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THE 
: LONDON, EDINBURGH, ayp DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 
| JOURNAL OF SCIENCE, 


A Ob 
— 


CONDUCTED BY 


SIR OLIVER JOSEPH LODGE, D.Sc., LL.D., F.B.S. | 
SIR JOSEPH JOHN THOMSON, O.M., M.A., Sc.D., LL.D., F.R.S. 
| JOHN JOLY, M:A., D.Sc., ERS. EGS. 
| ®#£«.GEORGE CAREY FOSLER, BA LL.D., F.B.S. 
AND 


WILLIAM FRANCIS, F.L.S. 


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


VOL. XXXILT.—SIXTH SERIES. 
JULY—DECEMBER 1916. 


ie, 


LONDON: 
TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET, 
SOLD BY SIMPKIN, MARSITALL, WAMILTON, KENT, AND OO., LP. 


SMITH AND SON, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN, 
AND YHUVE J. BOYVHAU, PARIS, 


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


. Cur spirent venti, cur terra dehiscat, 
Cur mare turgescat, pelago cur tantus amaror, 
Cur caput obscura Phoebus ferrugine condat, 
Quid toties diros cogat flagrare cometas, 

Quid pariat nubes, veniant cur fulmina ccelo, 
Quo micet igne Iris, superos quis conciat orbes 
Tam vario motu.” 


J. B. Pinelli ad Mazonium. 


CONTENTS OF VOL. XXXIV 


(SIXTH SERIES), 


-NUMBER CLXXXVII.—JULY 1916. 


Lord Rayleigh on the Flow of Compressible Fluid past an 
BUSH LC see NE 5 ULI IN STs 5, 5 wd ROMAN GUN Oe Uni alla bal 
Dr. J. R. Airey on the Roots of Bessel and Neumann 
fimmenons or tich Order. si... <4 ce tees oa eae os 
Dr. C. E. Weatherburn on Two Fundamental Problems in 
mcwlnconyiOr Mlasticity y fede... so) ease 
Dr. Manne Siegbahn and Mr. Einar Friman on the High- 
Frequency Spectra (L-Series) of the Hiements Tantalum— 
lemme Ce ate Wy eset es). « «tenement iy os a. ae 
Prof. D. N. Mallik and Mr. A. B. Das on Electric Discharge 
in a Transverse Magnetic Field. (Plate ID.)............ 
Dr. L. Vegard on Results of Crystal Analysis. (Plate IIT.) 
Mr. Sudhansukumar Banerji on Aerial Waves generated by 
PMO CMAN EAP wcll Yarn 6 Ser allacs ler eyalcs -« « «\eholenahafaenen ty) fhe. cio) cals 
Mr. H. H. Poole on the Dielectric Constant and Electrical 
Conductivity of Mica in Intense Fields................ 
Mr. H. Ikeuti on the Tracks of the « Particles from Radium A 
in Sensitive Photographic Films. (Plate IV.) .......... 
Mr. H. F. Biggs on the Decrease in the Paramagnetism 
of Palladium caused by Absorbed Hydrogen ............ 
Miss E. W. Hobbs on the Change in the Resistance of a 
Spastered-Hilm after Deposition... . Sei. os... s. 
The Harl of Berkeley and Dr. OC. V. Burton on a Semi-automatic 
inde eressure Installation 0... 2. cern ese. oe le 5 © 
Mr. G. H. Livens on the Mechanical Relations of Dielectric 
oe iacametic Molamisationy 4)... . . aeemes ale aids oe « 
Mr. B. C. Laws on the Strength of the Thin-plate Beam, held 
at its ends and subject to a uniformly distributed Load 
(‘Spoeeley Wase)), 16 cit Ae Reign ae 6 oo Hie oc ioe 
Proceedings of the Geological Society :— 
Mr. G. W. Tyrrell on 1 the Picrite-Teschenite Sill of Lugar 


(Ayrshire) and its Differentiation ................ 


Page 


1V CONTENTS OF VOL. XXXII.—SIXTH SERIES. 


NUMBER CLXXXVIITI.—AUGUST. 


Lord Rayleigh on the Discharge of Gases under High 
IGTESSUES 12). jessie « «ns WER IRE EE «0. oo oe 
Lord Rayleigh on the Energy acquired by Small Resonators 
from incident Waves of like Pernod .......... 2a 
Prof. A. Anderson on the Mutual Magnetic Energy of two 
Moving Point Charges age. th. .2..4....- 0 
Mr. G. H. Livens on the Principle of Least Action in the 
Theory ef Hlectrodynamits, >. 2.00.22... > - see 
Mr. G. H. Livens on the Hall Effect and Allied Phenomena. 
Mr. M. Ishino on the Velocity of Secondary Cathode hays 
emitted by a Gas under the Action of High-Speed Cathode 
INAYS a aks os oe ee ec See cae) rrr 
Dr. W. Makower on the Strageling of a@ Particles ........ 
Dr. W. Makower on the Hiiciency of Recoil of Radium D 
trom! Radimm C...... ee ek) er 
Dr. G. Green on a Method of Deriving Planck’s Law of 
Radiabion. .).. 0... 0 Es © 8a t ae ae oe 
Prof. G. N. Watson on Bessel Functions of Equal Order and 
ATP UMENG .%). 3 6. RNR. 6 is wk eee 
Dr. J. R. Airey on Bessel Functions of Equal Order and 
ATOUMOENG 2.055... s . Re ole. © @elees oh oe rr 
Drs. 8. Brodetsky and B. Hodgson on the Absorption of 
Gases in Vacuum-Tubestiy., 6 Jick. 252525 oe 
Dr. L. Vegard on the Electric Absorption of Gases in Vacuum- 
DOS. ish oii ap 5 4 se Pes Wools aces as ss oe rr 
Prof. W. M. Thornton on the Cause of Lowered Dielectric 
Strengthiin High-Frequemey Hields 2... 22). 5.2) eee 
Prof. H. I’. Dawes on Image Formation by Crystalline Media. 
Notices respecting New Books :— 
Dr. E. H. Barton’s An Introduction to the Mechanies of 
Phutds 2.55). Re ae oe 
R. P. Richardson and HE. H. Landis’s Fundameutal 
Conceptions of Modern Mathematics; and Numbers, 
Variables, and Mr. Russell’s Philosophy .......... 


NUMBER CLXXXIX.—_SEPTEMBER. 


Dr. L. Silberstein on Fluorescent Vapours and their Magneto- 
optic Properties... sca eee ee. HA le 2 oe tp 
Mr. EH. H. Nichols on the Diurnal Variation of Atmospheric 
‘Hlectrical Quantities . ) 2.2: Gp paeeles Lee ee balikee eene 
Mr. F. Tinker on the Vapour Pressures of Binary Liquid. 
Mixtures: Kinetic Theory based on Dieterici’s Equation. . 
. Mr. A. K. Chapman on the Hall and Corbino Effects ...... 


Sa 
ae 1 eee eT 


— 


262 


295 
3U3 


a 


CONTENTS OF VOL. XXXII.—-SIXTH SERIES. \/ 


Page 
Dr. E. J. Evans and Mr. C. Croxson on the Stark Effect of 
Eee) oeo Specunum lanes (Plate Vere hs 327 


Prof. R. W. Wood and Mr. M. Kimura on Scattering and 

Regular Reflexion of Light by an Absorbing Gas. (PlateVI.) 329 
Notices respecting New Books :— 

Dr. C. Chree’s Studies in Terrestrial Magnetism ...... 345 
Boambureh) Mathematical Tracts ..:i0 cates 2. eos. 346 
Dulletin of the Bureau of Standards, 2a Wee Slot 348 
Prof. Dayton C. Miller’s The Science of Musical Sounds. 350 

EH. T. Whittaker and G. N. Watson’s A Course of 

Modern Analysis: an Introduction to the General 

Theory of Infinite Series and of Analytic Functions, 

with an account of the -Principal Transcendental 
RRCENOMS, Sh ive sey nae AML Ts Ss 5 5: Ce MMMMER a sto copa sol 


NUMBER CXC.—OCTOBER. 


Lord Rayleigh on Vibrations and Deflexions of Membranes, 

_ESIES,, ile aed Bd SSN ae pe eae ee eR 303 
Prof. k. W. Wood on the Condensation and Reflexion of 

caceviolecules. (CPlate VIL) oo. . Coes ot ee 364 
Prot. H. C. Plummer on the Boiling-Points of Homologous 

DCD DOUICC TI Mel aA ee Aedes nec 2 3 371 
Prof. C. T. Knipp and Mr. L. A. Welo on a Wehnelt 

Cathode-Ray Tube Magnetometer. (Plate VIII.) ...... 381 
Prof. C. V. Raman on the “ Woltf-note ” in Bowed Stringed 

in POUONEIIUES 5 al Led eke 1.8) SAT Pee 3G. un aR an 391 
Mr. kK. H. Kingdon on some Experiments on Residual 

LO RNIZZEN GOLAN (VG We.) UE eae ie Ree) UC Ae Re 396 
Mr. R. W. Cheshire on a New Method of Measuring tie 

Refractive Index and Dispersion of Glass in Lenticular 

or other forrns, based upon the ‘ Schlieren-inethode” of 


PINON eh au ahsirsy ahr sap ak teas? tg wie ol od «oS RDN ag iy IC 409 
Dr. J. Robinson on the Photoelectric Effect on Thin Films 
Git LUE Tue CON PAAPE nce eRe Se Wn OREN ose aT ih Pee 421 


Prof. O. W. Richardson and Dr. C. B. Bazzoni: Experiments 
with Electron Currents in Different Gases. (1) Mercury 
PAO MeN ial scare «(SOE ay «at bych DM Reenp eee Po 426 


v1 CONTENTS OF VOL. XXXII.—SIXTH SERIES. 


NUMBER CXCIL—NOVEMBER. 


Page 
Mr. 8S. Ratner on the Mobility of the Nevative Ion........ 44] 
Mr. 8. G. Starling on the Equilibrium of the Magnetic 
Compass m Acroplanes) 32.2 )225..-. 4... >) 3c 461 
Prof. H. 8. Carslaw on Napier’s Logarithms: the Deve- 
lopment -of his Theory yee. 95. 8 a a 476 
Dr. L. Silberstein on Multiple Reflexion .......... 322 eee 487 
Dr. Manne Siegbahn and Dr. Einar Friman on an ee 
Vacuuin Spectr ograph: “Bees Ho eh a 494 
Dr. Einar Friman on the High-Frequency Spectra (L-Series) 
of the Elements Lutetium—Zine. (Plate XI.) .......... 497 
Messrs. E. B. Wood, O. A. de Long, and K. T. Compton 
on Diffusion Cells in Toniwed Gases .-.- 2... . oe 499 
Dr. L. Vegard on Results of Crystal Analysis.— ILI. 
(Plate; XL.) ..... +. Sek es oe). 505 


Prof. A. Oge and Mr. F. Lloyd Hopwood on a Critical Test 
of the Crystallographic Law of Valeney Volumes: a Note 
on the Crystalline Structure of the Alkali Sulphates .... 518 
Notices respecting New Books :— 
Alfred A. Robb’s A Theory of Time and Space ...... 526 
Proceedings of the Geological Society :— 
Mr. Arthur Holmes on the Tertiary Volcanic Rocks of 


Mozambique ./ ..Aiiy seen tee sk. Je). to 5 526 
Dr. A. Strahan on Cores from borings in Kent........ 527 
Mr. F. P. Mennell on the Geology of the northern 

marein of Dartmoomempa..)... 25:6... > lee 528 


NUMBER CXCII.—DECEMBER. 


Lord Rayleigh on Convection Currents ina Horizontal Layer 
of Fluid, when the Higher Temperature is on the Under 


Sider Ce Pes tn oR ne ene ese cee 529 
Mr. C. W. Raffety on some Investigations of the Spectra cf 
Carbon and Hydrocarbons (alate X01)... eee 546 
Mr. A. P. Carman on the Collapse of Short Thin Tubes. 
(Plate DOV 2). 3 icc \. RRs fee oc w eae) te) = eae 559 
Dr. C. B. Bazzoni: Experimental Determination of the 
lonization: Potential of Helium... 22.47). 2 5 oe ee 566 
Mr. H. Jeffreys on the Compression of the Earth’s Crust in 
(Cloyo) di Er SW Rien ger srs So 2 - 575 
Messrs. Miles Walker and W. Witcomb Stainer: An Inquiry 
into the Possible Existence of Mutual Induction between 
592 


VISSER HT Sh le Sekie oles. Qc RES nant het eae oa a Ce 


CONTENTS OF VOL. XXXII.—SIXTH SERIES. Vil 


Page 
Notices respecting New Books :— 
Oliver Heaviside’s Electromagnetic Theory .......... 600 
Prof. W. C. M°C. Lewis’s A System of Physical Che- 
DISTR Ue, copie CA) © ai eam MEE Le, 3 ee re 602 
Intelligence and Miscellaneous Articles :— 
Refraction of X-Radiation, by Dr. H. B. Keene ...... 603 
Additions to Prof. H. S. Carslaw’s Paper on Napier’s 
Logarithms in the November Number ............ 604 
ee ero 5 a rd cle «6, 565° ole. 0: a's ss crop Se Fine ne 605 
Pai ES: 


I. Ilustrative of Dr. M. Siegbahn and Mr. E. Friman’s Paper on 
the High-Frequency Spectra (L-Series) of the Elements 
Tantalum-Uranium. 

II. Mlustrative of Prof. D. N. Mallik and Mr. A. B. Das’s Paper on 
Electric Discharge in a Transverse Magnetic Field. 
III. Dlustrative of Dr. L. Vegard’s Paper on Results of Crystal 
Analysis. 
TV. Iustrative of Mr. H. Ikeuti’s Paper on the Tracks of the 
« Particles from Radium A in Sensitive Photographic Films. 
V. Ilustrative of Dr. EK. J. Evans and Mr. C. Croxson’s Paper on the 
Stark Effect of the 4686 Spectrum Line. 
VI. Illustrative of Prof. R. W. Wood and Mr. M. Kimura’s Paper on 
Scattering and Regular Reflexion ef Light by an Absorbing 
Gas. 
VII. Illustrative of Prof. R.W. Wood’s Paper on the Condensation and 
Reflexion of Gas Molecules. 
VIII. Dlustrative of Prof. C. T. Knipp and Mr. L. A. Welo’s Paper on 
a Wehnelt Cathode-Ray Tube Magnetometer. 
IX. lilustrative of Prof. C. V. Raman’s Paper on the “ Wolf-note” in 
Bowed Stringed Instruments. 
X. Illustrative of Mr. K. H. Kingdon’s Paper on some Experiments 
on Residual Ionization. 
XI. Illustrative of Dr. Kinar Friman’s Paper on the High-Frequency 
Spectra (L-Series) of the Elements Lutetium—Zine. 
XII. Illustrative of Dr. L. Vegard’s Paper on Results of Crystal 
Analysis. 
XIII. Illustrative of Mr. C. W. Raffety’s Paper on some Investigations 
of the Spectra of Carbon and Hydrocarbon. 
XIV. Illustrative of Mr. A. P. Carman’s Paper on the Collapse of 
Short Thin Tubes. 


GR Wasi 
LONDON, EDINBURGH, any DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES.] 


JUL29 1916 
oC 


7 : ; G 
JULY 1916. S PATENT OF 


I. On the Flow of Compressible Fluid past Obstacle: 
By Lord Rayieicu, 0.1, F.RS.* 


T is well known that according to classical Hydro- | 
dynamics a steady stream of frictionless incompressible | 
fluid exercises no resultant force upon an obstacle, such as 
a rigid sphere, immersed in it. The development of a 
“resistance ”’ is usually attributed to viscosity, or when 
there is a sharp edge to the negative pressure which may 
accompany it (Helmholtz). In either case it would seem 
that resistance involves something of the nature of a wake, 
extending behind the obstacle to an infinite distance. When 
the system of disturbed velocities, although it may mathe- 
matically extend to infinity, remains as it were attached to 
the obstacle, there can be no resistance. 

The absence of resistance is asserted for an incompressible 
fluid ; but it can hardly be supposed that a small degree of 
compressibility, as in water, would attect the conclusion. 
On the other hand, high relative velocities, exceeding that 
of sound in the fluid, must entirely alter the conditions. It 
seems worth while to examine this question more closely, 
especially as the first effects of compressibility are amenable 
to mathematical treatment. 

The equation of continuity for a compressible fluid in 


* Communicated by the Author. 


Phil. Mag. 8. 6. Vol. 32. No. 187. July 1916. B 


2 Lord Rayleigh on the Flow of 
steady motion is in the usual notation 


dp dp du _ dv , dw\_ 
a, we ea ta =0, . 5 
or, if there be a velocity-potential ¢, 
dpdlogp , dodlogp , dpdlogp 5 jee = 
Fi geen Pia i i or a +V7*p = 02. 12) 


In most cases we may regard the pressure p as a given 
function of the density p, dependent upon the nature of the 
fluid. The simplest is that of Boyle’s law where p=a’p, 
a being the velocity of sound. The general equation 


dp _ 
(“? = c—49, .. . es 
where q is the resultant velocity, so that 
q’ = (do/dx)’ + (db/dy)’ + (dgjdz)?. . - (A) 


reduces in this case to 
a* log p=C—3¢’, 
or 
a log (p/p)=—37, - »- + + + (5) 


if pp correspond to g=0. From (2) and (5) we get 


1 (dddq , dpdq , dbdd 
2h = SL soe AAS ea a 
Vib = 94 Te da 1 y dy Ge de fe (6) 


When q? is small in comparison with a?, this equation may 
be employed to estimate the effects of compressibility. 
Taking a known solution for an incompressible fluid, we 
calculate the value of the right-hand member and by in- 
- tegration obtain a second approximation to the solution in 
the actual case. The operation may be repeated, and if the 
integrations can be effected, we obtain a solution in series 
proceeding by descending powers of a*. It may be pre- 
sumed that this series will be convergent so long as q? is 
less than a?. 

There is no difficulty in the first steps for obstacles in the 
form of spheres or cylinders, and I will detail especially 
the treatment in the latter case. If U, parallel to @=0, 
denote the uniform velocity of the stream at a distance, the 
velocity-potential for the motion of incompressible fluid is 


known to be 
p=U(r+e/r) cos, . . - . « (7) 


the origin of polar coordinates (r, @) being at the centre 


Compressible Fluid past an Obstacle. 3 


of the cylinder. At the surface of the cylinder r=c, 
dag/dr=0, for all values of 0. 
On the right hand of (6) 


dg dq? 4 ap ag’ _ ap dq? ee a (8) 
5 : 


ee 


dx dx ~ dydy adr dr 
and from (7) 


- 1 d 1 /a 
=i (a) + a (3) ae =1+- = 00s ZO a) 
Also 


Ldg_ TOM (9) eT, 
U de =(1-; “eos 8 ToS (1+ S)sin 9; 


rd@ 
wedg?  . Ac" 4c? A it gy rts 5 
tT a ue + as Cos 20, Te nae = Garett 26. 
Accordingly 


2 2 
V7o= ae {=a (7) 008 8 + cos 30}. (10) 


The terms on the right of (10) are all of the form 
x? cos nO, so that for the present purpose we have to solve 
Be en 1 dp i Op a 

es He) dee de? ee gece eh CE) 

If we assume that ¢@ varies as +” cos nO, we see that 

m=p+ 2, and that the complete solution is 


n —n re ¢ 
o=cos nO 4 Ar + Br + mre ‘ (12) 


A aud B being arbitrary constants. In (10) we have to 
deal with n=1 associated with p=—5 and —7, and with 
n=3 associated with p=—3. The complete dolistion as 
regards terms in cos @ and cos 30 is accordingly 


=(Ar+Br7) cos 8+ (Cr? + Dr-*) cos 30 


2U3e? (ond ct cos 30 
+a [08 0(— gat age)“ | 08) 


The conditions to be satisfied at infinity require that, as in 
(1), A=U, and that C=0. We have also to make dd¢/dr 
vanish when r=c. This leads to 

13 UFc? Le 
B=U + —. 12a —7),2? Tmeliee? (14) 


Dez 


4 Lord Rayleigh on the Flow of 
Thus. 


Ce ae 2 pd 3 4 i 3 
o= Ee + nar fen + Sou 300s 30 


eee Gs es cos 30 5 
+ Ee 4 €08 a(— 278 =e a) = io . (15) 


satisfies all the conditions and is the value of ¢ complete to 
the second approximation. 

That the motion determined by (15) gives rise to no 
resultant force in the direction of the stream is easily 
verified. The pressure at any point is a function of g?, and 
on the surface of the cylinder g’=c~*(dd/d0)’. Now 
(db/d@)? involves @ in the forms sin” @, sin? 36, sin @sin 30, 
and none of these are changed by the substitution of r—6 
for @; the pressures on the ‘cylinder accordingly constitute 
a balancin @ system. 

There is no particular difhculty in pursuing the approxi- 
mation so as to include terms involving the square and 
higher powers of U?/a. The right-hand member of (6) 
will continue to include only terms in the cosines of odd 
multiples of 6 with coefficients which are simple powers of r, 
so that the integration can be effected as in (11),(12). And 
the general conclusion that there is no resultant force upon 
he cylinder remains undisturbed. 


The corresponding problem for the sphere is a little more 
complicated, but it may be treated upon the same lines with 
use of Legendre’s functions P,,(cos @) in place of cosines of 

multiples ‘of 6. In terms of the usual polar coordinates. 
(7, 0, ), the last of which does not appear, the first approxi- 
mation, as for an incompressible fluid, is 


$=U cos O(n £5) =U(rt+ S)Pr » (16) 


c denoting the radius of the sphere. As in (8), 


sth og dq _ dd dq’ ae ae d¢ — U3 (- alae | ve 
dx dx dvr dr 2¢0d0 { T 3p a) 


6e® 24e 
eo) ya ~+ 51) cy pete C (17) 


on substitution from (16) of the values of @ and q?. This. 
gives us the right-hand member of (6). | 


Compressible Fluid past an Obstacle. — . i) 
In the present problem 
gee Step ease 2 AG | 18 
ai ha he 8 dor 8 ee 
while P,, satisfies 


poe gen ns ede Eee 
<n 6 go\sin 0 8) nay alee Ot 3 CLO) 
so that 
7? FS) sce be) 
reduces to 
CO 2 dp Nl ace lL) ap : 
di” a r dr a ee eto 


The solution, corresponding to the various terms of (17), is 
thus 


gPt2P,, 

P= (+2) (p43) —mint 1)’ 
With use of (22), (6) gives 
G3 { - oP, Cle eat OC las Téa t 


(22) 


br | 2478 107? 10r° 17678 
+ ArP,+ Br-?P,+Cr2P,+ Dr-4P3, . . . (28) 


A, B, ©, D being arbitrary constants. The conditions at 
infinity require A=U, C=0. The conditions at the surface 
of the sphere give 


=5(U+ + oa) pe (a) 


and thus ¢ is completely determined to the second approxi- 
mation. 
The P’s which occur in (23) are of odd order, and are 
polynomials in «4(=cos 6) of odd degree. Thus dep|dr is odd 
(in w) and dd/d@=sin 6 x even function of w. Further, 


g’ =even function + sin? 6 x even function=eyen function, 
dq’ /dr =even function, 
dq’/d0 =sin 6 x odd function. 


Accordingly 


dq dq? dd dq? . 
ss a ny a nae = =odd function of p, 


$= 


a? 


and can be resolved into a series of P’s of odd order. ‘Thus 
not only is there no resultant force discovered in the second 


6 Flow of Compressible Fluid past an Obstacle. 


approximation, but this character is preserved however far 
we may continue the approximations. And since the co- 
efficients of the various P’s are simple polynomials in pt, the 
integrations present no difficulty in principle. 


Thus far we have limited ourselves to Boyle’s law, but it 
may be of interest to make extension to the general adiabatic 
law, of which Boyle’s is a particular case. We have now to 
suppose 


Di po—piPo)"s = | as (25) 
making 
dp _ YPo ey (2 Vy 
Se SS S| =a | — ) ai, Hae ete 2 
dp Po \Po Po ee) 
if a denote the velocity of sound corresponding to pp. Then 
by (3) A Nhe | 
wa 1S ay ae 
ee) =0-1;. . ia 
If we suppose that pp corresponds to g=0, C=a?/(y—1), 
and ; : 
Ragen ay eg 9 
3 = ee 
whence 
dlogp _ dgidx 
ae = 2@—(y¥—-De - ° . . (29) 
The use of this in (2) now gives 
1 dpdq* _ dbdq? _ dbd@ 
rap! Res 2 : wee Lee 
paar 2a?—(y—1)q? | da dx dy dy de de h, oy 


from which we can fall back upon (6) by supposing y=1. 
So far as the first and second approximations, the substitu- 
tion of (30) for (6) makes no difference at all. 


As regards the general question it would appear that so 
long as the series are convergent there can be no resistance 
and no wake as the result of compressibility. But when the 
velocity U of the stream exceeds that of sound, the system 
of velocities in front of the obstacle expressed by our 
equations cannot be maintained, as they would be at once 
swept away down stream. It may be presumed that the 
passage from the one state of affairs to the other synchro- 
nizes with a failure of convergency. For a discussion of 
what happens when the velocity of sound is exceeded, 
reference may be made to a former paper *. 


* Proc. Roy. Soc. A. vol. lxxxiv. p. 247 (1910) ; ‘ Scientific Papers,’ 
vol. v. p. 608. 


TI. The Roots of Bessel and Neumann Functions of High 
Order. By Joun R. Atrgy, M.A., D.Sc.* 


oe general formule for the higher roots of Jn(7), G,(2), 
&c. have been given by McMahonf. If m=4n? and 


=F (2n+4s—1), and «® represents the sth root in order 
of magnitude of the equation J,() =0, then 


‘ me Ane lim st) 
aaa: 3 (88) 
__ 32(m—1)(83m? — —982m +3779) - (1) 
a 15(88) eec@8 


A similar formula has also been found f{ for the roots of 
Y,,(x), where 
B= (Qn-+4s-+1)7—are fog my 

When 7 is large, however, the above expressions are not 
applicable to the calculation of the earlier roots. In calcu- 
lating the roots of J,(x), for example, to two places of 
decimals, the first root of Jy99(x) which can be found to this 
degree of accuracy by (1) is the 50th, and in the case of 
J 1000(x), the first root so obtained is approximately the 1000th. 

More suitable formule for all the roots of J,(#) can be 
derived from a result given by Debye §, of which the first 
two terms are as follows: 


J,(2) = (se53) [cos { n(tan o—)— - 


mn tan d 


* naval” cot?) eos Jn i(tan 6—¢) ~The. .@) 


where cos@= -. 
v 


This result can be obtained directly from 


re) — |, ° (esngd—nd)db . . . (3) 


* Communicated by the Author. 

+ Annals of Mathematics, vol. ix. (1895). 
t Proc. Phys. Soc. vol. xxiii. (1911). 

§ Math. Annalen, 67. Band (1909). 


lc lr 


8 Dr. J. R. Airey on the Roots of Bessel and 


by the method employed by Lord Kelvin * in evaluating 
the integral : 


w= | cos m|.«—tf(m) ]dm. 
0 


When z and n are great, the range of integration to be 
considered in (3) is in the neighbourhood of the places where 
xsing—nd is stationary with respect to ¢, i. e. within the 
small range ¢,—7 and ¢,+%, where 

x cos d;—n=0, or cos ¢,= ef 
Ke 

By Taylor’s theorem, for small values of 6— 4), 

“singd—nd=x sin gd —nd\— oe Be 


xsin dy 


a ao el iat Ow sin dy... a (4) 


Substitute 


(p—¢))?= zi Wr 


asin = 


Then 


2 


2 2 
do = cena du, 


and the limits of integration become ultimately —< and 
+a, 
Therefore (3) becomes, when 


& COs 
b= _ COs eas CC 1 


6 (5sin b) . 6 (# sin $j)’ 
y= sin $;—ngd,=n(tan d;—¢)), 
Ti ee aa Bis ne een ig 
n(“) = =| ian Cos (bp— p?— by + Cu :5« pe (5) 


and 


= ii ‘ 
= =) | 2 et Ei — 
i 2 if re ena 
= (; — ee in (72% bp — cit, 


* Lord Kelvin, Proc. Roy. Soc. Feb. 1887; Lord Rayleigh, Phil. 
Mag. Dec. 1910. : 


Neumann Functions of High Order. 9 
Taking the first integral of (6) and putting 


0 =p? + bui—cy'... 
we find, from 


(G1 c0s8d0=T(w) cos", 
« 9 2 


+ } 1) o@, 150? 3 
[seers toon (Jun + HET (Bost 


= 


1 ry L 3 as 
=1'(5)cos7 -- (5+; gcot” 6)P(5 ; )eos Se bee (7) 


From the second integral, a similar expression is obtained 
in which the sine occurs in place of the cosine in (7). 
Finally, we get from (6), 


sue) ats) [G(¥-9) 


‘1 . Z) ey 377 
(5+ 5 cot $s) sang, FP (p)e08(¥-Z) +] Brace) 


The next four terms in the bracket have been calculated, 
the coefficients of which are more conveniently ony essed in 
decimal form. 


It Ley AG) P(s+4) , aah le 
J(a)=— = oe 054 “1. 


then the functions A;(¢;) take the following values, / being 
written for cot? ¢, : 


Ao(¢,)=1. 

Ay(¢;) = 0°12500 + 0°20833k. 

Ao($,) = 0:02344 + 0°01215k + 0°11140k2. 

A3($1) =0°00488 + 0:05941k + 0:12310k? + 0°0683943, 

Aj(d,) =0°00107 + 0:02249% + 0°06521%? + 0°10673%3 
+0:04443h4. 

A;(¢1) = 0°00024 + 0:00780k + 0:04503h?+ 0:09702h8 
+ 0°08960k* + 0:02986k°, 


A first approximation to the roots of J,(«) can be found 
by making the first term in (8) equal to zero 


COS (« sin dy —nd\— a) =Cos [ n(tan di1—9)) — 4 =0. (10) 


10 Dr. J. R. Airey on the Roots of Bessel and 


Hence sibs 
n(tan d; — d4) — i “= Pp = is 


ad 


i n(tan g— $y) = PTY ) ie 


The errors in the corresponding values of ¢, for the first 
four roots of J,(z) are 9’ 8’, 1’ 19", 0’ 25”, and 0’ 11”. 
The roots p, are then given by 


Pp= SCC Dy. '. . 2). er 
Putting 


n(ton y— Gi) = SPD. . (8). 


it can be shown without difficulty that, when r is written 
for n tan q,, 


Ay—1.3.5. Ag" 4+1.3.5.7. 
Ap—1.3.Agr?+1.3.5 7A 


zi crer=s 


tan e= 


9. 
- , (14) 

The quantity ¢ is in general very small and decreases as p, 
the number of the root, increases. For the first four roots of 
Ji(a), € has the following values : 0°03596, 0°01839, 0°01248, 
and 0:00947. 

Any root of J,(#) can thus be calculated from (18) by 
successive approximations. Only in extreme cases is it 
necessary to go beyond the second approximation. In 
calculating the first root of J,(x), for example, the following 
values of d, and € were successively obtained : 


1 == 74° 43! 104: €=0-03631. 
74° 59' 93'"5 + ¢=0°03587. 
74° 59! 161-9, 


the last value of d; giving 3°8316 as the first root of J,(z). 
A number of errors have been discovered in Bourget’s Table * 
of the roots of J, to J;, seven of these occurring in the 
values of the roots of J;. The first root of J;iz) is 8°7715, 
not 8°780, and the ninth root 34°989, not 34°983. Other 
methods t give 8°77148 as the first root of J;(2). 

The first five roots of Jio(x), Jioo(x), and Jyoo(v) have 
been calculated from the formule (12), (13), and (14) : the 


* Ann. de l'Ecole Normale, 3 (1866); Lord Rayleigh, ‘Theory of 
Sound,’ vol. i. Table B. p. 330. 

if Lord Rayleigh, Collected Works, ve i. p. 190; Phil. Mag. June 
1916, p. 520. 


Neumann Functions of High Order. 


successive approximations for the roots of Jy (a) are as. 


follows: 
No. of root. dy. 
oh DANO! AD! +5 
Pao 4s <7, 
a3 14 48)"*5 
D2. BOR 327-5 


30° 13! 48'"°5 
p=3. 34°39 34 
34° 39! 38/2 
p=4. 37° 58! 35'"1 
37° 58! 55'-3 
p=5. 40° 39! 23-4 
40° 39’ 36-5 


Pp- 
108°767 
108°837 
108°836 
ase 
115°740 
121°561 
NPAC 75) 
126°861 
126°871 
1akSi7 
131°824 


0-02755 
0:02726 


0:01250 
000806 
0:00596 
000473 


Roots of J 40 (a) J100(x), and J 1000(2). 


Roots of Jio(x). 


i. 14°4755 
2. 18°4335 
3. 22-0470 
4, 25°5095 
oD. 28°8863 


Ji00(a). 
108°836 
115-740 
121°575 
126°871 
131°824 


Ji000(x). 

1018°62 
1032°76 
1044°39 
1054°74 
1064:24 


For very large roots of J,(x), ¢, is nearly equal to 90° 
and its value is not easily determined from (11). Using 
seven-figure tables, any root can be found to six significant 
figures when ¢@, is not greater than 85°. 

If ¢, exceeds this value, the sine of its complement 6, can 
be calculated from the formula 


immer Gillies 
sin =«+ > = =, + = Pane na t @LD) 
where 1 Alen 
NT rep eae 
il 2 19 
coset} = N— 5. — aaa laps (16) 
and 


Pp=ncosec Oj. 


A more accurate value of cosec 6; can be obtained from 


il 2 19 
Coe Ta Sa eine mio oil ad (17) 
where 1 aa 
TOM (5,2 4 saint) ek 


12 Dr. J. R. Airey on the Roots of Bessel and 


Taking an extreme case, the second root of J:(.) and 
applying (15), the following values were obtained :— 


Ke 63. P2- €. 
0:0553582 3° 10 Ai oe 6°0121 0°02065 | 
0°0551690 3°10’ 2"-64 6°0328 


the second approximation agreeing with the value of p, from 
equation (1). For ps9 of J;(x), (17) gives at once the value 
157°8626554. 

A simple expression for the roots of J,,() when 7 is large, 
can be found from (11) and (12). 


From (11) Celeste 
or 
tan gy=A+ = + 175 seg 
aes paaee DF 
—— : 
An 


Therefore, from (12), since 
ea do,  tan* dy, 
re pal CE A Fy | 


DW 
oie oC E igh eS (18) 


sec gd, =1+ 


Even for the small value n=5, this gives py=8744.. 
instead of 8°7715, and for n=100, p= 108" 77, pp=115° 72, 
and p,=121°58. From (18) it is seen that for very large 


values of 1, ie approximates to a ratio of equality. 
n 
dJ ,,() 
ax 
matical and physical problems as J,(“) =0. 
f 


The equation =( is nearly as important in mathe- 


7 ‘ = 2 a I 
=o (2n+4p—3) and HEA ey = ee — ae e- 


1 2 19 
cosec 8; =u — Fae BO 


and the higher roots of J,'(x)=0 are given by 
Pp=n cosec 6;. 


A simple and convenient formula for the roots of J¢ (2)=0 


Neumann Functions of High Order. ; 13. 


when n is large is : | 
£ 
hi [1+ 5 += ite fey | : 


where BL ea 
ac (1 sou, 

and 5 [3(4p—3)3r 
ie f An i 


Thus the first root of Jro00(2) = 0 is 1008°84. 
Lord Rayleigh has given an approximate expression > LOK 
the first root of J,/(#) =0, Viz. 
pi n+ 0: 51343. 


The Bessel function J_,(#) of negative order is represented 
by a formula similar to (9), y in this case being equal to. 
n(tand,—d,+7). Hence the first TBE soa atone of the. 
roots of J_»(v) can be derived from 


n(tan 5 —-¢,+ 7) = 


pp=n sec $1, 


; for the first root being found from the least positive value. 


of tan @,—¢,. Thus, for the first root of J_s(#), p= and 


(paw a 


and 


tan di—bi=5- The small quantity e can be determined 


as betore, leading to closer values of $, and p._ The first two. 
approximations of p; for J_s(#) are 1°802 and 1°868. 
The Neumann function G,(2) is given by 


1 
G.,,(2) = — = = a ts) sin {+ Vs + Sh. (19). 


Hence it follows that the roots of G,(#) can be calculated 
from formulee similar to (12), (13), and (14). In this case,. 
(13) is replaced by 


n(tan d; — o)=(2= arte. A elit? eh) 


A difficulty arises when p=1 in this expression, but the 
first roots of G,(x) of any order are’found from values of 


* Phil, Mae. Dec. 1910, 


14 Roots of Bessel and Neumann Functions of High Order. 


these functions when the argument and order are nearly 
equal *. The formula (18) holds tor the roots of G,(«) 
when n is large, if X takes the value 


a 3(4p—3)r72 
An 


The Neumann function Y,,(x) is equal to 
(log 2—y)J,,(x) —G,(z). 


Substituting (9) and (19) for J,(x) and G,(#),and proceeding 
as before, we find that 
| agige Utes) 9 
tan [ n(tan ¢;—¢) ] = (5) Besse ae (21) 
~where soe 
as a =0°0738043. 


Therefore 
n(tan @;—;) =0°71172764 (p—l)w . . (22) 


to a first approximation, p=1 corresponding to the first root. 
The first roots of Y,() are found as in the case of G,(z). 
For the second and higher roots, it is not necessary to go 
beyond the second approximation from (22). Thus, using 
five-figure logarithms, the first two approximations for the 
second root of Y,(#) are 5°330 and 5°3549f. The first five 
roots of Gyoo(w) and Yjo9{.v) given below have been calculated 
from (20) and (21), or by employing the G and Y functions 
-of nearly equal order and argument. 


Roots of Gioo(). Yi00(2). 
he 104°380 104°133 
ae 112°486 112°325 
Ds 118:°744 118°608 
4, 1 24S 1B ta | 
a 129383 129-266 


The approximation for the first root of Yyoo() from (22) 
gives $;=15° 44’ 15 and p;=103°90. 

Substituting this value of ¢, in (14) to calculate e, it is 
found that the series in the numerator diverges from the 
first, 15A,7* being greater than Ayr. Consequently 103-90 is 
‘the nearest. value of the first root of Yjo9(v) by this method. 


* Phil. Mag. June 1916, p. 520. 
+ Proc. Phys. Soe. vol. xxiii. (1911). 


bi dda 


TI. On Two Fundamental Problems in the Theory of 
Elasticity. By C. E. Wearuersurn, M.A. (Cantab.), 
D.Sc. (Sydney) ; Lecturer in Mathematics and Physics, 
Ormond College, University of Melbourne (Australia) *. 


§ 1. Introduction. 


) ae theory of vector integral equations, which I have 

treated elsewhere ft, finds an important and direct 
application in the fundamental problems of elastic equi- 
hbrium, requiring the determination of the displacement at 
any point of an elastic body when the value of the surface 
displacement is known or that of the surface traction. 
These will be referred to as the first and second boundary 
problems respectively. In the present paper I generalize 
the problems by the introduction of a parameter X, in the 
manner proposed by Poincaréf in his discussion of the 
problems of the potential theory. Moreover, by the use of 
vector analysis I construct, from Somiglianas integrals 
of the equations of equilibrium, dyadics which form the 
basis of displacement functions whose properties exactly 
resemble those of ordinary simple and double stratum poten- 
tials. The treatment of the fundamental problems in elas- 
ticity is thus recast, and will be found to run exactly parallel 
with that of the potential problems. Corresponding theorems 
are established concerning the magnitude and the reality of 
the singular parameter values, and the simplicity of the pole 
of the solution at each of these. The singular case of the 
potential problem for the inner region when the value of 
the normal derivative is known, finds its counterpart in the 
problem of equilibrium of an elastic body under given 
surface traction. . 

The integral equations that arise are vector equations with 
dyadic kernels and dyadic resolvents. Connected with the 
resolvent of the dyadic which forms the basis of a double 
stratum displacement is another, which, in a separate paper §, 
I prove to be a “‘ Green’s dyadic,” analogous to the Green’s 
function for Laplace’s equation vanishing over the boundary. 

Lauricella || was led, in order to escape the difficulty of 

* Communicated by the Author. 

+ “Vector Integral Equations and Gibbs’ Dyadics,” Trans. Camb. 
Phil. Soc. vol. xxii. pp. 138-158 (1916). 


Tt Cf. “La méthode de Neumann et le probléme de Dirichlet,” Acta 
Math. Bd. 20 (1896), 


§ “Green’s Dyadics in the Theory of Elasticity,” Proc. Lond. Math. 
Soc. 1916-17. 

|| Atte R. Ace. Lincer (5), t. 15,, pp. 75-83 (1906); also IZ Nuovo 
Cimento (5), t. 18 (1907), Four notes. 


16 Dr. C. E. Weatherburn on Two Fundamental 


a kernel becoming infinite of the second order, to discard 
the natural idea of surface traction for that of ‘‘ pseudo- 
tension,” a concept which has no physical significance. 
This is equivalent to eliminating the anti-self-conjugate part 
of the double stratum dy adic. I hope to show that this 
kernel does not become infinite of too high an order to be 
treated by the methods of my paper already referred So. 

The problems may be attacked from an entirely different 
point of view, by the use of the Green’s functions for 
Laplace’s equation. According to this method * the cubical 
dilation @ and the molecular rotation R are regarded for 
the moment as known, and the displacement D is found 
in terms of one or both of them by the aid of the Green’s 
functions for the region occupied by the body. Then, 
taking the divergence and the curl of the value of D so 
found, we deduce, in the case of the first problem, a athe 
integral equation for @ trom the solution of which the value 
of Dis deduced. The treatment of the second problem gives 
a pair of integral equations, one scalar and one vector, from 
the former of which @ is obtained in terms of BR; then this 
value of @ substituted in the second gives a single vector 
integral equation for determining R. 


§ 2. The Equations of Hquilibrium.—Confining our atten- 
tion to isotropic bodies we shall sustain no loss of generality 
by assuming the bodily forces zero; for it is well known 
that the equations of equilibrium can always be reduced to 
the form in which the terms representing these forces do not 
appear +. The ordinary equations of equilibrium are 


2/7 ae 
V vee 0; 
00 
Zine apps eat 2 ot 
V2 AM =: 
vot k=; 


while the conditions to be eunanied at the surfaces are ex- 


pressed by 


ta 0S +(e 
dn ordn Oydn Oxdn Bzdn 

* Cf. eae Atti Lincet, t. 16, (1907), pp. 248-255 and 441-450. 

+ Of, , Marcolongo, “ Teoria Mat. dello See dei Corpi 


Elastici.” Nien (1904), p. 233. 


Problems in the Theory of Elasticity. 17 


and two similar equations obtained from this by cyclic per- 
mutation of the variables. In these w, v, w are the rectan- 
gular components of the displacement of the particle at the 
point (wz, y, z) of the body, x the inward drawn normal, 
@ the cubical dilation, and L, M, N the components of the 
surface traction at the boundary point considered ; while 
the constant & is given by 


k= (O—0")/o" = (A+ #)[h, 


Q being the velocity of propagation of longitudinal waves 
and w that of transverse waves in the body. The constants 
r, # are those employed by Lamé*, mw being the rigidity 
=p. In the equations involving the surface tractions we 
have supposed the units so chosen that pw is equal to unity ; 
and we shall throughout adhere to this assumption. 

If we denote the vector displacement of the point p of 
the body by D or D(p), and the surface traction at the point s 
by T or T(s), the equations for the equilibrium of the 
particle at p are readily reduced to the single equation 


WD crac divi .D — QName Goel) 
or its equivalent 
Ge-Pl) erad div D— curl curl D0. 9") 


while for the surface-point s _ 


—T(s) =2 p+ (k—1)ndivD+nxXcurlD, . . (2) 
n being the unit vector in the direction of the inward 
normal. It wili be convenient to introduce the symbol F or 
F(s) to denote the surface traction with its sign changed, i.e. 


F(s) = —T(s), 

whose value is given by the second member of (2). For 
consistency of notation we shall throughout use the letters 
p, 7 to denote points within the region considered but not 
on the boundary ; while ¢, s; 3, 0 will represent boundary 
points, and dt, ds, &c. the corresponding elements of the 
boundary. The surface % bounding the bedy separates the 
finite inner region S from the infinite outer region 8’. We 
shall assume that & possesses everywhere a definite tangent 
plane and two definite principal radii of curvature. Singu- 
larities such as points and edges are excluded. 

* “TLecons sur la théorie math. de I’élasticité des corps solides.” 
Deuxiéme Lecon. Paris (1852). 

Phil. Mag. S. 6. Vol. 32. No. 187. July 1916. C 


a I rn i i a a 


18 Dr. C. E. Weatherburn on Two Fundamental 


§3. A Particular Integral—An unlimited number of 
particular integrals may be found to the equation (1). In 
fact, if B is any biharmonic vector function, it is easily 
verified that 


s= y’B-— peed div B 


satisfies the equation. Taking 4ar as the biharmonic func- 
tion, where a is any constant vector and r the ‘‘ radius vector ” 
measured from a fixed point p, we obtain the particular 
solution 


99 


rad div (ar)... 4) ae 


k 

Oo Od +k) ® 
This is the vector equivalent of the set of integrals due to 
Somigliana *.. There is no loss of generality in taking a as 
a unit vector. This integral becomes infinite at the pole p 
to the same order as 1/7. Hence it cannot be regarded as 
an actual solution of any physical problem for the elastic 
body embracing the point p; but we are able to construct 
other integrals with s) as a basis, which are finite and con- 
tinuous throughout the body. 

It is important for our argument to notice that s (pq), 
which has been defined as a function of the variable point g 
with p as its pole, is symmetricalin pand gq. For if r is the 
radius vector from p to q 


ae SAR aT ee 
PD TEEarC AG, 


ees k eae 
~,» 1th) Lr 2? + | 


which is unaltered if r changes sign, that is if 7 is measured 
from gtop. The expression is therefore symmetrical in 
pand gq, which may be expressed 


So( pq) = So( gp). 


In the potential theory = (; ) is an integral of Laplace’s 


equation forming the basis of double-stratum potentials. 
To find an analogous solution of (1) we shall determine 
by means of (2) the surface traction corresponding to the 


* “Sulle equaz. dell’ Elasticita,” Annali di Mat. (2) t. 16 (1888). 


Problems in the Theory of Elasticity. 19 


displacement 8). Calculating separately the different terms, 
we have 


© grad div ana“ ja. graa( )] 
=< fa. | grad( )r+ a 


ul 
=al (<)+a.mgrad(=)-2. 
Ie 


ye 
; 


aa 


d By ented 

dn ma (3 
ee dh bom 
1 

(;:)e 


—_ 


= os (- )ta. n grad (< ie grad (2) + na. grad-. 
Hence 
d k+2 They ae 
7 ae eee 


k it i d (1 dr 
een on grad | +na. grad : —3—(;) grad? (ia) (4) 


In the second term of (2) we have 


; 1 k , 
div 8) = a. grad = I +h) V’ (a- grad 7") 
my oat Phdey (i By Paiehy 
of phen 1+k me hs earls), ih? 
Lastly, in the third term 
a 
n x curls) =n x curl 
= in (grad » x a) 
=N.a sone —n. cede ana d.2 9:0G) 
r r 
Substituting in (2) the values given by (4), (5), and (6) 


we find for the value of the surface traction * F, (sp) due to 
the cere f (gp) whose pole is p, 


Fy (sp) = = =-( “ae ames (2 X grad: *) 
da u 
<a (grad alee sla) time 4i(h) 
* It will lead to no confusion if we speak thus of F= —T as the 


“ surface traction.” The symbol employed will always indicate which 
quantity is referred to. 
C2 


20 Dr. CG. E. Weatherburn on Two Fundamental 


§ 4. Vector Potentials of Elastic Strata.—If D and D' are 
two solutions of the equation (1) regular within the given 
region, and F, F’ the corresponding surface tractions, the 
bodily forces being supposed zero, Betti’s reciprocity theorem” 
may be put in the form 


{F(s).D'(s)ds—fF(s)-D(s)ds=0. . . . (8) 


Take the usual set i, j, k, of three unit rectangular vectors,. 
and consider the values of the particular integral sp of the 
previous section when the vector a is replaced successively 
by i,j, k. Denote its values in these cases by 81, 82, 83, and 
the corresponding surface tractions by F\(sp), F2(sp), Fs(sp) 
respectively. Then 
i k tee 
i= a1 +k) grad div (ir), 7 


2 ae eae 
S2 = Gigs an a a +) eee ee 


Is k 
8s = = = 2(1 +h) orad div (k 7) - p) 
We may in the formula (8) replace D’ by s;, s:, 83; in suc- 
cession provided the point p be isolated by (say) a small 
sphere Z with pas centre. It then follows that 


F © Xm\ - ds = 5 = : 
J. FC) -sa(ps) ds =| Bao) -DE)ds (m=1, 2,3) 


Multiplying these in order by i,j,k, and adding, we find a 
result which may be written 


{0 8, (ps) +j8o( ps) +ks3(ps) |]. F(s) ds 


= I D(s) -[ F, (sp) i+Fo(sp)j+F3(sp)k]ds,. . (10): 
e 24+Z 


where the expressions in square brackets are dyadics. In 
evaluating the integrals over the small sphere Z, we notice 
that the first member contributes nothing ; for sj, So, ss 
become infinite at p only of the first order, while the area of 
the small sphere is of the second order, Considering the 
second member, we take the values of F,, F,, F; given by (7) 


* “ Teoria della elasticité,” Cap. VI. 11 Nuovo Cimento, 1872; Annali 
di Mat, (6) 1875. 


oa ial 


Problems in the Theory of Elasticity. 21 


fer the particular values i,j, k of a. The first term of (7) 
substituted in ( a ap 


rH)? EA (<i +i(- ihe .|ds 
i d (1 
=, D(s) 5 (5 ) 4s 


an expression which becomes equal to —ArD(p)/(1+4) 
when the radius of Z decreases indefinitely. The next term 
of (7) contributes to the value of the integral 


- | ee Piet 
ne oat D(s) . lix (2x grad ~ iti x (nx grad \j+ _.. Jas, 


which vanishes along with the radius of the sphere ; for since 
n has the same direction as r it follows that n x grad = 


becomes infinite of order less than the second. Hence the 
integral vanishes in the limit. Lastly, from the third term 
we derive 


Taal Or Bat -(gradr grad r)ib'] 7) ds 


dk 
= Hl, D(s). . (grad » grad r) © (- ) ds. 


The limit of the value of this expression when Z decreases 
indefinitely is —4rkD(p)/(1+4). 

Substituting in (10) the Se thus found for the integral 
over Z we obtain the result 


4nD(p) =) Ds) . [F, (sp) i+ Fo (sp) j-+ F, (sp) k] ds 


-|_ [is, (ps) +] s.(ps)+ks;(ps) | - F(s) ds, . (11) 


which is the vector equivalent of Somigliana’s formule * for 
the components of the displacement at any point of an elastic 
isotropic body in terms of its surface values and the values 
of the surface tractions. The formula (11) is analogous to 


the relation 
Tl du 
ae) ( [ug al (is r dal & - 


giving the value of an harmonic function wu at any point p of 


* Annali di Mat. 17 (1889). 


22 Dr. C. E. Weatherburn on es Fundamental 


the region in terms of the boundary values of the function 
and those of its normal derivative. But this expresses the 
harmonic function as the difference of two others which are 
potentials of double and simple strata respectively. It is 
therefore an obvious suggestion to examine the two integrals 
in (11) from the point of view of stratum potentials. This 
has been done by Lauricella* for the case in which the first — 
integral is modified by using only the self-conjugate part of 
its dyadic. The elimination of the anti-self-conjugate part 
is the object for which his idea of pseudo-tension is intro- 
duced. We shall show, as Lauricella did in the modified 
case, that the two integrals in (11) possess properties and 
boundary discontinuities analogous to those enjoyed by 
the Newtonian and logarithmic potentials of double and 
simple strata. 


§ 5. First, it is evident that each of these integrals satisfies — 
the equation (1) which characterizes a displacement. For 
each of the vectors s,;(ps), 82(ps), 83(ps) as a function of p 
is a solution of that equation, and so also is D(p) by hypo- 
thesis. Hence the first integral of the second member of 
(11) must satisfy (1). But asa function of p this is inde- 
pendent of the boundary function D(s); hence the integral 
is a solution of (1) whatever be the finite and continuous 
function D(s). Indeed it is easily verified that the function 
F, (sp) satisfies (1) for all values of a. If for brevity we 
denote the dyadics in (11) by the symbols Tf 


ig d : 
Yep) = 3, Fulsp) i+ FaCsp) 5+ Bs (ep) &] | 


Piatt es fe 
®( pq) = 5— Lisi (pg) + is:(pq) + ks3(pq) | | 
then the integrals 


W(p) = fv (s) -W(sp) ds 
i) — | B(ps) .u(s) ds) 


in which u(s) and v(s) are finite and continuous vector 


(13). 


* Il Nuovo Cimento, loc. cit. pp. 155-174. 

+ I have chosen the notation to resemble that employed in my earlier: 
papers on the potential theory. Cf. Quarterly Journ. vol. xlvi. (1914- 
15). Three papers. 


Problems in the Theory of Elasticity. 23 


functions of the boundary point s, are solutions of the 
equation (1) of elastic equilibrium, and are finite and con- 
tinuous vector functions of the point p of the region bounded 
by =. The function W(p), whose dyadic V(sp) involves 
d 
dn 
elastic stratum of moment v(s) ; while V(p), whose dyadic 


(*), will be spoken of as the vector potential of a double 


i ines . 
is similarly related to = will be called the vector potential 


of a simple elastic stratum of density u(s). It will be 
observed that the density and moment as so defined are 
vector functions of the position of the point s of the boundary. 
When no ambiguity is possible these potentials will be called 
briefly potentials of double and simple strata respectively. 


§ 6. Vector Potential of a Double Elastic Stratum.—The 
double stratum vector potential W(p) defined in the previous 
section has discontinuities at the boundary exactly resem- 
bling those of the ordinary double stratum potential, and 
expressible in the form 


W(tt) = w(t)+ |v (s). Y(st) . 


14 
W(t-) =—v(t) + J v(s) » VY (st) ds ie 


t+ being a point of the inner region indefinitely close to the 
boundary point ¢ but not on the boundary, and ¢t~ the corre- 
sponding point of the outer region. ‘To prove this we shall 
consider separately the parts of W(p) due to the three terms 
of F,(s) given by (7). First the potential 


ovate ny yy 
Wi (p) = hey a 2 | v(s) e [i> @H ds 


1 "hy! nalvonl 
= sea IYO dala) 


is, to a constant factor, that of an ordinary double stratum 
of moment v(s). ‘This satisfies the known relations 


w(t) = VO+m© he 


BT te i 
w, (¢~) a ay) + wy (t) | 


The potential w.(p) arising from the second term in (7) is 


24 Dr. C. E. Weatherburn on Two Fundamental 


continuous at the boundary. For it istoa constant multiple 
equal to 


Silvis) ° E x (x xX grad “| as = {v (s) Xx | grad (=) x n | ds. 


It is clear from this form of the expression that w.(p) is 
continuous at the boundary. For as p moves up to ¢ along 


the normal and coincides with it, grad > dt remains a finite 
: ae / al 
vector in the direction of the normal, so that grad © ) xn dt 
vanishes in the limit. Hence there is no discontinuity due 
to the element dt, and w.(p) is continuous at the boundary. 
Lastly, the integral w;(p) arising from the third term of 
(7) 1s, as in § 4, given by 


Lk d (1 
w3(p) = oe eae (s)« [grad r grad? a ce ds. 


This is an integral with discontinuity at the boundary equal 
to one-third of that of the corresponding ordinary double 
stratum * ; so that 
k 
w(t") = TrEY @) + w(t) 
(16) 


k 
w3(¢-) =— aE Ot Ws) 


Combining the results for the three potentials w, (p), we (p)‘ 
and w3(p) we have the discontinuity for W(p) expressed 
by (14). 

The continuity of the normal derivative of an ordinary 
double stratum potential has its counterpart in a further 
property of the vector potential W(p), viz. that if the fune- 
tion v (s)is finite and continuous along with its first derivative, 
the surface traction due to a displacement of the particles 
represented by W(p) is continuous at >. This may be put 
more definitely as follows. Imagine a surface >’ close to 2, ~ 
and with its normal everywhere parallel to the normal to &. 
Then the surface traction due to the displacement W(p) of 
the body bounded by =’ remains continuous as >’ moves up 
to and passes through =. This theorem, which is due to 
Lauricella +, is true when v(t) is finite and continuous along 
with its first derivative. 

* Cf. Lauricella, loc. cit. pp. 161-164. 


+ Cf. Atti Lincet (5), t. 15 (1906), p. 429; also Annali di Mat. (1907), 
Cap. II. § 6. 


Problems in the Theory of Elasticity. 25 


§ 7. Vector Potential of a Simple Elastic Stratum.—Con- 
sider next the simple stratum potential 


V(p) =z [i S, (ps) +] 8» (ps) +85 (ps) | -u(s)ds 


= { (ps) .n(.)ds 5 Ca AO ves ves ota eI G)) 


The dyadic ®(ts) is conjugo-svmmetric, being both sym- 
metric and self-conjugate. It is symmetric because the 
function so(¢s) is symmetric. It is also self-conjugate ; for 
if expressed in nonion form it has the same coefficient for 
k: Or 

2(1+k) Ovoy 
conjugo-symmetric and enjoys all the properties established 
for such a kernel in the last part of my paper first 
referred to. 

We observe that the function V(p) is continuous at the 
boundary. ‘This is clear from the form of s)(ts) which 
becomes infinite at t=s only like 1/r. Further, V(p) satis- 
fies (1) and may therefore be regarded as the displacement 
of the point p of a body occupying the region bounded by 
some surface to be specified. The corresponding surface 
traction varies with this surface, and will now be shown to 
be discontinuous * at >. Consider a surface &’, either within 
or without 2, with its normal n’ everywhere parallel to the 
normal n to >. Lets, o denote points on &’; ¢, 5 on &. 
Because the function so(pq) is symmetric we have 


515) = 8; (Ss), Ae) ee (18) 


representing a displacement at the point s with 3 as the 
pole. The surface traction on &' due to this displacement 
is therefore F,'(s3), whose value is obtained from (7) re- 
placing d/dn by d/dn’, so that 


ij as for ji, viz. The dyadic is therefore 


ead Sl 1. 1 
1 Sis ——{ —} ———__ } Tad - 
1 (8?) als) rex x grad ) 
abd 7 V\ Oren 
+ rendu (5) (Se)emds - - U9) 


the point s (=&, 7, ¢) being the current point and $ the pole 
from which + is measured. Hence, since in (17) u(S) is 
independent of s, the surface traction at the point s of S' due 


* Cf. Lauricella, 12 Nwovo Cimento, loc. cit. pp. 166-174. 


26 Dr. C. E. Weatherburn on Two Fundamental 


to the displacement V(p) is given by 


F(s) = = | E F,/(s3) +j F,'(s3) +k F,'(s9) | -u(s)ds 


ae = u(9) « [Fy'(sS)i+- Fy’ (s3)j+F,y/(s9) k] ds 
= fu(s) «8 (98) 45, Lyk. sj 


the dyadic W'(sS) being derived from Y(s3) replacing d/dn 
by d/dn'. This function F(s) resembles the double stratum 
potential W(p) of the previous section, but the dyadic in 
(20) differs from that in the expression for W(s) in the order 
of the variables. The effect of this upon the boundary dis-. 
continuity of the expression appears thus. As &’ moves up 
to and coincides with &, s moves up to and coincides with a 
point ¢. But whether sis the point ¢* of the inner region 
or the point é¢~ of the outer region, the value of the ex- 
pression F,’(st) given by (19) is equal to —F,‘(¢s) = —F, (és) 
given by (7); for the change in the order of the variables. 
means a change in the direction of r. But it is to the 
element dt in the integration that the discontinuity is due. 
Hence the discontinuity in the function (20) is opposite in 
sign to that of W(p). If, then, F(¢*) and F(t~) denote the 
surface tractions for the inner and outer regions respectively 
due to the simple stratum displacement V(p), 


F(t+) =—u(t)+ | u(S). VES) a 
F(t-) = u(t)+ fu(s).Wes)jds) 


It should be observed in what relation the formule (21) 
stand io (14). Regarded as equations in v($) and u($) the 
first of (14) and the second of (21) are integral equations ; 
but they are not associated. Though the order of the vari- 
ables is different in the two kernels, the unknown occurs as 
a prefactor in each case. The equation associated to (14 a) 
is obtainable from (21%) by making the kernel the prefactor. 
The difference is vital because the dyadic is not self-con- 
jugate. Its value is neatly expressed by the formula 


ee a a i 
29r(L +k) V (sp) =a Gr x [ grad (-) x n| 
ad (lm 
+ 3k (grad r grad r) me G ) ae 


The first and last terms in this expression are self-conjugate 
dyadics, but the second is anti-self-conjugate. 


(21) 


Problems in the Theory of Elasticity. 27 
§8. The First and Second Boundary Problems.—I shall 


consider the fundamental problems of elastic equilibrium in 
a general form analogous to that proposed by Poincaré for 
Dirichlet’s and Neumann’s problems, an arbitrary parameter 
X being introduced into the prescribed boundary conditions.. 
We set before ourselves the determination of displacements. 
W(p) and V(p) for an elastic isotropic body corresponding: 
respectively to the boundary relations 


Ewe) — we) J-FL wey) tw) J= £0) 
r . (95) 


i[TV(t-)—TVv(et) | — * (rv) +TV (et) |=—f@) 


f(¢) being a given piecemeal continuous function of the 
boundary point, and TV(t*) denoting the surface traction at 
the point ¢ due to the displacement V(p) for a body occupy- 
ing the inner region, while TV(t~) has a similar meaning 
for a body occupying the outer region. The problems (25) 
will be called the first and second boundary problems re-: 
spectively. For the parameter values X= +1 they relate to. 
the inner and outer regions separately. 
Endeavouring to satisfy these by vector potentials 


W(?p) =|¥ (s) « V(sp) a 
V(p) =) ®(ps). u(s) ds)’ 


due respectively to a double stratum of moment v(s) and a 
simple one of density u(s), we find from the boundary 
properties of such that v(s) and u(s) are solutions of a pair 
of vector integral equations 


v(t)—A\ v(s) . V(st)ds=£(t) | 
wlt)—AJ (ts) -u(s)ds=£(t) J 


(26): 


(27), 
where 

Nes — Valor 
that is the dyadic conjugate to V(ts). These integral equa- 


tions are not, associated because V(ts) is not self-conjugate. 
The kernels of the equations (27), 


1 Mace : 
St) == = [F,(st)it F.(st)j + F3(st) k | 


and its conjugate, become infinite at the point s=¢ but of 
order less than two, and therefore Fredholm’s method of 


28 Dr. C. E. Weatherburn on Two Fundamental 

solution is available. There is no need to eliminate the 
; ‘ , P if . 

‘second term of F)(sp) in (7) involving a x (n x grad -). This 


term does not become infinite of the second order as p 
approaches the boundary point s along the normal. For 


ili : : 
grad— =—r/r*® ; but n has the same direction as r, so that 
Yh 
ux grad= becomes infinite of order less than two. It is this 
Yr 


fact that makes the integral w.(p) of §6 continuous at the 
boundary. 

Hach of the equations (27) has one and only one solution 
unless X is a characteristic number of the kernel of that 
equation. The solutions are expressible in terms of the 
resolvent dyadics H(ts) and H'(ts) of V(¢s) and y(¢s) respec- 
tively, connected with these by equations of the form 


H (ts) —V(ts) =A (H(¢S) . V(Ss)dS=2 | V(US).H(Ss)d3, (28) 


and a similar set (28') for H'(ts) and x(ts). In terms of 
these resolvents the solutions of (27) are 


v(t) =£(t) +2 £(s) -H(st)ds 8 (29) 


u(t) =f(t)+\ H'(ts) .£(s)ds 


and these values substituted in (26) give the displacements 
that satisfy the boundary problenis (25), viz. 


W(p) =f f(s). [ V(sp) +2fH(s9) . ¥(Sp) a ii 
V(p) = J [®(ps) +2J (pS) .H'(Ss)d3 |. £(s)ds) 


§¥. The resolvents H(ts) and H’(ts), in terms of which 
the solutions have been expressed, are known meromorphic 
{unctions of the parameter A, and either becomes infinite 
only when 2 is a root of its denominator D(X) or D'(A). 
These singular values of X depend only on the form of the 
boundary. The solutions (30) may be written 


W(p)= J f(s) -H(sp)ds 
Vip) = \T'(ps) - f(s)ds 
if we define the functions H(sp) and I''(ps) by the equations 
; H(sp) =WV(sp) +2 H(sS) - V(Sp)d5, 
IX( ps) =®(ps) +r | O(pS) -H'(3s)ds. 


(31) 


Problems in the Theory of Elasticity. 29 


The dyadic H(sp) so defined is an extension of the 


resolvent H(st) obtained from it replacing ¢ by a point p- 


not on the boundary. From the preceding relation it is 
easily verified that 


| V(s3) .H(Sp)d9= { H(s3). V(Sp)d8, 
so that H(¢p) is defined by the alternative relations 


H(tp)—WVitp) =) | H(‘s) : V(sp)ds=n \ V(ts) ~H(sp)ds. (32) 


An exactly similar pair of relations (32°) define the dyadic: 


H' (ip) in terms of y(tp). Then the dyadie I’(ps) defined 
as above may be extended, replacing s by another point g 
not on the boundary, the new function I”(pq) being 
specified by 


I’ (pq) =®(pq) +) B(ps) -H/(sq)ds. 


It is then easily verified that 
\T'(ps) : x(sq)ds= | D(ps) .H'(sq)ds : 
and thus the dyadic I’(pq) is defined by the alternative 
relations 
I” (pq) —®(pq) =r | O(ps) «H'(sq)ds=rf I"(ps) -x(sqds, . 


in which g may be replaced by a boundary point ¢. 
Similarly, if we define a dyadic I'(pq) by the equation 


(pq) =P(pq) +2) P(ps) H(sq)ds, 


it satishes the alternative relations 


I( pq) —P( pq) = r| @ (ps) -H(sq) ds =n I( ps) «V(sq)ds. 


§ 10. Singular Parameter Values—We may now prove 
that the characteristic numbers 2, and 2X/, of the kernels 
Wits) and y(ts) respectively are real, and in absolute magni- 
tude not less than unity ; also that each is only a simple 
pole of the resolvent involved. For these values the homo- 
geneous equations 


v(t) =A, |v(s) .W(st)ds 
u(t)=,' | x(ts) «u(s)ds 


admit each one or more non-zero solutions, Regarded as 


moment and density of double and simple strata respectively, 


(33) 


(33°) 


30 Dr. ©. E. Weatherburn on Zwo Fundamental 


these define displacements W(p) and V(p) satisfying the 
‘homogeneous problems 


We)— WO)=a, [ We) + ae 8) 
TV(¢~) —TV(t*) =A, [ TV(t-) + TV(e*) | 


The equations (34) are not associated, and the charac- 
‘teristic numbers of W(ts) are in general different from those 
of y(ts). We observe, for the following argument, that if 
‘U(p) is any regular displacement satisfying the equation (1), 
-and TU(t) the corresponding surface traction, the integral 


[Uj=—[U.T0 de... 


represents twice the potential energy of the deformed body, 
and is therefore a positive quantity, vanishing only for a 
translation or rotation of the body asa whole. Further, if 
‘V is another regular displacement satisfying (1), Betti’s 
theorem gives 

{[U.1V=¥eDU]de=0) 


To prove now the reality of the characteristic numbers *, 
consider for example one Ag of x(ts) for which the second 
‘equation (35) admits a solution Vip). If this parameter 
value is complex (=a+ib), so also is the potential V(p) 
(=U+7U,). Then, on separating real and imaginary parts 
in (35), we obtain 

(1—a)TU- —(1+a)TUt +6/TU,- +TU,+)=0 

(1—a)TU,- —(1+a)TU,*—d(TU- +TU™ )=0, 
where TU~ is written for TU(¢~),and so on. Multiply these 
equations scalarly first by U,; and U respectively, subtract 
and integrate over }; then by U and J, respectively, sub- 


tract and integrate as before. Then in virtue of (37) it 
follows that 


b{[U*]+[U,+]—[U-]—[U,- ]}=0 ic 
—a{|U*]+[0,7|—[U-]—[Uy ]}=[0" ]+[0,*]+( 0 iid 
(38) 
where 
[U-] =| Uru (e-)at, 


which by (36) is positive, as it applies to the outer region 


* The proofs in this § and the next follow closely those of Plemelj 
for the case of the potential theory. Cf, e. g., “ Potentialtheoretische 
Untersuchungen,” § 24. 


Problems in the Theory of Elasticity. 31 


for which the direction of the inward normal is reversed. 
Since, then, the second member of the second equation (38) 
is positive, and cannot vanish except in the degenerate cases 
already mentioned, the coefficient of @ cannot vanish, nor 
therefore that of 6. Hence 0 itself must be zero, making 
My real. Then V(p) must be real, giving U,=0, U=V, and 
@=2,. Thence by (88) 


A= {IV-J+IVA[V-J-[vt}}. 39) 


Thus the absolute value of ),' is greater than unity, except 
in the degenerate cases for which one of the expressions 
[ V~-] or [V*] is zero. 

Similarly, starting with the first equation (35), we may 
prove the same result for the characteristic numbers 2, of 
mes): 


§ 11. To prove next that each singular value XA‘) is a 
simple pole of the resolvent H’(ts). We have seen that it is 
a pole, and therefore also a pole of the density u(t) of the 
same order. If this order n be >1, u(t) may in the 
neighbourhood of A, be expressed in the form 


u(t) =p(t)/(A—Ao!)" + Bi(t)/(A—Ag!)" +006 


where p(t) does not vanish identically. Substituting this 
-value in the second of equations (27) written in the form 


u(t)—(A—Ag’) | x(ts) « u(s)ds—Ag J y(és) -u(s)ds=f(8), 
and equating to zero the coefficients of (A—A,) ” and 
(A—2,')~"**, we obtain the relations 
P (t) ro’ | x(¢s) - p (s)ds=0, 
Pi(t) —Ag’ | x (ts) « pi (s)ds= { y(ts) « p(s)ds=p(t)/r. 


If now we take p(¢) and p,(¢) as vector densities of simple 
elastic strata whose potentials are V(p) and V,(p) respec- 
tively, these equations are equivalent to 


P¥- —TV' —A)|(TV7+TVt )=0 . 
TV,~ —TV,+ —Ap(TV,- + TV,+) =(TV-—TV+)/Ay’. 
Multiply scalarly the first of these by V, and the second 
by V, subtract and integrate over %; then in virtue of (37) 


and the continuity of a simple elastic stratum potential we 
deduce 


[V+]+[V-]=0. 


32 Dr. C. E. Weatherburn on Two Fundamental 


Hach of the expressions [V*]| and [V7] is therefore 
identically zero. The stratum density p(t) must therefore 
vanish, because in the degenerate cases referred to the 
surface traction is zero, and therefore gives no discontinuity 
at the boundary. Since then p(¢) is zero for all values of 
n> 1, it follows that the poles of the resolvent H'(ts) are 
simple. The same may be proved of the poles A, of the 
resolvent H (ts). 

These resolvents may then, in the neighbourhoods of their 
poles Ay and Ay’, be expressed in the forms 


H (ts) =P (ts)/(X) —A) +K (Es) : 


| (40) 
H’ (ts) =P'(ts)/ (Xo —A) + K' (és) 


where K(ts) remains finite at Ag and K’(ts) at Ao’. Since the 
poles are simple the residues P(¢s) and P’(ts) are given by * 


P(ts)= > n,(t) v(s) | 
‘ae bys ie ata 
PGs) = 2 u(é)m,(2) | 


where n,(s) and m,(s) are the solutions of the homogeneous 
equations associated with (34), satisfying the orthogonal 


relations 
Js cia eal P= ae 
{m,(t) .u(t)dt 0, if 14). 


while £ and #’ are the multiplicities of the roots A, and Ag’ 
of D(A) and D'(A) respectively. 


§ 12. Solution in the Neighbourhocd of a Singular Value 
of %.—The solutions of the boundary problems (25) as 
expressed by (30) in general hecome infinite when 2X is 
equal to a singular parameter value )» (say) in the first case 
and A,’ in the second. In order that this should not be so 
it is necessary that the residue of the solution at this pole 
should vanish, that is 


{ f(s). P(st}ds=0 for v(t) . 
\P'(ts) .£(s)ds=0 for at 4) 


These are equivalent, in virtue of (41) and the linear 


* Cf. Plemelj, Monat. fiir Math. und Physik, Bd. 15 (1904), 8. 127- 


Problems in the Theory of Elasticity. 33 


independence of the functions n,(s) and of the others m,{s), 
to the separate conditions 


eGiem. sjds—— 0, 21, 2, 3. cua scons (6) 


by 
fi(s) nt Ses — On) — Rit) d,s mom Hole. Ula) 


which are the usual necessary and sufficient conditions for 
the existence of a solution to (27) at a singular parameter 
value. 

If these conditions are not satisfied, we can construct the 


function 
LC) =fG) —| f(s) = P(séldisp gine) v2) e (25) 


which satisfies the first of the conditions (43), and similarly 
the function 


£5(¢), =£(t) —{P'(¢s) of (s)\dsti rants 45458) 
which satisfies the second. ‘This is easily verified by means 


of the values of P(st) and P’(ts) given by (41). Thus there 
do exist solutions to the modified problems 


aL WO) — WEE) FLW) + WE) Sho) 
: , (46) 
£[TV(t-)—TV(t*)] > [TV (¢-) + TV(t*)] = —£,(t) 


which are regular at the characteristic value A, in the first 
case, and Ay in the second. The solutions of these problems 
are also expressible in the form (31). The poles of the 
dyadics H(tp) and I"(pt) are all simple; hence in the 
neighbourhoods of their respective poles, Ny and Ao, they 
may be written 


H(tp)=K (tp) + P(tp)/(Ao—A) : 
I" (pt) =@' (pt) + dQ! (pt)/(o’—d) J? 
where K(tp) and @’/(pt) remain finite at the poles A» and A, 
respectively. It can then be shown, exactly as T have done 
in the case of the potential theory *, that the solutions of the 
modified problems (46) are given by 
W(p)= J f(s). ie 
V(p) = J@'(ps) -f(s)ds J’ 


which are regular at the poles considered. 


(47) 


(48) 


* Proc. Roy. Soc. Victoria, vol. xxvii. pp. 169-170 (1915). 
Phil. Mag. 8. 6. Vol. 32. No. 187. July 1916. D 


34 Dr. C. E. Weatherburn on Two Fundamental 


Reverting to the equations (33) we see that the first 
integral, regarded as a function of p, is of the nature of a 
simple elastic stratum potential of density AH'(¢qg). From 
this it follows that 


$[ TI (t*g)—TY" (tg) | =AH'(ég), 
—$ [TI (t+9) + TI" (t- 9) ] =x(t— +2) x (ts) - (sq) ds 
== H' (iq). 
Adding and subtracting, we find 


TI" (t-g) = —(1+A)H’ (tq), 
TH 9g) =— al eet 


An exactly similar pair of equations (50) with undashed 
letters may be deduced from the first form of (33'). From 
the second integral in (33') which, regarded as a function 
of g, is of the form of a double stratum potential of moment 
ALT (ps), we find in a similar manner 


mE 8 6) . 
D(pt-)=(1— a0 (pt) J’ 


showing the nature of the boundary discontinuity of the 
dyadic I'(pg), which I shall show elsewhere to be the 
Green’s dyadic for the inner region corresponding to zero 
surface displacement. A similar pair of relations (51’) hold 
for the dyadic F’(pg). From the preceding results and the 
equation (47) the following relations may be deduced * :-— 


(51) 


TG! (tp) = —(1+A)K (tp) + P'(tp) | (52) 
; TG (t+ p) = —(1—A)K (tp) — P'(tp) 
an 
G(gtt)=(1 + ANE(gt) — Wn (53) 
G@(yt-) = (1 — A)G(yt) + AYQA(g2) 


§ 13. The First Boundary Problem for one Region only.— 
Consider now the boundary problems for the parameter 
values A:=-+1, which correspond to the inner and outer 
regions separately. Taking the first boundary problem for 
the inner region S, (A= —1), the moment v(s) of the double 
stratum whose vector potential solves the problem is given 
by the integral equation 


v(i)+ (vis). W(silds=f(). . . .- (58) 
To prove that X=—1 is not singular for the kernel V(ts), 
* Tbid., § 2. 


Problems in the Theory of Elasticity. 35 


it is sufficient to show that the reduced problem is incom- 
patible ; 2. e. that the homogeneous equation 


We) + ('v(s)-VGids=—0 27 2° 2 (52) 


does not admit any solution but zero. Suppose that it does 
admit one, v,(s). Then the potential W,(y) of the double 
stratum of moment v,(s) vanishes at the boundary of the 
inner region. and therefore identically throughout that 
region, because zero is the only regular solution of (1) 
vanishing over &. The surface traction for the displace- 
ment W,(p) is therefore zero for the inner region, and 
being continuous at the boundary is zero for 8’ also. The 
value of W,(y) must then be identically zero throughout 8’. 
It cannot be the displacement due to a simple translation or 
rotation, for a double stratum displacement vanishes at 
infinity. It follows that 


2v, (t) = W(t") —W,@-) =0. 


Thus the homogeneous equation (54’) does not admit any 
solution but zero, and A=—1 is nota singular value. The 
equation (54) then gives a unique finite and continuous 
moment v(s), which; defines a double stratum potential 
representing the displacement at any point p of the body 8, 
assuming the value f(t) at the boundary. As a particular 
case of (31) the solution may be written 


W(p)= Jj f(s) me GE OCS. c PA ee a 
where the suffix denotes the value of X involved. In virtue 
of (50) this may be given the alternative form 

W(p)=—4)\f() TL a(t pdt, 
the:index + having the usual significance. 
§ 14. If the body occupies the znfinite outer region S' the 
parameter value for the problem is X\=+1, and the dis- 
placement W(p) of the body, assuming the surface value 


—f(t), is expressible as the potential of a double stratum of 
moment v(s) given by the integral equation 


v(t) — | v(s).W(st)ds=f@). . . . (56) 

& Now the parameter value A=+1 is singular for the 
kernel W(st), the homogeneous equation 

v(t)= | v(s) « W(st)ds Seis a) be, (OG ) 


admitting certain non-zero solutions. To see this we revert 


Dee 


36 Dr. C. E. Weatherburn on Two Fundamental 


to formula (11). If instead of p we take a boundary point ¢, 
the first member must be replaced by 27D(t), and we may 
write the equation 


D(t)= | Dis). V(st)ds— | Olés) .F(s)ds. . (11) 


If, then, any displacement D(p) possesses zero surface 
traction, it satisfies the relation 


D(t) = | D(s) . V(st)ds, 


which is identical in form with (56’). But the only regular 
deformation corresponding to zero surface traction is a 
translation or rotation of the body as a whole*. Now there 
are three independent translations, i, j, k, and three in- 
dependent rotations, ix p, j x p, kK x p, where p is the position 
vector of the particle p referred to the c.m. of the body, and 
the unit vectors i, j, k are taken along the principal axes of 
the body. These, then, are the six independent solutions 
of the homogeneous equations (56’). It follows then that 
the associated homogeneous equation 


v(t)=(W(ts) -v@)de oo. 
also admits six linearly independent solutions 
a(s), (2=1, 2,..., 6); 


and in order that (56) may possess a solution it is necessary 
and sufficient that f(s) be orthogonal to each of the 
functions «,s); that is, it must satisfy the relations 
\£(s) <2,(s)ds=0, “= 1,2, .....5:6). ear 
If these conditions are satisfied (56) admits a solution v(s) 
which, when substituted in (26), gives a displacement for 
the outer region whose value at the boundary is —f(¢). 
Even when the conditions (58) are not satisfied the 
-problem may be solved for the boundary value —f(t) as 
near as a displacement of the surface asa whole. For by 


§ 12 there exists a solution to the modified problem corre- 
sponding to the boundary value —f,(¢), where 


£,(t)=f(é)— \£(s) - P_i (sé) ds 
=f (7) -= v,(z) ) £(s) -n,(s)ds 


=f (¢)=a—w Xp,. Sere a. 
* Cf.,e, g., Marcelongo, loc. ert. p. 196. 


Problems in the Theory of Elasticity. 37 


where a and are constant vectors. This proves the state- 
ment. ‘The solution W(p) to this problem is given by the 
double stratum potential 


W(p) =f £@) -Kastp)de, 


which, in virtue of (52), may be expressed in the alternative 
form 


a 


W(p)=—4) f(s) -TC,s(d-p)dt. 


§ 15. Second Boundary Problem for one Region only.— 
Consider next the second boundary problem requiring the 
determination of the displacement for a given value of the 
surface traction. In the caseof the ¢nner region the particular 
value of the parameter is X= +1, and the required displace- 
ment for a given surface traction f(¢) is expressible as the 
potential of a simple elastic stratum of density u(s) given by 


u(t)— y(ts).u(s)\ds=f(@). . . . (60) 


The value A= +1 is, however, a characteristic number of 
the kernel X(ts) ; for the homogeneous equation 


mii) — \iy(Cs) = wis) agen) |. aac) 


and its associated 
u(t) = | u(s) .y(st)ds= | W(st).u(s)ds . . (62) 


admit certain non-zero solutions. The last equation may be 
written 


u(é)= =| [ Fi (sf)i+ F,(st)j+F;3(st)k |] .u(s)ds, 


and that this is satisfied by any constant vector a is easily 
verified by considering separately the three parts wy,(¢), 
w,(t), and w,(¢) asin $6. We thus find 


wy(t)=a/(1+h), we(t)=0, w;(¢)=fa/(1+4), 


showing that aisa solution of (62). Further, if p is the 
position vector of the point p relative to the c.m. of the body 8, 
and the unit vectors i, j, k be taken along the principal axes 
of inertia, it can be similarly shown that ix p,j Xp, Exp are 
also solutions of (62). The only independent solutions of 
this equation are the six vectors i, j, k,ixp,jxp, k xp. 
Hence, in order that (60) may admit a finite and con- 
tinuous solution, it is necessary and sufficient that f(¢) be 


38 Two Fundamental Problems in Theory of Elasticity. 


orthogonal to each of these six vectors. The six conditions 
thus expressed are equivalent to the two relations 


(s@at=0, Jp@)xt@d=0, . > - Gay 


which are the conditions of equilibrium of the body S acted 
on by the surface forces f(¢). These relations must &@ prior 
be satisfied if the problem is to admit a solution. Then (60) 
admits a finite and continuous solution u(t) which, as density 
of a simple stratum, defines a potential V(p) representing 
the required displacement. This displacement is, by (81), 
equal to 


V(p) = ST, (pt) -£@)dt=4)T,, (pit) .£(dt. (64) 


§ 16. If the body occupies the infinite outer region 8’, the 
parameter value for the problem is X=—1, and the dis- 
placement of the body for a given surface traction —f(¢) is 
expressible as the potential of a simple stratum of density 
u(s) given by the integral equation 


u(t) + fu(s).Wits)ds=ft@). . . . (65) 
Now the corresponding homogeneous equation 
u(t) + \u(s).W(ts)\ds=0 . . . . (66) 


does not admit any solution but zero. For suppose that it 
admits.a solution u,(¢) ; then the simple stratum displacement 
V,(p) with this function as density has zero surface traction 
at the boundary of the outer region. ‘The function V,(p) 
for the region S’ must therefore represent one of the 
degenerate displacements ; and since it vanishes at infinity 
this displacement is identically zero throughout 8’. But the 
function is continuous at &, being the potential of a simple 
stratum. Hence it vanishes over the boundary of the inner 
region, and therefore aliso throughout that region. The 
surface traction at the point ¢* is therefore zero. Thus 


2u,(t) =F(¢-) —F(¢*) =0, 


which proves the statement. 

Since then (66) does not admit any solution but zero, 
(65) does admit a unique finite and continuous solution u(s); 
and this function substituted in (26) gives the required 
solution of the problem for the outer region 8’, which by 
(31) may be expressed in the form 


V(p) =) T_,(pt) .f(Qdt=4 fT, (pt-) . £(dt. 


[oem 


IV. On the HMigh-Frequency Spectra (L-Series) of the 
Elements Tantalum—Uranum. By Manne SIEGBABN, 
Dr. phil., and Hrxar Fran, Lie. phil. * 


[Plate I.j 


Introduction. 


‘@ a former communication f the writers have given an 

account of some preliminary researches on the high- 
frequency spectra of the elements gold—uranium. We then 
followed the line called «, being the one best determined 
from the measurements by Moseley f of the other elements. 

This research contains a somewhat complete representation 
of the L-series of the heaviest elements (from tantalum to 
uranium). We have succeeded in finding at least 11 dif- 
ferent line-groups. The measurements also indicate some 
rudimentary groups, the existence of which further investi- 
gations may decide. 

We have also examined the elements polonium and radium. 
In the case of polonium, two of the characteristic lines « and 
8, and several others were obtained. It is possible that some 
of these are due toimpurities. With radium, only a very weak 
a-line could be photographed owing to the small quantity we 
had at our disposal (0°l mgr.). These measurements defi- 
nitely confirm the ordinals of these elements as being 84 and 
88 respectively. 


Haperimental Arrangements. 


For these experiments the X-ray spectrometer, seen in 
Plate I., was used§. The lead-slits, the clockwork with the 
rock-salt crystal, and the plate-holder are mounted on a 
marble plate furnished with three set-screws. The width of 
the slits may be changed arbitrarily. | In this case the first 
slit was 0°1 mm. wide, and the second about 2 mm. The 
crystal, mounted on a small table, was adjusted by two 
screws, which respectively displaced the crystal and turned 
it about a horizontal axis. With the aid of the clockwork 
the crystal is turned round at a constant speed of 15° in an 


* Communicated by the Authors. 

+ Phil. Mag. xxxi. p. 403 (1916) 

t Phil. Mag. xxvii. p. 7038 (1914). 

§ Some of the elements were examined with a similar apparatus of 
wood. 


40 Dr. M.Siegbahn and Mr. E. Friman: High-Frequency 


hour. The plate-holder is fastened on the marble plate with 
two screws, and may, by loosening the screws, be made to 
approach to or to recede from the crystal. By this means it 
is possible always to focus the rays on the middle part of the 
range during examination. The adjustment of the plate- 
holder perpendicularly te the line: slit-centre-the crystal 
rotation axis, is made by an optical method. 

In order to screen off the photographic plate in the casket 
as much as possible from the scattered radiation a lead shield 
extending from the crystal to the casket was placed on the 
support seen in Pl. I. 

The X-ray tube had the form seen in fig. 1. The anti- 
cathode, consisting of a copper cylinder fastened on with 


Fig. 1. 


ee es 


Pump+« 


putty to the outside of the glass-grinding, was cooled with 
water. As window, a foil of 0°05 mm. of aluminium was 
used. The tube was exhausted with a Gaede molecular 
air-pump. To obtain a constant vacuum it proved advis- 
able to use a valve (fig. 2), placed between the two pumps. 


Spectra of the Elements Tantalum—Uranium. Al 


This consists of a micrometric [AA] adjustable cone [C] 
carefully ground. The air is then constantly streaming 
through the holes [BB] and the conic opening. The 
delicate regulation of the vacuum was done through altering 
the resistance of the air-pump motor. 


The attachment of. the substances on the anticathode has 
been already described in the case of Au-U*. Of the 
elements now examined Ta, Ir, and Pt in metal form were 
soldered on the anticathode, while Os as a salt, Ra as radium 
bromide, Th as thorium oxide, W and U in powder form were 
rubbed on it. Po was examined in the form of an electrolytic 
deposit on a piece of sheet-copper. 


Results. 


The results of the measurements are given in the Tables I.- 
XIV. In the first column the notation of most of the lines 
is to be found, and in the second the relative intensities of 
the lines under the assumption of the value 10 for the 
strongest «-line (#,). The third column contains the 
glancing angles of reflexion, and the fourth the wave- 
lengths calculated from the formula % = 2dsin g@, where 
log2d has been taken to be 0°75035. In the last column 


are the values of a/ = The accuracy of the measurements 


is estimated to about 0°3 per cent. 


* Siegbahn and Friman, Phil. Mag, J. ec. 


Dr. M. Siegbahn and Mr. E. Friman: High-Frequency 
TABLE I. 
Tantalum. 

Line. | Rel. Intens. g. X.108 cm. | fi 1052 
Caer a ees 10 15° 50° (207 1°536 0°807 

Do ediceoecns 3 15 44 55 1°528 0°809 

Ge ete sees 10 15 38 35 1:518 0812 
CT Cs area = 14 16 30 1°388 0°849 
ene se | 2 13 48 30 1:343 0363 

ye 8 13 35 45 1323 0869 | 

pene y 3 13 22 55 1303 0876 
ia ae 6 13°29 1-280 0-884 
ete es 3 ll 37 55 1-135 0939 
Ye visec sete i LA GAs: 50 LAGE 0953 
salem ie 1 11 14 40 1-097 0955 | 
Tape I: 
Tungsten. 

Line Rel. Intens. d d. 10% cm Whe 10-4 
Cites} s- 8 ean cia 1°539 0-806 
Cia 2i5.2) 10 15 49 55 1'535 0-807 

By ip 3 1b TS 325 1-481 0°822 
Zc ee 10 15 9 20 1-471 0325 =| 
Cie 15h ae. 9 14 16 15 1:387 0:849 | 
1 Wie as 3 13 (19915 1-296 0878 
| oan cesar nae 8 Laan is) vale 1:278 0-884 
oe cae 3 12 55 15 1-258 0-892 
| GR m ores 6 12 44 20 1°241 0-897 
Picacho 3 Hy Sook 1:095 0:956 
Wali oes i 10)) 5338 1°064 0-969 
at 1 logs 0 5 1-058 | 0-972 
TaBLE ITT. 
Osmium. 
Line. Rel. Intens, o. | d. 108 cm. /e ee 
| 
Gide aa 8 15° 51’ 20” | 1538 0-806 
Ova sce ais 10 15 49 20 1534 0:807 

TO ie ee 3 14 22 55 | 1:398 0846 

ae ha le 14 16 40 | 1-388 0°849 

(Bi eceenmank 3 1224 to 1:214 0°908 

ge 8 12 15 1194 0915 

Nec 3 12 3 50 1:176 0-922 

Aa cite duane 6 11 ya a3 1:167 0:926 

sf SME 2 | 3 1027 25 | 1-021 0:990 


Spectra of the Elements Tantalum—Uranium. 43. 


TABLE LV 
Iridium. 
Line. Rel. Intens. | r.108em. | Vs l Wie 
ee 
1 14° 44" 40” 1432 | 0-836 
Gg isc... 3 14 17 50 1390" (848 
Epes a 3 13 59 1-360)! | 0858 
ee 10 13 52 40 1:350 0-861 
ial .. 1 13) 345.40 13220) 0-870 
il: 6 138 28 15 13ib 0873 
eee. 3 TONS 45 Loy) 0-922 
BAe. 8 Tle 5Ounls Es | 0-931 
ee. 2 11 40 1138 0-938 
oe 6 11 86 995 1133 0:940 
BNE ck: 4 D2 7.:30 nei | 0-946 
ee 0 Lb 17 10 1:101 0:953 
Pees ik... 3 1l 16 1-100 0-954 
Pee ates 3 HO hsain 0-989 | 1-006 
A ie i 9 50 25 0:962 1-020 
eet c 1 9 46 50 0-956 | 1-023 
Se 1 9 22 45 0-917 1-044 
TABLE V. 
Platinum. 
| | Cai 
Line. Rel. Intens. d. A. 108 cm. | a = 10m 
lea aa 0 13° 58’ 1-358 | 0'858 
We ees ce ee 3 1352) 40" | se sae 0:861 | 
a a 3 18 35 45 Le ia 0:869 
BL 10 13 29 20 toes 0873 
2 13,95 130 1275 | 0:885 
Si. 2 12 44 50 [242 | 0:897 
Mie Ot bales 3 1152.30). eee 0-929 
PM, 4 11 42 20 LNG 0:936 | 
HG eony. ge. 2 ile yes ee) | 0:939 
a 8 TUT I 1120 | 0945 | 
es, 6 TT rel) 1-101 0:953 
Mpa. 1 11 14, 50 1:098 0-954. 
1 Ae 5 tO 1:083 0-961 
ean 2 10 58 40 1-072 0-966 
Hite et 0 10 8 15 0991 | 1-005 
FAME 3 9 48 0958 | 1:022 
pes, il 9 32 25 0-933 | 1035 
oleae cok 1 Oa On ea0 0-929 | 1:038 | 
Patan at 1 9 11 50 0-900 1-054 | 


44 Dr. M. Siegbahn and Mr. E. Friman: High-Frequency 
TasLteE VI.—Gold. 


Line. Rel. Intens. p. | A. 108cem. Ji Wee 
els eee 3 13° 10’ 30" 1:283 0883 
Chee 10 132 Sees On 7 Or 0:887 
Fi asc ee 0 126 eS a 91197 0-914 
Bem... 1 1 AT 0 1:102 0:952 
(ON cane 8 11837507) + 11-080 0962 
Bay | Sa come 6d 10 54 30 1:065 0:969 
(Conk Sea 2 10 35 50 1-035 0-983 
rh Soe 3 9 25 35 0-922 1-042 
Wath eax 1 9 11 0:898 1-055 
a, Oe 1 OGRE 35 0-894 1-057 
Wiens 0 8 52 45 0-869 1:073 


Taste VIJ.—Mercury. 


| i 
Line Rel. Intens. d | A. 10% cm. | Jt 10%. 
| 
Che 3 15° 52’ 30” | 1539 0-806 
Gila. 2-5: 10 14 18 55 1°392 0°847 
| 3 14 13 40 1383 0-850 
(Been | 2 12 50 20 1:251 0-894 
CRs 10 12 43 45 1°240 0:898 
a. 8 10 44 45 1-049 0-976 
(Seuae ose. os 6 10 40 5 1:042 0-980 
ok ee | 3 |) Ege ORB 0396 | 1-056 
TaBLE VIJIJ.—Thallium. 
Line. Rel. Intens. Q. Pe OF em: a/ 50-4 
ie mea aie tie 3 12028) ab” 1-215 0-907 
(Ta eRe eee 10 122 21 40" 1°205 O°911 
Te eee 0 1230 (95 1:124 0-943 
1 10 42 25 1:046 0-978 
Abela 2 10 36 30 1-036 0°982 
ee 8 10 21 15 1-012 0-994 
i 6 10 17 45 | 1-006 0-997 
ee 2 10 13 0-998 1-001 
BEeEe 2 9 59 55 0-977 1-011 
0 9 22 45 0-917 1°044 
WY igeeen ss | 3 8 49 55 0:864 1-076 
Fe, SS Se 1 8 37 40 0°844 1:088 
| 1 8 35 10 | 0840 1-091 
eee 1 8 15 25 0-808 1112 


Spectra of the Elements Tantalum—Uranium. 45, 
Tanne IX .Meade) -) 


Line. Rel. Intens. d. dX. 108 em. Aye Ose 


—_————= 


meee 3 12° 9! 45" 1:186 0-918 
: 10 12> 3 1-175 0:923 
7 ieee 0 11 10 30 1-091 0:957 

2 10 28 30 1-023 0:989 
a= 2 10. 10: %5 1-008 0:996 
a 9d 103 35 0-983 1-009 
5 2 9 54 5 0-968 1-016 

2 9 44 30 0:855 1-081 
oa ee 3 8 36 20 0°842 1-090 
= alee 1 8 22 45 0:820 1-104 
ce 1 8 20 15 0816 1-107 
ieee. ax. 1 8 5 20 0°792 1-124 


TABLE X.—Bismuth. 


Line. Rel. Intens. op. AX. 10° cn. We 10-4, 

8) 1POR On 251 L171 0°924 
Go 3 Hey 49r 5 1:153 0931 
GONE RS S25 10 aay Don | 1144 0:935 
TEE Sains 1 LOPRSO, Son) | 1:059 0:972 

2 10m 9e) 25 0:992 1-004 
etpedeee ssc 3% oi 2 9 59 55 0:977 1:012 
(C oges Geae 6 9 45 3d | 0:954 1:024 
Cee 8 9 43 10 0:950 1:026 
[ci Cee 2 9 34 50 0:937 1:033 
(Sees. 8s YY Ore 26. 20 0:923 1:041 
3 eee 3 S 16 40 0-810 L111 
oi) SHR RCEE 1 SG 0-794 E22 
Bn scrasian 2 8 4 15 0-790 1:125 
5 ee eee 0 7 46 45 0°762 1:146 


TABLE XI.—Polonium. 


