupied in the trace is, as in the preceding
picture, 27/p.

Epicycuics, &c.

If two adjacent pendulums having the same frequency
and amplitude form one pair, while the other pair are also
equal to each other in these respects, we can draw epicyclics
by releasing one pendulum ot each pair a quarter of its
period later than the other.

Friction Small.

Figures 19 to 30 include examples of well-known epicyclics.
To draw these figures, A was tuned in unison with B, and
CG with D. All four pendulums were initially supported
with their bobs in the outward position. If B liberates A
a quarter of a period after its own release, the pen is given
a uniform clockwise circular motion. If D similarly releases
C, the table describes circles in the same sense. The motion

the Duplex Harmonograph. AQ9
Index of Diagrams 19 to 35.

of the pen relatively to the table is thus compounded of two
uniform circular motions in the same direction. The trace
will be a direct epicyclic, the two motions being sometimes
spoken of as “concurrent.” If the pendulum C releases D
the trace is a retrograde epicyclic due to the combination
of “countercurrent” circular motions. The direct epicyclics
are all epi-trochoids or epi-cycloids, while the retrograde
epicyclics are hypo-trochoids or hypo-cycloids.

Frequencies in simple ratio.

In figure 19 all the amplitudes are equal. The frequency
of the pendulums C and D is three times that of pendulums
2K 2

500 Dr. J. H. Vincent and Mr. C. W. Jude on

Aand B. The circular motions are both clockwise. The
trace is the epitrochoid drawn by a point at a distance a
from the centre of a circle of radius a/3 rolling on a fixed
circle of radius 2a/3. In drawing such curves, one pair of
pendulums having been suitably released, the other pair was
started at a time most convenient to the experimenter, so
that the orientation of the diagram in such figures is not
determinate. Further, the commencement of the drawing
was also arbitrary since we put the pen down so as to prevent
the overlap interfering with an interesting portion of the
trace. If, then, we insert 6 and @¢ in the corresponding
pairs of terms, the equations to the trace may be written

x=acos | pt+é] +acos [3pt+¢],
y =acos [| p(t+7/2p)+0]+a cos | 3p(t+7/6p) +9].

In figs. 19 to 26 the duration of the trace is the period of
the slower pair of pendulums.

All else being unchanged, the circle drawn by C and D
is reversed in fig. 20. The trace is the hypotrochoid de-
scribed by a point distant a from the centre of a circle of
radius a/3 which rolls inside a fixed circle of radius 4a/3.
The equations to the trace are

x=acos [ptt+é] +acos | 3p(t+7/6p)+ 1,
y=acos | p(t+7/2p) +0] +acos [3pt+¢].
Owing to unequal damping, the curves in figs. 19 and 20
did not pass through the origin of coordinates as they would
have done had the conditions been ideal.
If, now, the amplitudes are made inversely as the fre-

quencies, fig. 19 will become fig. 21, while fig. 20 is replaced
by fig. 22. The equations to fig. 21 may be written
v=acos [pt+é@] +a/3 cos [Bpt+¢].
y=acos [p{t+7/2p)+@)] +a/3 cos [3p(t+ 7/6p)+ 4].
This is the well-known caustic by reflexion of parallel rays
from the inside of a circle of radins 4a/3. It is the epl-
eycloid due to the rolling of a circle of radius a/3 ona circle

of twice its radius.
The equations to fig. 22 are

w=acos [ptt 6] +a]3 cos [3p(t+ 7/6p)+¢].
y=acos | p(t+7/2p)+ @] +4/3 cos [Bpt+ 4].
The curve is a four-cusped hypocycloid, or astroid, traced by

a point on the circumference of a circle of radius al rolling
inside a circle whose radius is 4a/3.

the Duplex Harmonograph. AOE

Hpicyclic curves may be drawn with a definite orientation
if we provide a method of fixing the initial positions of the
radius vectors of the circular motions. Thus the astroid
could be drawn with its cusps on the axes of coordinates
by providing B with two rods so as to release C and A
simultaneously, as was done in fig. 18, C then similarly
releasing D. Counting time from the instant when Cand A
are released, the equations to the trace would be

x=acos | pt+7| + 4/3 cos [3pt+7],
y=acos[pt+7/2p) +7] +a)3 cos [3p(t—7/6p) +7].
or wis + uP = (4a]3)?”.

Since the evolute of an ellipse may be derived trom the
astroid by homogeneous strain, to draw the evolute of the
specified ellipse

FO +7 /b? = 1p
we must make the amplitudes of Aand C equal to
3(a,?—6,7)/4a, and (a,?—b,”)/4a, respectively,
and those of B and D equal to
3(a,7—b,7)/4b, and (ay?—6,’)/40, respectively,
while the method of release and the frequencies of vibration
are unchanged.

Figs. 23 “and 24 are drawn with the amplitudes equal,
the ratio of the frequencies of the pairs of pendulums being
1 to 2; fig. 24 is the direct and 24 the retrograde epicy relic.

The equations to tig. 23, in which the circles are both anti-
clockwise, are

2=acos! p(t+7/2p) +0] +a cos [2p(t-+7/4p) + 4],
y=acos [ pt+6] +acos | 2pt+¢).
This curve is the trisectrix, the epitrochoid traced by a point

distant a from the centre of a circle of radius a/2 rolling on
a circle of equal radius. The equations to fig. 24 are

v=acos [pt+6] +acos[2p(t+7/4p)+¢],
y=acos [| p(t+7/2p)+0]+acos [2pt+¢].

This hypotrochoid is described by a point distant a from the
centre of a circle of radius a/2 rolling inside another circle
of radius 3a/2.

Figs. 25 and 26 are related to figs. 23 and 24 like figs. 21L
and 22 are related to figs. 19 and 30. In fio. 25 the : ampli-
tudes of the motions of A and B are twice those of Cand dD,

502 Dr. J. H. Vincent and Mr. C. W. Jude on

the two direct circular motions having radii inversely as
their frequencies. The equations to fig. 25 are

w=acos [ p(t+7/2p)+86] +4/2 cos | 2p(t+ 7/4p) +9],
y=acos [ pt +6] +a/2 cos [2pt+¢].

The origin of coordinates was marked by the needle on the
prepared plate with the pendulums at rest. This curve is the
cardioide, the epicycloid traced by a point on the circum-
ference of a circle of radius a/2 rolling on an equal circle.
To draw fig. 26 one of the component circular motions of
fig. 25 must be reversed. The equations to the trace are

x=acos | pt+0] + a/2 cos [2p(t+7/4p) +9],
y=acos [ p(t+7/2p) +6|+a/2 cos | 2pt+—].

The curve is the tri-cusp, the three-cusped hypocycloid
drawn by a point on the circumference of a circle of radius
a/2 rolling inside a circle of radius 3a/2.

Frequencies slightly different.

In the foregoing examples of epicyclics the frequencies
have been in simple ratio. if this ratio is slightly departed
from the whole trace may be regarded as a family of curves,
each member of which approximates to that proper to the
simple ratio, while each successive member is rotated about

the origin, The only examples of this character which we
shall give have the ratio of the frequencies nearly unity.

In figs. 27 and 28 the frequencies of the first pair Aand B
are slightly less than those of Cand D. The amplitudes
are equal. ‘The anti-clockwise circular motion due to C and
D has slightly greater angular velocity than the clockwise
motion. The equations to the trace are

a2=acos [pt+] +acos|{p+h} (t+ 7/2{p+h}) +9],
y=acos [ p(t+7/2p)+@]+acos[{p+h}t+ 9],

in which h is small compared with p. The trace lasts for a

time 27/h in fig. 27 and zjh in fig. 28. The curve is the

hypotrochoid traced by a point distant a from the centre

of a circle of radius ap/{p+h} rolling inside a fixed circle of
radius a{2p+h}/{p+h}.

“This is intimately connected with the explanation of two
sets of important phenomena,—the rotation of the plane of
polarization of light, by quartz and certain fluids on the one
hand, and by transparent bodies under magnetic forces on
the other ..... It will also appear in kinetics as the path

the Duplex flarmonograph. 503

of a pendulum-bob which contains a gyroscope in rapid
rotation *.”’

Our apparatus was originally set up in order to illustrate
the vibration of a vibrating electron in a magnetic field +.
Tf we regard the radiation of energy from the vibrating
electron as a continuous process, the loss of amplitude is
roughly illustrated by the inevitable frictional decrement.

Fig. 29 results from the suppression of one of the con-
stituents of the motion. The equations to the trace are
obtained from those to fig. 27 by omitting the second
term in the expression for y, We may regard the motion
as being elliptical at any instant, the family of ellipses
having the origin as the common centre. The two straight
lines 7? =a’, and the two circles (+ yy =4a70? constitute
the complete envelope of the family. The pen remained
down during the same period as in drawing fig. 27.

Fig. 30 corresponds with fig. 27, but now the component
circular motions are in the same direction. In this instance,
the pen was placed on the table when the pendulums D atl
B were released. The simultaneous start of C and A thus
occurs when the pen isat (—2a, 0). The trace is at first
distorted by the vibrations of the pendulum-rods thus set
up. The tuning of the pendulums was that employed in
mo 21... Lhe equations to the trace drawn by the four pen-
dulums may in this case be written in the more definite form

«w=acos[pt+7| +acos[{p+thit+7],
y=acos | p(t+7/2p)+m7]|+acos[{ptht(t+7/2{p+hi)4+r],

where ¢ is counted from the release of C and A, the record
lasting shehtly longer than 7/h. The innermost loop of the
eurve is distorted by the backlash due to the play of the
jointed parts. The trace isa portion of an epitrochoid traced
by a point distant a from the centre of a cirele of radius
ap/{p+h} rolling outside a circle of radius ah/{p+ht}.

thiow3 lis similar to fio. 27, but now the amplitudes are
inversely as the frequencies. The equations to the trace are

“=a{p+ht/pcos[ pt+@] +acos[{p+h}(t+7/2{p+h})+ 9],
y=alpt+h}/pceos | pt+7/2p)+0]+acos| {pthtit+].
Tn this case the duration of drawing ts 27/h. The curve

is the hypocycloid due to the rolling of a circle of radius a
inside a circle of radius {2p+hta/p. The small blank area
at the centre of the figure should have been sensibly circular
and of radius ha/p.

* Thomson and Tait, ‘ Natural Philosophy,’ Part I.
+ Wood, ‘ Physical Optics,’ 1911, p. 506.

504 Dr. J. H. Vincent and Mr. C. W.-Jude on

Frequencies in Simple Ratio but Unequally Damped.

The design of the table allows of an adjustment in the
friction which enables one to damp unequally the motion of
the table and the tracing-point. For instance, let the tracing-
point and the attaching rods be kept as light as possible,
while the table is released from its suspending thread, loaded,
and supported on an oiled glass plate by the lower end h
(fig. 2) of its axial needle. The damping of the motion of
the table can be made as large as is wished with respect to
the damping of the tracing-point by adjusting the load.

If one pair of pendulums 1s clamped, say those bearing the
tracing-point, and the other two set to draw a circle, when
the tuning is accurately adjusted, and the friction small as.
before, the needle-point does not draw a recognizable spiral,

but ee the smoke away from an anoles surface. If,
however, the motion of the table be sensibly damped we get
fig. 32.

The chief feature of this curve is the regularity of the
spacing of the successive branches. The original was
measured in a random direction through the centre on a
travelling microscope, and it was found that the distance
between the branches was, as nearly as could be measured,
"100 em. No change in the distance could be detected until
one approached within a single turn of the point of rest.
The friction was in this case almost entirely due to that of
the axial needle in contact with the oiled glass plate. This
suggests that the trace is the orbit of a point moving with
an acceleration towards a fixed centre varying directly
as the distance, under a constant frictional resistance. A
solution to this dynamical problem is that the orbit may be
the involute of a circle of radius /, the acceleration towards
the centre being & times the distance from the centre, and
the retardation being 2/k. On this view, the circle of which
the trace is the involute has a radius of ‘016 cm., so that if
we regard the friction as being strictly independent of the
velocity, the tracing-point will come to rest ‘016 em. away
from the frictionless position of equilibrium. The motion
departs from one of uniform angular velocity about the
centre of force by a term involving merely the square of the ~
ratio of / to r, so that the tracing-point moves with sensibly
uniform angular velocity until it closely approaches the end
of its path.

Fig. 83. If two direct circular motions are compounded,
whose frequencies are in the ratio 1:2, the curve drawn is the
limacon of Pascal, cardioide, or trisectrix, according as their
amplitudes are in the ratios 3:1, 2:1, or 1:1 respectively,

- the Duplex Harmonograph. 505

the circle of higher frequency having the smaller amplitude.
If pendulums C and D are set to draw the involute as in
fig. 32, while A and B draw a circle in the same direction
whose radius is as nearly unaffected by friction as is practi-
cable, we may by arranging that the initial radii are as 3:1
draw a curve approximating very closely to the limacon of
Pascal. As the ratio of the radii sinks in value owing to the
damping of the motions of pendulums C and D, a stage 1s
reached when the shrinking radius is twice that of the circle,
and the cardioide is described. Passing through this stage,
the decrease in the radius of the slower motion progresses
until the radii become equal, which transforms the trace into
a trisectrix. The needle was then removed from the plate,
and when all had been reduced to rest a mark was made on
the prepared surface by the tracing point. This black speck
should be visible in the diagram on the trisectrix at the
inner vertex of its loop, but in our picture it is very slightly
displaced within the loop. The apses lie on a straight line
through the centre. This confirms the view that the motion
of the damped pendulums is isochronous through the range
covered in the picture.

The arrangements for fig. 34 are as for fig. 33, except that
D is fixed, so that the trace is drawn by a point with a uni-
form circular motion of slowly diminishing amplitude to-
gether with a simple harmonic motion which is rapidly
damped. The frequency of the motion in the circle is twice
that of the single vibration. The initial amplitude of the
single vibration is three times the radius of the circle. The
style was removed when the slower vibration had almost died
away, leaving the circular motion alone operative.

In fig. 35, we have the involute of the circle as in fig. 32
combined with a simple harmonic motion due to the vibration
of pendulum A, B being clamped. The needle was removed
when the damped pendulums © and D, drawing the involute,
had come to rest.

THREE FREQUENCIES : Two OpprosItE PENDULUMS
BEATING.

The Duplex Harmonograph lends itself readily to the
description of figures in which tbe combined rectilinear
motions are themselves subject to periodic change in
amplitude after the manner of beats in acoustics. Of this
kind of diagram we shall give a few examples divided into
four classes. In the first three of these classes we shall deal

only with three frequencies. In the first, the mean frequency

506 Dr. J. H. Vincent and Mr. C. W. Jude on

of the beating pendulums is equal to the frequency of the
motion at right angles. In the second, the mean frequency
is twice that of the third frequency, while in the third class
the mean frequency is half that of the remaining frequency.
The fourth class will contain examples involving four
frequencies such that simple relations hoid between the
means of the frequencies of oppositely-placed pendulums.

Index of Dice 36 to 46.

Frequencies | yr.34 Fats sail
N of beating | | ves iy Other Pendulums oe OL ie
0. of beating ona Initial Conditions.
| pendulums pendulums. | frequency. recording.
as |
| |
| 36 |) | | D releases C. x
eayibu te oo J B releases A.
38 r and p | p S |
; | A
39 ' p—h | = |
Ca S | | Displacements
me | Te | | sy /{ All starting | all negative
| 4] | and | 2p+2h pth 2 r together. \ except
‘ . 7]
42 J 2p+dsh | 20 | | pendulum C
is hs | : ) in No. 42.
a) r and “pth He icy = | C releases D.
ll a i a \ releases B
aes p | | a |. releases B.

The other Frequency their Mean.

In fig. 36 pendulums B and D are tuned in unison, while
A and C have frequencies respectively greater and smaller
by equal amounts. Band D release A and C as they pass
through their mean positions. All four amplitudes are
initially equal. In this case we have allowed the pen to be
in contact with the prepared surface from the commence-
ment of the motion of the pendulums Band D. The tracing-
needle begins by drawing a circle of radius 2a and is removed
when it has traced the approximate straight line along y.
If friction had been completely eliminated the length of this
line would have been 4a. Counting time from the simul-
taneous release of A and C, the equations to the trace are

e=acos[{p+h}t+7] +acos|{p—h}t+7r],
y=acos| p(t+7/2p)+7] +acos|p(t+7/2p)+7],
the record extending to =7/2h. The displacement along

the Duplex Harmonograph. DOT

being 2acosht cos [pt+7], the instantaneous value of the
amplitude may be regarded as 2acosft, whilst the oscilla-
tions are always in quadrature with the motion along y.
The trace is the projection of the motion of a point in a
circle of radius 2a with uniform angular velocity p, the
eircle rotating about a diameter with uniform angular
velocity h. Similar methods of representation are applicable
in other cases. Frictional decrement can likewise be repre-
sented on this view.

Fig. 37 was produced under identical conditions with the
exception that the motion of B was suppressed. It may be
regarded as the projection of fig. 36 on a plane at 60° to it
through the axis of w.

Fig. 38 is produced by altering the conditions of release
from those of fig. 36. The bobs of all the pendulums start
from their outward positions simultaneously. The equations
to the trace are

ew=acos|{pth\t+m]+acos[{p—h}t+7],
y=acos | pé+7] +acos[pt+7],

so that the displacements along « and y pass through their
zero values simultaneously. The trace fe from t= mh to
t=27/h. In this time the amplitude of the approximately
rectilinear simpie harmonic motion changes from its maximum

value 24/2a, passes through its minimum 2a, and again
attains its maximum. During this interval the icertion of

the resultant motion changes through a right angle, and
when t=27/h the motion is again at 45° to the axes. The
amplitude of the motion along the axis of w now begins to
decrease, and the direction of the resultant rectilinear vibra-
tion rotates in the opposite direction. This is illustrated in
fie. 39, in which the oscillation of the direction of the re-
sultant vibration can be readily traced owing to the loss of
amplitude by friction as the oscillation takes place. The
equations to the trace being as before, the record in fig. 39
lasts for the time 27/h from t=7/2h. If friction had been
inoperative both diagrams would have been bounded by
— 2G.
The other Frequency half their Mean.

The pendulums are now tuned so that the third frequency
is half that of the mean frequency of the pair of oppositely-
placed pendulums which produce beats together.

In fig. 40 the frequencies are in the ratios 2p+h, p+h,
2p+3h,p+h. All the amplitudes are equal, and the pen-
dulums are released together from their extreme outer

508 Dr. J. H. Vincent and Mr. C. W. Jude on

positions. The needle is allowed to trace from the com-
mencement of motion during half the cycle of operations, so
that the equations to the trace are

e=acos | {2p+h}t+m]+acos | {2p4+3hit+7],
y=acos [{p+th}t+r] +acos[{pt+hti+7],

from t=0 to t=a7/h. The trace commences with the
parabola y?+aa—2a?=0, and if we neglect the effects of
friction all the succeeding parabolas may be derived from
this by altering the scale of drawing and the sign of the
v-coordinate. ‘The points where all the parabolas cross the
axis of y are (0, ta\/2), coinciding with the cusps of the
envelope of fig. 54. Since the vibration along the axis of y
has a sensibly constant amplitude, the figures will be bounded
by the lines y= +2a and the parabolas 7? tar—2a?=0.

In the introduction we state that the figures may be re-
produced with fidelity. This extends even to cases in which
tbe iength of record varies, as the frictional loss is almost
the same whether the needle is in contact with the smoked
glass or not. This is illustrated in fig. 41, in which the
second quarter of the complete cycle of operations is
shown.

In fig. 42 we are again only concerned with three different
frequencies, these being identical with those of ngs. 40
and 41. Pendulum C was in this case released with its bob at
the extreme inside position, all the other initial displacements
being negative. On release the motion consists of a sensibly
rectilinear simple harmonic motion along the axis of y. This
changes into motion in Lissajous’ figure of eight. The style
was placed down when the amplitude along the axis of x had
first attained its maximum. ‘The equations to the trace are

v=acos [{2p+h}t+7]|+a cos {2p4+d3htt,
y=acos [{pthtt+ma] t+acos[{p+hit+r],

from ¢=97/2h to t=7/h.

The other Frequency twice their Mean.

The remaining examples we shall give involving three
frequencies have the third frequency twice the mean of those
of the oppositely-placed beating pendulums.

In fig. 43 the frequencies are in the ratios 2p+h, pth,
2n+h, p. The amplitudes are equal, and the pendulums
start together with their bobs displaced outwards. The pen

the Duplex Harmonograph. 509

traces during a quarter of a complete cycle. The equations
to the trace are

vr=acos [{2p+h}t+m]+acos [{2p+h}i+7],
y=acos [{p+th}t+7]| +acos [pt+r7],

from t=2r7r/h to t=37/h. On release, the inotion is along
the parabola with ihe maximum parameter. The component
of this motion along the axis of y dies down to zero and then
increases to a maximum, when the original parabola is again
traced. At this stage, the pen was permitted to record and
the trace was stopped when the motion had again become
sensibly rectilinear.

We have had an example in fig. 42 in which motion in a
Lissajous’ figure of eight degenerates into rectilinear motion
along one of its axes of symmetry. In fig. 44, the motion
becomes rectilinear along the other axis of symmetry. The
initial amplitudes of the four pendulums are equal while the
frequencies again fulfil the conditions of fig. 43, but h is
much smaller compared with p, so that the effect of the
frictional decrement in the amplitude of the motion along «
is very marked. A and © release B and D in mid-swing,
the bobs all starting from their outer positions. Counting
time from the instant when B and D commence to sw ing, the
equations to the trace are

=a cos [{2p+h}t+ 37/2 | +a cos [2p +h}t+37/2],
y=acos [{p+h}t+7] +a cos [pt+7],

in which the frictional decrement is neglected. The record
lasts until t=7/h. Instead of the floure being bounded by
the two straight lines «= + 2a its centr ral portions are sensibly
contracted, so that when the motion is reduced to oscillation
along w its amplitude is no longer 2a.

eg the trace had been gonial this straight line would
have opened out again into the fioure of eieht. This is
shown in fig. 45, in Salinels the trace commences when t=7/h.
A corresponding frictional effect is again evident. The
record again lasts for the time 7/h. ain fig. 46 a similar
cycle of events is recorded, the pendulums having been
slightly readjusted. In this case both the waning and the
waxing in the y-direction are recorded, the time occupied
being 27/h.

510

Dr. J. H. Vincent and Mr. C. W. Jude on

Four FREQUENCIES; PAIRS OF OPPOSITE PENDULUMS

BEATING.

We will now give a few examples from what we have
called the fourth class of those figures in which oppositely-

placed pendulums are beating.

In this class four frequencies

are involved, simple relations subsisting between the means
of the frequencies of opposite pendulums.

Index of Diagrams 47 to 54.

| l
| Freq. of | Mean read Diff. of

| Initial Conditions.

A & B negative
iC & D positive
A & B negative
O & D positive
A & B negative
C & D positive

Ail negative
but C.
All negative.

All negative.

H
i

All

starting

together.

|
)

All negative but D. |

| Freq. of | Mean freq. Diff. of
A&C. of AEC. AKC. BED. of B& D.| B&D.
| ee ee eee res | a ee eR ey nee 8 2 | eee
Qn Fe al lal negative.
and | 2p+h 2h | and 2p+h ) ee
2p+2h | | pth 2 |
i | |
pth pt2h
and p 2h and p 4h
prhk | p—2h l
2pth | pt+2h {
and 2p+2h Qh and pth. | oh
2p+3h | | Pp |

B releases C, D releases A.

a. Mean frequencies and differences of frequencies of
opposite pendulums as two to one.

To satisfy these conditions the ratio of the frequencies will
be 2p, p, 2p+2h,p+h. These frequencies are employed in

figs. 47 and 48.

equal.

bobs in the extreme outward position.

The initial amplitudes in both cases were

In fig. 47 the pendulums were started with their

The tracing-point

was put down on the plate when the bobs of the pendulums
Band D had first reached their inward positions, so that
counting time from this instant the equations to the trace

are

a=a cos | 2pt+7]+acos{2p+ 2hit+r],

y =a Cos pt

+ucos{p+htt,

the trace lasting until t=7/h. In this figure h is again so
small with respect to p that the effects of friction are notice-
Disregarding these effects, however, the excursions
in the approximate parabolic path are limited by the initial

able.

the Duplex Harmonograph. ou

parabola y?+av—2a?=0. At the beginning of the trace the
component rectilinear excursions along the axes diminish,
but the motion in the «x-direction dies down more rapidly,
and when it is zero its amplitude changes sign so that the
subsequently growing parabolas turn their concavities in
the opposite direction. The trace continues until the para-
bolic motion has dwindled to an oscillation along the axis
Ob.

Instead of releasing the pendulums so as to draw parabolas,
in fig. 48 they are started so that the suecessive curves are
all examples of Lissajous’ figure of eight. The pendulums
are released simultaneously, A and B from their outward
positions and C and D from their inward positions. As the
quivering of the pendulum rods again proved troublesome,
the style was placed on the prepared surface when its motion
was approximately rectilinear along the axis of y. The
equations to the curve are

w=acos | 2pt+m]+acos {2p+ 2h}t,
y=acos[pt+m]| +acosip+h}t,

the trace lasting from t=a/h to t=2/h. . While the motion
along the axis of y decreases in amplitude, the motion along
the axis of # increases. This motion attains a maximum,
and then both begin to die away together and reach zero
value simultaneously, when the needle was removed from
the plate. By the time that the combined vibration along «
had reached its maximum amplitude, frictional damping had
notably affected its amount. Otherwise the whole figure
would have been enclosed in the space bounded by the four
parabolas
(ya) +ax/2—a?=0,

which also constitute the complete envelope.

B. Mean frequencies equal and differences as in «.

In this case the frequencies will be p+h, p+2h, p—h,
p—2h. In fig. 49, using these frequencies, with initial
amplitudes equal, pendulums A and B are released from their
extreme negative position, while © and D start with them
from the corresponding positive position. The equations to
the consequent trace are

w=acos[{pth}i+7] +acos{p—Ahtt,
y=acos [{p+2h}i+m7]+acos{p—2htt,

the record extending from t=7/h to t=2m/h. The motion

ah? Dr. J. H. Vincent and Mr. C. W. Jude on

at any instant may be regarded as approximately a simple
harmonic motion in a straight line. The direction of vibra-
tion in this line oscillates through an angle 2tan712; when
t=/h the resultant amplitude is zero and its direction makes
an angle tan~* (—2) with the positive direction of the axis
of 2. This angle increases until, when the motion along y
is zero (4. e. when t=37/2h) the maximum amplitude in the
direction of w equals 2a. In its growth from zero the
amplitude passes through a maximum value of 5a/2 when its
direction of motion ne les an angle tan~* (— J 8/2) with the
axis of x. If we neglect the decrement in the oscillations
due to friction the excursions of the tracing-peint are limited
by the Lissajous’ figure of eight

e=Za sin ht,
y= 2a sin 2ht,
or

v—Ae2?—40y?=0.

y. Mean frequencies as two to one and differences equal.

Yhe remaining examples of figures drawn by four pen-
dulums in which the opposite pendulums are beating, all
obey the conditions that the mean frequencies of the opposite
pairs are as two to one and that the differences are equal, s
that the frequencies may be taken proportional to 2p-+h
pt+2h, 2p+3h,p. Using three frequencies only, we ha
shown how families of Lissajous’ figures of eight may be
drawn in which succeeding members increase or decrease
in one dimension, the curves in figs 44 and 42 degenerating
into two straight lines respectively at right angles. “In fig. 50
both these effects go on simultaneously and at the same rate.
To draw the figure the pendulums were started together
from positions of equal displacement, A and B outward and
Cand D inward. The needle was put down when the com-
bined amplitudes were at a maximum, the record extending

from t=77/2h to t=7/h, the equations to the trace being
x=acos [{2p+h}t+7]+acos {2p+3htt,
y=acos|{p+2h}t+7]+acos pt.

The complete trace would contain two families of similar
and similarly -placed curves, each family consisting of two
series in which the curves are increasing and decreasing
respectively in amplitude, the curves in each family being
described in contrary directions.

the Duplex Harmonograph. 513

In fig. 50 the combined oscillations of the oppositely-
placed pendulums are always in the same phase, while in
fig. 51 they are always in quadrature. The bob of pendulum
C was started from its inward position when the others were
released from their outward positions. A and C are thus
in opposition, while B and D conspire. The equations to
the trace are

v=acos [{2p+h}t+m]+acos {2p+ d3hte,
y=acos | {v+2h}t+7]+acos | pt+7],

from t=7/2h to t=a/h. In this case the successive curves
are still Lissajous’ figures of eight. The process of passing
from a straight line to another at right angles by transfor-
mation of the shape of the curve occupies the time 7/2h.
All the curves lie within the area enclosed by the parabolas

y? +2ax—4a? =0,

which constitute the envelope.

Returning again to the condition that the resultant vibra-
tions along the two axes shall be in the same phase, in
fio. 52 we set these resultant motions to draw a parabola
instead of a Lissajous’ figure of eight, as in fig. 50. To do
this the initial amplitudes of the two combined motions are
maxima, the equations to the trace being

w==acos |{2p+h}t+7]+acos[{2p+d3h}t+7],
y=acos [{p+2h}i+r]+acos [pt+7],

from t=7/2h to t=m/h. The style was removed when the
maximum parabola was being drawn. If the needle had
been allowed to continue tracing, its parabolic path would
have gradually decreased in size until the point had reached.
the origin. ‘The subsequent trace would have consisted of a
series of growing parabolas drawn with the vertex in the
opposite direction, the whole cycle of operations involving
the drawing of the two families of oppositely-placed para-
bolas, each consisting of an increasing and a decreasing
series.

This is illustrated in fig. 53, in which the conditions are
the same except that the tracing commences with the release
of the pendulums. The path of the pen is at first distorted
by the elastic vibrations of the pendulum rods. The de-
creasing half of the first family of parabolas and the in-
creasing half of the second family are shown on the trace,
the time occupied being w/h, half the period of the complete

Phil. Mag... 6, Vol. 29) No. 172. Apnt 1915. 2 L

See Prof. J. A. Pollock on the

cycle of operations. In figs. 52 and 53, if the timing had
been accurate and friction negligible, the parabolas would
have been bounded by straight lines at 45 degrees to the
axis of coordinates.

Still keeping the amplitudes equal and the frequencies as
before, the remaining initial conditions are now changed.
In fig. 54 A and C are released from their extreme outward
positions by means of rods carried on the pendulums B and D.
The rod on B strikes the amplitude-prop of C while a hooked
rod on D releases A. A and C are released simultaneously,
and in this case the style was in contact with the prepared
surface from the commencement. The equations to the trace
are

ex=acos[{2p+h}t+m] +acos[{2p+d3h}hi+7],
y=acos [{p+2h\t+37/2]+acos | pt+7/2].

Again neglecting frictional decrement we see that the com-
plete trace would be bounded by the circle #?+y?=4a?.
The pen was removed when t=7/2h, when the path was
approximately rectilinear along the axis of y. If the trace
bad been continued, parabolas would have been drawn with
their concavities in the opposite direction, the completed
figure being symmetrical about both axes. The parabolas
touch the curve

Aata'+ x (y*—40 a°y? — 32 at) —8(y? 207) =0,

the cusps of which are on the y axis at distances ta/2
from the origin, the apex of the figure that we have traced
coinciding with one of the two double points of the envelope.
The portions of the curve between these double points con-
stitute the effective envelope.

LIV. The Nature of the Large Ions in the Air. By J. A.
Potiock, D.Se., Professor of Physics in the University of
Sydney”.

N 1905 Langevint discovered that in addition to the
small gas ions, with a mobility of about 1:5, there are
in the air much larger ones which have a mobility of only
1/3000. Although our knowledge of the properties of these
large ions is very slight, yet from the few facts which are
known some deduction may be made as to the nature of the
ionic structure.
* Communicated by the Author.
+ Langevin, Comptes Rendus, cxl. p. 282 (1905).

Nature of the Large. Ions in the Arr. D15

The main facts are as follows :—

1. Ions with a mobility of 1/3000 under usual atmo-
spheric conditions form a well defined class; this was
Langevin’s original discovery. McClelland and Kennedy *
in their investigation found no evidence of any other
ions. In my own measurements there is no indication of
another type of ions with a mobility approximating to that
of the Langevin 1 ion, though there is a class of ions with a
mobility inter mediate between those of the large and small
i0ns.

2. Langevin and Moulin f mention that the simultaneous
variations in the numbers of the large and small ions in the
air are opposed in direction, a fact which is also shown in
the measures made on the few occasions when I have taken
continuous observations of both classes of ions at Sydney.
Langevin and Moulin further state that the number of the
large ions is the greater the more numerous the (dust) par-
ticles in the air, and they consider the large ions as created
by the Be achinant of small ions to these neutral particles.

From the results of C. T. R. Wilson’s f investigations on
the formation of clouds in closed vessels, it may be inferred
that these large ions do not exist in air recently freed from
dust, and that they are not developed in intervals of time
extending to days in dust-free air has been shown by workers
in this laboratory.

3. The mobility depends on the humidity§. The first
suggestion of the probability of a connexion between these
ions and the moisture in the air was made by Sir Ernest
Rutherford in his book on Radioactive Transformations.

These facts lead one to picture the Langevin ion as a col-
jection of water molecules surrounding a dust particle, the
whole being electrified by the attachment of a small ion.
Judging Pecan the mobility measurements, the size of the ion
at constant temperature depends on the vapour-pressure. If
any change of vapour-pressure occurs, the radius changes
until equilibrium is again established, and there is still
equilibrium when the vapour is saturated |, for cloud con-
densation experiments with unfiltered air show that the ion
does not grow to a visible drop until there is some slight
supersatur Aisa.

* McClelland and Kennedy, Proc. R. I. Acad. xxx. A. p. 71 (1912).

+ Langevin and Moulin, Ze Radium, iv. p. 218 (1907).

orn Wilson, Phil. Trans. A. elxxxix. p. 265 (1897).
Epotibele Journ. and Proc, Roy. Soc. N.S. Wales, xliii. p. 198 (1909).

; The term saturation will be used throughout to denote the condition
of the vapour when in equilibrium with a plane water surface having a
thickness great compared with the range of molecular force.

2L2

516 Prof. J. A. Pollock on the

The large ion thus affords an interesting example of the
adsorption of water vapour at a rigid surface. In connexion
with such an idea it is interesting to recall a statement of
Lord Rayleigh. Referring to the rise of a liquid in a
eapillary tube Lord Rayieigh says *: ‘ Above that point (the
meniscus) the walls of the tube are coated with a layer of
fluid, of gradually diminishing thickness, less than the range
of forces, and extending to an immense height. At every
point the layer of fluid must be in equilibrium with the
vapour to be found at the same level. The data scarcely
exist for anything like a precise estimate of the effect to be
expected, but the argument suffices to show that a solid body
brought into contact with vapour at a density which may be
much below the so-called point of saturation will cover itself
with a layer of fluid, and that this layer may be retained in
some degree even in what passes for a good vacuum. The
fluid composing the layer, though denser than the surrounding
atmosphere of vapour, cannot properly be described as either
liquid or gaseous.”

In the large ion, according to the foregoing suggestion,
we have similar conditions, modified perhaps by the electri-
fication, the equilibrium vapour-pressure depending on the
thickness of the adsorbed fiuid surrounding the rigid core.

To obtain some idea of the nature of the relation between
mobility and vapour-pressure which is to be expected in
connexion with such an ion, consider unit mass of a mixture
of ions and water vapour as the working substance in a
Carnot’s engine. A cycle may be performed involving only
reversible processes, so we have, for the mere change of state,
the well known relation,

a =—(5
dp/g —- \d@/,’

where @=the entropy,
p=the vapour-pressure,
=the absolute temperature,
>= the volume of the working substance.

Let o and o'=the density of water vapour at the saturated
pressure P, and at the pressure p, respec-
tively.

p, p' and X, A'=the densities and latent heats of vaporiza-
tion of water, and of the adsorbed fluid,
respectively.

+: ae Phil. Mag. xxxill. p. 220 (1892); Scientific Papers, ili.

p. 020.

Nature of the Large Ions in the Air. All

The relation may be written

a (aple= le — Naa)

Gi Gioy CaN G: ol .NGO Jc

where m refers to the mass of the vapour.
Changing the variable this becomes

nN LN (ap)
6 =(5-5)(35), e ° e ° ° (1)

If we now make; the assumption that the density of the
adsorbed fluid is considerable compared with that of the sur-
rounding vapour, so that 1/p' may be neglected in comparison
with 1/o', then taking p=o'R@, the equation becomes

dd ag

For water, making a similar assumption that 1/p may be
neglected in.comparison with 1/o, we have

r 1 dP
am, Pe.
| “(%)
so ho PN
thy ay
P\dée,
Putting \//A=1/n we may write
apy dP
pide nk dé”

If at two temperatures, 0, and @., corresponding values
are p, and P, p, and Ps, then integrating we have

(#2) ih n Py
Prism oe P,

That is, the mass of adsorbed fluid, and consequently the
mobility, will be the same at a temperature @) as at a
temperature 0, if the new vapour-pressure is

2 Ps
P2 aes P, e

This equation, according to the assumptions which have
been made, is the formula for reducing the observed mobi-
lities to a common temperature, and thus affords a basis for
a discussion of the observations.

518 Prof. J. A. Pollock on the

Table I.* contains the observed values of the reciprocals
of the mobilities, reduced to standard atmospheric pressure,
together with the temperature and vapour-pressure at the
time the observations were made.

TABLE I.
7 |
| i
1/u. be dogan | pte Door
| Cent. mm. mm. mm. |
Grouped results. |
fone Meo et) 1562. | O60 OG7 oo
Tst2oue ye tS:S | NGSTED i ipsa Ses: | 5:87 |
BOTT 8 He AS sui eet elles | 1490 |
3155 OS whee 2111 ol, 19 SGian, aveeleaas |
Beebe eee, |) 18420 | bea | 16:05 :
Single observations.
DS) ess} 12-95 6-60 8-85
2570 | 20-9 | 18-04 10-64. 10°24
Bey ea 2010. XY (417-22 11-02 Wl
2725 | O3-n) eel 92094 | 14-24 11-80
| ray

Values of the reciprocals of the mobilities of the large ions reduced
to standard atmospheric pressure, and the temperature and water-vapour
pressure at the time the observations were made, together with the
vapour pressure reduced to 20° C.

The single observations are those depending on one day’s
measures only, natural air being used. For these determi-
nations the humidity was calculated from the wet and dry
bulb readings; whereas in the case of the grouped measures
where the moisture in the air was artificially controlled, the
humidity was found by absorbing the water vapour and
weighing. The estimations of the humidity by the wet and
dry bulb readings now appear to be low, though from a com-
parison of the two methods of determining the humidity,
made at an early stage of the investigation, no constant
difference was apparent.

When the humidity changes, the ions do not at once reach
a stage of equilibrium with the new vapour-pressure con-
ditions, and in the determinations, when a change of humidity
occurred, 13 minutes were allowed to elapse before the
measurements for mobility were made. In reviewing the
observations, I think it is pessible that this interval may not
have been long enough in the case represented by the first

* Wor details see Journ. and Proc. Roy. Soc. N. S. Wales, xliii. pp. 61
and 198 (1909). A single observation, given in one of the tables pre-

viously published, has been omitted, as on looking. up the original
records the result is found to be unreliable.

Nature of the Large Ions in the Air. 219

entry in the table, so the estimated value, 1263, of the
reciprocal of the mobility for a humidity of 4 per cent. may
be toa large.

Asa determination of mobility requires the measurement
of a series of ionization currents, the investigation is a tedious
one in the case of the natural large ions where the ionization
is not under control and is subject to considerable variations.
The values given in Table I. represent the results of observa-
tions on the few occasions, during two and a half years or
more, when the ionization was sufficiently constant in this
laboratory for the purpose of the calculation. Under the con-
ditions a very exact comparison of the theoretical deduction
with observation is not possible.

It can be said at once, however, that the observations
show that X/A’, or m in the reduction formula, must be very
nearly equal to unity, and on the assumption that it is so,
the pressures corresponding to the mobilities in Table I.
have been reduced to a common temperature of 20° C., and

are entered in the table.
Fig. 1.

EEE NY =
caw
en

~
Q

ur Fress ureé—-Mmms
+
+
+

VGDO0
Q
i
|

O JOGO ION 2000 3000 4000
Feeciprocal of Mobility 20°C.
From these reduced observations fig. 1 has been drawn;
the curve therefore shows the relation between the reciprocal

520 Prof. J. A. Pollock on the

of the mobility, reduced to standard atmospheric pressure,
and the water-vapour pressure for a temperature of 20° C.
The single observations have been included, but they are
not to be considered as of equal weight with the grouped
measures.

A second approximation to the value of n is now possible
for the part of the curve in fig. 1 corresponding to a pressure
of about 15 millimetres. It appears here that the fit of the
points to the line is perceptibly better when n is taken as 1
than when it is taken as 1+°01, and noticeably better than
when it is put equal to 1+°03.

For the large ions, then, to a considerable degree of
accuracy the mobility remains the same as the temperature
varies, if the equilibrium vapour-pressure is a constant
fraction of the saturated vapour-pressure for a plane surface;
or, in other words, the mobility at standard atmospheric
pressure is a function of the relative humidity only.

This relation may be written

4 dleGile
oe Pada

and in this form is a to the relation found by Trouton*
between the equilibrium vapour-pressure and the mass of
contained moisture in the case of flannel and cotton-wool,
and independently discovered by Masson and Richards ft
during a very accurate investigation with this latter material.

The determination that the ratio of X to 2X is approxi-
mately unity in the case of the large ions enables a deduction
to be made as to the condition of the adsorbed moisture, for
with its latent heat nearly equal to that of ordinary water it
seems impossible to consider the adsorbed fluid other than in
the liquid state.

Trouton t has shown that there are two possible modes of
condensation of water vapour on rigid surfaces. If special
precautions are taken in drying the surfaces, on exposure to
water vapour, a gaseous form of condensation occurs, which
changes somewhat abruptly to the liquid form at a vapour-
pressure depending on the nature of the surface. In the
case of the gaseous form of condensation, one would imagine
that the ratio of the latent heat of water to the latent heat
of vaporization of the dense vapour, or n in the formula of
reduction, would be much greater than unity. Insofar as

* Trouton, Proc. Roy. Soc. A. Ixxvii. p. 292 (1906).

+ Masson and Richards, Proc. Roy. Soc. A. Ixxvii. p. 412 (1907).

t Trouton, Proc. Roy. Soc. A. lxxix. p. 388 (1907); Chem. News,
‘96. p. 92 (1907).

Nature of the Large Lons in the Aur. 521

the density of the vapour is small in comparison with the
density of the adsorbed moisture, the value of n found
necessary for the reduction of adsorption observations to a
common temperature might, therefore, possibly form a
criterion in determining the condition of the absorbed fluid.

This question of the condition of the adsorbed moisture is
even capable of somewhat more precise consideration, for in
the case of the large ions, and of uncharged drops of a similar
nature, which as shown by cloud condensation experiments
do not become unstable until the vapour becomes, at least,
slightly supersaturated, equation (1) becomes

ein ee a ale

7) iam (~ Vpydg
and no doubt the equation also holds for fluid adsorbed at a
plane surface.

Putting p'=p+6p with the sign of do undetermined, the
expression may be written

r ) —o dP )
(+B He)

Pp’ po dé
op
or Mae pa"
rn oe op
p

At atmospheric temperatures, as o is small compared with
p, we see, without requiring a knowledge of the density,
that when the vapour-pressure is that of saturation the
adsorbed fluid has a latent heat differing very little from
that of water. From the point of view of Laplace’s theory
it is difficult to see how this conclusion could fail to carry
ae the inference that the difference in density is also
small.

If it is experimentally found that (1/p)(dp/d@)m is equal to
(1/P)(dP/d@) for all ai. of p, a eae ‘

oO
Baa
r (ee

for all values of the vapour-pressure.
If the adsorbed fluid is in the liquid state, little change,
I believe, occurs in its density as the vapour-pressure alters,

Hae Prof. J. A. Pollock on the

so for low pressures o’/p’ may be neglected in comparisor

with o/p, and as a first approximation we have here

wr S=r (1 -- <),
p
At all pressures it would appear, then, that if the adsorbed
fluid is in the liquid state, it has a latent heat and a density
very little different from those of water.

It must be mentioned that the preceding thermodynamic
argument was first applied in this connexion by Trouton in
the earlier of his papers quoted ; I use it here, however, in
the converse form to that in which it was emplo syed by him.

The large ion, then, may be considered as a rigid core
surrounded by a film of water rather than by a dense atmo-
sphere of water molecules, though of course the transition
layer from water to vapour must be an important feature.

An estimate of the diameter of the ion may be made, in
the light of a knowledge of the mobility, by considering the
resistance which the moiecules of the surrounding gas offer
to the motion of the ion in an electric field; but, on account
of the assumptions which are involved in any such application
of the principles of the kinetic theory as outline below, the
result of the calculation must be taken as giving merely the
order of magnitude.

According to the work of Langevin®™ in this connexion,
amended by H. A. Wilsont, the velocity of an ion im a field
of unit intensity is given by the expression

u= 14 el,/myry,

where e is the ionic charge, /, the mean free path of the ion,
m, its mass, and vy, its mean velocity of thermal agitation.
The size of an ion, however, cannot be calculated merely
from a knowledge of the mobility, and we are forced to
_ follow the converse method of assuming size and mass and

calculating mobility. To obtain a suitable expression for
this purpose, let $1 be the diaineter of the ion, s, that of the
air molecule, and in general let the subser ipts 1 1 and 2 refer
to ionic and molecular quantities respectively.

From the kinetic theory we have ft

cam zai as Sq 14m , 2M
1 "Ts y)
UD) nae

* Langevin, Ann. de Chim. et de Phys. vii. p. 385 (1908).
igebe ne Wilson, Phil. Mag. xx. p. 385 Ce
t See Wellisch, Phil. Trans. A. ccix. p- 272 (1009) ; also Lusby, Phil.
Mag. xxii. p. 784 (1911).

Nature of the Large Ions in the Avr. 323

where ng, is the number of air molecules per cubic centimetre,
and M the mutual potential energy at collision of ion and
molecule due to the ionic charge. In the case of the large
ion, on account of its size, M will no doubt be very small
compared with mgvy, and the last factor in the above expression
will be neglected in this discussion as probably differing
little from unity.
Let Si ele Pa

then with well known substitutions the expression for the
reciprocal of the mobility may be written

1 PoP IN OAPI E Oa TLE
Pace) ho),

where N, is the number of air molecules per cubic centi-
metre and pg the density of air under standard conditions,
n the viscosity at the temperature ¢° C., and p the air pressure.
The modification of this formula so that it may include
the effect of the persistence of velocities after collision is
doubtless a small one, and may be neglected for the present
purpose.

The diameter of the ion as calculated by the preceding
expression does not vary greatly for moderate changes in
the density, so, in view of the lack of any knowledge of the
core, limits to the diameter of the ion according to this
method of estimation are set out in Table II. for various
values of the mobility, by assuming firstly that the ion is all
water, and secondly that the density of the whole is 2:5,
this figure being considered as possibly the value of the
density of the core.

If the resistance to the motion of the transition layer is
not negligible compared with that offered to the motion of the
ion as a whole, then the diameter of the liquid part of the
drop corresponding to any mobility will be less than that
given in the table, and the mass greater than that indicated
by the diameter.

The calculation has been made from the following data :—

Noe=1°29 x 10" E.s.U., sg=2°9 x 107° (diam. of air molecule),
ey 80 Xe LO Mo 4x On? (mass | 5, ‘s Ne

po=0°001293, volume of ion= ve ( ) ;

p=1-0132 x 10°.

524 Prof. J. A. Pollock on the

Tape II.
p/P 1/u. in ile 0' =25;
per cent. 5, /S- S)- i) sass se
ieee 1250 11-4 33x10-7| 9-4 27 x10-7
432 2000 12:5 3: | 104 3:0
83:5 3000 13:6" | 9/39 |» aes 33 ae
100-0 3440 Ie 44>, | gales Sa

Diameters of the large ion for various values of the mobility.

For equilibrium, under usual atmospheric conditions, the
value of the free surface energy must be influenced by the
attraction of the core, so the thickness of the surrounding
film of water cannot, from this point of view, be great com-
pared with the range of molecular force. The preceding
calculation makes the diameter of the whole ion about a
tenth of the value, 5x 107° centimetre, usually accepted for
this range. The nucleiof these large ions may, then, be very
minute, and although their actual size is unknown it is pos-
sible that they may be fairly uniform in diameter, as there is
evidence that the mobilities of the fully developed ions under
given atmospheric conditions lie within somewhat narrow
limits.

It is not quite clear how the electrical energy of the ions
is related to their diameter. The charge is, however, not
necessary for equilibrium, and it is not unlikely that the
conclusions as to the nature of the ions, only rendered pos-
sible by the happy chance of their electrification, may apply
with perhaps little modification to many of the far more
numerous class of unelectrified nuclei which exist in ordinary
air.
A detailed discussion of the large ion was published by
Sutherland in the Philosophical Magazine for September
1909. In his own words his view is as follows * :—‘To
account for the very small mobility of the large ion of
Langevin I have imagined the structure already described,
namely, a nucleus of (H,O), or (HO); or both in a state
very similar to that of a liquid surrounded by an envelope of
H,O vapour which is kept highly concentrated close to the
nucleus. This envelope is similar to the surface film of
vapour of H,O deposited on the grains of fine powders.
The number of H.O molecules per em.* close to the nucleus

* Sutherland, Phil. Mag. xviii. p. 366 (1909).

Nature of the Large Ions in the Air. 525

will have a value N., like a saturation value, and the number
will diminish with increasing distance from the nucleus till
it becomes N, where its influence has ceased.”

For the velocity u of the ion in an electric field he gives
the expression

dit _ .2928NT2(HN, +G+HN,)
ee ad INGE (CHIN 5 ai ote 9 Us

where 7 is the distance between the centres of a molecule of
H.O and one of air when in collision, g the number of ions,
and N, the number of air molecules, per cubic centimetre,
T the absolute temperature, and H, G, and H, on the right-
hand side of the equation, constants.

The form of the expression within the brackets was sug-
gested by a preliminary result of my own measurements that

1/u=1200 + 107°5 h,

where / is the humidity in grammes per cubic metre. Such
a form of relation only holds approximately as may be seen
from an inspection of fig. 1, and the numerical values in the
equation need modifying in ‘the light of the later measures,
and may require considerable alteration if it turns out that
the observations at low humidities require correction.

Stress is laid by Sutherland on the point that according
to his view of the movement of gaseous ions the mobility
should depend on the density cf jonization, Jn my obser-
vations the number of ions per cubic centimetre varied from
650 to 32900, but no dependence of mobility on this factor
is apparent.

Sutherland’s discussion has the great merit that it involves
no unreasonable assumptions. It was developed, however,
before the evidence in favour of a rigid core in connexion
with the large ions was fully appreciated, and from what I
have said in this paper I believe his investigation requires
modification.

A correspondence with this gifted author during the
progress of the experimental work was a source of ‘oreat
encouragement in a most tedious investigation.

Summary.

The large ions in the air, which were discovered by
eee in 1905, have a mobility which at constant atmo-
spheric pressure is a function of the relative humidity only.
At standard pressure the mobility varies from 1/1250 w hen
the humidity is 4 per cent., to 1/3440 when the pressure is
that of saturation.

526 Mr. W. Ellis Williams on the

2. The ions do not exist in dust-free air, so the picture
most readily formed is that of a collection of water molecules
surrounding a dust particle, the whole being electrified by
the attachment of a small ion. The ion thus affords an
interesting example of the adsorption of water vapour at a
rigid surface.

3d. A thermodynamic argument, based on the relation
between mobility and relative humidity, leads to the con-
clusion that the adsorbed moisture is in the liquid state
with a latent heat and density little different from those of
water.

4, The order of magnitude of the diameter of the ion, as
calculated on usual kinetic theory lines, varies from 3 to
4x 107‘ centimetre according to the atmospheric conditions.

The Physical Laboratory,

The University of Syduey,
November 25, 1914.

LV. On the Motion of a poe in a Viscous Fluid. By
W. Exuis Witttams, B.Sc., AF. AéS., University College,

Bangor™.
[Plate IX. ]

Novation :—
jy amin rests coefficient of viscosity.
v=p/o.. kinematic coefficient of viscosity.
NL eSonnate velocity of sphere.
GR ae EN radius of sphere.
R,©.... velocities along the polar coordinates r, 4.
u,w .... velocities along the cylindrical coordinates @, z.
UD Seas eae Stokes’s current function.
Tey ye pressure at a point in the fluid.

HE mathematical solution of the problem presented by
the motion of a solid body moving with finite velocity
through a viscous fluid has hitherto presented insuperable
difficulties, and no solution has been obtained even for the
apparently simple cases of a sphere or cylinder moving with
uniform velocity along a straight line. The complicated
nature of the equations of motion together with the difficulties
presented by the boundary conditions, which require that
both the normal and tangential velocities should have specified
values at the surface of the moving body, seem to place the
direct solution of the problem far above the reach of any
known method. he actual solutions of the problem which
are given in the current text-books of hydrodynamics are
* Communicated by Prof. EH. Taylor Jones, D.Sc.

Motion of a Sphere in a Viscous Fluid. 527

obtained by ignoring some of the terms in the general
equations of motion, and thus simplifying them to such an
extent that a direct mathematical solution is possible.

These solutions may be divided into two classes, which
may be looked upon as limiting cases of very high and very
low velocities respectively. The equations of motion of an
incompressible fluid as obtained by Navier and Poisson may
be written :—

On”

with similar equations for the other coordinates (X, Y, Z are
the components of the impressed forces).

The right-hand sides of these equations contain terms such

Ou : é :
aS Ua which are of the second power in the velocity, and

il Op LE Ou se Ou Ou

Hae: Uwe Toe ae a 8 sh WR)

also terms such as vV/?u, which are of the first power ; if,
therefore, in any case the velocity is large and the coefficient
of viscosity v is small, the terms vV/?u &c. may be neglected,
and we thus arrive at the so-called “perfect fluid” or
irrotational equations. These equations can be solved fora
large variety of boundary conditions, and it is these solutions
which take up the greater part of the current text-books of
hydrodynamics. It is found, however, that the actual motion
observed is, in general, very different from that given by the
solutions thus obtained, a difference which is perhaps most
glaringly shown by the fact that, according to the perfect
fluid theory, a body moving with uniform velocity experiences
no resistance to its motion.

This discrepancy is usually ascribed to the occurrence of
“‘eavitation.” At sharp corners and edges the theory makes
the velocity of the fluid infinite, the pressure in the fluid has
consequently a large negative value, a cavity is formed
around the edge, and the instability of the motion around
this cavity is supposed to cause a general breakdown of
the motion. Without denying the fact that cavitation may
occasionally occur, and that its occurrence may alter the
whole motion, a careful survey of the experimental facts
available will show that the above explanation is totally
inadequate. In the first place, the motion of the fluid sur-
rounding a moving body is (at ordinary velocities) the same
for air and for water, due regard being paid to the different
densities and viscosities, and the irrotational solutions are no
more applicable to air than to water. It is difficult, how-
ever, to imagine anything resembling cavitation taking place

528 Mr. W. Ellis Williams on the

In air, and, in fact, the lowest pressures experimentally
observed on aeroplane wings and similar bodies are never
more than a few cms. of water below atmospheric pressure.
In the second place, the difference is by no means confined to
cases in which sharp edges are present, but is quite as great
when the moving bodies are spherical or cylindrical so that
cavitation cannot take place even in heavy liquids, and, in
fact, the difference may be shown to exist in cases where the
absence of cavitation may be experimentally demonstrated.

The reason for the discrepancy appears to lie rather in the
nature of the boundary conditions and the impossibility of
satisfying them in the irrotational solutions. All the experi-
mental evidence available goes to show that for fluids such as
water and air the particles of fluid in the immediate neigh-
bourhood of a solid boundary have no motion relative to that
boundary, while the irrotational solutions cannot be made to
satisfy this condition owing to the fact that, viscosity being
neglected, tangential motion gives rise to no stress, so that
in “general the solutions indicate a large amount of slipping
at the boundary.

In the case of solids of “‘stream-line” or fish-like shape
the motion appears to be similar to that given by the
irrotational solution except in a layer of fluid in the neigh-
bourhood of the surface of the moving body*. In this
layer, which is relatively thin, the velocity eines rapidly
from the value given by the boundary condition to that
given by the irrotational solution. Within this layer the

Ou . <
value of a2 5 therefore large compared with w and the
=

term containing the viscosity is no longer negligible. The
high value of ibe space differential of The velocity gives rise
to a considerable tangential force on the surface of the body,
which is generally known as “skin friction.”

Tn most cases, how ever, the transition layer is not confined
to the surface of the moving body, but departs from it near
midsection and gives rise to a “wake” of eddying motion,
which persists in the fluid for a considerable distance behind
the moving body, completely altering the character of the
motion.

In certain cases the motion may be approximately repre-

sented by the assumption of surfaces of discontinuity in the
fluid, and the solutions obtained in this way by Kirchhoff and
Rayleigh, though far from giving an exact representation of
the observed motion, are yet a oreat advance on the older
theory. The methods developed do not, however, throw any

* Vide Prandtl, Handbuch d. Naturwiss. iv.

Motion of a Sphere in a Viscous Flind. d29

light on the way in which these surfaces arise or enable us to
decide in what cases they will occur.

If, instead of neglecting the terms of the form vV/7u in the
ao of motion, we retain these and neglect those of the

type us the equations reduce to a form which can be

solved for certain cases, and give solutions which are
applicable when the velocities are small and the viscosity
high.

The most important of these solutions is that for a sphere
moving in a straight line, obtained by Stokes in his memoir
on the motion of pendulums. It was shown by Rayleigh
that this solution may be expected to hold so long as Va/y is
small compared with unity, and experiments by Ladenburg*
and others show that when due account is taken of the
boundary conditions the resistance formule derived from
the solution agree with the experimental results to a very
high degree of accuracy. The limiting velocity thus defined
is, however, very Jow, and the practical applications of the
solutions are confined to motion in very viscous fluids.

An attempt has been made by W hitehead + to obtain a
second approximation to Stokes’s Holiion by expanding the
neglected terms in powers of the velocity, and taking
account of second powers only. The method, however, does
not lead to any definite results, as the vorticity becomes
infinite in certain parts of the field. This is taken by him to
indicate that the motion becomes unstable and breaks down
suddenly in the neighbourhood of the critical velocity, and
that eddying motion involving high values of vorticity is set
mp.) Lt will be seen later that this explanation cannot be
reconciled with experimental results.

It will be gathered from the above that nothing is really
known as to the natnre of the motion for values of the
velocity which are neither very high | nor very low, and the
investigations described below were undertaken with the object
of throwing some light on this problem, and in particular
to determine the way in which the motion changes as the
velocity is increased beyond the critical value.

The problem has been approached both from the experi-
mental and theoretical sides, and in order to simplify matters
as far as possible the investigations have been confined to
the case of a sphere moving with uniform velocity along a
straight line. The actual motion of the fluid surrounding

* Ladenburg, Diss. Munich, 1906.
t Whitehead, Quart. Journ. Maths. 1889, p. 148,

Phil. Mag. 8. 6. Vol. 29. No. 172. April 1915. 2M

930 Mr. W. Ellis Williams on the

the moving sphere has been observed, and it has been found
possible to photograph the motion and to map out the stream-
lines for velocities up to 720 times the critical value. At
the same time certain mathematical solutions have been
obtained which serve to throw some light on the changes
which take place in the form of the motion as the velocity
increases beyond the critical value. We shall first describe
the methods by which the stream-lines were experimentally
observed and measured, and then proceed to describe the
results and to compare them with the solutions obtained.
The greater-part of the experiments was carried out with
a sphere moving in water, but in order to obtain measure-
ments over as wide a range as possible it was found desirable
to use liquids of different viscosities. The advantage of doing
this lies in the fact that irregular currents are set up in the
liquid by the inevitable slight differences of temperature of
different parts of the trough, and accurate measurements are
impossible if these currents have a velocity which is an
appreciable fraction of the velocity of the sphere. It was
found that the iowest practicable velocity in water was about
1 cm. per see., which gave Va/y=23, and it was thus im-
possible to work in the immediate neighbourhood ot the
critical velocity. After trying a number of different liquids,
it was found that good results could be obtained by using
mixtures of glycerine and water in different proportions
according to the viscosity desired. For pure glycerine the
value of v is about 15 and all values between this and ‘013,
the value for water, may be obtained by suitably adjusting
the proportions of water and glycerine in the mixture. The
viscosity of the mixture was determined in each case by
means of an Ostwald viscosimeter.

The experiments were carried out with two rectangular
troughs of similar shapes but of different dimensions. The
smaller trough was of glass and measured 19 x10 x9 em.,
and was used for the glycerine experiments, while the larger
trough was of wood with glass face and ends and measured
46x18x17cm. The radii of the spheres used were 1°25,
°88, and 40 cm. respectively. The experimental results
obtained with different spheres and liquids can be correlated
by means of the dimensional theorem :—The motion in any
two liquids with geometrically similar boundaries is similar
when the quantity Va/v is the same in the two cases. In
what follows, therefore, the value of this quantity is given
for each result. In the experiments Va/v varied from ‘01
to 720.

Motion of a Sphere in a Viscous Fluid. 53)

The general arrangement of the apparatus is shown in
fig. 1. The trough T was placed on a table fixed to the
wall of the room. Above it was a long frame AB resting
on the rollers RR, and the sphere S was attached to a piece

Arrangement of apparatus.

of steel wire fixed to the middle of the frame, so that the
latter in moving along on the rollers carried the sphere from
one end of the trough to the other. The position of the
sphere was adjusted so that its centre moved along the centre
line of the trough. Motion is given to the frame by means
of a heavy bar, BD, hinged to “it and resting on the axle
of the pulley F, which is connected by a set of reduction-
pulleys to the axle of a small electric motor, M. The
velocity of the frame can be varied by varying the resistance
in the armature circuit of the motor and also, if necessary, by
varying the reduction ratio of the connecting pulleys. The
motor was kept continually running and the frame started by
dropping the bar on the axle. The frame being very light it
immediately took up the velocity of the axle, ‘and thus the
sphere moved along the trough with a uniform velocity.

The motion of the liquid is rendered visible by mixing
up a little aluminium powder with a drop of glycerine and
stirring it into the liquid in the trough. ‘The little aluminium
flakes remain suspended in the water for several hours, and
when illuminated by a beam of light from an are lamp show
clearly the motion of the liquid surrounding the moving
sphere.

A camera was arranged in front of the trough, and when
the motion is photographed the aluminium particles trace out
curves on the plates which serve to measure the velocity of
the fluid and enable the stream-lines to be plotted out.

2M 2

53? Mr. W. Ellis Williams on the

The arrangement of the illumination and exposure was
somewhat different for the two troughs. The small trough
being used chiefly for low velocities, the exposures required
were long enough to be made by hand, and the beam of light
was strong enough to illuminate the whole field of motion.
Considerations of symmetry show that the motion must he
the same in any plane passing through the axis defined by
the path of the centre of the sphere, and hence it is sufficient
to investigate the motion in one such plane. A beam of
light from the condenser placed in front of the are-lamp E
is focussed on a slit G and then rendered approximately
parallel by passing through a cylindrical lens formed by a
tall narrow beaker full of water; it is then narrowed down
by a vertical slit placed on the end of the trough, and.
passes through the trough as a thin vertical sheet of light.
illuminating the central “plane of the liquid in the trough.
The camera is placed direcily in front of the trough and is
focussed on the aes and the illuminated plane. Tfa plate
is exposed for a short time while the sphere is in motion,
each little aluminium particle will trace out a curve on the:
plate. This curve will be practically a short straight line,
and the ratio of its length to that of the trace of the sphere
itself will give the velocity of the particle, which, owing to
the lightnes ss of the particle, is also that of the surrounding
fluid. The photographs A—C reproduced in Plate 1X. give an
idea of the results obtained by this method.

The larger trough was used for the higher velocities, and the
above method could not be adopted with it as the beam of light
had to be made much more intense to get good photographs,
and hence only a small portion of the field could be illumi
nated, and it was necessary to take several photographs of
different parts of the field and then combine them together to
give a stream-line diagram. The beam of light in this case
passes through both a cylindrical anda convex clens and enters.
the trough as a convergent beam with a vertical-line focus
near the centre of the trough. In the neighbourhood of this
focus the beam gives a very strong illumination over a field
about 2 em. each way, the average thickness of the illuminated
plane being about2mm. By suitably inclining the beam and
altering its focus by moving the lenses, the illuminated
portion could be brought to any desired part of the trough,
and thus, by taking a “number of photographs, the whole of
the central plane can be covered. The camera is placed in
front of the trough as before, but the exposure is made by a
shutter attached a the lens which is automatically released

Motion of a Sphere in a Viscous fluid. D033

. by the motion of the frame when the sphere has reached the
centre of the trough. The duration of the exposure can be
varied to suit the velocity of the sphere, and the length of
the trace gives the velocity as before. About half a dozen
photographs were required to cover the whole field, and these
were combined and measured by the following method. A
circle of 2°5 em. radius was drawn in the middle of a sheet
of squared paper, which was then set up on a drawing-board
in front of a projecting lantern. One of the plates was
placed in the lantern and a magnified image of the plate
thrown on the paper, this was then adjusted so that the mean
position of the ball during the exposure coincided with the
circle drawn on the paper, the magnification being adjusted
so that the diameter of the image was exactly 5cem. The
length of each trace was then measured with a pair of
dividers and its direction marked on the paper. Hach plate
of a series was put into the lantern in turn, and thus the
whole field of motion was mapped out.

The motion is best represented by means of Stokes’s
current function y. If the motion be referred to cylindrical
coordinates 2, a, the line traced out by the centre of the
sphere being taken as axis of z and the corresponding

velocities being denoted by uw, w, then wy is defined by

TON v= — 1 ON
OG ola Oe:

WwW =

and hence the curves yr=const. vive the direction, and their

distances apart divided by @ give the magnitude of the
velocity at any point. The values of y for the median line
z=0 and for the surface of the sphere were first calculated
from the measurements of velocity, and starting from points
giving w="2, ‘4, &e., the curves were drawn in the direction
of the velocity as marked on the paper, and thus the stream-
line diagrams of figs. 2-8 were obtained. In each diagram
the straight line at the bottom is for y=0 and each suc-
ceeding curve is for values of yf increasing by an interval
of +2, the velocity of the sphere being always taken as
unity.

The results obtained are shown in the accompanying
diagrams and photographs which we may now proceed to
describe in detail.

The first two diagrams figs. 2 and 3and the corresponding
photograph A (PI. LX.) refer to a sphere moving with a

534 Mr. W. Ellis Williams on the

velocity of 13 cm. per sec. in glycerine. Fig. 2 is: for
a sphere of ‘8 cm. radius moving in the smaller glass trough,

Sphere in glycerine, Va/» =:0116 (Experimental).

while fig. 3 is for the same sphere moving in a cubical
trough measuring 10x10 10 cm., this latter being taken
for comparison with a theoretical result to be given later.
Fig. 3
}

Sphere in cube, Va/v=-0116 (Experiment a).

The kinematic eieent of viscosity w as found to fe 10 eee
units, and the value of Vajy is therefore -0116 or rather more
than 1 per cent. of the critical velocity. The motion should

Motion of a Sphere in a Viscous Flind. 53D

thus be accurately represented by a solution obtained by
neglecting the inertia terms. If, however, the lines of fig. 2
be compared with the diagram of Stokes’s solution given
on p. 932 of Lamb’s ‘ Hydrodynamies’ it will be seen that the
observed motion differs widely from the theoretical diagram.
This is obviously due to the effect of the containing vessel
on the motion, Stokes’s solution referring to a sphere moving
in a fluid extending to infinity. It may in fact be shown
that the effect of the boundary is appreciable even when the
sphere is small compared with the vessel. The velocity in
Stokes’s solution is everywhere in the same direction as that
of the sphere, while if we consider the flux across the median
plane perpendicular to the motion, it is obvious that the
total amount of fluid crossing the plane must be zero at
every instant (the sphere itself being reckoned as though it
were fluid) ; hence the forward motion of the liquid near
the sphere must be compensated by a backward flow in the
outer portions of the vessel. As the velocity of the fluid
only diminishes with the first power of the distance, the effect
is very marked even when the vessel is very large compared
with the sphere. This in fact is what we see in the photo-
graph ; in the immediate neighbourhood of the sphere the
fluid moves along with it, but as we go away from the sphere
we see that the velocity diminishes to zero and in the outer
parts of the vessel is mainly in the opposite direction.

Fig. 4.
|

Sphere in glycerine, Va/y=2°1 (Isxperimental).

Passing on to the diagram of fig. 4 and the corresponding
photograph B, Pl. [X., these refer to a velocity of *217 em.
per sec. in a mixture of glycerine and water of viscosity
v='09,. The value of Va/y is therefore 2°1 and the critical

536 Mr. W. Ellis Williams on the

value has therefore been passed. It will be noticed that the
form of the stream-lines has changed slightly. The diagram
is no longer symmetrical about the median plane, but the
point of zero velocity and the stream-lines which centre on it
have moved so as to be behind the centre of the sphere.
The velocity in front of the sphere is accordingly diminished
and that behind it increased. The change is much more
marked in fig. 0, Photo © (PI. [X.), w hich corresponds to a
velocity of -38 em. per sec. in the same mixture.

Sphere in glycerine, Va/yv=3°7 (Experimental ).

Va/v is now 3:7, and it will be seen that the point of zero
velocity has moved still further back and that the stream-lines
around it are somewhat elongated instead of being approxi-
mately circular as in the previous diagrams.

Fic. 6.

Sphere in water, Va/yv=23 (Experimental).

In fig. 6, Photo D (Pl. 1X.) we get the first of the results
for water. The velocity is ‘24 cm. per sec., and the value

Motion of a Sphere in a Viscous Fluid. D937

of Va/y is 23. In this figure it will be seen that point of
zero velocity has approached nearer to the sphere and also
moved still further back, the stream-lines around it being
nearly elliptical ; the increase of velocity behind the sphere
is very marked, the velocity of the fluid being nearly the same
as that of the sphere for a considerable distance behind it.
In the case shown in fig. 7 the velocity has been increased
to 1:3 cm. per sec., the corresponding value of Va/v being
125. The point of zero velocity has come almost into line

Sphere in water, Va/v=125 (Experimental).

with the top of the sphere. The velocity in front of the
sphere is very similar to that in the irrotational solution
(see Lamb, p. 137), and behind the sphere is seen a “ wake ”
of nearly constant velocity as shown by the parallel stream-
lines.

Bie. 8.

Sphere in water, Va/v=720 (Experimental).

Fig. 8, Photo E (Pl. [X.), represents a velocity of 7°5 em.
per sec., the corresponding value of Va/y being 720. In this
case the wake is considerably shortened and the whole motion

D398 Mr. W_ Ellis Williams on the

is altogether more like that given by the “ irrotational ”
solution ; in fact, at a distance from the sphere it is very
similar to that which would be produced by the motion
of a pear-shaped figure formed by the sphere and the fluid
which is moving along with it. ‘This is the highest velocity
for which stream-lines could be mapped : photographs were
indeed taken with velocities up to twice this value, but the
lines were tuo faint to be measured. They do not indicate
any important change 1 in the form of the stream-lines.

‘The motion shown in Photo F (P1.1X.) is altogether different
from the others and was obtained with a sphere moving in
water at the rate of ‘074 cm. per sec. giving Va/v=T- 1h
This is the lowest velocity for which photographs could be
obtained in water ; at lower velocities the motion is masked
by irregular currents in the water due to temperature and
other effecis. The peculiar form of the stream-lines in this
diagram may be due to some such effect, but the regularity
with which the spreading out of the stream-lines behind the
sphere appeared in a large number of different photographs,
seemed to suggest that it may be due to some kind of
instability appearing at this velocity.

It will be noticed that even at the highest velocity of
fig. 8 we do not get the irregular eddying motion which
has been observed in experiments in air and water channels.
This may be due to the comparatively small values of the
velocity employed in the present experiments, the eddies
being due to an instability of the motion which only appears
at values of the velocity beyond the highest used in these
experiments ; it is also possible that the turbulence of the
stream of air or water in the channel may affect the
motion. ;

If, however, instead of a sphere we take a flat plate with
its ple ne perpendicular to the direction of motion, eddies are

formed and are shown very distinctly in the photographs, of

which two are reproduced in G & H(PI1.IX.). In glycerine,
the motion is very similar to that produced by the sphere
under the same circumstances: in fact, the motion at low
velocities is, except in the immediate neighbourhood of a
moving body, practically independent of its shape. As the
velocity changes beyond the critical value, the stream-lines
change in very much the same way as for the sphere.
Photo G fora velocity "2 cm per sec. is of the same type
as C, but beyond this value the dev elopment of the motion
is very. different. Photo H shows the motion for a velocity
of +59 em. per sec., and it will be seen that we have here
a full development of eddying motion. The “wake” is

oe Se ee ee

Motion of a Sphere in a Viscous Flud. 539

separated from the rest of the fluid by a number of eddies
for which the stream-lines are closed curves around a
point of zero velocity. These eddies are in some respects
similar to the vortices whose motion is worked out in the
irrotational theory, and the dynamical effects must also be
similar, as the very remarkable calculation of the resistance
to the motion of plates and cylinders made by Karman
shows. <A close study of the ‘photograph shows, however,
that there is a very great difference between the eddies and
true ‘“ vortices,” for the eddies the central part of the
liquid moves more or Jess as a solid body, the velocity
diminishing towards he centre of the eddy where there
is always 2 point of zero velocity, whereas in a vortex the
velocity is inversely proportional to the distance from the
centre, becoming infinite at that point. It is hoped by
further experiments to trace out the gradual development of
the eddies and the way in which they die away.

We may now proceed to compare the above results with
those obtained by solving the equations of motion, and it will
be convenient to begin w vith the resuits for very low velocities
given in figs. 2 and 3. It was pointed out on p. 535 that
these cannot be directly compared with Stokes’s result owing
to the influence of the outer boundary. The motion of a
sphere inside a cylinder has been solved by Ladenburg *, and
the solution for a rectangular vessel might be obtained by
the method of images jee eloped by Lorentzt. The solutions
obtained are, however, so very complicated that the numerical
computation of the stream-lines would be exceedingly
laborious, and for the same reason it would be impossible to
use the solutions as a base of further approximation for
higher velocities.

A very simple solution may, however, be obtained for the
case of a sphere moving in the fluid contained in a con-
centric sphere, and the solution will apply with sufficient
approximation to the case of a sphere moving at the centre
of a cubical vessel which is represented in fig. 3,

The equations of motion with the inertia terms omitted
and in the absence of impressed forces may be written :-—

ult byeuk Op
at ip Ox

with two similar equations for v and w.
If we now introduce the Stokes’s stream function w defined

NY tl,

* Ladenbure, doe. evt.
+ Lorentz, ddhandlungen, i. p. 30.

540 Mr. W. Ellis Williams on the

by (2) and change to polar coordinates, the equations

reduce to
D(D= 20. 5 Ae
where D is the operator |
fo
Or 5 mee

for steady motion the equation reduces to

Dp =0 . .. 5).

The boundary conditions require that the velocity should
be zero at the surface of the outer sphere and equal to
V at the surface of the inner sphere. If aand b be the
inner and outer radii respectively, these give :

eo)

cosec 6. 5).

Atr=¢,
OY, = Va sin 8 cos, OY = Vasin® 4 a
Be Be v=0, Or
or

A
Let us assume w= a + Br+Cr?+ Dr* )sin?@ ; this value

satisfies (4) for all values of the constants A, B, C, D, and
on substitution in (5) we get four equations to determine A,
B, C,and D. In order to get a solution to compare with the
ease of a sphere ina cubical vessel given in fig. 3, we put
a=1, b=5°7, making the diameter of the sphere the same
as the edge of the cube, and we then obtain A= —°413,
-B=1237, co 325, D=:00339, and the stream function is
given by

y= (-= aa + 1:2377% —°32577 + 008397") V sin? @. (6)

The stream-lines given by this equation have been plotted
out and are shown in fig. 9 and it will be seen that they
agree very closely with the experimentally determined lines
of fig. 3. A test of the agreement is given by the position
of the point of zero velocity. This may be obtained from
the formula by putting oe =0, and we thus get r=5'6,
while the value measured on the diagram is r=35° 2 Also
if the diagram be compared with that of fig. 2. which
represents motion in the rectangular trough it ‘will be seen

Motion of a Sphere in a Viscous Fluid. O41

that there is no very great difference between the values of
the velocities in the neighbourhood of the sphere, so that the
above solution may be regarded as an approaimate repre-
sentation of the motion of a sphere in an elongated vessel,
the approximation being fairly close in the neighbourhood

of the sphere.

Fie. 9.

—

Sphere in sphere, Va/y=0 (Theoretical).

Taking the solution obtained above as a base, we shall
now attempt to obtain a second approximation in which the
terms containing the second power of the velocity are partly
taken into account.

Returning to the general equations of motion (1) and
writing them in terms of cylindrical coordinates z, o (sym-
metry about axis of z), we get for steady motion and no
impressed forces the two equations

u Wie Op u
ek +w ae +v( =u
Z a os”

oni) |
oe + w oe = or +vV ru.

Introducing the operator D defined above, these may be
written

542 . Mr. W. Ellis Williams on the

Eliminating p by differentiation, we get

vy D?rp= Ce ro Sn ~=") Dy.

Denoting, for brevity, the caiee on the right-hand
side by 3, this becomes :—

yD b= 3D 2.

We have to solve this equation subject to boundary
conditions of type (5).

Let r= Wy +h, where Wo is the solution for infinitely
small velocities, and yy, is to be looked upon as a correcting
term which is small compared with wo.

Substituting we have

vy Dry + yD, =SDy)+ 3D. 5 e 5 - (8)
Now since Wy is a solution of (4) vD%fo=0 and 3D,

is of the third power of the velocity and may therefore be
neglected. Further in the term $Dy, we may neglect wy,
(ocenrring in the operator 3) as compared with Wo, so that
SD contains yy, alone and is therefore a known function
of the coordinates. The equation thus becomes

yD =SDvy. . so) oe

and is a differential equation to determine yp.