Line. cu) r 108 cm. \/i 4 10-4, 

12° 43’ 40” 1:240 0°898 
led aye Py memes 12} 3) 835) 1176 0°922 
I il 45 35 1-147 0-934 
Ao aes 11 21 35 1-109 0:950 
10 46 50 1:055 0:975 
, lO 41 1:043 0:979 
BD Ghat 10 3 10 0°982 1:009 
| 9 48 50 0-959 1-021 
| [Bre stesnottatel 9 24. 45 0:920 1:042 
9 9 0-895 1:057 
8 56 45 0-875 | 1:069 

8 50 40 0:865 | 1-075 | 


46 Dr. M. Siegbahn and Mr. E. Friman: High-Frequency 


TasBLE XII, 
Radium. 
i Uae 
Line. g. A. 10% cm. WAR 10a 
Gi dictoces set 102° 207 307 1:010 0°995 


Taste XE, 


Thorium. 
Line. | Rel. Intens. p. | A. 108 em. | a/ i104 
pe 3 9° 54! 50! 0-969 | L016 
Gy wapioknet 10 9 47 25 0:957 1:022 
2 8 28 58 0-830 1098 
(ey seen 6 8 °F 45 0-797 1-121 
in ae 8 7 49 20 0-766 1143 
one 2 7 44 30 0°758 1149 
a Bae 4 6 40 30 0:654 1:236 
ae. 1 6 28 35 0635 1-255 
TABLE XIV. 
Uranium. 
| Line. | Rel. Intens. od. A. 108 em. /}- 10-8 
| po BE ale i 
Gs case 3 Soa oe 9:922 1:042 
Was Mon sie 10 SOAS 0-911 1-048 
| 2 Sait 40 0-786 1128 
occa. o. | 7 7 43 15 0756 1-150 
| 2 7 88 25 0°748 1156 
2 7.98 45 0-724 1:175 
Nong) 5 Sea i 7 20 50 0:720 1179 
cae 1 7 14 50 0710 1-187 
gee AN 4 6 16 25 0-615 1-275 
0 6 1k 25 0-607 1-284 
Pen et, 0 6) 1 25 0-596 1-296 


The wave-lengths of the 11 certain line-groups are put 
together in Table XV., and in fig. 3 the values of the ratios 


of : to the ordinals are graphically represented. 


47 


‘anium. 


Spectra of the Elements Tantalum-U1 


en 


TABLE XV. 
rd. 108 cm. 
| 3 | 

Element. ay. | ay. n By. B. [Bsc B3. Ba. Vie Vie | Y3° 
(lie 1528 1518 = 1:343 1:323 1:280 1-303 e [eVaieaes er 1G 1-097 
TEN ces EL 1-471 1:296 1-278 1-241 1-258 at 1-095 1064 | 1-058 
760s ...... 1898 | 1:388 s 1-214 1-194 1-167 1-176 . 1-021 | 
Mite. 860 | 1-850 eS 1-176 1-154 1:133 1-138 1-101 0:989 0:962 0:956 
7B Pt...) 1823 | 1318 1:24 1-142 1120-101 1-098 1-072 0-958 0933 | 0-929 
OMe, 1288 1°271 1:19 1102 1-080 | 1-065 1-065 1-035 0-922 | 0898 | 0-894 
80Hg......| 1:251 1:240 e a 1-049 1-042 Ss = 0'896 
Bitlis = 1215 1-205 1-124 1-036 1-012 1-006 0:998 0:977 0-864 0844 0-840 
82Pb ....... 17186 1-175 1-091 1-008 0-983 0-983 0-968 as 0:842 | 0:820 0:816 
S8Bi s.....| 2153 1144 1-059 0:977 0:950 0-954 0:937 0:923 0810 0794 | 0-790 
BDO | 1-109 e a 0920 
GB bie sencge <a 1-010 
90Th ..,..., 0969 | 0957 2s Kea 0-766 0-797 0-758 = 0654 0-635 
Oe | Ovee- | O011 as = 0°720 0756 0-710 i 0°615 0:596 
Cite on OORT | SO0128 oe £ ve 00140 
ee eee eee) 6:97 i i ‘5 9-92 


48 Dr. M.Siegbahn and Mr. EH. Friman: High-Frequency 


As the graphical representation shows, some of the groups 
(@, &, 82) form right lines over the whole range, while 
others (G1, 83, V1, Yo Y3) are slightly curved upwards. In 
the first case a Moseley relation, 


Jp = a(N—N,), 


is sufficient. The values of aand N, for these groups are 


Fig. 3. 


1,200 


1,!00 


O50 +3, 74 +75 °*96°*~«77~°*«S ~~ GSS. eS«G?« GS «BS Son SIE 
'b. R2. Tr. Uz 


Ae eee WN OS) IF me PLL. Au: Hg. i tek (et 


given in Table XVI., Ny seeming to be an integer*. Fig. 3 
shows, further, that «, and «, are nearly parallel, likewise 
B,, 83, and 71, Y2, y3) The two groups of strongest §-lines, 
f, and ,, intersect one another in the case of Pb. 


* Rydberg, Phil. Mag. xxviii. p. 144 (1914)... 


Spectra of the Elements Tantalum—Uranium. 49 


A comparison with the values of Moseley for tantalum- 
gold shows that his -, 8-, and y-lines correspond to our a, 
8, and ®8.. His values, however, are practically all about 
1 per cent. greater than ours. 

In their excellent research on the soft y-rays from radium B 
Rutherford and Andrade * have determined the spectrum of 
these rays after a similar method to the writers’. They have 
also shown that lead, the characteristic radiation of which 
was excited by the 6-rays from an emanation-tube, gave the 
same characteristic lines as radium B. In this way the 
isotopy of radium B and lead was directly experimentally 
proved. As Rutherford and Andrade, in the case of 
radium B, have found a great many lines, a comparison 
with the lines found by the writers for lead may be of 
interest. This is given in Table XVI. The agreement is, 
as seen, very good, especially for the strongest lines. 


TaBLE XVI. 
Radium B. | Lead. 
Soft y-ray spectrum |) X-ray spectrum 
(Rutherford and Andrade). |, (Siegbahn and Friman). 
| Angle of reflexion. | Intensity. || Angle of reflexion. | Intensity. 
1@) 1 ‘ 
12 16 mi Nate) 3 
We 8 12) 338 10 
ll 42 m 
! 1 ee 7 f LE hOss 
1 a 8 it | 
10 48 f 
10 32 m 10) 28:5 2 
10 18 m tOp 29a 2 
10 3 S 10 36 9 
| 9 541 2 
9 45 m | 9 44°5 2 
9 23 f | 
8 43 m 
8 34 m 8 363 | 3 
8 22:8 1 
8 16 | m 8 20°3 1 
S96 m Sivas 1 


For an evaluation of the numerical results, further investi- 
gations of a greater range are necessary. Such researches 
are at present going on in this laboratory. 


Physical Laboratory, 
University of Lund. 
Feb. 10, 1916. 
* Phil, Mag. xxvii. p. 854 (1914), 


Phil. Mag. 8. 6. Vol. 32. No. 187. July 1916. iv) 


eens 


V. Electric Discharge in a Transverse Magnetic Field. 
By Prof. D. N. Matum, B.A., Se.D., F.RS.E., and 
A. B. Das, M.Sce.* 


| Plate I. ] 


iP ie a paper in the Philosophical Magazine, Oct. 1908, 

one of us (D. N. M.) considered the behaviour of 
an electric discharge ina tube of De La Rive’s pattern, under 
gradually decreasing pressure. Jt was there shown that, 
starting with a pressure inside the tube equal to that of the 
atmosphere, the discharge is at first in the nature of a 
shower of sparks, filling the whole tube ; that these various 
streams combine into a single band or single thick stream, 
when the pressure is sufficiently low ; that only when the dis- 
charge is of this kind is there rotation under the action of a 
magnetic field, as in the usual De La Rive apparatus. It 
was also noted that, as the pressure is further reduced, the 
band is gradually changed into a shower again and the 
rotation ceases. 

2. In a subsequent paper (Oct. 1912) a theoretical ex- 
planation of the various phenomena was attempted, and it 
was shown that they are connected with the fact that the 
relative number of positive and negative ions present in the 
tube varied at these different stages. 

3. The object of the present paper is to discuss, in detail, 
the behaviour of the discharge when the rotatory stage is 
passed. It results both from theory and experiment that 
the phenomena are dependent on (1) the voltage of the 
primary of the induction-coil, (2) the nature of the coil, 
(3) the difference of potential between the electrodes of the 
discharge-tube, (4) their distance, (5) and (6) the pressure 
and the nature of the gas or vapour in the tube, and {7) the 
nature of the electrodes. 

It has been our object to investigate the manner in which 
the phenomena are severally dependent on them. 

4, That the variation in the character of the discharge 
depends on the induction-coil used to produce the discharge 
and the distance between the electrodes in the discharge- 
tube is clearly seen from the photographs (Pl. II. figs. 1, 
2, 3, 4). 


* Communicated by the Authors. Paper based on brief notes com- 
municated to the second and the third All-India Science Congresses 
(1915 and 1916). 


Electric Discharge in a Transverse Magnetic field. 51 


Fig. 1. “ Band” discharge which rotates when the electro- 
magnet is excited. (Pressure 55 millimetres. ) 

Fig. 2. “Glow” discharge at pressure 2 millimetres. 
(The induction-coil (A) used in these cases gives a 
spark-length in atmospheric air of 29 millimetres.) 

Figs. 3 & 4 are photographs of the discharge at the same 
pressures asin 1, 2 respectively, with the induction- 
coil (B) giving a spark-length in atmospheric air of 
13°4 millimetres. The discharge-tube (No. 1) is the 
same in both cases. In the latter group, no band 
discharge appears, and consequently the rotatory 
effect is absent throughout the entire range. With 
a smaller tube (No. 2), however, and the induction- 
coil (B) the discharge shows the various types referred 
to on page 50. 


5. It is a prior evident that there must be a definite 
relation between pressure in the tube, the voltage of the 
induction-coil, and the length of rotatory discharge. What 
the nature of this relation should be would appear from the 
following considerations. 

If n is the number of corpuscles per unit length (along 2) 
per unit area of cross-section of a discharge, then the 
equation of continuity (J. J. Thomson, ‘Conduction of 
Hlectricity through Gases’) is 


ghey @ | 


where wis the velocity of a corpuscle ; 

X=mean free path ; 

f is a function, which determines the number of 
collisions resulting in ionization, of the mean 
kinetic energy Xed of an ion given to it by the 
electrical field of intensity X ; 

8 is the fraction of collisions resulting in re- 
combination ; 

p=pressure ; 

= charge on an ion (+ and —). 


Now, for steady rotation both and ait must vanish, 


and we must have l Or 
{Xe Xe 
—})—C6=0 or es —8=0. 
j ( r ) 8 Nev ) B 


n == (9 yea 
H 2 


52 Prof. D. N. Mallik and-Mr. A. B. Das on Electric 
If we assume f’ to be of the form rane the X, p curve 


e e . . . DQ 
is a straight line, and if a is small, — = const. 
p 


As X and probably also f and 6 depend on the distance 
between the electrodes and the voltage of the induction- 
coil, there must be an exact relation between pressure &e. 
determining the condition under which the discharge be- 
comes rotatory. This exact quantitative relation is under 
investigation. 

6. Starting now from the stage at which the discharge is 
rotatory, we reach, when the pressure is gradually reduced, 
a stage at which the character of the discharge changes as in 
Hes. Dei, -( . 

7. Comparing the figs. 5 and 6, which give the initial and 
the final stages of a discharge at the same pressure, we 
observe that the illumination is very slight when the dis- 
charge is first passed, but that after a time it becomes much 
more marked. The effect is evidently due to the fact that 
when the discharge is first passed there is only slight 
ionization, but that, as the discharge is continued, ienization 
increases and the consequent illumination. 7 corresponds to — 
a lower pressure than 6. 

8. When this stage is reached, the ring-end of the dis- 
charge is found to spread over a finite length of the ring 
electrode, instead of being confined practically to one point 
in it. 

9. As the pressure is further reduced, there is a further 
spreading out as in fig. 7. 

10. If at these stages the electromagnet is excited, there 
is dispersal of the streams constituting the discharge, which 
is, at the same time, twisted about the Faraday dark space 
on either side of it. 

Figs. 8 and 9 show the effect of the magnetic field on 
fig. 6, which is a photograph of the discharge before the 
electromagnet is excited. 

Similarly figs. 10 and 11 correspond to fig. 7 of the non- 
magnetic field. In 8 and 11 the disk is the cathode, and in 
9 and 10 the ring is the cathode. 

11. If we admit that the twist is due to an angular dis- 
placement of ions about the axis of the electromagnet, this 
behaviour of the Faraday dark space | 10] must be due to the 
fact that it is a region practically devoid of ions. 

12. Fig. 12a, curve I. gives the amount of twist for 
different current strengths in the coil of the electromagnet. 
It is thus easily seen that since the magnetic force is 


Discharge in a Transverse Magnetic Field. 53 


proportional to current strength, within the limits of the actual 
experiment, the twist is also proportional to the intensity of 


3 ) T T; Saas Sr ar a) ae 1 
| s \ 
| | | | | | 
ee a) a | | oe ! ts pede AA 
eh eall | ae Wal 

———— Eg P| Pear Ate eens Use erates 20.1] 
| | | ‘| | 

| | | Ka | 
hase - r 1k fee Aol a] fe | 
| | | 1 | | RA i | 
| | | | | 
t | | | A\ t 
a oma aoe a Pe 4 . - 
| | fae) \ 
! | | | | lf | 
anv aL ere a imei So | 
| i | | /\ | 
| | | / | 
| | | 
! | | | T als j ] i 7 7 == 
| | A | 
| | 
opie ae = ~ 
SU ele | Waar ache | 
Fie = i = _f- eS eee } 
9} aa coe 
R) | | | | My { H 
{5/0 sea Las ab NESS Cele AE LN 
aeeiey a | Ica ina | 
2 | | | 
. : U os ———— i 
2 6; 7 i 1 yl ie 4 5 7 | 
1 | ( 
3 | | | 
‘S) i j j 7 Ta i WSS eS 
| Veal Weaeeaipe tl | 

= | aE | Rah ea Po 
| | \ [ | 
| | | 

{ 
—— toes File ah eae Sanna j= 
| aes Mod 
6 —-—-x - — = 
ety | 
\/ [ 
} i 
P | | 
5 | | wal Wea 
S | { 
| mia aia 1 
4 SET EE 
| 
| 

3 : yes 

30 40 50 69 70 80 go 100 


; ——~> Deflection in degrees. 

[Nature of the discharge.—W hen the induction-coil is started, nearly 
the whole space is dark, and then a bright band with a dark space 
near the cathode gradually appears. | 

the magnetic force. Fig. 12 b, curve II. gives the amount 

of twist for different current strengths at a lower pressure. 

It is seen that the corresponding twists are greater in this 

case than those at a higher pressure. 

ig 13. An approximate theory of the magnetic action of the 

excited magnetic field on these discharges can be worked 

out as follows :— 

, Using cylindrical coordinates, z, p, @ (where z is measured 

fromthe Faraday dark space [11], the equations of motion 

of’an ion may be written, if m is its mass, 


pees ie 1H ong eis) Yen loca D) 
fi pO) Apa Be panied Wel.) .\ (2) 


1d | 4 j ab) a ae 
pl gate} ARO EN eee OAS 


where A=coeff. of viscosity, B a coeff. to be determined, 


54 Prof. D. N. Mallik and Mr. A. B. Das on Eleetric 


Z=electric force in the direciion of z and R in the direction 
of p, while H=magnetic force, which we know is mainly 
in the direction of p (Phil. Mag. Jan. 1908). 


Fig. 126. Curve I. 


@ 


—— Current in ampr 
at : 
caimele elem 


| | 
| 
Slseriaatsd wae Sees 
Bee fees Es a ee 
| A | | | 
> eae vz ! 
ea I i | 
(Ae a eh 
7” | } } [ | 
7S een | | | : ~ 
10 20 39 40 50 60+ 70: '~ 80: ‘90 


——> Deflection in degrées. 


[Wature of the discharge.—Showery, with very fine strie. It takes a 
certain time for the strie and illumination to dev elop after the 
induction-coil is started. } 


Now, considering the equation (3) (to which, alone, we 


shall confine ourselves), if we have N negative and n positive 
ions per unit length, in any stream (masses m, and my 
respectively), we have, taking moments about the axis and 
summing up, 


{ (m,N +mgn) ds 2 (08) +) (A,N + Agn)p? ds 6 
+ | Bo(N t+n)dsO6=\Hpe(N+n)dsz. . (&) 


as the equation of motion of any stream of discharge. 
Now we may assume, as in Phil. Mag. Oct. 1912, the 
action between two streams of lengths ds, ds’ to be a repulsion 


2 
an ees [N=n) RY a eek alee 


ee g, g are the velocities of positive and negative ions, 
a, b their radii (assumed spherical), + the distance between 


ds, ds', K the 8.1.C. and V the velocity of light. 


(na?q —N6?q')?, 


es 


Mscharge in a Transverse Magnetic Field. ay) 
The third term of (4) will then be of the form 
Cf(a)(n—N)?8, 


the other terms (depending on the velocities) being neglected. 
Here C is a constant depending on the form of the various 
streams of discharge, and « the angular coordinate defining 
the position of the stream, whose equation of motion is given 
by (4), provided n and N are constant throughout the 
discharge, observing that in this case, alone, @ will be the 
same for all points. In any case, if n=N, the equation of 
motion is of the form 


10 +6 =) Hpids 


=the couple acting on the discharge due tothe magnetic 
action of the electromagnet, 


where I =|(m, + mz) n ds p’, 
since p=0, in the steady state. 


But this couple = 3M2, where M is the total magnetic 
strength ot induced magnetism (Phil. Mag. Oct. 1908). 
Therefore we have 


16 +6 =3Mi, 
where p= (Ay+ As)n\ p? ds. 


This is the same equation as was obtained in a previous 
paper by identifying the discharge (which is in the form of 
fig. 1) with an electric current. 

14. If the number of positive particles is small in com- 
parison with that of negative particles, the number of the 
latter will not necessarily be constant throughout any stream 
of discharge. In this case, putting n=O and considering 
the motion of a small element of a discharge, we have, 
when the steady stage is reached, 


BO=Hei or g=ae, ces ee ae a 


where B is a function of p, «, defining the position of the 
element of the discharge considered. This completely 
explains the twists described in para. 10. 

Comparing figs. 8 and 11 with figs. 9 and 10, one might 
be led to suppose that the twist is independent of the direction 


56 Prof. D. N. Mallik and Mr. A. B. Das on Elecéric 


of the discharge. This is not really the case, as is seen 
from the annexed diagram. 


A is the cathode, B is the anode, C the dark space. 


Curve IJ. is also easily explained by the fact that ata 
lower pressure 2 is greater than at a higher pressure. 

15. The equation (5) shows that the angular displacement 
of a corpuscle is proportional to H and is dependent on its 
distance from the axis as well as on its angular position. 
The latter explains the dispersal of the rays [10]. The 
equation (3) also shows that there will be a time-factor ; 
that is, the final effect will be reached only gradually. 
Experiment bears this out also. 

16. When the pressure is further reduced, the discharge 
becomes striatory (Pl. IL. fig. 13). On the introduction 
of the magnetic field, the Faraday dark space shortens 
in length, the number of strie increases, the distance 
between consecutive strive decreases, and they also bulge out 
(fig. 14). 

The effect of increasing the magnetic field is to increase 
the number of strize and further shorten the length of the 
Faraday dark space, as well as that between consecutive 
strie. 

17. On a still further reduction of pressure, all the striz 
disappear and the negative glow extends from the cathode 
to nearly the whole of the tube (fig. 15). 

At this stage, the discharge is luminous at first, but the 
luminosity decreases as the discharge continues to pass, in 
’ consequence, evidently, of a gradual recombination of ions. 

18. When the electromagnet is excited, the strie make 
their appearance, and the tube is illuminated with a reddish 
light (figs. 16 & 17). Fig. 17 shows the effect of increased 
magnetic field. 


Inscharge in a Transverse Magnetic Field. D7 


19. When the pressure is very low (about =a of a 


millimetre) the negative glow reaches the sides of the vessel 
and the whole tube becomes phosphorescent, owing evidently 
to the impact of corpuscles on the sides of the tube. 

On the excitation of the maynetic field, the phosphorescent 
effect becomes concentrated around the cathode, while the 
anode is surrounded by a red luminosity, the line of de- 
marcation between the two regions being clearly marked. 

20. A characteristic variation of the potential difference 
with pressure during these changes is (as was noted ina 
previous paper) the most marked feature of these experiments. 
In order to make a more detailed study of this variation than 
was attempted previously—through all ranges of pressure — 
it was necessary, in the first place, to standardize the E.M.F. 
available with the induction-coils A and B. For this, we 
observed the spark-distance between two brass spheres, each 
of radius 3 cm., when the voltage of the primary of the 
induction-coil was gradually changed, the spring of the 
interrupter being kept at a constant tension. The result is 
plotted in curve III., fig. 18, in which ordinates represent 
spark-lengths in millimetres and the abscissa the voltage of 
the primaries of the induction-coil A. 


Vig. 18. Curve III. 


——> Voltage in the primary of thé co/!. 


ae OOK length, in, mm. 


21, Comparing these with the result given in page 461 
of J. J. Thomson’s ‘ Discharge of Hlectricity through Gases,’ 


58 Prof. D. N. Mallik and Mr. A. B. Das on Electric 


and similarly plotted in curve IV., fig. 19, we come to the 
conclusion that the voltage of the secondary of an induction 
coil, under the conditions of these experiments, is fairly 
accurately measured by the distanee of the minimum spark- 
gap, across which it forces a discharge and 1s practically 
proportional to this distance. 


Fig. 19. Curve IV. 


WM me | | 2) aoa 71} 1 
100 = ie ia 


| 
| abel of | 
(¢) 4 2) ‘3 ‘4 “5 -6 7. -8 re) 
—> Spark length in em., with spheres 3cm. diam" — 


| | 
| 
| | | | 
j j ' | | 
| | | 
| | | 
| | | 
i | | ] 
| | i | | | | 
| | | | | 
Pe ee a A | 
| } ore 
: 1 
| j | | | 
1-0 


[Data from Baillie’s results—J. J. Thomson’s ‘ Conduction of Electricity 
through Gases,’ 2nd ed. page 461.] 


It follows also that the potential difference between the 
electrodes of the discharge-tube is measurable in terms of 
the minimum spark-gap with which it is parallel. In this 
way, the potential difference between the electrodes of the 
discharge-tube at different pressures of the contained air 
has been determined. 

22. Fig. 20, curve V. represents changes in the -P.D. 
with pressure, when the disk is the cathode, the spark- 
length of the induction-coil in this case being 13:4 mm. in 
atmospheric air. 7 


~~. * eo, 


Discharge in a Transverse Magnetic Field. 59 


Curve VI. represents the relation between these quantities 
under similar circumstances with the ring as cathode. 

These curves correspond to the series of discharges 
typically represented by figs. 3 and 4. The pressures to 
which these figures actually relate are, however, com- 
paratively low. 

Throughout this series of changes of pressure, the rotatory 
stage is absent. 


Fig, 20: Curves V. & Vi. 


se Eo Z| ie er 


——— > Pressure tii ri.in. 


5 | 15 2 25 S) 35 4 Folls 
—— P.0. Spark length in mm 


23. In fig. 21, curve VIL. represents changes in P.D. with 
ressure between the electrodes of the same tube with disk 
as the cathode, the spark-length of the induction-coil in this 
case being 29 mm. in atmospheric air. It corresponds to 
the series of discharges typified by figs. 1 and 2. 
It will be observed that the latter includes a stage at 
which discharge has, during a certain range of pressure, the 
appearance of a band, and that there 1s rotation, only when 
this state is reached, under the influence of a radial magnetic 


force. 


60 Prof. D. N. Mallik and Mr. A. B. Das on Electric 


The straight part A, B of curve VII. corresponds to the 
rotatory stage of which fig. 1 is the type. The bent part 
B, C of curve VII. corresponds to the range of pressure over 
which the discharge undergoes only a twist [10] under the 
magnetic force. 


Fig. 21. Curve VIL. 


TAT 


a 


(30) — > 


——+> Pressure in m.m. 


a) l Zz eS) 4 5 6 
—> P.D.- Spark length in mm. 


It will be seen that in curves V., VI., the straight portion 
correspending to A, B in curve VIL., is absent. 

24. We conclude, therefore, that only when there is “band” 
discharge (which alone rotates under the influence of radial 
magnetic field), is the ratio of votential difference to pressure 
constant. This is in accordance with art. 5. 

It is further worthy of note that the potential difference 
goes on decreasing as pressure decreases up to a certain 
point ; but after that, it increases with decreasing pressure, 
so that at very low pressures the potential difference is 
comparable to what it is at very high pressures. 

Tt follows also from these experiments that the potential 
difference at any given pressure, although it depends on the 


Discharge in a Transverse Magnetic Field. 61 


voltage of the induction-coil, is materially affected by the 
circumstances of the discharge. 

25. All these points seem to be capable of explanation on 
such considerations as the following :— 

Let V) be the voltage of the induction-coil; then the 
energy supplied per unit of time by the coil will be pro- 
portional to the Vo, say 79Vo, where zis the current in the 
circuit. 

Let V be the potential difference between the electrodes ; 
then the energy supplied to the electrodes per unit of time 
will be proportional to V=7'V. say. 

Therefore i) Vo=7 V + energy carried away by the positive 
and negative ions, thrown off from the electrodes, less the 
energy carried to the electrodes by positive and negative 
ions reaching them (per unit of time). 

But the energy carried off by an ion = Xed. 

Therefore, 


ioVo=uV+ Xe(Nq'r' +ngr)— Hi, 


where » and N are the numbers of positive and negative ions 
thrown off from the electrodes and occupying unit length 
of the discharge, and X, 2’ their mean free paths. 

In order to find H, we may proceed as follows: 

It can be shown that the equations of continuity in a 
discharge-tube can be written, in the steady state, 


ae =aNq' + nq, | 
i Ne tie) 
Sar =aNq'+nq, 
cogs ' Ri 
where o= yr (Xen —') { 


y= [F(Xea—@)] | 


and n, N the number of positive and negative ions per unit 
length of discharge, # being measured along the line of 
discharge. 

Therefore, we have 


Nq' + ng=const.= : : 


where 2 is the current carried by the discharge. 


62 Prof. D. N. Mallik and Mr. A. B. Das on Electric 


Again, the energy carried to the cathode by the positive 
ions may be written equal to 


Vs ee 
— e~*(aNq' +-ynq), 


where & is a coefficient determining the dissipation of 
energy during the passage of these ions. 

Also, the energy carried to the anode by the negative ions 
may similarly be written equal to 


eVa' 
d 


e- #2 (aNq' +ynq) 


(J. J. Thomson’s ‘ Conduction of Hlectricity through Gases,’ 
2nd ed. p. 490), 


where d=distance between the electrodes ; 
V=ditfference of potential between the electrodes 
assumed to vary uniformly. 


B= 7 Seng! + yngq)a(e"™+e-*") da. 


But from (5) and (6), if «, y be regarded as constant, 


ONG. Ong) 
A pak 


aNg'+yng=(aNigy +yngies" . . . . (8) 


if N=N,, n=n, at the cathode. 
Hence, 


(aN'q' + ynq) («—-y) =2 


Ve I ig | / 
K= Tas (aNigqy == yn491) we W2(e-Ft 4 Cae Von 
«0 


de@-y—h4 ela-y- k)dyaa if 


Ve 
dg ae pers y= 


| similar terms in k! 
ne (aN gy == y7491) Te say. 
If 2=y=0, the above equation reduces to H=0. This we 
may suppose to be the case during the rotatory stage in air 
[Phil. Mag. Oct. 1912]. Therefore, since in this case N=n, 


and the pressure varies inversely as mean free path, we get 
the equation | 


igVo—i/V =neXo' gi! 4-954) 


Discharge in a Transverse Magnetic Freld. 63 
But o=i +i=i'+ne(qtq'); 


; nr 
a CoO ai 
Noa Vek eee 

0 2 +ne(g+q) 


As, moreover, during this stage V is small compared 
with Vo, and ¢’ should be small compared with i,, we get the 
simple equation 


Xr 
x fT F957 
g+q 
This, as we have seen, is the case in air (curve VIL.) ; 
when, however, the pressure is sufficiently reduced, e, y are 
no longer zero. In fact, the terms in i become sufficiently 
effective in making V large, as is found to be the case 
(art. 24), since a, y, &, k' are all proportional to pressure, 
and it is reasonable to suppose «<r, remembering that 


x 
0 ==) Conse: nearly. 
p 


Be 


: d 
aK Ww and y & x 2 
26. Although it is not possible to work out completely the 
theory of this variation of pressure without a knowledge of 
a, y, k, k', we may get some insight into its nature in 
special cases by proceeding as follows :— 


From (8), we have 
(aNyqi’ tryniqije*—M = (@Nogo' +ynego) . - (9) 
if N=N,', n=n, at the anode, g=q, and g=4q'/q,’ ; 
but e(Noge! + ogo) =i=e(Nig’ +7191). . . (10) 


If, now, n»=0, 


gev—aa_ _Y = EN gy 
Df oa) / 
We have also 


io Vo—2' V = XeNqy'[ A’ —aP], since n=0; 


! 
and since z =small, 
0 
we get X as an exponential function of p. 

It is obvious, however, that the above investigation is not 
capable of giving a complete account of the variation of the 
potential difference, for we have assumed (1) that the 
potential varies uniformly from cathode to anode, and (2) that 
a, y are constant. As neither of these suppositions can be 
true always, it is not surprising that the curves obtained are 
more complicated than those given by theory. 


64 Electric Discharge in a Transverse Magnetic Field. 


27. It is not. without interest to follow the variation of 
current with pressure and potential difference. 

Curves VIII., IX., fig. 22, give the relation between 
current and pressure in a discharge- -tube (No. 1), the 
induction-coil used giving a spark-length, in air, of 13°4. 


Fig. 22. Curves VIII. & IX. 


(Disc-Cathode) 


T ——— 


1'-Ox!0 
= 
% 
aS 
aod 
S 
at 
3 | 
; 
"5 
10x10 
Oo 5 40 15 20 25 30 35 40 45 50 55 60 65 
——> Pressure in mm. i 
Fig. 23 Curve. X. 
_ lea T I at\ ia antl T rar 
| | i | 
| | it | 
| x 1 
Pama nam eS 4‘ —— ae 
5 | | X | { i 
S | in ii 
‘€ 3 | [hate ee . iT = | So 7 + 
cS (he S00 ee ee ie 
: ! t i \] } ; | 
<= H \ | ' 
Ee SS fe +——- a7 ob | 
ae aig | | \ Lah | 
x | | \ 4 | iiss 
= 2 ze | 4 t { } | 1 
Q } i | | | 
2 | | Nj ip | | 
I | | fee rs | ] | 
Q 1-5} — Ser —4{—} 4 SSS 
x [ \ t 
uy | | ! i a | ; H | 
| : ; Sy | 
| are [oe Sel 
Bp 
Ai gre | haa 
! | ; ! 1 op | 
Gi j A 1 : i ae esol 
1-0xio > Oxo + boxio73 


— > Current in arip: 


The curve X., fig. 23, gives the relation between the 
current and potential difference. 


Results of Crystal Analysis. 65 
From (9), (10) we have 
come ah se ! & i (a—y)d 
CN2Go= a—ry (Ma, + -)e TAR 


a=—O— 


We have also 
2Vo= Xe(Nyqy A’ + roqQor)—-E ; 
so that a third relation is required in order that we should 
know N,, ng, and 2. 
According to J. J. Thomson’s theory, when there is only 
one kind of ion in the tube, 


Viz=aV +b. 


It is our intention to discuss the experimental results in 
their bearing on this and other theories in a future paper. 

Our thanks are due to the authorities of the Presidency 
College, Calcutta, for affording us facilities for carrying gon 
the above investigations. 


VI. Results of Crystal Analysis. 
By L. Vecarpd, Dr. phil., University of Christiania *. 
[Plate ITI.] | 

$1. a a previous paper t I gave an account of the 

crystalline structure of silver, as determined by 
the Bragg reflexion method. The X-ray spectrometer was 
in principle the same as that constructed by Brage, only 
differing with regard to details which were mentioned in 
the paper. 

Since then the work has been continued, and some of the 
results obtained will be given in this paper. 

Besides some results concerning the structure of gold 
and lead, which were announced in a paper read _ before 
“Kristiania Videnskapselskap” on November 19, 1915, 
the present paper will chiefly deal with the more elaborate 
and complicated case, the determination of the structure 
of the Zircon group, represented by the minerals zircon 
(ZrSiO,), rutile (TiO), and tinstone (kassiterite) (SnQ,).. 
which are, as far as I know, the first cases of tetragonal 
crystals which have yet been analysed }. 

Not being aware of the fact that the Spinel group recently 
has been analysed by W. H. Bragg §, I have also made an 


* Communicated by the Author. 

t L. Vegard, Phil. Mag. Jan. 1916. 

{ An account of the analysis of the Zircon group was given in a 
lecture in Kristtania Vid. Selsk. March 10, 1916, 

§ W. H. Bragg, Phil. Mag. Aug. 1915, p. 305. 


0D?) 


Phil. Mag. 8. 6. Vol. 32. No. 187. July 1916. I 


66 Dr. L. Vegard on 


analysis of the structure of this group. The following spectra 
have been measured :—Magnetite (111), (110), and (100); 
spinel (111), (110), and (100) ; gahnite (111). The experi- 
mental results, as well as the lattice constructed from them, 
were in close agreement with the results of Bragg; so a 
more detailed account will be superfluous. 


§ 2. The Structure of (fold and Lead. 


The elements copper, silver, gold, and lead all have 
crystals which belong to the holohedral class of the cubic 
system, and from a crystallographic point of view we should 
expect gold and lead to have a similar space-lattice to that 
found for copper * and silver f; but still I think an actual 
determination will be of interest, as several lattices might 
give the right symmetry. 

The gold crystals used for the experiments were kindly 
lent me by Professor W. C. Brégger. ‘The one specimen 
had the form of an octahedron, but as it had linear dimen- 
sions of the order of only one millimetre, we did not with our 
instrument detect any reflexion from it. The specimen used 
had the common form of a thin plate, twinned about its 
principal face (111). 

The crystal plate being quite thin, we only got reflexion 
from the face (111). 

The lead crystal was produced artificially. Several 
methods were tried—e. g., a gradual cooling of the molten 
substance, and sublimation of the metal in an electric 
furnace ; but although crystals were formed, they mostly 
consisted of branches made up of small individual crystals, 
but we got no crystal face fit for our purpose. 

The method which proved most successful was to let 
lead precipitate on a piece of zinc from a solution of lead- 
acetate. 

In this way we got crystal leaves formed in a similar way 
to the gold plates with the principal face (111), which gave 
quite a strong reflexion. 

In fig. 1 are given the relative strength and the position 
of the reflexion maxima for the face (111) of gold and 
lead as observed with narrow slits (0°'4 mm.). The normal 
variation of intensity with increasing order shows that the 
‘“‘noint-planes’”’ parallel to the face (111) are equal and 
equidistant, and in the simple case of a cubic crystal with one 
sort of atoms there can only be one lattice, which satisfies this 
condition and gives the right glancing angle. 


* W. H. Bragg, Phil, Mag. xxviii. (1915) p, 355. 
* L. Vegard, doc, ert. 


Results of Crystal Analysis. 67 


For a given lattice we can, as shown by W. H. and 
W. L. Bragg, calculate the glancing angle (6) when we 
know the density (p) of the crystal, the atomic weight (A), 
and the wave-length () of the X-rays. 


Fig. 1. 


Gold CHI) 


1 142 15° 16° 30° 3/0 


Lead (Mi) © 


24° 


Settingangle of chamber 


Let the side of the elementary cube be a, and the number 
of atoms associated with a cube of side a be n, then 
a one oa eta) 
where N is the number of atoms in a gram-atom (N = 
Oo > 1077). 
Let the spacing of the (111) planes be 
dy, = ea, 
then r 
EN a D) 
a 2', S00 CG pm e () 
where 6, is the glancing angle of the first order. 
From (1) and (2) we get 


Le ee” fy IN) 
sind, = ea / EN 


68 Dr. L. Vegard on 


1 
For the simple cube lattice (a): e=1, €= fal 

1 
For the cube-centred lattice (0): n= 2, e=- ee 

: i 
For the face-centred lattice (¢): n=4, e€= We 
A lattice like that of diamond (n=s, o— =.) is excluded 

Nv 


on account of the normal distribution of intensities. 

Calling the glancing angles in the three cases @2, 0, 0, 
we get 

sin 6, =sin 6; sin 02.4 

In Table I. are given the glancing angles for gold and lead 
calculated for the three lattices, and also the observed values, 
which are in perfect agreement with the values calculated 
on the assumption of a face-centred lattice. 


TABLE [. 
Calculated. | Observed. 
Oa. Ob. | Oc. ax 
| | 
Solgar eecstk ck LT? 5B" 9 Sees | 1°28 | > ape 
oe eee go477 | 15°88" | 6°09 6° 09'-5 


Thus it 1s proved that gold and lead crystals have the same 
lattice as copper and silver. 


§ 3. The Structure of the Zircon Group. 


The mineral zircon is a compound with the chemical 
formula ZrSiO,. It may be considered as an addition 
product of equivalent portions of the two dioxides (ZrQ,, 
S102), or as the Zr-salt of an acid of Si corresponding to a 
formula Zr(SiO,). 

The zircon crystals belong to the tetragonal system of the- 
bipyramidal class. Isomorphous with zircon are found a 
number of substances, of which the following are the best. 
known : 


Rutile (Ti02)2, kassiterite (SnQ,)., and thorite (ThSiO,), 
the latter being analogous to zircon. 


The determination of the structure of these substances will 
be of special interest also for the reason that there are a 


Results of Crystal Analysis. 69 


number of analogous dioxides which show quite a different 
erystal form; and in some cases the very isame dioxide 
occurs in several forms. Thus the titanium dioxide (TiO,) 
is found in three modifications, viz., the two tetragonal 
forms rutile and anastase, and the rhombic bipyramidal 
modification brookite. 

Generally we should expect that the determination of the 
crystalline structure of isomeric substances would lead to 
important results, and especially in cases where we know 
the energy of transformation from one modification to 
another ; for this energy should equal the difference of 
potential energy of the system of atoms in the two modi- . 
fications ; and from this equality we may expect to be able to 
draw valuable information with regard to the law of the 
forces acting between the atoms, or the forces which con- 
stitute the chemical binding. 

The present paper will deal only with the determination — 
of the structure of the zircon group ; but [hope to be able to 
extend the investigation also to the other modifications, and 
to treat the more general atomic problem mentioned above. 

The specimens of zircon crystals at my disposal had only 
the faces (110) and (111) well developed. The rutile and 
kassiterite crystals, of which we had very fine specimens, had 
the faces (100), (119), (111), (101), but no face parallel to 
the base (001) *. , 

In cases where the crystal has no face parallel to the 
planes from which the reflexion is to be found, we have 
generally t been able to find an edge parallel to the plane in 
question, and the crystal has been mounted with this edge 
horizontal and the reflexion-plane vertical. 

In this way the reflexions from the zircon planes (100), 
(101), (001), and the plane (001) of rutile have been deter- 
minedt. In this case, however, we cannot claim quite 
the same accuracy for the glancing angle and the intensity 
as when the reflexion is found from a plane crystal surface ; 
but still the accuracy will be sufficient for our present 
purpose. 

§ 4. 

The results of observations are given in Table IJ. and 
graphically represented in fig. 2. For each of the three 
minerals the glancing angles and intensity for several orders 
have been found for the five reflexion-planes (110), (100), 
emit). (O01), (101) Tt. 

* All crystals were kindly lent me by Professor W. C. Brégger, of 


the Mineralogical Laboratory. 
+ With the exception of the (001) plane of Sn.Q,. 


Dr. L. Vegard on 


70 


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Results of Crystal Ana 


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72 Dr. L. Vegard on 


We observe the position of the chamber. If the position 
angle of the chamber for spectra of the order n and 7 he 
a, and a;, then 

ase’ an — Hy an — 
CobG; = . cosee —5 0b 5. - se 


a fed 


For each face the intensity of the strongest maximum is 
put equalto 1. Thus it is only the intensities of different 
orders corresponding to one and the same face which should 
be comparable. 

The intensities of the reflexion are measured in the 
following way :— 

The slit, which the primary beam has to pass before striking 
the crystal, is made quite narrow, while the slit in front of 
the chamber is kept open, and the ionization is measured for 
angles which are near to the glancing angle of the spectrum 
in question. 

The maximum ionization current (when the ionization of 
the “ white radiation” has been subtracted) is taken as a 
measure of the intensity of reflexion. 

In order to make certain that the intensities thus measured 
correspond to the same strength of the primary beam, the 
intensity measurements for each face were carried out rapidly 
and in symmetrical order. 


§ 5. Interpretation. 


The interpretation is based on the fundamental equation of 
Bragg, combining the spacing d, the glancing angle 0, and 


the wave-length 2 : 
nn =2edsin0,. .. . 5 ae 


From the values of @ given in Table II. we can calculate 
the spacing for any reflexion-face. 

In Table III. are given the absolute values of dyo9=dpoo 
and doo, as calculated from (4) (W=0°607 x 10-8). Column 3 

001 

A109 
crystallographic axes, as taken from P. Groth, Chemische 
Krystallographe. 

We see that in the case of zircon and rutile, 


gives the ratios and column 4 the ratios c'/a' of the 


doo Pid) ¢' 


A190 ican 


In the case of kassiterite the reflexion from (001) has not 
been found ; but from the similarity between the spectra of 


Results of Crystal Analysis. Cf 
the three substances, there can be no doubt that also for 
kassiterite the ratio de is equal to ¢’/a’. 

The elementary cell has a volume 
J 


2 3 C 
di09 door = Lio a? 


and the number of molecules n associated with this volume 
is 

aa Nip ¢’ 
=e. al Ore 3 = & C é F % ©) 


p is the density, N the number of molecules in a gram- 
molecule, M is the molecular weight; and in order to 
preserve the analogy between the three substances, we write 
the molecular formula for rutile Ti,O, and for kassiterite Sn,O,. 

The number n is given in the last column of Table III. ; 
and we see that within the limit of experimental error 
the number 7 is equal to 1/8 for all three substances, or 
in a rectangular prism with sides 2109, 2di99. and 2d 9, there 
should be just one molecule. 


TABLE III. 


| 

Substance. Gras. | Gl varie a a 1. 
Me em. ey em. C27 ea er 4 
ESO) veneesoct 2-30 107° | 1:47 107°} 0-640 | 0-639 | 0-124 
| TiO, .....e eee Qi2Gu- | LAGI E Ss 0646 | 0644 | 0-128 
SO O)e iss <conses OSB Ae niniatds cack) ca emmae eet ens 0123 | 


Before proceeding further in our attempt to arrange the 
atoms, we shall make a few remarks regarding our 
elementary lattice. and the formation ef compound lattices 
and some of their properties. 

Our elementary lattice will bea prism with one atom in each 
corner. Let the side of its square base be a, and height c; 
then the spacing of a simple elementary lattice would be 

a 


A199 = a, doo = ¢; diy) = Ws 


diy = peo i = —_{——, e e (6 a) 


74 * Dr. L. Vegard on 


In describing the compound lattices it may be convenient: 
to refer the lattices to a rectilinear coordinate system. We 
let the origin coincide with one of the atomic centres, and 
the z-axis be parallel to the tetragonal axis and the z- and 
y-axes parallel to the other sides of the lattice. 

Any other elementary lattice which may be made to 
cover the primary one by a simple translatory movement is. 
completely determined with regard to position by giving the 
coordinates of one of its points. Very often it is most. 
convenient to give the coordinates of the point nearest to. 
the origin, which we shall call the point of construction. 

Thus a face-centred lattice is made up of four simple lattices. 
with the following construction-points : 


(000), (a/2, af/2, 0), (0, a/2, ef2), (a2, 0, ef2), 


and the spacings : 


digo = a2, don = of2, dito = 


a a 
io.  —7 | - : 
d : d : 6 b 
a a 
2/1+(2) /2+(2) 
C C 


The lattice corresponding to the cube-centred lattice has. 
construction-points 


(000) and (a/2, a/2, c/2), 


and spacings 


digo = a/2, diy. = Gf2, dup = — 


a 
————— dy = — 
v 2 
s/1+(2) 2n/2+(“) 


A lattice analogous to that of diamond is composed of two: 
parallel face-centred lattices with the construction-points 


(000) and (a/4, a/4, e/4), 
and has the following spacings : 


a 
dio) = al4, dou = c/4, dio = 2/9 
a 


a 
dyo1 = dyi1 = ==)" A (6 dy 
a a 
an/1+(2) \/2+(2) 7 
C C 


ayo = 


Results of Crystal Analysis. 75 


Returning to our crystals, we shall first consider the most 
general type, zircon. In the cell (2d10) . (2d100) . (2110) 
there should be placed 


one atom of Zr, one atom of Si, and four oxygen atoms. 


The simplest way in which we might arrange the atoms of 
Zr and Si in the cell would be to suppose the Zr atoms. 
arranged in a simple elementary lattice, for which we put 
a=2do9 and c=2do, and the Si atoms in a similar lattice 
with construction-point (a/2, a/2, ¢/2); but such an arrange- 
ment would make the spacing of the (100) and (001) planes 
twice as large as thev are actually observed. 

The spacings being inversely proportional to sin @, we can 
easily make a comparison with the observed angles and 
those to be expected for the four types of lattices previously 
constructed, and we notice at once that the observed spacings. 
have a relation to each other similar to those of equation (6d) 
corresponding to the diamond type of lattice, and a calcu- 
lation will show that within the limit of experimental error 
the ratios expressed in (6d) are satisfied by observations, 
or 

SIN Ay992 SiN Ooor : SiN Ay190 2 Sin Oy<1 : sin O44; 


=4:4%,9 2:24 / ae (2) Gy 
; VA Lala| tN (7) 
=e O20 2 2ioare bike 2 ll. 


From equations (6d) we find for the sides of the 
elementary lattice : 
a= 4Adyoo, 


c= Ado. 


Now in a lattice of the diamond type there are 8 points: 
associated with a volume equal to that of the elementary 
lattice (a?c), and consequently in a cell (a/4, a/4, ¢/4) there 
should be 1/8, or exactly the number which is found for Zr 
or Si atoms in the elementary cell digodoo1. 

‘This leads us to the assumption that in zircon the Zr and 
Si atoms are arranged in a lattice of the diamond type. 

_ Regarding the relative position of the Zr and Si lattices, 
1 is not necessarily determined from symmetry. The only 
condition to be fulfilled is that corresponding points of the 
two lattices must be situated on the tetragonal axis, or the 
position of one must be derived from the position of the other 
by a translation parallel to the c-axis. If the construction- 
point for the Zr atoms has the coordinates (000), the 


76 Dr. L. Vegard on 


coordinates for the construction-point of the Si lattice will 
be (001), where 0<l<ce. 

To get further, we must take into account the distribution 
of intensities for different spectra. An important fact in 
this connexion is that the face (001) gives a normal distri- 
bution of intensities. This fact limits the possible values of 
1 to ¢f4 or ¢f2. 

The assumption J=c/4, however, leads to consequences in 
conflict with observations. Thus it cannot explain that the 
face (101) gives a nearly normal distribution of intensities, 
and we have as the only possibility : 


Cec. 


The arrangement of the Zr and Si atoms is shown in 
fig. 3. We might also come to this lattice in another way. 
Tt might be considered as composed of two lattices of the rock- 
salt type with points of construction (000) and (a/4, a/4, ¢/4), 
corresponding to the diamond type. 


Fig. 3. 


@ Zr atoms 
OSV ple aa 


From the isomorpnism of the three minerals we should 
expect the lattices of the metal atoms to be found by sub- 
stitution of the atoms in the zircon lattice with corresponding 


Results of Crystal Analysis. 6 


atoms of the other substances. In both cases—for Ti,O, as 
well as for Sn,O,—the Zr and Si atomsare to be substituted 
with the same sort of atoms. This will considerably alter 
the type of the lattice and make it much simpler. In fact, 
the metal atoms will be arranged in a prism-centred lattice 
with sides 2d1o99, 2d100, 2do01: but in order to preserve the 
analogy with zircon, we shall suppose the lattice formed in: 
the same way from an elementary lattice, a=4dyq9, c= 4do01. 

From the equations (6c) and (6d) we see that the ratios. 
for the spacings (100) (001) (110) (101) should be un- 
altered; but for the face (111) the spacing would in 
comparison be four times smaller than in the case of zircon, 
or the sine of the first-order glancing angle four times 
as large. 

As a matter of fact this does not occur. 

It is only the spectra corresponding to the orders 
1, 3, 5, &c. of zircon which have vanished for the (111) face 
in the case of TiO, and Sn,Q,, and the spacings have the 
following ratios : 


dy 99: door : dy10 ° dyo1 : ayy 


(@ 


ih 1 
=I: 1. Sst 5, (8) 
a el) ey 


This apparent discrepancy, however, is not fatal to the. 
correctness of the assumed arrangement of the metal atoms; 
for us a matter of fact the first-order spectrum for the 
face (111) in the case of TigO, and Sn.Oy is produced entirely 
by the oxygen atoms. 


96. Zhe Arrangement of the Oxygen Atoms. 


In the case of zircon the observed spacings for all the. 
reflexion planes considered are just the same as we should 
get from the lattice of Zr and §i without the oxygen atoms - 
hence it follows that the arrangement of the oxygen atoms. 
must be determined from the intensity measurements. Only 
in the case of rutile and kassiterite the maxima of order 1, 
3, 5, &e. for the face (111) should be due entirely to the 
oxygen atoms. 

The problem before us is to arrange the oxygen atoms in 
the Zr-Si lattice—with four atoms to each pair of (Zr-Si) 
atoms—in such a way that the distribution of intensities of 
the reflexion maxima is explained and the crystallographic 
symmetry accounted for. 


“78 Dr. L. Vegard on 


There are two different types of arrangements according 
‘to the view we take with regard to the chemical nature of 
‘the substances. The one view is expressed in the formule : 
Zr (SiO,), Ti(Ti0,), Sn(SnQ,); and if these were the right 
-expressions, the structure—even in the case of rutile and 

kassiterite—should distinguish between two sorts of metallic 
catoms. 

According to the other view, zircon is considered as a kind 
of addition product of two dioxides ZrO, SiO,; and if we 
‘substitute Zr and Si with either Ti or Sn, we should expect 
ito get a lattice where all Ti or Sn atoms are equal. 

In accordance with the first view, we should arrange four 
-atoms of oxygen round each of the Si atonis in accordance 
‘with the symmetry of the crystal. 

We have to consider the following three arrangements :— 

(1) The four O-atoms are placed along the tetragonal 
.axis through the 8i atom, and with the O-atoms symme- 
trically arranged on both sides of the Si atom. 

‘¥(2) The four O-atoms are arranged in a plane through 
‘the Si atom perpendicular to the C-axis with tetragonal 
‘symmetry with regard to this axis. There are two different 
arrangements. If we take the Si atom as origin the O-atoms 

will be situated in the (zy) plane and will either have the 


»coordinates : 
ce Dis (—J,l), (1; —l), (—C a) OL (2,0), (0,0), (0,—2), (—J, C), 


~where J is a parameter. 
(3) The oxygen atonis are arranged on the diagonals AG, 
BH, &c., fig. 3. Let the construction-points of the two face- 
.centred Si lattices be (000) and (a/4, a/4, c/4), then the con- 
struction-points of the oxygen lattices would be : 
(Ea, Ea, Ey), ( Ej, —€), €\C), (—ea, €)a, —€\C) 


(€,4, —€)4, —e,c), and 


a a c as a Cc 
(Gi ap €o0, 4 a+ Eo, 4 + cc ), (5 — €90, A —€E 0d, 4 + at), 


a oe C \ (G+ a C 
— —— Ent, = > Eg, 7 — E26 — + €o4, 5 — €9, | — EC}. 
4 ZONA: > A a a arog or Oil 2) 


To get the right spacing for the face (100%, e; and e, must 
“have the same numerical value; but still we have to distin- 
_guish between the two cases: 

(3a) & = é, 
(3 6) 6 =—&. 


Results of Crystal Analysis. 79 


This arrangement will be clear by noticing that if a==c 
the oxygen atoms would be arranged in the corners of tetra- 
hedra with the Si atoms at the centres. 

None of these arrangements, however, is able to explain 
the experiments. ‘Thus the types (1), (2), and (3a) would 
make the second order spectrum of the face (111) vanish, 
contrary to observations. 

The question whether the oxygen atoms might be arranged 
according to (36) is of fundamental importance with regard 
to the properties of the atoms. If the atoms were arranged 
in this way, the tetragonal lattice could be considered as a cubte 
lattice compressed in the direction of one of the principal axes, 
and there would be nothing in the geometry of the lattice to 
explain its tetragonal form. 

The compression of the lattice would be due to symmetry 
properties of the centres (atoms), and under the conditions 
present the atoms would exert a different force in the direction 
of the tetragonal axis from that in a direction perpendicular 
to it. 

The arrangement (30), however, cannot be accepted 
although it gives a finite value for the second order spectrum 
of the face (111) ; but no value of e will satisfactorily explain 
the actual intensity-distribution observed. Thus in order to 
account for the disappearance of the first order spectrum of 
(111) for rutile and kassiterite, we must put e=—1/8. With 
this value of ¢ the amplitude in the case of rutile would be 
given by the formula : 


Ay = 14138 cos n @, 


which would give a ratio of the intensity of the first to that 
of the second order equal to about 16:1, while the spectra 
actually observed are in the ratio 1:1. 

In a similar way we can show that we cannot arrange the 
oxygen atoms round each of the Zr atoms, when the Zr 
atoms are to take up acentral position in groups of four 
‘O-atoms. 

Let us then try to associate two oxygen atoms with each 
Siand Zratom. The O-atoms must be situated on a straight 
line through the Si or Zr atom considered and at equal 
distances on both sides of it; but the distance from a Si 
atom to the two neighbouring O-atoms may not be equal to 
the corresponding distance for the Zr atom. 

Then the determination of the lattice under these condi- 
tions would involve the determination of the two distances 
(parameters) and the orientation of the lines through the 


80 | Dr. L. Vegard on 


Zr and Si atoms relative to the axis of the crystal and 
relative to each other. 

With regard to the direction of the lines, let us first 
determine their position with regard to the tetragonal axis, 
and consider the following three cases :— 

(1) All lines are parallel to the tetragonal axis. This 
orientation is excluded because for zircon it would make the 
second order spectrum of face (111) vanish. 

(2) The lines belonging to one sort of atoms (Zr say) 
might be parallel to the tetragonal axis, the lines of the 
other sort perpendicular to this axis. The latter lines are 
divided into two groups in such a way that an individual 
of one group is perpendicular to one of the other. Such an 
arrangement would not explain the distribution of the. 
intensities of the face (111) for zircon and the disappearance 
of the first order spectrum of the (111) face of rutile and 
kassiterite. 

(3) All lines are perpendicular to the tetragonal axis. 
Let us consider the lines through the Zr atoms. The lines 
through the atoms belonging to one of the face-centred Zr 
lattices must be per pendicular to the lines through the atoms 
of the other face-centred lattice. In order to preserve the 
right spacing for the faces (100) and (111), the lines must 
be drawn so as to halve the angle between the sides (a) of 
the square base of the lattice. 

The lines through the Si atoms must be arranged in the 
same way. 


© Zr atoms 
Oa. 
With regard to the relative position of the lines through 


the Zr atoms and those through the Si atoms, there are ee 
different possibilities, which “will be apparent from fig. 4 


Results of Crystal Analysis. 81 


which gives the distribution of points in two consecutive 
planes (001). 

In a plane (001) either all lines may be parallel as shown 
in the figure, or the lines through Si can be drawn perpen- 
dicular to those through the Zr atoms. The latter arrange- 
ment is excluded as it would not explain the distribution of 
intensities of the (111) face. 

We shall then consider the arrangement represented in 
fig. 4, which, as will be seen, with a proper choice of the 
parameters ¢, and ¢é, will give the right lattice for the 
zircon group. This lattice is composed of 12 face-centred 
lattices with the construction-points which are given in 


Table IV. 


TABLE LV. 

Atom. x. Yy. Z. 

ar { 0 0) ) 
Po SA pedabuncocppdUdoDaneD AS a/4 a/4 o/4: 

: gi { a/2 0 0 
sche eta Nee a an a/4 o/A 

EG 6, 0 

O associated with Zr.. reed eee 0 


—a/t—6,a a/++e,a c/4 
a/2+6,a 6,0 0 


a/2—6,a | —Ee a 0 
a/++e,a a/4—e,a c/4 


—a/4+6,a a/4—e,a c/4 
( 
ajt— ea a/t+e,a c/4 


Curiously enough, this lattice does not apparently possess 
the same symmetry elements as the crystal ; thus the planes 
(100) and (010) are not symmetry planes with respect to 
the points of the lattice, and the lattice possesses no tetra- 
gonal screw axis. 

But still the lattice has the properties necessary to explain 
the symmetry of the crystals. 

Let a (100) plane containing Zr and Si atoms divide the 
lattice in two parts [ and II. The points of the mirror 
image of I do not coincide with equivalent points of I1; 
but they can be brought to coincide by a translatory motion 
along the three axes (w=a/4, y=al/4, z=c/4). 

Phil. Mag. 8. 6. Vol. 32. No. 187. July 1916. G 


82 Dr. L. Vegard on 


The distance moved being of atomic dimensions, the 
difference of position of the points II and those of the image 
of I is not to be detected as long as we regard the properties 
shown by a large number of atoms. 

Thus this asymmetry should not even be felt by ordinary 
light waves because one wave-length would cover some 
thousand atoms ; first when we come to a wave-length of 
the order of that of Rontgen rays, the want of symmetry 
with regard to the face (100) can be detected. 

From these considerations we arrive at the following 
general rule. 


Fig. 5. 


fo} 


Arrangement of atoms in 


(10) plane 
2 \ wit \ Oo 
> N ~ ‘ tS \ 
\ S. x ners x 
. \ of. , \ eo 
\ XN A SS ‘ 
x X XN = \ a 
Bhi Se a AN Se ee 
\ \“ S \ \ \ x SS 
\ \ ~ < \ N ~ N 
\ ~ > ‘ SS 
x ON: \ \ \ x 
nN \ x \ a N \ ‘ x 
@N A » ~~ O ~~ » \ ‘ ie) 
x \ » N . > » 
“ “ aS ES N 
bo \ 25N\ . 
\ \o— SN \oO=— 
\ SS . N S x = 
\ ‘ SS ~~ x 
~ NX ~ 
@ \ N > 6 e N \ \ ‘8 
peek fi Selita ode 
a \ d.< ‘ es > % 
De oS 
X bs WS N NS 
\ ae S S N NS 
O N Te) \ \ \ re) 
@ Zr atoms 
‘@) Si ” 
7 0 ” 


-~—- -kLines of intersection with (11) planes 


In order that a certain space lattice shall explain an 
element of symmetry ascribed to the finite crystal, it will be 
sufficient that the lattice possesses a symmetry of the foilow- 
ing kind: 

“Lhe lattice must be brought to coincidence with itself by 
performing on it the operation characteristic of the symmetry 
element in question and a translator "yY movement x = ea, 
y =e:b, and z=e;¢, where a,b, c are the sides of the elementary 
lattice, and €, €, €3 quantities not greater than unity. 


Results of Crystal Analysis. 83 


y 7. Calculation of Intensities and Determination of 
the Parameters. 


When the rays are reflected from a face with equivalent 
and equidistant planes the intensity distribution is said to 
be normal, and according to Bragg it is characterized by a 
gradual diminution of intensities with increasing order. 

The exact law for this variation is not known. Bragg * 
finds that the intensities corrected for the temperature 
effect as derived from the formula of Debye + are approxi- 
mately inversely proportional to the square of the order 
number (n). The cause of this variation is yet unknown. 

In spite of the fact that the rate of variation will var 
from one case to another, it will generally not be difficult to 
see from the observed intensities whether the spectrum is 
normal or not. The criterion is not so much the rapidity 
with which the intensity falls with increasing order, but 
much more a typical regular form of the intensity curve. 

The problem of finding the distribution of intensities in 
the case that not all the reflexion planes of the face are 
identical, has been treated by W. H. and W. L. Bragg. The 
calculation is based on the assumption that the amplitude 
reflected from a certain point-plane is proportional to the 
mass associated with unit area of the plane. 

In view of the theory of secondary radiation given by 
Sir J.J. Thomson f, it would be more natural to suppose 
the amplitude proportional to the number of electrons per 
unit area, and introducing the atom-model of Rutherford 
and Bohr we should put the reflecting power of an atom 
proportional to the atomic number. 

As the atomic number for most elements is approximately 
proportional to the atomic weight, it will make very little 
difference whether we use atomic weights or atomic numbers; 
but as it must be the number of electrons and not the gravi- 
tational mass which is concerned, we shall introduce the 
atomic numbers in our calculations. 

Let unit area of the reflecting plane be composed of 1, 
atoms of atomic number N,, v, atoms of atomic number N,, 
&c., then the number of electrons per unit area (4) will be 


p=vyN, + vgNot.... e ° ° e e (3) 
* W. H. Brage & W. L. Bragg, ‘X-Rays and Crystal Structure,’ 


p. 193. 
+ P. Debye, Verh. d. D. Phys. Ges. xv. 1918; Ann. d. Phys. 1914, 


ae am 
¢ Sir J. J. Thomson, ‘Conduction of Hlectricity through Gases,’ 
. orl, 

: G2 


84 Dr. L. Vegard on 


Let the face considered have a spacing d). A length 
equal to dy) on the normal to the face will be cut by a 
number of point-planes (7) with numbers of atoms per unit 


area Fi, Mo .-- Mr 4 j 
We select an arbitrary point (0) on the normal, and call 


the distances from this point to the (7) planes 
Dgidone Ape Oe 


The intensity of the reflected wave from such a face has 
been calculated by Bragg in the case of r=2 and a general 
geometrical method is given *. 

The general analytical expression for the intensity will be 


We hae oe 


k, is a factor which W. H. and W. L. Bragg put pro- 
portional to the intensities of the normal spectrum, and they 
give the following values 


w= 4 1-|213)415 
K,= | 100|20[7|3]1 


An is the calculated amplitude which is given by the 
formule : 


Dale sh jie 7) a= ny PO a 


. : = di; 
fi(n) = = wy; cos n 2a 

aC ) pa dy Po 
f(n) = Bue sin n 2qr di 
i=l do | 


In a great number of cases we can give the point 0 such 
a position that the 7 planes are symmetrically arranged 
with regard to this point, and as sin (—a«) =— sin a, the 
quantity f,(n) =0. . 

The lattice of the zircon group as given in Table IV. will 
give the spacing shown in fig. 6 for the five faces experi- 
mented upon. Of these the face (001) has identical and 
equidistant planes and should give a normal spectrum which 
is also in agreement with experiments. The intensities of 
the spectra of the four other faces should be given by the 
following expressions for f,(n) and f,(n) :— 


* W.L. Bragg, Proc. Roy. Soe. Ixxxix. p. 483 (1914). 


Results of Crystal Analysis. 85 


1 (O01) 


{ 
ee Pe Ne 


\ 
On +(—1)"N.+(Ni+N,4+4N3) cos 2 7 
Face (111) +2N; (cos na, + (—1)” cos n a). 


fa 2(n) =(N,—N,) sin n 3" 


ee (N, +N,+ 2N3)+N3 (cos n2a;+ cos n2a 

fil(n)=0. 

ie (2) =(N, + No) +2N; (cos nay+ cos nay). 
fAn)=0. 

ie (rn) =(N, +N.) +2N3 (cos 2na,+ cos 2nag). 
fo(n)=0. 


| 
| 
>. 
Es (12) 
(| 


110) 
) 


( 
(1 
( 


Ol 
100 


86 : Dr. L. Vegard on 


N,, N,, N3 are the atomic numbers of Zr, Si, and O 
respectively, or atoms which may substitute them in the 
lattice. 

The angles a, and a, are connected to the two parameters 


€, and e, (Table IV.) in the following way : 


a, =Atre,, dg=A477e. 


§ 8. Zarcon. 


To get the intensities of zircon we have in equations (9) 
to put 
N; = 40, .N, = 14, N= 8; 
and putting 
a, = 7—RB, 


we get 


/,(n) =40 + (—1)"144 86 cosn 2 + (—1)" 16 (cos nB 
+ COS nay). 


(111) 


Reh or 
Jo(n)=26 sinn > 


| 

(fi(r)=70+8 (cos 2n8+ cos 2na). | 
Ce jolie) =O. f 
) 


( f,(n) =544+16((—1)” cos n8B+ cos ney). 
(101) i ) +16 (( 321) Sar 2) 
var) =0- 
(A(r)=94+16 (cos 2nB+ cos 2na). 
|A@)=0. 

In determining the intensities we shall have to remember 
that the spectra from the faces (101) and (100) were deter- 
mined by reflexion from an edge of the crystal. Especially 
in the case of the (100) face the reflexion to be observed 
was very weak ; and under these conditions we must expect 
too low a value for the first order spectrum, because a 
smaller portion of the primary beam wili be reflected into 
the chamber when the glancing angle is sinall. 

The reflexion from the (101) face was better; but in this 
case also the first order spectrum is found too weak in 
comparison with those of higher order. 

The spectra for the faces (111) and (110), however, are 
very accurately determined ; but we see from the expression 
for f,(v) that the spectrum of the face (110) will be very 
nearly normal, and the position of the oxygen atoms will 
affect the intensities very little. Still, we notice from fig. 2 
that the intensity of the third order spectrum is too large as 
compared with that of the second order. | 


(100) 


(13) 


Results of Crystal Analysis. 37 


The spectrum (111), on the other hand, is very much 
influenced by the position of the oxygen atoms, and will 
form our main basis for the determination cf a, and ap. 

The values of f(r) and /,(n) for the (111) face are given 
Metaple Vv. tor n= 1, 2,... 6. 


TABLE VY. 

2. F{(2). f, (0). I obs. k. 

1 1:63 1:63 — (cos B+ cos a,) 0:80 1-00 
ee) 0 — 2-00 + (cos 2 6 + cos 2a,) 0°40 0°3U 
pas =-63) 0) | 1-63 — (cos 38 + cos 3a,) 0°40 0-12 
| # 0 | 8°75 + (cos 48 + cos 4a,) 1:00 0:05 

5 = Fos | 1:63 — (cos 56 + cos 5a,) 0:05 0-015 
| 6 0 | —2:00 + (cos63 + cos 6a,) 0 0:005 


In column 4 are given the relative intensities observed 
and in the last column the relative intensities of a normal 
spectrum. These values are somewhat different from those 
given by Bragg, because the spectra of the crystals con- 
sidered usually gives a normal spectrum with relatively 
stronger maxima of higher order. 

Comparing the values of the last two columus, we see 
that 8 and a, must be given such values that the intensity 
of the first order spectrum is diminished and that of the 
third increased. 

Consequently 
[sare ees, 


cos 5 must be positive (1.) 


cos 8 +cos a, = 2 cos 


® 


cos 38+ cos 3a = 2 vos 3/2(8 + &) cos 3/2(B—a,) 
must be negative (II.) 
As both 8 and a, must, be less than 7, condition I. gives 
that 
B+ a<T7. 
As the atoms must have a distance which is of the order 
of the length dy, neither a, nor 8 can have a very small 


value. Then we can suppose cos 3/2(@8—a,) positive, and 
from condition (II.) we get 


T 
aR <St+a<T. 


“ 


COS T1766 COS 77805 


88 Dr. L. Vegard on 


Now in order that the oxygen atoms may considerably 
reduce the intensity of the first order spectrum 8+a, must 
be considerably smaller than 7, and we should expect both 
a, and @ to be in the first quadrant, somewhere between 
Zowand 90>. 

To determine 8 and a, more accurately we could calculate 
the intensities for varying values of Band a, We can arrive 
at fairly good values more quickly by means of a graphical 
method. We can very quickly draw curves representing 
cos n8+ cos ne, for various values of 8, a, and n. In fig. 7 
the curves are given for 8=60° and for values of a, varying 
from 0° to 180°, and for n=1, 2, 3, 4, 5, 6. 


Fig. 7, 


In this way I have found that the following values will 
give the best agreement with observations : 


8 = ai Uh ap = 60°. 
In the formula for f(n) of the face (111) and also for the 


Results of Crystal Analysis. 89 


(110) and (100) faces, the values of 8 and a may be inter- 
changed without altering the value of f;(n) ; and consequently 
we might equally well ‘satisfy these faces with a,=30° and 
B=60°. The face (101), however, does not permit such an 
interchange, and the fact that this face gives a comparatively 
strong second order reflexion will make the combination 
8=60°, «= 30° impossible. 

With our present knowledge with regard to the laws 
governing the intensity variations we cannot claim a very 
great accuracy for the angles a, and 8. In the present case 
our experimental material is also somewhat limited—thus, if 
accurate intensity measurements for the (101) and (100) 
faces were available, we should probably reduce the possible 
errors in the values of 6 and a, still more. 

In Table VI. are given the values of A? and the calculated 
and observed intensities corresponding to @=30° and a,=60° 
for the faces (111) and (101). 


Darren VL. 
| 
(111) (101) 

N. A?. Teal 5 Tobs. A?, | Teal : Tobs. 
Wea. 2°73 91 80) | 95 1) 200 100 
| ee 4:0 40 AQe |. , Vice Sanne ag 
oes. BR all Wee 40 56 7 8 
Be 60:0 100 100 
ei 63 3 5 | 
Re. 4 07 0 
| 


The agreement between calculated and observed values is 
a very good one, and in view of the fact that the intensities 
of the other faces are also explained, we see that there can 
be no doubt that the lattice given in Table IV. is the right 
one, and that we have found very nearly the true values of 
a, and P. 

The values found for these angles give the following 
values for the two parameters of the lattice : 


SO eal pipe /o) 

; Aq Aap i TR 
Qo Hi 

Chi i = 


90 Dr. L. Vegard on 
The distances 1, and J, (figs. 4,5) from the Zr and Si 


atoms to the oxygen atoms will be 
i = eo A 2 ale 10. em: 
l, = ea V2 = 1:08 x 10-8 em. 


The distance from the Zr atom to one of the oxygen 
atoms associated with it is more than twice the corresponding 
distance for Si, a fact which may be due to the greater 
affinity between the Si and O atoms. 


§ 9. Rutile and Kassiterite. 
If the lattices of (TiO,). and (SnO,). belong to the type 
given in Table IV., the spectra should be derived from the 
formule (12) by inserting the corresponding values of the 


atomic numbers. 
In both cases Nj =N,=N, and for the (111) face we get 


( Alm) =N(1+ (—1)) +2(N+2N,) cos nF 


(111) + 2N, (cos na;+(—1)” cos na). 
alr) = 


Now the aoe ray analysis shows that spectra of 
uneven order with regard to n disappear ; consequently we 
have for all values of g: 

Ai(2q—1) =0=2N; (cos (2g— 1), — cos (2¢—1) ay), 
which gives 
Ay = 45 = &, 

This is an important result as it shows that in each of the 
two minerals all the metal atoms are identical as regards 
their relation to the oxygen atoms, and it is impossible to 
consider rutile say—as a titanum—titanate. 

The amplitudes for the four cases considered will be 

Ava N= (1) eee) 22 N,, cos 2a) 

Axio = N+N,+ Ns; cos 2na, 
Ayo = N+2N; cos ne, 

ieee = N +2N, cos 2na. 


The expression for Aj}, Ayo, 20d Ajoo will not be altered 
if we substitute w—a fora. Which of these two is the 
right value can be decided by means of the (101) spectrum. 


Results of Crystal Analysis. 91 


For (TiO,). we find as the best value #=111°°5, and for 
G50). 2=—112%5. 

In the case of SnOQ, it is only the (111) and (100) spectra 
which can be used for the determination of «; for the (110) 
spectrum is very nearly normal regardless of the value we 
give a, and the (101) spectrum is determined from a very 
small face, which will cause the observed first-order spectrum 
to be too weak. 

In Table VII. are given the values of A? and K and the 
calculated and observed intensities. 

The agreement is a very good one; thus the lattice given 
in Table IV. can explain the typical distribution of intensi- 
ties observed. 


TasLe VII. 
(Os)sai a= Thies 


(111) (101) 


100 2:99 100 100 || 100 | “ 1:02 LUO 100 


20 | 146 98 100 | 20 0:41 8 6 
| 0:14 03 ) iH | 5:15 35 11 
3 TDD TT 5 Be Ben 6 i 

| 
(100) (110) 
100 0:42 100 130 100 9:32 | 100 100 

20 2:08 98 85 20 TA Gao 50 
Fi 4-00 66 35 7 | EOE 14 16 
3 0:14 0 0 3 7-55 2-4 3 


(SnO>);,) 2 — lia ox 


(111) (100) 


100 2°92 18°5 17 LOO oreo 100 100 
30 52°5 100 100 30 9°82 51 60 
12 O'1 0 0 12 14°75 30 28 

9) 391 12°5 13 5 4°55 + 


92 Dr. L. Vegard on 


The results of the calculation are collected in Table VIII... 
giving the parameters e and the distances J (figs. 4, 5) for 
the three crystals. 


AG RaW elle 


Crystal. a. | e. d. 
(Zire) ..f// MALEOOON Vain lesan 271° x WO 
ZrSiO, 5 
icin GUS eee kOSere 
(COR Gia z1k. 82 DUIS 4 0155 1:99 ,, 
(SOs ets oak HOS Se Ota 208 ,, 


Photographs of a model of the zircon lattice are shown in 


Plate ULI. 
$10. The Molecular Structure of the Lattice. 


The Roéntgen-ray analysis has shown that crystals are 
built up of atomic lattices; and in a number of cases pre- 
viously treated-by W.H. and W. L. Bragg the arrangement 
of the atomic lattices has left no room for such a thing asa 
molecule. Suppose, for instance, that in a crystal of rock- 
salt we fix our attention to a certain Na atom, we cannot 
from the geometrical arrangement tell which Cl atom is 
associated with it. This fact, however, does not without 
further proof necessarily involve any fundamental change 
in our conception of the chemical binding as taking place 
between pairs of atoms (Na—Cl). 

The atoms might possibly be connected up in pairs in such 
a way that all requirements of symmetry were fulfilled. 

In the case of rock-salt and similar substances there should 
be four pairs of simple cubic lattices, and the lines connecting 
each pair should be arranged with cubic symmetry ; but as 
we have three equal directions in the crystal and four pairs, 
such an arrangement does not seem possible. 

If, however, we regard our lattice for the Zircon group, we 
notice that each of the Zr or Si atoms is associated with two 
oxygen atoms ; thus the groups SiO, and ZrO, form a kind 
of ‘molecular elements” of the lattice. This is not merely 
a way of regarding the geometrical arrangement of the 
atoms ; but we have reason to believe that the groups SiO, 
and ZrO, form chemically saturated compounds. First of 
all, the fact that the oxygen atoms are closer to the Si than 
to the Zr atoms goes to support this view, and, further, 


Results of Crystal Analysis. 93 


a peculiarity with the geometry of the lattice will make the 
assumption of molecules almost a necessity. 

Let us consider the arrangement of atoms in the (110) plan 
of zircon, fig. 5. The triangle (abc) has a Zr atom in two 
of its corners and a Si atom in the third. Now an oxygen 
atom d belonging to the Zr atom a will—on account of the 
geometry found for the lattice—have equal distances to the Zr 
atom 6 and the $i atom c(fig.5). Now the chemical-affinity 
forces must necessarily be different for (Zr—O) and (Si-O), 
and if such forces were acting between the oxygen atom d 
and any of the, atoms ¢ and 6, we cannot explain an equi- 
librium position of the oxygen atom d which makes the 


distances (d—b) and (d-c) equal. 


S11. The Structure of Thorite. 


As mentioned in the introduction, the mineral thorite 
(ThSi0,) also belongs to the Zircon group. This mineral, 
however, occurs in the so-called “‘ metamict”’ form, which 
indicates a state in which the outer appearance of a crystal 
is preserved, but the substance itself has in the course of time 
become isotropic. 

Thus tke crystals, when examined with polarized light, 
give no indication of an optical axis, and it seems as if the 
atomic arrangement in a lattice has become unstable. 

Now it would be a matter of interest to see how these 
crystals behave towards Rontgen rays. Do they give any 
reflexion? Or does a “ metamict”’ crystal possess any trace 
of its original lattice? 

I have made a series of experiments with a number of 
different minerals to investigate this point. A full account 
of these experiments will be given later. In this connexion 
I shall only mention that the mineral thorite gave no X-ray 
reflexion at all, although several very fine crystals were 
tested. Thus the Rontgen-ray analyses have shown that 
the lattice of thorite was completely broken down, and only 
the outer form has been preserved to indicate the atomic 
framework which once existed in the crystal. All symmetry 
properties, however, go to support the view, that the atoms 
must have been arranged in a lattice of the zircon type. 


§12. Remarks regarding the Intensities of the Normal 
Spectrum. 
W. H. Bragg and W. L. Bragg have found that the 
intensities of a normal spectrum gradually diminish with 
increasing order. As an average they put the relative 


94 Dr. L. Vegard on 


intensities equal to 100, 20, 7, 3 for the orders 1, 2, 3, 4. 
These numbers, however, contain the influence of tem- 
perature, which will tend to diminish the intensities of 
higher order as compared with those of low order. Cor- 
rected for temperature effect the intensities should be 
approximately inversely proportional to the square of the 
order number. As also apparent from the way in which 
Bragg has stated the law, it can only be considered as 
tentative and asa first approximation ; and I want here to 
give some facts which indicate that the intensity law cannot 
be quite so simple. 

In the case of silver we have to deal with well-defined 
crystals of only one element, and the two faces considered 
(100) and (111) give both a normal spectrum *; but the 
intensity falls off much more rapidly for the first than for 
the second face. 

The face (111) gives an abnormally slow rate of fall with 
increasing order. J have made a careful examination of 
this point, comparing the first and second order of the two 
faces, but could only confirm the result first obtained. Also 
the (111) faces of gold and lead show the same abnormally 
slow rate of fall. The results of the intensity measurements 


are given in Table IX. 


TaBLE IX. 
Order. 
Opening 
Hage. || ofaive 
| 1. 2. 3. 
(| C100): 5 al gama: 100 20 7 
x 4 100 25 
Silver 4 a 0:5 mm. 100 195 
apr tae: (111) 1 mm. 100 | 51 11 
ites, i 100 | 50 
Mi aes 05 mm. 100 35 
(111) = 100 | 35 
Gold jchiae. { 7 0-5 mm. 100 48 
GACH Cored ani. il) — 100 49 


| 


Also the face (110) of the Zircon group gives an 
abnormally slow rate of fall of the intensities, even when 
we take into account the effect of the oxygen atoms. Thus 
in the case of tinstone, where the oxygen atoms can have 
very little influence, the distribution corrected for the 


* Vegard, Phil. Mag. Jan. 1916. 


Results of Crystal Analysis. 95 


disturbing influence of oxygen should be 100, 62, 30, 8 
of a normal spectrum of the orders 1, 2, 3, 4. 

As, in the case of silver, the value of 9 occurring in the 
Debye temperature factor should be the same for both faces, 
the rate of fall of the intensity will be different for face 
(100) and (111) of silver, even if we correct for temperature 
in the way given by Debye. 

So long as we know so little about the cause of the 
intensity variation, it will be dificult to say anything definite 
with regard to the explanation of the observed differences. 

Jt might he possible that the temperature factor is different 
for different faces even in the case of a cubic erystal. On 
account of the different elastic properties along difterent 
point-planes, such an explanation might not be unlikely. 
But if it should be impossible to account for the difference 
of intensity variation by differences of the temperature 
factor, we should probably have to suppose that the atoms 
had different reflexion properties in different directions. 


Summary. 


1. The lattice of gold and lead has been determined, and 
found identical with that of copper and silver. 

2. The structure of the Zircon group has been completely 
determined. The Si as well as the Zr atoms are arranged in 
tetragonal lattices of the diamond type. In the case when 
Si and Zr are replaced by identical atoms Ti or Sn, we get a 
simple prism-centred lattice for the metallic atoms. 

d. The tetragonal structure is not produced by symmetry 
properties of the atonuc centres, but by the tetragonal arrange- 
ment of the oxygen atoms. 

A, The lattice has a sort of molecular structure with mole- 
cules of the type MO,, where M is an atom of Si, Zr, Ti, or 
Sn. The three atoms forming one molecule are situated on 
a straight line and with M in a central position. This line 
might be called the molecular axis. 

The positions of the oxygen atoms are determined, when we 
know the directions of the molecular axes and the distance to 
the central atom M (molecular distance). The fact that the 
molecular distance is different for different central atoms M, 
as also certain geometrical relations of the zircon lattice, goes 
to support the view that the groups MO, form chemically 
bound molecules. 

5. For all the minerals considered, the molecular axes are 
equally arranged and are always per pendicular to the tetra- 
gonal axis, which accounts for the fact that the ratio cla is 
smaller than unity and almost equal for all minerals. 


96 Mr. Sudhansukumar Banerji on Aerial 


6. The “ metamict” erystal of thorite gave no X-ray 
reflexion. The lattice is completely destroyed. 

7. Experiments on the intensities of the normal spectrum 
have shown that different faces of the same crystal give a 
different Jaw of the variation of intensities with increasing 
order. 


T have much pleasure in thanking Professor W. C. Brogger 
for lending me, from the excellent collection of his laboratory, 
the crystals necessary for this research, as also for the kind 
interest he has taken in the work. 

I am also very much pleased to thank Mr. H. Schjelderup 
for his most valuable assistance In making the observations 
on which the present work is based. 


Physical Institute, Christiania. 
March 31, 1916. 


VII. On Aerial Waves generated by Impact. By SuDHAN- 
SUKUMAR BANERJI, M.Sc., Sir Rashbehari Ghosh Research 
Scholar in the University of Caleutta*. 


1. Introduction. 


ERTZ, in his well-known papert+ on the collision of : 
elastic solids, shows that when two bodies impinge 
on each other with moderate velocities, the elastic distortions : 
are more or less entirely localized over the region of contact, 
and that the duration of impact, though in itself a very 
small quantity, is a large multiple of the gravest period of 
free vibrations of either body. It follows, therefore, that 
no appreciable vibrations of the solids are set up by the 
impact, and that all parts of the impinging bodies, except 
those infinitely close to the point of impact, move as parts 
of rigid bodies. 

In a recent paper Lord Rayleigh has investigated the 
circumstances of the first appearance of sensible vibrations 
in the case of two impinging spheres, and his results seem 
to show that if vibrations are excited at all, the leading term 
in the radial displacement at the point of contact. during the 


* Communicated by Prof. C. V. Raman, M.A. 

+ Hertz’s ‘ Miscellaneous Papers,’ English Edition, p. 146. [See also 
Love’s ‘ Treatise un Elasticity,’ Second Edition, p. 195. ] 

t Lord Rayleigh, “‘On the Production of Vibrations by Forces of 
Relatively Long Duration with Application to the Theory of Collisions,” 
Phil. Mag. vol. xi. pp. 283-291 (1906). (‘Scientific Papers,’ vol. v. 
pp. 292-299. ] Aig 


Waves generated by Impact. oT 


early part of the collision is given by an expression of the 


type 
_ Bw head ) 
= ye 008 ne + AP 


where & is the relative velocity of impact, a, is a certain 
constant which can be easily calculated from Lamb’s theory, 
and 


_4/2r.E 857 


r= = =i. 
9 p Var 
r being the radius of the sphere, E the Young’s modulus, 
and p the density. 

The leading term due to the end of the collision is obtained 
from this by changing nt to n(t—r), tT being the duration of 
impact. 

Also the ratio of the maximum kinetic energy of vibrations 
to the energy before collision is approximately given by an 
expression of the type 


R 


ky 


Ree oe 
~~ 50° 4 (H/p) 


Since “H/p is the velocity of longitudinal vibrations along 
a bar of the material of the solids in question, we see that, 
in general, the expression for @ is very small in magnitude, 
and that R is an exceedingly small ratio. 

Lord Rayleigh’s results show that under ordinary con- 
ditions, that is, unless the spheres are very large in size or 
the relative velocity of impact is very great, vibrations 
should not be generated in appreciable degree, and that the 
energy of the colliding spheres remains translational. More- 
over, even if vibrations be excited at all, the pitch of the 
gravest sound so produced would be very high, in fact almost 
beyond the range of audibility. For example, in the case of 
two mahogany balls of 6 cm. diameter, the frequency of the 
gravest vibrations excited would be about 37,000 per sec. 
We know, however, from experience that when two spheres, 
say two billiard-balls, impinge directly upon each other, aerial 
waves of considerable intensity are generated which are 
audible as the characteristic sound of impact. The investi- 
gation described in the present paper was undertaken to 
aseertain, both theoretically and experimentally, the origin 
and characteristics of the sound produced by such impact. 

Since, as we have seen, under ordinary conditions vibrations 
cannot be excited in any perceptible degree, practically the 


Phil. Mag. 8. 6. Vol. 32. No. 187. July 1916. di 


98 Mr. Sudhansukumar Banerji on Aerial 


whole of the sound of impact must be principally due to the 
impulse given to the fluid medium by the surfaces of the 
spheres, which undergo a sudden change of velocity as a 
result of the impact. “The only alternative explanation that 
might be suggested is some kind of action, namely, a sudden 
compression or rarefaction in the neighbourhood of the region 
of contact ; but this, it seems, can hardly be correct, as “the 
spherical shape of the balls and the smallness of tbe relative 
velocity of impact would not readily admit of any specially 
intense compression or rarefaction being set up in the medium 
round the region of contact. Probably some kind of local : 
reciprocating motion would be set up in this region, but 
this would not be of much importance. 

The first hypothesis suggested in the preceding paragraph 
regarding the origin of the sound can be fully tested by an 
experimental and theoretical investigation of the distribution 
of intensities in different directions round the colliding spheres, 
and by studying the manner in which the sound depends 
(1) on the duration of the impact, (2) on tne coefficient of 
restitution, (3) on the diameter of the balls, and (4) on the 
relative velocity of impact and possibly other factors also. 


2. Measurement of the Intensity. 


The distribution of intensities in different directions round 
the colliding spheres is found to possess many remarkable 
peculiarities “which would be very difficult to reconcile with 
any other hypothesis regarding the origin of the sound. 
Even by the unaided ear one can perceive that the intensity 
of the sound is greatest when heard in the direction of 
movement of the colliding spheres, and is comparatively 
quite feeble in the plane at right angles to this line. Inside 
a laboratory the reflexions from the walls of the room give 
some trouble. The contrast between the intensities in the 
two directions is therefore best appreciated by the unaided 
ear when the observations are made in the open air, so as to 
avoid such reflexions as far as possible. A rough estimate 
of the ratio of the intensities can be made by varying the 
distance of the colliding spheres from the observer. So far 
as could be judged, the sound in the direction of impact 
appeared at least three er four times more intense in one 
direction than in the other. Some uncertainty was caused 
by the difference in the character of the sound from various 
directions, this difference being so marked that by its aid 
alone the angle made by the line of collision with the direction 
of the observer could be judged with fair accuracy. Other 
remarkable peculiarities were revealed when it was arranged 


Waves generated by Impact. 99 


to obtain a quantitative measurement of tle relative inten- 
sities in actual experiment. It was then noticed that the 
intensity practically vanishes on a cone making an angle of 
about 67° with the line joining the centres. 

After many trials an apparatus has been devised which 
appears to satisfy the necessary conditions of extreme sensi- 
tiveness, suitability for quantitative work, and convenience in 
actual use. This apparatus, which is believed to be of a new 
type, is based on a ballistic principle. Its construction is 
quite distinct from that of the phonoscope invented by 
Dr. Erskine-Murray, or other similar devices in which the 
motion of a membrane or disk on which the sound-waves 
are incident deflects a pivoted mirror connected with it. 
As a matter of fact, a phonoscope of the ordinary type was 
given the first trial, but proved quite unsuitable for the 
present work, as the deflexion observed with it was too small 
and too sudden to be capable of measurement by visual obser- 
vation, or even for satisfactory photographic registration. 

The apparatus finally devised and employed consists of a 
small mirror attached to a pivoted axle whose free move- 
ment is controlled by a fine spiral spring. (In practice the 
balance-wheel and hair-spring of a watch proved very 
satisfactory, the mirror being attached radially to the wheel 
with a little cement.) The sound is received by a horn over 
the tubular end of which a mica disk is fixed. A sharp 
metal pointer is fixed normaily to the centre of the disk and 
its end lightly touches the pivoted mirror referred to above, 
but is not connected with it. The light from a slit illumi- 
nated by an arc-lamp is condensed by a lens on the pivoted 
mirror, the reflected light forming a sharp image of the slit 
ona distant graduated screen. Forthe production of impact, 
the balls are hung side by side by bifilar suspension from a 
framework which is capable of rotation round a vertical axis. 
The balls can be made to impinge on each other ina direction 
making any desired angle with the axis of the horn by simply 
rotating the framework. This angle can be read off on a 
graduated circle fixed below the frame. In order to obtain 
perfect regularity in the sound of the impact and to avoid 
unnecessary reflexions from closely contiguous bodies, an 
electromagnetic arrangement was used by which the balls 
could be automatically dropped on breaking the circuit. 

As soon as the balls collide, the sound-wave generated by 
the impact passes through the horn and impinges on the 
mica disk. The motion of the pointer attached to the disk 
gives a kick to the pivoted mirror, which moves off freely 
until it is brought to rest by the controlling spiral spring. 

H 2 


fel 


100 Mr. Sudhansukumar Banerji on Aerial 


The mirror then comes back to the pointer, which brings it to 
rest. The deflexion of the spot of light on the distant screen 
gives a measure of the kick given to the mirror. It is found 
that the apparatus is extremely sensitive, very faint sounds 
being sufficient to produce deflexions which can be read off 
by eye nearly as easily as those of a ballistic galvanometer. 
Moreover, the behaviour of the mirror is very regular, and 
its motion perfectly aperiodic. The mode of action of the 
apparatus described above can be verified by observing the 
motion of the mirror and the pointer under a low-power 
microscope. 

One valuable feature of the apparatus is that it is quite 
unaffected by any echoes of the original sound of impact 
from the walls of the laboratory-room in which the expe- 
riments are made. This is because the pointer attached to 
the mica disk ceases to touch the mirror long before the 
echoes from the walls arrive at it. The results obtained by 
its use have been verified by working at different points 
within the room, and also in rooms of widely differing shape 
and size. 

In order to fully understand the action of the mica disk 
and pointer, we have to study their forced vibrations under 
the influence of the sound-pulse. We shall presently come 
to this point. Meanwhile the results obtained by its use may 
be described. 

Observations have been made of the deflexions shown by 
the apparatus when two balls impinge directly upon each. 


Fig. 1. 


SSS sss 
ie ps Ae a a ae 


Observed Deflections —> 


) 20 40, a Aco. 80.100 120 140 160 iIs0 200. 
Inclination (in degrees) of the line of collision with axis of horn —> 

other with a given velocity in different directions with respect 

to the axis of the receiving horn. ‘The results are recorded 

in fig. 1. The curve exhibits quite a number of remarkable 

peculiarities. It shows that the intensity is maximum in the 


Waves generated by Impact. 101 


line joining the centres, and that it gradually diminishes 
until it practically vanishes at an angle of about 67° with the 
line joining the centres, when it again increases rather 
abruptly until it attains a second maximum value at an 
angle of 90°. The experiment has been tried with pairs of 
spheres of various materials, viz. (1) billiard-balls, (2) marble, 
(3) aluminium, and (4) wood, and analogous results are noted 
in all the cases. 

It is not difficult to see, in a general way, that the distri- 
bution of intensity shown in fig. 1 is in accordance with the 
hypothesis as to the origin of the sound with which we started. 
As a result of the impact the balls undergo very rapid changes 
of velocity in opposite directions. The case is somewhat 
analogous to the well-known effect of the zones of silence 
noticed in the neighbourhood of the prongs of a tuning-fork, 
but differs from it somewhat owing to the spherical shape of 
the balls, the non-periodic character of their motion, and their 
close contiguity at the instant of impact. 

That the results shown in fig. 1 are quite reliable has been 
further tested by three methods. By measuring the deflexions 
of the spot of light for impacts at different distances from 
the mouth of the horn, the balls being made to impinge 
always with the same velocity and in the same direction, 
the deflexions are found practically to vary inversely as the 
square of the distances from the mouth of the horn. The 


> | 
| 
2 Jee [ | 
I2 
Ce ae ea 


Observed Detlections —> 


[raat 
ge i |. | a 
4 ! oe 
0 (2 


4 6 8 10 i2 14 15 
Inverse sauare of the distance — 


results are shown in fig. 2, where the squares of the reci- 
procal of the distances have been plotted against the 
deflexions. The curve shows that over a very wide range 
the deflexions practically vary inversely as the square of the 


102 Mr. Sudhansukumar Banerji on Aerial 


distances, and it is only when we come towards the origin 
that the curve shows a tendency to assume a parabolic shape. 
' Further, measurement of the defiexions for different velocities 
of impact shows that within the range of the experiment the 
deflexions vary directly asthe squares of the velocities. The 
results are shown in Table I. When the squares of the 


Tasue LI. 
| Nos. Velocity of Impact. | Mean Deflexions. 

| 
1h aA | 23°06 8°63 | 
Di Se ots ig 5 | 21°32 - 753 
oh eon ae 19°50 6°82 
Dirud het se da eigee 17-75 | 5°20 | 
BM Medes 10:80 1:43 | 


velocities are plotted against the deflexions they give prac- 
tically a straight line passing through the origin. HExpe- 
riments have also been made with pairs of balls of the same 
material but of different diameters. In this case it is found 
that the deflexions vary practically as the fourth power of the 
diameters of the balls. The results for the case of three pairs 
of wooden balls are given in Table IL. All these results 
show, as we shall presently see, that the apparatus practically 
measures the intensity of the sound produced by impact. 


TABLE IT, 
| Nos. ae of the | Mean Deflexions. | 
| IAS noe ga 3 inches. 21:92 | 
ao. aaareeee sects 21 inches. 702 
Soe seme core ate 14 inches. 1:35 


3. Nature of the Wave-Motion. 


A complete theoretical investigation of the nature of the 
wave-motion started by impact, assuming that no vibrations 
are excited in the balls, is beset with considerable mathe- 
matical difficulties. We shall confine ourselves to the case of 
two equal balls. Asa result of the impact, the balls suffer 


Waves generated by Impact. 103 


changes of velocity U during the short interval of time 
known as the duration of impact, in consequence of which 
an impulsive pressure is communicated to the surrounding 
medium and a train of sound-waves is started travelling 
forward with a definite wave-front. As a simplification we 
shall assume that the change in velocity is instantaneously 
acquired by the balls. Asa matter of fact, if we examine 
the curve for the relative velocities of the centres of mass 
for the period the balls are in actual contact, we notice that 
the most rapid changes in velocity occur only at the epoch 
of greatest compression, and as the duration of impact itself 
is a very small quantity (usually less than the 2000th part of 
a second), we see that the effect of the duration of the impact 
on the sound-waves is generally not of very great importance. 
At any rate, we are not wide of the mark in taking the 
change in velocity as practically instantaneous. 


(1) The case of a single sphere. 


We shall first consider the case when a single sphere 
suffers an instantaneous change in velocity U. 

If M be the mass of the sphere, its equation of motion can 
be written in the form 


da 
aie 


where a is the radius of the sphere, p is the pressure at a 
point on the surface of the sphere, and dw an elementary 
solid angle. 

Also if denote the velocity potential of the wave-motion 
started, the condition of continuity of normal motion on the 
surface of the sphere gives 


M = —{\p cos 0 .a* de, Sun href) cea) 


oy ds ‘a 
hei Ohi. Gyvne tigre) enue, (2) 
The initial circumstances at time t=0 give 
ax 
220 and ars =|", Me Rte Os) 


Further, the condition of discontinuity at the spherical 
boundary of the advancing wave gives 


OF 2-0, [1 DUAR Ae) 


to be satisfied for r=ct+a, ¢ being the velocity of sound. 


104 Mr. Sudhansukumar Banerji on Aerial 
We can now assume for ¥ the following expression 


YO? (a7) 
View or° Yr 


and we can easily determine the arbitrary function involved 
in this expression by a method first given by Prof. Love * 
so as to satisfy all the conditions enumerated above. 

The method consists in assuming 


ft—r) =AAC 2 and  2—Ber. 


Jcos.0, ... .\ een 


and then on substitution in the boundary conditions (1) and 
(2), we notice that % satisfies a biquadratic equation, two of 
whose roots are zero. The constants A’s and B’s are then 
determined with the help of the remaining conditions. 