Now let py y=¢o(7) sin? O, Wi=¢)(7) sin? @ cos 8, where
wro(7) is to be determined so that yo satisfies the boundary
conditions and ¢,(7) is as yet undetermined.

Now D?¢,(r) sin? @ cos 0 =
( guin(v) — PPE) 4 24 PL) in? 005 8,
and $D¢o(7) sin? d=
ae ($e Ae Sok al “ee sin? 6 cos 8.
The term sin? @ cos @ thus divides out of (9) which becomes

Bs mae _ 2

ne 2h0(7) [# (y ‘ti suds om 20! (7) +0) (10)

yee ae

This is a differential equation of the fourth order to

Motion of a Sphere in a Viscous Fluid. p43

determine ¢,(), and the solution will contain four constants
which may be adjusted so as to satisfy four boundary
conditions such as (5).

If desired the process of approximation raay be carried
still further so as to include terms of the third degree in the
velocity. To do this we must first substitute the value of
$;(7) sin? @ cos @ in the operator 3 in the first term on the
right-hand side of (9). If now @,(7) sin*@cos@ be sub-
stituted in the last term 3D, which was previously neglected,
the resulting expression contains terms in sin 0, sin? @ cos 6,
and sin*@cos 0, and hence by adding to the previous value
of y a term of the form @¢,(7) [a sin? +5 sin? @ cos @4-
c* sin’ @cos@] and suitably choosing the values of the nume-
rical constants a, b, c, an equation giving @.(7) in terms of
@o{r) and ¢,(7) may be obtained, and the approximation is
thus corrected so as to include the terms of the third power
in the velocity.

In applying the above solutions to actual cases for com-
parison with experimental results, certain difficulties are met
with which are connected with the validity of Stokes’s
solution and are perhaps best dealt with here.

Reverting to the case of a sphere moving in an unbounded
fluid, Stokes’s solution may be written

3 bar Ninh
Va Vai (1- a) sin? @,
At a great distance from the sphere this becomes
3 igs
y= 1 Var sin? 6,

and the corresponding velocities wiil be

MM ates Bik a)
ee rsinO 00 ON ers
ane ts) 3 View.
0O= Ga TA — sin 0.

The velocity in the direction of motion of the sphere is

3 Va

2»

(cos? 8+ 4 sin? 6).

The total momentum: of the Auid in the direction of motion
of the sphere is obtained by integrating this over the whole

544 Mr. W. Ellis Williams on the

of space outside the sphere, and its value is

2 ra0 (0=2n (b=20
5 Va 1 ve r(cos* @+4 sin? @) dr dé dp
o=

_3y 37? : T=
) 2 [| f
a = r=a

This is evidently infinite even when V is infinitely small ;
now this momentum is produced by the force which acts
on the sphere maintaining its velocity against the resistance
of the fluid; and since this force is proportional to the
velocity and hence infinitely small, it must have acted for a
time which is of the second order of infinite quantities. For
the solution to be valid the sphere must therefore have
moved from infinity with the same infinitely small velocity,
otherwise the motion is not steady and equation (4) does not
apply.

Now it is evident that if the velocity is not infinitely
small, steady motion cannot be established, even when the
sphere starts from infinity, as the time during which the
force acts is now infinite of the first order only. The
precise extent of the departure from steady motion may be
found by considering the case of a sphere starting from rest
and proceeding with a small uniform velocity. This case
has been solved by Bassett (Hydrodynamics, ii. p. 286),
whose solution may be written

“Ton? vi+2anr V vt/7 +4(a2@—9r?) Je" dn.

When r is small and ¢ great the second term vanishes and |
the motion is the same as in the steady state, but when 7 is
large the integral does not vanish even for very large values

Bn aye ? f
of ¢; for the motion is only steady when —_ vanishes, and
‘ ; t

hence at poinis very distant from the sphere the motion
never becomes steady. The conditions are not very much
improved when the fluid is confined by an outer boundary 4
so that only small values of 7 need be considered, for the
time during which motion is possible without completely
altering the boundary conditions diminishes in the same
proportion as the dimensions of the boundary. The only
case where a steady motion is experimentally realisable is
when the outer boundary is a tube or elongated vessel, for
then the motion of the fluid is confined to the portion of
the tube in the neighbourhood of the moving sphere, and ;

Motion of a Sphere in a Viscous Fluid. D45

the boundary conditions are not seriously altered by a con-
siderable movement of the sphere. Now we have seen that
the motion given by the sphere in sphere solution of (6) also
represents very closely the motion in an elongated rectan-
gular vessel, and a second order approximation based on this
solution may be expected to show the way in which the
motion in this case alters with increasing velocity, even if
it does not vive an exact representation of it.

There is, ‘however, another cause which limits the validity
of Stokes’s solution at a great distance from the sphere,
which has been pointed out by Oseen and Lamb*. To get
really steady motion it is necessary to consider the sphere
as at rest In an infinite stream of fluid, or to consider the
origin as moving with the sphere. in either case the
velocity of the fluid relative to the origin has a fixed con-
stant value at infinity. Now the validity of Stokes’s solu-
tion depends on the possibility of neglecting terms of the

2
form 1 ot compared with terms of the form ee In this

Ox
Ou ORs a ees 6 i
one and a2 diminish indefinitely as the distance from
we ae

the origin increases while « remains constant, hence how-

ever small w may be, it is impossible to neglect me when
ris great. ow

Oseen has obtained a solution in which this fact is taken
into account, but as it only differs from the ordinary solu-
tion at considerable distances from the sphere, it is not easy
to test it experimentally, nor does it seem possible to base a
second approximation on it.

If, instead of referring the motion to an origin that moves
with the sphere, we refer it to a fixed origin, coinciding at
any instant with the centre of the sphere, the velocity of the
fluid far from the origin will be zero, and hence the term

0 can be neglected over the whole field. On the other

hand, ihe motion is not strictly steady except when quan-
tities of the second order are neglected. For let the sphere
be moving with velocity V and let & be the distance of its
centre from a fixed point, then if yw be the stream function,

Y= ite +&, a

di Mi:

* Lamb, Phil. } “i ew ble (LOT):
Phil. Mag. 8. 6. Vol. 29. No. 172. April 1915. 2-N

546 Mr. W. Ellis Williams on the

The last term is of the second order in the velocity, and when
~ = 0 when or 0, so that the motion
is steady with respect to both fixed and moving origins ; it
is in fact immaterial which system we choose. When
these terms are not neglected this is no longer the case, but
if reference is made to fig. 2 it will be seen that in the
immediate neighbourhood of the sphere the stream-lines are

these are neglected

all parallel to the axis of z, and therefore OF is zero, and

the second term is therefore nowhere very important, and
the results obtained by referring the motion to a fixed
origin may be expected to give a better representation of
the results than if the origin is taken to move along with
the sphere. This was in fact found to be the case, and the

calculations being similar for both cases only those referred

to a fixed origin are given below.
The solution for a sphere in a concentric sphere gives

b(n) = + Br+ Or? + Dr.
Substituting in (10) we obtain
: 9 tl 22 lifes. D)
NS (otra = (") si een) = - [- ah Br-+Cr?-+Drt |;
a particular solution of this is
Lape AB
br) =5| 5,

+3Br+$CBrP°—3DBr' |,
and the complementary function is
l BS) aD
2 nr ar 1-10,
where J, m,n, and p are constants to be determined by the

boundary conditions.
Thus the complete solution of equation (9) is

Ap & + Br+Cr?+ Dr‘) V sin? @-} [(- — + B’r
r 2v -
: l |
ee Bor— BDr*) +z m+ nr +p] V? sin? @ cos @..
The boundary conditions to determine /, m, n, and p are

$ (7) = 2h") —O at t=. ra:

—————

Motion of a Sphere in a Viscous Fliad. 5A7

For the case of an outer sphere of radius 5°7 we get on
substituting for A, B, C, and D, and solving the boundary
equations for Uni, Ts

l=—-0171, p=— "824, m=0224, n=-000166.

To compare with the experimental result of fig. 4 the values
of y have been calculated for Va/v=2, and the resulting
values of ware plotted in fig. 10. It will be seen on com-
paring the two diagrams that the theoretical solution is a

Fig. 10,

Sphere in sphere, Va/y = 2 (Theoretical).

very fair representation of the observed result. The second
term containing sin?@cos@ is unsymmetrical with respect
to the median “plane, and accounts for the backward trend
of the stream-lines. A quantitative measure of the agree-

ment between the two diagrams may be obtained by mea~

suring the displacement of the point of zero velocity from
the median line. In the theoretical diagram this is °8 em.
and in the experimental *9 cm. The approximate solution

a
is not applicable to values of ——- much higher than 2, on
v

attempting to apply it for a value of 3 it was found that
the stream function became negative in portions of the field.

As has been already: explained, the above solutions are
only applicable to motion in elongated vessels, as steady
motion is impossible in other cases. “The equations of motion
may, however, be solved in certain cases without the re-
striction to steady motion, and we may thus get a solution
applicable to a sphere moving in an infinitely extended fluid.
This can be done if we apply the method of approximation

2N 2

548 Mr. W. Ellis Williams on the

given above to Stokes’s solution of the motion of a spherical -
pendulum-bob given in his memoir of 1850.
When the motion is not steady equation (8) takes the form

D( Dy— 7 Sh) = 5 “3D. a

Let us assume that the position and velocity of the sphere
are exponential functions of the time and write, following
Stokes,

Wee ce

‘hen if we write ro =ce**’'hy (7) sin? 0,

where

3 3 Gis 3 AN -A(r—a) ‘1?
po(") = 24 +4 (1+ ra +a) (1+ rr) }. Ca

a) is the solution for infinitely small velocity and satisfies
the left-hand side of (11) equated to zero.
As before let y=wWo+, be the solution of (11), then on

substituting and neglecting third-order terms we get

D(Dy- ot) =" sD aes

the right-hand side being a function of wo only which may
be written

xv (1) ce?°t sin? 8 cos 8,

x (7) being a known function obtained by operating on ¢o(7).
To solve (13) we assume

yy = Cer db, (r) sin? 6 cos 8,

XN, and ¢,(7) being for the present undetermined.
Substituting in (13) we have

D} (,''(7) i - f1(7) —Ary (7) jer’! sin? 6 cos Qt
=y(rjer" sin? 6 cos'G..

et us denote the function in the brackets by &(r), then the
above equation becomes

D{E(r)ed™ sin? 8 cos 0} = y(r)e?* sin? 6 cos 8,

e (Ep — 9 Er) odie = (non oe ay oe

Motion of a Sphere in a Viscous Fluid. j49

This equation together with

1") — Slr) AG, =EC)

serve to determine A, and ¢,(7), the arbitrary constants
being determined by the boundary conditions, and the
problem is thus reduced to that of the solution of the two
linear equations (14) and (15).

To solve (14) we must evidently have \,= 2A, and then
the equation becomes

E"(0) — SE(r) =y(r).

A particular solution of the left-hand equated to zero is
&(7) = Br?, and hence the complete solution is

g=e[o+ nS 4 (Freer). len

We have now to solve

6 5
On (E)) — 29") —2r*bi(") = Er).

The equation

6 . 5
or ie 2 bi(7)— 27 di (7) = 0

is solved by

gi(r) =e NN (20°4 Bo a =):
Trot

if this be written (7) for brevity, then the solution of the
complete equation is

blr) = BLE YP + fac (se ()Edr]. 7)

The solution for vr may now be written
Qyt cy. Re 2) 2yt .
ye = co ho() sin? 6 + 20h, (7) sin? @ cos 8
= V¢o(r) sin? A+ V*d,(r) sin?Acos 8. . . (18)
In ecaleulati ting this solution for particular cases the inte-
grals occurring in the expressions for @;(1) are best evaluated
by numerical awetharle: In order to reduce the labour of

calculation, it is necessary to use an interpolation formula
so as to be able to earry out the calculation with a small

550 Mr. W. Ellis Williams on the

number of tabulated values of y(r). This function is of an
exponential form except when 7 is small, and it was found
that the usual interpolation formule (Newton and Gauss) did
not give sufficiently accurate results, since, being based on
Taylor’s theorem, they fail when the successive differential
coefficients do not diminish fairly rapidly. A very simple
interpolation formula may, however, be based on the assump-
tion that the functions are exponentials with slowly varying
indices. Thus suppose a, b, are two consecutive values of
the variable in the table, and /(a), f(b), the corresponding
values of the function which isto be integrated, and let ath
be the value of the variable in the step a—b, h varying con-
tinuously from 0 to (b—a) ; then we may put

fla th) = fraje™,
k being a different constant for each step in the table.
Hos

i 2

i Ve +h) = ee)

and the index & is given by

1 aot alg aoe 1D)
= ee ay.

ee \ Hath) ae

hence

The step (6—a) is usually unity, and hence & is found at
once by subtracting the logarithms of the two consecutive
terms in the table, and the step of the integral is got by
subtracting the two terms and dividing by &. The inter-
polation can thus be carried out without any very laborious
computation.

The calculations have been carried out for two values of X,
namely A="09 andA=1. The first represents a very slowly
changing velocity and is not very different from the case of
steady motion, while the latter value gives the case of very
rapidly accelerated motion. The results obtained are given
in the accompanying table showing the values of ¢o(7) and
o,(7) for different values of r. The value of w at any point
is obtained by substituting in (18).

In figs. 11 and 12 these values of y have been plotted out
for X=1 and for two values of the velocity, one above and
one below the critical value. It will be seen that the change

Motion of a Sphere in a Viscous Fluid. dd]

d= 09. h=10.

Te
(7) | gi(7). (7) $,(r)

1-0 5 000 500 00
12 73 002 670 00051
14 90 | -0078 780 00146
15 1-02 0129 845 00260
1°8 11h 0198 ‘900 00336
2-0 118 0277 920 00406
22 LaF 0544 *935 00474
2-4 136 | -0398 935 00469
2-6 1-44 0446 995 00489
28 1:52 0490 ‘915 00415
3-0 159 0524 -900 00406
4-0 1815 0586 "785 00350
5:0 1592 ‘0668 “670 00300
6:0 196 = = 0624 ‘575 00214
7-0 1-94 0630 495 00152
3-0 1-90 0630 436 00115
9-0 1:84 0670 389 | 0008

10-0 1-76 060 350

20:0 1-088 058

30-0 73 003

Fig. 11.

aa
eae
<2

Accelerated motion of sphere, Valv =) (Theoretical).

in the form of the stream-lines is of a similar character to
that in the steady motion; with increasing velocity the
motion becomes unsymmetrical with respect to the median

552 Dr. A. O. Rankine on the

plane, the velocity of the fluid increasing behind and dimin-
ishing in front of the moving sphere.

Fig. 12.

Accelerated motion of sphere Va/y=38 (Theoretical).

The solutions thus obtained are of course only applicable
Va

to values of slightly above the critical value ; the form

of the stream-lines in figs. 7 and 8 seems to show that a solu-
tion for these cases may be obtained by starting with the
discontinuous motion worked out by Kirchhoff and Rayleigh,
and some encouraging results have been obtained which will
be given in a later paper.

LVI. Note on the Relative Dimensions of Molecules. By
A. O. Rankine, D.Sc., Fellow of and Assistant im the
Department of Physics in University College, London*.

T is well known that the knowledge of the viscosity of a
gas makes it possible to calculate upon the kinetic
theory the mean free path of the molecules, and hence their
dimensions. According to Maxwell the relations are as
follows :-—
n=0°307prAG, :°. ). -)
where 7 is the viscosity, p the density, \ the mean free path,
and G the root mean square velocity of the gas molecules.

The value of G is ay) bo where p is the pressure of the gas.

Further, the equation
L Si
= inne? co lw.

* Communicated by Prof. A. W. Porter, F.R.S.

——— ee ee eee

Relative Dimensions of Molecules. 553

where N is the number of molecules per unit volume and o
is the radius of the molecule. gives the connexion between
the mean free path and the molecular radius.

As was pointed out by Sutherland*, however, the calcu-
lation of « by means of these two equations leads to incon-
sistencies, because the viscosity of a gas is not, in fact,
proportional to the square root of the absolute temperature.
Hence the value of o so estimated becomes smaller and
smaller as the temperature corresponding to the viscosity
datum increases.

The modification in the theory introduced by Sutherland
was to take into account the forces of attraction which the
gaseous molecules exert upon one another; and he showed
that this involved that the mean free path was smaller than

that estimated for forceless molecules in the ratio 1: 1+ ae

where C is a constant and T is the absolute temperature.
Upon this basis Sutherland showed that the viscosity ofa
gas, instead of being proportional to the square root of the

T
absolute temperature, was proportional to ——. This
eee
T

modified theory has been found to correspond with expe-
riment very well, at any rate much more accurately than
the simple theory.

In applying these results to the calculation of molecular
dimensions we have, according to Sutherland, to diminish
the molecular radius as estimated from the simple theory in
C
rT
for the molecular attractions make the molecules behave
from the point of view of frequency of collision as though
they were larger than they are in reality. ;

For this purpose, therefore, we require to know not merely
the viscosity of a gas at one temperature, but also the variation
with temperature, so that the constant C can be found.

The author has recently made such measurements for a
considerable number of gases, and the molecular dimensions
deduced therefrom exhibit some points of interest. The gases
in question are three members of the group of inert oases,
viz. argon, krypton, and xenon, and the three corresponding
members of the halogen group, viz. chlorine, bromine, and

Gh
2 2 e e ze
the proportion (1+ \ : 1, in order to obtain the true radius,
a

* Sutherland, Phil. Mae. vol. xxxvi. p. 507 (1893).

DD4 Dr. A. O. Rankine on the

iodine. These two sets of gases are adjacent in pairs in the
Periodic Table.

The molecular radii of these gases, calculated in the way
above indicated, are shown in the following table:—

TABLE I.
| |
| Molecular Gas Molecular Rane
Gas. ‘radius X 10° cm. as | radius x 10° em. 10,
Chlorine me 1-60 NEP OMS 22: 1-28 1°25
Bromine 3.4... 1-71 Krypton ... 1:38 1°24

Gdine: Aso... 188 MEMO 2.5. 1-53 125

The figures reveal the notable fact that the dimensions of
each corresponding pair of gases in the two groups are in
constant proportion, the numbers in the last column being
practically equal. In other words, we may say that the
radius of the molecule of a halogen gas is 1°24 times as
great as the molecular radius of the corresponding inert gas.
This statement is, of course, based upon the assumption that
all the molecules are spherical.

The cube of 1:24 is 1°91, or practically 2. This means
that the halogen molecules have practically twice the volume
of the corresponding inert molecules.

The molecular masses are also approximately in the pro-
portion of 2: 1, for the atomic masses of corresponding gases
are nearly equal, the halogens being diatomic and the inert
gases monatomic. We should thus expect the densities of the
molecules of, for example, iodine and xenon to be equal, and
a similar equality for the other pairs. This is set forth in
detail in the following table.

~ ees

Relative Dimensions of Molecules. De

It is, perhaps, worthy of note that the case where the ratio
of densities differs most from unity is that where argon is
involved, and that argon has, from the point of view of the
periodic arrangement of the elements, an abnormally high
atomic weight, the effect of which is to give the above-
mentioned ratio a low value.

Another point of interest in connexion with the molecular
dimensions of these two groups of gases may be based upon
the interpretation of Sutherland’s constant C previously
referred to. This constant is a measure of what Sutherland
called ‘‘ the potential energy of two molecules in contact with
one another,’ but which would be better defined as “the
work done against attractive forces in separating to an
infinite distance two molecules originally in contact.”

The following table shows the relation which exists between
these quantities of work for the pairs of gases with which we
have previously dealt.

TaBLe III.
Pair of Gases. Values of C. Ratio CO halogen), |
C (inert)
Chlorine: Argon ... 325, 142 23 |
Bromine: Krypton. 460, 188 2-4
Todine : Xenon...... 590F 252 23

The numbers in the last column are constant to an extent
certainly within the accuracy with which the values of C are
known. We may therefore say that the work done in sepa-
rating toa great distance two molecules of a halogen gas
originally: in contact j is 2°3 times as great as for two molecules
of the corresponding inert gas.

The attempt to go further and investigate the law of force
upon the basis of these figures has been ‘made by the author,
but has been found to eae to inconsistencies. One is driven
to the conclusion that a single law of force depending in a
definite way on mass and distance only cannot apply to
molecules differing widely in internal constitution.

rae. |

LVI. On the Precision Measurement of Air Velocity by means
of the Linear Hot-Wire Anemometer. By Louis VEssor
Kine, B.A. (Cantab.), Assistant Professor of Physics,
McGill Unwersity, Montreal *.

[Plate X.]

Section 1. DETAILS OF CONSTRUCTION OF THE LINEAR
Hor-WireE ANEMOMETER.

fy a paper recently published by the writer f, the theory
I of the convection of heat from small cylinders cooled by
a streain of fluid was extensively studied, the results com-
pared with experiment and applied to the development of a
precision anemometer intended to be of service in studying
complex problems of gas-flow. The special type of instru-
ment referred to may be called a “ linear anemometer ” in

contradistinction to several forms of integratin @ instruments

which have already been described f. Detailed specifications
are given in the memoir referred to for the construction
of this instrument. Use is made of the Kelvin Bridge

* Communicated by the Author.

+ Read before the Royal Society of Canada, May 28,1913. ‘ On the
Conyectiun of Heat from Small Cylinders in a Stream of Fluid: Deter-
mination of the Convection Constants of Small Platinum Wires, with
Applications to Hot-Wire Anemometry,” Phil. Trans. Roy. Soc. London,
vol. ecxiv. A. pp. 378-482, 1914; abstract in the Proceedings of the
Roy. Soc. A. vol. xc. 1914, pp. 563-570.

{ Preliminary experiments on the use of a platinum wire heated by an
electric current for the measurement of wind-velocity were carried out
by G. A. Shakespear, at Birmingham, as early as 1902, but were discon-
tinued for lack of facilities in the erection of a suitable whirling table
for the calibration of the wires. Ilectrical anemometry was indepen-
dently suggested by A. IX. Kennelly in 1909 (A. E. Kennelly, C. A.
Wright, and J. S. Van Bylevelt, Trans. A. I. KE. E. 28, pp. 363-397,
June 1909), and although the actual application to anemometry appears
to have been made as early as 1911], the results have only recently been
published (A. E. Kennelly and H.5. Sanborn, Proc. of the American
Phil. Soc. 8, pp. 55-77, April 24,1914). Electrical anemometry was
also developed independently by U. Bordoni (paper read before the
Societa Italiana per il Progresso delle Scienze, Oct. 13, 1911; published
in the “‘ Nuovo Cimento,” series 6, vol. ii1. pp. 241-283, April 1912; see
also ‘ Electrician,’ 70, p. 278, Nov. 22, 1912), and by J. T. Morris (paper
read at the British Association, Dundee, Sept. 27, 1912; published in
the ‘ Engineer,’ Sept. 27, 1912, the ‘ Electrician,’ Oct. 4, 1912, p. 1056,
and Nov. 22,1912, p.278). A form of integrating hot-wire anemometer
has also been described by H. Gerdien (Ber. der Deutschen Phys. Ges.,
Heft 20, 1913). The use of a hot-wire anemometer in the measurement
of non-turbulent air currents is described by C. Retschy in a series of
short papers published in Der Motorwagen, vol. xv. March—July, 1912.

Precision Measurement of Air Velocity. D97

connexions shown in fig. 1; the ratio-coils are adjusted so
that a/b=«/8, in which case a fundamental property of this
arrangement is that when a balance is obtained on the

Big. 1.

Diagram of Kelvin Bridge Connexions employed in Precision
Hot-Wire Anemometry.

galvanometer, A/B=a/b=e/@ independently of all connect-
ing- or contact-resistances. The resistances A and B refer
respectively to the resistance of the anemometer-wire between
potential terminals permanently fused to the wire and to
that of a manganin resistance. The resistances a and b were
made equal and about 500 ohms, while « and 8 were
adjusted to equality at about 250 ohms. In order to protect
the anemometer-wire from accidentally burning out, a key
KK, was inserted by means of which it was always short-
circuited except when observations were actually being taken ;
a double-contact key K, was inserted in the galvanometer
circuit in such a way that contact was first made through a
high resistance in the preliminary adjustments ; it was also
found convenient to connect the galvanometer through an ad-
justable shunt. The resistance B was constructed of No. 23 B.
and 8. gauge manganin wire wound non-inductively on an
asbestos frame so as to dissipate a maximum amount of heat ;
its resistance as measured between potential terminals soldered
to the wire was adjusted to about four times that of the
anemometer-wire at room temperature. By means of a fine-
adjustment rheostat R, the current in the anemometer-wire
can be adjusted until a balance is obtained on the galvano-
meter. It is advisable that the rheostat be always readjusted
to the position of minimum current to avoid overheating the
wire should the velocity of air-flow suddenly diminish ; this

5598 ~=-Prof. L. Vessot King on Precision Measurement of

may easily be accomplished by means of a spring control.
In taking a measurement of velocity the key K, is pressed
down and the current as read by the ammeter slowly in-
creased until on pressing down the key K, a balance is
obtained on the galvanometer. From the reading of the
current 2 the velocity V may readily be obtained from a
calibration curve corresponding to the formula

PoigthvV, . . . ann

or from a conversion-table connecting 2 and V calculated
from the above expression.

The ammeter employed by the writer was a Weston
direct-current instrument of range 2 amperes; the scale
was equally graduated over this range, each division repre-
senting 0°02 ampere ; by estimation the current could be
read to 0°002 ampere. If the conditions of air-flow are
sufficiently steady and it is required to resolve smali velocity
differences, the use of a Weston Laboratory Standard
ammeter is recommended; the scale covers a range of
15 amperes and is uniformly graduated directly to 0-01
ampere ; by means of a diagonal scale it is possible to sub-
divide each division directly into fifths and by estimation to
twentieths, so that it is possibie to read the current to 0-0005
ampere.

The galvanometer employed was a Weston portable instru-
ment with jewel-bearings, of resistance 277 ohms and capable
of detecting a current of about 10~-° ampere; this degree of
sensitivity is, in fact, ten times more than is necessary.
When employed in connexion with hot-wire anemometry the
constants of damping are very important in determining
the rapidity with which observations can be made. It was
found that equally sensitive galvanometers varied within
wide limits in this respect.

A convenient form of fork suitable for holding in position
the anemometer-wires, and offering a minimum of disturbance
to the flow of air inits neighbourhood, is illustrated in fig. 2.
Fastened to a block of ebonite are the two arms of the fork
consisting of steel strips about 5 mim. in width. At the end
of each is soldered a small brass block drilled to receive two
fine needles fastened about 1 cm. apart. Threaded through
the eyes of these two needles is a 3-mil platinum wire
having its extremities firmly clamped in the brass block just
mentioned. The ends of the anemometer-wire are threaded
through these two loops and secured in position by being
twisted a couple of turns around the wire; the fundamental
property of the Kelvin Double Bridge already referred to

;

Air Velocity by means of Hot Wire Anemometer. 599

only requires the electrical contact at these points to he
moderately good. The tension in the wire is adjusted by a
fine silk thread carried down from each of the brass blocks
to an adjustable screw in the centre of the ebonite block ;

Fig. 2.

Details of Fork for holding Anemometer-Wires and Potential
Terminals.

[The wire is shown in position over an end of a channel of rectangular
cross-section, and illustrates the guard-ring effect obtained by the
use of potential terminals fused to the wire. |

this thread is also effective in preventing lateral vibrations
of the fork. Carried up from each end of the ebonite block
are two thin steel strips crossing each other to the opposite
arms of the fork, insulated from each other and also from the
fork by means of thin mica strip. These steel strips, which
are held in position along the arms of the fork by two
lashings of fine waxed silk cord, serve to brace the fork and
at the same time serve as potential leads; at each end is
soldered a small brass block drilled to hold a fine needle at
the extremity of which is soldered a short length of 6-mil
platinum wire. To these are soldered one extremity of the
1-mil platinum potential terminals, the other being fused to
the anemometer-wire; this is most easily accomplished by
connecting the wire to the bridge connexions and adjusting
the current until it is at a bright red heat; the potential
wires are then brought to the required position and wound
twice around the anemometer-wire; by applying a slight
tension while this is being done, a satisfactory fused contact
will be effected. The free end should then be broken or
cut off close to the anemometer-wire, so as to diminish the
cooling effect of the potential leads. The heating to which
the wire is subjected during this operation serves to anneal
it sufficiently well for permanent use.

In the course of experiments by the writer on the flow of
air between parallel planes, considerable experience has been
acquired as to the most suitable method of employing the

560 Prof. L. Vessot King on Precision Measurement of

linear anemometer in precision measurements of air-velocity.
In the following sections are set out in greater detail direc-
tions as to the most efficient method of using the instrument
and data now available as to the resolving power, upper and
lower limits of correct velocity registration, life of wires,
sources of error, &c., which may be of use to experimenters
wishing to employ the linear anemometer in aerotechnical
investigations.

Section 2. ON THE SELECTION AND CALIBRATION
oF ANEMOMETER- WIRES.

Platinum wire 24 or 3 mils in diameter is found to be
most suitable for the purposes of hot-wire anemometry.
The metal should be as pure as possible as judged from a
determination of the specific resistance and temperature
coefficient. The wire employed in platinum thermometry is
especially suited to the purpose in that its ele*trical constants
are usually specified with great accuracy. The wire should
be drawn down to 24 or 3 mils, and if found to be satis-
factory on microscopic examination, a considerable length
should be reserved for the purposes of anemometry ; if
ossible the final diamond die employed should be reserved
solely for drawing down anemometer-wires. The wire having
been mounted and annealed in the manner already described
and the potential terminals fused in position, the manganin
resistance should be set to the value previously determined
as that to which the anemometer must be heated by the
electric current in order to attain the temperature best
suited to the type of velocity measurement to be undertaken.
For most purposes a temperature which corresponds to a
dull red appearance of the wire is most suitable. Wires
may be set roughly to the same temperatures by adjusting
the manganin resistance so that a balance is obtained for the
same current when the anemometer-wire is in stagnant air,
carefully protected from draughts. The potential terminals
are generally fused to the anemometer-wire at a distance of
2-5 em. apart, although for some purposes the writer has
worked with a distance as small as 1 cm. The distance
between potential terminals could be made very much less
if the wire is calibrated directly, though at a considerable
loss in galvanometer sensitivity ; a sensitivity of 10-° ampere
would probably be sufficient for use with a wire of 24 mm.
between potential terminals; as the velocity measured is
practically that over this distance, it is seen that velocities
can approximately be measured at a point with a minimum

disturbance of flow.

Air Velocity by means of [Hot Wire Anemometer. 561

It is preferable to calibrate the wire directly by means of
a rotating arm, the velocity being corrected for *‘ swirl” in
the manner already described *. Wires are usually calibrated
by the writer over the range V=60 to 800 cm./sec.,
enabling the constants of the formula

PLES ls et eG

+o be determined ; experiments are discussed below showing
that this formula may with fair accuracy be employed in
the determination of velocity very considerably above and
below these limits. Kor precision work a series of about
ten determinations of current and velocity should be taken,
and the points y=?" and «=,/V plotted on accurate section
paper in order to eliminate possible gross errors or accidental
mistakes of reading. The line of closest fit to this series of
points should then be determined by calculation ; the scale
is altered by multiplying y or wv by a suitable power of ten,
so that these coordinates are expressed by numbers of the
same magnitude. Under these conditions the line of closest
fit may conveniently be taken to be the major axis of inertia
of the system of points regarded as masses of equal weights.
If there are n points, the inclination of this line, whose
equation may be written y=yot« tan @, 1s easily seen to be
given by the formula

4a top 2[Say)/n— ay] (9
OT Seine py
wand y being the coordinates of the centre of gravity given
by =%(2)/n, y==(y)/n. From this formula we find tan 0

and yo, from the formula yy=y—w tan @; hence reducing
back to the original scale we obtain 7)? and #. The caleula-
tion is semewhat facilitated by taking n=10. Theagreement
of two independent determinations of the calibration constants
computed in this way is well illustrated in the case of
wire 17, Table II.

While velocities corresponding to an observed value of 2
may easily be obtained from the calibration curve correspond-
ing to formula (1), a considerable saving of time can be
effected if more than 100 current observations have to be
reduced to velocities by calculating out a conversion-table
corresponding to the formula

Peer Ti eiut p-eay ys CED

at intervals of 0°01 ampere over the range required. The

—
~
—
-

* Phil. Trans. paper, pp. 888, 428; in the sequel this paper wi!
referred to as reference (1).

Phat, Mag. Si 6h Mole2 on No. 172: April L9Ld: 2 ©)

562 Prot. L. Vessot King on Precision Measurement of

calculations are very quickly carried out by the use of :

table of squares, Crelle’s Multiplication Tables anda pee

lating machine ; a 150-eniry table can easily be computed

in an hour. Calculations should be entered with one signi-
ficant figure more than is to be employed finally, and first
differences should be tabulated for convenience in interpola-
tion and as a check on the accuracy of the work.

If the anemometer is to be employed i in the measurement

of very low yelocities (of the order of 10 cm./sec. or less).

the disturbing effect of the free convection current set up by
the heated wire may become sensible. From data derived in
the course of the experiments by the writer already referred
to, the “effective velocity’ of the free cony ection current
eh up by a 3-mil wire at 1000° C. is estimated at about
15 em. Isec., is ae reduced to about 8 cm./see. at 200° C., and.
does not demiich materially with the diameter of the wire*

The effect of the free convection current on the deter mination

of velocity from formula (1) at low velocities would demand a
separate investigation. Some information on this point: ean
be derived from the experiment described under fig. 5, in

which the distribution of the flow of air into a slit in a plane:

is measured and compared with the distribution calculated

from theory. The evidence there discussed points to the:
fact that the anemometer registers velocities as low as.

sy em./ see. with an accuracy of about ten per cent. For

use in low-velocity measurement it is, however, more satis—
factory to calibrate the anemometer under ee conditions.

that the air-velocity makes the same direction with the

vertical as in the experiment in which the instrument is to-

be employed.
The linear relation expressed by formula (1) has been
tested experimentally for velocities as low as 17 em./see.T, and

was found to hold good within limits of experimental error..

Theoretically the linear relation mentioned is the asymptote
to a transcendental curve expressing the true relation

between heat-loss and velocity : 1t is shown, however, that to:

an accuracy of 23 per cent. a linear formula of the ty pe (1)
may theoretically be employed when the velocity is as low
as that given by the relation Vd=1:87x10-7, V being
expressed i in cm.|sec., and the diameter d in emf For a

1-mil wire this limiting velocity is as low as V=2°9 em./sec.,.

probably much lower than the lower limit imposed by the
disturbing effect of the free convection current.

* Reference (1), Table VIIL., p. 424.
+ Reference (1), Table III., p. 416.
t Reference (1), Description of Diagram L, p. 426.

Air Velocity by means of Hot Wire Anemometer. 568

: 2
Fic. oO.

Pin SS “Sols v6) / a 3 ad SD mr.

2 /
Velac ity f 1 ;
ie ae eee | eee
: Go §
&e
ii

CE ee
(\
iH

Test of Anemometer Readings at Low Velocities: 3-mil Wire No. 7.

A short rectangular channel oi width 0°75 mm., having a plate at
right angles to its length fitted flush with its upper extremity, was
set up in the manner ilustrated in the lower part of the figure. By
means of suitable connexions to a gasometer, air from the room was.
drawn into this channel under a constant pressure-difference of 2°35 cm.
water. The distribution of flow at a sufficient distance from the opening
of the channel is approximately that which would be set up in a perfect
fluid by a distribution of sinks along a line coinciding with the opening
of the channel into the plane mentioned. In the neighbourhood of a
plane bisecting this slit at right angles, where the velocity is measured
by the portion of the anemometer-wire between potential terminals, the
distribution of velocity is approximately radial. Taking a set of axes
(v, y) having as their plane the diametral plane just mentioned, with
origin at the centre of the slit, and measuring the axis of y along the
direction of the channel, the velocity at any point (x, y) is approximately
given by V=V VY yl(y?+2), V, being the maximum velocity at «=0.
The anemometer-wire, represented by A in the figure, was set by means
of 2 micrometer-screw in various positions in the plane y=3 mm.; the
observed velocity distribution thus obtained was compared with the
theoretical by choosing V, to agree with the experimental value at
v=0. The figure shows that the readings of velocity are fairly accurate
for velocities as low as 12 cm./sec.; the deviations are possibly due to
the limitations of the simple formula employed in calculating the
theoretical velocity distribution.

ri Oy

264 Prof. L. Vessot King on Precision Measurement of

For the study of velocities lower than 17 em./sec. the
writer has found the whirling-table method unsatisfactory,
as elaborate precautions have to be taken against the dis-
turbing effect of draughts, perfect air- stillness being very
difficult to secure in an ordinary laboratory, while the
correction for “‘swirl” is not easily made and theoretically
and practically is proportionally greater at small velocities
than at large ones*. A more promising method of calibra-
tion is to employ a horizontal or vertical rail along which a
carriage can be driven at a measured velocity and arranged
automatically to move smoothly backwar ds and forwards
with constant speed over its range; the rail need only be
moderately long (2 or 3 metres), as the anemometer has no
appreciable lag and little more than a second is required to
obtain a balance on lhe galvanometer f.

At high velocities the linear relation (1) was tested for
velocities as high as V=900 em./sec., the usual upper limit
at which anemometer-wires are calibrated by the writer in
practice. Hrom experiments on the flow of air between
parallel planes, 1t appears that the calibration formula may
be extended to velocities as high as 2800 em./sec., making
use of a 3-mil wire. An example of a test on this point is
discussed under fig. 4: the distribution of velocity of a
stream of air issuing from a channel 0°75 mm. in width was
measured at intervals of 0°05 mm. When the resulting
curve, which attained a maximum of about 2800 cm./sec.,
was integrated to obtain the total flow in cm.® per sec.,
a fair agreement was obtained with the value of the total
flow measured from the rate of fall of the gasometer, the
pressure-difference over the length of the ehannel remaining
the same in the two cases. Above this velocity it was found
difficult to work with wires of the usual leneth of about
5 em., as transverse vibrations are liable to be aa up which
invalidate the readings and tend to break off the potential
leads, while the tension on the wire required to destroy
synchronization with the free period of a stretched wire in
an air-current is near the breaking-point of the wire. The
upper limit of velocity measurement might be increased
almost indefinitely by shortening the wire or by using
stouter wire; this step results, | however, in diminished

* Reference (1), Description of Diagram IL., p. 108,

+ For horizontal calibration a photometer-bench and carriage could
well be used, while for vertical work the anemometer may be mounted,
suitably counterpoised as one of the weights of an ordinary Atwood’s
Machine, the moving system being electrically driven. An excellent

design for the purpose is the moving lamp photometer described by
Trowbridge and Truesdell (Phys. Rey. iv. p. 290, Oct. 1914).

a ses a

Air Velocity by means of Hot Wire Anemometer. 565

Fig. 4.

a ek
HEE ERREEEEEEE
rte oe aa ae

cae!
ae ee
aul aa ae

ae
mas +t
Tales in

= f nee PEM OT aes | Eos | ate
A a A OMe aA” 5 Oe ita

Test of Anemometer Readings at High Velocities: 3-mil Wire No. 7.

The distribution of velocity illustrated in fig. 4 was measured by
setting the anemometer-wire in various positions by means of a
micrometer-screw in a plane at a distance y=3 mm. trom the upper
extremity of the channel described under fig. 3. The dimensions of
the channel were :—width, 26 = 0:75 mm. ; breadth, d = 5-08 em.;
length, /=5'06 cm. Integrating this velocity-distribution between the
limits «= +092 mm., the total flow per unit breadth of channel is
182 cm.’/sec. under a pressure-difference of 11°5 em. water. From a
series of observations on the rate of fall of the gasometer for various
pressure-differences, the total flow for breadth d was obtained, and
hence the flow per unit breadth, assuming approximately uniform
distribution of conditions over the breadth. From the curve connecting
this flow with the pressure, the value per unit breadth corresponding to
a pressure-difference of 11°5 em. water came out to be 196 em.°/see., in
fair agreement with the value obtained from the anemometer measure-
ments ; the latter is probably an underestimate, owing to the destruction
of momentum of flow as the jet travels through the stagnant air in
this region. The low velocities beyond the edges of the channel are
due partly to a “ diffusion” of the jet owing to its dragging action on
the quiescent air through which it is flowing, and at a greater distance
to an indraught of air from the surroundings into the moying air of the
jet. The distribution of velocity over the channel approximates fairly
well to a parabola.

566 Prof. L. Vessot King on Precision Measurement of

sensitivity and resolving power, while in the use of- stouter
wire the working cur rents increase rapidly and the apparatus
has to be specially designed to meet the resulting heating
effects.

Section 3. On THE RESOLVING PoWER OF THE Hot-WIRE
ANEMOMETER.

By differentiating formula (3), we obtain

VisV = 40/0) A—i,7/"). ~~. 2 eee
We can conveniently define the “resolving power” of the
anemometer by the ratio V/6V, where 6V is the increment
of velocity just observable at the velocity V. Formula (4)
then expresses the resolving power of the anemometer in
terms of that of the pone for current. In the ammeter
employed by the writer the scale was uniformly graduated,
and could be read to 6¢=0°002 ampere; on this basis
the resolving powers of typical 23- and 3-mil wires are
ealeulated for convenience of reference in Table I.

TaBue I.
Resolving Powers of 24- and 3-mil Anemometer- Wires.

24-mil wire No. 17, | 3-mil wire No. 8. |
ViE-@— 0-564)? x 27674. | V=(?—081)?x 1166.
ci=0-002 amp. | 6¢=0-002 amp.
V/eV = 71/0701 —0°564/0?). ] V/V =7t/di(1—0°81/7"), |
i We eV /OVe| GON allo). |.. Verde WOVgRl am |
pach easel pe ho ek Beet face eek al aa |
ainperes.| cm./sec, cm./sec. “amperes. cm./sec. | | em./sec.
0-90 167 342 0-48 | 1-10 18 45°4 | 0-40
1-00 524 | 545 | 096 || 1-20 46 | 655 | 070
1°50 785 141 a6 1-80" |. 684 "| 170)” aes
2:00 | 3260 215 fot 24) |, 3150 264 ee

In most applications the steadiness of velocity conditions
is not sufficiently great to warrant making use of the full
resolving power we the instrument ; in experiments carried.
out by the writer on the flow of air between parallel planes
an extremely good pressure regulation was maintained by
means of the Tépler tilting manometer shown in Plate X.*,
and it was found possible to make use of the full resolving
power of the instrument. The observed resolving power at

* For a description of this instrument, see a paper by A. Tépler, Anz.

d. Phys. lvi. p. 610, 1895; also Miiller~Pouillet, Lehrbuch der Physik,
1906, vol. 1. p. 462.

i
\
}

Air Velocity by means of [Hot Wire Anemometer. 567

various velocities was found to agree fairly well with the
calculated values. If conditions are sufficiently steady, and
a higher resolving power than this is required, the use of a
W Pon Labor: atory Standard ammeter for which 6¢=0:°0005
ampere will give a fourfold increase in resolving power.

lt is shown theoretically * that the disturbing effect of a
thin wire in a uniform stream of fluid is extremely small ;
ata distance of ten radii from the centre of the wire the
velocity is only disturbed by 1 per cent. of its value, so that
the velocity is practically measured at a point.

The anemometer employed by the writer was mounted on
a micrometer-screw in the manner indicated in the photo-
graphic reproduction given in Plate X. Hach division of
the divided head corresponded to 1/100 mm., and it was
found that in very steep gradients of velocity a movement of
the wire of this magnitude resulted in an easily detectable
euange of velocity. The limit of the resolving power in a
steep. gradient is e: asily estimated by referring to fig. 4; a
movement of the wire over a distance of 9°05 mm. resulted
in an increase of velocity from 1570 to 1970 em./sec. As
the resolving power V/SV of the wire employed is about 160,
the change of v elocity just detectable at the lower velocity is
about 10 cm./sec. ; so that it is easily seen that a change of
velocity of this amount can be detected in a distance of
1/800 mm. It is thus possible, by the use of the linear
hot-wire anemometer, to measure velocity gradients as high
as 80,000 em./sec. per em. In extremely “sharp ovadients,
in which the velocity changes very greatly over a distance
comparable to the diameter of the wire, a correction for the
disturbance of flow and the effect of the gradient on the
heat-loss might be appreciable. The correction is difficult
to determine theoretically, but may, if necessary, be deter-
mined experimentally by measuring the same gradient by
means of anemometer-wires of different radii. The fact
that the total flow nee by integrating the velocity
distribution of fig. 4 agrees with that determined directly,
indicates that in ore as high as 8 x 10*em./sec. per cm.
the corrections mentioned are probably small.

Section 4. Sources or Error in Hor-Wire ANEMOMETRY.
(.) Disturbing Effect of the Anemometer-Fork and Wire.
The special design of fork required to hold in position the

anemometer-wire will depend largelv on the nature of the

work to be undertaken. By employing fine steel needles to
* Reference (1), p. 405.

968 Prof. L. Vessot King on Precision Measurement o7

hold in position the wire and_ potential terminals, the
disturbing effect at the point of measurement is reducee toa
minimum : : moreover, the velocity actually measured is that
over the interval between the potential terminals whose
disturbing effect is practically nil, so that a “ guard-ring ”
effect is obtained. The disturbing effect of the anemometer-
wire itself has already been discussed in the preceding section,
and is seen to be extremely small. As the wire is calibrated
in a uniform stream in which the velocity is constant over
the interval between potential terminals, care must be taken
in using the instrument that the flow is also uniform over
the measuring wire; this condition is easily judged if the
wire be employed at a high temperature so that it glows:
dull red, as in this case small variations of velocity can be
detected with great accuracy by the unequal brightness of
the wire. If the changes otf velocity are too rapid to allow
of an approximately uniform flow over the wire being brought
about, the distance between potential terminals should be
hunted until this condition is realized. In precision work
eare should also be taken that the conduction losses in the
wire are the same under conditions of calibration as in actual
use. This condition may be secured by so disposing the
potential terminals that they oceupy the same _ position
relatively to the direction of flow under conditions of
calibration as in the distribution of flow which it is required
to investigate.

Gi.) Lifect of Vuriations of Atmospheric Conditions.

It has been shown experimentally by the writer * that the
heat-loss per unit length from a wire in a current of air of
velocity V is given by a tormula of the form

H =r? = B+(y+BVV)(0—%), - . - (By

r being the resistance of the wire per unit length when
heated by a current 72 to a temperature O—@) above that of
the atmosphere 6). HE represents the radiation-loss in watts:
per unit length, calculated according te the formula fF

B = 22x 0-514(@/1000)*2, . . a

* Reference (i), pp. 399-401.
+ The constants of this formula are derived from the results of
eae and Kurlbaum fcr polished plativum (Verh. Devt. Fhys. Ges.,.
Berlin, xvii. p. 106, 1898).

Air Velocity by means of Hot Wire Anemometer. 569

to* will show that even at very low velocities and high
temperatures the radiation-loss is only a small fraction ‘of
the total loss, and may therefore be neglected in studying
the effect of small variations of atmospheric conditions on
the heat-loss. The constants y and @ have small temperature
coefficients given by
¥=Yo[1+0°00114(8— 4) ] and B=Py[ 1 +0:00008(8—6,)],
(7)
which may also be neglected in discussing the problem in
hand.
According to the theoretical formula developed by the
writer. and Shown to be in substantial agreement with the

results of experiments on the heat-loss ree a evlindrical
wire of radius a, we have

Va kg ane) Gy = D/, TSR ALe i oe hr a)

x, being the thermal conductivity of air, so its specific heat
per unit mass (at constant volume), and oy its density ; the
suffix , refers to the values of these constants under atmo-
spheric conditions and at temperature One

Comparing the approximate theoretical formula

re yg 9.47 V KO Go) > wiv ve 20 (2)
with the ealibration formula
Ga Mdigg RHONA Vegi tt devs NNel toni ae ct GLO)
we have

ry = yl(P@—O>) and rk= 8,(@—@).. . C1)

An anemometer-wire is calibrated under given atmospheric
conditions which do not attect the m: inganin resistance,
and therefore also leave unaftected the resistance 7 and the
temperature @ to which the wire is raised: it is required to
determine the small correction 6V which must be added to
the velocity V corresponding to the observed current 7 when
the atmospheric conditions are slightly altered. These are
separately discussed under the headings of pressure, humidity,
and temperature,

(2) hyect of Vanations of Atmospheric Pressure.—
The study of the variation of the convection constants

with pressure has recently been undertaken by Kennelly

* Reference (1), Tables III. & IV., pp. 416 & 418.

270 ~=Prof. L. Vessot King on Precision Measurement of

and Sanborn*. The experiments were carried out over
a range of pressures from 4 to 3 atmospheres, and show
that that part of the heat- loss depending on the velocity V
varies as ,/o,V over this range of pressures. This result is
in agreement with what we should expect from the theoretical
equation (9) ; and as it is well known that the thermal con-
ductivity of a gas is independent of the pressure, it follows
that the only factor in this equation which depends on the
density is By). We thus have by differentiating (9) with
respect to o), the current i being given, the relation

OPo/Bot+ s0V/V = 0.

From (3) 885/Bo= $95 o/eo=5)%/Po, Where po refers to
atmospheric pressure, we obtain finally

OV/V=—opolpo: os 4. ee

Jt will thus be seen that ordinary variations of pressure
have a very small effect on velocity determinations ; if they
should be sufficiently large to affect appreciably the measure-
ment of velocity, the corresponding correction may be made
by the application of formula (12).

A compensating arrangement to correct for pressure
changes might, if necessary, be devised: an inspection of
formula (9), together with the experiments of Kennelly and
Sanborn, show “that the anemometer measures the product
of the density and the velocity, that is, ihe mass-flow o¢ a
gas. From many practical points of view the measurement
of mass-flow is the desider atum, and compensation is in these
cases unnecessary.

(B) Effect of Variations of Atmospheric Humidity.—The
extent to which the presence of water-vapont affects the

rarious factors involving thermal conductivity and specific
heat which enter into the theoretical formula (9) is diffi-
cult to foresee. As the proportion of water-vapour mole-
cules even at saturation is relatively small, their influence

* A, KE. Kennelly and H. 5. Sanborn, “‘ The Influence of Atmospheric
Pressure upon the Forced Thermal Convection from Small Electrically-
Heated Platinum Wires,” Proc. American Phil. Soc. vol. viii. 1914.
The results of experiment are examined in the light of Boussinesq’s

formula, II = 8A (sox Va/7)(O—8,).

It would be interesting to examine these resuits in the light of the
emendation of Boussinesq’s formula; small discrepancies at. high and
low pressures might thus be explained. The results given in the paper
mentioned are not published completely enough to enable this to be
done ; moreover, an absolute comparison w ould not be satisfactory, as
the velocities are considerably affected by “swirl,” which would be
difficult to allow for without a special determination.

Air Velocity by means of Hot Wire Anemometer. 971

may be judged to be small on the heat loss from the heated
anemometer-wire. In fact this point was specially tested
in the experiments of Kennelly and Sanborn referred * to
with the result that no appreciable effect on the heat-loss
due to this cause could be detected, aithough a small effect
was thought possible. No disturbing effect ou velocity
determinations due to this cause has been noticed by the
writer : it is hoped, however, to definitely settle this point
by a special series of experiments.

(y) fect of Variations of Atmospheric Temperature.
Differentiating equation (2) with respect to @, the current

i being given, we easily obtain by making use of (10)
and eit),

OG 2 280),\(OSE) MU Sa AI ks oan ie)

In the case of a 24-mil wire employed at 1000° C., 7,7=0°5
approximately, while for a range of velocity grea D0 to
2300 em.jsec. i varies between 1 and 2 amperes, so that the
jacuor (1 —2,"/2?)~* varies irom 2°00. to L14. It will be seen
that by employing a wire in the neighbourhood of 1000° C.

variations of room temperature of 42°C. give rise to errors
of velocity determinations which at most are less than one
per cent. If fluctuations of room temperature exceed this
amount, the corresponding correction can easily be made by
Pe tomate (Jk).

In the experiments carried out by the writer, the room
temperature rarely fluctuated more than by + 9 C. Jy vend
as the wire was employed at a high temperature, it was not
thought necessary to adopt temperature- compensating de-
vices. The Kelvin Bridge connexions lend themselves
extremely welltoa compensating arrangement, which we pro-
ceed to describe. The fundamental relation Ay Be¢/b=als
will remain valid at all temperatures if the resistances
(B, >, 8) are constructed of manganin, while (A, a, «) are
of platinum or of a wire or combination of wires having the
same equivalent temperature coefficient e¢ as the platinum
anemometer-wire A. The coils (a, 4) are so disposed that
they can readily attain the temperature of the air-stream
whose velocity it is required to measure. Under these cir-
cumstances if (a. %) refer to O°C., while for convenience
temperatures are measured on the platinum scale, we have

A = Aj(1+€@,) and a= ay(1 + €A%,). ae)
When a balance is obtained on the bridge we have A= Ba/8,

pac ocw. Ny Co.

372 Prot. L. Vessot King on Precision Measurement of

so that the temperature @, attained by the anemometer-wire
Ais given by

Ag(1+e0,)=(B/B) .a(1+ eJ,);
or more conveniently by

6, —0,,=(B/B) - [4o/Ao—B/B] (1+). » (15)

To the order of approximation employed in (9) the law
expressed by this formula will not be greatly altered if the
temperatures are measured on the platinuin scale, the effect
being to alter the convection constants yp and By to slightly
different values yy and By : hence if / denote the length of
the anemometer-wire between potential terminals, we may
write (9) in the form

AP= My! + By’ Vv V)(O,—%,)s

which gives, on making use of (14) and (15),

(B/8)a.(L+ €8o,)? =1(B/B) « [a /Ao—8/B] (yo + Bo'./V) (1+ €8,) §
or finally,
7=1(Y¥9 + Bo'/V) [%o/Ao—8/Bl. . . (16)

This formula indicates that to the first order of small
corrections the determinations of velocity will be independent
of room temperature.

In the practical realization of this system of compensation
the ratio-coils (a, 2) may be made of platinum or of some
metal having the same temperature coefficient as platinum,
but more conveniently by combining two resistances, either
in series or in parallel, so that the equivalent resistance is of
the required value and the equivalent temperature coefficient
that of the anemometer-wire. It is easily proved that by
employing wires whose temperature coefficients are respec-
tively less and greater than that of platinum, the required
combination can always be obtained and_ predetermined.
The paralle! combination is to be preferred, as its greater
current-carrying capacity would materially lessen the danger
of heating from “this source. This point should be car efully
tested for before these coils are inserted in the Kelvin
Bridge. The current in the coils (2, 8) and (a, 6) are
easily calculated in terms of the current through the anemo-
meter-wire *,,and these coils should be so wound as to dissi--
pate heat as readily as possible in order that the change of ;
resistance due to current-heating be entirely neglible for the

* Neference (1), p. 58.

Air Velocity by means of Hot Wire Anemometer. 573
Y OY

maximum current employed in the velocity determinations.
Tf the coils be wound bare in the form of an open grid, the
eurrent-heating may easily be predetermined from the data
obtained from the writer’s analysis of Langmuir’s observa-
tions on the free convection of heat from small wires *.

Jt may be noticed that Pa en eOn compensation may
also be obtained by constructing (a, b, «, 6) of manganin,
and B of a wire, or combination of wires, having the same
temperature coefficient as the anemometer-wire and of suffi-
cient current-carrying capacity to be unaffected by the
heating effect of the measuring current. It is easily seen
that this arrangement is not as satistactory as that already
discussed, since this resistance must be easily capable of
adjustment as different anemometer-wires are inserted in the
circuit.

(iti.) alyeing and Life of Anemometer-\Vires.

The comparative immunity of the hot-wire anemometer
from serious corrections due to such fluctuations of room-
temperature as may ordinarily occur in a laboratory, is
secured by the use of a wire at a high temperature, this
being rendered possible for accurate work by the employ-
ment of the Kelvin Bridge connexions. The limitation in
the direction of high temperatures is a les progressive
increase in the resistance of the wire due to ‘ ‘evaporation ” i
the effect which will be referred to as “ageing” increases
for very thin wires, and in practice sets a lower limit to the
diameter of the wire which it is possible to employ for any
considerable length of time at about 24 mils. Experience
has shown that in the case of a wire of this diameter the
ageing becomes distinctly noticeable after the wire has been
employed to measure about 1000 velocities. In Table H.
are given two calibration formulee for the same wire sepa-
rated by 1060 velocity determinations, showing that at the
high temperature of 1000° C. employ red the ageing becomes
distinct only after extended use. [For this reason it is
desirable to employ the wire at as low a temperature as is
consistent with the source of error represented by (13) being
considered sufficiently small for the purpose in hand ; also
if durability is required to use a larger wire. In precision
measurements of velocity it is advisable to recalibrate the
wire at intervals of about 500 observations; the necessary
whirling arm is easily improvised with materials which are

available in a laboratory +, and the necessary calibration only

* Reference (1), Description of Table VIII, p. 424.
+ Reference (1) p. 428, Diagram [i.; also Plate 8 (a).

d74 Prof. L. Vessot King on Precision Measurement of

occupies about an hour’s time. The necessary corrections
for ageing are easily applied by dividing the observations
into groups of 100 and applying proportional velocity-
corrections to each group. In the case of wire No. 17%
given in Table II., it was found that the correction for
‘ageing’ corresponding to 100 readings was a little less
than the change of velocity just detectable by the in-
strument.

The life of the anemometer-wires depends to a large ex-
tent on conditions of service. In the measurement of steep
gradients of velocity the risk of burning out is considerable,
and during the course of the writer’s experiments was ihe
cause of failure of most of the wires. It is seen from
Table II. that the life usually ranges from 500 to 1000
velocity observations.

TABLE IH.
* | | iDRe in num- |
Bie yn . | Maximum ber of velocity Calibration formule
No. | NOURI velocity. | determina- | for Wire 17.
| | tions.
A a OREM
| ee
¢ | 8mil | 2930cn./sec. | 500 (1) V=(- 0-564)? x 276-4
; | (2) V=(2—0°545)2x 2863
op 3 2500 | 540 | (3) V=(? —0°535)? x 286°3
lebih teoce | | Formule (1) and (2) were
)- |
Taira” me ae | separated by 1060 obser-
r | vations and show the
} 9. > |
“y PO x PLL | ADD effect of ageing. Formula
a Ae 2 sy. _ (3) was derived from ob-
Mi oe eee | WUT oagafiome immediately
| | | following (2), and shows
| | _ the agreement obtained |
| by calculating the con- |
| ' stants from the line of
| | | | closest fit.
Rots
|

For purposes of continuous recording it is necessary to
employ a wire at a somewhat lower temperature : : the result-
ing error due to changes of atmospheric temperature can be
eliminated by the use iat the compensating ratio-coils already
discussed. In this way the life of the wires should be
considerably lengthened and the ageing diminished ; the
margin of galvanometer-sensitivity is ample to meet the
resulting Ahead of sensitiveness. The writer has found
in many cases that the use of a wire at a dull red tempera-
ture in ‘allowing conditions of flow to be readily judged by
inspection, is an advantage which compensates in large
measure the disadvantage of being obliged to recalibrate the
wire at interyals to determine the correction due to ageing.

Air Velocity by means of Hot Wire Anemometer. 375

Section 5. Norges oN VARIOUS APPLICATIONS OF THE
Hor-Wire ANEMOMETER.,

The high resolving power, comparative freedom from
serious corrections, towe ether with extreme sensitiveness at
low velocities, make the linear anemometer a very suitable
laboratory instrument for use in studying various problems
of gas-fow. In particular the instrument has recently been
employ ed by the writer in a detailed investigation on the
flow of air between parallel planes, with especial reference
to the study of criteria of stability of laminar flow ; in fact,
the system ‘of precision anemometry described in the present
paper was evolved with special reference to this problem-
The detailed analysis of velocity gradients furnishes a new
method of attacking problems “of gaseous viscosity, while
investigations on the heat-loss from a wire at different
velocities. pressures and temperatures promise to throw much
light on phenomena relating thermal conduction in cases.

In the course of the investigation referred to, and which
it is hoped to publish shortly, it was found necessary to take
many thousand ea ; although it was found possible
to make as many as 100 velocity determinations an hour,
and to reduce fel very rapidly in the manner already
indicated, it is easily seen that more extended investigations
of this type will require very considerable routine labour.
The equipment necessary for work of this kind is not usually
available in a physical laboratory, and the prosecution of
research of this type is better suited to the resources and
personnel of modern aerotechnical laboratories. In ie
field the linear anemometer described would seem to have
wide field of usefulness as a standard instrument, the One;
for which is several times emphasized in the 1912-13 Tech-
nical Report of the Advisory Committee for Aeronautics *
There is no doubt that a compilation of results obtained in
this way would assist very materially in the development of
a rational theory relating to many problems of aerodynamic
resistance. A recording form of instrument is now under
investigation by the writer, and it is hoped by this means to.
very materially lessen the labour of taking observations ; it
is also hoped to do away with the necessity of calibrating
anemometer-wires by arranging that previously calibrated
wires be issued from a reliable firm of instrument-makers :
the Kelvin Bridge connexions makes the insertion of such
calibrated wires, with potential terminals already fused in
place, possible without risk of introducing error due to im-
perfect contacts.

* Darling & Son, London, 1914, President’s Report, p. 16.

D76 Prof. L. Vessot King on Precision Measurement of

The non-compensated Kelvin Bridge and anemometer can

easily be employed to analyse oas-temperatures as well as
Petocities The special feature of these connexions already
mentioned eliminates the error due to variation of tempera-
ture along the length of the current- and potential- -connecting
wires and the same advantage is obtained as in Callendar’s
system of compensated leads in platinum thermometry.
When using the anemometer-wire as a linear thermometer,
it is necessary to employ a very small measuring- current
and a more sensitive galvanometer than in velocity measure-
ments; the resistance B is reduced to the proper value by
shunting by means of a suitable manganin resistance, and
fine adjustments are made by including in parallel an adjust
able high resistance such as that which forms the essential
part of the Kelvin V arley potentiometer. If temperature
distribution is to he measured in a stream of air at high
velocities correction should be made for the ‘kinetic
heating,” 2.e., the heating-effect due to the impact of the
air-molecules on the wire. From some experiments of the
writer, this kinetic heating-effect 1s roughly proportional to
the square of the velocity, and is about 1° C.at 1500 em./see.
in the case of a 25 mil-wire*. If very accurate tempera-
tures are required the measuring-current should be varied
and temperatures extrapolated to zero current. By pro-
ceeding in this manner the writer has succeeded in measur-
ing temperature as well as velocity distributions to 1/100 ofa
degree C., making use of a galvanometer of sensitivity 10~°
ampere. ‘Lhis additional ‘property renders the hot-wire
anemometer useful in investigating conditions of heat-transfer
from gases and liquids to solid surfaces; data on these
points would be of importance in many technical problems
relating to methods of air-cooling of internal-combustion
engines, oil-cooling of electrical transformers s, and the ven-
tilation of electrical ma ichinery.

In connexion with the modern types of oscillograph the
employment of the anemometer in the measurement of
variable gas velocities and temperatures might furnish results
of value in many engineering problems.

In technical problems relating to the measurement and
recording of gas-flow in pipes, the anemometer employed in
the form considered would seem capable of useful develop-
ments; its property of easuring the mass-flow at all
pressures has already eee Bane’. In particular, the
application to the measurement of steam-flow would give

* Compare the experiments of Joule and Thomson, Trans. Roy. Soe.
June, 1860; Kelvin, ‘Collected Works,’ vol. i. pp. 900-914.

Air Velocity by means of Hot Wire Anemometer. 577

recorded data from which might be obtained the efficiency
of any steam-operated engine. In the case of the steam
turbine, the possibility of obtaining a record of performance
analogous to the indicator-diagram of the ordinary recipro-
cating engine might make the subject worthy of a special
investigation,

In closing, the writer has much pleasure in_ thanking
Professor H. 'T, Barnes, F.R.S., Director of the Macdonald
Physics Laboratory, for the kind way he has facilitated the
present work by every means in his power.

SUMMARY.

(1) Specifications relating to the construction of the
linear hot-wire anemometer are given in detail ; by means
of the Kelvin Double Bridge connexions it is possible to
employ wires at high temperatures, thus making velocity
determinations practically independent of ordinary fluctua-
tions of room temperatures.

(2) The most suitable methods of determining the con-
stants of the calibration formula 7?=%?+h,/V are discussed,
together with evidence as to accuracy of registration for
velocities less than 50 em. [sec. and greater than 900 em./sec.,
these being the limits usually employed in calibrating
anemometer-wires.

(3) If the change of velocity just detectable by the in-
strument is denoted by 8V, the ratio V/6V which defines
the resolving power of the anemometer is expressed in terms
of that of the ammeter employed in connexion with the
apparatus ; it is shown that with an ammeter reading to
0°002 ampere the resolving power of a 24-mil wire at
V=3800cm./sec. is about 140, and the change of velocity
just detectable is about 6 cm./sec..

(4) Various sources of error are considered in detail ;
effects of variations of atmospheric pressure, humidity, and
temperature are dealt with and are shown to be negligible
under ordinary circumstances if a high-temperature wire be
emploved. Data are given on the ageing and life of anemo-
meter-wires. It is shown that the anemometer measures
the mass-flow of a gas; a compensating arrangement of
ratio-coils is described which eliminates variations of room
temperature as a source of error.

(5) Applications of the hot-wire anemometer to physical
and technical problems are described.

McGill University,

Nov. 19th, 1914.

Phil. Mag. 8. 6. Vol. 29. No. 172. April 1915. 2 P

eos

LVIII. On the Coefficients of Self and Mutual Induction of
Coazial Coils. By 8S. Burrerwortu, J1.Sce., Lecturer in
Physics, School of Technology, Manchester*.

ih LTHOUGH many formule have been given ior the

mutual induction of coaxial circles and solenoids,
little seems to have been done on ihe mutual induction of
coils for which the ratio of the inner and outer diameters
differs considerably from unity. The present investigation
is to supply suitable formule for such cases.

The method adopted is to find the mutual induction
between two mutually external semi-infinite coaxial coils
having zero core diameters and unit winding density (?.e. the
number of turns per unit area of channel section is unity),
and then by applying the laws of combination of mutual
inductances to find the mutual induction between finite
hollow coils.

The results are extended so as to include self-induction.

The semi-infinite coils of the nature indicated will, for
brevity, be referred to as ‘‘solid coils.” If we take the

radius of the larger coil as the unit of length, then only two
variables are involved, viz.: the radius of the smaller coil
ranging from zero to unity id the distance of the coil faces.
Dimensiona! considerations will give the correct formula
when the radius of the larger coi] is not unity.

2. For any magnetic field possessing circular symmetry
about an axis, the 1 magnetic potential Q satisfies the equation

9 )
Cee OM hoe <9, OL
op’ © p Op a
in which z represents the distance along, and p the distance
from the axis of symmetry. A solution of this equation is

1 = (" pre To(rpyar a

£40

reducing when p=0 to

0 =("gayemar, PO

If ® is the stream function (7. e. the magnetic flux through

* Communicated by the Author.

Coefficients of Mutual Induction of Coaxial Coils. 579
a cirele of radius p and centre at 2, 0), then

i O
o— —2ar | "pe dp

S00)

=2np | GOON CNS Ch) rei uy CAS
0

or on expanding J,(Ap) in ascending powers of p and making

use of (3),

pees os N72 (2 a L 1 aie 2
Pg a4 ) (5) be ja +l dz2t! yD
so that ® can be found at all points if the potential at all
points along the axis of symmetry is given.

Now let a solid coil of radius + extending from z to infinity
be placed with its axis along the axis of symmetry. The
aumber of linkages that the old makes with this coil is

(-)e a 2°Q, a

] aE ORS 2 ‘ £6
2 2Qn+3)in Invl d2” (6)

3. If Oo is due toa second solid coil of unit radius extending
meom c—(0 to z-=— oa) then

Os an (/p+2—p)dp
0
i au

= NuTmes LOO

shienerc?— Was)
jon g is large then (7) may be expanded in inverse powers
Ol Zz o giving
a ee eae) 2s 1
an 22 (2843) |s 's+1(22)”
so that
1 @O 1S (-) P@ts) 1
| "= > == | ES)

(eat i eet (28+ 3) is’ ls ita 1}. (Ga

s=0

2p s

580 » Mr. 8. Butterworth on the Coefficients of

Inserting (9) in (6) and rearranging so as to express N
In inverse powers of ¢,

p=@

ek ——)|2 2p n=p . gen
179" = (22)? = 1 (2n+3)(2p—2n4B) in |p —n nk Ip—n+1
-, =.

or on expansion

(22)° o7 Z
Fries tO) 26 eae a! )
+ @>s(a3+ 9” aia 0

BS iste. Wie) elm iawn co re

From this formula, N (which represents the mutual in-
duction between the two solid coils) can be obtained to 1 in
100,000 if z>3.

4, When z is small (7) may be expanded in direct powers

of z giving
0 if ile}
= prac ((

+> so a

In this formula the factor —4z* log z requires special
Seger 2 5 | P
treatment. Denoting it by w/2z,

1 do 3 1 dx@ 2n-—3
> = —(log e+ D See

2a dz” Dy Or dz — aanapis

Hence by (6), if x represents the number of linkages due

to @,
1 ve 5)
eerie ae” airs ft log z+ 5)
Ay | a ned “ p AQ 2 2

— =): j2n—3 (3) st
2 oh (2n +3) iD in+1 2~ fos ie (12)

—F

This converges so long as z>r.

For the remaining portion of ©, it is preferable to retain
the finite form (7) because of the slow convergence of (11).

Self and Mutual Induction of Coawial Coils. a81

Hence asa working formula for z>r we have

ee = A—(B—a2)r?+Crt—Dr® . . . (B)
in which |
Sey Slog =e + ob
10B = log +2F — =.
140 =— (+a) ' (B’)

atl Behe SSB
ee) ae nt )
CG ammeo SVS

2 9nt=D (— yea 21—3 2ir |
a = =) > aT
( 1a, (Coa) n Tr 2) )

If we use the series formule (8) and (9) for 3A and
40 B, when z>2 the range of formula (B) is 4>2z>7.

5. When =z is less than r, the method followed in the
preceding sections fails to give a convergent series, because
of the logarithmic term in auby For this case the method
adopted is to find some simple magnetic distribution which
will give rise to an axial potential containing the term
Te log 2 z and other terms for which the preceding method is
applicable. The linkages due to this distribution are calcu-
lated by direct integration, while those due to the terms not
containing log are found by the preceding method. The
difference between the two results gives the linkages due to
mz log z

Let there be a linear distribution of poles on the axis ahs Z
extending from z=0 to <=—e, and having a density 7
The potential at 2 due to this distribution is _

=@ + {2 es gf oe) Ses)

in which as before o= —az? log <

Since we only require the linkages n corresponding to @,
we can choose ¢ to have‘ any convenient value. It will be
supposed that ¢ is very large. Then, by the method of

582 Mr. 8S. Butterworth on the Coefficients of

Section 2, if 7’ are the linkages a to a’,

n' min i os Cm .
Is = 315 log (e+2)—cr + 7 1 log (e+ es

+ terms which vanish when ¢ is ian a ee

To obtain the linkages x’ by direct integration we must
find the work to be done to remove the linear magnetic
distribution from the field due to the solid coil of radius »
whose (south-seeking) pole is at z, 0.

Now the axial potential at a distance « from this pole is

by (7) py eek |
>) dP + J 7 = e tad = hater, aan | we. ;

— O42, = —a(0° log ———_—— +7 Vi¥ +a? —2re),(15)

ra | ram \9 2)

aon nla—a (c—2)0)(a,rde . . - (16)

e/”

By direct integration,

2 (e=2)%(6, 9) ae ;

a) Ave l 3B A2 1 To 492 4 a?
aa ia 5 ate+ aa: 0g

=(5s)r°— 3" #2 )log vr MP a
20 7

i oh, SN RM 2a)
+7 V/ 4 +x (79° meres Te. e pte 2+ SMe

A ane) |
=2r(wta grt see), ae

) ey
]
L. .
=— 57'z (when c=().
. »
Putting . =c+-<, and supposing ¢ large, (17) becomes
) ‘ Ie ~ 4
- 2-7 los (e+ <¢) — =P Der 2 ’

+ terms which vanish when ¢ is infinite. . (18)

When w is small, the integration (17) can be obtained in
serles by using thie: form (11) for Oo, and integrating term
by term, “the oonatant of integration being — }74z to correspond

with (18).

———

Self and Mutual Induction of Coaxial Coils. 583

Using this method for the lower limit of (16), and com-
bining with (14),

a 2 (- EAN)

27? 3 \2

ean Gulch a) Gee 3) Dee an Hiisrene Name Tl
ee coe

ao ete (—)” |2n—3 ih ,2n

r. (19)

Using this to replace (12), we have instead of (B),

N 2 ye 1 ‘ ; | |
Qa styles + A’—(B'—£)r? + Cr*—Dr®. (C)

in which

ee ane oO ek 2
BAT = Fe i. Ok g ge
B'=log 2(1 Be? |
40B' =log 2(1+¢)— et 50° | i
} 2 ‘ & (C’)
SB ea, aa ae ee = :
oN 3 Fewer a0 tS a |
pola (- —\ 2n—3 22n |v?” |
7) ot ee el [jn PV n+1) Tn 3) \ }
6. When z=0, (19) becomes
n 1
gues 1 AG io (oe, — 5): Ue pee 21)
Hence by (6) and (11),
Noss ay i AG) te
Jet 6 ee (los, est dete (2
where
N=o \? 2n— 3 |2n pana
T= 22 2-2(In 4-3) |v [a |n—1 [n+
iyi toes Lior On +- 37 yr
a a Gas Oe a Qn+4/) (Qn+ 7)(2n+3)(n+4: 3)(n+ 15)
. Asa check on this result, the case of <=0 will now be

Pied by another method, inv ‘olving elliptic integrals. The
method is to start with {he known elliptic integer ey formula
for the mutual induction between two coaxial eolenwae and,

a8 Mr. 8. Butterworth on the Coefficients of

by integration, to find the mutual induction between two
solid coils.

Certain reduction formule are required, and these will be
dealt with first. If E and K are complete elliptic integrals
of the first and second kind to modulus «, then

CB Hea dik K es

dar a? de «a1—zx*) nee

From (21) we readily derive the following reduction
formule,

(42) (eBdr= or B+ Kae i . 6 oc) Sen

(n+ 2)? fe m2 Kdae=a OB (n+ 2)a"1(1—a?)K

+(e? (o "Kade! « i0 ae
‘E 1 E
| pde= (5-«)K-2 =), 9 0) er

by means of which {ewae, fokae can be expressed in

terms of EH, K | Kae and (5. if n is any integer positive

or negative.
In the succeeding work we are particularly concerned
with

2 1
Hee ti Kdz, and c= Kae . sy ea
OU

Expressing K in series,

eg mt Mec 4 2n—1\? x7
5251 a(S) Sh :
a 2 a +2 FBAG =. 2n 2n-- 1. (26)

usu a : eames | ? 6d0
: ; (l—z’ sin? 6): > sind

=2(1- 3 = — --)=1'83193.. . (27)

Self and Mutual Induction of Coaxial Coils. 585

Also
mas dx sin 6
ly aa w(l—a? sin? u(1—.? sin? 0): ae a) ee y
= T Eda > add ae 6 cot 610 8s
2). ya-y)t J, sind | J, —a? sin? 62 28)

by integration by parts.
Performing the last integral of (28) by expanding

(1—.? sin” @)-= in ascending powers of v, and integrating
term by term,

ag |() ui eS (FS Danie ee NS aoe (2
eo 2 | Same tet. Ore

8. The mutual induction between two semi-infinite co-
axial cylinders of radii a and 4, external to each other and
with their ends in contact, is

m= smal (e+ 0)H—(e—0)Kt, .. (30)
the linear winding density being unity, and the modulus of
E and K being 0/a, with a>.

The mutuai induction between a solenoid of radius a and
a solid coil of radius 6 (a>6) with their ends in contact is

eR to
: Ta | (2? +P)E—(?—&)K le IU)
which on applying the reduction formule becomes

dette 3
ma gray (4 +2) | B—5 (1-2) 7K wae : (32)

When b=a,

My, =

= Mo= : abt ao = 3). (33)

The mutual induction between a solenoid of radius 6 and
a solid coil of radius a (a>6) with their ends in contact is

a
M3 = M_+ ( mda

ue ellen ayes (= 33 3) K

Me yaa(otun) b. (34)

586 Mr. 8. Butterworth on the Coefficients of

Finally, the mutual induction between two solid coils of
radii a and & with their ends in contact is

=a {lir(14 °° )E—r(14 +377) 1-2) KK q
—3u—37?(v+u)}. . (85) .

in which r=b/a.

When r=1,

ae Bu;)=2°4094 w (by (27)). . (36)

Inserting the series for E, K, u, v in (35) we find

N , 3 ee
Ie = & [is 50? *(log.- - +35) + 37 to}. (37)

LS i ee yon ae ‘
ele w26-2n+4/ (2n+7)(2n4+3) (n+3)(n+1)° (38)

This result is identical with formula (D).
When r= 1, we find (to five figures)

N=274094 0°,

where

which is in agreement with (36).

9. By means of formule (A), (B), (C), and (D) it is 4
possible to evaluate N for all values of z, and all values of 7 q
up to unity.

The formule have the following ranges :-—

(Cay) eo ay (B) 4>2>7,
(CG) Pr ae > 0: CD) 2-0:

Table I. shows the agreement of (A) and (B) when -=4 ;
Table II. shows the agreement of (B) and (C) when s<=7:

N 3
- being the numbers tabulated.
Dar77? = rT

; : N :
In Table II. are tabulated €=>—,,, for -<4, and in
SAT
R iy IN ji :
Table IV. aré N= G — 13> the forms € and 7 being
chosen as being the most suitable for graphical interpolation.

For interpolation in 7 it is convenient to notice that 7 is ¢
almost linear in 4° ene Wel par

Self and Mutual Induction of Coarial Coils.

_——

, Formula (A)| 0:0137566 | 0:0137

APA hn Li:

—_

0-2.

| OA: |

0-6.

0:8. 1-0.

419 | 0:0137172

0°0136836 | 00186406

(B) 010137564 ols 0-0137172 | 0:0136834 | 0:0136405
| |
TAB Es

| | | My
| eo | 02, | 0-4. | 06. 08. ra) |
__¢ Pormula (B), 0117413 | 0087850 | 0-067835 | 0-054688 0-01547 |
pe (C)) O-117417 _ 0087348 | 0-067834 | 0:054686 0-045476 |
Roc. | | ao eNews, | |

Teen TH es

Iacpinia JOB a oe Int, 3

BONA ee Oey a 0-2. 0-4. DEY |

10 | 07122063 0:09469 0:075971 0-062649
| 08 0°134215 010182 | 0:080307 0:065405

06 0°145694 010835 | 0084106 0-067835

O-4 0°155685 0:11378 0:087350 0-069746

02 | 0:163293 | O-LI741 0:089389 0:070970
meer «| et08. | 1-0. 2-0. 4-0.
1-0 0:052866 = @045477-—«| «(0026022 v-013641

08 | 0054687 | 0:046722 0 026297 0013684

06 | 0:056260 | 0:047778 0026520 0-013717

04+ | 0057477 0048585 0:026685 0:013742
Ge / 9-058249 0:049093 0:026785 0 013756
ee ‘ Ee

: E AL New
AeA pe,
SNM oan,

P, eA, DROS iho ah aie Ta
Po 0001986 1292 0906 0514 °| 0331 |
Pcs | 1642 1066 0746 0493 | 0271
| O6 1375 0888 0620 0351 0225
pe 0'4 1176 0760 05380 0300 0192

02 TOHS aii GESS 0476 0269 0173

288 Mr. 8. Butterworth on the Coepicients of

10. ALutual inductances of jinite coils.
(a) Non-overlapping coils.

By definition, N gives the mutual induction between two
semi-infinite aot ‘coils of unit winding density, having
radii + and unity, the distance of their faces being <. It
the radii of the coils are a and /, the separation ec, and the

winding densities 2, 7, then by dimensions the mutual
induction is

nynoa®N ( © “) 2 2

ae
with b<a.
If b>a, then from the reciprocal property

= /@ b Cc
IN = ad —_— ye? e e . . (
on(<, A wN(5, 4). (40)

Tf the coils are hollow, and the inner and outer radii are
ty, da; 6,, b, respectively, then from the laws of combination
of mutual inductances, the mutual induction (M) is given by

ey Ry et NO ye AND
M/n,7.=a,° < N & 7)N @ at
B Ug ds Ug hy:

Dye e (ae c ae ae.
a UN(s. ae N(_: ea

Li dy
When the coils have the same radii (41) becomes (using

(40))
M/nins=asN (©, 1)— 2a5N(£ =) bain (7 1) . (tla)
As = ds
Jf in addition, the coils are in contact
M/nyng= (a9? + a) N(0, 1)— 2uiN(0. s . (416)

Now let the coils be finite and of lengths 2l;, 2/,, the
distance of their mid-points being h. From the laws of
combination of mutual inductances we find

M—IM(h hy ==4,) 4- M(h-+ 1, +1.)
—M(h—/, +1.) —M(h+l,—-1,), ° = (42)

in which M(c) is given by (£1).
When the coils Shes e the same length

M=M(h—2l)+M(h+21)—2M(h).. . (42a)

Self and Mutual Induction of Coaxial Coils. D989
(L) Overlapping coils.
Regard the field inside the outer coil as made up of two
portions,

(1) the uniform field calculated by assuming the coil to.
be part of an infinite coil;
(2) the field due to the polarity of the coil faces.

If M,, M, are the linkages through the second coil due to
these two fields, then M=M,+M,; M, is given by (42)
and (41) (no regard being paid to the sign of ¢c), and

M,= Sm nyrok(a—ay)(Os8—b,), . . (43)
oOo

k being the length of the overlap.
When the coils have common centres, h=0, and (42)
becomes

M,=2{M(,+1,.)—-M(,—i.)}.. . . (495)
If in addition they have the same length (20),

Mp 2{Mi(22)—M(O) Pc.) 2 |. (420)
11. For the purposes of calculation, it is convenient to
alter the notation as in the following example.
Let the coils have the following dimensions :—

Outer radi) e Gs — 0) entea 05 ==) 4.em:
nner Tracie =) Cie i= AK
Lengths, 22a 2 =o emes2/.—445 em.

Displacement of centres=h=21 ecm.

Then since the amount of overlapping is 4 cm., we have
from (43)
9
ogy, = 3 x4x 5x (44—23) =746°7.
Again,
he — == Cie C—O Cai 2; =O5
h+l,+l,=co=46, €5/ Qa == 25 = 46, C5/ 0, —Zg =9°2
h—l, +l,=c,=40, Cee €/@7—es one
hth—h=c= 2, C4/Ag=2,= 02, ¢4/a,=2,' =0°4
2

bofag=7,=0°4, b)/a? = =0°2, 6,/a;=7,=0°8, b,/a,=7,' =0°4.

* Maxwell, ‘ Electricity and Magnetism,’ vol. ii. p. 312.

290 Mr. 8. Butterworth on the Coeffietents of

Therefore by (41) and (42), using the notation & of the
Tables and putting

Enq for E(zp, rq), E'pg for Elzp, 7q’)
= a4 8 (En + &1— £91 — Er) — 7 (E 12 + E92 — Ego — Ena) f
<=) (LP 5 p,'3 (E15) + 49 — E59) —E'n) — 1" 3(E yo + Eno E a5 — Ep)
5 As
= eye 282 )
10°( me x 0° ti oe

277 ny Ne

0
—5'(— 2 x 0036524 = x 0103077)

~~
= —104°4.
* M=M)4 My = 27? io( 7467 — 104°4) = 12, 68040.

12. Selj-inductances.

Since the self-induction of a coil is the same as the mutual
induction between two coincident coils, we have by the
method of section 10,

b= 1,
in which J, is the self-induction calculated by assuming the
coil to be part of an infinite coil and Ly are the linkages due
to the polar field of the coil.

If the coil has length c, outer radius a, inner radius b,

winding density n,
Ly =srva's(1—ryr +2r+ 3), . oes
in which z=c/a, r=b/a, and no allowance is made for the

space occupied by insulation.
Also Ly, is given by the M, of (42 ¢) in which (41 a) holds,
so that, using the present notation,

Ly=2{M(c)—M(0)}. . . . . (45)

ee LNG a N( i) : (46)
M(0)/n?=a@ { (1+2°)N (0, 1) —2N(0, r) }
or using the notation &, 7 of the tables,
Mie )[2rena? = Ele, 1) — 2 (2; 7) 19°F G, 1)
fe

when << 4,

(47)

M(cje 2 mena = (1) —H( 2, Lae °"n(<, 1) —F ‘a(7 1) :
when z<>4, |

and

\
;
}
M(0)/Qr2n2u® = (1 + )E(O, 1) —27°E (0, 7). )

Self and Mutual Induction of Coaxial Coils. Sob

When z2>4, we obtain an accuracy of 1 in 10,000 if we
use

Dat(1-2+8-%),. 2. . us)

in which «, 8, y are functions of + and are tabulated below.

Darn

| | . | |
Tr a. | p. | y: |
CAN | De He
0-0 0-73238 033333 «=| «(00953
0-2 0-73699 033719 | 0:09738
0-4 0°75574 035579 | 01071
0-6 078447 | 0:39042 | 0:1306
0-8 0-81718 04890610) |) O-1701 |
10 | 084883 | 050000 (02806

| |

The values of «, 8, y have been calculated from the
formule

qa=2{(1+7°)E(0, 1) — 27°E(0, 7), |
ik
i Pe.
ae cate +. (49)
a 16 1) — 27° verre 1) 5 |
iL
q=3 (1—1)?(1 + 2r 4 37°),
with z=4 in the expression for y. Hence the tabulated

values of y are only correct for <=4. Howey er, for larger
values of z, the error in the final result js alway s less fran
1 in 10,000.

The neve formule fail when 7=1, but in this case the
coil becomes a thin cylinder, the self-induction of which is

: eee 4 Ul, JL NS a BCS SE 30 JOUR
Bol bean eas ties east) OO

so that for n==1,

i Key gu us mak ®) 30
3a’ een tee bak
* Russell, Phil. Mag. vol. xiii. p. 420 (1907) ; Havelock, Phil. Mag,

vol. xv. p. 3832 (1908).

D992 Coefficients of Mutual Induetion of Coaxial Coils.

13. In order to illustrate the method of working for short
coils, take a coil having

outer radius=a=4 cm.,
inner radius=b=2 em.,
length=c=4 cm.
Then s=c/a =, h—=0/@—\7 oe
By (44) ti 2arcn2a? — (07229 lode
By (47 a)
M(c)/2a?n?a’ = &(1, 1) —2(0°5)°E(1, 0°5) + (0°5)7E(2, 1),

= 0045477 — 5 x 0°048216+ me x 0:026022,

ae
=():0347a6)
ets 1 : i Bek:
M (0) /22r?n?a°? = (1 + =) x 07122062 — fi x 0°150930,

= (088144.
Therefore L=bL,—2M(0)+2M(c)
=a (Ou 2351)
TON.
The Stefan- Weinstein * formula for the same coil gives
L=2459:5n?,
so that the error in using the latter formula for this coil is
0:23 per cent.

14. Conclusion.

1n applying the formule and tables, their range of appli-
cation should be borne in mind. They are intended to be
used only when the inner and outer diameters of the coils
differ appreciably (b/a<0°8), and when the coil-lengths
are not too small (c/a>0:2). An exception to this rule is
Table V. which (with graphical interpolation) holds for all
values of b/a. For coils whose dimensions are outside these
limits the usual solenoid or circular filament formuiz are
more suitable, the geometric mean distance correction being
applied to the channel section.

It should also be noted that no allowance is made for the
insulation space of the winding.

Finally, by successive differentiation of the formule for
the function N, many known formule for the mutual induc-
tion between solenoids, flat coils, and circles may be obtained.

* Fleming, ‘Principles of Electric Wave Telegraphy, p. 140 (2nd
edition},

f 6593).

LIX. The van der Waals Formula (and the Latent Heat of
Vaporization). ByT.Caruton Surton, B.Sc., Government
Research Scholar in the University of Melbourne*.

7” the Journal of the Chemical Society, 1914 (p. 734),

Applebey and Chapman derive a “ formula for the latent
heat of vaporization’ which gives results in good agreement
with those of Mills and Young f.

The impression given by their paper is that the Mills-
Young values are ‘ observed,” and may be used as experi-
mental data confirming theoretical results. This is so, only
in the sense that the Mills-Young values are “calculated
from observed” values of pressure, temperature, and volume
by a process very similar to that which Applebey and
Chapman themselves employ (see Appendix).

This considerably modifies the inferences that should be
drawn from their work.

The chief difference in the processes is that Millst uses
the Biot’s formula

log p=A+ Bat + Ce!

to express the relation between vapour-pressure and temper- |
ature, where ABC «8 are five arbitrary constants chosen to |
suit the measured values of the vapour-pressure, ¢ is the
temperature, and p the saturation vapour-pressure ; whereas
Applebey and Chapman prefer to use a form of van der
Waals’ equation

(p+ “:) (v—b)=Rt,

in which 6 is to be treated asa variable. ‘They find after-
wards that 6 varies uniformly with the temperature. Con-
sequently, the equation may be put in the form

(p =F “.) (v—b.—yt.—t) = Re,
where ¢, is the critical temperature, y= ay =constant, and

at
b, is the value of b at the critical temperature.

* Communicated by Prof. Sydney Young.
+ Sci. Proc. R. Dublin Soc. 1910, p. 412.
+ Journal of Physical Chemistry, 1902, 1905 et sqq.

Phil. Mag. 8. 6. Vol. 29, No, 172. April 1915, 2 &

594, Mr. T. Carlton Sutton on the

As shown in the Appendix, this leads to the results

L=¢(7—v,){B log a. 4'+Clog B. Be}
and

i me Ri log iam b 1 1 ) db

u—b +Re(— —b is Vo—O dt ; te

respectively, where L, v, and v, are the molecular latent heat,
molecular volume of liquid and molecular volume of vapour
at temperature ¢, d is the van der Waals’ constant closely
connected with the volume actually occupied by the molecule,
and R is the gas constant.

The following table compares the results so calculated
with such experimental measurements as have been actually
made. The latter are meagre, and have been made at the
boiling-points (under one atmosphere pressure) only.

L (per grm.) Caloulated |

92°°4 75°95 75°6 79:0 ‘| Brown, 1908.

van der Waals Formula. 595

The modified van der Waals’ formula
(p+ “) free eee
db
where Y= 7 =constant,

leads to results very close to the measured results. In many
ways it is to be preferred to the Biot’s formula

log p=A+ Bai + CB.

(1) It contains three adjustable constants, whereas Biot’s
formula contains five.

(2) Hach of these three constants has a physical signi-
ficance.

(3) The agreement with the measured values of p, v and
t is very close—the discrepancies being of the same order of
magnitude as when Biot’s formula is used.

(4) The derived equation (B) gives values of the latent
heat of vaporization in satisfactory agreement with the
measured values (e. g. at the boiling-point under one atmo-
sphere pressure—see Table), whereas Bakker, using the
unmodified van der Waals’ equation, obtained results 20 per
cent. too low*.

(5) The tables in Applebey and Chapman’s paper (pp. 739-
742) show that the agreement between the two series of
calculated latent heats is still better at higher temperatures.

The fact that 5, s is found to be constant, suggests that the

molecule ee cae with increase of temperature.

[A uniformly expanding atom has been assumed by Schott
in the Adams’ Prize Essay for 1908+, and by Richards in
some researches published in the Zets. Phys. Chem. xlii.
and subsequent vols. |

Applebey and Chapman remark (p. 742-3) ‘ for halogen
derivatives” of benzene “the calculated values are all much
too high, the mean percentage differences...... for chloro-
Eeuzene, bromobenzene, and iodobenzene being respectively
aiae, 0:04, 5°3227 There are no experimental results with
which to compare these two series of calculated values. It
seemed desirable, therefore, to obtain Mill’s constant, pw
for each of these substances.

* Bakker, Zeits. Phys. Chem. 1895, p. 519.
+ Electromagnetic Radiation (Schott), Camb. Univ. Press, 1912.
t See ‘Journal of Physical Chemistry,’ 1905 and 1909.

2Q2

596 Mr. T. Carlton Sutton on the

Chlorobenzene.

Miiw’s Constant.

Calculated by Biot’s formula. Calculated by |
Temperature. — “~ =~ Applebey’s |
Reeale. from formula.
————EE Cale. 1905. revised data 1909. |- — |
EER Cherm 79°46 79:9 85:9
TOOT ee wheast cc 80°83 80°9 " ( S86 |
OO Sh ie: ce 81°34 81:5 84°5
SAS health 81°84 82:2 83'9 |
DAQS) Aim a eeen. 83:06 82°8 84:0
Dr Doe) eee | 84:93 82:5 85-4 |
Bromobenzene.
1OOSIC SS. 56°93 ses | 59°8
to ee | 54:10 54-6 59-6
1800) Ble | 54-98 55°0 59°6
QO csaen 55:9 55-7 59:2
28! | Aa ae 56°25 56:0 58:7
CE a esas 56°44 55:8 58:8"
DEO?) Fe eek | 56°92 56°1 58°7
DOCH, bec: | 57°15 56°2 586
Todobenzene.
19OS Gs Pate: 41°87 42°9 45°9
IA), 0 a in re 43°64 43°5 460
AS ASUS a Aa 44°38 43°6 46:5
OCs PEA) 45°10 43°2 46°9

Mills has twice calculated the values of yw’, employing
Biot’s formula to represent the connexion between saturation-
pressure and temperature. The calculations of 1905 show
irregularities in the values obtained in these same cases,
chlorobenzene, bromobenzene, and iodobenzene. The recal-
culations of 1909, based on Young’s revised data, show
irregularities that are smaller, but are still much greater
than those given by other substances. When the modified
van der Waals’ equation is used, these irregularities still
occur. It will be seen that they are smaller than in the
previous cases, though only slightly so. What seems more
important is the fact that, in general, maximum values
obtained by means of Biot’s formula correspond to minimum
values obtained by the van der Waals’ relation, and so on.
This suggests that the error is due to some change in the
liquids not accounted for by either of the formule.

It seems therefore that in the case of these non-associated

van der Waals Formula. 597

liquids, the differences between the two series of calculated
values of latent heat are quite as likely to be due to a
divergence of Biot’s formula from the facts as to such a
divergence of the modified van der Waals’ relation. |

Consequently it will be seen that this modified van der Waals’
relation, and the derived expression (B) for the latent heat
of vaporization, hold good for ali the non-associated liquids
examined and for none of the associated liquids ; hence the
agreement between the two series of calculated latent heats,
and the variations in the value of Mills’ constant (see above)
give delicate tests of association in a liquid.

My thanks are due to Professor Young for supplying
some data I had overlooked, and for kindly offering to read
the proofs of this paper.

The University,
Melbourne.

APPENDIX.

In the following it is shown in detail that the methods of
calculation used by Mills-Young and by Applebey-Chapman
are essentially the same.

The data employed in the Mills-Young and the Applebey-
Chapman calculations are the measurements of the vapour-
pressure and the vapour-density of thirty-two pure liquids
made by or under the direction of Dr. Sydney Young.
These have been collected and revised, and are published in
the Scientific Proceedings of the Royal Dublin Society,
1910, pp. 412-443. With them appear Dr. Mills’ series of
calculated latent heats that we are about to discuss.

In each case the process of reduction has consisted of—

(1) Finding some formula connecting the temperature
with either the vapour-pressure or the specific volume
of the saturated vapour. Determining the constants
of this formula over a range extending to the critical
temperature.

(2) Ditterentiating this formula and substituting the value

dj : i Me
of . so found in the Clausius-Clapeyron relation

ax
L=t(v,—v,)
( 2 1) dt >
where v2 and 1 are the molecular volume of vapour and

liquid respectively, ¢ the absolute temperature, p the satura-
tion-pressure, and L the molecular latent heat.of vaporization.

598 The van der Waals Formula.

(1) For the vapour-pressure formula, Mills* has chosen
Biot’s form

logp=A+ Bai +C@e +. 55a

involving five arbitrary constants.
Applebey and Chapman prefer to use a form such as
van der Waals’ equation (J. C. S. 1914, p. 734),

[ses =) (v—b)= = ie

but treat b asa variable (p. 735). Later (p. 736) they find that
ois a constant, and is to be chosen to suit the data (see
method p. 737).

This is equivalent to employing the three-constant formula

a ao, oe
(0+ %) (0-0-5; 8) =Re 2 Nan

where a, b, and Gare adjustable constants and R is the gas

constant, b is the value of the van der Waals’ constant at the
critical temperature, and 6¢ the difference between the critical
temperature and the absolute temperature considered.

The constants of the formule (a) and (}) are chosen so as
to suit the same series of measurements—that 1s to say, 80
that their graphs may agree as closely as possible with the
experimental graph. They are, therefore, the same relation
expressed in different forms.

(2) By differentiating to obtain @, and substituting its
value in the Clausius-Clapeyron relation, Mills gets

L=t(r,—%){ Blog a. ai +C log B. B'},

while Applebey and Chapman by an ingenious but mathe-
matically similar process (pp. 735-6) obtain

1 lap
2 == 5
2) +R (— - Saale

These two formule are derived from relations which
approximately represent the same exjerimental results, by
means of the same mathematical processes ; and sioalds
therefore (it the approximation is good), give concordant
values of L.

* Journal of Physical Chemistry, 1902, 1904, &c.

——S -

The Boiling-Points of Homologous Compounds. 599
Applebey and Chapman obtain (p. 737) the value of =
by applying the law of rectilinear diameters, and the

relation
(5? _ 2K
ot y Ve

where the suffix c denotes that the measurement is made at
the critical point.

Mills, on the other hand, uses the law of rectilinear
diameters to obtain the critical constants, chooses the con-
stants of Biot’s formula to fit the values so found, and then

shows that the result
(5?) 7h
Ob ve

is true—a process equivalent mathematically to that used by
Applebey and Chapman.

LX. The Bowling-Points and Critical Temperatures of Homo-
logous Compounds. By ALLAN Fsercuson, D.Sc. (Lond.),
Assistant Lecturer in Physics in the University College of
North Wales, Bangor*.

“hee present communication explains a new empirical

formula which appears to represent the relation between
the boiling-points and molecular weights of the normal
paraffins with considerable accuracy over a wide range, and
also establishes, for the same series, empirical formule
showing the relation between critical temperature and
chemical constitution—a relation which does not, hitherto,
appear to have attracted very much notice.

It would serve no useful end to disturb the dust which
has collected over formule more than a generation old: in
discussing previous results it will be sufficient for our pur-
poses to consider the boiling-point equations proposed by
Walker f, by Ramage f, and by Young §.

Walker’s formula is of the type

Gi GMO MN Ro raat ae) Tiley st =, Sore ce
where @ is the absolute boiling-point, M the molecular weight,

* Communicated by Prof. E. Taylor Jones, D.Sc.

+ Trans. Chem. Soe, Ixy. pp. 193, 725 (1894). (For earlier references
see this paper.)

t Camb. Phil. Soc. Proc, xii. p. 445 (1904).

§ Phil. Mag. Jan. 1905, p. 1.

600 Dr. A. Ferguson on the Botling-Points and

and a and é are constants (for the normal paraffins a=37°38,
b=0°5. The boiling-points in this, and in all other cases
considered, are supposed to be measured at a pressure of
760 mm.).

This formula represents the observed facts with fair
accuracy over a limited range, but does not lend itself well
toextrapolation. The reason ea in the fact that the formula,
as is obvious, assumes a linear relation between log @ and
log M, which is by no means the case, as the graph of these
quantities shows a slight, but quite distinct, curvature. In
this case, therefore, as in all other cases in which a straight
line is assumed to coincide with a limited portion of a curve
of slight curvature, the agreement between calculated and
observed quantities is very good over the mid-portion of the
range chosen, but the differences become more and more
marked the further one extrapolates beyond either the upper
or lower limit of the range chosen.

The formula given by Ramage,

G=0(M0—2-")};,.. 2 ae
where a is Waiker’s constant, and n the number of carbon
atoms in the molecule, gives notably improved results for
the lower paraffins. But above decane or thereabouts, the
factor 2—” becomes negligible, and the formula then becomes
identical with Walker’s, equally implying a linear relation
between log 6 and log M, which relation, as is seen by the
curve between these quantities, is only approximately fulfilled.
From n=i14 onwards, the faeeence between theory and
experiment becomes steadily greater.

The formula proposed by Young is of a different type.
If A be the difference between the boiling-point @ of any
homologue and that next above it in the series, then

A=arygs + +e. sy ee eeu

where for the normal paraffins (and for certain other series),
c=144:86 and d=0148. Thus the boiling-point of any
paraffin can be calculated. provided that of the homologue
next below it be known. This restriction apart, the formula
gives very consistent results, and the observed and calcu-
lated results do not show, at the upper and lower limits of
the range considered, that tendency to a gradually increasing
divergence so characteristic of those formule which assume
approximate coincidence between a straight line and a curve.

It should be noticed in passing, as we shal! have need to
use the principle later, that equation (iii.) is really a dif-
ferential equation. Young’s A is really the change in

Critical Temperatures of Homologous Compounds. 601

6 per unit change in x, and (iii.) may therefore be written
dn @aNve e ° e . ° ° e

if we treat n as a continuous variable. This equation can—
theoretically—be integrated, yielding a relation between @
and » which could then be compared with those of Ramage
and of Walker. Unfortunately the resulting integral,
though apparently simple in form, does not integrate in
terms of any of the simple functions, and the series-form
into which it does integrate is somewhat unwieldy to
handle *.

Tt will be seen that in Walker’s formula 6 is assumed to
be proportional to a definite power of M; in the formula
now to be proposed a similar relation is assumed to exist
between the logarithms of 6 and of M—that is

loon GO = k(lom AM) as. deer)
or, what amounts to the same thing,
Geic(log Ms STy eo) . g a(vls))

When @ is measured on the absolute scale, and logarithms
are taken to the base 10, we obtain for the normal paraffins
k=1:929 and s=°4134. These values for the constants /:
and s hold with considerable accuracy over the range n=4
to n= 17, and were obtained by treating the observed boiling-
points over this range by the method of least squares. The
last two columns in Table I. f show the boiling-points as

TABLE I.
| Boiling- Walker. | Ramage. Young. | Ferguson.
earatin: |) POW ee | a —_ a
| (Obs.). | Cale. Die. Cale.| Diff. | Cale.| Diff. | Cale. | Diff.
pe ee leans 163 non
io) O C9) fe) ° fe) C fe) °
C,Hy5...| 2740 | ... |... | 2756} +16 | 2726 | —1-4 | 274-7] +07
OH...) 3093 | ... |... | 8122) +2°9) 309-4 |.+0-1 | 310-2 | 40:9
C.H,,...| 8420 |... | 1. | 3439) 41:9 | 342-0 | +00 | 341-9 | —O1
C7H,,...| 371-4 | 3738 | 43:4 | 3723 | +0-9 | 371°3 | —0-1 | 3707 | —0-7
C,H,,...| 2986 | 399°1| +0-5 | 398-3 | —0:3 | 398-1 | —0-5 | 397-2 | —1-4
C,H,...| 4225 | 4229] +0-4 | 429:5 | +00 | 422-9 | +0-4 | 421°8| — 0-7
C,,H.. .| 4460 | 445°5| —0:5 | 445-2 | —0°8 | 445-9 | —O-1 | 4448 | - 1:2
C,,H,, .| 467-0 | 466°8| —0-2 | 4668 | —0-2 | 467-4 | +0-4 | 4665 | — 05
C,.H,, .| 487-5 | 487-2) —0:3 | 487-3 | —0:2 | 487-7} +0:2 | 487-0 | —0°5
C,;H,, ., 5070 | 507°3| +03 | 507-0 | +0-0 | 5068 | —0-2 | 506-4) —06
©,,H3, | 525°5 | 526-0) +0°5 | 526-0 | +05 | 525°0 | —0'5 | 5251 —O-4
C,;H;, .| 5435 | 544°1| +0°6 | 544-2 | +0°7 | 542°3 | —1-2| 5429! —06
C,,H3, -) 560°5 | 561-9} 41-4 | 5619 | +14 | 558-9 | —1-6 | 560-2 | —0°3 |
Gi Hiss | 5760 | ... | ... | 5790} +3°0 | 5747 | —1:3 | 576-4 | +0-4]

* For assistance in elucidating the properties of this integral I am
indebted to the friendly counsel of Mr. G. B. Mathews.
{ The observed boiling-points are those given by Young, Phil. Mag. lc,

602 Dr. A. Ferguson on the Boiling-Points and

calculated from equation (v.) and the differences between
the observed and calculated values. The remaining columns
show the results of similiar computations using equations (i.),
(i1.), and (ii1.).

A fair idea may be obtained of the relative accuracy of
the various formule by computing the average error, regard-
less of sign. Thus

for rangen=7to n=16, average error, W=071,
average error, R=1°:03,
2 ” Y=0°58,
He Oe Oe

Thus it appears that, over the fee taken, the new formula
is more accurate than either (i.) or (ii.), and, whilst having
the advantage of not being a difference formula) is only
slightly less exact than that of Young. Further, the dif-
ferences between the calculated and observed values are
much the same in magnitude at any point of the range.

We now turn to the consideration of the relation between
critical temperature and constitution. Granted sufficient
exact experimental data, it would not be a difficult task to
find directly various empirical relations connecting critical
temperature and molecular weight in the normal paraffins.
In the present paper an indirect method is followed, which
will be seen in the sequel to lead to results of a fairly high
order of accuracy, while several interesting relationships
will be elucidated by the way, which would be obscured by
the direct method of attack.

It has long been known that ditterent substances are ap-
proximately 1 in ‘‘ corresponding states” as far as temperature
is concerned, when at. their boiling-points (under normal
pressure) ; were this exactly true, the ratio of the critical
temperature to the boiling-point, when measured on the
absolute scale, would be a constant for all substances. Now
while it is the case that this ratio does not vary very greatly
for substances whose boiling-points are so diverse as those of
oxygen and aniline, there is, nevertheless, a slight variation,
and this variation is, for the normal paraffins, a perfectly
regular one. Calling the ratio @,/@ for any given paratin R,
and the corresponding number of carbon atoms sine
molecule n, it can be shown graphically that the relation
between log R and log n is very accurately linear, leading
therefore to the relation

for rangen=4to n=17

Rnl=h, “ TotGyaigy

whe re g and fh are constants. For the normal paraffins the

Critical Temperatures of Homologous Compounds. 603

values of these constants, as determined from the graph
between log R and log n, are g="120 and h=1°841. The
closeness with which equation (vii.) fits the observed values

is exhibited in Table IT. below.

Average percentage difference neglecting sign =0°33 per cent.

It seems therefore that, apart from the first member of
the series, which, as usual, is anomalous, equation (vii.) fits
the observed values very exactly. It can easily be deduced
from (vii.) that if Rx» and R,41 be two successive values of
R for two paratins having n and n+1 carbon atoms in the
molecule, then

R, =( ay ist
Re 1+ Oba Ct ae (vill.)
bringing out quite clearly the observed facts that the value
of K decreases as n increases, but at a decreasing rate. In
fact, if (vil.) can be assumed “to hold over any wide range,
(viii.) shows that for large values of n the corresponding
values of R tend to become equal.
Writing (vil.) in the form

enPehOs es MOS > Leg

it is clear that 0 can be ue between a ) and any of
the various formule [(i.), (ii.), (iii.), and (v.)] proposed to
represent the relation between boiling-point and constitution.
We thus obtain empirical formule for @, which will vary
in form according to the particular boiling-point relation
chosen.

Thus, eliminating @ between (ix.) and Walker’s equation (i.)

* The observed values of R are given by Young, ‘ Stoichiometry,’
p- 188.

604 Dr. A. Ferguson on the Boiling-Pownts and

we obtain, after giving to the various constants their numerical
values,

— 68°30 “7M ° — es * See (x3

120

nv
Similarly, using Ramage’s equation (11.), we find that

68°80 :
0, yy [M2 . . Gi)

As Young’s formula is a difference formula, a different
method has to be followed in effecting the elimination.
Assuming n to be a continuous variable and differentiating
(ix.) with respect to n, we have

i =n + O6,9gnie.-. 2.

Oi ee ao Re : Be oe
Hliminating ae between (xii.) and (iv.), writing ane as
A. in accordance with Young’s notation, and substituting

the numerical values of the constants in the resulting equation,
we obtain

¢ 267 +12
Age 1 266°7 120 8, \

n 120 Gols 6 ~ 7738

21 Ging
A; is the ditference between the critical temperature of any
given paraffin and that of its homologue next higher in the
series ; 0, 0, und n refer to the given paraffin. So that,
knowing the boiling-peint and critical temperature of any
given paraffin, the critical temperature of its next higher
homologue can be calculated from (xiii.). In fact, the
method of use of (xiil.) for calculating critical temperatures Is
strictly analogous to that of (iii.) for calculating boiling-
points. ane ss

If we take equation (v.) as our boiling-point formula,
then, taking logarithms of (ix.), and eliminating log @
between this and (v.), we find that

log @.=log h+h(log M)*—glogn,. . (xiv.)
a relation which is equivalent to

hM*Cog va
Gea [a oa Ses nee (xv.)

Substituting the numerical values of the constants in (xiv.),

Critical Temperatures of Homologous Compounds. 60d

which is the most convenient form for computation, we
have

log 0.='2650+ 1:929 (log M)*4—-120 log n, (xvi.)

as the empirical equation which gives the relation between
the critical temperature and molecular weight of the normal
paraffins.

It now remains to compare the values calculated from
formule (x.), (xi.), (xiil.), and (xvi.) with the observed
values of 9,. Unfortunately the data for critical tempera-
iures, like those for boiling-points, are of very different
value, and one has either to adopt certain more or less arbi-
trary canons of selection or rejection, or to take the average
value of the various figures given for any one substance by
different experimenters—a process about as likely to result
in the true critical temperature as the averaging of half-a-
dozen slightly erratic public clocks is to result in Greenwich
mean time.

In comparing, therefore, the calculated and observed values
of @,, I have restricted myself to the accurately-determined
values for pentane, hexane, heptane, and octane given in
Young’s Royal Dublin Society paper*. The comparison is
shown in Table ITT. below.

TaBue ITI.

fe) ° fe) ° fo) fe) Oni}

C,H,,...| 470-2 |[481-2] +110] 473-7 +35] ... |... | 470-7] +05)
C,H, ,...| 507-8 |[514-6] +68] 510-6] +2:8| 5083} +0:5| 507°6| —0-2|
C,H,,..-| 539-9| 5448) +4-9| 549°6| +2°7| 541-2} +1°3| 540°3| +0-4|
C,H, ,...| 569°2| 572°5| +33) 571:-4| +2°2| 569°7| +0°5| 569°6} +0-4)
OAs vot) OCT Saree tl) OOTCS | ce. 596'0 |)... | 5965 |
P|) 18 620 2 MME DION Ge ay lee | GALS |

The first two values in the third column are bracketed,
since Walker’s boiling-point formula only has reference to
the paraffins between C,Hy, and C,,H3,. It will be seen that
the values given by formula (xiii.) deduced from Young’s
boiling-point equation are in close agreement with the ex-
perimental numbers, and that the values calculated from
equation (xvi.), which is deduced from the boiling-point

* Proc, R. D. 8, xii. 31. p. 374 (1910).

606 ~~ Dr. A. Ferguson on the Boiling- Points and

equation proposed by the present writer, agree still more
exactly with the observed values. Making due reservation
for the paucity of the experimental data, it would seem that
(xvi.) can be used with some confidence to calculate the
critical temperatures of the higher members of the series of
normal paraffins. 3

Proceeding on similar lines to those followed in deducing
boiling-point formule for the paraftins, it would be possible
to obtain empirical equations for other homologous series.
But it is perhaps of more interest to consider certain of these
series as derived from the paraffins by substitution, thus
obtaining some insight into the question of replacement
values. Taking the primary alkyl bromides as an example,
defining the term “replacement value” as meaning the
difference between the boiling-point of the paraffin and that
of the corresponding bromide, and denoting it hy p, we find
that the graph between log p and log n is very closely linear.
This leads at once to the relation

pn = 1875,
or since
} p=O2—9p,
we may write
oe Op = a PERO eae” 0 (xvii. )

where n as usual is the number of carbon atoms in the mole-
cule (the numerical values of the constants were read off
directly from the graph). From (xvii.), therefore, we may
calculate at once the boiling point of a given primary alkyl
bromide, if we know that of the corresponding paraffin.
The agreement between the observed and calculated values
is shown in Table IV. below.

Tapunlyi.*

| Bromide. @p obs. 6 obs, | 63 cale. Difference. |

| | 9 Q ° |
|CsH,Br ......| 2280 343'8 3441 | +403
|C,H,Br......) 2740 3740 | «= 3760 | 2S 420
C,H, ,Br......| 3093 4025 | 4022 | _083
C;1,,Br......| 3420 rr a (os

| CrH, Bee): avis S520 ag ABE | 08

| C,H, Bris... 398°6 474-0 | 4743 | = 408 |

Average difference regardless of sign =0°-73,

* Observed values from Young, Phil, Mag. Z. ¢.

Critical. Temperatures of Homologous Compounds. 607

It is of interest to see whether analogous formule can be
applied to those series in which the influence of association
is manifest—the primary alkyl alcohols, for example. If
the graph between logo and log be plotted for this series,
it will be seen that whilst the relation between log p and
log » is still approximately linear, the points are scattered
rather irregularly about the line passing through their mean
position. Calculating for this line, we find that

dole ae
Oy= O04 ie Peotone: (VALS)

giving the boiling-point of the alcohol (@,4) in terms of that
of the corresponding paraffin (@p). The results are shown

in Table V.

TABLE V.*
Ta meee (Al <n LL pRheRIe MURA eG, OL ties .. 0) satan j
Alcohol. Op obs. 6a obs. @4 cale. Difference.
ese. ig ae ° O Cie ore
CH OH)... 228°0 370:2 3740 +5'8
Cyr OH, ».: 2740 389°9 391°8 +1°9
Can OH... 309°3 411:0 409:0 —2:0
©, GOH *..: 342:0 431:0 429-0 —2:0
Cre OF «.:.| 371°4 449:°0 449-0 +0:0
Cone OH”. 398°6 464:°0 468°8 +48
Cl OT, 429°5 486'5 486'8 +0°3
Cyr Ol. “446:0 504:0 505:5 +1°5

Average difference regardless of sign =2°°04.

Whilst, therefore, the equation for the bromides is in close
agreement with the observed facts, that for the alcohols shows
much greater divergences. A similar divergence has been
noted by Young, who, using equation (ii1.), has calculated
values for A for each of the series considered above. In the
case of the bromides the average difference regardless of
sign between the observed and calculated values of A is,
over the range taken, 1°25. In the case of the alcohols the
average difference between the observed and calculated
values of A is 4°28. Thus, while the average values of the
errors in A are considerably larger than those in the formule
(xviii.) and (xvii.), the ratios of the errors are very approxi-
mately the same, for in each case the error for the alcohols
is about three times the corresponding error for the bromides.

* Observed data from Young, J. c.

608 The Bowling-Points of Homologous Compounds.

Summary.

(1) A boiling-point formula is proposed for the normal

paraffins
log 0=1°929 (log M)"™ . =e

which covers the range from n=4 to n=17 with an average
error of 0°64.
(2) An equation

an = 1$4:1.\.. 0 See (B)

is shown to hold for the normal paraffins betwen ethane and
decane with an average error of 0°33 per cent.

(3) By eliminating 0 between (8) and any of the various
boiling-point formule, empirical equations are obtained for
the critical temperatures of the normal paraffins. In par-
ticular, the equation

log 0,=*2650 + 1:929 (log M)**—120 logn . (gy)

is shown to be in very close agreement with the observed
facts.

(4) The boiling-points of the primary alky] bromides are
connected with those of the corresponding paraffins by the
equation

187-5 :
6p=Opt+ “488? ° ° e ° e (6)

the mean error over the range n=3 to n=8 being 0°73.
(5) A similar formula for the primary alcohols, viz.:

331°1

CRSP crise ’ e s 3 @ e (€)

gives the boiling-points of the alcohols over the range n=3
to n=10 with an average error of 2°04.

The writer hopes, with wider data, to extend these
formule, especially as regards critical temperatures, to other
allied series.

University College of North Wales,
Bangor.
January 1914.

r 609 J

LXI. On a New Form of Sulphuric-Acid Drying- Vessel.
By the Karl of Berkey, PA.S., and HE. G. J-
Hartuey, B.A.”

i the course of a prolonged series of measurements of the

vapour pressure of aqueous solutions, some of which
have already been published f, the method employed has
been to saturate a current of dry air with the vapour of the
solution and pure solvent respectively, The problem of
drying the air-current has thus arisen, and some experi-
ments to determine the relative efficiency of phosphorus
pentoxide and sulphuric acid as desiccating agents seem to
be worth recording, since comparatively few quantitative
results on this subject appear to have been published f.

Our experiments were designed to throw light on two
separate points, and proof was thereby obtained that :—

(1) Sulphuric acid is capable of drying air as completely
as phosphorus pentoxide, at least to the extent that is
required in this class of work.

(2) By employing suitable apparatus it is not necessary to
bubble § the air through the acid, but only to lead it over its
surface; thus avoiding both ie risk of introducing acid
spray into the air-stream and uncertain changes in the volume
of the air-current. In vapour-pressure measurements by
this method the latter source of error is of considerable
importance ||.

In addition to the foregoing it was shown that the vapour
pressure of the sulphuric acid itself, to which Mr. J.J Manley
has called attention 4], is too small to be of importance.

For these experiments a new form of phosphorus-pent-
oxide vessel was devised having some points of advantage
over the usual tube ; we give a ‘description of it as ie may be
of use to others. This vessel is shown in figs. and 2;
legs are sealed in at the sides at a, thus a blikg it to
stand on the balance or elsewhere without support ; it is fitted
with a ground-on cap } and a ground-on detachable mercury
cup ¢. “Tt may be mentioned here that we have found this

* Communicated by the Authors.

+t Proc. Roy. Sve. 1906, A. vol. xxvii. p. 156; Phil. Trans. A. vol. ceix.

p. 177.

t Morley, Am. J. Sci. vol. xxx. p. 140, vol. xxxiv. p. 199; Shaw, Phil.
Trans. A. 1885, vol. clxxix. p. 84.

§ Moistened pumice is frequently used, but for accurate work great
precautions are required to obtain suitable pumice and to avoid spray.

| ‘Nature,’ July 1905 ; Proc. Roy. Soc. doe. cit. p. 165.

q Phil. Trans. A. vol. cexii. 19138, p. 248.

Phil. Mag. 8. 6. Vol. 29. No. 172. April 1915. 2 ik

610 The Harl of Berkeley and Mr. E. G. J. Hartley on a

mereury-cup joint very satisfactory for vapour-pressure work
up to 30° C., where the pressure of the mercury 3s still insigni-
ficant. There have been no signs of leak even when the joint
has been under water, but in the latter case it is advisable to
put rubber lubricant on the ground glass that holds the cup ;
this lubricant is dissolved off with petrol before weighing
the vessel.

The most convenient way of filling the vessel is to pass
the inlet tube just through a rubber plug of the right size to
fit the mouth of the phosphorus-pentoxide bottle; on in-
verting the bottle and gently tapping, the drying agent runs
freely into the vessel, thus avoiding exposure to external
moisture. Fora large number of experiments it 1s better

Fig. 3. Fig. 4.
\e ec

Ignited
asbestos

pice

Fig. 1. Fig. 2.

to purchase stoppered bottles, each containing about one
charge of the pentoxide, and keep them over sulphuric acid
in a desiccator.

After filling, the vessel was heated to about 240°C. ina
stream of dry ozonized air until ozone issued freely. By this
treatment Mr. Manley * has shown that the lower oxides of
phosphorus are completely oxidized J.

The sulphuric-acid vessel is shown in figs. 3 and 4; it is
fitted with detachable mercury cups, and an inverted U tube

* Private communication.

+ It should be noted that of the various samples of phosphorus pent-
oxide tried, only Kahlbaum’s was treated successfully in this way. All

the others contained such a large proportion of impurities that even after
many hours’ rur practically all the ozone in arapid current was absorbed.

New Form of Sulphuric-Acid Drying- Vessel. 611

joins it to the pentoxide tube. The four horizontal branches
(each 22 cm. long with an external diameter of 2 cm.)
are about half filled with acid, which is thus contained in four
separate compartments, so that the moisture from the wet-air
stream is nearly all absorbed in the first compartment. A
long series of experiments, which need not be detailed here,
have proved that with such a vessel containing 70 c.c. of acid
(10 c.c. in the first compartment), 20 c.c. of water can be
abstracted from an air-current without allowing any to pass
—or put in another way, air saturated with moisture at
30° C. will be completely dehydrated, although passed for 5
days at a rate such that 0°15 gramme of water is taken up
per hour.

In the first experiment, a current of moist air was drawn
through a soda-lime tube* to remove the greater part of
the moisture and then through a Winkler sulphuric-acid
drying-vessel of large size connected to the pentoxide tube.
The air-bubbles were formed in the Winkler at the rate of
50 in 14 seconds. After 52 hours the pentoxide tube was
welghed and had gained :0010 gramme. As no special
precautions had been taken to dry the tube connecting the
Winkler to the pentoxide, the same experiment was repeated
immediately after the above weighing, and the air passed for
160 hours. The gain in weight of the phosphorus pentoxide
was now only 0002 gramme, showing that the sulphuric
acid had allowed practically no moisture to pass by.

In all these experiments the pentoxide tube was weighed
against a glass counterpoise of about the same external area,
and, before weighing, both vessels were washed and dried
in as near a similar manner as possible, the weighings being
reduced to a vacuum.

In the next experiment the Winkler was replaced by the
new form of sulphuric-acid vessel. After a week’s run, when
about 600 litres of air had passed, the sulphuric acid had
absorbed 5°52 grammes of water while the pentoxide tube
had gained only -0001 gramme.

Further evidence of the efficacy of this form of vessel
is found in a great number of experiments in which air was
passed over water, in a weighed vessel, and then over the
sulphuric acid. The loss in the former should equal the gain
in the latter. Without entering into lengthy details the
following gives some idea as to the magnitudes involved—
the vessels were similar to those described by us in Proe.
Roy. Soc. A. vol. Ixxvii. p. 158 (1906).

* Cp. note on page 618.
Dn
2 R 2

612 New Form of Sulphuric-Acid Drying- Vessel.

Lay f

In a run at 30° C. nearly 7 grammes of water was passed
over 23 c.c. of acid (4 ec. in the first branch) at the rate of
‘11 gramme per hour without any loss ; the spent acid in the
first Ginlet) branch was then replaced by 4 c.c. of fresh acid
and the run continued at arate of ‘(075 gramme per hour
until another 7 grammes had passed—again there was no loss.

The azdvantages of the new form over the Winkler seem
therefore to be as follows :—

(1) There is no constriction in the air-current.

(2) There is no danger from sulphuric-acid spray.

(3) Even after a somewhat prolonged use the drying
agent is still effective; with the Winkler, however, the water
absorbed dilutes the whole of the acid, and the issuing air is
saturated up to the vapour pressure of the solution.

(4) As by far the greater quantity of water is taken up
in the inlet branch, it is only necessary to renew the acid in
this branch for the vessel to be again efficient.

(5) For any prolonged run it is easily seen that less acid
is required in the new form than in the Winkler.

In view of Mr. Manley’s observations that sulphuric acid
itself has an appreciable vapour pressure *, being, for example,
found condensed on the lids of desiceators, experiments
were made to ascertain the magnitude of any error that
might be introduced by neglecting this effect.

Air at the laboratory temperature was passed over sulphuric
acid in a vessel very similar to that already described and
then over water. At the end of six days the water was
tested with a drop of barium chloride solution, which, after
standing some few minutes, gave a slight turbidity. By
comparison with the turbidity produced by known amounts.
of sulphuric acid, the water was estimated to have contained
‘0001 gramme of acid. The same experiment was repeated
with the train of vessels immersed in a bath at 30° C., the
air being passed for three days; again the amount of acid
carried over into the water was found to be about
‘000L gramme.

It may be mentioned that during the operation of weighing,
air, dried by passing over sulphuric acid, is circulated through
the balance-case. So far there are no signs of any harm
having been done to the balance. But an alternative method
for drying the air seems to be to pass it over stick potash,
as the following experiment shows :—

Air derived from outside the laboratory, during very

* Loc. cit.

Method for examining Optical Qualities of Glass Plates. 613

rainy weather, was passed for 102 hours through a soda-lime
tube * (1°5 metres long), the last third of which was filled
with small pieces of potash, and then over a weighed
sulphuric-acid vessel, with the result that the latter gained
‘0041 gramme.

In this connexion it may be of interest to note that quan-
titative measurements show pure anhydrous CuSQ, to be a
very efficient drying agent for air containing little more than
traces of moisture (it will! take up about 0°05 per cent. of its
own weight). It has the advantage that it can be used over
again after heating to 210°-220° C. in an air-stream. The
dehydrating property of CuSO, seems difficult of explanation,
for, according to theory 7, the substance should not absorb
water-vapour unless the partial pressure is greater than that

of the hydrate CuSO, 1 Aq.

LXII. Note on a Sensitive Method for Examining some Optical
Qualities of Glass Plates. By the Harl of BERKELEY,
F.R.S., and D. FE. Toomas, M.A., B.Se.t

N the course of a research on the relation between the

concentration and the partial pressures of the vapours

of miscible liquids, we have used a Rayleigh interferometer

for determining the refractive index of the vapours and their

mixtures. The new method for the examination of glass

plates arose out of certain troubles experienced in the
adjustments of this instrument.

Lhe interferometer is one of Zeiss’s, designed for working
with columns of gas one metre in length, and was modified
so as to double the optical path as shown in figs. 1
and 2.

Vig. 1 is a plan of the apparatus as set up for the exami-
nation of glass plate X—the arrangement differs from
that used in examining gases only in the removal of the

* These soda-lime tubes have proved to be very convenient; they are
made in the shape of a very broad S (as seen in plan), the bends of the
S are turned up at an angle of 45° with the horizontal and are fitted with
ground-in glass stoppers for rapidity in filling. The vertical inlet and
exit tubes are relatively narrow and carry mercury cups, and the whole
tube can stand on the bench without supports.

+ Cp. Lehfeldt’s translation of van’t Hoff’s Lectures on Theoretical
and Physical Chemistry, Part I. p. 60.

{ Communicated by the Authors.

614 The Earl of Berkeley and Mr. D. EH. Thomas on a

gas-tubes ; the Vincent mercury lamp *, M, gives a narrow
vertical source of light, which is focussed on the slit, 8, of
the collimator, C. On the collimator, which contains Wratten
filters for isolating the green line, is fixed a metal plate, P,
cut so as to have two vertical slits, each 4 mm. wide, with their
inner edges 8 mm. apart. Light emerges from the colli-
mator in two parallel beams, which, striking a half silvered

Fig. 1.

mirror, B,are reflected through the jacket, J, to a mirror, N,
fully silvered on its front surface. The reflected beams are
then brought to a focus by the objective of the telescope, T,
a lens of 30 cm. focal length. The interference pattern thus
produced is viewed by means of a cylindrical lens of very
short focal length.

Fig. 2.

Fig. 2 gives an elevation of the apparatus, omitting the
collimator.

* This lamp was given to us by the Silica Syndicate Co., to whom
our thanks are due. The lamp is in the form of a capillary U tube (bore
about 1 mm.) made of fused silica—the light is generated in the capillary
by heating the mercury there until vapour is formed, it will then run
for hours on a direct current (100 volts) of 0:09 amp. Being very small
and compact (one form, giving a horizontal source of light, could be
carried in a waistcoat pocket), it can be placed exactly where wanted.

i ote ae a =

Method for examining Optical Qualities of Glass Plates. 615

The upper part of the two beams, just before entering the
telescope, is displaced downwards by the glass plate, B, with
the result that, if the latter is properly adjusted, there will
appear in the field of view a sharp horizontal line separating
the upper and lower interference bands.

The lower part of the beams passes through the two thin
glass plates, G and G,; the former is fixed, and the latter is
capable of rotating about a horizontal axis through its upper
edge. The rotation of G,, actuated by a long lever and
micrometer screw, causes a change in the optical path of the
lower right-hand beam and thus produces a bodily shift of
the lower set of bands ; the upper bands, asin Lord Rayleigh’s
original instrument, are merely a set of reference marks.

Testing the Plate.

The plate is placed at 45° to the optic axis as shown in the
diagram (this we will call position “ a’’), with its upper edge
as near as can be to the horizontal separating line. The
upper and lower bands are now brought into coincidence and
the reading of the micrometer head noted ; at the same time
the path of the two beams through the plate is located. The
plate is now rotated threugh 180° (position “b’’), and placed
so that the light still passes through the same parts of the
glass, the bands are again brought into coincidence and the
reading noted ; the difference between the readings is a
measure of the difference in the optical paths in the two
positions.

The optical arrangements were such that the central bright
area of the interference pattern was traversed by four sharp
black bands, and settings were always made on a certain one
band. After a little practice it was found that the error of
setting for either observer did not exceed half a scale-division
from the mean of a number of settings. Since 50 scale-
divisions correspond to one band interval, the shift of the
lower band system can be measured to the 1/100 of a band ;
this corresponds to a retardation of one beam on the other
of the 1/100 of the wave-length of the light used.

The following are a set of readings made in this manner :—

P _ Position: Readings. Rare pp
Observer. of plate, | {uae Alte Ga) Means. Diff.

IBS es a 188:0 1880 187°8 1879

Bre ener hb 1920 192'0 192°2 192°1 4:9

OS a 187:0 187°0 186°5 186'8

ct Shin b 1911 L911 19i1 Lots! 4:3

616 The Harl of Berkeley and Mr. D. E. Thomas on a

Other measurements made at different times and on
slightly different parts of the plate confirmed these results.

If we assume, as an approximation, that placing the plate
at an angle of 45° increases the relative retardation in the
two beams by about 13 per cent., we get for the plate at
right angles to the optic axis a retardation (425-55 )/50 = "074
of a band. Hence, for a single transmission, the difference
in paths D(uw—1)="037 2.

Obviously this difference may he in a variation in D or
w—l1 or in both, and we are unable to distinguish; but we
would draw attention to the fact that the condition teria
perfect echelon spectroscope plate is precisel) y that D(w—1)=
constant, while Michelson’s method”, which is generally
used for testing the figuring of these plates, only gives in-
formation on the product Dp; over and above this, the
method here described is some 30 per cent. more sensitive.

As regards sensitiveness, it is most probable that if the
cylindrical lens were fitted at its focus with a fine vertical
fibre just wide enough not to cover completely the bright
space between the black bands in the upper field of view,
then the lower bands could be fitted to the upper with extra-
ordinary accuracy—this arrangemei is the analogue of that

used by Dr. C. V. Burton f in his micro-azimometer, where
single settings were made with a probable error of 1/800
part of the width of the central bright band of a diffraction
pattern.

The Practical Application of the Method.

Kchelon plates are usually obtained by cutting up a large
plate, which has been figur ed under the Michelson test, into
strips, and these are then cut into the requisite lengths. Mi
is obvious that, short of building a very large Rayleigh
interferometer, the initial plate ‘cannot ‘be placed in our
instrument. We would suggest, however, that if the above-
mentioned strips were e made somewhat wider than necessary
they could be examined separately by our method and then
refigured.

There are two ways by which such an examination may be
made ; both involve a comparison with a standard plate.

(1). By fitting the interferometer with a system of mirrors
or prisms so as to separate the two beams widely enough
(as outlined in fig. 3), we can get an “optical contour”
of the strip by the direct comparison of its different parts
with the standar d.

Astrophysical Journal, vol. viii. 1898, p. ov.
t Phil. Mag. vol. xxiii. 1912, p. 385.

Method for examining Optical Qualities of Glass Plates. 617

(2) Using the instrument as it is* (or better with a
rather larger collimator and telescope and shorter air-path),
the different parts of the strip may be compared with one
another and finally with the ends, whose “ optical thickness”
can be separately referred to the standard.

Unfortunately, the first method, which is the more efficient,
involves somewhat elaborate and substantial fittings for
adjusting and fixing the mirrors.

Should it be found essential to figure the whole plate
before cutting it up, the great advantage of the present
instrument in having reference bands which are part of the
same optical system as those under examination will have to
be sacrificed. Burton’s signal, mentioned above, placed
on a travelling micrometer might be used, or another set

Fig. 3.

‘STANDARD
PLATE ~s—t

of reference bands could be obtained by modifying the
arrangement shown in fig. 3, so that part of the two beams
pass clear of the mirrors A and B to be reflected into the
telescope by mirror C.

* A geometric fitting for ensuring that the standard plate and strips
always have the same orientation would have to be provided.

i GAS]

LXII. The Photoelectric Effect—Ul. By O.W. Ricwarp-
son, F. R.S., Wheatstone Professor of Physics, University
of London, King’s College, and F. J. Rogers, Associate
Professor of Physics, Stanford University, California”.

He, the Philosophical Magazine, vel. xxvi. p. 549 (1913),

Dr. K. T. Compton and one of us published data for the
metals platinum, aluminium, sodium, and cesium, showing
the relative photoelectric efficiency when a given amount oF
energy fell upon them in the form of light of different wave-
lengths. It was intended to reduce ‘the measurements to
absolute values, but this was not possible at the time owing
to the absence of a suitable radiation standard. This de-
ficiency has since been remedied and the results of the
measurements are given below. They are to be taken in
conjunction with the numerical data and curves published
in the paper already referred to.

To reduce the current measurements to absolute values,
the only new data required were the capacities of the electro-
meter and the variable condenser which was added to the
quadrants in some of the experiments. The capacity of the
electrometer and its connexions was found to depend toa
considerable extent upon the voltage on the needle. This
was therefore determined separately for each of the different
voltages which had been used. Both the method of mixtures

co)

and a leakage method were employed, and the values obtained

when plotted against the potential of the needle were found

to increase regularly with the applied potential. The new
values also agreed with the values in terms of the variable
capacity which had been used in reducing the measurements
described in the former paper, showing that the capacity of
the electrometer with a given potential on the needle had not
changed in the interval.

In the energy calibration in the former paper, the mouo-
chromatic light from the illuminator was allowed to fall on a
linear thermopile provided withaslit adjusted tothe same effec-
tive width as the strip used in the photoelectric measurements.
The steady thermo-electromotive force developed under the
influence of this radiation was balanced against the drop of
potential produced by the flow of a known small current
through a known small resistance. In order to reduce the
former observations to absolute values, it was therefore
necessary only to determine the electromotive force yene-
rated in the thermopile when a beam of radiation of known
energy density was allowed to fall on it through the same
slit. The source of radiation of known energy density was
a standard incandescent-lamp obtained from the Bureau of

* Communicated by the Authors.

On the Photoelectric Lfect. 619

Standards. This was set up at such a distance from the
thermopile as to give rise to an electromotive force con-
venient to measure and comparable with those obtained with
the monochromatic illuminator. The directions supplied
with the standard were carefully followed. As a result of
these measurements it was found that the unit of energy per
unit time previously employed was equal to 0:404 erg per
sec. For example, to reduce the values of Hi in Table IV,
column 6, p. 562, to ergs per second, it is necessary to
multiply each number by 0°404.

At this time the illuminator was readjusted and the energy
distribution in the quartz-mercury arc-lamp spectrum re-
determined. The values obtained were not sufficiently
different from those given in the table just referred to, to
eall for comment.

We can now consider the results for each of the elements
separately. The value of N, the quantity of electricity
liberated when unit quantity of radiant energy falls on the
metal, will be expressed in terms of the unit 1 coulomb per
ealorie. ‘lo reduce the numbers given to electrostatic units’
of quantity per erg, it is only necessary to multiply them by
72. The quantity N may also be appropriately termed the
photoelectric yield, although it would be better to restrict
the term photoelectric yield to the case in which the light is
completely absorbed. The value of the photoelectric yield,
in this sense, can be obtained from the data given, together
with a knowledge of the reflecting power, in the different
parts of the spectrum, of the metals concerned.

Platinum (loe. cit. p. 561).
Unitorm N= 9°05 x10" coukjeak
Greatest value of N (at v=1°5 x 10" sec.~*)
= 3:0 x 1075 coul./cal.
Alununium (loc. cit. p, 562).
Unit of N=1°92 x 10~* coul./cal.
for curves 1, 3, and 4.
Curve 1 (taken immediately after scraping the aluminium
and setting up).
Maximum value of N (at v=1°36 x 10 sec.~!)
==2 Do lO -* eoul foal:
Curve 3 (72 hours later).
Maximum value of N (at 1°42 x 10" sec.~+)
= 1-01 x 10~* coul./cal.
Curve 4. This was for a flat strip at perpendicular
incidence and was taken about 6 hours after scraping and
setting up.
Maximum value of N (at 1°38 x 10! see.~!)
= 1:28 x 107‘ coul./cal.

620 Profs. O. W. Richardson and I’. J. Rogers on

Curve 5. This was taken somewhat later than curve 4,
with the same flat strip set so that the light was incident at
approximately 15°. Owing to the inclination of the strip
the effective width of the beam of radiation falling on it was
less than in the previous cases. Allowing for this, the unit
of N is 1:98 x 10~® coul./cal. and the maximum value of N
(at p= 1°37 x 10" sec.)

= 9°7 x 10-5 coul./eal.
Sodiuni (loc. cit. p. 563).

Curve 1. The observations used in constructing this curve
were commenced about one hour after distilling the sodium
on to the strip and took over an hour to complete. There
was a rapid photoelectric fatigue. This was corrected for by
extrapolation ; so that all the observations were reduced to
the values corresponding to the instant of the first observa-
tion. This extrapolation cannot be made quite exact since
different parts of the curve decay at different rates. This
difference was allowed for in so far as it could be ascertained ;
so that Curve 1 may be taken as representing the emission
from a surface of distilled sodium one hour after distillation.
The unit of N for this curve is 2°56 x 1075 coul./cal.

Curve 2. This curve represents the actual condition of
affairs 24 hours after distillation. There is some doubt,
arising from a possible inaccuracy in one of the data used
in correcting for fatigue, about the position of the curve for
values of yx 10~ greater than 125. Relative to the rest of
the curve on the left-hand part of the diagram, the point at
vX 10~%=125 is correct ; but the true position of the maxi-
mum at yx 10~-8=133 may be 10 per cent. below the value
on the curve as drawn, and the end point at vx 10-8=150
as much as 30 per cent. below the value shown, intermediate
points dropping by regularly increasing percentages. The
unit of N for this curve is 1°31 x 107° coul./cal.

Curve 3. This represents the state of affairs 18 hours
after preparation, when the rate of fatigue was extremely
slow. The unit of N for this curve is 6°25 x 10-6 coul./cal.

The values of the quantity of electricity emitted per unit
energy of incident (isotropic) radiation are collected in the
following table. The values at t= 0 have been derived on
the assumption that the decay of the photoelectric effect is
exponential for the first 24 hours. As there is a certain
amount of experimental evidence indicating that under
certain circumstances, not yet properly understood, the
emission may show an initial increase followed by decay
according to an exponential law—as well as other types of
behaviour under different conditions—these numbers may be

the Photoelectric Liffect. 621

altogether wrong. For this reason, and as they are extyra-
polations and not actually observed values, they are enclosed
in square brackets.

Sodium.
Time from Value of N Value of N
preparation of for First for Second

surface, Maximum (a). Maximum (6),

[0] hours. [AT si; coul ical) oa.< L0>*] coul:/eal:
inane Wee tccii 2) MOMs
25. 3-6 amir’ a 8x10-4*

HAS fe) is, O59x10-4 DAMS MOet 00),:

Cesium (loe. cit. p. 4564).

Owing to an oversight the capacity used in the original
measurements with this substance has not been recorded,
and as we had not time to repeat the measurements before
leaving Princeton, the data we are able to furnish have
only a qualitative significance. It appears from the
possible values of the capacity, and the other data, that
the value of N for the maximum point on the curve at
y=1°2 10" sec.~! must lie between the limits :—

1x1075 and 5x10~ coul./cal.

The value of the unit of N varies of course between
corresponding limits.

These numbers, although very wide apart, are of consider-
able interest. They show that cesium under the conditions
of the experiments previously described is very inefficient
photoelectrically. It is very much worse than sodium, con-
siderably worse than aluminium, and comparable with
platinum in this respect. In view of the highly electro-
positive character of czesium, this result seems very
surprising and rather points to the view that the czesium
used in these experiments was already much fatigued before
the observations commenced. ‘This position is supported by
the following additional considerations :—

(1) The absence of fatiguing during the observations
which was remarked in the previous paper.

(2) The absence of the expected first hump, which, if the
experiments with sodium may be taken as a guide, dies out
more quickly than the second.

(3) If, as seems to be the case in general, photoelectric
fatigue is more rapid the more electropositive the metal, we
should expect it to be accomplished more quickly with
cesium than with sodium; so that it well might be practi-
cally complete before the actual observations commenced.

In our opinion it is desirable to examine the photoelectric

* This number may be 10 per cent. too high (see above).

622 On the Photoelectric Effect.

spectrum of czesium under better conditions from the stand-
point of preventing fatigue.

Previous measurements of absolute photoelectric efficiency
have been made by S. Werner * with sputtered films of gold,
platinum, silver, bismuth, and copper, and by Pohl and
Pringsheim f with surfaces of calcium, sodium, potassium
and potassium amalgams of different concentrations. For
a platinum film deposited in hydrogen at 0°6 mm. pressure,
Werner finds at v=1°36X107 that 1 calorie of radiation
liberates 5°6X 107+ coulomb. ‘The value for platinum at
this frequency found in the present paper is 2 x 10~° coulomb
per calorie. Werner’s value is for complete absorption of
the radiation, whereas our value is reckoned per unit incident
energy of (isotropic) radiation.

Werner also made some rough measurements of the emis-
sion from surfaces of ordinary (not sputtered) platinum, and
it appears from the data given by him that the value for
complete absorption by the hydrogen sputtered film re-
quires to be reduced by the following factors to obtain
the value for radiation incident on a surface of ordinary
platinum, viz. :—

on account of the reflected light ;

Ob bo)

on account of the observed greater sensitiveness of
hydrogen films compared with films sputtered in
nitrogen ; and
1 e e e
— on account of the greater sensitiveness of the nitrogen
20 . s e
films compared with surfaces of polished platinum.

According to these figures the total reduction factor is

= 2 x = =: so that under the conditions of our ex-
periments Werner’s numbers would give a sensitiveness at
y= 1-36 X10" sec.-? of about 5°6 x 10 * coul./cal. | Wie me
only just over one-fourth of the value given by our
measurements.

The values for sodium one hour after distillation are
practically the same as those found by Pohl and Pringsheim
(17x1074 as compared with 15—19x10~‘ coul./cal.) so
far as the first maximum is concerned. For values of
v>10- sec.~' the curve does not agree with their typical
curves but is more like that given by the figures in the last
column of Table I. of their paper (loc. cit. p.176). According

* Ark. f. Math., Fys, 0. Astr. Bd. viii. Nr. 27, Upsala (i912), Diss,
Upsala (1913).

+ Verh, der Deutsch. Physik. Ges. xv. Jahrgang, p. 111, p. 173, p. 431
(1913).

‘ The Contact Differencce of Potential of Distilled Metals. 623

to the observations of Compton and Richardson, the relative
magnitude of the two humps is determined by the amount of
photoelectric fatigue which has occurred. The data of both
pairs of observers can be harmonized on the following
assumptions :—

(1) That Pohl and Pringsheim’s typical curves cor respond
toa state of less advanced photoelectric fatigue. This in-
volves the assumption of the attainment of better vacuum
conditions in their experiments, which seems to be borne out
by an examination of the general character of their results.

(2) That the first hump (the ‘‘ resonance” hump (a)) is
present from the beginning, or at least is formed very
quickly, and does not suffer much alteration in the earlier
stages of the photoelectric fatigue.

(3) That the Tae hump is small initially, and increases
to a maximum value during the early stages of fatigue.
According to Compton and Richardson’s experiments, this
initial increase is followed by a decay, which is slower
than that of the first hump, as the fatigue progresses.

It is not claimed that this is the only possible explanation
of the observed differences; it is put forward as a possible,
and, on the whole, rather probable one.

LXIV. The Contact Difference of Potential of Distilled
Metals. By FERNANDO SANFORD*.

ir has seemed to the present writer that some of the results

of the experiments published under the above caption in
the September number of this Journal, may perhaps be
accounted for ina manner not taken into consideration by
Mr. Hughes. I refer especially to the observation that after
a film of zine or bismuth had been condensed from the metallic
vapour upon a very thin film of platinum on a glass plate in
a high vacuum, the condensed metallic film was at first much
less electropositive to the platinum film than it became after
standing for some time, and that the change to the more
electropositive condition was hastened by admitting a very
small quantity of air to the vacuum.

In some experiments which I have made for another
purpose I have observed that when a section of a glass tube is
heated to a temperature of 100 degrees, or even less, it becomes
plainly electronegative to the colder parts of the same tube.
The change is necessarily slow, since, on account of the low
conductiy ity 0 f glass, the electrons require considerable time
to gather in ie heated parts of the glass. It would seem that

* Communicated by the Author.

624 Notices respecting New Books.

in Mr. Hughes’ experiments the glass must have been consi-
derably heated over the regions upon which the metallic films.
were condensed. If this heating process was kept up for
some time the glass in these regions probably became nega-
tively electrified, and accordingly lessened the electropositive
inductive effect of the metallic films upon the plate connected
to the electrometer. As the glass cooled off, its charge
diffused slowly and the metallic films appeared to become
more electropositive. The admission of a small amount of air,
by lowering the insulation of the high vacuum, would then
enable the negative charge on the glass to diffuse more rapidly.

Stanford University, Cal.
Dec. 26, 1914.

LXV. Notices respecting New Books.
Bulletin of the Bureau of Standards. Vol. X. (1914).
Washington: Government Printing Office.

pas volume of the Bulietin (Nos. 1, 2, 3 and 4) exhibits the
creat activity of the Bureau. It contains amongst other
papers the following :—(i.) Constants of Spectral Radiation of a
uniformly heated enclosure, by W. W. Coblentz, in which are
described experiments with enclosures with white and with black
walls, which yield as mean values of Planck’s constants: C=
14456 +4 micron deg.; A=2911 micron deg. (results of 94 energy
curves). (ii.) Melting-points of the refractory elements of atomic
weight from 48 to 59, by G. K. Burgess and R. G. Waltenberg.
The summary results for the probable melting-points of the pure
elements are: nickel 1452 + 3, cobalt 1478+5, iron 1530 + 5,
manganese 1260+20, chromium 1520 to > iron ?, vanadium 1720
+ 30, titanium 1795 +15. (in.) Latent heat of fusion of ice,
by H. N. Dickinson, D. R. Harper, and N. 8. Osborne. Final
result (mean of 21 determinations) 79-68 cal,; per gram. Mean
of experiments by electrical method, 79°65, by method of mixtures
79°61 +02; electrical method (second set, ice at — 3°-78), 79°65.
(iv.) Melting-points of some refractory oxides, by C. W. Kanolt.
(v.) The pentane lamp as a working standard, by E. C. Crittenden
and A. H. Taylor. (vi.) Comparison of the silver and iodine
voltameters and the determination of the Faraday, by G. W.
Vinal and St. J. Bates. Results: E. Ch. Eq. of iodine 131502;
value of Faraday (iodine=126-92) 96515; (Ag=107°88) 96494.
Recommended value for general use, 96500. (vii.) Production of
temperature uniformity in an electric furnace, by A. W. Gray.
(viii.) The silver voltameter, by E. B. Rosa, G. W. Vinal, and
A. §. McDaniel. (ix.) Flame standards in Photometry, by E. B.
Rosa and E. C. Crittenden.
The Bureau publishes these papers in separate form; also
a set of technological papers. Amongst its recent circulars are
one on the testing of barometers and a valuable one on Polarimetry
(i. e. of sugars). It has also just published a decennial index te:
the Bulletin (Vols. 1-10).

Phil. Mag. Ser. 6, Vol. 29, Pl. VIIL

Fie. 36.

Fie. 42.

Fie. 40.

)

Fra. 48.

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Phil. Mag. Ser. 6, Vol. 29, Pl. X.

Kine.

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| THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

[SIXTH SERIES.

MA ¥ 1915. OY
PME IS SATIN PO TE

LXVI. The Scattering and Regular Reflexion of Light by Gas
Molecules—Part I. By C. V. Burton, D.Se.*

fi; if HAVE been so much interested in reading a memoir

by Lord Rayleigh+ in the February number of the
Phil. } as that time has had to be found for the completion
of this paper, which had been laid aside since last August.
The “simple aérial resonator vibrating symmetrically is
undoubtedly more mathematically tractable than a radiating
molecule ; and in this paper also, problems relating to such
resonators (or, more generally, secondary vibrators) are con-
sidered by way of introduction. But, as will appear in
Part II., the difficulties arising from our ignorance of the
mechanism { of radiation can for the most part be evaded.
Something has, indeed, to be assumed as to the influence of
the orientation of the molecule on its response to incident
radiation, and, by way of illustration, two alternative
assumptions have been made, representing perhaps the ex-
tremes of possibility. Much of the analysis—for example,
that which deals with reflexion from a three-dimensional
multitude of vibrators—is equally applicable to the acoustical
and to the optical case: whichever case is in question, there

* Communicated by the Author.

+ “Some Problems concerning the Mutual Influence of Resonators
exposed to Primary Plane Waves,” Phil. Mag. Feb. 1915, pp. 209-222.

t If, indeed, mechanism is not a wholly improper term in this
connexion.

Phil. Mag. 8. 6. Vol. 29. No. 173. May 1915. 28

is an essentially similar transition from diffuse to regular
reflexion as the distribution of the vibrators becomes denser.

2. The present investigations deal exclusively with vibrators
which scatter, without absorbing, wave-energy of definite
frequency, though their extension to molecules which absorb
some part of the incident energy should present no difficulty.
Wood ™* has lately emphasized the importance of determining
experimentally to which category the resonant atoms of
mereury vapour belong, and in the second part of this paper
some tentative suggestions are made towards that end.

3. The case of an isolated ‘‘ simple aérial resonator, excited
by plane waves,” has been dealt with by Rayleigh in the
paper already cited; and the result (with a changed notation)
may be stated as follows. Let the primary waves be defined
by the velocity-potential

=A cos (pt—vr),; . . . 2

where p/2@ is the frequency and 27/v the wave-length ;
then the secondary disturbance due to a resonator at the
origin is

C
as — y a ye
ea 7 008 (pt—ur=—y), - iss) 4) ee
where y is the lag in phase and
ees Ger
C= pane Yy- e . e . . e (3)

This last relation is deduced from the sole assumption
that the resonator merely scatters sonorous energy without
changing its total amount.

4, Consider next a square, forming part of the piane of
yz and having for its sides y= +3), ex=+4b. Let simple
Helmholtz vibrators be distributed over the surface of this
square with complete irregularity like the molecules of a gas,
the average number of vibrators per unit of area being co.
For the moment, the only restriction made regarding o is
that the aggregate surface occupied by the vibrators is
insignificant in comparison with the spaces between them.
It is simply postulated that all the vibrators are sending out
vibrations of the same amplitude and phase, represented
typically for the nth vibrator by

Cee

Vn= cee (pt-=0T,,) 3, see

The manner in which the vibrators are kept going is not the
* Guthrie Lecture, Proc. Phys. Soc. xxvi. p. 185 (1914).

Regular Reflexion of Light by Gas Molecules. 627

immediate object of inquiry ; what has to be determined is
the relation between the energy of the plane waves pro-
pagated in the direction (say) of # decreasing and the energy
diffusely scattered. ‘To find the plane-wave energy, imagine
a sufficiently large (acoustical) lens of focal length f placed
in front of the square bx}; an “image” will be formed at
the principal focus, and will take the form of a diffraction
pattern whose scale is determined by fA/b; % being the
wave-length concerned, and equal to 2a/v. It will be con-
venient to suppose / so large that the lens, having degenerated
into a sensibly flat plate, may be removed, and so need
trouble us no further.

5. No matter how sparsely the vibrators are spread in the
yz-plane, we can postpone the discussion of more delicate
points by taking the square 0x6 so large that it can be
divided up into numerous elements, each containing many
vibrators ; 7 will then, perhaps, be enormous, but that does
not matter : we can apply the methods familiar in physical
optics to determine the distribution of disturbance in the
diffraction pattern. In any case it conduces to simplicity
that 6: should be large. In the plane w= —/, taking 7’, z
as current coordinates, y'=0, 2'=0 is the position of the
geometrical “image, >and at that point the amplitude * due
to ob? vibrators, a nt acting in complete agreement of phase, is
ob’a ; where a is the amplitude due to a single vibrator
at distance 7. The distribution of amplitude (g) over the
diffraction pattern is thus known to be

sina sin 8

B’

g=ol'a

where a=qba'/Af, B=mby'/rf.

6. As an arbitrary measure of the energy of the “image’
we may take the surface-integral of g? over the Hea
pattern; that is, in effect (since “dar =)jfda/ab, dy'=AfdB/7b),

o2b2a2h ef?
2b74, ae Ne se aE ads

ev —©o

22 92 2 £2 ya stra sin
LSE { ne ee dad

7 B
Sia OtaNey 2
The energy diffusely scattered has Snext to be con-
ea With the origin as centre let a spher e of radius 7

* Amplitude of pressure-variation for example.

28 2

628 Dr. C. V. Burton on the Scattering and

be described ; the plane 2= —/ touches this sphere, and over
the smail area effectively covered by the diffraction pattern
the sphere and the plane are indistinguishable. Excluding
from consideration that small area, and a Jike area at the
opposite pole (7, 0,0), take some definite point P on the
surface of the sphere. If, then, Q is any point within
the square )X6, the length PQ may lie anywhere between
limits which differ by a large number of wave-lengths ; and
if a vibrator placed at Q is sending fortha disturbance (4) of
prescribed phase, the phase in which this disturbance reaches
P may be any whatever: under the conditions of the problem
all phases are equally likely, and this is true for each vibrator
independently of the other vibrators. Hence it follows that,
on an average, the (amplitude)? at the point P is equal to a?
(that due to a single vibrator) multiplied by ob}? the number
of vibrators*. The expression oh’a?, being constant over
practically the whole spherical surface, has only to be
multiplied by 47/? to furnish the total diffusely radiated
energy, on the arbitrary scale already used. Thus, finally,
the plane-wave energy, reckoned in one direction only, bears
to the scattered energy the ratio

ota)? f?/Anobt'aft=onl4ar=melv. . . . (5)

8. No matter how thinly the plane of yz is besprinkled
with vibrators, this result is perfectly definite, provided only
that we can deal with a sufficiently extended area to be able
to assign a definite value to co. We are led to the conclusion
that the motion given out by the vibrators can be sharply
divided into two categories: plane waves and irregular
disturbance. If the vibrators are restricted to a finite area,
the plane waves and the irregular disturbance become sorted
out from one another at great distances, or at a more
moderate distance with the help of a lens. Their energies
are, of course, simply additive.

9. It is otherwise evident that a portion of the resultant
disturbance from a plane distribution of synchronous vibrators
must be assignable to plane waves of unique specification.
For if the activity of the vibrators is due to the incidence
of primary plane waves, these waves must pass on with
diminished amplitude, and (in general) with altered phase ;
from a knowledge of which things the amplitude and phase
of the secondary plane waves emitted by the vibrators could
be written down. From considerations of symmetry we
should then know likewise the amplitude and phase of the

* Of. Rayleigh, “‘ Wave Theory of Light,” § 4. Encycl. Brit. vol. xxiv,
(1888) ; Collected Papers, vol. iii. art. 148.

Regular Reflexion of Light by Gas Molecules. 629

regularly reflected waves, and the ratio of their energy to
that of the incident train. Now in a medium such as we
consider, plane waves retain their simple character as we
trace them back, even up to the plane of vibrators in which
they originated. At points close to, or in, that plane, there
will naturally be immense inequalities of disturbance, but all
these inequalities belong to the irregular motion, and have
nothing to.do with the regular waves. A recognition of
this fact leads to great simplification in the problems which
here concern us: for example, a three-dimensional swarm of
vibrators can be divided up into laminee, each of which, in
regard to normally incident plane waves, behaves in a very
simple manner.