If we assume that the ratio of the mass of the air 
displaced by the sphere to its own mass is a very small 
quantity, we see that the expression for w can be written 
in the simple form 


4 (4427) 


WMO VIN...) 2a? orpe (tta—r_1 
mAB (Jone Ae Bon ade 
oe 


where A is an indeterminate constant. 

The first term in this expression is a degenerated function 
wrich does not satisfy the usual differential equation for 
wave-propagation, and consequently does not represent a 
wave-disturbance. This term arises from the subsequent 
motion of the sphere with a nearly constant velocity which 
involves only a local reciprocating motion of the neigh- 
bouring air. 

The wave-motion produced is therefore given by the 
expression 

(ct+a—r 
a) 


Bay 4/20 ec ae ctta—r 1 
=) es oo mi ah E ‘ cos (SFE — Fr) Joos 8. (7) 


Thus we see that the wave-motion generated by an 
instantaneous change in velocity of a single sphere is of 
the damped harmonic type which is practically confined to 
a small region near the front of the advancing wave. 


* Love, “Some Illustrations of the Modes of Decay of Vibratory 
Motions,” Proc. Lond. Math. Soc. (2) vol. ii. p. 88 (1904). [See also 
Lamb's ‘ Hydrodynamics,’ Art, 295. ] 


Waves generated by Impact. 105 


(2) The case of two spheres. 


The solution for the case when both the spheres undergo 
instantaneous change in velocity U cannot, however, be so 
easily obtained. In a recent paper* published in the 
‘Bulletin’ of the Calcutta Mathematical Society, I have 
given a method by which a solution for this case can be 
obtained. For our present purpose, however, we see from 
symmetry that if we take as our origin the point of contact 
of two equal impinging spheres, the velocity potential of the 
wave-motion which satisfies the boundary conditions over 
the surfaces of both the spheres can be written in the form 


ae ct—r 2 { A (“—) 
w= Ay _ pas jy A,Ua' (2) } ce . 1*P.(cos 8) 


2n Non (— 
+ &e.+ Ao, Barn no ) 5 aie l . 7°” Pea(cos 0) 
ror ( 7 ) 


oye. ERIM IN as mene eb! er SON MOR a 


due regard being paid to the dimensions of both the sides. 
Ay, As, &e.; Ap, Ao, &e., are certain constants not 
depending on the radius of the spheres to be determined 
by the boundary conditions. 

At a great distance from the source of sound we can 


neglect all powers of - in this expression beyond the first. 


So that at a great distance we have approximately 
2 cl—r ct—7r 
= we | Aoe Xie) + Aorp2e2 Gan P,(cos 8) +e. | eae) 


Also, since the sound-pulse is practically confined to the 
wave-front, and also since in this region we have either r 
equal to ct or less than ct by a few diameters, we see that 
the expression within the bracket may be regarded as a 
simple function of the time and the inclination 6, independent 
of the radius of the spheres or the distance r. 

The intensity of the sound for large values of r therefore 

; Ua 
varies as —5-. 


* On “Sound-waves due to prescribed Vibrations on a Spherical 
Surface in the presence of a rigid and fixed Spherical Obstacle,” 
Bulletin of the Calcutta Mathematical Society, vol. iv. 


106 Mr. Sudhansukumar Banerji on Aerial 


We thus arrive at the following results :-— 

(1) The intensity of the sound varies as the square of the 
change in velocity of the colliding spheres. 

(2) It varies inversely as the square of the distance from 
the point of contact of the spheres. 

(3) It varies as the fourth power of the radius of the 
spheres. 


The truth of these results has already been verified 
experimentally, provided we assume that the apparatus 
measures the intensity, which we shall presently see it. 
does. 

We can easily study the forced and the free vibrations 
of the mica disk under the action of the sound-pulse. The 
forced and sree vibrations of a membrane and those of a 
telephone plate have been studied by various writers. 
Without entering into mathematical details, we see that 
the disturbance produced by impact travels forward as a 
sound-pulse of the damped harmonic type which is sensible 
only within a few diameters from the inner side of the 
boundary of the advancing wave. Its action on the mica 
disk, which has usually a smaller natural frequency of 
vibration than that of the waves, is so very sudden and 
lasts for so short a time that the whole effect partakes of the 
character of an impulsive pressure in consequence of which 
the mica disk suddenly acquires a velocity and free vibrations 
of considerable amplitude are excited in it. Assuming that 
the mica disk is not displaced considerably, we can easily 


see from elementary considerations that since p = is the 
pressure per unit area on the mica disk, where y is the 
velocity potential of the sound-pulse and p the density of 
air, the initial velocity communicated to the mica disk would 
_be practically proportional to the quantity 


ea 

\\e eae 
t) being the instant when the sound-pulse meets the mica 
disk. When this velocity attains the maximum value, the 
mirror leaves the pointer and moves with that maximum 
velocity. This velocity is therefore proportional to the 
quantity [ev], the instant t being so chosen that this 


expression has the maximum value. If we denote this. 
quantity by v and the deflexion of the mirror by @, then we 


Waves generated by Impact. 107 
must have v and @ connected by the relation 
v=aé + 667, 


where a and 6 are two constants depending on the elasticity 
of the spring and its initial strained condition. In actual 
practice the mirror is initially in contact with the pointer 
with a sensible pressure, and as the deflexion is usually very 
small, the second term in the above expression is nearly 
negligible in comparison with the first. v° is, therefore, 
practically proportional to a@; in other words, the angular 
deflexion of the mirror is approximately proportional to the 
intensity of the sound incident on it. 

On account of mathematical difficulties, it seems to be 
a hopeless task to attempt a numerical calculation of the 
distribution of intensities in different directions round the 
colliding spheres. But the analogous problem of two 
vibrating spheres whose distance apart varies periodically 
presents features similar to this problem, when the wave- 
length of the disturbance produced is sufficiently small. 
For this case, however, we car approximately calculate the 
distribution of intensities in different directions by the 
following method. 

If we take as our origin the point symmetrically situated 
between the two spheres and the line joining the centres as 
our initial line, it is easy to see that the velocity potential 
_ of the wave-motion will be given by 


r= | Anjan) +As aes 8) 
iG { (ka)? fo(ka)} | 


seh, ~ pia P,(cos 6) + &e. Je, 
£ {(kayiffka)} 
where 
met n(n+1) n—1)n(n+1)(n+2 
i(O)= Sar enti {14 mage. wh 2 An aes 
Lape ia. 27 


+ 0 01h eee aaNet 


2h 
and Teas the wave-length. 


The unknown constants Ap, A», &c. have to be determined 


108 Mr. Sudhansukumar Banerji on Aerial 


by means of the condition of continuity of normal motion 
on the surfaces of the spheres. We see from this expression 
that the disturbance produced at any point due to this 
system of two vibrating spheres will be the same as that 
due to a prescribed vibration given by 


[Ao + AP. (cos 0) + AgP, (cos 0) +.. .]e**, 


on the surface of a single sphere of radius equal to that of 
either sphere and having its centre at the origin. Now, from 
a consideration of the nature of the motion produced in the 
immediate neighbourhood of the two spheres, we can easily 
ascribe approximate values to the constants Ap, As, &e., 
which will conform as nearly as possible to the true state 
of affairs. As a first approximation, we can represent this 
disturbance by U cos 26. e*“, which is the same as 


U[4P. (cos 6)=-4Po(cos 6) | e*, 


so that Ag= —4, A,=¥% and the rest vanishes. This type of 
vibration shows that while the two caps bounded by the 
parallels of latitude of 45° and 135° are moving outwards 
the intermediate zone is moving inwards and vice versa. 

Now, if we assume various values for the quantity ka 
which will determine the wave-length for a particular pair 
of balls, we can easily calculate the values of the intensity 
at a great distance from the source of sound. First suppose 
that ka=1, then since at a great distance 


ier 
Tnlkr — (ki (kr)*} 2 


we have 
Ao ‘ As Vs 
= — E (l—z)— gq (9 — 81) Po(cos 0) 


Ay ] e'k(ct—r—a) 
f fs 


POE 
TP agees = 296 —5612)P4(cos 8) + Ke. 


Denoting the real and the imaginary parts of the expression 
within the bracket by F and G, we have 


5As 296A 
F=3A,-— “39 ?P2(cos 0) + {09337 P,(cos 0) + &e., 
SA 561A 


G=—tAy+ nL P,(cos 8) — = P,(cos 0) + Ke. 


402337 


Waves generated by Impact. 10% 


Since the intensity is proportional to F?+G?, we see from 
the above expressions on substituting the values of Ag, 
As, &e., that the intensity is maximum in the direction of 
the line joining the centres, and that it gradually decreases. 
and assumes the minimum value in the perpendicular 
direction. 

Now, if we further diminish the wave-length—that. is,. 
if we assume ka=2, we get 


25 —502 44+ 62°52 
wr =— 100 .. +A, 5842-95. P.(cos 0) 


aioe mI0.60 4) gik(et—rba 
$-1534329-36 1 1(C089) +&e. | ev Ge 


+A 


> 


+ 
30 


[By (8 —16/) + 3A,(7°53 + 10°71) (cos 8) 


th(et—r-+a) 
+ &e. Cn i 
Hence 
F=—8+30°12P,(cos @) + &e., 


G=16+42°8P,(cos 0) + Ke. 


We thus see that in this case F?+G? is maximum at 0° 
where its value is 3941°44 nearly, and that its value 
gradually decreases and assumes the minimum value nearly 
116 at an angle of 61°, and that it again increases and 
assumes a second maximum value at an angle of 90°, where 
its value is 558°16 nearly. If we further decrease the wave- 
length we get results analogous to the above case, namely, 
that the intensity is maximum in the line joining the centres, 
and that it assumes a minimum value at some angle inter- 
mediate between 0° and 90° and a second maximum value 
at 90° which is much less than the first maximum. We can 
easily proceed to a second approximation by determining 
the values of the coefficients in the expression for the 
prescribed vibration on the surface of the imaginary sphere 
so as to agree as closely as possible with the actual state of 
affairs. 


4, Leperimental study of the character of the 
sound-wave. 


The experimental results described before would be very 
difficult to explain on any hypothesis other than that which 


—EOoeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeyeEeyeyeyeEyeyeEeyEyEyEyyEeyeyeEEEEeeEeEEeEeEeeeEeeeeeeeeeeeeeeeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeElll ne 


110 ’ Mr. Sudhansukumar Banerji on Aerial 


we have assumed for the origin of the sound. This hypo- 
thesis may therefore be regarded as confirmed by expe- 
riment to the practical exclusion of any others. It was, 
however, considered that further experimental study of 
the character of the sound-wave emitted in different direc- 
tions by the colliding spheres would be of very great 
‘interest. 

Attempts have been made to obtain photographie records 
of the motion of the mica disk under the action of the sound- 
wave by various methods. The first, due to Siegbahn *, was 
that of optically recording the motion of a pointer attached 
‘to the disk by the use of two microscopes focussed on it, one 
-on either side. Another method which was used was simply 
to fix a small mirror to the mica disk at the place of greatest 
angular deflexions and to photograph on a falling plate the 
motions of the image of an illuminated slit formed by 
reflexions from the surface of the mirror. While both of 
‘these methods gave results confirming the broad indications 
-of theory, the photographs obtained could not be regarded 
as satisfactory records of the character of the sound-wave 
owing to the free vibrations of the mica disk excited by the 
‘sound of the impact which continued for an appreciable 
period. Even the first two or three swings, which were 
much larger in amplitude than the others, showed the free 
vibrations somewhat prominently. This was evidently due 
to the highly impulsive character of the sound-wave. Some 
‘improvement was obtained by using a mica disk stiffened by 
-attachment to a wire stretched in front of it under tension. 
(This was taken out from an old gramophone.) The motion 
of the wire was recorded photographically. Even with this 
‘arrangement, however, the free vibrations of the receiving 
apparatus were prominent. Unless a sufficiently sensitive 
cand at the same time strongly damped recorder is found, 
there does not appear much hope of obtaining a satisfactory 
direct record of the character of the sound-pulse. The 
writer hopes shortly to try the use of an acetylene-gas_ 
manometric flame and will also make further experiments 
‘with receivers of various types for obtaining an accurate 
record of the character of the sound-wave. 

Experiments have also been made with the bailistic 
apparatus described in the first part of the paper to compare 
the effects of the impacts of balls of the same size but of 


* Phil. Mag., May 1914. 


Waves generated by Impact. aL 


different materials. On trying the effect of a pair of wooden 
balls and those of a pair of billiard balls of equal size, the 
latter were found to give a deflexion about twice the 
deflexion due to the former. In this we may trace the effect 
not only of the larger coefficient of restitution of the billiard 
balls, but also probably of the shorter duration of impact 
which would be more effective in setting up an impulsive 
wave-motion in the fluid. 


5. Summary and Conclusion. 


The intensity of the sound generated by the collision of 
two solid spheres varies very greatly in different directions 
relative to the line of impact, and the character of the sound 
shows a similar pronounced variation. This observation was 
first made with the unaided ear and communicated to me by 
Prof. C. V. Raman, and the present work was undertaken at 
his suggestion to investigate this effect in detail, both theore- 
tically and experimentally. A newtype of apparatus in which 
the ballistic principle is utilized has been used to investigate 
the intensity of the sound in different directions. The results 
show a maximum intensity in the line of collision, practically 
zero intensity on the surface of the cone of semi-vertical 
angle 67°, and a second, but feeble maximum in the plane 
perpendicular to the line of impact. These results combined 
with the indications of theory and further observations on 
the character of the sound-wave show that, practically 
speaking, it is produced entirely by the accelerated motion 
of the spheres during the impact. The law of variation of 
the intensity of the sound with the velocity of impact and 
the radius of the balls has also been found and tested 
experimentally. The investigation was carried out in the 
Laboratory ot the Indian Association for the Cultivation 
of Science. The writer hopes later cn to carry out further 
work on the subject, particularly in the matter of getting 
direct records of the character of the sound-wave and ob- 
serving the effects of oblique impact and the impact of sphercs 
of unequal diameters. 


Calcutta, 
28th January, 1916. 


[ caLiieaah 


VIII. On the Dielectric Constant and Electrical Conductivity 
of Mica in Intense Fields. By H. H. Poors’. 


Introduction. 


N view of the possible analogy between electric polariza- 
tion and magnetization, it seems not unreasonable to 
suppose that some variation in the Dielectric Constant of a 
dielectric might be observed as the potential gradient 
approaches the sparking value. In the following paper an 
account is given of some experiments on mica (Muscovite),. 
which was chosen on account of the ease of obtaining it in 
the form of uniform sheets of very small thickness. The 
method finally adopted consisted in charging a small mica 
condenser to a known voltage by means of a Wimshurst 
machine, and then discharging it through a suspended-coil 
ballistic galvanometer, suitable precautions being taken to 
avoid piercing the insulation of the latter. Asa satisfactory 
electrostatic voltmeter for reading the pressures involved 
was not available, another condenser of nearly the same 
capacity and charged to the same pressure was simultaneously 


oO 
discharged through a second similar galvanometer. In 


g 

many of the experiments a small leyden-jar was used for 
this purpose. As the potential gradient in the comparatively 
thick glass of the jar was only a very small fraction of that 
in the mica, it was assumed that the capacity of the jar 
remained constant. To remove any uncertainty on this 
point, an air-condenser was finally constructed and used in 
the latest experiments without, however, any appreciable 
effect on the results obtained. It was found that the ratio 
of the throws of the two galvanometers was almost inde- 
pendent of the time of charge if this was not less than 
20 seconds: this condition was always fulfilled. 

It soon became evident that, at high pressures, a very 
noticeable leak occurred through the mica, so the arrange- 
ment was modified so as to include a third galvanometer to 
measure this leakage current, the possibility of surface leak 
being eliminated by means of an earthed guard-ring. 


Apparatus. 


The connexions are shown diagrammatically in fig. 1. A 
Wimshurst machine A is used to keep the system charged 
to an approximately constant potential, which is roughly 
measured by the electrostatic voltmeter B. A large leyden- 
jar C makes the pressure regulation easier by adding to the 


* Communicated by the Author. 


Dielectric Constant and Electrical Conductivity of Mica, 113 


capacity of the system. The multiple key D, when in the 
position shown, connects the small leyden-jar E and the 
mica test-condenser FT to the high pressure source. As 
already mentioned, E was subsequently replaced by an air- 
condenser. Any leakage through the mica is measured by 


Fig. 1. 


the galvanometer G, provided that the potential difference 
across the mica is constant. When D is in the position 
shown, the galvanometers H and [I are short-circuited. On 
releasing a trigger the key discharges EH and F throngh H 
and I respectively, the short-circuit connexions having first 
been broken. J and K are two condensers, each being about 
5 microtarads, L and M are two self-inductances. This 
arrangement prevents large potential differences from oc- 
curring between the terminals of the galvanomeiers during 
discharge, while not affecting the quantities passing through 
them. A similar arrangement protects the galvanometer G 
in the event of the mica in F being pierced. N is an earthed 
guard-ring which prevents any surface leak over the mica 
from reaching G. The point O is connected to earth, P 
and @ are two high resistances to prevent surges, as 
explained later. 

The first step in the construction of the test condenser was 
to obtain a uniform sheet of mica about 6 or 7 cm. square 
and from 0°01 to 0°02 mm. thick. The sheets, obtained by 
eareful splitting, were examined under a microscope between 


Phil. Mag. 8. 6. Vol. 32. No. 187. July 1916. I 


114 Mr. H. H. Poole on the Dielectric Constant and 


crossed nicols, and a suitable one selected, free from flaws 
and variations of thickness. Owing to the almost perfect 
cleavage these latter always occur abruptly, forming escarp- 
ments across the sheet, due to faulty splitting. These 
escarpments were at once visible between crossed nicols, 
owing to the variation in tint on either side. The thickness 
was at first measured with a spherometer reading to about 
0:001 mm., but errors due to abrasion of the surface by the 
spherometer leg, and possibly also to slight buckling of the 
mica sheet, made it hard to be certain of the thickness to 
within 0°002 mm. Accordingly an optical method was 
adopted. 

A Nernst filament was mounted a few centimetres in front 
of the sheet, and the image of its reflexion in the mica was 
thrown on the slit of a grating spectrometer by means of a 
lens, the arrangement being such that the reflexion was very 
nearly normal. Under these conditions the spectrum is 
crossed by a number of dark bands due to interference. 
The condition that a wave-length » should be absent is 
2ut=nnr, where t is the thickness, wu is the index of refraction 
for the wave-length X, and n is any integer. If 2 is the 
wave-length of the centre of the next dark band towards the 
red, then 


or, if OX be the width of one complete band, 


d KB 
Op eee: = 
a alk): I 


SO Xr 
(= 


98 (H _ tu 
200. (5 an | 

As the wave-lengths of a great number of dark bands can 

be accurately measured, we can find 6 with very fair 

accuracy for any required part of the spectrum. If we 


mn 
knew pw and a 
given part of the spectrum we could find ¢, but, as there 
was some uncertainty as to the dispersive power of mica, the 
method was only used as a comparative one. A sheet of 
mica about 0:06 mm. thick was mounted in front of the 
spectrometer, and the value of 6A found for various parts of 
the spectrum from red to green. The mica was then cut up 
into twelve pieces the total thickness of which was measured 


for the given specimen of mica and the 


Electrical Conductivity of Miea in Intense Fields, 115 


with a micrometer-gauge and found to be 0°774mm. So, the 
thickness of this sheet being known to be 0°0617 mm., we 
can find the thickness of any other sheet by comparison, 
since for a given part of the spectrum, 6 varies inversely 
ast. The thickness of any sheet measured by this method 
always agreed with that obtained with the spherometer 
within the limits of error of the latter. The results were 
apparently reliable to within about 0:0001 mm. 

The conducting sheets of the condenser were tinfoil disks 
about 1°15 cm. in diameter cut out with a sharp cork-borer. 
They were stuck on the opposite faces of the mica with 
seccotiue, being pressed into close contact by means of a 
roller, care being taken to bring them exactly opposite to 
each other. When the seccotine was perfectly dry, the free 
surface of the mica was well washed under a tap to remove 
all traces of seccotine, and finally rinsed in distilled water 
and dried by gentle heat. The mica sheet was then stuck 
on the iron guard-ring with seccotine, as shown in fig. 2. 


Fig. 2. 


The insulated wire shown pressing lightly against the lower 
disk was eventually connected to the galvanometer G, while 
the small wire tripod standing on the upper disk was con- 
nected to the key D (fig. 1). In the earlier experiments the 
surfaces of the mica surrounding the disks were exposed to 
the air. In this case the capacity of the condenser was 
found to increase with pressure for voltages greater than 
1000. This increase apparently depended on the hygro- 
metric state of the air, as it was considerably reduced, 
though not entirely removed, by placing the condenser ina 
19 


116 Mr. H.H. Poole on the Dielectric Constant and 


desiccator. By carefully watching the condenser in a per- 
fectly dark room, a faint flash was seen surrounding the 
tinfoil disk at the instant of discharge, indicating that the 
effective area was increased by a brush discharge over the 
surface. To prevent this the mica and the tinfoil disks were 
covered with a thick layer of shellac dissolved in absolute 
alcohol, and the whole gently warmed for several days. 
Great care was necessary not to overheat the condenser, as 
this softened the seccotine. Both sides of the mica were thus 
treated as indicated in fig. 2. The iron ring which formed 
the support was placed on an earthed metal stand in a 
desiccator. The base of the latter was formed of paraffin- 
wax, through which the various wires passed. As the 
diameters of the disks were from 600 to 1000 times the 
thickness of the mica sheet, the edge correction was very 


iP kde ; ; 
small, so that K= ie where K is the capacity of the con- 


denser in E.S.U., & the dielectric constant. d the diameter of 
the disks, and ¢ the thickness of the sheet, both in ems. 

The resistances P and Q were not used in the earlier ex- 
periments, but it was found that the mica was sometimes 
pierced by sudden connexion to the high-pressure source, 
even though the pressure of the latter was, if gradually 
applied, insufficient to cause breakdown. ‘This was pro- 
bably due to a surge, which might cause the pressure in the 
condenser to rise considerably above that of the source. To. 
prevent oscillations in a condenser circuit, it is necessary 


that R?> a where Ris the resistance, L the self-inductance,. 


and K the capacity. It was not very easy to estimate the 
value of Lin this case, but, as K was so small, it appeared 
that a considerable value of R was necessary to prevent risk. 
The resistances used were carbon lamps whose resistances. 
when cold were about 1200 ohms. Wires were soldered to 
the contact pieces of the lamps, and paraffin-wax was then 
run into moulds surrounding the lamp caps to prevent the 
risk of sparks passing across from one terminal to the other. 
At first these were inserted between the high-pressure source 
and the key. This prevented surges at charge, but it was. 
found that the act of discharge pierced a condenser which 
had stood the applied voltage for over half a minute. When 
this happened, the throw of the galvanometer I was much 
greater than it ought to have been, in fact off the scale. 
Apparently the condenser was pierced by being charged in 
the reverse direction to a somewhat lower voltage than it 


Electrical Conductivity of Mica in Intense Fields. 117 


had previously survived, an occurrence which is possibly ot 
some interest in connexion with theories of polarization. 
To prevent this from happening again the resistances were 
ultimately connected as shown in fig. 1, thus preventing 
surges either at charge or discharge. 

The dischar ge-key, shown diagrammatically at D in fig. 1, 
was formed of stout copper wires dipping into mercury cups 
in a block of paraffin-wax. The wires were mechanically 
connected by a rod of sealing-wax. <A rubber spring held 
the key in the discharge position, but the key could be held 
in the charge position, as shown in the figure, by means of a 
trigger, on release of which the key very rapidly flew over 
to the discharge position. As the voltages employed ex- 
ceeded 5009, it was found necessary to make the gap at the 
key at least a couple of centimetres long, as otherwise the 
mercury sometimes splashed up far enough to enable a spark 
to pass, thus discharging the large leyden-jar into one of the 
galvanometers. As it was necessary that as short a time as 
possible should elapse between the disconnexion of the con- 
densers from the source an: their connexion to the galvano- 
meters, the mercury cups connected to the source were made 
about 5 cm. deep and filied with mercury to within about 
2 cm. of their tops, which were only about 0°6 cm. wide. 
The wires of the key, when in the charge position, reached 
nearly to the bottom of the cups, so that, on releasing the 
trigger, the key had moved about 3 em., and thus had 
acquired a considerable velocity, before it broke connexion 
with the source. This made it possible to combine rapidity 
of action with a sufficiently wide gap. The mouths of the 
mercury cups were made narrow to reduce splashing. 

As the leakage through the mica is relatively large at the 
higher pressures, 1t is necessary to ascertain whether a per- 
ceptible loss of charge would occur during the time of 
operation of the key. To estimate this interval a standard 
2 microfarad condenser was charged to a suitable pressure 
and discharged by means of the key through a ballistic gal- 
vanometer. A resistance of 0°106 megohm was then con- 
nected as a shunt across the condenser and the reading 
repeated. The leak of the condenser through the given 
resistance in the interval between disconnexion from the 
source and discharge, is measured by the ratio of the two 
throws. Measurements made in this way gave values of 
this interval ranging from 0:0070 to 0°0093 sec. for the side 
of the key connected to the mica condenser. The figure 
0:008 was taken as a mean, and a very small correction pro- 
portional to the leakage current was added to the observed 


118 Mr. H. H. Poole on the Dielectric Constant and 


value of the charge on the condenser. This correction might 
have been neglected without serious error, as it was always 
very small. 

The galvanometers H and I were similar and of the 
D’Arsonval type, their resistances being about 800 ohms 
each. As the throws to be observed were large the scales, 
which were 50 cm. long divided in mm., were set up so 
that the luminous images were at the ends of the scales when 
the galvanometers were short-circuited. The scales were 
set so that perpendiculars to them at their middle points 
bisected the mirrors. A careful standardization was neces- 
sary to test the relation between throw and quantity for 
each galvanometer. This was carried out with a standard 
4 microfarad condenser charged to various pressures up to 
about 10 volts, measured on a Weston voltmeter, and dis- 
charged through the gaivanometer, the key D being used. 
It was also necessary to know the capacity of the condenser 
Ei (fig. 1), or, what comes to the same thing, to find what 
throw will be produced in the galvanometer H by charging 
Eto some known potential. By combining a Tucker hygro- 
metric battery giving about 600 volts with a set of small 
dry cells of the flash-lamp type and with the continuous 
current supply of the laboratory, pressures up to about 
1000 volts could be obtained. These pressures were measured 
in sections by means of a Kelvin multicellular voltmeter 
reading up to 300 volts. Knowing the variation of sensi- 
tiveness of the galvanometer H with the throw, a seale of 
voltage was constructed. The characteristics of the two 
galvanometers are shown in the following table. Here @ 
is the throw, V the factor by which the throw of H must be 
multiplied in order to find the voltage of the source, and Q 
the factor by which the throw of I must be multiplied in 
order to find the charge on the mica condenser in micro- 
coulombs. 


} 

Q. | V. Q. | 

50 12:21 | 671x1007 ia 
100 12:29 | 6-78 
150 | 12°37 | 6-84 

200 12:44 | 6-9 

250 150 | 6-96 
300 and upwards. 12°56 6-99 


| 


The figures in the V column refer to the air-condenser 


Electrical Conductivity of Mica in Intense Fields. 119 


finally used; with the leyden-jar the figures were about 
3 per cent. smaller, 

The galvanometer G was of the Ayrton & Mather sus- 
pended-coil type, the coil being enclosed in a silver tube, so 
that the galvanometer was dead-beat even on open circuit. 
Its resistance was about 296 ohms, the constant was found to 
be 8:0 x 107° microampere per scale-division, the deflexion 
being very nearly proportional to the current over the range 
employed. 

The Wimshurst machine, having vuleanite plates and 
insulation, generally excited itself without any special warm- 
ing or other precautions, but tt was impossible to say which 
way it would build up, so that the condensers were some- 
times charged the reverse way accidentally. For determi- 
nations at voltages exceeding 2000 it was found best to run 
the machine at a nearly constant speed by means of a motor, 
and regulate the voltage by the pin-point method. This 
consists in fixing an earthed pin-point near the plate of the 
machine. ‘The best position seemed to be just in advance of 
the collecting-comb connected to the condensers. By moving 
this pin in and out by means of a rack and pinion, the 
voltage could be adjusted over a wide range, and kept 
approximately constant for a considerable time. At voltages 
less than 4000 the electrostatic voltmeter was used for 
approximately indicating the pressure before discharge, 
above that the galvanometer G served the same purpose. 
The actual value of the pressure was obtained from the throw 
of H on discharge. The electrostatic voltmeter was of the 
vertical needle type, reading to the nearest 100 volts. As 
the divisions were very close together towards the upper 
part of the range, it could only be “yead rather roughly. 


Haperimental Results. 


The first definite results were obtained with a mica sheet 
about 0-017 mm. thick. ‘he value obtained for the di- 
electric constant / at voltages not exceeding 1000 was about 9. 
At higher voltages the value of & steadily increased. It was 
subsequently found that this increase was due to a brush 
discharge, as already mentioned. The conduction current 
through the mica was inappreciable for potential gradients 
less than 0°5 megavolt per cm. At higher oradients it 
increased rapidly, : as shown by the following figures, which, 
with the exception of the last, are means falcen fon ane 
curve obtained by plotting the results. The last figure is 
the highest obtained in this series. 


120 Mr. H. H. Poole on the Dielectric Constant and 


X. C. C. 

1:0 0-01 0-01 
15 0 O04 0:05 
2°0 0-15 0-15 
2°5 0°50 0-45 
2°83 1:06 097 


The figures in the first column are the potential gradients 
in megavolts per cm., those in the second column the current 
density in microamperes per sq. cm., while those in the 
third column are the current densities calculated from the 
formula 


log C=log X +. 3'286+0-794 . X, 


which was found to agree with the later results. It will be 
seen that the agreement is good at the lower gradients, but 
that the observed current is too large at the higher gradients. 
This is almost certainly due to the brush discharge increas- 
ing the effective area at high potentials. 

This condenser. stood a pressure of 4750 volts or 2: 83 
megavolts per cm.; it was eventually pierced by being inad- 
vertently connected to the high-pressure source when the 
pressure of the latter had risen too high. The thickness of 
the mica sheet was subsequently measured optically and 
found to be 0°0168 mm. The figures given above are cal- 
culated on this value. 

A new condenser was then made out of a sheet 0:012 mm. 
thick. The results obtained agreed fairly well with the 
previous ones, both as regards dielectric constant and con- 
duction current. The latter seemed to be a function of the 
potential gradient only, as the voltage required to produce a 
given current was directly proportional to the thickness of 
the sheet. Further experiments on this point are, however, 
desirable, as the sheets employed did not differ very much 
in thickness. The sparking potential through an insulator 
is known not to be a linear function of the thickness, so 
possibly a similar effect might oceur with the conduction 
current. 

This condenser was in use when the brush discharge was 
discovered. ‘I'he mica surfaces were then covered with a 
thick layer of shellac dissolved in absolute alcohol, and the 
condenser gently heated for several days in front of an 
electric radiator. The upper part of the jar H was also 
coated with shellac. The capacity and the leakage conduc- 
tivity were both found to have increased by about 35 per 
cent., while the capacity was now almost independent of 
the voltage throughout the entire range. ‘This increase in 


Electrical Conductivity of Mica in Intense Fields. 121 


capacity was probably due to overheating, which may have 
caused the seccotine to soften and creep along the surface, 
thus increasing the effective area. Only a few deter- 
minations were made, as the condenser was pierced by the 
discharge from a pressure of 2680 volts, or 2°23 megavolts 
perem. The mica may possibly have been injured in some 
way during the heating as the condenser had previously 
survived 2800 volts. This sheet of mica was not measured 
optically, as it had been covered with shellac before the 
optical method was devised. 

A third condenser was then made out of a sheet of mica 
0:0173 mm. thick, measured optically. It was coated with 
shellac as before, but was only gently warmed for several 
days over a carbon-filament lamp. The resistances P and Q 
(fig. 1), which had previously been connected between the 
key and the high-pressure source, were transferred to the 
pesition shown in the figure, thus preventing surges at 
discharge. 

Since the tinfoil disks were 1:15 em. in ciameter the 
capacity K of the condenser 


‘ Me ee 
~ 16 x 0:00173 x 9 x 10° 


where k: is the dielectric constant. Hence 

() 

= soa x i Oe y 

where Q is the charge on the condenser in microcoulombs, 

and V is the pressure in volts. Q and V are obtained by 

multiplying the throws of the galvanometers I and H by the 
appropriate factors, which have already been given. 


(>) 
Several sets of observations were taken using the leyden- 


jar E. Hach day’s readings agreed well amongst them- 
selves, but small variations, amounting to a few per cent., 
occurred from day to day. The value found for k was about 
9°0 ; it was apparently independent of the potential gradient 
within the limits of experimental error up to a gradient of 
3145 megavolts per cm., which must be very near the 
sparking value. At this gradient the leakage current 
amounted to 1°8 microampere per sq. em. ‘This current 
agreed well with the formula 


ie rae = 


=95°315x10-°x&k microfarad, 


or 
log C = A+log X+ BX, 
where © is the current density, X the potential gradient, 
a,b, A, and B are constants. From these determinations 


122 Mr. H. H. Poole on the Dielectric Constant and 


the values found for A and B were 3:25 and 0:80 respec- 
tively, the logs being to the base 10. 

Experiments were also made on the value of & when 
the pressure was falling. The condenser was charged up to 
about 5000 volts, and allowed to leak down to the required 
potential and then discha arged. A decided rise in the value 
of k was observed, evidently due to a hysteresis effect. This. 
was not a permanent change caused by the high pressures, 
as, on again charging the condenser up to some voltage and 
then discharging, the same value was obtained for k& as 
before the high voltages had been attained. The conduction 
current appeared to be about the same whether the pressure 
was stationary or falling; this could not be accurately deter- 
mined, as the calvanometer G only measures this current 
correctly when the pressure is stationary, or nearly so. 

As some doubt was felt as to the constancy of the leyden- 
jar Ii, an air-condenser was constructed. This consisted of 
three co-axial copper cylinders, of which the inner and the 
outer were earthed, and the intermediate one supported 
midway between them on vulcanite insulators. As the 
length of the insulated cylinder was about 26-4 em., and its. 
mean diameter 21°3 cm., the air-gap being about 0°625 cm., 
the capacity ought to be about 450 cm. or 5x 1074 micro- 
farad. The value found by experiment was 5°25 x 10-4. 
This is about 3 per cent. less than the capacity of the 
leyden-jar. 

A set of readings obtained with the air-condenser are 
shown below. Here V is the voltage, obtained by multi- 
plying the throw of H by the proper constant, Q is the 
charge on the mica condenser, similarly obtained from that 
of I, kis the dielectric constant thus found, X is the potential 


gradient in megavolts per cm.= C is the conduction 


Vv 
ESO 
current in microamperes per sq.cm., and C’ the current 
calculated from the formula log C=3'286 + 0'794. X+log X; 
D is the difference between the last two columns. 

The first two readings are rather low. The first one was 
obtained by charging with a battery in the course of the 
voltage calibration, and it is possible that the steadiness of 
the voltage may have caused & to havea lower value. It 
appeared, both from these and the previous experiments, 
that, when the voltage was falling, the value of & was in- 
creased, probably due to a hysteresis effect. With the 
machine it was hard to keep the voltage quite steady, and 
to this some of the variations observed in / must probably 


Electrical Conductivity of Mica in Intense Fields. 123: 


be ascribed. It is possible that these fluctuations might, on 
the average, tend to increase the value obtained. The 
readings marked (a) were obtained after the condenser had 
been charged for 5 minutes; it will be seen that, at this 
pressure, there is no evidence of any increase in either the- 
dielectric constant or the conduction current with time. 


Vv. Q. ke. xe oh CH OM De 
Stan. Ord | 8:33 0-49 
riayl 0338 8:42 O45 | 
1060 | 0478 | 850 061 
HHO > 0574. | 8:50 0:70 
1428 | 0650 | 858 0-83. | 
1591 0-726 | 861 0:92 | 0-01 0-01 0:00 
apis | 0712) 851 091 | O01 0-01 0-00 
1874  O875 8:80 1-08) 7 0;01lis 0-015 0:00 | 
2184 | 0-996 | 859 126) || OOP ame 025 0-00 
9245 | 1:057 | 8-85 1:30 | 0:025 | 0-025 0:00 
245() 1118 | 859 > 00s 0°035' | —0:005 | 
Die Ve dels! 8°65 1:45 0:04 0:04 0:00 | 
2804 | 1:287 8:66 162 0-06 0-06 0:00 | 
2996 1366 | 858 73 | OO Os —0:005 
3176 1468 | 8-70 EeS4ay | Oa OOS | ORO | 
3369 509 | 8-72 1:95 | 0-14 0-135 | +0:005 | 
3532 1624 | 866 04a) Onli OA165; |) 9==0;005))| 
3752 1713 | 860 DAT a | ODL Oe —0:005 
4272 1-952 | 863 DAE Oe) OARE | OO. | 
|) 4518 | 2°060 | 8:60 2-61 0:58 0595) = 001m 
736 2152 | &56 274 | 07a Ono) 1220-06515) 
4849 | 2217 | 8-63 2°81 0:92 092 0:00 
HONOM 2268 1 | 8-54 2-90 Lal leer 0-005 00 
5040 | 2290 | 856 2°91 1205. aloo 1-20; 0G1 0 
5165 | 2:339 | 853 2:99) 133 asco ny 0025". 
SP0d 2862 |. 8:56 3-01 LARS es | 0-02) 
5256 | 2401 | 862 | 304 | 151 | 1-415 | —0-005 | 
5205 1 v2433) |- 8:81 301 1-49 1-415 | +0:075 | 


The last readings were obtained after the condenser had been 
charged for two successive periods of 5 minutes each. The 
condensers were discharged at the end of the first period, 
but a spark at the key prevented a reading of V being ob- 
tained ; they were immediately charged again and, at the 
end of a further period of 5 minutes, the figures given were 
obtained. It will be seen that a slight increase in both 
k and C was obtained, but the changes were almost within 
the limits of possible error and further measurements are 
desirable. They are, however, attended with considerable 
risk of piercing the mica. The hysteresis effect is clearly 
shown in the following figures, which were obtained by 
charging the condenser to about 5300 volts and allowing, 


124 Mr. H.H. Poole on the Dielectric Constant and 


it to leak down to the voltages given. The leakage-currents 
are not given, as, at the pressures at which they could other- 
wise be accurately read, the fall in potential was so rapid 
that the lag of the galvanometer prevented any reliance being 
placed on the readings. 


Vik ee: ie 
3878 | «1819 8-84 
3188) 1 7) gaualeaes 8:90 

eres: 1-338 9-03 

|; - 2463 7 | ee otean 9:04 

2146 1-040 9125." 
1704 0:815 898 | 
1266 0615 Sy 
807 0-384 9-01 


It would appear as the result of these observations that up 
to a gradient of 3 megavolts per cm. / is constant to within 
a few per cent., and also that a small hysteresis rise in 4, 
amounting to about 4 or 5 per cent., is noticeable when the 
pressure is falling. 

Turning to the leakage-current: all the evidence would seem 
to point to this being a true conduction current, as distinct 
from an effect due to increase of polarization. In the case 
of the high-voltage readings given above, the current through 
the condenser averaged 1°56 microampere for about ten 
minutes, during which time the quantity transferred through 
the condenser must have been about 935 microcoulombs. 
The actual charge on the plates available on discharge 
was only 2:4 microcoulombs, and may have increased by 
0°05 microcoulomb in that time. Moreover, no appreciable 
reverse current could be observed through the condenser 
after discharge, as should be caused by a recovery from 
polarization. 

The agreement with the formula given is within the limits 
of error over the entire range, that is for currents ranging 
from 0°01 to 1°5 microampere per sq. cm.; as the current 
is extremely sensitive to small changes of voltage near the 
upper limits, the actual errors are naturally greater in that 
part of the scale, though the relative errors are probably 
smaller. 


Electrical Conductivity of Mica in Intense Fields. 125. 


This close agreement over so large a range is remarkable, 


and suggests that there may be some definite physical basis. 


for the formula. However, it may be argued that the 
formula can only be an approximation, as, if it were a 
general one applicable to all fields, we would expect it only 
to contain odd powers of XA, whereas it, or its equivalent 
C=aXe*, contains both odd and even powers. The formula 


C=aX(e*+e-*), which only contains odd powers, is in- 
distinguishable from the preceding one over the range: 
covered, as the second term is negligible. The first of these. 


formule gives a value 5°3 x 10'* ohm ems. for the specific 
resistance of mica in weak fields, in which Ohm’s law 


would be approximately obeyed, while the second gives. 
2°65 x 10‘ ohm ems.: in this case the agreement with Ohm’s 


law would be much closer, and would extend over a much 
greater range, as will be seen from a comparison of the 


curves 1n fig. 3. As the value given by Kaye and Laby for 


the specific 1 resistance of mica is 9x 10! ohm em., it seemed 


to be desirable to make some determinations of the con- 
duction current in weaker fields, using an electrometer to. 


measure the current. 


Accordingly some experiments were carried out in this. 


way, the current being measured by the rate of rise of 
potential of a standard 4 microfarad condenser, the potential 


being measured by means of a Dolezalek electrometer whose. 


sensitivity was about 100 scale-divisions per volt. The 


s] 


values of w or the specific conductivity are plotted against. 


x 


X in fig. 3.. Here, as before, C is in microamperes per sq.. 


em., and X is in megavolts per cm. The upper curve has 
for its equation C=aX(e*+e-°*), and the second curve 


the equation C=aXe*, where a and 6 are the values ob-. 


tained from the previous results, so that either of these 


curves will fit the values obtained with large values of X. 


It will be seen that they both fail to represent the current 
correctly at small fields, though giving results of the right 


order of magnitude. The simpler formula differs less from. 


the observed value than the other. The measurement of 


these small currents is rendered very uncertain by soakage. 


and polarization effects, which make it hard to know how 
much of the observed current is to be ascribed to conduction. 
The reading marked @ was obtained after the voltage had 
been on for an hour; it lies slightly above the value prev viously 


obtained at this pressure, indicating a small rise of the. 


} 


cain 


126 Mr. H. H. Poole on the Dielectric Constant and 


leakage-current with time. The lowest reading plotted in 


the figure represents a current density of about 107° micro- 
1 e 
ampere, or 55, of the greatest current observed with the 


galvanometer. For these small gradients the value of the 
-specific resistance found from the curve is about 1°36 x 10”. 


Fig. 3. 


O4 x 


“There was some evidence that, at still smaller gradients, a 
further increase in the specific resistance occurred, but the 
readings were rendered most uncertain by soakage currents. 

On earthing the tinfoil disk which had been connected to 
the pressure source, a small reverse current was always 
noticeable. A reverse current about 3 x 107‘ microampere 
per sq. cm. was observed immediately after the mica had 
been subjected to a gradient of 0°536 megavolt per cm. 
‘(which caused a current 16°8x10~*). This reverse current 
died away rapidly at first, but even after several days a 
current about 4 x 10-‘ microampere per sq. cm. was observed. 


Hlectrical Conductivity of Mica in Intense Fields. 127 


This effect, though looked for, was never detected in the 
galvanometer experiments, showing that this reverse current 
does not increase with the field to at all the same extent as 
the direct current. 

The low value found for the specific resistance in weak 
fields may be due to a permanent change in the mica caused 
by the intense fields to which it had been subjected. On 
the other hand, it seems equally probable that large varia- 
tions in the resistivity may occur in different specimens. 
Some experiments on the insulation resistance of the standard 
condenser showed that the resistivity of the mica used in 
its construction was not less than 5x10!*. This value was 
obtained when the potential difference across the condenser 
was anything between 20 and 30 volts. With smaller 
pressures considerably larger values for the resistance were 
obiained, but in all cases the interpretation of the readings 
was rendered difficult by soakage effects. 


Meaning of the Results obtained. 


No evidence has been obtained in favour of a theory of 
electric polarization analogous to the molecular theory of 
magnetism, if we except the small hysteresis effect observed, 
as no certain variation of k with X has been established, 
even in the most intense fields. On the other hand, the 
resuits do not furnish any real evidence against such a 
theory. According to Sir J. J. Thomson’s doublet-chain 
theory *, polarization and conduction are due to the forma- 
tion of chains of electric doublets under the influence of the 
field. If there are N doublets per c.c. arranged in chains 
and the electric moment of each is M, the total electric 
moment per c.c. will be NM. Now, taking the highest of 
the figures given, the value found for this moment was about 
6240 E.S.U. If we assume that each doublet consists of 
a+ anda — charge each 48x 107" E.S.U. at a distance 
1078 em. apart, the value of M will be 4°8x107!8, and there- 
fore N will be 1:3x 107. Taking the formula of Muscovite 
as K,O,3A1,0,,68i10,,2H,O, there are about 2°3 x 107! mole- 
cules per c¢.c., and each contains 42 atoms. It is there- 
fore probable that, even in the most intense fields employed, 
the number of doublets arranged in chains formed only a 
small fraction of the whole, and so it is not surprising that 
no evidence of saturation was obtained. 


* J. J. Thomson, ‘The Corpuscular Theory of Matter,’ p. 86. Also 
Phil. Mag, July 1915. 


128 Dielectric Constant and Electrical Conductivity of Mica. 


It seems possible that the exponential term in the ex- 
pression for the current in the strong fields may be due to. 
a distribution of electronic velocities in accordance with 


Maxwell’s law *. According to this the number of electrons 
3E 


possessing a given energy E should contain e 25 asa factor,. 
where EH is the mean electronic energy. Now, in the case 
of insulators, few, if any, electrons normally possess enough 
energy to escape from the atoms to which they are attached, 
and so take part in the conduction current. In a very 
strong field, sufficient energy may be contributed by the field 
to enable some of the most energetic to escape. In order 
that this may be possible, it is necessary that the electron 
should possess energy not less than H,— E., where EH, is the 
energy required to escape from the atom, and KE, is the 
energy contributed by the field during the escape. We may 
write H,= Xed, where ¢ is the electronic charge, and d some 
distance of atomic magnitude. Hence the number of avail- 


able electrons, and so the current, might be expected to 
3Xed >» . 

contain e* “8 asa factor. Comparing this with the results 

obtained, we find, after reducing X to H.S.U., that 


ded 


— =5°5x10-*. These results were obtained at about 


2H 

9° C., so, if we assume that the average electronic energy is 
the same as that for gaseous moleculest, we find that 
H=2°8x10-"4, so as e=4:8x 107 we find that d is about 
2x 1078, which is of the right order of magnitude. In the 
case of the strongest fields, the energy contributed by the 
field would be nearly four times the mean electronic 
energy f. 

If this explanation of the occurrence of the exponential 
factor is correct, and if, as assumed above, the mean elec- 
tronic energy is proportional to the absolute temperature, 
the results obtained at different temperatures should vary 
considerably. It is proposed to modify the arrangement so 
as to enable determinations of the conduction current to be 
made at varying temperatures. 


* See Richardson, Phil. Mag. August 1915. 

+ Thomson, loc. ert. 

t The internal field due to the polarization is neglected here. As, 
however, we may assume that it is proportional to X it will not affect 
the form of the result. 


Tracks of « Particles from Radium Ain Sensitive Films. 129 


Summary and Conclusion. 
The results obtained may be summarized as follows :— 


(1) No variation in the dielectric constant could be de- 
tected up tu gradients of 3 x 10° volts per cm. 

(2) The conduction current increases very rapidly with 
the gradient when the latter is large. The formula 
log C=A+log X + BX represents the current within 
the limits of error over a wide range. A possible 
explanation of this formula is suggested on the basis 
of a distribution of electronic velocities in accordance 
with Maxwell’s law. 


In conclusion I wish to express my gratitude to Mr. H. 
Thrift, F.T.C.D., for the assistance he has so kindly given 
me in reading the galvanometers. 

Physical Laboratory, 

Trinity College, Dublin. 
April 24, 1916. 


— 


IX. The Tracks of the « Particles from Radium A in Sensi- 
tive Photographic Films. By H. Ikeurt, Research Student 
Imperial University, Tokyo”. 

[Plate IV. | 


ae a point source of 2 rays is placed on a photo- 
graphic plate, the paths of radial @ rays can be traced 
in the film. If the rays are homogeneous, all the tracks will 
terminate on a spherical surface, whose radius is equal to 
their range in the substance. The consequence is that the 
radial tracks appear asa halo. Haloes obtained in this way 
with radium C have already been illustrated in a previous 
paper by Mr. Kinoshita and myself ft. Although no parti- 
cular difficulties were encountered in obtaining them with 
radium C, we were not able to get them with radium A. 

Since the above paper was published, I made further 
experiments and obtained lately some photographs on which 
the a-ray tracks of radium A are clearly visible. The 
method employed consisted simply in knocking a plate at 
the film side with a small iron ball, which had just previously 
been exposed to radium emanation for a few minutes. The 
plate was developed usually half an hour later, and examined 
under a microscope on drying. In the present experiments, 
Ilford Process Plates only were used. 


* Communicated by Prof. H. Nagaoka. 
+ S. Kinoshita & H. Ikeuti, Phil. Mag. March 1915. 


Phil. Mag. 8. 6. Vol. 32. No. 187. July 1916. K 


130 Tracks of « Particles from Radium A in Sensitive Films. 


A microphotograph of one of the plates is reproduced in 
Pl. IV. fig.1; enlarged 120 diameters. Irregular dark areas 
seen in this figure are the spots at which the plate was struck 
in the process above mentioned. Around these areas there are 
to be seen a number of spots, each of which consists of a set 
of a-ray tracks radiating from a common centre. In these 
cases, fine dust particles which had adhered to the ball must 
have been set free by the shock and have settled down on 
the plate, thus forming the nuclei of the e-ray radiation. 
Rubbing the ball with a fine emery-paper before exposure 
to the emanation was found to be effective in obtaining a 
larger number of spots. 

In order to show the spots more clearly, two of them, 
marked P and Q in fig. 1, are reproduced in figs. 2 and 3 
respectively, magnified 345 diameters. The halo in fig. 2 
has a mean radius of ‘0507 mm., and is, as we have shown 
in the previous paper, due to radium C. When closely 
examined, another concentric halo is to be seen inside this 
halo, evidently due to a set of homogeneous a rays from 
radium A, so that the outer one appears as a corona, resem- 
bling a pleochroic halo. The halo due to radium A is more 
clearly illustrated in fig. 3, in which, however, the halo due 
to radium C is feeble. In the above two cases, the radii of 
the outer and inner haloes are ‘0507 and -0348 mm. respec- 
tively, so that their ratio 1:°686 is very nearly the same as 
that of the ranges of the respective a rays in air, viz. 
6°94 cm.:4°75 cm. or 1:°685. This result shows that the 
radiant nuclei can be regarded practically as points. 

It may be remarked that haloes produced simultaneousl 
on the very same plate are somewhat different ; the halo due 
to radium C is most conspicuous in P while it is very feeble 
in Q. This singularity seems to have been due to the 
manner of exposing the iron ball to the emanation. In 
the above experiments, the emanation was preserved in a 
glass vessel inverted over mercury, through which the ball 
was introduced. Under this circumstance, the active deposit 
found on the ball would consist of two different parts: one 
deposited directly from the emanation during the exposure, 
and the other, a portion of the deposit which had already 
been accumulated on the surface of the mercury in contact 
with the emanation, and which contained radium C in sucha 
proportion that it was in equilibrium with radium A. It 
would be natural to suppose that the second part of the 
deposit covered the ball not uniformly all over the surface, 
but adhered to it rather irregularly. Consequently, which 
of the tracks of the a rays from radium A or radium C 


| 
, 
! 
I 


Decrease in the Paramagnetism of Palladium. 131 


predominated would largely depend on the portion of the 
surface of the ball whence the nucleus was detached from. 

This interpretation seems to be confirmed by the following 
experiments. When the iron ball was first placed in the 
glass vessel and exposed for a short time to the emanation 
which was introduced subsequently, a similar process gave a 
normal result; in all the haloes produced on a plate, the 
relative numbers of the tracks of « rays from radium A and 
radium C were the same. 

In conclusion, I wish to thank Professors Nagaoka and 
Kinoshita for their kind interest in this experiment. Ihave 
also to thank Professor Tawara of the Metallurgical De- 
partment for his kindness in allowing me to use the 
microphotographic apparatus. 

March 1916. 


X. The Decrease in the Paramagnetism of Palladium 
caused by Absorbed Hydrogen. By H. F. Bieas, B.A., 
Assistant Lecturer in Mathematics tn the University of 


Manchester *. 
CONTENTS. 
I, Introductory. 
II. Method. 
. Principle of the Method. 
. The Electromagnet and its Circuit. 
The Design of the Pole-pieces to give a constant Pondero- 
motive Force. 
The Torsion-system. 
. Winding and Measurement of the Coils. 
. Measurement of the Magnetic Force. 
. Adjustment and Measurement of Balancing Current. 
IIL. Particulars of the Experiment on Palladium. 


eS 


NID Ot 


REFERENCES. 
(1) Grauam.—Chemical and Physical Researches, p. 297. (Edin- 
burgh, 1876); Proc. Roy. Soc. xvii. 212, 500 (1869). 
(2) CuriE.— Ann. Chim. & Phys. 7 sér. v. p. 850 (1895). 
(8) Honpa.—dAnn. d. Phys. xxxii. p. 1043 (1910). 
(4) Lanecevin.—Ann. Chim. § Phys. 8 sér. v. p. 71 (1905). 


I. InrRopucroryY. 


: | ‘HE importance of determining the magnetic properties 
of hydrogen, whether in the form of molecules, atoms, 
or bare nuclei t, suggested a measurement of the effect of 


* Communicated by Sir Ernest Rutherford, F.R.S, 

+ Hoitsema (Zerts. phys. Chem. xvii. p. 1, 1895) has shown that the 
hydrogen in palladium is monatomic for small hydrogen-content. 

Ki. Newbery and the author made a rough experiment to see whether 
the hydrogen given off from palladium carries a charge, but could 
observe no such effect at pressures of a few hundredths of a millimetre. 


K 2 


132 Mr. H. F. Biggs on Decrease in the Paramagnetism 


absorbed hydrogen on the susceptibility of palladium, an 
effect believed on the authority of Thomas Graham “ to be 
a strong increase to the paramagnetism of palladium. A 
preliminary experiment, «however, contradicted Graham's 
result, showing that palladium when saturated with hydrogen 
becomes almost neutral. Since the method adopted is con- 
siderably modified from that used in previous absolute 
determinations of susceptibility, the experimental arrange- 
ments are described in detail. 


Il. Meruop. 


L. Principle of the Method. 


A disk of palladium foil is placed so that it fits within a 
small coil (coil A) carried on the arm of a torsion-balance,, 
which swings in a non-uniform magnetic field. A current 
passed through the coil is then adjusted so that the force 
exerted by the field on the current balances the force on the: 
palladium and on the matter of the arm itself. The process 
is repeated with the palladium removed. The difference of 
the balancing currents, multiplied by the total area of the 
coil, gives directly the magnetic moment, M, of the palladium. 
The intensity, H, of the field at the same spot is deduced 
from the throw of a galvanometer connected to the same 
coil when the circuit of the electromagnet is broken, this. 
deduction being based on the comparison between the throws. 
given by another coil (coil B), 


(a) when the circuit of the magnet is broken, 
(b) when the coil is snatched out of the field. 


The mass-susceptibility (that is, the magnetic moment per 
gram per gauss) is then K,,=M/Hm, where m is the mass of 


the palladium. 


2. The Electromagnet and its Circuit. 


The magnet used was an “ optical magnet” with cylindrical 
coaxial cores, 7 cm. in diameter, whose distance apart could 
be adjusted by screw collars. To the ends of these cores. 
were screwed pole-pieces designed to give a field in which 
the ponderomotive force would be parallel to the axis of 
symmetry of the field, and roughly constant for all points on 
or near this axis. This form of field, essential in the 
preliminary experiment, is also useful in this method, as it 
renders unnecessary the exact replacement of the disk of 
foil in the same position in the coil—a difficult matter in the 
ease of palladium-foil, which expands when charged with 


of Palladium caused by Absorbed Hydrogen. 133 


hydrogen and becomes much crumpled on being charged 
and discharged. ‘The design of the pole-pieces is given in 
the next section. 

Several points of the circuit supplying the magnet could 
be connected at will with a voltineter, and, by using currents 
which brought the voltmeter pointer exactly to a division of 
the scale, the current could be kept accurately constant over 
a range of | to 7 amperes. 

The fine adjustment of the current was made by a weighted 
wire passing over a groove in a wide brass tube. The wire 
used for Bowden brakes was found suitable, and a run of 
about 10 feet was enough for the adjustment. 

The voltmeter had to be placed at some distance to avoid 
the influence of the electromagnet on its field, and was read 
through a field-glass, an arrangement which had _ the 
advantage of automatically preventing parallax error in the 
reading. 

As it was important for the ballistic measurements that 
the time taken to break the circuit should be constant, a 
mercury key actuated by a spring was included in the 
circuit. The core of the electromagnet was earthed to get 
rid of electrostatic effects. 


3. The Design of Pole-pieces to give a constant 
Ponderomotive Force. 


Suppose the field to be symmetrical about the axis of a. 
Let x, 7 be cylindrical coordinates. Then, if the pondero- 
motive force is constant, 


HH, oH/02= const. 
Z 
he const. <a; 
Again, the flux across any section of a symmetrical tube 
of force is constant, 


H,r?=const. for any line of force. 


Therefore the equation of a line of force will be 
SP (OU 

Let @ denote the inclination of a line of force to the axis, 

ik 
aie: | 

Then, since the lines of force are normals to the equi- 
potential surfaces, the curvature of the latter at the axis will 
obviously be = (3°) =— es where is now considered 

ri Pp or r=0 Ag 


as a function of w# and ~. 


ie 
then tan d= ao 


134 Mr. H. F. Biggs on Decrease in the Paramagnetism 


If, then, the faces of the pole-pieces are spherically curved 
with radii of curvature differing by four times the distance 
between the centres of the faces, these faces should coincide 
nearly enough with such equipotential surfaces, and should 
give a field with approximately constant ponderomotive force 
for points on or near the axis. Further, the radii 17,, 72 of 
the faces (measured perpendicular to the a should 
correspond to the radii of a single symmetrical tube of force. 
Thus we should make 1,/7y=(a»/v;)4. In the preliminary 
experiment referred to, the palladium was balanced against 
weighed bits of bismuth in a field thus designed. The 
apparatus was very simple, and no knowledge of the field 
was necessary. The method gave results correct to about 
10 per cent. In the final experiment larger pole-pieces. 
were used, whose dimensions were those given in fig. 1. 


Mig: 


The edge of the concave pole-piece was rounded as shown 
to avoid disturbance of the field by sharp edges. To test 
the uniformity of the force a glass rider was made with a 
bit of bismuth at one end and a bit of aluminium at the 
other. The rider was placed on the coil in such a way that 
the ends lay in the axis of the field, and could be turned 
either with the bismuth in the strong field and the aluminium 
in the weak field or wee versa. By noting the difference of 
the balancing-current in the coil, when the rider was 
reversed, it was found that the ponderomotive force only 
varied by about 1 per cent. over 1°3 cm., the distance 
between the ends of the rider. 7 


oS 


of Palladium caused by Absorbed Hydrogen. 135 


4. The Torsion-system. 


The torsion-head was fixed to a bridge built on a slate 
bench, the space under the bridge being enclosed in a 
draught-tight chamber with a glass back. The space 
between the magnet-coils was also enclosed in a chamber, 
whose top came close under the bench. The torsion-stem 
passed through a hole in the bench, lined by a tube passing 
into the magnet-chamber close to its back wall. 

R (fig. 2) represents the flat end-piece of the rod of the 
torsion-head. Pt.Jr.is the torsion-wire of platinum-iridium, 
a substance which has been found to give a very constant 
zero, Its diameter was ‘05 mm., its length about 5 cm. 


Bie, 2) 


The torsion-wire was connected to one end of the fine wire 
of the coil A. The other end of the coil-wire passed outside 
the tube g, making a couple of turns round the tube on the 
way, and was connected to a phosphor-bronze spiral, Ph. Bz., 
enclosing the torsion-wire, and waxed at the top to R. Into 


136 Mr. H. F. Biggs on Decrease in the Paramagnetism 


the lower end of the tube g was fixed the drawn-out end of 
a glass rod g’, whose weight was enough to keep the whole 
system nearly vertical when suspended. To g’ was fixed 
an arm a of a narrow glass tube, through which passed the 
leads of coil A slightly twisted together. The object in 
having the leads twisted, as also in having the spiral Ph. Bz. 
at the top instead of at the bottom (as in a galvanometer), is, 
of course, that the linkages of the lines of the field with the 
circuit of coil A shall be confined to the coil itself. The end 
of the arm a was bent into the are of a circle, fitting coil A, 
which was glued to it and thereby stiffened. 

The image of the ftlament of a pocket flash-lamp in the 
mirror was projected on a scale at a distance of about 60 cm. 
The paper vane worked in a narrow metal chamber placed 
on the bench. This arrangement confined the motion to 
about 3° and provided the required damping. For the 
highly paramagnetic uncharged palladium in the stronger 
fields the equilibrium was found to be unstable ; closer stops 
were therefore made by bending a tin wire so that its ends 
nearly met on either side of the vane which projected 
slightly above the top of the metal chamber. 

Pb is a bit of lead wire whose free upper end was bent so 
as to bring the centre of gravity of the system on to the 
axis of g' and g. c is a small glass cross-arm on whicha 
little lead rider was placed when the palladium was removed 
from the coil, thus keeping the centre of gravity in the same 
vertical. The arm c also served to clamp the system by 
being raised slightly on a small lever while the palladium 
was being inserted in, or removed from, the coil. 

It was found necessary to earth the torsion-head as well 
as the core of the magnet to get rid of electrostatic effects. 


5. Winding and Measurement of the Coils. 


Coil A, the coil on the torsion-arm, and coil B, the 
exploring-coil, were made as nearly alike as possible, being 
wound as follows: 20 turns of No. 47 (‘04 mm. diam.) 
silk-covered copper wire were wound on a layer of thin 
paper slightly waxed to a glass tube of diameter nearly 2 em. 
The coil was wound in 3 layers (of 10 turns, 6 turns, and 
4 turns, respectively), each layer when finished being thinly 
coated with seccotine. When dry it was slipped off the 
tube, and the paper cut away from each side of the coil. 

To obtain the total area (mean area of a turn X number 
of turns) of the coil, two points were marked on the parts of 
the wire not wound on, and the distance between these two 
points was accurately measured, (1) before the coil was 


of Palladium caused by Absorbed Hydrogen. 137 


‘wound, and (2) after the coil was wound, the ends being 
held out in a straight line. From the length thus found of 
wire wound on, the total area of the coil can be deduced 
thus :— 

If J be the length of the wire wound, n the number of 
turns, A the total area, we have, to a close approximation, 
if the normal to the wire always makes a small angle with 
_ the radius, the bar denoting mean value, 

(2a 
ys [? 
A=nrr=ntr’ = —— . 
Ant 

The materials of this coil (either the copper wire or the 
seccotine or both) proved to be badly chosen, the coil being 
almost as strongly paramagnetic as the uncharged palladium. 


6. Measurement of the Magnetic Force. 


As already mentioned, a similar coil B was wound, which 
was used as an exploring-coil to measure the value of the 
field at a certain spot in the usual way. The coil was waxed 
into a piece of ebonite fixed to the end of a light lever-arm, 
weighted just beyond the fulcrum, which, when released, 
raised the coil quickly out of the field. A change in the 
rate of raising the arm was found to make no difference to 
the galvanometer-throw, and it was therefore assumed that 
the movement was rapid enough to give correct readings. 
A piece of plate-glass was wedged flat against the lip of the 
concave pole-piece, and the ebonite piece containing the coil 
slid against this, so that the coil came accurately back to the 
same position each time. This position was so chosen that 
the coil was coaxial with the field and halfway between the 
pole-pieces, and was, as nearly as could be judged, the same 
as that afterwards occupied by coil A. Measurements of 
the field were then taken for four values of the current from 
1-7 to 7 amperes. The current for each vaiue, before 
measurements were taken, was reversed 50 times each way, 
after which the throw was constant, and the same for both 
directions of the current. The flux through the coil when 
raised out of the field was found, by breaking the exciting 
circuit, to be negligible. 

In the gaivanometer-circuit were included the coils of a 
Hibbert standard inductor (and for the stronger fields used, 
an extra resistance) and the galvanometer was thus stan- 
dardized for each value of the resistance. The strength of 
the field at one particular spot is thus found for all the 
values of the current for the steady state of the magnet. 


138 Mr. H. F. Biggs on Decrease on the Paramagnetism 


Further, for one particular value of the current the throw 
was observed when the circuit was broken by the spring 
break. The strength of the field at the spot actuaily occupied 
by the palladium and coil A is then determined by a 
comparison between the throws given by coils A and B 
respectively for the same exciting current when the circuit. 
is broken, the galvanometer being standardized in each case 
by means of the standard inductor. In fact, if A, B be the 
total areas, Hy, Hp the values of the magnetic force for 
the same current, ©,, Og the galvanometer-throws on 
breaking the circuit, Sa, Sg the throws given by the standard 
in series with coil A, coil B respectively, we may assume. 
(since none of the quantities differ by more than a few per 
AH, Og/Sa 
BHg  @,/Sz" 

While the magnet was excited, coil A was kept in position. 
by turning the torsion-head. 


cent. as between A and B) that 


7. Adjustment and Measurement of Balancing Current. 


The current was taken from a storage-cell and adjusted by 
means of a pair of dial-resistances in series, and when the. 
adjustment was reached the two resistances were varied, 
their sum being kept constant until the voltage-drop over 
one of them balanced against a standard cell. 


Ill. PARTICULARS OF THE EXPERIMENT ON PALLADIUM. 


The palladium used was prepared electrolytically by 
Messrs. Johnson and Matthey. Traces of iron were found 
in this, but they can hardly have affected the measurements 
of the effect of hydrogen, though the determinations of the 
susceptibility of palladium itself are probably too high on 
this account. The disk of foil weighed -17 gram. 

To control the hydrogen-content of the palladium, the. 
charging was done in a sodium hydrate electrolytic cell in 
series with a rough hydrogen voltameter. ‘The actual 
measurement of the gas, however, was made by heating the 
palladium and pumping off the gas evolved into a tube 
calibrated for the purpose. 

The observations were planned to answer twoquestions. Let 
M,, M, be the magnetic moment in the same field, H of the pal- 
ladium with different hydrogen-contents v,, v2 em.?at N.T.P., 
and let the (negative) quantity (M,—M,)/H(v,—v,)=K, be 
called the “ susceptibility-gradient”’ due to hydrogen. Then 
the questions to which an answer was sought are :—(1) Is. 


of Palladium caused by Absorbed Hydrogen. 139° 


the susceptibility-gradient independent of the field H ? 
(2) Is it independent of the hydrogen-contents 7, v, ¢ 
Both these questions seem to receive an affirmative answer,, 
as far ag the accuracy of the experiment goes. The Tables. 
summarize the results. 


H | ae 
eae 1040 | 1905 | 2807 | 3519 
— 10° K TAG 1718) canon aT 
| 108K, (Pd) 660. | 5:42 | 584) 5:20 


Susceptibility-gradient due to absorbed hydrogen for various. 
streneths of field. Hydrogen-content :— 
550 x volume of palladium 
to 4x 
Susceptibility of palladium, 


Vol. of hydrogen | | 
vol. of Pd. | 
v/v DO ROM Meer lowland 
v/v Dit hh BOL: NOM Re TO 
- 10° K, 761 7-01 788 7-92 
i | | 


Susceptibility-gradient due to absorbed hydrogen for various 
hydrogen-contents. 
“H=1303 oauss. 


The volumes v,, vg are not accurately known, as is their 
difference, but the values given serve to indicate the ranges. 
over which the observations were taken. Incidentally the 
susceptibility of uncharged palladium was found, the values 
agreeing with previous determinations. It may be noted 
that the irregular variations observed by Curie® in the 
product KT (where T is the absolute temperature) might 
possibly be accounted for by supposing that the palladium,. 
which was placed in a closed vessel, gave off and re-absorbed 
traces of hydrogen when the temperature was raised, but it 
seems more likely that they are due to slow reduction of 
iron impurity and subsequent re-oxidation, since similar 
irregularities were observed by Honda‘) in the case of 
impure palladium, but notin the case of a very pure sample. 


140 =Decrease in the Paramagnetism of Palladium. 


The temperature throughout the experiment varied within 
a few degrees of 15° C. 

It will be seen from the Tables that the figures for what 
has been called the susceptibility-gradient show no systematic 
variation, in spite of the fact that palladium itself shows a 
marked decrease in susceptibility with increase of field. 
This result might lead us to suppose that the diamagnetism 
of the hydrogen absorbed is simply added algebraically to 
the paramagnetism of the palladium, if it were not that the 
value obtained for the susceptibility-gradient due to the 
absorbed hydrogen, 7 x 107%, is probably much greater than 
the diamagnetism of hydrogen gas. It is true that the 
latter has never been satisfactorily determined *, but it is 
probably beyond the range of the methods used. Again, on 
Langeyin’s“ theory, if the diamagnetism of hydrogen itself, 
whether in the torm of molecules or atoms, were equal to 
the value here found, the radius of the orbit of the electrons 
in the hydrogen molecule or atom would be 2°4 x 1077 ¢ cm., 
which is certainly too large in either case. The observed 
effect is therefore probably due to some interaction of the 
hydrogen and the palladium, whereby the latter’s para- 
magnetism is reduced. It should be noticed that if the sus- 
ceptibility-gradient due to the absorbed hydrogen remains 
constant till saturation is reached, since palladium will 
absorb 900 times its volume of hydrogen, the resulting com- 
bination should be diamagnetic. Unfortunately no pains 
were taken to saturate the palladium, and the apparatus 
was taken down to make way for an experiment on the 
susceptibility of hydrogen gas before this point was noticed. 


I wish to express my best thanks to Sir E. Rutherford 
for allowing me the use of all the apparatus required for 
this research. 


* Kammerlingh Onnes and Perrier (4k. Wet. Amst. xx. p. 81, 1911) 
have made a rough determination of the susceptibility of liquid hydrogen ; 
also Pascal (Ann. Chim. Phys. viii. sér. 19, p. 28, 1910) deduces a value 
for the susceptibility of hydrogen from that of its compounds. Both 
these determinations give a value of about 2-4x107!° for the suscepti- 
bility per cm.’, which would mean that the radius of the electronic 
orbit in hydrogen is about 1:°2x107* cm. Bohr’s theory (Phil. Mag. 
xxvi. pp. 5, 868, 1918) gives °5x 107° cm 


aaa ea 


XI. On the Change in the Resistance of a Sputtered Film 
after Deposition. By Miss E. W. Hosss, B.Sc. (Bristol)*. 


CONSIDERABLE amount of work has been carried’ 
out by various investigators on the electrical resist-. 
ance of metallic films deposited by cathodic sputtering. In 
the main, the films dealt with have been previously aged by 
exposure to air and high temperature after their removal 
from the discharge-vessel, but in some instances arrangements. 
have been made for measuring the resistance of a film in 
position during and after deposition. Thus in 1912+ 
Kohlschiitter and Noll published some results of experiments 
on thin films of silver deposited in hydrogen, nitrogen, and’ 
argon, in which measurements were made in situ of the 
changes in resistance with time which occur when a film is. 
left in vacuo and in various gases after deposition. The 
following is an account of experiments on thin films of 
platinum and palladium carried out on somewhat similar 
lines, in which a fuller study of these changes was made. 
In particular, the effect of admitting gas into the discharge-. 
vessel was investigated in detail; it 1s this that forms the 
chief subject of the paper. 


APPARATUS. 


The apparatus did not differ in essentials from that usnally 
employed for the deposition of films. The films, 1:2 cm.. 
long and 1 cm. wide, were deposited on polished strips of 
fused silica, which rested on the anode. In order to measure: 
their resistance, leads were introduced into the discharge- 
vessel and contact was made with the film through thick. 
end deposits, in some cases by soldering wires to them and 
in others by means of a mercury pellet. The latter method 
was found to be by far the more convenient, and was almost. 
invariably employed. It was quite satisfactory if the mercury- 
were covered by mica disks in order to protect it from 
electrical bombardment. An exploring point connected to 
a Braun electrometer was introduced for measuring the. 
cathode fall of potential during deposition, and the current 
was controlled by a cadmium-iodide amyl alcohol adjustable. 
resistance. While adjustments were being made, the silica 
could be screened by a mica shield, rotated by a ground- 


glass joint, but it was not necessary to use this in most of 
the work. 


* Communicated by Dr. A. M. Tyndall. 
T Zeitschrift fiir Llectrochemie, vol. xviii. p. 419 (1912). 


(142 Miss 8. W. Hobbs on the Change in the 


RESULTS. 


The mechanism of cathodic sputtering at low pressures is 
probably similar to that of the production of colloidal solu- 
tions of metals in Bredig’s method. The metallic particles 
are molecular aggregates possessing a negative charge, which 
they lose when, or shortly after, they strike the plate on 
which the film is deposited. Hence a semi-stable layer of 
‘particles is formed, having an electrical conductivity which 
depends on the conditions during deposition and on the 
thickness of the layer. In particular, Kohlschiitter and Noll 
(loc. cit.) have shown that the nature of the residual gas 
during discharge has a great influence on the fineness of the 
particles deposited, and thus on the initial resistance of the 
film and on the subsequent changes which occur. 

In the matter of adherence to the silica, films may vary 
from a soft film easily rubbed off to one of a hard metallic 
nature. An important factor in this connexion is the 
-eathode fall of potential, which controls the speed with 
which the particles impinge on the plate, though the limits 
of adherence vary with the nature and pressure of the 
residual gas in the discharge-vessel. 


The Ageing of Films. 


It is well known that films deposited by cathodic sputtering 
undergo a process of ageing, entailing a change in the 
“properties of the film. In 1898* Fawcett showed in this 
laboratory that the change may continue for months, but 
that it is hastened by rise of temperature. 

The change in condition after deposition is made apparent 
by measuring the electrical resistance, which decreases with 
time. An investigation was made of the effect of ageing 
in vacuo on the electrical resistance of platinum and pal- 
ladium films. The films were deposited at pressures varying 
from *14 mm. to *4 mm. in air, and with a modified form of 
apparatus from ‘5 mm. to *8 mm. in hydrogen; the pump 
was put on immediately discharge ceased, and resistance- 
time measurements were taken. 


Deposition in Air. 


A number of results obtained for platinum films deposited 
“in air (the case most fully investigated) are given in fig. 1, 


* Phil. Mag. xlvi. p. 500 (1888). 


Resistance of a Sputtered Film after Deposition. 143 


Fig. 1. 


RESISTANCE 


CHANGE IN 


TIME 1h Pa! nee 


Fig. 


‘350000 MeRW Une 
FRR aoa 
mm. a 


CHANGE IN RESISTANCE (‘%) 


TIME IN MINUTES 


in which resistance-time measurements for films A to F are 
plotted as full curves. Similar results for palladium in air 
are shown in the early portions, as far as the point marked 


144 Miss E. W. Hobbs on the Change in the 


(a) or (A), of the full curves in fig.3. The ordinates express. 
the resistance of the film as a fraction of the first value in 
each set of observations, which were always started about a 
minute after the discharge was cut off. Though the results. 


Pigs. 


©) 
H 


®Y 
5 & 


CHANGE IN RESISTANCE 


ee ae ae 
2 eee Se 
20 


40 60 80 i0o 120 140 160 
TIME IN MINUTES 


of the first 90 minutes of ageing are alone shown, the readings. 
were in many cases extended over many hours; these confirm 
the view that the film is settling down toa steady state, which, 
however, may not be reached for days. 

On the question whether the rate of ageing of a film 
depends on its thickness, there is no evidence one way or the 
other. This is made apparent on reference to Table I., in 
which all the results for platinum, including those plotted in 
fig. l,are given. Ry is the resistance of the film when the 
pump is put on after the discharge has ceased, and R its 


resistance one hour afterwards. It will be seen that = 
| 


Resistance of a Sputtered Film after Deposition. 145 


is never greater than one, but varies in a haphazard manner 
from film to film. This is no doubt due to the impossibility 
of ensuring a constant texture of film throughout a given 
‘series. 


TABLE I. 
Weehines Rige |B iN RG 1 OMIM ett 2) 
Film) obms. Be | Film} olins. | RS Film|) ohms. | Ry 
243 Oot 4000 0-85 14,140 0-81 
278 0:98 487 0:92 14,980 0°64 
C 407 0-90 )d50 0-85 15,110 0°88 
1070 O72 || 7900! || 0:89) Je) 15,930 | 0-74 
B. 1545 098 | L. 7600 0:95 || M. | 16,270 0°51 
D.| 1790 0:80 | E. 9200 RZ Zs 20,500 0:60 
1970 O87 |! 11,800 , 0-89 29,000 0°53 
A 3100 COs ie Bia) 125550), |). 0-95 68,400 0°84 
K 3400 1-00 | | 


The reason for the metastability of a newly deposited film 
cannot be definitely stated. The most feasible explanation 
is that of Kohlschiitter and Noll (loc. c2t.), who suggest that 
owing to surface-tension the finely ivided particles in a 
fr eshly deposited film undergo a process of coalescence, by 
means of which the gaps between the particles are lessened 
in number, and so in general the resistance falls. Further 
evidence that such a process occurs was obtained from an 
examination of the optical properties of films. Kohlschiitter 
and Noll point out thatif the conditions are such that coa- 
lescence can only go on with increase of distance between 
the particles, the resistance will rise and may reach infinity. 
As evidence of this they cite the rise in resistance that they 
observed if silver films were deposited in hydrogen and left 
to age im vacuo, when it was found that the resistance 
decreased to a minimum and then rose to infinity. From a 
number of different observations they conclude that the 
metal is deposited in hydrogen in a coarse form, and hence 
that less material is required for a film of given resistance than 
if the deposition had occurred in nitrogen or argon ; this in- 
-ereases the chance of a break in the film. In the author’s expe- 
riments on platinum and palladium films, despite the extreme 
thinness of many of the films, this rise in resistance was never 
observed—the resistance always decreased with time when 
the film was left to age in vacuo, whether the depvusition had 
taken place in air or in hydrogen. 

The phenomenon may be viewed somewhat differ ently by 


Phil. Mag. 8. 6. Vol. 32. No. 187. July 1916. L 


146 Miss E. W. Hobbs on the Change in the 


supposing that the material originally deposited in an amor- 
phous state undergoes a slow process of crystallization ; in 
other words, that this is another example of the metastability 
of metals observed by Cohen and his co-workers. Unfortu- 
nately, no light is thrown upon the true nature of the change 
by an analysis of the ageing curves, as they are not expressible 
in terms of any simple formula. 

On both views, the effect of heat, unless it caused a break 
in the film, would be to bring about a marked fall in resistance, 
in the former case owing to the decrease in viscosity and the 
consequent acceleration of the process of coalescence, and in 
the latter because of the increased velocity of the allotropic 
change at higher temperatures. Fawcett (loc. cit.) showed 
that the decrease in resistance which occurred in his expe- 
riments when a film was heated was accompanied by a 
marked hardening of the surface of the film. This effect was 
observed in the present work. 

It is important to note that the ageing must be qeaeeetine 
during deposition, and consequently i it is another factor influ- 
encing the nature of the film obtained at the cessation of 
discharge. Again, while the discharge is taking place, films: 
being deposited near the cathode will be affected to some 
extent by its temperature, which Hodgson and Mainstone* 
have shown is a function of the current through the 
discharge-vessel and the cathode-fall of potential. All 
attempts, therefore, to obtain quantitative results of the 
rate of laying down of the film by measurements of resist- 
ance during deposition are doomed to failure. 

Deposition in Hydrogen. 

An investigation of the change in resistance with time was 
made for films deposited in hydrogen, and the ageing curves 
in this case were found to be similar to those obtained with 
air as the residual gas during discharge. In fig. 1 the dotted 
curves G and H represent two ageing curves for platinum 
films (Ry = 8400 ohms and 9300 ohms respectively) deposited 
in hydrogen, and the early portions of the dotted curves in 
fig. 3 give similar results for palladium. It wiil be noted 
in the case of palladium that rates of aveing are shown which 
are far greater than those obtained under an y other conditions 
(vide curves W and Y, fig. 3). There is strong evidence that 
in many instances this change is a continuation of a change 
which has been proceeding at a very rapid rate during the 
discharge itself, and this probably applies to platinum films as 
well, though their case has not received such full investigation. 


* Phil. Maz. vol. xvi. p. 411 (1918). 


Resistance of a Sputtered Film after Deposition. 147 


Thus a number of films have been observed in which the 
resistance during deposition has fallen at a very rapid rate, 
far greater than in air, although it is known that hydrogen 
is not a gas specially favourable to cathodic sputtering. 
Moreover, the resulting film had atransparency which sug- 
gested a much higher resistance. It is probable that this 
rapid change is set in action by the sorption* of hydrogen 
during deposition; and this view receives support from 
results given below on the effect of admitting hydrogen to 
the discharge-vessel; it will be discussed again in that 
connexion. 


Liffect of Addition o, Gas. 


It is known that films of sputtered metal take up gas during 
deposition. Owing to the relatively large surface, some 
sorption of gas in a film will occur when gas is admitted into 
the discharge-vessel after deposition; an investigation of its 
effect on the electrical resistance was made. 

In the first experiments, dry air was admitted to the vessel 
containing a film which had been ageing for some time. 
The pressure reached a constant value (atmospheric) within 
a few seconds after the stopcock, through which the air 
entered, had been opened. ‘Typical results for a number of 
films are shown as full curves in fig. 2. The ordinates are 
changes in resistance expressed as a percentage of the 
resistance of the film at the moment that the gas is admitted. 
The abscissee are values of time in minutes. The initial 
parts of the curves give the later stages of ageing, the air 
being admitted at the point (a) in every case. The films to 
which the full curves shown here refer are those for which 
the complete ageing curves are given in fig. 1; they are 
designated by the same symbols. 

Kohlschiitter and Noll (loc. cit.) record some results of 
admitting nitrogen to silver films, deposited in nitrogen, 
immediately after their deposition. In one case only do they 
appear to have observed a rise in resistance, but state that in 
general the ageing is apparently stopped, though a mea- 
surement of the resistance of one film 15 hours after the 
admission of the gas shows that in reality the process is 
delayed and not brought to a complete standstill. On the 
removal of the nitrogen, the ageing proceeded. 

In the author’s experiments a rise in resistance with time 


* The term sorption has been suggested by McBain to include under 
one title the phenomena of adsorption (surface condensation) and ab- 
sorption (solid solution). 


L 2 


148 Miss E. W. Hobbs on the Change in the 


is shown in every case on the admission of air*. Except in the 
curve EF, which 1s separately discussed below, the rise continued 
for not less than 25 minutes, and for most films still longer at 
a slow rate. There is no fall in resistance in any of the cases 
given to indicate that the process of ageing, which is shown 
to a marked extent in curves A, B, and C, continues after the 
admission of gas; but in some of the films investigated the 
initial rise was followed by a very gradual decrease in 
resistance. In such cases, the ageing was evidently still 
continuing though at a decreased rate. It may therefore 
be present in curves A, B, and C though swamped by the 
gas effect, and consequently the effect of the admission of gas 
on these films cannot be examined with any certainty, since 
the effect of ageing is quantitatively unknown. Fortunately 
investigations have been carried out on films in which the 
ageing was negligibly small. Curves D and E are examples 
of these. Since they no doubt represent the true effect of 
the air and, moreover, show extremes of magnitude, the 
experimental numbers are given in Table IT. 


TABLE IT. 
l 
Film D. | Film E. | 
| 
Time in mins, ie eee Resist. in ohms. Time in mins. Resist. in ohms. 
0:0 | 3100 0:0 1545 
15:0 | 3090 5D 1537 
30°0 | 3090 Tere) 1534 
40:0 | 3090 BRS) 1526 
41:0 Air admitted. 23°2 1520 
41°75 | 3187 44-7 | 1516 
42°3 3200 60°2 1514 
44°4 3218 82°5 152 
45:3 3223 | 105'5 | 1511 
47°5 3229 106:5 | Air admitted. 
49°4 3234 | 107:0 | 1514 
527D 3229 108°5 1516 
58:25 3245 | 110°0 1517 
62:0 3248 1160 1519 
73:0 3254 123°8 | 1520 
84°7 3258 1360 1521 
102°5 3262 154°3 1522 
138°0 3267 4 hours later. | 1524 
16 hours later. 3296 Next morning. 1526 
Bal AN ie) 4’, 3297 | 


* It is possible that this might be missed if the experimental method 
were such that the ageing was proceeding at a rapid rate at the moment 
when the gas was admitted. 


Resistance of a Sputtered Film after Deposition. 149 


The main point to notice in these results is the rapid 
initial rise in resistance followed by a more gradual rise 
which may be prolonged for a long period of time. Again, 
no relationship between the change in resistance and the 
thickness was obtained, the effect of the gas depending on 
the nature of the film. 

In film F, the pump was put on 2 minutes after the air 
had been admitted. The result was to produce a rapid fall 
in resistance changing to a more gradual fall. it is seen 
that the effect of the air is not immediately counteracted by 
putting on the pump, and that, as in the experiments of 
Kohlschiitter and Noll with nitrogen, the ageing continues in 
a film placed in vacuo after a treatment with air. 

It would have been interesting to have obtained sorption 
curves by measuring the change in pressure with time in 
order to compare them with the resistance-time curves ; but 
owing to the smallness of the amount of platinum in a film, it 
was quite impossible to dothis. Indirect evidence that there 
is an Intimate connexion between the two phenomena may, 
however, be given. Of the various experiments on the 
sorption of gases by solids, those of Bergter* on charcoal 
are of particular interest, as they were carried out under 
conditions of temperature and pressure comparable with the 
above. The curves that he obtained by plotting volumes of 
gas taken up with time are very similar to the resistance-time 
curves shown here. 

This connexion between sorption and change of resistance 
if real is interesting. A mere cessation of ageing could be 
attributed to a decrease in surface-tension accompanying the 
adsorption of gas. But the fact that a rise in resistance 
occurs, and moreover continues for a long time, suggests that 
the result of admitting gas into the discharge-vessel is not 
solely a surface-tension effect. It is more possible that some 
change in orientation of the individual units taking part in 
conduction is brought about by the air adsorbed on the film 
and absorbed in the molecular clusters themselves. In the 
present state of the theory of conduction in metals, further 
discussion of this point would be unprofitable. 


Admission of Hydrogen. 

The effect of adding hydrogen to platinum and palladium 
films is not in the general case a simple one. There may be 
(1) the normal rise in resistance following the admission 
of gas; (2) a fall in resistance due to the evolution of heat. 


* Ann. d, Phys. XXxvil, p. 472 (1912). , 


150 Miss E. W. Hobbs on the Change in the 


accompanying the sorption of hydrogen which involves a 
marked change in the nature of the film; and (3) a fall in 
resistance due to the combination with evolution of heat 
between the hydrogen and any traces of oxygen present in 
the film under the catalytic action of the finely divided metal. 
The relative values of these three effects may vary greatly 
with the condition of the film and with its previous history. 
Evidence of (2) is given in the ageing curves of films depo- 
sited in hydrogen, shown as dotted curves in figs. 1 and 3. 
It will be noticed that in the hydrogen curves, both for 
platinum and palladium, the initial portions are much steeper 
than those of the air curves, which often show a lag in the 
ageing rate at the beginning (e. g. curve C, fig. 1). 

Still greater changes have been observed in the short 
interval of time that elapsed before the pump was put on 
after the cessation of the discharge. For instance, in 
eurve H (fig. 1) the film was deposited in hydrogen at a 
pressure of *5 to ‘6 mm., and was left for about | minute 
at this pressure. During that time its resistance fell from 
something greater than 30, 000 to 9300 ohms. 

Striking examples of the effect wrought in a film by 
hydrogen are given in the curves of fig. 3. They refer to 
palladium films, aged zn vacuo, to which dry hydrogen was 
admitted, the point of admission being marked (h). In the 
case of curve Q, the film was previously supplied with air at 
the point (a) and the pump was put on at the point (p). 
Films P and Q were deposited in air; films T, W, and Y in 
hydrogen. In the films P and Q the initial rise in resistance 
at (h) may be taken to represent the direct effect of the gas. 
The subsequent rapid fall, which sometimes almost swamps 
the initial rise, is attr ibutable to the chan ges produced in the 
film by the heat evolution which accompanies the large 
sorption of hydrogen in palladium. Further complications 
may be introduced by the combination between hydrogen 
‘and traces of oxygen present inthe film. This effect in the 
‘case of the curves T, W, and Y, deposited in hydrogen, will 
‘be absent. Some hydrogen will have already been taken 
up by the films, and the consequent evolution of heat will 
have taken Peak so that on admission of hydrogen at the 
point (h) the ordinary rise in resistance due to the sorption 
of gas alone is shown. 

In most instances, when hydrogen is admitted to a 
palladium film, the percentage rise in resistance due to the 
direct effect of the gas is much greater than that produced 
by letting air into the discharge-vessel. This might be 


Resistance of a Sputtered Film after Deposition, 151 


expected from what is known of the SEL sorption of 
hydrogen and air in palladium. 

The latter stages in the examination of film P deserve 
notice. After exposure to hydrogen for 48 minutes, the 
film was placed in vacuo for 21 minutes and hydrogen was 
again added. The results are shown as P’ at a lower level. 
The second dose of hydrogen, it will be seen, only caused a 
small rise in resistance and no succeeding marked fall. It 
may be assumed that at this stage the film has been trans- 
formed into a state in which the sorption of hydrogen is 
only slight. The rate at which palladium takes up hydrogen 
is known to vary not only with the form of palladium 
employed, but also with different samples of the same 
Kind. 

The result of adding hydrogen to platinum films deposited 
in hydrogen is similar to that with palladium, and needs 
no further comment. The addition of hydrogen to platinum 
films deposited in air gave changes in resistance which varied 
In magnitude and in sign for different films. The following 
examples may be cited of films, deposited in air, to which 
hydrogen and air were added alternately, the pump being 
applied between the additions. 


(1) Film K. (Resistance 3400 ohms.) The first addition 
of hydrogen gave a small rise of 0:8 per cent. in 
14 minutes ; subsequent additions gave small falls 
from 2 per cent. to 0°5 per cent. in 10 minutes. Air 
always gave a rise in resistance. 

(2) Film L. (Resistance 7660 ohms.) Small rises, from 
2 per cent. to 0:2 per cent. in 25 minutes, followed 
every addition of hydrogen. The rise given by the 
first addition of hydrogen is shown in curve L (fig. 2). 
It will be seen that the change of resistance-time curve 
is similar to those for the addition of air. As before, 
a rise in resistance always followed the admission 
of air. 

(3) Film M. (Resistance 4900 ohms after having been 
subjected to several treatments of air but none of 
hydrogen.) The resistance fell to 1900 ohms in 
80 minutes on the addition of hydrogen. Subsequent 
additions of hydrogen and air both gave slight rises, 
in the former case followed by a fall. 

If it be assumed, as ssems probable, that the relative 
values of the direct effect of hydrogen on the resistance 
and the indirect effect due to evolution of heat in sorption 


152 Change in Resistance of Sputtered Film after Deposition. 


and in chemical action may vary widely with different films, 
the results given above are not necessarily conflicting. 

In no case examined, whether the film was deposited in 
air or hydrogen, has air produced a fall in resistance, though 
in many instances the film had been previously treated with 
hydrogen which had probably not been completely removed 
by the pump. It is clear, therefore, that in such cases, 
namely when air is in excess, the effect—if it be present at 
all—of the combination of hydrogen and oxygen in the film 
under the catalytic action of the platinum must be negligibly 
small. 

Kohlschiitter and Noll state that the catalytic power of a 
platinum deposit varies with the nature of the residual gas 
in the discharge-vessel during deposition. It is great if the 
gas be argon, when the deposit consists of very fine particles, 
and less if the gas be hydrogen, when a coarser film is 
obtained. 

It is noteworthy that the electrical properties of a film 
deposited by cathodic disintegration are not the same as 
those of a powder. Thus Goddard* has shown that the 
resistance of the powders he used decreasd when gas was 
admitted, and increased when the gas was pumped out. 
Some rough experiments of tke author on palladium-black 
gave qualitatively similar results. The above results show 
that with sputtered fiims the converse is true. 

McClelland and Dowlingy have observed some curious 
effects by applying a transverse electrical field to a thin 
layer of conducting powder. On removal of the field it was 
found that the powder had in general a greatly increased 
conductivity. There is no evidence that this occurs in the 
case of sputtered films. Thns in a few cases the author tried 
the effect of placing a plate parallel to and at a distance of 
3 or 4 mm. from the film and charging it to 1600 volts at 
atmospheric pressure. On removal of the field no change in 
conductivity was observed to have taken place. 


Summary. 


1. A study of the physical changes occurring in a metallic 
film after deposition by cathodic sputtering has been carried 
out by measurements of the changes with time in the electrical 
resistance of the film when retained in vacuo. 

2. The direct effect of admitting gas into the vessel cen- 
taining the film under observation has been shown to be a 

* Phys. Review, xxviii. p. 405 (1909). 
+ Proc. Royal Irish Acad. vs]. xxxii. A. No. 5 (1915). 


On a Semi-automatic High-Pressure Installation. 153: 


rise in. resistance which may continue for a considerable 
time. Evidence is given that ‘this phenomenon is intimately 
connected with the sorption of gas by the film. 


The above experiments were carried out in the Physical 
Laboratory of the University of Bristol, under the direction 
of Dr. A.M. Tyndall. I am greatly indebted to him for the 
help and encouragement that he has given me throughout the- 
work. My thanks are also due to Mr. J. D. Fry for his help 
in the earlier stages of the work, particularly in the con-- 
struction of the apparatus. 

The expenses of the research were defrayed by a grant 


from the University of Bristol Colston Society. 
April 20th, 1916. 


XU. A Semi-automatic High-Pressure Installation. 
By The Earl of Berkevey, £.R.S., and C. V. Burton, D.Sc.* 


— the primary object of the installation was to deter- 

mine the compressibilities of solutions, we add to the 
description of the pressure-apparatus itself details of two 
devices found useful in this connexion. 

Previous work} on the subject has shown us that when. 
altering the pressure, the alteration must take place slowly;, 
otherwise, the mercury rising in the capillary of the piezo- 
meter will leave behind it an appreciable quantity of the 
solution clinging to the walls, and thus vitiate the estimation. 
of the change of volume. The method by which this is 
overcome is given in some detail on p. 159. Again, when 
the pressure has been changed, itis essential that the thermo- 
dynamic heating or cooling brought about thereby should 
have time to dissipate before the level of the mercury 
meniscus is read. In other words, the new pressure must 
be maintained constant for a period of one to two hours ; an 
automatic device for doing this is shown in fig. 6. 

The general arrangement.—The working fluid is castor oilf, 
and the pressure is applied by means of a pump made bY 
Messrs. Schaffer and Budenberg. As this pump is of « 


pattern regularly made and catalogued by the makers, its 


* Communicated by the Authors. 

+ This research is not yet published. 

¢ Ordinary “cold pressed” castor oil becomes somewhat gelatinous 
under high pressures. If, however, it is cooled to 0° C. until no further 
precipitate comes out and then filtered, the oil will remain uncbanged 
for quite a long time. 


154 The Harl of Berkeley and Dr. Burton on a 


construction and working need not be referred to in detail. 
For raising the pressure rapidly to (say) 600 atmospheres, 
there is a low-pressure reciprocating plunger operated by a 
lever and working in conjunction with automatic suction 
and discharge-valves in the usual way. For the attainment 
of higher pressures, the low-pressure pump is then shut off 
by screwing down a needle-valve, and a high-pressure 
plunger of small section is gradually forced inwards by 
means of a screw provided with a star-handle. 

The arrangement of pump, tubing, gauges, and valves 
is shown diagrammatically in plan and elevation in fig. 1. 


Fig. 1. 
Elevation. 


To auxiliary 
reciprocating 
pump 


The gauges (G,, G,, G3) are of the ‘‘dead-weight” type, 
the essential feature being an accurately cylindrical piston 
of hardened steel fitting as closely as possible without actual 
rubbing contact in a mild steel cylinder, so that, whilst 
the combination forms an almost oil-tight joint, the mecha- 
nical friction opposing a lengthwise motion of the piston is 


Semo-automatic High-Pressure Installation. 155 


in any case small, and under working conditions is reduced 
to a negligible amount by keeping the piston in rotation. 

The pressure-ranges for which the three gauges are 
designed are— 


0 to 200 atmospheres for Gy, 
200 atmos to 800 . Go, 
SU0K.5, to 1500 EP G3. 


The corresponding piston-areas being approximately— 
0°10 sq. in. for Gy, 0:02 sq. in. for Gp, 
and 0°01 sq. in. for Gs. 