10. Suppose, now, that there is a completely irregular
distribution of secondary vibrators over the plane of yz, the
number per unit of area being «; and for the moment
suppose o to be small enough to justifv the assumption that
all the vibrators send out disturbances of the same amplitude
and phase when excited by the primary waves (1). In § 24
it will be shown that this restriction can be removed.

11. Let the secondary disturbance due to the vibrators be

p=¥ 008 (pt—vra—9) Lievb tabetha Ne GOD)

then it is the plane waves comprised in (6) that have to be
determined. In the diagram O is the origin, P a point (2, 0, 0),

p=v(y?+27) the distance from O to any point Q in the
plane of yc, and PQ=s. As in the figure, draw two circles
with O as centre and radii p, p+dp; the annulus between

630 Dr. C. V. Burton on the Scattering and

them contains 2acpdp=2mosds secondary vibrators, whose
contribution to the velocity potential of emitted plane waves
is, for # positive,

dyfp"= 2a sds . : cos ( pt—us—y),

whence

9 Sy

42470

Wy = — sin (pt--us—¥)
The upper limit is written as s=R instead of s=oo for the
sake of definiteness, the sheet of vibrators being limited toa
circle of such great radius p,; (say), that R or piv (1+ 27/p,;")
is as nearly as we please constant over those values of x» with
which we concern ourselves. Thus, within a constant,

2araoC
f= — To cos (pt—vw—y+ 7) We : (7)

v
12. In particular, if the vibrators are tuned to resound to
the frequency of the primary waves, y=$7 and

s=2L

?
POLAT EO I
U

For points in the yz-plane, the total regular disturbance is
represented by
(Wt), <0=(A—2aC/v) cos pt,

and for the amplitude-constant C of the disturbance ema-
nating from each resonator, we have by § 3 (since the ampli-
tude of the exciting waves is no longer A but A—2zaC/v),

C=A/u—270C/v?,
9
v U

Y= — a cos (pt —vu.r)

that is

so that finally

(9)
where w= 21a/v"

Similarly for negative values of #2, that is to say in the
reflected wave,

ap" aE

15. The energy per unit area of the primary waves
{arbitrarily represented by unity), diminished by the energy

l+w

COs (piu)... °. oe eae

Regular Reflexion of Light by Gas Molecules. 631

of the transmitted waves +p” and of the reflected waves
ap’, gives for the energy diffusely scattered

De Ue NG
ee ee eee
The energy of the reflected waves, referred to the same
standard, is w?/(1+w)?; and the ratio of regularly reflected
to scattered energy is w/2=7o/v’, in agreement with (5).
14. The more general case where the vibrators are not
tuned as resonators (y=437r) need not be discussed at length,
since its solution can be derived (§ 23) from that of a still
more general problem. But it will be useful to write down
the expressions for the secondary plane waves when 27a[v?
is negligible in comparison with unity. From (2), (3), and
(7) these are readily seen to be

9)
0 Eee ere
corresponding to the primary waves (1).
15. It will now be convenient to introduce complex
quantities. When, in place of (1), we write for the primary

waves
AE TAG exe) UG =U ie eye sth aay te ee (CLO)

the plane waves emitted by o secondary vibrators per unit of
area in the plane of yz are

ap", pl =
ak Mvexplpicnue)y i) ote nw ea CEB)

2
ae ne —y). a esectig (Cee

16. Suppose, now, that in the space between the planes
v=0 and «=L there is a statistically homogeneous swarm
of secondary vibrators, the average number ot vibrators per
unit of volume being y. For tie most part, no restriction 1s
imposed on the value of v, but when vy is small enough for
the aggregate bulk of the vibrators in any considerable
volume to be but an insignificant fraction of that volume—
the vibrators being then distributed like the molecules of a
gas—some of the results alr eady obtained become applicable.
Consider the lamina bounded by the planes wv’, a'+da!; da’
being in any case very small compared with the wav e-length
of the primary disturbance (10) ; for the moment let it also
be chosen so small that 2avde’]u? is a negligible fraction.

27a A sin ihe te
— a ey t{ pt+ve—y+4r}

where k=

632 Dr. C. V. Burton on the Scattering and

This will enable us to make use of the formula (11), (12)
provided only that the value of v conforms to the restriction
already indicated ; the vibrators in the lamina dz’ being
sensibly a plane distribution for which o=vdz'. Writing
xydz' in place of k in (12) we have

_ 2avsiny

“ni (la—v)
a PGT) | ae

provided y is not too great ;_

expressing, as to intensity and phase, the relation between
the resultant plane waves incident on the lamina dz’ and the
plane waves emitted by the vibrators contained in that
lamina.

17. But even if v is too great to allow of a gas-like distri-
bution of vibrators, so that (13) no longer holds good, the
homogeneousness of the swarm of vibrators still leads to the
conclusion that y is a (complex) constant; or, in other
words, that the waves emitted by an elementary lamina da’
have an amplitude proportional jointly to dz’ and to the
amplitude of the resultant incident plane waves, with a phase
differing from that of the incident waves by a constant.

18. Understanding, then, that v is quite unrestricted, let
the waves originally incident on the slab O<z<L be repre-
sented by (10), and let the saves given out by the lamina 2’
to #'+dzx' be
B'expi{pt—v(#e—z') \dz', B'expi{pt+vu(@—2') da’, . (14)
where B’ is a function of 2’, and is in general complex. At
any plane #=2’’, for which 2’ lies within the limits 0, L
the total disturbance arriving is

A exp 2( pt—ve"’) +| ; B’exp i{pt—v(a@"’ —2')} da!
0

L
+| _ Blexp iipt +u(a"” —2') da"

=H" exp tpt, says 5 ae
so that EK” is a complex function of 2”.
19. Now by definition of y the waves emitted by the
lamina dz" will be

—XE"exp {pt —u(e@—a")}, —yE"exp ifpt +u(a—2")} ;

and these (on replacing single by double accents) must be
identical with (14) ; that is

xB + io

Regular Reflexion of Light by Gas Molecules. 633
or
x 1B’ + A exp i(—va"') +{ Blexpiv(—al' +a )dea'
0
i
+) Blexp i u(a —2’)da'=0.

The double accents being dropped, this may be written
g

xy 'B+ A exp (—i&)+vu7l exp (i) | B’ exp E'dé'
0

0
+v-lexp |, B’ exp (—78')dé’=0; (16)
where E=v0=277/N, Q=vln . =. 7)

20. To (16) add the equation obtained by differentiating
(16) twice with respect to € The definite integrals are
eliminated, and we get

CEO 2iy
een ( a DB

of which the solution is
B=O;, exp ipé+C,exp(—t&);. - . (8)
where b=Vile Divi), oe. (19)

and C;, C, are constants, to be determined by substituting
the expression found for B in (16). This now becomes

O=y-{C; exp mE + Cy exp (—iwE)} + A exp (—2€)
5 SS . / . |
+u exp (— it) {C, exp iwé’ + C, exp (—ipé’)} exp 2&'dé'
0

W/] e . /
+u-lexp el, £C, exp ipe! + Ce exp (—iné!)} exp (—i€')dé'; (20)

which must hold good for all values of § from 0 to 7. Thus,
when §=0

‘ ica
a =f : Seer iigess A (2=1)E
0O=A+x1(C,4+ Ce) + Cv Gea eee
We exp —n(ace De nae
ae —i(ut1) r=

or
— A(w?—1)=C, | (uw? —1)y-!—2v-" wt 1) exp?(w— Ln +7 (w+ 1) |

+C, { (uw? —1)y~1 +207}(u—1) expf —2(u+ 1)n} —tv (wu — 1) ie

(21)

634 Dr. C. V. Burton on the Scattering and
Similarly, when =n (20) becomes

A(w—1)=C1 | (w—1)x7? exp aut 1)9 + tv“ 1) expi(u+ 1a
—iv-(u—1) }
+2 { (u?—1)y7 exp{—é(w—1) } —tv- Nu +1) exp{ —i(u—1) 9}
+iv-\(w+1)}. 2. © (22)

21. The constants C,, C. are thus determined, and the
solution of the proposed problem, in terms of the single
complex constant y, is fully indicated. The results, more-
over, are not limited in their scope to the acoustical type of
problem which has so far claimed our attention ; in the form
(18), (21), (22) they would be equally applicable to a cloud
of light-scattering molecules or particles, whether the scatter-
ing is accompanied by absorption or not: it would only be
necessary to assign to y its proper value in each case.

22. There are two cases in which the results of § 20
assume a specially simple form; in the one case L is very
small compared with A, that is, 7 is very small; in the other
case L (or 7) is infinite. When 7 is very small, it is most
convenient to go back to the integral equation (16), which
now tukes the form

0
x IB+A eof B'dé'=0.
0

This shows that, to our degree of approximation, B is a
constant, so that the above definite integral =By=BuL,
and we get

AX
~ 1+yL
The waves emitted by the total of the vibrators are now

by (14)

a’, ay’ = BL exp 2(pt vx)

AyL Aig
=e MET) | os
vL or 27L/X being small. |

23. If we now introduce the condition that through the
lamina 0<«#<L the secondary vibrators are distributed like
gas-molecules, (13) holds good and (23) takes the form

WW" =— Aufl +wexp i(dr—y)} exp i(ptF uat d—y) -
where w=2re sin y/v",

and o=vL, the number of vibrators per unit area of the

-
|

Regular Reflexion of Light by Gas Molecules. 635

lamina. The last written result can be readily put in
the form

we" a’ = —Aw(1+ 2w sin y +.w”)-? exp (pt fatta—yte)

sn _ weosy, 1+wsiny

Wier eos) AVG + 2w sin y+ w)

24. If the vibrators are tuned as resonators, y=47 and
(24) becomes identical with (9), (9a). Now in the deduction
of (24) no limit has been imposed on the closeness of packing
of the vibrators, except the condition that their aggregate
bulk is but a small fraction of the space through which
they are distributed. If we can conceive of the vibrators as
indefinitely small, and as retaining always their property of
scattering without absorbing wave-energy, the number o
per unit area of the lamina may be as great as we please
without invalidating (24) or its particular form (9), (9a).
The restriction provisionally imposed in § 10 is thus found
to be unnecessary.

25. From (9) together with (5) a good idea is gained of
the change of behaviour of a sheet (or thin lamina) of
resonators as the number o per unit of area is gradually
increased. The proportion of the incident energy contained
in the regularly reflected beam is

Aqr*a?/v*
(1 + 2are[v?)”’
which gradually approximates to unity as o is increased.
At the same time the proportion of the incident energy which
becomes diffusely scattered is v?/wo times this expression,
that is
Anau?
(1+ 2ma/v?)”’
which becomes insignificant both for very small and for very
large values of o; attaining its maximum when 27a/v°=1;
that is when this scattered energy is half the energy of the
regularly reflected train.

26. A further point should now be remarked; if the
secondary vibrators become so closely crowded together that
the freedom of position for any given one is sensibly restricted
‘by the presence of the others, the irregularity of the distri-
“bution will no longer be complete, and the proportion of
energy scattered will be less. As the swarm of vibrators
becomes more and more compressed, though retaining as
complete an irregularity as still remains possible, the distri-
bution will resemble that of the molecules of a liquid rather

636 Prof. J. A. Pollock on a

than of a gas, and the diffusely scattered energy may then
be small compared with v?/mro times the energy regularly
reflected. Thus the lamina, with increasing density, may be
expected to approximate in behaviour to a specular reflector
more rapidly than is indicated by the theory in its simplest
form: true absorption, as before, being absent.

27. None of the incident plane-wave energy is diffusely
scattered by an extensive ordered arrangement of secondary
vibrators, such as that investigated by Lord Rayleigh * (who
points out, however, that the scale of the arrangement must
not be too great if we wish to avoid complications from
spectra of various orders). ‘This leads to a considerable
divergence between the properties of plane assemblages of
vibrators, according as the distribution is ordered or wholly
irregular. For in the jatter case, when the vibrators are
tuned as resonators, (9), (9a) indicate that the secondary
plane waves emitted are always directly opposed in phase to
the incident train. Since the vibrators ex hypothes: absorb
no energy, there must evidently be another energy-term in
question ; and this term, as we have seen, corresponds to the
disturbance irregularly scattered. When the arrangement
of resonators is orderly and not too open, so that the only
secondary disturbances are normally propagated plane waves,
the phase of these must always be such as to keep the total
wave-energy unchanged.

The second part of this paper deals in some detail with
the aspects of the problem of § 18 when the thickness IL is
infinite, as well as with the modifications which have to be
made in the various formule when it is desired to apply them
to the solution of optical probiems.

Boar’s Hill, Oxford,
6th March, 1915.

LXVII. A New Type of Ioninthe Air. By J. A. PoLLock,
D.Se., Professor of Physics in the University of SydneyT.

Introductory.

N an address to the members of Section A of the Austral-
asian Association fer the Advancement of Science at
Brisbane, in 1909, 1 mentioned that observations of atmo-
spheric ionization, made at the Physical Laboratory of the
University of Sydney, indicated the presence in the air of
* Phil. Mag. loc. cit.
+ Communicated by the Author.

New Type of Ion in the Air. 637

an ion with a mobility intermediate between that of the
small gas ion and that of the large ion of Langevin. Under
average atmospheric conditions this new ion has a mobility
of about 1/50, and like the Langevin ion its mobility depends
on the hygrometric condition of the air. The large ion,
however, judging from cloud condensation experiments,
retains its stability even if the vapour-pressure becomes
slightly greater than that of saturation for a plane water
surface, irrespective of the temperature, whereas the ion
of intermediate mobility dizappears if the vapour-pressure
exceeds a certain value, less than that of saturation for
summer temperatures.

In a paper on the Nature of the Large Ion*, recently
published in the Philosophical Magazine, I put forward the
view that in the Langevin ion we have an instance of the
adsorption in the liquid state of water-vapour by a rigid
nucleus, as from the relation between mobility and vapour-
pressure it was deduced that the adsorbed fluid had a latent
heat very little different from that of water. In this paper
I propose to show that the ion of intermediate mobility
consists of a rigid core surrounded by adsorbed moisture
which, on the whole evidence, is certainly not in the liquid
condition.

Trouton}, in 1907, made the interesting discovery that
there are two modes of condensation of water-vapour on
rigid surfaces. If special precautions are taken in drying
the surfaces, on exposure to water-vapour, adsorption occurs
as a dense atmosphere of water molecules, in a state, no
doubt, intermediate between that of a gas and that of a
liquid. At any rate, a change to the liquid condition some-
what abruptly takes place, in these circumstances, when,
according to Trouton, the humidity is about 50 per cent. in
the case of glass, and about 90 per cent. in that of shellac.

In the intermediate ion, the state of the fluid, doubtless,
corresponds to that of the moisture condensed at low pressures
on carefully dried surfaces in Trouton’s experiments. The
two classes of ions thus appear to illustrate in a somewhat
striking way Trouton’s discovery of the two modes of con-
densation. Further, the intermediate ion is not to be found
when the vapour-pressure exceeds 17 millimetres, and it
seems not unlikely that at a critical pressure, by a change
in the state of the fluid surrounding the nucieus, it develops
into the large ion of Langevin.

* Pollock, Phil. Mag. April 1915.

+ Trouton, Proc. Roy. Soc. A, lxxix. p. 383 (1907); Chem. News,
xevi. p. 92 :1907).

638 Prof. J. A. Pollock on a

Measurement of Mobility.

The mobility of the ions wes determined by passing a
steady stream of air through a cylindrical condenser, and
measuring the leak between the electrodes for various dif-
ferences of potential between them, as in Zeleny’s* investi-
gation of the mobility of the small ions, and Langevin’st
original determinations of the mobility of the large ones.
The condenser consisted of a brass tube, 164 centimetres
long, provided with an axial electrode of the same length;
the diameter of the inside of the tube was 3°65, and that of
the inner rod 0°66 centimetre. The inner electrode was
divided into two sections, insulated from each other: the
portion at the mouth of the tube had, on different occasions,
the lengths 3°8,7°7,and 25 centimetres, the distance between
the two sections being 4 millimetres in all cases. This brass
tube with its inner electrode will be called the testing-pipe.

In my experience the ionization seems more uniform if
the air passes through some length of tubing before being
used, and for the great majority of the observations given
here, the air, before entering the testing-pipe, travelled first
through 28 metres of iron piping and then through 9 metres
of galvanized iron pipe, the diameters of the pipes being 4°5
and 7°7 centimetres respectively. In all but the determi-
nations at humidities about 90 per cent., the air was from
the compressed supply of the laboratory, which is fed by a
Sturtevant blower worked by a motor and storage-cells.
The blower was open to the air of the laboratory workshop,
which in turn was kept open to the outer air, the measure-
ments being ordinarily made at night. Control experiments
with air drawn directly into the testing-pipe showed that the
piping impressed no peculiarity on the mobility determina-
tions, though, no doubt, diminishing the ionization.

The estimation of mobility requires the determination of
a critical voltage in connexion with a series of ionization
currents, and although the natural ionization is very variable,
at times the measurements agreed among themselves suffi-
ciently well for the purpose of the calculation. Fig. 1
represents one of the best examples of the type of results
obtained on these occasions when the long section of the
inner rod was attached to the electrometer, while fig. 2,
drawn from similarly accordant measures, shows the form of
the plot of the observations when the electrometer was
joined to the short section of the inner electrodet.

* Zeleny, Phil. Trans. A. excy. p. 193 (1900).

+ Langevin, Comptes Rendus, cxl. p. 232 (1905).

t For further observational detail see Journ. and Proc. Roy. Soc.
N.S. Wales, p. 61 (1909).

New Type of Ion in the Atr. 639

Ines, ab,

ine
-

a EE

/ON-CURRENT

Bank VOLTS |

Owing to the variable nature of the ionization the lie of
the lines in the diagrams is ordinarily subject to some un-
certainty, but the determinations of the humidity are not
sufficiently satisfactory to make any attempt to weight the

640 Prof, J. A. Pollock on a

measures advisable in the present instance. The mobilities
have been calculated, without correction, from the formula

_ Clog. b/a)Q
arta, On eRe

where b/a is the ratio of the radii of the tube and inner rod,
Q the air-stream in cubic centimetres per second, V the
critical potential difference between the electrodes corre-
sponding to a special value of X, the distance of an end of
an electrode from the mouth of the tube.

In the curves like that in fig. 1 there are two critical
voltages. The upper one A, the minimum potential difference
for which the current has its constant value, represents the
voltage for which the extreme ions of a certain class just
reach the further end of the long electrode, counting from
the mouth of the tube, the extreme ions being those which
enter the tube at a distance 0 from its axis. The ions here
are the large ions of Langevin.

The other critical potential B gives the voltage at which
the extreme ions of another class just fail to reach the near
end of the same electrode.

When the short section of the inner rod, at the mouth of
the pipe, was attached to the electrometer, only one critical
potential occurs, as shown in fig. 2. The value here is the
voltage when the extreme ions of a certain class just reach
the further end of this short electrode. It was found that
the critical voltages in the two latter cases refer to the same
class of ions, the calculated value of the mobility being in-
dependent of the particular arrangement of apparatus which
was used. It is the ions of this class which form the subject
of this paper.

The values of the mobilities which have been determined
are given in Table I., T being the temperature in centigrade
degrees, p the vapour-pressure, p/P the relative humidity,
and N the number of these ions per cubic centimetre under
the circumstances of the experiment. A considerable number
of measurements were made with artificially dried air when
the value of the humidity was about 33 per cent. To keep
the table within reasonable limits only the extreme measures
for these humidities have been given, but the omitted results
are included in figs. 3 and 4. The numbers have been re-
duced to standard pressure on the assumption that at constant
temperature the mobility varies inversely as the density of
the air, but this and the corresponding temperature correction,
calculated on whatever basis, are too small to be of the
slightest consequence in connexion with the present measures.

New Type of Ion in the Air. 641

The positive and negative signs in the table indicate the
electrical class of ions to which the respective observations
refer, but it may be stated here that no definite difference
between the mobilities of the positive and negative ions can
be deduced from the results.

TABLE I,
Length
f
Titre, dle Le p/P. N, eceae
cm.
Mee per cent.
15:0. '- 21°8 O78 4 130 3°85
15-24 20°8 0-73 i. 158 \
17-6— 19°7 0:68 is 78 A
30°9— 18:4 4-97 31:5 AN 160
49-24 248 coo 3 ce TG
43:0— 20°0 592 34:0 Bae 3°85
53:0 — ies 6-41 iy oa. is
49:8— 20:9 11:67 64 bu 160
53:0 — ?20°9 10°82 59 Bu fi
534+ 24°5 14:18 62 Bie 25
54:8 — 15:3 6°66 5)\I aes 160
5d 1+ 20-4 13:29 "5 426 4
5672+ 20°1 11:29 64 114 a:
6071-4 216 14-11 73 1 156 2
63°3+ Dai) 15°66 all ‘ 25
653+ 22°6 15°43 76 291 160
68:44 21-0 13:16 71 281 c
86°44 24°3 15°58 69 se 25
91-04 24-2 14-40 64 ms 4
110°3+ 22°1 13°89 70 514 3°85
124-2 — 222, 14°35 42, 1346 a
on — 22:2 15:43 78:5 1144 3°85
139°9 — 24-5 16°67 ie 174 TE
1561 — 23-4 14:87 69°5 458 -
157'°3+ 22°8 14:06 68 1165 ch
350+ 19°5 16°88 89:1 re 25
407+ 19°7 17:09 x ths A

Discussion of Results.

In my previous paper, on the nature of the large ion, I
showed by a simple thermodynamic argument that a formula
of reduction for adsorption observations at different tempe-
ratures is contained in the expression (p/P2)m=(Pi/Ps)!”.
p and P are, respectively, the values of the pressure of the
vapour in equilibrium with theadsorbed fluid,and the saturated

Phil. Mag. 8. 6. Vol. 29. No. 173. May 1915. bs

642 Prof. J. A. Pollock on a

vapour-pressure for a plane water surface at the same
temperature, and z is the ratio of the latent heat of vapori-
zation of water to that of the adsorbed fluid surrounding
the core of the ion. mis the mass of the adsorbed fluid, or
in the present application, the mobility of the ion reduced to
constant air density. ‘The suggestion was also made thata
clue to the condition of the adsorbed fluid might be obtained
in the value of n found necessary in any instance for the
reduction of adsorption observations.

In the case of the large ions, unity, to an accuracy perhaps
greater than one in a hundred, is the value of n which gives
the best fit to a line of the points representing the mobilities,
determined at different temperatures, when plotted against
vapour-pressures. In other instances mentioned in the paper
nis also unity. No heat change due to a variation of sur-
face energy is involved in the value of n, so in these cases
where n=1, as the heat per unit mass necessary to annul a
temperature change due to the mere alteration of state is the
same as that required to keep the temperature constant when
water evaporates, I think we may definitely conclude that
the molecules in the contained or adsorbed fluid are in the
same condition of aggregation as those of water.

Now with regard to the intermediate ion, its mobility also
depends on the hygrometric condition of the air ; this ion,
then, as well as the large one is composed, partly at least, of
water molecules. But the mobilities are as sixty to one,
so, if no other cause of difference exists, the moisture
forming the ions must be in verv different states in the two
instances.

If such is the case, the observations of the mobilities of the
intermediate ion should only fall into line in relation to
the vapour-pressure if reduced according to the formula
( Pi/ P2)m= (P1/P2)””, with some value of n greater than unity.
Unfortunately the results are not accordant enough to enable
the value of n to be determined in this way with any
accuracy, and as (P,/P,)”, within the limits of the observa-
tions, is so nearly equal to unity for even small integer
values of n, all that can be done is to compare plots of the
measures in the extreme cases, firstly when n is taken as
unity, and secondly when it is put equal to some large
number. In the first instance, when n=1, by the preceding
formula the mobility will be constant if p,/P;=p./P,. Ac-
cordingly the mobilities are to be plotted against the relative
humidities, as in fig. 3, and if the points fall into line it is
to be taken as evidence that the adsorbed fluid is in the
liquid condition. On the other hand, when 2 equals some

New Type of Ion-in the Air. 643

large number, the mobility will be constant only if p;=po,
so the mobilities are to be entered against the vapour-pres-
sures. If the plot here is better than in the previous case it

80’
Oy 1)
COLON a ya
x
ae ae Oo
ay 60 O
1
= O
= {
=) ;
=
ma 'o
'
to i
= 140 |
= J
<=
~~ /
pa CGXODO
aie (CCG ODD»)
/
4
20 ;

a 5,0 00 is
MOBILITY—RECIPROCAL
is to be considered that the fluid is in the state of a dense
vapour rather than in that of a liquid. Such a plot is shown
in fig. 4. vs
An inspection of the figures shows that the fit to a line is,
for certain groups of observations at least, better in fig. 4
than in fig. 3. The fit might, perhaps, be slightly improved
if n were taken equal toa smaller integer for the determi-
nations at higher, than for those at lower pressures, which
would indicate that the latent heat of the adsorbed fluid
becomes greater as its density increases. But the general
want of accord amongst these pioneering observations is so
considerable that the present line of argument is not con-
clusive, and cannot be taken as more than supporting the
assumption that the intermediate ion consists of a rigid
nucleus surrounded by a dense atmosphere of water-vapour,
rather than by water in the liquid state. Further evidence
is available, and the assumption is greatly strengthened by a
consideration of the circumstances connected with the

disappearance of the ion.

27T2

644 Prof. J. A. Pollock on a

Instability of the Ion.

The curve in fig. 4 shows that at a pressure of about
fifteen millimetres the mobility of the ion increases very
rapidly with increase in the value of the vapour-pressure.

VAPOUR PRESSURE

| On< to
Om
Oo
10 7
; ;
| of
5 O”

Co 5,0 1,00 1,50
MOBILITY—RECIPROCAL

On many occasions, as already mentioned, successful simul-
taneous observations of the intermediate and large ions were
obtained, but with vapour-pressures exceeding seventeen
millimetres, while the observations of the large ions were
equally good, all trace of the intermediate ion disappeared.
To be quite definite, above this pressure no evidence was
ever found of any class of ions with a mobility between 1/50
and 1/3000. Disintegration of the ion at a critical vapour-
pressure is unlikely, and it is much more probable, assuming
a rigid nucleus, that the adsorbed fluid is in the condition of
a dense vapotr, and that at the critical pressure it changes
its state to that of a liquid, like the moisture adsorbed by
glass and shellac in Trouton’s experience.

Such a change means a decrease in the energy of the
aggregation, and is to be expected when the molecules of

New Type of Ion in the Air. 645

water-vapour round the nucleus become sufficiently tightly
packed. The advent of a liquid surface involves a diminished
rate of molecular escape; rapid condensation will therefore
occur, with a decreasing unit-surface energy, until further
increase in the size of the ion means an increase in the total
energy of the mixture of ions and vapour. The final result
is no other than the large ion of Langevin, where, as I have
shown, the surrounding moisture is in the liquid fate.

There is independent evidence from cloud condensation
experiments that the large ion has a rigid core, but, as yet,
no such evidence exists in the case of the intermediate ion.
If, however, the intermediate ion becomes the large one by a
change of fluid state only, it must have the same nucleus as
the larger aggregation. The mobility which the core alone
would have may be estimated by extrapolation in connexion
with the curve in fig. 4. Jndging from the comparatively
large value which is indicated the nucleus may be, at most, a
collection of not many molecules. In this connexion it is
interesting to remember that the mobilities of the fully
developed large ions, under given atmospheric conditions,
appear to lie within narrow limits; the explanation depends,
no doubt, on some characteristic of the nuclei.

Sutherland, in his paper on the Ions of Gases in the
Philosophical Magazine for September 1909, definitely makes
the suggestion of an ion of intermediate mobility “‘ consisting
of an envelope of vapour, such as that of H,O, surrounding
a small ion which is the central nucleolus,” and he applies
the conception to the experiments of Moreau on the cooled

gases of flames sprayed with electrolytic solutions. The
discussion in this present paper is on different lines to that so

ably developed by Sutherland.

Summary.

A description is given of certain characteristics of an ion
in the air with a mobility intermediate between that of the
small gas ion and that of the large ion of Langevin.

The mobility is found to depend on the water-vapour
pressure rather than on the relative humidity.

Both the intermediate and large ions exist in the air at
the same time provided the vapour-pressure is below seven-
teen millimetres. Above this pressure only the large ion is
found.

On the whole evidence it seems probable that the inter-
mediate ion consists of a rigid nucleus enveloped by a dense
atmosphere of water-vapour. The mass of the ion becomes
greater as the vapour-pressure increases, until at a critical

646 Miss M. O. Saltmarsh on the

pressure the adsorbed fiuid assumes the liquid state, and the
ageregation develops, by the rapid condensation which
ensues, into the large ion of Langevin.

The intermediate and large ions thus appear to form
a somewhat striking illustration of Trouton’s discovery of
the two modes of condensation of water-vapour on rigid
surfaces.

The Physical Laboratory,

The University of Sydney,
January 8, 1915.

LXVIII. The Brightness of Intermittent Illumination. By
M. QO. Satrmarss, B.A., Demonstrator in Physics at Bedford
College, London™.

HEN light from a constant source falls on a surface,
the illumination over a small area is uniform. If
the source is screened from the surface at regular intervals
of time, the illumination will appear intermittent and flicker-
ing, unless the interval of time during which the light is
screened, be small enough. For a particular value of this
interval of time, the flickering of the illumination will just
cease, and in this case the visual impression received when
the screen is illuminated will be just carried on with equal
intensity over the period of darkness, so that the illumination
is apparently uniform. If the time of complete darkness is
less than this value, and the visual impression which is carried
on from one time of iilumination overlaps that received
during the next time of illumination, the eye, while still
retaining the effects of one stimulus, will be acted on by
another.

Is the brightness, therefore, that corresponding to the
new stimulus alone? or, does the residual effect of the pre-
ceding one influence it in any way ?

It was with a view to settling this point that the observa-
tions described in this paper were made.

A photometer bench, about 2°5 metres long, was used and
it was fitted with a smoothly running upright carrying a
Bunsen photometer. An electric lamp was placed at either
end of the bench, ard in front of one,a black cardboard disk
was rotated about a horizontal axis through its centre by
means of an electric motor.

The number of revolutions of the disk per second was

* Communicated by Sir J. J. Thomson, O.M., F.R.S.

Brightness of Intermittent Illumination. 647

determined by a revolution counter working on the shaft to
which the disk was attached.

Hiqual portions of sectors were cut out of the disk at even
distances round a ring, forming a number of equal apertures,
and the same number of equal shutters; the apertures were
not necessarily equal to the shutters in size. The outer
diameter of the ring was 40°3 cm. and the inner 24°3 cm.
The disk was placed in such position that as it revolved, the
lamp was alternately completely open to or completely
screened from the photometer.

The fraction of the light eut off when the disk revolves is
the ratio of the size of a shutter to the size of a shutter and
aperture together. Since the lamp itself was not a line of
light, the times of complete darkness or full brightness were
less than that taken for a shutter or aperture to pass
respectively.

The time curve of the flux of light on the photometer for
a disk with equal apertures and “shutters is of the shape
shown in the diagram.

|
|
|
i

Flux of Li,

Time

AB is the time taken for one edge of a shutter to pass
from one side of the light to the other, and so completely
obscure it; CD is the time taken for one edge of an aperture
to pass from one side of the light to the other, and so com-
pletely open it. It was easily determined by measuring
what fraction of an aperture was occupied by the luminous
filament of the lamp.

The time taken for this fraction to pass in front of the
lamp was subtracted from the time taken for a shutter to

648 Miss M. O. Saltmarsh on the

pass, and this gave the time of complete darkness, BC.
The time DE, during which the illumination was uniform,
was obtained in the same manner; and the total time of
illumination was given by CF.

The observations were made as follows:—The lights were
adjusted at equal heights from the bench and at a distance
of about 190 cm. apart; the photometer was set between
them at the same height in the position of equal illumination,
and the distances from the lamps measured to +5 mm.; the
sides of the spot were reversed, and the photometer was set
again. This was repeated independently three times for the
same distance apart of the lights, and the means of the dis-
tances a and b of the lamps from the photometer were taken
for one side of the grease spot, and the means a’ and 0! for
the other side were taken.

!
The ratio a = = was calculated. The light I, was then
2
moved in 10 cm. and the process repeated, giving another

value for ty The mean of the two was taken.

iL
2

For the whole set of observations the mean difference of
a value from the corresponding mean value was *d7 per cent.
This represents the average error of the observations. If

the brightness, when the disk is rotating, is of what it is

when the photometer is illuminated by the full light from
the lamp, the observation will give a value of x.

If the residual visualimpression does not alter the apparent
brightness of the surface of the photometer, « will also re-
present the fraction of the whole light which is transmitted
through the rotating screen. If it does, « will not be the
same as this fraction, and the illumination will appear
flickering when the residual visual impression only partly
overlaps the next stimulus. Thus flickering would disappear
for a certain speed of opening and closing the shutter, and
would reappear again for a higher speed.

In the case of equal-sized shutters and apertures, the ratio
of the duration of full brightness to total darkness is 1.
With shutters which are smaller than the apertures the
ratio is greater than 1,and there would be more overlapping;
while with apertures smaller than the shutters the ratio is
less than 1, and there would be less overlapping.

In the tables I, is the intensity of the unscreened light and
I, that of the screened. I, was about 18:7 candle-power, as
compared with a 10 candle-power pentane standard.

649

When the speed of the disk was not great enough the
appearance of the photometer was flickering. It was found
possible to obtain consistent values when the flickering was
not too pronounced. In these cases, there was no overlapping
of the visual impressions. Those observations in which the
appearance was flickering are marked with an /f.

Table I. shows results obtained with a disk having 4 shutters
and 4 openings al! equal insize. The fraction of light trans-

mitted by the disk was °500.

Brightness of Intermittent Illumination.

TABLE I.

Time of total | ime of full | oe |
darkness. brightness. I, cl, | *
0144 sec. | 0144 sec. | +7694 1-549 497 f
‘0116 ‘0116 7723 IES 5G | 497 f

Table II. shows results obtained with a disk having

8 apertures and 8 shutters of equal size.

light transmitted is °500.

The fraction of

TABLE IT.
Time of total| ‘Time of full | I, I,
darkness. brightness. ih cl, a
00847 20. 5 00847 sec. 8211 16255 y 504 fu.
00653 00653 "8211 1°646 "497
Mean... 301 mea

In the two above sets of observations, no reappearance of
flicker was observed as the speed of the disk was increased ;
but in both cases the time of darkness was the same as the
time of full brightness, and the duration of visual npression
might be dependent upon the time for which the eye is ex-
posed to light, in such a way that, if the latter were diminished,
the former would be also, and there would be less chance of
overlapping with shutters and apertures of equal size. A
disk in which the apertures were 3 times the size of the
shutters was therefore used, there being 4 apertures and
4 shutters. The fraction of light transmitted by this disk
was *700.

650 On the Brightness of Intermittent Illumination.
The results are shown in Table ITI.

Tasie ITT.

r | |
Time of total | Time of full | sy ie 3
darkness. | brightness. | 1 | ck, 5 |

006: OGL cena ningagateee |) seise memory TA f

002738 =| ~—--01269 8185...) 1-020. | eee

00241 =| = 01483 7904 0 1.) 1054 40a
| | | | Mean. TAT |

If the apertures are smaller than the shutters there might
be less chance of overlapping: The results in Table LV.
were obtained with a disk with 4 shutters and 4 openings,
the shutters being 3 times the openings; the fraction of light
transmitted through this disk is *250.

TasueE LV.
Time of total | Time of full | Li I. |
darkness. | brightness. | Ty | ral “ |
| 0326 sec. | ‘00748 sec. 7904 7) 3175 249 f |
| 0135 00310 “7904 | 3148 "251
bo 0118 | *00270 7904) coli sows 251
| | Mean... "2503 |

No reappearance of flicker was observed.

It will be noticed in the tables that the time of total
darkness, when the appearance is flickering, becomes less
when the fraction of light transmitted through the disk, and
therefore the illumination, is Shae This “1s? am agreement
with the results obtained by Porter *

Conclusion.

From the above values of « it can be concluded that the
residual visual impression carried on from one time of
illumination to the next and overlapping it makes no difference
to the apparent brightness.

I should like e record my thanks to Dr. Womack, of
Bedford College, for suggesting this matter as a subject for
investigation.

Bedford College,
Regent’s Park, N.W.,
March a 1915.
* Porter, Proc. Roy. Soc. 1902.

cote

LXIX. Remarks regarding the Series Spectrum of Hydrogen
and the Constitution of the Atom. By L. Vucarp, Dr. Phil.,
Lecturer of Physics at the University of Christiania *

N the number of the Phil. Mag. for Jan. 1915, Dr. H.
Stanley Allen has published two interesting papers,
where he considers the case in which the circulating electrons
of the atom, in addition to the electrical forces, are acted on by
a magnetic field equivalent to that of an elementary magnet
placed at the centre and with its axis perpendicular to the
plane of the orbit.

Generally, he finds that the magnetic effects “are not in
themselves sufficient to account for more than a small fraction
of the effect that would be necessary to give the observed
distribution of lines in spectral series.”

In the case of hydrogen, however, he tinds that the de-
viation from the Balmer formula as Sora by Curtis would
be explained, when in certain states of motion the electron
was acted on by the field of an elementary magnet with a
moment of 5 or 6 magnetons and placed at the centre,
and he states that “in support of the view that the core
contains 5 magnetons we have the fact first pointed out
by Chalmers that the magnetic moment produced by an

co)
electron moving in a circular orbit with angular momentum

be be 5 q 4 99
of 9, 18 exactly 5 magnetons.

Regarding this last point I should like to make a few
remarks.

The 5 or 6 magnetons which are necessary to explain the
deviations from the Balmer formtla in the way proposed by
Dr. Stanley Allen must be due to a magnetic system near
the centre, and are of course not to be identified with the
5 magnetons produced by the light-emitting electron in the
normal state of the atom; and if the explanation of
Dr. Stanley Allen is correct, it would have important con-
sequences with regard to our conception of the inner
nucleus. |

The inner magnetic system might either be produced by
circulating electrons or by the “rotation of the positive
nucleus.

The angular momentum yp of a sphere is 2 Ma,’#, where M
is the mass, a, the radius, and w the ang ular velocity. The

* Communicated by the Author.

652 Dr. L. Vegard on the Series Spectrum of

magnetic moment M,= and the kinetic energy tuo”.

le
ata
If the mass of the emg a of purely electromagnetic

origin, its radius is equal tae _-, When the charge is supposed

3M
to be on the surface of the nucleus. For a volume distri-
bution we can assume the formula to give a radius of the
right order, or about 10-16 em. for the hydrogen nucleus.

In order to get a magnetic moment of 5 magnetons we

should have to assume an angular momentum of 1800 my
and the number of rotations (v) in unit time would be
4:6 x 101, and the kinetic energy 2°8x10*® erg, or about
10!° times the energy of the outer electron in the normal
state of the atom. As long as we know so little about the
interior of the atom we are “perhaps not allowed to say that
the existence of such rotations and the enormous store of
energy are impossible, although there seems no special reason
for the assumption unless we would suppose that the internal
energy of the atom which is brought to light through the
atomic disintegration is to be of a rotational Son ine e, and that
the rotations are preserved also for the lighter elements like
hydrogen.

Let us next consider the case in which the inner magnetic
system is composed of (N—1) electrons circulating round

the nucleus with a positive charge +Ne. If the angular
momentum of all inner electrons = ma,?w, is equal to 5_?
7

1
they would produce a magnetic moment of 5 magnetons.
Let all electrons form one ring, then

ils a |
ce (N—1)*(N—Z8yn-4)
atid W,=(N—18y_.)(N—1)W.

a; and W, are radius and energy of inner ring, a and W

the corresponding quantities for the light-emitting electron
in the normal state. |

In order that the inner system shall act electrically on
the outer electron as a single charge +e, = must be a small
quantity. In fact, “ diminishes rapidly with increase of N.
Thus with an inner ring of 4 electrons = 61, and W,=940W,

* Ttis supposed that the inner nucleus can be treated asa charged solid
body. :

Hydrogen and the Constitution of the Atom. 653

and we see that the energy of the magnetic system also in
this case would be very great compared with that of the
light-emitting electron in the normal state.

Hven if we take it for granted, however, that the assumption
of aninner magnetic system isa legitimate one, we should still
meet with the difficulty that, according to Dr. Stanley Allen,
the magnetic moment must vary considerably with the state
of motion of the light-emitting electron. In fact, it is
assumed that for the state of motion corresponding to an

en :
angular momentum of — the magnetic moment of the inner
T

system is equal to zero, while for the stationary circles of
greater momentum the magnetic moment is 5 magnetons,

Dr. Allen gives no indication as to how the passage of the
electron from one stationary circle to the next can increase
the magnetic moment from 0 to 0 magnetons. With certain
modifications of Dr. Allen’s assumptions we might, however,
in quite a formal way explain the formula of Curtis through
the effect of an internal magnetic field.

We suppose the inner magnetic system to be produced by
circulating electrons, and that the inner magnetic system
and the outer electron maintain a constant difference of

I
momentum equal to —,

L- Bi= =?
uttine hr
p g p= 5
2Qar

h
(p= 9,,(T— 4),

and the magnetic moment of the inner system would be
ye
I~ hare?
where c is the velocity of light”.
Now Dr. Allen has deduced the following general formula
for the magnetic influence on the spectrum:

= (r—2) =5(rT—2) magnetons,

Vv 1 A,
N \ Bs’ =)
(2+ = (1+ =
where
16 wr’ me?® 4 qr e4
Jee M,=—3>-(T—2).
As g he (T )

* Mg and e are given in electrostatic units.

694 Series Spectrum of Hydrogen and Constitution of Atom.

Putting A a7"
5 B= apie and 7,=2, we get B,=0 and

We A ake i
fs pet eee

This formula is somewhat different from that found by
Dr. Stanley Allen, but it will equally well represent the
observed facts.

For the six lines considered in Dr. Allen’s paper we get:

eh ee Ave Se 51 7. eam

{

| es Gee 1/9 | 1/8 | 3/25 | 1/9 | 5/49 | 3/32 |
| | | |

a a oe
It happens that for these lines eae comes out

practically constant equal to 1/9, and we get the formula

Hive UN ud 1

Noh cans
which has exactly the same form as the empirical formula
found by Curtis. Using the values et and

e=4:78x10-, we find B,./9=5°9x 1078, while the corre-

sponding constant in Curtis’ formula is equal to 6°9x 10~°.
Thus the deviation from the Balmer formula would be

satisfactorily explained through the magnetic influence of the

; : h
inner core, when a constant difference of momentum of —
T

issupposed to be maintained between the outer and inner
system.

It may be granted that in dealing with atomic structure
we have a fairly great freedom for making assumptions, but
still I think we ought to hesitate in assuming any connexion
between the outer and inner system which would change the
magnetic moment of the latter from zero to 5 magnetons
when the outer electron passes from the circle corresponding
to r=2 to that for which 7=3.

We have previously seen that the magnetic core of 5 mag-
netons, whether it consists of a rotating nucleus or a system
of electrons, would store an energy which is enormously

Electron Theory of Optical Properties of Metals. 655

greater than the kinetic energy of the outer electron ; and it
seems hardly possible to suppose a system with so much
energy to be essentially affected by the passage of the outer
electron from one stationary circle to the next; for it must be
kept in mind that in the passage of the electron from T=2
to t=3 the inner system should take up the whole energy
involved in the production of the magnetic field of 5
magnetons.

As, however, the assumption of a mutual connexion
between the inner core and the outer electron seems essential
for the explanation of the correction term of the Balmer
formula through magnetic influences as proposed by Dr.
Stanley Allen, it seems that we shall have to seek another
explanation for these deviations.

Christiania, Feb. 8, 1915.

LXX. On the Electron Theory of the Optical Properties of
Metals.—Il. By G. H. Livens *.

i. [NTE ODUCTION.—The explanation of the phenomena
associated with electrical conduction in metals based

on the electron theory, originated by Drude and Lorentz f,
was generalized for application to rapidly alternating fields
such as those associated with radiation first by J. J. Thom-
son ft, and then subsequently in greater detail by Jeans §
and H. A. Wilson ||. The general method of attack adopted
by the last two authors differs essentially from the more
direct methods employed by Drude and Lorentz and in the
hands of Wilson, who alone works it right through on the
statistical basis, it leads to formule which differ essentially
from those obtained by Lorentz, whose results, in the opinion
of the present writer, represent the only complete formule
to be obtained from the theory as usually specified. Ina
recent paper on the electron theory of metallic conduction,
exception was taken to the deductions of the formula for
the electrical conductivity put forward by Thomson and
Wilson. A correction vf considerable importance was made
in Wilson’s detailed analysis which renders the final formula
to be obtained from it more consistent with Lorentz’s original
results. In the present paper it was intended to carry out a
promise given in my former paper and to extend this same

* Communicated by the Author.

+ Vide Lorentz, ‘The Theory of Electrons.’

t Phil. Mag. Aug. 1907.

§ Phil. Mag. June & July, 1909,

|| Phil. Mag. Nov. 1910.

656 Mr. G. H. Livens on the Electron

correction throughout the whole analysis for the optical
properties of metals along the lines laid down by Wilson
and Jeans; but on attempting the problem along these lines
I found that the fundamental differential equation on which
the theory is constructed turned out to be identical with the
equation used by Lorentz to determine the velocity distri-
bution function. It was therefore preferred to adopt the
more direct method of attack constructed on the basis of
certain remarks bearing on this subject in a former paper ™, in
order to exhibit clearly the very general validity of the method.

In addition, the opportunity will be taken to introduce a
modification of an entirely different character into the general
theory, which has long been considered necessary in a proper
treatment of the subject but which has, as far as Iam aware,
never yet been introduced.

2. General basis of the theory.—We shall for the present
assume, with all previous writers on this subject, that the
whole of the electrical and optical properties of any metal
arise from the fact that there are a large number of electrons
in the metal free to move about in the space between the
atoms. The atoms and electrons will be presumed to be
perfectly elastic spheres, at least so far as concerns their
interaction in collision: we shall also presume that the
atoms are comparatively of such large mass that the magni-
tude and direction of the velocity of any atom and the
magnitude of the velocity of the electron are unaffected by
a collision between the two.

In the absence of any external field the atoms and elec-
trons will be moving about in a perfectly irregular manner,
and the velocity distribution will thus be exactly that
specified by Maxwell’s law, so that if N is the number of
free electrons per cubic centimetre of the metal, then the
number in the same volume with their velocity components

between (&, 7, 6) (E+dE, n+dn, €+dZ) is given by
sN=Ny / = evdédndt,

wherein we have used
WSP+7?+C
and g for a constant connected with the mean value u,? of u?
for all the electrons by the relation
3

a Da?

* “Qn the Electron Theory of Metallic Conduction,” Phil. Mag.
March 1915,

Theory of the Optical Properties of Metals. 657

When an electric field (of intensity H) is applied in any
direction all this alters; the electrons will be pulled about
by the field in such a way that they will acquire momentum
at a rate of eH parallel to the direction of E. We shall
assume, however, that each collision between an electron
and an atom completely removes all effects imparted by the
electric field during the previous free motion, so that the law
of distribution of the initial velocities for the free paths being
pursued at any instant will be precisely that given as above
by Maxwell’s law. It seems necessary to make some such
assumption as this in order to ensure, for instance, the possi-
bility of the existence of a steady state when the electric
force is uniform and constant in time ; and the present one
is probably the most general assumption we can make™® as it
involves no detailed specification as to the dynamica] nature
of the collisions, and also enjoys the comparatively wide
range of generality possessed by Maxwell’s law itself. |

It is assumed that collisions between electrons are too
infrequent to be of any importance in the theory: this
appears to be a legitimate assumption on account of the
extreme smallness of the size of an electron. We shall also
assume for the present, that the motion of any charged atoms
is unaltered by the field, so that there will be no contribution
to the current on this account.

3. The instantaneous velocity distribution when the electric
field is in action.—On the basis of the assumptions mentioned
in the last paragraph, it is possible to calculate the actual
instantaneous distribution of velocities at any instant when
the electrons are subject to the action of an electric force EH,
which will be presumed to be applied parallel to the z-axis
of coordinates chosen for the analyses. In fact the number
of electrons which started their paths with velocity com-
ponents between (&, m, >) and (&+d&, mo+dm, +40)
at {imes which lie in the small interval between t=7 and
t=(t+drT) previous to the instant ¢ is

BN, =8Noe Fat,

where
g — Quy
sNy=Na/ Ge dE dn dh,

* [April Ist, 1915.] It is, as a matter of fact, involved in the
assumption of hard elastic spheres for both atoms and electrons,
+ Vide Lorentz, ‘The Theory of Electrons,’ p, 308.

Phil. Mag. 8. 6. Vol. 29. No. 173. May 1915. 2U

658 Mr. G. H. Livens on the Electron

where also
2 £2 2 2
Uy? = Er +7," + So

and T, is the mean value of 7 for all the SN, electrons and
which is identical with the mean time between two encounters
for an electron with velocity wp.

The velocities of these electrons at the instant ¢ are
given by

ie
E=f)+ “( Edt’, =m, c=%;

where ¢’ is used as an auxiliary time variable for the
integration. We write the first of these equations in the
form

ep
E a: Mm
so that
az
o— Edt’.
t’=t—r

Thus if we interpret the function SN, in terms of (&, y, €)
instead of (&, 4, )), we find that the number ot electrons
at the instant ¢ with their velocity components between
(£, n, 6) and (+d, »+dn, €+d£) and for which the
value of 7 is as specified is
| Fi glance 2
IN,= on / Le o( ) ™ d&dndtdr,

wherein we have neglected squares of the small quantity ¢,
as is usually done in these theories, and used

2
Tl a ea
To the same order of approximation we may also write

Ny Ughe ak ie, gee
SN a Ge (1+ MS®) dtdndtar,

On integration over all values of r from 0 to « we find the
total number of electrons per unit volume at the instant ¢
with their velocity components between (& 7, ¢) and
(£+ dé, n+dn, €+d€) in the form

fe ue OusumMeee (oe a2 da,
sN=Na/ Se (1+ came be ra ddd

Tm )t'=t—r

Theory of the Optical Properties of Metals. 659

which is the generalized form of a result already given for
two special cases of limited applicability.

If E is constant for a time large compared with the mean
time between two collisions, this “reduces at once to

Laat) oem pe 2
pN=N4 / Se ‘ E see i Te rm dr |
Us Cs OC Namiararms
ang/ Be [4 EB)
3 2

wherein, if we write, what is pou true,

we recognize Lorentz’s well-known general law of distribu-
tion of velocities of the electrons under the action of a steady
field.

If we put again, in a simple harmonic field,

Dy Bye”,
we get
Sone gr? ) cE oO upt i ip(¢ _ in) an
IN=Nq / Loe E as veh Eo ea) Cy ie dr dEdndg
0 ip Tr

pile A : pt
Ge (or 2qeEHy tp e oP.
=n / Se 1+ ee (e" a] aa aera d&dndt

oe e, ace 2qeETm Byee"

T° m 1+iptn

al dEdndé,

which agrees with a result obtained directly in another
paper.

It appears, however, on due consideration that this last
result is restricted for application to problems in which the
field is represented by a simple harmonic train of stationary
waves in which the wave-length is very long compared with
the mean free path of anelectron. ‘To remove this restriction
we must take account of the change of phase in the vibrations
of the field from point to point in the metal, and to do this
we must introduce a more general type of field. We may
assume quite generally that the field is propagated in the
direction of the z-axis as a plane simple harmonic wave-train

with the velocity - ae “*, so that the electric force K, which is

* cis the velocity of radiation in vacuo.

2U 2

660 Mr. G. H. Livens on the Electron

presumed to be polarized parallel to the z-axis of coordinates,
depends on the coordinates of time and space by the ex-

‘ (° Q =) so that

He eg ia

If we now confine our attention to all the electrons which
at time ¢ lie in or at least infinitely near the plane z=z (say
between <=z and z=z+dz), then we shall have for any one
of them which has been moving for a time 7 since its last

collision previous to the instant ¢

io(«—=) ee nZ

EB c wire l—

ages hii ”)
6

wherein (&, 7, €) are used as usual to denote the velocity
components of the electron at time ¢.

Thus the distribution of velocities among these electrons |
near the plane z=z< is such that there is at the instant ¢, the
number

was Zz | 2gckE ("( —ipr(1—nZ) —<dr
cua ve : mee mip(1—ng) J, ars Je a apne

per unit volume with their velocity components between

(é, UE ¢) and (E+dé, n+dn, oe dg). This gives

at Seg eae acta)
sN=Nq/ Lie [1 cs m({1l+ip(l—n€)ta | dednds,

which determines completely the velocity distribution among
the electrons in and near the plane z=z. This is the result
which will be of greatest use to us for our future work: it
might easily have been obtained directly from the previous
result for stationary waves by a simple application of
Doppler’s principle.

It must be insisted that the various formule here obtained
are of very general application, in no way less general than
Maxwell’s lawitself. The final results involve no assumptions
as to the rapidity or otherwise of the variation in the field E,
and will in fact be generally applicable for the most rapidly
alternating fields. It is interesting, however, to notice that
for these very rapid alternations the velocity distribution is

ponential factor e

Theory of the Optical Properties of Metals. 661

practically identical with that given by Maxwell’s original
law, which is just what we should expect, as in such circum-
stances the alternations of the field are too rapid to take
effective hold on the inertia of the electrons.

4. The expression for the current density.—We are now in
a position to calculate the conduction current density under
the assumptions we have specified above. ‘This is done
exactly along the lines laid down by Lorentz, and is therefore
given by its components (1,, I,, L.) parallel to the coordinate
axes, where

L=e{£6N, I,=e\ndN, L=e( con

the integrals being extended over all values of (&, 7, €) for
all the electrons per unit volume.
On inserting the value of 6N we find that this gives

2 Ti
+0 ("to (7+ -90,2N £2 Say pay te i cmthe i
= a \_ iy | ee é vi fe \ Mee Bad | dédndt

C=t—-7
_ 2Ne*q ee +0 (+e oie a0 (" - = dr (=r 1
i Tie: Te: —2 \ — i LE eo ‘ Tm t! Pas d&dnde

waled,—1—0.

This gives us the general formula for the current at time ¢.
Particular cases are worth noticing.

(i.) If E is constant for a time large compared with the
mean time between two collisions, then we have at once

, =e +o a Bae Be 7 dédndt
a m T° 2A ee a 0 7

_ INeg y, g | an

mr hin \ =i ae Se

Now noticing that we can put, as a good approximation,
bin

Up ae
m y?

+o — gu
Erne d&dndé.

we see that we can at once transform the triple integral with
respect to (£, 7, €) by a spherical polar transformation. into
-a single integral with respect to w from 0 to infinity, if we
put

2]
U ia

dédndf=4arwdu and &= 3°

662 Mr. G. H. Livens on the Electron
and. we thus get

ik pana BBL, (ea 4 nf ee ee :
anes an

and using v=qu? this reduces to

AnrNe? i lay ee J _
I= 3mq E(" i de,
4Ne71,, / 2)
ay =( 3m 2

which is Lorentz’s result for steady currents.
(ii.) Again in the case of stationary waves of period p and
comparatively long wave-length, we may use

tpt
E=EKye?” 5

and then we deduce as above

ee 87Ne? St Ne*glin af Eki ies Ra ie Fe.
a tuner ae

2 14 Bln,
U

which agrees with the results of my former paper.

(iii. ) nie however, we use the real case of progressive
simple hoanatonie waves we must, as above, put

H= EKoe ai e)

and then we find firstly that

1 — INC ng vot ae (eee ge" d&dnd&
m be ae, Fgh = rn | u
Or again by a spherical polar transformation in which
E=ucos0, C=ucos dcosd,

the triple integral reduces to

Sa ee Tv 27
{ oat), wean | cos?@ sino do { - Le
0

0 J (1 + Bin) a £— "sin 6 cos @

Theory of the Optical Properties of Metals. 663

Now we know that

se pln Pm)
" ag Oe GOO ) Rc a
pln He : ( ‘Dey prniln “7 oh, 9
: (1+ ais sin @ cos : Ihe a) eee 0 cos*h
which ce by a well-known method and gives

ou (1+% hm Wn

mm» 2 2] 2
Vie (1+ ne [ (1+ ia), ais ae @|
a 5; 2 OEY MO) °
ve (1 4- ‘P| i 2 _— sin? 0

The integral with respect to @ is then
cos? @ sin 6 dé

9) Tv
47 | ip Le 2 HOG lise :
0 ~/ Ce P=) + io Ge sin?@

which by the transformation

lim \? aise
ae 039 = 4/(1+ tie aad aR

is directly evaluated in the form

2] 2 :
aie 1b? ig nln | prlm
c sin-} } :

vintlsg ah ( ey p nlm?
C? mi prey ea our }

(1 + Pyne

i. apn \” pn ln?
(1 i 7) nga
Calling this expression
dary (u),

we have then on insertion in the last integral,

en . NS
SarNe7lm io — gue an ip| é— P5
i= ee J Ef wy (we du | Kye ( )

664 Mr. G. H. Livens on the Electron

from which all the circumstances of the problem can be —
deduced, even though it is impossible to evaluate the last
integral except perhaps approximately. We notice, however,
that if the velocity u for the majority of the electrons is

considerably less than = the velocity of radiation in the

metal, which condition is, I presume, nearly aiways fulfilled
at all attainable temperatures, this formula reduces as a first
approximation to that given above for stationary waves.
The theory for stationary wave-radiation is therefore of
rather a surprisingly wide generality, in spite of its more
apparent restrictions. It will not, therefore, be necessary

for us to examine the present case in any further detail, even
if that were possible.

5. On the fundamental differential equation of the Jeans-
Wilson theory.—Jeans, and Wilson following him, adopt
rather a different mode of attack, based on a calculation of
the rule of increase in the momentum of certain specified
groups of electrons. Wilson’s analysis is slightly the more
general and detailed of the two, and I shall therefore confine
my attention to his equations alone.

Wilson assumes that the number dN of free electrons per
unit volume with their resultant velocities between wu andu+du
remains practically constant, although particular electrons
are continually entering and leaving the group. He there-
fore attributes to each such group a definite permanent
existence whose average motion under the action of an
electric force may be specified by a certain differential
equation which he finds to be of the form

£ (me dN)=EHedN —Bw dN,

wherein w is used to denote the average velocity of the
group in the direction of the applied electric force EH and
8 is some function of u, m, and e. On integration of this
expression over all the groups and using

{= e\w dN,

Jeans’ equation for the electric current density I is obtained
in the form

di Ne

am 1 oe

In the theories of Jeans and Wilson this differential

Theory of the Optical Properties of Metals. 665

equation is fundamental, but in the opinion of the present
writer it is by no means complete asitstands. The following
deduction of the equation may perhaps make this point clear,
If 5N denote, as before, the number of electrons per cubic
centimetre with their velocity component between (&, 7, €)
and (€+dé&, n+dn, €+df) then the component of the
momentum of this group parallel to the z-axis is m£6N, and
if E acts in the same direction we have

Now we know that

m = =) Dp
whilst according to Wilson the change in ON in the time dé
is equal to the number of collisions which take place in this
group in the time dt with the sign changed, and this is

*
d(6N)=—dN \". Sine us
Nee
ee
Ta
so that
d oN
dt (CRD Re

° les : ‘
and thus using t,== — we have the above equation in the
form i
méudN
UP ;

and on integration over all values of (&, 7, €) we reproduce
the Jeans- Wilson equation given above.

But the assumption that the change in 6N is due entirely
to the collisions is not valid. In fact 8N is itself a function
of (&,7, €) and therefore changes on account of the variations
in these quantities. In the present instance we ought therefore
to write : momentum equation in the form

a (EON) =e (N+ &8N)) - a,

* This neglects the contribution to the momentum of the group by
the electrons coming into it. This does not, however, affect the tinal
result as on integration these terms go out by themselves.

666 Mr. G. H. Livens on the Electron

and it is then an interesting verification of our present
analysis to show that if 5N is given by our previous general
law, viz.,

a8 y 9 NS 3 uxt
sN=Na/ Tye [14 a | : el dt! |g dn a,
7 m 0

Tm Jt =t—r

then the equation is identically satisfied.

6. The polarization currents and the total electric field at
a point in the metal.—It must now be remembered that in
the case of most metals to be dealt with, there is usually
a contribution to the total current of electricity not only as
the result of the motion of the various free electrons, but also
as aresult of the relative displacement of the neutralizing
charges in each atom, caused by the electric field pulling the
opposite charges in opposite directions. This part of the
current is easily calculated, as has already been explained in
great detail in a previous communication on absorption
in dielectric media, and turns out to be of the form

dt’
where P is a vector defining what is analogous to the
polarization in dielectrics and whose intensity is given by

A
Sone
where
kes e?/m

n2—no? tinny ’
wherein =} denotes a sum taken over all the electronic
resonators in the atoms per unit volume for the typical one
of which np is the period of free vibration and mn’ the
coefficient of the damping force in its equation of motion.
The constant a is a numerical constant whose value in an
ideal case is } and in any real case is at least of this order of
magnitude.

In addition to this current there is, of course, as always,
to be added the ethereal displacement or polarization
current, which is measured by

dK
dt’
as in the Maxwellian theory.
Now we know that the resonance electrons in the atoms

Theory of the Optical Properties of Metals. 667

are effective in modifying the electric field at any point
Inside the metal, so that they will by such means also in-
directly affect the currents of conduction. In fact, at any
point in any homogeneous medium polarized to intensity P,
there is an additional electric force of intensity

in the direction of P, arising solely from the distribution of
the immediately surrounding polarized molecules or atoms.
We must therefore include this part of the electric field in
the general expression for E, which therefore now becomes

K-+aP.

This is the complete expression for the total electric force
which is effective in driving the electric current.

7. The electro-optical equations—The fundamental equa-
tions of the optical theory are the generalized Maxwell
equations which, expressed in their differential form and
using the Hertz-Heaviside system of units, are simply, for
non-magnetic media

le rN Ona ohalcs

qi =Curl F, Cap TOME
wherein EH, H denote as usual the electric and magnetic
force vectors at any point in the field, I’ the total current
density at the same point which is, inside the metal ex-
pressed by

di} ab
dt Bae i

Let us now examine the propagation of light in a medium
where these equations are satisfied. In order to simplity the
equations we adopt the standard convention and consider as
previously the propagation of plane homogeneous waves

taking place in the direction of the axis Oz, so that the
components of E, H, and I’ involve the coordinates of space

Wee ita =

f NL
and time by the exponential factor e aN c), where v is in
general a complex quantity, a function of p, the frequency
of the light disturbance used.
Since in this case all differentials with respect to v@ and y
vanish, the general pou reduce to
— Hy = = =i a ni, —— Hy

c c

668 Mr. G. H. Livens on the Electron

and thus for the propagation of this wave-train to be possible
in the medium we must have

e 9 al i]
Hore Dye

which is the general relation to be satisfied between the
electric force and current components.
If we put
N=b—iK

and assume that w and « are real the exponential factor is

and thus — is the velocity of the disturbance in the medium,

and therefore m is the ordinary refractive index. The
absorption is determined by «, which is therefore called the
absorption coefficient.

8. The equations in the metal—Now we know I, in terms
of E,, and can therefore at once proceed to an examination
of the equations as far as they concern the propagation in the
metal itself.

If E, is, as above, of the form He” Ge) then we know
that
TAME eile
dt oa di

where C is used for the generalized form of the conductivity
which we have found to be

eee ela Aa d fy ury(u)e—* du,

x(u) having the same significance as before ; but

dK, Gee
ae ==iphi,, ar =p;
AF,
and P.= iL —aX

as before, so that

-=[eo ad oe s = ;

Theory of the Optical Properties of Metals. 669
We therefore conclude that

or again
F C A
ees Vp AE Sse INES at iss See
Cer (aA) (1 ie ie)

which is the fundamental equation of the theory in its most
general form.
If we write

hen cays ch
(Ho — 2K) aa?

so that “y and x) would be the refractive index and absorp-
tion coefficient respectively of the metal if the free electrons
were extracted, and if also we can separate C into real and
imaginary parts in the form

C= C; + Co,

then we may conclude that
C
2 2 2 2 2 *
be K Ko Ko + ean:

whilst
K = nko 4 OTA
pe = Fo 0 ‘al ai

The whole question thus turns on the determination of C
and its separation into real and imaginary parts. We have
above given the most general form for CU obtainable on the
basis of the present theory, but it is probably too complicated
to be of any assistance in the present instance, and we must
have resort to the-first approximation furnished by the
analysis for stationary waves. In fact, if the velocities of
the greater majority of the electrons are considerably smaller
than the velocity of propagation of radiation in the medium,
we may write with a sufficient approximation even for
progressive waves

1 4 (Pim

om ar

* This separation is not correct since A is also complex; but it is
probably sufficient in any real case to take A in the terms in C with its
real value alone.

670 Mr. G. H. Livens on the Electron

so that ERY
Me Sa Ne?qlin @ { we du
Oi oe aan GN ct, Ge
Lh Nae
and :
8arNe’ql?, p g? (wet du
CamiGutar one ot Ny eh wen
3 7 pe,
ol ee 2

which appear to be the most generally applicable results
which it is possible to determine from the present form of
theory.

It must be noticed that C, is the ordinary expression for
the conductivity and can be written in the more usual form,
if we use

v= qu,
Gels 2 Ne71,) : ve" dv
jae ee p2l.2q°
T MU, Pn Y
7 0 it =i Aaa

9. The emission of light by the metal and the complete
radiation formula.—The emission of light by the metal as a
result of the electronic motions taking place in its interior
has been fully discussed by Lorentz and subsequently by
Thomson, Jeans, Wilson. and others. Under the funda-
mental assumptions on which the present theory is based, it
is found that the emissivity of a thin plate of any metal of
small thickness A and for light of period p is

5 NE ge q (~ ve—*dv
1

emg 7 44 =

the notation being exactly the same as in the present paper.
The coefficient of absorption for such a thin plate is shown

by Lorentz to be equal — where o denotes the con-

ductivity, so that it is

Ara tt NC mA al ve" dv
1

36mg 7
0

Theory of the Optical Properties of Metals. 671

so that we have, with Lorentz,

E 8a Hy’ He pm

A ¢ A’ Ae*arq’
or using 3 2are
g= ————
q Upsets t x 9

we have - Owrmu,,2

ti Shah) Phe

The present analysis thus verifies completely what was
surmised in a former communication, viz. that Kirchhoft’s
law, which affirms that H, is independent of the particular
metal under consideration, is absolutely true on the basis of
the present theory. The incompatibility of the previous
results rested in the formula adopted for the conductivity
which has in the present instance been considerably modified.

It thus appears that the variation from the above law
obtained by Wilson in his analysis is not, as he says, due to
the neglect of the motion of the atoms and the mutual in-
fluence of the electrons, but rather to an incompleteness in
his analytical investigation of the statistical motions of the
electrons themselves.

The reason why the present form of theory fails to give an
appropriate form of EH, has been fully discussed in another
place and need not detain us any longer here.

10. On the dissipation of energy.—Before closing this paper
I think it is necessary to point out an apparent discrepancy in
the present form of theory, which at first escaped my notice.

According to Thomson the energy dissipated per second in
a unit volume of the metal is equal to the work done by the
electric force on the electrons during the whole of their free
path motions, and according to a calculation I made of this
work in the case when the electric force H is given by

H = Ey cos pt,
I found that the energy dissipated per unit time and

volume is

i ° fg a
H= Ai 2 Ne%l Hy’ (" cin cues
ey ee 27 2.9
3 = My», Bhan Wau Pelm™g
.

which is taken to be equal to the mean value of o'E?, which

* Phil. Mag, January 1915.

672 Electron Theory of Optical Properties of Metals.
is }o'E,”._ We thus conclude that

Go 2 Ne’ln (2 e-*dy
vin IT MU», un Gg
o 1 +*——

In the above discussion and also in a previous paper}l
find, however, that the more consistent value for o is

2 aia
ets 2 Ney, " ve—°dv
il

3 MUm abe
Uu te Er - gq

Wilson obtains by both of his methods formule which

differ from the first of these by a factor 2 but his deductions
are I believe both incomplete.

There is unfortunately a mistake in my previous work
which considerably affects the final formula. The result of
the integration at the top of page 182 of the paper mentioned
should have its denominator squared. On making this cor-
rection a formula for the conductivity is obtained which is
perfectly consistent with that given above, so that there is
no real discrepancy*.

11. Conclusion.—It thus appears that a detailed analysis
of the motion of the electrons in the metal along lines
suggested by Thomson and Lorentz enables us to formulate
a complete theory for the action of the most rapidly altern-
ating fields in the metal. Moreover, this theory is perfectly
consistent with the special deductions given by Lorentz in
the particular case of steady fields, and in this respect shows a
distinct advantage over the form of theory proposed by
H. A. Wilson.

A discussion of the actual bearing of the present form of
theory on the fundamental] question as to the number of free
electrons taking part in conduction will be reserved for a
future communication.

The University, Sheffield,
Nov. 14, 1914.

* 'Chis mistake was kindly pointed out to me by Dr. Bohr. I must
also add that through the kindness of Dr. Bohr I have, since the above
was written, had the pleasure of reading his Dissertation on the present
subject, published in 1911. Reference will be made to this work in
future papers.

f° 673° J

LXXI. Lead and the End Product of Thorium. (Part II.)*
By Artuur Houses, A.K.CS., B.Se., £.GS., Imperial
College, London, and Ropert W. Lawson, M.Sc., Radiwm
Institute, Vienna f.

CoNnTENTS.

§ 9. Constant of Disintegration of Thorium E.
§ 10. Bismuth as a Possible End Product of the Thorium Series.
§ 11. Further Evidence from Atomic Weight Estimations.
§ 12. Remarks on the Selection of Material for Atomic Weight
Determinations.
§ 13. The Possible End Product of the Actinium Series.
§ 14. Conclusions.

§ 9. Constant oF DISINTEGRATION oF THorIUM H.

HE results we have so far obtained show indirectly
that thorium E is an unstable element, even though
the radiation from it has not hitherto been detected. In
order to obtain information as to its half-life period we must,
then, resort to some method other than those usually used.
The present section gives a more complete treatment of a
method outlined recently by one of the authors{. Before
passing on to this, however, the suggestion made recently by
Fajans in this connexion must be considered §. Using the
results of an analysis by Boltwood || of a thorium mineral
poor in uranium (thorite), and assuming that all the lead
estimated had its origin in thorium, Fajans gave 2.10" years
as the maximum possible value of the half-period of thorium
EK. Absence of knowledge regarding the age of the mineral
referred to, renders it difficult if not impossible to determine
by how much this value is too great; and in the case of
another thorite analysis (by Holmes, loc. cit. 1911), also used
by Fajans, where the age is known, it is certain that not
more than 12 per cent. of the total lead found can be due to
thorium. Using this fact, it can be readily shown that
the Th E/Th ratio is 8.10~*, from which the half-period of
thorium E is found to be about 1°2.10° years. The thorium
content of this mineral has been given in Table I. No. 3
(Part [.).

Since thorium E is unstable, it is clear that definite and
reliable results for the half-period of this element can only
be obtained from analyses of minerals rich in thorium. A

* Part I. of this paper was published in this Magazine, vol. xxviii,
pp. 828-840 (Dec. 1914).

+ Communicated by the Authors.

{ Lawson, ‘ Nature,’ July 9, 1914, p. 479.

§ Fajans, Heidelberger Akad, Ber. . Abh. xi. p. 12 (1914).
|| Boltwood, Am. Journ. Sei. xxiii. p. 88 (i907).

Phil. Mag. 8. 6. Vol. 29. No. 173. May 1915. 2X

674 Messrs. A. Holmes and R. W. Lawson on

fortuitous choice of analyses would be useless for our purpose.
The effect of thorium lead must be sufficiently predominant
to exert a marked influence in spite of the unavoidable
presence of uranium lead. Moreover, it cannot be lightly
assumed that in all the minerals given in the above mentioned
table, the amount of original lead is the same; and in the
event of thorium lead being of relatively short life, it is
clear that negligence of this inequality would lead to indefinite
or even conflicting conclusions regarding the halt-period of
thorium EK. |

Careful consideration of the analyses made it clear that
the only results which could be used with confidence were
those of Nos. 3, 6,7, and 12. The instability of thorium
lead, and the large percentage of uranium in mineral No. 12,
makes it certain that the effect of thorium E in affecting the
Pb/U ratio of this mineral must be very small and, practically
speaking, negligible. The value (0°041) for this mineral can
thus be taken with great probability as representative of
Devonian minerals. Also, since these four minerals are
similar in type, comparable in composition, and from the
same locality, it is possible that their contents of original
lead are not very different. Vogt* has given 0:000z gram
as the average amount of lead in 100 grams of rock. If we
assume this value, and give z the value 5, the corrected Pb/U
ratios for those minerals poor in thorium show a better
agreement than previously. Complete agreement is net to
be expected from what has already been said, but the mag-
nitude of this value is evidently of the right order. The
value 0°0005 gram will thus be taken as the amount of
original lead in 100 grms. of the minerals used for the caleu-
lation of the half-period of thorium HE by the first two of the
following three methods to be described.

The total amount of experimentally found lead is evidently
equal to the sum of the following three constituents :—
(a) Original lead; (6) Uranium lead; (¢c) Thorium lead.

This statement can obviously be expressed in the following

form— Pb,=Pb, +k.U;+m.Th,,

where Ph;, U;, and Th; are the present percentage contents
of the minerals in lead, uranium, and thorium respec-
tively. Pb, is the quantity of original lead present per
100 grams of the mineral; & is the amount (constant) of
uranium lead associated with one gram of uranium in these
Devonian minerals; and m is the equilibrium amount of
thorium lead associated with one gram of thorium.

(a) In the first method for the determination of the value

* Vogt, Zeit. fiir prakt. Geol. 1898 ; Holmes, loc. cit. p. 253,

Lead and the End Product of Thorium. 675

of m, the amount of original lead is assumed as above, and
the values of & and m are obtained by solution of two equa-
tions obtained as follows. The results of any two of the
four analyses Nos. 3, 6, 7, 12 are substituted in the above
equation, and the two equations obtained by different com-
binations of the analyses give on solution the required values
of kand m. In the following table is given the series of
results so obtained :—

TABLE VI. a.

Combination of Value of /. Value of m.
Analyses. |
Nos. 3: 6 0-043 5.10-5 |

Sey 0:044 a0. 2
5 880 0-042 7.10-°
age 7 0045 I Oe
Monin le 0-042 ent
Mean=6. 107°

The value obtained from 6:7 is undoubtedly too low, and
is due to these two analyses being almost alike.

(6) In the second method we assume Pb,=0:0005, and
the value of & to be 0:042 for Devonian minerals, and find
the value of m from each of the four analyses Nos. 3, 6, 7,
and 12. Before passing on, there is one point to be men-
tioned in connexion with the value of k above stated. The
Pb;/Um ratio for No. 12 expressed in terms of the time-
average value of uranium is 0:041. In this discussion the
actual amount of uranium present is being used in the cal-
culations, and hence we must use the ratio between the
present lead and uranium contents of the minerals Pb/U.
For mineral No. 12 this is equal to 0:042. As in the previous
ease, we require the equation Pb,=Pb,+4£.U:+m. Th; for
the purpose of calculating m, which is the only unknown
present. The following table gives the results obtained by
this method :—

TaBLe VI. 0d.

No. of Analysis... oy | 6. | > |

91 10s |

~J
—
eS
—
=)
_
(=)
|

DVin GUO 22eisa stele ota

Mean=8'7 . 107°
ee Se ee eee ee ee

676 Messrs. A. Holmes and R. W. Lawson on

There is as good an agreement between this mean and that
found by the previous method, as was to be expected from
the value of the lead-uranium ratio there found.

(c) The value of m can also be calculated from the four
analyses already used, and without any assumption as to the
distribution of lead. The unknowns in the expression
Pb:=Pb,+h4.U;+m.Th; are clearly Pb, k, and m. By
insertion of the results of three of the chosen analyses, three
equations are obtained which can be solved for the unknown
terms. From the four sets of three equations obtained by
use of the analyses 3, 6, 7, 12, the values of m given in the
following table were obtained. The mean value of m=7.10~*
here obtained is practically the mean of those obtained by
the two preceding methods, and corresponds to a half-period:
for thorium EK of 7:10->x 1-5. 10-=1:05.. 10° years. it
is interesting to note that the present method is quite inde-
pendent of the stability or instability of thorium EH. More--
over, since the lead producing power of thorium is only 0°4
that of uranium, it follows that, were thorium E a stable
isotope of lead the value of m would be 0:4 x 0:042=0°017,,
instead of 7.10~° as found (0:042 being the uncorrected
lead-uranium ratio for minerals of Devonian age). The
wide difference between these two values of m leaves little
doubt that thorium E is relatively unstable. With the object
of fixing the value of the half-period of thorium E more:
definitely, it is the intention of the authors to examine other
suitable minerals of different ages, and to apply to these the
same method fer the evaluation of the halt-period. That
the results for the unknowns given in Table VI. c¢ differ from

TABLE Vl.e.

ear Wvalucveriebe. | | siValuctor vs Value of m,

Nos. 3, 6,7 0:0010 0-044 2.10-5
ps oy 7, Le 0:0000 0042 Liem
asp oe 0:0001 0-042 10.10-5

» 6,7,12 0:0026 | 0-042 Bes
Mean=7'1.107°
ihe mean more than in the other cases is only to be expected,

since any slight error in the values of the substitutes will
most probably be magnified during the process of solution

Lead and the kind Product of Thorium. 677

of the equations. Our previous assumption regarding the
magnitude offisconfirmed. ‘lhe mean value of Pb,=0:0009
is not far removed from that assumed in the first two methods.
The result 0°0026 obtained from Nos. 6, 7, 12 is certainly
too high, the reason most likely being that the analyses 6
and 7 are practically the same.

§ 10. BismutH As A PossinLE Enp PRopUcT oF THE
THORIUM SERIES.

The results obtained in the present section have an im-
portant significance in relation to another possible end product
of thorium. Were thorium EH, with a half-period of about
10° years, to emit a rays, these should, according to the
Geiger-Nuttall law, have a range of about 3cm. This fact
renders it highly improbable that we have here to do with
an a-ray product, because « rays with the above range would
hardly have escaped detection. This would appear to exclude
the possibility of the end product being an isotope of mercury
in Group II. 6 of the Periodic Classification. It seems more
likely that the disintegration of thorium E is accompanied
by the loss of a 8 ray, which would bring the resulting pro-
duct into the position of bismuth in Group V.b. In sucha
case, we are again faced with the task of deciding whether
what we may by analogy call thorium-bismuth is a stable or
an unstable product. Here again the method used in the
present section might be applied. A systematic examination
of thorium minerals for bismuth and thorium would be
necessary. If the end product of thorium is a stable bismuth
isotope, the ratio Bi/Th for minerals of the same age should
be constant, whereas for minerals of different ages it should
vary ina similar manner to that found in the case of the
Pb/U ratio. On the other hand, if the Group V. 0d product
of thorium is unstable, the suggested analyses would serve
to determine its disintegration constant. The following ex-
pression would be used in this connexion—Bi,= Bio+n. Thy.
Insertion of the results of two suitable analyses would result
in two equations from which the amount of original bismuth
(Bi,) and the equilibrium constant (mn) between thorium-
bismuth (thorium F) and thorium could be found. From
the value of n so obtained, the half-period of thorium IF could
be calculated, and evidence adduced as to whether the sue-
ceeding change takes place with loss of an a@ ray or of a
Bray. In the former ease, the succeeding product would be
an isotope of thallium with atomic weight 2U4°4 (Group IIT.8),
and in the latter an isotope of polonium (Group VI. 0).

678 Messrs. A. Holmes and R. W. Lawson on

Hofmann* has given the results of his analyses of two
samples of Bréggerite, from which the following percentage
constituents have been calculated. It is clear that if the end

TABLE VII.

| ratio}, ratio.
ae 67-40 | 8°61/ 4-10! 0-30 | 72:35| 4:27 1000 1490

LS Sree 67°08 | 8-49 oii 0°33 | 71:96] 4°82 990 1440

1 The lead ratios agree with those of the minerals given in Table I].
Part I. p. 835, Moss district, Norway, Age pre-Jatulian.
product of the thorium series is a stable isotope of bismutht,
we may use the Bi/Th,, ratio to determine the age of the
mineral. Moreover, the age as calculated in this manner
should agree with that as found from the Pb/U,, ft ratio.
The two expressions fer the age of the mineral will be
respectively Bi,/Th,, x 2°09 . 10° years, and Pb,/Um x 8°58. 10°
years. Th” and Um are the respective time-average values
of the thorium and the uranium contents of the mineral, and
the ages as found in the case of the mineral under considera-
tion are given in the last two columns of Table VII. In
consideration of the relatively small percentages of thorium
and bismuth present, and the difficulty of their estimation,
it might be thought that the agreement between the ages were
sufficiently satisfactory to favour the view that thorium-
bismuth (“thorium F”’) might be a stable product.

Opposed to this view, however, is the fact that in the
collection of analyses of thorite and allied minerals cited
by Hintz in his ‘ Mineralogie,’ vol. i. p. 1675, there is not a
single determination of bismuth. Moreover, the Devonian
minerals already examined by us have been tested for bis-
muth with negative results. Hillebrand (Bull. 220, p. 114,
U.S.G.S., 1903) gives two analyses of uraninite which

* Hofmann, Ber. d. d. Chem. Ges. xxxiv. p. 914 (1901).

+ The part played by original bismuth in such minerals is almost
certainly quite negligible. For instance. Vogt (doc. cit.) suggests
000000 x grm. as the average amount of bismuth in 100 grams of rock
(cf. Holmes, Joc. cit. 1911, p. 253).

t If actinium lead is unstable, the results obtained for the age of a
mineral from the Pb/U ratio will be rather low, and if the end product.
of actinium is bismuth, the age obtained from the Bi/Th ratio will be
slightly high.

Lead and the End Product of Thorium. 679

include determinations of bismuth, but the latter is clearly
independent of the thorium present in each case. More
recently Soddy* has examined Ceylon thorite for bismuth
and has failed to find an appreciable amount. The evidence
is thus fatal to the view that an isotope of bismuth might
be the end product of the thorium series.

§ 11. FourtHer Eviprencs rrom Atomic WEIGHT
ESTIMATIONS.

An initial objection to the view that the end product of
the uranium family is lead, was that the theoretical atomic
weight of the end product radium G did not agree with the
atomic weight of lead (207:1) as closely as could be desired.
The recent discovery of isotopic series of elements has de-
monstrated that chemical identity does not necessarily imply
equivalence of atomic weight, and the objection has now
lost its original force. Radium G ought to have an atomic
weight of 2062, calculated from the atomic weight of
uranium (238°2) 7, or 206-0 if calculated from the atomic
weight of radium (226:0)¢. The difference between these
two values is very small when we consider the intrinsic
difficulties of the atomic weight determinations; and the
results bear excellent testimony to the careful and exact
atomic weight determinations of Hénigschmid. Fortunately,
a number of determinations of the atomic weight of lead
extracted from radioactive minerals is already to hand, and
the results are very gratifying. Thanks to the energy with
which investigators are attacking the problem, the amonnt
of evidence of this nature will soon be considerable, so that
further conclusions with regard to the lead disintegration
products and their stability will be possible. We may now
take the theoretical value for the atomic weight of radium G
to be 206°2 without appreciable error. Quite recently
Honigschmid and Fraulein St. Horovitz§, Richards and
Lembert||, and Maurice Curie{], have published the results
of their experiments in this connexion. Hénigschmid and
Friln. St. Horovitz determined the atomic weight of lead

* Nature, vol. xciv. Feb. 4th, 1915, p. 615.

+ Honigschmid, Wren. Anz. 22nd January, 1914.

{ Honigschmid, Wren. Sitsungsber. cxx. p. 1617 (1911) ; exxi. p. 1978
(1912) ; exxi. p. 2119 (1912).

§ Honigschmid and St. Horovitz, Wien. Anz. 12th June, 1914; Zert?.
fiir Elektroch. xx. p. 819 (1914); C. &. elvili. p. 1797 (1914).

|| Richards and Lembert, Journ. Am. Chem. Soc. vol. xxxvi. 7.
p. 1829 (1914); Zert. fiir Anorg. Chem. \xxxviii. p. 429 (1914). See also
Fajans, Hed. Akad. Ber. A. Abh. xi. (1914).

q Maurice Curie, Comptes Rendus, clviii. p. 1676 (1914).

680 Messrs. A. Holmes and R. W. Lawson on

extracted from the Joachimstal (Bohemia) uranium residues,
and found the value to be 206°736+0:0U9. Richards and
Lembert used lead chloride which had been obtained by
working up (a) North Carolina uraninite, (6) Joachimstal
pitchblende, (c) Colorado carnotite, (@) Cornwall pitch-
blende, and obtained the following values respectively:—
206°4, 206°57, 206-59, and 206°86. ‘The result for (a) is of
particular interest, because uraninite is less likely to contain
ordinary lead than pitchblende, which is a secondary form of
uraninite, and in which ordinary lead may probably be present
as impurity. The Joachimstal pitchblende is non-crystalline
in structure, and generally contains within its mass veins of
other minerals such as galena, and iron and copper pyrites.
The uranium residues, obtained from large quantities of the
ore, would thus be expected to have a larger content in
ordinary lead than selected samples of the pitchblende. This
fact is probably the cause of the difference in the atomic
weights 206°74 and 206°57 obtained for lead from Joachimstal
pitchblende by Honigschmid and St. Horovitz on the one
hand, and by Richards and Lembert on the other. Tor lead
which had been extracted from carefully selected samples
of pitchblende free from galena, the former experimenters
obtained an atomic weight 206°40*. For ordinary lead
treated by the same method as their other materials, Richards
and Lembert obtained the value 207:15+0-01, a figure in
close agreement with that given in the International Table.
The preliminary results given by M. Curie vary between
206°36 and 206°64 for uranium minerals, whilst for ordinary
lead from galena he found the value 207°01.

The results so far obtained clearly show that the lead
extracted from uranium minerals (in all the above minerals
the quantity of thorium present was very small) has a lower
atomie weight than ordinary lead, though the value is higher
than is to be expected from theory. It is quite possible that
the discrepancy is due to the presence of ordinary lead in
the minerals used, as well as to the presence of actinium lead,
of which the atomic weight is considered by Fajans+ to be
about 207, and by other workers as high as 210. The pre-
sence of one or both of these elements would give an
increased atomic weight in the right direction, and so at
least in part explain the discrepancy.

Still more recently Hénigschmid and Fraulein St. Horovitzt
have determined the atomic weight of the lead extracted from
(a) Uraninite from Morogoro in German East Africa, and

* Honigschmid and St. Horovitz, Wien. Anz. 9th July, 1914.

1 Fajans, loc, cit. p. 11 (1914).
{ Honigschmid and St. Horovitz, Wien. Anz. 15th October, 1914.

Lead and the End Product of Thorium. 681

(b) Bréggerite from Norway. In the first case they obtained
a mean value 206°04, and in the second case 206°06. These
results would appear to indicate that the presence of ordinary
lead as impurity was responsible for the high results obtained
with other minerals. The purity of the first mineral (a), and
the fact that the second mineral (0) is an old primary un-
altered mineral, lend support to the view that these contained
a minimum quantity of ordinary lead. The above results
show that the lead extracted from the minerals mentioned
was practically pure radium G.

In the case of thorium lead the evidence is as yet less
clear than in the case of uranium lead. If we assume the
atomic weight of thorium to be 232°4, then the atomic
weight of thorium lead, derived by loss of six helium atoms
from thorium, is to be expected to be 208°4*. Soddy
and Hymant+ have extracted lead from thorite (Ceylon),
and in the two preliminary determinations of the atomic
weight they have already published, they arrived at the
values 208°3 and 208°5. From the known rates of disinte-
gration of uranium and thorium, it can readily be seen from
the analysis of the mineral used by the authors cited, that if
all the lead were of radioactive origin and thorium lead
stable, there should be about thirteen times as much thorium
lead present as uranium lead. The theoretical atomic weight
to be expected can thus be shown to be 208:24, a result less
than that found experimentally. Unfortunately, the evidence
in the present case must still be regarded as inconclusive.
Richards and Lembert (loc. cit.) have determined the atomic
weight of lead separated from thorianite from Ceylon. This
material contained 60 per cent. thorium and 20 per cent.
uranium, so that if thorium lead be stable, the atomic
weight to be expected would be 207-40. The value they
actually obtained was 206°82, suggesting at once the insta-
bility of thorium lead. That this result is higher than
those found for the other minerals they used, cannot without
further evidence be assumed to be due to the presence
of a stable thorium lead. The presence of ordinary lead
as an original constituent of the mineral, or as an infil-
trated secondary product, would very well explain the dif-
ference. M. Curie obtained for lead extracted from monazite
sand the value 207:08. Here again we are noi certain of the
role played by ordinary lead, and furthermore, this mineral
always contains a certain quantity of uranium which would in-
troduce a disturbing factor owing to the presence of radium G.

* Tf the atomic weight of thorium is 232°2, as determined by Honig-

schmmid (1914), then that of thorium lead would be 208-2.
t Soddy & ILyman, Trans. Chem. Soe. cv. 1914, p, 1402.

682 Messrs. A. Holmes and R. W. Lawson on

From the evidence of atomic weights, we can safely decide
that the existence of uranium lead with a lower atomic weight
than that of ordinary lead has been proved. Moreover, the
most recent results of Hénigschmid show that this has an
atomic weight in almost perfect agreement with that theore-
tically to be expected. In the case of thorium lead the
evidence is much less certain, a fact undoubtedly following
from the instability of thorium lead (thorium E).

The following table summarizes the evidence from atomic
weights up to the present time.

TABLE VIII.*
Atomic Weights of Lead from Radioactive Minerals.

aye. Ws Experimenter. Source of material. ee : ae
Ordinary lead) Honigsclmid: > 7) 9 eeeeee 207°12 |207:18
uf pa a eviclaards vale een nO ies ota 207:1 (207-15
Lembert. ; :
a » |M. Curie. Galena. 207:1 |207-01
Uranium lead. | Hénigschmid and| Uranium residues >206°2 |206°74
St. Horovitz. (Joachimstal),.
bs ‘ a Selected pitchblende >206°2 |206°40
(Joachimstal).
is 5 Uraninite.? 2062 |206:04
(G. E. Africa).
he Y H Broggerite (Norway).* 206°2 (206-06
. » | Richards and Uraninite (North 206°2 |206 40
Lembert. Carolina),
i As Pitchblende (Joachim- |>206:2 |206°57
stal).
a is js Carnotite (Colorado). >206°2 |206:59
‘3 oA a Pitchblende (Cornwall). | >206:2 |206°86
» | Maurice Curie. ‘| Pitchblende. >206°2 |206°64
a & A Carnotite. >206°2 |206°36
ma ? Yttrio-tantalite. >206°2 |206°34
Thorium lead’.| Soddy & Hyman.} Thorite (Ceylon). 208°24 |208°40
Thorium lead+| Richards and Thorianite (Ceylon), 207°40 206°83
Uranium lead. Lembert.
fs » | Maurice Curie. Monazite sand. % 207°08

* If thorium E and radium G are isotopic with lead, and are the respective
stable end products of the thorium and uranium families of radio-elements.

2 International atomic weight.

° Marckwald, Centralblatt fiir Min. u. Geol. 1906, p. 761; Chemisches
Centralhlatt, 1907, i. p. 869.

4 This result would appear to point to the stability of thorium E, if the
atomic weights of thorium EH and radium G are respectively 208-4 and 206:0.
Remembering, however, that in all probability a small quantity of original
lead is present in this mineral, it is clear that we must be cautious in drawing
conclusions regarding the stability of thorium E from such a result, where
the mineral contained only 4 per cent. of thorium. Moreover, the theoretical
atomic weight of radium G, 206, is likely to bea minimum value.

° Theoretically 90 per cent. pure.

* A revision of Table V. in Part I. p. 839.

Lead and the End Product of Thorium. 683

§ 12. REMARKS ON THE SELECTION OF MATERIAL FOR
Atomic WEIGHT DETERMINATIONS.

If the conclusions reached in the foregoing sections are
correct, we are given a reliable means of guiding the selection
of material for the determination of the atomic weights of
uranium lead and thorium lead respectively. It is obvious
in the former case that the most suitable material is a mineral
rich in uranium—such as uraninite—and practically free.
from thorium. In order to eliminate the effect of original
lead in controlling the atomic weight of the extracted lead, it
is further clear that the older the mineral, the nearer to the
theoretical value 206-2 should be the value of the atomic
weight actually found. Since thorium lead is relatively
unstable, any effect which the presence cf thorium in the
mineral] might exert will also be eliminated by using geolo-
gically old minerals.

The case of thorium lead, however, is not so definite.
Here, clearly, rich thorium minerals with as small a per-
centage of uranium as possible must be used. Further, the
best results will be obtained for geologically young minerals,
since once the equilibrium amount of thorium lead has been
formed, additional time only results in addition of more and
more uranium lead, which tends to give a lower value than
the theoretical value 208°4 for the atomic weight of thorium
lead. The magnitude of this effect is clearly shown by the
following example. Suppose we have a mineral with 60 per
cent. thorium and 0-4 per cent. uranium, and that we assume
it is of post-Cretaceous age, with a Jead-uranium ratio of
the order 0°01. On the results of the previous sections

follows that the amount of uranium lead present per
100 grams of the mineral will be 0°-4x0:01=0-0040 gram;
the amount of thorium lead present will be 60 x 7°10~° =0-0042
gram; andthe amount of original lead 0°000z gram. Thus in
such a case—and a thorium mineral more free from uranium
will be difficult to find—the atomic weight obtained, instead of
being equal to the theoretical value 208° 4, would lie at about
207-2. This number is practically equal to the atomic w eight
of ordinary lead, so that in the present case this element
would be without appreciable effect. Thus it would appear
to be a difficult matter to obtain a thorium mineral which
would give a higher value for the atomic weight of its con-
tained lead than about 207, and the inevitable presence of
ordinary lead in minerals prohibits exact calculation of the
atomic weight of thorium lead from that found for the lead
mixture.

In the case of minerals of greater age than that assumed

684 Messrs. A. Holmes and R. W. Lawson on

above—and all the minerals hitherto examined are actually
very much older—the atomic weights to be expected would
be somewhat lower than 207°2. It is interesting to notice
that the actual determinations by Richards and Lembert on
Ceylon thorianite, and by Curie on monazite, give results in
complete harmony with this conclusion. This experimental
verification adds still further support to the view that thorium
lead is unstable.

$13. Tae PossisLE Exp Propuct oF THE ACTINIUM SERIES.

When uranium is present in a mineral, actinium lead will
always be a disturbing factor. Should actinium lead be a
stable product, the minimum value of the atomic weight of
lead extracted from a thorium-free uranium mineral would
be 206°26 if the atomic weight of actinium lead is 207, and
206°50 if the atomic weight of actinium lead is 210. On
the other hand, if actinium lead is of relatively short life,
and if about 8 per cent. of uranium Is transformed along the
actinium series, then the high degree of stability of uranium
lead renders it certain that the atomic weight of the latter
would be quite inappreciably affected by the presence of
actinium lead, even if this has an atomic weight of 210.
The recent determinations of the atomic weight of lead from
crystalline pitchblende and bréggerite by Hénigschmid,
indicate either that the atomic weight of actinium lead is
207, or that if it is 210 then actinium lead is an unstable
product. In the case thatit is unstable, evidence with regard
to its half-period might be obtained by determining the
atomic weight of lead of radioactive origin, obtained on the
one hand from geologically young, and on the other hand
from geologically old, uranium minerals. It is clear that
a higher value should be obtained in the former case than in
the latter.

Regarding the end product of actinium there would appear
to be three possibilities, each of which we will now consider
in turn.

As mentioned in a previous paragraph, the atomic weight
determinations of radium G by Hoénigschmid indicates that
if actinium lead has an atomic weight of 210, it must be
unstable, otherwise instead of obtaining a value 206°04 for
lead from Morogoro pitchblende, the value 206°50 should
have been found. It would thus appear certain that the end
product of actinium cannot be a stable lead isotope unless
this has an atomic weight nearly the same as radium G.

The second possibility is that the end product is an isotope

Lead and the End Product of Thorium. 685

of bismuth. There is ground for believing that if actinium
lead is unstable, it will disintegrate with the loss of a 6 ray
and the resulting formation of a bismuth isotope. As to
whether or not this product is stable could readily be
tested. We require the bismuth and uranium contents of
a series of uranium minerals. The fact that about 8 per cent.
of the uranium in a mineral disintegrates along the actinium
series gives a direct means of testing whether the amount
of bismuth actually found is of the order of magnitude
of that to be expected on the assumption that actiniuin
bismuth is stable. Should actinium bismuth be unstable,
then clearly the Bi/U ratio should be about the same
for all fresh unaltered minerals of whatever age. If, on
the other hand, it is a stable product, the Bi/U ratio
should be constant for minerals of the same age, but it
should vary directly with the geological age of the mineral.
It is worthy of notice that if actinium bismuth is a stable
product, so that appreciable quantities of it can be extracted
from uranium minerals, it should be found to have an atomic
weight of 210 or 206, both of which values differ by two
units from the atomic weight of ordinary bismuth. This.
method would appear to be a hopeful one for the question in
hand, since a result lying near 210 or 206 would not only
point to bismuth as the end product of actinium, but it would
also give a means of deciding between 230 and 226 as the
atomic weight of actinium. ‘The occurrence of bismuth in
radioactive minerals has already been commented upon, and
it is to be hoped that in the near future more definite
evidence for or against bismuth as the end product of actinium
will be forthcoming.

The occurrence of thallium in radioactive minerals has
also been noted. For instance, Exner and Haschek* found
it to be present spectroscopically in appreciable quantities in
pitchblende from Cornwall. The fact that if actinium
bismuth is unstable, disintegrating with loss of an « ray, we
should have a thallium isotope, renders thallium a_ possible
end productof actinium. Theanalysis of a series of uranium
minerals for uranium and thallium would be necessary to
test this suggestion. The method of showing whether this
thallium product is stable or unstable is the same as that
suggested in the case of bismuth last treated. Further, in
this case, the atomic weight of the thallium should be 206 or
202, instead of the atomic weight 204 of ordinary thallium.
It is doubtful whether we are justified in laying very much

* Exner and Haschek, Wien. Sitzwngsber. cxxi, p. 1077 (1912).

686 Messrs. A. Holmes and R. W. Lawson on

stress on the occurrence of thallium in the Cornwall pitch-
blende, since this is of a very impure nature. Professor
H6nigschmid informs us that he has been unable to find
appreciable traces of thallium in selected pitchblende from
Joachimstal. Now the atomic weights of the lead from
pitchblende from these two districts are respectively 206°86
and 206°40, results which indicate the greater purity of the
Joachimstal material. It would thus appear that the sug-
gestion regarding thallium as the end product of actinium is

highly improbable.

§ 14. ConcLusions.

(a) In Part I. of this paper it was shown from the uranium,
thorium, and lead contents of four series of radioactive
minerals:

(1) That the lead-uranium ratio is remarkably constant in
minerals of the same age, and varies sympathetically
with the geological age of minerals of different
antiquities, so that radium G may be regarded as
stable or practically so. It appears impossible that
any slight instability that may exist could ever be
definitely detected. The slight deviations in the
values of the Pb/U ratios are such as can be readily
accounted for by the presence of traces of original
lead, by the presence of the unstable thorium-lead,
and by the possible alterations which the minerals
may have suffered since their original crystallization.

(2) That thorium-lead does not tend to accumulate in
geological time, 2. e. thorium LE is an unstable
product.

(3) That the Pb/U ratios may be used as before and with
greater certainty for the determination of geolegical
time and the gradual construction of a complete
geological time-scale.

(b) On the assumption that the total lead present in the
minerals is made up of three constituents: (a) Uranium
lead (radium G); (b) Thorium lead (thorium ) ; and (ce)
Original lead, it has been shown how the results obtained
can be applied to find the half-period of thorium E. This
has been found tentatively to be about 10° years.

(c) It has been suggested that the disintegration of
thorium E may be accompanied by the loss of a @ ray, so
that the resulting product would be an isotope of bismuth.
Evidence has, however, been brought forward which suggests
that the latter cannot be a stable end product. Should this

Lead and the End Product of Thorium. 687

Group V.b product of thorium be unstable, the method here
used for thorium E could be applied to find its disintegration
constant, when the bismuth and thorium contents of suitable
minerals are known. Information could thus be obtained as
to the next succeeding product, which would be a thallium
isotope if the bismuth isotope disintegrates with loss of an
a ray, or a polonium isotope if the change takes place with
loss of a B ray.

(d) The evidence of atomic weight determinations of lead
extracted from radioactive minerals also points to the
stability of radium G, and the instability of thorium HE. It
has been shown that the most suitable mineral for the
determination of the atomic weight of radium G is a geolo-
gically old, primary, fresh, uranium mineral. Using such a
mineral, broggerite, Hénigschmid and St. Horovitz found
the value 206-06. Should actinium lead have an atomic
weight of 210, the above result for bréggérite would indi-
cate that it is unstable. Experiments to throw light on the
stability of actinium lead are suggested in §13. IE the
atomic weight of actinium lead is 206, we are unable to say
whether it is unstable or not. Methods of attacking the
question of the end product of actinium have been suggested,
und it has been shown how one of these might ‘incidentall
throw light on the atomic weight of actinium. For thorium
ff it would appear that no very definite conclusions can be
drawn from atomic weight determinations, owing to the
instability of this element. In the most favourable cireum-
stances, a value of about 207 would be obtained, instead of
the theoretical value 208-4. Determinations of the atomic
weights of bismuth and thallium from uranium and thorium
minerals, would serve a useful purpose in supplying definite
information regarding the end products of thorium and
actinium.

(e) Atomic weight estimations can now be used to correct
the crude determination of the age of a mineral by means of
its present lead and uranium contents. Corrections must be
applied both for original lead, which may be considerable
(e. g. Nn some specimens of thorianite from © eylon), and for
a small equilibrium amount of thorium lead. These cor-
rections appear to make very little difference to the time-
scale as at present constituted. That this is true for the
Devonian minerals of Norway we have already seen. The
pre-Jatulian minerals of the Moss district of Norway require
practically no correction, for broggerite from that district
contains uranium lead in an almost uncontaminated state, as
shown by the atomic weight of lead prepared from the

688 Dr. J. R. Wilton on Ripples.

mineral (206:06 found ; 206°2 theoretical value). As more
evidence accumulates, the ratios for other geological periods
will be similarly tested. At present it would seem that the
age of the older intrusive rocks of Ceylon, as deduced from
Pb/U ratios in thorianite and thorite, is probably too high.
These ratios are nearly always greater than 0:2, giving an
age exceeding 1600 million years. Zircon from the same
pegmatites, however, gives a ratio of 0°164 (1370 million
years), which seems to be more probable, in the light of
atomic weight estimations. Zircon is much less likely to
contain original lead than thorite or thorianite, and apart
from the difficulty of estimating the very small quantities of
lead which have accumulated, zircon represents one of the
most valuable minerals for age determinations.

Finally, in addition to our previous acknowledgments
to Professors Strutt and Mache, we wish to express our
thanks to Professor Stefan Meyer, for his kindly interest
and encouragement during the progress of this work.

LXXIT. On Ripples. By J. R. Witton, M.A. Dise
Assistant Lecturer in Mathematics at the University of

Sheffield *.

LL carrying the approximation to the form of a wave to
such an extent as is done in my paper on “On Deep
Water Waves” (Phil. Mag. Feb. 1914, pp. 385-394), which
will here be referred to as “‘ Waves,” it is important to make
certain that the sense of accuracy thus obtained is not
illusory. The present paper therefore takes up the con-
sideration of the corrections which have to be applied. We
shall, however, still suppose that the wave (or ripple) is
formed under ideal conditions,—that there is no wind, no
secondary disturbance of any kind,—that the “ocean” is
“deep” (a depth of ten centimetres will be ample for the
ripples we shall actually consider) and of unlimited extent.
With this understanding there are three things for which
we have to make allowance, namely :—
(1) Surface Tension,
(2) The formation of waves in the air,
(3) Viscosity.
Now the first order approximation to the velocity when
(1) and (2) are taken into account is known to be given by

29h Pap, 20 7
2a p+po ®& p+p”

* Communicated by the Author.

Dr. J. R. Wilton on Ripples. 689

in which p is the density of water, p’ of the air, T is the
surface tension, and the rest of the notation is that of
“Waves.”

It is assumed in obtaining the above value of c? that the
air is incompressible, but the removal of this restriction
will rather lessen than ivcrease the effect of air waves.

Since p'/p is small the effect of p' is to multiply one term
in c? by 12—p’/p, the other by 1—p’/p. Now

nearly. Hence the correction is of the order 1/400 of the
uncorrected value. We shall consider this as negligible.
In fact, in such a (relatively) high wave as that of “ Waves,”
fig. 1, the ratio of the last term retained to the first is of
the order

A,,/A,= 1/150.
If, then, we omit p’, we have as a first approximation
Qarc? Ba wee
car Gr gp

which we shall, for brevity, write in the form

ea KS
But
T/op (4 /JoL—O7To,

so that «, the correction due to surface tension, is appreciable
for waves of length less than about 25 cm., and for very
short ripples it may become very large.

Finally, we have to take account of viscosity. It is shown
in Lamb’s ‘ Hydrodynamics,’ § 332, p. 566 (Third Edition),
that, if A/°0048 cm. may be considered large (say 10 or
more), the effect of viscosity is to introduce a time factor
e —2(2m/4t which does not affect the form of the wave, and to
introduce a correcting factor to the form of the wave of the
order of magnitude

22m fry’ _ _. 2vgT*2n/a)§
gh, 2a T VitKc ”
AAT Nes (p

vp being the coefficient of dynamical viscosity.
Phil. Mag. 8.6. Vol. 29. No. 173. May 1915. 2Y

690 Dr. J. R. Wilton on Ripples.

The following table will show that surface tension is
always of considerably greater effect than viscosity *.

Wavelength, |SupeeTenwon | Views | natn

r. K. e% K/T.
DAVEMs igseehonee ‘033 0004 80
GiGi a Pe cecseeee ‘075 ‘0011 68
SH bah beenasee “30 ‘006 50
ERR sie Maton ea 1-2 03 40
5 eee (A Reman ee ce 4°8 ‘ll 44
EUs Wat esses nee 19 "36 53

Thus x/7 is never less than 40, so that we may neglect
viscosity even for small ripples without risk of serious error,
provided always that the condition that A/:0048 is to be
‘‘large”’ is not forgottent. But for longer waves the correc-
tion due to the formation of air waves is of the same order
of magnitude as «: thus, when A=35 cm.,

x=1/400,

nearly. Hence, if we include T but omit the other two
corrections, we must apply our results only to ripples and
waves of from, say, 1 mm. to 20 cm. in length, so that the
form of waves which ordinarily occur in the open sea
will not be affected by any of these considerations. We
shall, actually, apply our formule only to ripples of from
5 mm. to 25 mm. in length. With this understanding we
proceed to determine the form of a wave when surface
tension is taken into account.

Let R be the radius of curvature of the wave: R will be
reckoned positive when the concavity is upwards. Then, if
II is the atmospheric pressure, the pressure along the free
surface, within the water, is

p=ll—-T/R,

* It will be observed that for ripples, down to half a centimetre in
length, we may still speak of the correction due to viscosity, but it is
absurd to speak of the “ correction” due to surface tension, for the latter
is the predominating influence in determining the form of ripples of
2 cm. length, or less.

+ The viscous time factor is, however, far from being negligible for
short ripples.

Dr. J. R. Wilton on Ripples. 691

and the surface condition becomes *
preg ogt Wak bo) i) ye) C)
To determine the form of the term T/pR we expand 1/R
in a series of cosines of multiples of & by means of the

equations for w and y on p. 392 of “ Waves.” Since on the
free surface '=£, we have

1 2ri(dnd en dn (2 dn! ele
Roy (a ae ae ae) [aga

The sign is determined by the fact that R is positive at
the trough of the wave where £=0. A,, =—a, is negative,

and therefore the predominant term in 1/R is 2rra/D.
which is positive.

On substituting in (1) we find, as in “ Waves,” p. 387,

O=fut+ | C+ A, cos ng} J (14+ SnA, cosné)?+ (Ena, sin ne}
1 1 t
Kk ) aA, cos nEtS WA?
il 1

a S > mn(m+n)AmAn cos (m—n) &

ma=l1n=m+1

1 + > nA cos n€)? + Zn, sin n&)? re Kumi)

where, as before,

As a first approximation we see that
C+5u=0,
A,(1+2C+«)=0,

so that A, is arbitrary,

and

as it should.

* The explanation of any notation not given in this paper will be
found in “ Waves,” pp. 385-6.
22

692 Dr. J. R. Wilton on Ripples.

To obtain closer approximations we must expand the right
hand side of (2) and re-arrange in cosines of multiples of &,
The coefficient of cosn& is then to be equated to zero. It
would be possible to obtain the general form of the equation
thus derived, but it would be extremely complicated and
there would be no advantage in doing this. We shall there-
fore write down the resulting equations only in so far as
they are necessary to obtain the approximation we desire.
We shall determine each approximation accurately so far as
T is concerned.

The equations derived from (2) are, if we retain terms of
the fifth order and reject those of higher orders,

pt20(14 Ay’t 4A.) 4+2A)74+4A.2+4A/7A,

+ (AG + 8 Ge <a 2A,*As -+ RA,”) = 0, ° e . (4)

20(Aj + 9A A, +6A,A3) + A, 4 AP +3A,A,45A,A.+ 6A, Ae one
3 SUL D y 5 g 3 15 2
= K(A, Fag gAn + 3A,A, aa 64. Ay te 4 Ay As Tae 8 A,?As rae 5A, A» + 15A,As3)
=O 0 Con
20(2A,+3A,A3) + A.+A)?4+3A,2A,+4A,A;
+ (4Ag—% Ai? + 7 At 2A PAs GAyA,)=0, . eenGm

25 15 15

+ «(9Ag+ Ap SAA, 5, Abt Ava; ee

—2 AAP +10A,A)=0, . (1)
8CA,+ A,+4A,A34 2A,?

: »)
+ K ( 6A, = Tq = 4A,” a 3A,7As — 6A,A3) = O ° (8)

10CA;+A;+5A,A,4+ 5A,A,

35 25 A5 15
+ (255+ Fy¢ As > ASA, t ~ A2A,+ “5 ArAv— 10A,Ay—15 A; As)

=0. . (9)

Dr. J. R. Wilton on Ripples. 693

From these equations, remembering (3), I find by

successive approximation :—
J a are

: 1 30«K°—71K?+17xn—8

—, 5S yeti 4
95 2c—17 16 (2«-—1)?(8«—1) GINO ae en)

mes 21k +8),
mem 16 Oc 1) Berl)”

4 18248«! — 53640«* + 63260«° —29010K? + WWile~1216

768(2«—1)5(3«—1)?(4e—1) Tr Sa a
_ 18«?—183«7+361e—128 ,
Bee 1) co BD
A. — 32884 — 4680! + 189804? — 247860? +1109 14-1600 , e
meeco l) Nee 128k DGe Doc). *, + (13)

Me 1 ee e416
Gee 2 ig Deni

1 240° + 220«K4—2422K? + 470 1K? — 2858K4 704
a‘, (14)
e256 (2*—-1)° (3«—1) J
De ea oy
cele ih Sn
1 24n°—164«4— 56642 + 1821K?—1322« +440 i
T 198 Cee) 2 ee eo

I have also calculated, independently, the values of these

constants in the particular cases «=1 am) eee

When «=1, Ay= —a,

A= gat tot a Sinise
an et nite!
When «=2, A,=—a,

= at A= oe e

694 Dr. J. R. Wilton on Ripples.

It will be found that equations (10),...(15) agree with
(16) and (17), and that they also agree with the known
result when k=0. Hence it is improbable that there is any
undetected error in calculation.

The most interesting thing about the constants whose
values are given in equations (10),...(15) is the unexpected
form of the denominators. It is easy to see that this form
is general; for the coefficient of A, in the equation which
determines it is

2nC+1+n%« = n’e+1—n(1+k)
(n—1)(ne—1),

to a first approximation. Hence the denominator of A,

contains as a factor (n—1) 0 (re—1). Now we have seen

that we are justified in poalagting the effect of air waves
if « is greater than about ‘01; so that for a considerable
range of values of n we have to consider the possibility
of values of « of the form «=1/n, where n is a positive
integer. To the consideration of these values of « we shall
return later. On the other hand, since we are not justified
in neglecting the effect of air-waves if « is less than about
"01, we cannot, as might appear at first sight from equations
(10), ... (15), conclude that however small « is, so long as it
is finite, the value of one of the coefficients A, becomes
illusory, and therefore that the ordinary theory, in which
«=(, is incorrect.

Let us take first the case of a ripple whose form is largely
determined by surface tension—say that for which «=10,
and therefore A=‘54 cm. From equations (10),... (15)
I find, for this ripple,

A= —<a. A,='21a?—:0072a',
A3;= —'033a?+:0049a°, <A,="003la*, A;=°00023a",
w=11—1°43a?+ :014a'.

The largest value of a which we may safely insert in these
equations is * a=1°'5. We then find

Ai;=—15, A,=*44, A,=—-075,
Ag "016," ) Aj=-0007,
Lis os ine... C— 20) i) Cm.) Sees
A='94 cm.

* It may be verified that R is positive for this value of a when €=0,

Dr. J. R. Wilton on Ripples. 695

The corresponding values of « and y are (“ Waves,”
p. 392) given by

ct— «="086(E—1°5 sin € + ‘44 sin 2E—:075 sin 3&
+016 sin 4€+°0017 sin 5€),
y='086(1'5 cos E—-44 cos 2E+ °075 cos 3&
—°016 cos 4E—-0017 cos 5).

Corresponding values of « and y are given in Table L.,
and the ripple is drawn in fig. 1.

TABLE I,

Ripple of wave-length *54 cm.

Eg ct— x. y

0 0 095

7/6 “008 094

a/3 ‘009 O77

m/2 013 037
27/3 ‘037 "039
57/6 "128 —'129

T 272 —°175

Bie I

Ripple of length :54 cm.; amplitude -27 cm. ; velocity 25°7 em./sec.

N.B.—The scale of this figure is just twice that of those given below :
it is about 7:1 to 1.

~ ~~ aoe ‘
Let us now take the rather longer ripples «== and
[Pema .
First, when «=2 the constants Aj, We. are given Dy
equations (17). In the particular case when a=1, which

696 Dr. J. R. Wilton on Ripples.
is a relatively high ripple, we have
K=2,)) (A= trem). hea ee
Aj=—f, *vAg=0083)) A;="06l, ) Ay — 00
A;= —‘0087,
p=2°18, c= 20°3 cm./sec.

Corresponding values of 2 and y are given in Table IL.,
and the ripple is shown in fig. 2.

TaBueE LI.
Ripple of wave-length 1:22 cm.

E ct — xX. y

0 0 ‘181

7/6 018 165

a/3 038 et

1/2 ‘096 001
27/3 "239 —110
d7/6 "421 —'167

7 ‘611 — 185

Fie. 2

Ripple of length 1:22 cm. ; amplitude ‘37 cm. ; velocity 20-3 cm./see.

Again, when «=1 we find A,, &c. from equations (16).
Taking in this case a=1/3, we find
k=1, ear em... A/2a= 7208:
A,=— 333, A,=—*043, A;=:0037, A,=:0029,
A;='0004,
p= 185; e==1'9°4. cm/sec.

Corresponding values of x and y are given in Table ITI.
and the ripple is drawn in fig. 3.

Dr. J. R. Wilton on Ripples. 697

Tasxe III.
Ripple of wave-length 1°31 cm.

eS ch— x. y

0 0 ‘O77

7/6 068 068

m/3 144 031

x /2 "258 — 009
27/3 "389 — 039
57/6 522 — 054

T 653 —°060

Fig. 3.

Ripple of length 1:31 cm. ; amplitude ‘14 cm.; velocity 19:4 cm./sec.

We come now to the most interesting portion of our
inquiry,—the consideration of the form of those waves
for which « is the reciprocal of a positive integer n other
than unity. When « is not actually equal to 1/n, it is
always possible to choose a sufficiently small to insure
the convergence of the series for A,, A3,.... For it is
easy to satisfy oneself that the index of the power of n«—1
in the denominator of any coefficient is less than the index
of the power of a which it multiplies. Hence if we put
a=(nke—1)b, some power of ne—1 will divide every co-
efficient A,,, and it is now manifestly possible to choose
a value of 6 which secures convergence of the series. If
ne—1 is small a will be small, 2. e. the amplitude of the
wave will be small; but as A, and the succeeding co-
efficients become relatively important the form of the wave
may be very different from that of a simple cosine curve.

When «=1/n the case is different. The ordinary method
of approximation breaks down altogether, and we have to
start again ab initio. We shall consider in particular the
case K=4$.

When «=}, to a first approximation C= — 3, and therefore
equation (6) leads to

3A,?+ terms of order higher than the second = 0.

Hence A, cannot be of the first order unless A, is of the

698 Dr. J. R. Wilton on Ripples.
same order. But if A, is of the first order, the second
approximation to (5) is
ba Oe Ss iy
and substituting in (6) we have, to the second order,
—3A,?+3A,* = 0,

a. Cry As = +A}.

From equations (7),(8), and (9) we see that, if A; and A,
are both of the first order, Az; and A, are both of the second
order, A; and A, are both of the third order, and so on.

If «=1/3, it is easy to see that we may take the orders of
the successive coefficients to be

Vee ml LAO aii ee wea nae Gera a a ee,
W hile in the general case, when «=1/n, the orders are
1,2,3,..:-n—1; n—2,n—1,7...2n—3;
2n—4, 2n—2,...38n—5; 3n—6,...
It is only in the particular case of 4«=, 7. e. n=2, that
there is any ambiguity in the form of A,.

Let us return now to the consideration of the case c=4.
To the first order we have

A,=—a, A,= +a.
And, on substituting these values in the other equations,
we find as a first approximation,

A,=—a, c= +ta, A3,= +30’, A,=0, A;=0,
C=— it fa, pw=st+Za.
As a second approximation I find, after rather long
analysis,

A,=—a, ji\ == +4a— ta’, Aga + ga? +13a*,
Jat ermal ae _ + 3,3
A,=+3a*, A;=0, Ap=+5 $2",
eee) ATES) 590 2 ES By A Sy
C=—itgatse, p= Phia—jza

In particular, when a="2 the two sets of values are :——
Gyo Ay= ="2,,) As=-095, A,=:045, A, =—008
A;=0, A,= —*0003,
p=159, c=24°6 cm./sec.
(2) Ay=—°2, A,=—'105, A3;=—'015, A,=°004,
A;=0, A,=°0003,
j= L29) c= 22-2 em./secs

Dr. J. R. Wilton on Ripples. 699

_ The two corresponding sets of values of # and y are given
in Tables IV.a and IV. 6, and the ripples* are shown in
figs. 4a and 4), respectively.

Taste IV.
Ripple of length 2°44 cm.
al b.
E o£ y (Ae ct — x. Yy
OAM NG 025 0 Ovi ees
7/ONn | 21 050 | /6 12 088
ae 37 087 ae 30 014
7/2 52 038 /2 54 —-042
Qn/3 val —-039 Qn/3 78 ~-052
57/6 96 — 086 57/6 1:00 — 046
T 1-22 —°095 T 1-22 — ‘O44 |
x ae |
Fig. 4a

Ripple of length 2°44 cm.; amplitude -182 cm.; velocity 24°6 cm./sec.

Fig. 4 6.

ee We mem

Ripple of length 2°44 cm. ; amplitude 175 cm. ; velocity 22:2 em./sec.

It is possible that the dimple at the crest of the second
ripple is due only to the neglect of terms of higher order ;
but it seems very unlikely that the form of the first ripple
can be due to this cause. One is tempted to say that 4a is
probably unstable, 4 probably stable. In any case there is
room for experimental investigation of the forms of ripples
of this particular length, and also of the form of “high”
ripples of very short wave-length, such as that shown in
fig. 1. It would also be interesting to obtain experimentally
the forms of the ripples whose lengths are given by «=1/3,

K«=1/4, &e.

* [have not strictly adhered to the customary distinction between
ripples and waves.

700 Dr. F. Li. Hitchcock on the Operator V in

It must, however, be remembered that although viscosity
does not, to any appreciable extent, affect the forms of these
ripples, it does very rapidly damp them out. Thus the
amplitude of the ripple in fig. 1 is halved in less than one fifth
of a second, so that it must be sought within a very few
centimetres of the generating source. But, if the ripples of
fig. 4 could be produced, they might be expected to travel
some twenty or thirty centimetres without any serious dimi-
nution of amplitude.

LXXIII. On the Operator V in Combination with
Homogeneous Functions. By FRANK L. Hircncocn, Ph.D.*

ds. MONG the uses of the Hamiltonian operator V there

are three which are particularly remarkable. First
is the use of V to distinguish the character of fields of
force, fluid motion, and other vector fields. Second is its
use to express integral relations having to do with space-
integration over surfaces and volumes. Third, when VY is
combined with functions which are homogeneous in the
point-vector p, many new results are obtained.

To recall the leading facts under the first category:—If a
vector function I of the point-vector p satisfies the relation
VV F=0, its rotation vector or “‘ curl” is zero, and its distri-
bution is lamellar. If SYVF=0, the “divergence” is zero,
and the distribution solenoidal. If both these relaticns hold,
so that VF=0, the distribution is Laplacean. If F is
everywhere at right angles to its own curl, we have
SEVE=0; as I am not aware of any name for such a
distribution, I shall venture to call it orrHOGYRAL?}. The
most significant property of an orthogyral vector is that it
becomes lamellar when multiplied by a suitably chosen
variable scalar ft.

Under the second category fall the quaternionic forms of
the theorems of Gauss and of Stokes on multiple integrals,
which have been greatly extended by the late Profs. Tait
and C. J. Joly and by Dr. Alex. McAulay.

My present object is to develop somewhat further the uses

* Communicated by the Author.

+ Pronounced ortho ji’ral.

{ Such a characterization of vector fields by means of differential
operators may be greatly extended. Thus the four fields to which names
are above given are characterized by the linear operators VV, SV, V, and
SFV, special cases of the general linear quaternion function of vy, which
in these combinations is, analytically, both vector and differentiator.
I have considered the general question in a former paper (“The Double
Nature of Nabla,” Phil. Mag. Jan. 1909).

Combination with Homogeneous Functions. 701

of V in the third of the above ways,—in combination with
homogeneous functional operations. A few facts are already
Known: chief of which is Huler’s theorem, written in qua-
ternion form as

SOEs omar one 2 ees Gals)

where Fp is any function of p (scalar or vector), homogeneous
of degree n inp. Aside from Huler’s theorem, most of the
known results on homogeneous functions in connexion with
V are combinations of V with linear vector operators, and
are due to the writers above mentioned. For example, if
dKo=d¢ddp, and if ¢' is the linear vector operator conjugate
to d, then

Pie — Pai NAVE Onion, ae cn Ce)

where « is any vector not acted on Divganv ace

2. Before proceeding to the proof of new theorems, it
will be necessary to enter briefly into a few elementary
considerations. First, with regard to notation, I shall write,
for brevity, To=r and Up=u, so that p=ru.

Next, as to the definition of a homogeneous function, it is
most natural for a vector algebraist to write

Ao OD Pht Bas racs ban tae ooh a Cee)

as the definition of homogeneity, either for scalar or vector.
This is of course precisely equivalent to the usual definition,
and much moreavailable. In words: A homogeneous function
of p is one that can be factored into a power of 7 (that is Tp),
and a function of uw (that is Up), alone.

The differentials of Tp and of Up are important, and may
be expanded in many forms (Tait, Art. 140). For the
present purpose we may take as most convenient for the
tensor of p,

Cie OL, 5). CL we CRO Tee

and for the unit vector
Ge MO USUAD aia) aL) We
Again, we often need to apply V to a function of wu alone.
This is achieved by writing
EU Eth s,s ney ae ta
We then have
dku=ddu
=d(dp+uSudp)r—, by (5).

* Tait, ‘Quaternions,’ 5rd Ed., Arts. 185, 186.

702 Dr. F. L. Hitchcock on the Operator V in

We now obtain VFu from dFwu by changing dp into p’, at
the same time writing V’' to the left of the whole. That is

V Fu=V'd(p +uSup)r7, . . =e

where the accents indicate that V acts only on the accented p.
The expression V/ Fu therefore stands always for a function
homogeneous of degree —1. This holds when Fp is either
scalar or vector, since the foregoing identities depend only on
the linear character of @ and not upon its dimensionality.

Finally, if Fp is homogeneous of degree n in p, V Fp is
homogeneous of degree n—1. For

V Fe=V (r’Fu), by definition,
=nr1V7r . Fu+r*V/Fu, by distributing V;

but, in the first term on the right, Vr=u (by Tait, Art. 145),
and in the second term, VVFu, as has been shown, is homo-
geneous of degree —1. Hence the right side may be factored
into 7"-! anda function of u alone. It is therefore, by defi-
nition, homogeneous of degree n—1. ‘This, also, holds for
scalar or for vector.

3. I shall now prove the following theorem in regard to
solenoidal vectors:—

Any homogeneous vector may be rendered solenoidal by
adding to it a term of the form pt, where t is a properly
chosen scalar; exception must be made of vectors of
degree —2.

For consider the effect of V upon the vector pSV Fp,
where Fp is a vector homogeneous of degree nin p. We
have

V (pSV Ep)=Vp-SVEp+VSVEp.p, by distributing V.

But, in the first term on the right, Vp=—3. Furthermore,
scalars are commutative, so that if we take the scalar part of
both sides we may write

SV (pSV Fp) =—38V Fp+8pV.SVEp.

Now the scalar SVFp, as already pointed out, is of
degree n—1. We may therefore apply Euler’s theorem to
the right-hand term, and have, (by (1)),

SpV .SVFp=—(2—-LDSVF p.
By combining terms, therefore,

SV (pSV EFp)=—(n+2)SVFp. . . . (8)

Combination with Homogeneous Functions. 703

The following identity will now be evident,
pSV He \ _
SV 4 Fp+ eV =O nominee)

Fp being a vector homogeneous in p of degree other
than —2. This identity proves the theorem and shows
how to find the scalar ¢.

The term pé is uniquely determined. Lor if there were
two values, their difference would be a scalar multiple of p
and would be solenoidal. Call this difference pt, But by
the same order of reasoning as above, SV (pt,) = —(n+ 2) ty,
which cannot vanish unless ¢;=0 or n= —2*.

4, Asa simple, but important, extension of the foregoing
theorem, let us suppose (what is frequently the case) that a
non-homogeneous vector can be written as the sum of several
vector terms, each homogeneous in its own degree, e. g. let

Fo=F p+ F.p+...+¢F,pt...,

where the subscripts denote the degrees of their terms. By
applying the theorem to the separate terms, we see that F'p
may be rendered solenoidal by adding the vector

F ip F F.p }

The series concerned may be infinite, provided they are
convergent.

Conversely (as an example of integration with Y), if the
convergence of a vector, (SV Ip), be given, we can write
down a value for the vector itself, which shall be a scalar
multiple of p, provided we can expand the convergence as a
sum of homogeneous scalar functions of p lacking a term of
degree —3. For example, if we have given

convergence=SV Fp=t)+t+t.+...+,

where the subscripts denote the degrees of their terms, then
a possible value of Fp having this convergence is

to ty ty
Fp=—p{ 3+ i 5 tanh

a flux directed toward the origin.
5. These very simple results on the solenoidal character of
vector fields may naturally lead us to inquire whether there

* In a similar manner we may show that a term of the form pf, if ¢ is
a scalar of degree —3 in p, is always solenoidal,

704 Dr. F. L. Hitchcock on the Operator V in

are not analogous facts in regard to lamellar vectors. The
following is the case :—

Any homogeneous vector may be rendered lamellar by
adding to it a term of the form Vprt, where Tt is a properly
chosen vector; exception must be made of vectors of
degree —1.

for consider the well-known vector identity (Tait, Art. 90),

VaViy=ySaB—BSay.
Writing p for a, V for @, and Fp for y, this identity
becomes
VpVV¥p=Fp' .8pV —V'Splo'", 2 ae
where, on the right, accents indicate that V acts only on Fo.
By Euler’s theorem, Fp’. SpV/=--nFp. Also,
VSpFhp=VSpKo+V Spr p’, - 2 aaiy
by distributing V. (Unaccented V acts on all that follows

in the same term.) But the first term on the right of (11)
is the same as —Fp, by Tait, Art. 146. Whence (11)

becomes
Voetp—— Vo VW Spl.» i4 ee (12)

By adding (10) and (12) and solving for Fp we therefore
have the identity Ya oar:
He ek Bp Velie
n+1 n+1 ’
Fp being any vector homogeneous in p of degree other
than —1l. The right-hand term is obviously lamellar.
Stated in words, (13) shows that any homogeneous vector
field (exception noted) may be taken as the sum of two
fields, one lamellar (irrotational), the other at right angles
to the point vector. By transposing, and operating with VV,
(13) becomes

a Pave
VV { Bp+ POV | Wo. Oo. ail

(13)

This latter identity proves the proposition, and shows how to
find the vector 7. The identity (14) may be verified by
direct operation™. ;

* The method used above for obtaining (14) is not quite parallel to
that by which the analogous (9) was proved. Indeed, (9) might have
been proved by first establishing the identity, analogous to (18),

(n+2)Ep= -pSVEp—VVVpFp,
by expanding the last term on the right by the formula, Phil. Mag. June
1902, p. 579, (6). We then have (9) by the operator SV; or we have
(14) by writing VVFp in place of Fp which is any homogeneous vector,
so making 2 become x—1.

Combination with Homogeneous Functions. 705

The term Vopr is uniquely determined. For if there were
two possible values their difference would be of the form
Vp7,, and would be lamellar. Now the term Vp7, is of
degree n, hence 7, is of degreen—1. We therefore have
identically*

VV Vpen= —(n+1)7,;—pSV7, 5 r (15)

If n does not equal —1, 7, does not have degree —2, and
may be rendered solenoidal by a term in p without altering
the value of Vpt, Hence we may suppose SV7,=0, and
the right side of (15) cannot vanish if 7, does not vanish and
nis not —1; that is, the term Vopr is uniquely determined.

6. From the identity (14) may be easily deduced a second
example of inverse operation (integration) with V7. Suppose
a rotation vector, (VV Fp), to be known at all points of a
given region, and to be expressible as a sum of vectors each
homogeneous in p, lacking a term of degree —2. For
example, let

rotation vector= VV Fep=m)+7+72+...,

where the subscripts denote the degrees of their terms.
A possible value of Fp then is

— — HO ie De
Fp=—Vp{ 424% tab a Wigley
a vector everywhere at right angles to p. The vector Fp is
often called the vector potential of its derived vector VV Fp.
Thus (16) shows how to write down a possible vector potential
for any assigned solenoidal vector whose components are either

olynomials or other sums of homogeneous terms (exception
noted). (16) may be directly verified by expanding the right
side with the aid of identities like (15)f.

* Phil. Mag. Joe. crt.

+ It is well known that possible values for a required vector potential
can be found by partial integration with respect to the point-coordinates
x,y, andz. The above method illustrates how yv may replace partial
integration,—a principle probably more far-reaching than any appli-
cation which has yet been made of it. As another illustration, let
Xdv+ Ydy+Zdz=dP=0 be an exact differential equation. X, Y,andZ
are components of the vector VP. Suppose P=Spl'p. If VP can be
written as a sum of homogeneous vectors

VPSo7-- teat. 3)

we may write down P by the formula

1 alae SN acc A RN
Gane st
proved by multiplying both sides of (13) by p and taking scalars.
Phil. Mag. 8. 6. Vol. 29. No. 173. Alay 1915. 2Z

706 Dr. F. L. Hitcheock on the Operator V7 in

7. Most of the foregoing results are extensions of familiar
properties of linear vector functions. The identity (13),
expressing a homogeneous vector at the sum of an irrota-
tional vector and a vector perpendicular to p, appears in the
linear case as the familiar

dp=op+ Vep (Tait, Art. 186), . . . (17)

where w isa self-conjugate linear function and e¢is a constant
! LE ii,

vector. Here ape EES and Vep= —— whence

the identity (17) may be written

Laver magia a 8)

To bring out the analogy between (13) and (18), put, as
before, dFp=¢dp, so that ép=nFp by Euler’s theorem, and
VSpk p= —¢'p—Fp by (12). By (2), VpVV Fp=($'—9)p.
By substitution of these values, (13) becomes

(6—¢$')p , ppt+nd'p,
n+1 7 nest lL). + (19)

which evidently reduces to (18) for the case n=1. The
right-hand term may be taken as an extended wp; and, just
as wp is at all points of space normal to the general family
of central quadric surfaces Spdp=const., so this term is
normal to the cubic or higher surfaces SpFp=const. Again,
just as the axes of wp possess the special property of being
mutually at right angles, so the axes of its analogue have a
specific configuration ; but the consideration of axes lies
outside the scope of the present paper. |

8. The identity (13) also throws a good deal of light on
the nature of orthogyral vectors. To distinguish these
sharply from other vectors, we may say that an orthogyral
vector is one satisfying the two following conditions :-—

1. Neither the vector nor its curl vanishes identically.

2. The scalar product of the vector and its curl vanishes
identically.

Lamellar vectors are thus excluded from the company
of orthogyral vectors. It will also be convenient to distin-
guish two cases, according as SpF op does, or does not, vanish
identically. If an orthogyral vector Fp is everywhere at right
angles to p, the family of surfaces normal to Fp consists of
cones. The right-hand term of (13) disappears, and the vector
may be said to be conical. If, on the other hand, SpF p does not

Combination with Homogeneous Functions. 707

vanish identically, an orthogyral Fp may be said to be mized.
These two types are clearly not restricted to homogeneous
vectors. I shall speak of the scalar SpF as the associated
scalar of the vector Fp.

From (13) we may now deduce a variety of simple con-
sequences. First, any homogeneous vector whose associated
scalar vanishes identically is orthogyral. Thisis evident from
the mere form of (13) except when n=—1. In this case it
can be shown by actual expansion of VV Vopr, as in (15).

Second, if a homogeneous orthogyral vector be divided
by its associated scalar, the resulting vector is lamellar, for we
have

VeVV Ep + VSplp ‘ oD
vv 4 = =i),

by actual expansion, if SFVF=0, 7. e. if F is orthogyral.
iixception must be made of conical vectors.

It further appears from (13) that the theory of orthogyral
vectors may be connected with that of algebraic plane curves.
Suppose n, the degree of the homogeneous vector Ip, to be a
positive integer, and let Fp be orthogyral of mixed type,
and let its components be polynomials. Let w, y, and z, the
usual coordinates of a point p in space, define a point ina
plane in homogeneous coordinates. Then the associated
scalar, SpFp, will define, by its vanishing, a plane curve, of
degree n+1. I shall now show that if Ip is orthogyral the
curve defined by its associated scalar must have x double
points. The condition that Fp shall be orthogyral is that
the scalar product of the two vectors VV Fp and VSpf'p
shall vanish identically. Call the components of VSpFp
X, Y,and Z, and those of VW Fp X,, Y,,and Z;. Expanding

the scalar product we have the condition
XX,+ YY,+ ZZ,=0, identically.

These six components define six curves. Wherever X and Y
both vanish, either Z or Z, must vanish. But Z, meets X
or Y at most in n(n—1) points, the product of their degrees*.
Hence at the remaining n?—n(n—1) intersections of X and
Y we must also have Z vanish. That is, VSpl*p vanishes at
n points, and the curve SpFp=0 has n double points.

For example, let n=1. The associated scalar, being of
the second degree, defines a conic. It must have one double

* If Z,, X, and Y happen to have a common factor, we may make a
new choice of our coordinate system. It is easy to complete the formal

‘proof.
: 272

708 Operator V in Combination with Homogeneous Functions.
point, hence consists of two straight lines. We may then put
Spi p=SapsBp,
where aand Bare constant vectors. Operating by V gives
VSpFp=—288p —B8ap.

The vector VV/Fp, being now of degree zero, is constant,
and must be parallel to Va@. Hence the most general
orthogyral linear vector is of the form

aVpV28+b(aS8p+ BSzp),

where a and 0 are constant scalars.

If we let n equal 2, the associated scalar defines a cubic
with two double points, hence degenerate.

If n=3, the most general orthogyral vector has for its
associated scalar a quartic of deficiency zero.

In asimilar manner, if we start with any two homogeneous
scalars we may write down orthogyral vectors in the form (13).
For the vector VVuV/v is solenoidal, whatever scalars uw and
vmay be. Hence

aV pVVuVvtbV (uv)

is orthogyral if u and v are homogeneous, a and b being
constants. That is, to any pair of algebraic plane curves
corresponds a two-parameter family of orthogyral vectors.

9. Inconclusion it may be said that the differential and
integral relations of this paper are extensions to space of the

nd eras, In fact, most

{ ae

dx
of the preceding results reduce to these, or to identities, if we

one-dimensional formulas for

a
put p=iwv and Wits That a calculus with V is worthy

of systematic and extensive development there can be no
doubt. We should naturally expect greater variety and
complexity in proportion as the geometry of space is many-
sided in comparison with that of one dimension. It would
be essential to consider next the values of n treated above
as exceptional cases—not a difficult matter, but leading
to logarithms and other non-homogeneous functions, beyond
the special domain of the present paper.

ly 7050

LXXIV. Radiation from an Electric Source, and Line Spectra.
The Hydrogen Series. (Preliminary Note.) By L. SinBer-
STEIN, Ph.D., Lecturer in Natural Philosophy at the
University of Rome*.

|e the source, i. e. the seat of impressed electric force,

be a sphere, of radius a, having the permittivity K,
while that of the surrounding medium is unity. Then the
intensity of electromagnetic radiation emitted by the source

is, at distances great in comparison with a,

aD ra CNR el rae care tice CB)
where A is the wave-length in vacuo and / is a certain
function of X and K(X) which I will not write out here f.
For large values of K, viz. when the refractive index
e/v=K"? is of the order 10°, the spectrum (1) of our
source consists of an infinite series of very thin and sharp
“* lines,” corresponding to the maxima of J, say Jj, Js, etc.,
arranged after the descending wave-lengths Ay, Ay, etc.; the
reciprocals of the latter are the successive roots of a com-
paratively simple transcendental equation. The requirement
that these “lines,” or exceedingly narrow bands, should lie
within the visible spectrum, reduces the source to molecular
dimensions {. The intensities of the successive spectrum-
lines are, with a high degree of approximation, proportional
-to the square of the wave-length, 2. e.
Ceara ue Tney AU SeN A area Ns Bhs VON 2EK) yo ACER)
In other words, the lines, from red to violet, become fainter and
fainter.

The permittivity of the source, K, may be, in the general
treatment of the problem, any function of A, which function
may be said to define the intrinsic or the atomic dispersion.
In accordance with the essence of the method of investi-
gation employed (ef. loc. cit.), I have strictly avoided any
attempt to enter into the mechanism of the ‘“ source,”
which may consist of many electrons or other subatomic
entities. Guided by the analogy of molar lumps of matter,
I assume k=47°a?K to be, in general, of the form

je Hao Seal Br) cca eae UR | cca (3)

* Communicated by the Author.

+ This function and the corresponding details of the problem are
given in a paper communicated to the Royal Society, 22nd March, 1915.

t The case of small K, corresponding very nearly to the (normal)
continuous spectrum, is treated in the paper quoted above. In that case

—5
a is, at all accessible temperatures, greater than 107° em,

710 Dr. L. Silberstein on Radiation from an

where «, 8;, y; are constants, which will have to be deter-
mined, for each individual spectrum, on the ground of
experience, such as the dispersive properties of molar lumps
of the substance in question or the characteristic features of
the spectrum itself. But we need not enter here upon
details of this kind. |

The purpose of the present note is shortly to report on
certain results already obtained by means of {1) in the
simplest case of the dispersion-law (3), viz. for k=1.

Then
B

k = OT nae ° e ° e e (3 a)
If k) be the static value of k, that is, for an invariable
impressed force, or for A/y=«, we have e+B= hy.
Introducing (3a) into the full formula (1), I obtain, for
the wave-lengths of the successive spectrum-lines,

ky gainer Dy / 1 ne aE Nn ue

Mm a(vt Ta) + Lat a) ~ ay @
where wu; (i=1,2,3,...) are the successive roots of the
transcendental equation mentioned: above*. The relative
intensities of the ‘“‘lines” are given by (2), and since
K,.>K,, K;>K,, etc., the successive lines become not only
fainter but also sharper. The values of the successive roots,
Uy, Ug, etc., Increase indefinitely, so that the lines become
more and more crowded from the red towards the violet end
of the spectrum, and, by (4),

Ne iy. 6 rr
Thus, the lines constitute a series having its convergence-point
at N=y¥.

These being exactly the characteristic features of the
so-called “first ’’ hydrogen spectrum f, it seemed interesting
to try to determine the constants 2, 8, y (and the order of
magnitude of K), so as to represent the beautiful series
of that gas by the formula (4). My calculations, which
now extend over many days, are not yet completed. The
following tables contain some of the results of my first
attempts. In each of these tables, the first column contains
the order-number (i) of the root; the second column,
A; calculated by means of (4), in microns; the third, the

* Owing to the large values of Ke, and, a fortiort, of KA, ZY,

' : K | cos
this transcendental equation becomes — ++ a

— cos u. ue gu)
+ Which, in its essence, is also shown by aluminium and thallium.

=0, where g(w)=sin w/w

Electric Source, and Line Spectra. roe

observed wave-lengths (to five figures only), with the usual
short denominations of the hydrogen lines ; and finally, the
fourth column, the differences, A=Acaic.—Aobs,. in Angstrom
units. The symbol Ag stands for Balmer’s “ theoretical
end,” if we are to translate literally the name used by
Germans. The constants, at the top of each table, are given
in microns. I should like to remark that the tables are
numbered, not according to the importance to be attributed
to the degree of agreement attained, but simply in their
chronological order.

PAB EVI
i "36605. 3 kyp=4°1200 5 c= —3°2398.

u A; cale \obs. a
3 4336, 4340, Hy 39
4 ‘4097, ‘4102, 5 —4°7
5 3965, 2970n 50
6 3884, 3889, ¢ —48
i 3831, 3835, —4:0
8 3795, 3798, 9 26
9 3769, ‘BaT0y 23
10 3750, -aio0an Uke +02
at et 3735, 3734, +13
te 3724, EE, $25
Weds 3715, oTOu +38
14 3708; =u yee
15 3702, 3704, ¢ —15
16 3697, S8975) % 5 +04
17 3693, ee es
18 3690, BURT Lge ~15
19 36875 3687, p +0-2
20 3684, 3682, ¢ +18
00 3660, 3661, Hy —0'4
nes 3646,=,

{

It will be noticed that the theoretical “ lines”’ Ayy and dj;
are superabundant * ; such would be also AXg1; Age coincides

* The first of these may also be coordinated with Hy, giving
A=1+4:'3.

712 Dr. L. Silberstein on Radiation from an

nearly with H>, 2. e. with H,, (Balmer’s m=21; H, is Hs),
og With H,;, and X,, very nearly with H3;, the last observed
line of the hydrogen series. In a word, from 2=20 or so
the formula (4) is “ getting out of step,’ with the above
constants that is. The reason of the superabundance of
theoretical lines is, of course, the choice of y just behind
the last observed line H3,, which is certainly before the
convergence-point of the series (15°1 A.U. before Ag).
Owing to this circumstance the theoretical lines begin
to crowd, as it were, too early. This is one defect of
Table I. Another, more serious, defect is that the above
constants y, a, ky give, for 1=1 and 1=2,

NiO Zz0l 4) WNa — 4 oao.
whereas the first two hydrogen lines are (to four figures)
H, = 6568, H, = -4861,

so that, the corresponding differences are A= —296 (!) and
—22 A.U. They have, therefore, not been incorporated
into Table I. In fact, the constants used in that table have
been calculated without any regard to H,, Hg. (And i
the reader tries to determine a and &) from these
two rebellious lines, so as to make them fit exactly, or
nearly so, he will at once spoil the rest of the series, from
A; till the end.) But apart from these two defects, the
agreement of the two columns of Table I, and more
especially for 12=3 till z2=13%*, seems remarkable. Notice
that the decreasing intensities of the successive lines, as
expressed by (2), and not given in the table, may serve as a
supplementary corroboration of the formule here proposed.

In Table IL, in which y is very little smaller than in
the preceding table, the constants ky), « have been deter-
mined from postulating >, = Hs, A; = H,. Hence the
absence of A for these two lines, calculated only for the
sake of control. But the exceedingly close agreement
of Ag, Az, Ag, Ag With Hy, etc., seems very remarkable. The
results for ~;, A, are as bad as in the first case, but the
agreement for the remaining part of the series is, in general,
more close.

The coordination of the theoretical and observed 2’s

* The lines being here less crowded than in the lower part of the
table, there is little probability for the agreement being a “chance-
agreement.”

‘Electric Source, and Line Spectra. 713

TasieE II.
y="366063 ky 3934503; «=—4:0819.

| i. | A; cale, | Moba: aN
| 3 4338, | 4340, H, 26
4 LUO. VN eeelaph 3 0:0
| 5 3970, 3970, ¢ 0-0
6 3888, 3889, ¢ 03
rl 3835, 3835, 4 —01
8 3798, 3798, 9 405
9 37725 TT Oaa ue: $155
10 3752, 3750, 42:5
Data ‘3737, ‘3734, +3'4
12 3726, Ph) +49
13 ‘37173 oleae) +52
14 3709, ‘3704, x +5°7
© 3660, oot Hon ~06
3646, =),

becomes, after 2=13 or 14, ambiguous. There are again
superabundant lines, owing to the smallness of the interval
Reo —Hs;. Compare the remarks made above. ‘To prevent
the premature crowding of the theoretical lines and to take
in, at the same time, the hitherto ignored lines Ha, Hg, the
convergence-point 7.=y is pushed hack, nearly as far as
-Balmer’s limit. Some of the results of the corresponding
calculation, which is just started and is being assiduously
retouched (with regard to the values of «@, fy), are collected
in Table III. Its chief purpose here is to show that the
enormous value of A for H, can well be reduced, although
at the price of spoiling to a certain extent the successive
lines, and that the superfluous theoretical! lines can be entirely
abolished. It will be enough to give here the wave-lengthis
to four figures only.

The numbers (1), (2), ete. inserted in the third column
‘stand for “ first, second, etc.” observed line of the hydrogen
series; they may facilitate the testing of coordination. It
is quite possible that the simplest dispersion formula (3 @)
will turn out to be too narrow for the purpose of reducing

714 Dr. H. Stanley Allen on an

TABLE III.
y= "36493; ky=5°0500; a=—2*5000.

¢ A; cale d obs. =
1 6499 (1) 6563 H, —64
2 -4909 (2) 4861 +48
3 4363 Gy 43417 +22
4 4107 (4) -4102 93 4
6 3881 (6) 3889 ¢ 18
8 ‘3789 (8) -3798 9 =
12 ‘S715 (12) 3722 =F
16 3687 (16) 3692 5
21 ‘3672 (21) 3674 4 mei)
28 3662, (28) 3662, H,, + 01
29 36615 (29) :3661, Hs: {5 irl
00 3649, p= "3646, a,

the greatest A of Table III. to one or a few A.U. only ; but
before deciding on the introduction of two new constants, by
using formula (3) with k=2, I shall not spare further efforts
to succeed with the simpler dispersion law (3a). The results
of my further calculations, together with some general con-
sequences of the proposed theory, will be reported in a later
publication.

London, April 8, 1915.

——— eee
SS

LXXV. An Atomic Model with a Magnetic Core. By H.
SranLtEy ALLEN, 1/.A., D.Sc., Senior Lecturer in Physics
at University of London, King’s College *.

es spite of their limitations models have been of great
service in the development of physical theories. The
two models illustrating the structure of the atom that
have attracted most attention are those that have been
suggested by Sir J. J. Thomson and Sir EH. Rutherford
respectively. Thomson’s atom consists of a sphere con-
taining a uniform volume distribution of positive electricity,
in which a certain number of negatively electrified corpuscles
are distributed. In Rutherford’s atom there is a central

* Communicated by the Author.

Atomic Model with a Magnetic Core. 715

nucleus of small dimensions carrying a positive charge, and
this nucleus is surrounded by electrons in orbital motion.
According to Nicholson coplanar rings of electrons are not
possible in such a case, and the model can neither be of the
“planetary ” nor of the “Saturnian” type; but, provided
the electrons are in one plane, can only possess a single ring
of electrons.

In these models only the electrostatic forces due to
the positively charged portion of the atom are taken into
account. It has been pointed out by the present writer*
that it may be necessary to consider not only electro-
static but also magnetic forces in the immediate vicinity
of the atom. According to this view the atom would
consist of a magnetic core which is electrically charged,
surrounded by electrons in orbital motion. Whether it is
possible for the electrons to form concentric rings in this
case is a point deserving the attention of mathematicians ;
experimentally such rings appear to have sufficient stability
to allow them to be directly observed as in the striking cases
recorded by Birkeland +. In these experiments photographs
were taken of the discharge through a large vacuum-tube
with a magnetized sphere as cathode. Rings were formed
round this globe resembling the rings of Saturn, in some

Oo
cases as many as three distinct rings could be observed.

The Scattering of the Alpha Rays.

The scattering of alpha rays by atoms of matter has
afforded results from which Rutherford has formed an
estimate of the size of the nucleus. In the theory of
scattering which he has proposed only electrostatic forces
are considered. In this case the scattering depends on the
inverse fourth power of the velocity of the « particle. If we
consider an « particle moving in the equatorial plane of
a simple magnet, it appears that the scattering would depend
on the inverse square of the velocity. In the general case
of a charged particle projected in any direction in a com-
bined magnetic and electrostatic field, it is probable that
some intermediate law would be obeyed. The experiments
of Geiger and Marsden{ agree moderately well with the

* ‘Nature,’ vol. xcii. p. 6380 (1914).

+ Birkeland, C. &. vol. cliii. p. 938 (1911).

t Geiger & Marsden, Phil. Mag. vol. xxv. p. 620 (1915). “ Several
experiments were made, and in every case the scattering was found to
vary at a rate more nearly proportional to the inverse fourth power of
the velocity than to any otherintegral power. Owing to the comparative
uncertainty of the values of the velocity for small ranges, however, the
error of experiment may be somewhat greater than appears from column VY,
of the table.” The tabulated numbers vary from 22 to 28.

716 Dr. H. Stanley Allen on an

first law, but the point is one of so much importance that it
may repay further examination.

The scattering of « rays by the magneton has been dis-
cussed by Hicks *, who has caleulated a number of trajectories
in the equatorial plane similar in character to those investi-
gated by Stgrmer f. The difficulty of forming an estimate
of the amount of scattering is very considerably increased
in the general case when the « particle is projected towards
a magnetic atom in any direction whatever, but the results
obtained by Hicks go to show that scattering of the right
order of magnitude can be obtained by postulating a
reasonable number of magnetons in the core.

There is no idea of calling in question the mathematical
investigations of C. G. Darwin +, who dealt with the motion
of a charged particle under the action of a central force
varying as some power of the distance. The only question
at issue is whether the experimental results on scattering are
of so decisive a character as to prohibit the introduction of
magnetic forces, and to lead with certainty to the conclusion
that the nucleus of a heavy atom cannot have a radius much
exceeding 1078 cm.

The Size of the Nucleus.

Two arguments only have been advanced in favour of the
extremely small diameter assigned to the nucleus of the
Rutherford atom. The first is derived from the wide-angle
scattering of « particles, and, as we have seen, the argument
is inconclusive because no account has been taken of the
possibility of a magnetic field being associated with the
atom. The second depends on the assumptions that the
whole mass of the nucleus is electromagnetic in origin and
that this mass arises from a structureless charge of magnitude
Ne, where N is the atomic number. Now we know that
both « and 8 particles are derived from the nucleus of radio-
active substances, and it is hardly thinkable that these should
exist in the nucleus save as discrete particles. We conclude
that at least for the elements of high atomic weight the
nucleus must possess a structure, though it remains to some
extent doubtful whether the two elements, hydrogen and
helium, at the beginning of the periodic table, are to be
looked upon as possessing a simple or a complex nucleus.
According to the views advanced by Nicholson from a study

* W. M. Hicks, Proc. Roy. Soc. vol. xe. p. 356 (1914).

+ Stormer, Archiv for Mathematik, Christiania, vol. xxviii. p. 36
(1906).
’ t C. G. Darwin, Phil. Mag. vol. xxv. pp. 201--210 (1918).

— Atomic Model with a Magnetic Core. (iy:

of nebular and coronal spectra, the nucleus for each of these
elements is probably complex. It is only to the simple
nucleus that we need attribute the small size necessary to
account for the large mass.

The properties of a terrestrial element connected with its
atomic number N can be explained just as in the case of the
Rutherford atom, if the resultant positive charge of the
complex nucleus amount to Ne. In particular Bohr’s theory
of the hydrogen spectrum remains unaffected if we attribute
a complex character to the nucleus, provided the resultant
positive charge of the nucleus be +e.

The interesting results obtained by Nicholson in his paper
on Electromagnetic Inertia and Atomic Weight * appear
to the writer to indicate a diameter for the nucleus of an
atom of radium or of thorium considerably greater than that
formerly assigned. Thus it is suggested in the paper that
the « particles in a thorium atom have a mean distance apart
comparable with the radius of an electron, 10-8 cm. Now
as the atom of thorium must contain the equivalent of about
58 « particles, it would appear that the radius of the com-
plex nucleus of thorium must be considerably greater than
105 em.

The view to which we are thus led is that the central
portion of an ordinary atom may contain @ and £6 particles,
or hydrogen nuclei in orbital motion. This motion would
give rise to an external magnetic field. Butas the velocities
in question must presumably be less than that of light, the
radius of the magnetic core must be greater than that of the

simple nucleus of Rutherford, and is perhaps of the order of
NOG), cm.

Magnetism.

The views of magnetism that are widely accepted at the
present time are those developed by Langevin and by Weiss.
An electron in orbital motion may be regarded as equivalent
to an elementary magnet. According to the theory of
Weiss there is a certain elementary magnet, the magneton,
which is common to the atom of a large number of different
substances. It was pointed out in a discussion on this
theory at a meeting of the German Naturforscherversammlung
in 1911, that there may be a connexion between Planck’s
“universal constant” h and the magnetic moment of the
magneton. Nicholson regards this constant as an angular
momentum. McLarent} identifies the natural unit of

* Nicholson, Phys. Soc. Lond. Feb. 26, 1915.
+ McLaren, ‘ Nature,’ vol. xcii. p. 165 (19138).

718 Dr. H. Stanley Allen on an

angular momentum with the angular momentum of the
magneton. According to Bohr’s theory* the angular
momentum of a “bound” electron is constant and is h/27.
Conway +, using a different model, obtains the value h/z.
Let us suppose that an electron ( charge e, mass m) is moving
in a circular orbit (radius a) with angular velocity o.
Then its angular momentum is ma*w, and the magnetic
moment of the equivalent simple magnet is eam. Thus the
magnetic moment is equal to some constant multiplied by
he/m. Taking the angular momentum as //27, we obtain
eae 92°7x10-” u.M.U. as the value of the magnetic
moment. This is exactly 5 times the magnetic moment of
the magneton of Weiss. This numerical relation was first
pointed « out by Mr. Chalmers f at the discussion on Radiation
at the Birmingham meeting of the British Association.
The magnetic moment of the magneton is found by dividing
the magnetic moment of the atom ; gram 1123°5 by Avogadro’s
constant. Weiss used the value of this constant found by
Perrin, but if we take the more recent value given by
Millikan (60°62 x 10”) we obtain as the magnetic moment
of the magneton 18°54x 10~”, which is exactly 1/5 of the
number given above.

These commensurable numbers may be of significance in
connexion with the structure of the atom. The magneton
may arise as a difference effect. The way in which this may
come about may be illustrated by a simple model. Suppose
we have a uniform sphere of positive electrification of

radius A rotating in the same sense as an electron with
angular velocity ©. Outside this, suppose we have a single
ring of mean radius @ containing n ‘electrons, The remaining
negative electrification required to produce a neutral system
may be supposed concentrated at the centre without rotation.
Then the magnetic moment of the rotating sphere § may be

* Bohr, Phil. Mag. vol. xxvi. p. 1, p. 476 (1913).