The needle-valves V,, V2 (fig. 1) are used for shutting off 
the gauges G,, G. when these are not in use. No such 
valve is required for the high-pressure gauge G3, which 
carries a sufficient load to keep it from lifting when the 
pressure is low enough to be measured by G, or Gp. 

Details of gauges.—A piston intended for high-pressure 
measurements must have a small sectional area if the weights 
are to be kept within manageable limits, and it must be very 
closely fitted. The oil which supports the rotating piston 
must consequently be under a considerable pressure before 
the lubrication is free enough to render the arrangement 
sufficiently frictionless. It is for this reason that any 
single gauge is only serviceable over a restricted range 
of pressure. 

The gauge Gp, used for intermediate pressures, is shown in 
more detail in figs. 2 and 3. The upper end of the piston* 
has the form of a spherical cap, upon which rests the 
hardened steel pin P spherically hollowed to a somewhat 
greater radius. This pin is inserted in, and thus supports, 
the tubular weight-carrier C. To the lower end of C an 
annular plate Q (fig. 3) is pivottally connected, with freedom 
to tilt about a horizontal axis A A, while two pins B project 
downwards from Q and engage in radial slots in the 
worm-wheel V. The worm-wheel is kept in rotation, and the 
weight-carrier and weights must necessarily partake of this 
rotation, the transmission of the motion being such that the 
forces applied to the carrier are sensibly equivalent to a 


* The pistons and cylinders were made for us by Messrs. Elliott, and 
are copies of those used at the National Physical Laboratory. The 
designs were given us by Mr. Jakeman of that laboratory, to whom our 
est thanks are due. 


156 The Earl of Berkeley and Dr. Burton on a 


couple. The rotation of the weight-carrier is communicated 
to the piston in much the same way, except that the equality 
between the constituents of the driving couple depends, in 


o 


\ 
See 
(ZZ A 
ANTAL 
i ' \ 
SS SP A WE EWE UW UU WG | c SON 
1 ‘ 


L 


Lee 
S 


\ 


X 
WEN 
WY 


[ 
\ 


SN 
\ 


| 
NEN 
\ 
ae 


A 
Rd 


— 


D Seen / 


this case, on the approximately equal flexibility of two flat. 
springs F F attached to the piston-head, and engaging with 
pins H H as indicated in the upper part of fig. 2. 

Description of joints—The various tees, elbows, and ex- 
tensions (there are some 40 joints in all) used in this 


Semi-automatic High-Pressure Installation. ToT 


apparatus are made tight in a simple way which we have 
used extensively for some years, and have found to be 
entirely satisfactory. For example, A and B (fig. 4) are 
two tube-ends which have to be 
Fig. 4, joined together. One of them, A, 
is recessed, and the recess is filled 
by a ring of dermatine* C, which 
before the joint is assembled stands 
alittle proud of the tube. Itis now 
only necessary to bring the tubes 
firmly together, and a fluid-tight 
joint results. 

In the figure the tube-ends are 
held together by an ordinary union, 
but the same form of joint is equally 
serviceable when, instead of this, 
the tubes are provided with flanges 
which are bolted together ; and it 
is equally applicable to large and 
to small bores. It can, of course, 
be used where one of the pieces to 
be coupled has a bore smaller than 
the other or is a blank. ‘ Red” 
dermatine is rather harder than 
“black,” and is correspondingly 
easier to machine ; it stands oil or 
water. 

The advantages of this joint 
are:—It is fluid-tight from the 
outset, before any pressure has been applied, and remains 
tight up to any pressure which the couplings are strong 
enough to stand. It can be taken apart and remade ft 
a great number of times without renewing the dermatine 
ring, and is extremely simple to make, requiring only 
ordinary workshop accuracy. On the other hand, where two 
pieces have to be coupled fluid-tight and in accurate relation 
to one another, this result is readily secured, as the pieces 
are in direct contact. If under working conditions the tubes 
or couplings are subjected to mechanical strain, the joint 
itself is unaffected. 


* Gutta-percha also makes an excellent temporary joint, and is to 
he preferred to dermatine should it be necessary to use very thin (in the 
radial direction) rings. It is not acted upon by castor oil. 

+ This is essential, as the apparatus must be cleaned out at intervals, 
because the castor oil gets contaminated by small metallic particles 
(possibly scale), and also appears to have a slow action on the steel. 


158 The Earl of Berkeley and Dr. Burton on a 


The semi-automatic valve —This is shown in fig. 1 at S, and 
in fig.5in section. It isused to shut off the pump from the 


Fig. 5. 


= 
= — 


Vi 


(REBT CRO RIEL ees rar 
Daa ta [hii 


‘Bi 


Wan 
— SS 
Sa) 
Yo Dead weight 


rest of the pressure-system when the plunger has been pushed 
in to its limit, and has therefore to be run out again so as to 
draw a fresh lot of oil into the pump. This valve must 
accordingly be efficient up to the highest pressures used 
(1500 atmospheres), and we have not found needle-valves of 
the ordinary type satisfactory. The connexions to the pump: 
and to the pressure-tubes are as indicated; and in the view 
shown there is free communication between them because 
the valve W is withdrawn from its seat. The valve-face is 
formed by a ring of dermatine, sectioned in black. The 
slow leakage of oil which would otherwise take place around 
the unpacked valve-stem S is prevented by the auxiliarv 
needle-valve N. It is only while the valve is being altered 


Semi-automatic High-Pressure Installation.’ 1059 


from the open to the closed position that any such leakage 
is possible. To close the valve, the needle N is opened and 
the valve-stem S is thus free to rise and seat itself securely, 
the light spring R becoming compressed at the same time. 
It will be noticed that the dermatine ring is so arranged 
that it becomes automatically tighter under the pressure 
which it has to hold. The screw plunger-pump can now be 
refilled with oil, and, after closing N, the valve will auto- 
matically reopen when the pump-pressure has been brought 
toa value a little in excess of that in the remainder of the 
pressure-system. 

Method of altering the presswre-—As already stated, in 
much of the work for which this installation is used it is 
necessary to avoid rapid changes, and especially rapid in- 
creases of pressure. For this reason, when a weight has to be 
added to the load carried by the rotating piston of the gauge 
(Gy, G., G3), means must be provided for allowing the extra 
weight to come on gradually. The plan adopted is to 
suspend the weight from three triggers on a cage carried 
upon a ball-bearing at the end of a fairly extensible spring.. 
The suspended system is lowered by the slow unwinding of 
the cord to which it is attached; and as the unwinding 
proceeds, the extra load is placed gently in position and its 
weight gradually transferred from the cage to the pile of 
weights below it, the spring shortening meanwhile. As 
soon as the cage has been relieved of the whole of the 
weight, the three triggers withdraw automatically, leaving 
the cage clear for lifting. The gradual removal of weight 
is effected by the same mechanism ; but in this case the 
triggers are differently set, so that they are adapted for 
picking up instead of for releasing. 

To obtain an increment of pressure less than that due to a 
weight. unit, a light container is placed on the top of the 
weights, and above it is suspended a tin funnel containing 
the requisite quantity of shot. On opening the orifice of 
the funnel the shot runs into tne container at a rate which 
depends (within limits) only on the area of the orifice and 
not on the head of shot. 

Automatic compensation for leak.—It is sometimes necessary 
to maintain a pressure sensibly constant for a considerable 
time. To this end the head of the pump-screw is fitted witha 
ratchet-clutch, Q, fig. 1 (the “free-wheel” clutch of a bicycle),. 
to one member of which is attached a horizontal lever L. 
It happened to.be most convenient to arrange this lever to 
the right hand of the pump, which meant that the down-stroke- 
of the lever would be the operative one. As it was desired 


160 The Harl of Berkeley and Dr. Burton on a 


-to reciprocate the lever by means of a wire rope, a sufficient 
weight was hung on to it, and the rope was carried up to a 
ball-bearing crank-pin, set in the face of a worm-wheel. The 
worm is driven through a magnetic clutch, by means of 
-which the automatic control of the pumping is effected 
whichever of the three gauges G,, Gy, Gs is in use; the fall 
of the gauge-piston below a certain level closes the circuit of 
‘the electromagnet and causes the pump to work ; the rise of 
the piston above a slightly higher level breaks the circuit 
.and stops the pumping. 


\e Ni 
E-Qp-£ 
SECTION THROUGH A.B 


PLAN 


The (continuously-driven) pulley on the worm-spindle 
yprovides a choice of speeds, the aim being to arrange 
-matters so that even with the greatest expected leak only 
intermittent pumping will be needed to keep up the pressure. 
_A detail of the contact-maker is shown in fig. 6. 


Semi-automatic High-Pressure Installation. 161 


The roller R is journalled in one end of the lever L, which 
is so loaded that R always rests in light contact with the 
undermost of the pile of weights. When the weights fall 
the lever L is tilted clockwise and afew cubic centimetres of 
mercury contained in the glass tube T attached to L run 
down to the right-hand end of the tube and make contact 
between two iron electrodes EK. The tube T is exhausted 
to a Fleuss vacuum and hermetically sealed, the electrodes 
being closely fitted through ground conical tubulures, and 
made tight with sealing-wax. | 

It may be worth while noting that this type of vacuum 
contact-maker is capable of carrying currents up to some 
5 to 7 amperes at 100 volts D.C. without sustaining injury. 

Sensitiveness of the gauges.—It is well known that in this 
form of gauge the sensitiveness depends largely on the rate 
at which the pistons are rotated. Several experiments were 
made to find out the most suitable rate, 2. e. the slowest rate 
compatible with sufficient sensitiveness. The outcome of 
these being that G3 should revolve once in 6 seconds, Gz once 
in 26 seconds, and G, once in 45 seconds. At these rates it 
was found that with the pistons of both G, and G; floating 
undera pressure of 800 atmospheres, but with the latter 
slowly falling while the former is slowly rising, an addition 
of 20 grams to G, (equivalent to a change of pressure of 
0'17 atmosphere) the movements would reverse, and G, 
would fall while G; rose. 

Pressure release-valve.— Whilst working with this in- 
stallation it was found an advantage to be able to release 
the pressure somewhat rapidly, but at the same time to do 
so without jerks, An ordinary needle-valve is not suitable 
because the small opening soon gets clogged by any solid 
particles there may be in the oil (or even any material of 
greater viscosity?), and on enlarging the opening the 
obstruction flows away, and a rapid variation ensues. 

The following device successfully avoided the trouble, 
especially at the higher pressures, where it is most needed. 

A cylinder, closed at one end by a needle-valve, was bored 
out to ¢ inch and tapped 26 threads to the inch. A plug 
carrying the same thread was loosely fitted into the cylinder 
and the whole joined on to the pressure-system. On opening 
the needle-valve, the plug travels forward until there is a 
firm contact between the adjacent thread surfaces, thus 
forming a narrow channel through which the oil can flow. 
The length of the channel (some 50 inches) acts asa resistance 
to the flow, but its depth is such as to avoid clogging. 


Phil. Mag. §. 6. Vol. 32. No. 187. July 1916. M 


XIII. On the Mechanical Relations of Dielectric and 
Magnetic Polarization. By G. H. Livens*. 


il HERE still appears to be some uncertainty as to 

the exact expression for the mechanical forcive of 
electric or magnetic origin acting on the elements of a 
polarizable medium in an exciting field of force, and as to. 
the consequent reduction of this forcive to its representation 
by means of an applied stress system. 

The original procedure of Maxwell t, based, for the case of 
magnetic media, on the most elementary physical ideas 
respecting the nature of the polarization involved, leads to a 
definite expression for the forcive on any medium which is 
independent of the law, or even of the existence of a law, for 
the induction of the polarization by the exciting field, and 
which should therefore hold in the case of all substances in 
which the polarization is of the nature of that described, even 
if this polarization involves hysteretic qualities. This, the 
original, procedure has been elaborated and extended in the 
more recent developments of the theory of electrons, and 
the general validity of the result obtained is thereby fully 
confirmed and substantiated both for dielectric and magnetic 
media. 

A more general mode of discussion based on the method of 
energy, but avoiding molecular theory, has been originated by 
Korteweg {, formulated in general terms by von Helmholtz § 
and further developed by Lorberg, Kirchhoff ||, Hertz, 
Cohn **, and others. The expression for the forcive on the 
polarized medium obtained by this method is, however, in-- 
consistent with that obtained by Maxwell on his simpler 
form of the theory. The discrepancy is regarded by some 
authors as due to incompleteness in Maxwell’s formulation, 
but Larmorf{ has shown that, at least in the simplest case of 
isotropic media, it is due mainly to fundamental errors both 
in the physical assumptions on which the energy method is 
based and also in the analytical processes by which that 
method is developed. This criticism appears, however, to- 
have been entirely overlooked and the energy method is still 

* Communicated by the Author. 

+ Treatise, vol. ii. §§ 641, 642. 

t Wied. Ann. ix. (1880). 

§ Wied. Ann, xiil. (1882); Abhandlungen, i. p. 798. 

|| Wied. Ann. xxiv., xxv. (1885) ; Abhandlungen, ‘ Nachtrag,’ p. 91. 

@ Wied. Ann. xli. (1890); ‘Electric Waves’ (English Edition), 
pp. 259-268. | 

** ‘Das electromagnetische Feld, Ch. viii. 

tt Phil. Trans. A. exc. p. 280 (1897). 


Relations of Dielectric and Magnetic Polarization. 163 


~ very generally reproduced and it has subsequently been 
further developed and extended by Cohn”, Gans t, Sano tf, 
and others. 

It seems therefore desirable to examine in closer detail the 
criticism of Larmor, applying it to the more general case 
when the law of induction is no longer linear or isotropic, 
but with the view rather to emphasising the underlying 
physical principles than of adding anything original to the 
criticism itself. To aid in the examination a short account 
of Maxwell’s theory based on a correct interpretation of the 
energy method is added, and it is thus hoped to establish on 
a firmer footing the only consistent specification of the 
problem which has so far been put forward. 

The discussion is confined entirely to the case of dielectric 
media, and the methods and notations of the vectorial calculus 
will be employed throughout. 

2. According to the theory of Maxwell as elaborated and 
extended in the theory of electrons, a body polarized to 
intensity P in a field of force of intensity E requires on 
account of the polarization induced in it an amount of 
energy per unit volume at any place expressed by 


which together with the energy of the free charges makes 
up a total equivalent to a distribution throughout the whole 


field of density i at each place. 


This energy does not, however, arise as a result simply of 
the action of mechanical forces applied to the media as a 
whole, and is not therefore compensated by a reduction of 
the available mechanical energy associated with these forces ; 
for part of it has arisen from the store of internal elastic or 
thermal energy in the medium, which may for the purposes 
of the present argument be regarded as of effectively non- 
electric nature. ‘he separation of these two fundamentally 
different parts is effected by the usual method, which consists 
in imparting a small virtual variation to the general con- 
figuration, both geometrical and electrical, of the medium 
and calculating the work in each part separately. In this 
way it is easily seen that the part 


W,=— {, " (PdB) 
0 


of the total electric energy per unit volume is associated 


* L.c. p. 512. t Ann. Phys. xiii. p. 634. 
{ Phys. Zeitschr. iii. p. 401 (1902). See also the articles by Pockels 
and Gans in the Encyclopddie der mathematischen Wissenschaften, Bd. v. 


ad 


164 Mr. G. H. Livens on the Mechanical 


mainly with the mechanical forces on the medium as a 
whole, for it is the part of the total that remains when the 
internal configuration determined by the polarization is main- 
tained constant during the establishment of the system. 

The function W,, mav now be regarded as the potential 
function of the required mechanical forces. The expression 
for the forcive on the element on account of the polarization 
in it then follows as a matter of course, and its linear 
constituents are represented per unit volume by the com- 
ponents of the vector | 


— grad W,,= grad (PB), 
the differential operations not, however, affecting P; the 


angular components are similarly determined and are 
represented by the vector 


[PE 
When a potential of force exists the former vector is 


equivalent to 
(PV )E, 


wherein YV is the usual Hamiltonian vector operator. Thus 
in the case of isotropic media, when the induction of the 
polarization follows a linear law so that 


Aqr 
the forcive per unit volume is completely specified by the 
vector 


AS? grad ie 
The additional forcive which acts on the medium on 
account of the distribution of free charge of density p 


throughout it is represented by a force on the element of 
yolume at any point whose intensity per unit volume is 


pi, 
and this, combined with the above forcive arising on account 


of the polarization, is completely represented in the most 


general possible case by an applied stress system whose nine 
components are of the types 


ete” T=), 1, Eoe 


wherein 
Deans Pp 
Arr 


is the total displacement vector of Maxwell’s theory. 


Relations of Dielectric and Magnetic Polarization. 165 


The remaining part of the total electric energy of the 
polarizations, viz. 


—(" aa), 


represents the energy supplied from the total store of internal 
energy of other than electric type in the medium during the 
process of setting up the polarization. This expression with 
sign changed thus represents the increase in the energy of 
non-electric nature in the dielectric medium consequent upon 
the induction of its polarized condition. 

3. In the theory of von Helmholtz the whole argument is 
placed on a different footing. No distinction is now drawn 
between the fundamental constituents, sether and polarizable 
matter, of the dielectric field, which is regarded as consisting 
of a single uniform medium capable of transmitting the 
electric actions in the same manner as an ordinary elastic 
solid transmits mechanical forces ; the electric force at any 
point of the field is the straining force and the total electric 
displacement in Maxwell’s sense represents the strain 
produced. The displacement D is subject to the usual 
characteristic equation of the theory, viz. 


div D=p, 


and the electric force exciting the displacement is derived 
from a potential. 

The first step in the formulation of the theory is to express 
the energy per unit volume in terms of the field vectors and 
the potential in such a form that the variation of the integral 
which represents the energy for the whole volume leads on 
integration by parts to the characteristic equation for the 
displacement as one of the conditions of internal equilibrium ; 
the integral is then asserted to be in its normal form, which 
means that it represents the actual distribution of the energy 
of the medium as well as its total amount. Its variation 
with sign changed, owing to change of material configuration, 
should then give the extraneous forcive that must be applied 
in order to maintain mechanical equilibrium ; the variation 
with respect to the electrical configuration being null, so 
that electric equilibrium is provided for by the characteristic 
equation already satisfied. The variation without change of 
sign should thus give the mechanical forcive of electric 
origin that acts on the medium. 

The organized energy in the medium is known to be 


D 
we { ee { (EdD), 
0 
where div D=p amc: B= —grad d; 


166 «© Mr. G. H. Livens on the Mechanical 


so that the forces acting will be derived from the variation 
of W; the variation with respect to ¢@ leads to the electric 
forces, and that with respect to the material configuration 
leads to the mechanical ones. The problem is to determine 
the mechanical forces when there is electric equilibrium, that 
is, when the variation with respect to ¢ yields a null result. 
The form of W above expressed does not lead to this null 


result. We can, however, by integration by parts derive 
the form 


w=| dv bdo, 
0 


the essence of this transformation being that in the new 
integral the distribution of the energy among the elements 
of volume dv of the medium has been altered. This form 
does not satisfy the above requirements either, but by 
combining the two forms we obtain 


w= {ar [os— |" (bax ], 


whose variation with respect to @ is null as required ; 
although, as integration by parts is employed, the variation 
is not nuil for each single element of mass. This integral is 
then taken to represent the actual distribution of the 
organized energy in the medium when in electric equilibrium 
and not merely its total amount, and variation of it with 
respect to the material configuration should thus give the 
actual bodily distribution of mechanical forcive, not merely 
its statical resultant on the hypothesis that the system is 
absolutely rigid. Now, in finding the variation of W arising 
from a virtual displacement ds of the polarized material, we 
have to respect the condition that the free charge pdv is 
merely displaced, so that by the equation of continuity 


Spt (V, pds) =0, 


and also that each element of the material is moved on with 
its own elastic constants, so that if \/’ denotes the vector of 
space variation taken without reference to the variation of H, 


6D+ (és, V')D=0, 


while things have been arranged so that a variation of } 
produces no result,—but only, however, no aggregate result 
on integration by parts. ‘The transitions at interfaces are 
supposed to be gradual, so that the volume integrals can all 
be extended over ‘he entire field, without the necessity for 


Relations of Dielectric and Magnetic Polarization. 167 


the introduction of surface integrals. We have therefore 
ray — ba 2) 
sw= (de| p—| (sD. 4B) | 
e/ e 0 


> E 
=— lacy, pos)dv + ( af ((8s, \7')D, dB). 
Now R : 


B °E 
{ * ((85, V')D, dB)=—(D, (85, V)B) + (6V) | (DaB), 
e090 « 0 
so that on integrating the first integral by parts, we get 
H 
ow= { adv | (as) o— — (D(6sV/) £) + (6sV) { (DaB) | y 
0 


The vector coefficient of 6s with sign changed is now taken 
to determine the linear components of the mechanical forcive 
on the ae medium ; its v-component for example is 


— pt +(D$ =) Oo. (ban). 


This expression is far more complex than that given by 
Maxwell's theory, and it leads to a stress whose specification 
differs from that of Maxwell in its leading terms, which are 
now of the type 


E 
esi Ie ( (Dd). 
ew 0 


These general formulze were first given by Cohn for the 
particular case of isotropic magnetic media for which a 
general law of induction is valid; they are here seen to be 
in no way limited by these assumptions. In the most 
general case, however, their form alone suggests an obvious 
difficulty. The type of forces which are repr “esented in them 
are much more general than is the case in all known 
mechanical systems, in so far as the force on any element of 
the medium is a function not only of the conditions in that 
element at the instant when it is under examination, but 
also of the whole of its past history. In any ordinary 
physical theory the existence of such forces is entirely 
excluded as it would, for instance, result in a force on an 
unpolarized element of matter formed by the combination of 
two opposite but equal polarized elements with different past 
histories. 

We need not, however, dwell upon the difficulties of this 
kind inherent in the present form of the theory, as it will 
soon be seen that the deduction given is entirely fallacious. 


168 Mr. G. H. Livens on the Mechanical 


4. On close inspection it will be seen that the analysis of 
the preceding paragraph determines in reality something 
quite different from that which is the ultimate object of 
search. In calculating the mechanical forcive on the element 
of the dielectric medium by the variational method, care 
must be exercised to determine the change in the energy of 
the moving element of the medium, because it is this change 
which is brought about by the action of the mechanical 
forces when the constitution is maintained constant. But in 
the argument of the preceding paragraph, the work done by 
the bodily forces acting on the material element during its 
displacement is equated to the change in the energy in the 
element of volume originally occupied by the element before 
its displacement, which is quite a different thing from the 
change in the energy of the moving element itself. The 
analysis is therefore fundamentally unsound and will require 
considerable modification. To obtain its proper legitimate 
form we must confine our attention to a finite portion of the 
dielectric medium and follow it in its motion. We enclose 
the portion by a surface 7 on its outer boundary and then 
notice, with Larmor, that in the displacement a space is left 
unoccupied on the one side of the surface and a new space 
is occupied on the other; the result is that the above 
expression for variation of W, or at least that part of it 
referring to the portion of the dielectric medium inside /, is. 
now represented by 


E 
SW = \dv | $V, pos) —(D, (osV )h) == (dsV) | (Ddk) | 


0 
if (a[p9— { , (Dd) | S8n, 
0 


wherein the volume integral is taken throughout the portion 
of the medium enclosed by the surface f and the surface 
integral over its outer boundary or / itself; ds, represents - 
the outward normal component on / of the virtual displace- 
ment given to the medium. The volume integral of the first 
term in the first integral now transforms by integration by 
parts into the integral 


\p(ds7) ¢, 


together with a surface integral over f equal and opposite in - 
sign to the first part of the surface integral already present 
in OW ; and the integral of the last term in the first integral 
is equal but opposite in sign to the remaining part of this 


Relations of Dielectric and Magnetic Polarization. 169: 


same surface integral: thus finally we get 
SW = J dv[p(5sV)d— (D, (8sV)E)] 


as the expression for the variation of the energy in the 
specified portion of the medium followed in its motion. As. 
the result now applies for any arbitrary surface /, the inte- 
grand represents quite properly the distribution of the 
variational energy throughout the material dielectric ; the 
coefficient of 6s in it therefore gives properly the linear 
constituent of the forcive on the element; its « component 
per unit volume is 


y 
pee i (DS) = ple (x8, 
provided a potential of force exists. This type of forcive is,, 
however, quite an impossible one, as it points to the existence 
ef a bodily forcive of amount 


grad 3) 
on the elements of free sether, which could not therefore be- 
in equilibrium. 

The final form of this result deduced by an elaboration of 
Larmor’s argument can be verified by a simple direct: 
calculation if it is noticed that it is in reality the variation 
of W with respect to the electrical potential, when the 
variation of this is brought about by a virtual displacement 
of the matter, that is really required ; for it is the work of 
the mechanical forces balancing the electrical attractions 
that is to be determined; and if the elastic conditions of the- 
dielectric medium specified completely by the vector D 
remain unaltered in any displacement, we may be sure that 
ail the work done is used up as purely electrical energy of 
configuration, none of it having then been absorbed by the 
medium into its store of internal energy of effectively non- 
electric nature. In fact, if we perform the variation in this. 
way with D, and consequently also p, constant we get 


SW = \dv[p(dsV) $= (D(SsV)E)], 


exactly as above. 

This simple mode of treatment also emphasises what it is 
that really determines the internal electrical coordinates of 
the medium. Helmholtz asserts that they, or at least the 
conditions in them, are determined by the potential distri- 
bution, and for the simple case examined by himself and 


170 Mr. G. H. Livens on the Mechanical 


Larmor, where the medium is isotropic and the law of 
inducticn linear, this is probably true; but the present 
argument shows that it is the electrical displacement in 
Maxwell’s sense that more properly defines degrees of 
electrical freedom in the dielectric medium. 

The fact that in the aggregate for the whole field the 
above expression for 6W vanishes, which Helmholtz is at 
such pains to secure and which results from the characteristic 
equation for D, viz. 


div D=p, 


merely verifies that there is on the whole no resultant force 
on the infinitely extended system. 

The second term in the expression for the forcive just 
deduced is exactly analogous to the corresponding term 
depending on the polarization of the elementary theory of 
Maxwell. It is only to be noticed that the whole cireum- 
stances summed up in the vector D are subject to the virtual 
displacement of the matter, whereas in Maxwell’s theory the 


i a) . ral . e.e ee 
part 1,4 of this vector defining conditions in the ether 


does not partake of motions of the matter, the remainder 


i =p-¢ being the only portion that so moves. Herein 


lies the second cause of discrepancy between the two theories, 
and it would appear that the method described by von Helm- 
holtz must be radically unsound: it would be valid if there 
were only one medium under consideration, of which W 
is the energy function ; but there is here in the same space 
the ether with its stress and the polarized matter with 
its reacting mechanical forces, and there is no means of 
disentangling from a single energy function in this way the 
portions of energy which are associated with these different 
effects. 

To sum up, we may say that the theory of electric stress 
formulated by Helmholtz, Hertz, Cohn, and others is funda- 
mentally at fault. In its correct analytical form it involves 
physical assumptions which have been long regarded as 
wholly untenable, besides leading to an impossible type of 
dielectric body forcive. We are therefore virtually thrown 
back to Maxwell’s original and simple theory, which is the 
only one that has really proved to be consistent with all the 
facts, among which it must not be forgotten to include those 
fundamental ones concerning the electrodynamic properties 


Relations of Dielectric and Magnetic Polarization. 171 


of moving dielectric media, which necessitated in the first 
instance the introduction of the electron theory in its main 
aspects. 

5. Before concluding this discussion reference must 
perhaps be made to a small discrepancy of a different kind, 
which occurs in the form of the theory given by Pockels in 
his article on ‘ Magneto- and Hlektro-striction”’ in the 
Encyklopidie der mathematischen Wissenschaften. It will 
be observed that the stress system obtained on Maxwell’s 
theory is not in general self-conjugate, as must necessarily 
be the case with all stresses in material media without polar 
molecules, in order that the energy principle may be verified. 
Now Pockels obtains by von Helmholtz’s method a self- 
conjugate stress in which the corresponding cross-terms are 
each half the sum of the corresponding terms in Maxwell’s 
stress ; he secures this by including the effect of a general 
virtual displacement possessing rotational as well as trans- 
lational qualities. He attempts to assign the discrepancy 
in Maxwell’s determination to a neglect of the rotational 
part of the displacement, and supports his contention by a 
reference to Hertz’s interpretation of the energy method of 
von Helmholtz, as given in Lorentz’s article on “ Maxwell’s 
Theory” in the same work. It appears, however, that 
Pockels criticism really redounds on himself, inasmuch as he 
has neglected to include in the hypothetical expression of 
the virtual work the effect of any possible applied couples 
such as are necessitated by Maxwell’stheory. His reference 
to Hertz’s work in no way helps the matter for, as is clearly 
recognized by both Hertz and Lorentz, this methed only 
determines the sum of corresponding cross-terms of the stress 
matrix and not each of them independently. Besides, the 
difficulties encountered above and involved in the method of 
application of the integration by parts also occurs in Pockels’ 
work, and in such a way as to obscure what in reality comes 
to a mutual cancelling of the two additional parts of the 
variation, one of which alone is included by Pockels, arising 
on account of the rotational terms. 


The University, Sheffield, 
April 15th, 1916. 


Lele 


XIV. Strength of the Thin-plate Beam, held at its ends and 
subject to a uniformly distributed Load (Special. Case). 
‘By B.C. LAWS; Bsc syA hase Al Cee 


| a recent communication + the author gave the solution 

of a special case of the thin-plate beam, in which the 
uniformly distributed load was taken as acting downwards, 
2. €. so as to press the beam against the broad supports. 

As therein explained the case is met with in those struc- 
tures as e.g. ships, and floating docks, built to withstand 
hydrostatic pressure, where the shell plating is riveted to. 
and supported at frames or girders forming the foundation 
of the structure ; and if we take a section of the plating in 
the direction perpendicular to the frames we obtain the 
profile of the beam forming the subject of discussion. 

In the present case the problem is considered wherein the 
load acts in the reverse direction; the beam is not now 
supported directly by the frames but by the rivets which 
secure it to the latter. 

The appearance of the beam between consecutive frames 
is of the nature indicated in fig. 1,in which F, F, denote 
the frames, and 7, 7, the lines of rivets. 


Fig. 1. 


Plate beam with a uniformly distributed load w per square inch. 


Such cases are commonly met with in practice, as e. g. in 
the plating of the inner bottom and bulkheads of floating 
structures, and give rise to stresses in the material different 
from those obtained in the previous case. 

As before, consider a strip of plating of unit width and 
thickness ¢, constituting a beam of length r7;=2a, and 


* Communicated by the Author. 
t Phil. Mag. vol. xxxi. April 1916. 


Mr. B. C. Laws on Strength of the Thin-plate Beam. 173 


subject to a uniformly distributed load w per unit length 
due to a head of water h. 

Take the axis-of X at the mid-surface of the plate, 7. e. in 
a plane distant t/2 from the faces of the supporting frames, 
the origin O midway between the supports, and the axis of 
Y downwards. 

The diagrammatic representation of the beam is shown in 


fig. 2. 


Fig. 2. 


Distribution of forces acting on beam subject to load w 
per square inch. 


The solution is worked out on lines similar to those indi- 
cated in the previous paper, and we are led to the fundamental 
equation :— 

v(@?—2?) | 


Ny eae 5) 


B.1. 2! 


where P denotes the pull along the beam. 
Put P= im? Hy. Y. 


The solution to this equation is i-- 


Q i NUL = Wego 
BT. a3 4 iti aah (a) 


d Lae 2 Ma —mM.a 
Xv 1K é _ 
0 €o 


giving the bending moment at any point of the beam, where 
é9 is the base of the natural or Naperian logarithms. 
The deflexion y at any point of the beam is given by :— 


; M.a —M.a\ = ( pm. —m. ae 
Pomme 2 Perr eo Me oe), ola? at) 
gle = pMedCrert2) 


Ma — mM. 
é mn M2. 2 


0 
eo eye e) 
The greatest bending moment occurs at d and is obtained 
by putting «=a in equation (a); and the maximum deflexion 
yo.is obtained by putting «=0 in equation (0). 


174) =Mr.B.C. Laws on Strength of the Thin-plate Beam. 
Now the elongation of the beam 


— ———————— i ts——> 


or . 
Mi. ek 
yy = Neat ° e ° . e e (c) 
2/2 
Fig, 8 

Q 

20 
mule 
2 
eS 
z 
NS 
£10 
Y 
\ 
N 
) 
S85 
N 
x 
t 
S 

re) 


TWICKNESS OF PLATE (tN INCHES. 


e o 


By assuming a value of m,¢ may be obtained by com- 
bining equations () and (c), thence by the aid of (a) and 


Geological Society. 175 


the relation P=m?E.I the stresses due to bending and 
stretching. 

Again, taking the case where a=11 ins., w=:007 ton per 
inch run, H=13500, and m=°1, we obtain :— 


i =ira0oun. 
Max. stress due to bending = 16°86 tons/in.? 
is » stretching = 1:05 ye 
Maximum tensile stress =a ieout 


99 


By varying the value of m other values of ¢ and stress. 
may be obtained, from which data a curve showing the 
‘“‘ variation of stress with plate thickness” may be constructed 
as indicated by B in the diagram, fig. 3. 

The curve A—drawn for comparison—in the same diagram 
is reproduced from the previous paper. 

Taking an allowed stress of 10 tons/in.’, the corresponding 
thickness of plating is -418 in. as compared with °376 in. in 
the previous case. 


XV. Proceedings of Learned Societies. 


GEOLOGICAL SOCIETY. 


[Continued from vol. xxxi. p. 572.] 


April 5th, 1916.—Dr. Alfred Harker, F.R.S., President, 
in the Chair. 


TIXHE following communication was read :—— 


‘The Picrite-Teschenite Sill of Lugar (Ayrshire) and its Differ-. 
entiation.’ By George Walter Tyrrell, A.R.C.Sc., F.G.S. 


This sill occurs near the village of Lugar in East Central Ayr- 
shire, and is magnificently exposed in the gorges of the Bellow and. 
Glenmuir Waters, just above the confluence of these streams to form 
the Lugar Water. It has a thickness estimated at 140 feet, and 
is intrusive into sandstones of the ‘ Millstone Grit.’ The contacts: 
consist of a curiously-streaked and contorted basaltic rock, passing 
at both margins into teschenite. The upper teschenite, however, 
becomes richer in analcite downwards, and ends abruptly at a sharp. 
junction with fine-grained theralite. The lower teschenite becomes. 
somewhat richer in olivine upwards, but passes rapidly into horn- 
blende-peridotite. The central unit of the sill isa graded mass. 
beginning with theralite at the top and passing gradually into. 


176 Geological Society. 


‘picrite, and finally peridotite, by gradual enrichment in olivine and 
elimination of felspar, nepheline, and analcite. 

The field detail of the Bellow, Glenmuir, and other sections is 
given in Part 2 of the paper; and the petrographic detail, with 
‘several chemical analyses, in Part 3. A unique rock, named 
lugarite in 1912, with 50 per cent. of analcite and nepheline, 
occurs as an intrusion into the heart of the ultrabasic mass of the 
sill. Part 4 deals with the special significance of this sill in 
petrogenetic theory. The mineral and chemical variations are 
described and illustrated by diagrams. It is shown that the 
average rock of the sill, obtained by weighing the analyses of 
the various components according to their bulk, is much more 
basic than the rock now forming the contacts. Hence, assuming 
that the sill is a unit and represents a single act of intrusion, the 
main differentiation cannot have occurred zm situ. Other special 
features of the sill are the identity and banding of the contact- 
rocks, its asymmetry, the density-stratification of the central ultra- 
basic mass, and the sharp junction between the upper teschenite 
-and the underlying theralite. 

The theory is advanced that the differentiation units were pro- 
duced by the process of liquation, but that their arrangement within 
‘the sill took place under the influence of gravity. ‘There are sharp 
‘interior junctions between a unit consisting mainly of calcic ferro- 
magnesian silicates, and a unit consisting mainly of alkali-alumina 
silicates with water, the former giving rise to the central ultrabasic 
stratum, and the latter to the teschenites. These partly immis- 
-cible fractions arranged themselves according to density. Then 
within the central ultrabasic stratum there was a_ subsidiary 
_gravity-stratification—due to the subsidence of olivine-crystals, 
giving rise to the graded mass described above. If differentiation 
had occurred subsequent to the arrival of the sill in the position 
that it now occupies, the contact-rocks should have the same com- 
position as the average rock of the sill. This, however, is not the 
case, as the average rock has the composition of an almost ultrabasic 
theralite, entirely different from the teschenites of the contacts. 
Hence it is believed that, after forming contact-sheaths of theralite, 
and undergoing gravity-stratification subsequent to lquation, the 
intrusive activity was renewed, and the sill was moved on along 
bedding-planes into cold rocks, leaving its contact-sheath behind 
‘adhering to the old contacts, and establishing new contacts with 
its upper and lower teschenite-layers. Here crystallization began, 
and, by the subsidence of olivine, the subsidiary gravity-stratification 
of the central ultrabasic layer was effected. The extraordinary 
flow-banding shown by the contact-rocks affords confirmation of 
the renewed movement thus postulated. 

In conclusion, the sill is compared with five other teschenite- 
pierite sills in Scotland, those of Ardrossan, Saltcoats, Blackburn, 
Barnton, and Inchcolm. 


Sregpann & Fray. Phil. Mag. Ser. 6, Vol. 32, Pl. I. 


——— 


i 
Pra) mw tag 


i 
= 
re 
Jj 
Rs 
natin 


Phil. Mag. Ser. 6, Vol.'32, Pl. IL. 


Mari & Das. 


Vie. 1, 


Fic. 5. 


Fre. 6. Fra. 7. Fie. 8. 


Fra, 9. Pra. 10. Pig. 11. 


Fie. 15. Fig. 16. Fro. 17. 


Phil. Mag. Ser. 6, Vol. 32, Pl. III. 


VEGARD. 


Zircon lattice. 


the d 


-AXIS. 


irection of the a 


aim 
b 


C-AXIS, 


” 


” 


” 


Fig. 3. 
fied 345 diameters. ) 


|, Mag. Ser. 6. Vol. 32, Pl. IV. 


Phi 


fied 120 diameters. ) 


° 


onl 


(Ma 


Erte. 2. 
fied 345 d 


IKEDTI. 


Veni 


‘ 
c 


(M 


umeters. ) 


le 


enl 


(Ma 


a on iy 
a 
A a 


ae ay 


THE 
LONDON, EDINBURGH, anv DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES. ] 


AP OrG, LS Tg ko: 


XVI. On the Discharge of Gases under High Pr ressures. 
By Lord Rayurren, OM, F.R.S* 


HE problem of the passage of gas through a small 
aperture or nozzle from one vessel to another in which 

there is a much lower pressure has had a curious history. 
It was treated theoretically and experimentally a long while 
ago by Saint-Venant and Wantzel f in a remarkable memoir, 
where they point out the absurd result which follows from 
the usual formula, when we introduce the supposition that 
the pressure in the escaping jet is the same as that which 
prevails generally in the recipient vessel. In Lamh’s 
notation {, if the gas be subject to the adiabatic law (ppv), 


EEE 12) e ya = 
‘ 2—9 l— = Ona 5 il 
? ae y-1 at Po Dore ay ake 


where g is the velocity corresponding to pressure p; 
Po. Po the pressure and density in the “discharging vessel 
where g=0; c the velocity of sound in the gas when at 
pressure p and density p; ¢, that corresponding tO Poy Po: 
According to (1) the velocity increases as p diminishes, but 
only up to a maximum, equal to ¢o4/{2/(y—1)}, when p=0. 


* ep cmmunicated by the Author. 

‘Mémoire et expériences sur l’écowement de lair, déterminé par 
au différences de pressions considérables,” Jowrn. de [Ecole Polyt. t. Xvi. 
p. 85 (1889). 

{ ‘Hydrodynamics,’ §§ 28, 25 (1916). 
Phil. Mag. 8. 6. Vol. 32. No. 188. Aug. 1916. N 


178 Lord Rayleigh on the Discharge of 


If y=1°408, this limiting velocity is 2°214¢). It is to be 
observed, however, that in considering the rate of discharge 
we are concerned with what the authors cited call the 
‘reduced velocity,” that is the result of multiplying g by 
the corresponding density p. Now p diminishes indefinitely 
with p, so that the reduced velocity corresponding to an 
evanescent p is zero. Hence if we identify p with the 
pressure p; in the recipient vessel, we arrive at the im- 
possible conclusion that the rate of lischarge into a vacuum 
is zero. From this our authors infer that the zdentijcation 
cannot be made; and their experiments showed that from 
pi==0 upwards to p;='4py the rate of discharge is sensibly 
constant. As /, still further increases, the discharge falls off, 
slowly at first, afterwards with greater -apidity, until it 
vanishes when the pressures become equal. 

The work of Saint-Venant and Wantzel was fully discussed 
by Stokes in his Report on Hydrodynamics*. He remarks 
‘These experiments show that when the difference of pressure 
in the first and second spaces is considerable, we can by no 
means suppose that the mean pressure at the orifice is equal 
to the pressure at a distance in the second space, nor even 
that there exists a contracted vein, at which we may suppose 
the pressure to be the same as at a distance.” But notwith- 
standing this the work of the French writers seems to have 
remained very little known. It must have been unknown to 
O. Reynolds when in 1885 he traversed much the same 
ground f, adding, however, the important observation that 
the maximum reduced velocity occurs when the actual 
velocity coincides with that of ‘sound under the conditions 
then prevailing. When the actual velocity at the orifice 
reaches this value, a further reduction of pressure in the 
recipient vessel does not influence the rate of discharge, as 
its effect cannot be propagated backwards against the stream. 
If y=1°'408, this argument suggests that the discharge reaches 
a maximum when the pressure in the recipient vessel falls to 
*527 pp, and then remains constant. In the somewhat later 
work of Hugoniot $ on the same subject there is indeed a 
complimentary reference to Saint neue and Wantzel, but 
the reader would hardly gather that they had insisted upon 

the differenco between the pressure in the jet at the orifice 
and in the recipient vessel as the explanation of the im- 
possible conclusion deducible from the contrary supposition. 

In the writings thus far alluded to there seems to be an 


* B. A. Report for 1846; Math. and Phys. inten 1. pag 


+ Phil. Mag. vol. xxi. p. 185 (1886). 
{ Ann. de Chim. t. ix. p. 383 (1886). 


Gases under High Pressures. 179 


omission to consider what becomes of the jet after full pene- 
tration into the receiver. The idea appears to have been 
that the jet gradually widens in section as it leaves the 
orifice and that in the absence of triction it would ultimately 
attain the velocity corresponding to the entire fall of pressure. 
The first to deal with this question seem to have been Mach 
and Salcher *, but the most elaborate examination is that of 
R. Emdenf, who reproduces interesting pictures of the 
effluent jet obtained by the simple shadow method of Dvorakf. 
Light from the sun or from an electric spark, diverging from 
a small aperture as source, falls perpendicularly upon the jet 
and in virtue of differences of refraction depicts various 
features upon a screen heldat some distance behind. <A per- 
manent record can be obtained by photography. Emden 
thus describes some of his results. When a jet of air, or 
better of carbonic acid or coal-gas, issues from the ne? ale 
into the open under a pressure of a few millimetr es, it is seen 
to rise as a slender column of the same diameter to a height 
of perhaps 30 or 40 cm. Sometimes the column disappears 
without visible disturbance of the air; more often it ends 
in a small vortex column. When the pressure is raised, 
the column shortens until finally the funnel-shaped vortex 
attaches itself to the nozzle. Ata pressure of about one-fifth 
of an atmosphere there appears again a jet 2 or 3 cm. long. 
As the pressure rises still further, the jet becomes longer and 
more distinct and suddenly exhibits thin, bright, and fairly 
equidistant disks to the number of perhaps 10 or 12, crossing 
the jet perpendicularly. The first disks have exactly the 
diameter of the nozzle, but they diminish as the jet attenuates. 
Under still higher pressures the interval between the disks 
increases, and at the same time the jet is seen to swell out 
between them. These swellings further increase and oblique 
markings develop which hardly admit of merely verbal 
description. 

Attributing these periodic features to stationary sound 
waves in the jet, Emden set himself te determine the wave- 
length (X), that is the distance between consecutive disks, 
and especially the pressure at which the waves begin to 
develop. He employed a variety of nozzles, and thus sums 
up his principal results : 

1. When air, carbonic acid, and hydrogen escape from 
equal sufficiently high pressures, the length of the sound 


* Wied. Ann. Bd. xli. p. 144 (1890). 
+ Wied. Ann. Bd. lxix. pp. 264, 426 (1899). 
$ Wied. Ann. Bd. ix. p. 802 (1879). 

N 2 


180 Lord Rayleigh on the Discharge of 


waves in the jet is the same for the same nozzle and the 
same pressure. 

2. The pressure at which the stationary sound waves begin 
to develop is the same in air, carbonic acid, and hydrogen, 
and is equal to ‘9 atmosphere. 

This is the pressure-ewcess behind the nozzle, so that the 
whole pressure there is 1-9 atmosphere. The environment 
of the jet is at one atmosphere pressure. 

Emden, comparing his observations with the theory of 
Saint-Venant and Wantzel, then enunciates the following con- 
clusion :—The critical pressure, in escaping from which into 
the atmosphere the gas at the nozzle’s mouth moves with the 
velocity of sound, is equal to the pressure at which stationary 
sound waves begin to form in the jet. So far, I think, 
Hmden makes out his case; but he appears to over-shoot 
the mark when he goes on to maintain that after the critical 
pressure-ratio is exceeded, the escaping jet moves everywhere 
with the same velocity, viz. the sound-velocity; and that 
everywhere within it the free atmospheric pressure prevails. 
He argues from what happens when the motion is strictly in 
one dimension. Itis true that then a wave can be stationary 
in space only when the stream moves with the velocity of 
sound; but here the motion is not limited to one dimension, 
as is shown by the swellings between the disks, Indeed the 
propagation of any wave at all is inconsistent with uniformity 
of pressure within the jet. At the surface of the jet, but not 
within it, the condition is imposed tliat the pressure must be 
that of the surrounding atmosphere. 

The problem of a jet in which the motion is completely 
steady in the hydrodynamical sense and approximately 
uniform was taken up by Prandtl*, both for the case of 
symmetry round the axis (of z) and in two dimensions. In 
the former, which is the more practical, the velocity com- 
ponent w is supposed to be nearly constant, say W, while 
wu and v are small. We may employ the usual Eulerian 
equations. Of these the third, 


dw dw. dw dw _ 1 dp 


“ode. wa de 7 ods 
reduces to 
(7 eo mel dp : 
W oan Ee . +) en 


when we introduce the supposition of steady motion and 


* Phys. Zeitschrift, 5 Jahrgang, p. 599 (1904). 


Gases under High Pressures. 181 


neglect the terms of the second order. In like manner the 
other equations become 


aa as) Lap mad 0: > 1 ap : 
\ es ae Say AG ane) (3) 


Further, the usual equation of continuity, viz. 


De: | eee 
da dy dz ‘ 


here reduces to 
du , dv~, dw : P= 
Gita ae 1. Sp Sea ac 
If we introduce a velocity-potential ¢, we have with use 
of (2) 
Wido Wadi Wea" a’ 
Ee  eemereria ee 


Ui 


V*o=— 
: Qe 
where a, = V (dpjdp), is the velocity of sound inthe jet. In 
the case we are now considering, where there is symmetry 
round the axis, this becomes mo 


dnb 1 dd +(1-— nT) 
dr my 


—=5 =i) Ba ae AE 
dy” a; \es j ep) 
and a similar equation holds for w, since w=d¢Jdz. 

If the periodic part of w is proportional to cos Bz, we have 
for this part 


Dea arn J2 \ 
a7u 1 dw 1¢ —1) Bw=0, . ae 


dr? yr dr Oe 


and we may take as the solution 
w= W + Hos 82.Jo{ «/( W*—a’) Bria}, . (9) 


since the Bessel’s function of the second kind, infinite when 
r=(, cannot here appear. The condition to be satisfied at 
the boundary (r=R) is that the pressure be constant, equal 
to that of the surrounding quiescent air, and this requires 
that the variable part of w vanish, since the pressure varies 
with the total velocity. Accordingly 


Jing VCW?—a"). Siyjat=0)° < . . * (10) 


which can be satisfied only when W >a, that is when the 
mean velocity of the jet evceeds that of sound. The wave- 


182 Lord Rayleigh on the Discharge of 


length (A) of the periodic features along the jet is given 
by A=27/8. 

The most important solution corresponds to the first root 
of (10), viz. 2-405. In this case 


RV (W2/a?—1 
Ee |. an 


The problem for the two-dimensional jet is even simpler. 
If 6 be the width of the jet, the principal wave-length is 
given by 


N=2V(Wee—H1). 4. ... ee 


The above is substantially the investigation of Prandtl, who 
finds a sufficient agreement between (11) and Emden’s 
measurements *. 

It may be observed that the problem can equally well be 
treated as one of the small vibrations of a stationary column 
of gas as developed in ‘ Theory of Sound,’ §$§ 268, 340 (1878). 
It the velocity-potential, symmetrical about the axis of z z, be 
also proportional to e@+8), where & is such that the wave- 
length of plane waves of the same period is 27/«, the equa- 


tion is § 340 (3) 


oo = SSeS )6=0, . ae 


and if k>8 
P= eCtrs) Seay (kh? — 67) Jr}. Se 


The condition of constant pressure when 7=R gives as 
before for the principal vibration 


V (i2--62).B=2:405. . . . . (5) 


The velocity of propagation of the waves is ka/8. If we 
equate this to W and suppose that a velocity W is superposed 
upon the vibrations, the motion becomes steady. When we 
substitute in (15) the value of &, viz. WB/a, we recover (11). 
It should perhaps be noticed that it is only after the vibrations 
have been made stationary that the effect of the surrounding 
air can be properly represented by the condition of uniformity 
of pressure. To assume it generally would be tantamount 
to neglecting the inertia of the outside air. 


* When W<a, 8 must be imaginary. The jet no longer oscillates, 
but settles rapidly down into complete uniforinity. This is of course the 
usual case of gas escaping from small pressures. 


Gases under High Pressures. 183 


The above calculation of X takes account only of the 
principal vibration. Other vibrations are possible corre- 
sponding to higher roots of (10), and if these occur appre- 
ciably, strict periodicity is lost. Further, if we abandon the 
restriction to symmetry, a new term, r~*d?/d@?, enters in 
(13) and the solution involves a new factor cos (n@+e) 
in conjunction with the Bessel’s function J, in place 
of Jo. 

The particular form of the differential equation exhibited 
in (13) is appropriate only when the section of the stream is 
circular. In general we have 


aoe id 
* oy? 


the same equation as governs the vibrations of a stretched 
membrane (‘Theory of Sound,’ § 194). For example, in 
the case of a square section of side b, we have 


2 (Ae Oe ie ile) 


p= cos. cos TH, eilint +62), lh ses ealanieligia) 


vanishing when «=+46 and when y=+34b. This re- 
presents the principal vibration, corresponding to the 
gravest tone of a membrane. The differential equation is 
satisfied provided 
Ke Be Jaren wh aes) ken eS) 

the equation which replaces (15). It is shown in ‘Theor 
of Sound’ that provided the deviation from the circular 
form is not great the question is mainly one of the area of 
the section. Thus the difference between (15) and (18) is 
but moderate when we suppose 7R* equal to 6”. 

It may be worth remarking that when V the wave- velocity 
exceeds a, the group- -velocity U falls short of a. Thus in 


(17), (18) 


bea nd BNG dk _ Ba, 
Nie ih aaa de i de j 


so that 


OP ie. SN OR aE (19) 


Returning to the jet of circular section, we may establish 
the connexion between the variable pressure along the axis 
and the amount of the swellings observed to take place 


184 Lord Rayleigh on the Discharge of 
between the disks. From (9) 

d= \ wdz=W2z+ H8"!sin Bz. Jo{ Y(W?/a?—1) . Br}, 
and 


(GP) =Hy(W/a?—1) . sin Az. Jo/(2405). . 20) 
\ 7R 


The latter equation gives the radial velocity at the 
boundary. If dR denote the variable part of the radius 
of the jet, 


Ey kos Rea H cos Bz W? A 


. (21) 
Again, if dp be the variable part of the pressure at the axis 
(r=0), 


P04 ?=C'—4 wv? = — Wow, 


where p is the average density in the jet and dw the variable 
part of the component velocity parallel to z. Accordingly 


°P = WH cos Be; + 6 eee 
and 
SR __ Jo'(2405),/(W?/a?— 1) 
in a 


In (23) we may substitute for @ its value, viz. 


2°405 a 
Ry/ (W? —@?)’ 


and for Jo (2°405) we have from the tables of Bessel’s 
functions —0°5191, so that 


oR ° ai =z =v 

Saas 0°2158R(a W-?). . aaa 
As was to be expected, the greatest swelling is to be found 
where the pressure at the axis is least. 

A complete theory of the effects observed by Mach and 
Emden would involve a calculation of the optical retardation 
along every ray which traverses the jet. For the jet of 
circular section this seems scarcely practicable ; but for the 
jet in two dimensions the conditions are simpler and it may 
be worth while briefly to consider this case. As before, we 


Gases under High Pressures. 185: 


may denote the general thickness of the two-dimensional jet 
by J, and take b+ 7 to represent the actual thickness at the 
place (z) where the retardation is to be determined. The 
retardation is then sufficiently represented by A, where 


3(b+n) *3(0+n) e 
a={ (p—pijdy= hy — $pi(b+n), . (2d) 
0 0 


p being the density in the jet and p, that of the surrounding 
gas. The total stream 


1 


3(0+n) (3(b+n) 20 
= p(W +8e)dy=W | pay +p dw dy; 
0 20 0 


and this is constant along the jet. Thus 
= O0—taq— 2 wi, en Lois 8 ie CZGo 


C being a constant, and squares of small quantities being 
omitted. 
In analogy with (9), we may here take 


dw=H cos Bz. cos {8y,/(W2/a?—-1)}, . . (27) 


and for the principal vibration the argument of the cosine is 
to become $7 when y=43b. Hence 


128 H cos 82 


: oe een Meh (28) 
Also 
o= | wdz=W:+67H sin Bz .cos{ By / (W?/a?—1)}, 
a Th ) = THY {Wea 1} sin Be, 
Thus 
Por a dd Hae Ecos Beer Wa). age 
aman ak i BW 
Accordingly 


Rata ecos 82 72/2 nN P , ( 
we CU BW [piv i" (es 1; ay ea (29) 


so that the retardation is greatest at the places where » is 
least, that is where the jet is narrowest. This isinagreement 


186 Lord Rayleigh on the Discharge of 


with observation, since the places of maximum retardation 
act after the manner of a convex lens. Although a complete 
theory of the optical effects in the case of a sy mmetrical jet 
is lacking, there seems no reason to question Emden’s 
opinion that they are natural consequences of the consti- 
tution of the jet. 
But although many features are more or less perfectly 
explained, we are far from anything like a complete mathe- 
matical theory of the jet escaping from high pressure, even 
in the simplest case. A preliminary question is—uare we 
Justified at all in assuming the adiabatic law as approximately 
governing the expansions throughout? Is there anything 
like the “ bore” which forms in front of a bullet advancing 
with a velocity exceeding that of sound?* It seems that 
the latter question may be answered in the negative, since 
here the passage of air is always from a oreater to a less 
pressure, so that the application of the ‘adiabatic law is 
justified. The conditions appear to be simplest if we suppose 
the nozzle to end in a parallel part within which the motion 
may be uniform and the velocity that of sound. But even 
then there seems to be no reason to suppose that this state of 
things terminates exactly at the plane of the mouth. As the 
issuing gas becomes free from the constraining influence of 
the nozzle walls, it must begin to expand, the pressure at the 
boundary suddenly falling to that of the environment. 
Subsequently vibrations must set in; but the circumstances 
are not precisely those of Prandtl’s calculation, inasmuch as 
the variable part of the velocity is not small in comparison 
with the difference between the mean velocity and that of 
sound. It is scarcely necessary to call attention to the 
violence of the assuimption that viscosity may be neglected 
when a jet moves with high velocity through quiescent air. 


On the experimental side it would be of importance to 
examine, with more accuracy than has hitherto been attained, 
whether the asserted independence of the discharge of the 
pressure in the receiving vessel (supposed to be less than a 


certain fraction of that in the discharging vessel) is absolute, 
and if not to ascertain the precise law of departure. To this 
end it would seem necessary to abandon the method followed 
by more recent workers in which compressed gas dis- 
charges into the open, and to fall back upon the method of 
Saint-Venant and Wantzel where the discharge is from 
atmospheric pressure to a lower pressure. The question is 

* Proc. Roy. Soc. A. vol. Ixxxiv. p. 247 (1910); Scientific Papers, 
vol. v. p. 346. 


Gases under High Pressure. 187 


whether any alteration of discharge is caused by a reduction 
of this lower pressure beyond a certain point. To carry 
out the investigation on a sufficient scale would nead a 
powerful air-pump capable of absorbing the discharge, but 
otherwise the necessary apparatus is simple. In order to 
measure the discharge, or at any rate to determine whether 
it varies or not, the passage of atmospheric air to the nozzle 
might be somewhat choked. The accompanying diagram 
will explain the idea. A is the nozzle, which would be varied 
in different series of experiments ; B the recipient, partially 
exhausted, vessel; C the passage to the air-pump. Above 


the nozzle is provided a closed chamber E into which the 
external air has access through a metal gauze D, and where 
consequently the pressure is a little below atmospheric. 
F represents (diagrammatically) a pressure-guuge, or micro- 
manometer, whose reading would be constant as long as the 
discharge remains so. Possibly an aneroid barometer would 
suffice; in any case there is no difficulty in securing the 
necessary delicacy*. Another manometer of longer range, 
but only ordinary sensitiveness, would register the low 
pressure in B. In this way there should be no difficulty in 
attaining satisfactory results. If F remains unaffected, 
notwithstanding large alterations of pressure in B, there are 
no complications to confuse the interpretation. 


Terling Place, Witham. 
June 10, 1916. 


* See for example Phil. Trans. cxevi. A. p. 205 (1901); Scientific 
Papers, vol. iv. p. 510. 


eee 


XVI. On the Energy acquired by small Resonators from 
wncident Waves of like Period. By Lord RAYLEIGH, 
OMIT AIS 


i discussions on photo-electricity it is often assumed that 

a resonator can operate only upon so much of the 
radiation incident upon it as corresponds to its own cross- 
sectiont. Asa general proposition this is certainly not true 
and may indeed differ from the truth very widely. Since 
1875 { it has been known that an ideal infinitely small 
acoustical resonator may disperse energy corresponding toan 
area of wave-front of the primary waves equal to )2/7, an 
efficiency exceeding to any extent the limit fixed by the above 
mentioned rule. The questions of how much energy can be 
absorbed into the resonator itself and how long the absorption 
may take are a little ditterent, but they can be treated without 
difficulty by the method explained in a recent paper §. The 
equation (49) there found for the free vibration of a small 
symmetrical resonator was 


2 2 
P+ up+4mrar'(1—ikr) SP =0, . Me kL 


¢ 
Me Gia 
in which p denotes the radial displacement of the spherical 
surface from its equilibrium value 7, M the mass, w the 
coefficient of restitution, o the density of the surrounding 
gas, and k=2a7—+wave-length (A) of vibrations in the gas. 
The first of the two terms containing o operates merely as. 
an addition to M. If we write 


WeaM+4aor*, . . . Sees 
(1) becomes 
d? a , 
M’ a + pp—t Anakrt eno «> ee ies (3) 


Thus, if in free vibration p is proportional to e”*, where n is. 
complex, the equation for m is : 


n?(—M'+7i.4ackr*)+m=0.. . . . ©) 


* Communicated by the Author. 

+ See for example Millikan’s important paper ona direct determination 
of Planck's constant “h”; Physical Review, vol. vii. March 1916, p. 385. 

T ‘Theory of Sound,’ § 319; X=wave-length. 

§ Phil. Mag. vol. xxix. Feb. 1915, p. 210. 


Energy acquired by Resonators from incident Waves. 189 


The free vibrations are assumed to have considerable per- 
sistence, and the coefficient of decay is e~*, where 


g=2rokr V/ (u/M")=2mapkr/M, 2. |. OO) 
if pa p/M’ : 
We now suppose that the resonator is exposed to pri- 
mary waves whose velocity-potential is there 


P= cel, Me gL AE Ae (6) 


The effect is to introduce on the right hand of (3) the term 
A4nr’ca.ipe; and since the resonance is supposed to be accu- 
rately adjasted, p?=p/M’'. Under the same conditions id?p/dé? 
in the third term on the left of (3) may be replaced by 
—pdpldt, whether we are dealing with the permanent 
forced vibration or with free vibrations of nearly the same 
period which gradually die away. Thus our equation 
becomes on rejection of the imaginary part 


jv! d?p dp . 
M ae + 4aaphrt 7+ Ep= — sonics oSunyay§ Wee (0) 


which is of the usual form for vibrations of systems of one 


degree of freedom. For the permanent forced vibration 
M'd?p/dt? + pp=0 absolutely, and 


dp __—s a sin pt 


dt 77? Md coh te (8) 


The energy located in the resonator is then 


Me? 
2 j.2)¥? 0 Nie Fea tee Sean (9) 


and it may become very great when M is large and 7 small. 

But when M is large, it may take a considerable time to 
establish the permanent regime after the resonator starts 
from rest. The approximate solution of (7), applicable in 
that case, 1s 


_ 4 COS pt 


ae le a jnstn-gis 10) 


q being regarded as small in comparison with p; and the 
energy located in the resonator at time ¢ 


lp \? Ma? 
aim (2) = Bees an 


190 Prof. A. Anderson on the Mutual 


We may now inquire what time is required for the accu- 
mulation of energy equal (say) to one quarter of the limiting 
value. This occurs when e~“=4, or by (5) when 
2 ae loo 2. MT’ 


i ee 


The energy propagated in time ¢ across the area S of primary 
wave-front is (‘ Theory of Sound,’ § 245) 


Sa by | is! ai Me ia(de} 


where a is the velocity of propagation, so that p=ak. If we 
equate (13) to one quarter of (9) and identify ¢ with the 
value given by (12), neglecting the distinction between M 
and M’, we get 

7 ow 


= ae _—_ 2 * 
| Qlogeee” Sr los 2 “sae Co 


The resonator is thus able to capture an amount of energy 
equal to that passing in the same time through an area of 
primary wave-front comparable with 7/7, an area which 
may exceed any number of times the cross-section of the 
resonator itself. 


XVIII. On the Mutual Magnetic Energy of two Moving 
Point Charges. By Prof. A. ANDERSON f. 


HE method here given of finding the mutual magnetic 
energy of two moving point charges of electricity is 
elementary. It does not claim to have the elegance of 
Heaviside’s method. It is, however, a confirmation of his 
result by simple mathematics, in which there are no vector 
potentials, no curls of any kind, and no trouble of adjusting 
the vector potential to make the space integral of the scalar 
product of it and the magnetic force vanish for the polar, or 
displacement, currents in the medium. 
Consider, first, the case of two charges, e; and e,, at A and 
B moving along parallel lines AL and BM with velocities 
w, and w,, AB being perpendicular to AL and BM (fig. 1). 
Let P be any point in space. Draw the plane PLM 
perpendicular to AL and BM and join PA, PB. 


* log 2=0°693. 
t+ Communicated by the Author. 


Magnetic Eneray of two Moving Point Charges. 191 
The mutual magnetic energy at P per unit volume is 


[eyez ;W> Sin O sin 6! cos LPM 
Agr?” 


ro 


ao 


Let angie APB=a2, PAB=A, and PBA=B; also, let 
AD = ¢. 


By means of a spherical triangle it is easily seen that 
cos a=cos 6 cos 6' + sin 6 sin 4’ cos LPM. 
Also, cos@=sinAcos@ and cos =sin Boos ¢, 
where ¢ is the angle between the planes APB and LABM. 


The mutual energy per unit volume is consequently 
[Le e3W1 We (cos e—cos* sin A sin B)/4arr?r’?. 


Take an element of area dS in the plane APB at P ; then 
the element of volume at P may be taken 


pd¢ dS, where p is the perpendicular from P on AB. 


If the triangle APB be rotated about AB through a com- 
plete revolution, the mutual energy in the ring traced out 


by dS is 


C1€9W WW 
re ds (27 cos a—sin A sin B ie cos *h ry) 
ial 
_ Me egy W 
= Te —1, dS (2p cos a—p sin A sin B). 


Since rr’sinae=pe and sinAsinB=p*/rr’, this i 
equal to 


! 


ae AS [= cos a sin” sin =o 


192 Prof. A. Anderson on the Mutual 


This must be integrated over the whole half plane on one 
‘side of the base AB. 


There ars two integrals to be found, 


sin 3a 


ase 


dS and = 


‘2 cosa sin 2x 
P 


They present no difficulty, the first being obtained by req 
ducing to polar coordinates, A being pole and AP radius 
vector. The second is very easily got by t taking as element 
dS the difference between the areas of two segments of 
circles on AB, one containing an angle « and the other an 
angle a+da. Their values are respectively 4c and 2c. The 


total mutual energy is, therefore, 


Me eg Wo 
ae 


Consider now the case of two charges, ¢; and é, at A and 


V2 . E Om 


B both moving in the direction AB (fig. 2) with velocities 
4, and vz The mutual energy per unit volume at P is 


Hej C01 V9 


Arr 
And the mutual energy in the ring traced out by dS at P is 


sin 8; sin 0,/7;?77. 


Me €oU U2 1V2 
as Hees 


a 


_ gy Heseaeit2 a aS. Pe eotyVg sin 2a 
— . 9 r 37759 9 e 3 ° 


sin 6, sin 65/7;775" 
ip 1 1 


But | sin Soi" 267 
, Me €oVy V2 5 
Hence the total mutual energy = = ae that is, double 


what it is in the former ease for the same values of charges 
_and velocities. 


Magnetic Energy of two moving Point Charges. 193 


We are now in a position to solve the general case. 
The component velocities of e, at A (fig. 3) are uw, %, wy, 


Fig. 3. 


and those of e, at B wu, v2, wa. The origin is taken at C the 
middle point of AB, the axis of y being along CB. 


2 
rP=r+yet+ 7 and 


2 
G : 
rer? —yet t? where AB=c, as before. 


The components of magnetic force due to e, are 


C 
UjeZ—W, (y + 5) 


a1 = 


ry? y) 
Wk Uy 2 
=e} ——__>— 
By 1 re y) 
; G 
Utyt+ 9 —Uj{x 
ad 
Vi et 13 > 
il 
and those due to é are 
( Cc 
Ube Wo | Y ae 
i 2 
oe. 3 ) 
12 
B Wat — Uge 
V} 29) re ) 
Cc 
Us Oise 5 —s Ug 
Yo es a) . 
"2 


Phil. Mag. 8. 6. Vol. 32. No. 188. Aug. 1916. O 


194 Mutual Magnetic Energy of two Moving Point Charges. 
os 1°09? (ety et9 + By Bs + 112) 
= 0? (UyUq + WW) + Y? (UUs + Wy Wy) + 27(UyVq + UyUe) 
— 2y (010, + Wyvy) — Se + UjWe) — vy(UzVs + Ugry) 


5 2 U1Ws —WV2) + = aC jUg —UyVo) 


— : (uytty + WW). 


The total mutual magnetic epeBy may be found by 
evaluating the integrals (ae -, GC... ., Dub ib Sinan 


necessary to do this. ee those which evidently 
vanish, and observing that 


ae dy dz 

Pit? an ee ipa = 

it is clear that the total mutual magnetic energy is 
A(2vyt9 + UyUlg + Wye) + Byes + Wyo), 

where A and B are constants to be determined. 


Let 2, to, Wi, Ws all vanish, the energy is 2Ar,0». 
Hence, by the former result for this case, 


Heres 
Ss Sais 


Again, let 2, we, v1, ve all vanish, the energy is 
(A+ B)w,w., 


and, therefore, by the former result, 


__ e122 
Bi ae 
thus B=0, and the energy is given by 


€1€ 
a z (20109 + UyUg + W We), 


or, if V, and V, are velocities of e, and ¢é, ¢ the angle 
between their directions, and @:, 6, the angles between 
these directions and AB, by 


se (cos e+ cos 8; cos 2). 


E1950 a 


XIX. On the Principle of Least Action in the Theory of 
Electrodynamics.—I. By G. H. Livens*. 


T is a fundamental problem in the general theory of 
electrodynamics to formulate a scheme fer the description 
of the general phenomena of the electromagnetic field on an 
analogy with the behaviour of some hypothetical system 
obeying the ordinary laws of. generalized mechanics. Now, 
any dynamical problem can be enunciated in a single formula 
as a variation problem, and thus if the laws of electro- 
dynamics can be derived in a minimum or variational 
theorem, the formulation is virtually complete; there remain 
only such interpretations, explanations, and developments as. 
will correlate the corresponding integrals relating to known 
dynamical systems. 

In this form the problem has been solved in slightly 
different ways by Larmorf, Lorentz {, Macdonald §, and 
others. The integral which is the subject of variation in 
each case is virtually the same as 


to 1 1 
) [u +{{Eao) ay ae} de dt, 


wherein L is the part of the Lagrangian function of the 
system not depending on the conditions of the field; A is 
the vector potential of the magnetic field ; C is the total 
current of Maxwell’s theory and includes both the true 
electric and fictitious ethereal fluxes, and E is the electric 
force intensity at the typical field-point. It is assumed that 
the system consists entirely of free sether and electrons, and 
the integration with respect to v is taken over the whole 
field. 

On analogy with the known properties of electrostatic 
fields it is assumed that the term in EH? represents the true 
potential energy of elastic strain in the ether, both as 
regards its distribution and total amount; it then follows 
that the integral 


1 
x { (AC) de 


represents the totality and distribution of the kinetic energy 
of electric origin in the ethereal field. 


* Communicated by the Author. 
+ ‘ Auther and Matter’ (Cambridge, 1900), ch. iv. 
{ ‘Lia théorie électromagnetique de Maxwell, &c.’ (Leiden, 1892), 
§§ 55-61. 
§ ‘Electric Waves’ (Cambridge, 1902), App. C. 
O 2 


196 Mr. G. H. Livens on the Principle of 


But in all other applications of the general theory of 
electromagnetism it is {ound more convenient to assume that 
the magnetic energy, which for other reasons is regarded as. 
of kinetic type, is distributed throughout the field with a 
density at any place expressed by 

1 
7 
where B is the vector of magnetic induction. 

Now, these two expressions for the magnetic energy do 
not in the most general case agree, even in total amount, 
for, as Macdonald points out*, in the derivation of the one 
from the other by the method of integration by parts, an 
integral over the infinite boundary is brought in which is 
not generally negligible. It thus becomes a question whether 
the results of the dynamical theory can be used in the more 
usual formulations and developments of the subject. An 
attempt to justify such usage so far as concerns the 
expression for the force on a moving charge has been made 
by Larmor ft by the examination of a special problem with 
restricted conditions, but some doubt may still exist as to 
the general validity of the argument thus employed. 

The question is settled inthe theory of relativityt, where it 
is verified that the usual expression for the force is consistent 
with the second expression for the magnetic energy ; but 
insofar as this verification is based on the differential invariant 
theory associated with Minkowski’s four-dimensionai analysis 
of the genera] theory, it can hardly be said to throw much 
light on the physical bearing of the problem. 

It seems therefore desirable to attempt a direct formulation 
of the general dynamical theory on the basis of the second 
and more usual expression for the kinetic energy of electric 
origin in order to confirm, if possible, the result obtained by 
Larmor under special circumstances, which is generally used 
without hesitation or restriction in either form of the theory. 
The object of the present note is to show that such a formu- 
lation can easily be effected and the results derived from it, 
insofar as they are identical with those deduced on the older 
basis, fully substantiate Larmor’s conclusions from his special 
problem. 

The principle of least action is applied, in the manner 
already elaborated in full detail by Larmor, to determine the 


* [.e. p. 33. 

+ haves. broc, 1915. 
Cf. Cunningham, ‘The Principle of Relativity’ (Cambridge, 1914), 

58. 


B2, 


+ 
a 
p: i 


Least Action in the Theory of Hlectrodynamics. 197 


sequence of changes in a system consisting entirely of free 
electrons and ether, but for which the Lagrangian function 
is 
pol 
L +| —— (B?— E?) dv. 
OT 

The main difficulty experienced in using this form of function 
is that it is not on either account explicitly expressed in 
terms of the independent coordinates of the system, which 
in the present case are usually taken to be the ethereal 
displacement mee and the-position coordinates (wv, y, 2) of 
the various electrons. The main claim for the alternative 
expression for the magnetic energy is that it is so expressed, 
but even then the difficulty still persists as regards the 
electric energy. It appears, however, that any such explicit 
interpretation of the functions can be avoided by the use of 
Lagrangian undetermined multipliers, as, in fact, is done by 
Larmor tor the potential electric energy. In this way the 
variations of H and B can be temporarily rendered in- 
dependent of each other and also of the coordinates of the 
electrons, the form of the undetermined functions thus 
introduced being, however, finally chosen to secure the 
correct relations of dependence between the differentvariables. 

Now the functions E and B are connected with one 
another and with the coordinates of the electrons by the 
relations 


{ diy Hidv—4aDe=0, 


{ (con B— i “*) dvu— BE eC 
e dt G 


wherein } represents a sum taken over all the electrons, the 
typical one having a charge e and velocity v (#, y, z). The 
second relation being a vector one is in reality equivalent 
to three independent equations. We now introduce four 
Lagrangian undetermined multipliers ¢, A,, A,, Az, all of 
which are functions of position in the field. The last three 
may be taken to be the rectangular components of a vector A. 
It is thus the variation of 


ts ° 
: dt E + eel dv | B?— Hh? + 2¢ div E 
ty 
—2 (A, Curl B— 5 a) } —Siget+3- (Av) J, 


that is to be made null, afterwards determining the forms of 


198 Mr. G. H. Livens on the Principle of 


@ and A to satisfy the restrictions which necessitated their 
introduction. The complete variation is easily effected and 
is, in fact, 


{= [1a G2)— 5205p 2052) } & 
eee poy +4... }oet+ “(A or+A bj +A85) | 


eu (" dt { iv | (BBB) — (HSE) +4 div 8B 
ge i 
—(A Curl a ie (soa 


On removal by integration by parts of ae various differ- 
ential operations affecting the independent variations and 
noticing that the time differentiations affecting the coordinates 
of an electron are total differentiations following the motion 
of that electron, we get finally for the total variation 


i ae, Ae ($2) -& Oa “ne oP _§ “ie a0 Curl A], 


e oa, 
oe Bed (payee | 


tel “aif doll ABs Curl A, 8B) —(B+gradd4 2S 5H) | 


ee 


1 ie eo 
-2{ ‘atl dS{63E,—[A8B],} 

\tg 

i by (Aede-+A,by +A.82)~ 7 (ASE) |" 


(Cy 


The integral with respect to S is taken over the infinitely 
extended surface bounding the field. 

The variations 62, dy, 6z, which give the virtual displace- 
ment of an electron e, and the variations dE, 6B, which 
specify the condition variations in the ether, can now be 
considered as all independent and perfectly arbitrary ; hence 
the coefficients of each must vanish separately in the 
dynamical variational equation. We conclude that ¢ and A 
must tend to zero regularly at infinity and then at each 


point of the field 
1 oA das 
H+ ; oe) 


B— Curl A=0, 


Least Action in the Theory of Electrodynamics. 199 


whilst for each electron there are three equations of type 


Gi ($5) — Sa 7 gle Cul Alt § S2 +0$2 <0. 


With the exception of the second equation, these are 
identical with Larmor’s equations deduced from the older 
alternative basis. It thus appears that the dynamical 
equations determining the strain conditions of the ether 
and the motion of the various electrons are identical on 
either form of the theory. 

It also appears that the scalar and vector multipliers } 
and A, which were originally introduced into the theory as 
undetermined functions, are, in fact, the scalar and vector 
potentials of Maxwell’s theory ; but they are here relegated 
to their proper position as auxiliary functions introduced to 
procure analytical simplicity in the relations of the theory, 
and are so far without direct physical significance. 

The present discussion does not, of course, help in the 
elucidation of any difficulties experienced in considering a 
definite choice of one of the specified forms for the magnetic 
energy distribution and the consequent rejection of the 
other, except in so far as it has removed one of the difficulties 
in the way of a general acceptance of the more usual 
expression in terms of the magnetic induction. It does, 
however, show that, dynamically interpreted, the two different 
forms of the expression are identical, and this covers the 
whole field of observable physical activities, with the possible 
exception of those associated with the phenomenon of 
radiation. The exclusion of radiation is important, and 
emphasises the essential difference between the two forms of 
the theory, as wellas the reason for the similarity in their 
dynamical aspects. It appears, on general grounds, that the 
purely radiation portions of the field, representing as they 
do detached portions of the general dynamical system which 
have become isolated from the remaining parts with a 
definite quota of the total energy, will be completely 
inoperative as regards the dynamical relations of the electrons 
and of the ether in their immediate neighbourhood ; and 
this view is supported by the known properties of such fields, 
inasmuch as the energy in them, being equally divided 
between the potential and kinetic types, is not represented 
at all in the generalized Lagrangian function of the system. 
Thus, from the point of view of the dynamics of the electron, 
and this is all we are ultimately concerned with, we may 
omit altogether the purely radiation portions of the field ; 


200 Mr. G. H. Livens on the 


and this procedure would secure the analytically necessary 
localization of the fields, which is essential to the mathematical 
developments of either form of the theory, and. it removes 
the origin of the discrepancy between the two estimates of 
the magnetic energy in the field. It would thus justify a 
further simplification of the problem by restricting it to 
stationary or quasi-stationary motions, and secure full 
confirmation for the generality of the procedure adopted by 
Larmor. 


The University, Sheffield, 
April 15th, 1916. 


XX. On the Hall Effect and Allied Phenomena. 


To the Editors of the Philosophical Magazine. 
GENTLEMEN, — 


‘ie the April number of your Magazine, in a short review 

of my paper on the ‘“ Electron ‘Theory of the Hall 
Effect and Allied Phenomena,’ Mr. A. W. Smith directs 
attention to two points in my work which may have given 
rise to some misunderstanding. In reviewing the apparent 
discrepancies in the sign of the four effects I stated, what 
appeared to be a general rule, that the relative signs of the 
four phenomena concerned are always the same. This 
statement was based on the only evidence I had access to 
when compiling the paper in question; but [am not yet 
inclined to modify it to any extent. On page 110 of 
Baedeker’s “* Die elektrischen Hrscheinungen in metallischen 
Leitern,” a table of the four coefficients for twelve metals is 
compiled from the work of Zahn. Of these twelve all but 
silver, copper, and antimony conform to the rule. To this 
list Mr. Smith adds four more substances of which three are 
irregular. There is therefore, on the whole, still a balance 
of 10 to 6 in favour of my original statement. 

However, apart from statistics, I quite agree with Mr. Smith 
that the rule in all its simplicity cannot possibly be infallible ; 
but this is only what we might expect. The necessity for 
simplicity of the physical hypotheses required in the analysis 
of a mathematical theory puts that theory quite out of keeping 
with the actual facts, which are concerned in the present 
instance with substances whose composition is known to be 
extremely complex and irregular. The constitutional irre- 
gularities of matter are in no way susceptible of exact 
mathematical specification, and have therefore always to be 


Hall Effect and Allied Phenomena. 201 


excluded in a strictly analytical theory, except in so far as in 
such cases where the average aspect of the irregularities may 
be conveniently determined and expressed. Nevertheless, 
the general compatibility or otherwise of the results of an 
ideal theory based on the hypothesis ofa suitable average may 
provide some indication of the validity of the general theory; 
and in this way only is the theory proposed in my paper to 
be used. 

The same point may be further emphasised by reference 
to Mr. Smith’s second difficulty. In reviewing the influence 
of temperature on the Halleffect in the ferromagnetic metals, 
I stated that as the temperature increases the Hall effect 
increases exactly parallel with the magnetic permeability, 
until the critical temperature is reached, when it decreases 
rapidly to a value more akin to that found in the simpler 
metals. I conclude that this is evidence for the explanation 
of the irregularities of these metals by the assumption of 
strong local magnetic fields proportional to the polarization. 
My use of the words ‘‘ exactly parallel’? was perhaps un- 
fortunate, as | did not intend to imply that the constant of 
proportionality was independent of the temperature, or in 
other words that the magnetization and Hall effect depend 
on the temperature in the same way. The phenomena of 
magnetization and conduction are fundamentally distinct, 
and the average influence of the strong local molecular fields, 
which now seem to be generally regarded as surrounding 
the molecules of most substances, will in all probability be 
very different in the two cases. The proportionality of the 
strength of the local field with the polarization has been found 
to provide a generally suticient explanation in most cases of 
the irregularities of the phenomena concerned ; but it is to 
be remembered that it is an averaged field that is operative, 
and the type of average may be very different in the separate 
cases or at different temperatures, so that the mode of 
dependence of the various phenomena on the temperature 
need not necessarily be the same. This implies that the 
constant Ain Mr. Smith’s equation may quite consistently 
be a function of the temperature, without in any way affecting 
the general conclusions drawn in my theoretical paper. 

Very truly yours, 
The University, G. H. Livens. 


Shetheld, 
May 26th, 1916. 


| e024 


XXI. On the Velocity of Secondary Cathode Rays emitted by 
a Gas under the Action of High-Speed Cathode Rays.. 
By M. Isuino, Rigakushi, of the Imperial University of 
Kyoto, Japan * 


CONTENTS. 
. Introduction. 
. Apparatus. 
. Experimental Procedure. 
Experiments on the Positive Ions. 
Experiments on the Action of the Metallic Gauzes. 
Arrangement of the P.D. between the Gauzes. 
Effect due to Rontgen Rays. 
. The Results, The Distribution of the Velocities of 
the Secondary Rays. 
. Examination of the Residual Currents. 
. General Conclusions. Summary. 


p27) CDANDA LO? LOAM I 
DID OU C9 bo 


tr 
oS Oo 


§ 1. Introduction. 


HE ionization of various substances in different states. 
has been investigated by many observers. The initial 
velocity of the particles during ionization may have an 
important bearing on the theory ‘of ionization and on that of 
the structure of the atom. Lenard t was the first to esti- 
mate the velocity of the secondary cathode rays from a 
metal. The metal was bombarded by slow cathode rays 
produced by the action of ultra- violet light on a metallic 
plate. He observed that the greater proportion of the 
secondary cathode rays left the metal with velocities less. 
than ‘“‘that acquired by the corpuscle under a fall of potential 
of 11 volts” (or merely for brevity “a velocity of 11 volts”). 
v. Baeyert, Gehrts§, and Campbell || and others investi- 
gated the same problem. They used a retarding electric 
field to determine the velocity of the secondary cathode 
rays, 2. e. so-called 6 rays. ‘The results showed that the 
velocity of the 6 ravs depended upon the velocity or the 
kinetic energy of the primary rays, provided that the kinetic 
energy was not large ; but that the velocity did not depend 
very much on the kind of metal from which 8 rays were 
liberated. They have shown that the greatest velocity of 
the 6 rays did not exceed 30 volts, notwithstanding the fact 
that different velocities of the primary rays were used. 
For the case of primary rays having very high speeds of 


* Communicated by Sir J. J. Thomson, O.M., F.R.S. 
+ Lenard, Ann. d. Phys. xii. p. 449 (1903). 

t Ve Baeyer, Phys. Zeits, x. p. 176 (1909). 

§ Gehrts, Ann. d. Phys. xxxvi, p. 995 (1911). 

| Campbell, Phil. Mag, xxv. p. 803 (1913). 


On the Velocity of Secondary Cathode Rays. 203: 


several thousand volts, Fiirchtbauer* observed, that for 
positive rays with velocity 2100 to 4500 volts the velocities 
of most of the 6 rays from a metallic plate were between 
27 and 30 volts, and were independent of the velocity of 
the primary rays. He found that this was also true when 
the positive rays were replaced by cathode rays with velo- 
cities of 1920-4400 volts. He used a magnetic field to 
determine the velocity of the 6 rays. Campbell { found 
that the initial velocity of the 6 rays was less than 40 volts, 
and was nearly independent of the velocity and the nature 
of the high-speed primary rays—which He be a or 8 par- 
ticles;—and also of the nature of the metals which emitted 
the d rays. Bumstead{, however, showed that the 6 rays 
produced by @ rays Ao polonium have velocities which 
range from 0 to 2000 volts. He pointed out the existence 
of tertiary rays which were produced by the 6 rays and had 
generally slow velocities, and he made use of a plan to 
eliminate them. Campbell§ suggested that the uniform 
nature of these 6 rays from metals may be due to the ioni- 
zation of a gaseous layer condensed on the surface of the 
metals, and not to the ionization of the metals themselves. 

So far as I know, there are not many direct experiments 
on the nature of the secondary cathode rays ejected from 
the gaseous molecules themselves. Sir J. J. Thomson | 
measured the velocity of secondary cathode rays ejected by 
slow cathode rays. He used a retarding electric tield, and 
determined the maximum velocity possessed by those secon- 
dary rays whose velocities were sufficiently high to produce 
an appreciable effect upon the luminosity of the residual 
gases. Potential differences up to 1500 volts were applied 
to a Wehnelt cathode. He found the maximum velocity of 
the secondary rays to be about 40 volts, and that the energy 
of the rays was independent of the energy of the primary 
rays and of the nature of the atom emitting them. 

The present investigation was undertaken with a view to 
ee eriaine the distribution of velocities of the secondary 
cathode rays produced from gases by high-speed cathode 
rays, and to find some relations between this distribution 
and the speed of the primary rays and nature of the gaseous 
atom. 


* Furchtbauer, Phys. Zeits. vii. P. 748 (1906). 

+ Campbell, Phil. Mag. xxii. p. 276 (1911), xxiv. p. 529 (1912), xxiv. 
Paleo (1912). 

t Bumstead, Phil. Mag. xxvi. p. 255 (1918). 

§ Campbell, Phil. Mag. xxviii. p. 286 (1914). 

|| J. J. Thomson, Camb. Phil. Soc. xiv. p. 541 (1908). 


904 Mr. M. Ishino on Velocity of Secondary Cathode Rays 
§ 2. Apparatus. 


The final apparatus used is indicated in fig.1. The primary 
cathode rays are produced in an eronecne glass discharge- 
bulb D by means of an induction-coil. The high- speed 


Bye. AL. 


r? Te gasreservorr etc. 


4{t{t{---1] 1] 1 ]----- 
“| 
[afelajale 


cathode rays are allowed to pass into an ionization-chamber U 
connected to the discharge-bulb by a narrow aluminium 
tube I of diameter 0'4 mm. ‘The primary rays ionize the 
residual gas in C, and the gas molecules emit the secondary 
cathode rays. 

To douaminins the intensities and velocities of the secondary 
rays, the method of retarding electric field due to Lenard 2 
was used. Three concentric gauze cylinders of fine brass 
wire, V;, V2, and V3, are supported by two pieces of ebonite, 
PP, around the path of the primary rays. The gauzes are 
connected to secondary cells and can be raised to any 
potentials. A tubular brass electrode E is situated outside 
the gauzes and concentrically with them. The charge on 
those corpuscles which arrived at this electrode is measured 
by means of a Dolezalek electrometer. A stout brass wire, 
insulated from the ionization-chamber with a piece of amber 
M, holds the electrode EH, and the amber is protected by an 
earthed guard-tube. 

The ionization-chamber is 8 cm. long and 8 em. in 
diameter. The innermost gauze is 1°6 cm. in diameter. 
The distances between the successive gauzes and that between 
the outermost gauze and the electrode are3 mm. T is a 
Faraday cylinder with one end fitted to the ionization- 
chamber. The other end of Tis closed by a glass window W 


* Lenard, Ann. d. Phys. viii. p. 188 (1902). 


} —— - Pump, Gauge,etc- 


lean 


produced froma Gas by High-Speed Cathode Rays. 205 


covered with willemite. The direction of the primary cathode 
rays could be thus observed and adjustments made with a 
small magnet. A horseshoe magnet NS is placed around T 
to prevent the entering rays, and any rays excited pate, 
from escaping. It was experienced in preliminary investi- 
gations, that when the ebonite pieces were struck by the 
rays, some charge accumulated on the ebonite pieces and dis- 
turbed the electrometer. ‘To get rid of this effect, short 
protecting tubes B, B’ are fitted at each end of the gauzes, 
and any unnecessary surfaces of the ebonite pieces are 
covered with earthed foils. The gauzes and the electrode E. 
are covered with candle-soot (§ 7). 

To the ionization-chamber are connected two U-tubes, U, 
and U,. A Gaede pump, McLeod gauge, spectrum-tube, and 
P.O; drying-tube are connected to Uj, and a gas reservoir 
and drying-tubes to U,. The U-tubes are cooled with liquid 
air or a mixture of solid carbonic acid and alcohol. This 
cooling condenses any mercury vapour coming from the 
pump and also any wax vapour from the joints. The whole 
apparatus was arranged to get rid of wax-joints as much as. 
possible. 

§ 3. Huperimental Procedure. 


The separate parts of the apparatus before being put 
together were washed with water and alcohol, boiled with 

then for about one hour, again washed with diculied water,. 
and finally dried. Glass parts before being thus treated 
ware washed with nitric acid. The electrometer-electrode E,. 
the gauzes, inside surfaces of the ionization-chamber aad) 
Faraday cylinder were covered with candle-soot. 

The whole apparatus was exhausted by the Gaede pump 
and charcoal tube cooled with liquid air. To drive out any 
gas adhering to the electrodes of the discharge-bulb, the bulb 
was kept discharging while the app: aratus was exhausted. 
A sinall quantity of air or hydrogen gas, previously dried 
by means of P,O; and CaCl,, was then inarouuced from a 
reservoir. The apparatus was exhausted to a desired pres- 
sure. The pressure was as low as possible, since, otherwise, 
the secondary cathode rays and the positive ions produced 
at the same time with the secondary corpuscles would ionize 
the residual gas by their passage to the electrometer-elec- 
trode. The same reason necessitated reducing the successive 
distances between the gauzes and the electrode to a value 
much smaller than the mean free paths of the corpuscles. 
and the positive ions. The value, as mentioned above, was 
3mm. However, owing to the hardening of the dischar oe- 


bulb, the pressure had to be kept above a certain value. 


206 Mr. M. Ishino on Velocity of Secondary Cathode Rays 


The actual measurements were made at the pressures between 
‘0003 and 0:0008 mm. of mercury. At this low pressure, 
the current of the secondary corpuscles was diminished, but 
was quite measurable owing to the form and position of the 
electrometer-electrode and to the highly sensitive electro- 
meter employed. The sensitiveness of the electrometer was 
1500 seale-divisions per volt at 1:5 metre distance when the 
needle potential was 200 volts. 

The different discharging potentials in the bulb were 
obtained by slightly varying the pressure, or by changing 
the primary current of the induction-coil. The potential 
was measured with a spark-gap micrometer having two 
brass spheres of equal diameters of 3°5 cm. connected with 
the discharge-bulb in parallel. 

Hy drogen gas was prepared by the electrolysis of a solu- 
tion of barium hydroxide, and the ionization-chamber was 

rinsed with a stream of the hydrogen for a considerable 
time. The purity of the hydrogen gas filling the chamber 
was examined. This was done by connecting - a small spec- 
trum tube to the ionization-chamber, passing a discharge 
‘through the tube, and measuring the wave- lengths of the 
spectral lines from the gas by a Hilger wave-length spectro- 
meter. ‘The examination of the spectrum showed that the 
ionization-chamber was filled with quite pure hydrogen 

gas. The same examination also proved that the cooling by 
means of U-tubes was absolutely necessary in order to ‘free 
the gas in the chamber fromany wax vapours. Measurement 
of the current due to the secondary rays was made to at 
least half an hour after applying the cooling material to 
‘the U-tubes. 

A great difficulty experienced, however, was in geting a 
constant deflexion of the electrometer, which measured the 
‘onization current inside under a definite condition. Much 
time was spent in trying to effect this. The induction-coil 
worked uniformly, and the electrometer very satisfactorily. 
The McLeod gauge showed that the pressure did not suffer 
marked change. But the variation of the electrometer de- 
flexion amounted sometimes to 15 per cent. The cause of 
this variation might be either a small fluctuation of the 
induction-coil, or a small trace of wax vapour inside the 
jonization-chamber. 

A method of taking readings of the electrometer de- 
flexion illustrated in the following table was adopted to 
-overcome the difficulty above mentioned. Here v was the 
potential applied to one of the gauzes, and was varied to 
-any desired value, while the other gauzes were kept at 


produced froma Gas by High-Speed Cathode Rays. 207 


definite potentials. The reading for v=0 was taken both 
before and after taking the reading for v= a desired value, 
v' say, and the mean of these two readings for v=0 was 
taken as representing the value for v=0 at the time of 
taking the reading for v=v’. The reading for v=v' was 
expressed in a percentage of the mean reading for v=0. 


Deneen Mean of Deflexion 

v. mogre Deflexion as percent. 
i f for v=0., of this mean. 

0 13°6 cms. | 100°0 
0:3 13-2— ow 96°3 
0 13°8 
0:7 {pied 18°95 796 
0 14¢1 | 
0°5 12°0 14:2 84:5 
0 14:3 | 


When the deflexion for v=0 changed too much, the ex- 
haustion of the apparatus was renewed. The deflexions 
were observed for time varying from 10 seconds to 60 seconds. 
The final results selected for the percentages of the currents 
were the averages of a considerable number of observations. 


§ 4. Heperiments on the Positive Ions. 


Positive ions are produced in the gas at the same moment 
as the negative corpuscles are ejected from the gas atoms 
under the bombardment of the primary rays. The initial 
velocity of the positive ions therefore must be due to the 
recoil caused by the corpuscles. Hence we have 


MV = NV, 


where M and mare the masses, and V and v the velocities 
of the positive ion and the corpuscle respectively. Since 
their kinetic energies are 


iIMV?=He, and 4mv’?=H’e, 


where E and E’ are the potential falls corresponding to the 
velocities V and v respectively, we have 


nel 

K = me e 
Taking the initial velocity of the corpuscle, in accordance 
with Sir J. J. Thomson’s result, as that corresponding to a 
potential fall of 40 volts, then in the case of a hydrogen 
atom, m/M=1800, and we have H=0:02 volt appr oximately. 


208 Mr. M. Ishinoon Velocity of Secondary Cathode Rays 


Thus the initial velocity of the positive ions should be very 
small. Hence it was expected that the positive ions would 
not affect the measurement of the distribution of velocities 
of the secondary cathode rays. Measurements have been 
carried out with two gauzes. To them the following poten- 
tials were applied: V,=0, V.=—v volts, where v was 
varied from 0 to several hundred volts. The currents 
arriving at the electrometer-electrode were measured. The 
results are shown in fig. 2. 


Hise2, 
S2/ON UIA 
oos OOz oo9 OOS OOv 
i =f =e] ao gl oe 


: oO 
) re) 
Negative Current in Yo 


of Ut QUISIN IAIQES2N/ 


_ 


| | | | { | 
Oo 20 so 1600 2900 300 400 


Yin Vo/ts 


The form of the curve shows that ionization occurred for 
large values of v. This ionization should be attributed to 
the positive ions, because almost all the negative corpuscles 
must be stopped at the gauze V, by such high potentials. 
But the positive ions, diffusing into the region between V, 
and V,, are accelerated there, and get a velocity high enough 
to be able to ionize the residual gas; and the corpuscles thus 
produced are driven to the electrode E by the electric force 
between V, and E. Using different fields, such as V;= —10 


a 


\QUISIND AAIQEBIY 


produced from a Gas by High-Speed Cathode Rays. 209 


volts, V,= —10—v volts, where v is varied as before, a repe- 
tition of the experiments yielded similar results. 

To see what potential difference is required to stop the 
positive ions, the following experiments were carried out 
with three gauzes. Keeping the potentials of V, and V; at 
some constant values, the potential of V, was varied from 
0 to +30 volts, the idea being to stop the positive ions with 
the potential difference between V, and V2. The results for 
three cases, viz. (V;=0, V3=0), (V,=0, V;=—10 volts), 
and (V,=0, V;=—30 volts), are given in fig. 3. 


Fig. 3. 


Orel x2 5 10 IS 20 30 40 
See Vouiri Fositive Vo/t 


The curves show that nearly all the positive ions are stopped 
by a potential difference of 1 volt. The current, however, 
increases very slowly till V,= +8 volts, and then in the case 
of curves II. and III. becomes constant, while it continues to 
increase in curve I. 

Hence in order to stop the positive ions completely, we 
should use a potential difference greater than 8 volts against 
them. 


§5. Ewvperiments on the Action of the Metallic Gauces. 


Some of the secondary cathode corpuscles will hit the 
wires of the gauzes. The effect of the hits is to diminish 
the number of the secondary cathode rays which would 
otherwise arrive at the electrode. The hits would probably 
also cause the emission of tertiary cathode rays, and perhaps 


Phil. Mag. 8. 6. Voi. 32. No. 188. Aug. 1916. E 


210 Mr. M. Ishino on Velocity of Secondary Cathode Rays 


of soft Réntgen rays. It might be expected that the hits 
would occur oftener when the gauzes are uncharged or 
charged positively, than when they are strongly negatively 
charged, as v. Baeyer * and Compton t+ suggested. As the. 
intensity of the electric field at a point just outside a wire is. 
4aa, where c denotes the surface density of the electricity 
on the wire, it is clear that, when the gauze is negatively 
charged, a negative corpuscle will be deviated from its path, 
so as to tend to pass around a wire, when it comes near it. 
Keeping constant potential differences between the gauzes,. 
while the values of the potentials were changed, the follow- 
ing measurements of the current to the electrometer were: 
made :— 


Vee Wie: Ves Defiexion. 
0 volt. +6 volts. —110volts. 2°75 cem./15 sec. 
—2 +4 —112 3°3 
—4 2 —114 3°6 
—6 0) —116 375 
=(s) —2 —118 3°85 
—10 —4 —120 3°9 
—16 —10 —126 4:0 


These results confirm the above idea, and show that it is 
necessary to charge the gauzes negatively, in order to prevent 
the corpuscles striking them. 


§ 6. Arrangement of the Potential Differences 
between the Gauczes. 


From what precedes it is clear that we must have two. 
electric fields: 


(1) One variable electric field to measure the distri- 
bution of velocities of secondary cathode rays ; and 
(2) One electric field to stop completely ail the positive 
ions. 
The condition necessary to prevent hits of the secondary 
corpuscles on the gauzes is that the gauzes must be nega- 
tively charged. | 
Various arrangements of the fields between the gauzes 
were tried, and finally the following was selected : 


V,= 12 volis, V,=—2 volts, V;=—12—v volts, 
where v was varied from 0 to 2000 volts. The first potential 


* v. Baeyer, loc. cit. 


7+ Compton, Phil. Mag. xxiii. p. 579 (1912). 


produced from a Gas by High-Speed Cathode Rays. 211 


difference between V, and V, stopped all the positive ions, and 
the second potential difference between V, and V3; measured 
the number of the secondary cathode rays, this field being 
the main one. The third potential difference between v, 
and Hi served to prevent the reflexion of the corpuscles ats 
the electrode E. This potential difference was already shown 
to be necessary by the experiments of v. Baeyer * and after- 
wards of Compton Tt and others. 

Before describing the results with this arrangement, it 
might be interesting to give some results obtained by an- 
other arrangement, in which V i=0; V;=+10 volts, and 
V. was varied from 0 to —500 volts. The results are shown 
in curve I., fig. 4. The curve cuts the V.-axis at about 


Fie. 4 


30 


aAIqeEsoyy 


P_ 


BAIZISOF = ¢— ——> 7UAIID 


fo) 


> V2 in Vol 
ane if ee 160 200 240 280 320 360 400 


—120 volts, and then the positive current increases gradually 
with V,. With this arrangement, positive ions were not 
stopped in the region V, Vo, but in the region V, V3. They 
were therefore accelerated in the first region, and acquired 
velocities large enough to ionize the residual gas. The 
positive currents for lar ge values of V, are partly due to 
this ionization. Moreover, the positive potential of V3; with 


* vy. Baeyer, loc. cit. t+ Compton, loe. cit. 


212 Mr. M. Ishino on Velocity of Secondary Cathode Rays 


respect to the electrode E did not prevent the reflexion of 
the secondary rays from the electrode. This defect accounts 
in part for the positive current. This arrangement was 
abandoned as unsatisfactory. 


§7. Lfects due to Réntgen Rays. 


Rontgen rays would be produced from the gas molecules 
under the bombardment of the high-speed primary cathode 
rays. Some effects due to the Réntgen rays were observed. 
With a brass electrode not covered with soot, the experi- 
ment described in § 6 was tried. The results are shown in 
eurve IJ. tig. 4. The amount of the negative current is 
always less ‘than that in the curve I. obtained with soot- 
covered electrode. Thisis explained to be due to the emission 
of some rays, when the Réntgen rays hit the electrometer 
electrode. 

The other effect caused by Réntgen rays will be considered 
in the discussion of the final results ($9). 


§ 8. The Results. The Distribution of the Velocities 
of the Secondary Cathode Rays. 


The results obtained for air are tabulated in Table I., the 
values of the potential differences apphed to the gauzes 
being V,;= —12 volts, V,.=—2 volts, and V;= —12—v 
volts. The first column gives the values of v representing 
the variable retarding fields for the secondary cathode rays. 
These results are calculated by noting the number of second- 
ary cells used, each cell being considered to have a potential 
difference of 2 volts. The kinetic energies of the primary 
cathode rays are written in volts in the first row. The 
figures in each column (in the second row and below) are 
the currents representing the relative numbers of the second- 
ary cathode rays which have velocities greater than v volts 
in the corresponding rows. 

As the table shows, the currents decrease very rapidly 
with increase of v up to 40 volts, then slowly, and after 
v= 900 volts they become almost constant and small, though 
not negligibly so. The currents observed in the case of the 
primary rays with a velocity of 10,500 volts are plotted in 
eurve II., fig. 5, where the abscissee indicate the values of 
v in volts. 


produced from a Gas by High-Speed Cathode Rays. 213 


Velocity of 


| 14,500 


Primary Rays. | volts. 


volts. 


70 
0-3 


70 
110 
150 
190 
270 
390 
590 
790 
990 

1190 
1390 
1590 
1990 


100-0 
94-0 
90°) 
788 
76°5 
63°7 
60:0 
526 
49-9 
45:2 
41:0 
364 
347 
32-0 
29'0 
25 0 
20°6 
175 
16-0 
13°6 
11:8 
11:07 
10°72 
10°62 
10-11 

9°35 
9°13 
8-20 
8 85 
8°68 
8°50 
8°68 


‘aan ie. 
LxGOO!, P1200 


volts. 


100-0 
9+£6 
SES 
oa 
766 
65°1 
61:0 
558 
52°4 
45°3 
41°5 
39°2 
36°0 
33°1 
29°3 
24°6 
19-6 
17°6 
15°5 
128 
11°3 
10°34 
10°34 

9°37 
Gale, 
8-94 
8-24 
8-04 
8:00 
8:00 
8°00 
8:00 


volts. 


100-0 
95°6 
91-0 
85°77 
75°6 
66°8 
61:5 
54°8 
50°6 
44-7 
41-7 
378 
36'1 
BUI) 
28:2 
24:7 
18°9 
16°6 
Lov 
12:0 
10°78 
10°42 

9°82 
9°15 
8°75 
8°36 
(08 
TAD 
7-58 
741 
2D 
T 41 


10,500 
volts. 


1000 
96°6 
94-0 
878 
Ts 
65°0 
592 
543 
50°6 
43°5 
41°4 
370 
34:3 
31°9 
28°8 
25°0 
WS 
16°8 
14°5 
116 
10°10 

9-04 
8:60 
8:02 
Ue 
721 
6°83 
6°66 
O71 
6'64 
6-71 
(I f/ 


8100 


volts. 


1C0°0) 
97°5 
91:0 
846 
770 
64-4 
61°6 
530 
50-2 
44-5 
39°5 
37°3 
30'1 
32°1 
20 
23°1 
18°4 
159 
12°7 
10°75 
9°68 
8-90 
824 
T51 
6°96 
6°69 
6-23 
6:04 
6:02 
o'84 
5°93 
5°93 


7100 


volts. 


100-0 
95°d 
93°0 
83°5 
75°6 
64°6 
59-0 
54°5 
51°6 
46°8 
41:0 
36°1 
339 
29°4 
25'8 
22°9 
17:3 
15:1 
13°3 
10:25 

8°46 
eel 
6:68 
6°24 
6-01 
5°51 
5°18 
5°08 
5°04 
4°84 
4°84 
5°04 


The results obtained for hydrogen gas are given in Table IT. 


§9 


. Examination of the Residual Currents. 


The nature of the residual currents obtained for the values 
of v greater than 1000 volts was repeatedly examined. If 
the residual currents are due to the secondary cathode rays 
with very iarge velocities, the currents should decrease as 
v increases, because it seems quite improbable that a gap 
occurs between 1000 and 2000 volts in the distribution of 
But there was no evidence of such decrease. 


the velocities. 


214 Mr. M. Ishino on Velocity of Secondary Cathode Rays 
Tasie I].—Hydrogen. 


Velocity of ‘ 15200" ~* DL400 7900 

Primary Rays. j volts. volts. volts. 
volts. 

v=0 100:0 100-0 100-0 

0:3 95°1 95°5 97:0 

05 845 85:0 85°5 

O77 761 790 776 

1:0 66:0 66-2 66°0 

15 584 56°0 579 

20 52°3 d1°5 52-1 

2°5 48:8 47-7 45°8 

3 41:3 44°3 42°8 

4 39°4 39°5 39-1 

5 39°6 35°8 36°2 

6 342 32°2 ot'4 

7 316 30°0 31°3 

8 29nk 282 28°5 

10 23-4 20°6 23°6 

14 22°1 20°6 20:2 
20 17°6 Wi-4 174 

30 16°5 14°8 134 

40 14:2 12°7 114 
70 11:0 9°62 8:50 
110 9°53 S41 7°25 
150 9°15 7:58 6°34 
190 8°52 124 3°83 
230 8:37 6°91 5°68 
270 8°05 6°74 5°47 
310 7:93 6°67 5:23 
390 (ETE 6°35 5°04 
490 759 6:21 4:91 
590 745 6°04 4-7) 
690 7-20 6°03 4°50 
790 7:05 2°68 4°40 
990 6:90 5°53 4°20 
1190 672 5°38 4°18 
1390 6°75 5°54 4°16 
1590 6°69 5°38 4:15 
1990 6°70 0°38 4:15 


Again, if they are due to the ionization of the residual gas 
by corpuscles of large velocities or Roéntgen rays, they 
should increase with very large values of v; which was also 
not observed. 

The primary rays strike the gas molecules in their paths, 
and by these bombardments they may produce Rontgen rays 
of some hardness. The Rontgen rays would strike the 


produced from a Gas by LTigh-Speed Cathode Rays. 215 


metallic gauzes, and might cause them to emit 6 rays with 
very high speed of the same order as that of the primary 
eathode’ rays. To test whether the residual currents are due 
to these 8 rays the following experiments were made. 
The innermost gauze V, was covered with a thin celluloid 
film of a thickness of about 0-0015 mm. (0-2 mg. per sq. em.), 
and the current to the electrometer was measured, taking 


V,=—12, V.=—2, V;=—12—v volts. 


o 


Deflexion Deflexion 
v. cm./20 sec. V. em./20 sec. 
volts. volts. 
) 1-4 400 Lays) 
10 135 600 1:3 
40 1:3 1000 14 
200 115335) 2000 1°35 


The currents are thus practically constant through a wide 
range of the potential differences up to 2000 volts. Next, 
the same film was applied to the outermost gauze V3, and 
the currents disappeared. 

According to Lenard’s * law of absorption-density, we can 
assume tail the absorption coefficient of celluloid (density 
1°35 gr./em.*) for cathode Pays is nearly equal to that of 
paper (density 1°30 gr./cem.*), which is’ 2690 cm.~'* for 
& rays with a speed of 30,000 volts. Then, by the law of 
inverse fourth power of velocity {, the coefficient of celluloid 
for cathode rays with a speed of 10,000 volts (which is the 
average velocity of the primary cathode rays used in my 
experiments) is 2690 x 3* em.7!. Hence the intensity I) of 
the cathode rays of 10,000 volts js reduced to the value I on 
emergence from the celluloid film, where 


Tae ASS ee 
—aijvecs = Ire 


Therefore, with this film we can practically stop ail the 
cathode rays with the velocity of 10,000 volts. The film, 
however, can be easily penetrated by Roéntgen rays, even by 
those which have a hardness corresponding to 500 volts, as 
the experiment of E. Laird + showed. Hence the residual 
currents cannot: be attributed to the secondary cathode 
rays produced by the primary rays or the scattering of the 
primary rays themselves ; but they are most probably due 
* Lenard, Wied. Ann. lvi. p. 255 (1895). 
eer J. Thomson, ‘Conduction of Electricity through Gases,’ p. 379, 
t K. Laird, Ann. d. Phys. xlvi. p. 605 (1915). 


216 Mr. M. Ishino on Velocity of Secondary Cathode Rays 


to the B rays produced from the gauzes by the Réntgen 
_ rays. 

Consequently, the currents due to the secondary eathode 
rays only can be obtained by subtracting the residual 
currents from the observed currents. The results so cor- 
rected are given in Table III. for air, the mean value of the 
currents over the range v=1100 to 1990 volts being taken 
as the residual current for each case. The current for each 
value of v indicates the relative number of the secondary 
corpuscles whose energies are greater than the v volts con- 
cerned, the total number of the corpuscles being taken 
as 100. 

TaBLE [I].—Air. 


14.500 12,600 11,200 10,500 8100 7000 


Velocity of 
Primary 


ae volts. volts. volts. volts. volts. volts. — 
volts. 
v=0 100:0 100°0 100-0 100°0 100°0 100-0 100-0 
0°3 94°5 95-1 96°2 97°4 98:4 96°3 96°3. 
0°5 89-0 OT2 90-4 93°6 91-2 92°6 90°9 
0-7 769 82°7 84:5 87°0 836 82°6 82°9 
1:0 74:3 74:6 73°6 798 756 74:4 T4A7 
15 60°4 62°71 64:1 62°5 62:1 62°7 62°3 
20 56°2 576 58'5 5674 59-2 56°9 STD 
2°5 48°] HES By eet TET 50-0 52uk a Vary 
3 45:2 48:2 46°6 471 47-1 49°] 47:2 
+ 40-0 41:0 40°3 39°3 41-0 44] 41-0 
5 304 364 oil 37°0 35°7 380 366 
6 30°4 309 32°8 32°6 33°4 32°8 320 
i 28°5 30°4 31:0 oo 30°8 30°0 30:1 
8 25°38 27°3 QA: ara 209 25°8 26°8 
10 22°3 23°2 PAT) 23°8 22°4 22:0 227 
14 sees) 18:1 18-7 Gere 18°83 18°9 18°6 
20 1371 12°6 12°4 14°1 13°2 130 1st 
30 9°66 10°38 9°88 10°90 10°65 10°65 10°38 
40 8:02 8:16 8°31 8:44 7°24 8°75 8°32 
70 5°44 Deed 4:97 5°38 4-89 5°59 5:25 
110 3°20 3°35 3°64 372 3°99 371 3°60. 
150 2°62 2°54 3°25 2°58 3°16 2-60 2°79 
190 2°24 2°54 2°61 Dy 2°46 Ea 2°25 
270 ZA 2 17, 1°88 1:49 1:67 1°37 78 
390 eSr¢ 1°24 1°45 1:18 1:09 4-15 1-28 
590 0°955 1-02 1:03 0°62 0°81 0-60 0-839 
790 0494 0251 0411 0-214 0319 02538 0324 
990 071338 . 0°044 0-151 03032 — Oty 0°147 0-104 
1190 0 0 0 0 0 0 
1390 


ye) Si) 


0 0 0 0 0 0 
1590 0 0 0 0 0 0 
0 0 0 0 0 0 


produced from a Gas by [High-Speed Cathode Rays. 217 


For the case of hydrogen, the same examination was tried, 
and the same correction was made as in the case of air. 
The corrected results are given in Table LV. 


TasLe I1V:—Hydrogen. 
Velocity of \ 15,500 11.400 © 7500 


Primary Rays. volts. volts. volts. Mean. 
volts. 
v=0 100:0 100-0 1006 100:0 
0-3 94:8 96°0 968 99°9 
0-5 83°4 84-2 84°8 84:1 
O7 TEED) USE Carat 75°8 76:0 
10 63:°5 64:3 64:4 641 
15 59'5 53°5 56:1 500 
2°0 489 48:8 50°0 49-2. 
2° 45:1 sey) 43°4 44-4 
3 oT) ze ieall 40°3 39°5 
4 35:1 35'9 364 39°8 
i) 31°0 32'1 Jda'4 32°2 
6 29°5 23'3 31°6 29°4 
7 26°7 26:0) ees 27-0 
8 24-9 ell 25°4 248 
10 20°1 21:4 20°3 20°6 
14 16°5 16:1 16°7 16°5 
20 ley 12:7 13°8 PPh 
30 10°41 9°90 9°65 9:99 
40 7:96 770 (is)s) TT4 
70 4°59 4°44 4°53 4-52 
110 3°08 307 3°22 3:12 
150 2°60 2:29 2°28 2°38 
190 1-93 1:92 74 IS 
230 1:73 1°58 1:59 1°63. 
270 1-45 1:40 1°37 1-41 
310 1°30 1:32 1-12 1°25 
390 1:13 0:985 0-919 1:01 
490 0:935 0-835 0-783 0°84. 
590 0-785 0656 0-575 0-67 
690 0515 0645 0°355 0505 
790 0354 0-275 0:251 0-293 
990 0193 0-116 0-042 0-120 
1190 0 0 0 0 
1390 0 0 0 0 
1590 0 0 0 0) 
1990 0 0 0 0 


§ 10. General Conclusion. 


As is seen from Table III., the currents for each value of 
v are nearly equal. Hence we can conclude that the distri- 
bution of the velocities in the secondary cathode rays from 


218 Mr. M. Ishino on Velocity of Secondary Cathode Rays 


the atoms in air is nearly independent of the energy of the 
primary rays. This result agrees with those obtained for 
6 rays from a metallic plate, and with that obtained for the 
secondary rays from gas. 

The mean values of the corrected currents for each value 
of v were calculated. These are given in the last column 
in Table III. The mean values may be taken as giving the 
most probably true distribution of the velocities. These are 


ine 
Fig. 5 
"S3/0A Ul 22UeJALUIG JERUERCA « is 
OOS! OO0tI oo¢! o0oz! coll ooo! 006 cos OOoL 009 ocs oleh 4 ooe coz Os! CO CS O | 
T 7 , —— [I 


| S -4I 
50 N SLI 
« 0-02 
40 S2-1Z 
35 G-Zz 
if 
30 
x } | | | | | 
ta es alee io 
PRECCOC Peer goo” 
: } | | BT ee 
ae | s 
© 15}-+-# == = See) 
x | Se Shhneee OCA | ae 
5 -i— 
ee ee | “tote 4-4-4-4f-+-4--b-} ft et tt 
G j Zo isis 5 10 iS 20 25 aa 30 


———> Fotential Difference in Vo/ts 


plotted in curve I., fig. 5. It is clearly seen that corpuscles, 
whose velocities are very large up to 1000 volts, exist in 
the secondary rays. This result does not agree with that 
obtained by Sir J. J. Thomson in the case of slow primary 
cathode rays ($1). It is noticeable that the number of 
the secondary corpuscles whose energies are greater than 
40 volts is only about 8°3 per cent. of the total number of 
the secondary corpuscles. The fact that a small fraction 
of the secondary cathode rays have large velocities agrees 
with Bumstead’s result obtained in the case of « rays. 

The curve I., fig. 5, resembles a hyperbolic curve over the 


produced from a Gas by High-Speed Cathode Rays. 219 


range v=40 to v=700 volts. Assuming an equation 


y=Czx~4, calculation by the method of least square gives 
Uive= Aketeraen Yee 


The Table IV. for hydrogen leads us to the same conclu- 
sions as in the case of air. The mean values in the last 
column in this table are plotted in fig. 6. 


Fig. 6. 


SI/Of Ul aDVAaLELSIG JE1IUIIIJ < 
OCS! Cot! COC! CCz! Coil coc! Gs cos CoOL 909 90S 00b  OO0¢ 00z2 oo! oS O 


HYDROGEN 


» —— 9 Yu QUAIING 


oO | Qo Sy ae VS ife} 15 20 25 30 
——*Potential Difference in Volts 


The mean values are very close to the corresponding values 
for air. Calculation gives an equation : 


eed ee 


over the range v=40 to 700 volts. The difference between 
the two exponents of w in the equations for air and hydrogen, 
while not negligible, is not large. These results seem to 
support strongly the fact that the distribution of velocities 
with which these secondary rays are emitted does not vary 
much with the nature of the atoms emitting them. ‘This 
property also agrees with that of 6 rays from metals (§ 1). 
For potentials lower than 30 volts, currents caleulated 
from the above equations do not agree with those obtained 


220 Mr. M. Ishino on Velocity of Secondary Cathode Rays 


by experiment, the observed values being much greater than 
the caleulated values. This is explained by supposing that 
there are some tertiary rays with slow velocities. 

The relative number of the secondary rays which are 
emitted with any given velocity can be ebeuael by finding 
the ditferential coefficient of the curves of distribution at that 
corresponding point. The differential curves are plotted in 
the figures 5 and 6 in dotted lines. The maximum number 
of the secondary rays is found at v=0°7 volt for air, and at 
v=0°6 volt for hydrogen. These values cannot be deter-. 
mined very accurately, "becanee the currents at small values 
of v may be the sum of the secondary and the tertiary rays. 

According to Sir J. J. Thomson’s * theory of ionization, 
the kinetic energy Q communicated toa corpuscle of mass m, 
at rest in an atom, by a particle of mass mm. moving with 
velocity V, is given ‘by the equation : 


4mymy 
Q (m+ my, 


4d? my, ih 
ni oe ‘ee + iN5 


where d is the perpendicular distance from the corpuscle my 
to the initial direction of the moving particle mg, T is the 
kinetic energy possessed by the moving particle, and HB 
and e are the charges on the particles m,and m, respectively. 
For the case in which the moving particles are the cathode 
rays, m,;=m, and H=e, and also the equation is simplified 
as follows 
Q=— 
= =: 
ae 


The theory assumes that (1) the force which keeps the 
corpuscle to the atom is negligible compared with the action 
of the moving corpuscles, and (2) if the energy communi- 
cated to a corpuscle in an atom exceeds a certain value, 
which may be a characteristic for the atom in question, the 
corpuscle is liberated from the atom. Suppose the energy 
necessary to ionize the atom is go, then the ionization will 
occur for the values of d less than d) given by the relation 


ie! di \ 
ected = Ue 


* J, J. Thomson, Phil. Mag. xxiii. p. 449 (1912). 


produced from a Gas by High-Speed Cathode Rays. 221 


Hence, if the moving corpuscle passes the stationary cor- 
puscle at a distance d,, which satisfies an equation 
4 


side) ana h ¢ 
a= (S57 
Jorg i 


the stationary corpuscle will be ejected with a velocity 
corresponding to the energy g. The number of the corpuscles 
ejected per centimetre of ie paths of the moving corpuscles 
will be proportional tod?. Therefore the ratio of the ejected 
corpuscles, whose kinetic energy is not less than q, to the 
total number of those ejected at the same time, is given by 
the equaticn 

de 2 — 90 Go 


dy” D0! Geman 


Now, if T is very iarge OU eee with go and q, this ratio 
is approximately equal to g,/g+q. Again, at g is large 
compared with go, this ratio becomes g/g =C J 

The corrected currents in the Tables II. a TV. are the 
quantities corresponding to g. It is seen that the values of 
**a” in the equations representing the experimental results 
are nearly unity. The agreement between theory and ex- 
periment is thus fairly good. 


Summary. 


The distribution of the velocities of the secondary cathode 
rays emitted from air and hydrogen under the action of 
high-speed cathode rays over the range 7000 volts to 15,000 
volts has been studied. 

(1.) It was found that the corpuscles were emitted with 
velocities varying from 0 up to 1000 volts, but the great 
majority of the corpuscles—about 90 per cent. of them—had 
velocities less than 40 volts. 

(II.) The distribution of the velocities of the secondary 
cathode rays appeared to be nearly independent of the 
velocity of the primary cathode rays, and also approximately 
the same for air and hydrogen. 

(III.) It was found that the distribution curve of velocity 
can be expressed with an empirical formula y=Cw-*¢ over a 
range of z=40 to c=700 volts, where y indicates the relative 
number of the secondary cathode rays emitted with velocities 
greater than « volts, and a is a constant differing slightly 
from unity. This is in accordance with the theory of ion1- 
zation given by Sir J. J. Thomson. 

(IV.) Some similarity between the secondary cathode 
rays from a gas and from a metal was pointed out. 


222 Dr. W. Makower on the 


(V.) The effect of the positive ions produced at the same 
time as the secondary corpuscles was examined, and the 
potential difference to stop them determined. The effects 
due to Réntgen rays produced from gaseous molecules by 
the primary rays were also investigated. 


In conclusion, I desire to express my best thanks to 
Sir J. J. Themson for his many suggestions during the 
course of the investigation, and also for his kind permission 
to carry out this research at the Cavendish Laboratory. 


The Cavendish Laboratory, 
University of Cambridge. 


XXIT. The Straggling of « Particles. 
By W. Maxowrr, VW_A., D.Sc.* 


fae variation in the number of @ particles from radium C 

and polonium near the end of their ranges in air was 
first studied by Geigert. Later, Taylort repeated the 
observations with radium C and extended the investigation 
to the cases of hydrogen and helium. The method adopted 
by both investigators was to count the number of scintil- 
Jations on a zine sulphide screen ke;t at a fixed distance 
from the radioactive source and to vary the ee of the 
gas through which the « rays passed. The results obtained 
by Geiger and Taylor for the variation in the number of 
ot particles from radium C in air are in substantial agree- 
ment, but the cia thus indicated near the end of the 
range is considerably larger than is predicted by theory §. 
It seemed, therefore, desirable to test the point further by 
an independent experimental method to see whether there 
is any defect inherent in the scintillation method giving rise 
to the discrepancy between theory and experiment. 

The method adopted in this investigation consisted essen- 
tially in substituting for the zinc-sulphide screen a photo- 
graphic plate which, after development, could be examined 
under a microscope. ‘The density of the silver grains in 
the film gives a record of the number of « particles which 
fell upon the pl: It is therefore only necessary to count. 
the number of grains visible in a definite area in the field 


* Communicated by the Author. 

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

{ Taylor, Phil. Mag. Sept. 1913. 

§ Flamm, Sttzwngsber. d. K. Akad. d. Wiss. Wien, Math -nat. Kl. 
exxil. [I]. A (1914); Bohr, Phil. Mag. Oct. 1915. 

|| Kinoshita, Proc. Roy. Soe, A, vol. lxxxiii. p. 432 (1910). 


Straggling of « Particles. 223 


of view of the microscope to determine the number of 
a particles which impinged on that area. 

The main difficulty of the method consists in obtaining 
plates capable of development without showing under the 
microscope a number of blackened grains, even when the 
plate has not been exposed to light or any radioactive source; 
but it was found that this difficulty coald be overcome by 
using Schumann plates, which possess the additional advan- 
tage rot having such thin sensitive films that an e@ particle 
str riking nor mally or at a small angle to the normal usually 
meets only a single silver grain in its passage through the 
film. Moreover, when viewed even under a high-power 
microscope, all the silver grains, being nearly in one plane, 
are simultaneously in Beenie. a cine nnarnee which greatly 
facilitates counting. To obtain accurate results the grains 
affected by the & particles must not be too close together, 
and it is important not to over-develop the plates, for if the 
silver grains become too large, several neighbouring grains 
may be fused intoa single large particle, and if this takes 
place the counting haa imeenabe. 


Fig. 1 (a). Fig. 1 (0). 


P 


d 


The experimental arrangement used is shown in fig. 1 (a) 
and (6), drawn in plan andseetion. The thin platinum wire 
W, rendered active by exposure for several hours to radium 
emanation and freed from adhering emanation in the usual 


way by heating, was passed thr ough holes bored in the two 


2274. Dr. W. Makower on the 


thick brass plates, A and B, which were parallel and fixed 
slightly Jess than a millimetre apart. The wire was held in 
position by wax. The apparatus was placed in a magnetic 
field sufficiently intense to deflect B rays, but too weak to 
deflect « rays appreciably. Thus only a rays could reach 
the Schumann plate P, which was 8°5 cm. long and 1*2 cm. 
wide and situated 6°2 cm. below the wire. After a suitable 
exposure the plate was removed and developed, and mounted 
on a movable microscope stage in such a way that it could 
be displaced parallel to its length. Cross-wires, suitably 
placed in the eyepiece of the microscope, defined a definite 
area within which the number of silver grains could be 
counted, and by altering the position of the plate the number 
of « particles reaching different sections of the film could 
be compared. 

In order to deduce from such observations the variation 
in the number of « particles with the distance from the 
screen, it Is necessary to apply certain corrections. For 
consider the active wire W (fig. 2) at a distance WN 
=6'2 cm. from the photographie plate. The whole photo- 
graphic effect must be confined between the points A and B, 


io 2 
Bie. 2, 


W 


By Boe XxX OA 


where WA= WB=the range in air of the « particles from 
radium C. But quite apart from any stoppage of « particles 
by air, the density of the silver grains on the photographic 
plate at any point X will diminish as that point recedes 
from N, both because WX continually increases and because 
the obliquity of the rays to the photographic plate changes. 


Straggling of « Particles. 2295 


It is easy to see that on account of these two causes the 
density of the silver grains will be inversely proportional 
to WX*. If, therefore, N be the number of grains observed 
at any point X, the variation of the product N(WX)® gives 
the required falling off in the number of @ particles in their 
passage through the air intervening between the wire and 
photographic film. 

The results of a series of experiments is shown in fig. 3, 


eurve I., which has been arbitrarily drawn to give an extreme 


Number cf Particles. 


Ob 7 | : 
GO -1 -2 +3 -4 -5 -6 -7 +8 +9 7:0 
Centimetres 


rangeof 7 cms. The absolute measurements of the distances 
involved have not been made with sufficient accuracy to 
improve upon the accepted value of the range for radium C. 
Hach point on the curve repr esents the counting of a large 
number of silver grains, varying from about 7000 for the 
points at the maximum to some 400 near the end of the 
range. ‘The number of grains outside the a-ray photograph 
which amounted to about 3 per cent. of the maximum 
intensity was subtracted from each observation. 

In the same figure (curve Il.) is given the theoretical 
curve calculated from Bohr’s formula 


1 ,-(y 
W(s)ds ae ren. ds, 
which gives the probability that the range R has a value 
between. Roi +s) and Roi+s + ds). The value of p has 
been taken equal to 1:6x10-?. Curve III. represents the 
experimental results of Geiger. Those obtained by Taylor 
have not been plotted, for he states that his results obtained 
for air agree with those of Geiger, though it should be stated 


Phil. Mag. 8. 6. Vol. 32. No. 188. Aug. 1916. Q 


226 Dr. W. Makower on the Efficiency of 


that the curves given by Taylor appear to be slightly 
steeper. 

It will be seen that the results of the investigation approach 
considerably more closely to the requirements of theory than 
those previously obtained, but, as might be expected, the 
agreement between theory and experiment is not perfect. 
The facts that the photographic film, though thin, is still of 
finite thickness, and that there may be imperfections of the 
surface emitting the radiation, lead to the anticipation of 
straggling greater than is demanded by theory. Moreover, 
as the end of the range is approached the silver grains 
become feebler and feebler and consequently more difficult 
to count, and it seems not unlikely that the same difficulty 
in the case of the scintillation method may lead to even more 
serious errors. For, whereas a photographic film can be 
studied again and again at leisure, scintillations have to be 
counted at the moment of impact of the @ particles. If, 
when giving only feeble scintillations near the end of their 
range, some of these escape detection, erroneous results will 
be obtained. But, however this may be, it will be seen that 
the photographic method gives a more rapid falling off in 
the number of « pari at the end of their range than was 
previously indicated by the scintillation method and tends to 
confirm the law given by Bohr’s theory. 


Physical Laboratories, 
The University, Manchester. 


XXII. The Efficiency of Recoil of Radium D from 
Radium C. By W. Maxkownr, M.A., D.Sc.* 


VHERE is evidence that when radium A is deposited on 
a platinum surface half of the radium B formed is 
shot into the metal so that the radium C produced from it 
is situated at varying depths below the surface. The recoil 
stream of radium D subsequently emerging traverses dif- 
ferent thicknesses of platinum, and therefores escapes with 
all possible velocities t. On this account the efficiency of 
recoil of radium D from radium C should be low, and it is 
of interest to determine its value. 
It has been pointed out to me by Dr. N. Bohr that the 
problem is a very simple one, for if the recoil-streams have 


* Communicated by the Author. 
+ Wood and Makower, Phil. Mag. Dec. 1915. 


Recoil of Radium D from Radium C. 220 


definite ranges *, and if radium A is distributed uniformly 
on the surface of a flat plate, the radium B and therefore 
the radium © subsequently formed will be uniformly dis- 
tributed through a depth a equal to the range of the recoil- 
stream from radium A in the plate. For, consider atoms of 
radium Aon the surface of the plate at A (see figure). The 


radium B produced will be shot downwards into the plate 
and distributed uniformly over the surface of a sphere of 
radius a. The fraction confined within the surface of a 
segment of the sphere of thickness 6v at a depth w below 


the surface is —, since this is the ratio of the area of the 
a 


segment to that of the hemisphere. Radium C is now pro- 
duced in situ from the radium B, and if ¢ is the range of the 
recoil-stream from radium C, the fraction of atoms coming 
from the segment considered whicl: emerges from the plate 
can be seen to be — for this represents the ratio of the 
area of the cap of the sphere of radius ¢ outside the plate to 
the total area of the sphere. Consequently, the probability 
that an atom of radium B will be projected from the surtace 
of the plate toa depth x and then give rise to an atom of 
c—ax bu 

oe 


radium D which escapes from the plate will be 3, 


The efficiency of the recoil, as usually defined, will have half 
this value, since half of the atoms of radium D must of 
necessity be projected away from the surface of the plate. 


* Wertenstein, Théses presentées & la Faculté des Sciences, Paris, 


1913. 
Q2 


228 Efficiency of Recoil of Radium D from Radium C. 
Hence the total efficiency E of the recoil is given by 


provided that a is less than c, an assumption which is cer- 
tainly correct, since the range of the a@ particles from 
radium A is less than that of the « particles from radium C., 

When a=0, H=1, and when a=c, E=34, below which 
value the efficiency cannot fall unless the surface is rough 
or contaminated with grease. 

The determinations of the efficiency of the recoil of 
radium D were carried out as follows:—A clean platinum 
plate was exposed for a short time as a cathode in an electric 
field to a large quantity of radium emanation and then re- 
moved without passing through mercury or in any way con- 
taminating the sur face. The platinum was mounted with 
wax ona piece of mica with a hole in the middle, and after 
sufficient time had elapsed for the radium A to decay, the 
mica was placed on a piece of aluminium foil. In this way 
the active platinum surface was brought near the aluminium 
without touching it. A potential difference of 100 volts 
was maintained between the platinum and the aluminium 
which was the negative pole. The electric field was main- 
tained for several hours, during which the radium C decayed. 
Some of the radium D thus produced remained on the 
platinum, whereas some reached the aluminium by recoil. 
By measuring after some months with an a-ray electroscope 
the amounts of polonium on each plate, the efficiency of recoil 
could be deduced. 

Two separate experiments were made, in both of which 
the platinum plates were exposed to the emanation for 
10 minutes. The recoil was started in one case after another 
25 minutes, and in the other after 28 minutes. It is easy 
to calculate the fraction 6 of the maximum quantity of 
radium J) found before the recoil began under the special 
conditions of the experiments. If P is the quantity of 
polonium subsequently detected on the platinum plate, and 
A the amount on the aluminium plate, the efficiency is 
given by 

2A 


4 (Peay 8), ee 


Two sets of measurements of the activities of the plates 
were made at an interval of three months and the results 


On Planchk’s 


were found to agree well with each other. 


Law of Radiation. 


229 


They are given 


below :— 
Date of Exposure to Emanation, 2Ist Sept., 1915. 

| Platey i: | Plate 2. 
Date of | MM | Mean E. 
eee Pin |) Ain ys vega UGE 

div./min. | div./min. ‘a ie: ‘diy./min.|diy./min. . a 
81.1.16 | 2°18 "83 ‘161 | -657 1:93 ‘78 "186 | -707 

| 687 
OG, | 3d4 1-46 | "161 | -696 3°37 1-315 | -186 | -690 


Inserting the value H=-68 


7 in equation (J) we obtain 


= Seog 
¢ 

The ratio of the ranges of the « particles from radium A 
and radium C in platinum seems not to have been deter- 
mined, but the value of this ratio in gold can be deduced 
from the experiments of Marsden and Richardson to be 
*72* ; and since the atomic weights of gold and platinum 
are nearly equal, this number must apply with considerable 
accuracy to the case of platinum. There is therefore more 
difference between the ranges of the recoil-streams from 
radium A and radium O than of the corresponding « particles, 
and there can be no doubt that this difference is real, for 
experimental errors such as contamination of the platinum 
surfaces would tend to reduce the efficiency of recoil and so 
increase the value of the observed ratio a/c. 

Physical Laboratory, 
The University, Manchester. 


XXIV. A Method of ae Planck’s Law of Radiation. 
By Guoreu GREEN, 1).Sc., Lecturer on Natural Philosophy 
in the University of Glasgow ales 


HE following method of deriving Planck s law was sug- 
gested by Lord Rayleigh’s paper on * The Dynamical 
Theory of Gases and of Radiation ” t. We take the radiating 
body to be a gas contained in a perfectly reflecting enclosure 
in thermal equilibrium at temperature 6; and the investiga- 
tion is based on the following assumptions :— 
* Marsden and Richardson, Phil. Mag. Jan. 1913. 


+ Communicated by Professor A. Gray, BRS 
t Scientific Papers, vol. v. p. 248. 


230 Dr. G. Green on a Method of 


Let the gas consist of molecules having internal modes of 
vibration which are identical with the modes of vibration 
of unit volume of ether. 

Let no molecule emit. radiation in any mode until the total 
energy in that mode is hf; A being a constant, and / the 
frequency of the mode. When the limiting energy is 
reached let the whole quantity hf or any portion of it be 
emitted at a discharge. 

Let Maxwell’s Jaw of distribution of velocities hold 
throughout the gas. This implies an equilibrium condition 
within the enclosure maintained by collisions, statistically 
regular absorption, and sudden emission in each mode when 
the limiting energy is reached. 

Consider now the energy distribution amongst the N mole- 
cules per unit volume in a single vibrational mode. If we 
adopt the view that each internal mode corresponds to the 
motion of an electron in an orbit whose plane is fixed relative 
to each molecule, the velocities to be considered in estimating 
the kinetic energy of a molecule in a given mode are clearly 
velocities relative to the centroid of each molecule. Hence, 
in assigning for any mode the number of vibrators having a 
given velocity, we have only to consider all phases in a plane 
orbit as equally possible—and we are not concerned with the 
directions of the velocities in space or with the orientation of, 
the orbital planes. On the above view, or on any other which 
allows us to deal with the velocities in any internal mode asa 
two-dimensional system, we have according to Maxwell’s law 


Ay ad 
(., Re 
(A’e Be ede) for any assigned vibrational mode, 
Number of molecules having energy in range (e—de, e) 


=Ae “de =) ee 


where A depends on the number of molecules and on the 
temperature of the enclosure, and /@ is the mean total 
energy of a molecule in any mode, assuming all velocities 
from zero to infinity to be possible in that mode. In the 
above equation, since in each mode the potential energy is 
on the average equal to the kinetic, we may take ¢ to represent 
total energy, instead of kinetic energy alone. This involves 
only an alteration in the value of A, which we now determine 
to suit the case where ¢ represents total energy of a molecule 
ina given mode. Integration of (1) gives 


Number of molecules in energy range (0, €) 


Sa) ie 


Deriving Planck’s Law of Radiation. 231 
Hence for the mode whose limiting energy is hf we have 


je Ses JS nae aan es 


poli i) , 


the N molecules per unit volume being all in energy range 
(0, hf). 

From (8) and (1), for the total energy of the N molecules 
in unit volume in mode 7 we easily obtain the value 


Nko — Ee cri 


But the “law of equipartition of energy” (assumed applic- 
able to the combined system of matter and ether in unit 
volume, the free modes of each being identical,——and not to 
matter or ether separately) requires that the total energy 
per unit volume in each mode be independent of f. Accord- 
ingly the radiant energy contained in unit volume of ether 
must be 

iy 
Ne hf 


(1-.-#) 


No additional terms involving @ alone can enter, in accord- 
ance with Wien’s law. The same point is clear from the 
consideration that the total kinetic energy in any mode must 
equal one-third of the kinetic energy of the translational 
motion of the centroids of the N molecules,}(N40@/2), as stated 
in Lord Rayleigh’s discussion. Hence the mean energy per 
molecule contained in mode f in unit volume of sether is 


] > 
a . e s e e e (6) 


and the number of modes of vibration in the frequency 
range df, as given by Lord Rayleigh’s paper referred to 
above with the correction indicated by Professor Jeans, is 
8277df/V*, where V is the velocity of light in ether. We 
have therefore as the total energy in ether in the frequency 


range df 


NER eee Chea 


8 hf = 
sal af oe se 


( 1) 


232 Prof. G. N. Watson on Bessel Funetions of 


or, in terms of the wave-length A, 
8a Vh dn 


(8) 


in agreement with Planck’s law. 

In the second assumption aboye the essential requirement 
is the existence of a limiting energy, hf,in each mode. For 
this it is not necessary ee emission of energy should take 
place suddenly when the limiting energy is reached. Emission 
and absorption may be continuous processes, and the attain- 
ment of the limiting energy condition may involve some 
other action than that postulated, such as the disruption of 
the molecular system with possibly very little transference 
of energy to the ether. 


XXV. Bessel Functions of Equal Order and Argument. By 
G. N. Watson, M.A., Assistant Professor of Pure 
Mathematics at University College, London. 


To the Editors of the Philosophical Magazine. 
GENTLEMEN ,— 
Y attention has been directed to a paper under the title 
“ Bessel and Neumann Functions of Hqual Order 
and Argument” by Dr. J. R. Airey (Phil. Mag. June 1916. 
pp: 520-528). The ultimate object of that paper is to 
obtain a number of numerical results concerning Bessel 
functions ; with these results | am not at present concerned. 
But in the theoretical work which Dr. Airey gives at the 
beginning of his paper, on which his numerical results are 
based, there are a number of statements for which more 
complete investigation appears to be necessary. Some of 
these statements contain errors which seem to me to be 
serious from the theoretical point of view, while others are 
merely liable to be misinterpreted by the casual reader. 
Before enumerating the statements to which exception 
may be taken, I wish ‘to emphasize the fact that these errors 
do not, so far as I can see, affect the numerical results con- 
tained in the paper. 
(1.) Graf’s approximation for J,(m) is incorrectly given 


(p. 520) as 
Jn (n)= Dao 


the correct formula is 


I'\(2)6s 
nbda(a) ~ gers, 


but this is obviously an oversight. 


Equal Order and Argument. 22> 


C1.) Dr. Airey’s formula (2) seems to me to be mis- 
leading. In it he equates the Bessel function J,(z) to an 
asymptotic expansion which is not that of the Bessel 
function but is the asymptotic expansion of Airy’s integral ; 
it is well known that this integral is only an approximation 
mnd.(2), and..im tact, the asymptotic expansions of J,(z) 
and of Airy’s integral agree in their first two terms. ‘This 
fact appears to be. implied by Dr. Airey’s statement: “‘ The 
third and following terms are correctly given in (14)”; 
tank that alvather more: definite statement would heave 
been desirable, but this is acomparatively trivial matter. 

(III.) In finding the asymptotic expansion of J,(n), 
Dr. Airey uses Bessel’s integral, and proceeds thus”: 


Cn) = = | "cos {n(sin w—w)} dw 
0 
aS ses Gh taeal,. 20° w" *y 
— A cos ae ‘a Z {90 + sn = 5 .) } dw, 
and gives a reference to Lord Rayleigh’ s work (Phil. Mag. 
Dec. 1910). Now the last expression is really 


a Mee) 
—\ cos {n(w—sin w)} dw, 
71 “0 
and this last integral is not even oscillatory but is quite 
divergent; at any Tate when » is an integer, as Is supposed 
to be the case. tor, if we take k to be a (large) integer, 


we have 
i Qkhr f kh nar 
at cos {n(w—sinw)}du= a ‘7 cos {n(w—sin w)} dw, 


because cos {n(w—sin w)} is a periodic function of w with 
period 27 ; since, in addition to being periodic, the function 
under oonetdlenauton is an even function, we have 


Tv 7 
i cos {n(w—sin w)} dw=2 cos {n(w—sin w)} dw, 


—TT « 0 
and so 
1 2kir ; 2 fe te : 
=) cos {n(w—sin w)} dw= pa cos {n(w—sin w)} dw 
= Jil ae 


and this obviously tends to infinity with 4. 


* It is implied that the infinite integral is an approximation to Jn(m). 
The fact that the evact expansion (11) is derived trom this result which 
purports to be merely approximate indicated to me the desirability of 
making this investigation. 


234 Prof. G. N. Watson on Bessel Functions of 


Since we have just proved that 


i tok 
| cos {n(w—sin w)} dw=2kJn(n), 
0 
it follows that in no sense can 


1 ic.e) 
a ( cos {n(w—sin w); dw 
vd 


be said to be an approximation to J,(n). 

The fact is that Dr. Airey has made an error of the 
same nature as one which occurs in the course of Lord 
Rayleigh’s work ; but, as Lord Rayleigh’s error appeared in 
only a single line of his work, his final result was not 
vitiated. 

Lord Rayleigh’s analysis was as follows :— 

“Tf z=n absolutely, we may write ultimately 


Jn(n) = a { “eos {n(w—sin w)} dw 
0 


= = cos {n(w—sin w)} dw 
0 


ine nw* VAN Ra 
= — cos —-dw= — C) cos a® da 
T 6 ™ \n 
0 10 
= V4)2-s3 empties. 


| OF course, at the present time, most mathematicians 
would employ a notation which has been introduced in the 
last few years and would replace the symbols = in the 
second and third lines by ~. | 


to 
For the reasons explained above, | cos {n(w—sin w) } dw 
us 
0 
is not an approximation to } cos{n(w—sinw)}dw. But, 


in the ranye O< wm, it is permissible, Jor purposes of 
approximation, to replace w—sinw by 4w*; though it must 
be stated that it is somewhat difficult to give a completely 


rigorous proof of this. And, since cos (nw?/6) dw ts con- 
0 
vergent, it is easy to prove that it 1s an approximation to 
T 
cos (nw?/6)dw when n is large. Consequently, Lord 


0 ° " e 
Rayleigh’s work may be made strictly accurate by replacing 


Equal Order and Argument. 235 


the expression (which occurs in the second line of the 
analysis quoted above) 


~( cos {n(w—sin w)} dw 
ee) o 


a! 


r T 
PB | cos (nw?/6) dw. 
7 9 

To return to Dr. Airey’s paper, his formula (9), which is 
obtained by taking a new variable =n(w—sin w) in Bessel’s 
integral, becomes an ewact result if the « in the upper limit 
of the integral is replaced by nz, so that the formula reads 


seer ON 7.6 1 / BXe 
i — i ( iG (“) mae 7 aX ( ) > ( ) ‘ % ere js) 'e d jo 
) TY ay ‘ 60 \n i 840 \n/} “ Ee ae 


ne) 


Next Dr. Airey substitutes [(p)cos3pm for } w?-'cosxda 


0 
wherever it occurs, quite disregarding the fact that this 
integral diverges except when 0<p<1, and that it is only 
when 0<p<1 that the equation 


{ x2?) cos edx=I'(p) cos par 


0 
is true. 
In point of fact, by integrating by parts, it can be shown 
‘ne 
that | 2’-*coswdu differs from I'( p)cos$pm by an ex- 
/0 


pression whose asymptotic expansion, when n is an 
integer *, is 


{ (p—1) (n)P-?— ( p—1)(p—2)(p—3)(mm)?-4 +...) cos nm, 


and so the integral of each term in the expansion given 
above for J,() differs from the expression which Dr. Airey 
substitutes for it by a term which may be written 


cos na X O(1/n?). 


Now it so happens that the aggregate of these terms 
cancel one another. But it seems to be impracticable to prove 
this by employing Bessel’s integral alone. 

To prove the correctness of Dr. Airey’s expansion (11), it 


* When x is not an integer, a second expansion of which sinzz is a 
factor has to be added to this. 


236 Bessel Functions of Equal Order and Argument. 


seems necessary to make use of Schlémilch’s contour 
integral 


gn2(t—1/t)¢—n—-1 dt, 


on 
— 
e 
wee” 
| 
| HS 
———, 
| ~ 
8 Oo 
oS ~—- 


‘or the derived Bessel-Schiaffi integral 


ar ea ; sin nw {°~ 
Jr(z)= —J cos (n6—z sin 6)dd— eo" Oe 
T T 
0 20 
from which can be derived the formula 


2 


a i ee 
J(2)= aired emg—2z sin 6 9 
< } 

aT 
the integral being taken round a contour consisting of three. 
sides of a rectangle whose corners are 


—T+H1, —T, TT, T+, 


As @ describes this contour, e=n(@—sin @) describes a 
similar contour whose corners are 


—NT +01, —nT, nt, nT+n1; 


to avoid a branch point at the origin, it is convenient to. 
suppose that the @ contour passes above it, and then the 
« contour encircles it one and a half times. 

But a contour of this nature is easily deformable into. 
the contour used by Debye in his investigations of Bessel 
functions of high order; and the problem of obtaining the 
range of validity of the expansion seems to be much more 
difficult with the contour consisting of three straight lines. 
than with Debye’s contour. 

The fact is that Bessel’s integral is extremely ill- -adapted 
for obtaining the asymptotic expression of J,(z) when 7 is 
large ; while Debye’s beautiful contour integrals, which 
(like Bessel’s integral) are special cases of Schlémilch’s 
contour integral, yield theoretically complete expansions 
without great difficulty. The real reason for this les in the 
fact that the integrands in Debye’s integrals are positive 
and monotonic while the integrand in Bessel’s integral is 
oscillatory. 

I might remark that I have obtained the first two terms 
of the expansions of Jy(n) and Jxn’(n) by using Bessel’s. 
integral, though it was necessary to use Tannery’s s theorem * 


for integrals in the course of the analysis. Lo obtain the. 


* Bromwich, ‘Infinite Series,’ p. 443. 


Bessel Functions of Hqual Order and Argument. 237 


third and following terms, exceedingly intricate algebra 
would be necessary. 

(IV.) The explanation of the fact, mentioned by Dr. Airey 
on p. 523, that all the formule for J,(n) etc. are true when 
n is not an integer will be seen from (III.) above; for these 
formule really have to be derived from the Bessel-Schlafli 
integral, or some such result, which holds whether n is an 
integer or not. 

(V.) It would have added considerably to the interest of 
Dr. Airey’s paper if he had given soine indication as to how 
formula (16) on p. 526 for J_,(7) is obtained without making 
use of the methods employed by Debye. 

I am, Gentlemen, 
Yours faithfully, 


Trinity College, Cambridge. G. N. Watson. 
June 17th, 1916. 


XXVI. Bessel Functions of Equal Order and Argument. 


To the Kditors of the Philosophical Magazine. 
GENTLEMEN,— 


S Prof. Watson surmises, the object of my paper was 

to obtain formulee for the calculation of J,(n), Yn(n), 

ete. for integral values of the order and argument and was 

not intended in any way as a contribution to the theory of 
these functions. 

Formule for Jn(n), J_n(m), Yn(n), ete. had already been 
given, and for the purposes of the paper, the calculation of 
further terms was all that was required. I can corroborate 
Prot. Watson’s statement that intricate algebra is necessary 
for the computation of the third and following terms. 
However, Bessel’s Integral appeared to afford a simple 
method of deriving the asymptotic expansion of J,(a) in 
the case under consideration. Although divergent integrals 
were employed and errors of small order introduced thereby, 
it is interesting to note that, owing to these errors cancelling 
one another, results were obtained which were in agreement 
with those derived from the Bessel-Schlafli integral and the 
contour integrals of Sommerfeld and Schlomilch. 

From the Bessel-Schlifli integral, Prof. Nicholson found 
that, to order = J,(z) can be equated to the asymptotic 
expansion of Airy’s integral as in formula (2). Debye’s 
formule for J,(a) and J_,(wv) derived from Sommerfeld’s 


238 Bessel Functions of Equal Order and Argument. 
integral are given in (7) and (16). The following expressions 
were obtained by Graf from Schlémilch’s integral : 

When x<1, 


e—a(y— tanh y) 


it 
VC) ~ 
(A, / 2a tanh y’ a oe a 
When eal, 
Ge ey 
Tile) ~ 52 
al) massa =a, 
When z>1, 


We 


2 cos | 2(tan d—6) 


Tig pion 
alae) a/ 2rrx tan § 


Graf proceeded to throw the last expression into the form 


J0)~AJ = 00s E (a+ 5) | ‘ 


and found that 8,, the nth root of J.(z), is represented 
approximately by 


1 
-, Where cos o=.=. 
2 


5 1 
b= ( 2n +a— ae 
overlooking the simple formula 


8, sec , 


where ae ee 
(ano pee, 


and which is especially useful in caiculating the roots of 
the functions .J,(~%) . of high order. This result is given 
in detail in the Phil. Mag. July 1916, where, as in the paper 
on “The Bessel Functions of equal order and argument,” 
the integral 


( 67-1 cos 9d0 =T' (a) cosa 

0 = 

was employed for values of w outside the limits 0<a<1. 
Perhaps, in this case also, the correct result may be due 

to the errors being mutually destructive. In the expression 


(14) for tane, 7 is equal to a not ntan®@ as stated in 
the paper. tes 
I am, Gentlemen, 
Yours faithfully, 
Joun R. Arey. 


Ve 239) a 


XXVIII. On the Absorption of Gases in Vacuum- Tubes. 
To the Editors of the Philosophical Magazine. 


(GENTLEMEN,— 
W* shall be much obliged if you will permit us to add a 
note to our paper on “Absorption of Gases in Vacuum- 
Tubes, &c.,’ published in the May issue of the Philosophical 
Magazine. Arter the publication of our paper we received 
copies of a paper by Dr. L. Vegard entitled “ Uber die 
elektrische Absorption in Entladungsréhren,” Videnskaps- 
selskapets Skrifter, I. Mat.-naturv. Klasse, 1913, No. 7, 
Kristiania. We regret that we had no access to this paper, 
as Dr. Vegard to a certain extent anticipated some of our 
results. The preliminary account of Dr. Vegard’s work was 
published in the Philosophical Magazine, xviil. 1909, p. 465. 
However, owing to the form of the title—* Hlectric Discharge 
through HCl, HBr, and HJ,”—the possibility that the paper 
might have any reference to our work escaped our notice. 

As affecting the subject of our paper, we should like to 
mention that Dr. Vegard found that there was no absorption 
—except fora little which he calls ‘‘ non-conservative ”— 
with normal cathode-fall, whilst with abnormal fall the rate 
of absorption increased with increasing fall for nitrogen, 
oxygen, and helium, but not for hydrogen. Dr. Vegard 
also suggests that there is some connexion between absorption 
and disintegration of the cathode. We have much pleasure 
in acknow edging Dr. Vegard’s priority in these results. 

Dr. Vegara suggests that the absorbed gas enters into 
combination-—“‘ due to a new kind of combining power 
called ‘electric affinity ’ ’’—with the disintegrated particles. 
of the cathode. 

Yours faithfully, 
Physics Department, S. BRoDETSKY, 


University of Bristol. z 
Mey 22nd, 1916. B. Hopeson. 


XXVIII. On the Electric AD of Baik in 


Vacuum- Tubes. 


To the Editors of the Philosophical Magazine. 
GENTLEMEN,— 


N the number of the Phil. Mag. for May this year 
Dr. 8. Brodetsky and Dr. B. Hodgson have published 

an interesting paper on the absorption of gases in vacuum- 
tubes and allied phenomena. In this connexion I should 
like to call the attention of the authors to some papers 


240 Dr. L. Vegard on the Electric 


dealing with the same subject, which I have published some 
ears ago. 

As these papers seem to have been overlooked by the 
authors, I should like to state briefly some of my results. 

In a paper appearing in the Phil. Mag. for Oct. 1909 
I published some experiments on absorption in oxygen and 
hydrobromic acid, showing that the absorption was mainly a 
function of the cathode-fall. As long as the cathode-fall was 
small the absorption was very small indeed ; but if in any way 
—either by increase of current or seater of pressure —the 
cathode-fall was raised the absorption rapidly increased until 
wt reached a maximum of the order of the electrochemical 
equivalent. 

It was concluded that the absorption took place at the cathode 
and was closely connected with the disintegration of the cathode, 
and to explain these results I put forward the hy pothesis that 
the Ei caon was produced by the rapidly moving positive 
ions (positive rays) before the cathode. 

Further experiments were carried out at the Cavendish 
Laboratory in 1909 on the absorption inoxygen, helium, and 
hydrogen, and the results given in a paper which was put 
46 the care of the Society of Science at Christiania, in 1910. 
In this paper I introduced a distinction between two kinds 
of absorption. One quite definite and lasting, which I called 
the conservative absorption, qadtanother which jeamem non- 
conservative, and which is to designate a number of absorption 
phenomena which may have various causes and essentially 
depend on the state of the tube. 

When the bulb has been run for some time the conservative 
‘absorption becomes definite and can for the same tube very 
nearly be regarded asa single-valued function of current and 
pressure. This function was studied, and it was found that 
the absorption per coulomb was e essentially dependent on the 
cathode-fall in the way shown in my first paper, and set in 
when the cathode-fall was raised above a critical value. 

For a cathode-fall. C which did not greatly exceed the 
critical value Cy and a constant current, the absorption per 
coulomb was found to be approximately given by the expression 


G=kO—O)).. . .° 2 
Lalso calculated the number (yz) of atoms absorbed per coulomh. 
In the table are given the valnes a where N is the number 


of molecules in a cubic centimetre of gas at atmospheric 
pressure and 0° C. 


Absorption of Gases in Vacuum- Tubes. 241 


w/2N. 
Eleetrodes. Absorption. | Disintegration. 
Old | eye detent | 1-25 x 10—* (C—320) | 95x 10-* (C—495) 


lacinumay .tewecsee- 0:90 x 10-4 (C—400) | 0:81 x 10-4 (C—495) 


The absorption was determined for gold and platinum 
electrodes. In the last column are given the corresponding 
disintegration values for discharge in air as caleulated from 
results of Holborn and Austin*. The table shows the 
intimate relation between absorption and disintegration. 

The considerable difference between the absorption with 
gold and platinum electrodes was used to prove, by alternately 
changing the metal of the anode and cathode, that the con- 
servative absorption was connected with the cathode. 

The relation (1) only holds approximately for cathode-falls 
up to about 1000 volts. When the cathode-fall is increased 
still more the absorption increases less rapidly and reaches, 
as already mentioned, a maximum value. 

In helium the absorption was smaller ; but there was a 
distinct conservative absorption to be observed. 

In hydrogen, however, no conservative absorption was found, 
but only absorptions of the more conservative type, which 
very much depended on the state of the buib and its previous 
history of running. This want of conservative absorption in 
hydrogen gave an explanation of the effect of the residual 
gases of the bulbs to reduce the velocity of absorption. 

Regarding the view taken about the mechanism of the 
process, I give the following statements from my paper. 

“ Whatever may be the view we take about the mechanism of 
the process, we cannot escape from the assumption that gas is 
attached to the metal and carned away with the metallic 
deposit.’ And further, “The previous results suggest that the 
electric discharge imparts to the gas a new kind of combining 
power, ‘electric affinity, which probably is a function of the 
cathode-fall and which is a power attached to ions moving with 
a high velocity.” 

The attachment of the gas to the metal was considered as 
something between an occlusion and a real chemical combi- 
nation, and was compared with the attachment of helium to 
the radioactive minerals. 


* Wissenschaftl. Abh. d. Phys. Techn. Reichsanst. B. iv. p. 101 


(1904). 
Phil. Mag. 8. 6. Vol. 32. No. 188. Aug. 1916. R 


242 Prof. W.M. Thornton on the Cause of Lowered 


An extensive series of experiments were further under- 
taken in Christiania in 1910. The absorption was studied in 
pure nitrogen with a number of electrodes of various metals, 
and found to follow essentially the same law as already stated 
for oxygen. Further, the rate of disintegration of the 
cathode was measured with the same tube as that used for the 
absorption experiments and for gold and platinum electrodes. 

The results of my experiments on electric absorption were 
published in the Proceedings of the Christiania Academy™*. 

Reading the paper of Dr Brodetsky and Dr. Hodgson, 
I was very glad to notice that they have confirmed the 
results which I obtained in 1909. As regards the view 
taken about the mechanism, we also agree in the essential 
points. Thus they assume the absorption to be due to the 
velocity (energy or momentum) of the positive ions before 
the cathode, and they suppose the gas to be attached to the 
disintegrated metal. 

Still Edo not quite agree with the assumptions on which 
they base their theoretical deduction of the absorption curve, 
for both assumptions would for a constant current make the 
velocity of absorption increase continually with increase of 
eathode-fall, while all experimental evidence goes to show 
that there is an upper limit for the velocity of absorption. 

Yours faithfully, 


University of Christiania, L. VEGARD. 
May 12, 1916. 


XXIX. The Cause of Lowered Dielectric Strength in’ High 
Frequency Fields. By W.M. Tuornton, D.Sc., D.Eng., 
Professor of Electrical Engineering in Armstrong College, 
Newcastle-upon- Tyne F. 


1. "EXHE breakdown strength of dielectrics falls with rise 

of frequency of the field, up to the frequencies used 
in wireless telegraphy. The experimental evidence for this 
is not extensive, but it is found in solid, liquid, and gaseous 
media. For example, glass having a strength of 320 kilo- 
volts per centimetre at 50 cycles a second bears only 128 
kilovolts at 8500 cyclest. A light petroleum oil failing at 
170 kilovolts at 60 cycles, fails at 67 kilovolts at 90,000 


* “Uber die elektrische Absorption in Entladungsréhren.” Kristiania 
Vid. Selsk. Skr. Nr.7 (1918). 


+ Communicated by the Author. 
t Moscicki, Z. 7. Z., 1904, p. 527. 


Dnrelectric Strength in High Frequency Fields. 243 


eycles a second*. In air the change is less, the breakdown 
voltages at 60 and at 40,000 cycles a second being respec- 
tively 22°5 and 19°5 kilovoltst. The effect is not to be 
accounted for by temperature change, for during the test 
this does not rise appreciably. An explanation ft has been 
given in general terms based on the fact that on the appli- 
cation of the field there is polarization at the velocity of 
electric waves in the medium, followed by a further polariza- 
tion caused by the interattraction of the separated molecular 
charges. This secondary polarization is not instantaneous, 
and the breakdown conditions are determined by the super- 
position of the two fields, the elastic restoring force being, as 
always in dielectrics, proportional to the displacement. 


2. Let F be the intensity of the applied field, and «x the 
electrical displacement. When, as in paraffiin-wax, inter- 
attraction is negligibly small, be =F. When present it has 
been shown § to be approximately proportional to «?, so that, 
when fully established, 


beet + ca? | em Net 


\ 


During the growth of the internal field || we may write, when 
F is unidirectional, 


by ca7(, | = 8 aaa rns ak) (C2) 


where « is constant and small relative to the time of test, or 
to the periodic time when the field is alternating. From this 


| b= 20m ies) = — acare-t = 0) 


Or da acne 
ey) 


dt b—2ee(1—e-*) ° 


This is infinite, and the material breaks down, when 


b 
© a(n) Me dl'i on (AP) 


§ “The Electric Strength of Solid Dielectrics,” loc. cit. p, 180. 
__ || See “The Polarization of Dielectrics in a Steady Field of Force,” 
Phil. Mag. March 1910, p. 401. 


R 2 


244 Prof. W. M. Thornton on the Cause of Lowered 
The field at which this occurs is 
1)? 
Ree (5) 


= 


and if time is given for interattraction to be complete 
Hye 
tee 
When the applied field is alternating let F=F sin pt ; 
then 
b? 

Fy Sim Bi aaneie 6-2)? . se) ea 
where pt, is the phase at which failure occurs. Since 
p = 2nr/T, where T is the period, and @ is always much 
less than T, take t;}=q/2, g being a small fraction. Then 


, in a A ores + 


pg - 
When / is small, 
a 


b?c 
Fop = Aeq(1—e-*) = . : . Se D (8). 
This is constant, so that if breakdown occurs, as it always 
does, in an early stage of the maximum possible polarization 
(that is, the maximum which would be reached in a steady 
field if the structure did not break down), Fop is constant, 
and the breakdown voltage is inversely proportional to the 


Jrequency. 


3. Polarization at a voltage well below that of breakdown 
takes many hours to reach its final value of equilibrium in 
the field, and this value is independent of the external field *. 
The influence of time of application of the field on break- 
down strength is very largely controlled by the rate at which 
interattraction is established. 

A full experimental examination of the influence of time 
of application on breakdown voltage has been made recently 
by Ma. F..W. Peek, Jdriqe) gale has arrived at the empirical 
formula 


F,=A+BT 2 


where I’, is the breakdown gradient and T the time of its 


* Phil. Mag. loc. cit. 1910, p. 398. 
T Loc. cv; page: 


Dielectric Strength in High Frequency Fields. 245 


application. This may be compared with equation (5) above, 
and the graphs of both are given below, taking 


A= 2/4; a=" ao. 


_ b?/4e ; 
l—_e77t 


Nea Th. |! EF ae, 


Curve I. = 


O 5 10 iS 20 es 
: MINUTES 


The close resemblance of the curves would appear to 
justify the view that the observed influence of time of 
application on breakdown strength is largely, if not entirely, 
caused by the rate of growth of interattraction, and that 
Peek’s formula might be replaced by a simpler exponential 


type. 


4. The nature of dielectric hysteresis has long been under 
discussion. That it is a time and nota static phenomenon 
was recognized from the first, but it would appear that the 
use of the term “viscosity” instead of hysteresis, or as indi- 
eating the cause of it, is not tenable. The retarding force 


246 Prof. W. M. Thornton on the Cause of Lowered 


due to viscosity, if taken as usual to be proportional to rate. 
of displacement, leads to results not in keeping with the 
experimental facts. 

Writing equation (1) with a viscosity term 

c +be—ca=Vosin pt. .. . eee 

This equation was submitted to Professor A. R. Forsyth, 
who very kindly gave the following note upon it :— 

“The equation is not integrable in ‘finite terms.’ It 
belongs to the Riccati type, and its integral can best be 
expressed by means of series. Take 


ee Oe b 


— 2 a ° ° 2 . (10) 
the equation becomes 
ad7u cK . b? 
ie (= sin pl — iz) u=(. . 2 gees 
1 : ses 
Kei pt — 37 +20, and write sin?@?=2z; the equation 


becomes, first 


ae? ap” 


7 


d7u 4cK < b? 
ae ( %eos 20 — a)" = 0, > 5 aes 
and then 


Y; On ek du | (cKk)—ib? 2cF,z 
z(1 2) 752 + (5 =a “)o a a ap )x = 0. (13) 


Of this equation there are two distinct integrals proceeding 
in ascending powers of z, which is positive and not greater 
than 1. One of these is of the form 


lpkethe +... =t, . ae 


and the other 
Ze [,z9/? + J,29/2 + cco — ibs. . 5 ° 6 3D) 


and the most general value is 
i = Ade BIT,, 4 y+) os Td 


where A and B are arbitrary constants. Thus the value of 
« would be 


RE ap tls 
me a dt dt 17) 
= Gols eg BT 9 anal 


which really contains only one arbitrary constant...” 


Dnelectrie Strength in High Frequency Fields. 247 
cF,—iv? 2cF 


Writing mea and ep =” and substituting 


for win equation (13) the series A(L+hyz+h 27+ ...), we 
obtain 


Z kg +(m—1)k,—n i 


ks + (m— es +3)a—nk, f 


iS aae a 
Gomme nee 3 


bf oo) 
Vn eae 


ae 24 4i.4 (m—IF3FB)lg—nky \ 


Seg tee ORE SUS Mame einer i eS Oe ae TBE ai (18) 


Hach term vanishes identically and the first series becomes 


9 
T, = 1—2mz + 3 { (m—1)m +n} 2? 


iy 9 . 
— , | (m4) 3 { (m—L)m-+n +2me | Zee LO) 
In the same manner the second series gives 
3 1 
yeJ 4-2 ey aint 
i ics sly+ (i mn) +o} 


¢ OW) 
“4 ge (2. ls + (m— a) 4—n } 


post a” 
ae ie ee ; I,+(m— =) I,— nl, } 


In this = — is 2 and 
Mm — 1/2 = 3/2 2) md/2 
at 62 oe CO ima cray WO wi)istn fe [eG 
(21) 


Making the necessary substitutions and finally obtaining 
a series for aa the conditions for breakdown are given by 
¢ 


equating to zero the denominators of successive terms, each 


4 : ax . , 
of these then making sie infinite. 
¢ 


= 


248 Prof. H. F. Dawes on Image 


This gives, in order of approximation, 


= F 62 ap b? " tiaZp? 
C 


eae he"? | a ee Sete 

In these p never occurs in the denominator, so that, as 
might be expected, viscosity of displacement of charge would 
increase the field required for breakdown as the frequency 
is raised, and this is not the case. 

Dielectric hysteresis must therefore be considered as 
neither a static nor a viscous retardation of displacement, 
but as a quasi-elastic adjustment of equilibrium of the field 
of interattraction between opposite charges in molecules 
whose (fixed) axes are in perfect confusion of alignment ; 
the molecules being partly polarized, and interattraction 
started, by the application of the external field. 


a = (36? + 28a2p?)8, 


XXX. Image Formation by Crystalline Media. By H. F. 
Dawes, Professor of Physics, McMaster University, Toronto, 
Canada *. 


il 1877 Stokes t discussed the question of the position of 

the image (the ‘‘apparent depth” of an object-pvint) 
due to the refraction of light from a crystal to air through a 
cleavage-plane or other plane refracting surface. ‘This paper 
was written with the object of developing a method of testing 
crystals by applying the then recently discovered microscope 
method of determining the index of refraciion, and was 
tested out for a great many specimens by Sorby f. 

Stokes’s investigation does not seem to have been very 
generally followed up or applied from an optical point of 
view, although the results are particularly interesting in the 
study of the optical properties of crystals. In fact, the 
only record ot the application of the principles developed 
which I have been able to find is in Clay’s admirable 
collection of Experiments in Light. Independent of these 
sources, however, the writer developed in 1908 a simple 
laboratory experiment of the type under consideration for 
use in his laboratory classes at the Physical Laboratory of 
the University of Toronto. In addition to the phenomena 
considered by Stokes there are further cases of image 
formation which are interesting and important, and the 
writer proposes to consider certain of these. 


* Communicated by Professor J. C. McLennan, F-.R.S. 


+ G. G. Stokes, Proc. Roy. Soc. xxvi. p. 885 (1877). 
t Sorby, eid. p. 384. 


(22) 


Hormation by Crystalline Media. 249 


General Considerations. 


The ordinary light in uniaxial crystals is propagated in 
precisely the same way as in ordinary isotropic substances, 
so that the laws for image-formation by this light are w ell 
known and need no further discussion. It is the extra- 
ordinary images in these crystals and both images in biaxials 
which follow more complex laws,—due to the more compli- 
cated shape of the wave-tront and the fact that the rays are 
not necessarily at right angles to the waves. In investi- 
gating the laws for the image-for mation in such eases, there- 
fore, it is necessary to take into account the curvatures of 
the incident wave as well as the nature of the refracting 
surface and the change of velocity on refraction. The image 


co) 
is in general astigmatic, so that the investigation involves 


the determination of the positions and directions of two 
focal lines. 

As an example of the method which must be employed, 
consider the formation of the image of a point-source within 
a uniaxial crystal by the extraordinary light refracted into 
air through a plane surface which is normal to the optic 
axis of the crystal. Such an image is intended to be seen 
by the eye or a low-power microscope, so that the pencil of 
rays coming from object or image-point is of small angular 
aperture. Hor this reason the ‘standard approximations of 
the “first order” discussion of imave-formation are justified 
and will be assumed without comment. ‘The procedure here 
followed is not the same as that of Stokes, although it 
amounts to the same thing in the end. 

As is well known”, this extraordinary wave in a uniaxial 
crystal travels out from a point-source in the form of a 
spheroid whose axis of revolution is the optic axis of the 
erystal. The following are geometrical and optical properties 
of such a wave-front. 

The principal indices of refraction are m equal to V/a, 
and n, equal to V/c, (1), where V is the velocity of light in 
air (or vacuum), a is the velocity of the extraordinary, wave 
along the axis, and c its velocity at right angles to the axis ; 
a is also the velocity of the ordinary wave. 

If a wave has travelled a distance v along the axis it has 
also travelled a distance y at right angles to the axis such 
that y=cw/a, (2), so that the semi-axes of the wave-surface 


* Drude, ‘Theory of Optics,’ Properties of Transparent Crystals, 
from (55) setting b equal to c. LTurther relations are quoted or are 
deduced very directly from results in the same chapter; they will be 
referred to by the number of the equation which represents them. 


250 Prof. H. F. Dawes on Image 


are wv, y,y. The radius of curvature of the spheroid in any 
normal section at its pole is 


r=yp/e=Cxla=nrx/nz’, (3). 


The diagrams are drawn to correspond to a crystal of 
principal indices nyj=4/3 and n»=2. The trace of the wave 
is shown at incidence and just after refraction; also the 
circle of curvature of the elliptic wave is shown in certain 
cases. The traces for actual wave-positions are indicated by 
continuous lines and for “ virtual”? wave-positions by dotted 
lines. ; 

For the example under consideration the source of light P 
lies within the crystal at a distance p from the refracting 
surface at M (fig. 1). The ordinary waves form an image 


igs A; 
Pr 


at a distance g from M such that p/q is equal to 7, (4), as 
in ordinary isotropic substances. The extraordinary waves 
also form an image, but it is not situated at Q, although the 
two sets of waves travel with the same velocity along PM. 
(This is contrary to the statement usually made in discussions 
of this subject, viz., that only one image 1s seen on looking 
along the axis into a crystal. It is true that the eye seems 
to see only one image, but this is because the one image is 
vertically beneath the other-and the depth of focus of the 
eye is very great compared with the distance between the 
images. The presence of the two images may, however, be 


Formation by Crystalline Media. 251 


readily demonstrated by means of a microscope ; for example, 
in the case of calcspar it is necessary to move the microscope: 
along its axis a distance of about one-seventh of the depth 
of the object-point in the crystal in order to change the focus. 
from the one image to the other.) 

The formation of this second image arises from the fact 
that although the waves have equal velocities along the axis 
they have not the same curvatures, and the position of the 
image is determined not only by the change of velocity on 
refraction but also by the change of curvature. Since the 
_wave has travelled a distance p along the axis, the radius of 


curvature of the wave-front is pa? as it reaches the 
refracting surface, by (3). Now, when refraction takes 
place at a plane refracting surface, the ratio of the curvatures 
of the incident and refracted waves is equal to the ratio of 
the velocities of light in the two media. Hence the radius 
of curvature of the wave as it enters the air is given by 


9 
s Cp 


cpa =a i «aang 


and since the image in air is situated at the centre of 
curvature § of the waves coming from it, the distance s must 
be the required distance from Five retracting surface to the 
image. In the diagram, P'M is the incident wave, R'M its 
circle of curvature, and 8’M the refracted wave. 

These results may be used to measure the principal extra- 
ordinary index cf the crystal by the microscope method. 
Thus, s/p which is equal to c?/aV is also equal to n/n”, (5), 
by (1). The image is therefore at the same depth as it 
would be in a substance of index n/n, and the value of n, 
is given by 


ng=— or — by (4). 
( 


The values of p, g, and s will then be found by focussing 
the microscope in turn on P, Q, and 8, and finally on M. 

No one appears to have called attention to the fact that 
the two images cannot be distinguished from one another by 
the usual polarization test if the eye or microscope is placed 
symmetrically over the axis of the crystal through the 
source. For, in the wave-front of the extr aordinary. pencil 
the vibration directions radiate out equally in all directions 
from the axis, so that an analyser will transmit the same 
amount of light in all azimuths and the source will appear to 
be of the same brightness for all a of the analyser. 
On the other hand, in the wave-front of the ordinary pencil 


252 Prof. H. F. Dawes on Image 


the vibrations are uniformly transversal about the axis as 
centre, and therefore the image formed by this pencil w ill 
also seem of constant brightness as the analyser i is rotated. 
If, however, there is any departure from the symmetry of 
the position of the eye, the brightness of the images will 
change as the analyser i is rotated, so that the images may be 
distinguished. Even in this case there is no position of the 
analyser which will produce complete extinction of either 
image. 


IMAGE FORMATION UNDER OTHER 'YPES OF 
CIRCUMSTANCES. 


1. Image of a point in air by light refracted into a uniaxial 
crystal through a plane interface normal to the optic axis.— 
‘The position of the image in this case may be determined by 
a process similar to the above. Thus, in fig. 2, with the 


Fig. 2. 


source at P, the curvature of the wave P’M before refraction 
is 1/p; after refraction it will be a/Vp. But after refraction 
the wave will proceed as part of a spheroid R’M whose 
principal curvature has this value, the rays radiating from 
the centre R of the spheroid. Since the required image is 
at the point from which the light thus radiates, its position 


oO 


is this centre and its distance from the refracting surface is 


the corresponding semi-axis of the spheroid. In accordance 
with (8) this distance is ~~~ _ 


or, in terms of the indices, n,*p/m. This distance also is 


Formation by Crystalline Media. 253% 


the same as if the light were being refracted into a substance. 
of index 77/1}. 

2. Image formed by direct refraction through a plane 
parallel plate of uniaxial crystal of thickness t, the faces being 
normal to the awis.—The light is refracted into thé erystal in 
accordance with the law developed in the preceding section, 
and is refracted out according to that of the first example 
considere:!). Thus, if the source of light is at P in fig. 2 at 
the distance p from the first surface M, the virtual source of 
the refracted pencil is R at the distance nj 2»/n, from that. 
surface (as in Section 1) and cone a at the distance: 
t+n,?p/n, from N at the second surface of the plate. The 
image formed by the second refraction is therefore by (5) 
situated at § at the distance p+in/ny from N. 

The distance from the final image to the original source is: 
t(1—n,/n»”), (7), so that the image is ‘drawn up” toward’ 
the observer by this amount, which is the same as for an 
isotropic plate of index n,?/n, and is independent of the 
distance of the source. 

3. Image formed by direct refraction from one uniaxial 
erystal to another through a plane surface of separation which is 
normal to the axis of each crystal. —The set of waves forming 
the image is extraordinary in both crystals, since the plane. 
of incidence and refraction is a principal section in each. 

One may deduce the law for the image position in this 
case from the laws developed for refraction from air to- 
crystal and from crystal to air. Suppose for the moment 
that the two crystals are separated by a plane parallel air-. 
space of thickness t, and that the light is refracted from 
crystal 1 into this air-space and then into crystal 2. Accord- 
ing to (5) the image formed by the first refraction is situated 
at aM distance »,'p/n? from the first refracting surface, and 
therefore (¢+7'p/n.) from the second. Since this image 
forms the source with reference to the second surface, it 
follows by (6) that the distance of the final image from this 
surface is (¢-+my'p/i_”) ng! /n4". 

Now the direction of the final part of any of the rays 
forming this image is independent of the thickness of the 
air-space, and is therefore the same as if the thickness were 
indefinitely diminished, 7. e., as if the crystals were in optical 
contact. Hence the required image position, which is deter- 
mined by the direction of the rays for optical contact, is the 
limiting position of the above image when the value of ¢ is 
reduced to zero. Its distance from the surface of contact 
will therefore be 


254 Prof. H. F. Dawes on Image 


As in other cases considered, this distance is just the same as 
if the light were refracted from a substance of absolute 
refractive index ng */ny’ to one of index n,''?/n4"’. 

4. Image formed by direct refraction through a series of 
plane parallel plates of uniawial erystal all having optic axes 
wn the common normal to the surfaces.—The extraordinary 
wave in the first crystal will be extraordinary i in ali suecessive 
plates and the ordinary will remain ordinary, so that only 
two images will be formed at each stage. 

By the ar gument of Section 3 the image-position at each 
stage and therefore the final i image-position will be the same 
as if the light were refracted into and out of a thin air-film 
of negligibl e thickness separating each pair of plates. By 
(7) the i image is “drawn up”’ toward the observer a distance 
#(1—ny/n2), on account of refraction through any plate of 
thickness ¢ and indices m, and ng, and this ‘drawing up” 
process will be repeated for each plate of the series. Hence 
the image will be nearer to the observer than the original 
source by the sum of all such quantities as ¢(1—m,/n,’). 
The case of some plates being isotropic is of course included 
in this, the corresponding displacement terms being obtained 
by setting n; equal to ng. 

D. Tmage- -formation by “direct refraction” through a lens 
of uniaxial crystal with optic axis lying along the geometrical 
avis.—Recall that for refraction from a medium of index n’ 
to one of index n” through a surface of curvature 1/r, the 
object and image distances are related by the law 


n'! n' 


»  U me)» 
where & is the dioptric power of the surface and is equal to 
(n''—n’)/r, (9). Note that this is fundamentally a relation 
‘between o curvature 1/u of the incident wave and tne 
curvature 1/v of the refracted wave. 

In the case under consideration, the light-wave P’M from 
a pomt lon the axis 4i@ 73) /is incident on the first 
surface M; is refracted as a spheroidal wave R’M whose 
centre of ae vature is a certain point Q and centre of radiation 
the centre of the spheroid R; is incident on the second 
surface N as a wave with centre of curvature at a point § ; 
is finally refracted into air as a homocentric pencil of 
centre T. Tis the required image-point. 

Since the light travels along the axis of the crystal wal 
the velocity cor rresponding to the index n, the powers of the 
first and second surfaces will be &' and k’’, equal to (ny—1)/r" 


cand (l—7)/r", by (9). In accordance with (3) 
RM 7), 5 QM/n,? and RN =," 4 SN/n,?, (LON: 


Formation by Crystalline Media. 255 


Draw the ray PABC whose successive directions make 
the angles a, 8, y with the axis; the line QA makes the 
angle S with the axis. These angles, much exaggerated in 
the diagram, are sufficiently small to admit of the ordinary 
approximations of the standard “ first order ” discussions of 
the principles of image-formation. 


Fig. 3. 


The line SB is parallel to QA. For the time required to 
travel from any one waye-position to any other is the same 
for all rays ; and for any given ray this time is proportional 
to the distance between the waves. Hence, since A lies on 
the wave-front R’M from R and B lies on the corresponding 
wave-front through N, it follows that RA: RB:: RM: RN. 
But QA : SB in this same ratio by (10), so that the triangles 
RQA and RSB are similar. 

The distances from A and B to the axis are 


of =e NI = MSO — Tee 
and 7) IN ai =19 IN > O= TING amelie): 


and, in view of these values, standard relations for the first 
and second refractions may be written from (8) as follows: 


nO—a=k'y', (12), 

and y—njo=k''y", (18). 
The problem now resolves itself into the elimination of 
the quantities which are involved in the use of particular 


rays and of the intermediate image formed within the medium 
of the lens. By this process the law for the i image-position 


256 Prof. H. F. Dawes on Image 


is determined in terms of the optical constants of the system 
and the position of the object. The method corresponds to 
the standard one of dealing with a thick lens. Thus: 


y—a=h'y' +k''y", by addition, from (J2, 18), 
= hy! + hlly' +k"tQ, by (11), 
=hly'+kl"y' +k" tn 78/nq? 
=k! y! + ky! +k tnyk’y'/n,? 
HAM tnya/ng?, by (12). 
Hence 
yla=K .PM+(1+k"tny/n?), (14), 


where K is the power of the system and is equal to 


ki + kh +h R'ény/ns?, (15). 
Similarly, 


afy=—K . TN+(1+ ktn,/ns2), (16). 


(14) and (16) state the fundamental general law for the 
image-position by expressing the distances from the refracting 
surfaces to object and image in terms of the ratio y/«, which 
may be considered an arbitrary parameter. These formule 
are similar to the standard formule for a lens of isotropic 
substance *, the modification consisting essentially in that 
the distance travelled within the crystal is divided by n/n, 
corresponding to division by the index in ordinary lenses. 
The positions of the cardinal points of the lens and the lens 
formulee of standard type may therefore be written down on 
this basis. 

In addition to the image system formed by the light 
which traverses the lens as the extraordinary wave-front, 
there is an image system produced by the ordinary wave- 
front, and although the light traverses the lens in the 
neighbourhood of the optic axis the two image-systems are 
not coincident. The fundamental formule corresponding to 
(14, 16, 15) are for this system, 


a ~~ PM+ (L4+4''t/n), C17); 
aly=—K’. WN+(14+4't/n), (18), 
where K’' the power of the lens is 
k+k' +Kk't/ny, (19). 


It will be seen that the difference in the two sets of 
foimulee lies in the terms which involve the thickness of the 
lens, so that the two images of an object coincide when and 


* See Hermann, ‘Geometrical Optics,’ p. 57 4 


Formation by Crystalline Media. 201 


only when the thickness is negligible. In consequence of 
this it will not be possible with thick lenses of crystal to 
focus sharply either on a photographic plate or in the focal 
plane of an eyepiece. ‘The two most important cases in 
which this may affect the efficiency of the lens are in its use 
as objective of telescope or collimator. In these cases it is 
the separation of the one set of principal foci which is 
important. This separation may be estimated as follows :— 

The distances from the lens to the second principal foci 
are determined by setting the arbitrary parameter equal to 
zero in (16) and (18), giving 


O=—K .FN+(1+#'tn,/n,”), 
O=—K’. HYN-+ (14+ Mt/n,), 
from which the distance between the foci is equal to 


k’t(ny?— ng?) [nn KK’. 


In a quartz lens of say about 20 cm. focal length and 
surfaces of equal curvature, this amounts to about ‘001 of 
the thickness, and in a calcite lens of the same power to 
about *037 of the thickness. The separation is thus un- 
important in quartz lenses of less than about °5 em. thickness 
except for the highest magnifications. 

This separation of the second principal foci disappears 
when the first surface of the lens is plane, so that plane 
waves are unchanged in shape until the second surface is 
passed, but in this case the separation of the first principal 
foci is proportionately greater on account of the fact that the 
whole of the dioptric power is produced at the one surface. 
Similar considerations would show that the separation of the 
first principal foci would be zero for a lens with the second 
surface plane. The fact of double refraction requires, 
therefore, that the objectives of collimator and telescope be 
plano-convex and that the lenses be placed with plane 
faces out. 

6. LImage-formation by direct refraction by uniaxial crystals 
cut so that the optic axis is at right angles to the geometrical 
awis of the refracting system.—Tor such examples of refraction 
of the extraordinary wave, the part of the spheroid presented 
to the refracting surface hes in the neighbourhood of zero 
latitude. The curvatures in normal sections in this part of 
the wave-front are unequal, so that on refraction by a 
symmetrical system an astigmatic image will be formed. 
Since the principal curvatures lie in and at right angles to 


Phil. Mag. 8. 6. Vol. 32. No. 188. Aug. 1916. S 


258 Prof. H. F. Dawes on Image 


the plane of the meridian, the focal lines, which are deter- 
mined by these curvatures, will be at right angles to and 
parallel to the axis of revolution, and hence at right angles 
to and parallel to the optic axis of the crystal. Thus in 
fig. 4, which represents the extraordinary wave from a 


Fig. 4. 
D4 


| Meridional 
Section 


_Axis of Grystal = . 


point P, the optic axis of the crystal is parallel to the x-axis 
and the geometrical axis of the refracting system lies along 
the z-axis. The wave travels with the velocity a along the 
optic axis and with the velocity ¢ at right angles to it, so 
that the section by the zy-piane is the equatorial circle and 
that by the za-plane the meridional ellipse. The focal line 
corresponding to the former section 1s parallel to the w-axis 
and intersects the z-axis at the image-point, as determined 
for a spherical wave of the same curvature travelling with 
the same velocity in an isotropic substance. Similarly, the 
focal line corresponding to the meridional section is parallel 
to the y-axis, and its point of interseetion with the z-axis is 
the same as if this wave were spheroidal about this line as 
axis of revolution. The position of this point of intersection 
will therefore be determined im accordance with the principles 
which we have been considering in previous sections ; the 
necessary modification of the formule will consist simply in 
interchanging the indices n, and ng, sinee the velocities a 
and ¢ are now interchanged with reference to the incident 
wave-front. 


Formation by Crystalline Media. 259 


Keeping these observations in mind, the following con- 
clusions may be easily deduced :— 

(1)* For an olject-point within a crystal at a distance p 
from a plane refracting surface. 


Distance from surface to : 
(a) focal line parallel to axis . . p/ng; 
(6) focal line at right angles to axis, pig/n,? ; 
te) erdinary mage.) lee Nias oily 
(2) For a lens of crystal. 


The powers of the first and second suPEwES and of the lens 
as a whole are :— 
(a) for pencils of ordinary light: 
=a fk 
oe n= —, K=h, + kot kykot/ny3 
(>) for focal lines parallel to the optic axis of the crystal, 
as governed by the circular section of the wave-surface: 


lb — 1 L- 
i= = - = —, K! =k! + ho! + hy h't/ng s 


(c) for focal lines at right angles to optic axis, governed 
by elliptic section of wave-front : 
Ny o—1 1—n 
ky" = : ’ hey! = Ss - 


Ts 


Kl = hey! + hel + ky hel tng ny. 


Knowing the values of the powers for these cases, the 
positions of the sets of cardinal points of the lens and the 
formule relating object to image positions may be written 
down in accordance with the standard formule for these 
quantities,—it was shown in the discussion of Section 5 that 
these formule in a modified form apply to the refraction of 
the elliptic wave. 

It will be seen that the powers of the individual refracting 
surfaces are the same for both the principal sections of the 
wave-front. The powers of the lens as a whole differ, 
however, in the term which involves the thickness, so that 
the amount of the astigmatism is governed by the thickness; 
if the thickness is negligible the focal lines degenerate into 
a point and the emerging pencil remains homocentric. The 
powers for the ordinary and extraordinary waves are, how- 
ever, essentially different for each surface and for the lens 
as a whole. There are therefore always two image series 
for a lens of crystal cut as described in this section. 


* Included in Stokes’s discussion. 


260 Image Formation by Crystalline Media. 


Crystalline Components in a Lens System. 


Whenever a crystalline component,—lens or plane-parallel 
plate,—which produces two image series is incorporated into 
a lens system, the system as a whole will of course produce 
two series of images. For example, for a doublet consisting 
of a lens of crystal of powers K’ and K" and a lens of glass 
of power K, the combination will have the powers K’+ K 
and K'’+K. It is evident that the relative values of the 
powers of such a combination may be made almost any- 
thing one wishes ; one of the values may even be made zero 
by choosing the glass lens of power equal and opposite to that 
of the selected crystal power. In this case the other power 
will be the difference K'—K" of the powers of the crystal 
lens. 

Again, it will not be possible to produce an achromatic 
doublet which will at the same time correct the chromatic 
aberration of both powers of a lens of crystal. If o’, o”, o, 
are the dispersions for the ordinary and extraordinary powers 
of the crystalline lens and of the glass lens combined with 
it, and K’, K”, K the corresponding mean powers, the 
chromatic aberrations of the powers of the combination are 


o K’ poekeeand 1K" +o0K. 


If the values of w and K are so chosen that one of these 
is zero in accordance with the ordinary principles of 
Geometrical Optics, the value of the other will be 


wo K”—o'K’, 


and this will not likely happen to be zero. In the case of a 
lens of Iceland spar, of which the principal indices are about 
1:66 and 1:49 and the dispersions for the D-—F lines -014 
and °0088. this outstanding chromatic aberration will be 
—:0051 (1/r—1]s), or about *02 of the power of the com- 
bination if the glass Jens has the index 1°62 and _ the 
dispersion *02. 


Summary. 

Laws for the image position are developed for a number 
of typical cases of image formation by the extraordinary 
rays in uniaxial crystals. 

It will be noticed that in the examples of light travelling 
along the axis ef the crystal, the thickness of the crystal 
traversed enters into the expressions for image position as if 
the index were n.*/n; as compared with isotropic substances; 
nm, and ny are the principal indices of the crystal. This 
quantity may be considered a sort of pseucto-index applicable 
to such cases. 

It is shown that lenses of crystal will in general form two 


Notices respecting New Books. 261 


series of images corresponding to the two sets of waves ; 
the laws for the image series due to the extraordinary 
waves correspond to those for lenses of isotropic substance, 
and the expressions are found for the dioptric power. The 
question of the use of crystal components in lens systems 1s 
also considered briefly. 

Physical Laboratory, 


University of Toronto, 
May 2, 1916. 


XXXI. Notices respecting New Books. 


An Introduction to the Mechanics of Fluids. By Evwin H. Barron, 
D.Sce., F.R.S.E. Pp. xiv+249. With diagrams and examples. 
Longmans, Green & Co.: London. 1915. 6s. net. 


JT is somewhat difficult to write a book on elementary mechanics 

which shall present any verv novel feature. But there are traits 
of this one which are certainly novel, and, which is still better, novel 
in an attractive way. One of these characteristics is that while 
the calculus is avoided it is replaced, not by the very artificial 
methods with which we are all so familiar, but by a method of 
summation which forms one of the best possible introductions to 
the calculus itself. This is a very great gain. The most inter- 
esting part of the book is that dealing with applications of the 
mechanics of fluids in technical practice. We have never come 
across so complete and satisfactory an account of the different 
forms of suction and compression pumps, rams, Bourdon and 
other gauges, water-wheels, presses, lifts, cranes, turbines, drills, 
brakes, &c. We commend this part in particular to school 
teachers ; nothing could be better calculated to stimulate and at 
the same time to instruct the youthful mind. 

The introduction to mechanics is in general very logical. One 
or two very minor points made the reviewer hesitate for a moment. 
The first page scarcely brings out the fact that a liquid opposes 
no permanent resistance to change of shape. To talk of very 
small resistances scarcely meets the case. In consequence of 
this inadequate statement it will not be evident (as it is stated 
to be) that viscosity ‘“‘need not enter into our account when 
considering any fluids at rest in equilibrium.” 

Again, is it a matter of such indifference what definitions are given 
of mass and force? The inadequacy of Newton’s treatment of 
mass arises from the fact that although the masses (or quantities 
of matter) of the same substance may be taken as proportional 
to their volumes, we cannot pass from these cases to cases in 
which the material is different. 

It must further be remembered that those who are driven to 
define force with reference to our muscular sensations are not 
only illogical, but are wrong; because, as every psychologist 
knows, our sensations of force are not proportional to the equivalent 
forces measured mechanically. 


262 Notices respecting New Books. 


We wish to emphasize, however, that the method of definition 
given in detail by Professor Barton, and which apparently is pre- 
ferred by himself, is pertectly logical. He has moreover put the 
matter very clearly—at least considering that he only professes to 
give an introduction to the subject. 

We recommend the volume very strongly to every teacher, and 
we are confident that it will be well received. 


Fundamental Conceptions of Modern Mathematics. By Roser P. 
RicHarpson and Epwarp H. Lanpis. Chicago and London. 
The Open Court Publishing Company. 1916. 


Numbers, Variables, and Mr. Russell’s Philosophy. By R. P. 
RicHaRDson and E. H. Lanpis. (Reprinted from the ‘ Monist’ 
of July 1915.) The Open Court Publishing Company. 1915. 


Tue book and pamphlet naturally go together, part of the 
criticism in the latter finding its place in the former. The 
purpose of the book is best explained by some quotations from the 
preface. It is— 


“to examine critically the fundamental conceptions of Mathe- 
matics as embodied in the current definitions .... In ex- 
pounding our own views we have often been obliged to find 
fault with those of others; but we have not gone out of our 
way for the sake of mere criticism ; we have merely cleared 
away false doctrine preparatory to replacing it with true. ... 
The keynote of our work is the distinction we find it necessary 
to make between quantities, values, and variables, on the one 
hand, and between symbols and the quantities or variables 
they denote or values they represent, on the other.” 


The theme is attractive, and is to be worked out in thirteen 
volumes, of which this is the first. It will be useful, meanwhile, 
to see how the authors put their maxims into practice. 

Consider, for example, the meaning of Variable. We are prac- 
tically told that all mankind since Newton have, in their endeavour 
tea define a variable, fallen into an estate of mathematical sin and 
misery. After much criticism of these endeavours our Penn- 
sylvanian authors say (pp. 154-5) that “‘ while we have made 
plain what a variable is not, and have described in what manner 
it is constituted, care has been taken to avoid any statement as to 
what a variable is..... Any attempt to give a precise account 
of the definition of the term ‘variable’ would require a somewhat 
lengthy consideration of the philosophical theory of the categories, 
which cannot be given in this place.” We had hoped for bread, 
and they give us a loaf-ticket! After all, did it never strike them 
that the mathematicians they criticize were in the same difficulty, 
and that their so-called definitions were intended to be simply 
‘first notions which experience would amplify and even correct as 
the mathematical student advanced in knowledge ? 