+ Conway, Phil. Mag. vol. xxvi. p. 1010 (1918).

t See ‘ Nature,’ vol. xcil. pp. 630, 687,713 (1914). The same relation
was noticed independently by Dr. Bohr (Richardson, ‘The Electron
Theory of Matter,’ p. 895).

§ This is a particular case of a more general theorem. Since the
magnetic moment arising from a charge é mov ing in a circular orbit of

radius a with angular velocity @ 18 eda, the mag onetic moment arising
from a volume distribution’ of electricity rotating about an axis is

32pdv7?Q, where p is the electrical density and dv an element of volume.
Assuming p constant, the magnetic moment

=2p03r*dv
= ip0Vk?
=3E#70O,

Atonue Model with a Magnetic Core. 719

taken as }HA?’O, where FE is the total positive charge, which
we shall assume equal to Ne. We have no direct evidence
as to the value of A’Q, but if, for convenience, we assume
that it has the same value as a*w for an electron in the ring,
the magnetic moment cf the rotating core becomes +Nea?a.
But a magnetic moment of Jea’w is equivalent to 5 magnetons.
Consequently the magnetic moment of the core is equivalent
to 2N magnetons. ‘he resultant magnetic moment for the
atomic model would be the difference between the 2N
magnetons of the core and the 5n magnetons of the ring,
Thus the magneton may be introduced as a unit for measuring
magnetic moments without necessitating the existence of a
single magneton as an independent entity.

It is not intended that this model should do more than
serve as a crude illustration of the structure of an atom, for
there can now be little doubt as to the complex character of the
core at least in the case of the heavier elements. In par-
ticular a spherical or spheroidal distribution is not an essential
feature of the proposed atomic model. It may be that all
parts of the core must move in one plane. There are
obvious outstanding difficulties such as the way in which
the parts of the core hold together so as to form a stable
system. Passing over these difficulties, the resultant mag-
netic moment of the atom with a spherical core would be
either the sum or the difference of 2N and 5n magnetons,
according to the relative directions of rotation of the core
and the ring.

It would seem that the diamagnetic properties of such an
atom would depend mainly on the ring, if @ is much larger
than A. For the expression for the magnetic susceptibility
‘would consist of a series of terms of which the most important

nena?
Am
Pascal * has shown that the molecular susceptibility of a

would be k= —

large number of chemical compounds can be calculated by

where V is the total volume, & the radius of gyration for a uniform
distribution of mass, and E the total charge of the rotating system.
|, Thus both for a sphere and for a spheroid rotating about an axis of
symmetry, the magnetic moment is } HA*Q.

We may note here that if the electrical distribution is associated with
a proportional distribution of mass, the total mass being Yt, the angular
momentum is Mth°?Q. If we assume that this is a multiple of A/2z,
th
4°
* Pascal, C. &, vol. clil. pp. 862-865, 1010-1012 (1911),

. E
say th/27, the magnetic moment may be written mm X

720 . Dr. H. Stanley Allen on an

adding together the appropriate multiples of the atomic
susceptibilities and a constant term depending on the
structure of the molecule. Further, he has shown that the
elements chlorine, bromine, iodine, and fluorine (and some
others) contain a common aliquot part in the specific
susceptibility. Certain compounds of the halogens show ~
less susceptibility than would be expected from the additive
law. This diminution can be expressed in terms of the same
aliquot part, whose value is 0°2468x107". If we identify
this quantity with the effect of a single electron in the ring,
we are led to the conclusion that the radius of the ring is
about the same for the elements in question, and that its
value is of the order of magnitude to be expected.

The difficulties connected with the explanation of para-
magnetic properties are, of course, left untouched by these
suggestions. No one has as yet explained how the orbits
become tilted when under the influence of an external
magnetic field. We may note that a similar difficulty is
found in connexion with Ritz’s theory of the Zeeman effect,
which is attributed to a precessional motion of the elementary
magnet, no explanation being forthcoming of the way in
which the precessional motion is set up.

The Quantum Theory of Spectral Series.

The success of Bohr’s theory in explaining the ordinary
Balmer’s series in the spectrum of hydrogen, and especially
in obtaining close agreement between the observed and the
calculated values of Rydberg’s constant, raises a strong
presumption in its favour. ‘The essential feature of the
theory is the emission of exactly one quantum of energy as
monochromatic radiation in the passage between one steady
state of motion and auother. This leads directly to an
expression for the frequency, v, as the difference between
two “sequences,” the form of the expression being

AN Vo
oD 2) Des :
When, however, an attempt is made to apply the theory to
the spectral series of elements other than hydrogen, serious
difficulties are encountered. These have been discussed by
Nicholson * with special reference to the spectra of helium
and of lithium. In the first place, it is necessary to suppose
that every electron concerned in the emission of radiation
emits one quantum, instead of supposing that one quantum

* Phil. Mag. vol, xxvii. pp. 541-5€4, vol. xxviii. pp. 0-103 (1914).

Atomic Model with a Magnetic Core. 721

only is emitted in passing from one state to another. This
is required both for ordinary spectra and for X-ray spectra.
In the second place, it appears necessary to assume that
there is no force between bound electrons, so that any one
of these electrons is independent of the others. This
supposition may be related to Sir J. J. Thomson’s con-
ception of tubes of force. A bound electron may have the
tube (or tubes) of force originating from it attached to the
nucleus, and if all the electrons in question are connected
to the nucleus in this way, they cannot exert force on one
another.

Perhaps it may be necessary to suppose that a bound
electron has both the ends of the double tube of force
belonging to it attached to a definite part of the core in such
a way that the attraction on the electron is proportional to e?
instead of to (Ne).

A further difficulty in applying the theory of Bohr to actual
series lies in the fact that the denominator of a sequence
contains terms which are not simple integers. Thus in
Rydberg’s formula we have m+yp where m is an integer,
pa fraction, in the formula of Moggendorf and Hicks we
have de a/m, in the formula of Ritz m+p+ B/m’.
Nicholson * has shown that the theory in its original form is
insufficient to account for such additional terms when electro-
static forces only are considered. The development which
I have given +, supposing the electron to be under the action
of magnetic forces, yields a formula containing a term of the
form B/m2, where B is proportional to the magnetic moment
of the core, but does not account for the fractional part p.
In attempting to apply this formula to actual elements, it is
found that in general the value of @ obtained by Ritz is
much too large to be due to a small number of magnetons.
Further, in the case of hydrogen, if we suppose the fractional
part due to a term of the form B/m?”, it is necessary to assume
different values for B in the two sequences, implying the
existence of two types of state in the core. This suggested
that the core itself might be intimately concerned in the
emission of radiation, and that the term in yg, and perhaps
also terms such as a/m and B/m?, might depend upon the
angular momentum of the core.
This line of thought, associating the constants in the
formula with the core of the atom, may be supported by

mo Ocw tt.
+ Phil. Mag. vol. xxix. pp. 40-49, 140-1438 (1915).

Phil. Mag. 8. 6. Vol. 29. No. 173. Alay 1915. 3A

722 Dr. H. Stanley Allen on an

several recorded results. According to Hicks * both wu and e
depend on the atomic weight or atomic volume of the
element, «/u being a pure number and equal to 0°21520.
Birge ¢ points out that the coefficients in the formula of
Ritz increase with increasing atomic weight, being propor-
tional to the atomic volume in the case of sodium, potassium,
rubidium, and csesium.

Several investigators have drawn attention to relations
between frequencies and atomic weights, Ramage and
Marshall Watts in particular having obtained relationships
involving the square of the atomic weight. This of course
implies that the constants in the spectral formule depend
upon the atomic weight of the element in question.

Hicks finds that the change necessary in the value of p to
account for the observed differences in the frequencies of
doublets and triplets can be expressed in terms of a quantity
which he calls the ‘oun’ depending on the square of the
atomic weight.

Further arguments may be drawn from the “ combination
principle of Ritz.” The formula of Ritz may be written

Eero BU ae NG Teale 0
[m+w+B(A—n)]?’
which is usually abbreviated as

n=A—(m, p, §).

The values of w and of @ differ in different “sequences,”
and Ritz shows that frequencies corresponding to definite
lines can be obtained by taking the difference between
various sequences.

In the case of a principal sequence ~ may have two values
#, and py, the corresponding values of 8 being @, and >.
Ritz proves that other lines may be obtained by taking
(m, [1— fs, Bi; —2) as a sequence. In the case of the triplets
of the alkaline earths, differences such as @,—, for the
principal series are the same as the corresponding differences
calculated from the diffuse series, so that evidence of a new
combination exists here.

According to the simple form of the theory put forward in
my former paper, @ is proportional to the magnetic moment
of the core. From this point of view, it is not difficult to
understand how we can get combinations such as 8i— Pe,
regarding the core as composed of positively and negatively
electrified particles in orbital motion.

* Hicks, Phil. Trans. vol. ccx. p. 85 (1910) ; vol. cexii. p. 33 (1912) ;
vol. cexiii. p. 328 (1913).

t Birge, Astrophys. Journ. vol. xxxii. p. 112 (1910),

en ee

Atomic Model with a Magnetic Core. 723

But the significant point is that accompanying B,—A»2, we
have “,—/s, suggesting that the quantity ~ also depends on
the revolution of parts of the core in such a way that the
effects can be combined by simple addition or subtraction.

For this to be the case, w would have to be proportional
to the first power of the angular velocity, i.e. to the angular
momentum or to the magnetic moment of the part of the
core with which it is associated. ‘Thus it should be possible
to express ys In terms of the magnetic moment, and it might
be possible to obtain some relation between p and £.

Rydberg has suggested that the correct expression for the
frequency of a line in a spectral series is some function of
T+p, where 7 is an integer and wis fractional. This view
has received strong support from Thiele, who maintained that
the wave-length was some function of (t+ p)?, where 7 could
take all integral values, both positive and negative. Nichol-
son’s recent critical investigation of the spectrum of helium
shows conclusively that in this case the frequency is a
function of T+

In Bohr’s theery of the hydrogen spectrum, the angular
momentum of a bound electron is assumed to be constant and
equal to th/2m. In order to obtain a theory applicable to
the spectra of other elements, it appears necessary to assume
that the angular momentum of the electron is (T+ p)h/27.
In order to account for the presence of u in this expression,
we assume that we must include with the angular momentum
of the electron that of the core, or more probably that of the
part of the core which is specially related to the electron.
‘Thus we make the ¢otal angular momentum of the electron
and the part of the core equal to th/27.

Then mro =- 1QO = th/2a.

So mre = th/27 + 1O
Sa (Ge ae VO ere

where jo = 2rIO/h.

Proceeding on the lines of Bohr’s theory we can then
obtain Rydberg’ s equation.

The extension of the principle of the constancy of angular
momentum from the electron to the core, receives a measure
of support from the work of Bjerrum and others. Bjerrum
assumed that the energy or the momentum of a rotating
molecule could be expressed in terms of h. The experimental
results obtained from the absorption of infra-red radiation
by gases are In agreement w ith the results of his theory.

The supposition that jy corresponds to the angular
momentum of only a part of the core was suggested by the
numerical values found in spectral series. It is intelligible

3A 2

724 On an Atonuc Model with a Magnetic Core.

from the point of view of tubes of force, for we may suppose
that a tube of force with one end on the electron has the
other end attached to a certain part of the core which carries
an equal charge of opposite sign. The suggested arrange-
ment appears to be in partial agreement with the views of
Stark with regard to the structure of the atom.

In order to secure agreement with known facts as to
spectral series it is necessary to regard was constant for one
type of state of motion, but as possessing different values
corresponding to the different types of state.

It cannot, of course, be claimed that these suggestions
constitute a theory of spectral series, but an attempt has been
made to see what modifications may be required in the
original assumptions of Bohr in order to obtain a formula
such as that of Rydberg for elements other than hydrogen.
The introduction of the coefficient ~ renders the structure of
the atom somewhat indefinite, for w~ depends on (at least
two) factors both at present unknown. It may, however, be
possible from a study of the numerical values of this quantity
in the case of particular elements, or elements belonging to
the same chemical group, to throw further light on the
character of these factors.

Conclusion.

The atomic model whieh is suggested in the foregoing
pages consists of a ring or rings of electrons surrounding a
central core, having a radius considerably greater than the
nucleus of the Rutherford atom and in consequence capable
of producing appreciable magnetic forces in its vicinity.
The total charge of the core must be equal to Ne, where N is
the atomic number. The magnetic moment of the core arises.
from the orbital motion of the discrete electrified particles
(a particles, 8 particles, hydrogen nuclei or positive electrons)
of which it is composed. The diamagnetic properties of the
atom arise mainly from the external electrons revolving in
orbits whose radius is of the order 10-*cm. The magneton
is regarded not as an independent entity, but as a unit
convenient for measuring magnetic moments introduced
in consequence of the principle of the constancy of angular
momentum.

A consideration of the laws of spectral series suggests
that the quantity w, the ‘phase’ of the series, is connected
with the angular momentum of the particular part of the
core specially associated with the external electron concerned
in radiation. A further study of the values of this quantity
may lead to a more complete knowledge of the structure of
the core.

[754

LXXVI. The Absorption of Homogeneous B Rays.
By K. W. Varver, M.A. (Cape) *.

VENHE experiments described in this paper were made to

investigate as accurately as possible the form of the
lonization absorption curve when very homogeneous 6 rays
pass through a standard substance like aluminium, and also
to test whether there is any simple relation connecting the
absorption with either the velocity or energy of the f
particle.

The apparatus used is shown in fig. 1.

Janae, I

Ret}
TO ELECTRO

Bisa brass box about 3 cm. wide and 10 cm. high. It
was placed between the poles (17 x 10 cm.) of alarge electro-
magnet. The position of the pole-pieces is denoted by the
broken line MM. Rays from a thin-walled tube A filled with
radium emanation describe circles in the magnetic field; a
pencil of homogeneous 8 rays passing through the slit S
could be concentrated at C. The possibility of obtaining very
homogeneous 8 rays by this method has been shown by
Rutherford and Robinson f, who employed it to determine
the 6 ray spectrum of radium B and C. In the present
experiments the relative positions of the source A, the slit 8,
and the opening C were arranged to obtain a concentrated
and nearly homogeneous beam at C. ‘The rays there passed
through a rectangular slit 1 em. x 2 mm. in a brass plate
and through the mica window D. <A ground-glass plate P
placed on the top of the box made it airtight. In order to
prevent scattering and diminution in intensity, the air-
pressure in B was reduced to about 1 em. of mercury. The
window D had a stopping power corresponding to about
2 cm. of air at N.T.P. In this way a strong pencil of 8 rays

* Communicated by Sir E. Rutherford, F.R.S.
+ Rutherford and Robinson, Phil. Mag. Oct. 1918.

726 Mr. R. W. Varder on the

was obtained for which the average value of Hp, where H is
the magnetic field and p the radius of curvature, could be
accurately determined. The value of p was fixed at 3°19 em.
in all the experiments, and readings could be obtained up
to about Hp=12,000 gauss em. From the dimensions of the
source, slit, and opening, it was calculated that the variation
of Hp for the i issuing rays was less than 3 per cent.

The lead blocks L ‘helped to screen the ionization apparatus
from the effects of the yrays. HE isa hemispherical chamber
of 2 cm. radius made of coarse copper gauze covered with
copper leaf and thin tissue-paper. ‘This vessel was charged
to +200 volt. F is a brass cylindrical vessel charged to
—200 volt. A copper electrode G passed through two
ebonite plugs at the ends of IF’ and terminated at one end
in E. ‘The other end of G was connected to a Wilson-Kaye
electroscope, which was screened by the lead block T. K is
a key for earthing G, and R a guar ‘d-ring. A wooden frame
W, coated on the inside with celluloid, minimised the effect
of reflected B radiation. The vessel F was put in to balance
partly the y ray ionization in E. By altering the position of
the lead block O relative to the vessel F, the « y ray lonization
in F could be varied so as to neutralize nearly the y ray
ionization in E. Pieces of celluloid U were placed in front,
below, and at the back of Eas shown. This was done to
minimise the reflexion effects of the @ rays.

The magnetic fields were measured by comparing the
throws of a ballistic galvanometer when an exploring coil
was removed from the field, with the throw produced when a
known current was broken in the primary of a standard
mutual induction. The current in ihe primary of the
mutual induction was measured on a standardized ammeter,
and the coils used had also been standardized in some
previous work. In this way the fields were measured to
about one part in 400. By means of a fluxmeter it was
found that the fields were practically uniform over the
region traversed by the rays in the box. A given field could
alwa ays be reproduced by setting the current in the magnet-
coils to a certain value and~ reversing it a number of
times.

In taking a reading, the current through the magnet-coils
was set ata particular ‘value and reversed a number of times
to reach a cyclic staie. The vessel B was exhausted by
means of a Fleuss pump. The block O was adjusted so that
the motion of the electroscope-leaf was slow when a thick
aluminium plate was placed over the flat face of E. Readings
were taken of the rate of motion of the electroscope-leaf

Absorption of Homogeneous 8 Rays. 727

when various thicknesses of foil were placed over EH. The
current was then reversed so that no @ rays passed through
C, and similar readings were taken. A curve of the type
AA (fig. 2) was obtained when the field was direct, and one
of type BB when the field was reversed; the two curves

Fig. 2.

ee
HP= 2535

IONIZATION
o

ps
oO

7 23 & “7 5

THICKNESS IN GMS. Bee ie

coincided for a very thick absorbing screen. ‘The curve AA
represents the combined 8 and y¥ ray effect in H, the curve
BB represents the y-ray effect alone, so that the curve CO,

which is obtained by subtracting curve BB from the curve

AA, represents the curve of § ray absormtion: The inter-
pretation of the curve BB will be discussed later.

After a small thickness of matter had been traversed, an
approximately linear relation (see curve CC) was found to
exist between the ionization and the thickness of the ab-
sorbing screen. The curves CC for various velocities are
shown in figs. 3 and 4.

728 Mr. R. W. Varder on the

The initial drop of the curves varied with the velocity ;
this is very probably due to the partial absorption of some of
the rays at the edges of the opening C (fig. 1). After this
initial drop, the curves are similar in type. In the diagrams
the thickness of aluminium is expressed in grams per sq. em.
This thickness in the diagrams is uncorrected for the thickness
of the mica window D. The initial rise of the curve BB (fig. 2)
is due to 8 radiation excited in the screens under the condi-
tions of the experiment by the y rays from A. The current

Fig. oe
>
SN

O
ALUMINIUM
‘)
Q

IONIZATION

GRAMS PER CM?

increased with the thickness of the absorbing screens fairly
rapidly at first and then very slowly. This effect is very
important when the ratio of the @-ray ionization in E to the
y-ray ionization issmall. In the experiment with Hp=2000
gauss cm.—the strong part of the spectrum—this ratio was
about 2. In this way absorption curves were taken from

Absorption of Homogeneous B Rays. 129

Hp=1380 to Hp=11500 gauss em. Curves for slower rays
are shown in fig. 3, and for faster rays in fig. 4. particles
could, however, be detected with certainty up to Hp=16000,
when they were so swift as to be able to pass through more
than 1 em. of aluminium. Readings at this point were
inconvenient on account of the high currents in the magnet-
coils necessary to produce the required field.

Fig. 4.

ALUMINI

>)
(e)

IONIZATION

p
o_

6 : : “K2
GRAMS PER CM2

By producing the linear part of curve CC (fig. 2) to cut
the axis of D, we get the thickness of matter, OD, which
would be traversed by the 8 particles if the law of absorption
had continued as in the linear part of the curve. The
distance OD might, on analogy of the Bragg «-ray curves,
be called the “range” of the 8 particle in aluminium. It
gives the thickness of aluminium in which most of the
B particles are stopped. Such a quantity may prove useful
in some experiments in deducing the velocity of § rays
from absorption curves. A curve showing the range of
8 rays in aluminium at various velocities is given in
fig. 3.

730 Mr. R. W. Varder on the

In discussing this question with Dr. Bohr, he informed me
that he had deduced from theoretical considerations that the

ro) N

RANGE IN ALUMINIUM GRAM CM2
: =

12,000

“4000 6000 8000
HP IN GAUSS CM.

loss of energy (dT) of a B particle in going through a
thickness of matter (da) should be given by

Dey AK
oT ae
where i moer((1 = 3?) 7 — 1)

is the kinetic energy of the particle.
B=ratio of velocity of 8 particle to the velocity of light,
K is a function of the velocity which increases slowly
with increasing velocity. If we assume K is constant, and

——

Absorption of Homogeneous B Rays. 73

integrate to find the thickness of matter R in which the-
kinetic energy is destroyed, we get

R B 1 ‘
{ ia={ — K £7dT
0 0
C 21

: 7
={"-Seeaia—2y},

c7Mo 1 ae
R= LU?) + (1-8) *— 2]
ee
= A,
where: A=[(1—6?)?+(1—6")?—2].

In the following table in column I. is given the value of

Hp in gauss cm. In column II. the value of @ deduced
from the relativity formula

Slay

Mo
; m ; ee
and =~; (c=velccity of light).
— =1:772 x10! e.m. units.
Mo
if II. Til. IV. V. Alea
Ho. (gy. T/moc*. A. R(obs.).| B/A. |
1380 632 "290 0651 | (018) GSO TI
1930 | -752 516 | ‘176 | 064 | 369 |
25385 831 “799 "O00 "124 ‘oto 0) |
3170 "882 1121 584 "189 "325
3790 9129 1-456 863 "279 "3238
4400 9331 1°782 1142 *360 “315
5026 ‘9476 2151 145 440 “304
6230 *9650 2°810 20% -580 “P80
7490 "9753 3°531 275 1 689 "285
8590 ‘9811 4-165 3369 =| 925 275
11370 A) 5:30 1:36 "26

The values of R(obs.) are expressed in gm. per sq. em.

In column ITI. the value of T/mpc? is given. In column LY.
the value of A. In column V. the value of R, the range ob-
served from experiments, and in column VL, R (obs, JA.

132 Mr. R. W. Varder on the
aCe ought to be equal to ¢?m/K. With the exception

R(obs.)

of the first value, we see that decreases slowly with

increasing velocity, as should be expected from theory since
K increases slowly with increasing velocity.

The fact that the aluminium absorption curve is approxi-
mately linear must result from a chance balancing of the
opposing effects of scattering and diminution of velocity. If
we had no scattering or straggling we should expect to have
a curve similar in shape to a Bragg a-ray curve.

For a substance like paper, which contains only elements
of low atomic weight, the effect of scattering is less important
than for aluminium, so that a greater fraction of 8 rays will
penetrate a given thickness of matter. This is seen from the
fact that the absorption curve for filter-paper (fig. 6) is

Fig. 6.

PAPER
ALUMINIUM
TIN
PLATINUM

IONIZATION

: “ ght Vee A BE ah Beis a
THICKNESS IN GMS. PER CM2

concave to the origin. For substances, like tin and platinum,
of high atomic weight fewer 6 particles penetrate a given
thickness, and the curves are convex towards the origin

(fig. 6).

Absorption of Homogeneous B Rays. 733

W. Wilson * has discussed the question of absorption of
swift 8 rays by aluminium in two papers. ‘The results given
in his first communication agree approximately with those
given in the present paper. He examined this question again
in asecond investigation, in which he believed that he obtained
more homogeneous @ rays. In this paper he finds that the
absorption curve for fast rays in aluminium rises to a
maximum for a small thickness of matter and then decreases
in an approximately linear relation. His method of measuring
the ionization due to the @ rays was to subtract the reading
of the electroscope with a thick screen from the reading with
the screen in which the absorption of the 8 rays was ‘under
examination. Thus he assumes that the y radiation produces
the same quantity of secondary @ radiation in a thick plate
as in a thin plate. In general this is not the case. ‘The
conditions of his experiment are not sufficiently ciear to
form an idea of the magnitude of this correction ; for it will
depend on what material he placed in front of his ionization
vessel (behind the aluminium screen). In my experiments,
with celluloid in this position, the secondary @-ray effect
was greater with a thick aluminium plate than with a thin
one, as is shown in curve BB (fig. 2). If this were the case
in Wilson’s experiment, the rise of his absorption curve to
a maximum receives a simple explanation. ‘This is rendered
the more probable since he mentions that the y-ray ettect
was relatively strong, thus giving a relatively great import-
ance to the secondary 8 rays from the absorbing screens. I
have tried to obtain his results by narrowing the slits to
produce greater purity of rays, but have been unsuccessful.
In Wilson’s experiment the air was not exhausted from the
apparatus. To test the effect of this, readings were taken
with air in the box B (fig. 1), but the only obvious effect
was a diminution in intensity of the @ radiation.

In conclusion I wish to thank Prof. Sir Ernest Ruther-
ford for suggesting this research and for many valuable
suggestions “during its progress. I also wish to thank
Dr. N. Bohr for his great interest and help in connexion with

the theory of absorption of B particles.
The Physics Laboratory,

Victoria University of Manchester,
March 4, 19165.

* W. Wilson, Proc. Roy. Soc. A, lxxxii. p, 612 (1909) ; Ixxxvii. p.310
(1912).

pocesae 4

LXXVII. Photo-electric Constant and Atomic Heat. By
T. CarLtTon Surron, B.Sc., Government Research Scholar
in the University of Melbourne”.

r§.HE photo-electric constant, /, as used in this paper
is defined (Jeans +) according to the relation

kv = 4mv? + wo,

where dmv’ is the kinetic energy of an electron driven out
of a metal by a radiant beam of wave-length vy, and wy
(a constant for any particular metal) is the energy required
to move the electron from within the atom to a point outside
the sphere of action of the atom.

Hence, kv is the total energy required to expel an electron
from the atom with velocity v.

The figures given in the accompanying table (column 6)
show that the values of & do not vary greatly from that of
one quantum, namely 6°6x10~*" erg; though there are
deviations (the values are invariably low) which have been
said to depend in some way on the atomic volume f.

Some such idea as an indivisible unit or ‘“‘ quantum” of
energy is suggested by the fact that the atomic heats of all
-elements have approximately the same value ; that is to say,
it requires the same amount of heat-energy to change the
temperature of any atom a given amount, irrespective of
the nature of that atom. ;

Here, again, there are deviations from the mean value
64.

When the divergences of the atomic heats are compared
with those of the photo-electric constants, a remarkable
regularity is noticed. Thus, when the atomic heat is high,
the photo-electric constant is low, and vice versa, except in
the case of tin and of certain elements of low atomic weight
(magnesium, aluminium, copper), which show an ano-
malously large change of atomic heat with temperature.
In these exceptional cases, both values are low at ordinary
temperatures, but as the temperature rises the atomic heat
increases and the said regularities appear (see Table).

That this relation holds good is shown in the last column
of the table, where the product has been taken of the atomic
heat and the photo-electric constant, and has been found to
give in almost every case a value close to 35:5.

* Communicated by Prof. T. R. Lyle, F.R.S.

{ Jeans, “ Report on Radiation and the Quantum Theory,” p. 59.
{ Hughes, Phil. Trans. 1912.

735

Photo-electric Constant and Atomic Heat.

|
Weieht Specific Heat. Observer. ae eae a | Observer. (AL. ioe ae 107.) |
a | eeemee ae |
@ngmitm cel 1104 0-055 | Voigt (1893). 6-18 5-67x10-27 | Hughes. 35-1 |
TAN DE a ee Se 65:4 0:93 Various. 6°08 Diteley = iy - | 357 |
MOA ese ce = OTE 0-031 | Behn (1898). 6-41 550 ,, z | 862 |
WalGiumiteccresecnes: 40:1 0-180 | Bunsen. 7°22 A Oley, S ES 4
HIS MU th eeretnen sw eeecs|e- 208; 0 0030 | Voigt (1893). 6°24 iGo, 4 B52
Antimony ............, 120-2 0:0508 | Gaede (1902), 6-20 ee) . | 855
Mesenie-.a gases.) 750 0-083 | Bettendorf & eh ee : | 855
Wullner.
SodiUnrearertewst ates: 23:0 0:297 | Bernini (1906). 6°83 ie eas aa One 355
Platinum 2k 1952 00316 | Gaede (1902). 617 585, = | 96-1
rere Ore (| 200 G, Oot Vey PA oes, Hughes. |] 3598 at 295° O.
Bice | or | RC bed. | ee) i kee
Gopiere oe. 63°57 4 at 20° C. 0-094 | Various. 5°98 } 3:8 | { ASD
at 900° C.0:126 | Richards (1893). | 8°01 ” | 30°4 at 900° O. |
Mi tee eee ches LOO 0:055 | Various. 6-55 49, ; Sera

Hughes, Phil. Trans, 1912, Richardson and Compton, Phil. Mag. vol. xxiv. (1912)

736 Photo-electriec Constant and Atomic Heat.

The physical interpretation of this relation is difficult at
present. It would seem, however, that |

(1) Both phenomena are due to the “ultimate discon-
tinuity ” of energy as postulated in the quantum
theory ;

(2) When a limited number of quanta are absorbed by a
molecule, the various properties of the molecule are
not necessarily affected to the same extent—as
some quanta may function in one way, some in
another ;

(3) For any particular metal, the proportion that affects
the atomic heat is related to the proportion that
affects the photo-electric constant; and if the
nature of the metal is changed, this relation is one
of inverse proportion.

The work of Nernst and Debye* shows that the atomic
heat at constant volume is a more reliable physical constant
than the atomic heat at constant pressure (the quantity
usually measured). The atomic heats at constant volume all
tend asymptotically to the same limiting high-temperature
value, 5°95; whereas the atomic heats at constant pressure
approach values which are different for different elements.
The magnitude of the difference between the values of the two
atomic heats is dependent on the amount of work required
to compress the heated solid block to the volume it occupied
when unheated ; that is to lessen the amplitude of swing of
the minute vibrating systems.

Consequently, the results given in this paper suggest that
the divergences of the values of the photo-electric constant
from the value of one quantum, 6°6 x 10~*, are due to the
work done in changing the amplitude of swing of the newly-
charged atoms left within the metal.

My thanks are due to Mr. I. O. Masson for kindly
correcting the proois.

The University, Melbourne.

* Nernst and Lindemann, Zeitsch. f. Elektrochemie, 1911, p. 817;
Debye, Ann. der Physik, 1912, p. 789.

THE
LONDON, EDINBURGH, ann DUBLIN

PHILOSOPHICAL MAGAZINE

AND

JOURNAL OF SCIENCE.

ee
(

[SIXTH SERIES.]

SO NE NO ia.

LAXVIIT. Ona Tidal Problem.
By Prof. H. Lams and Miss L. Swain*.

(ae object of this note is to illustrate the theory of the

tides in a very simple case, viz., that of an equatorial
canal of finite length, the tide-generating body (say the
moon) being supposed to revolve uniformly in the plane of
the equator. Simple as the question is, the results are hardly
intelligible without detailed numerical or graphical inter-
pretation. Moreover, the problem is at present almost the
only one which can be used to exemplify a point of some
importance in tidal theory.

On Laplace’s dynamical theory, as on the equilibrium
theory, there is necessarily exact agreement (or exact oppo-
sition) of phase between the tidal elevation and the forces
which generate it, in the case of an ocean covering the globe
or bounded by parallels of latitude, the depth being supposed
either uniform or a function of latitude only. The con-
spicuous and varied differences of phase which are observed
were accounted for in a general way by Newton J, as due to
the inertia of the water combined with the irregular con-
fiouration of the actual oceans. On the other hand, Airy, in
his ‘ Tides and Waves’ (1845), attached great importance to
the action of friction, and appears to have regarded the
phase-differences in question as attributable mainly to this
cause. This view seems to have met with wide acceptance,

* Communicated by the Authors.
+ Principia, lib. i,, prop. xxiv.

Phil. Mag. 8. 6. Vol. 29. No. 174. June 1915. e1

“

_—

_ _——— eee

738 Prof. H. Lamb and Miss L. Swain on

favoured no doubt by the prominent part which was being
assigned to tidal friction in various cosmical theories.
Qualitatively there is of course nothing to be said against
Airy’s explanation. The chief example ‘considered by him,
viz., that of an equatorial canal encircling the globe, is
merely a particular case of the now familiar theory of forced
oscillations with damping. If ¢ bea normal coordinate of
a dynamical system we have, on the simplest assumption as

to the nature of the dissipative forces, an equation of the type

d+igtup=o. .....

For the free oscillations
o= Ae! cos(ct-+¢), . 2.) ees

where T=2/k, o=,/(p—ih’) ss. . ee
whilst if P= C cos pe 6.) ee

we have the forced oscillation

C 3
$= pc0s( pt—B) ee

provided
Rceos8=p—p”, Resin 6=kp.. 2 Sa

There is here a retardation of phase given by

tan B= aes Hole bleed Me tl)
pop pt(p/p’—1)

In the tidal problem p= 2n, where n is the moon’s angular
velocity relative to the rotating earth.

The question remains, however, whether the frictional
forces which are operative are sufficient to account for i
observed differences. It was pointed out by Helmholtz *
1888 that the influence of viscosity on large-scale motions of
the atmosphere must be absolutely insignificant, and it was

easy to inter that the same conclusion must hold @ fortiori as
regards tidal oscillations of water t, where the kinematic
viscosity is much less. ‘This point was afterwards full
developed by Hough ¢, who showed in particular that with
even so moder: ate a depth as 200 metres the modulus of
decay (7) of free tidal motions of semidiurnal type would
be at least three years. It may indeed be urged that in
places where the tides are greatly exaggerated, as in narrow
channels and estuaries, there may be turbulent motions with
a local dissipation of energy far exceeding what takes place

® Berl. Sitzb., May 31, 1888; Wiss. Abh. Bd. iii. p. 292.

ye: Hydrodynamics, 2nd ed. (1895) p-. 045.

f Proc. Lond. Math. Soe. vol. xxviii. p. 287 (1896).

a Tidal Problem. 739

in the smooth regular motions postulated in the calculations
referred to, and it is possible that such tidal retardation of
the earth’s rotation as is taking place under present condi-
tions may be mainly due to this cause. It may fairly be
assumed, however, as a matter of physical intuition, apart
from calculation, that the damping of free oscillations of the
ocean of semidiurnal type would hardly be sensible until
after the lapse of a considerable number of periods.

If this be granted, it follows from (7) that the phase-differ-
ence produced by friction in an endless equatorial canal would
be insignificant. With such depths as occur in the ocean p
is considerably less than p”, and tan @ is therefore comparable
in absolute value with ‘I'/r, where T( =27/p) is the period of
the forced oscillation. The modulus of decay being assumed
to be large compared with 12 hours, 6 must differ 1 very little
from 180°. A phase-difference of 90°, such as is postulated
in some numerical illustrations of the theory of tidal friction,
could only arise exceptionally, by ‘‘ resonance,” in the case
of finite areas of water having a free period very closely in
accordance with the forced period.

It seems clear that the influence of friction on ordinary
tidal phenomena is unimportant. It was pointed out by
Hough in the paper referred to that phase-differences must
arise in another way, from the causes indicated by Newton,
in limited canals or oceans*. He remarks also that an
example in illustration of this is furnished by the problem of
the finite canal which had been treated, but not fully
examined, by Airy himself.

Moreover, it should not be overlooked that a mere equi-
librium theory, when ‘‘ corrected” on the principles explained
by Thomson and Tait, would also give differences of phase f.
Consider for example the case of a canal a few degrees
in length lying along the equator. W hen the moon (or
antimoon) i is in the zenith the differential changes of level
are everywhere slight, the disturbing force being nearly
vertical and uniform. When the moon is on the horizon,
the changes are again slight, since moon and antimoon now
nearly counteract one another as regards the horizontal foree.
Hence at the ends of the canal there will be high or Jow
water for some intermediate position; the theory shows in
fact that the corresponding hour-angle is 45°. At the centre
the range is comparatively small, and high water coincides
with the moon’s (or antimoon’s) transit.

* In seas whose breadth as well as length has to be taken into account
the question is further complicated by the “ gyrostatic”’ effect of the
earth’s rotation.

{ Thomson and Tait, Art. 810; ‘ Hydrodynamics,’ 3rd. ed, p, 541.

o De

740 Prof. H. Lamb and Miss L. Swain on

The calculations which follow will serve to illustrate the
foregoing remarks. The formule were worked out originally )
in response to an inquiry addressed to one of the writers from

abroad, and in ignorance, or more probably in forgetfulness
of the fact that the matter had already been treated to some
extent by Airy, and referred to by Hough.

We consider the case of an equatorial canal of uniform
depth /, the moon being supposed to revolve in the plane of
the equator. If @ denote longitude measured eastwards
from a fixed meridian, and nt the hour-angle of the moon
west of this meridian, the dynamical equation is of the form

¢ Bie
OF cor sin 2(nt4+-6),
where a is the earth’s radius, c’=gh, and & denotes hori-
zontal displacement eastwards*.

For an equilibrium theory we neglect the term 97&/0??.
If the origin of @ be taken at the centre of the canal, we find

6 e .
a We eee { sin 2nt cos 22+ - cos 2nt sin242—sin 2(nt + 6) } :
a

(9)
for this expression satisfies the differential equation, and
makes €=0 at the ends (@=+2). For the surface-elevation

we have
nee! ty J cos 2(nt + 0)—- me 0 AF 00s ant}

. 2
where H=/jag. This quantity H measures, on the sani
brium theory , the maximum range of the tide in the case of
an ocean covering the whole earth f.

At the centre (9 =0) we have

in 2
n= 5 Heos ne(1— | iia

If a be small, the range here is very small, but there is not
a node in aie. strict sense of the term. The times of high
water coincide with the transits of the moon and antimoon.

At the ends (@= +2), we find

= n= 3 H{( 1— a =) cos Q(nt-ta)= 2 —— =e sin 2 2(nt-ta) b
Mime Dao

sin 4a : 1— cos 4e
Ro cos 2¢9=1— PR, R, sin 2¢)>= — =e (13)
* ¢ Hydrodynamies, Art. 180.
+ ‘Hydrodynamics,’ Art. 179.

a Tidal Problem. 741

Here R, is the ratio of the range of the tide to the quantity
EL, anh dy) denotes the hour-angle of the moon W. of the
meridian when there is high water at the eastern end of the
canal; itis also the hour-angle K. of the meridian when
there is high water at the western end*. When a is small
we have

2
p= 2a, o=—yrt Ones A ° ° (14)

approximately.
The values of Ry and ¢o for a series of values of ranging

from 0 top m are given in the table at the end of this

paper, on the assumption that h=10820 feet.
When the inertia of the water is taken into account,
we have

&= Ls 2 [sin 2(nt + 0) — Ne { sin 2(nt + «) sin 2m(O + «)
A (m1) 0? | sin 4me

— sin 2(nt— a) sin 2n(@—a) LI], ot NCES

where ages For this satisfies (8), and vanishes for

O=+at. Hence

Deoe
mn Ae
ee | |
2 n= | eee 20 EO) panier 2(nt +a) cos 2m(O + a)

— sin 2(nt—«) cos 2m(a—a)\]. (16)

An a alent form is

1 = [ oe
= cos 2(nt + 8) — ——— < cos 2(nt+m@) sin 2(m+ 1)
a an z sin me ( (
=
— cos 2(nt—mé@) sin 2(m—1'a b Ix CES)
oer
If we imagine m to tend to the limit 0 we obtain the
formula (12). of the equilibrium theory. It may be noticed
that the expressions do not become infinite for m—>1, as in
the case of a canal encircling the globe. In all cases, how-
ever, which are at all comparable with oceanic conditions, m
1s considerably ¢ greater than unity.

* The phase-difference is 26). This angle, reckoned in degrees from
0 to 860°, is called by Darwin and Baird the “ lax” of the tide, Proc.
Roy. Soe. vol. xxxix. p. 185 (188) D).

Tt Cf. Airy, ‘ Tides and Waves,’ Art. 301.

T42 Prof. H. Lamb and Miss L. Swain on

At the middle point of the canal we have

A ne m sin 2e
gee REY eee Soi” 2 = |) ee
4 aan 2nt( 1 sin a) (i
As in the equilibrium theory, the range is very small if @
be small, but there is not a true node.
At the ends d=-+« we find

ik (@ sin 4a
2(nt
Uy 2 m?—I “a1 sin4dmaz 1) coer

ae (cos dna — cone) a 2(nt say lh. » (19)

sin 4ma

| er taeeeeae - irene

if msin4a—sin4me )

Ry cos 2¢,= (m?—1) sin4dma ’

(21)

m(cos 4ma— cos 4a)
(m?—1)sin 4ma) |

R, sin 2, =

The significance of the quantities R,, d, is the same as in
the equilibrium formula (13) *. When @ is small we have

if 2
eee d= — -7+ 4

+ 8) a

approximately, as before.

The values of R, become infinite for sin4dm2z=0. This
determines the critical lengths of the canal fer which there
is a free period equal to “at/n, or half a lunar day. The
limiting value of d, in such a case is given by

m(cos 4ma — cos 4a)

tan 2
ee m sin 4a— sin 4ma

= — cot 2a, or tan 2a, . (23)

according as 4mze is an odd or even multiple of z.

For purposes of numerical illustration we have taken
m=2°5, If w/n=12 lunar hours, this implies a depth of
10820 feet, which is of the same order of magnitude as the
mean depth of the ocean. The corresponding 5 wave-velocity
¢ is 360 sea miles per lunar hour. The first critical length

is 2160 miles («= i r), and the second is 4820 miles.

* See the footnote on p. 741.

a Tidal Problem. 743

The table gives results for a series of lengths varying from
0 to 5400 miles. The unit in terms of which the range is
expressed is the quantity H, whose value for the lunar tide
is about 1°80 ft. The hour-angles dy and ¢; have been
adjusted so as to lie always between +90°, and the positive
sign denotes position W. of the meridian in the case of the
eastern end of the canal, and H. of the meridian in the case
of the western end.

The diagrams show successive forms of the wave-profile on
the dynamical theory in the case of 2«=18°, corresponding
to a length of 1080 miles. In fig. 1 the curve a corresponds

Fig. 1.

eo >

to the instant when the moon (or antimoon) is over the centre
of the canal, and the following curves b, c, d, e, 7 represent
the profile at intervals of one-twentieth of a period, or 36
minutes. Only one quarter of a complete cycle is shown ;
the remaining curves might be obtained by reflexions with
respect to vertical and horizontal axes through the centre.

—— ee ee eee ee

ee eee

TAA Ona Tidal Problem.

In fig. 2 the neighbourhood of the centre is represented on
a larger scale, with a view to showing how the phase-
difference rapidly varies from 0 at the centre towards the
value 77° which obtains at the ends,

Fig. 2.

|
EQuiLiprium THEory. | DynamicaL THmory.
On aa bee ee ap | se ee a,
(degrees). | (miles). Pee aN (degrees). eeneeoleenant (degrees).
0 0 0 0 AS 0 O —45
igs) 270 ‘O01 ‘079 OHO) ‘001 | ‘O80 —J43'5
9 540 ‘004 157 — 42 004 | ‘165 —419
13:°5 810 009 234 0-5) 7010 | :266 AOD
18 1080 016 eoilell = bY) 7018 | :3896 = Ono)
29-5 1350 “025 "2.86 GHG 029 | -588 = Boy
Ca 1620 037 ‘460 ge) 044 | -941 oa
p15) 1890 "050 Holl —34O 063 | 1°945 Si)
36 2160 || 065 | -e01| —33 089 | { me
40°5 2430 ‘081 "668 = 31:6 "125 | 1:°956 165°2
45 2700 ‘100 oe —380°1 174 | -987 BS
49°85 2970 120 "795 — 28:7 945 7 “711 + 85:3
54 3240 "142 *853 = BPD °354 | ‘660 —839
58'S 3510 "165 ‘908 = 2S 540 | -780 i
63 3780 "190 *O59 —244 918 | 1141 —65°1
Giro 4050 "216 | 1:007 == 2S 2-067 | 2:294 — 689
2 4320 || -243 |7-051| —2r6 || «o | © { a
76°5 4590 272 | O91 —20:2 2:564 | 2°302 + 40°3
81 4860 Ol Wels 7 — 18:9 1-459 | 1:112 +445
85'5 5130 cone) \atss LAS 1035 | :715 +49°4
90 5400 ‘300 | 1°185 Ore °864 | °513 +559

ei el

LXXIX. On Topic Parameters and Morphotropic Rela-
tionships. By Wituram Bartow, /.R.S., and WILLIAM
JACKSON Port, F.R.S.*

| eas parameters were introduced by Becke (Anz. d.

Kais. Akad. d. Wiss., Wien, xxx. 1893, 204) for
defining, as between crystallographically related substances,
the lineal dimensions of the corresponding crystal structures ;
in a substance crystallizing in one of the rectangular systems,
the topic parameters, y, wy and a, are calculated as

x=/ (Ve) 3 p=ylai o=wpe,
a, b, and c being the crystallographic axial ratios and V the
molecular volume of the substance.

The topic parameters of a set of isomorphously related
substances define the actual changes in dimensions of the
point system which attend the passage from one crystalline
substance to others isomorphous with it. During a number
of years past, however, it has been customary to calculate
and record the topic parameters of series of substances the
members of which exhibit any chemical or morphotropic
relationship ; so far as we are aware, no fact or conclusion
of importance has resulted from the application of the topic
parameters to cases of morphotropic relationship as distinct
from those of isomorphism.

That topic parameters, as hitherto applied to the quanti-
tative description of morphotropy, are without physical
significance is well illustrated by a consideration of the
particular case generally chosen in explanation of the subject.
The instance in question refers to the morphotropic relation-
ship between ammonium iodide and its tetramethyl, tetra-
ethyl, and tetrapropyl derivatives, and derives authority
from the important position assigned to it in a number of
our most valued text-books (Groth, ‘ Hinleitung in die
chemische Krystal)ographie,’ 1904. p. 32; Groth, ‘ Chemische
Krystallographie,’ i. 1906, p. 171; Nernst, ‘ Theoretische
Chemie,’ Siebente Aufl., 1913, p. 373; Roscoe and Schor-
lemmer, ‘ Treatise on Chemistry,’ vol. ii. 1907, p. 220; Barker,
in same, vol. ii. 1913, p. 220); more recently Groth has again
referred to the importance of this example (Zeits. f. Aryst.
liv. 1914, p. 68). The data were collected by Slavik (Zeits.
f. Kryst. xxxvi. 1902, p. 268), and his interpretation of the
meaning of the topic parameters was confirmed and amplified
by Wagner (Zerts. f. Kryst. xliii. 1907, p. 148). Slavik

* Communicated by the Authors.

746 Mr. W. Barlow and Prof. W. J. Pope on

determined the following axial ratios, &c., in connexion with
the subject under discussion.

Crystal System. Wi: Axial Ratios.
su] 8 by Bap Cubie. Stead
N(CH3),1 ... Tetragonal. 108-70 a : c=1 : 0-7223.
N(C.H;),1 ... Tetragonal. 162°91 @a@:c=1 : 0°5544.
N(C;H;,),1 ... Rhombic. 235°95 a :6:c=0°7761 : 12 @ieaee

The following table, giving the topic parameters calculated
from the above data and stating the changes which they expe-
rience throughout the series, is taken from Nernst (/oe. cit.).

| |

fe engine cA tds GA TR, ON Micra (mA eal Nn ete | A NPr,I.
| | | an oe aah ——

| Vie. S751 | 5119 | 10870) 5421 | 162-91) 73:04 | 235-95
| yes 3860 | 1459 | 5319 | 1329 | 6648 —0:555 | 6-093
| yee 3860) 1-459 | 5819 | 1929 | 6648 |.1103 | 7-851
} |

lets SeeNe 3860 |—0-018 | 3842 ee, 3°686 1°247 4-933
| | |

The conclusion drawn by Slavik from this table, and-
repeated in the text-books mentioned, is that on passing from
the cubic ammonium iodide to the tetragonal tetramethyl-
ammonium iodide one dimension of the crystal structure,
that of w, remains sensibly unchanged, and that the main
increase in dimensions of the crystal structure occurs in the
two directions of x andy. Similarly, in passing from the
tetramethyl- to the tetraethyl-ammonium iodide, the
dimension » again changes but little and, as before, the chief
increase in dimensions of the structure occurs in yx and wp.

This interpretation, which has been so widely adopted as
illustrating the value of the topic parameters in elucidating
morphotropy, appears to us entirely erroneous for the
following reasons :—The method adopted for the description
of the tetragonal tetramethylammonium iodide gives to the
chief forms present the indices {100$ and {111}; the
descriptions given by Slavik and by Wagner show that
the salt crystallizes in combinations of {100} and {111},
resembling rhombic dodecahedra of the cubic system in that
the angle ‘110: 111=44° 23/30". Ibis clear, therefore, that
in accordance with edorow’s practice of assigning the ‘most
simple set of indices to the forms exhibited, the indices
{100} and {111} should be changed to 110} and {101}
respectively ; when this change is made the axial ratio
becomes

a:c=l1: cotan. 44° 23’ 50”=1 : 1:0214.

Topic Parameters and Morphotropie Relationships. 747

This mode of interpreting the goniometrical data, which is
a more rational one than that applied by Slavik and Wagner,
shows that the salt is markedly pseudo-cubic. So that in
passing from the cubic ammonium iodide to the pseudo-
cubic tetragonal tetramethylammonium iodide, the three
equal topic parameters of the first, V5T5=3°9, change
almost equally to approximately 7 108°7=4°8. This necessary
alteration in the table of topic parameters reproduced
above entirely destroys the sequence as between ammonium
iodide, tetramethylammonium iodide, and_ tetraethylam-
monium iodide; the table in question is thus valueless as an
illustration of the use of these parameters in exhibiting
morphotropic relationships.

A broad view of the morphotropic relationships holding
between the ammonium halides and their tetraalkyl derivatives
is obtained by considering the axial ratios and crystal systems
presented by a number of related compounds which exhibit
high crystalline symmetry. .

TSC Ree ne ae Cubic.

INJGUID 5 aso Bs

(CHa WINC o.c.. Letraconalia nance li Ort On:
(OER WIN Br.) 00... i aries OT 13h
(Cla). Cie aae a Qe C= Oni 22o%
(Clea Sle ae Hexagonal,” a: c= y 1-422.
ON Tetragonal, a: c=1 :0°5544.
(C5 Jet dag Genera Trigonal. Qe 1s 124719.
(Glee NBr \ 00... x ae: C= e406.
CGE VEEN Ts... Lemagonalky)) a? cb 1467
(CoH;),;HNCI...... Hexagonal. ai e— li O-otoL.
(Coie). EUN Br, .:. an Gs G= 1S 0:87 40.
(Cpe a CH. INit. . \Tetragonal.),\a :¢=1+: 0:5536.

The table shows that on replacing the metallic atom in
potassium iodide by the ammonium radicle, N Hy, the system
retains its cubic symmetry ; on replacing each of the four
hydrogen atoms in the ammonium halide by methyl, pseudo-
cubic symmetry results in the manner described in detail
above. The general applicability of the principle thus stated
is demonstrated by the similar axial ratios exhibited by
tetramethylammonium chloride, bromide, and iodide; simi-
larly, on replacing one of the hydrogen atoms in the
ammonium iodide molecule by a methyl group, so as to

748 Mr. W. Barlow and Prof. W. J. Pope on

produce the tetragonal methylammonium iodide, the axial
ratio, c/a = 1:467, indicates that pseudo-cubic symmetry
again results, this value of c/a being comparable to that of
2c/a in tetramethylammonium iodide.

On substituting an atom of antimony for one of nitrogen
in tetramethylammonium iodide, so as to obtain tetramethy]-
stibonium iodide, (CH;),SbI, the tetragonal symmetry
changes to hexagonal with the axial ratio, a : c=1 : 1°422.
An hexagonal system showing an axial ratio of this order
of magnitude is closely related to the cubic system for the
following reasons :—If a cube diagonal, which is a trigonal
axis of symmetry, is taken as the vertical axis-c of a system
of crystallographic coordinates of a trigonal or hexagonal
kind, one of the two obvious horizontal translations is twice
the distance between a cube corner and the centre of an
opposite cube face; taking the cube edge as of unit length,
the diagonal is 4/3 and the distance from the cube corner
to the centre of an opposing face is (3/2). Theaxial ratio
corresponding to this mode of referring a cube to a set of
trigonal coordinates is, therefore, a : c= 4/(3/2):V73=1:
14142. The trigonal or hexagonal tetramethylstibonium
iodide, tetraethyl-phosphonium iodide and -ammonium
bromide obviously approximate closely to these dimensions.

The hexagonal axial ratio alternative to the value,
az3c=1 : 14142, isa: c=1 : 1°6330 (Trans. Cheme See
xcl. 1907, p. 1157) and the values recorded for triethyl-
ammonium chloride and bromide are rather more than
one-half of this, namely, a : c=1 : 0°8165; these two sub-
stances may therefore also be regarded as morphologically
related in a simple manner to the cubic ammonium halides.

The axial ratio, a : c=1 : 0°5544, of the tetragonal
tetraethyl-ammonium iodide is also, though somewhat less
distinctly, that of a pseudo-cubic substance, because on
multiplying unit length along the axis-c by two, it assumes
the pseudo-cubice form, a: c=1:1:1088. Ample justifi-
cation can be given tor altering this axial ratio in this
manner ; this substance is of the same type as the hexagonal
tetramethylstibonium iodide and tetraethylammonium bro-
mide which have already been shown to be morphologically
closely related to the cubic system.

A comparison of the measured angles on tetraethyl-
ammonium iodide with principal measured angles on the

Topie Parameters and Morphotropic Relationsups. 749

other two compounds mentioned shows that a very complete
correspondence exists.
(C,H,),NBr. (O,H,),NI. (C,H,),PI. (C,H,),(CH,)NI.
Snipe tes 52° 52) 11 P11 O—51° 54” 31 N18 152° 55’ 111711 t=5le56'
Lei GC Wa ele OO Ye eee i 7s
ite G2 54 1112100 6L 8-310: 111 63 2

It thus appears that the basal plane (111) on the trigonal
compounds corresponds to the pinacoid (001) on the tetra-
gonal one, the rhombohedron (100) and the pyramid (311)
on the former also corresponds to the pyramid (111) and the
prism (110) on the latter. The tetragonal tetraethyl-
ammonium iodide is therefore morphotropically closely
related with the two trigonal substances, the dimensions of
which have been shown to be pseudo-cubic.

The values for the tetragonal methyltriethylammonium
iodide, with a : c=1 : 0°5536, are included in the above
table because, for the reasons just quoted, this substance
must also be referred to the cubic system.

Without at present entering into detail as to the manner
in which the mode of interpreting morphotrophic relation-
ships which we have previously developed (Trans. Chem.
boc ocx O00). 16is 5 xev L907, p.11l50; xen 1908,
p- 1528 ; xevii. 1910, p. 2308) is applicable to the present
case, we may regard the following conclusion as definitely
established. The method of representing the morphotropic
relationship which exists between ammonium, tetramethyl-
ammonium, tetraethylammonium, and tetrapropylammonium
iodide advocated by Slavik and Wagner, and generally
adopted in the text-book literature, has been proved to be
incorrect and should be abandoned.

The fact that so many of the alkyl derivatives of the
ammonium halides affect types of crystalline symmetry
and axial ratios closely related to the cubic system is
important ; it seems to be generically connected with the
fact that the ammonia in aluminium ammonium alum can
be replaced by hydroxylamine, methylamine, ethylamine,
and trimethylamine, with retention of the original cubic
symmetry.

The Chemical Laboratory,
The University, Cambridge.

ee Dia

LXXX. Construction of Cubic Crystals with Theoretical
Atoms. By Aupert C. Crenore, Ph.D.* (From the
Department of Physiology of Columbia University.)

| i{ Plate XI.]

ie a former paper upon this subject} there was developed

a general expression for the instantaneous mechanical
force which one electron revolving uniformly in a circular
orbit exerts upon a second electron revolving in a different
circular orbit. This is based upon the well-known equations
for the mechanical force that one moving electrical charge
exerts upon another; but the velocity of transmission is
taken as infinite to simplify matters, and reasons were given
why it seems probable that the results obtained with these
simpler initial equations would not be different if the more
complicated general expressions were employed. ‘The
mechanical force between two rings of electrons is shown
to be independent of the relative phase angles, and any re-
tardation of the transmission velocity would be likely to
affect only these phase angles, and not change the mechanical
force. :

In that paper the integral of the general equation was
obtained in one special case only, namely, when the axes of
revolution of the two electrons are parallel to each other and
the angular velocities of orbital revolution are identical ft.
The work of integrating for average velocities has now been
completed in the most general case, when the two axes of
revolution make any angle « with each other, and the
frequencies of revolution are either incommensurable or
equal to each other.

The General Equation.

Using the same notation as in the paper referred to, the
total instantaneous mechanical force is expressed as the sum
of four vector components. The first or electrostatic com-

ponent is
I

F, = KR»: mie Grea y DOr aS (1) :
Here e and e! denote the two charges, each having the same
sion, K the specific inductive capacity of the medium, and
R the scalar instantaneous distance between the two charges.

* Communicated by the Author.

t+ A. C. Crehore, Phil. Mag. July 1913, p. 25.

{ Loc. crt. Equations (42) and (44). Note an omission in the co-
efhcients B, corrected in a note, page 325, Phil. Mag. Feb. 1915.

Construction of Cubie Crystals with Theoretical Atoms, 751

R is the vector from the charge e to e’ varying with the time,
and F, is the first component force which the second charge
e' exerts upon e. Using the abbreviations

S=sin (of +0); S’=sin (wit+ 0); C=cos (wt + 8);
C’=cos (wt+6'), . . (2)

where » and o’ are the angular velocities of the charges and
6 and @' their phase angles respectively, referred to fixed
rectangular axes, it follows that

R=(xe#—aS)it(y—aCt+a/C)j+2k+a'8', . (3)

where 2, j, and f, 2’, 4’, and k’ form two systems of rect-
angular axes, referred respectively to the centres of the
orbits of e and e’. & and #’ take the directions of the axes
of revolution of the electrons, each being clockwise when
observed from the positive side or pole. 7 and 7’ each take
the direction of the line of intersection of the planes of the
orbits, the positive direction along each being defined by the
vector kxk'. and i! lie in the planes of the orbits respec-
tively in such ee as to make the two systems of axes
each have the conventional cyclic order 1,7, k and 1’, 7’, k’,
in the counter-clockwise rotation when viewed from the
positive side of each.

Hence Ley ees (Meet OD) raat voc AM em AS. Su((1)
where sis a constant,
Sava erat ot ey a). ee)

w, y, and z being the coordinates of the centre of the orbit
of the second charge, a and a the radii of the two orbits

respectively, and w a function of the time such that

2D
u=— ( —aaS ~ ayC+a'z8! sina+ta'yC'+a'vS! cos «
Ss? e e
—aa'SS' cos « —aa'CU"), ere 5)

If the force in (1) is resolved into three rectangular com-
ponents along the 2,7, and /& axes, which may be done by
taking the direct or dot products w vith i, j, and kin turn, we
obtain, observing that 2'.2= cos a, and 2 ie == STN OR

!
Ce QI . i—
=— pe (e—aS+a'S' cos a)i, ae, CG
=~ KR ( a 1)
/

B= — is (ye +4), . Orie et (>)

iy — fia(ete's' sina )h i Wie Ws! SUSE)

752 Dr. A. C. Crehore on the Construction of

Expressing all distances in some convenient small unit a,,
such as the radius of the orbit in the single electron atom,
instead of in centimetres, and letting m and n represent the
radii of the orbits and xy, yz, and z, the coordinates in these
units, we have
E=Ay0es Y=AxgY ys 2ZHAy2y3 A—Ma,y; O —neg
The value of s in (5) becomes
S=dy(xv2 +y2 +22 +m’ +n’),
or, denoting the radical by A-!, we have
Se Nae ° e e e e . (11)
and from (4)
R?=a;2 A(1+u)-2.). =.

Hence, substituting in (7), (8), and (9),

fh, =— = -A?(rg—mS + nS! cos «)(1+ “) 3g 5, Ueiap
! om

ee = — Kai A?(y,—mC+nC' \(l+u) 77, - . GA)
! \ 3

F, =— — A3(z,-+n8' sin «)(1+u) 77h. 0

This completes that part of the mechanical force which
arises from the first or electrostatic component. It remains
to consider the magnetic effect, which is derived from the
three component forces given by equations (2), (3), and (4)
of the former paper *.

It has been previously shown ¢ that the last two of these
terms when integrated contribute nothing to the transla-
tional force of the one charge on the other, so that the total
force will be obtained by considering, in addition to the
preceding, the force

/
Fo=+Sq.q)R. ..-. . (16)
Here the new quantities are pw, the magnetic permeability of
the medium, and g and q’, the vector velocities of the charges
respectively. Jt was shown in the former paper that

q.q’ =aa' ao’ (CC’cosa+8$8'),. . . (17)

* Loe. cit. p. 58.
1 Loc. cit. p. 63-

Cubic Crystals with Theoretical Atoms. 753
and putting w= = where ¢ is the velocity of light, we find

/ , i
ee’ aa ww
Ke?R?

Denoting the ratio of the velocities of the charges to that
of light by 8 and @’ respectively, resolving the force along
the three rectangular axes as before, and substituting the
value of R® as a function of the time, we find

Fo= +

(CCcosa+SS)R. . . (18)

By + (6% BBA (CO cos a +88)(74—mS + nf! cos 4)(1+u) Hi, (19)

Tene BAA (C! cos a + SS8'\(yy—m0 nC 4u)-27, . (20)
y

iNG@2
fF; =-+ — BB'AXCC'cosa+SS8')(z,+n8’sina)(1+u)"?k . (21)

The two sets (13)-(15) and (19)-(21) are the complete
general equations for the instantaneous values of the electro-
static and magnetic components of the mechanical force
exerted upon the charge e by the charge e', omitting the
third and fourth components as above mentioned, which
when averaged over a long period of time give zero. The
only quantities in these equations dependent upon the time
are the simple periodic functions 8, 8’, C, and C’, u being a
polynomial of seven terms each containing some of these
quantities. The expansion of (1+ u)~% into infinite series to
five terms is

1—1.5u+1. 875u? —2.1875u?+ 2.4609375ut—.... (22)

Average Values.

The process of finding the average values of these forces
over a long time T, integrating each equation with respect
to t between the limits of time 0 and T, and dividing by T,
is to multiply each term of the series (22) by the quantities
in parentheses and integrate each separately, adding the
resulting integrals. It may be shown that the series (22) is
rapidly convergent for large values of wy, yy, and zy, due
to the factor 2 in the expression for wu (6); and experience
has shown that there is no gain in including any terms
above the sixth power of the distance. For this reason the
next terms of (22) involving the fifth and higher powers of
u are not required. It is evident, however, that the number
of terms to be integrated arising from the fourth power of u

Phil. Mag. 8. 6. Vol. 29. No. 174. June 1915. auw

754 Dr. A. C. Crehore on the Construction of

alone is very large, and were it not for the fact that the
definite integrals of so many of them are zero it would be
impracticable to employ the process indicated at all. As it
is, the resulting equations which have been derived are
rather too long to publish here.

Average Translational Force of Atom on Atom.

The next process is to use these integrated equations to
derive the force that one atom exerts upon another, each
consisting of a single ring of electrons and a positive charge
equal and opposite in value to the sum of the charges of all
the electrons for a neutral atom, the centre of mass of the
positive charge being at the centre of the orbit of the electrons.
In so doing we may use the same equations for determining
the force that an electron e of the one atom exerts upon the
positive charge of the other atom by simply changing the
sign of the force because the product (—e) x (+e’) becomes
negative, whereas it was positive for two electrons, and by
making the radius of the orbit n of the positive charge equal
to zero.

We have also to consider the force that each electron e’
in the second atom exerts upon the positive charge of the
first atom by making the radius m equal to zero. Fortu-
nately, when the three sets of forces so obtained, first, the
electrons on electrons; second, the electrons of the one atom
on the positive charge of the other atom and the electrons
of the other atom upon the positive charge of the one
atom; and third, the positive charge of the one atom upon
the positive charge of the other, are added together, all the
terms involving the even powers of the radii m and n
cancel out. This drops a large number of terms and leaves only
those containing the product mn’, and gives the final result
which applies to any two atoms with certain reservations.

These special cases to which the general results do not
apply are those in which each atom has one single electron
or two electrons revolving at the same angular velocity in
each atom; for the integral equations differ when the
angular velocities are equal. Then the phase angle between
the two electrons comes into the account. This limitation,
however, is restricted to the case where the rings contain
one or two electrons. If there are three or more the phase
angle disappears in all terms up to and including the sixth
power of the distance, and we obtain precisely the same
equations for the force of atom on atom by integrating for
synchronous revolution as we get for incommensurable
velocities when the number per ring is three or more.

Cubic Crystals with Theoretical Atoms. 755

Practically the only special cases we have to study are,
therefore, where the two atoms are of the same kind, and
where they must have rings with one or two electrons.
Hydrogen on hydrogen, containing a single electron in the
atom, is one of these exceptions and it has been treated as
a special case, the equilibrium position of two hydrogen
atoms forming a molecule of hydrogen having been found,
as given ina ‘subsequent section.

The general equations for the mechanical force of the
second atom A’ upon the first atom A, with the exceptions
aed are as follows:—

merits = = mo = no iG 3X cose+:7T5Zsina—3°75X Y?cosa

m? > nil + 9-375 X

Pe
+7:5Zsin« cosa +13°125Xcos?a—13°125(X? + 5X Y?4 2XZ? sin? «
+5X*cos?a+7X?Zsine cosa+ Y*Zsina cose + X Y*cos? «)
+59°0625(+ X°Z?sin?a + X°Y?+ X°cos?a+ XY?Z?sin?a-+ X Y4

+ X°Y?cos?e +2X4Zsina cosa+?X?Y?Zsinaw cosa) | oat Deh (23)
2
F=+2 5 KG! 5 ate EmeaSnio'| + 3Y cosa—3'75 Y2cosa—3'75 XYZsin a
P P!

—3°75K2Y¥ cos a |pt— > m> n? [ + 20°625Y +1°875Y cos? «

Bic) Be
—13°125(2X?Y + 6Y°4 2YZ?sin?a2+4+4X7Y cos?2+ 6XYZsina cosa)
+59°0625(+ X°YZ?sin?« + X?Y?+ X4Y cos?a+ Y°Z?sin?« + Y°+ X?Y% cos?
+2X?YZsina cose+ 2X Y¥*Zsin« cos a) fon bi ai) Shot ai Poe (2B)

2 2
+ { + ze > mo > no! | + 15Zecosa+'75 Xsina—3°75Y*Zicosa
Ka? C Pp Pp’
—3-15XZ?sina—3'T5 XZ coset | yt— Sm? Sn? ia 56252
Bey Be
+3°75Zsin?a+75X sine cose + 1°875Zcos?a—13°125(+ X°Z
4+ 4Y°?Z.4+ Z3sin?a+3X?Zcos?a+4XZ?sina cos a+ X°Zsin?a
+ X3sina cosa+ Y*Zsin?a + XY’sina cose) +59°0625( + X°Z8sin’ x
4+. X?Y?Z4+ XZ cos?a+ Y2Z'sin?a+ Y!Z + X?Y°Zcos?a
+ 2X°Z?sina cose + 2X Y°Z?sin «cos “) | vot Fear i Re Ua
a2

756 Dr. A. C. Crehore on the Construction of

F,, F,, and F, denote the three components of the force
along the rectangular axes 7, 7, and k, referring to the atom A,
k being along the line of the axis of rotation of the electrons
so that they appear to revolve in the clockwise direction when
viewed from the positive side of the axis. j and 2 are in the
plane of its equator, the positive direction of 7, along the line
of intersection of the equatorial planes of the two atoms,
being the vector kxk’. a, is the unit in centimetres in
which all the other distances are expressed, and ¢ is the
velocity of light. The summation mo, m being the radius
and w the angular velocity of some electron in the atom A,
is to be extended to the P electrons in that atom; and
similarly =n?o' to the P’ electrons in the second atom.
X, Y, Z are the direction cosines of the position of the
second atom referred to the i, 7, and & axes at the centre
of the first atom, and a is the angle between the axes of
rotation of the two atoms. v is the distance between the
centres of the two atoms, measured in ay units.

Up to this point no particular hypothesis as to the structure
of the atom has been introduced. The equations apply as
well to the central nucleus theory of the atom as to any other
theory where electrons are moving in circular orbits about a
common centre, The test of different theories is in the
results obtained when numerical values of the different
quantities are introduced, especially the radii of the orbits
and the frequency of revolution of the electrons.

The nature of the equations shows that they are parti-
culariy well adapted to the equal moment of momentum
hypothesis for each and every electron, since }m?o in the
coefficient of v~* is proportional to this moment of momentum,
and equal to it if multiplied by the mass of the electron. If
we denote by @x the ratio of the velocity. of the electron
in hydrogen or in the single electron atom to that of light,
and let the unit a, be the radius of its orbit, and wx its angular
velocity, then we have for each electron in every atom

mo=n’o'=w,, and
mo = Pex; >n?o'=P'ex; Sm?odn?o'=PP'’o2. (26)
p’

P P’ P

e a * 2 2 / . .

The coefficient —>- 2m?wXn*o' in the equation may then be
P ik

replaced by the quantity PP’@2, where 62 is a constant
2 2
quantity, since 82 = a The introduction of this hypo-

thesis therefore makes the equations more easily applicable

Cubic Crystals with Theoretical Atoms. ag

to any twoatoms. It should be noted that there are no terms
in the coefficient of v~* which do not contain the factor 62,
whereas there are many such terms in the coefficient of v~°.
In fact, the terms containing @2 in this coefticient are added
to the terms without @2: and since 62 is a very small
quantity, all these 8x terms have been omitted from the
coefficient of v-* as they do not affect the value of F ina
perceptible degree.

The study of these equations is by no meuns completed.
When the three component forces are resolved along the
radius vector joining the centres of the atoms to get the total
attraction or repulsion between them, and equated to zero,
the result gives the locus of all points where there is no force
between the atoms along the radius vector. For any given
angle « between the axes of the two atoms we obtain a
surface in space surrounding the atom which varies in shape
continuously with achange in a it seems likely that these
surfaces possess important mathematical properties, and may
prove to be of considerable interest to the mathematician.
Only a few of the sections of these surfaces by a plane through
the origin have as yet been worked out, but they have proved
to be of considerable interest because it has been possible by
means of them to demonstrate the complete stability of a
simple crystal on the cubic system, such as rock-salt er
potassium chloride.

When using the numerical values of 8, and the dimen-
sicns of the positive electron given in my theory of the atom,
T obtain dimensions for a crystal which agree within the
limits of error with the experimental work of Bragg and
others and confirm his opinion that there is but a single atom
at each corner of the cube in the crystals mentioned. Fig. 1
shows a portion of such a crystal and indicates tle direction
that the axis of rotation of each atom must assume to produce
a stable equilibrium structure, Hach axis takes the direction
of the long diagonal of some cube ina manner to be described
in a subsequent section.

Special Case, «=0.

When the axes of rotation are parallel in the same direction,
and «=0, the equations are much simplified. The axes of
reference may then be chosen so that the two atoms lie in
the i, & or the x, z plane and y=0, also Y=0. Since Y isa
factor of F,, this force vanishes, showing that the total foree
lies in the 2, * plane. The forces may then be resolved along

758 Dr. A. C. Crehore on the Construction of

the radius vector joining the atoms and perpendicular to this
vector, giving the simplified equations*

mae OE: 1(8 —19X2) B2 72

eons =— + 16 Kar PP (8 HOAX \B 2a

psp + 3m?Snt(—404-200X?— 1754) b -
Pe sksP

3 e” :
Fyerp. = + i6 Ka2o sin 2X { +4P P82
HEWEAOHOKDL, ss QB)
Pee.

where > is the angle of latitude that the line joining centres
of atoms makes with the plane of the equator, and X=cosX.
If the force in (27) comes out positive it denotes an attraction
between the atoms, a repulsion if negative. If the force in
(28) comes out positive it indicates that the second atom is
forced in a direction toward the positive pole of the first
atom. Equating each force to zero and solving for v, we find

a=0 40 — 200 X?4+175 X4\E he
hsBxv=( 8-12 X2 ) : (29)
and hp8s0=4(40—70X2)8,. . . see
where h=(PP'=Sm?dn’)3, .
Pee:

The equations (29) and (80) are plotted as curves in fig. 2 in
terms of k,@xv as radius vector; and since k,8x is constant
for a given pair of atoms these radii are proportional to the
actual distance between the atoms. In this case the complete
equilibrium surface in space is obtained by revolving all the
curves about the axis of the atom A, the k axis, giving a
surface of revolution. A surface of revolution is obtained
only when the axes of the two atoms are parallel in the same
or opposite directions.

The factor sin 2 in (28) shows that in addition to the points
on the curved surface obtained from (30), the perpendicular
component force is also zero at all points on the & axis or on
the equator, that isin the 7, 7 plane. The arrows in the figure
indicate the directions of the along- and perpendicular-forces

* These identical equations have also been obtained from the
instantaneous values of the force when integrated for synchronous

rotation, which shows that they are true for either synchronous or
non-synchronous rotation.

Cubie Crystals with Theoretical Atoms. 759

exerted by the central atom upon the second atom at the
position of the arrow. They show that the loop curves above
and below the equator are stable positions of equilibrium for
small displacements, the force tending to restore the atom to
the curve for both the along- and perpendicular-forces. The
infinite branches approaching the asymptotes shown by dotted
lines are positions of unstable equilibrium for the along-force,
the along-force being a repulsion throughout all the shaded
region on the chart, and an attraction in the clear regions.

If the loop curve of the along-force intersected that of the
perpendicular-force at any point, this point of intersection
would be a position of stable equilibrium for all displacements
in the z, k plane; but there is no such intersection, and hence
no position where on/y two such general atoms unite to form
a molecule when the axes are parallel. This statement doas
not apply to the special cases above noted where there are
one or two electrons only in some of the rings.

Special Case, a=.

When the axes of the two atoms are parallel but in
opposite directions, we obtain equations which differ from
(27) and (28) only in the sign of the 82 v? term. This might
have been foreseen, by observing that the part of the force
arising from the first or electrostatic component is not
altered by changing the direction of rotation, but that the
magnetic or second compvnent is changed in sign. This case

gives instead of (29) and (30)

be ii sole AE a be
a= bo = (<a) i BD)

Ty ah (AO XE VL aes

The resulting chart in fig. 3 has a very different appearance,
due to this change in sign, giving loop curves for both the
along and perpendicular components along the equatorial
direction instead of along the direction of the axis.

The distances to the maximum points of the loops are in
the two cases,

Pecic) distance for along-force, h,Byv = V5 =2°236 direction of

| Sf a Gerp.-rorees!1\,,, . = 410 —3:162 J AXIS.

a=nf a 8 along-force, “A adie V3-15= 1:936 . direction of
‘5 »» perp.-force, ,, = 7:5 =2°739§ equator.

760 Dr. A. C. Crehore on the Construction of

Special Case, i=.

When the axes of the two atoms are perpendicular to each
other the resulting surface is not a surface of revolution, and
the section of the surface by a plane containing the axis of
the first atom A differs for the different positions of the inter-
secting plane. Such a section by a plane through the two
atoms, the axes of both atoms lying in the plane, is shown in
fig.4 (Pl. XI.). The force equations for atoms so situated are

= Ante; ck
a ae r) v?
nae + i Kae 12 PP’B2 (sinA cos A)v

“2 A SS? (+20—175 sin? d cos? 2) \ i (ee
es 2
OF twenty Fish ps

i ig Kaz esin X4 +8PP’B2 (sin A cos Ar)’

vin
a= 7 + Emin —20-+ 140 int W cost d) VL - + (35)
PY Ge.

Equating each to zero, we find
has Epi + 20—175 a
Aled : o=( 12 sin Xcos Xr

+20—140 sin? A cos? A\2
EBs Guan ccd OT )

There are now loop curves of stable equilibrium having their
maximum points in a direction at 45° with thez and & axes,
besides infinite branches of instability which approach the z
and k axes as asymptotes. The maximum distances of these
loop curves are

a { Along-force koBxv=1°990 | 45° with the
2 | Perpendicular-force ,, =1:937 axes.

(36)

If the axis of the second atom A’ is reversed in direction,
pointing toward the left instead of to the right as shown,
this has the effect of reversing the positive direction of the
j axis, since the vector k xk’ is the positive direction of this
axis. To keep the cyclic order of the i, 7, and & axes correct
the 2 axis must be reversed, which would change the sign
of X, and produce the effect of rotating the loop curves into
the first and third quadrants instead of leaving them as they
are in fig. 4. An additional reversal of the axis of rotation
of A has the effect of restoring the loop curves to their
original position in the second and fourth quadrants.

Cubic Crystals with Theoretical Atoms. 761

Assemblages of Atoms.

It may easily be demonstrated as a general proposition from
the force equations (23), (24), and (25) that when the
direction of rotation of each atom is reversed, the total
force of the one on the other is not changed.

When the axes of the two atoms make other angles with
each other than 0, . , and rr, the equations are not so simple,
and the labour of calculating curves is considerably greater.
There are two simple arrangements that may be made with
atoms of two different kinds, or of the same kind, using
these formulze, where all the atoms and their axes of rotation
lie in the same plane. Fig. 5 shows such an arrangement
of two kinds of atoms in rows and columns, adjacent atoms
alternating in kind and direction of axes. The formula
for the case 2=0, the stable equilibrium distance being
k,8<v= 2°236, applies to all vertical columns, and for e=a
and k,8yv=1:936 applies to all horizontal rows. It is
evident that each atom in the plane is rigidly held in its
position by the action of all the others. Along a diagonal
line the atoms are of the same kind alternating in direction,
and the formula where a= applies, showing that although
they are not at the stable equilibrium distance the force of
any atom upon the central atom is exactly balanced by a
corresponding atom on the opposite side of the central atom
at the same distance.

The diagram is merely illustrative of the process of building
up a solid structure with atoms. Of course the force perpen-
dicular to the plane is shown by the formule to be zero, but
they also show that for any displacement perpendicular to
the plane there is no restoring force, and without other planes
of atoms it is evidently an unstable arrangement.

There is another important consideration to be taken into
account in any arrangement. There are forces which de-
termine the directions of the axes of rotation independent of
any consideration of the translational forces upon the whole
atom. These are the third and fourth component forces *,
which are magnetic components contributing nothing to the
translational force but giving an internal turning moment.
These forces acting upon one atom are parallel to the plane of
the equator of the second atom, the one taking the direction
opposite to the velocity and the other opposite to the

= Locwete. p58.