Consider, also, their discussion of quaternions and vectors. 
Their view, we are told, is not precisely that of Hamilton. No 
objection an be taken to such a position. But what of their 
treatment of Hamilton? They quote on p. 47 his definition that 


Notices respecting New Books. 263 


‘4 right line considered as having not only length but also direction, 
is said to bea vector.” But this, they say, is obviously inadequate, 
‘* since a linear velocity at a point is a vector, and of course a velocity 
is not aline. Further, not all straight lines possess the attributes 
requisite to vectors.” These reasous are quite irrelevant. For 
Hamilton never said that a velocity was a line in their assumed 
sense of the term; nor did he even say that a velocity was a 
vector. Also Hamilton never said that all straight lines, in any 
sense in which other writers might be pleased to regard them, 
were vectors. Nor does his definition imply it. He said that 
right lines considered as having not only length but also direction 
were vectors—a totally different statement. Obviously, however, 
our critics are psychologically incapable of understanding the 
plain meaning of Hamilton’s words; for two pages further on 
they make the following remarkable utterance: ‘“ Even so high an 
authority as Hamilton states that a right line is a vector; a view 
which is quite untenable.” Hamilton certainly never made such a 
statement, except when he carefully guarded himself by saying 
that the right line was considered us having not only length but also 
direction. Following this second emasculated statement wrongly 
ascribed to Hamilton, the authors go on to declare that a right 
line remains the same so long as its length remains constant, even 
though it rotate like the spoke of a wheel into a new position. 
Now—so runs the argument—the vector of this rotating “ right 
line ” does not remain the same. Therefore, they conclude, this 
“right line” cannot be a vector. But whoever said it was? 
Their statement that Hamilton said so is absolutely ludicrous. 
In the course of two pages these critics of Hamilton, with 
wondrous wisdom, drop out of the definition certain all-important 
words, and then, with superb simplicity, develop their attack by 
first giving to the term right line a meaning which, as their own 
previous words prove, was not Hamilron’s meaning. They commit 
the double sin of misquotation and bad logie. 

We are nowhere told distinctly what they themselves consider 
a vector to be. They seem to follow the ordinary custom of 
making it apply as a class name to certain very different physical 
quantities, such as displacement, velocity, acceleration, force, 
angular velocity, moment of momentum, and the like—a use which 
compels them to talk profoundly of vectors of different sorts. 
For example, on page 76 they say: “In Quaternions, vectors of 
described straight lines and of points constitute one sort of 
applicate quantities; linear velocities at points, another; linear 
accelerations at points, a third; and so on.” In Hamilton’s 
Quaternions this is not so. Must we repeat it? A vector isa 
right line considered as having not only length but also direction. 
This is the opening statement of the Elements of Quaternions, 
and is the foundation of the whole calculus. Vectors, so defined, 
are found to satisfy an important law, the law of vector addition. 
Hamilton does not speak of a velocity or an acceleration as being 
a vector. When he applies his calculus to dynamies and physics 
he finds certain quantities which can be represented or symbolized 


264 Notices respecting New Books. 


by right lines regarded as having not only length but also direction, 
and which, when so represented, satisfy the law of vector addition. 
These representative lines are Hamilton’s vectors. Analytically 
there is only one sort. He speaks of the vector of a point, the 
vector of a velocity, the vector of a force; but he nowhere speaks 
of the point, or the velocity, or the force, as being the vector. For 
example, what is commonly called the vector product of two vectors 
has various meanings according to the kind of quantities symbolized 
by the vectors. Thus, if b be the vector of the force acting at 
the point whose vector is a, then in Hamilton’s view, ab is a 
quaternion, whose vector part, Vab, represents the vector of the 
moment of the force about the origin, and whose scalar part, Sab, 
measures what Clausius subsequently called the virial. Again, if 
a is the vector of the angular velocity, the expression Vab is the 
vector of the velocity of the point whose vector is b. In their 
misapprehension and misrepresentation of Hamilton’s meaning 
and method are not our authors confusing symbol and quantity ? 
On page 132 we read: “ Quaternions is defective in that it is 
not possible to multiply a vector by a quaternion. Thus, if the 
relation of one vector to another is the quaternion q, it does not 
necessarily follow that we can multiply a third vector by g and 
obtain a vector.” The second clause of the first sentence is simply 
not true. As regards the second sentence, which is probably 
meant to explain the first, why should we expect, or why should 
we desire, that any quaternion multiplying any vector should give 
a vector? The truth is that a quaternion multiplying a vector 
gives, in general, another quaternion. Why should a calculus be 
called defective because by the fundamental laws of its being it 
cannot always satisfy a law which it satisfies occasionally and 
under specially restricted conditions? If the quaternion is 
admitted at all, why desire to get rid of it by setting it in product 
combination with any vector? In uniplanar quaternions this 
law is satisfied; but then uniplanar quaternions is so defective a 
form of the calculus as to cease tv be quaternions at all. 
Whatever important additions this volume may make to mathe- 
matical thought, the authors’ claim to be the children of the light 
is seriously menaced by their misquotation and misrepresentation 
of the meaning of the words of the author they criticize, and by 
their elaboration of an argument which is either irrelevant or a 
begging of the question. Nevertheless they have a real appre- 
ciation of the value of the quaternion and speak warmly of the 
great services rendered by Hamilton. That they are also capable 
of doing good things is shown in their discussion of the meaning 
of Function, which occupies a comparatively small part of the 
volume, and of which more will be given in the succeeding parts. 
Here, we are glad to say, they give a definition, which in regard 
to length and style recalls Euclid’s famous definition of proportion. 
In this discussion also, as in the discussion on variables, almost 
every mathematician who is named is found sadly lacking in 
accuracy ; but the ordinary mathematical reader will be cheered 
by the thought that should he likewise fail into error he will still 
be in the company of the immortals. 


THE 
LONDON, EDINBURGH, anv DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. — 


c : 
B® LLY 
[SIXTH SERIES.] SEP 1Y - 
Y 
SS PA 


SEPTEMBER 1916. ee 


XXXII. On Fluorescent Vapours and their Magneto-optic 
Properties. By L. Stuperstein, Ph.D., Scientific Adviser 
to Adam Hilger, Ltd.* 


1. General Remarks. 


‘HE fluorescence of vapours, such as iodine vapour, 
excited by light of a particular, appropriately chosen, 
frequency N, gives what Wood calls a resonance spectrum, 
consisting of a series of lines, whose frequencies are ny>=N 
and, say, 74, 2, 13, etc., where the latter stand in some 
relation to the fundamental frequency N. 

Now, an ordinary oscillator or resonator, of free period 
T=27/N and relaxation- or extinction-time r=2/h, i.e. a 
system obeying the familiar differential equation 

Dyate ete N 2 — Ohm sae tases a 20) (CELL) 
when acted upon by an external force of frequency n, gives 
only oscillations of the same frequency n, and of no other in 
addition. (And, assuming T/7 small, the resonator answers 
vigorously only when n==N.) Since this simple behaviour 
is due to the assumption of Hooke’s law for the restitutive 
elastic force, 2. e. force proportional to displacement 2, we 
can appropriately. call a system obeying the equation (H) al 
Hlookean resonator. On the other hand, if N?z is replaced 
by a non-linear function of a, an exciting force of frequency 
N will generate oscillations of frequency N and also an 
infinite variety of other frequencies. 

* Communicated by the Author. 


Phil. Mag. 8. 6. Vol. 32. No. 189. Sept. 1916. rc 


266 Dr. L. Silberstein on Fluorescent Vapours 


Keeping this in mind, we can mathematically describe 
the excitation and emission of fluorescent spectra by saying 
either 

Ist, that the atoms of the vapour behave so as if each 
contained a Hookean resonator under the simultaneous 
action of forces of all the frequencies npx=N, 1, 29, 13, etc., 
2. e. by writing 


ee . INE inyt , t 
e+kie+t Ne=cme + ee" +e" +..., . . (1) 


or, 2nd, that each atom contains an appropriate non-Hookean 
resonator, acted upon by ee” only, i. ¢. that 


i? + he +N xtf(x)=oe™, . . . . (2) 


where f(z) is some non-linear function of the displacement. 

The equivalence of the two methods of treatment is 
manifest. In fact, the non-Hookean resonator will be the 
‘“‘ appropriate” one when, and only when, the supplementary 
term —/(x) ultimately reduces to eye’ + c,e"" +... 

If the frequencies n,, mg, etc., accompanying the funda- 
mental one N, and the ratios c;: ¢9, etc., are all assumed to 
be known (from experience), then we do not require the 
form (2) at all, and we can study the properties of each line 
of the spectrum separately by writing «=a)+x,+... and 
splitting (1) into 


ee ° in t . 
H+ ket, NP a= Ge", ca0, 1,2, 


and introducing into each of these equations, and of the 
analogous ones for y;, %, the well-known supplementary 
terms due to a superposed magnetic field. It is only when 
we wish to make a guess as to the law of sequence of the lines 
in the spectrum that we require the form (2). This will 
occupy our attention for a few moments in the next section, 
in which we shall assume hypothetically a particular form 
of f(x), and test it experimentally ; but in all the remaining 
sections of this paper we shall avail ourselves only of the 
equation (3) and of ordinary electromagnetism, so that all 
‘results obtained in these sections will be entirely independent 
of that hypothesis. 


2. Law of Constant Frequency Intervals. 


The simplest case of a non-Hookean resonator is well- 
known from acoustical problems in which use is made of a 
quadratic supplementary term, const. Xa’. Such a term 
would, obviously, not answer our purpose, since it gives, in 


and their Magneto-optic Properties. 267 


addition to N, its octave 2N as the next frequency, then 3N, 
ete., while the lines* of Wood’s iodine spectra, excited 
by green or yellow light, follow upon one another in intervals 
of 60 to 85 A.U. only. What we require is a fractional 
power of w, but little differing from unity. 

Let us write, therefore, f(v)=a.a?, where p is a real, 
positive, constant, whose value is for the moment undeter- 
mined. Then (2) will become 

e+ he +No®=oe*—az®. . . . . (4) 
Whatever the value of p, the difference 1—p will be a 
certain measure of departure of the resonator from Hookean 
elasticity, and it will be responsible for the series of lines 
accompanying the fundamental one. ‘The coefficient a is a 
real constant. If symmetry around the position of equi- 
librium is required, then 2? is to be taken as + {a#?|, according 
as «>0 or «<0. 

As to ¢, if the exciting agent be (polarized) light, ¢o is 
the product of the amplitude of the electric force in the 
incident wave, and of g/m, the ratio of the charge and mass 
of the resonator. 

Now, proceeding by the well-known method of successive 
approximations, write, first, 


a=ben’, 
and neglect az?. Then #+Na?=0, and 
ae Nae SBC TRNT (G5) 


Substitute the first approximation in 2?, so that (4) will 
become 


2+ he + Nev = ce —abPe'?™ ; 
thus, as a second approximation, 
a Eb 
where 6 is as above, and 0}, is determined by 

by EN? — p?) + 2kN pl) == Saba iilen « 1(9;) 


Thus, 6; is a complex magnitude, say 6,=pye""; 6, is the 
phase-difference with respect to the exciting oscillation, and 
pi the amplitude of the oscillations of frequency n;=pN. 
The former is of no interest here, and as regards the latter 


pt 
x=be eit 


* To speak only of the chief lines of groups or “ orders,” each of 
which contains in addition a number of fainter lines. 


D2 


268 Dr. L. Silberstein on Fluorescent Vapours 


it is enough to note that p; is proportional to | 6”|, and 
therefore, by (5), 
Co P 
p~( 7x) . 


Otherwise the coefficients 0, 6,, and the following ones are 
of no interest in the present connexion, since we are con- 
cerned here with the emitted frequencies only. Substituting 
the second approximation in the term aw? in (4) we have 


(be™ i pe r= pent ape bee 
EP eee 2 4 
Thus, as a third approximation, 
cabo 4 bc Ph 4 peek 4. pcr INE ens.) 


where 0, b, are as above, and bg, bs, ete., are easily determined. 
With regard to these coefficients it will be enough to remark 
here that as long as |6;| contains a positive power of e/kN, 
the corresponding line will be strong enough to be visible ; 
now, if 

r—l 


5 e 
4p 


(7) 


where 7 is a positive integer, the last condition will be still 
fulfilled for the 7th line of the spectrum (the fundamental 
line being the zeroth line). Thus, if p satisfies (7), we can 
have, theoretically, as many as r lines, not counting the 
fundamental one. 

Returning to (6a), we see that the third approximation 
gives a series of lines whose frequencies are 


m=N, m=pN, n=(2p—LN, n= (38p—2)N, ete. (8) 


Pushing the process to the fourth approximation we should 
obtain, in addition to (8), such frequencies as p?N, etc., 
satellites of the above, whose interest will be shown in a 
future communication. For the present we shall confine 
ourselves to the third approximation, 7. e. to the series of 
frequencies (8). 

This series can, obviously, be written 


ng=N=1—p)N, #20423, >. 


so that its members should succeed one another in constant 
frequency-intervals, 


p> 


dn=(l—p)N, 2. 2. 


the series extending from the fundamental line either towards 


and thew Magneto-optic Properties. 269 


the red or the violet end of the spectrum according as 
ps1". 

This result seemed at first discouraging, since the discoverer 
of the resonance spectra, Prof. R. W. Wood, speaks in his 
Clark University Lecture (of 1909, published 1912) and in 
his paper in Physik. Zeitschr. xii. p. 1204 (1911) of constant 
wave-lengths, and not frequency-intervals. But a glance 
at Wood’s numerical table, given in the last-mentioned 
paper (p. 1207), for iodine vapour has sufficed to notice the 
tendency of 6X to increase towards the red, so that a detailed 
comparison of the above law (9) with Wood’s measurements 
has seemed worth undertaking. The result is given in the 
following Table, in which A, B, C are Wood’s three resonance 
series of iodine vapour excited by light of wave-length 
X= 5461:0, 5769°5, 5790°5 respectively. The n-columns 
contain the observed frequencies, 2. e. the reciprocals of 
Wood’s wave-lengths reduced to vacuum, and 6n are their 
differences. Since Wood’s wave-lengths are, according to 
himself, correct only within about +1...U, the last figures 
in our n-columns are unreliable. It may be well to remark 
also that the lines made use of for this Table seem to be the 
chief lines of small groups or “orders” given by Wood ina 
later publication (Phil. Mag. xxvi. p. 828, 1913). To the 
fainter lines accompanying the chief ones in each “order,” 
I hope to return at a later opportunity. 

Yhe fundamental or exciting frequencies N, at the head 
of each of the three series, are printed in clarendon. At the 
bottom of the Table are given the mean values of én. 

The constancy of dn along each series is quite satisfactory. 
Since the observed 2X’s are uncertain almost within +1A.U., 
the deviations from 6n are well within the limits of experi- 
mental error. 

Using the above mean values for én in (9a), we have for 
1—p, which in some way is a measure of deviation from a 
pertectly Hookean resonator, for the three series, 


bp), — 00 hp, — OLS ye sO) 0) 
and for p itself 
= 9889. — 9882) Puaoola sy 0a) 


* Wood's series of iodine, A, B, C, consist of 17, 13, and 13 lines, 
respectively, on the red side of the corresponding fundamentals, and only 
of éwo ultra-N lines, in each case, thus showing a marked tendency to a 
unilateral development. The ultra-N lines can easily be accounted for 
by an extra term 2 ”, but since they are but two, we can avoid 
complication by disregarding them for the present. 


270 Dr. Li, Silberstein on Fluorescent Vapours 


| Series A. Series B. 
eee - 
N. | On. 2. 
Oeste 18307 17328 
| 207 
1 Rea 18100 17118 
210 
a ee 17890 16916 
21 
unease 17673 16705 
20 
Ae 17468 | 16488 
20 
Dicenae 17261 16283 
212, 
6 17049 16080 
204 
Teese 16845 15879 
205 
Or aides 16636 15687 
205 
= Ngee 16431 15476 
200 
Op cae : 16231 15273 
| 198 
11g lesa 16033 15091 
196 
1) OPE 15837); (fee) Ce 
204 
Aes ae 15633 14681 
} 194 
TAN et el eee ere es 
194 
ee 15245 
191 
PS sk ok eee 
| 191 
Iie leh RS 14864 
Ay a ea 
Mean, 62 =202'53 62n=203'61 


Series C. 

N On 

| 17265 
216 

| 17049 
207 

| 16842 
205 

| 16637 
206 

16641 
204 

16227 
205 

16022 
210 

15812 
196 

15616 
PAD 

15404 
205 

15199 
188 

15011 
} 208 
208 

14595 

6n=205'38 


Thus, the required deviations from unity are small and 
very nearly equal for the three series of iodine. 

Finally, let us remark that, p being greater than 0°98, we 
could have theoretically, by what has been said in connexion 
with formula (7), more than 90 lines, and since 6z is roughly 
200, each of the series could extend even to the very end of 
the spectrum (n=0, or X=). This would, obviously, be 


the case also for smaller values of p. 


Thus, the condition 


connected with (7) contains in reality no limitation, anda 
non-Hookean resonator with the above p proves to be com- 
petent to emit the observed resonance-series. 


and their Magneto-optic Properties. 271 


Notwithstanding this we shall not use it any further, but 
shall avail ourselves of the form (1), as split into the 
equations (3) for each line separately. The two procedures 
are equivalent to one another. The coefficients ¢,, ¢,, etc., 
are complex, say, ¢c:=p,c'", etc. ; but since we are not con- 
cerned with the phase-difference between the higher and 
the fundamental line, we can imagine 6; thrown on e’""’, by 
shifting the origin of ¢ for each 7 separately. 

Thus, our further developments will be based on the simple 
equations 

i+ ha + Na oe"", 5=0,1,2,.09 
where ¢; are real constants. For the sake of simplicity we 
shall henceforth drop the suffix ; and write, for each line of 
the spectrum separately, 

Did keg ce IN? == 6) eV Minh tlhe) CLL) 
with similar equations for y, z, keeping in mind that n=N, 
my=pN,and soon. At first sight it would seem that these 
simple equations can hardly yield anything new or important ; 
but the following sections will prove the reverse. 


3. Emission of Resonance Spectra in a Magnetic Field. 

Let the fluorescent vapour be placed in a uniform magnetic 
field of intensity H. Let us take the z-axis along the lines 
of the field, and the w and y axes equally inclined to the 
electric force of the exciting light, which is supposed to be 
rectilinearly polarized. Then, for each line of the spectrum, 
the coefficients ¢ in (11) will be the same for the 2- and the 
y-equation. Rigorously speaking, the values of the c’s them- 
selves will be slightly changed by the magnetic field*; but, 
for the present at least, these modifications can be disregarded. 
Thus, the presence of the magnetic field will give only the 


oF Hy dint. pee 
familiar supplementary terms ar Hy, Bua Ty Hw for the 


right sides of the x- and the y-equations respectively. The 
equation for z will remain unaltered. It will be enough 
to consider the case of exciting electric oscillations perpen- 
dicular to the magnetic field. Then we can write, simply, 
z=(, and we have, for each line of the spectrum taken 
separately, the two equations 


&+ N2a + ha— Ly = ce, ; : (12) 
ytN*y thyt+Ziz=ce™, J 
eae 
mV 


* Especially if we would look on (11) as the equivalent of a non- 
Hookean system. 


where Z= H. Remember that n stands here for the 


272 Dr. L. Silberstein on Fluorescent Vapours 


frequency of the given spectral line in ordinary circumstances, 
2. €. in absence of the magnetic field. Now the solution of 
(12) is, apart from damped vibrations, independent of ¢, 

c= Aer y= Be, ... 2c 
where A, B are complex constants. 

Thus we have, to begin with, the negative but nevertheless 
noteworthy result that there should be no splitting of lines of the 
fluorescent spectrum, 1. e. no ordinary Zeeman effect. 

As far as I know, this conclusion is not contradicted by 
experience. In Woodand Ribaud’s paper on ‘* The Magneto- 
optics of Iodine Vapour” (Phil. Mag. xxvii. pp. 1009-1018, 
1914), which to my knowledge is the only publication 
bearing on the subject, there is no mention of any splitting 
of the lines of fluorescent iodine, although the authors have 
disposed of a field of 20,000 gauss *. ; 

The reader will notice that the ordinary equations of 
Lorentz’s elementary theory of the Zeeman effect for common 
spectrum-lines differ from (12) by the absence of ce” (and 
of the terms kx, ky). This is the reason why Lorentz’s 
equations give a splitting, while our equations refuse to give 
any trace of it. | 

However, although none of the lines of a resonance- 
spectrum can be expected to be split by the magnetic field, 
each of them will undergo a change of intensity and of 
character, 7. e. of its state of polarization. In fact, substi- 
tuting (13) in (12) we have, for the constants A, B, 


A=t [Nn +in(k+Z)], | 
: 


B= 2 [N*—n? +in(k—2Z)], Ce 


where D=(N?=—n’ + ikn)?-— Lien 

These are the (complex) values to be inserted in (13). 

The intensities will be given by |A| and|B|, and this will 

occupy our attention a little later (cf. Section 6). The state 

of polarization, which interests us here, is entirely determined 
by the quotient y/z=B/A, and this is, by (14), 

y N= tint —Z) 

we N?—n?+in(k+Z) 

* No direct mention is made, in the quoted paper, of the absence or 

presence of splitting of the fluorescent lines. Wood and Ribaud’s state- 


ment (p. 1015) “ that if the Zeeman effect exists, it is less than 0’01 ALU. 
for a field of 20,000 gauss” refers to the absorption lines of iodine vapour. 


= pe"; say. . 7. (a ee 


and ther Magneto-optic Properties. 273 
The meaning of this equation is obvious. Let the vector 


E denote £ xineident electric force, apart from the factor 
ée*, and let r be the fluorescent light vector (x, y, 2) 
Then, in absence of the magnetic field, we have y/v«=1, 
2. e. @=0 and p=1, which means rectilinear polarization 
and r parallel to E. Owing to the magnetic field these 
rectilinear fluorescent oscillations become, in general, elliptic, 
and the major principal axes of the ellipses are turned away 
from E through certain angles €,in the plane 2, y. These 
angles € are easily expressed by p alone, and the ratio of 
axes, say b:a, by @ alone. In fact, if € be the angle 
contained between E and the a-axis, we have 


* 


1l—p b 


6 
Gt Bass 2H) Ue ANG) 


2 


tan’ G— ==)| tam 


And since, by (15), both p and @ are functions of n 
(n=, 1, N, ete., for the fundamental, first, second, etc., 
lines), the ellipticity and the rotation € will be different for the 
different lines of the resonance-spectrum, having ny=N for 
its fundamental. The sense of the rotation round the 


magnetic field is easily ascertained. Since Z= i it is 


nV 
one way or the other according as the resonator, which need 
not necessarily be an electron, is positively or negatively 
electrified. Details concerning € and b/a, at least for the 
zeroth and the first line of the series, will be given 
presently. 

Such then should, according to our theory, be the behaviour 
of the resonance lines in a magnetic field. As far as I know, 
nothing of the kind has yet been observed, but I have reasons 
to expect that the predicted rotations € will be detected even 
with moderate fields H. 

The reader may be tempted to simplify (15) by neglecting 4, 
—at least, for the first and the higher lines. I must warn 
him, however, not to do so, unless he is willing to neglect Z 
as well. In fact, we shall see further on that, say, for 
H=10,000 gauss, |Z| and & are not only of the same order 
but very nearly equal to one another. 


* In our present case 2=0. 


274 Dr. L. Silberstein on Fluorescent Vapours 


4, Magnetic Rotation of the Fundamental Line. 


For the fundamental line of the resonance spectrum we 
have n=N and therefore, by (15), 


a keh 
and, by (16), b/a=0 and 
tan — a 6i @ | Je) yo) Aeaeenie (165) 


Thus, the fundamental line will continue to be rectilinearly 
polarized, but its plane of polarization will be rotated round 
the magnetic field through an angle €)=(r, E)= arc tan (Z/k). 

It will be remembered that lea where H is the 
absolute value or the intensity of the magnetic field. Thus. 
Z is positive or negative according as g>0 or g<0. This 
settles the question of the sense of rotation. Whether the 
charge g is positive or negative, can only be decided by 
experiment. 

Let us, henceforth, write shortly 


q/mV =n, 
so that Z= Hy, and 


tan pene - : e ° ° . (165) 


If exists, and is great enough (for H of the order 10*) 

to be measured, the ratio 7/k, and therefore the product 

Ua: 
will be found experimentally (where 7=2/k is the relaxation- 
or the extinction-time of the resonator). In order to find 7 
and 7 separately, some further connexion is required, which 
might be obtained from the first or the higher lines of the 
resonance-spectrum. 

Before passing on to these it will be convenient to distin- 
guish here a certain, particular, intensity H, viz. that 
intensity which makes €)= +405°, i. e. that intensity which 
makes 


La ote 0, 
Z=—k for q<0.5 


Let us denote this particular intensity of the field by H,, 
so that 


(17) 


1S oo h7y eee ene iegea tc (aL) | 


We shall see later that, for iodine vapour, H, is of the 
order of 10* gauss. 


and their Magneto-optic Properties. Ph 2 te 


5. Magnetic Rotation and Kilipticity of the First 
and the Higher Lines. 

For any line of the resonance series, and for any field H, 
pand @ are determined by the general formula (15), which 
gives at once 

ny [ (N?—n?)? + (2? — Z?)n? |? + 42? n?(N? —n?)?\8 
a (N?—n?)? + (kt Zn? 
2Zin( N?—n?) 
(N?—n?)? + (= Zn? Saher Nh aie (19), 


Here we have to insert n=Np for the first, n= N(2p—1) 
for the second line of the spectrum, and so on. Introducing 
-(18) and (19) into (16), we obtain the corresponding rotations 
€ and the ratios b/a of the axes of the ellipses described by 
the end-point of the fluorescent vector r, for any line and for 


any field H. 
It wiil be enough to develop the final formule for &,. 


» (18) 


fang —=— 


for the special intensity H=H, of the magnetic field, 


when the above formule are considerably simplified. In. 
fact, if the resonator carries, say, a negative charge”, then,, 
by (17), Z+4=0, and (18), (19) become 
2nk iL 
tan 0= x35» Pe jeos 8) 

Substituting these values in (16) we have, finally, with a 
magnetic field H, (which would produce a rotation of the 
fundamental line through €)=45°) for any of the higher 
lines of the spectrum : 


1+ |cos@{’ a 2 ‘ 
where nh ge (20)) 
tan C= Nemes é 


Thus, all the lines, beginning with the first, will be 
elliptically polarized, and their a-axes will make with E the 
angles (, , etc., expressed by (20) with n=Np, N(2p—1), 
etc., respectively. All of these angles will differ more or 
less from €&)=45°. 

If either € or b/a could be measured on any one of these 
lines, in the field H,, then @, and therefore by the third of 

* If it is positively charged, Z=A, and we obtain the same final 
formulee if the réles of 2, y are exchanged, as will be seen at once from 


(15). Thus, the amount of rotation and the shape of the ellipse will be 
the same, but with reversed sense. 


276 Dr. L. Silberstein on Fluorescent Vapours 
2 
equations (20), the value of k= — would be found. And 


since nt can be measured on the fundamental line, we should 
have 7 as well as r. 

To illustrate (20) a little further, take the first line of the 
resonance-spectrum, n=pN. Let T be the fundamental 
period of oscillation, and + the extinction-time, as before. 
Then 0= 6, is given by 


tan 0, = aed. a y 
ml—p*) 7 

For the second line we have only to replace p by 2p—1, 
and for the third line, by 37—2, ete. Thus 


(21) 


mee tO eh "arr Ae ae 
9 eB 
fangs — aie?) Z (21") 


Teepe tr 
and so on. ‘Thus in the case of the iodine series A, for 


which p=0°9889, 
tan @,=28'51-; tan Qp= 14-18; tan @,=9396-, 


and so on. Now, T/r is certainly a small fraction, so that 
6, is a small angle, @, nearly one-half of 6;, and so on. 
Thus, by (20), the angles (, &, etc., become, apart from the 
sion, 70,", 10,7, and so om, 7 e. of the order 10*gage 
3.10°(T/r)?, etc. And as for the ratio b/a, this can now be 
written 


bja= z = z tan 0, 


4. €. 2 
bi _ 4-96 Ly eee 1-70. Se 
Qy T ao T a3 C 


and may, if t does not greatly exceed 10°T, be just detectable. 
The decision must be left to the experimental physicist. 
At any rate, b/a would be for the second line only about 
one-half, and for the third line one-third of what it is for 
the first line, and so on. 

It will be remembered that (20), and therefore also (21), 
(22) are valid for H=H,, as defined above, and this field is, 
as we shall see in the next section, very likely equal to about 
10* gauss. For other field intensities of this order the 
behaviour of the lines nj, 79, ete., will be very much the same. 
For any field we have the rigorous equations (18), (19) with 
(16), determining completely the “character” of each line 
of the resonance-spectrum. 


and their Magneto-optic Properties. 247 


6. Destruction of Fluorescence by Magnetic Field. 


I have taken the liberty of copying the title of this section, 
literally, from Wood and Ribaud’s paper already quoted *, 
first, because it is a short title and, second, in order to better 
recall the reader’s attention to that experimental paper. It 
will be well to quote also a few lines from Wood and Ribaud’s 
paper. On p. 1017 the authors say :— 


““Under these conditions we estimated the reduction of 
‘‘intensity [of iodine vapour fluorescence | to amount to fully 
“90 per cent. with a field of 30,000 gauss, and it is probable 
“that with a field of 50,000 the fluorescence would be 
“practically destroyed.... No obvious explanation of the 
“effect of the field in reducing the intensity of and ultimately 
“practically destroying the fluorescence suggests itself” T. 


Now, such a dependence of intensity of fluorescence upon 
the magnetic field has already been hinted at, just after the 
equations (14). We shall now see that these equations not 
only account for the observed effect, but, bringing it into. 
close connexion with the magnetic rotation € (and 
especially), make the explanation so plain that it can be 
seen almost immediately. In fact, let I be the intensity of 
any line (taken separately), when the vapour is placed in a 
field H. ‘Then, disregarding ellipticity, 

7) == |Ve\lea (161, 
where A, B are given by (14). It will be enough to consider 


here, in detail, the fundamental line only. Writing, in (14), 
n=N, we have 


ei eA =, igi 
Al=x > PTL fe [Bi N° 472? 
whence, for any H, 
ee 
WON Ze 


and, therefore, for H=0, [=c,?/N?k?. Thus, the ratio of 
the two intensities, Jo=Iqy : 1, 
i 

Jigs Sa 23) 

0 1+ (Z/k)?? - (49) 

where the suffix is to remind us that the fundamental line 

is concerned. Thus, with increasing intensity of the magnetic 

field the fundamental fluorescence tends to nothing. 
* “Destruction” stands, of course, in general, for “diminution of 


intensity.” 
+ The few words are italicized by the present author. 


278 Dr. L. Silberstein on Fluorescent Vapours 


This seems very simple by itself. But the matter becomes 
even more simple when it is remembered that, by (16p), 
L/k=tan &, where € is the rotation or the angle E, r, so 
that (23) becomes 

Jo= Cos’ Ge... - - %: ee 


Now, this is as familiar as the most common of our 
everyday-life mechanical experiences. In absence of the 
rotating-agent (field H) the force E pulls r in its own 
direction, and is thus most effective ; and when these make 
with one another an angle &, then Ecos & only is operative, 
whence, and so on. The limit of magnetic rotation is 
€5=90°, but before this is reached, the fluorescence vanishes. 
We might even have started the whole investigation by such 
a line of reasoning. 

At any rate, it seems now indubitable that the magnetic 
“‘ destruction”’ of fluorescence (at least as far as the funda- 
mental line is concerned) is intimately connected with a 
magnetic rotation, and since the former is fully established, 
IT have but little doubt that the rotation will also be found 
by appropriate experiments. Im fact, if Jo=+/5, as in Wood 
and Ribaud’s experiments *, with H=30,000, then, by (24), 
the angle of rotation can be expected to be as huge as 

Gy l1>-565. 


Again, by (23), we should have, for the same magnetic 
9, 


held a Zi)? —= 9, 4.1e: 
L= +3k, 
and therefore for H=10,000 gauss, 
L= +k. 


Thus, our above H, would be 10,000.gauss, or, at any rate, 
very nearly so. The signs correspond to 7=0, as in (17). 
For the relative intensity J, of the jirst line of the series 
we have, by (14), writing n=Np, the rather complicate 
formula 
yes [ N?(1—p?)? + k? || N?(1—p?)?—2?—Z? | 
™ N4(1—p?)*— 2N?(1 — p?)?. (2? — hk?) — (k? + 22)?” 
for any magnetic field H. The corresponding values of Ja, 
J3, etc., for the second, third, etc., lines are obtained from 


(29) 


* Properly speaking, Wood and Ribaud’s experiments, dealing with 
the whole fluorescence, authorize us to assume not J,=1/10, but 
Sl g) ia if 
ZlIig=o) 10 | 
that the fundamental line taken by itself is reduced, nearly, in the same 


ratio. fi would be interesting to investigate spectrophotometrically the 
-dimunution of intensity of each line separately. 


; but in absence of better information we assume here 


and their Magneto-optic Properties. 279 


(25) by writing, instead of p, (2p—1), (3p—2), and so on. 
in particular, for = H,—10,000, ». e. for Z= +k, we have, 
neglecting (‘I/7)*, 


1 TS? 
a a ie} sles denaye eG yey 


and similar expressions for Jo, etc., with (2p—1), etc., 
instead of p. Returning to (20), and remembering that 
6,, 0, etc., are small angles, we can put (25 a), etc., into the 
more suggestive form 


oi Sm Gua. sin? ; 
Jj=l— p 5 Jo=1— (Qp—1)”” eter (26 a) 

If the angles of rotation, &, &, etc., are small (as it would 
follow from our above considerations) then Jy, J», etc., would 
differ but little from unity. That is to say, a field which 
reduces the fundamental line to about 4), of its intensity 
would weaken but little the remaining lines of the spectrum, 
while Wood and Ribaud’s experiments seem to indicate that 
the fluorescence is strongly reduced as a whole. Thus, our 
theory seems to be just to the fundamental line, but not 
so to the higher lines of the resonance-spectrum. The 
reason of this outstanding difficulty seems obvious. In fact, 
formule (25), etc., for the first and higher lines have been 
deduced from the equations (12) under the assumption that 
the coefficients ¢,, cz, etc., have the same values in the presence 
of H as in the absence of any magnetic field*. Now, as 
has already been mentioned, this is only approximately true, 
and may well require a considerable correction when such 
strong fields as H-=10,000 gauss are in question. The 
results, however, of my theoretical investigations concerning 
this matter are better postponed until spectrophotometric 
measurements of the magnetic reduction of the separate lines 
of the series are available. 

Such measurements would also enable us to find the time 7, 
and since ¢) gives 77, we should beable to estimate y=g/mV, 
which need not, thus far, be the electronic ratio. Mean- 
while, it may be interesting to see what 7 is like 27 7 is of the 
order of the electronic ratio. Now, returning to (23), and 
taking again for Jo Wood and Ribaud’s value ,, for 
H=30,000 gauss, we have 10,000|7|==2/r, @. e. 


DIO =i 
PS HERS 


* The reader will notice that this concerns only ¢, ¢2, ete., but not ¢o, 
the coefficient for the fundamental line. 


== A elireeeccr «1 CAP) 


280 Dr. L. Silberstein on Fluorescent Vapours 


and if T be the fundamental period of the iodine series A 
(A=5461 A.U.), 


“= 1-7 . 10>*; of» electronic. .. xray 

On the other hand, if » approaches more the value of the 
electrolytic ratio, say, for hydrogen, then T/r would be of the 
order 10-°. In the former case 7 would amount to about 
5900 T, and in the latter toa hundred million T. Inter- 
mediate and other values are also possible. The decision 
must be left to the experimental physicists. 

In connexion with this subject one more remark. In 
Wood and Ribaud’s paper we read, p. 1016, loc. cit. : 
“The effect of the magnetic field in reducing the intensity 
of the fluorescence becomes more marked as the vapour- 
pressure of the iodine is diminished.” This would mean, 
by (23), that with decreasing pressure the time t is lengthened 
(or the ‘friction ”’-coefficient diminished), and this seems 
quite plausible. In fact, there are reasons to believe that 
extinction is due not only to emission but also due to the 
encounter of the molecules or atoms, and then 7 ought to 
contain a term proportional to the mean time elapsing 
between successive encounters. Now, the latter time is 
certainly lengthened when, ceteris paribus, the gas or vapour 
is rarefied. 


7. Short Note on the Amount of Polarization. 


In the above treatment of the subject we have tacitly assumed 
that the atoms or molecules of the fluorescent vapour, in which 
the resonators are embedded, are fixed, 7. e. non-rotating, 
with respect to the apparatus, and therefore with respect to 
the direction of the exciting electric vector E. The result 
has been that rectilinearly polarized incident light has given, 
in absence of magnetic field*, fluorescent oscillations r 
parallel to E, 2. e. totally polarized. On the other hand, 
Wood has found experimentally a partial polarization only 
(Phys. Zeitschr. xii. p. 1209), viz. for iodine vapour only 
17 per cent. and for sodium vapour 20 per cent., ‘and in a 
very rarefied vapour and at a low temperature even 30 per 
cent.” 

Now, it occurred to me that this state of things could be 
accounted for by assuming that the atoms or molecules 
carrying the resonators are endowed with rotational motion, 

* To this simple case will our attention be confined here. The same 


problem with a superposed magnetic field will be taken up at a later 
opportunity. 


and their Magneto-optic Properties. 281 


provided that the mean time of revolution is comparable 
with r, and, of course, that the axes of rotation are hap- 
hazardly distributed. 

Without entering upon the details of my calculations, 
which in vector language have assumed a very simple form, 
I shall quote here the final result only. Let, for the sake of 
simplicity, the absolute value of the angular velocity, w, be 
the same for all atoms or molecules, but let all directions of 
axes of rotation be equally represented. ‘Then the amount 
of polarization P, ranging from 0 to J, is nil for rays parallel 
to the rav n of the exciting (unpolarized) light, and, for any 
ray perpendicular to n, for the fundamental line of the 
spectrum, 


P= A=OT, . 6 ° 2 (28) 


1+ 3a? + feat” 
where 7 has the same meaning as throughout the paper, and 
a is the angle through which the atoms are turned during 
the time 7. [In reality, the angles a will not be equal but 
distributed round their average «, say, according to the 
error-law. But, for the present, we shall content ourselves 
with the above formula, keeping in mind that 7 is not 
precisely the averagewT|. Forw=0 we have P=1, i. e. full 
polarization. When o (or, better, w7) increases, P decreases, 
tending to nothing. If, as in one of Wood’s more recent 
experiments with iodine vapour, P=-064, then, if the whole 
of the missing 93°6 per cent. can be thrown upon molecular 
rotation, formula (28) would require that 


On— AiO: TAC an Saree 20) 


This is a huge angle (warning us not to neglect «* in 
the above formula). If 7 is of the electronic order, or + 
equal to about 5900 T, as in our example of section 6, 
then wT =0° 2’ 23’. That is to say, during one period of 
luminous oscillation, of the fundamental period, the atoms 
would turn round through less than 23 angular minutes, so 
that their revolutions would still be very slow in comparison 
with the fluorescent oscillations, although comparable with 
the quickness of their extinction. Owing to the small value 
of wT, the equations leading to (28) have been considerably 
simplified. Further details concerning this subject will be 
given in a later publication. Here but one more remark. 
If the ratio g/m has the electronic value, then, by (27) 
and (29), 


@=3'81.10" radians per second. 
This seems a prodigious angular velocity. Now, it is very 


Phil. Mag. S. 6. Vol. 32. No. 189. Sept. 1916. U 


282 Mr. E. H. Nichols on the Diurnal 


remarkable that the law of equipartition of energy of the kinetic 
theory of gases gives an angular velocity of the same order. 
In fact, taking for the mean translational velocity of iodine 
molecules at 0° C. the value 1°64.10* cm./sec. and for the 
molecular radius 2 .10~* em., and attributing to the molecules 
two exchangeable rotational momentoids, against three trans- 
lational ones, I find for the average angular velocity 


Oia e m2 
which is well comparable with the above value. 


I gladly take the opportunity of expressing my thanks to 
Professor R. W. Wood, who has drawn my attention to 
these beautiful phenomena and has encouraged me to take 
up the subject of resonance-spectra from the theoretical 
standpoint. 


July 27, 1916. 


XXXII. The Diurnal Variation of Atmospheric Electrical 
Quantities. By KH. H. Nicnots, B.Se., A.R.C.Sc.* 


1 ee diurnal variation of the electrical conductivity has 
been investigated recently by Kahler tf and Dorno { 
by means of a continuously recording dissipation apparatus, 
while Gockel § has obtained eye observations of conductivity 
and electric charge. For the investigations here detailed, 
two Ebert Electrometers || have been used to record oe 
respective positive and negative electric charges, while 
the Wilson Compensating Gold Leaf EHlectroscope J 
measured the conductivity and air-earth current, the 
potential being obtained from a Kelvin Water-Dropper 
apparatus. The method employed for the Wilson instru- 
ment was a variation of the original method adopted by 
C. T. R. Wilson. There are three operations involved :— 
(a) The charge on the test-plate is first found by removing 
the cap, and replacing it quickly after momentarily earthing 


* Communicated by the Director of the Meteorological Office. 

+ Ergebnisse der Met. Beob. im Potsdam, 1909, 1911. 

t “Studie tber Licht und Luft des H ochgebir ges,” 1911 (Davos). 

§ Mache u. von Schweidler, Atmosphdrische Elektirizitat, 1909, p. 90; 
A. Gockel, Met. Zeit. xxiii. pp. 58, 339 (1906); xxv. p. 9 (1908) ; 
Archives des Se. phys. et nat., Sept. 1913. 

|| Phys. Zevt. viii. 8. p. 246 ; vii. 16. p. 527; x. 8. p. 201. 

q Proc. Camb. Phil. Soe. xiii. p. 184 (1906). M. 0. Geophys. Mem. 
Novi: 


Variation of Atmospheric Electrical Quantities. 283 


the plate. The reading of the gold leaf is taken (g,), and 
also after again earthing (g)). During this operation the 
compensator must be fully out. The charge on the plate is 
represented by (9;—4)- 

(b) The charge lost in five minutes from the test-plate is 
obtained by removing the cap and adjusting the compensator 
until the reading of the gold leaf is again gg, and reinains 
at go for the five-minute period. The cap is then replaced 
and the compensator fully drawn out. The gold leaf is 
again read (ga) and also after earthing (q'). The leakage 
in the test-plate due to the air-earth current is represented 
by (92-90). 

(c) The first operation (a) is repeated to find the charge 
on the plate (93 — go’). 

The conductivity is given by 


N= ABHG—a0!)! (or +93— 9o—9o )) X 10-* e.m.u., 


where 5°89 is a universal constant, and no instrumental 
calibration is involved *. The formula may be proved as 
follows. The surface density of the charge on the test- 
plate is given by 


o = electric force /4ar = = e@.S.U. 


for a potential gradient of 100 volts per metre (1 volt 
per cm.). Supposing the loss of charge on the test-plate 
is p per cent. per minute, then the current is equal to 


1 p 1 
1200m ~ 100 * goose Per second = 1°473p x 10-! amp./sec. 


For a potential of v volts per cm., the current 
=1-473pv X 107" amp./cm.? 
The percentage loss per minute (for five-minute intervals) 
=p=100(g2—go') /[5 X 241 + 935—Jo~g0!") 
=40(g2—go')/(gr +93—Jo—Jo )- 


al a a, ° . . . . . 
* This assumes the constancy in scale value for the divisions in the 
eyepiece. 


U2 


284 Mr. E. H. Nichols on the Diurnal 


Therefore the current 
=5°89v{(g2—go)/ (91 +93—9o—go )} X 10~-* amp./em.”, 


and the conductivity 
=5'894(g—90') |(9it+9s—Go— go) } x 10-* e.m.u. 


The mobility, or the ionic velocity under the influence of 
unit gradient of 1 volt per cm., can be calculated from the 
conductivity (A) and the positive charge (H,). It is con- 
sidered that the conductivity measured by the Wilson 
electrometer is that due to the positive ions*, so that 
X,=E,,, where w, is the mobility of the positive ions. 

Observations of the diurnal variation of the electrical 
elements were commenced at Kew Observatory in May 
1914, and continued during June and July on four days 
of each month (Table I.), with an additional day in August 
(Table III.). During May and June hourly observations 
were taken, from 7 a.m. to 8 P.m., each extending for about 
15 minutes, while during July and August the times were 
from 4a.m.to 8p.m. The weather conditions were excep- 
tionally favourable, only a very few days being affected by 
rain. As the instruments must be suitably mounted in the 
open, any rain beyond a slight drizzle prevents continued 
observation. Synchronous values of the potential were 
obtained from the Kelvin water-dropper, a tactor being 
applied to convert the values to the potential gradient in 
the open. 

A few hourly results are missing for the different ele- 
ments. In these cases the mean value for the preceding 
and following hour has been used, while in one or two 
instances hourly values have been extrapolated by the 
usual method. 

The mean summer diurnal variation for Kew obtained from 
twelve daysin May, June, and July is shown in fig. 1. There 
is a maximum for the positive charge at 11 a.m. and for the 
conductivity at 2p.m. The air-earth current shows no im- 
portant variation, but the mobility gives a decided increase 
afternoon. The potential curve indicates a definite minimum 
in the afternoon, and agrees closely with the normal for 1898— 
1912 ¢ for those months, thus proving that the»variation of 
the elements obtained should be generally applicable to 
electrically quiet days. 


* Lutz, Luftelektrische Messungen am Miinchen, 1911. 
+ C. Chree, Phil. Trans. A.cexv. p. 141. 


Variation of Atmospheric Electrical Quantities. 285 


Fig, 1.— Kew, 1914. 
Mean Diurnal Variation (May, June, July) (unsmoothed). 


we 
i 13 14 $5 ey fie Tf i) 23} 


(+ Charge 
perc.c. yx10"° 
in €.m.u,. 


Conductivity xX 107 
ine. 7. u. 


bab Gl 
bora 
a 

a 
fCHE 


(Air-Earth 6 
Current)x10 
in amp. cm? 


at) J 
Sa 
eee rae 


volts/metre 


ippissestentasees 
rh octet eT 


San ee 

250 pa 

Potential Otis che 
oe PEE — 


Normal 


(1898-1912) ‘°° 


a 
| lees 
reves 


Monthly Means jor May, a and 1914. 


From the results for May given in Table L., there isa 
minimum for the positive charge at 9 a.m. and two almost 
equal maxima at 11 a.m. and 2 Pp.m., while the conductivity 
shows a well-marked minimum at 8 a.m. and maximum at 
4pm. The air-earth current shows a minimum at 8 A.M., 
which corresponds with a maximum of potential. The po- 
tential is low during the afternoon with a minimum at 1 P.M., 
while the mobility increases gradually during the day. For 
June the positive charge has a maximum value at noon, and 
two minima at 9 A.M. and 7 P.m., which correspond with the 
times for minima in the conductivity. As usual, the maxi- 
mum for the positive charge precedes that for conductivity 
by a few hours. The air-earth current varies somewhat 
similarly to the potential, and the mobility increases during 
the day to a maximuin at 5 P.M. 


286 Mr. E. H. Nichols on the Diurnal 


The results for July give a more detailed account of the 
electrical conditions, as the negative charge was also mea- 
sured and observations commenced three hours earlier. The 
range for the conductivity is considerably less than for the 
other months, there being a maximum at 1 P.M. and a 
minimum during the night. There is a curious well-marked 
minimum at 11 a.m. which corresponds to a maximum 
electric charge. Individual results show a minimum at 
11 a.m. for July 23, 28, and 30, while for July 21 there 
is one at 9 a.m. The minimum at 11h. appears in the 
3-monthly means, and the depression from 10 a.m. to 1 P.M. 
is more pronounced still in the Eskdalemuir results discussed 
below, so that it is probably a real phenomenon. 

Although the data for the individual days cannot be dis- 
cussed in detail, a few points may be noticed. ‘There 
appears to be no definite effect of precipitation on the elec- 
trical quantities as a general rule, but there are cases in 
which a considerable decrease in electric charge or con- 
ductivity has been observed to follow rain (see Table III.). 
In the few measurements made before and after thunder- 
storms, the electrical values obtained show nothing abnormal. 
Usually there is a similarity between the variation of the 
conductivity and the positive and negative charges, the two 
latter showing a close agreement. There is also a definite 
minimum about midday in the ratio of the positive and 


negative charges (5), which is obviously associated with 
the solar effect. z 

A partially successful attempt was made on June 29-30, 
1914, to obtain the electrical variation for the complete 
24 hours, observations being commenced at 7 A.M. on 
June 29 and continued hourly till 7 a.m. June 30 (see 
Table II.). After 9 p.w., however, the Wilson electroscope 
failed owing to insulation trouble, which is a great diffi- 
eulty in night work. For such work an electric torch must 
be used for reading, oil lamps being fatal to results, because 
of the production of conducting gases. The diurnal varia- 
tion of electric charge gave some indication of a double 
period with primary maximum and minimum about 4 p.m. 
and 4 A.M. respectively. The values for both charges were 
lower at night than during the day. The ratio of the charges 


Gr) shows a remarkable maximum at 3 A.M,.and a mini- 


mum at 12 noon. Such an outstanding maximum might 
appear fortuitous, but similar examples of high maxima 
occurred on July 23 at 4a.m., July 28 at 5 a.m, and 
August 4 at 5 A.M. 


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‘UOHVLIvA [VUINIG Loj son[eA A[YQUOP UvePT “FTE ‘may—y] HTAV I, 


Mr. E. H. Nichols on the Diurnal 


288 


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289 


mtities. 


Electrical Qua 


7 Atmospheric 


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290 Mr. E. H. Nichols on the Diurnal 


Observations at Hskdalemuir during August 1914. 


As Kew Observatory is only a few miles from London 
and practically at sea-level, it will be advantageous to 
compare the electrical variation with that for Eskdalemuir 
(237 metres above M.S.L.) in Dumfriesshire, removed from 
disturbing artificial conditions. The importance of the elec- 
trical investigation was increased by the fact that it was 
possible to include a day of partial solar eclipse, and thus 
compare the diurnal variation obtained with that for the 
normal day. Observations of the conductivity variation were 
made on six days, and of the electric charges on seven days, 
viz. August 13, 14, 17, 20, 22, 28, and 31. As the eclipse 
day was August 21, it was considered important for compa- 
rison to observe on the preceding and following day, andalso 
to have the other days well balanced with respect to August 21. 

Comparing the monthly means at Kew and Eskdalemuir, 
the conductivity is nearly twice as great at the latter place 
(see Tables I. and IV.). There is a corresponding early 
morning minimum about 5 a.m. for both places, and the 
July depression at Kew about midday is repeated for 
August at Wskdalemuir. The most remarkable difference 
in the variation is the sharp increase in conductivity at 
Eskdalemuir between 8 and 10 a.m., there being only a small 
rise at Kew. This is certainly to be associated with the 
rapid disappearance at Hskdalemuir of the morning mist, 
which was frequently dissipated during these hours. 
Because of the decreased relative humidity the charged 
ions in the air would lose a percentage of their water 
molecules, thus increasing their mobility and, consequently, 
the conductivity of the air. 

It is noticeable that the charges at Hskdalemuir varied 
less than at Kew. The mean positive charge is about 
6x10-8%e.m.u. at both places. For the electric charges 
at Eskdalemuir there is a maximum about noon, and a 
minimum during the night. There is also an indication of 
a secondary maximum and minimum in the early morning. 


The ratio of the charges (a) shows a decrease during 


the day with a minimum at 1 P.m., which agrees well with 
the Kew results. The air-earth current is somewhat irre- 
gular, and there is no well-marked variation. It should be 
noted that the conductivity and potential were both high at 
Hskdalemuir, so that the air-earth current is high, while the 
mobility shows 100 per cent. increase from 4 to 10 a.m. and 
maintains an almost steady value for the rest of the day, 
being about double that obtained at Kew. 


Variation of Atmospheric Electrical Quantities. 291 


Ficlipse Observations, August 21, 1914. 

instrumental readings were taken every hour from 4 A.M. 
to 10 a.m., and then every half-hour to & p.m. The duration 
of the eclipse was from 10 4.M. to 1 P.m., with a maximum 
phase of 0:7 for Eskdalemuir. The meteorological con- 
ditions were not ideal, as in the early morning there was 
a wet mist, which made observation very difficult. About 
8-9 a.m. the air became drier, but it was still dull, the sun 
being only visible for short periods, and then dimly through 
thin Stratus and Strato-Cumulus. At 2p.m. and 2.40 P.M. 
a little rain fell, but not sufficient to prevent observation. 
The diurnal variation observed is quite different from that 
obtained on ordinary days. In order to compare the results. 
more conveniently, they have been represented graphically 


in figs. 2 and 3. 
Fig. 2.—Lskdalemuir, 1914. 


Noon 
Ae Ge aie Oe OM eit eter sans ee ana 
140 
120 | al +— Bee | 
clase Feyiod 
100} 
80} = 
(+Charge ~ s-l-7 
per c.c.) X10 Ge 
in e./71,uU. 40 ae eee 
(Charge /, a a 
per c.c)X10 
iq €.m. . ——= = —= ~ 
10 t + 
Ratio S zi 
of Charges 4 
E+ /e~ 
2 Se 
ee ee 
pe eal Ce 
120} bet ate 
= Sq 5 
re 
80 aT + aay Zn J tL 
27 Tigres = | 
Conductivity X/0 20 ae ase : 
-— 47 @.7.,U. i cel | 
ee | On aa: 


August mean diurnal variation (smoothed). — ~~ 
Eclipse day variation [(Unsmoothed)..... 


As the normal variation for August is only obtained from 
6 or 7 days, it appeared desirable to smooth the values given 
at+2b+e 
= - Lhe 


in Table IV. by means of the formula q 


292 Mr. KE. H. Nichols on the Diurnal 


eclipse day variation is, however, unsmoothed. ‘There isa 
considerable decrease in conductivity and electric charges 
from 10 a.m. to 5 P.M., commencing soon after the primary 
phase of the eclipse. The maximum for the day, instead of 
appearing in the early afternoon, is retarded till about 5 P.M. 
There is a decrease in the mobility which might be explained 
by the fact that increased relative humidity due to the fall 
in temperature during the eclipse causes condensation of 


Fig. 3.—LEskdalemuir, 1914 (continued). 


00 


e Pe fod) 


(Air-Earth 2 
Current)X/0 ~ aoo 
; 2 
i amp./cm. 
Pa. 300 


Potential 
volts per metre 309 


August mean diurnal variation (smoothed) ~~ == 
Eclipse day variation (unsmoothed)..._. === 


water molecules on the ions. ‘There is but little indication 
of an increase in the ratio of the electric charges, which has 
been found by some observers, during or near the totality 
stage, and ascribed to the reduction in the number of the 
negative ions”. 

Superficial observation might connect the depression in 
the electrical quantities from 10 a.m. to 1 p.m. with the 
eclipse, but a similar depression is obtained in the August 


* Leyst, Luftelektrischen Beobachtungen, 1907. 


er 


Variation of Atmospheric Electrical Quantities. 293 


mean, and is more or less distinctly apparent in the indi- 
vidual days except August 31. ‘Thus it seems to indicate a 
general phenomenon. It was often noticed at Eskdalemuir 
during this month that an early morning ground-mist was 
dissipated between 8 and 10 a.m. and was followed by bright 
sunshine with cloudless sky lasting a short time but be- 
coming overclouded towards noon. Although at 10 a.m. 
the mist had quite cleared on the slopes near the Obser- 
vatory, it was often visible in the valleys. It is possible 
that the dissipation of the valley-mist to higher levels might 
cause a decrease in the mobility of the ions, and so in the 
conductivity. 

In order to study the eclipse effect more closely, the 
results for August 20 and 22 may be considered. The 
general form of the variation for both days was similar 
except that on August 22 the increase in the conductivity 
and the electric charges from 7 to 10 a.m. was very marked, 
and also that instead of a gradual fall from 4 to 8 p.m. there 
was an increase, this being especially marked in conduc- 
tivity. The mean for the two days shows a deeper depres- 
sion in conductivity between 10 a.m. and 1 p.m. than is 
obtained on August 21. This reduces the probability that 
any eclipse factors produced the diminution of electrical 
quantities on the day of eclipse between these hours. 
But when we consider the retarded maximum in the con- 
ductivity and both charges, there is some justification for 
considering it to be the result of an indirect solar effect. 
Some observers have considered it sufficient in eclipse work 
only to investigate the electrical variation during the actual 
phases, but a lag in producing any electrical effect would 
make later hours important. Wiechert’s reasoning * with 
respect to solar radiation and the maintenance of atmo- 
spheric ionization, indicate the probability of a lag in any 
eclipse effect which may be due to the suspension of the 
ionizing influence of ultra-violet light. Although most 
eclipse observers in the past do not appear to have noticed 
any striking electrical changes, it is to be hoped that a 
longer series of observations will be possible in the future at 
stations situated near the line of totality, with sufficient 
observations on preceding and following days to obtain the 
normal form of the diurnal variation. 

The few results obtained by G. Dobson + and other ob- 
servers | show generally a diminution of conductivity and 

* “Tes Recherches sur l’Electricité Atmosphérique,” Arch. des Sc. 
Phys. et Nat. 1912, p. 385. 


+ Q. J. Roy. Met. Soc. 1913, p. 221. 
t Ludeling & Nippoldt, Abhand. d. Kon. Preuss. Met. Inst. 1908. 


294 Diurnal Variation of Atmospheric Electrical Quantities. 


number of ions per ¢.c. during the eclipse period. For the 
1912 eclipse at Kew the conductivity and number of ions 
showed. a decided increase during the first phase, but this 
was followed by a well-marked depression. for Hskdale- 
muir, no marked change in potential was found, and this 
agrees with the results from other eclipses. 

It is well to emphasize that while small instrumental 
differences may have existed between the three Ebert 
electrometers used, as we were at that time dependent 
on values supplied for the capacity and anemometer 
readings by the makers, these uncertainties cannot affect 
the character of the diurnal variation described above, 
because for any one element only one instrument is 
involved. 

There is some difference of opinion as to the best method 
of treatment of results from the Wilson electroscope. Some 
experiments of G. Dobson * at Kew appear to show that the 
conductivity as measured usually with the Wilson instrument 
on a tripod-stand 1°3 metres from the ground is smaller than 
that measured at ground-level, owing to the crowding to- 
gether of the equipotential surfaces above the test-plate and 
the production of a saturation current. The correction for 
the conductivity to bring it to ground-level values was 
+ 20 per cent. The mean for these corrected values is 
similar to the total conductivity recorded by the Ebert 
instrument. As the individual results ditfered so widely, 
and owing to the insensitiveness of the Ebert apparatus 
many negative values for mobility and conductivity were 
obtained with it, the deduction is not convincing that the 
corrected conductivity obtained from the Wilson apparatus 
corresponds with the total conductivity measured by the 
Ebert apparatus. Under existing conditions, it has been 
thought advisable to continue the old mode of reduction 
untii a more satisfactory method is devised. ‘The results 
detailed above are, however, quite comparable, as the 
observational conditions were uniform. 

In conclusion, I desire to acknowledge my indebtedness 
to several members of the Observatory staff at Kew and 
Eskdalemuir for assistance in carrying out the experi- 
mental work involved, especially to Dr. C. Chree, F.RS., 
Superintendent of Kew Observatory, under whose super- 
vision the work has been carried out, and also to Lieutenant 
C. D. Stewart, who performed a number of the earlier 
observations. 


* Proc. Phys. Soe, 1914. 


fr 29500 | 


XXXIV. The Vapour Pressures of Binary Liqud Miatures: 
Kinetic Theory based on Dieterici’s Equation. By FRANK 
Tinker, M.Sc.* 


i a series of papers published at intervals during the last 

sixteen years}, Dieterici has advanced a much more 
fundamental equation of state than the better known Van der 
Waals’ expression. ‘The equation in question, viz., 


a 


nee ait SNE ea kenanees ait 8 


U— 


{Qe 


in which the constants a and 5 have a meaning similar to 
that given them by Van der Waals, is based on the following 
assumptions { :— 


G.) The pressure 7 within a fluid is given by the equation 
TOO) — Ra ea a eee 2 


2. @., the product pressure x free space is given by the 
perfect gas equation; being independent of the size of 
the molecules, or of the forces between the molecules, and 
the same for all fluids at the same temperature. 

(ii.) The pressure at the surface of the fluid is always less 
than that in the interior because of the inward pull exerted 
by the forces of cohesion. Assuming that the boundary 
pressure is proportional to the number of molecules which 
have kinetic energy enough to overcome the inward pull, 
Dieterici showed that this boundary pressure (») is related 
to the pressure within the fluid (7) by the exponential 
equation 

A 


=e, EL. hema r Eo | 


where A is the work done by the molecule in reaching the 
surface. Combination with equation [2] gives immediately 


eee 
Le ag RE ea eines eek [4] 


and making the further assumption that A is proportional 
to the density of the fluid, and equal to where a is a 


* Communicated by Principal Sir O. J. Lodge, F.R.S. 

+ Ann. Phys. u. Chem. xi. p. 700 (1899); Ann. der Phystk [4] v. 
p- 51 (1901) ; 2bed. xxv. p. 269 (1908) ; zb¢d. xxxv. p. 220 (1911). 

{ The assumptions ate also similar to Van der Waals’, but are given 
a different quantitative expression. The two equations become identical 
at low and medium pressures. 


296 Mr. F. Tinker on the Vapour Pressures 


constant for the substance in question, the equation [4 | 
becomes identical with equation [1]. 

Dieterici himself has shown that his equation gives ex- 
cellent results when applied to critical data in the usual 
way. Recently others also have taken up the development 
of the subject. In particular McDougall has lately demon- 
strated * that the pressure of the saturated vapour over a 
liquid is also given by the equation to a high degree of 
aecuracy. 

In view, therefore, of the interest which has been revived 
in the causes for the deviations of the vapour pressures of 
binary mixtures from the well-known law of admixture in 
molecular proportions, and on account of the importance of 
the question to the subject of osmotic pressure, it seemed 
that it would be of interest to extend the application of 
Dieterici’s equation to such mixtures also. Van Laar has 
already studied the subject from the thermodynamic side. 
Tt will be seen from the treatment which follows that 
Dieterici’s equation leads to expressions for the partial and 
total vapour pressures identical in form with those obtained 
by Van Laar with the aid of the thermodynamic potential ; 
but that the kinetic treatment enables the subject to be 
developed further, and gives a very simple relationship 
between the variation of the vapour pressure of the mixture 
from the theoretical, and the variation of the latent heat of 
vaporization from the theoretical value calculated by the 
mixture rule. 


(a) Relation between the Partial and Total Liquid Pressures 
in a binary nuxture and the relative molecular concen- 
trations of the two components J. 

Let X and Y be the two components of the liquid mixture, 

and let N molecules of X be mixed with n molecules of Y. 


* Journ. Amer. Chem. Soc. xxxyiii. p. 528 (1916). Mr. McDougall 
shows that 


Ris 
aN Ta ea ee 
where v, and v2 refer to the specific volumes of the liquid and vapour 
respectively, and where a and 6 have the values given by the equations 
Me ee 
xo Vi Un’ V+. 

+ In what follows I use the term liquid pressure (7) to denote the 
bombardment pressure exerted by the liquid molecules on either side of 
a plane of unit area placed anywhere within the liquid. It is of course 
different from the internal or intrinsic pressure due to cohesion. In 
another place (‘ Nature,’ vol. xevii. p. 122 (1916)) I have also called it 
the “ diffusion pressure ” to distinguish it from the latter. 


of Binary Liquid Mixtures. 297 


Also let m@,=liquid pressure of pure X, 
To 9 9 29 2 

7, and 72 =partial liquid pressures of X and Y in 

: the mixture, 
7 =,’ +77.’ =total liquid pressure in mixture, 
V,and V,=molecular volumes of pure X and Y, 
Veaeancn ey — a Ok oan Nein tae 
mixture. 

In general, V,’ and V,' will be slightly different from 
V, and V, owing to the small volume changes which take 
place on mixing. 

Consider first the partial liquid pressure 7,’ of the com- 
ponent X. Before the N molecules of X are introduced 
into the mixture, the liquid pressure of the pure X is given 
by the relation 

RT 
Ty N( Vi 5) 9 A e 5 ° ° [5} 
where the constant R refers to the mass of N molecules. 
Now add the n molecules of Y to the N molecules of X. 
We have, volume of free space in mixture 


= N(V,/ = b) + n( Vo’ —bp). 


Hence partial pressure 7,' of component will be given by 
the equation 

i LE ee 

© NCW —6:) + n(V,'—5,) ~ (N+n)(Vi=3)’? 1? 


since the two “free spaces’”’ (V,'—0,) and (V»'—4,), being 
in the same mixture, are equal to one another. | 
Combining equations [5 | and [6] we obtain 


ite N V,—5}, a N 1 
mame Neon Wb, Nea yma ces «4 


where ¢, denotes the fractional increase in the “free space” 
of the molecule of X by the mixing. 
Similarly 


! 


oa — (Ley een et) 


TT) Nin 
If instead of a and ue we substitute the molar 
N-+n N+n : 


fractions v and (1—w) in the usual way, equations [7] and 
[8] become 
w,'=17,4(1—e,), ° . “ : ° . [9] 
% =m, (l—a)(l—e).  . . . . [10] 
Phil. Mag. 8.6. Vol. 32. No. 189. Sept. 1916. X 


298 Mr. F. Tinker on the Vapour Pressures 


We may usually neglect the volume changes of liquids on 
mixing, so that e; and €, are practically zero. In this case, 
the total liquid pressure 7 becomes 


T= +7, =mMe+m(l—z), . . . [11] 


Hence when there are no volume changes on mixing, the 
total liquid pressure can be calculated from the two partial 
liquid pressures by the law of admixture in molecular pro- 
portions. For the partial liquid pressure of each component 
is then proportional to its molar fraction. If, however, there 
is a volume change, the law is only approximately accurate. 


But the ratio see = (l—e«) will in general be so nearly 
ee eel 


equal to unity that there will be an appreciable deviation 
from the mixture law only in very abnormal cases. 


(b) Relation between the Partial and Total Vapour Pressures 
ina binary mixture and the relative molecular concen- 
trations of the two components. 


We can now apply Dieterici’s equation proper to the 
questions in hand. 
Let p, and p,=vapour pressures of pure components X 
and Yo 
Dy TOC iy — E of X and Y in mixture, 
p=p1 + .'=total vapour pressure of mixture, 
A, and A,=work done when a molecule of X or Y 
is evaporated from the pure liquids, 
A,' and A,’/=work done when a molecule of X or Y 
is evaporated from the mixture. 
N, n, x have same values as before. 


From equation [3] we have 


bat RT 
P= TW1é 5) 
_ 
py) =m eo! 
Hence 
| EN OARS 
o 1 1 
(EET R 
P1 1 


Combining this equation with equation [9] and neglecting e, 
we have 
Gi) 


pr = pire Hel 5 San iney eta se we [12] 


of Binary Liquid Miatures. 299 


_ (Ag'—Aa) 
RT 


Similarly 
po! = ps(l—a)e 

; t (A,'—A)) fs (A,’—Ad) 

P=Ppi tpo' = pite RT +p2(1l—a)e ; 
[13] 


The last equation is of the same form as the equation 
Van Laar has obtained for the total vapour pressure of a 
binary mixture on thermodynamic grounds. Van Laar’s 
expression 1s 


and 


_ (1'=1) _ (u2'= M9) 


p=pere = +p(l—xe * 


where yu; and py are the thermodynamic potentials of the two 
components *. 
Hquation [13] can also be written in the more convenient 
form 
os _ OAs 
pe pee p(n er (ea 


where 0A; and QA, represent the increase in the work done 
by evaporating a molecule of X or Y from the mixture over 
the work done when the molecule is evaporated from the 
pure liquid. 

Furthermore, it will be seen that 9A, and OA, are 
practically identical with the excess of the molecular 
latent heats of vaporization OL, and OL, of the two 
components in the mixture over their molecular latent heats 
in the pure state. Hence we may write t+ 


_ Oly _ OL, 
P = pPive el + p(1—wx)e RT 5 * 6 [15] 


We may develop equation [15] still further. Expanding 
OL; _ OL, 
e ®T and e #7 we have, since QL, and Ol, are small, 


p= =p2(1-$ San) tpt x) (1- oe) 


it 
= pre + pa(1—2) — ap} PwwTa +72(1—a) OL, |. 


Now if po is the vapour pressure of the mixture calculated 


* Zeit. Phys. Chem. \xxii. p. 723 (1910) ; zbed. Ixxxiii. p. 599 (19183) ; 
and other papers. 

{ Equation [15] indicates that 0#=0L, or that as regards the vapori- 
zation of liquids, increase in thermodynamic potential is equivalent to 
increase in latent heat of vaporization. 


a 


300 Mr. F. Tinker on the Vapour Pressures 


according to the law of admixture in molecular proportions, 
and if L denotes the molecular Jatent heat of vaporization 
of the perfect mixture *, we have 


Po= pit t+p1—-2) 
and pl=picl,tp(l—z)lL,, .. « 2) sei 
since the two components evaporate in the ratio of their 
partial pressures. 


Hence also 


ytd Ly + p2(1 —x)d L, = pool. 


L 
Consequently p= pit + px1—x)—po ar 
OL 
= Po —Po RT b) 
1. @. 


p=po(1— Sn): 2°. {i 


or in words, “the total vapour pressure of a binary mixture 
is equal to the vapour pressure calculated from the law of 
admixture in molecular proportions, multiplied by a factor 


(1-97). where OL is the excess of the molecular latent 


heat of vaporization of the mixture over ithe {theoretical 
value calculated from the mixture rule.” “72nyage Se 


This simple relationship shows that the variation of the 
total vapour pressure from the straight line law is deter- 


mined almost entirely by abnormalities in the latent heat of 
vaporization fT. _ ate 


Proceeding from equation [17] we have’ three possible. 
cases :— 


(i.) If the actual molecular heat of vaporization of the: 


* The actual molecular latent heat of the mixture would be the 
heat of evaporation or condensation of one gm. molecule of vapour,. 
?.e. the heat required to produce 22-2 litres of the mixed vapour at 
ren The theoretical value L is of course calculated from equation 
16]. 

+ Volume changes on mixing will of course have a slight effect on the- 
vapour pressure also. Allowing for these, equation;| 17] becomes 


L ; 
p=p,(1— a) —(pi'e,+p2'es). 


of Binary Liquid Mixtures. 301 


mixture is equal to the theoretical value (QlL=0), the 
vapour-pressure curve is a straight line (Diag. ].). 


Diagramil. Diagram IT. 


aL negative 


Molar fractions of two components. 


(ii.) If the molecular latent heat of vaporization is less 
than the theoretical (¢. ¢. OL is negative) the vapour pressure 
is greater than the theoretical (Diag. II.). 

(ii.) If the molecular heat of vaporization is greater than 
the theoretical (7. e. OL is positive) the vapour pressure is less 
than the theoretical (Diag. III.). 


Diagram ITI. Diagram IV. 


| dL positive | 


If, as the relative molecular concentration of the two 
components is varied, the value of 9L changes from 
positive to negative or vice versa, the curve will cross over 
the diagonal (as in Diag. IV.). In this case OL is at first 
positive, and then changes over to negative. 

As yet there have been no measurements made by which 
the above equation [17] can be subjected to a quantitative 
test. But its general soundness can be shown indirectly 
from surface-tension data. It is well known that there is a 


302 Vapour Pressures of Binary Liquid Miztures. 


very close connexion between the latent heat of vaporization 
of a liquid and its surface-tension, liquids having a high 
latent heat, for instance, also having a high surface-tension *. 
Abnormally high vapour pressures are therefore to be asso- 
ciated also with abnormally low surface-tensions, and vice 
versa. FE. P. Worley has already advanced this view on 
experimental groundst. Corresponding to the three cases 
just given for the differences in latent heats, this investigator 
has given the three following rules for the connexion between 
the vapour pressures of binary mixtures and their surface- 
tensions :— 


“(i.) If at any given temperature the vapour pressures of 
mixtures of two liquids agree with the values calculated by 
the rule of admixture in molecular proportions, the surface- 
tensions of the mixtures agree with those calculated by the 
formula 


8 = Vi8; =F Wopss af 


“‘(ii.) If the vapour pressures are greater than those 
calculated, then the surface-tensions are less than those 
calculated. 

*‘(i1.) If the vapour pressures are less than those calcu- 
lated, the surface-tensions are greater than those calculated.” 


By comparing his own and other surface-tension curves 
with the vapour-pressure curves of Zawidski and others, 
Worley has shown that every pair of liquids so far investi- 
gated comes under the rules laid down. Mixtures of benzene 
and’ ethylene dichloride, for instance, form perfect mixtures 
and obey the first rule; carbon bisulphide and acetone, 
benzene and carbon tetrachloride, ether and carbon bisul- 
phide obey the second rule; whilst in the same way, water 


and the alcohols, and pyridine and acetic acid obey the 
third. 


* So much can be deduced from the Laplace Theory of Capillarity. 
Also Walden has given the following empirical formula for this 
relationship, based on a large number of experimental data : 


L= 364vy, 


where L=latent heat per gm. in calories; v=sp. vol. in c.c.; y=surface- 
tension in dynes/cm. 

+ Journ. Chem. Soc, Trans. cv. [1] p. 273 (1914). 

t Vi and V;2 are the volumes of the two liquids expressed fractionally. 


ha c0 sna 


XXXV. The Hall and Corbino Effects. 
By AuBeRT K. CHAPMAN ™. 

FEW years ago Professor Corbino observed that when 
a uniform radial current flows through a circular disk 
of metal placed in a magnetic field normal to the plane of 
the disk there is produced a circular current, the density 
of which is inversely proportional to the radius (Physikalische 
Zeitschrift, xii. pp. 561, 842, 1911). His experiments were 
confined principally to bismuth and antimony, in which this 
current is relatively large. Professor Adams (Phil. Mag. 
Feb. 1914, p. 244) has developed an expression for its 
magnitude on the assumption of the simple electron theory 
and this, of course, predicts that the effect has the same sign 
in all metals. Ina recent research (Phil. Mag. Noy. 1914, 
p- 692) Professor Adams and the writer quantitatively in- 
vestigated in this respect a number of metais and found that, 
in every case, the sign is the same as that of the Hall effect. 
The method used consisted in measuring the current induced 
in a coil placed near the disk when the radial current, and 
hence the circular current, is reversed insign. ‘This induced 
current was balanced by the current in the secondary of a 
known variable inductance, through the primary of which 
was passed the same current that traversed the disk. Upon 
reversing the magnetic field the circular current changes 
sign and may again be balanced by an alteration of the 

variable inductance. We then have the relation 


MC=ml, 


where C is the circular current in the disk, I is the radial 
current, M is the mutual inductance between the current C 
and the coil placed near the disk, and m is the reading of 
the standard mutual inductance. When the magnitude of 
the circular current permitted, observations for various 
fields were taken and the values of m/H given. Asa result 
of this investigation there was given also a table comparing 
the magnitudes of the Hall and Corbino effects, each being 
referred to copper, which was taken as standard. In general 
these numbers are of different orders of magnitude : how- 
ever, if the values of m/H referred to copper are multiplied 
by the ratio of the specific resistance of the metal in question 
to that of copper, they are brought into much better agree- 
ment. This is to be expected, since, in the Hall effect, 
differences of potential are measured, while in the Corbino 
effect actual currents are sought. The numbers for the 
Hall effect were taken from the Recueil de Constantes 
Physiques (1913), published by the French Physical Society. 
| * Communicated by Prof. KE. P. Adams. 


304 Mr. A. K. Chapman on the 


Field. Hall Effect. | Corbino Effect. 
Copper tierce ueeecs ind. H —1-0 —10 
Hel Brey cere Sree een 3700 21°9 16°4 
Aluminium +) 22.25-2.. ind. H —0O73 | —0°48 
SBismithes eee 1650 So TOU; | —ZOT00: 

Be eerste 8 ee 3930 — 12200 — 16300° 
ANTIMONY 22. -see sere 1750 421° 557° 
Platinum eee Sine en (AG. 7 —0°56 
Goldie seas tee tee ind. H 27 4 ae 
Silver...... a eee ind. H —160 | —1°61 
Nickel Recess... 1700 —23°'7 — 32:0 
ANALG eee ea ae, eS 063 | 0°53 
Cadmium: 25:2..2: ie Sneed 1:06 Lor 
Cobaltiger ees. iec ts | 38460 461 | 5:86 


The numbers given herewith under “ Corbino Effect” are 
values of m/H referred to copper and multiplied by the ratio 
of the specific resistance of the particular metal to that of 
copper. Multiplication by this factor brings the two 
phenomena to the same order of magnitude, but the agree- 
ment is not close except in one or two cases. Without 
doubt this is due, in part, to the fact that the values were 
determined for totally different specimens in the two 
instances: it will be shown later, however, that there 
remains an outstanding difference between the two effects 
when measured in the same specimen. Whether this dis- 
crepancy is due to the free boundaries necessary in the Hall 
effect, to some difference in internal behaviour, orto both,seems, 
in the present state of our knowledge, impossible to say. 

Assuming that, in the Corbino effect, the function of the 
magnetic field is to produce an electromotive force at right 
angles both to the magnetic force and to the primary current, 
and that this electromotive force is proportional to the vector 
product of H, the magnetic force, and HE, the primary 
electromotive force, we have 

Fi=cV Ek, 
where V indicates the vector product, E, is the electro- 
motive force of the Corbino effect, and cis the proportionality 
factor (E. P. Adams, Proc. Am. "Phil. Soc. vol. liv. no. 216, 
1915). Applying this to a disk of the form used in these 
experiments, we have 


O= 5 —log— SLE, 


where C is the total es ue I is the total radial 
current, 7, 1s the external radius of the disk, and 7, is the 
internal radius of the disk. We may then call 


2arC 
C=. ae 


the Corbino constant. Ht log r} 


Halil and Corbino Effects. 


The Table below gives the values of this constant for the 
metals investigated in the previous experiments. 


CoprEr. Tron. 

Field. ex10"° Field. Echo 
7310 2°38 8450 alts 
6750 2-51 7620 6°60 
5920 | 2°61 6750 6°58 
4220 2°40 4890 6°64 
2820 3°36 3300 6°55 

2060 8:27 
1390 7°45 
ALUMINIUM. 
GOLD. 
7970 0-770 | a 
BIsMuUTH. ANTIMONY. 
f | 
7720 381° 7210 | io 
6540 415: 6180 38°35 
5820 443° 4380 40°7 
4690 520° 2590 40°9 
2630 613° 
953 666° 
419 Tpke SILVER. 
107 aos rt 
30 allt: 
6900 4:78 
6240 5:05 
N ; 5510 4°86 
ee 4580 4-84 
6930 11°76 
5970 13:0 CoBALT. 
4810 14:6 = 
3730 14:5 757 3 
; 10 2:48 
2570 14:4 6690 9-6] 
408 53860 2:63 
CADMIUM. =a 
z . ZINC. 
| 
7210 | 0-715 
3 i 7470 0°3438 
PLATINUM. 
7056 0:199 ; 


305 


306 Mr. A. K. Chapman on the 


It is the purpie of the present paper to report on the 
variations of the Corbino effect with the magnetic field ina 

number of metals and alloys and, at the same time, to 
compare these variations with measurements of the Hall 
effect in a disk exactly similar to the one used for the 
Corbino effect. 

The experimental method used was the same as that 
described in the former paper. In the earlier experiments 
some difficulty was experienced with the rotating commutator. 
It was in series with the galvanometer, and any inequality 
in the pressure of the brushes or any undue pressure of the 
brushes always resulted in undesirable electromotive forces 
which made the galvanometer very unsteady. By lubricating 
the brushes with a mixture of graphite and oil, and by taking 
care that each brush had a smooth, broad, but not too heavy 
contact, a great deal of this trouble was overcome. 

The specific resistances of the metals under investigation 
were determined by measuring the potential difference 
between the central wire and the circumference of the disk 
when a known current was flowing radially through it. To 
take account of any non-uniformity, four points on the 
circumference, ninety degrees apart, were chosen, the 
potential difference between each point and the centre 
measured, and the mean taken. In all cases the disks were 
of the order of 0-9 mm. thick, so that such measurements by 
means of a Wolff potentiometer are accurate only for metals 
of considerable resistance. The values for copper and silver 
were simply taken froma table. The disks were allimmersed 
in a well-stirred oil-bath during this process. 

Circular plates with four integral electrodes, ninety degrees 
apart, were used for the Hall effect. The electrodes by 
which the primary current was led into the plates were 
heavy ; the others were much smaller and were connected 
to the galvanometer circuit. These plates were placed in a 
hard rubber mounting and introduced into the magnet in 
the same position as that occupied by the disks used for the 
Corbino effect. They were made circular in order to include, 
as nearly as possible, the same flux. In every instance 
where curves for both effects are given, the two disks were 
either cut from the same sheet of metal or cast from the 
same mass of molten metal, as the case may be. 

Copper.—It was thought advisable to repeat the measure- 
ments on copper under more favourable experimental 
conditions. The disk, cut from a sheet of commercial copper, 
was 0°9 mm. thick and 5:0 cm. in diameter. The radii, 
ry and 7, were 5:0 cm. and 0°266 cm., as they were for all 


Hall and Corbino Effects. 307: 


the disks investigated. The radial current in every case was 
about twelve amperes, and the value of m/H for each field 
is the mean of a number of determinations. One-half the 
difference between the balancing inductance readings for 
positive and negative fields is denoted by m. 


The Corbino Effect in Copper. 


igh m. m/TH. eX107. 
2590 133i) 0:00469 Dalelt 
4280 19°9 0:00465 2°09 
6130 32.0 0:00522 DESY) 
7590 39°8 0:00524 2°36 
8490 37°53 0:00442 1:99 
2510 el | 0:00484 2°18 
4060 188 | 0:00462 2:08 
5890 28-7 | 0:00481 DPA 
7450 34-2 0:00460 2:07 
85380 43°] 0 00505 DT 


Summarizing :— 


| Field. m/H. 

2500 | 0-00476 

4200 0:00463 

6000 0-60501 

7500 0-00492 

8500 0:00473 

lWiteanan eee seer 0:00481 

—1/208 


Mean c=2:1710-". 


It will be observed that the values of m/H for the five: 
fields do not depart by more than 4 per cent. from the 
mean, so we may say with safety that the value m/H=1/208 
is accurate to 2 per cent. There is no evidence of any 
variation of m/H with the field. 

Bismuth.—Two disks, from the same body of metal, were 
cast to exact size in graphite moulds; one for the Corbino 
effect and one for the Hall effect. The specific resistance, 
determined by the method above, was found to be 1572 x 107%, 
while the value used for copper is 17 x 1077. 


308 Mr. A. K. Chapman on the 
The Hall Effect in Bismuth. 


a 
H. EX104. K=74. Kon: 
208 | 0:84 7:49 14130 
486 | 1°81 6-90 13020 
1050 | 3°68 6:53 12320 
1690 | 5:3D 6:12 11550 
3150 | 8-25 4°85 9150 
3600 | 9-10 4-66 8790 
4670 hes) Fea shee@ 4:37 8250 
5680 se aye 3°99 7530 
7060 | 139 3°63 6850 
7310 | 146 3-69 6960 
8320 15°7 3°49 6580 
10400 Lif ee 3-11 5870 
10570 | 18:1 3:16 5960 
—————— { 
The Corbino Effect in Bismuth. 
| EL m/f. (m/F) oy: | R ex10". 
| 
156 0-673 1400 | 12950 3038 
510 0-680 i4i4. | 13080 — |) a a06 
1125 0-657 136%) | ~. 12640) eee: 
1720 0-593 1140 | 11410 267° 
2020 0°569 1184 | 10950" aaa: 
2330 0-549 1142. | . 10560 +; eee 
3320 0-464 065°. |, 8980 .\, eames 
4050 0-442 p19 | 8500 199- 
5I7O ~ | 107394 B20). |. .7580>. AL 
580 1. O86r a673.. |» "7060. | aie 
6720 | - 0:342 mA | 6850 > aaa 
8210 ||) 0838 W030.) 6500 | Saas 


In the above tables d is the thickness of the plate 
(0°0919 cm. for the bismuth plate); E is the potential 
difference across the plate ; I is the primary current flowing 
through the plate; H is the magnetic field: K is 530 x 10-® 
for copper; m/H is 1/208 for copper; (m/H)ou is m/H for 
bismuth divided by m/H for copper ; Ris (m/H)o, multiplied 
by the ratio of the specific resistances of bismuth and copper. 

A similar notation is used throughout the remainder of 
this work. 

Fig. 1 shows the plot of the two effects against the values 
of H. It is seen that both decrease very quickly with 
ancreasing field, the Corbino effect falling off the more 
rapidly. 


Hall and Corbino Effects. 309) 
Fig. 1. 


eR CorbinoE tect. 


o (K}., Hall Effect. 


Alloys of ean and Tin.—It has been found by von 
Hitingshausen and Nernst, among others, that the sign of 
the Hall effect in certain alloys of bismuth and tin is reversed 
with increasing field. To compare this phenomenon with 


Fig. 2. 


mee | ae 
ee se 
Se ee 
PE TS A 
pte Ace | 


BECP SSEEEEE EE 
ae ee i eee 


o (KL, Halt Etfect. 
e R CorbinoEffect. 


5000 €00 TO an ay wD 


the Corbino effect, two plates were cast from the same alloy 
and the two effects measured under like conditions. The 


310 Mr. A. K. Chapman on the 


Table gives the numbers, and fig. 2 is a plot of both for an 
alloy containing 1 per cent. tin; the numbers are all 
referred to copper, which is taken as unity. 


The Hall Effect in a Bismuth—Tin Alloy 
(about 1 per cent. tin). 


' Rd 

H. Ex 104. K= 57° (Dea 
307 0-91 551 10400 
869 2-48 | 5:27 9940 
1490 3°88 | 4-79 9040 
2180 510 | 4-29 8100 
3120 6:00 | 3°53 6660 
4050 6:35 | 2-89 5450 
4720 6:45 | 2-51 4740 
5850 615 | 1-94 3660 
6355 5:90 | 1-71 3230 
7030 545 | 1:43 2700 
8600 4:30 0917 | 1730 
9110 3°88 0-784 | 1480 


d=0°0915 cm. 


The Corbino Hffeet in a Bismuth-Tin Alloy 
(about 1 per cent. tin). 


H. mH. | (m/H)op- R. cx 107. 

390 0142 | 295 7670 63-8 

925 0132) rer 7150 593 
1500 0121 | 252 6550 54-4 
2160 0-101 21-0 5460 45°4 
2700 0:0859 17-9 4650 38-8 
3540 | 00673 | © 140 3640 30-2 
4330 00544 11:3 2940 OA-4 
5455 00370 77 2000 16°66 
6130 0:0286 5-95 1550 12:87 
6800 0:0195 4-06 106U 8:78 
3495 0-0043 0:89 231 1:93 
8915 00000 0:00 ss, a 


It is seen that both effects decrease rapidly with increasing 
field, while the sign is that of the effect in bismuth. The 
‘two curves are nearly parallel, the Corbino effect being the 
smaller and falling to zero at 8910, where the Hall co- 
efficient has a value of 1550 referred to copper. 

In the case of an alloy containing about 2 per cent. of 
-tin the conditions are somewhat different, as may be seen 


from the Table and fig. 3. 


olK), Hall fi ttect, 


oR Covbine E tech 


Hall and Corbino Effects. 


Fig. 3. 


Sea Se ae ee ee 
‘J See eee eee 
one SC ao ur G00 


9000 


The Hall Effect in a Bismuth—Tin Alloy 
(about 2 per cent. tin). 


H. 


304 
805 
1485 
2185 
3600 
4670 
5970 
6750 
7425 
8150 
9845 
10450 


Ex 10?. 


0303 
0-600 
1:03 
1:27 
1°39 
1°33 
1-21 
0°85 
0-73 
0-485 
—0:240 
—0°545 


ee 


= FL 


1-594 
1-400 
1-294 
1:087 
0-721 
0°521 
0°406 
0:234 
0-183 
O-111 
—0:045 
— 00976 


(K)oy: 


3010 
2640 
2440 
2050 
1360 
983 
766 
442 
345 
209 
= BD 
—184 


The Corbino Effect in Bismuth-Tin Alloy 
(about 2 per cent. tin). 


m/H. 


0:0800 
0:0661 
0:0497 
00454 
0:0279 
0:0212 
0'0126 
0:00905 
0:00428 
— 000319 
— 0°00560 


(m/H)ou- 


16°64 
13°75 
10°34 
3°44 
5°80 
44} 
2°62 
1:88 
0:89 

— 0°66 
—1:16 


ex 10%. 


mhobo bo ce 


| 


WIE MON ONOS 
Oe ood ocr 09 WO © 


Oo 09 te mI Or Or 


| 


H 
40000 


al? Mr. A. K. Chapman on the 


Here the curves are no longer parallel; the Corbino 
effect has a value 4187 and the Hall effect 3000 times that 
of copper for a field of 250. The former falls off the more 
rapidly, crosses the Hall effect curve at a field of 4250, 
becomes zero at a field of 7900, and reaches a value of 
—291 at 9110. On the other hand, the Hall curve goes to 
zero at 9325 and is —184 at a field of 10450. 

By some physicists it has been suggested that this reversal 
is the result of the superposition of two or more effects. In 
the usual method of evaluating the Hall coefficient such a 
condition might arise through the simultaneous occurrence 
of the simple Hall effect and the Ettingshausen effect ; the 
latter giving rise to the usual galvanomagnetic temperature 
difference at the boundaries of the plate, which in turn, 
because of the Thomson effect, produces a potential difference 
which appears in the potential reading taken as a measure 
of the Hall effect. To be sure, under such conditions, 
the galvanomagnetic potential difference obtained at an 
given field depends upon the ratio between the Hall and 
Httingshausen coefficients, and it is conceivable that the 
peculiar phenomena in bismuth-tin alloys might be due to 
this circumstance. It is at once evident, though, that the 
Corbino method entirely precludes the possibility of the 
occurrence of the Thomson effect, since there are no free 
boundaries present. Hence one may say with certainty 
that the reversal obtained in the Corbino curve is not due 
to the coexistence of the Hall and Ettingshausen effects. 

Because of the junction of dissimilar metals at the centre 
and circumference of the disks employed, the Peltier effect 
may enter, producing a radial heat-flow and consequently a 
circular current. In a recent paper Zahn (Ann. d. Phys. 
xlvii. 1. pp. 49-82, May 11, 1915) has shown that such an 
error in ordinary experiments on the Hall coefficient amounts 
actually to a very small fraction of the whole, and it may 
safely be assumed that such is the case here. 

And finally, the best guarantee that such complicating 
effects do not play any considerable part is the reversal of 
the radia] current, which took place twenty times per second. 
As is well known, a very definite time is necessary for tha 
Httingshausen and Httingshausen and Nernst effects to 
establish themselves, since each depends upon the rise either 
of a transverse or of a longitudinal temperature difference. 
The brief interval between reversals is not adequate for the 
building up of any great temperature difference. 

Nickel.—The disks were cut from a sheet of metal supplied 
by Messrs. Eimer and Amend. 


Hall and Corbino Effects. 
The Hall Effect in Nickel. 


H. E x 107. 
1100 | 0:0335 
2305 0:0682 
3420 | 0:1068 
5700 0°1839 
7750 0:2060 
9150 0°2104 

10200 0:2073 


313 


0:01491 
0:01448 
0:01526 
0:01581 
0:01505 
0:01127 
0:00998 


— 
A 
Ss 
Q 
=] 


| 


ebobpochris 
OFA OVS ONTO 
bronTe Bo Deal 


The Corbino Effect in Nickel. 


| H. | m/H. 


(m/H oy: R. Exon 

915 0:01689 3°51 MS) 7°60 
18380 0:02355 4-90 27:3 10°60 
2810 0:02399 4:99 27°8 10°79 
4580 0:02434 5:06 28-2 10:97 
6550 | 002362 4:91 27°3 10°62 
7950 |  0:02099 4:37 24:3 9°45 
9025 '- 0:01873 3°90 2177 8°44 

Fig. 4 


R Corbino Effect. 


(K), Hall Ettect. 


A a a ea 
_ Sane ease eee 
a Een od ff 


Fig. 4 shows the plots of these numbers. In the case of 
the Hall effect the points for low fields do not lie well on 
the curve; this is due to the difficulty of accurately 


eave Meg. o. 6. Vol, 32. Now L8G Sen 1916, Y 


314 Mr. A. K. Chapman on the 


measuring the small potential differences obtained in a plate 
0:0398 em. thick. The Corbino effect rises to a maximum 
at a field of about 4500, and decreases rather rapidly to 
fields of 9000 lines, which is as far as it was practicable 
to carry the readings. It is likely that the Hall effect also 
has a maximum at approximately 4000, and it certainly 
decreases for increasing field at nearly the same rate as the 
Corbino effect, the two curves being almost parallel from 
6500 to 9500. | 

The Influence of Thickness——The theory mentioned above 
predicts that the Corbino effect is independent of the thickness 
of the disk employed. This theory has been confirmed in 
this one respect, within the limits of experimental error, for 
three different nickel disks of quite different thicknesses. 
They were supplied by Messrs. Baker & Company, and were 
of thicknesses 0:0505, 0°1526, and 0:20315 cm. respectively. 
Here one must take into account the change in mutual 
inductance between the coils and the disks of varying 
thickness. ‘To obtain an accurate measure of this change, 
the method used in the previous paper was employed. The 
coils were placed against a circular plate of brass, 5 cm. in 
diameter and having a hole of the same size as the central 
wire of the other disks drilled in the centre. A sector of 
small angle was cut out of this disk, to the edges of which 
heavy leads were soldered. The coil and disk were then 
placed between the poles of the magnet and their mutual 
inductance compared with a known standard. A _ heavy 
sheet of mica was then interposed between the coil and disk, 
and the mutual inductance again measured. In the first 
instance the distance from the centre of the coil to the disk 
was 0°0438 cm., while the mutual inductance proved to be 
21000 e.m.u. ; in the second the distance was 0°1317 ecm. 
and the inductance 17500 units. Assuming, then, that the 
change of inductance with separation is linear for the changes 
used with the three disks above, we may calculate the 
inductance between the coils and each disk. Taking the 
induetance measured with the brass disk of 0:0894 em. 
thickness as standard, we find :— 


The inductance increase for the thin disk 3°90 per cent. 
$ s decrease ,, medium ,, 6°32 
55 + a jeee thick Sb Gaye 


39 
39 


Below are the values of 6/F for the three disks, where 6 
is the change in balancing inductance, measured in degrees 


Hall and Corbino Effects. 315 


of rotation of the secondary coil. F is the fluxmeter reading 
on reversing the field. 


Thickness of disk. 18, 0/F. 
0°0505 em. 165 15°3 
01526 _,, 166 14:7 
0°20315 ,, 169 15:0 

Mean 15:0 


The agreement is very good, especially when it is 
considered that, in all probability, the disks were cut from 
different sheets of metal. | 

The mutual inductance between the coils and the Corbino 
disk was also determined by a graphical method. This 
method, which was purely analytical, gave results in good 
agreement with those obtained experimentally, as is shown 
below :— 


M (ealeulated) © 11790 e.m.u. 
M (experimental) 11860 e.m.u. 


This graphical method was of such a nature that one could 
investigate the most advantageous form of coil for any given 
conditions. 

It was tacitly assumed that the flux calculated for a plane 
passing through the middle of the coil and parallel to the 
disk is a mean value of the flux in the surface of the coil 
next to the disk and the flux in the opposite surface. That 
this assumption is justified may be interred from the agree- 
ment of the experimental and calculated values of the 
inductance. 

A similar assumption was made in the calculation of the 
inductances for the nickel disks of varying thickness. That 
this is legitimate is also borne out by the above theoretical 
determination, since the change in thickness of the disks 
was only about half of the actual thickness of the inductance- 
coil used. 

Iron.—This disk was made from a bar of Norway iron, ag 
was the plate used for the determination of the mutual 
inductance of the two coils and the disk. ‘The inductance 
was investigated both with and without the external field ; 
there was no change due to the presence of such a field 
except for large values beyond 10,000. At this particular 
point the change was less than 1 per cent. In the study of 


Y 2 


316 Mr. A. K. Chapman on the 


this same disk last year, nothing of unusual interest was 
found. Upon carrying the fields below 1390, which was 
the lowest used in the previous experiments, a remarkable 
effect came to light. The value of m/H is zero up toa 
field of about 760, rises to a sharp maximum at 950, falls to 
half the maximum value at 1500, and then decreases slowly 
as the field increases up to 10,000, which was the largest 
field used. 

In order to investigate this region accurately, it was 
necessary to read small differences in field with a certainty 
greater than that afforded by the fluxmeter. Accordingly, 
a standard ohm, immersed in an oil-bath, was put into the 
magnet circuit and the current flowing determined by taking 
the potential drop across it; a Wolff potentiometer was 
employed, together with a standard cell prepared by Professor 
Hulett of the Department of Chemistry. By taking corre- 
sponding current and fluxmeter readings for several fields of 
very different magnitudes, it was found that, for any given 
region, the field 1s proportional to the current. Having 
determined the factor of proportionality, it was then possible 
to fix accurately small differences in the field by the 
corresponding current readings. 

During the course of preliminary experiments on this 
metal, some difficulty was experienced in duplicating the 
numbers obtained at various times. In view of the fact 
that the magnetic properties of 1ron vary rapidly with the 
temperature, as does the Hall effect, the disk together with 
the inductance-coils was put into a brass box, so arranged 
between the poles of the magnet that a constant stream of 
water could be kept flowing through it. In this way the 
temperature was, at all times, kept stationary to within one 
degree, which proved to be sufficiently constant for the work 
on hand. 

Residual magnetism also interfered seriously with the 
measurements at low fields. This was eliminated by applying 
an alternating current to the magnet before the desired 
direct current was passed through it. In every case the 
initial alternating current employed was much larger than 
the direct current to be used and was gradually reduced to 
zero, thus effectively demagnetizing the magnet and disk. 

Having taken these two precautions, it was possible to 
obtain fairly consistent results, as shown by the table. 


Hall and Corbino Effects. 317 
The Corbino Effect in Iron. 


H. 28/H. x10", 
142:3 0:00035 1-05 
172°8 0:00033 | 0:00 
221-1 0-0014 4:2 
305:3 —0-0003 0:99 
378-0 0-0008 On4 
496°7 —0-00020 —0-60 
7580 —0-00013 — 0-39 
798° 0:00138 416 
815: 0-00160 | 4-82 
826° 0:00182 5-48 
848° 0-00289 8:7 
856" 0-00304 | 9-2 
865° 0:00358 | 10:8 
887 000411 12-4 | 
902 0:00432 13-0 | 
911 0-00461 13-9 
919 0:00490 14:8 
930 0:00516 15:5 
044 0:00540 | 16-2 
962 000530 | 16-0 
980 0:00510 15:4 
996 0:00492 14:8 

1014 0:00473 14-2 
1053 0:00427 12-9 
1081 0:00388 11:7 
1117 0:00331 10-0 
1162 0:00310 9:3 
1180 0:00313 9:4 
1197 0:00317 9:5 
121 0:00306 9:9 
1927 0:00334 101 
1304 0-00291 8:8 
1392 0:00296 8-9 
1425 0-00309 93 
9195 0-00245 7-4 
3530 0:00263 7-9 
3580 0:00241 73 
5500 0-00233 7-0 
5680 0:00225 68 
7190 0:00222 67 
7290 0:00219 66 
8270 0:00220 66 
8640 0:00209 6:3 


Fig. 5 gives the plot of m/H against H, the external 
field; the maximum is very sharp and somewhat resembles 
a resonance peak. The point at 760, where the effect first 
appears, also seems to be very definite, while the falling off 
of the curve beyond 2000 is gradual. 


318 Mr. A. K. Chapman on the 


cx 10° 


Fig. 6 is a plot of the values of m against H; m is 
directly proportional to the circular current. It is seen 
that, in one region, the circular current actually decreases 
for an increasing field ; it rises quickly to 5-1 and falls to 
3°7. Beyond 1179 the curve is a straight line. 

It is important to observe that the curve for m/H remains 
practically of the same form when the internal field is 
introduced instead of the external field. We have 

H;=H—NI, 
where N represents the demagnetizing factor; H,; the 
internal field ; H the external field; and I the intensity of 
magnetization. For a disk of the above form N comes out 
12°25, which is so near 4m that the external field and the 
induction are practically the same. We may write the 
above equation for H,, 


ihe 


Node 
1+ ———— 
An 
where yw is the permeability. 

Since, for this particular specimen of iron, no values of w 
were at hand, the numbers given in the Recueil de Constantes 
Physiques par Abraham et Sacerdote (1913) were used. 
From this H was determined and 2m/H plotted against H 
as in fig. 7; this curve is plotted on a much larger scale 
than the previous ones. 


Hall and Corbino Effects. a19 
Fig. 6. 


Ge ey a 
es 3 EY 
OS: |S a ey a a 

| 2 ae 
Fig BEE EE ME 
Gal 


10 


G 


a 
cn 
Ee 
PN 
an 
ae 
aie 
eS 
S 
Ss 


25 Corbino Current. 


§ « Gorbino Current. 


2SSNOR 


320 Mr. A. K. Chapman on the 


That the form of these curves does not depend upon the fre- 
quency of reversal of the radial current is demonstrated by the 
entire agreement of experiments made with twenty reversals 
per second with those where five or six per second were used. 

In view of the possibility that these peculiar effects in the 
iron disks might arise through the close proximity of the 
heavy pole-pieces of the magnet, through the fact that one 
of them is traversed by a one-centimetre hole, or through 
some other unnoticed circumstance, it was thought that a 
repetition of the experiments would be of value, with the 
disk in a solenoid. For this purpose a suitable solenoid, 
137 cm. long and 22 em. in diameter, was constructed. It 
was wound on a brass tube of 7 cm. internal diameter, and 
consisted of three sections having respectively 4347, 3450, 
and 3490 turns of number 10 copper wire, thus giving 
fields of 40°22, 31°88, and 32°30 lines per square centimetre 
per ampere. In series with each section was a one-tenth 
ohm standard resistance immersed in an oil-bath; the 
potential drop across these standards could be read with a 
Wolff potentiometer, thus giving an accurate measure of 
the current, and hence of the field due to each section. At 
the higher fields the large currents produced a marked 
heating of the apparatus, so that it was necessary to allow a 
liberal stream of water to flow through the inner brass tube 
in which the disk was placed. The disk and inductance- 
coils were exactly the same as before except for the fact 
that they were enclosed in a water-tight brass box. In this 
way the temperature of the disk was kept constant to within 
a very small range, even though the solenoid itself became 
warm. As before, readings on the variable inductance were 
taken with the magnetic field in both directions, and the 
determinations of the current flowing in the sections were 
made both beforeand afteradjusting the balancing inductance. 
The numbers given are the means of series of readings. 

The solenoid has the advantage of affording a uniform 
field over the whole disk, together with the possibility of 
dealing accurately with much lower fields and with much 
smaller variations in higher fields than is possible with the 
usual type of magnet. It is obvious, from the figures given 
for the number of lines per square centimetre per ampere, 
that it is easy to vary the fields to the greatest nicety and 
also to determine them with an accuracy more than amply 
sufficient for this purpose. There is another advantage of 
great importance in the manipulation. In the arrangement 
used, there are always coils of very high inductance in series 
with the galvanometer and, at the same time, in the magnetic 
field, so that any unsteadiness in the magnetizing current, 


Hall and Corbino Effects. avail 


producing a corresponding change in the field, is immediately 
apparent in the behaviour of the galvanometer. With the 
magnet there is a relatively high field produced per ampere, 
so that any inconstancy in the magnetizing current is very 
troublesome in producing fluctuations of the galvanometer. 


The Corbino Effect in Iron (solenoid). 


2&§=<Corbino Current. 


H. 0/H. eX 
20:2 O:000 CON ee aia heater ts! 
38°5 —0:00519 —31°3 
45°9 —0'01089 —65'6 
576 —0°00977 —55°8 

1346 — 000446 —26°9 
139-0 —0:00360 —21°7 
1843 —0:00176 —10°6 
249°5 0-00000 0:0 
3178 0:00094 5°69 
3o9'2 000206 12:4 
448°5 0:00256 15-4 
4752 0:00500 18-1 
D7, 0:00352 21:2 
042°7 0:00541 20°6 

70°3 0:00307 185 

585'1 000273 16°5 
613°5 0:00224 13°5 
683°5 0:00197 Ig) 
780'1 0-00192 11:6 
8480 0-001L77 10°7 

108 


0:00179 


On the other hand, for the solenoid, the field per ampere is 
comparatively low and, hence, small variations in the current 
have little influence on the galvanometer. In fact, while 
using the solenoid, no such difficulty, on account of small 
changes in the current, was encountered. | 

As will readily be seen from fig. 8, in which the balancing 


322 Mr. A. K. Chapman on the 


inductance is plotted against the field, the same peculiar 
maximum occurs, although the field at which it takes place 
is different and the absolute value of the peak is altered. 
In all probability the occurrence of the maximum at a 
distinctly lower field is due to the fact that, when the magnet 
was used, one of the pole-pieces was traversed by a hole, 
which so distorted the field that the density of lines was very 
low over the central part of the plate. It is evident from 
the formula fer the circular current, 


G P92 
C= oe 


that its magnitude is inversely proportional to log I/7;, and 
hence, if the region of small radii is excluded from the field, 
that a proportionately stronger field is needed to produce a 
given circular current. Obviously, the field over the whole 
‘disk is uniform in the case of the solenoid, and so, since the 
region of small radii comes under its full influence, it might 
be expected that a given peculiarity in the curve would take 
place at a lower field strength. 

On pursuing the investigation to values of H below 300, 
another remarkable effect was brought to light, as is shown 
by the same curve. Below this point the circular current 
is negative. So far as the writer is aware, this is the first 
instance in which such a change in sign in the Corbino 
effect has been detected in a pure metal. At first sight it 
seems reasonable to suppose that this reversal might be due 
to residual magnetism, but such is not the case. A current 
of 33 amperes was passed through two coils of the solenoid 
connected in parallel in such a way as to produce a positive 
flux, then readings of the Corbino effect at a positive field 
of 100 were obtained and found to be negative. Next, an 
equally high negative field was applied for a time and a 
balance again sought at 100 positive. The Corbino effect 
was negative as before, showing that the reversal is not 
due to residual magnetism, for the current of 33 amperes was 
considerably higher than any actually used for the plots 
given. It should be remarked though, that, after taking a 
set of determinations for a curve, going from low to high 
fields, the disk was always carefully demagnetized by reversals 
before proceeding again to lower fields. This was purely 
a matter of precaution, as the effect of residual magnetism 
was actually very small. 

In fig. 9 we have a plot of the circular current divided by 
the field ; here the maxima and minima are very sharply 


ied 


Hall and Corbino Effects. 323 


defined. The minimum is made very conspicuous because 
of the small value of the field and the quite appreciable 


Fig. 9 


iene 


ce | ae 
HEHE 


ae 
| 
ae 


OO 


iS 100 £00 oo 400 $00 «6600 TOO 6800) «(90K 


value of the circular current. This makes m/H very large 


negatively. The maximum at 525 is much the same as that 
found in fig. 5. 


The argument used in the case of the bismuth-tin{alloys 


324 Mr. A. K. Chapman on the 


against the possibility of the other two transverse effects 
seriously interfering with the accurate determination of the 
Corbino effect applies equally well here. There is the 
additional assurance that isothermal conditions were approxi- 
mately realized by the use of the water-bath with both the 
magnet and solenoid. 

Unfortunately it is impossible to attain high values of H 
with such a solenoid as the one used in this research. For 
that region one must rest content with the curves obtained 
with the magnet. At low values of H, for various reasons 
mentioned above, the results of the experiments with the 
solenoid must be considered much the more trustworthy. 

The Corbino Kffect in Bismuth at Low Fields.—In certain 
metals, notably bismuth and antimony, the Corbino effect is 
large enough to admit of investigation at low fields. This 
was done more or less thoroughly for bismuth and roughly 
for the 1 per cent. alloy of bismuth and tin. It may be 
observed from fig. 1 that the three points for low fields all 
correspond approximately to the same value of c. That this 
is actually the case for this region is shown by the accom- 
panying table. The value of ¢ up to a flux of 800 lines per 
square centimetre is constant within the limits of error. 
While the constant ¢ is of the same order of magnitude as 
that found with the magnet, there is yet, as in the case of 
iron, a marked difference between them. It is likely that, 
as before stated, this is due to the distortion of the field 
occurring when the magnet is made use of. 


The Corbino Effect in Bismuth (solenoid). 


: 
ie 2¢/H. | ex 10", 
44-0) 0-027 | 116-4 
85-8 0:0276 | 119: 
138-5 - 00258 LiL 
182°6 0-268 116: 
269-4 0-0267 115° 
314-9 0:0267 115" 
397°1 0 0277 119: 
441-6 0-0272 117° 
519:4 0-0271 117° | 
562°5 | 0-0270 116: 
631°7 | 0-0269 Ge 
678-2 | 0-0267 115: 
736-5 | 0-0272 117: 
786-5 0:0266 115: 
838-2 | 0-0264 114: 
| Mean 115-9 


Hall and Corbino Effects. 325: 


Below are the values of ¢ for the bismuth-tin alloy (1 per 
cent.) ; they must be considered rather unreliable on account 
of the extremely small effect obtained at these fields. 


The Corbino Effect in a Bismuth—Tin Alloy 
(about 1 per cent. solenoid). 


H. | 26/H. cx 107. 
380 | 0:8 9-1 
499 | S11 9-5 
618 | 1:25 8:7 
743 | 1-6 9:3 
876 | 2-0 9:8 

| Mean 9°3 


The Effect of the Lack of Symmetry of the Field in the 
Electromagnet.—As was explained before, in takin g measure- 
ments such as the present ones, the balancing inductance is 
adjusted so that the galvanometer shows no deflexion. This 
is done first with the field in the positive direction ; then. 
the field is reversed, the inductance again adjusted for no 
deflexion, and half the difference between the two readings 
on the adjustable inductance is taken as a measure of the 
Corbino current. In every case where the magnet was 
employed, the two positions of the balancing inductance. 
were asymmetrical with respect to its position for zero 
field. It was at first supposed that this was due to the lack 
of exact uniformity of the disks and a consequent circular 
component of the primary radial current. However, upon 
experimenting with the solenoid, this lack of symmetry was 
found to disappear, showing that some other reason must be 
sought for its presence when the magnet is used. The 
possibility at once suggests itself that a distortion of the 
field might be responsible ; it is seen that such a condition 
could easily arise through the lack of parallelism of the pole- 
faces, or through a lack of coincidence between the centre 
of the disk and the centre of the hole traversing one of the 
pole-pieces, or by a failure to place the disk perpendicular to 
the flux. In order to test this, a bismuth disk was so 
arranged as to be movable between the pole-shoes while 
everything else, including the magnetizing current, remained 
constant. Approximate settings were then made with the 
disk in varying positions ; the movements were such as to 
produce exaggerated representations of changes in position 


326 On the Hall and Corbino Effects. 


that might occur during the progress of ordinary experi- 
ments. ‘These changes would arise, in practice, through 
opening the magnet, changing disks, and adjusting the 
inductance-coils. The figures below show that such changes 
do alter the aspect of the readings for positive and negative 
fields as regards their symmetry about the balance-point for 
zero field. 


Ps pie pe P,. 
33 37 
31 37 
28 33 
345 32:5 
35°5 365 
36 34 
30 36 


P,—P. is the difference between inductance balance-points 
for zero fields and positive fields, and P_—P, is the corre- 
sponding difference for zero and negative fields. 

These rough determinations show that, by a simple change 
of the position of the disk in a field so distorted, the aspect 
of the figures as regards symmetry may be totally changed. 
This is partly taken account of by using half the difference 
between inductance settings for positive and negative fields 
as a measure of the Corbino effect. But this does not take 
into consideration the symmetrical distortion of the magnetic 
field and, as pointed out above, the experiments with the 
solenoid are to be regarded as much the more satisfactory 
in that regara. 


The writer is greatly indebted to Professor E. P. Adams, 
under whose direction this research was undertaken, for his 
generous help and unfailing interest throughout the progress 
of the work. 


Palmer Physical Laboratory, 
Princeton, N.J. 


Raat al 


XXXVI. Preliminary Note on the Stark Effect of the 4686 
Spectrum Line. By H. J. Evans, D.Sc. (Lecturer im 
Physics, Manchester University), and C. Croxson, B.Se.* 


[Plate V.| 


HE effect of an electric field on spectrum lines which 
was discovered by Stark f is obviously closely connected 
with the structure of the atom, and it was with the object: of 
further testing existing theories as to the origin of the 4686 
spectrum line that this research was undertaken. The Stark 
effect in the case of hydrogen was examined from the 
theoretical standpoint by Warburg f{, and also by Bohr §, 
who based their work on the view put forward by Sir E. 
Rutherford that the hydrogen atom consisted of a single 
electron revolving round a positive nucleus. Bohr deduced 
that in an electric field each of the hydrogen lines of the 
Balmer series should consist of two components polarized 
parallel to the field, and showed that Stark’s results for the 
separation of the two strong outer components polarized 
parallel to the field for the first five lines of the Balmer series 
agreed approximately with the values calculated from his 
theory. Further, in the case of an electron of charge e 
revolving round a nucleus charge Ne in an electric field of 
strength E, he deduced the following expression for the 
frequency difference Av between the two outer components 


ay foe 
Av= {2 eNeii (no? —n,’), SPP omhinitel ts (Vo (1) 


where h and m are Planck’s constant and mass of electron, 
and nv, and n, are the numbers representing the position of 
the spectrum line ii a series of the type 


where K is the Rydberg constant. 

Stark’s experiments had shown that the effect of an 
electric field was far more complicated than the deductions 
from the simple theory mentioned above, and this complexity 
was emphasized in connexion with theories on the structure 
of the atom. Recently, however, Bohr’s theory has been 


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

t Electrische Spektralanalyse Chemischer Atome, by J. Stark (Leipzig, 
S. Hirzel, 1914). : 

{ Warburg, Verhandi. d. Deutsch. Phys. Ges. xv. p. 1259 (1918). 

§ Bohr, Phil. Mag. xxvii. p. 506 (1914); xxx. p. 404 (1915) 


328 Onthe Stark Effect of the 4686 Spectrum Line. 


generalized by Epstein *, who, employing Sir E. Rutherford’s 
model; of the hydrogen atom, has obtained results for the 
positions of the components of the H,, Hg, H,, and H; lines 
of the Balmer series in complete agreement. with Stark’s 
experiments. 

The complete theory of the effect of an electric field on 
the 4686 line, which on Bohr’s theory is due to a helium 
atom consisting of an electron revolving roand a nucleus 
charge 2e, has not yet been published, but the ratio of the 
separation of its components to that of the Hs line of hydrogen 

can be readily deduced from Bohr’s simple theory. If, in 
equation (1), for Hg we put N=1, n»=4, ny=2, and for the 
AGs6gimme NE 2) in, — 4 _ is found that the ratio of 
the separation of the outer components of the 4686 line to. 
that of Hg is as 7:24. ‘This ratio of the separations will 
also be approximately true for wave-length. 

Since the experimental results obtained by us at present 
are not complete, it is proposed to examine the question 
further, either with a spectroscope of higher power or by 
the application of stronger electric fields. 

The experiments were carried out by Lo Surdo’s + method: 
in which the spectrum of the dark space in a helium vacuum 
tube excited by a steady voltage was photographed. The 
voltage was obtained from a 2000-volt dynamo connected in 
series with a battery of small cells giving about 3000 voits. 
The spectrum was photographed by a direct-vision prism 
instrument which gave a bright spectrum and had sucha 
dispersion that 1mm. on the photographic plate at > 4700 
represented 10°8 A.U. Several photographs were taken, one 
of which is reproduced in Plate V. During this exposure - 
the constant voltage on the tube was 4000 and the pressure 
of the helium was 1:96 mm. 

The strong lines which appeared on the plate consisted of 
the lines of the ordinary helium spectrum, the Hg line (4861), 
the mercury line (4358°5), and the 4686 line, which has 
practically the same intensity in the dark space as in the 
portion of the discharge immediately above. ‘The Hg line, 
the second member (4472) of one ‘of the diffuse series of 
helium, and the second and third members of the other 
diffuse series at 74922 and 4388 showed the well-known 
effect of the electric field which has been studied in detail 
by Stark, Lo Surdo,and Brunettit. The maximum distance 


* Epstein, Phys. Zeit. xvii. pp. 148-150 (1916). 

i Lo Surdo, Estratto da [’ Elettroteenica Giornale ed Atti deli’ Assoct- 
azione Elettr oteenica Italiana, Anno i. no. 25, Ottobre 5, 1914. 

+ Brunetti, Rendiconti della R. Accademia der Lincei, xxiv. (19%); 


Scattering and Regular Reflexion of Light. 329 


between the extreme components of the 4388 line was 
15-7 A.U., and, according to Stark’s measurements in the 
cease of an electric field of 28,500 volts. per cm., this separation 
would correspond to a field-strength of about 37,000 volts 
percm. The 4686 line showed a definite broadening in the 
dark space, but no definite components could be identified. 
The line in the strongest photographs appeared to be 
broadened nearly symmetrically, and the intensity of the 
outside edges was less than the middle portion of the line. 
The total width of the line in the dark space was about 
3°7 A.U., and the corresponding distance between the 
components of Hg was 7°2 A.U. Hach of the components of 
Hg was double, but the lines were too diffuse for an accurate 
estimate of the distance between the outside components. 
However, by making use of Stark’s results, it could be 
calculated that this distance was approximately 10 A.U. 
For the sake of comparison the widths of the 4686 and Hg 
lines were measured in the region above the dark space, and 
found to be in each case about 1:4 A.U. Although it is 
impossible from these results to test quantitatively the simple 
theory of the Stark effect for the 4686 spectrum line, it 
seems evident that if the broadening of the line is altogether 
due to the electric field its magnitude is decidedly less than 
one-half that of Hg. It also follows that it is difficult to 
explain by means of the Stark effect the great width of the 
line in comparison with the .hydrogen lines when they are 
generated by a condenser-discharge. 


University of Manchester, 
July 1916. 


XXXVI. Scattering and Regular Reflexion of Light by an 
Absorbing Gas. By R. W. Woop and M. Kiuura™*. 


[Plate VI.] 


i previous paperst by one of us, it has been shown that 

mercury vapour, at room temperature, in an exhausted 
quartz bulb, when illuminated by the light of a quartz 
mercury arc, re-emits diffusively a monochromatic radiation 
of wave-length 2536, which has been named resonance 


* Communicated by the Authors, 
+ R. W. Wood, “Selective Scattering and Absorption by Resonating 
Gas Molecules,” Phil. Mag. May 1912. 


Phil. Mag. 8. 6, Voi. 32. No. 189. Sept. 1916. Z 


330 Prof. Wood and Mr. Kimura on Scattering and 


radiation (Plate VI. fig. 3: lower spectrum, mercury are; 
upper, resonance radiation). This radiation appears to be 
wholly free from polarization, even when the exciting 
radiation is plane polarized. If ‘the density of the vapour is 
increased, the diffuse radiation (which is at first a volume 
radiation) is confined to a shallow layer of molecules lining 
the front surface of the illuminated bulb, and, with further 
increment of density, disappears entirely, being replaced by 
regular reflexion, much as if the inner surface of the bulb 
was silvered. 

In the present investigation an attempt has been made to 
clear up some of the “doubtful points mentioned in the 
previous papers, and ascertain, if possible, how the diffuse 
scattering passes over into regular reflexion. 

In the earlier werk it was found that the resonance radiation 
had its maximum intensity at the moment of starting the 
quartz arc, and that it became almost negligible after the 
lamp had been in operation for eight or ten seconds, asa 
result of self-reversal of the exciting 2536 line, and conse- 
quent absence of the exact frequency necessary for stimulating 
the mercury molecules in the bulb. In our work, we have 
used a water-cooled quartz mercury arc of the type described 
by Kerschbaum™*, in which the are is driven against the 
front wall of the tube by a weak magnetic field. This reduces 
the self-reversal to a minimum, for the cooler, non-luminous 
absorbing layer is “squeezed out,” so to speak, the current- 
carrying vapour being in contact with the quartz wall of the 
tube. The spectrum of such a lamp is quite unique in 
appearance; for the 2536 line is so much brighter than 
any of the other lines, that it is enormously overexposed, 
appearing much like a photograph of a distant arc light 
taken at night. Bright diffraction rays radiate from it inall 
directions, causing it to stand out in the photograph with the 
conspicuousness of a first-magnitude star in the Milky Way. 
A photograph of the spectrum is reproduced as a negative on 
Plate VI. fig. 4.. The wavy lines joining the two spectra 
were caused by the elevation of the plate between the two 
exposures. 

The lamp consists of a straight tube of quartz with the 
negative electrode (mercury) below, and a positive electrode 
of tungsten above. It was made to order by the Cooper- 
Hewitt Co., and operates on 110 volts, with resistance 
sufficient to hold the current down to about 3°5 amperes. 
During operation the anode is at a full red heat, but the 


* ‘Electrician,’ 72. p. 1074 (1914). 


Regular Reflexion of Light by an Absorbing Gas. 331 


rapid circulation of water in the brass jacket keeps the tube 
quite cold. Asa resonance lamp we have used a quartz tube, 
closed by worked plates of fused quartz, containing a drop of 
mercury, and highly exhausted. This tube was mounted in 
front of, and close to, the crystalline quartz plate which formed 
the window of the water-jacket of the lamp. It is important 
to have the rays of the arc traverse the mercury vapour as 
near to the front window of the resonance lamp as possible, 
since it has been shown in one of the previous papers that 
the intensity of the resonance radiation is reduced to one-half 
of its value by traversing alayer of mercury vapour, at room 
temperature, only 5 millimetres in thickness. A screen of 
black paper, perforated with a hole, cuts off stray radiation 
scattered by the walls of the resonance lamp, and it is advan- 
tageous to cover the further end of the tube with a small cap 
of black paper, or provide some other suitable black back- 
ground. 

If the invisible light from the resonance lamp is focussed 
upon a sheet of uranium glass by means of a large quartz lens, 
we obtain a bright spot of yellow fluorescent light, and can 
render visible the vapour rising from a warm drop of mercury 
by holding it close to the screen in the path of the rays, the 
shadow of the vapour cast on the uranium glass appearing 
like a column of black smoke, as shown in one of the photo- 
graphs published in an earlier paper. 

In the present investigation we have used, for the study of 
the reflexion of the light by the mercury vapour, an exhausted 
thick-walled bulb of quartz closed by a plate of slightly pris- 
matic form, as shown by fig. 1. 


liven Ik, 


By means of this arrangement the rays reflected from the 
inner surface could be studied uncontaminated by the reflexion 
from the outer surface. 

In the earlier work, the curved surface of the bulb was 
used, which made experiments on the polarization of the 
reflected light impossible. 

Z 2 


332 Prof. Wood and Mr. Kimura on Scattering and 


We first investigated the diffuse resonance radiation as a 
function of the density of the mercury vapour in the bulb. 
The light of the quartz arc was passed through a quartz 
monochromator arranged to give a convergent cone of 2536 
monochromatic light. It was simply a roughly constructed 
quartz spectroscope with a very wide slit and no telescope- 
tube, shown in diagram in fig. 2. The image of the slit 


Fig. 2. 


artZ 
Quarnera 


formed by the 2536 rays was located in space by means of a 
strip of uranium glass, and the bulb mounted in such a 
position that the image fell upon the centre of the prismatic 
plate. The dispersion was sufficient to remove the other 
images of the slit from the bulb, which obviated the use of a 
second slit and lens for obtaining the monochromatic illumi- 
nating beam. The quartz bulb was mounted over a chimney 
of thin sheet iron, with a Bunsen burner at its base, and the 
temperature determined by a nitrogen-filled mercury thermo- 
meter, the bulb being in contact with the upper surface of the 
quartz bulb. A camera of very simple construction, furnished 
with a quartz lens, was focussed upon the bulb, the process 
consisting in first focussing it with uranium glass upon the 
image in space of the 2536 line formed by the monochro- 
mator, and then measuring the distance between the lens 
and the image in space. This gives us the proper distance of 
the bulb from the lens to secure a sharp focus. 

The arrangement of the apparatus in this experiment is 
shown in fig. 2, the rays reflected from the two surfaces 
of the prismatic plate falling to one side of the lens. In this 
way we obtain only an image of the scattered resonance 
radiation from the bulb. | 

The photographs obtained at different temperatures are 


Regular Reflexion of Light by an Absorbing Gas. 333 


reproduced on Plate VI. fig. 1. The temperatures of the 
mercury vapour are as follows :— 


Temp. Pressure of Hg vapour. 
CMM ORES SBA. yt "00168 
DAO Os. ene A 00574 
CROWES 8G sT a0 Te ae! ‘0750 
OO ies 28 Su a cae "270 
@ AN SSNs aaeerneen ass de 11-00 


The bulb at room temperature (a) appears more or less 
filled with the resonance radiation. This, as has been shown 
previously, is due to the fact that the radiation from the 
molecules which lie in the path of the primary beam excite 
to resonance the entire mass of vapourin the bulb. At 40° 
(b) the pressure has increased about 3°4 times, and the 
radiation comes chiefly from the front part of the buib. At 
76° (c) it is confined chiefly to the inner surface of the plate, 
though a slight haze to the right of the image indicates that 
some radiation capable of exciting resonance still penetrates 
to a depth of a millimetre or so. This radiation is without 
doubt of wave-length slightly greater and slightly less than 
that of the centre of the exciting line, in other words the 
edges of the 2536 line. The size of the patch of resonance 
radiation contracts rapidly as the temperature goes up, owing 
to the inability of the radiation to spread out and excite 
secondary resonance. At 175° (e) it has shrunk to the 
dimensions of the image of the slit thrown upon the bulb by 
the monochromator. It will be noticed that in case (d) where 
the density is 16 times as great as at room temperature, there 
is still a slight broadening of the image, due to the secondary 
resonance. The spreading is of the order of half a millimetre, 
which is about what we should expect from the known 
stopping power of the vapour at room temperature. A further 
increase of temperature causes a rapid diminution in the 
intensity of the scattered resonance radiation, the energy of 
the primary beam passing off as a regularly reflected wave. 
The intensity is a maximum in the vicinity of 100° (2. e. ata 
pressure of about 0°3mm.). At 150° (pressure about *3 mm.) 
the intensity has decreased to about half its maximum value; 
at 200° (pressure 18 mm.) to about one quarter, and at 
250° (pressure 76 mm.) to perhaps one tenth. At 270° 
there is absolutely no trace of the scattered radiation. The 
estimates were made from a series taken under similar con- 
ditions, with a slit somewhat broader, so that a better deter- 
mination of the relative densities could be made. Hqual 


334 Prof. Wood and Mr. Kimura on Scattering and 


exposure times (40 secs.) were given and the images 
developed simultaneously. Itis evident that the scattered 
resonance radiation decreases (replaced by true absorption 
probably) long before regular reflexion commences. 


Haperiments on Selective Reflexion. 


In our first experiment on this subject we placed the 
quartz bulb a little inside of the focus of the monochromator, 
so that the incident radiations came to a focus after re- 
flexion from the prismatic plate. A plate of uranium glass 
was mounted in such a position that the two reflected images. 
were focussed on it. The image formed by reflexion from 
the outer surface was noticeably brighter than the other, 
owing to absorption by the fused quartz plate, which was 
twice traversed by the rays reflected from its inner surface. 
On heating the bulb to a red heat with a Bunsen flame, the 
latter image brightened up until it appeared to be about three 
times as bright as the image reflected from the outer surface. 
In this way itis possible to demonstrate the selective reflexion 
of the vapour to a small audience at close range. A sheet 
of heavy plate glass must be used as a protection against a 
possible explosion of the bulb as the pressure may rise to 15 
or 20 atmospheres. 

The reflecting power was next determined quantitatively 
in the following way :— 

The total radiation from the water-cooled are was reflected 
from the inner surface of the prismatic plate into a small 
quartz spectrograph, the slit of which was opened rather 
wide, and shortened to a length of about 1 millimetre. 
The spectrum lines thus photographed as small rectangular 
patches, and twenty or thirty exposures could be made ona 
single plate. The exposures were made by a slow swinging 
shutter of the pendulum type. We first made a series of 
exposures of continuously increasing duration by operating 
the shutter once, twice, three times, &c., with the bulb at 
room temperature. This gave us a record of the reflecting 
power of the inner surface of the quartz plate. The bulb 
was then raised to a red heat, and another series of exposures. 
were made in the same way. 

The plate showed that the rectangle representing the 2536. 
line had the same intensity for an exposure of five seconds 
for quartz reflexion (bulb cold), and one second for mercury 
reflexion (bulb hot). 

Since the reflecting power of a surface of fused quartz is. 
roughly 5 per cent. in the ultra-violet, this experiment shows. 


Regular Reflexion of Light by an Absorbing Gas. 335 


us that the reflecting power of a surtace of dense mercury 
vapour for the light of the 2536 line is not far from 
25 per cent., nearly that of most metals in the same region 
of the spectrum. The next point to determine was the 
density at which selective or metallic reflexion commenced. 
Various methods were tried, the following being the one 
finally adopted. 

It is of course desirable to repress as completely as possible 
the ordinary, or vitreous, reflexion of the quartz surface. 
This can be done by polarizing the incident beam with its 

electric vector horizontal, and setting the prismatic face of 
the bulb at the polarizing angle. Under these conditions the 
intensity of the two beams reflected from the outer and the 
inner surface is reduced nearly to zero, and a very slight 
increment in the reflecting power of the inner surface due 
to the mercury vapour becomes at once apparent provided 
the reflecting power is increased regardless of the direction 
of the plane of polarization, as proved to be the case. 

The divergence of the two reflected beams was so great 
that both were not received by the quartz lens of the camera, 
consequently the one reflected from the outer surface was 
made nearly parallej to the other by reflexion from a piece 
of platinized glass at nearly grazing incidence. Two small 
images of equai intensity were thus recorded simultaneously 
on the photographic plate, one above the other. The upper, 
due to reflexion from the outer surface of the prismatic plate; 
the lower, representing the reflexion from the inner surface. 
The temperature of the bulb was now gradually raised, and 
a number of exposures of equal duration made at different 
temperatures, the plate being moved slightly between the 
exposures. Hig. 5, Plate VI. shows the result of the final 
experiment. The first three exposures, “‘a,” “6,” and “c,”’cor- 
responding to temperatures of 180°, 210°, and 235°, showed 
both images of equal intensity; in other words, no increment 
of reflecting power resulted from the presence of mercury 
vapour up to a pressure of 50 mm. (at 235°). Exposure ‘d,” 
taken ata temperature of 279°, showed the lower image con- 
siderably brighter than the upper one. The pressure in this 
case was about 120 mm.; and since we can infer that the 
effect would be noticeable at a slightly lower pressure, we 
are sate in saying that the first appearance of specular 
reflexion by the vapour takes _ place at a pressure not very 
far from 10 em. Exposures “e” and “f” were made at tempe- 
ratures of 300° (pressure 25 cm.), and one other still higher 
temperature which was beyond the reach of the thermometer. 
On the print these have practically the same intensity, but 


336  ~=Prof. Wood and Mr. Kimura on Scattering and 


on the original negative the density of ‘“‘f”’ is certainly double 
Gabor << e.” 

As subsequent experiments showed that the dense vapour 
reflects polarized light in much the same way as a film of 
metal, it was of some interest to see whether the reflecting 
power of the quartz surface in contact with the vapour 
passed through a minimum before beginning to increase with 
increasing vapour pressure. In the case of metallic deposits 
on glass, the reflecting power of the glass is considerably 
diminished by very thin layers of metal, when the reflexion 
is from the glass side; in fact it is reduced nearly to zero if 
the thickness of the metallic film is just right. If a number 
of strips are silvered cathodically on the face of a prism of 
small angle (five to ten degrees) with exposures to the 
discharge of from say one minute to ten minutes, one or 
more strips will be found which appear quite black in 
reflected light when viewed through the glass, though all 
reflect much more powerfully than the glass when viewed 
from the silver side. 

The mercury vapour was examined for a similar phe- 
nomenon in the following way:—The convergent 2536 beam 
from the monochromator was reflected from the prismatic 
plate of the bulb, and the two reflected beams received ona 
plate of uranium glass. The temperature of the bulb was 
then gradually raised and the fluorescent images on the 
uranium plate watched. If the vapour behaved exactly as a 
metal film of increasing thickness, the image formed by 
reflexion from the inner surface ought to fade away gradually 
and then rapidly brighten. No trace of such a phenomenon 
was observed. The two images remained of the same in- 
tensity until a temperature of 250° was reached ; above this 
point the image due to tke inner reflexion rapidly brightened 
as the temperature rose, reaching its maximum brilliancy in 
the neighbourhood of 300°, at which temperature it appeared 
to be four or tive times as bright as the other image. 

This specular or metallic reflexion of the light by the 
vapour occurs only when there is exact synchronism between 
the luminous vibration and the free period of the system 
which causes the 2536 line. This fact is emphasized because 
there is another type of selective reflexion which occurs 
when the synchronism is not exact, and which is the result 
of the refractive index of the vapour. This will be discussed 
presently, after the polarization experiments have been 
treated. 


Regular Rejleaion of Light by an Absorbing Gas. 387 


Polarization Experiments. 


Renewed attempts have been made to detect traces of 
polarization in the scattered resonance radiation, but without 
success. Even when the vapour is illuminated with plane 
polarized light, and the spot of surface luminosity (which we 
have at say “100°) 3 is photographed through a Savart plate or 
Fresnel double prism of R. and L. quartz with a polarizing 
analyser of quartz and Iceland spar, no trace of the fringes 
appear. The same thing occurs in the case of sodium vapour 
illuminated with a sodium flame. This is very remarkable, 
since strong polarization has been found associated with the 
stimulation of the frequencies corresponding to the band or 
channeled spectra of sodium and iodine vapour, the polari- 
zation showing not only in the line directly excited by the 
monochromatic light (resonance radiation), but also in all of 
the other lines (lines of the resonance spectrum *). 

In our present search for traces of polarization we employed 
the Fresnel double-image prism of right- and left-handed 
quartz. This gave, with polarized green mercury light, on 
analysation, six black horizontal bands, while with the light 
of the 2536 line about thirty bands could be counted. They 
were very distinct, however, in the photographs made with 
the quartz camera. Asan analyser we used a double-image 
prism of Iceland spar and quartz, which we made by grinding 
and polishing a prism of about 8° from a small piece of spar, 
securing direct vision and fair achromatization by com- 
pensating it with a small quartz prism of about the same 
angle. Both prisms were ground and polished in less than 
an hour. 

The double-image prism was mounted in front of the lens 
of the camera, and the aperture considerably reduced by a 
diaphragm to secure sharp definition of the fringes. In 
some of onr experiments we employed a small Foucault 
prism as a polarizer, but the loss of light was considerable, 
and we accordingly cut a 60° prism of quartz perpendicular 
to the axis; this, when mounted about 10 em. behind the 
monochromator, in the converging beam of 2536 light, gave 
two brilliant polarized images on the uranium glass plate, 
separated by a distance of about 3.em. One of these was cut 
off by a screen, and the rays diverging from the other 
Uluminated the prismatic plate of the bulb. 


* R. W. Wood, Phil. Mag. July 1908, and Oct. 1911, p. 480. 


338 Prof. Wood and Mr. Kimura on Scattering and 


The arrangement of the apparatus is shown in fig. 3, 
which explains itself. The camera is of course focussed upon 


aS SSS C=) Double Im. 
a i : 
Crud Prism, 
{ So es STS Double Rerr. 
‘ Prism. 
aR \\ 
RUN 
\ 
RS \ 
\ \ 
FAS, A N Fresnel RL. 
\\ \ Prism 
sa 


the Fresnel prism, by adjusting the distance as described 
previously. An exposure of four or five minutes was sufficient 
to give the two images of the spot of resonance radiation 
formed by the double-image prism. The presence of two or 
three per cent. of polarization would be indicated by faint 
traces of the horizontal dark bands in the images. 

At first we obtained distinct traces of the bands, but their 
appearance and intensity were very variable, and we finally 
found that they were due either toa slight deposit of mercury 
globules on the inner surface in the high temperature expe- 
riments, or to a slight cloudiness of the inner surface of the 
prismatic plate near one edge. We finally got things so- 
adjusted that no trace of them appeared at any temperature 
between 20° and 200°, above which the scattered resonance 
radiation practically disappears; and we feel quite certain 
that there is no trace of polarization in the scattered radiation, 
even when the incident light is plane polarized, and the 
density of the vapour is so great that we are approaching the 


stage at which selective metallic reflexion begins. ‘This. 


seems very remarkable, since, as we shall see presently, if 
the incident light is polarized, the metallically reflected wave 
is polarized ne 

This makes it appear probable that we shall have to reject 
the idea that the reflected wave can be accounted for by the 
application of the Huygens principle to the waves emitted by 
the resonators. Moreover, as the vapour density increases, 


the scattered resonance radiation practically disappears some 


Regular Reflexion of Light by an Absorbing Gas. 369 


time before the appearance of the reflected wave, which is 
additional evidence against such aview. We had hoped to 
find that, as the density increased, an increasing percentage 
of polarization would be found in the scattered radiation, but 
such does not seem to be the case. 

The complete absence of polarization appears to be rather 
remarkable, and not easy to explain. <A rapid rotation of 
the molecule would probably act as a depolarizing factor; 
and if the vibrations of the resonator were constrained to 
take place along fixed lines or planes in the molecule, these 
would be Suleanoll in all possible positions, and we chore 
expect, from the most elementary considerations, less than 
50 per cent of polarization, even with the incident light 
plane polarized. This was found to be the case with the 
resonance spectra of sodium and iodine vapour; but in: 
the present case, where we are apparently dealing with a 
vibration of much simpler type, there is no trace of polari-. 
zation whatever. 

If we are to regard the metallically reflected wave as the 
resultant effect of vibrations emitted by a closely packed 
system of resonators in perfect synchronism, we must show 
how the polarization results, for the reflected wave is plane: 
polarized as we shall now see. 

In the case of the reflexion of light from metal surfaces, if 
the plane of polarization is parallel or perpendicular to the 
plane of incidence, the reflected light is also plane polarized. 
lf, however, the incident light is polarized in an azimuth of 
45°, the EAleated light is usually more or less elliptically 
polarized, due to the phase difference between the two 
reflected components. 

We examined all three cases, but need mention in detail 
only the one in which the incident light is polarized in 
azimuth 45°, The light of the water-cooled are was passed 
through a small Foucault prism arranyed to transmit vibra-. 
tions inclined at 45° to the vertical. It was then reflected at 
the polarizing angle from the inner surface of the prismatic 
plate of the bulb “into a quartz spectrograph. Between the 
slit and the collimating lens, and close to the latter, we: 
mounted the double-image prism already referred to, arranged 
so as to transmit horizontal and vertical vibr ations, the images: 
formed by it lying one above the other. It may be worthy 
of mention that in working with a quartz spectrograph it is 
important to analyse the polarized light before it enters the 
lenses, on account of the natural rotation of the latter. 

The rays which entered the spectroscope were of course 


340 Prof. Wood and Mr. Kimura on Scattering and 


plane polarized by reflexion from the inner surface of 
_the prismatic plate, or, in other words, the incident 
polarized light was split into two components, one cf which 
was wholly transmitted, while the other was in part re- 
flected, with its vibrations parallel to the reflecting surface. 
The light thus reflected gave a single image after passage 
through the double-image analyser. The slit of the spectro- 
graph was opened somewhat and its length contracted 
to such a degree that the spectrum-lines of the mercury 
arc appeared as small squares on the plate. Exposures 
were made with the bulb at room temperature (fig. 6, 
Plate VI., upper spectrum), and at a red heat (lower 
spectrum). In the latter case, owing to the metallic re- 
flexion of the 2536 line, hoth components are reflected with 
equal intensity, and unite into a plane polarized resultant in 
azimuth 45°. This is doubly refracted by the analyser, 
yielding two images of the square for the wave-length in 
question. It will be noticed that all of the other spectrum- 
lines (squares) are represented by single images only. 

This experiment does not prove, however, that the reflected 
light is plane polarized, for we should have a similar appear- 
ance if it was depolarized or circularly polarized by the 
reflexion. 


To prove that it is plane polarized we must rotate the 


double-image prism into such a position that one of the 
two images disappears. It was assumed that this would 
occur for a rotation of 45°, and we accordingly turned the 
prism through an angle of about 40°, and then made a 
number of successive exposures, turning the prism through 
a small additional angle each time. One of these exposures 
showed the second image completely absent, proving that 
elliptical polarization was not present, or at all events that 
it was too small to be detected. 


Selective Reflexion and Refractive Index. 


Selective reflexion of another type occurs at the boundary 
surface separating quartz from dense mercury vapour. This 
occurs in the case of frequencies slightly higher than that 
of the 2536 line. The mercury resonators in this case emit 
no scattered radiation, and there is practically no loss by 
absorption. In some earlier work *, in studying the reflexion 
of the light of the iron are by mercury vapour, it was found 


* R. W. Wood, “Selective Reflexion of Monochromatic Light by 
Mercury Vapour,” Phil. Mag. xviii. p. 187 (1909). 


Regular Reflexion of Light by an Absorbing Gas. 341 


‘that an iron line one Angstrém unit on the short wave- 
length side of the 2536 mercury line, was much more power- 
fully reflected than a pair of iron lines on the long wave-. 
length side situated at 0-1 and 0:4 A.U. from the mercury 
line. No explanation of this was given in the paper, but 
the suggestion was made later in Wood’s ‘ Physical Optics’ 
(second edition), page 432, that it undoubtedly resulted from 
the sudden change in the refractive index of the vapour in 
the vicinity of the absorption-line. 

“The 2536 line shows powerful selective dispersion and the- 
refractive index, in its immediate vicinity on the short wave- 
length side, is much below unity, probably as low as 0°5, 
or even much less close to the line. In the case of light 
going from a rare to a dense medium, a high value of the- 
refractive index for the latter is accompanied by strong 
reflexion. When, however, the ray goes from dense to rare 
(quartz-mercury vapour) as in the present case, a low value 
of the index for the latter is accompanied by strong 
reflexion. 

“On the long wave-length side, for a region very close to 
the line, the index of the dense mercury vapour may rise to a 
value as high as that of quartz, in which case there will be 
no reflexion at all.” 

It would appear then that, if we could employ light of two 
frequencies, one slightly higher and the other slightly lower 
than the frequency of the 2536 line, the former would be 
powerfully reflected and the latter not atall. This condition 
was realized by employing as our source of light a quartz- 
mercury are operated at a potential just sufficient to distinctly 
double the 2536 line by self-reversal. On Plate VI. fig. 7 
we have four views of the 2536 line taken with a small Fuess 
quartz spectrograph, very accurately focussed. This line has 
a faint companion on the short wave-length side, indicated 
by an arrow on the photographs. If the light is first passed 
through mercury vapour in a heated quartz tube, the main 
fine is weakened or removed by absorption, and the faint 
companion remains, as shown by fig. 7 (a), in which the 
upper and lower figures represent the line without and with 
mercury absorption. Fig. 7 (6) shows the appearance of the 
Jine when the quartz arc, designed to operate at a potential 
drop of 170 volts, is run at 30 volts, while (c) and (d) show it 
reversed at 60 and 80 volts. 

We made our experiment as follows:—The light of the 
lamp running at 80 volts was reflected from the inner surface 
of the prismatic plate of the quartz bulb into the quartz 
spectrograph, the slit of which was reduced to a length of 


342 Prof. Wood and Mr. Kimura on Scattering and 


1mm. by a diaphragm which could be raised by a micro- 
meter-screw. An exposure of one minute was given: the slit 
diaphragm was then raised 1 mm., the quartz bulb raised to 
a red heat by a Bunsen burner, and a second exposure 
of fifteen seconds made. Fig. 2 (Plate VI.) shows the 
result of the experiment. The reversed 2536 line appears 
as a doublet and is indicated by an arrow, the faint com- 
panion on the short wave-length side appearing to its left. 
This was the exposure made by light reflected from the 
cold bulb. Above it we have the exposure made with the hot 
bulb. The light reflected from the hot bulb is seen to consist 
solely of the short wave-length component of the doublet 
(widened and reversed 2536 line), for which the reflecting 
power of the quartz-mercury vapour surface is very slight. 
The long wave-length component has disappeared entirely, 
owing to the very low value of the reflecting power for 
this frequency. The width of the doublet is about 0°8 ALU. 
It is perhaps worthy of mention that we have here a rather 
efficient method of isolating from the total radiation of a 
quartz-mercury are running at a moderately, high tempe- 
rature, a single line of wave-length about 0-4 ‘A.U. less than 
that of the 2536 line of a similar lamp running at a low 
temperature. 

This might be very useful in certain special investigations. 
One could of course make the difference even less than 
U4 A.U. by operating the lamp at a Jower voltage. 

The reflecting power of the dense vapour for the light 
forming the continuous background of the spectrum of the 
cadmium spark was also investigated. It was found that 
the spectrum of the reflected light showed a bright line in 
coincidence with the 2536 line of mercury. No cadmium 
line appears at this point, and the continuous background 
is absent or very faint on the long wave-length side of the 
bright line, due to the low value of the reflecting power 
for this region. An enlargement of the photograph is re- 
produced on Plate VI. fig. 3. The quartz bulb was at a full 
red heat during this experiment. It is evident from the 
photograph that the spectroscope lacked sufficient resolving 
power to show the minimum to advantage. The bright line 
with its dark border on the right, due to the powerful and 
feeble reflexion of these wave-lengths in the light of the 
continuous background of the spark, is indicated by an arrow. 
Above is the 2536 line of mercury for comparison. 


Regular Reflexion of Light by an Absorbing Gas. 348 
The Structure of the 2536 Line. 


The interesting work of Malinowski (Ann. der Physik, xliv. 
p- 947, 1914) has made it appear extremely probable that 
the 2536 line is, in reality, not a single line, but a very close 
multiple line. On this account we have investigated the 
structure of the resonance line with a quartz Fabry and 
Perot interferometer. 

Malinowski found that, if the resonance lamp was placed 
in a magnetic field, and excited by a lamp outside of the field, 
the intensity of the resonance radiation increased rapidly with 
an increase of field strength, reaching a maximum at 1000 
gauss, and then passing through successive minima and 
maxima. 

The first maximum is easily explained as follows :-— 

Assuming the 2536 line of the exciting lamp to be more or 
less reversed, as is sure to be the case with a lamp of the 
ordinary type, it is clear that the magnetic components of 
the resonating vapour in the magnetic field will respond to 
the frequencies to the right and left of the centre of the 
reversed line where the intensity is greater. 

The fact that successive maxima and minima occur indicates 
that we are not dealing with a single line, and a normal 
Zeeman effect. Malinowski obtained a curve of similar type 
by placing the resonance lamp ina magnetic field and passing 
its radiation through a quartz cell containing mercury vapour 


at room temperature. As ordinates he took the ratio 2 in 
0 

which J is the intensity after passage through the absorption 
cell, and J, the intensity (for a given field) without the cell. 
This compensates for the variations of Jo with the field, and 
the curve shows the transmitting power of unmagnetized 
mercury vapour for the resonance radiation of mercury 
vapour in a field of varying strength. A similar curve was 
also obtained when the absorption cell was placed in the 
magnetic field and the resonance lamp outside of the field. 

These results indicated that the absorption line and emission 
{vesonance) line had a similar complicated structure, and 
behaved in a similar manner in a magnetic field. 

Malinowski showed also that the secondary resonance 
radiation, excited in a bulb outside of the field by the light 
of a resonance lamp in the field, gave a curve, the maxima 
of which occupied the positions of the minima of the curve 
obtained with the absorption cell, showing that the energy, 
which was refused passage by the cell, was re-emitted. It 
was not possible to draw any very definite conclusions 


344 Scattering and Regular Reflexion of Light. 


regarding the exact structure of the line from these results, 
though Malinowski was able to make some interesting calcu- 
lations as to the probable widths of the component lines. 

We have made a preliminary investigation with the inter- 
ferometer of the structure of the 2536 line of the water- 
cooled mercury arc and the resonance radiation of the vapour 
at room temperature. The results are not as satisfactory as 
we hoped for, owing to the low reflecting power of the 
metals in this region of the spectrum. Mirrors of silicon 
were tried, but if thick enough to exhibit the high reflecting 
power found by Hurlbut, their absorption was too great. 
With mirrors of cathodically deposited cadmium we obtained 
fair results, however. 

The interference fringes disappeared completely when the 
Fabry and Perot plates were separated by a distance of 6 mm., 
and re-appeared with a separation of 12 mm., both for the 
water-cooled arc and the resonance radiation. 

If we consider that the resolving power of the plates is 
no greater than that of a Michelson interferometer, our 
inference from this result would be that we were dealing 
with a doublet with a separation of about 0°03 A. 

Asa matter of fact, the resolving power was somewhat 
greater, as the fringes obtained with small path difference 
were distinctly narrower than those of a system formed by 
two interfering beams. The finite widths of the components 
of the assumed doublet may, however, be sufficient to account 
for their disappearance with a 6 mm. path difference, and 
their diminished visibility at 12 mm.; so that, for the present 
at least, we are justified in considering the line a doublet. 

It is obvious that a further study of the line with a higher 
resolving power will be necessary before any positive state- 
ment can be made. This we intend to do in the autumn 
with a plane grating of 90,000 lines in the 10th-order spec- 
trum, which should give usa resolving power of about :0028. 

On looking over the plates made in this region with the 
very powerful quartz spectrograph of Professor Weiss witha 
resolving power equivalent to that of 13 large 60° quartz 
prisms, it seems as if the study of the line with this instru- 
ment would scarcely be worth while, for the resolving power 
is only about 033. On this account it will be necessary to 
use a large grating unless we succeed in getting something 
better in the way of a metal film for the interferometer. 


In this investigation we have been greatly aided by a 
substantial grant from the Solvay Institute of Physics of 
Brussels. | 


B35 


XXXVIII. Notices respecting New Books. 


Studies in Terrestrial Magnetism. By C. Caren, M.A., Se.D. 
LL.D., Superintendent of Kew Observatory, etc., etc. Mac- 
millan’s Science Monographs. Macmillan & Co., Ltd.: London, 
1912. Pp. xi+206. 


HIS volume is “intended to give a connected account” of the 
author’s ‘“‘own original work” in terrestrial magnetism, 
without aiming at being a text-book or even at summarizing the 
results of other investigators in those branches of terrestrial 
magnetism with which it actually deals. The author gives us 
‘almost entirely facts, or supposed facts.” Absence of theory is 
by no means due to lack of curiosity as to “ the causes of things,” 
but to the author’s well-founded belief that for the time being 
theorising is less promising than the extension of “positive 
knowledge.” 

The book being mainly concerned with results derived from 
magnetograms, the author gives in Chapter I. a useful explanation 
how such records are obtained and interpreted. The next chapter 
is dedicated to Secular Changes of the magnetic elements ; the 
subject is illustrated by tables and a diagram extending from 1860 
up to 1910. Chapter III. deals with Non-Cyclic Chanves, and 
Chapter IV. with Diurnal Inequalities, ohtained by taking in 
suecession the difference of each hourly value from the mean 
of the twenty-four; here, as in the remaining chapters, many 
numerical data and several diagrams are given. Chapters V. 
and VI. are dedicated to Diurnal Inequality on Ordinary, and on 
Disturbed Days, respectively. The next chapter treats of the use 
of Fourier series for the representation of diurnal inequalities. 
Although this method is to be employed with discernment, the 
author agrees with the well-established opinion of investigators of 
other branches of terrestrial physics that ‘‘ even if there should 
be no natural force answering to each term of a Fourier expansion, 
a study of individual terms of the series may prove of marked 
utility.’ Chapter VIII. deals with Annual Variation, and 
Chapter IX. with the Absolute Daily Range, 7. e. the excess of 
the largest over the smallest value during the twenty-four hours. 
Comparatively much space is dedicated to Antarctic Magnetic 
Results (Chap. X.). Chapters XI. and XII., dealing with Mag- 
netic Storms and their “sudden commencements,” contain a 
number of very interesting curves which are likely to attract 
the attention even of non-specialists. Of equal interest will be 
the Comparison of Arctic and Antarctic Disturbances, treated in 
Chapter XIIT., in which the author gives us numerous Kew-curves 
juxtaposed with Birkeland’s arctic, and corresponding antarctic 


ial, Mag. S. 6. Vol. 32. No. 189. Sepes 1910. 2 A 


346 Notices respecting New Books. 


curves. The remaining chapters of this valuable Monograph are: 
XIV. Sunspots and Terrestrial Magnetism ; XV. Wolf’s Sunspot 
Formula; XVI. Nature of Sunspot Relationship ; XVII. General 
Conclusions,—in which the eontents of the preceding chapters 
are shortly resumed. It is to be regretted that the author has 
taken no notice of Zeeman and Winawer’s researches in solar 
magneto-optics in connexion with the spectroscopic properties 
of sunspots. 


Edinburgh Mathematical Tracts. London: G. Bell & Sons, Ltd., 
1915 :— 

No. 1. A Course in Descriptive Geometry and Photogrammetry for 
the mathematical laboratory, by EK. Linpsay Incr, M.A., B.S8c., 
Pp. vil+79. 2s. 6d. 

No. 2. A Course in Interpolation and Numerical Integration 


for the mathematical laboratory, by Davip Grips, M.A., B.Sc. 
Pp. vin+90. 3s. 6d. 


No. 3. Relativity, by A. W. Conway, D.Sc., F.R.S. Pp.438. 2s. 


No. 4. A Course in Fourier Analysis and Periodogram Analysis 
for the matbematical laboratory, by G. A. Carsn, M.A., D.Sc. 
and G. SHearer, M.A., B.Sc. Pp. viii+66. 3s. 6d. 


No. 5. A Course in the Solution of Spherical Triangles for 
the mathematical laboratory, by Hrrsert Bett, M.A., B.Se. 
Pp. viii+66. 2s. 6d. 


No. 6. An Introduction to the Theory of Automorphic Functions, 
by Lester R. Forp, M.A. Pp. vii+96. 3s. 6d. 


1. The first Tract of this carefully edited and beautifully 
published series begins with a very clear and interesting expo- 
sition of the purpose and the methods of Descriptive Geometry. 
The subject is introduced by a plain description of Monge’s 
method, of the fundamental properties of projections, and of the 
methods of “Contours” and of “ Perspective.” Some useful 
hints concerning Laboratory Methods are given. The intro- 
ductory chapter closes with an account of the evolution of 
descriptive geometry, from Vitruvius up to Lambert. Chapter II. 
treats of the straight line and the plane in orthogonal projection, 
and contains a good number of fundamental problems with their 
solutions, explained on clear diagrams. A few numerical examples 
accompany each of these problems. The chapter closes with a 
collection of appropriate exercises. Chapter III. is dedicated to 
curved surfaces and space-curves, in orthogonal projection, the 
subject being again developed chiefly upon a series of interesting 
problems concerning cylindrical surfaces and solids of revolution. 
In the two concluding sections, the method of contours is applied 
to topographical surfaces. Chapter IV. initiates the reader in the 
perspective representation of regular solids, of plane and of twisted 


Notices respecting New Books. 347 


curves, and of curved surfaces, and closes with an exposition of 
the relation between the method of perspective and that of 
Monge. Chapters III. and IV. are each provided with appro- 
priate exercises. The last chapter of this exceedingly useful and 
fascinating Tract is dedicated to the so-called Photogrammetry or 
Metrophotography, whose aim is to obtain from photographs a 
correct metrical representation of the object photographed. 


2. Contains a collection of useful hints for the practical pro- 
cesses of Interpolation and Numerical Integration. The first 
two chapters give some theorems in the Calculus of Finite Differ- 
ences, and formule of Interpolation, e.g. those of Lagrange, 
Newton, Stirling, Gauss, and Bessel. The subject is illustrated 
by numerous concrete examples. Chapter LII. treats of the con- 
struction and use of Mathematical Tables. Many mathematical 
and physical students will be particularly grateful to the author 
for that chapter, in which they will find details on the subject, 
otherwise not very accessible. The last chapter of this very 
useful ‘‘ Course ” is dedicated to Numerical Integration. 


3. Professor Conway gives in the four chapters of “ Relativity ” 
the substance of his four lectures delivered before the Edinburgh 
Mathematical Colloquium : I. Hinstein’s Deduction of Fundamental 
Relations. II. Transformation of Electromagnetic Equations. 
Til. Applications to Radiation and Electron Theory,—with a 
short investigation concerning relativistic dynamics (pp. 28-34). 
IV. Minkowski’s Transformation, —containing a very short expo- 
sition of Minkowski’s “ geometrical” representation of the Lorentz 
transformation, and of the meaning and use of Minkowski’s 
four- and six-vectors. This little “ Tract” can be recommended 
as a rapid but easily intelligible introduction to the theory of 
relativity. 


4, The object aimed at by the authors is “ not to discuss the 
series of Fourier from the theoretical standpoint, but rather to 
provide what is necessary for the practical application of the 
subject.” Chapter I. contains generalities on Fourier’s Theorem, 
illustrated by a number of useful and elegant examples. From 
- Chapter IT. the reader learns the art of the practical evaluation of 
the coeficients of a Fourier expression; here the 12-ordinate 
and the 24-ordinate Arithmetical Methods are fully explained, a 
concise computing form, devised by Prof. Whittaker, is annexed, 
and an account of Perry’s, Wedmore’s, Harrison-Ashworth’s, and 
Beattie’s graphical methods is given. Chapter III. is dedicated 
to Periodogram Analysis, a subject which “has not received 
adequate treatment hitherto in any English text-book.” ‘This 
chapter will be very helpful to the physicist, as well as to 
the astronomer and meteorologist. The last chapter is a brief 
introduction to Spherical Harmonic Analysis. The exposition of 
the subject is based on Whittaker’s general solution of Laplace’s 


348 Notices respecting New Books. 
equation \/?7V=0, 


V= j s(e+24 cos u+ry sin u, u)du, 
0 
f being an arbitrary function of its arguments. ‘Thence spherical 
(and, more especially, zonal) harmonics are deduced. Further, 
their integra] properties are demonstrated, which lead at once 
to the expansion of a function as aseries of surface harmonics. 
Stress is laid (pp. 57-66) on the practical evaluation of the co- 
efficients, Bauschinger’s scheme for the calculation of the auxiliary 
coeflicients being fully explained by means of two numerical tables. 


5. The aim of this Tract is to give an account of various methods, 
both numerical and graphical, of solution of Spherical Triangles. 
Every teacher of mathematics will gladly share the author’s 
opinion that this subject is of importance not only in view of its 
applications, but ‘‘ has also considerable value from the educational 
point of view.” After a few remarks on the use of Logarithmic 
Tables (Chap. I.), the fundamental formule of spherical trigono- 
metry are developed in Chapter II. im an easily accessible and 
attractive manner. Chapter III. treats of the numerical solution 
of the Right-angled, and Chapter IV. of that of the General 
spherical triangle. The following chapter contains special appli- 
cations of the most important and instructive kind: Great Circle 
Sailing, Finding the Longitude at Sea, Reducing an Angle to the 
Horizon, and Conversion of Star Coordinates, with numerical 
examples. The last chapter is dedicated to Graphical Methods of 
Solution of spherical triangles. 


6. The last of the Edinburgh Tracts hitherto published contains 
a most fascinating and easily intelligible introduction to the theory 
of Automorphic Functions which will, doubtless, greatly contribute 
to the diffusion of this beautiful but by no means very accessible 
branch of mathematics. The reviewer, tempted to enter upon the 
numerous attractive details of this beautiful work, regrets the 
necessity, dictated by reasons of ‘‘ space,” of limiting himself to a 
mere quotation of the chapters. These are: I. Linear Trans- 
formations. II. Groups of Linear Transformations. III. Auto- 
morphic Functions. IV. The Riemann-Schwartz Triangle 
Functions. V. Non-Euclidean Geometry. VI. Uniformisation. 
—A very full Bibliography of automorphic functions (pp. 88-96), 
up to 19138, will be helpful to those who desire to pursue the subject 
further. 


Bulletin of the Bureau of Standards. 
(Washington: Government Printing Office, 1915.) 


Amoynest the publications of scientific interest during 1915 we 
note the following :— 

In the March number is described a determination by the bomb 
method of the heats of combustion of Cane Sugar, Benzoic acid, 


Notices respecting New Books. o49 


and Naphthalene by H. ©. Dickinson. The object of these deter- 
minations is to provide standard values to aid in the calibration 
of the various forms of bomb calorimeters in use in commercial 
laboratories as well as by scientific investigators. The following 
table shows the degree of agreement with other recent observers. 
The results are expressed in 15° calories. 


Authority. Naphthalene. | Benzoic acid. Sucrose. 
Bere (UO) sateen tes ec asl) + i wseaet 6318 
Riirede (O10) eee 9633 6323 3952 
athe (1910), ox ees. 9643 
Tiaroux (1910) ees... 9631 
Dickinson (1910-1912)... 9612 6323 3945 


In the same number appears a paper by D. R. Harper on the 
specific heat of copper in the interval 0° to 50°, with a note on 
vacuum-jacketed calorimeters. The value obtained (correct to 
third significant figure with temperature coeflicient correct to 
10 per cent.) is 


eles 
+59 
e=0°883, +0-00020(t— 25) Gee seg 
or c=0'0917 + 0:000048(¢ — 25) wee 


In the May number appears an investigation on the insulating 
properties of dielectrics, dealing with the volume resistivity and 
surface leakage of a number of ‘insulating materials with the view 
of ascertaining their suitability for replacing hard rubber. The 
volume resistivity (in ohm-centimetres) is over 5000 x 10” for 
special paraffin, ceresin, and fused quartz; it is about 1000 x10” 
for hard rubber. Mica comes next with a value 200 x 10”, and 
then sulphur with a value 100x10". The most reealate 
variety of bakelite has a value of only 20x10", while many 
varieties of bakelite le very low (e. g. No. 140, 20x10°). A 
large number of determinations are made on surface resistivity 
as affected by light and moisture. These are represented by 
curves. ‘This is due to a surface film, usually of water or oil, on 
the insulator, and is generally the important factor in determining 
the leakage between two conductors. However, for insulators 
having a vol. resistivity less than 10* ohm ems. placed in an 
atmosphere having a humidity less than 25 per cent., the greater 
part of the current may flow through the insulator. 

The same number contains a determination by W. W. Coblentz 
of the absorption, reflexion, and dispersion constants of quartz, 
an accurate knowledge of which is so necessary in order to 
determine spectral energy curves of a black body when a quartz 
prism is used. Data are given extending from the ultra-violet to 


350 Notices respecting New Books. 


3u in the infra-red. Quartz is practically transparent from the 
ultra-violet to 1‘8y. Tabulated data are given for allowing for 
the effect of the absorption which enters beyond this point. The 
results show that (within the errors of observation) in unpolarized 
light, the transmission is not affected by the direction in which 


the radiations pass through the material with respect to the optic 
axis. 


The Science of Musical Sounds. By Prof. Dayton C. Mitizr, D.Sc. 
10s. 6d. net. New York: The Macmillan Co. London: Macmillan 
& Co. Ltd., 1916. Pp. viii+286. 


THis work will be heartily welcomed by those who are fascinated 
by the various problems and methods of harmonic analysis and 
its applications in connexion with musical sounds. It presents, 
substantially as given, a course of eight experimental lectures 
delivered by the author at the Lowell Institute during January 
and February 1914. As befits a course of lectures intended for a 
general audience, the subject matter consists in large part of 
elementary and familiar material, but selected and arranged to 
develop the main purpose. But, in accordance with the design 
of all such lectures at the Lowell Institute, however elementary 
their foundation, they meet the legitimate expectation that they 
shall include the most recent progress of the science under review 
and experimentally illustrate this to the fullest possible extent. 
It is somewhat difficult to adequately present in book form this 
demonstration aspect of a course of lectures, But in this case 
the lavish use of diagrams and photographs of special apparatus 
and their records, or of arrays of forks, pipes, &c., leaves nothing 
to be desired. 

After two preliminary lectures on waves, vibrations, and tones, 
we have a lecture devoted to methods of recording sound waves. 
After passing in review the time-honoured methods, the author 
introduces his own device the phonodeik or sownd-demonstrator. 
This consists essentially of,a horn and glass diaphragm for re- 
ceiving the sound, and a connected mirror for demonstrating its 
amplitude, frequency, and quality. The mirror has a few silk 
fibres from the diaphragm passed once round its axle and then 
held tight by a spring. The light reflected from the mirror may 
be used for demonstration purposes or for yielding permanent 
photographie records. 

The next lecture deals with the analysis and synthesis of har- 
monic curves, the fifth with the influence of the horn and 
diaphragm on the sound waves. ‘The sixth and seventh lectures 
are occupied respectively with the tone qualities of musical instru- 
ments and of vowels. 

The history of theories of vowel sounds is given clearly and 
compactly. It is interesting to note that the results of the 
author’s work here described are in entire agreement with the 
fixed-pitch theory of vowels due to Helmholtz. 


Notices respecting New Books. 351 


After a concluding chapter, there occurs a valuable bibliography 
of over a hundred references, general and special. 

To mathematical treatment this work makes no pretence, scarcely 
a couple of pages of equations occurring throughout the text. Its 
strength lies in its experimental and graphical treatment, and in 
the photographie records giving analyses of sounds from the 
simplest vibrations of forks up to vowels and the vibrato notes of 
operatic artists. 

Both for showing what has been accomplished up to date and 
for the stimulation of further research on this subject, this work 
may be heartily recommended. 


A Course of Modern Analysis: an Introduction to the General 
Theory of Infinite Series and of Analytic Functions ; with an 
account of the Principal Transcendental Functions. By EH. T. 
Wuirtaker, F.R.S., and G. N. Watson, M.A. Cambridge: 
At the University Press. 1915. Second Edition, completely 
revised. 


THE first edition of this well-known work was reviewed in these 
pages in 1904 (Phil. Mag. vil. p. 605). In the preparation of the 
new edition Professor Whittaker has had the assistance of his 
friend and former pupil, Mr. G. N. Watson, to whom are due 
the new chapters on Riemann Integration, on Integral Equations, 
and on the Riemann Zeta-Function. In addition to these, however, 
important changes have been made throughout the work. New 
paragraphs have been added, certain discussions formerly included 
in one chapter are now rearranged and distributed in two, and in 
many cases the demonstrations have been altered so as to be in 
line with the growing demands for greater rigour. The main 
character of the treatise remains the same, Part I. treating of the 
Processes of Analysis, and Part II. of the Transcendental Functions. 
In the Second Part there are introduced, in addition to the Zeta 
Function already named, special discussions of the Mathieu 
Function and the Theta Functions. There is no doubt that all 
these changes are of the nature of improvements, and have led 
naturally to a marked increase in the size of the volume. Nor is 
this increase merely to be measured by the greater number of 
pages, practically one half more than the number in the first 
edition. An important feature of the new work is the use of 
large and small type to distinguish what might be termed the 
main stream of discussion from what is more subsidiary; and 
by this means more matter is packed into the page. Numerous 
examples are appended to the chapters, and many of these are 
quite as important as the illustrations given inthe text. By such 
methods the volume is kept within reasonable dimensions, and yet 
the ground covered is amazingly extensive. The work abounds in 
historic references and indications to the student for more extended 
reading in both original memoirs and treatises on special branches. 
It is interesting to note that recent work done in Professor 


BO2 Notices respecting New Books. 


Whittaker’s own mathematical institute in the University of 
Edinburgh is explicitly recognized as adding to the sum of mathe- 
matical knowledge. Through his inspiration the ‘ Proceedings of 
the Edinburgh Mathematical Society,’ which for many years were 
mainly devoted to geometrical developments, now contain research 
papers on trauscendental functions, certain results of which find 
their place in this valuable treatise on Modern Analysis. The 
centre of gravity of mathematical development in the United 
Kingdom has appreciably shifted since Professor Whittaker 
flitted from Cambridge to Edinburgh by way of Dublin. 

It should be said in conclusion that the book is beautifully 
printed, and that its usefulness is greatly enhanced by a very 
full general subject index as well as a complete list of authors 
quoted. 


Diophantine Analysis. By Ropert D. Carmicuant. New York: 
John Filey & Sons. London: Chapman & Hall. 


Tus is No. 16 of the mathematical monographs edited by 
Mansfield Merriman and Robert 8. Woodward. The most 
familiar Diophantine problem is that connected with the property 
of the right-angled or Pythagorean triangle, and may be stated 
in these words: to find two integers the sum of whose squares is a 
square. ‘Theorems of this nature have had, and no doubt will 
always have, a great fascination for all interested in the theory of 
numbers; and the fascination is probably the greater on account 
of the lack of a general method of investigation. Fermat is 
universally recognized as the great master of Diophantine analysis. 
He left behind him many theorems without proof; and some of 
these still await a demonstration. Fermat’s Last Theorem that, 
if n is an integer greater than 2, there do not exist integers , y, z, 
all different from zero, such that #”*+ y”=z", is one of those for 
which no general solution has yet been found. The Academy of 
Sciences of Gottingen holds a sum of a hundred thousand marks 
to be presented as a prize to the person who first gives a rigorous 
proof of the theorem. Professor Carmichael devotes the fifth 
chapter of his book to a discussion of this equation, the earlier 
chapters being taken up with the consideration of equations of the 
second, third, and fourth degrees. The sixth and last chapter 
deals with what is called the method of functional equations, a 
method of considerable use in the investigation of Diophantine 
problems. In particular, it is applied to the solution of another of 
Fermat’s problems, namely: To find three squares such that the 
product of any two cf them, added to the sum of those two, gives 
a square. The book is clearly written and fully carries out the 
aim of the author, which is ‘to systematize, as far as possible, a 
large number of isolated investigations and to organize the frag- 
mentary results into a connected body of doctrine. The principal 
single organizing idea here used and not previously developed 
systematically in the literature is that connected with the notion 
of a multiplicative domain.” 


Evans & Croxson. Phil. Mag. Ser. 6, Vol. 32,:PL. V. 


He 5047— 
He, SiO le> 


He 4922—- 


al SSG ia 


He 47|3— Sau 


He 4472— 


He 4437-3 


Phil. Mag. Ser. 6. Vol. 32, Pl. VI. 


Woop & KIMURA. 


Fig. I. 


THE 
LONDON, EDINBURGH, anv DUBLIN | 


PHILOSOPHICAL MAGAZINE 


AND 
JOURNAL OF SCIENCE. ~—s, 
| Pe a 
V4, Vad ws C. “i 
[SIXTH SERIES.] TO: se 
OCTOBER 1916. tn ee 
rr ip Oe ar 


XXXIX. On Vibrations and Defleaions of Membranes, Bars, 
and Plates. By Lord Rayutrien, O.M., FAS. * 


Be ‘Theory of Sound,’ § 211, it was shown that “ any con- 

traction of the fixed boundary of a vibrating membrane 
must cause an elevation of pitch, because the new state of - 
things may be conceived to differ from the old merely 
by the introduction of an additional constraint. Springs, 
without inertia, are supposed to urge the line of the proposed 
boundary towards its equilibrium “position, and gradually to 
become stiffer. At each step the vibrations become more 
rapid, until they approach a limit corresponding to infinite 
stiffness of the springs and absolute fixity of their points of 
application. Itis not necessary that the part cut off should 
have the same density as the rest, or even any density 
at all.” 

From this principle we may infer that the gravest mode 
of vibration for a membrane of any shape and of any variable 
density is devoid of internal nodal lines. For suppose that 
ACDB (fig. 1) vibrating in its longest period (7) has an 
internal nodal line CB. ‘This requires that a membrane with 
the fixed boundary ACB shall also be capable of vibration 
in period 7. The impossibility is easily seen. As ACDB 
gradually contracts through ACD’B to ACB, the longest 
period diminishes, so that the longest period of ACB is less 
than tr. No period possible to ACB can be equal to 7. 


* Communicated by the Author. 


Phil. Mag. 8. 6. Vol. 32. No. 190, Oct. 1916. 25 


354 Lord Rayleigh on Vibrations and 


If we replace the reactions against acceleration by external 
forces, we may obtain the solution of a statical problem. 


Fig. 1. 
S 


B 


When a membrane of any shape is submitted to transverse 
forces, all in one direction, the displacement is everywhere 
in the direction of the forces. 

Similar conclusions may be formulated for the conduction 
of heat in two dimensions, which depends upon the same 
fundamental differential equation. Here the boundary is 
maintained at a constant temperature taken as zero, and 
“‘ persistances”’ replace the periods of vibration. Any closing 
in of the boundary reduces the principal persistance. In 
this mode there can be no internal place of zero temperature. 
In the steady state under positive sources of heat, however 
distributed, the temperature is above zero everywhere. In 
the application to the theory of heat, extension may evidently 
be made to three dimensions. : 

Arguments of a like nature may be used when we consider 
a bar vibrating transversely in virtue of rigidity, instead of 
a stretched membrane. In ‘Theory of Sound, $184, it is 
shown that whatever may be the constitution of the bar in 
respect of stiffness and mass, a curtailment at either end is 
associated with a rise of pitch, and this whether the end in 
question be free, clamped, or merely “ supported.” 

In the statical problem of the deflexion of a bar by a 
transverse force locally applied, the question may be raised 
whether the linear deflexion must everywhere be in the 
same direction as the force. It can be shown that the 
answer is in the affirmative. The equation governing the 
deflexion (w) is 


ca (Bae) = oe. 


where Zdz is the transverse force applied at dx, and Bisa 
coefficient of stiffness. In the case of a uniform bar B is 
constant and w may be found by simple integration. It 
suffices to suppose that Z is localised at one point, say at 


Deflexions of Membranes, Bars, and Plates. 300) 


x=b; and the solution shows that whether the ends be 
clamped or supported, or if one end be clamped and the other 
free or supported, w is everywhere of the same sign as Z. 
The conclusion may evidently be extended to a force variable 
in any manner along the length of the bar, provided that it 
be of the same sign throughout. 

But there is no need to lay stress upon the case of a 
uniform bar, since the proposition is of more general appli- 
cation. The first integration of (1) gives 


d (—~d*w £ 
— —_ = } ° ® o o- 7 
ees, { cea 2 


and \Zdx-=0 from «=0 at one end to r=6, and takes. 
another constant value (Z,;) from «=6 to the other end at 
x=l. A second integration now shows that Bd?w/dz? is a 
linear function of « between 0 and 0, and again a linear 
function between 0 and J, the two linear functions assuming 
the same value at e=b. Since B is everywhere positive, it 
follows that the curvature cannot vanish more than twice in 
the whole range from 0 to J, ends included, unless indeed 
it vanish everywhere over one of the parts. If one end be 
supported, the curvature vanishes there. If the other end 
also be supported, the curvature is of one sign throughout, 
and the curve of deflexion can nowhere cross the axis. If 
the second end be clamped, there is but one internal point 
of inflexion, and again the axis cannot be crossed. If both 
ends are clamped, the two points of inflexion are internal, 
but the axis cannot be crossed, since a crossing would involve 
three points of inflexion. If one end be free, the curvature 
vanishes there, and not only the curvature but also the rate 
of change of curvature. The part of the rod from this end 
up to the point of application of the force remains unbent 
and one of the linear functions spoken of is zero throughout. 
Thus the curvature never changes sign, and the axis cannot 
be crossed. In this case equilibrium requires that the other 
end be clamped. We conclude that in no case can there be 
a deflexion anywhere of opposite sign to that of the force 
applied at «=, and the conclusion may be extended toa 
force, however distributed, provided that it be one-signed 
throughout. 

Leaving the problems presented by the membrane and the 
bar, we may pass on to consider whether similar propositions 
are applicable in the case of a flat plate, whose stiffness and 
density may be variable from point to point. An argument 
similar to that employed for the membrane shows that when 
the boundary is clamped any contraction of it is attended 


2B2 


356 Lord Rayleigh on Vibrations and 
by a rise of pitch. But (‘Theory of Sound,’ § 230). the state- 


ment does not hold good when the boundar y is free. 

When a localised transverse force acts upon the plate, we 

may inquire whether the displacement is at all points in 
the same direction as the force. This question was 
considered in a former paper™ in connexion with Fig. 2. 
a hydrodynamical analogue, and it may be convenient C 
to repeat the argument. Suppose that the plate 4} 
(fig. 2), clamped at a distant boundary, is almost 
divided into two independent parts by a straight 
partition CD extending across, but perforated. by a 
narrow aperture AB; and that the force is applied 
at a distance from CD on the left. If the partition 
were complete, w and dw/dn would be zero over the 
whole (in virtue of the clamping), and the displace- 
ment in the neighbourhood on the left would be 
simple one-dimensional bending, with w positive 
throughout. On the right w would vanish. In 
order to maintain this condition of things a certain 
couple acts upon the plate in virtue of the supposed 
constraints along CD. 

Along the perforated portion AB the couple required to 
produce the one-dimensional bending fails. The actual 
deformation accordingly differs from the one-dimensional 
bending by the deformation that would be produced by a 
couple over AB acting upon the plate, as clamped along CA, 
BD, but otherwise free from force. This deformation is 
evidently symmetrical with change of sign upon the two 
sides of CD, w being positive on the left, negative on the 
right, and vanishing on AB itself. Thus upon the whole a 
downward force acting on the left gives rise to an upward 
motion on the right, in opposition to the general rule 
proposed for examination. 

If we suppose a load attached at the place where the force 
acted, but that otherwise the plate was devoid of mass, we 
see that a clamped plate vibrating freely in its gravest 
mode may have internal nodes in the sense that w is there 
evanescent, but of course not in the full sense of places 
which behave as if they were clamped. 

In the case of a plate whose boundary is merely supported, 
i.e. acted upon by a force (without couple) constraining w to 
remain zero}, it is still easier to recognize that a part of 

* Phil. Mag. vol. xxxvi. p. 354 (1893); Scientific Papers, vol. iv. 
p- 88. 

+ It may be remarked that the substitution of a supported for a 
clamped boundary is equivalent to the abolition of a constraint, and is 
in consequence attended by a fall in the frequency of free vibrations. 


oS ag ee 
a8) 


Oo 


Deflexions of Membranes, Bars, and Plates. OM 


the plate may move in the direction opposite to that of an 
applied force. We may contemplate the arrangement of 
fig. 2, where, however, the partition CD is now merely 
supported and not clamped. Along the unperforated parts 
CA, BD the plate must be supposed cut through so that no 
couple is transmitted. And in the same way we infer that 
internal nodes are possible when a supported plate vibrates 
freely in its gravest mode. 

But although a movement opposite to that of the impressed 
force may be possible in a plate whose boundary is clamped 
or supported, it would seem that this occurs only in rather 
extreme cases when the boundary is strongly re-entrant. 
One may suspect that such a contrary movement is excluded 
when the boundary forms an oval curve, 2. e. a curve whose 
curvature never changes sign. A rectangular plate comes 
under this description ; but according to M. Mesnager *, 
“M. J. Résal a montré qu’en applicant 1 une charge au centre 
dune plaque rectangulaire de proportions convenables, on 
produit trés probablement le soulévement de certaines 
régions de la plaque.” I understand that the boundary is 
supposed to be “supported” and that suitable proportions 
are attained when one side of the rectangle is relatively long. 
It seems therefore desirable to inquire more closely into this 
question. 

The general differential equation for the ieee of a 
uniform elastic plate under an impressed transverse force 
proportional to Z is f 


Gt (GA ldn?  d2/ cy?) 10 ee aie en rh) (od) 


We will apply this equation to the plate bounded by the 
lines y=0, y=7, and extending to infinity in both directions 
along x, and we suppose that external transverse forces act 
only along the line z=0. Under the operation of these 
forces the plate deflects symmetrically, so that w is the same 
on both sides of e=0 and along this line dw/de=0. Having 
formulated this condition, we may now confine our attention 
to the positive side, regarding the plate as bounded at a=0. 

The conditions for a supported edge parallel to x are 


ha a aaa AO en Oe 6 os AM ae 9) 
and they are satisfied at y=0 and y= if we assume that w 


as a function of y is proportional to sinny, n being an 


* C. R. t. 162, p. 826 (1916). 
+ ‘Theory of ad? §§ 215, 225; Love’s ‘Mathematical Theory of 
Elasticity,’ Chapter xxii, 


358 Lord Rayleigh on Vibrations and 
integer. The same assumption introduced into (38) with 
L=0 gives 

(Pida—n)*w=0, .. . eer 
of which the general solution is 


w={(A+Bz)e-74+(C+Da)e™} sinny, . . (6) 


where A, B, C, D, are constants. Since w=0 whenv=+oa, 
C and D must here vanish; and by the condition to be 
satisfied when ec=0, B=nA. The solution applicable for 
the present purpose is thus 


w=A'sinny(V-pnz)e™. .. ee 


The force acting at the edge e=0 necessary to maintain 
this displacement is proportional to 


dV ?w d? dw Sy 
7 tae ae ual that is da simply, . (8) 


in virtue of the condition there imposed. Introducing the 
value of w from (7), we find that 


GCwjde—2e A sin ny, .\°. | een 
which represents the force in question. When n=1, 
w=Asiny.(l+2)e-*; . . 0 ee 


and it is evident that w retains the same sign over the whole 
plate from x=0 to r=. On the negative side (10) is not 
applicable as it stands, but we know that w has identical 
values at +2. 

The solution expressed in (10) suggests strongly that 
Résal’s expectation is not fulfilled, but two objections may 
perhaps be taken. In the first place the force expressed in 
(9) with n=1, though preponderant at the centre y=47, Is 
not entirely concentrated there. And secondly, it may be 
noticed that we have introduced no special boundary condition 
at v=a0. It might be argued that although w tends to 
vanish when x is very great, the manner of its evanescence 
may not exclude a reversal of sign. 

We proceed then to examine the solution for a -plate 
definitely terminated at distances /, and there supported. For 
this purpose we resume the general solution (6), 


w= sin ny{(A+Ba)e-+(C+Dza)e"}, . (11) 


which already satisfies the conditions of a supported edge 
at y=0,y=a7. At x=0, the condition is as before dw/de=0. 


Dejlexions of Membranes, Bars, and Plates. 359 


At «=I the conditions for a supported edge give first w=0, 
and therefore d?w/dy?=0. The second condition then 
reduces to d?w/dx?=0. Applying these conditions to (11) 
we find 


D=Be-2, C=—e-*(A421B). . . (12) 


It remains to introduce the condition to be satisfied at 
x=0. In general 


dw 


== sin ny| e~™"{ —n(A + Ba) +B} 
+e{n(C+D2)+D}];. . (13) 
and since this is to vanish when r=0, 
—nA+B+nC04+D=0. ... . (14) 
By means of (12), (14) A, C, D may be expressed in 


terms of B, and we find 
dw nBsinn 
dx 7 eet [e~™*{—at (21—a)e-?™"" 

+ e~nl-2)f_ (2) — a) + ae") |, (15) 


In (15) the square bracket is negative for any value of « 
between 0 and J, for it may be written in the form 


— ve" {1 —e-2nel—2)} ke (2l—x)e7?2""{ e?*@ —e-™} , (16) 
When 2 =0 it vanishes, and when x=/ it becomes 
—2Qle-?(er’ —e-™), 


It appears then that for any fixed value of y there is no 
change in the sign of dw/dx over the whole range from 
x=0 to x=l. And when n=1, this sign does not alter 
with y. As to the sign of w when #=0, we have then 
from (11) 


n(er ae eo) 9 


w= sin ny(A+C)=Bsinny® 


so that dw/dz in (15) has throughout the opposite sign to 
that of the initial value of w. And since w=0 when v=, 
it follows that for every value of y the sign of w remains 
unchanged from v=0 to e=/. Further, if n=1, this sign 
is the same whatever be the value of y. Every point in the 
plate is deflected in the same direction. 

Let us now suppose that the plate is clamped at «= +/, 


360 Lord Rayleigh on Vibrations and 
instead of merely supported. The conditions are of course 
w=(, dw/dz=0. They give 
D=e-*™4{2nA + B(2ni—1)} . .. see 
C=—e—™{ A+ 2nl) + 2n?B}. . . . G8) 


The condition at =O is that already expressed in (14). 
As before, A, C, D may be expressed in terms of B. For 
shortness we may set B=1, and write 


Helse etCnl—1).  . 9 


We find 
—nA +1=2n?Pe—™/H, 
D=(2Qnl4+1—e7?™) e-?/H, 
aC + D=—e ™, 2r7P/H. 
Thus 


oe = sin ny|e7-”*(—nA+1—nz) +67 (nC+D+nDz) | 
= H7-'sin ny. e7™ [2n7Pe-™ —nef1 +6e-?"(2nl—1) }] 
+H" sin ny. @—-9)| —2n7l? + na{2nl + 1—e""1], 
vanishing when #=0, and when r=. 
This may be put into the form 
as = — H~!sin ny[2n7l(l—2)e-?""(e* — e-™*) 
+ na e~™(1—e7 2) (en C—-2) el) 1. (20) 
in which the square bracket is positive from #=0 to z=. 


It is easy to see that H also is positive. When nl is 
small, (19) is positive, and it cannot vanish, since 


ees >1— Pai. 


It remains to show that the sign of w follows that of 
sinny when z=0. In this case 


w=(A+C)sinny; . . . . 2 ee 
and 
—e-* (2—Inl+An'l?) +e- 4" 
= eW2nl(g2nl 4 g—2nl_9 4 9nl—An?l?). . (22) 
The bracket on the right of (22) is positive, since 


Dees t5 3 An??? 1 6n'4l* 
em em tm=9 (14 5 -- A! 4.0). 


n(A+C) a 
+ = 


We see then that for any value of y, the sign of dw/dx 


Deflexions of Membranes, Bars, and Plates. 361 


over the whole range from =0 to w=1 is the opposite of 
the sign of w when e=0*; and since w=0 when x=, it 
follows that it cannot vanish anywhere between. When 
n=1, w retains the same sign at «=0 whatever be the 
value of y, and therefore also at every point of the whole 
plate. No more in this case than when the edges at r= +1 
are merely supported, can there be anywhere a deflexion in 
the reverse direction. 

In both the cases just discussed the force operative at 
2z=0( to which the deflexion is due is, as in (8), proportional 
simply to d*’w/dz*, and therefore to sin ny, and is of course 
in the same direction as the displacement along the same 
line. When n=1, both forces and displacements are ina 
fixed direction. It will be of interest to examine what 
happens when the force is concentrated at a single point on 
the line x=0, instead of being distributed over the whole of 
it between y=0 and y=z. But for this purpose it may be 
well to simplify the problem by supposing / infinite. 

On the analogy of (7) we take 


> NL yen Sin RYE Ws) (23) 
making, when «=0, 
wid — > ny A) sin yee a alee (240) 


If, then, Z represent the force operative upon dy, analysable 
by Fourier’s theorem into 


L=Z,siny+Z,sin 2y+ Z3sin3y+...,. . (25) 


we have 
yo Na 2 
Li =( Zsin ny dy= An SIMA es) (20) 
if the force is concentrated at y=. Hence by (24) 


oe a 0) 
so that 
7, LEO 
= Sie ny 7) = cos n(y +) e-™(1+nz2), (28) 


where n=1, 2, 3, &c. It will be understood that a constant 
factor, depending upon the elastic constants and the thickness 
of the plate, but not upon n, has been omitted. 

The series in (28) becomes more tractable when differ- 
entiated. We have 


dw x Ly, cos n(y—n) — cos n(y +n) 
paca Nea i —nz. 29 
dx 2a > n CREAR.) 


* This follows at once if we start from v—/ where w=0. 


362 Lord Rayleigh on Vibrations and 


and the summations to be considered are of the form 
Sn cosmigie=™. |.) ie (30) 
This may be considered as the real part of 
Rie, .0.. 
that is, of 
— log (=e e)). . .- Aaa 
Thus, if we take 
Sneak 4+ 4Y, |. - ls 
e-X-"¥ =] —e- 9), and e-XH¥a]—e-@+ 8), 
so that 
eX Se Be" cos 8. - |. eae 
Accordingly 
=n! cos nB e~™ = —4 log (1+ e-2*—2e-* cos B); (35) 
and 
dw «Ly F 1+e—*—2e~ cos (y—n) 
dx Am ©? 1te-*—2e-* cos (ytn)- 
From the above it appears that 
W=2 log {1 +e-*—2e-* cos (y+) }salogh 
must satisfy V*W=0. ‘This may readily be verified by 
means of 
V7 logh=0, and V?W=2 V? log t+ 2d log h/da. 


We have now to consider the sign of the logarithm in 
(36), or, as it may be written, 


(36) 


e* + e-*—2 cos (y—n) 
8 64 e-*—2 cos (yty) ~ (3 


Since the cosines are less than unity, both numerator and 
denominator are positive. Also the numerator is less than 
the denominator, for 

cos (y—)— cos (y+) =2sinysinn=+4, 

so that cos(y—7)> cos (y+). The logarithm is therefore 
negative, and dw/dx has everywhere the opposite sign to 
that of Z,,. If this be supposed positive, w on every line 
y=const. increases as we pass inwards from x=oo where 
w=0 to «=Q. Over the whole plate the displacement is 
positive, and this whatever the point of application (7) of 
the force. Obviously extension may be made to any 
distributed one-signed force. 

Jt may be remarked that since the logarithm in (37) is 
unaltered by a reversal of 2, (36) is applicable on the 
negative as well as on the positive side of c=0. If y=y, 


Deflexions of Membranes, Bars, and Plates. 363 


«=0, the logarithm becomes infinite, but dw/dz is still zero 
in virtue of the factor «. 

I suppose that w cannot be expressed in finite terms by 
integration of (36), but there would be no difficulty in 
dealing arithmetically with particular cases by direct use of 
the series (28). If, for example, 7=47, so that the force is 
applied at the centre, we have to consider 


Pasha nm . sinny. c= (lpn), <8) 


and only odd valuesof nenter. Further, (38) is symmetrical 
on the two sides of y=47. Two special cases present 
themselves when «=0 and when y=3z. In the former w 
is proportional to 


sin y— yesin dy + asin DY aa thes el Deh Cad) 
and in the latter to 


1 


ge (L+3e) + me m1 +5a)+.... (40) 


e~7(1+a)+ 
August 2, 1916. 


Added August 21. 


The accompanying tables show the form of the curves of 
deflexion defined by (39), (40). 


Yy. (39). Y. (39). 
0° 0000 50 ‘7416 
10 1594 60 "8574 
20 3162 70 "9530 
30 4675 80 1:0217 
40 6104 90 10538 
a (40). Sie (40). 
0:0 10518 30 "1992 
0°5 9533 4:0 ‘0916 
10 "7435 5:0 0404 
2:0 ‘4066 10:0 ‘0005 


In a second communication * Mesnager returns to the 
question and shows by very simple reasoning that all points 


* CR, July 24, 1916, p. 84. 


364 Prof. R. W. Wood on the Condensation 


of a rectangular plate supported at the boundary move in 
the direction of the applied transverse forces. ; 

If z denote Vw, then \/?z, = V/*w, is positive over the 
plate if the applied forces are everywhere positive. Ata 
straight portion of the boundary of a supported plate z=0, 
and this is regarded as applicable to the whole boundary 
of the rectangular plate, though perhaps the corners may 
require further consideration. But if Vz is everywhere 
positive within a contour and ¢ vanish on the contour itself, 
z must be negative over the interior, as is physically obvious 
in the theory of the conduction of heat. Again, since V/7w 
is negative throughout the interior, and w vanishes at the 
boundary, it follows in like manner that w ig positive 
throughout the interior. 

It does not appear that an argument on these lines can be 
applied to a rectangular plate whose boundary is clamped, 
or to a supported plate whose boundary is in part curved. 


P.S. In connexion with a recent paper on the “ Flow of 
Compressible Fluid past an Obstacle ” (Phil. Mag. July 1916), 
I have become aware that the subject had been treated with 
considerable generality by Prof. Cisotti of Milan, under 
the title “Sul Paradosso di D’Alembert”’ (Attz A. Istituto 
Veneto, t. xv. 1906). There was, however, no reference to 
the limitation necessary when the velocity exceeds that of 
sound in the medium. I understand that this matter is now 
engaging Prof. Cisotti’s attention. 


XL. The Condensation and Reflexion of Gas Molecules. By 
R. W. Woop, Professor of Experimental Physics, Johns 
Hopkins University *. 

[Plate VII. | 


N a previous communication t I have shown that a jet of 
mercury vapour, in which the molecular motion is 
restricted to one dimension, is reflected from a flat plate 
of glass approximately according to the cosine law. The 
mercury vapour after reflexion was condensed as a metallic 
film on the wall of the bulb, which contained the reflector at 
its centre and was immersed in liquid air. 

Removal of the bulb from the low-temperature bath caused 
the film of solid mercury to melt and collect into small drops, 
which made measurements of its thickness in different gases 
impossible. 

* Communicated by the Author. 


+ Phil. Mag. August 1915, “ Experimental. Determination of the Law 
of Reflexion of Gas Molecules.” 


and Reflexion of Gas Molecules. 365 


During the past winter the work has been continued, 
cadmium being used in place of the mercury. Permanent 
films can be obtained in this way, and measurements made 
with the photometer. The chief point of interest in con- 
nexion with the work, however, has been the discovery of the 
fact that there is a sort of critical temperature for each 
substance, below which condensation appears to take place 
at the first collision, in other words the chance that reflexion 
takes place is zero. This is true, however, only in case the 
wall is of some substance other than the metal in question. 
The critical temperature is surprisingly low. For mercury 
it is in the vicinity of —140°, for cadmium about — 90°, and 
for iodine about — 60°. 