762 Dr. A. C. Crehore on the Construction of

acceleration of the electron. When the planes of the two
atoms are parallel these moments of force evidently vanish;
and when the two axes of rotation are not parallel there is a
moment of force to restore them to the parallel condition, the
moment being a function of the angle between the axes of
revolution. In fig. 5 the two adjacent atoms in a horizontal
row on either side of a given atom with axis downward each
tend to turn the given atom upward, if slightly displaced,
while the adjacent pair above and below tend to turn it
downward. The sum of the turning moments before dis-
placement is zero; but, unless the two sets of moments after
displacement show a restoring moment, there is no stable
equilibrium. In fig. 5 itis not evident without calculation
that the moments after displacement show stability in the
plane, since different formule apply and the distances are
different. The calculation has not been made.
Another arrangement having all axes and atoms in the
same plane isshown iu fig.6. This is based upon the formule
T
where a= 5.

Here there is an arrangement of perfect

squares. the same formule applying to all adjacent atoms in
both rows and columns. The diagonal atoms have axes in
the same straight line alternating in direction, and a=7.
Adjacent atoms along the diagonals have equal and opposite
translational effect on the central atom. The moment of the
forces to turn the axis by an adjacent horizontal pair of atoms
is exactly counterbalanced by an adjacent vertical pair of
atoms, the one pair turning clockwise and the other counter-
clockwise by an equal amount. If the axis of any atom is
displaced in the plane of the paper so as to bring it more
nearly into the direction of the adjacent vertical pair, the
turning moment due to this pair is decreased. At the same
time the turning moment of the adjacent horizontal pair is
increased, and the sum of the moment is, therefore, in a
direction opposite to the displacement, thus proving that the
equilibrium is stable for moments.

The adjacent diagonal atoms, however, all four tend to
turn the central atom in the same direction as the displace-
ment; but the rate of change of the moment is of the second
order of smallness because the axes are nearly parallel,
whereas, the rate for adjacent horizontal pairs is a maximum,
their axes being at right angles. The total equilibrium for
moments is, therefore, stable.

Cubic Crystals with Theoretical Atoms. 763

A Culie Crystal.

If we now attempt to build up a solid by placing such
planes one above the other, a possible way is to place an
exactly similar plane above this one with all axes reversed,
making the odd planes like fig. 6 and the even planes like
fig. 7. A different formula applies to the distance between
the planes, for, in adjacent atoms a=, whereas in the
T
2 3

It is evidently necessary to seek further for the proper
arrangement in a cubic crystal. All three principal planes
of atoms mutually perpendicular to each other should be
identical in character, a condition which cannot be secured
when the axes of all atoms in one plane lie in the same plane.
Fig. 1 shows an arrangement of the axes in a cubic crystal*
that satisfies all the required conditions. All axes of rotation
lie along some long diagonal of the cube, and a plane of
atoms parallel to any face of the cube is similar to all other
such planes parallel to any face. A study of the figure
shows that the axes of any two adjacent atoms along an
edge of the cube lie in the same plane, namely the plane
through the two atoms and through the centre of the cube
to or from which the axes point. Moreover, the angle
between the directions of the axes of rotation of every two
adjacent atoms in the whole structure is the same, equal to
cos-'4=70° 317, being the angle between any two ad-
jacent long diagonals of the cube. We have to study only
four different sorts of cubes shown in figs. 8 to 11 from
which the complete structure may be built. The lower left
front corner cube in fig. 1 is like that shown in fig. 8; the
next adjacent cube to the right in the front row is like
fig. 9; and the cube immediately above this is like fig. 10,
and the one just back of that, being the central cube in
fig. 1, is like fig. 11. Fig. 8 shows axes of all atoms point-
ing towards the centre, and fig. 11 all away from it. Fig. 9
shows four axes pointing to the centre of an adjacent cube
on one side, and four towards the centre of the corresponding
cube on the opposite side; while fig. 10 simply reverses the
directions of these arrows. In each of these figures any
two adjacent axes of atoms along an edge of the cube lie in
the same plane, namely, a plane containing the two diagonally

plane a= 5, and we do not get a perfect cubic crystal.

* Since this paper was communicated it has been found that the axes
of rotation of atoms in the odd planes parallel to the hexagon fig. 12
should all be reversed in direction.

764 Dr. A. C. Crehore on the Construction of

opposite parallel edges of a cube. This plane is always per-
pendicular to one brag of the cube and makes angles of 45°
with each of the other faces.

Equilibrium for Translational Forces.

As far as the translational force upon the atom is con-
cerned, its value for all adjacent atoms along any edge may
be found by substituting in the general formulee (23), (24),

and (25) the one set of values Y=0, X= — me fy fe ve

cos a= ; a=T70° 31"7, which gives the ee

F.=+ -— 2 { PP B2 (—1:089) v-4#+ Sm? Sn? + BO1T)o-* | 2, (38)
Pip
COs a=t

2
F,= + Kat ee (-+°19245)o-4 + Sin? Sn¥( + 1-686)o~* t k.(39)

Multiplying F, by X and F, by Z, and taking the sum to

find the total force along the “ney joining the atoms, we find

F,= +

cos a=
oO

e? ! € = —
~~ | PP 8204+ Sm Zn%(—190)r (40)
Whence, equating to zero we find the distance for stable
equilibrium to be

hoBev=1:378. .. 9... ee

For the value of ky see (31) above.

This result includes both cases where the axes point as in
fig. 8 and as in fig. 11, since we have shown above that a
reversal of both axes of rotation does not change the force.
The same remark applies to figs. 9 and 10, and includes all
adjacent atoms along edges throughout the whole structure.
This result proves that for a small displacement of any given
atom in any direction there is a strong force brought to bear
upon it to restore it to this position due to the six face-
centred atoms above and below, to the right and left, and to
the front and back of it.

This statement does not apply to the other surrounding
atoms. ‘They produce a zero resultant translational force,
but neither a restoring force nor the opposite for small dis-
placements. It is seen that any pair of atoms adjacent to
the given central atom along a diagonal line either of a face
or of a cube has axes parallel to each other. The general

Cubic Crystals with Theoretical Atoms. 765

formule (23), (24), and (25) show that when « remains the
same and the direction cosines X, Y,and Z are each re-
versed, each component and, therefore, the total force
changes sign. The force of each and every pair of atoms
situated at equal distances along any diagonal line upon the
central atom is therefore zero; and this completes the
demonstration of equilibrium of the whole for translational
forces.

Stable Equilibrium for Translational Forces.

Moreover, the central atom is in stable equilibrium for
small displacements. It has already been shown to be stable
for the six adjacent face-centred atoms. The restoring force
per unit of mass and distance for one such atom may be

found by differentiating (40) with respect to v, giving

dF,
dv

e” x “
=taG { —4PP'B20 5411 -dy Emin | (42)

and substituting the equilibrium distance (41) we get

2
dF, LEN ON ewer Sine ee, (43)
Pp P

ado 1h Ka2v!

For the opposite face-centred atom we get the same value,
for a small displacement of the central atom toward the one
face-centred atom and away from the other makes the force
of each toward the original position, changing sign when
passing through the origin. For a pair of such atoms we
must then double (43).

It is different with two atoms along a diagonal of the
cube, for example. In the case where the axes are all in
the same straight line, the atoms being of ditferent kinds,
the formula where «= applies, showing a repulsion at all
distances. By differentiation of the force equation for e=7
and addition of those for two adjacent atoms, we show that
the rate of change of the force is zero for each pair of such
atoms, producing a uniform field of force of zero value at
the central atom. This reasoning applies to all the other
atoms except the six face-centred adjacent atoms ; and we
have, therefore, completed the demonstration of stability
with the exception of the non-translational forces, which
produce only turning moments to control the directions of
the axes. These will now be considered.

766 Dr. A. C. Crehore on the Construction of

The Equilibrium of Turning Moments.

It may be shown thatthe total sum of the turning moments
of all the atoms in the structure fig. 1 upon any selected
atom is zero, and there is no tendency to turn its axis. Any
two atoms tend to turn so that their axes are parallel and
the turning moment is a function of the angle a between
the axes. The moment of atom “a,” fig. 12, upon A at the
centre is counter-clockwise in the plane aAtk, when viewed
from h. The line AA is perpendicular to this plane, and the
turning moment of a@ upon A may be represented as the
vector Ah. The moment of the opposite atom 6 whose axis
is parallel with a’s, being an atom of the same kind at the
same distance, is equal to that of a and in the same direction.
The moment of the pair a+0is then 2Ah. The moment of
eon A is such that it is counter-clockwise viewed from n,
and represented by An, which is perpendicular to the plane
eAot. The axis of the opposite atom 7 is parallel to e at the
same distance, and this atom being of the same kind doubles
the moment, so that e+f gives 2An. The atom d givesa
counter-clockwise moment when viewed from sg, the line As
being perpendicular to the plane Adtp, and the atom ec
similarly doubles the moment, making that of e+d give 2As.
The sum of the three vectors Ah+ An+ As therefore gives the
sum of the turning moments of the atoms a, b, c, d, e, and f
upon the central atom A. Itis evident that this sum is zero,
for these three lines le in the plane Alsjng, being a hexagon
made by sectioning the large cube, and they are, moreover,
120° apart. As they are equal in magnitude their sum is zero.

Now consider the atom & upon A. ‘This is represented by
the vector Aj, since viewing from 7 the rotation is counter-
clockwise in the plane Afta. The four atoms h, 2,7, and k
are similar in kind and have parallel axes. Hence the
turning moment of the four is 4Aj. The effect of o on A is
represented by Al, perpendicular to plane Aoée, and the four
similar atoms /, m,n, and o givea moment 4A/. The effect
of p on A is represented by Ag, perpendicular to plane Apéd,
and the four atoms p,q,7, and s give 4Ag. Since Aj, Al,
and Ag lie also in the same hexagon and 120° apart, the sum
of the moments of h, i, 7, &, 1, m, n, 0, p, g, r, and s is zero.
The only remaining atoms are those along a diagonal of the
cube. Since these, t, u, v, w, 2, y, Zz, and B, are all similar
atoms with axes in the opposite direction to that of A the
turning moment of them all is evidently zero, though for a
small displacement the moments of these would give in-
stability. The control of the position for stability lies with
the other atoms.

Cubic Crystals with Theoretical Atoms. 767

Stable Equilibrium for Turning Moments.

The stability of the equilibrium may be demonstrated by
considering the nearest face-centred atoms a, 6, c, d, e, and f.
Suppose the axis of A is slightly displaced in the plane
Aeto, so as to become more nearly parallel with e and /,.
The moment of force due to e and f is diminished and may
be represented by the arrow An’, a little less than An but
in the same direction. The moments due to the other pairs
a,b,c, and d are scarcely affected in magnitude, as the
angle between their axes and that of the central atom A
is only changed by a comparatively slight amount. The
direction of Af is slightly rotated out of the plane of the
hexagon toward the front side, and of As similarly rotated
toward the back side. The resultant of Ah+As then main-
tains approximately the same magnitude and direction as
betore, namely Al. The resultant of the moments of the
six atoms a, b,c, d, e, and fafter displacement of A is then
the sum of A/ and An’, namely AM in the direction of AJ.
The resulting moment AM tends to turn the atom A in the
counter-clockwise direction when viewed from J, and hence
acts against the direction of the small displacement which
was in the opposite direction, so as to make A more nearly
parallel with e and f. The original position of the axis of A
is, therefore, one of stable equilibrium for such a displace-
ment. Were it displaced in any other direction we would
arrive at a similar result. A similar process of reasoning
may be applied to the twelve atoms h, 2, 7, k, 1, m, n, 0, p, 9,
7, Ss, arriving at a similar conclusion. |

This completes the proof of stability both of the directions
of the axes of rotation and the translational position of each
and every atom in the whole structure in a cubic crystal.
Before giving numerical values of the distances between
atoms it seems best to consider a special case, that of the
hydrogen molecule.

The Hydrogen Molecule.

Assuming that the two atoms in the hydrogen molecule
are alike, each having a single electron revolving at the
same speed, we have to find the average force between two
synchronously revolving electrons, resolved along the line
joining centres. This has been done in a former paper* for
the case where the axes of rotation are parallel. This gives

* Loc. cit. equations (42) and (44). Note an omission in the co-

efficients, corrected in a note at bottom of page 825, in a subsequent
. } . ~\ ~ 4
paper, Phil. Mag. vol. xxix. Feb. 1915,

768 Dr. A. C. Crehore on the Construction of

to the sixth power of v when the radii of the orbits are each
equal to m in ay units,

Piatong= +729 | — 9-2-4. m¥(1—T\(8—4:5X2)0--++ "(1 —F)°(— 75 487
; i 32°3125K4)u-#t, (44)

Dene = =p bk (45)

lalong?
!
Pipeep = + qesin2A i L5m?(1—T)on* + m4(1—-T)*(—75 + 13-1 25X90 b,

Ke2
REE a4 Aion
Py ctp. = Pt Bigs 1 eee. ak te! 0

Here I denotes cosy, and y+ is the phase difference between
the instantaneous positions of the two electrons being con-
sidered, and is constant for synchronous rotation. When
we add the. forces of the two electrons, one in each atom,
upon each other, the electrons on the opposite positive
charges and the positive on the positive, the force of one
hydrogen atom upon another with axes parallel and when y
is zero degrees 1s
Di ibe

ig ion a 5 82 vt (—16 + 24X2)0? + m%(20—100X2 4 87-5X%)}
*

3 e*m? ’ o

sin 24.4 80? +-m?*(20—35X") i. oe

pew. 16 Kav’

Stable Equilibrium for Translational Forces in the
Hydrogen Molecule.

Equating the along force to zero we find
Byw=[15— 2-25 X44 (— 19 $225?) | ee

The plot of this curve is shown in fig. 13. Equating the
perpendicular force (49) to zero gives sin 2\=Q, denoting
the vertical and horizontal axes on the chart. Since the
brace in this equation contains no # term there is no curve
equating this factor to zero. ‘The values of v obtained from
this are very small and have no meaning, since the equations
only hold for large values. The series from which it is
derived is not convergent for small values. The arrows
indicate the directions of the two component forces at various
locations on the chart, and show that there is a stable position
of equilibrium for the two positions on the axis, one above
and the other below the central atom.

Cubic Crystals with Theoretical Atoms. 769

- Stable Equilibrium for the Phase Difference in the
Hydrogen Molecule.

It remains to be shown that when the phase angle y is
zero the two electrons are in stable equilibrium as to phase
for small displacements along the orbit. It is, first, evident
by considering the instantaneous forces when y is exactly
zero that there is no force to accelerate or to retard the
second electron. The static force due to the positive charge
of the opposite atom on the electron may be resolved into
two, the one perpendicular to the plane of the orbit and the
other along the radius of the orbit, neither of which gives
any tangential force along the orbit for any position of the
electron. The only other force is that due to the second
electron upon the first. The instantaneous values of the
four components of this force when resolved along the
tangent to the orbit is given by equations (53) to (56)
of the former paper. These all vanish when y=0 and
«2=(, as it does when the atom is on the axis, which
completes the proof of phase equilibrium of the electrons
when y=0. 7

That the equilibrium is stable may be shown by slightly
displacing the second electron along its orbit. The effect of
the positive charge gives no force along the orbit as before.
We need only consider electron on electron. For small
displacement a small component of force is obtained along
the tangent for the first, second, and third components of
the instantaneous force, but not for the fourth. (See
equations (1), (2), (3), and (4) of former paper.) The fourth
component depends upon the rate of change of the distance
Ri between the electrons which is zero. ‘The first and second
components give respectively

2 (22
Te ee fA get i a Ui

where Ay is the small displacement angle, ax the radius of
the orbit, and R the distance between electrons, also between
atoms. The second force is negligible in comparison with
the first because G2 is a small quantity. These two com-
ponents alone show instability because they tend to increase

Phil. Mag. 8.-6. Vol. 29. No. 174. June 1915. 3D

770 Dr. A. C. Crehore on the Construction of

the displacement angle Ay. The third component* gives
2R 2

F=-ya 4 +:

a force tending to restore the electron and produce stability.
The ratio of I’, to F, gives

2 P2
pan ete =— Bie. . .) Jae
1 *

If this ratio exceeds unity when the distance R is that for
stable equilibrium of the translational forces, the moments of
the forces are also stable.

The distance between the atoms at the position of equi-
librium is found by making X=0 in (50), which gives, taking
the negative sign, }

Spe ge or v= v3, wy: (55)

This proves that the ratio in (54) is equal to three, and that

the restoring force of the third component is three times

greater than the electrostatic force of the first component.

The total restoring force for a small displacement along the
orbit is then

eee (56)

=— pets: on =

where As is the linear displacement.

Vibration Frequencies in the Hydrogen Atom.

We will now find the force with which one hydrogen atom
is restored to its original position when displaced along the
line joining the atoms. Let the common centre of mass of
the two atoms be taken as origin, a point halfway between
the atoms which remains fixed, and let each of the two
atoms be displaced to an equal distance away from the origin.

The force equation (48) becomes, when X=0,

e? Hae og Hoot ened u i
Piguet =! + jhxe —TGtee Bab a hewn |, (57)
where z is the distance in centimetres from the mid-point to
one atom. Differentiating to find the rate of change of the

* The factor (1+«)-2 was omitted from the text, p. 73, in giving the

formula (55), but it appears in the brackets in the alternative value there
given.

Cubie Crystals with Theoretical Atoms. tia

2
| EBEY (58)
Ax K 32 ae y
omitting the term in #~" as being small.
This expression gives the force acting per unit of distance
and per unit of mass. Hquating to the mass times the acce-

leration per unit of distance, we derive the frequency of
oscillation

ituti ae tl uilibrium distance
force, and substituting 7= 2 Be 1e eq ;

COL SUM ON ah ee tHe NEON

ey
OFT MIA 2

Naleng =

By a similar process we find from (49) the frequency ot
vibration of the mass m perpendicular to the line joining the
atoms to be

Nalong HEIR IP PUAN FAIRS ( 6 0)

These were published as equations (13) and (15) ina recent
paper”, without the process of derivation.

Comparison of Experimental and Theoretical Valuest.

We are now ina position to use the experimental deter-
minations of the distances between atoms in cubic crystals,
and of the frequencies of vibration in the hydrogen atom as
determined by the light spectrum, to compare the results
ef the theory with known experimental values. For many
points in the theory reference must be made to a recent
papert. The fundamental constant b, the radius of the

* Phil. Mag. vol. xxix. p. 526, Feb. 1915.

+ Since this paper was communicated the force of every atom in a
cube, of edge four times the length of the elementary cube, upon the
central atom has been calculated. There are 26 atoms surrounding the
centre in the first cube, and 98 more in the next cube, making a total of
124 atoms besides the central atom. With the central atom there are
125=5° atoms. The approximate result in equation (66) has been
modified a little by this work so that it comes into good agreement with
the atomic theory in (70). A study of monometric crystals of various
compounds has made it possible to find the values of Ym? characteristic
of individual atoms. This is an additional property of the atom itself,
as characteristic as the atomic weight or the atomic number, and has
proved to be of importance in determining the compounds into which the
atom enters. It has in fact made it possible to predict the arrangement
of the atoms in some crystals in advance of the published investigation
by means of X-rays, and it will prove interesting if these predictions are
later contirmed by such a study,

te eoG. G2e,

3) D2

{ie Dr. A. C. Crehore on the Construction of

positive electron, which is the same as that of the hydrogen
atom, is determined from the fact that the number of electrons
per gram is constant for any substance, equal to 6 x 10”
approximately, together with the approximation that the
volume of all the atoms per gram is 10~!” cubic centimetre,
the reciprocal of the ether density. ‘This gives

b='135x 10-7 cm. =

The fundamental constants wx=2z7s, the angular velocity
and frequency of the single electron in the hydrogen atom,

are determined from Planck’s constant together with the
above value of }, to be

= e 6) ie 19
Seay sor) oJ K0) A >) Seka (62)
where A is Planck’s constant, 6°5 x 10-2", ande the electronic
charge 477x107". Hence

o,=150x10". .°.. eee

The distance between a sodium and a chlorine atom in
rock-salt along the edge of a cube is given in a, units in the
theory by (41) above. In centimetres this is

=m?>dn?
/

1=1378— —Teee

using the value of k, in (31) and writing p=, c being

the velocity of light. Equating J to 2°814x10-%em., the
distance in a rock-salt crystal, we find

a/ SD n= 102/ PP es
ie

The theory gives for sodium P= 23 and for chlorine P’=35,
each being equal to the number of electrons in the atom.
Hence for sodium and chlorine atoms

ay div Dn? = 2895...)

23 35

The values of the radii of the orbits of the electrons within
the atoms, m for sodium and n for chlorine, are measured in
ax units, where ax is the radius of the orbit in the single
electron atom, hydrogen. We cannot arrive at the theo-
retical value of the radical in (66) until this unit is determined

Cubic Crystals with Theoretical Atoms. 173

in centimetres. A rough approximation to it was given in
the former paper, but a more accurate way is to derive it
from the hydrogen spectrum, assuming that (59) gives the
vibration frequency of the fundamental term in Balmer’s
series. This may be written

e 5
i lons alan ee bhrary metres me (67)

All the quantities in the right member of this are known
except ax. Hence, by equating the frequency to 823 x 1,
the constant in Balmer’s series, we find ax,

Tie, = DNA Se MEL iat,
and AxWx

By=

: = -OOROSS Ui wy ates, HOS)
The value of this unit in the former paper, determined in a
different way, was °285x107-2%cm. From (63) and (68)
the universal constant angular moment of momentum of
every electron is seen to be

MOdi—=" OA UOm ea ates shoe 4) Od)

185,000 times smaller than the value of this constant given
by Bohr’s theory* of the central nucleus atom.

If we assume that the electrons are distributed in a
chlorine atom according to the scheme shown in fig. 3 of the
former papert, having rings of 16, 12, 6, and 1 electrons,
and also that the positive charge has a spherical shape, its
radius is 2:-4x10-%cm., or oe =116 ay units. Taking
Mo=11, 1 =7'5, ny=5, and n; negligible, we find =n?=2800

approximately. The electrons in the sodium atom are not
shown in the figure referred to, but the radius of the positive
charge is 2°09x107~™ cm., =10'lay units. An estimate of
the positions of the electrons in three rings 13, 8, and 2,

gives about 1500 as the value of Sm? for sodium. The
23
theoretical value is, therefore, approximately

nf Sm?Sn? = 2050, aitiedow eal > ACgO)

23. 35
which is to be compared with 2895 in (66). It is needless
to say that these figures are in agreement within the limits

* N. Bohr, Phil. Mag. July 1918, vol. xxvi. p. 16.
+ Loe. cit. p. $28.

774 Construction of Cubic Crystals with Theoretical Atoms.

of error of our knowledge of the distribution of the electrons.
If the radius of the outside ring of electrons in chlorine
were about 13 instead of 1las units above, and that in sodium
about 11 instead of 10, this would make the values in (69)
and (66) agree.

There are many fundamental questions that have probably
suggested themselves to those who have followed the subject
here presented; and the belief that they will receive a satis-
factory answer inereases with the progress of the work.
Other forms of crystals than the cubic system must be
explained, as well as the combinations of elements known to
chemistry. ‘This latter, of course, need not be restricted to
the neutral atom, but ‘the former most likely is. It is
probable that alternative forms of stable crystals will be
found with the same combinations of atoms, and the most
stable of these is likely to be the one that exists in nature,
though sometimes there may be more than one.

One important matter that should be emphasized is made
evident by a study of the cubic crystal in fig. 1. There
is the same number of chlorine atoms in roek-salt that have
their axes pointing in any given direction as in the exact
opposite direction, and so of sodium. The total magnetic
effect of the crystal is, therefore, very small. Were this not
the case, and if there were an excess in one direction more
than another, there would be an aggregate giving a strong
unbalanced magnetic force, and the crystal would be magnetic.
It is to be noticed that the magnetic component in this crystal
varies as the inverse fourth power of the distance from the
atom while the electrostatic varies as the sixth power. At
great distances, therefore, only the magnetic force can prevail,
the electrostatic being a negligible quantity in comparison;
and if the axes of the atoms are so turned on the whole as to
cancel their magnetic components, the structure is non-
magnetic.

The ability to magnetize the substance by an outside force
must depend upon the degree of stability of the turning
moments which have been considered in detail in the paper.
It is quite possible that in iron we have a peculiar case where
the translational equilibrium is very stable, but the turning
moments are in almost neutral equilibrium. Such a con-
ae will explain the unique properties of iron, nickel, and
cobalt.

The fact that the action between two magnetic poles in
iron is inversely as the square of the distance instead of

Principal Series in Spectra of the Alkali Metals. 779

inversely as the fourth power of the distance, seems to show
that the atom of iron must come under one of the special
cases above mentioned, having a centre ring with only one
or two electrons. The force in this case may be inversely
as the square of the distance. This is shown by the equations
for the hydrogen atom, (48) and (49) above, where the first
term due to the magnetic force contains the inverse second
power of the distance, showing that for hydrogen atoms
at a great distance, having parallel axes, the attraction is
inversely as the square of the distance. This suggests that
possibly the rings of electrons in fig. 3* of the former
paper are correct, for there iron and cobalt have a single
central electron, and nickel two central electrons, which will
give rise to the same law of variation of the force with the
distance as that in hydrogen.

In this paper the positive charge in each atom is supposed
to be at rest free from vibration of its whole mass, and long
waves of radiant heat are attributed to the vibration of this
mass. ‘The period of this vibration in the hydrogen molecule
was given in the former paper. It should be considered,
therefore, that the work in this paper applies to the state of
matter at the absolute zero of temperature.

LXXXI. On the Principal Series in the Spectra of the
Alkali Metals. By W. Marswaty Warts, D.Se.f

N the Philosophical Magazine for 1908 (xvi. p. 945)
Wood describes the absorption spectrum of sodium
vapour in which he measured 48 lines of the principal
series forming the most extended “ Balmer series” hitherto
observed.

Bevan t applied Wood’s method to the other alkali
metals, and observed extended series in lithium, potassium,
rubidium, and cesium. He also discusses the representation
of these long series by formule, obtaining very close results
by the use of the formula employed by Mogendorff in 1906
and by Hicks in 1910, viz.,

)9675
On aoe 2 eA
m+ w+ <) 4

* Loe. cut.

+ Communicated by the Author.

{ Proc. Roy. Soc. lxxxiti, p, 428 (1910), lxxxiv. p. 209, lexxv.
pp. 54, 55, Ixxxvi. p. 800 (1911).

al Series

incip

Dr. Marshall Watts on the Pr

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117

in the Spectra of the Alkali Metals.

‘C.QPF[P UozIMyYyorT UoA pur ‘9).OFFLTF SOU ‘TL-SPFTP Aouonbe.sz-oouds0au00 Oo} svy od

Jills

60:—

99-F 1G
LO-PLVG
6L-F1FG
16. FIV]
G0-C1VG
61-G1VG
GE-G1PG
0¢-G1VG
69-G1G
88:S1FG
60:-91FG
16-91FG
Ca-91G
18-916
60-L1 FG
OP-LIFG
VL: LIbG
IL-S81¥G
TG-814G
96-816
CP-61VG
00-066
69:0GFG
66: [16TG
F0:-G6G
06-GGFG
R8-S6PG
66-FCPG
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68-G1PG
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6E-91VG
99-9146
O8-9LFG
OL-L1VG
88-L1FG
TL- L1G
60-816
PP-81FG
00-6146
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CO-0GFG
09-0GFG
66: 1GPG
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GL: LEVG

778 Dr. Marshall Watts on the Principal Series

Other formule which have been employed (besides that of
Kayser and Runge, which is not sufficient) are the Ritz
form

Oc ee
ane
(m+ m+ aD
and that of Lohuizen
109675

O.F.=C.F.— 9°
C a
(uta or)

A comparison of the results given by these different formulze
is calculated to afford useful guidance in the examination ot
less extended series.

The case of Sodium has been examined at length by
Birge *, who gives wave-lengths on the Fabry and Perot
scale reduced to vacuum, and employs the Ritz formula
with 109678°6 as the value of N instead of 109675.

In the table on pp. 776-7 I give the wave-lengths calculated
from the formula

O.F.=41448°67 — oe 279922
0313285
(m +°147408— ea

for the less refrangible component, and

109675

i
vi—1

O24 144857
( m+ 148204 —

for the more refrangible component of the pairs of lines.

In Potassium the formulee
109679

OiF.— 35005 bee eS a
(m 4993076 — a

—Il

O.F.=35005°56— — poeta ie
7 A) ORB Re
(+ 296228 — oS)

give the following values :—

* Astrophys. Journ. xxxii. p. 112 (1910).

in the Spectra oj the Alkali Metals.

The Principal Series in Potassium.

Ramage.

(Flame.)

{ 7697

7664
4047-39
4044-33

3447°56
3446-55
3217

{ 3217°36

7699°32

4
i

4047571
4044°599

Exner & Haschek.

Are.

4047-42
4044°36

{ 3447-54

8217-75

( 3446°51

7665-29 Hermann.
Jewe

ll,

Observed.

Spark. Schillinger.
7699°41
7665°56

4047°30| { 4047°35

4044°30| | 4044°31

34476 3447°52

3446°60 { 344653

32173 3217-54
3103-05

2992°4 2992°47
2963:37

f 4047-4
| 40443

oT on Saunders.

Kayser &
Runge.
Are,
7969°3 x.
{ 7665°6
4047°36 4
\4 4044: 99 |
3447°49
5446°49
{ 3217-76
| 3217-27

3102°37
3102°15
303494
2992°33
2963°36
29428

Bevan.

Absorption.

2928°0
2916°6
2907°6
2900°4
28946
2889°7
2885°9
2882°9
2880°3
2877°9
2875'S
28741
2872°5
28711
28700

769941
7665°39

Eder & Valenta.

Calcu-
lated.
| Mm.

9 { 7699:10
76693°60
3 { 4047°51
4044°13
4 { 3447°59
3446°56
5 { 3217°83
3217°36
6 { 3102:27
3102:00
7 ( 3035:09
| 3035-00
8 | 2992 43
9 | 2963°53
10 | 2942°99
11 2927°86
12 | 2916°38
13 | 2907-45
14 | 2900°38
15 | 289467
| 16 | 2890°00
Pelle 2886°12
118 | 2882:87
WLP eed: 12
20 | 2877-77
21 2875°75
22 | 2873°99
23 | 2872°46
94 | 2871:12
25 | 2869°93
Eder & Valenta,

779

Observed—Calculated.

Birge.

7701-92
776854

Lehmann.

Birge has convergence-frequency 3500529, and Hicks 35006:21.
For Rubidium, the formule

O.F.=33688-20—

O.F.=33688'20 — -

(m4-368 365934— ges N os

m—1

O73868
m—lL

~°073542\2?

Flame.

780

give the following results :—

Dr. Marshall Watts on the Principal Series

The Principal Series in Rubidium.

Are. Spark.

Eder &
Valenta.

7950°4
( 18062

Ramage.
| 421563
_4202°04
{ 309186

322818

Birge has the convergence-frequency 33687°82,

Kayser &
Exner & | Runge.
Lehmann.| Saunders.| Haschek.| (Arc.)
7950°46 | {7947-6 7950
7805°98 | | 7800-2 7811*

Exner &
Haschek.

(4215-75 | | 4215-73 | ( 4215-72
| 4202-06 | 4201-97 { 4201-98
3591-74 359174
{ 358721 | 3587°23
F 3351-00 3351-03
3348°89 1 3348-86

Bevan.
(Absorp-
tion.)
{ 3158°25t
| 3157-69
3112°95
3082°39
3060°62
3044°33
3032:20
3022°70
3015:16
3009°03
300411
2999°96
2996°51
2993'52
2991-12
298894
2987-01
2985°45
2984 05
2982-68
2981:51
2980-52
2979°62
297881
2978:°10
2977-39

m. \Caleulated

Observed—Calculated.

Birge.| Hicks.| Bevan.

9| { 794903, +°01 | 0 0
{ 480313 +04 25
3 eee =O tO) 0

4201°99 | +-01 0
A eae +04 |+:20] 0

3587°22 | +02 )
5 | | 3800°94| +-06 | +16] 9

| 3348-93 On + 06
g| { 322927 | —-01 | —01 |+ -02

| 3228-10 | +:08 04
7 | ( 3158:39 | +--08 | —-22 |41-00| —19

\ 35767 |02 + -27| —03
81 811295) 10. eae =i
9| 3082:34| +:05 | —-65 +01
10} 38060°56| +-06 | —-41 —0]
11] 3044-47] —-14 | —-40 =O)
12)" 3032-251) 0556 =i
13| 3022-74) —-04 | =.38 —-09
14] 3015-20 )\=-04 |is = iil
15| 80v9:11] —:08 | —-08 —15
16|, 300412) —-01 |= 7 —-08
17] 2999-98] —-02 | —:03 —-08
18| 299651} O | —:26 —06
19), 2993:57 |= 059) acon a
20| 2991-06} +-06 | —-09 Ah
21| 298889] +:05 | —"18 — ‘Ot
92} 2987:02| —-O1 | —14 —-06
93| 2985°38| +:07 | 0 +02
941. 2983:94) ---11 | 45:14 +°06
25| 298267| +-01 | +:41 —-05
96| 2981-53} —-02 | +53 —-08
7| 298053) —-01 =r
98| 297962) 0 —07
29| 297881} 0 —07
30| 2978:08/ +-02 05
31| 2977-42] —-03 — 9

Eder & Valenta (Arc).

‘a { T94L7-7
7800°3

and Bevan 33687°5.

For Cesium the formulee
O.F.=31404:31—

( m 4418202 —

+ 3158°7 Saunders.
Hicks 33687-50,

109675
ee)
m—1

in the Spectra of the Alkali Metals. 781
109675
° 2 eee
(m+-449969— “28

m—1
give the following results :

O.F.=31404:31—

The Principal Series in Cesium.

|
Observed. mM. Calculated Observed-Calculated
Kayser & | Exner & is Bevan.
eae ee Runge. | Haschek. Guna) (Absorp- Bevan.| Hicks.
care ) | (Are) | (Are) ) | tion.)
8945-0 | ( 8949-92 9 | £ 8949-92 0
8522°4| | 8527-72 “| ) 8527-72
4593°34 | ( 4593-29 | { 4593-30 2 (003 19 —-02
4555-44 | | 4555:34 >} 4555-46 | 4555-29 +:04
3888°83 | { 3889°1 | [ 3888-75] ( 3888-80 || ees 66 104) | -EAo
| 876°73| | 3876-7 || 387631] | 3876-53 3876'34 ISI)
3617-08 3617°49 | | 3617-56 || eae 3] +13 | —-09
{ 3611-84 1 3611-70 1 3611-70 3611-54 4:05 | —-22
re 6 ey 09/419] -¢-09 | —-05
3477-25 | | 8477-08 47688 |4+:15| —-10
340018 |) 7 | £3400-05/4-08} 0 |—14
3398-40 | | 3398-27 3398-06 |+-21| +-18
toa | {| 3348'84 3348-93 |—-09 | —-09 | +-67
334872 { 3347.56|| 8 { Baar: 63 |—-07 | —12
a (ss1s3s g | { 831418 |—-02| —-09 | +-30
3313-28 3313-26 |-+-02 | —-05
3289-25 3289-43 |—-18| —-25 | —2-1
3287 1 39886 68 || 1° (3 3288°76 |—-08| —16
f 3271-14
$270.66 |! 11) 4 3970-64 |4+-02| —-16

3256'86 |—-09) —19
: 3246-43

8246 02 | 13 { 324619 ~-10| —-16
aes ' 3237-83

3237°58 || 14 8287°58) 0 | —07
Lae 3230-88
ee) | 823068 4-01 —-09
we 3225-20

822511] 16] { 359° 03 ho 0
3220°34 || 17 | 3220-40 |—-06 | —-09
3216-45 ||18| 3216-46|—-01| —-03
3213-02 |/19| 3213-11|—-09| —-09
3210-18 |/20] 321025|—-07| —-08
3207-78 || 21] 3207-78] 0 | —-02
3205-64 || 22] 3205°63|+-01| —-02
3203-80 || 23] 3203-76 |-4+-04| +-01
320213||24| $20212|4-01| —-02
200-71 || 25} 3200-66 4-05 | +-01
3199-45 || 26| 3199:37|-4-08| +-03
3198-25 ||27| 3198-22|+-03| —-01
3197-28 | 28] 3197-19 |-+-09| 4-05
3196-2029] 3$196-27|—-07| —11
3195°42||30| 3195-43|—-01| —-05
3194-59] 31} 3194-67 |—-08| —-13 |
3193°24 || 382] 3193:98}—-04| —-10 |

{
Flame. Are. Spark. |Absorp- Hes Observed--Calculated.
— ee ion: 4|| cee
Kayser| Exner | Eder | Exner |
Ramage.| & & as Ore Bevan. | 7%. Bevan.| Hicks. Birge.
Runge. | Haschek.| Valenta. |Haschek.)
| |
6708-0 | 6708-2 | 670810) 6708 07} 6708-09 | 2|6707°94+:06| © 0 0
3232-82 | 3232-77| 32328 | 3232-98] 3232-80 3 | 3232°39 —-09| 0 0 0
2741-43 | 2741:39) 2741°37 | 2741°57| 2741-32 4.|2741:34/+-08| O | 4-07 | —:07
2562°60, 5 | 2562'44'4+-16| +07] 0 |—O1
2475°13) 6 | 2475°15|—--02 | —-11 | +20 | —-23
2425°55 7 | 2425:52)4+--03| —-08 | +-01 | —"19
239454 8 | 2394-46 +-08 | —-02 | +03 | —15
2373°9 || 9| 2373'65,+°25|+:09 |} +10] 0
2359°4 || 10 | 2359-03 +°37 | +°22 | +-22 | +:15
23485 || 11 | 2348-33 4-17 | 4-03
2340°5 |] 12 | 2340-27/+-23 | +08
| 2334-3 || 13 | 2334-04) -+-26 | +-12
| 2329-0 || 14.| 2329-12 —+12| —-28
| 2325-2 || 15 | 2325-17 +-03 | ~-18
2321-9 || 16 | 2321-97/ —-07 | —-20
2319°3 || 17 | 2319-31/—-01 | —-15
2317'1 || 18 | 2317-08|+--02 | —-12
| 2315-2 || 19} 2315-21/—-01 | —-15
2313°6 || 20 | 2313°61,—-01 | —-15
23122 || 21 | 2312-23) —-03| —-17
2311-1 || 22| 2311-04 +-06| —-08
23100 || 23 | 2310:01/—-01 | —-15
2309-0 || 24 | 2309-10 —-10 | —-24
| 23083 || 25|2308:30| 0 | —14
2307-5 || 26 | 2307759 —"09 | —-23
2306-90)| 27 | 2306-96 —-06| —18
2306-48] 28 | 2306-39 +09 | —-03
2305'87|| 29 | 2305:88| —-01 | —11
| 2305°41)| 30 | 2305:42|—-01 | —-11
| 2304-99] 31 | 2305:01 —-02| —-11
2304-63'| 32 | 2304-63) O | —-15
234-29 | 33 | 2304°30/—-01 | —-10
| 2304-00 | 34.| 2303-99'-+-01 | —-07
| | 2303-73| 35 | 2303-70 -+-03 | —:06
| | 230346)! 36 | 2303-44|+--02 | —-12
| | 2303-24 | 37 | 2303-20 +-04 | —-05
2303-03 | 38 | 2302-98)+-05 | —-08
| | 2302-83] 39 | 2302-77/ +-06 | —-03
| 2302:59.| 40 | 2302:58/+-01 | —-08
| 41 | 2802-41] —-03| —*12
|

782 Principal Series in the Spectra of the Alkali Metals.

In the case of Lithium the results are not quite so
satisfactory. Hicks finds the convergence-frequency to be
43482°20+1:18. Bevan uses 43482°98, but points out that his
latest measurements indicate a higher convergence-trequency.

In the following table J give the results calculated trom

109675
081 RR\2
(m — 04929 + eee

m—L

O.F. = 4348518 —

The Principal Series in Lithium.

42 | 2302°24/—-04 | —°13

Tonization by Positive Rays. 783

If we take the mean observed value of the first line
6708°12, which corresponds to the oscillation-frequency
14903°26 in vacuo, and, applying the Rydberg—Schuster
rule, add 2857971, Hicks’s mean value for the convergence-
frequency of the subordinate series, we obtain 43482°97
for the convergence-frequency of the principal series.

LAXAXXIT. Jonization by Positive Rays.
By NoRMAN CAMPBELL*.

ie | yas chief facts concerning the ionization of gases by

electrons seem to be now thoroughly established, but
comparatively little is known concerning ionization by the
impact of atoms except when the atoms are those constituting
a-rays. The first direct attack on this problem is described
in the recent papers of v. Bahr and Franck { and of
Pawlow ft, who come to the surprising conclusion that the
positively charged particles from hot platinum and hot
phosphates can ionize gases when their energy is less than
‘that required by electrons. This result is of the greatest
importance for theories of the structure of the atom, for it
would seem to indicate that the “ionization potential” for
electrons has not the fundamental significance which has
otten been attributed to it; it appears that further experi-
ments, especially on the variation of the ionizing power of
positive particles with their speed, energy or charge, are
desirable.

It is not easy to conduct such quantitative experiments
on a gas, into which rays of such low speed as are here con-
sidered have practically no power of penetration. But recent
work has tended to show § that the liberation of electrons,
which takes place when ionizing rays of any kind fall ona
metal surface, varies with the properties of those rays in
precisely the same manner as the ionization which the rays
cause ina gas. Indeed it is highly probable that, when the
surface of the metal has been recently polished, the electrons
are actually liberated from a layer of gas on the surface of
the metal. Hxperiments on such metal surfaces are free
from many of the difficulties which attend experiments on
ionization in a gas which has any appreciable volume.

* Communicated by the Author.

a vy. Bahr and J. Franck, Deutsch. Phys. Gesell. Verh. xvi. 1. p. 57
’ t W. Pawlow, Roy. Soc. Proc. A. vol, xe. p. 898 (1914).

Hh Satan N. Campbell, Phil. Mag. June 1915, p. 803, and Mareh.
0, p. vd,

784 Dr. Norman Campbell on

2. The liberation of electrons at metal surfaces under the
action of positive rays has been studied, under the name of
‘secondary cathode radiation,” by several investigators.
The only investigations which seem to threw any light on
the number of electrons which are liberated by each positive
ray are those of Fiichtbauer*, Baerwald +, and Koenigsberger
and Gallus ft. Fichtbauer found that when positive rays
(probably hydrogen) with an energy of from 31,300 to
15,000 volts fell on a copper plate, the number of electrons
liberated by each positive particle varied from 1°36 to 0°89,
while the proportion of the positive particles reflected varied
from 0:083 to 0°15. Baerwald, using positive rays of
hydrogen and helium falling on an aluminium plate, found
that the number of electrons liberated by each particle did
not change appreciably between 30000 volts and 900 volts,
but that the ionizing action ceased abruptly at this lower
limit. Koenigsberger and Gallus found that the number of
electrons liberated from brass by positively charged rays, of
which the nature and speed were not precisely determined,
lay between 2 and 4.

Finally Bumstead §, working with e-rays, found that the
variation of the ionizing power of the rays at a metal surface
varied with the speed in almost exactly the same manner as
the ionizing power ina gas. Hauser ||, in agreement with
other workers, finds the number of electrons liberated at a
metal surface by each a-ray to be about 60.

3. In the experiments about to be described, which were
conducted in an apparatus of which the‘essential portion is
shown diagrammatically in fig. 1, the positive particles were
not (as in the work just mentioned) those of these canal-rays,
but those liberated from heated phosphates as in the work
of v. Bahr and Franck and of Pawlow. The use of such
particles is necessary if their effect is to be investigated for
very low potentials at which a discharge will not pass ; it
has the additional advantage, even at higher potentials, that
the observations can be made in a very low vacuum and the
ionization completely localised at the metal surface; it has
the disadvantage that greater uncertainty exists as to the
nature of the particles. The particles liberated at the heated
strip S, which was of platinum coated either with sodium

* OC. Fiichtbauer, Phys. Zett. vil. p. 153 (1906).

+ H. Baerwald, Ann. d. Phys. xli. p. 648 (1918).

t J. Koenigsberger u. A. Gallus, Deutsch. Phys. Gesell. Verh. xvi. 4.
p. 190 (1914).

§ H. A. Bumstead, Phil. Mag. xxii. p. 907 (1911).

|| Hauser, Phys. Zeit. xii. p. 466 (1911).

Ionization by Positive Rays. 785

phosphate or a mixture of sodium and aluminium phos-
phates, passed through the tube T with an energy cor-
responding to the difference of potential (V) between 8 and
A, and fell on the copper plate C at the same potential as A.

ies

iS}

g
iS)
WY
A
vt
Leseee i
Ny
B
(@)

The reflected rays and the electrons liberated at C fell on the
cylinder Band plate D, which were coated with soot to avoid
further reflexion.

All the portion of the apparatus shown was enclosed in a
glass tube, covered with metallic foil to avoid electrostatic
disturbances and exhausted by connexion through a wide
tube to charcoal immersed in liquid air throughout the
observations.

The values of V between 0 and 400 volts were obtained
from cells ; no higher potential from this source was avail-
able. From 2000 volts upwards they were obtained either
from an influence-machine or an induction-coil. With the
machine, V was varied by varying a high resistance placed
as a shunt across its terminals ; with the coil, V was varied
by varying the current in the primary, a unidirectional
secondary current being secured by the use of the usual
valves. V, when not greater than 400 volts, was read on a
voltmeter ; the higher values were determined by means of
a spark-gap between balls of 2 em. diameter, the readings of

Phil. Mag. S. 6. Vol. 29. No. 174. June 1915. 3E

786 Dr. Norman Campbell on

the spark-gap being interpreted by the tables given by
Landolt and Bornstein.

The current was measured by the method, adopted in all
my recent work, which depends on the use of an electro-
meter and a high resistance. It was always of the order of
10-” amp.; if it was steady, it could be easily measured to
1 part in 1000.

4, Let I, be the positive current carried to C by the
primary positive particles, I, and I; the positive currents.
carried away from it by the reflected positive particles
and the liberated electrons respectively. Then R=I,/I,
will be termed the reflexion coefficient of the rays,
P=—J,/I, will be termed their ionizing power. If the
incident positive particles carry each a single electronic
charge, then P will be the average number of electrons
produced by each positive particle.

If i, is the current received by B, C, and D connected
together, then 2,=1,. If 2, is the current received by C
when B and D, connected together, are kept at a potential
slightly higher than that of C, so that all the electrons
liberated at C, as well as the reflected primary particles, are
absorbed in B and D, then 7,=1,—I,—I;5. If, on the other
hand, B and D are kept at a potential lower than C, the
difference of potential being greater than that of the fastest
electrons liberated at C, then none of the electrons can leave
C andi;=I,—I,. These statements will be true only if the
difference of potential (v) between B and UC is not great
enough to alter the path of the primary or reflected positive
particles. An examination of the relation between 7, or 73,
and v shows that, if V was not less than 400 volts, this
condition could be fulfilled toa high degree of accuracy.
In measuring 2, and 73, v was +4 and —40 volts respectively;
any uncertainty in P in the results owing to the condition
not being fulfilled exactly certainly does not amount to more
than 2 per cent. We may tnen take

pees
ty

P]
Roe
ZI

When V is less than 400 another method of measuring P
was adopted. C was pushed forward till it occupied the
position C’ indicated by the dotted lines. D was maintained
at a potential 400 volts higher than that of A, B, and C’,
which were all at the same potential. This difference of
potential was, of course, sufficient to ensure that no reflected
rays fell on D; it appeared from preliminary experiments.

Tonization by Positive Rays. 787

that it was also sufficient to ensure that all the electrons
liberated at GC’ were absorbed in D. If % is the current
which with this arrangement is received by D, then
i,=13 and P=—i,/i,;. 1, was measured as before by with-
drawing the plate to U. No precise measurements of R
could be made in this case, but everything pointed to it being
not greater than 0:02.

5. For values of V which were obtainable by means of
cells, the homogeneity of the rays was investigated by
noting the variation of 7; when the potential of B, C, and
D was varied relative to A. It was found that, when B, C, D
were raised from the potential of A to very nearly that of S
the value of 2, did not decrease 10 per cent. ; accordingly at
least 90 per cent. of the rays entering B and striking C
must have had an energy not differing by more than 1 or
2 volts from V. Similar experiments were difficult when.
the influence-machine or induction-coil were used to obtain
V, for the potential obtained in this manner is not perfectly
steady; their variations caused large induced currents in C.
But there is no reason to suppose that if a small steady value
of V gives homogeneous rays, a large steady value should
give heterogeneous rays—so long, of course, as there is no
sion of an ordinary discharge passing between S and A.
The current between S and A was approximately saturated
when V was 250 volts, but continued te increase slightly up
to the highest potentials used.

VOLTS O 19000 20000 30000 40000 500g0
6. The results obtained are shown in figs. 2, 3, 4.
Figs. 2 and 3 give the results for P, fig. 4 those for R.
3 EH 2

0:5

788 Dr. Norman Campbell on

Fig. 3 merely shows the part of the curve for small values
of V on a scale larger than that of fig. 2. The continuous
and dotted curves in fiz. 2 (and the marks o and x, for

Fig. 3.

0:04

a-0l

VOLTS O 100 200 300 400

the corresponding observations) refer respectively to the
results obtained with the induction-coil and the influence-
machine. Not all the observations are plotted, but those
which are given are thoroughly representative.

Fig. 4.

VCLTSO {0000 20000 30000 40000 50000

It will be observed that there is some difference between
the observations with the influence-machine and those with
the induction-coil, the values of P for the latter being some-
what lower. The difference cannot be due to the coil giving
a “reverse current,” for the presence of such a current
would decrease i; and increase 7, (since cathode rays also
cause ionization) and so increase P. I am inclined to think
that the smaller values of P with the coil are due to a lack
of homogeneity of the positive rays and the presence of some
possessing energies considerably lower than the maximum
indicated by the spark-gap, a conclusion which is enforced by
the fact that the results were much more regular and con-
sistent when the influence-machine was used. In an attempt
to extend the observations with the machine to higher values

- o .
of V,a serious accident to the apparatus occurred. Since the

Lonization by Positive Rays. 789

observations must necessarily be interrupted for some time,
and since it is proposed so to rearrange the apparatus that
direct observation of the speed of the particles can be made
by the use of a magnetic field, it appears better to give
the results already attained than tv wait for a complete
elucidation of this matter which, I feel sure, will not affect
the main conclusions.

A limit to the values of V which could be investigated
was set by the occurrence of a discharge between S and A.
If this discharge had been regular it would probably not
have made measurements impossible; but, as might be
expected in so high a vacuum, the discharge appeared as a
flickering green phosphorescence on the glass walls, while the
current through the tube varied wildly. Work at higher
potentials will probably be possible only if the continual
evaporation of gas from the walls of the vessel is obviated by
the metheds employed in the making of X-ray bulbs. The
highest reading of V for which any satisfactory reading was
obtained was 54,000 volts (equivalent to a spark-gap of
2°7 cm.) ; the corresponding value of P was 0°5; it is not
shown in fig. 2.

7. No very great interest attaches to the values of R
plotted in fig. 4. The readings obtained are somewhat
irregular, but it must be remembered that an error of
1 per cent. in 2, or 723 means an error not less than 10 per
cent. in the value of R. The greatest value of R obtained
was about 0-1, agreeing with the value assigned by Fiicht-
bauer. KR does not decrease very rapidly with the energy
of the primary rays until that energy is jess than 1000 volts.
For V=400 I could detect no difference between 2, and 7; ;
R was certainly less than 0°01. When V was greater than
30,000 volts, the readings for 7; became so irregular that no
satisfactory values for R could be obtained; but the curve
has been extrapolated beyond this limit for reasons which
will be noted immediately.

8. Fig. 3 shows that the curve for P approaches the axis
of zero ionization asymptotically. There is, therefore, no
definite ionization potential such as is found for cathode
rays. This conclusion agrees with that of v. Bahr and
Franck for the ionization of a gas ; but while they find an
appreciable ionization of a gas for potentials as low as
10 volts, I have been unable to detect any sign of ionization
below 40 volts. For 80 volts P is about 0°0005; for
40 volts it is certainly less than 0°0001. The difference in
this respect between a metal and a gas may be due only to
the possibility of detecting smaller ionization in the case of

790 Dr. Norman Campbell on

the latter: the authors quoted do not state the lowest value
for the ionizing power which they were able to detect. It
was probably lower than that which could be detected in
these experiments, but it is impossible to say whether this
difference is sufficient to account for the difference in the
limits of appreciable ionization.

9. The main interest of the experimental results lies in
fig. 2, which shows that the ionization increases to a
maximum at about 38,000 volts, and thereafter falls rapidly.
This conclusion is so surprising that something should be
said about the reliability of the measurements. There are
three possible sources of error.

(1) The measurements made with the largest values of V
were generally somewhat irregular and some of them were
rejected altogether. The measuring apparatus was shielded
effectively from the effects of the induction-coil; but as
soon as B or C was connected to it, the electrometer
fluctuated, even when 8 was not heated and there was no
appearance of a discharge from S to A. This effect may
have been due to electrostatic induction through the tube
opening into B, or, more probably, to charges creeping
along the inner surface of the glass vessel with which A, B,
and D were in contact only ata few points. The measure-
ments of ¢;, when B and D were connected to the measuring
apparatus, were much more irregular than these when
onty C was so connected ; indeed, measurements of 2, were
practically impossible when V was greater than 35,000 volts.
Though this difficulty was troublesome, there is no reason to
believe that it had any effect but to make the observations
vary somewhat widely about the true mean. All the
measurements which in other respects seemed at all reliable
agreed in showing a decrease in P for the highest values of
the V.

(2) It is uncertain that the energy of the rays falling on C
is actually that corresponding to the potential indicated hy
the spark-gap. There appears to be no method of removing
this source of uncertainty other than measurements in
a magnetic field which will permit the veloeity of the
rays to be found. Such measurements are about to be
undertaken. But it is to be noted that the decrease in P
will not be wholly illusory unless the average energy of
the rays decreases notably as the equivalent spark-gap
increases.

(3) It has been mentioned that at the highest values of V,
2, could not be measured. The values of P plotted are those

Tonization by Positive Rays. 791
given by

lg—t3

tata
% and 23 being measured while the values of R were taken
from the extrapolated part of fig. 4. This procedure doubt-
less appears rash; but it must be remembered that, if the
real values of R are greater than the assumed, the real
decrease in P will be more and not less marked than that
shown. If, on the other hand (and it will be noted presently
that this alternative is more probable), the real values of R
are less, then, since R cannot be <0, the values of P given
will be at most 10 per cent. too small. It is impossible by
any assumption concerning KR to explain away the apparent
decrease of P after the maximum. It may be added that
any serious error on this account is highly improbable, for
the measured values of 73 were constant, so far as could be
detected, for all values of V greater than 20,000 volts so
long as the conditions of S (temperature and so on) were
maintained constant. ‘The decrease in P shown in the curve
was due to a decrease in 7g, not to a change in 23, and must
therefore be attributed to a change in the electronic current
from C. |

10. Assuming that the form of the curve relating P and
V is in the main correct, I think that we can obtain some
idea of its significance by the light of a very interesting
suggestion due to Ramsauer * Noting that the velocity at
which #-rays have their maximum ionizing power is nearly
the same as the velocity at which electrons have their
maximum ionizing power (about 10° cm./sec.), Ramsauer
suggests that the ionizing power of a charged particle is a
function of its velocity rather than of its ener gy, so long
us this velocity is above a certain value. He shows that the
form of the Bragg ionization curve for a-rays can be deduced
with considerable accuracy by assuming that the ionization
of a-rays of a given velocity is always proportional to (about
10 times greater than) the ionizing power of electrons of the
same velocity. The limiting velocity, at which this relation
ceases to hold, is the velocity of an electron (about 2 x 108
em./sec.) corresponding to its ionization potential (11 volts
in hydrogen) : particles which have less than this velocity,
whether they be electrons or atoms, cannot penetrate within
the atom, and the ionization (if any) which they produce is

* K. Ramsauer, Jahr. d. Rad. uw. Llek. ix. p. 515 (1912).

792 Dr. Norman Campbell on

determined by actions quite different from that which results
in the ejection of an electron from an atom into which the
ionizing particle penetrates.

Now the experiments which have been described do not
confirm completely Ramsauer’s suggestion. Garratt * has
shown that of the positive particles emitted from a heated
phosphate some 10 per cent. are hydrogen atoms singly
charged. According to Ramsauer, such particles ought to
begin to display the high penetration and high ionizing
power of «rays when their velocity is that of 11-volt
electrons ; such a velocity they would attain by falling
through about 20,000 volts. Assuming, then, that Garratt’s
conclusion applies to the conditions of these experiments,
there ought to have been found, according to Ramsauer, a
rapid increase in the value of P for values of V greater than
20,000. No such increase was found, and so far the more
advanced developments of Ramsauer’s theory are rendered
doubtful.

On the other hand we may, perhaps, express the funda-
mental idea of Ramsauer’s theory in the following way :—
The great difference in penetrating power and in ionizing
power between a-rays and canal-rays is physically similar to
the great difference in the same respects between electrons
of which the energy is and of which the energy is not greater
than 11 volts. In both cases he supposes the difference to
arise from a difference in the power to penetrate the atom.
Now in my experiments it is clear that the values of V (if
there are such) for which the positive particles investigated
become a-rays have not been reached ; accordingly it is to
be expected that the relation between P and V should
resembie the relation between these quantities for electrons
of which the energy is less than 11 volts rather than that of
electrons of which the energy is greater than 11 volts.

This expectation is fulfilled. From yv. Baeyer’s work fT
we know that as the energy of cathode rays falling ona
metal plate is increased from 0, the number of electrons
leaving the plate increases to a maximum at 5 volts and
then falls to a sharp minimum just before 11 volts, where
the rapid rise begins.

It is generally believed (for reasons which seem perfectly
adequate) that these electrons leaving the plate when the
energy of the incident cathode rays is less than 11 volts
represent, not an ionization of the plate, but a reflexion of
the incident rays. Nevertheless it is possible that the

* A. I. Garratt, Phil. Mag. xx, p. 573 (1910).
Tt O. v. Baeyer, Phys. Zeit. x. p. 176 (1909).

lonization by Positive Rays. 793

analogy between the maximum with cathode rays at 5 volts
followed by a fall, and the maximum with positive rays at
38,000 volts also followed by a fall, is not wholly faise.
The cause of both falls may be the same, namely the
beginning of the stage at which the rays penetrate the
surface of the plate.

11. It is suggested then that the fall of P for values of V
greater than 38,000 volts represents the beginning of the
stage at which the positive rays acquire one of the properties
of a-rays—the power of penetrating atoms. And if this be
so, we should expect that it would be followed shortly by the
acquirement by the rays of the other property of «-rays, a
great lonizing power ; we should expect that P after falling
to a minimum should increase rapidly to a value charac-
teristic of a-rays, and the form of the curve suggests,
perhaps, that this rise should occur for some value of V less
than 100,000 volts. It may, then, be possibly capable of
observation.

If this view is correct, we should expect R as well as P
to decrease after the maximum of P is attained. Unfortu-
nately, as has been said, no evidence on this point is available.
It is not to be expected that a similar fall in ionizing
power would be found it the observations were made ina
gas in which the rays are completely absorbed. For if, in
the process of absorption in a gas, the speed of the rays is
reduced to zero, it is almost impossible that faster rays should
possess a smaller total ionizing power than slower rays : they
could only possess (as in the case of a-rays) a smaller ionizing
power over some portion of their range.

It remains to consider whether the view suggested is
supported or refuted by any known facts. A definite
penetration of atoms by the positive particles of canal-rays
does not seem to have been observed. Goldsmith * has
described observations which appear to show that such
particles can make their ways through sheets of mica about
0:005 mm. thick, even when their energy is as low as
10,000 volts. But, since he could find no trace of rays after
penetration, even when the initial energy was 35,000 volts,
itis probable that such particles make their way between
the atoms rather than penetrate through them, as do a-t “ays.
On this side, then, there is no evidence that their penetration
does not begin at about 38,000 volts.

On the other side, it is almost impossible to fix the least
energy which confers on particles the properties of an a-ray.
The least velocity of such rays which has been measured is

* A.N. Goldsmith, Phys. Rey. ii. p. 16 (1918).

794 Lonization by Positive Rays.

about 5x 10° cm. sec., corresponding to an energy (for a
single electronic charge) of 5 x 10° volts; such rays are about
3 mm. from the end of their range in air. Their energy
when they cease to penetrate must be considerably less, for
it falls rapidly towards the end of their range, but no definite
limit can be assigned. Moreover, it is clearly rash to assume
too great a similarity between the properties of doubly
charged helium atoms and singly charged hydrogen atoms.
But at least nothing is known inconsistent with the view
that the first stages of the conversion of canal-rays to a-rays
begins at about 38,000 volts and is not complete at 50,000
volts. It is hoped that more light may be thrown on the
matter by an improved apparatus in which higher values of
V can be investigated (it does not seem impossible that 10°
volts might be attained), the reliability of the measurements
somewhat increased, and more information as to the nature of
the particles obtained by deflecting them in a magnetic field.
12. In conclusion a numerical coincidence may be noted
which is probably without theoretical significance. The
maximum value of the ionization per incident particle ob-
served in these experiments is within 10 per cent. the same
as the maximum value observed with cathode rays, namely
that at about 280 volts energy. The copper plate was
throughout in the “state A” described in a recent paper”.

Summary.

The liberation of electrons from the surface of a copper
plate struck by the positive particles from heated phosphates
has been studied.

Figs. 2 and 3 give the relation between P, the number of
electrons liberated by each positive particle, and V, the P.D.
through which the particles have fallen.

Fig. 4 gives the relation between V and R, the reflexion-
coefiicient of the positive rays.

The significance of the results shown in fig. 2 is discussed.
It is suggested that the fall in P for the largest values of V
may be due to the beginning of the stage at which the
particles can penetrate the surface-layer of the metal, exhi-
biting one of the properties which distinguish a-rays from
canal-rays. At the highest potentials which could be studied
no indication was obtained of the acquirement by the positive
particles of the other distinctive property of a-rays, the great
ionizing power.

Leeds, March 1915.

* N. Campbell, Phil. Mag. xxix. p. 869 (1916).

LXXXITI. Phe Quantum-Theory of Radiation and Line
Spectra. By Wittiam Witson, PA.D., University of
London, King’s College *.

N his able report on Radiation and the Quantum-Theory
Prof. Jeanst, dealing with theories of line spectra,
remarks that Bohr’s assumption is ‘‘not inconsistent with
the quantum-theory and is closely related to it.” The
possibility therefore of deducing the results of Planck and
Bohr from a single form of quantum-theory naturally suggests
itself. Such a theory is developed in the present paper, and
it will be seen that it contains that of Planck (in one of its
forms) as a special case and, while formally distinct from
Bohr’s theory, leads to the same results when applied to the
Rutherford type of atom in which an electron travels in a
circular orbit round a positively charged nucleus.

This theory is based on the following hypotheses :—

(1) Interchanges of energy between dynamical systems
and the ether, or between one dynamical system and another,
are “catastrophic” or discontinuous in character. That is
to say, each svstem behaves as a conservative one during
certain intervals, and between these intervals are relatively
very short ones during which definite amounts of energy
may be emitted or absorbed.

(2) The motion of a system in the intervals between such
discontinuous energy exchanges is determined by Hamil-
tonian dynamics as applied to conservative systems. It will
be convenient to speak of a system, during such an interval,
as being in one of its steady states.

(3) Let q, Go, -- P1,P2,... be the Hamiltonian positional
and impulse coordinates of a system in one of its steady
states, and let L be its kinetic energy, expressed as a function
of gi, Jz,-..and gj, go,.-... This function is homogeneous
and of the second degree in gq, gs,..... If L contains
products gy gs (73s), we sball suppose them to have been
removed by a substitution of the form :

r= Colep qu + dp go! alrite st One Gil
and we have therefore
L= tAiqy a tA.g? Sip tcnelnts gAngn’s
* Communicated by Prof. J. W. Nicholson.

+ J. H. Jeans, Phys. Soc. Report on Radiation and the Quantum-
Theory, p. 51 (1914).

796 Dr. W. Wilson on the Quantum- Theory

and further

eb Noi S) Gis
2L=q, —— +...4¢9, ——
ag he aaa +q Oar

and consequently

2L, = MP1 )
2, =Y |
2 2Pe2
r (1)
SRS pee |
Zi = One )
where L,=4A,97’, L,=4Aoq,”, de.

We assume that the system in one of its steady states has a
pe ee oan | :
period a corresponding to q1, ae corresponding to gz, and so
1 2

on. From the equations (1) we get

2 | Ldt=| pidq

and similar equations containing L,, L;, &e. Our third
hypothesis can now be stated as follows :—The discontinuous

energy exchanges always occur in such a way that the steady
motions satisfy the equations:

Jrrda = ph |

ome) > 0: (eh oe) “6. (see g)

where p, o, T,...are positive integers (including zero) and
the integrations are extended over the values ps; and q; corre-

sponding to the period a . The factor his Planck’s universal
v

constant. It will be convenient to denote these integrals by
H,, Hg, ... respectively. Shan ae .
We shall now consider the statistical equilibrium of a
collection of N similar systems of the type specified above.
Let Noor... be the nnmber of systems for which Hi=pA,
H,=och, H;=th, and so on; and No'o'r’... the number of
systems for which H,=p'h, H,=o'A, H,=7h...~ Leta

further write

Noor...
foot... = yy - £2 io" (3)

of Radiation and Line Spectra. 197

so that we have

bey pore: . . ° (4)

o=0 o—0 rT=0
For the sake of brevity we shall say that Noor... systems
are on the locus (pat ...), No'o’r'... on the locus (p'a't' . . .),

and soon. Wehave for the energy of the whole collection
of systems the following expression:—

p=O 6=0O T—D

H=N .» + Epor...foor... : (5)

where Epor... is the energy of a system on the locus
(pot ...). If P is the number of ways in which N systems
can be distributed, so that Noor... lie on the locus
(Gam. .), No'c.. on the locus (@o/7 ...), and so on, we
have
N!
oy (Noor...)! (No’o’r’...) A SCOSD AE nae T (6)

We shall call P (after Planck) the ‘ thermodynamic
probability ” of the distribution in question, and identify the
quantity

with the entropy of the assemblage of systems; the quantity
k is the entropy constant. We may assume Noor... ,
No’o’r’..., and a forteorti N to be individually very large
numbers, and therefore, by Stirling’s theorem,

nX
(N Noor we Oe i ) uP
[Mores be DROME ae ;
This last equation, together with (3) and (7), leads to the

following expression for the entropy of the collection of
Systems;

2) ie.e)
vN <= » ?
p=—kN>22... Soar... log oor... ° : (8)

The condition for statistical equilibrium is expressed by
dd=0,

where the variation is subject to the total energy being

798 Dr. W. Wilson on the Quantum- Theory

constant, and also to equation (4). We easily obtain, by the
usual variational method,

i log fpor... of Boor... -y=0

or foorie. = Ae Peers... |
The value of A is determined by equation (4). The
et where T is the
absolute temperature of the collection of systems. From

(8) and (9) we find

quantity 8 can be shown to be equal to

lee)

j= EN DS NA ae EP Or ake (log A— Poor...)

00 0

or $= —INlog A+ieN=E2., oi Alber. |-e0) aaa

SISA

and therefore by (5) and (9)
g=—kNlop A+kBE. . . oe Gy
On differentiating with respect to 8 we get

ED ws AN 8S E+E, te ci

aa ds
and since
PLAS SS GB Eye
. 0 0 0
we have
da l SSS PE ey
= 5+ A>... Epor...? ‘ «
dlogA EH
or A iahan sy

Substituting this in equation (11) we see that

dp tg iB
or = = k@.

Therefore £6 = st The law of distribution of the systems

of Radiation and Line Spectra. 799

among the different loci is therefore expressed by

OO one
Aum ete) 1 ;

0, 00 00, Epor...
el Men annyen 7
000

when equilibrium has been attained.
Hiquations (5) and (12) give us, for the average energy
of a system,

Hoor...

SS EAE
Ea a Wie Or Wel om
; TT ONORN0
R= oO 00, Qn IBDORP oo (13)
pajpe) ps 2 aaa inn zeapl
OFONO

Theory of Radiation.

The foregoing results are very general. We shall show
that they include Planck’s theory (one form of it at any
rate) as a special case. We may write the equation of
motion of one of Planck’s oscillators, when in a steady state,
in the form

EG) Were
mT + Kq=0.

The most convenient form of solution for our purpose is
G— cos(2 mys 0), enn yes nly

where R and @ are the constants of integration, and there-
fore
p= —2mrvmRsin@mvi—@). . . . (1d)

The energy of such an oscillator is easily shown to be
SPANO Rc thee ail dy fe a ODS
Now we have, from (14) and (15),

t+ = i+ -
(ag =47?y?mR? | sin?(2rvt —O)dt,
t t

and consequently
ph=27?ymR’*,
and therefore, by (16),
1p 7) Ae MO Ne aa aL esa).

800 Dr. W. Wilson on the Quantum- Theory

On substituting this value of Ep» in (12) we find that
Ve ee
(A ey it) e MY, . {ee
Therefore the law of distribution of the oscillators among
the different loci is precisely that given by Planck *.

From (13) and (17) we deduce for the average energy of
an oscillator

H= ae 5 ° . ° ° e (19)

a well known result in Planck’s theory. It may therefore
be said that the proposed theory includes that of Planck (at
least in one of its forms).

We may regard the ether as a collection of oscillators
which, through the medium of matter, exchange energy
with one another. The number of these, per unit volume, in

the frequency range between v and v+dy, has been shown
by Jeans and others f to be

Sarv'dv

3 b)

where c is the velocity of radiation in the ether. The most —
probable distribution of these oscillators among the loci men-
tioned above, i. e. the distribution corresponding to maximum
entropy, is one which makes their average energy

hv
hy

Pe)

and therefore we get for the energy within the frequency
range v to v-+dy,

Srhv’® dv
U,dv= 3 i h P)
ekl —j

whichjis Planck’s radiation formula.

Theory of Line Spectra.

We shall now show that Bohr’s t assumptions, in so far at

any rate as we restrict ourselves to the type of atom or

* M. Planck, ‘Theorie d. Warmestrahlung,’ p. 139, equations (220) and
(227), second edition.

+ J. H. Jeans, Phil. Mag. x. p. 91 (1905) ; M. Planck, Joc. ct. p. 175.

t N. Bohr, Phil. Mag. xxvi, p. 1 (1913).

of Radiation and Line Spectra. 801

emitting system to which Bohr’s theory has been applied
with some measure of success, can be immediately deduced
from the theory outlined above. ‘The systems which he
assumes to emit the hydrogen, helium, and other spectra are
characterized, in their steady states, by constant kinetic
energy, and by one positional coordinate q. ‘The hypothesis
expressed by equations (2) takes, for such systems, the form

t+}
21. | dt=ph
t
or L= ee, a Society! yaa trea 20)

and, since L in these systems is numerically the same as
Bohr’s* W, we see that (20) expresses Bohr’s principal
hypothesis. A further assumption made by Bohr is that the
energy emitted by an atom, in passing from one steady state
to another, is exactly equal to hvy;, where v, is the frequency
of the emitted radiation. Now according to the foregoing
theory, since the energy of the ether vibrations of frequency
v must be equal to rhy (equation (17)), where r is a positive
integer or zero, it follows that the energy emitted by an
atom (like those assumed by Bohr) must. be equated to

ryhyvy, -{- rehvy+ es ey ° ° ° ° (2, 1)

where the 7’s are integers, not necessarily all positive, and
¥;, vo... are the frequencies of the corresponding ether
vibrations. The present theory therefore includes ae
second assumption of Bohr’s as a special case. 7

The conclusion that energy emissions to the aillen are
represented by an expression of the form (21), and are not
necessarily monochromatic in all cases, receives some support
from Prof. Barkla’s experimental work on X- radiation 7.
It is noteworthy that Barkla finds that the energy absorbed
from the primary radiation, during the production of the
“ fluorescent” radiations, is equal to

dmv? + hryg thy
per electron emitted, the first term representing the kinetic

energy of the emitted electron and vx, v;, the frequencies of
the ‘‘ fluorescent ” radiations.

* N. Bohr, loc. cit.
+ C. G, Barkla, ‘ Nature,’ 4th Mar. 1915.

Phil. Mag.S. 6. Vol. 29. No. 174. June 1915. 3F

802 Mr. K. K. Smith on Negative

The main object of this paper is to show that the form of
quantum-theory which seems necessary to account for line
spectra is not really distinct from that originally proposed
by Planck, and the subject of its further application to line
spectra and other phenomena may be left for a future
publication.

In conclusion I wish to express my thanks to Professors
J. W. Nicholson and O. W. Richardson for their advice and

criticisms.

Wheatstone Laboratory, King’s College,
March 1915.

LXXXIV. Negative Thermionie Currents from Tungsten.
By K. K. Suira, A.B., Fellow in Physics, Princeton

University *.
Introduction.

TENE emission of negative electricity from an incandescent

metal or carbon filament has been the subject of several
investigations t. ‘The number of electrons carried from the
filament to a neighbouring positively charged electrode
increases very rapidly with the temperature. The exact
quantitative relation between the number of electrons emitted
and the temperature of the filament was established by
Richardson, and has been verified by the experiments of
others. It was assumed that the emission is determined
simply by the number of electrons whose kinetic energy is
sufficient to overcome the forces tending to prevent their
escape from the metal.

This relation is expressed by the formula
aa
a= aes D,

where i is the saturation (maximum) current in amperes per
square cm. and T is the absolute temperature. ‘The quantities
a and b are constants, the latter being proportional to the
work done by an electron in escaping from the metallic
surface. On this view, a pure metal in a perfect vacuum
would give a thermionic current which would be a function
of its physical properties only. In any actual case the
presence of traces of impurities or gases would presumably

* Communicated by Prof. O. W, Richardson, F.R.S.

+ Richardson, Camb. Phil. Proc. vol. xi. p. 286 (1901) ; Phil. Trans.
A, vol. cci. p. 497 (1903) ; H. A. Wilson, Phii. Trans. A, vol. ccii. p. 248
{1908) ; Deininger, Ann. d. Phys. xxv. p. 804 (1908).

Thermionic Currents from Tungsten. 803

have secondary effects, but would not be required to explain
the existence of the current.

In view of some recent experiments”, which seemed to
east doubt upon the above explanation of thermionic currents,
the following investigation was undertaken. Professor
Richardson has already published the results of the earlier
experiments t. The object has been (1) to make a detailed
quantitative investigation of the negative thermionic emission
from tungsten over a large temperature range, and (2) to
discover, if possible, more evidence as to the conditions
which determine the emission.

Haperimental Arrangements.

The filaments used in these experiments were all taken
from a spool of pure ductile tungsten furnished by the
General Electric Co., Schenectady, N.Y. They were
‘0-041 mm. in diameter, and the lengths used in the different
lamps varied from 2°5 cm. to about 9 em. ‘They were
electrically welded in hydrogen to copper leads, which, in
turn, were welded to platinnm wires. Then the wires were
mounted axially in cylindrical glass tubes and sealed in.
The tubes were 3°2 cm. in diameter, and contained copper
gauze cylinders 2°6 cm. in diameter. Separate wires con-
nected these anodes to the outside of the lamps.

Wel (C

x)
|
|

Vig. 1 shows the arrangement for exhausting and heatin g
the lamp LL, in the vacuum furnace. ‘The tioure is purely
diagrammatic and is not drawn to scale. The U-tube, U

* Pring and Parker, Phil. Mag, vol. xxiii. p. 192 (1912); Pring, Proe.
Eth Soc. London (A), vol. Lxxxix, p. 344 (1918) ; Fredenhagen, Deuisch.
Phys. Gesell. Verh. vol. xiv. p. 884 (1912).
+ Richardson, Phil. Mag. vol. xxvi. p. 845 (1913).
ee hee

804 Mr. K. K. Smith on Negative

could be surrounded with a Dewar flask containing liquid
air. The pressure indications were read on the McLeod
gauge, G. By raising or lowering the mercury at T, the
lamp and gauge could be shut off trom the vacuum-pump, or
connected to it. The tube C was half filled with coconut
charcoal, and could be surrounded with liquid air when it
was desired to reduce the pressure as much as possible. A
bulb, P, containing phosphorus pentoxide, was attached
between © and the Gaede mercury pump, and auxiliary
pump. The furnace itself was evacuated by a separate
pump. |

The lamp was exhausted and heated at the same time.
The furnace reached a maximum temperature of about
600° C., which was maintained for several hours in order to
get rid of absorbed gases. At first considerable quantities.
of gas were given off, but the pressure gradually decreased
with prolonged heating, and finally became quite small.
The furnace was then allowed to cool slowly, liquid air was:
applied at U and ©, and the filament was heated by an
electric current. The cylinder was charged positively, so
as to be bombarded by electrons from the filament. At
intervals the amount of gas still being driven off was
estimated by closing the trap T for five minutes, and noting
the increase of pressure, if any. The details of some of
these experiments, which have already been published *,
show that the observed thermionic currents were too large
to be ascribed to the evolution of gas from the filament, or
to an action depending upon impacts between the gas mole-
cules and the filament. After the pressure indicated on the
gauge had become practically inappreciable, the lamp was:
either sealed off immediately and removed, or the observa-
tions were taken on the unsealed lamp.

Measurement of Currents.

The lamp filament was usually made one arm of a Wheat-
stone’s bridge. The positive terminal of a battery was
connected to the receiving electrode, and the other terminal
to the positive end of the filament. An electrometer having
a sensibility of 790 divisions per volt, and a capacity of 130
electrostatic units, was used to measure the smallest currents.
A condenser whose capacity could be varied from 0-001
microfarad to 1 microfarad was used in parallel with the
quadrants when necessary. With larger currents, resistances
varying from 100 ohms to 1 megohm were put in the

* Richardson, Joc. cit.

Thernionic Currents from Tungsten. $05

thermionic circuit, and the quadrants were connected to the
ends of these resistances in turn. For currents greater than
1 microampere a unipivot galvanometer with variable shunts
was used. With these arrangements, it was possible to
measure the emission over a range from 107’? amp. up to
lampere. Owing to the high melting-point of tungsten,
it was possible to obtain thermionic currents of the order of
magnitude of the heating currents. In what follows, the
thermionic current will usually be expressed in terms of unit
area, otherwise it is to be taken as the total current from the
filament.

Measurement of the Filament Temperatures.
Pp

Through the kindness of Dr. Irving Langmuir, a curve
was obtained with the wire, showing the temperature of the
filament asa function of the current carried by it. This
curve was determined by photometric measurements on a
special lamp, using a piece of the same wire that was used
in these experiments. As a check on the temperature deter-
minations some smal] lamps were made from these filaments,
and were used in an optical pyrometer of the Holborn-
Kurlbaum type, constructed in this laboratory. These lamps
were calibrated by observations on a black-body furnace at
the melting-points of copper and nickel. The results of the
two methods were in satisfactory agreement. In the earlier
experiments it was customary to determine the resistance
each time an observation was taken, since the bridge galva-
nometer was of course much more sensitive than the
ammeter to small variations in the temperature of the
filament.

The effects produced by a thermionic current in the
ordinary Wheatstone’s bridge circuit have been considered
by Richardson and Cooke*. As far as the present experi-
ments are concerned these effects are of no importance, so
long as the thermionic current is small in comparison with
the heating current. Tor very high temperatures, however,
the large thermionic currents cause the temperature estima-
tions to be much too high. In fact, even if the bridge is
not used at all, the temperature of the filament will be over-
estimated if a large thermionic current is flowing, and,
furthermore, the two ends will be unequally heated. Fora
given cur rent per square centimetre these disturbing effects
increase with the length of the filament.

* Phil. Mag. vol. xx. p. 173 (1910).

806. Mr. K. K. Smith on Negative

EXPERIMENTAL RESULTS.
Lam ip lis

Lamp 1 contained a filament 5 cm. long, and a cylindrical
anode. After the lamp had been heated in the furnace for
9 hours, liquid air was applied at U and C, and the filament
was glowed for an hour and a half at various temperatures
above 2500° K. The thermionic currents varied from 3 to
40 milliamperes. The trap T was closed at intervals, and
the increase in_ pressure during five minutes was noted. In
the first period the pressure rose from 0:005 to 0:050 micron :
in the last period the increase was from “0.” to 0:003 mnicron.
The furnace (still above room temperature) was then opened,
and the lamp was sealed immediately. The pressure i

cated was “0.”

Professor Richardson measured five series of thersaleeae
currents greater than 1 microampere, the potential difference
between the anode and the cathode being 120 volts. The
results are shown in fig. 2, in which log 7 —1/2 log T
is plotted against 10000/J'. Series 1 A, 2A, and 3A were
taken with decreasing temperatures, the others with increasing
temperatures. It will be noticed that the points for the 3rd
series are shifted to the left of the preceding ones. At first
sight, this would seem to indicate less current ata given
temperature is the heating was continued. Jt is believed,
however, that this is not to be ascribed primarily toa change
in the emitting power of the surface, but rather to a change
in the resistance of the filament, owing to excessive heating.
After the first series of readings, the filament was heated for
over an hour at a temperature above 2500° K. ‘The ther-
mionic current was turned off to prevent unequal heating,
which is likely to burn the filament outat one end. Between
the 2nd and 3rd series the filament was heated to a still
higher temperature for 20 minutes, and the resistance in-
creased from 1625 to 1670 units, or about 2°8 per cent. The
same thermionic current was obtained with R=960 after
overheating as with R=945 just before. ‘These resistance
changes as the filament evaporates cause the temperatures
to be overestimated, and are sufficient to account for the
above variations in thermionic current.

In series 4, thermionic currents (not shown in fig. 2) up
to 320 milliamps. were obtained, or 8 times as much as had
been obtained before the lamp was sealed off. Immediately
after this large current was obtained, it was found that at

T hermione Currents from Tungsten.

807

2480° K. the current was only 22°5 milliamps., whereas it
had been 58 milliamps. just before this. Series 5, taken

-3

al:
4

|
|
|
if
{
\

Fig. 2.

Lamp |/
Series| |
A I-A

5 P2,

@ 3

eee

cd +
5

immediately afterwards, shows the effects of this severe
heating in a more striking manner.

The filament was about

808 Mr. K. K. Smith on Negative

300° hotter than usual before the current was large enough
to be read on the microammeter. At this higher temperature,
1890° K., the current was now only 0:014 milliamp. per
sq. cm.,as compared to 5:36 milliamp. per sq. cm. at the
same temperature in series 2. In other words, the thermionic
current had been reduced to 1/380th part of what it had
been. It began to increase very slowly, although the tempe-
rature was constant. ‘The other points of series 5 were
determined at once, without waiting to see if the current
would become steady at the lower temperature. The current
increased with the temperature much more rapidly than
before, and approached the preceding series so that at the
highest temperature it was 60 per cent. of the current at
the same temperature in series 4. About 2500° K. a blue
glow appeared after continued heating, and the copper anode
became red-hot.

The above experiments were performed before Dr. Lang-
muir’s paper * on thermionic currents was published. They
confirm his conclusion that the effect of residual gases is
to decrease the thermionic current, especially at low tempe-
ratures. The fact that considerably larger currents, at the
same temperatures, have been obtained by the writer is
undoubtedly owing to better vacuum conditions. According
to the above paper, the normal vacuum curve was obtained
at a pressure of 0°07 micron, and gives the following values :
a=34x10° amps. per sq. cm., b=55,500. By the use of
charcoal and liquid air, as described before, it has been
possible to keep the pressure as low as 0:001 micron, or
less.

In Lamp 1, so long as the thermionic current did not
exceed those employed before the lamp was sealed, the
vacuum was practically perfect, and very large currents
were obtained. At 2000° K. 26 milliamperes per sq. cm.
were measured. Later, the excessive heating and bombard-
ment of the anode by electrons liberated occluded gas, which
reduced the thermionic current. The presence of gas is
proved by the blue glow which appeared. There is no
reason to suppose that any appreciable amount of gas was
present previous to the excessive heating. Tests have been
made at various times with an induction-coil, on similar
lamps which had not been overheated, and no indications of
gas were ever found. The potential difference and the size
and shape of the anode were such as to permit the normal

* Phys. Rev. ii. p. 450 (1918).

Thermione Currents from Tungsten. 809

current to flow unlimited by any space charge effect *. This
is proved by the fact that the currents were practically
saturated at 120 volts. Even with a blue glow in the lamp,
the points at the upper end of curve 5 fall far to the left of
the straight line. Hence this deviation cannot be the result
of a space charge effect, but is undoubtedly caused by the
large thermionic currents, which were of the order of mag-
nitude of the heating current f.

Lamp 2.

Lamp 2 contained a filament 85 cm. in length, and a
cylindrical anode of copper gauze. It was heated for 14
hours in all, and liquid air was applied at C, but not at U
(at first). The filament was glowed for about 34 hours, the
thermionic current during the last hour being 10 milli-
amperes. In the first 5 minute test with the trap T closed,
the increase of pressure was from 0:001 to 0:050 micron ;
in the last 30 minutes the increase, if any, was not more
than 0°0001 micron from “0.” Liquid air was then applied
to U to condense the mercury vapour, but no change in the
thermionic current could be detected. The lamp was then
sealed, and the thermionic currents were measured from
1050° K. up. The results (plotted in fig. 3) show that

within the limits of experimental error, the equation

aj gee
1=aT’e-T.

is satisfied throughout. (In all the figures the current is
expressed in amperes per sq. cm.)

In order to determine how the thermionic current depended
upon the voltage, tests were made at the following tempera-
ames lOTO 1350, M65) 1590, 7 lO, and (1825) 12
reduce the results to the same scale, the current with 200
volts has been taken as the unit measure in each case, and

* Lanomuir, loc. cit.

+ Since these experiments were completed, Dr. Dushman has published
(Phys. Rev. iv. p. 121, 1914) the results of some experiments in which
he observed the decrease in the thermionic current caused by bombard-
ment of the anode. The temperatures were above 2000° Kx. and the
maximum currents obtained appear to be about the same as those
observed by Dr, Langmuir. The values of the currents given for the
Coolidge X-ray tube (Phys. Rev. ii. p. 409, 1918) are larger, but not so
large as the writer's.

810 Mr. K. K. Smith on Negative
Fig. 3.

PSEEEEE pre aa

Poe timer ee aa
wt fa

ms Se Oe
eee ee

FOUR
Cee nea
we ca
a Se ae me
Coe ee

the ratios, which were the same for all the temperatures, are
shown in the following table:—

Current.
Volts. Current with 200 volts.
25
50 0°35
100 Or On.
300 1:07
400 1:14
500 121

Thermionic Currents from Tungsten. Sil

Table I. shows the values of the currents which best re-
present the emission observed under good vacuum conditions.
in Lamps 1 and 2. They are given by the values :

a = 6°7 x 10° (amps. per sq. cm.), & = 54760.

TasLe I.
Temperature, | Thermionic Current.
aa RR TNT NOUR er esRRTTTN tat
1050° K. 0:000000285 microamps.
1100 0:00000256 per
1150 0 0000227 sq. cm.
1200 0000169
1250 0:00106
1300 000583
1350 00282
1400 0-122
1450 0°476
1500 1:70
1550 5°60
1600 Wf
1650 49:0
1700 132°
1750 337°
1800 809°
1850 1880°
1900 4120:
1950 8730
2000 17800°
2050 35200:
2100 67000-
2150 124000:
2200 224000-
2300 674000:

2250 | 394000-

Lamp 3.

Lamp 38 contained a filament 5°25 cm. long and a copper
gauze anode. It was given the usual treatment and sealed
off, but the seal cracked while cooling. The apparatus was.
then arranged so that the thermionic currents could be
measured with the lamp in place in the furnace, and connected
to the Mcleod gauge, etc. After the lamp had been re-
heated and exhausted to a low pressure, the furnace was.
opened, liquid air was applied at U and ©, and a series of
observations was taken at once, beginning at low tempera-
tures. Ihe potential difference was 190 volts. At several
points it was noticed that the current was unsteady, and that
it increased with the time. In these cases, readings were
taken every minute for 5 minutes or so, and the average.
reading was taken. The pressure was “0” atthe beginning,

812 _ Mr. K. K. Smith on Negative

and did not exceed 0:002 micron until the current was as
large as 0°130 amp. (1°95 amp. per sq. cm.). With this
current the pressure increased (with the trap T closed) from
0:002 to 0°060 micron in 6 minutes. Currents as large as
0°350 amp. were observed, while the pressure increased to
0:600 micron, and then decreased to 0'400 micren. The
temperature was at least 3000° K., and the filament burned
out a few minutes later.

The results are shown in fig. 4. Points in the middle of
the curve fall along a straight line from which the value
b= 69,500 is calculated. At the lowest temperature (1950°)
the current is about 1/600 of the current observed with
Lamp 1 at the same temperature. At 2500° K. the corre-
sponding ratio is about 1/3, while at the highest temperature
the two currents are practically identical. ‘These results will
be discussed after they have been compared with those of
the next experiment.

Lamp 4.

This lamp was an exact duplicate of the preceding one.
After the usual treatment the filament was given its initial
heating (by a current) at 2100° K. This was continued for
34 hours, during which time the thermionic current increased
from 3 to 133 microamperes. The observations of the last
40 minutes are shown in the following table :-—

Time. | Pressure. Total Current.
10.42 p.m. 0-006 micron. 16
45 18
AT | 21°5
00 | 27°5
Sie | 36:4
53 42
05 66°5
nT ' 0-001
8 109
9 114
11.00 | 16
133

11.20 | 0-0001
| |

With a potential difference of 210 volts ihe 1st series of
observations at different temperatures was taken, after which
the thermionic circuit was broken, and the filament was
glowed for 40 minutes at temperatures varying from 2700°
to 3000° K. The 2nd and 3rd series were then taken in
close succession. During all these series the pressure was
“* 0,” according to the gauge, whether the trap T was closed

Thermionic Currents from Tungsten. 813

or not. This was true even after the filament was finally
burnt out. All the observations are shown in fig. 4, for
comparison with those of Lamp 3.

-2

-3

i
aN

1
OV

1
—)

&
Si foci ld ad

!
60"

3S
Taken together these two experiments show that the initial
negative thermionic current is very much less than that
obtained on subsequent heating. The actual difference would

be more obvious if the currents, instead of their logarithms,

814 Mr. K. K. Smith on Negative

had been plotted. This conclusion was confirmed in every
-one of the subsequent experiments. Under some conditions,
as in the last experiment with Lamp 1, and in some others
to be described later, it is possible that the current may show
_a decrease, but no emission less than the initial emission has
been observed.

The first observations on the two lamps (3 & 4) are in
good agreement at low temperatures, but at a certain point
the Ist series (Lamp 4) shows that the current is increasing
much more slowly with the temperature. The heating of
the filament at very high temperatures, with the thermionic
-circuit broken, was followed (series 2) by much larger
currents than before. At 2000° K. curve 2 bends sharply
to the left, and then again to the right, while curve 3 is
regular. Heating the filament would tend to remove im-
purities and absorbed gases from the filament. It is known, »
for example, that the oxide volatilizes in a vacuum without
the evolution of gas. On the hypothesis that initially there
-existed a surface layer which hindered the emission of
-electrons, we should expect to find larger currents in the
second series. ‘The falling offin the rate of increase with
the temperature, which occurred at about 2000° in the first
‘two series (Lamp 4), is caused by the bombardment of the
-anode. At low temperatures, and hence with smaller
-currents, the latter effect is not apparent. Heating the
filament to a high temperature without the thermionic
-current does not bombard the anode and free it from gas,
-although it does rid the filament to a large extent of what-
ever is hindering the normalemission. Heating the filament,
. -and allowing the thermionic current to flow, cleans both
| anode and cathode.

Lamp 5.

.
: This lamp was like the preceding ones, except that the
; filament was 6 cm. long. The observations were extended
| -over three days, and during this time there was always a
| supply of liquid air about U and C. Out of ten series of
observations the largest currents are plotted in fig. 4. The
: smallest currents were found when the filament was first
heated, and this agrees with the results of the other experi-
ments. With this lamp complete saturation was obtained,
| whereas the results already given for Lamp 2 show that the
- currents always increased somewhat with the voltage. When
voltages as high as 600 volts were first applied the results
were as follows: at 1800° K. the current was saturated at
from 200 to 400 volts, and then decreased with increasing

Thermionie Currents from Tungsten. 815

voltage. The measurements were repeated immediately
afterward, and it was found that the current had been
reduced one half. At the same time the pressure had in-
creased from 0:010 to 0°014 micron, owing, no doubt, to the
evolution of gas from the anode.

The gas pressure, however, was evidently not an important
factor in determining the thermionic current. For example,
the same pressure (0°001 micron) was indicated in series 1
and 8, although the currents were considerably larger in the
latter case. On the other hand, series 3, 8, and 9 agreed
quite well, although the corresponding pressures were 0-001,
0:010, and 0:100 micron respectively. Repeated heating
did not increase the emission beyond what is shown for this
lamp in fig. 4. In view of these facts it was believed that
the great reduction in current, as compared with the other
lamps, must be caused by condensable vapour, probably
water vapour. The lamp had been heated in the furnace to
500° C., or more, but there was the possibility that water
vapour had entered afterwards from below the furnace, even
though liquid air had been around the U-tube all the time.
This was the first lamp which had not been sealed shortly
after the furnace was opened. A slight change in the
apparatus was therefore made before the next lamp was set
up. The glass tube leading to the lamp was bent into a U
inside the furnace, so that liquid air could be applied close
to the lamp. Any vapour which might arise from the un-
heated tubing below the furnace would then be condensed
before it could enter the lamp.

Lamp 6.

This lamp contained two parallel filaments, 1°7 em. apart.
One filament ‘A’ was 8°5 em. in length, and the other
“B” 7-2 cm. It was then unnecessary to use a cylindrical
gauze anode, and thus one possible source of gas was
removed. All the platinum and copper connecting wires
were completely covered with melted glass before they were
sealed in the lamp, so that the only surfaces exposed on the
inside of the lamp were of tungsten and glass. At the
same time that the lamp was being heated in the furnace,
the two filaments were glowed in series for six hours at
temperatures varying between 2000 and 2500° K. As soon
as possible after the hot furnace was opened, liquid air was
placed around the new U-tube next to the lamp. Liquid
air had previously been placed around the other U-tube and
the charcoal tube as usual.

OE ss ——_ —_——_— |

816 Mr. K. K. Smith on Negative

Using filament “A” as the hot cathode, three series of
observations were made, and the results of the first and
last are shown in Table II. Following the 2nd series, the
cathode had been heated to 2700° K. for twenty minutes.
After this the currents were as large, at the same tempera-
tures, as the currents which had been observed in the sealed
lamps. The pressure was “0” during the whole time.
This experiment shows that the thermionic emission from
tungsten is not a secondary effect arising from the presence
of gas or condensable vapour, but must come from the metal
itself.

Teen Le
Current. Temperature.

(Amps. per sq. cm.) « A,” series 1. | “A,” series 3. ‘ B,” series 1.
DADA ORS mesh alae | aalal Anan a 1540 1695
ig) SRA ek PL ee ee iy Lai5S 1575 1740
Ars) Pee ARR. ONL! hah oe | 1795 1615 1785
2) wo Nich lr oh ah oe | 1840 1650 1835

SN Whee Ory see: Wy Paean | 1895 1695 1890

Immediately after the 3rd series with filament “A” as
cathode, filament ‘‘ B”’ was made the cathode, and “A” the
anode. ‘The results of the first series of observations under
these conditions are also shown in Table II. It will be seen
that the emission under the same vacuum conditions depended
upon which filament was used as the cathode. As stated
above, both filaments had been glowed in series, and other-
wise they had been treated alike, except that ‘‘ A” had been
heated to 2700° K. for twenty minutes. Above 1900° K.
the currents were limited on account of the small dimensions
of the anode, and after a certain value was reached the
currents could not be increased, with constant voltage, no
matter how much the temperature was raised.

The liquid air was then removed from the gauge and
furnace U-tubes, the trap T being closed. The pressure
increased from “0” to 0°25 micron, and the resistance of
the glowing filament, “ B,” began to increase slowly. This
was caused by the oxidation of the filament by the water
vapour released from the U-tubes. The resulting ionization
in the gas neutralized the space charge effect, and the current
(T=2180° K.) began to increase. Series 4, taken at once,
showed that below 1950° K. the currents had been decreased

Thermionie Currents from Tungsten. 817

by the formation of the oxide, but above this temperature
they were larger, for the reason just given. The following
table shows how much the currents at 2040° were increased
by the removal of the liquid air.

Volts. Total Current in Microamperes.
Liquid Air on. Liquid Air off.
P=0-U001 micron. P=0°250 micron.
20 1:05 =
35 oe 18
40 4° ; 350
60 O35 362
80 AT5 365
100 28-2 ha
120 40-4 368

Below 2000° K. the currents were steady, but above this
it was noticed that they increased with the time. The
following table shows one instance of this, as well as the
effect of replacing liquid air about the U-tube near the gauge.

Time. Total Current
(= ZO S key:
0 mins. 350 microamperes,
1 368
15 375
2:0 388
30 410
3°5 P=0 220 micron.
4:5 450
Liquid Air replaced.
50 102
5D 93
6:0 84
6°5 84
GOie LP =O 1705) ek6
8:5 86
9 86
constant.

The increase was probably caused by the more rapid re-
moval of the oxide at this temperature, or to the increase in
temperature due to the filament burning ; while the decrease
was caused by the absence of ions after the condensation of

Phil. Mag. S. 6, Vol. 29. No. 174. June 1915. 3G

818 Mr. K. K. Smith on Negative

the water vapour. The resistance of the filament remained
constant when liquid air was about the U-tube, otherwise it
increased gradually. The liquid air was removed once more,
and repeated measurements showed, as before, that the
currents increased rapidly with the voltage when liquid air
was around the U-tube; if it was not, the currents were
saturated. After the liquid air had once been removed the
vacuum conditions were never so good as before, but the
pressure was always so low (<1 micron) that ionization
by collision had no effect.

Lamp 7.

_ This experiment was undertaken in order to measure the
currents before and after the lamp was sealed. The filament
was 2°6 cm. long, and the anode was a cylinder of copper
gauze. ‘The lamp was heated 10 hours on one day, and
6 hours on the next day. After the first series of observa-
tions had been taken, the trap T was closed, and was kept
closed until the lamp was sealed 6 hours later. The pressure
was “0” before any measurements were taken, and no indi-
cation of pressure could be seen up to the time that the lamp
was sealed. After sealing, the liquid air was removed from
the U-tube, U, and three or four minutes later the pressure
in the gauge was 0°370 micron, but it did not increase
after that. This was the pressure of the gases that had been
condensed.

The results of the typical series are shown in fig. 5. The
first currents were less than those that had been observed in
the first heating of the other filaments. Afterwards they
increased, as can be seen in series 4. This condition was
not permanent, however, for after the filament had been
kept at a temperature of about 2000° K. for two hours, it
was found that the currents were smaller than in series 4,
although they were still much larger than in the Ist series.
These results (series 6) have not been plotted, since they are
represented well enough by series 9. After continued
heating the emission was again about the same as in the
Ath series, and the lamp was then sealed.

A few days later measurements were taken on the sealed
lamp. Without any preliminary heating whatever, series 9
was taken with increasing temperatures, followed immedi-
ately by series 9 A with decreasing temperatures. It is
evident that a marked change has been produced by simply
heating the filament once. The final results have been
plotted, and it will be seen that the points lie quite accurately

Thermionic Currents from T: ungsten. 819

along a straight line through the whole range of tempera-
ture, which was from 1760° to 2520° K. The agreement

between the observed and calculated currents can be seen in

Table III.
The value of 6 (59700) is larger and the currents are

3G 2

OO EE EEE oe

§20 Mr. K. K. Smith on Negative

smaller for this lamp than for the other sealed lamps. The
only known difference in treatment to which this can be
ascribed is that this filament was never heated to a tempera-
ture so high as 2700° K., and to avoid the risk of burning
out the filament, the temperature was seldom raised above
2500° K., except for short periods while observations were
being taken. The close agreement indicated in Table ILI.

TaBLE III.
Lamp 7.
inh | Current in microamperes per sq. cm.
|

Observed. Calculated.

NIGOOIMKS feel cleat 37°5 33'8

NOD ste cok ses heuer 75°0 78:0
i Sis | A ae A ar 150° 157°
DSO ee cece cg 300° 295°
BOAO ee SEs deol eee: | 600° 607°
MOOD TE oe weacscucees 1200° 1260-
AU) a ean emcee Ae 2400- 2460:
OBO We ie wenn eee ceca 4800: 4730°
7 US ee aoe ain a one ae er 9600: 9680-
PASO) Mics: oc -Ueeesesmasets 19200: 19300°
22S USES Ses dee seer ce 38400° 37800:
DOOD ny sricbce ecechocenet: 76800" 76700-
DSM) opin coee tue st eon 154000° 157000:
7a 8 A 2a i hm eee 308000: 311000-
BADD bern teanesebeueniens 616000- 597000°
DOA" WC onsen csices sox 1230000- 1190000:

a = 44x 10° amperes per sq. cm.
b = 59700.

does not, of course, prove that the emission was characteristic
of the pure metal alone. An oxide, for example, could
remain upon the wire indefinitely if the temperature were
not raised too high. Four months after the lamp was
sealed, experiments showed that the emission was unchanged.
The filament was then heated to 2900° K. for three minutes,
with a potential difference applied to prevent the thermionic
current from flowing, but this produced little, if any, change
in the emission at 1800° K. The temperature was then
raised with the thermionic current flowing, but unfortu-

nately the filament was burnt out near one end before any
observations could be taken. |

y
a
i

Lhermionic Currents from Tungsten. 821

Discussion oF RESULTS.

Figure 5 may also be used to illustrate the typical results
of all the experiments. When the observations are plotted
it is found that they fall into four groups, represented by
curves I., IT., III., [V., as follows :—

I. The first currents in four lamps (3, 4, 5, 7) (cf. also
fig. 4). The first currents in the other lamps were
not measured.

I]. (a) The last series with lamp 1 (ef. fig. 2).
(6) The currents after the liquid air was removed
from lamp 6.
(c) The largest currents in lamp 5 (cf. fig. 4).
(d) The first currents after lamp 7 was sealed.

III. The permanent emission in lamp 7, which was not
heated much above 2500° K. This would also
include the currents from filament “B,”’ lamp 6,
before the liquid air was removed.

IV. The currents observed after heating the filaments to
a very high temperature, 2700° K. or more, under
the best vacuum conditions (lamps 1, 2, 6 “A”’).

These variations in the thermionic emission seem to be the
results of progressive changes in the surface conditions *.
When the filament is first heated its surface is probably
slightly oxidized, and otherwise contaminated by impurities.
No special precautions to clean the filament were taken
before it was sealed in the lamp. After a good vacuum has
been secured, heating the filament for a sufficient time to a
temperature above 2700° K. removes the impurities, and
then the emission is represented by IV. This condition is
permanent thereafter provided the vacuum is maintained,
but under the following circumstances the emission changes
to I]., which is not much larger than the original I.:
(a) if gas is liberated inside the lamp by excessive heating
and bombardment of the anode (lamp 1), or (b) if the fila-
ment is oxidized by allowing small amounts of water vapour
to enter the lamp, as in 6.

The removal of impurities from the surface seems to be
indicated quite clearly by the observations made when the
filament in lamp 7 was first heated above 2000° K. after
sealing. This changed the emission permanently from II.
to LII. Since the lamp had been sealed under good vacuum
conditions there was no fresh supply of gas by which the

* Cf. Langmuir, loc. crt,

822 Negative Thermionic Currents from Tungsten.

filament could again be contaminated. It should be recalled
that no emission greater than IIL. could be obtained in lamp 5,
and it seems quite likely that continued heating was futile
in this particular case, because the impurities were being
renewed as rapidly as they were driven off.

Barring lamp 3, which was burnt out at the end of the
first series, lamps 5 and 7 were the only ones in which the
currents represented by IV. were not obtained, and it seems
significant that these were the lamps which were not heated
much above 2500° K. Moreover, as already shown in
Table II., one filament in lamp 6 gave the large currents IV.,
but immediately afterwards, and under the same vacuum
conditions, the other gave currents approximately the same
as III. The former had been heated to 2700° K., whereas
the latter had not been heated above 2500° K. This is what
might be expected if the filaments contained an impurity
which could not be removed except at very high tempera-
tures. If the temperature were not raised too high the
impure surface would give continuously currents such as III.,
but after the impurity had been driven out, the permanent
thermionic emission, 1V., would be characteristic of the pure
metal.

One of the most interesting results of these experiments
is the enormous range of validity established for the emission
formula

b
i= ATe-2,

when the conditions are such that no change in the character
of the emitting surface is believed to occur. The formula is
shown to hold good whilst the current is varied by a factor
of 10”, the corresponding range of temperature being from
1050° K. to 2300° K. An equally good agreement is shown
with the formula 1=CT2e"%T, with C and d constants and
d equal to 52000; so that It is not possible to distinguish
between the relative merits of these two formule with the
data so far obtained.

This investigation was begun under the direction of
Professor Richardson, and I wish to express my appreciation
of his assistance and advice during its progress. My thanks
are also due to Dr. Irving Langmuir for supplying the wire
used, and for furnishing the results of special experiments
with it. To Dean W. F’. Magie I desire to acknowledge my
indebtedness for helpful criticism.

Palmer Physical Laboratory,
Princeton, N.J.

LXXXV. Note on the Higher Derivative of a Function, the
variable of which is a Function of an independent variable.
By I. J. Scawatt*.

N E. R. Hedrick’s translation of Goursat’s work
‘A Course in Mathematical Analysis’ appears the fol-
lowing problem (p. 32, 6.):—
Show that the nth derivative of a function y=¢(u), where
wis a function of the independent variable x, may be written
in the form

(a) = =A.p'(u) + O2 To? +. Stes =e & a TRO, a (x),
where

Co) A=

Gre ne) Au (eo). deus 7
SSS SS SS U :
cetigg ee We ee ant eZ Gor

AF (<1) eu FE (ex OA e Bee ))-

[ First notice that the nth derivative may be written in the
form (a), where the coefficients A,, As,...., An are inde-
pendent of the form of the Funetion @(u). To find their
values, set d(u) equal to u, w’,...., u” a and
solve the resulting equations for Aj, Ag,...., An The
result is the form (6) ].

I have quoted the problem and the suggestions in full, and
shall now proceed to give several proofs for it, in the hope
that these proofs will illustrate certain operations with series
which might be useful in similar work.

I. Let y=¢(u), wherein wu is a function of 2.
Then

AU Ma GY diy win. wae

dix dude ~? ) da

2y d*u du
dx? TC u) at (w) da ay

d°y au We AU 788 lu
T= ow) Gea + 8G") Se Ts +60) (T).

* Communicated by the Author.

824 Dr. I. J. Schwatt on the
We now assume
ae od A,

4 An |
Age i? (a) + 5) 22 g!"(u ee 3 a (u)+ + 7 "(u), (1)
wherein the A’s are functions of u and 2, but independent
of d(u).
To show that (1) is true, we must show that it holds also

for the n+ 1st derivative.
Differentiating (1) with respect to 2, we have

d™tly d A, v Avda. ivd Ay
ye =¢'(v )Fe TI +o (2 ule 1! dz dxz2\
As du, d A; 2 des
aa ee 1 n— nr eee
met IE daz ee o (u) Pay 2)ld ‘7 dx Great
n( aay du ie A, du
ce Peg Mae Tea re (ode de
yee

a ote a1? Mt 5 Sig" (U4. : Pr gu ) + ae
which is of the same form as (1).

We shall now determine the coefficients represented by
the A’s in (1). Since the A’s are independent of y, they
will have the same value whatever @(u) might be.

Letting therefore:

du
a
ie. 39 = == 2uA,+A,; 3
a du? 5
NSP + Tee A,+3uA,+ A;;
‘ OO Wk aye ss
yu 9 bP] d xn ae i U ‘ha U Ag +.. ae © wheat Be

Solving these equations for the A’s, we obtain

d™u
Ay= dae
Sole = VAN) ids
A, = By yn (G Uu da”
Lagirins ae, (3) ou
Bi di yn Cee 1 de 2 da™

gery

ee ee
SN a ae a et,

ges

Ss <<
ae a
eee a oe,

=
Le 2

Fligher Derwative of a Function. 825

Let us now assume

fun A[ x ae pal
A,= 2(-1)) (Xe Sou ee Es eb (78)

Now if y=u*t1, then from (1) follows

Ny ,K+1
Jal = (“T" ay + S Tuts te eel ie vet Ants

n, K+1 K
and ae au Tei CAea ee

da” Nal

But ie

XxX Y
A,= BS tal) ie )ur yey 9

for values of X from 1 to « inclusive.
Therefore

Ny K+1 K Ny A—
ee s $ (at Ae ge- Aty dy

ax” A=1 y=0 dak

Denoting the double summation by 8 and letting X=«+1—a,
and in turn «+7y= 8, we have

oo s we (Ee

o— les —o p—o da”

gen ales p-of K+1\(e+1l—a\ , d™usti-P

Bee Nie. ae Gs
since for B=«+1, CY aah

da”
K K K B
Now > > M,, A= S > M,, Be
al 6—a (Fes C=)

This can be shown in the following way:

ii Ma

> Me= Mi1+ Mi,2+ Mi,3+ OO) Gic +M,

fps
+MootMost+....+Moe,.
+ M3,3+ Par Way yo + M3 ,
mete ah Gea ae wits

826 Dr. L. J. Schwatt on the
Adding by columns we have

> > M, p= Mi, 1+ (Mi,2-+ Ma») + 4 (M, +e eee +M,,«)

a=1 Bp=a

By means of (4) the form (3) changes to

K B ao/K+1\(ke+1l—ea\ , dwuttt-B
s= 3 ub (r ane) ae
K ps Beale a Pie. |
ie B wl
roe ce) ub cae 2! 1) ( oO B—« e
Now

ac ame ee stig B!
ot B-—a J— (cx+1l—«)! al’ («+1—£)! (B—a)! B!

=Grieay! pr yo (i “8 He)
Therefore
aS (—1°(“f" ae a -D in

But $ ay (2)= SpE )-1= 0-012

and (5) becomes

é Pa apa
= ==) 8 SEN
e Fa 1) B ue dan
Hence
i, ad ukth K e( K+1 d®utti-B
JA dar + 2 aie B yu? dx”
eS Waa vaca \ mate oa ae tu ae
= 3 (-1) ( Behe , since —“_— =0,

and A,+; is of the same form as the one assumed for A, in

(2)

II. The following is another method for proving the
given theorem.

Higher Derivative of a Function. 827

Let « successive operations u— = each on (uw), that is -

[eS Nu g)uge) «++ operations | $ (u

wherein wu and ie are not permutable, be designated by

du
(u 5) 6 (0,

49> LEE enar At

d\ S d \”
Let =e aoa. =e ase primer (phos
e u é, then ( == We Ga
dx

therefore

== 5 3(-1) iC )(e- ay eh (e?).

dx” k=1 K+ a=0
But (K— a)” Ca GAG —e a)” e(k—a)z
on q? ehk—a)z

ei adam
ea d”uk-4 ;
1 Ghee

Hence

a"p(u) _

ON Weg af K at
aan = pal = ( —1) (f) ws Pee p(x).

II]. A third way of proving the theorem is as follows :—

dy _ dy du
dx du’da’

Gy du dy ye

dx? ~ dx?" du dix) du?’
but (S) = Ie = du? (q)x et |
du) ~ 2! da? ‘dat }”
therefore Cy S 1 i a :) dyke q®y .
dx? eel eee ( Uda dus” * Q)

EE —————< << - —

828 Dr. I. J. Schwatt on the

Again d®y ss dy _.@u dud’y (4 3 dy
ae a ae dx dz du? * =) dus”
Since Bu? 24 du- As deu
dx® da? dx 4 a?
and du du d?u du\?
eee Ta + 18u ne ()

therefore .d’u du 1 fe (1) =)

Fa See IN aie
and (5") = 1 dul (Tess (5) o du
die sale NY dese 2 eh
and hence
cues d°u me fda? ale duy d*y
x? ee
Ey pau? aye ae (3) 9 du ay

dx da du TOI
Tay: _dx? Ye dx?_| du3?

or written symbolically

TET ea Wtotaee ae 0 °u" see
ee ee) a) ae

da? K=] K ! a=0

Let us assume that this form holds for all values of n from
1 to n inclusive, that is

n n k—1 eo No, K—a JK
ay > = > (—1) Oe ae a0 oN (3)

gt erie a ot dx” Gia
We shall then prove that this form holds also for
Differentiating (3) gives

n+1 K— N,,K—a jK+1
ee 353 =1)'(f)u ou d y du

dunt dar du*t! dz

d”™uk—«
=F ae S(- 1) e Au dun }.

qd?+1
dnt >

Now LY palaces 5 a/K d d”™yk-« |
2) 20 (it dai a
nae o du Ch aS, =e iC en oes a
— > — Ql ae :
K=2 Pak see dz dz” eS iam ae “dan tl

Higher Derivative of a Function. 829

Therefore
a SS oan Liner. il du dty
age = 2, hk ye aus den dx” du®
Oo Neste e 1 du d*y
aah ee Mame ee
2 | = Ya filet De Kk! dx dak
n K—1 a ake 1 d*
5 mcs) eat =
=I | ,( V GE darth” aie Lie a am at ©)
But ( K jee 1 —1> 1
Crate Lie! =| ot Vroenyt
therefore
d” n+1

a(n\ dw 1 dud™y
Fae = Cay (a )e |e

du” jn! dadu"t!

n the he — Caine ae a d*y
+e mel ("Ge cpa mh gee ee

We must now show that
1 du de*ly dni
n! dx dutt} a6 (es) c Jue ie |

1 qd” n+1 ex s is (ey ue pacenul men
~ n+l dart! Ce cit

1. Gs

a Ny, N—B $ n+1,n+1—-a
1) S (= 1) (i Oe dun * de _ ST ks x u

da dz = ~ dgnti ?

or if we let n—a=—, we must show that

du dub is n+ 1 qztly,btl
ay 3 (- , (a) 0 We 6 we dem mega: "(pei aa dat Bs

a) one Cee
Now

Q(2tlyb+l qd” du
dan} =(8+1) Tal “)
and (6) reduces to
a)
bs LaN pet \ a: ptudru® — & nC (3) eg a (w =)
oat 4) (js)e-* dada” Wren Tae

dau" OM

830 Dr. I. J. Schwatt on the

By Leibnitz’s theorem for the derivative of a product we

have
2 (-1 (a Ju = Tk i= Cana, =) -@

In order to prove (7), it is only necessary to show that
it holds for w=, when it becomes

(pb) !

Pr-n-+p—]

" 5. 0 y= Ge Be —B8)1(n—y) ! (pB—n+y) p= eer
which is evidently cae of w But when @ is a

constant, n is also a constant, and therefore (7) holds for all
values of u. So that:

ise af7 idud™ n+1 i a eee
(n+1) % (-1) (“)w = — == (- i) MD ree 2 ’

and therefore

n n Pas nt+l,«—a jk
Tea = 5 2 (-1)' (eS

GEN Th aa hel a da” Vda

The following examples will illustrate the theorem.
To calculate
ee aon het?
(i. dak (1— a")?
By Leibnitz’s theorem for the derivative of a product of
two functions,
d* Wee Bf FOLK —. ap d* — an)\—p
da* (l—a”)? 3 (a= dae“ ey)
ym—Kta qt

=I 5 en): gala)?

yn kta ad
m gmk a7?
ai US emer e ence cor inl) j

Let a=u, and (l—u)-?=y, then

Gaye @ nip—ypPPt+D(Mt2) + B=)
ie SABLE Rinment Ce Care
ie te 8) @B=y)! <(p+8=)! | 2a
= 2a, oh Ge oaan y-a)! (p-1)l Gaanpr

Higher Derivative of a Function. 831

Therefore
cersay hima? 2A
SEM) RE) a-rel
“s)

qn ko.

B= te

=F ey (Co) a Ee

Gr exec eee))

(L222
(ii.) aa Let 22—a’?=u and (l—u) ?=
F dat (1—a3 + at)? Paiiwer Js
then
dy _ ri $ iL S i yo
dae = Fie ge ‘ -1)"( ar dic” cas
Now

d* d* +K—1 pep
S57 (1— =e eee a= —u) Wade

and
ad” K-@ dl” K—@ ae K-@
bien =a (a — 2") sola Caen
a= agie—e) = cag

ada”

as : 3x—3a+ 48 pok— 3a —n
=n! (-1) ie 8 a és Vai 3a+48—n

Therefore
dy nm iff k-l alk a
Giaale a BCD ( ) (eat
3K —3a+48 ek Ts +K—] =
("5 8 aN Jo yok 84+ 48— cl(? i ) dat +aryes

2 2) 3) Nei Ges i)
~ ot l—at +2")? x=1a=0

linc Gar jE — iv")
a To ee"

n! pe eae Ve =

— ==55— SES
a (l—a +2")? ae ae a
— xt) *yt48

x hoch \Civtaay: Tae — a+ 27 )*

University of Pennsylv ania,
Philadelphia, Pa., U.S.A.

ak — Bie

ROE 4d

LXXXVI. On Self-Intersecting Lines of Force and Equi-
potential Surfaces. By G. B. Jerrery, M.A., B.Sc.,
Assistant in the Department of Applied Mathematics, Uni-
versity College, London*.

Y a well-known theorem due to Rankine, if n sheets of
an equipotential surface intersect at a point of equi-
librium, they make equal anzles zr/n with each other. The
object of this paper is to give some simple extensions of this
theorem and some analogous results for lines of force. We
will confine our attention to the cases in which the lines of
force can be defined in terms of a force-function, 2. e. when
the field is either two-dimensional or has an axis of symmetry.
The case of a two-dimensional field admits of a very
simple treatment. Both the potential @ and the force-
function wy satisfy the differential equation
“gah Ms
ae -- oy ==(): o §e | er (1)

Taking the point of equilibrium in question as origin, the
potential in its neighbourhood can be expressed in the form

6=H,4+ HayitHnyet -..., | pees (2)

where H, is a homogeneous function of the coordinates of
degree n. Hach term of this series must be a solution of
(1), and hence, in polar coordinates,

H,=Ar" sin (nO—a@). .. .
The tangents to the branches of the equipotential surface
at the origin are given by the roots of H,=0, 2. e. by
a etn a+27 a+(n—1)r

To n 5 n . e e e . e *5 n

Hence the n sheets of the equipotential surface make
equal angles 7/n with each other. It is obvious that in this
case a precisely similar theorem holds for lines of forcef.
It remains to find the relation between the lines of force and
the equipotential surfaces at the same point of equilibrium,
@ and w are connected by the relations

OG Lod ‘Lop vot
Or roe’ rod or’

* Communicated by the Author.

+ A proof of this property for any curve defined by a solution of (1)
was given by Stokes in a note appended to Rankine’s paper, Proc. R. S.
xv. 1867.

Innes of Force and Equtpotential Surfaces. 833
and from (3), by the aid of these relations,
w= Ar" cos (n8—a) + terms of higher degree in r.
The directions of the lines of force are therefore given by

tata 2a+3mT 2a+(2n—1)7r

[; nae ae everest | Oe) PLONE ay! Olt er ep Le
De? Dred)? 3 2Qn

It appears that at a point where n equipotential lines
intersect, » lines of force will also intersect. Further, these
lines of force make equal angles a/n with each other and
bisect the angles between the equipotential lines.

We pass on to the case of a three-dimensional field having
an axis of symmetry. Let P be a point of equilibrium at a
distance a from the axis of symmetry. Take the foot of the
perpendicular from P to the axis of symmetry as the origin
of cylindrical coordinates a, z. ¢ and y no longer satisfy
the same differential equation, but

CO NOD Ore

J 4 20a 02

22 eT)

and
Oa ON Onva.
asics yes TOE aT Bitchy RAE UN TS (5)
Take polar coordinates through P in the meridian plane,

the initial line being parallel to the axis of symmetry.

Then

2— I COs to, Git 7 SUMO.) er) 2. laa (O)

Transforming (4) and (5) to the new coordinates, they
become
ea Seon hos
(a+rsin0)(5 +35. + 2593]

where the upper sign is taken for the potential and the lower
sign for the force-function.
In the neighbourhood of P let ¢ be expressed in the form

Wy) =P7"Sn + Hear baa Se Petar er situ aReo oon

RUMETE Vpeicdjsets ice plas « are functions of @ only. Substituting
in (7) the coefficients of the various powers of r in the
expression so obtained must vanish separately. Hquating
the coefficient of »”—? to zero, we have

— +n?In=0).
Phil. Mag. 8. 6. Vol. 29. No. 174. June 1915. 3 H

+ (sin 92. + cos 9 S4)=0, (7)

834 Mr. G. B. Jeffery on Self-~Intersecting
Hence, S,=A sin (nd —a)
and
o=Ar' sin (nO— 2) + terms in higher powers of r. . (8)

The tangents in the meridian plane to the branehes of the
: 5 aan lean: Rifts:
equipotential surface through P are in the directions

ae a+o at+(n—1)7r
== as sn Fouhiel olen vein 5

ea) a
i VL Tt

which agrees with Rankine’s theorem.
@ and wy are connected by the relations

Oe Loe Oo 1 ear
Do oOo, ‘Os bla ioe

which, by the aid of (6), transform into

od 1 ov lod 1 ov

Or r(at+rsind) 00°’ rd06@ atrsind or’

From these relations, together with (8), it is easy to
obtain

p= — Aar” cos (nf —a) + terms in higher powers of r. (9)

The tangents to the lines of force through P are therefore
in the directions

ee en 94437 24+ (2n—1)r

ee a, 5) Seon nea § 3 e . e e ’ LS) = i a

2n 2n

2n :

Hence, at a point of equilibrium not lying on the axis of
symmetry, the intersecting equipotential surfaces make
equal angles with each other; the lines of force also make
equal angles with each other, and bisect the angles between
the equipotential surfaces.

If the point of equilibrium lics on the axis of symmetry,
these results are no longer true. The potential in the
neighbourhood of P can be expressed in terms of zonal
harmonics,

p= Anh) $3 Bat LP eh) os og oe (10)
where p=cos@.

The tangents in the meridian plane to the equipotential
surfaces through P are, therefore, given by the roots of
P(e) =0.

‘his equation has n distinet roots between w= +1 exclusive.

Lines of Force and Equipotential Surfaces. 835

Thus there are n sheets of the surface intersecting at P, but
these do not make equal angles with each other. The axis
of symmetry cannot be one of the equipotential lines, for
P,() cannot vanish for 0=0 or 7. When n is odd p=0
is a root of P,(#)=0, and therefore when an odd number
of sheets intersect one of them is normal to the axis of
symmetry.

The force-function corresponding to (10) is well known
to be *

v= auc ogi |) We Jia mee) higher powers of r.

The lines of force en P are in the directions given
by w= +1, 2. e. the two parts of the axis of symmetry,
together with the roots of £ Plu) = 0. This equation has

one and only one root between any two consecutive roots of

Fig. 1.

L}
4
'
Q
1]
y
Q
Q
By
\ yy

ee ee ee ee

The Intersection of Two Lines of Force and Two Sheets of an
Iquipotential Surface.

P,,(u)=0, and hence it has n—1 roots between p=
exclusive. The axis of symmetry 1 1S, theref fore, alw ays one
of the intersecting lines of force; there are n—1 others

* Lamb, ‘ Hy dredynamics s,’ p. 120.
»

3 hy

836 Prof. R. R. Sahni on the

making n lines of force which intersect at a point where
n sheets of an equipotential surface intersect. These, again,
will not make equal angles with each other, but one line of
force will lie between every two consecutive sheets of the
equipotential surface.

Fig. 2.

The Intersection of Three Lines of Force and Three Sheets of an
Equipotential Surface.

The cases when two and three sheets intersect are shown
in figs. land 2. The firm lines represent the equipotentials
and the broken lines the lines of force.

LXXXVII, The Photographie Action of a, B, and y Rays.
By R. R. Saunt, I.A., Professor, Government College,
Lahore *.

[Plate XII.]

EVERAL investigators have studied the action of

a particies on sensitive films. Thus, Kinoshita ft has
shown that when such a film is exposed to an a-ray source,
the photometric density of the film, on development, is
proportional to the number of e& particles incident on it;
also that a single « particle produces a detectable effect on

* Communicated by Sir E. Rutherford, F.R.S.
+ Kinoshita, Proc. Roy, Soc. ser. A, Ixxxiii. p. 4382 (1910),

Photographie Action of a, B, and y Rays. 837

the photographic plate. Reinganum* was the first to
obtain tracks of single « particles in sensitive films; Michl f
and, subsequently, Baisch t and Mayer § also studied, in
some detail, the photographic action of # particles. The
tracks obtained by Reinganum and Michl were mostly
eurved, but, as Michl points out, this was probably due to
the particular way in which the gelatine film contracts
on drying. Walmsley and Makower || were the first to
publish a microphotograph of a-ray tracks. Some inter-
esting microphotographs of a-ray tracks have been published
in a recent paper by Kinoshita and Ikeuti {.

In the course of an investigation by the present writer,
into certain properties of « and @ rays, by the photographic
method, it was found that most photographic plates, on
development, show, under the microscope, a large number
of blackened grains, even when they have not been previ-
ously exposed to light or some other stimulus, such as
radiations from an active substance. It is clear that with
a plate free from this defect, the examination of photo-
graphic films under the microscope should afford a very
sensitive way of studying the properties of « rays, and the
method would seem to be applicable to all the phenomena
which have hitherto been investigated by the method of
scintillations. The method has, in fact, been used by
Mayer to study the scattering of « particles by metallic
films. Both Reinganum and Mayer recommend the photo-
mechanical and the Sigurd dia-positive plates, made by
R. Jahr (Dresden), as being better than any other plates
used by them; but, as Mayer points out and as his results
also show, even with these plates the number of grains
visible under the microscope on an exposed plate was too
large to permit the counting of « particles with accuracy.

In this short paper it is intended to describe a plate
suitable for various kinds of investigations with the photo-
graphic method, and briefly to refer to some of the experi-
ments which have been undertaken with it.

Selection of plate.— After considerable search with
different kinds of plates, sensitive films, and papers with
suitable developing solutions, it was found that Wratten
and Wainwright’s lantern-plate presented an absolutely
clear surface. For a temperature of 18° C., which was

* Reinganum, Phys. Zeit. xii. pp. 1076-77 (1911).

t+ Michl, Akad. Wiss. Ber. Wien, pp. 1431-1447 (1912).
t Baisch, dan. der Phys, xxxv. p. 565 (1911).

§ Mayer, Ann. der Phys, xli. (1913).

| Walmsley & Makower, Proc. Phys. Soc. xxvi. (1914).
q] Kinoshita & Ikeuti, Phil. Mag. March 1915.

ge ee

838 Prof. R. R. Sahni on the

considered to be the most convenient, a development of
about 24 minutes was the best. For lower temperatures,
this period was much longer. The other plates made by
the same firm gave unsatisfactory results. The Ilford
Process and the Imperial Sovereign plates come next after
the Wratten lantern-plates. The following developer was
found most suitable to be used with the plate :—

Solution A.
. Quinol, 25 gm.
. Potassium metabisulphite, 25 gm.
. Potassium bromide, 25 gm.
. Distilled water made up to 1000 c.e.

He Oo NO

Solution B.
eS Ole 0) com:
2, Distilled water up to 1000 c.c.

Solutions A and B to be mixed in equal proportions
immediately before use.

A 20 per cent. solution of “ Hypo” was used for fixing.

It is of importance to exclude even red light as much as
possible during all the operations. It is also important to
use water as free as possible from dust nuclei. [or this
purpose the water for washing should be filtered or previ-
ously boiled and cooled.

Some experiments with « particles—In the experiments
here referred to, both with « and £ particles, the active
sources employed consisted of polonium (containing some
radium E), thorium active deposit, and radium © on the
tip of fine sewing-needles and on suitable metal plates of
different dimensions. In the case of the needles, only the
tip was effective, the rays from the rest of a needle, if acti-
vated, being screened off by means of a small cardboard disk.

The microphotograph in Plate XII. fig. 1 was obtained
with a very weak thorium active deposit needle, the tip
being exposed for a few seconds to the sensitive plate.
Only a portion of the microphotograph is here shown. The
centre is about the point marked C. Fig. 2 (a, 6, and c) was
obtained with a fairly strong polonium needle. Fig. 3 isa
portion ofa similar microphotograph obtained with a thorium
active deposit needle. In all these cases the activated tip
of the needle was held in contact with the photographic
plate. Similar results were obtained with other a-ray
sources. It will be noticed, at once, that a marked feature
of all a-ray photographs is the presence of straight tracks
radiating from a common centre, which is either the point

Photogvaphic Action of «, 8, and y Rays. 839

of contact of the active tip with the sensitive plate, or, when
the needle is held at a short distance above the plate, a point
immediately below it. When the period of exposure is
a sensible interval, a dark circular nucleus is formed.
A careful examination of the radiating tracks will show the
presence of an occasional track which is bent as the result
of scattering. The « particles appear to suffer only one
encounter, no tracks with more than one bend being visible.
The per ipheral tracks appear to be longer than those nearer
the centre, but this is simply an optical effect, the central
tracks being foreshortened. Fig. 1 shows this clearly.

It may be mentioned that, as was to be expected, the
tracks produced with thorium active deposit were longer
than those obtained with polonium. The longest track with
thorium active deposit measured 48°15 y, while the av erage
length of the ten longest seen near the periphery was 32° A lee
The longest track observed consisted of only 15 silver grains,
while another track which measured 42:0 jo was made up of
24 silver grains. The average number of grains in the ten
longest tracks observed was 16°8. In all the above cases,
care was taken to make sure by an examination with an oil
immersion lens that the tracks did not consist of two
separate ones in continuation of one another. Michl gives
the length of the longest track with 9 grains as 32. He
obtained tracks with 13 grains, Baisch mentions bent tracks
with 15 grains, while Tipe hie and Ikeuti observed with
radium C tracks with as many as 16 grains.

The action of B particles—In 1911 C. T. R. Wilson *, in
his well-known condensation experiments, made the tracks
of 8 particles through moist air visible. The trails in this
case were indicated by droplets of water condensed on the
path of the 8 particles, thus forming cloud-lke streaks
which were immediately photographed. No one, however,
seems to have obtained a photographic impression of separate
8 particles or studied a 8 photograph. A comparison
between the # and @ photographs reveals certain points of
interest and practical i importance.

The three active sources used in the experiments with
a particles were also employed with @ particles, the
a particles being, in all cases, sifted off by wrapping the
sensitive plate in paper of suitable thickness. Figs. 4 and 5
are mnicrophotographs with @ rays, the first with a thorium
active deposit needle, and the second with the radium EB
needle. In both cases, the plate was wrapped up in black
paper, in which it is ordinarily packed, as well as in one

* C. T. R. Wilson, Proc. Roy. Soc. ser. A, Ixxxy. (1911).

840 The Photographic Action of a, B, and y Rays.

thickness of aluminium foil weighing 7°5 mgs. per cm.’, the
active point being held next to the foil. A still greater
enlargement of the edge of a photograph has also been
studied with camera lucida sketches, noting down carefully
the grains which became visible at different depths of the
film. The distribution of the grains is so irregular that it is
dificult to say with certainty whether there are definite
tracks due to individual 8 particles. But if such tracks are
present, they are certainly not straight.

It will be noticed at once that the 8-ray photographs are
quite different in character from those obtained with @ rays.
The following points of distinction may be noted :—

1. The distinct radial character of the # photographs is
altogether absent in the 8 photographs. Compare specially
a photographs fig. 2 (a, b, and ¢) with 8 photograph fig. 5.
They were produced with the same active needle, and the
magnification is also about the same.

2. Ina #-ray photograph no straight tracks are visible.

3. In the case of a 8 photograph there is no well-defined
centre, the dark nucleus, when formed, being irregular and
nebulous.

Photographs have been taken by interposing an increasing
number of aluminium foils of known thickness between the
active source and the sensitive plate, activated plates as well
as needles being used. The general character of the B-ray
photographs, as indicated above, is maintained in all cases,
the only difference noticed being a decrease in the photo-
metric density or of the number per unit area of separate
silver grains. A platinum plate which had been exposed to
radium emanation for a couple of hours a month previously
showed clearly the emission of both e and 6 rays due to the
growth of the long-period active deposit.

The method is also being used for studying the weak
activity of potassium and rubidium. Preliminary experi-
ments show that they give off @ rays*. It may be possible
to discover other substances of still weaker activity by
sufficiently increasing the period of exposure. Experiments
in this direction are proceeding.

It will thus be clear that the Wratten lantern-plate affords
an easy method for determining the emission or otherwise of
a and £ particles from a given substance, as well as of
studying the complex character of 9 rays.

The action of y rays.—A sensitive plate was exposed to
64 mgs. of radium contained in a thick glass tube at a
distance of 4cm. In one case, the rays had to penetrate

* Campbell & Wood, Proc. Camb. Phil. Soc. xiv. p. 15 (1907).

Geological Society. 841

through a lead plate 4 cm. thick; in another case, a lead
plate, 3 mm. thick, was interposed; in a third series of
experiinents, a varying number of aluminium foils covered
the sensitive plate; while in a fourth case, the sensitive
plate was exposed directly to the rays. All the photographs
thus obtained resemble a 8-ray photograph. In cases 3 and
4 the grains are apparently equally crowded, in 2 they are
less crowded, while in 1 they are much thinner. Fig. 6
represents a photograph obtained in the second experiment
given above. As expected, the y-ray photographs were
found to be similar to the @-ray photographs of the same
magnification, since it is known that all the effects of y rays
are due to secondary @ rays, which they excite.

The experiments described in this paper were preliminary
to the use of the photographic method tor determining, with
accuracy, the scattering of @ particles by gases.

In conclusion, the writer wishes to express his indebtedness
to Dr. W. Makower for suggesting this research, and to him
and to Dr. N. Bohr for their continued kind interest and
advice during the progress of the experiments described in
this paper. He is grateful to the Governing Body of the
School of Technology for kindly placing the excellent
resources of that institution at his disposal, and to Mr. R. B.
Fishenden personally for kind and constant advice through-
out the photographic operations. He also wishes to acknow-
ledge the kindness of Dr. Hickling, of the Geological
Department of this University, who was good enough to
allow him the use of his microphotographical apparatus.

The Physical Laboratory,
Victoria University, Manchester.
March 1916.

LXXXVIII. Proceedings of Learned Societies.
GEOLOGICAL SOCIETY.
[Continued from p. 208. ]
January 6th, 1915.—Dr. A. Smith Woodward, F.R.S., President,
in the Chair.
jae following communications were read :—
1. ‘ The Silurian Inlier of Usk (Monmouthshire).’ By Charles
Irving Gardiner, M.A., F'.G.S.

The Usk inher hes a few miles north of Newport (Mon.).
Between the coalfields of South Wales and the Forest of Dean the
Old Red Sandstone is bent into an anticline, the axis of which runs

SS —

—

842 Geological Society :—

very nearly north and south. This has been denuded away to the
west of Usk, and Silurian beds have been exposed, the rocks seen
being of Ludlow and Wenlock age.

In the southern part of the inlier the Silurian rocks are arranged
in two anticlinal folds, the axes of which run nearly north and
south and dip southwards. ‘These folds are separated by a fault.
The western one is named the Coed-y-paen Anticline, the eastern
one the Llangibby Anticline. The Old Red Sandstone is believed
to rest unconformably on the Ludlow Beds along mueh of the
margin of the Coed-y-paen Anticline, and beneath the Ludlow
Beds, which are about 1800 feet thick, come 35 to 40 feet of a
Wenlock Limestone, which covers Wenlock Shales: of these latter
some 850 feet are seen. It is impossible to separate the Ludlow
Beds into an upper and a lower series, owing to the absence of
the Aymestry Limestone. They are composed mainly of sandy
shales and sandstones above, and of sandy shales with layers of
ealeareous nodules or of calcareous bands below.

Dayia navicula is a common fossil up to 240 feet from the
top of the Ludlow Shales, and Holopella gregaria and H. obsoleta
occur only in the uppermost beds.

At their base the Ludlow Beds seem to pass conformably down
into the Wenlock Beds, and the Wenlock Limestone is probably
not at the summit of the Wenlock Shales. The Wenlock Lime-
stone occurs either in irregular layers separated by sandy shales, or
in massive beds largely made up of crinoid fragments. Corals are
rare in it.

The Wenlock Shales below the limestone are divisible into an
upper portion, which consists of sandy shales and sandstones, and a
lower portion composed of mudstones. The Coed-y-paen Anticline
has been much affected by pressure, the hard Wenlock Limestone
Bed has been fractured in no fewer than twelve places, and portions
of it driven in on to the soft underlying shales.

In the northern part of the area there is much alluvium and
drift; consequently, although no Wenlock Limestone is now to be
seen beyond the Wenlock Shales, it is possible that the limestone
may occur beneath the drift, as, when last exposed, the Wenlock
Shales are seen dipping north-eastwards, and beyond the drift
Ludlow Beds are observed near Clytha. The Liangibby Anticline
extends as far north as Cwm Dowlais, and shows Ludlow Beds
resting upon Wenlock Limestone, the anticline ending against an
east-and-west fault. North of Cwm Dowlais nothing but Ludlow
Beds are seen between the Coed-y-paen Anticline and the Old Red
Sandstone, from both of which they are faulted.

Anaccount of the Ludlow Beds along the western and eastern
sides of the inlier is given, and a large amount of evidence with
regard to the ages of the rocks at numerous exposures is produced
in the form of lists of fossils.

The fossils have all been named by Dr. F. R. Cowper Reed,
who contributes an appendix in which several new species and
varieties are described in detail.

On Cone-in-Cone Structure. 843

2. ‘Some Observations on Cone-in-Cone Structure and their
Relation to its Origin.” By Samuel Rennie Haselhurst, M.Sc.,
F.G.S.

In a brief review our state of knowledge is summarized, and the
deductions of other investigators are analysed.

The author then outlines the phenomenon of megascopic
pseudostromatism, and certain tectonic features which are
always associated with cone-in-cone structure in areas where it is
greatly developed. He points to the disadvantage accruing from
many observers not having seen it 2m sztw on a large scale, and
shows how a simulation of horizontality in stratification masks what
he takes to be the key to the diagnosis of this structure.

Two typical areas are described :—

(a) The St. Mary’s Island-Tynemouth district of the D, Coal-Measures
of Northumberland.

(6) The Hawsker—Robin Hood’s Bay—Ravenscar district of the North
Riding of Yorkshire.

The specimens collected in these areas are unique, and some
dozen types from other areas, including Sandown, Portmadoc,
Olney, Somerton, Lyme Regis, and Merivale Park are examined in
detail with reference to :—

(a) Evidence furnished by distorted fossils.
(6) Chemical composition.

(d) Microscopic structures.

The author critically examines the accepted hypothesis that
cone-in-cone structure is something essentially due to
erystallization.

He describes the results of some high-pressure mimetic expert-
ments, aided by a Royal Society grant which he now gratefully
acknowledges. These experiments were designed to produce this
structure, and reveal what the author believes to be many new
points on the origin of concretions and cone-in-cone in particular.
The experiments are new, inasmuch as the media used, namely :
brittle, semi-plastic, and plastic, are enclosed in tunics of varied
design, and then subjected either to a high uniform hydrostatic
pressure or to a direct thrust. The results are in many ways
analogous to those of Ewing, Goodman, and Daubrée who, it is
remarked, did not attempt to explain cone-in-cone structures.

The author concludes from the evidence :

(i) that cone-in-cone is not due to crystallization, but is a mechanically
produced structure due to great and localized pressure ;
(i) that it is closely allied to the phenomenon known as pressure
solution;
(ii) that cone-in-cone structure is closely associated with other rock-
structures which are mutually indicative the one of the other, and
also of their mode of origin,

Padang

INDEX to VOL. XXIX.

——pS—_

A BSORPTION spectra of organic
substances, on the, 192.

Adolian tones, on, 433.

Air, on the nature of the large ions
in the, 514; on a new type of ion
in the, 686; on the measurement
of the velocity of, by the linear
hot-wire anemometer, 556.

Alkali metals, on the principal
series in the spectra of the, 775.

Allen (Dr. H. 8.) on the magnetic
field of an atom in relation to
theories of spectral series, 40; on
the series spectrum of hydrogen
and the structure of the atom, 140;
on an atomic model with a mag-
netic core, 714.

Alpha particles, on the velocity of
the, from radium A, 259; on
tracks of the, in photographic
films, 420.

rays, on the photographic
action of, 836.

Anemometer, on measurement of
air velocity by the linear hot-wire,
556.

Atomic heat, on the photoelectric
constant and, 734.

model with a magnetic core,
on an, 714.

Atoms, on the magnetic field of, in
relation to theories of spectral
series, 40; and molecules, on the
gyroscopic theory of, 310; on the
series spectrum of hydrogen and
the structure of, 140, 332, 651;
on the construction of cubic crys-
tals with theoretical, 750.

Baly (Prof. E. C. C.) on light
absorption and fluorescence, 228.
Barlow (W.) on topic parameters
aud morphotropic relationships,

745,

Berkeley (the Earl of) on a new
form of sulphuric acid drying-
vessel, 609; on a sensitive method
for examining the optical qualities
of glass plates, 613.

Beta rays, on the absorption of homo-
geneous, 720 ; on the photographic
action of, 836.

Bohr (Dr. N.) on the series spectrum
of hydrogen and the structure of
the atom, 382.

Boiling-points of homologous com-
pounds, on the, 599.
Books, new :—Kippax’s The Call of
the Stars, 206; Silberstein’s The
Theory of Relativity, 335; Scud-
der’s The Electrical Conductiyity
and lonization Constants of Or-
ganic Compounds, 482; Bulletin
of the Bureau of Standards, 624.

Bragg (Prof. W. H.) on the relation
between certain X-ray wave-
lengths and their absorption co-
efficients, 407.

Burton (Dr. C. V.) on the scattering
and regular reflexion of light by
gas molecules, 625.

Butterworth (S.) on the coefficients
of self and mutual induction of
coils, 578.

oa on the X-ray spectrum of,
407.

Campbell (Dr. N.) on the ionization
of metals by cathode rays, 369;
on ionization by positive rays, 783.

Carbon filaments, on the thermionic
emission from, 362.

Cathode rays, on the ionization of
metals by, 369.

Coils, on the coefficients of self and
mutual induction cf coaxial, 578.

Condensation nuclei produced by the
action of light on iodine vapour,
on, 415.

Conduction, on the electron theory
of metallic, 178, 425.

Contact difference of potential of
distilled metals, on the, 623. _

Crehore (Dr. A.C.) on the gyroscopic
theory of atoms and molecules,
310; on the construction of cubic
crystals with theoretical atoms,
750.

Critical temperatures of homologous
compounds, on the, 599.

Crystals, on the construction of cubic,
with theoretical atoms, 750.

Derivative of a function, note on the
higher, 823.

DTN D EEX. 845

Dispersion, on the theory of, 465.

Drying-vessel, on a new form of
sulphuric-acid, 609.

Dunlop (Dr. A.) on a raised beach on
the southern coast of Jersey,
207..

Earth’s crust, on the average thorium
content of the, 483.

Ehrenfest (Dr. P.) on a simplified
deduction of the formula which
Planck uses as the basis of his
radiation theory, 297.

Electricity, on frictional, on insula-
tors and metals, 261.

Electron, on the motion of the
Lorentz, 49.

theory of metallic conduction,
on the, 173, 425; of the optical
properties of metals, on the,
655. i

Energy, on the law cf partition of,
383.

Equipotential surfaces, on self-inter-
secting, 832.

Evans (E. J.) on the spectra of
helium and hydrogen, 284.

Ferguson (Dr. A.) on the boiling-
points and critical temperatures
of homologous compounds, 599.

Field of force,on motion in a periodic,
15.

Fleck (A.) on the condensation of
radium and thorium emanations,
3a7.

Fluid, viscous, on the equations of
motion of a, 445; on the two-
dimensional steady motion of a,
445 ; on the motion of a sphere in
a, 526.

Fluorescence, on light absorption
and, 223.

Fokker (Dr. A. D.) on Einstein and
Grossmann’s theory of gravitation,
Gi

Fraction, on the separation of a, into
partial fractions, 63.

Frictional electricity on insulators
and metals, on, 261.

Function, note on the expansion of
a, 65; note on the higher deriva-
tive of a, the variable of which
is a function of an independent
variable, 828.

Gamma rays, on the photographic
action of, 886.

Gardiner (C. I.) on the Silurian in-
lier of Usk, 841.

Gas molecules, on the scattering and
regular reflexion of light by, 625.

Gases, on the ideal refractivities of,
28.

Geological Society, proceedings of
the, 207, 844.

Glass plates, on a sensitive method
re mae optical qualities of,
6138.

Gravitation, on Einstein and Gross-
mann’s theory of, 77.

Gyroscopic theory of atoms and
molecules, on the, 310.

Harmonograph, on the duplex, 490.

Hartley (E. G. J.) on anew form
of sulphuric acid drying-vessel,
609.

Haselhurst (S. R.) on cone-in-cone
structure, 843.

Helium, on the series spectrum of,
284.

Hitchcock (Dr. F. L.) on the
operator vy in combination with
homogeneous functions, 700.

Holmes (A.) on lead and the end
product of thorium, 673.

Hopwood (F. Ll.) on the plastic
bending of metals, 184; on a
qualitative method of investigat-
ing thermionic emission, 362.

Hot-wire anemometer, on measure-
ment of air velocity by the, 556.

Hydrogen, on the series spectrum of,
140, 284, 332, 651; on the line
spectra of, 709.

Hydrostatic pressure, on the varia-
tion of the triple-point with, 148.
Ikeuti (H.) on the tracks of the alpha
particles in photographic films,

420.

Illumination, on the brightness of
intermittent, 646.

Induction, on the self and mutual,
of coaxial coils, 578.

Induction-coil, on the most effective
adjustment of an, 1.

Insulators, on frictional electricity
on, 261.

Iodine vapour, on condensation nu-
clei produced by the action of
light on, 418.

Tonization of metals by cathode rays,
on the, 869; on, by positive rays,
783.

Ions, on the nature of the large, in
the air, 514; on a new type of, in
the air, 686.

846 LANGDUE KS

Jeffery (G. B.) on the equations of
motion of a viscous fluid, 445; on
the two-dimensional steady motion
of a viscous fluid, 455; on sell-
intersecting lines of force and
equipotential surfaces, 832.

Jones (Prof. E. T.) on the most
effective adjustment of an induc-
tion-coil, 1.

Jones (W. J.) on the ideal refracti-
vities of gases, 28; on a theory of
supersaturation, 3d.

Jones (W. M.) on frictional electri-
city on insulators and metals, 261.

Jude (C. W.) on the duplex har-
monograph, 490.

King (Prof. L. V.) on the precision
measurement of air velocity by
means of the hot-wire anemo-
meter, 556.

Jxinoshita (S.) on the tracks of the
alpha particles in photographic
films, 420.

Lamb (Prof. H.) on a tidal problem,
(37.

Larmor (Sir J.) on the pressure of
radiation on a receding’ reflector,
208.

Lawson (R. W.) on lead and the end
product of thorium, 673.

Lead and the eud product of thorium,
on, 678.

Light, on condensation nuclei pro-
duced by the action of, on iodine
vapour, 413; on the scattering
and regular reflexion of, by gas
molecules, 628.

absorption and fluorescence,
on, 228.

Lindemann (Dr. F. A.) on the theory
of the metallic state, 127.

Line spectra, on radiation from an
electric source and, 709; on the
quantum-theory of radiation and,
795.

Lines of force, on self-intersecting,
832.

Liquid drop suspended in another
liquid, on the form of a, 149, 190.

Livens (G. H.) on Lorentz’s theory
of long-wave radiation, 158; on
the electron theory of metallic
conduction, 173, 425; on the law
of partition of energy and New-
tonian mechanics, 383; on the
electron theory of the optical pro-
perties of metals, 659.

Log” (1+.), on the expansion of,
65,

Lorentz electron, on the motion of
ive we

Lorentz’s theory of long wave radia-
tion, on, 158.

McCleland (N. P. K. J. O'N.) on the
absorption spectra of organic sub-
stances in the light of the electron
theory, 192.

Magnetic field of an atom, on the,
40.

Makower (Dr. W.) on the magnetic
deflexion of the recoil stream from
radium A, 253; on the velocity of
the alpha particles fromradium A,
259.

Mallik (Prof. D. N.) on the theory
of dispersion, 465.

Mercury, on an anomalous Zeeman
effect in satellites of the violet
line of, 241.

Metallic conduction, on the electron
theory of, 173, 420.

state, on the theory of the,
127.

Metals, on the plastic bending of,
184; on frictional electricity on,
261; on the ionization of, by
cathode rays, 569; on the contact
difference of potential of distilled,
625; on the electron theory of
the optical properties of, 655; on
the principal series in the spectra
of the alkali, 775.

Molecules, on the syroscopic theory
of atoms and, 310; onthe relative
dimensions of, 552.

Morphotropic relationships, on topic
parameters and, 745.

Morrow (Miss G. V.) on displace-
ments in certain spectral lines of
zine and titanium, 394,

Nabla, on the operator, in combin-
ation with homogeneous functions,
700.

Nagaoka (Prof. H.) on an anomalous
Zeeman effect in satellites of the
violet line of mercury, 241.

Nichrome filaments, on the therm-
ionic emission from, 367.

Nutting (P. G.) on the visibility of
radiation, 301.

Onnes (H. Kamerlingh) on a sim-
plified deduction of the formula
which Planck uses as the basis of
his radiation theory, 297.

PND a:

Optical properties of metals, on the
electron theory of the, 655.

qualities of glass plates, on a
sensitive method for examining
the, 613.

Organic substances, on the absorp-
tion spectra of, 192.

Palladium, on the spectrum of, 154;
on the X-ray spectrum of, 407.

Paratlins, on the boiling-points and
critical temperatures of the, 599.

Partington (J. R.) on the ideal
refractivities of gases, 28; on a
theory of supersaturation, 35.

Paulson (Dr. I.) on the spectrum of
palladium, 154.

Pealing (H.) on condensation nuclei
produced by the actiou of light on
iodine vapour, 415.

Photoelectric constant and atomic
heat, on the, 754.

effect, on the, 618.

Photographic action of alpha, beta,
aad gamma rays, on the, 836.

films, on tracks of the alpha
particles in, 420.

Plastic bending of metals, on the,
184,

Pollock (Prof. J.) on the nature of
the large ions in the air, 514; on
anew type of ion in the air, 636.

Poole (J. H. J.) on the average
thorium content of the earth’s
crust, 433.

Pope (Prof. W. J.) on topic para-
meters and morphotropic relation-
ships, 745.

Porter (Prof. A. W.) on the variation
of the triple-point of a substance
with hydrostatic pressure, 145.

Positive rays, on ionization by, 785.

Quantum-theory of radiation and
line spectra, on the, 795.

tadiation, on Lorentz’s theory of
long wave, 158; on the pressure
of, on a receding reflector, 208; on
the visibility of, 301; from an
electric source and line spectra,
on, 709: on the quantum-theory
of, and line spectra, 795.

theory, on Planck's, 297, 383.

Jiadium A, on the magnetic deflexion
of the recoil stream from, 253; on
the velocity of the alpha particles
from, 259.

emanation, on the condensation

of, 337,

847

Raman (Prof. C. V.) on motion in a
periodic field of force, 15.

Rankine (A. O.) on the relative
dimensions of molecules, 552.

Rayleigh (Lord) on the mutual
mfluence of resonators exposed to
primary plane waves, 209; on the
widening of spectral lines, 274;
on zolian tones, 433.

Refractivities of gases, on the ideal,
28.

Resonators, on the mutual influence
of, exposed to primary plane waves,
209.

Rhodium, on the X-ray spectrum of,
407.

Rice (J.) on the form of a liquid
drop suspended in another liquid,
149.

Richardson (Prof. O. W.) on the
photoelectric effect, 618.

Ripples, on, 688.

Rogers (Prof. I. J.) on the photo-
electric effect, 618.

Sahni (Prof. R. R.) on the photo-
graphic action of alpha, beta, and
galuma rays, 836.

Saltmarsh (Mass M. O.) on the
brightness of intermittent illumin-
ation, 646.

Sanford (I’.) on the contact difference
of potential of distilled metals,
623.

Schott (Dr. G. A.) on the motion of
the Lorentz electron, 49.

Schwatt (Dr. [. J.), note on the
separation of a fraction into partial
fractions, 63; note on the expan-
sion of a function, 65 ; note on the
higher derivative of a function,
the variable of which is a function
of an independent variable, 823.

Shand (Prot. S. J.) on tachylyte
veins in the granite of Parijs,
207.

Shorter (S. A.) on the shape of
drops of liquid, 190.

Silberstein (Dr. L.) on radiation
from an electric source and line
spectra, 709.

Silver, on the X-ray spectrum of,
AO7.

Smith (IX. K.) on negative therm-
ionic currents from tungsten,
802.

Southwell (R. V.) on the collapse of
tubes by external pressure, 67.

848

Spectra, on the absorption, of organic
substances, 192; on radiation
from an electric source and line,
on, 709; of the alkali metals, on
the principal series in the, 775;
on the qnantum-theory of radia-
tion and line, 795.

Spectral lines of zine and titanium,
on displacements in, 394.

series, on the magnetic field of
an atom in relation to theories of,
40.

Spectrum, on the series, of hydrogen,
140, 284, 332, 651; of palladium,
on the, 154; of mercury, on an
anomalous Zeeman effect in the,
241.

lines, on the widening of, 274.

Sphere, on the motion of a, in a
viscous fluid, 526.

Sulphuric-acid drying-vessel, on a
new form of, 609.

Supersaturation, on a theory of, 35.

Surveyor’s tape, on the correction
for rigidity for a, 96.

Sutton (T.C.) on the van der Waals
formula and the latent heat of
vaporization, 593; on the photo-
electric constant and atomic heat,
734.

Swain (Miss L.) on a tidal problem,
737.

Takamine (T.) on an anomalous
Zeeman effect in satellites of the
violet line of mercury, 241.

Tape, on the form of a suspended,
including the effect of stiffness, 96.

Thermionic currents from tungsten,
on negative, 802.

emission, on a qualitative
method of investigating, 362.

Thomas (D. E.) on a sensitive
method for examining optical
qualities of glass plates, 618.

Thorium, on lead and the end pro-
duct of, 673.

content of the earth’s crust, on

the average, 483.

emanation, on the conden-
sation of, 337.

Tidal problem, on a, 737.

Titanium, on displacements in
spectral lines of, 594.

INDEX.

Topic parameters and morphotropic
relationships, on, 745.

Triple-point, on the variation of the,
with hydrostatic pressure, 143.

Tubes, on the collapse of, by external
pressure, 67.

Tungsten, on negative thermionic
currents from, 802.

Tunstall (N.) on the velocity of the
alpha particles from radium A,
259.

van der Waals formula, on the, 593.

Vaporization, on the latent heat of,
593,

Varder (R. W.) on the absorption
of homogeneous beta rays, 725.
Vegard (Dr. L.) on the series spec-
trum of hydrogen and the structure

of the atom, 651. .

Vibrations maintained by a periodic
field of force, on, 15.

Vincent (Dr. J. H.) on the duplex
harmonograph, 490.

Viscous fluid, on the equations of
motion of a, 445; on the two-
dimensional steady motion of a,
455 ; on the motion of a sphere in
a, 026.

Walmsley (Lieut. H. P.) on the
magnetic deflexion of the recoil
stream from radium A, 253.

Watts (Dr. W. M.) on the principal
series in the spectra of the alkali
metals, 775.

Waves, on the mutual influence of
resonators exposed to primary
plane, 209.

Williams (W. E.) on the motion of
a sphere in a viscous fluid, 526.
Wilson (Dr. W.) on the quantum-
theory of radiation and line

spectra, 795.

Wilton (Dr. J. R.) on ripp’es, 688.

Wire, on the form of a suspended,
including the effect of stiffness, 96.

X-ray wave-lengths and absorption
coefficients, on the relation between,
407.

Young (A. E.) on the form of a
suspended wire or tape including
the etfect of stiffness, 96.

Zinc, on displacements in spectral
lines of, 394.

END OF THE TWENTY-NINTH VOLUME.

Printed by TayLor and Francis, Red Lion Court, Fleet Street.

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