We will take up first, however, the law of reflexion. 

In my first communication, I drew attention to the sup- 
posed circumstance that there was practically no reflexion for 
directions which made small angles with the reflecting surface; 
in other words, there was a zone of clear glass just above the 
circle cut by the plane of the reflecting surface in the wall 
of the bulb. Above this clear zone the gradually increasing 
thickness of the deposit indicated that the law of reflexion 
approximated the cosine law, and was independent of the 
angle of incidence. 

As soon as it was found that permanent films of cadmium 
could be obtained, quantitative experiments were commenced. 


Fig. 1. 


The form of tube used is indicated by fig. 1 (a) and (0). 
(a) is the type used for demonstrating the one-dimensional 
motion of the cadmium molecules, to which attention was 
first drawn by L. Dunoyer in the case of sodium. The tube 
must be kept in communication with a Gaede pump during 
the experiment as an extremely high vacuum is necessary, 
and the cadmium is heated by a gas-flame 3 or 4 mm. high 
burning at the tip of a glass tube drawn down to a fine 
capillary. 


366 Prof. R. W. Wood on the Condensation 


If the bulb is kept at room temperature no trace of a 
deposit appears, for reasons which will be given presently ; 
but if the wall opposite the constriction in the tube is cooled 
with a small pad of cotton wet with liquid air, a small 
circular deposit of the metal at once forms, showing that the 
molecular stream shoots across the vacuum-bulb, without 
spreading out laterally. If the vacuum is sufficiently high, 
prolonging the experiment for fifteen or twenty minutes 
causes no appreciable increase in the diameter of the circular 
spot, which becomes distinctly visible in ten seconds, after the 
cadmium has reached the proper temperature (which is only 
a little above the melting-point). If the vacuum is inferior, 
the diameter of the spot may be much greater, owing to the 
deflexion of the cadmium molecules by the residual gas. 

A photograph of the bulb with the circular deposit is 
reproduced on Plate VII. fig. 1. 

The most instructive way of performing the experiment is 
as follows. 

The bulb and tube are kept at room temperature until the 
cadmium melts. Ifnow the side of the bulb is touched at X 
with a pellet of cotton wet with liquid air, a large deposit of 
irregular shape at once forms, showing that the bulb is filled 
with cadmium vapour (with 38-dimensional motion). It 
seems rather paradoxical that we can have a bulb at room 
temperature filled with cadmium vapour, which apparently 
can be condensed only by the application of liquid air. If the 
bulb is touched by the cotton pellet for only a second or two 
the deposit forms, and continues to increase in thickness 
indefinitely, after the removal of the cotton, for the chance 
of reflexion at room temperature from a cadmium surface is 
zero, 2. €. condensation occurs at the first collision. 

If now the wall of the bulb opposite the constriction is 
touched with liquid air, the small circular spot at once forms, 
and goes on building up indefinitely after the removal of the 
cotton. No further deposit occurs on the side of the bulb, 
for the circular deposit, once formed, serves as a trap for the 
one-dimensional molecular stream which impinges on it. 

These phenomena will be more fully discussed later on. 

Tube (6) was designed for obtaining a deposit after reflexion 
of the molecular stream from a flat polished surface of glass. 
In order to obtain films thick enough for quantitative mea- 
surements, it is necessary to keep the reflecting surface at a 
moderately high temperature for at least 30 minutes. This 
was accomplished by a small electrical heating-coil introduced 
into the tube which carried the reflecting surface. The tube 
was closed ina flame and the closed end ground flat and 


and Reflexion of Gas Molecules. 367 


highly polished. It was cemented in position with sealing- 
wax, with the reflecting surface at the centre of the bulb. 

The bulb was wrapped in cotton, which was kept soaked 
with liquid air throughout the experiment, which lasted about 
thirty minutes. 

A photograph of the deposit is shown on Plate VII. fig. 2; 
and itis evident that the clear zone is absent, the deposit 
coming quite down to the plane of the reflecting surface. 

Measurements of the thickness of the film for different 
angular distances from the line normal to the centre of 
the reflecting surface, made by observing photometrically 
the transmission of red light by the film, showed that the 
cosine law was obeyed to within the probable error of the 
experiment. 

During the progress of the investigation, however, the 
exceeding ingenious method devised by Knudsen was com- 
municated by him to me in a letter, and as this method was 
so obviously superior to that above described, the quanti- 
lative investigation of the law of reflexion was abandoned in 
its early stages. 

Knudsen’s method, which was published in the Annalen 
der Physik in the early part of 1916, is based upon the fact 
that for reflexion from a spherical surface under certain con- 
ditions a film of uniform thickness will be formed if the 
reflexion from each element of the spherical surface follows 
the cosine law. This he found to be the case. 


The general subject of condensation and reflexion as a 
function of the temperature was next investigated. 

It is apparent, from the experiment with the bulb which 
I have described, that cadmium vapour, at least when highly 
attenuated, will not condense as a homogeneous film on a 
glass wall at room temperature, whereas it will on a surface 
of metallic cadmium. 

In the vicinity of the heated bead of the metal we may 
have a metallic film produced, but here the nature of the 
deposit is very variable, depending on the condition of 
the glass surface. Black splotches sometimes form, granular 
deposits of white metal and mirror surfaces may alternate, 
and in fact no two tubes act alike. 

In the bulb, however, the conditions are different, and the 
phenomena can be reproduced at will. 

I first attempted to get some idea of how many collisions 
with the wall were possible without condensation, by sub- 
stituting for the bulb a glass tubo bent at a number of right 
angles, as shown on Plate VII. fig. 3. The end of the tube 


368 Prof. R. W. Wood on the Condensation 


was blown out into a small thin-walled bulb, the surface 
of which was touched with a pellet of cotton wet with 
liquid air. A film formed on the cooled spot, as shown in 
the photograph, proving that the vapour could traverse the 
bent tube without condensation. Tubes with over a dozen 
bends were used, proving thata large number of reflexions 
were possible, the number of bends being the minimum 
number of reflexions sufficient to carry the molecule to the 
condensation bulb. 

It was found that a much longer time was required to form 
a film in the bulb when a2 large number of bends were intro- 
duced, which indicated that a portion of the molecules had 
‘been condensed on the way down the tube. In the case of 
the tube shown on Plate VII. fig. 4 the deposit in the bulb 
was produced in fifteen minutes. The cotton pellet was then 
applied at the bend marked A, and a deposit of equal density 
was obtained in two minutes. Transferring the cotton to the 
point B gave an equal deposit in about 10 seconds. 

It is evident that the density of the cadmium vapour is 
considerably decreased by its passage down the bent tube, 
but as no visible deposit could be seen on the walls, it was 
not at once evident what became of it. 

A glass stopcock was introduced between the bulb and the 
straight tube containing the cadmium, which was then heated ” 
above its melting-point. The stopcock was then closed and 
at the same instant the cotton pellet wet with liquid air was 
applied to the bulb, but no film was produced, even after 
many trials. This seemed to show that the vapour was in- 
capable of existing as vapour in a bulb at room temperature 
for any appreciable time. 

A very thin-walled bulb of exceptionally clean glass was 
then blown ona tube, and after forty minutes heating of the 
cadmium, it was removed from the pump and examined in a 
strong light against a black background. The wall of the 
bulb diffused a small amount of light, as if a light smoke had 
deposited on it, and examination under the microscope with 
qz-inch oil-immersion objective showed widely separated 
aggregates of the metal which appeared to be clusters of 
crystals (fig. 2a). More frequently, however, deposits such 
as shown by fig. b were obtained. When we consider the 
eacessive thinness of metal films which are partially trans- 
parent to light, it is clear that an amount of cadmium 
sufficient to form a film of moderate opacity over the entire 
surface of the bulb, if collected into a single crystal, would 
probably not be much larger than one of these small 
ag oregates. 


and Reflexion of Gas Molecules. 369 


Experiments were next made to determine the temperature 
necessary for the formation of a homogeneous film of cadmium. 


Fig. 2. 


The buib was cooled by the application of alcohol a little 
above its solidifying point (with a consistency of Canada 
balsam), but no trace of a film could be detected at the end 
of five minutes. Solid alcohol, however, caused a film to 
appear in ten or fifteen seconds. As the alcohol contained 
some water, its freezing-point was in the neighbourhood of 
—100°, which can be considered as the approximate critical 
temperature for cadmium. Mercury gave no film in a bulb 
cooled to —130°, but a film immediately appeared if the 
temperature was reduced to —150°. The critical tempe- 
rature for mercury is therefore in the neighbourhood 
of —140°. 

I am not yet certain how sharply these temperatures can 
be determined, or what the nature of the deposit will be for 
prolonged action at a temperature a few degrees above 
the critical temperature; but it seems probable that in this 
case a granular deposit will be formed made up of exceedingly 
small grains very close together. 

As I have said before, if any portion of the inner surface 
of the bulb has received a low-temperature film deposit, a 
similar type of condensation continues even at room tempe- 
rature, the rest of the bulb remaining clean. 

These experiments show pretty clearly how condensation 
takes place on metal and glass surfaces. In the case of 
surfaces composed of the same metal as the vapour, the 
molecules stick at the first impact, the probability of reflexion 
being zero, even at room temperature. The same is true 
for a surface, such as glass, if it is cooled below a certain 
critical temperature which is characteristic for each sub- 
stance investigated. If the temperature is above this, the 
probability of reflexion is large, and only a few molecules 


Phil. Mag. 8. 6. Vol. 32. No. 190. Oct. 1916. 2C 


370 Condensation and Reflexion of Gas Molecules. 


are fixed at the first impact. These, however, act as traps 
and gather others about them, just as does a small area 
of film. A granular deposit therefore forms, the grains 
increasing in size until they are finally visible under the 
microscope. 

In working with cadmium it was found that a film formed 
in the bulb before the cadmium had reached the melting- 
point. This film was black or neutral grey by transmitted 
light. If, however, a piece of cadmium which had been 
heated above the melting-point 7m vacuo for several minutes 
was used over again in a fresh tube, the deposit was blue by 
trausmitted light, resembling a film of silver. Apparently 
the cadmium contained some impurity which distilled off at 
a comparatively low temperature. 

To determine the nature of this substance the deposit was 
condensed in the bulb of a small end-on vacuum-tube with 
external electrodes of tinfoil. 

The tube was sealed off from the cadmium tube, heated, 
and the spectrum of the discharge photographed. The 
cadmium and mercury lines were found on the plate. The 
experiment was repeated with a long U-tube immersed in 
liquid air between the pump and the tube, to prevent diffusion 
of mercury vapour; but again the black deposit was obtained 
which showed the mercury lines. Evidently the cadmium 
contains a trace of mercury, which distils off first, carrying 
some cadmium with it. 

The critical temperature for the alloy of cadmium and 
mercury has not yet been determined. It is probable that we 
are dealing with molecules of the alloy, for it has been found 
that mercury vapour is capable of dissolving less volatile 
metals, in the same way that the dense vapours of certain 
organic compounds are capable of dissolving non-volatile 
compounds, such as potassium iodide. 

Iixperiments were also made with iodine, for which the 
critical temperature is in the neighbourhood of —60°. Above 
this temperature a black granular deposit forms, below it a 
deep red film. The iodine crystals at one end of a tube bent 
at two right angles were kept at 0°. The other end was then 
plunged into a bath of known temperature, and the nature of 
the deposit observed. When working in the vicinity of the 
critical temperature, deposits were sometimes obtained which 
were bright green by transmitted light. These probably 
were made up of exceedingly minute granules, analogous 
to the coloured films and fogs obtained with sodium and 
potassium. 


On the Boiling-Points of Homologous Compounds. 371 


The whole subject is one that merits an extended in- 
vestigation. I have not yet examined the impact of the 
vapour of one metal against a film composed of another 
metal, nor have I studied the behaviour of non-metallic 
surfaces other than glass. 

Some very extraordinary results were obtained in the case 
of sodium condensed on surfaces cooled with liquid air, the 
colour of the film changing as soon as the liquid air was 
removed. I believe that there is a change in the structure 
of the film with rising temperature, but the phenomena were 
too complicated and not sufficiently studied to make their 
discussion worth while at the present time. 

It is my plan to continue the study of the subject with 
improved apparatus ; for it is obvious that we must be able 
to control the temperature of the metal, the distillation-tube, 
and the bulb. It will also be necessary to arrange matters so 
that the behaviour of superheated vapours can be studied. 


XLI. On the Boiling-Points of Homologous Compounds. 
By H. C. PLumMer *. 


1. J N discussing the relations which subsist among the 
boiling-points of series of homologous compounds, 
without the guidance of any physical theory, the difficulty 
is to avoid the representation of a large mass of data by a 
number of unconnected formule on the one hand, and on the 
other the too hasty generalization of some particular type 
which happens to hold for some special class of substances. 
The first difficulty is the greater because the observations 
relate to very short series in many cases, and these are 
naturally affected by more or less considerable errors. The 
second difficulty is well illustrated by the early experience 
of Kopp. 
As an example of the way in which a short series may be 
represented by a very simple formula, the isoparaftins may 
be chosen. The formula in this case is 


T=430"5 log n, 
where n is the number of carbon atoms. The comparison 
with observation is shown in Table I. This is satisfactory 
* Communicated by the Author, 
2C 2 


372 Prof. H. C. Plummer on the 


TABLE I. 


Isoparaffins : C, Hen 42. 


nD. Abs. B.P.* Cale. O-—C, 
x 259°7 259°2 405 
5, 300-95 300-91 40:04 
6. 335:0 335-0 0-0 
7. 363°3 363°8 Los 


enough as far as it goes. But obviously it does not go very 
far. The more general formula of this type, however, 


T=a log (bn+c) 


or its equivalent 
A .107=6, const., 


may prove more widely useful in connexion with similar 
series, This is suggested by some experience to be described - 
later. 

As another example the small class of hydrosulphides 
available may be quoted. Here the simple formula 
suggested is 


T—30°-93 (n +8), 


and the comparison with observation is given in Table II. 


TMasnn IT. 
Hydrosulphides : C,,H2,4i1—SH. 
N. Abs. B.P.tT Cale. O—C. 
if 279-0 O78:4 406 
2. 309:0 3809°3 —0°3 
3. 840-5 840°2 +03 
4, 370°5 371-2 —0-7 


Again the range is too small to make any wide deduction 
possible. 


2. Nevertheless, the linear formula, though pushed beyond 
its legitimate limits by Kopp, certainly has its area of useful 
application. This is to be seen in the case of the alcohols, 
the boiling-points of which are fairly represented by the 
formule : é 

Normal primary : T=18°64 (n+17), 
Normal secondary: 18°64 (n+16), 
Primary iso : 18°64 (n+16°5). 
* Young, Scientific Proe. R.D.S., xv. p. 97 (1916). 
t+ Young, Phil, Mag. ix. p. 6 (1905); J. de Chimie Physique, ili. p. 245. 


Boiling-Points of Homologous Compounds. 373 


Tasxe III. 
Aleohols: C He,4,—OH. 
Normal Primary. Normal Secondary. Primary Iso. 
n.. T. . Cale, O—C. T. Cale. O—C. T. Cale. O—C. 
1, 3877 3355 +22 
2. 351:3 3542 —2'9 a a 
3. 3870°2 3728 —26 355°45 35416 41:3 z : yi 
Seeds Wadia lhe) 3730 37238 +02 381°05 38212 —1-1 
5. 411:0 41071 +0°9 3920 3914 +06 405°05 400-76 +43 
6. 4310 428°7 42:3 409:0 4101 —-11 4230 4194 +36 
7. 4490 447-4 +1°6 4375 428:'7 +88 4370 438:0 —1-0 
8. 4640 4660 —2-0 4520 4474 +46 
9. 4865 4846 +1°9 
10. 5040 5033 +0°7 


The calculated are compared with the observed values in 
Table III. A serious error in one of the latter is plainly 
evident in the secondary series, recognized by Young. The 
run of the residuals in the iso-group may point to curvature. 
But as a whole the experimental results appear to be in 
better accord with a linear than any other formula. The 
constant 18°64 agrees with Kopp’s original ideas, and it 
may he noticed—though this may be accidental—that, as in 
the case of the hydrosulphides, the constants additive to n 
are simple. This point is perhaps further illustrated by the 
aldehydes and amines, which can be represented by : 


Aldehydes: T=26°76 (n +10), 

Amines : 26°76 (n+9), 
as shown in Table IV. The lowest member of the aldehyde 
series is quite anomalous, and the amines may show a slight 


curvature. But the simple linear formula is clearly not 
without its uses in representing the facts. 


TABLE IV. 

Aldehydes : CrHenii—CHO. Amines: C,H»,41:—-NH2. 
Nn. "ih Cale. @O=-C be Cale. O—C. 
2520) 2676 156 a P ; 
1. 293'8 294°4 — 06 267°0 267'6 —0°6 
ats 322°0 SAAN + 09 291°7 294°4 — 27 
3: 347:0 347°9 — 09 322'7 Sy i +1°6 
4. 376'0 3746 + 14 348'5 347°9 +0°6 
5. 4009 4014 — 05 3760 3746 +1-4 
6. 428'°0 428'2 — 0:2 402:0 401°4 +0°6 
fis 427:°0 4282 —1:2 


374 Prof. H. C. Plummer on the 
3. The ethers form a double series, C,H»2,4;—O—C,Ho,41, 


for which a considerable number of boiling-points have been 
determined, and it seems desirable to connect them if 
possible by a single formula. Since the composition of 
these substances is symmetrical the absolute boiling-points 
T,.s=T,, are symmetrical functions of r and s. Hence a 
function of the form 


Tps=at+b(r+s)+e(r+s)?+d(r—s)? 
is suggested, and a simple calculation gives 
T,s=189°°75 + 32°42 (7 + s) —0°°557 (7 + 5)? + 0° 138(7 —s)?, 


the last term allowing for some variation among isomeric 
compounds. The numbers resulting from this formula are 
given in Table V.(a), and the result of subtracting these 


TABLE V. (a). 
Ethers : C,H».4;,—O—C,H.,4;. Ty. cale. 


rT. s=1) vie 3. 4. if 8. 

oO Oo (@) (9) (9) oOo 
1. 252°4 282°1 éll-1 Sa en 41655 443°2 
2. 282°1 310°5 3381 BuO os. 45079 463°2 
ay olde 3381 364:2 Sou) =. 4605 482-4 
T- 39°2 3643 389°5 AIS:5 ... - 480°2 500°8 
a. 418°5 439°9 460°5 4802 ... 5384°5 550°9 
3. 443°2 463°2 482-4 HOUSE a | -5ai9 565°9 

TaBuE V. (0). 
Hthers: O-—C. 

T. S==il. 2. os 4. he 8. 

(e) () J o oO °o 
he —3°0 +1°7 +0°8 +41 ... +43 +2°8 
O47 oases 0a) OS eeeeiee 
3. +0°8 —1°5 —()°5 +06 ... +01 —2°4 
Ae +41 0d eG | LOAN es eee 
ae +43 —0°3 +0°1 SG ae at OA +0°9 
8. +2°8 —1:0 —2°4 27 Sa. 09 -12 


from the observed boiling-points, which it is unnecessary to 
reproduce, is shown in Table V.(%). The comparison may 
be considered satisfactory on the whole, though when 
r (or s)=1 the discrepancies are not altogether negligible. 


ae Pie ele be = 
Te 
Rie) Dat oe gin 


Boiling-Points of Homologous Compounds. 375 


4, This tolerable success in representing the boiling- points 
of the ethers by a single formula suggests a similar treatment 
for the esters. Here the composition, C,H2.4;-—-COOC,Hy+1, 
is no longer symmetrical in r and s, but a formula of the 
same type may be tried. Numerically this takes the form 


Ts = 277° 68 + 25°55 (r+ s) —0°°368 (7 +s)? +0°°1153(r—s)?, 
which leads to the numbers given in Table VI.(a). When 


these are subtracted from the corresponding observed values 


TaBLE VI. (a). 


Ksters : (OAs Perey = COOC,Ho-+1- We, cale. 

0) IM 8 ad, cy 5 AG MT 8) 
ie) e) ty) ° fe} oO [@) fe) te) 
303°5 3273 3511 3744 3963 4196 4414 462°7 483°5 
327°3 351:1 3740 3964 4182 439°5 460-4 480°7 500°5 
352'1 3744 3964 4177 4386 459°0 4789 4982- 517-1 
3875°8 3963 4182 43886 4585 4779 4968 5152 533-1 
399°1 4196 489°5 459:°0 4779 4963 5143 531'7 548-7 
4219 4414 4604 4789 4968 5143 531:3 5478 563:7 
444-2 4627 480°7 4982 5152 531°7 5473 5632 578-2 
4659 483° 500° 5171 53831 548°7 5637 5782 592°3 


= 


One SUF OS ee See ai 


Tassie VI. (0). 


Esters: O—C. 

s=0) ~.1, % 3. 4, 5. 6. ie 8. 

ce) fe) e) {e) {e) fe) Oo fe) fe) 
+14 429 +16 413 +40 4380 +37 +432 +430 
—05 —10 —22 —35 -—05 +01 —03 -19 00 
8) 01 =I) —20 1:9" —0:5 +05 —05 
sea eee 0 2am Or 0:3)) — Oro talc oem —alcy 
eee ICO eae Ona? 
edly TNS) 7 oan, =O) OD ye aN. 
+55 +16 403 00 414 +07 -02 —04 
ep Odea 1-9 Ob | 075) eOrene-- O17 


By 


ORNS Soot ee 


the differences shown in Table VI. (0) result. On the whole 
they may be regarded as satisfactorily small. Buta sensible 
and systematic discrepancy again appears in the first line 
and column, of the same kind as that suggested in Table V. (6). 
The boiling-points of the initial series are rather higher than 
the general law indicates. This is not surprising, however, 
because these series are known to be otherwise abnormal in 
their physical properties. The convenient assumption of 
symmetry in the formula does not seem to have had any 
serious detrimental effect on the representation. 


376 Prof. H. C. Plummer on the 


5. But by far the most important single series is presented 
by the normal paraffins, both on account of the number of 
consecutive members and the special care with which the 
boiling-points have been determined. It has also received 
the greatest amount of discussion. Table VII. conveys the 
results of considering this series from more than one point 
of view. In the first place, the boiling-points in the third 
column are calculated by the formula 


T=— 69°04 184°65,/n— 6°89 n. 


The comparison given in the next column shows that this 
formula successfully represents T as a function of n. In 


TaBLe VII. 
Normal Paraffins: C,,Ho,+2. 
nm T. Cale. O—C. AT. Cale. O—C.Cum.sum.| (Young.) 
Cc O° Oo Oo oO 1@) ie} fe) 
1. 1083 1087 —04 7 705 +12 —02 +1:2 
2. 1800 1783 +17 490 514 -24 +410 +19 
3. 2290 2301 —I11 438 427 411 —14 —1:2 
4. 272-8 27277 +01 864 36°4 00 —0°3 —0°2 
5. 8092 3094 —02 328 325 +03 —03 —0-6 
6. 341°95 3420 0-0 294 293 +01 0-0 —O-4 
T. ott 3713 4-0 27:2 268 404 +401 —0°3 
8. 3986 3982 +04 249 248 401 +405 +01 
9, 423°5 423°0 +40°5 22°39 23:0 —05 +06 +02 
10. 446:°0 446-0 0:0 21:0 216 —06 +01 —0-2 
ll. 4670 4676 —06 205 204 +01 —0°5 --0°8 
12. 4875 4879 —0-4 195 193 +02 —04 —06 
13. 507:0 507-2 —02 185 183 +02 —02 —02 
14. 525° 5256 —O1 180 173 +07 0-0 +01 
lo. 543°5 5428 +07 170 165 +05 +07 +0°8 
16. 560-5 5594 -+1-5 155 158 —03 +12 +1:2 
HM 5760 575:2.. £08 140 152 -—}2 409 +09 
18. 590°0 590°'4 —04 130 146 -16 —03 —0:3 
19. 6030 6050 —2:0 —19 |+|§ —19 


the second place, the observed boiling-points have. been 
differenced, following the ideas of Prof. Young, and 
represented by the formula 


20648 
AT T1366 13°82. 
The numbers calculated in the sixth column by this formula 
are based on the observed values of T in the second column. 
The residuals are shown in the seventh column, and the 
cumulative sum of these in the eighth column, the ‘first term 
being. chosen so that the sum of the column is nearly zero. 
This column fairly shows the result of building up the sertes 
by successive differences and should be comparable with the 


Boiling-Points of Homologous Compounds. Ott 


fourth column. Finally, for the sake of comparison, are 
added in the last column the residuals which follow similarly 
from Young’s own formula 


ING a AU ao 


The author’s differences (being C—Q) have been reversed 
in sign and also altered by the addition of a small constant 
(0°-4) in order to make their mean value zero, an adjustment 
practically equivalent to a small change in the initial point 
assumed. 

A comparison of the fourth, eighth, and ninth columns of 
Table VII. shows that the representation by three distinct 
methods is not only equally good, but that after the first 
term or two it is almost identical in detail. Strictly within 
the range considered the three formule are virtually the 
same, and it would appear that the common residuals do 
represent approximately the actual errors of observation 
reduced to a homogeneous system. On the other hand, the 
facility with which the series of numbers can be represented 
in ways mathematically different, makes it impossible to lay 
stress on any particular functional form or to pass beyond 
the precise limits covered by the experiments. 


6. It is useful to consider the locus of the points v=n, 
y=T, as a curve. According to Kopp this curve is a 
straight line, but the truth of this is at best limited. 
According to the formule found in this paper, the curve is 
more generally parabolic in shape. Again, the locus of the 
points z=T, y=AT may be considered as a curve. For 
the normal paraffins it has been shown that this curve may 
be treated as hyperbolic. Now, according to Young, this 
curve is approximately the same for all substances which 
are not associated, so that AT is a function of T only, 
independent of the particular substance. Let the curves be 
T=f(n) and AT=4(T), and consider the f curves of two 
different series. By a displacement parallel to the axis 
of xv (or n) they may be brought into coincidence at some 
point T. But by Young’s law AT, being a function of T 
only, is the same for both curves. Hence the curves will 
coincide at the next point, and the next, andsoon. Thus the 
two curves will coincide altogether, and it follows that if 
the equation of one in its original position is T=/(n), the 
equation of the other must be T=/(n+a), where a is a 
constant. This principle should be very useful, but it proves 
not to be so. The reason for this is that comparatively small 
errors in AT, if systematic, are fatal to the satisfactory 
representation of /(n). The law, though confessedly only 


378 Prof. H. C. Plummer on the 


approximate, is good and useful in its own sphere. A 
residual of 10 per cent. in a difference is often innocuous 
and smaller than the effect of experimental error. But a 
residual of 1 per cent. in the absolute temperature is 
generally unsatisfactory. Hence the determination of f(n) 
and (1) are in practice quite different problems, and the 
former is the more exacting. 

Thus we have been led to treat differently three classes of 
compounds. The first, typified by the alcohols, with marked 
chemical association, 1s represented by a linear formula. 
The second, an intermediate type containing the ethers and 
esters, has slight molecular association and the boiling-points 
show a slight degree of curvature which is fairly represented 
by a parabolic formula. Finally, the very complete series 
of normal paraffins has been taken as the type of compounds 
free from association, and the well-developed curve formed 
by their boiling-points has also been represented by a 
parabolic formula of another kind. In one or another of 
these three classes probably all series of boiling-points can 
be more or less well represented. Thus to the first class 
may be attributed : 


Cyanides (C,Hen4,—CN) : T=19°14 (n+17'5), 
Nitro-compounds (C,H241:—NO.) : T=16°9 (n+21), 
Ketones (CH;—CO—C,,Hon4i) : T=23°52 (n+ 13), 

and to the second class: 

Acids (C,H241—COOH) : T=368°83 + 23°34n—0°'643 n?. 


The comparison, which suggests some irregularities in the 
experimental data, need not be given. 

So far as the parabolic formula found to represent the 
normal paraffins can be regarded as strictly typical, it applies 
to series of hydrocarbons which are for the most part very 
fragmentary. Two examples are given in Table VIII. In 
the first series the constant additive to in the formula is 0, 
and the calculated values are taken without change from the 
third column of Table VII. In the second case the constant 
added to n in the formula is 6°4. 


TABLE VIII. 
Ce AoC = CHC,Hs,. CgHsC, Hen+y. 
N. T. Cale. O-—C. nN. Mt. Cale. O-C. 
ie) G 3) Oo Oo ie) 
4, 2740 272°7 +13 if 383°6 382°4 +1:2 
5. 809°5 309°4 +01 2. 409:°0 408°4 +06 
6. 341:0 342°0 —1:0 2 431°5 437-45. — 09 
hse 371°0 371°3 —0°3 4. 4530 454°9 —19 
5. 


4745 4760 —1°5 


Boiling-Points of Homologous Compounds. d79 


7. More extensive material is provided by the halogen 
compounds, and since the differences between the boiling- 
points are fairly represented by Young’s law, it follows that 
the boiling-points themselves can be represented approxi- 
mately by the normal paraffin curve. The necessary constants 
to be added to n are about : 


CHCl - a= +24; C,,Hon41Br of +32; 
Cr Hen+il > a= + 4°2, 
But the residuals, though fairly small, show a marked 
systematic tendency, which may otherwise be expressed by 
an increase of the constant a with n. Hence it may be 


opportune to illustrate the utility of another formula already 
suggested, namely 


T=a log (bn+c), 
which may be taken to have the numerical forms : 
C,,H»,4101 : T=1000° log (0°155n + 1°6209), 
C,Hen4iBr: 1000 log (0°155n+1°7446), 
Creare : 1000 log (0°155n+1-:9082), 


where something of individual accuracy is sacrificed to 
uniformity and simplicity. This is not serious, however, as 


TABLE IX. 


C,Hon+1Cl. C,Hen4iBr. CrHon+il. 
nm TT. Cale. O—C. T. Cale. O-C. T. Cale. O-—O. 
le) Oo {e) O Oo Oo e} (oe) e} 

249°3 2494 —O0'1 2775 2787 —1:2 315°8 3145 +4+1°3 
285°5 2853 —03 3114 312-7 -—1:3 345°5 3460 —0°5 
319°0 3193 —0°3 343°8 344°3 —0°5 375°5 3753 +02 
351:0 350°2 +0°8 3740 3733 +02 403:0 402°8 +02 
3796 379°5 +0:1 402°5 401°3 +12 |. 429°0 428°7 +403 
406:0 406-7 -~0°7 429:0 4273 +17 4520 453:°0 —1:0 
433'0 482°3 +0°7 452;0 451°7 +0°3 

4570 4565 +0°5 4740 4749 —09 


is shown by the comparison given in Table 1X. The three 
constants ¢ may be expressed in the forms : 


CrHenyi1Cl: e-=1:509+4+ 35°46—318, 
C,H»n41Br: 1°509 + 79°92—339, 
and though the divisors are not quite the same this suggests 


at least some relation between the constant and the atomic 
weight of the halogen element entering into the compound. 


AGO FeO alarmist 1°) diem 


380 Boiling-Points of Homologous Compounds. 


8. It may be noticed that the logarithmic ferm of f(x) is 
capable of a considerable range of adaptability. Thus when 
6 is small in comparison with c the differences become nearly 
constant, and the linear formula is practically reproduced. 
Again, it satisfies the halogen compounds, as shown in 
Table [X., and in another form it represents the fragmentary 
series of iso-paraffins, as shown in Table I. The corresponding 
difference equation is 


AT=4(T)=alog(14+0b.107”). 


If AT is strictly a function of T only, a and 6 must be 
absolute constants. But in practice the merely approximate 
truth of this law is probably consistent with a certain range 
of variation in these numbers. The values used for the 
halogen groups were a=1000°, 6=0°155. It is more 
difficult to compare the logarithmic with the hyperbolic 
difference equation for the normal paraffins. But the values 
a= 815, b=0:223 make the two nearly coincide from 400° 
to 600°, and below 400° they diverge more and more. 

Hence it is impossible to represent the normal paraffin 
series as a whole by the logarithmic formula. And yet the 
relation is worth examination. Hxperience suggests that it 
is more important to represent the higher members than to 
imitate the whole series by an indifferent compromise, for the 
greatest difficulties are always met with in the lower terms. 
When this is done the lower members show marked residuals 
which can, however, be easily represented by an empirical 
correcting term. ‘The result is to give the formula : 


T = 800° log (0:2323 n + 1290) —70°/2”, 


which is compared with experiment in Table X. The 
formula is certainly artificial, but its type is suggested by 


TABLE X. 
Normal Paraffins: C,Hoen+2. 
v7) Tt, Cale. O-C. N. aril, Cale. O-—C. 
e) (eo) ie) oO e) (9) 
0. 20°4 18°5 +19 10. 4460 4462 —0°2 
If 108°3 111-0 —2°7 ib 467°0 467°9 —09 
2. 180°0 177°8 +2°2 12. 487°5 488°3 —0°8 
or 229:°0 229°7 =F 7 18. 507-0 507°6 —06 
4, 272°8 272°6 +02 14. 525°5 5258 —0°3 
ay 309°2 3093 ~*~ —01 15. 543°5 543°1 +0°4 
6. 342°0 341°9 +0°1 16. 560°5 5506 =« +09 
is 371°4 371°2 +02 Le 5760 575°4 +0°6 
8. 3986 398'1 +05 18. 590-0 590°5 —0°5 
a: 423°d 4231 +04 19. 6030 6049 —1°9 


On a Wehnelé Cathode-Ray Tube Magnetometer. 381 


the analogous formula of Ramage and may perhaps be 
justified on similar grounds. The interesting point, however, 
is that this formula does what no other equation suggested 
has succeeded in doing. Ii leads the series right down so as 
to include the boiling-point of hydrogen as the zero paraffin. 
The wish to do this constituted the original incentive to the 
work contained in this paper. It was suggested by reading 
the two papers by my colleague, Prof. Sydney Young, to 
which reference has been made and from which all the 
experimental material used above has been taken. To his 
kindly interest this paper is entirely due. 


Dunsink Observatory, 
July 31, 1916. 


XLII. A Wehnelt Cathode-Ray Tube Magnetometer. By 
Cras. T. Kyrep, W.A., PhD., Associate Professor of 
Experimental Electricity, and L. A. Wuio, M.S., Research 
Assistant in Astronomy, Unversity of Illinois, U.S.A.* 


[Plate VIII. ] 
IL. Introduction. 


pp a recent paper { the authors described an apparatus for 

determining the horizontal intensity (H) of the earth- 
magnetic field, depending on the measurement of the mag- 
netic and electrostatic deflexions of a beam of cathode rays 
and on the assumption of the ratio of the charge to the mass 
of the electron. It has since been suggested to the authors 
ihat it might be well to deduce H by comparing the de- 
flexion produced by the earth’s magnetic field with the 
deflexion produced by a known calculated field, thus 
avoiding the assumption of the ratio e/m. It was for the 
purpose of carrying out this suggestion, and also in the hope 
of obtaining a more compact and convenient instrument, that 
the work now to be described was undertaken. 


Il. The Apparatus and the Manipulation. 


The discharge-tube is a glass Jar about 40 cm. in height 
and of 12 cm. diameter, clamped at the bottom to a brass 
plate, which is in turn fastened to the widened part of a 
brass pipe about 5 cm. in diameter. The last fastening is 


* Communicated by the Authors. 
+ Knipp and Welo, Terrestrial Magnetism and Atmospheric Elec- 
tricity, vol. xx. 1915, pp. 53-68. 


382 Prof. C. T. Knipp and Mr. L. A. Welo on a 


adjustable so that the axis of the discharge-tube can be made 
to coincide exactly with that of the pipe, which turns in 
holes cut in a wooden frame. The pipe thus serves as a 
bearing, permitting the discharge-tube to be turned about 
a vertical axis. The rotation about the vertical axis is a 
necessary feature of the instrument, for the earth’s magnetic 
field cannot very well be eliminated for a zero reading 
of the deflexions, but they must be observed for various 
orientations of the tube. 

The tube is exhausted through a glass stem inside of the 
brass pipe. One end of the stem is waxed into a hole at 
the bottom of the jar, and the other is fitted by a long and 
well-ground joint to another glass stem which connects to a 
Gaede capsule-pump, to a bulb containing the drier P,Os, 
and to the charcoal-liquid-air bulb for further improving 
the vacuum. 

The cathode shown at OC, fig. 1, is of a form previously 
described by the authors * but has, in this case, been some- 
what simplified and made more compact. The object of 
using the Wehnelt cathode ratber than the cold aluminium 
cathode is to secure electrons having a low velocity with 
consequent measurable deflexions for the relatively weak 
earth-field, and at the same time get a definite beam of 
sufficient range without the use of diaphragms, as was 
found possible when the spot of lime (Bank of England 
wax) does not exceed 0:02 cm. in diameter. The direction 
that the beam takes is uncertain, and two adjustments of the 
cathode are necessary, one in the plane of the drawing and 
one perpendicular to it. J£ the beam should take some 
such direction as that indicated in the drawing, fig. 1, it can 
be made to take a vertical direction by turning at the joint 
Jo, and a similar adjustment in a plane perpendicular to the 
plane of the drawing can be made by turning at J3. 

The anode is an aluminium ring of nearly the internal 
diameter of the jar, lying at its bottom, and connected by a 
wire to mercury at the bottom of the glass stem below the 
joint J;. The mercury is in contact with a sealed-in platinum 
wire which is connected to the positive terminal of a high 
potential battery. Since the cathode and the wire leading 
to the anode from the inside of the exhaust-tube are very 
close together, a mica screen is inserted at M to prevent a 
direct discharge which might take place if the vacuum 
were at a favourable stage, with possible injury to the 
battery. 

The deflexions of the beam of particles are recorded on a 

* Loe. cit. 


Wehnelt Cathode-Ray Tube Magnetometer. 383 


photographic plate held near the top of the tube in a simple 
form of plateholder and shutter. The holder, which rests on 
a brass ring waxed to the inside of the tube, is a disk of 


Fig. I. 


| 
~Y 


SS 


i ml 
: y Ne ~ 
2 = = s 
~. : 
S D 
S 
ri ~ oS ) 
SWANS ES SUR SUULESS SESE BSEBEE TLTeLieiesien S\ s 
Ps = = = : 
F eee gens : ee 
i i Seal} 
aoa t : 
S ‘ 
a oY) Sel 
We e ‘ 
| i Biers NE Se 


ie 


S—Standard Coil, to Battery. 
H—Heating Circuit, to Battery. 
DIS—Discharge Circuit, to High Potential Battery. 


384 Prof. C. T. Knipp and Mr. L. A. Welo on a 


aluminium with a small segment cut away. On the under 
side of this disk a shallow box is built. One end of the box 
is slightly undercut on the inside so that the photographic | 
plate can be wedged in, and a wedge-shaped clamp actuated 
by a spring in the wall at the other end secures the plate in 
position, not only from falling out but presses it up against 
the disk, so that its position is never in doubt. The cover is 
a thin aluminium plate, hinged at a side, bent over at the 
edges to prevent light from entering edgewise, and is 
backed by a spring so that it will stay closed. The lid 
is opened by the winch shown at J,, which winds a string 
hooked to the handle on the lid. The discharge-tube is 
closed at the top by a piece of plate-glass, sealed on with 
half-and-half (beeswax and resin) wax. 

The platinum strip on which the lime is placed is heated 
electrically. To do away with troublesome winding of the 
wires as the tube is turned, a hard rubber plate containing 
two rings of mercury was arranged, immediately beneath 
the brass pipe bearing. The rings are connected to the 
terminals of a battery by binding-posts leading up under 
the rings through the rubber. From the rings, wires lead 
up the inside of the brass pipe but outside the exhaust stem, 
through the brass pipe above the wooden frame, and thence 
to sealed-in platinum terminals set in tubes perpendicular 
to the plane of the drawing and indicated by the dotted 
circle at D. Wires on the inside of the cathode-mounting 
lead to the cathode itself. One terminal of the heating 
circuit leads to the negative end of a high potential battery, 
while the positive end of the high potential battery is con- 
nected to the sealed-in platinum wire below J, as has 
already been stated. 

The coil of six turns, wound so close that the current may 
be considered a circular one, is wound on a built-up and 
accurately turned wooden core. It is pivoted on the hori- 
zontal support, shown above the discharge-tube, so that it 
hangs vertically, but is prevented from turning about a 
vertical axis by an adjustable clamp (not shown) on the 
support. A pointer on the coil which moves over a gradu- 
ated circle on the support gives the orientation. A pointer 
on the brass supporting-pipe and a graduated circle on the 
wooden frame gives, it might be added, the orientation of 
the tube itself. When it is desired to remove the cover from 
the discharge-tube, the coil is lifted off and the support 
swung back about the pivot P. 

When making a determination, the pump is generally run 
for over an hour, the charcoal bulb being gently heated during 
the while to drive off gas absorbed at ordinary temperatures. 


Wehnelt Cathode-Ray Tube Magnetometer. 385 


When the bulb has cooled it is immersed in liquid air. The 
heating circuit may soon be closed, the resistance adjusted 
until the platinum glows with a red heat, and 1000 volts now 
applied between the cathode and anode. It is well to leave 
the glowing catnode thus for, say, five minutes. If the 
heating current is then slowly increased a diffused spot will 
show on the plateholder-lid which is covered with willemite. 
On further heating, the diffused spot becomes either a well- 
defined spot or a sharp line in a plane containing the 
platinum strip. After the cathode has been adjusted so 
that the spot is east of the centre of the plateholder by an 
amount equal to the estimated deflexion (and in the plane of 
the coil), and the orientation of the tube noted, the plateholder 
is opened and an exposure made for 5 to 15 seconds, de- 
pending on the intensity of the beam. The switch of the 
coil circuit is then closed so that the beam is deflected still 
more, and the exposure at the new spot made. Then the 
plateholder is closed, the tube turned to 180°, the plate- 
holder again opened and the process repeated. We thus 
get four spots or lines on the plate, which should be arranged 
along a straight line. Half of the distance between the two 
inner spots is the deflexion due to the earth’s field alone, 
while half the distance between the outer two is the deflexion 
with the earth’s field increased by the standard one. 


Ill. The Declination. 


_ Itis unnecessary to determine the declination by another 
instrument in order to set the coil in a plane perpendicular 
to the meridian. The instrument itself is indeed a suffi- 


ciently accurate declinometer for the purpose of setting 
the coil. 


Fig. 2. 


A test photographic plate is first taken with the coil in 
any position, which is noted on the graduated circle, and 
four spots obtained. Fig. 2 represents the general case. 
OA is the deflexion due to the earth’s field alone, and 
OC that due to both fields. But OC is the resultant of OA 


Phil. Mag. 8. 6. Vol. 32. No. 190. Oct. 1916. 2D 


386 Prof. C. T. Knipp and Mr. L. A. Welo on a 


and the component AC of the coil-field making an angle ¢ 
with BA produced. On this test-plate, then, the line BA is 
drawn and produced; draw the line AC and measure the 
angle @. The coil must be turned through the angle ¢ in 
order that it be perpendicular to the magnetic meridian. 


IV. The Formula. 


Vig. 3 shows the relative positions of the cathode C which 
is taken as the origin, the photographic plate P ata dist- 
ance 21, the centre of the coil O at a distance 7, the undeflected 


Fig. 3. 


path of the beam along the diameter taken as the 2 axis and 
the deflected paths, with and without a current in the coil. 

Calling z, the deflexion produced by the earth’s magnetic 
field alone, the equation * 


Be 
eee ; H(«,—2)da eer oO 6 (1) 


2 
holds when the deflexions are so small that (=) may be 


neglected. When a current flows in the coil so that the two 
fields have the same direction the resultant field is 


1 Ge Sei 


* J. J. Thomson, Phil. Mag. vol. xviii. pp. 844-845 (1909). 


Wehnelt Cathode-Ray Tube Magnetometer. 387 


Letting 2,, be the deflexion then produced we have, 
similarly, 


a et Ki 
<ee— <{ (H + ETE} (w,— v)da. ° Tl Mics (2) 


H is constant so that, integrating (1) and the first term of 
the right-hand member of (2), we have 


m2 
ge Es EEE OOO TA Ape ar) 


mv 2 
and 


y oe [at al? TEA ode]. ne 


2 


Because of the low velocity of the electrons and the high 
vacuum used, the factor “ is constant and is eliminated by 


dividing (4) by (3). aie for H the formula appears 


Ee oe =," a H,(a,—a)d 
vy Se see 


In the circular coil, used because of ease of construction 
and to secure compactness, the field H, is not constant along 
the path of the beam, but must be expressed as a function of 
the distance from the cathode. 

Calling 2’ the distance from the centre of the coil to the 
point in question, we have, per unit current*, 


HL eh 


re, ees 


but «’=x—r, so that we have for the field, in terms of the 
distance from the cathode, the centre of coordinates, 


2aran seg DF 1.3\71(#—r)! 
= a aaa -(3) Cae hk | N OCA aan 3 | 
Rise a NM a) 
M =| H.(v,—x)dex 
0 


and substitute the value of H, from (5), we have, using four 


* A. Russell, Phil. Mag, vol. xiii. pp. 420-446 (1907). 
2D 2 


where 


388 Prof. C. T. Knipp and Mr. L. A. Welo on a 


terms of the series, 


71 dx 1\?1 («—r)dzx 
La aman [z=G@aa— (3) ac 
i a A OS aa ete) = — 
4A) 3ata’—(a—r)? \2.4.6/ 5a'a’—(a—r) 
LAL 21 x(a—r)'dz 
oo i 2—(a—r) (5 aman 
ao aa —— pl Ae S| 
2.4) 3a'a’—(a—r) \2.4.6/ 5a8a?—(a—r)?]: 
Introducing a new variable y=x-r, we have, after col- 
Jecting terms wherever possible, 


ui Hey Re y ae -(5) 3 As ag 
MS aman | (m r) ame 2) ae (v,—7) ay? 


V\? 1 a? dy ale y dy 
15) Cc = yea! We —y" 


4 — te y? dy “ee a y® dy 
( 30) 672g? NEL b sae a—y? 
+E 7 pe) 

2.4.6) 5aSa?—ys° 


Integrating, collecting terms, restoring the variable « and 
putting in the limits, we have, finally, from the four terms of 
the series with whieh we started, 


=e ys oe ae oe ae es ieee (at+a,—r)(at+7r) 
2.4/3 2.4.6 “(a—a,+7r)(a—7r) 
~${2 -(5)- Ge Ss) 5 — io 2) 5 | loeecs a? —r? 
Z 2.4 Daw °a—(a,—7)? 
ey al ‘1 Bye BN? 1 
+ 85"((5) +a) at Gee) sla 
ipa. Clavel : 3.5) 
~=[(5) +(3) s+ z, =)5] * 3 
+ (53) 5+ (S22) a] ae, 
gO N24) 3 XO aS 3 
LT/1. BYR 7163. eG, 7) 
Seales a) 3 +(5 | [<7 


A = |S NO. 7) 19 (Gina ae 
a’ 2 6 
bs, 


M=27ran 


5 5 
-=| Ae 2) - 2 a CeO \ 
a® A era Z 


(6) 


Wehnelt Cathode-Ray Tube Magnetometer. 389 


The law of the series is apparent. The convergence is 
slow, but the coefficients are easily evaluated if it is noted 
that the series of the first two terms in (6) when multiplied 


by = is the E series of the elliptic functions when the 


modulus is one. Its value is then unity, so that 


1\2 (1.321 ae CRE on 
1-(5) (5a) sc) gg OREO 
ame 3\!1 7.3 521 
(3) + ($4) 3 +(G-rg) s+ +--+ = 100000 — 0°63662 = 0-36338, 


and so forth. 

Then, neglecting terms of the power five and higher 
because the fourth significant place in M is not affected, 
there results 


fa,-r (a+a,—1r)(at+r) 
a) eR So Taro UC a RY 
M= 27ran ane 0°63662 . log, Czer Gn 


1 e— 7? 
hk 0°63662. log. =— Graeay 
PEN FE358. 7) = + | 0-36338 2 
a a 2 


3 


Be NO 
= (Hees aa 


9 
(9) 


ele ae 
7 1388) er : 


It should be remarked that in evaluating 
M =|° H.(a,—2)dz 
0 


we have assumed the same distribution of the field along the 
actual path of the beam as along the diameter of the coil. 
It was the plan at first to use opposed fields so that the 
beam would follow the diameter very closely, but it was 
noted on Plate No. 26 and two previous ones where de- 
flexions were taken in both directions, that, so far as 
measurements indicated, there was no difference in the 
distances from either of the spots to the middle one corre- 
sponding to no current in the coil. The assumption as to 
the distribution is thus justified. 


390 On a Wehnelt Cathode-Ray Tube Magnetometer. 
V. The Results. 


Measurements were made on seven photographic plates, 
numbered 27-34, number 29 being unexposed, for the 
platinum strip was accidentally burned out while the 
heating current was adjusted. Photographs 28, 30, 33, 
with number 26, are shown in Plate VIII., and the data from 
the measured photographs are collected in Table I. 


TABLE I, 
| Plate | M. ca it Ze Zee | 2ec—Ze H Hee 
No. cm. |amperes.| cm. | cm. em. c.g.8. m 


— ——— ee ———_—————————— | 


27 | 48°61) 33:13) 1-402 | 1-059] 1:886; 0827 | 01590 | +0:0007 
28 | 4861) 33°13) 1463 | 0993) 1812) 0819 | 0-1571 | —0-0012 
30 | 48°61 | 33:13) 1:4382 | 1:024| 1-863} 0-839 | 01548 | —0:0035 
31 | 4861) 33:13) 1-423 | 1-032] 1846; 0-814 | 0:1598 | +0°0015 
32 | 48°61) 33:13| 1:417 | 0-980] 1'759! 0-779 | 0:1579 | —0-0004 
33 | 4861) 83:13, 1323 | 0952 1652; 0-700 | 0:1594 | +0-0011 
84 | 4861) 33:13) 1-473 | 1-009] 1:831| 0-822 | 0-1602 | +0-0019 
0:1583 | +0-0005 


The constant M was calculated by the formula of the 
preceding section, and agrees satisfactorily with the value 
determined graphically. The diameter of the coil, the 
distance between the cathode and the photographic plate 
and the vertical distance of the latter from the edge of the 
coil, were measured with a cathetometer, and it was deduced 
that a=43°46 cm., #j=33':13 cm., and r=5:01 em.’ It wall 
be remembered that the coil has six turns. 

The current was measured with a Siemens & Halske pre- 
cision ammeter, removed from the magnetometer, to 1 part 
in 1500, with no observed fluctuations during the time of 
exposure. The deflexions tabulated are the half means 
of five measurements of the distances between the outer two 
and the inner two spots on each plate. The precision 
attained in measuring deflexions is indicated by taking 
Plate no. 33 as an example, where the maximum devia- 
tion of a single measurement of <, from the mean of five 
measurements is 1 part in 316, and for z,, it is 1 part in 230. 

The deviations from the mean of the values of H are tabu- 
lated and the relative probable error of +0°00046 or aa H 
calculated, but the performance of the instrument is not 


The “ Wolf-note” in Bowed Stringed Instruments. 391 


fairly indicated because of the presence of electrical circuits 
in the neighbouring rooms of the laboratory and a changing 
distribution of iron about the building. 


VI. Summary. 


1. A magnetometer has been designed and built, depending 
on the property of the cathode rays being deflected by a 
magnetic field, the deflexion being compared with that 
produced by a known standard field. 

2. A Wehnelt cathode is used to impart a low velocity to 
the electrons, resulting in measurable deflexions for the weak 
magnetic fields. The deflexions are recorded on a photo- 
graphic plate. 

3. The elimination of the earth’s magnetic field for a zero 
reiding of the deflexion is avoided by mounting the tube so 
as to turn about a vertical axis, and deflexions observed for 
different orientations. 

4. The magnetometer is also a declinometer of sufficient 
precision for setting the coil in a plane perpendicular to the 
magnetic meridian. 

5. Account is taken of the fact that the field of the standard 
coil is not uniform along the path of the rays,and an analytical 
solution of the problem is found. 

6. Seven determinations of H have been made with a rela- 
tive probable error in the fourth significant place. 

The general design and construction of the apparatus is 
that of the senior author, who also did a part of the pre- 
liminary work. The junior author supplied some of the 
details of construction, deduced the formula, and made the 
final observations and the necessary calculations. 

Laboratory of Physics, 


University of Illinois, 
June 9, 1916. 


XLII. On the “ Wolf-note” in Bowed Stringed Instruments. 
By ©. V. Raman, IA." 


[Plate IX. ] 


T has long been known that on all musical instruments 
belonging to the violin family there is a particular note 
which it is difficult to elicit in a satisfactory manner by 
bowing. This is called the “ wolf-note,” and when it is 
sounded the body of the instrument is set in vibration in an 


* Communicated by the Author. 


392 Prof. C. V. Raman on the ‘‘ Wolj-note”’ 


unusual degree; and it appears to have been realized that the 
difficulty of maintaining the note steadily is due in some way 
to the sympathetic resonance of the instrument*. In a 
recent paper tf, G@. W. White has published some interesting 
experimental work on the subject, confirming this view. The 
most striking effect noticed is the cyclical variation in the 
intensity of the note when the instrument is forced to speak 
at this point. White suggests as an explanation of these 
fluctuations of intensity that they are due to beats which 
accompany the forced vibration impressed on the resonator 
when the impressed pitch approaches the natural pitch of 
the system. ‘The correctness of this suggestion seems open 
to serious criticism. For, the beats which are produced 
when a periodic force acts on a resonator are of brief 
duration, being merely due to the superposition of its forced 
and free oscillations, and when, as in the present case, the 
resonator freely communicates its energy to the atmosphere 
and the force itself is applied in a progressive manner and 
not suddenly, such beats should be wholly negligible in 
importance, and should, moreover, vanish entirely when the 
impressed pitch coincides with the natural pitch.. In the 
present case the essential feature is the persistency of the 
fluctuations of intensity and their markedness over a not 
inconsiderable range; and it is evident that an explanation 
of the effect has to be sought for on lines different from those 
indicated by White. I had occasion to examine this point 
when preparing my monograph on the ‘‘ Mechanical Theory 
of the Vibrations of Bowed Strings,’ which will shortly be 
published, and the conclusions I arrived at have since been 
confirmed by me experimentally. 

From the mechanical theory, it appears that when the 
pressure with which the bow is applied is less than a certain 
critical value, proportionate to the rate of dissipation of 
energy from the vibrating string, the bow is incapable of 
maintaining the ordinary mode of vibration in which the 
fundamental is dominant, and the mode of vibration should 
progressively alter into one in which the octave is the pre- 
dominant harmonict. In the particular case in which the 
frequency of free oscillation of the string coincides very 
nearly with that of the bridge of the violin and associated 
masses, the mode of vibration of the string is znitially of the 
well-known type in which the fundamental is dominant. 


* Guillemin, “The Application of Physical Forces,” 1877. 

+ G. W. White, Proc. Camb. Phil. Soc. June 1915. 

{ Compare with the observations of Helmholtz, ‘ Sensations of Tone,’ 
English Translation by Ellis, p. 85. 


in Bowed Stringed Instruments. 393 


But the vibrations of the string excite those of the instrument, 
and, as the vibrations of the Jatter increase in amplitude, the 
rate of dissipation of energy increases continually till it 
outstrips the critical limit, beyond which the bow fails to 
maintain the usual type of vibration. As a result of this, the 
mode of vibration of the string progressively alters to a type 
in which the fundamental is subordinate to the octave in 
importance. The vibration of the beliy then begins to decrease 
in amplitude, but, as may be expected, this follows the change 
in the vibrational form ot the string by a considerable interval. 
The decrease in the amplitude of the vibrations of the belly 
results in a falling off of the rate of dissipation of energy, 
and, when this is again below the critical limit, the string 
regains its original form of vibration, passing successively 
through similar stages, but in the reverse order. This is then 
followed by an increase in the vibrations of the belly, and 
the cycle repeats itself indefinitely. The period of each cycle 
is approximately twice the time in which the vibrations of 
the belly would decrease from the maximum to the minimum, 
if the bow were suddenly removed. 


The foregoing indications of theory are amply confirmed 
by the photographs reproduced in Plate IX., which show the 
simultaneous vibration-curves of the belly and string of a 
’cello at the wolf-note pitch. It will be seen that the form 
of vibration of the string alters cyclically in the manner pre- 
dicted by theory, and that the corresponding changes in the 
vibration-curve of the belly follow those of the string by 
an interval of about quarter of a cycle. That the two sets 
of changes are dynamically interconnected in the manner 
described is further confirmed by the prominence of the 
octave in both curves at the epochs of minimum amplitude. 
The explanation of the cyclical changes given above is also 
in accordance with the observed fact that they disappear and 
are replaced by a steady vibration when the ratio of the 
pressure to the velocity of bewing is either sufficiently 
increased or sufficiently reduced. In the former case the 
string vibrates in its normal mode, and in the latter case 
the fundamental disappears altogether and the string divides 
up into two segments. 


Effect of Muting on the “ Wolf-note.” 


Since the pitch of the wolf-note coincides with that of a 
point of maximum resonance of the belly, we should expect 
to find that by loading the bridge or other mobile part of the 
body of the instrument important effects are produced. 


394 Prof. C. V. Raman on the “ Wolf-note ” 


This is readily shown by putting a mute on the bridge. The 
pitch of the wolf-note then falls immediately by a considerable 
interval. On the particular ’cello I use, a load of 17 grammes 
fixed at the highest point of the bridge lowers the wolf-note 
pitch from 176 to 160 vibrations per second. A larger load 
of 40°4 grammes depresses it further to 137 vibrations per 
second, and also causes two new but comparatively feeble 
resonance-points to appear at 100 and 184 respectively, 
without any attendant cyclical phenomena. An ordinary 
brass mute has a very similar effect. 


The Formation of Violin-tone and its Alteration by a Mute. 


The positions of the frequencies of maximum resonance of 
the bridge and associated parts of the belly for notes over 
the whole range of the scale are undoubtedly of the highest 
importance in determining the character of violin-tone, and 
the explanation of the effect of a mute on the tone of the 
instrument is chiefly to be sought for in the effect of 
the loads applied on the frequencies of the principal free 
modes of vibration of the bridge and associated parts of the 
belly. The observations of Dr. P. H. Edwards on the effect 
of the mute * are evidently capable of explanation on the 
basis of the lowering of the frequencies of maximum reso- 
nance by the loading of the bridge. But a more detailed 
understanding of the dynamics of the problem requires 
further theoretical and experimental investigation. Recently, 
I have secured an extensive series of photographs showing 
the effect on the motion of the bridge in its own plane pro- 
duced by fixing a load on it at one or other of a variety of 
positions. The close parallelism between the effect of loading, 
as shown by these photographic curves and as observed by 
the ear, seems to show that the motion of the bridge in its 
own plane determines the quality of violin-tone to a far 

reater extent than might be supposed from the work of 
Giltay and De Haast. A detailed discussion of this and 
other problems relating to the physics of bowed instruments 
is reserved for a separate communication. 

This investigation was carried out in the Laboratory 


of the Indian Association for the Cultivation of Science, 
Calcutta. 


20th May, 1916. 


* P. H. Edwards, Physical Review, January 1911. 

t Giltay and De Haas, Proc. Roy. Soc. Amsterdam, January 1910. 
See also E. NH. Barton and T. F. Ebblewhite, Phil. Mag. September 1910, 
and C. VY, Raman, Phil. Mag. May 1911. 


Wt 


in Bowed Stringed Instruments. 39: 


Note dated the 8th of August added in proof. 


Since the paper was first written, several other interesting 
effects have been noticed, of which the following is a 
summary :— 

(a) Cyclical forms of vibration of the G-string and belly 
of a ’cello may also be obtained when the vibrating length is 
double that required for production of the wolf-note, that is, 
when the frequency is half that of the wolf-note. In this 
case, when the pressure of the bow is sufficient to maintain 
a steady vibration, the second harmonic in the motion of the 
belly is strongly re-inforced. When the pressure is less 
than that required for a steady vibration, cyclical changes 
occur, the principal fluctuations in the motion, both of the 
string and the belly, being in the amplitude of the second 
harmonic. In this, as in all other cases, the cyclical changes 
disappear and give place to a steady vibration, when the 
bow is applied at a point sufficiently removed from the 
end of the string. In this particular case, a large, almost 
soundless vibration may be obtained by applying the bow 
rather lightly and rapidly at a point distant one-fifth or more 
of the length from the end; the octave is then very weak in 
the vibration of the string, but may be restored, along with 
the tone of the instrument, by increasing the pressure 
of the bow.: 

(0) The ’cello has another marked point of resonance at 
360 vibrations per second. The pitch of this is also lowered 
by loading the bridge. 

(c) When the vibrating length of the G-string or A-string 
of the ’cello is about a fourth of the maximum or less, cyclical 
forms of vibration may be obtained at almost any pitch 
desired, by applying the bow with a moderate pressure rather 
close to the bridge. 

(d) As the frequency of vibration is gradually increased 
from a value below to one above the wolf-note frequency, 
the phase of the principal component in the “ small” motion 
at the end of the string, that is also of the transverse hori- 
zontal motion of the bridge, undergoes a change of approxi- 
mately 180°. ‘This is in accordance with theory. 


| 396 | 


XLIV. Some Experiments on Residual Ionization. 
By i. H. Kinepon, M.A., University of Toronto*. 


[Plate X.] 
I. Residual Ionization in Acetylene. 


7NTRODUCTION.—If dry air be enclosed in a clean 
zine vessel and removed from the neighbourhood of all 
ordinary ionizing agents, it is found, on measuring the con- 
ductivity of the gas, that ions are being produced in it at the 
rate of about 8-7 per c.c. per second. The production of these 
ions is called natural ionization. If, however, the zine vessel 
be surrounded by a water-screen the natural ionization is found 
to decrease ; and if the measurements be made over a con- 
siderable body of water, such as the ocean or the great lakes 
of America, the ionization in air is found to fall to a definite 
minimum of about 4°4 ions per c¢.c. per second, which we 
may call the residual ionization. The difference between 
these two rates of ionization has been shown to be due to a 
penetrating radiation from the earth’s surface, which can be 
cut off by a screen of water if sufficient thickness be used. 
The possible components of the residual ionization appear 
to be:-——(1) a radioactive impurity in the gas, (2) a radio- 
active impurity in the walls of the ionization-chamber, 
(3) ionization by the collisions of thermal agitation. In an 
effort to determine to which of these sources residual ioni- 
zation should be attributed, McLennan and Treleaven+ 
measured the residual ionization in several gases, the results 
obtained being as follows :— 


Gas. No. of ions per c¢.c. 
per second. 
Carbon Dioxide ............ 4°83 
Hydrogen... -sceeeeee ree il Li 
thy lene... <2: peeeeeeeeee 6°32 
Nitrous Oxides; eee 502 
Acetylene .:..):.eeeeee 27°00 


It will be noticed that in the above list the residual ioni- 
zation in acetylene is very much higher than in any of the 


* Communicated by Professor J. C. McLennan, F.R.S. 
t J.C. McLennan and C. L. Treleaven, Phil. Mag. xxx. p. 415 (1915). 


Experiments on Residual Ionization. 397 


other gases, a fact which certainly cannot be accounted for 
on the ground of its density. The present investigation was 
therefore undertaken to see whether this large ionization 
could be traced to either of the sources (1) or (3) above. As 
a result, it has been found that in acetylene made from 
calcium carbide there is present a slight trace of radium 
emanation, and this it has been shown accounts for the high 
residual ionization in the gas. 

Experiments.—The gas used at first in the experiments 
with acetylene was taken from a commercial Prest-o-Lite 
tank. The ionization-chamber was made of zine because it 
has been shown that this metal contains smaller traces of 
radioactive impurities than any other. The external dimen- 
sions of the chamber were :—diameter 11°6 cm., length 
22-8 cm., and its volume was 2167 c.c. The thickness of the 
walls was about 3mm. ‘The chamber was carefully scoured 
with emery, and washed with dilute hydrochloric acid and 
water, to remove radioactive deposits. During all the course 
of the work the chamber was absolutely air-tight. It was 
provided with a zinc electrode which was connected to a 
sensitive electrometer in the usual manner. The wall of 
the chamber was kept at a potential of 240 volts, which 
ensured that all the currents measured were saturation 
currents. 

The ionization-chamber was filled with acetylene which 
had been carefully dried and freed from dust, and the number 
of ions made per c.c. per second was found to be about 20. As 
this number was considerably less than that previously found 
by Mclhennan and Treleaven, the zinc Wolf electrometer 
(PI. X. fig. 1) used by them was filled with acetylene from 
the Prest-o-Lite tank, and the number of ions made per c.c. 
per second was found to be only 12. The difference between 
this number and that found with the zinc ionization-chamber 
was probably due to a radioactive impurity in the walls of 
the latter. It was thought that the low value of the ionization 
was due to the commercial acetylene not being pure, and to 
test this some acetylene was made in the laboratory from 
calcium carbide, and both the Wolf electrometer and the zinc 
ionization-chamber were filled with it. The number of ions 
made per c.c. per second (n) was measured at intervals during 
a period of about 19 days. The following set of readings was 
obtained with the Wolf electrometer, and a similar set with 
the zine ionization-chamber. 


398 Mr. K. H. Kingdon on some 


T (hrs.). a. 


0 27°0 
22 25°2 
40 24:1 
88 Zt 

113 19°38 
139 18-0 
209 16°7 
310 16:0 
454 ose: 


A curve plotted from these readings is shown in fig. 2. 
Fig. 2 


nn SHE: sgt Hee 
: SeEEeeeeeverit 
2B = Perry ryt 


BEREEES 
SRSEEoOo 
Ho rr San Rees ee 

Saeens S000 Se) EEE EEE LOO 
rT Pee SEER0 GREE E eee eee EBRSEESEER See es 
rr Sane GeRe0 0000000005000 5000n 50000 50n50GRRGE SHERERE 
POEL BOS Sones GeeeReaee Pty 


CT Ce 2 DEES oes See eee 
€ a2 ! rH Co FE SEcEEe SESeTCEETEOEEEE seoererevensere 
sriniiiart siiasatiiaeiftsiti EEESEEEEEEE EEE HEE EEE EEE 
seesesast ce nee ree eee 
Secetteesiesats 
I) scszesesadueratearaerers sEetec su fouuadtesedenens covsaeas eeensneesreeesseee ateaiitt 


HEEHEn aot srssestasssrssset ECCEEEe ey 
HEEA Yt EECEH — Tt 


aa 


10) 


eee ees FE 


Fi “1 Seeuns ieee siitccuiess eee 
SSEHH EEEGEPeoeEEas POD EETOSEHeaaEserdarifcssstosafaaate augasus cose scser25! 

re) a 100 150 290 250 300 350 400 450 500 
Time in hours 


No of ions perc.é. per Second 


The decrease in the ionization clearly indicates the pre- 
sence of some radioactive impurity in the gas. From the 
shape of the curve it may be estimated that the number of 
ions made per c.c. per second would finally fall to about Lo. 
From this the number of ions per c.c. per second initially due 
to the radioactive impurity was about 12. As this number 
of ions had decreased to 6, or to one half, in about 90 hours 
(3°8 days), it may be concluded that the impurity was radium 
emanation. In the gas from the Prest-o-Lite tank this 


Experiments on Residual Ionization. 399 
P 


emanation had had ample time to decay, and so the number 
of ions made per c.c. per second was quite low. 

To make certain of the source of the emanation the calcium 
carbide was tested for radioactivity in the following manner. 
A metal cap, fig. 3, was made which could be screwed on 

Fig. 3. 


io} 


\» 


in place of the drying-tube of the Wolff electrometer. A 
small brass cup, which was to contain the carbide, was sup- 
ported ona rod screwed into the end of thecap. ‘The figure 
shows the cap in place, the top of the cup being flush with 
the wall of the electrometer. The following measurements 
were made. The cup was put in place, and the electrometer 
filled with fresh, dry air. ‘The number of ions made per c.c. 
per second, n, was then measured. The cup was removed, 
filled with powdered carbide, and the carbide was covered 
with aluminium foil -0003 cm. thick. The cup was then 
replaced in the electrometer, and the number of ions made 
per c.c. per second again measured. The next day two more 
measurements of the ionization were made, the first with 
both brass cup and carbide in place, and the second with them 
both removed. The air in the electrometer was the same 
as that used on the first day, and had probably received 
slight traces of radium emanation from being in contact 
with the carbide. Hence the values of n obtained on the 
second day are considerably greater than those obtained on 
the first day. However, the difference between the two 
readings taken on either day, which is the important thing 
here, is almost the same for both days. 


First day. 2. Second day. 2. 
Brass cup in place ....... ak 8-7 | Carbide and cup in place ...| 12:2 
Carbide and cup in place ...| 10°6 | Both removed .................. 


Difference .............+: 19 Difference ...........0008 18 


400 Mr. K. H. Kingdon on some 


These figures clearly indicate that the carbide was slightly 
radioactive. Any trace of radium emanation present would 
make itself very apparent in the gas generated from the 
carbide, and it is to the emanation that the high value of 
the residual ionization previously obtained was due. 


II. On the possibility of a Portion of the Residual Lonization 
in Gases being due to the Collisions of Thermal Agitation. 


Introduction.—In the first part of this paper the three 
possible components of the residual ionization in gases were 
stated to be :—(1) a radioactive impurity in the gas, (2) a 
radioactive impurity in the walls of the ionization-chamber, 
(3) ionization by the collisions of thermal agitation. Now, 
for the gases air, carbon dioxide, hydrogen, and nitrous oxide, 
McLennan and Treleaven* have shown that if a clean zine 
ionization-chamber is used, and if the gases are properly dried 
and filtered before admitting them to the chamber, uniform 
and reproducible values of the residual ionization are obtained 
for each gas. Hence it does not seem probable that the 
residual ionization is in general due to the presence of traces 
of radioactive emanation in the gas. The part played by the 
second of the above-mentioned components has been quite 
fully investigated. Recently McLennan and Murray f have 
shown that by constructing the lonization-chamber of ice, it 
is possible to obtain a very low value for the residual 
ionization in air. Their experiments show the great effect 
of radioactive impurities in the walls of the chamber on the 
ionization, for with one ice-chamber the number of ions made 
per c.c. per second was 2°6, while with another the number 
was 5°5. Although from these experiments it appears pos- 
sible that the residual ionization, in air at least, may be due 
entirely to radioactive impurities in the walls of the ionization- 
chamber, yet it seemed worth while to make some experiments 
to test for the presence of ionization due to the collisions of 
thermal agitation. The method was to vary the temperature 
or the density of the gas in the ionization-chamber, and from 
the resulting changes in the ionization to see if it was possible 
to detect the presence of any ionization produced by the 
collisions of thermal agitation. The results of the expe- 
riments in which the density of the gas was varied show 
that only a part of the residual ionization can be due to the 
collisions of thermal agitation. The results of the expe- 
riments in which the temperature of the gas was varied give 


* Loe. cit. 
+ J.C. McLennan and H. G. Murray, Phil. Mag. xxx. Sept. 1915. 


Meperiments on Residual Ionization. AOL 


mdications of a small number of ions per c.c. per second 
produced by the collisions of thermal agitation, but more 
refined experiments are required to confirm this point. In 
addition a formula has been derived for the number of thermal 
collisions in a gas per c.c. per second producing ionization 
which agrees with the experimental results if the number of 
such collisions is small. 

Theory.—The question of ionization by the collisions of 
thermal agitation has been investigated theoretically by 
Langevin and Rey *. In this paper the authors obtained an 
expression for the operas of collisions in a gas per c.c. 
per second for which the relative velocity of ane colhding 
molecules normal to the sphere of shock was greater than an 
arbitrary standard. If we denote fe number of these 
“ effective” collisions by K, then 


eu —tihme? 
K —yer72 hme : 
where y=total number of collisions per c.e. per second, 
fee dye 2 02 x 10-" 
— fel” PN ONOU KS SSS Ways ab 5 
v =arbitrary minimum velocity. 


According to this formula K would vary very rapidly 
with the temperature, a prediciion which is contradicted by 
experiment. 

Hxeeption was taken to Langevin’s work by Wolfke f, 
who suggested that the potent factor in producing ionization 
at the collision of two molecules was not their relative 
velocity normal to the sphere of shock, but rather their 
relative velocity tangential to it. Indeed he suggested that 
the normal component would rather prevent ionization by 
pushing the electron further into the atom, although it is 
difficult to judge of the value of this suggestion on account 
of the very oneal nature of our ‘knowledge of the 
mechanism of an atom. However, on this ground Wolfke 
suggested that the number of effective eollneious would 
depend on the relative velocity of the molecules normal to 
the sphere of collision being less than a certain value, v. 
The formula obtained for the number of effective shocks is 


pole ensety). 
where the symbols have the same meaningas before. I*rom 
this Woltke calculated that if the collisions in air produce 


* Langevin and Rey, Le Radium, x. p. 142 (1918 
¥ Wolfke, Le Radium, x. p. 265 (1913). 


Phil. Mag. 8. 6. Vol. 32. No. 190. Oct. 1916. 25 


402 Mr. K. H. Kingdon on some 


4 ions per c.c. per second at 17° C., they will produce 2 at 
130° C., and 6 at —20° C. It should be noted that the 
above formula only includes the “negative” condition for 
an ionizing collision, 27. e¢. the normal velocity must be below 
a certain value. A factor representing the “‘ positive ” con- 
dition should also be introduced, 7. ¢. the tangential velocity 
must be greater than a certain value. To do this we may 
proceed as follows :— 

The expression for the total number of collisions per e.c. 
per second is obtained by Boltzmann as follows. We assume 
the presence of two kinds of molecules of masses m and m, 
respectively ; » and 2, are the numbers of each kind per c.c,; 
dw and d@, represent the products of the velocity components 
for each kind; and /, 71, represent for the two kinds of mole- 


cules the values of the function 


3733 
n ie mV" p— hme? 
ps 


The conditions of a collision between a molecule m and a 
molecule m,; can be characterized by the two parameters 
6 and a defined as follows (fig. ta) :— 


Fig. 4 a. Fig. 40. 


M, is the centre of the molecule of mass m,. The molecule 
of mass m moves with a relative velocity g parallel to M,G. 
and the projection of the centre of this molecule on the plane 
P drawn through M, perpendicular to MjG hes at M. The 
line M,Q represents the intersection of the ,pianes P 
and GM,X. ‘Then M,M=é, and the angle MM,Q=a. The 


number of collisions per c.c. per second is then 
v=\ffigbdo dw, dbda; 
or integrating for a from 0 to 27, 


p= 2ur | bdb\ gffido do. 


EHeperiments on Residual Ionization. 403 


Langevin has carried out four of the remaining integrations 
in such a way as to obtain the result 


Da? 2 a (Ce ae 

= j € ) 

Ss ? ( bab \ gre 2 adg, 
hm Zhm }o Jo 


where o isthe radius of the sphere of action of the molecule 


and 
reat hems 
ie T° 


If now we let 0 (fig. 4.0) be the angle between g and the 
normal to the sphere of action, then v the velocity normal to 
this sphere is equal to gcos@ and b=csin@. Making these 
substitutions in the above integral, we obtain for the total 
number of collisions per c.c. per second for which the relative 
velocity normal to the sphere of action is less than a certain v, 


7 
2 


ue = a ° hmv2 
2a k?o7 a s@) =— == 
L= 3) inn 2 cos2@ 16 
———— 5 var He ( 
hm 2him Jo PEC OSC, 
2m k?0? 7 ({? sa 
= = —— ¥ ye army 
» ») C 
fe i Z2hm } > 


31.2.2 
—_— SoS). oe (1 — e-ahn?) 
hem? 2hm 


ee fl 
= 7)20 N/ a 


ip —thmv? ) , 


To obtain the number of collisions per c.c. per second for 
which the relative velocity tangential to the sphere of 
collision is greater than a certain u, put w=gsin @, and 
6=csin @ in the expression for vy above. Then the number 
of such collisions is 


2 k?G* wien. -( 2icos 
M= ——— Bers) ae —a € 2sin20d0 
hm 2hm Ny Jo sin’é 


Da kic? oT Aico) : : 
—— oe hoe Ue 2eMre™ du 
h?m* Zh), 


el 2a h2G? ea 
hem? 2hm 


a 
=m’ oc” ve ea akmu2 
Asm 


ean? 


2B 2 


404 Mr. K. H. Kingdon on some 


Hence the probability that for any collision the relative 
velocity of the molecules normal] to the sphere of action shall 


be less than v is (L—e72""”), since n’o? nf is the total 


number of collisions per ¢.e. per second. Similarly the pro- 
bability that the relative velocity tangential to the sphere of 
action shall be greater than wu is e-2*™°, Therefore the 
probability that both these conditions are fulfilled for a par- 
ticular molecule is 


a oa BG eee 


and the number of such collisions per c.c. per second will be 
N=n’°-c" 2T yam? (1 p—Bime?), 
hm 

Euperiments.—The only experimental work which has 
been done on this subject is by Patterson *, and by Devik fT. 
In Patterson’s work the gas was contained in an iron cylinder, 
and as iron usually contains some radioactive impurity, the 
- number of ions generated per c.c. per second was quite large 
(i= 6). Ele failed to detect any effect of temperature on 
the ionization up to 400° C., but it is possible that the effect 
might have been masked by the largeness of the currents 
measured. Also, as the air in the receiver was always at 
atmospheric pressure, its density would decrease as the tem- 
perature was raised; this decrease in density would decrease 
the ionization current due to the earth’s penetrating radiation 
and also that due to a radiation of the 6 or y type coming 
from impurities in the walls of the chamber, both of which 
form part of the total current measured. 

In Devik’s experiments the gas was momentarily heated 
by an adiabatic compression, and the ionization measured at 
the moment of greatest compression. The only gas which 
showed any signs of ionization caused by the high temperature 
(estimated at 900° C.) was antimony hydride. 

In view of the methods of the above experiments, it was 
thought worth while to carry out another investigation in 
which the following conditions should be satisfied :—(1) the 
ionization-chamber should be airtight: (2) the residual ioni- 
zation should be as low as possible so that any change would 
make itself more apparent; (3) the temperature should be 
kept constant during the time of each reading. Unfortu- 
nately, in order to fulfil these requirements the range of 


* Patterson, Phil. Mag. vi. p. 231 (1903). 
+t Devik, Cie ob 1. Heid. Akad. Wiss. xxiv. (1914). 


Haperiments on Residual Lonization. AO5 


temperature had to be reduced considerably. The zine 
ionization-chamber previously described was used. It was 
covered with thin asbestos, then wound with nichrome 
resistance-wire, and packed in magnesia. By passing 
currents up to 1°5 amperes through this wire, the chamber 
could be maintained at any temperature between 10° and 
100° C. for as long a time as desired. The chamber was 
absolutely airtight, the wax joints around the electrode 
and guard-ring being kept cool with a _ water-jacket. 
The temperatures were calculated from the changes in 
pressure. 

Variation of Ionization with Pressure.—The gases used 
were carbon dioxide, acetylene, and hydrogen. Several sets 
of readings were taken with each of these gases at room 
temperature to show the connexion between ionization and 
pressure. The readings and curve (fig. 5) shown were 
obtained with carbon dioxide, and are typical of the others. 
The ionization showsa slight maximum at 650 mm. pressure, 
due presumably to a soft radiation from the walls of the 
chamber. 


p (mm.). Nd. 

764 17-0 | 
726 166 | 
667 16-7 | 
626 165 | 
579 16°0 | 
519 14:2 
450 pal 
366 i 6 
262 89 
141 5d 

60 56 


We may proceed as follows to see whether this curve 
gives any indication of the presence of lonization by collision. 
The possible components of the ionization are—(1) the ioni- 
zation due to the earth’s penetrating radiation, which from 
the experiments of McLennan and Treleaven will be about 
5°1 ions per ¢.c. per second at 760 mm. pressure; (2) that 
due to a possible ionization by thermal collisions, which from 
the same experiments cannot be more than about 4°8 ions 
per c.c. per second at 760 mm. pressure; (3) that due to any 
radioactive impurity in the walls of the receiver. Now 
component (1) will vary directly as the pressure, and may be 


406 Mr. K. H. Kingdon on some 


represented by the straight line OP in the figure. If we 
diminish the ordinates of OA by the corresponding ordinates 
of OP we obtain the curve OB, which is the ionization- 
pressure curve for the remaining components. 


Fig. 5. 


_—— - 
3 a a 


eoee GeGne GHGKauauancunnE Seoeeeegeusesnesuanesuasa: i 
FEPeEEEEeEEeECEeEe eee cee teeeeeeeee 
ean Perret z Pore ror i BpaneaaeReaenat 
1S Hee HEE scesaee oy Poet a 
r oa! y a GSES es Paes Beet Y 
Berea Geaee cage” EuEEESUEECGEESEE pee be oh 
SEEEESoart aastecstesteeessstes fess east eonttesst ecst esti tevattecstfest?=#z¢ 
Hy S0008 SERS8 SE000RREEE CORRE SEER ESSRR SH ——~ aan 
—— Ty aes oe ! Baa : Si — SSS 
Suzan oonea bates cuueecnabssnonsonnesnni=1_ases Cenceenenetttt 
BEEEEEEH Pocus cevo4 coats aosaasaaas soeaeesaes vatsaeeces bares snaesiatsessaes 
BRGGE SogEH Seney SoSeGHEGES SocueSEaEe QeEoeeubee Gunna dune seaunbeage cece: 55555 
rot Aap SSGG BEGGS GES 2 anEGGSGEGSERSGSoeons scoe 
Elia tiiirti tenntipctnnin iment 
aoe s 
on a 
is 


sessrasasdint idee avparcarett cess eeetittet 


Zi, 
TaeEs PAP USE Ee Sees Sn SEoEEGEeeE 
JERE 4p SSSEE PSSER SEES ESSE! 

i, 


ae a 
Re ee 
‘Bi 2eGer ohn 


No of fons per c.c per second. 


sed ibvsccscevevetseaeresetarsarerstesrsreteerereerere 


6 160 260. .64060)5400 500 600 700,maeE 
Pressure in mm. of Hg 


Again, on the above theory of ionization by thermal 
collisions, the number of ions produced per c.c. per second 
varies as the square of the number of molecules per e.c., and 
therefore as the square of the pressure. Hence the pressure- 
ionization curve for this component may be represented by 
the curve OQ. By diminishing the ordinates of OB by the 
corresponding ordinates of OQ, the curve OC is obtained, 
which is the pressure-ionization curve for the radiation from 
the walls of the chamber. This curve shows a very pro- 
nounced maximum ata pressure of 550 mm.; and it is evident 
that if it is possible for a curve such as OC to represent 
correctly the pressure-ionization curve for a soft radiation, 


Haperiments on Residual Ionization. AQT 


then it is possible that about 9 ions per c.c. per second are 
produced in the gas by the collisions of thermal agitation. 
Now it isa well-known fact that when a radiation produces 
ions in a gas, the total number of ions made is the same so 
long as the radiation is totally absorbed in the gas ; also the 
effect of recombination will be least, and ther fore the 
current will be greatest, when the gas is at such a pressure 
as Just to absorb the rays. Hence it is possible that for a 
certain range of pressures ‘the current will increase as the 
pressure tear eases, on account of the effects of recombination. 
The conditions of this experiment are particularly favourable 
for a large recombination effect, since the direction of the 
rays 1s perpendicular to the direction of the electric field, 
and also since carbon dioxide isa heavy gas. Yet in spite 
of these arguments it seems improbable ‘that the effects or 
recombination could cause a rise from 7 to 9 ions per ¢.c. per 
second, or an increase of about 33 per cent. in the current, 
as is the case here. These results would therefore seem to 
show that the residual ionization cannot be wholly due to the 
collisions of thermal agitation. They cannot, however, be 
said to exclude the possibility that a smaller number of ions 
than 5 per c.c. per second may be due to these collisions. 
Variation of Lonization with Temperature —The gases used 
were carbon dioxide and acetylene, and the range of tempe- 
rature was from 18° to 100°C. Great difficulty was ex- 
perienced in getting reliable sets of readings; for as each set 
required a period of from six to eight hours, it was quite 
possible that the leak of the electrometer might vary during 
this time. As a determinat*on of the leak required that the 
ionization-chamber be exhausted, and as also it was found 
that in order to obtain consistent readings the gas had to be 
allowed to stand in the chamber for some time before com- 
mencing readings, it was only possible to obtain one leak 
reading for each set of temperature readings. All the sets 
of readings, however, agree in showing that the ionization is 
practically unchanged from 18° to about 80° C. For tempe- 
ratures from 80° to 100° some sets of readings show a very 
marked increase in the current, while in others this increase 
is very small. The two chief sources of possible error would 
appear to be thermo-electric currents, and the driving off of 
minute quantities of radioactive emanation from the walls 
of the ionization-chamber as its temperature was raised. 
These errors were guarded against by eliminating possible 
thermo-junctions, and by exhausting the ionization-chamber 
while it was heated to 100°. To test for the presence of 
thermo-electricity the junction of the brass electrode and the 


408 Heperiménts on Residual Lonization. 


brass connecting wire at the top of the ionization-chamber, 
which junction was the only one that could possibly serve as 
a hot junction, was heated with a small flame, the fame was 
removed, and the earth connexion to the quadrants of the 
electrometer broken, but no change could be detected in 
the normal ionization current flowing to the electrometer. 
It seems likely though that the rise in current sometimes. 
noted at temperatures of about 100° was due to small 
quantities of emanation being driven off from the walls of 
the chamber. 

It might be interesting to see if the proposed formula for N 
agrees at all with the results of the soepore iture experiments. 
The experiments on pressure show that the number of ions 
produced per c.c. per second in carbon dioxide at 20° C. and 

760 mm. pressure is probably less than jour. In the ex- 
pression for N we have then to assign values to u and v so 
that the following conditions may be fulfilled: 

(1) N must be about 4 at 20° C. and 760 mm. pressure. 

(2) N must change slowly with the temperature, at least in 

the region of 20°. 

The only physical condition suggesting itself which will 
fulfil the above requirement is that, for a collision to produce 
ionization, it must be almost perfectly tangential (this will 
make the total number of such collisions small), and that the 
arbitrary minimum tangential velocity of each of the colliding 
molecules must be about equal to the most probable velocity 
for a temperature of 20° (this ensures that N shall change 
slowly with the temperature in this region). Then for carbon 
dioxide, if we put the minimum relative tangential velocity 
i= D2 Xo AS X LO ema per second, and the maximum 
relative normal velocity i =8-97 x 107!" cm. per second, we 
find that at 20° C., N=4: and at 100°, N=5-1. - Wha 
changes very slowly : as the temperature is raised, which is in 
qualitative e agreement with the experimental resuits. A more 
exact application of the formula for N does not seem worth 
while at the present time, since, for the reasons stated above, 
the accuracy of the readings does not warrant it. It may, 
however, be of interest to note that using the above values 
for wu and v, and making changes in h, p,and 7 to correspond 


te) 
to the rise in temper ate e, at 302° C. the value for N is 6:4. 


Summary. 


(1) It has been shown that the high residual ionization 
in acetylene prepared from calcium carbide is due to the 
presence of slight traces of radium emanation. 


Measuring Refractive Index and Dispersion of Glass. 409 


(2) Tt has been shown that only a portion of the ultimate 
residual ionization in gases can be due to the collisions of 
thermal agitation. 

(3) A formula has been devised for the number of collisions 
per c.c. per second producing ionization which is in qualitative 
agreement with experimental facts. 


In conclusion the author wishes to express his thanks. 
to Professor J. ©. McLennan, who suggested the problem, 
and whose assistance and encouragement have been most 
valuable. : 

Physical Laboratory, 

University of Toronto. 
May 8, 1916. 


XLV. A New Method of Measuring the Refractive Index and 
Dispersion of Glass in Lenticular or other forms, based 
upon the “ Schlieren-methode” of Tépler. By R. W. 
CHESHIRE, B.A.* 


HH methods most commonly employed for the accurate 
refractometric examination of optical glass for com- 
mercial purposes involve special preparation of the specimen 
under test. The Pulfrich refractometer, which is probably 
used in optical workshops more frequently than any other 
instrument, is designed to give an accuracy in the deter- 
mination of n, of one unit in the fourth decimal place, and 
in the partial dispersions of two or three units in the fifth 
decimal place. Asa preliminary to the examination on this 
instrument the glass must be cut in the form of a right- 
angled prism and, of the two faces forming the right angle, 
at least one must be plane and well polished and the other, 
in order to secure good definition through the telescope, 
must meet it in a sharp knife-edge. The commercial 
aceuracy of determination in the optical constants specified 
above is sufficient for practical designing purposes, and is, 
In any case, greater than the accuracy with which the 
refractive index of successive meltings of glass can be 
reproduced on the large scale. 

It may happen, however, that one is confronted with the 
problem of determining the optical constants of a piece of 
glass in a form in which it is not possible to examine it on 
the refractometer. The glass, for instance, may be a small 
component lens forming part of a microscope objective and 


* Communicated by the Director of the National Physical Laboratory. 


410 Mr. R. W. Cheshire on a New Method of Measuring 


too small therefore to permit of a prism being cut from it 
with which to conduct the ordinary examination; on the 
other hand, it may be considered undesirable fo femeemee 
injure the lens in any way which would militate against its 
subsequent use in an optical system. One of the most 
obvious methods of solving the problem is to measure one or 
more of the optical lengths associated with the simple lens, 
the principal focal length or the back focal length for some 
specified wave-length, “and to supplement this information 
with measurements of the radii of curvature of the lens 
surfaces and the axial thickness at the vertex. A knowledge 
of these quantities is theoretically sufficient to enable one to 
caleulate the index of refraction vf the class for the wave- 
length employed, but a little consideration will suffice to 
show that the limitations of practice restrict the accuracy 
possible in the determination of the refractive index to a 
figure appreciably higher than one in the fourth place of 
decimals, with a corresponding greater relative inaccuracy 
as regards the partial dispersions. In the first place, the 
ordinary type of spherometer in which the three fixed points 
lie on a circle of diameter 1} in. to 2 in. is, in many cases, 
useless for the measurement of the curvatures of the surfaces. 
The small lenses used in the construction of microscope 
objectives, eyepieces, and the smaller sized telescope ob-~ 
jectives require to be examined on a special optical sphero- 
meter designed to deal with these small apertures. Various 
optical devices involving the use of the lens surfaces as 
reflectors have been suggested and empioyed from time to 
time, but none of these methods aim at or secure an accuracy 
of the order of 1 in 5000 which is the present desideratum. 
And it should further be remembered that even this accuracy 
is not sufficient for the determination of the partial dispersions. 
Secondly, as regards the determination of focal lengths or 
of back focal lengths, the presence of spherical aberration in 
the simple uncorrected lens ver y seriously limits the reliance 
to be placed upon the observations. In the most favourable 
case, that of a convexo-plane lens, the ratio of the longitudinal 
spherical aberration for the extreme ray to the focal length 
may be taken to be approximately represented by (///)?, 
where 7 denotes the focal leneth and h the semi-linear 
aperture of the thin lens. But,in order to secure reasonably 
good definition and an absence of depth of focus, a semi- 
aperture of at least 7/40 is advisable, corresponding to axial 
spherical aberration amounting to about 1 part in 1500. 
When all these factors are taken into consideration it will 
be seen that the accuracy to be expected in the determination 


the Refractive Index and Dispersion of Glass. 411 


of n, is of the order of one in the third place of decimals 


rather than one in the fourth. 

The method here described was devised with a view to 
obtaining a direct measurement of the refractive index of 
the glass, thus obviating the necessity for making intermediate 
observations which, by their unreliable nature, greatly limit 
the accuracy finally obtainable. The rationale of the method 
consists in immersing the glass under examination in a 
liquid whose refractive index fer the specified wave-length 
ean be varied continuously until an equality is attained 
between the index of the glass and that of the surrounding 
liquid. The ordinary process can then be applied to measure 
the refractive index of the liguid on the Pulfrich refracto- 
meter. It should be noted that the method may he employed 
for the refractometric examination of glass in any irregular 
shape, and not solely of glass in lenticular or prismatic form. 
It is not claimed that the method is an absolute one for the 
determination of the refractive index of the glass, as the 
ultimate accuracy is, of course, that set by the measure- 
ments on the refractometer; but it is possible with care to 
equalize the refractive indices and partial dispersions of the 
glass and the immersion fluid to at least as high an order 
of accuracy as that with which they can subsequently be 
measured on the refractometer. 

The ‘‘Schlieren-methode”” invented by Topler and de- 
scribed in detail in Wied. Ann. vol. cxxxi. p. 33 (see also 
R. W. Wood, ‘ Physical Opties,’ p. 78) affords an extremely 
sensitive means of detecting the presence in an otherwise 
homogeneous transparent medium of strize or other optical 
non-homogeneity, and is therefore well suited to the present 
purpose. The arrangement of the apparatus employed is 
shown diagrammatically in plan in fig. 1. 


Fig. J. 
1g’ 
| 
Cc IE | 
A D ieees, 
i } ir ae i 
B H 
K Ki 


A vertical straight edge behind which is placed a source 
of monochromatic light A is represented in plan by B. An 
image B’ of this straight edge is projected by means of a 
telescope objective C (focal length 125 em., aperture 8 em.) 
into the plane of a second straight edge, F, disposed as 
shown in front of an observing telescope H, so as to cover 
about one-half of the full aperture of its objective G. The 


412. Mr. R. W. Cheshire on a New Method of Measurina 


various parts of the apparatus are mounted rigidly in any 
convenient manner, and the two edges and the observing 
telescope E should be approximately centred with the axis 
of the principal Jens C. The observing telescope, situated 
at a distance of roughly 5 metres from the lens C, has an 
aperture of about 3-5 cm. and a magnitying power—in 
normal adjustmeni—of x24. The edge F is secured toa 
rack-and-pinion motion providing for a slow transverse 
movement across the front of the observing telescope in a 
right- or left-hand direction. The desired parallelism of the 
two edges may be checked either by removing the eyepiece 
of the observing telescope and looking down the tube, or by 
examining with a small pocket magnifier the images of the 
two edges projected into the Ramsden circle of the: telescope. 
A small rectangular glass cell, H (see below), containing the 
immersion fluid and the elass under test, K, is adjusted in 
the usual way on the prism, D, of the refractometer (not 
shown in diagram), which is placed just in front of the 
lens C and at such a height that the cell is approximately 
opposite the centre of the lens. Suppose, for the sake of 
simplicity, that a prism of small angle is being examined 
and that it is immersed in the fluid contained in the cell 
with its refracting edge vertical. In general the refractive 
indices of the olass prism and the surrounding liquid will 
differ, and the glass will therefore exert a prismatic effect 
upon light passing dive it. The image of the edge B 
will therefore present a doubled appearance, the image 
formed by light passing through the glass being displaced 
laterally by a small amount relatively to the rest of the 
image ‘depending upon the magnitude of the difference 
between the two indices. Consequently, on traversing 
slowly the edge F across the objective of the observing 
telescope, which will be supposed to be focussed upon the 
prism, it will be found that the surface of the prism gradually 
darkens and finally becomes completely black, whilst the 
side faces and the rest of the cell will show up brightly. 

On continuing the steady motion of the edge F these faces 
also will begin to darken, until finally no light passes into 
the observing telescope and the whole field ee black. 

The order of these two phenomena is determined by the sign 
of the difference of the refractive indices of the fiuid and 
prism. If now the refractive index of the immersion fluid 
be changed somewhat, it will be at once obvious, from the 

relative interval at which these phenomena occur, whether 
the difference in the refractive indices has been increased or 
decreased by the alteration. In the course of a few trials it 


the Refractive Index and Dispersion of Glass. 413 


will be found possible so to adjust the refractive index of 
the liquid, that on moving the edge slowly across the 
observing telescope the whole field, including the faces of 
the prism and those of the cell to the right and to the left 
of the prism, darkens simultaneously and uniformly. Under 
these circumstances the prism clearly no longer exerts any 
prismatic action—in other words, an equality between the 
indices has been attained, and a reading for the refractive 
index of the liquid as it stands upon the Pultrich prism may 
be taken at once. 

Tt will be convenient now to describe the form of cell and 
the immersion fluid used in these experiments. Before 
doing so, however, it may be worth while to point out what 
are the conditions which it is desirable should be satisfied. 
In the first place it must be possible for light to pass into 
the cell, to strike the upper surface of the base-plate at 
grazing incidence and to be refracted down through the 
base-plate and thence into the prism of the refractometer. 
It will be evident from elementury geometrical consider- 
ations, that if the width of the base-plate be equal to the 
aperture of the Pulfrich prism upon which it stands, then, 
owing to the finite thickness of the base-plate, a considerable 
portion of the aperture of the Pulfrich prism will not be 
filled with hght unless the glass forming the base-plate be of 


C7 


b 


high refractive index. A glance at fig. 2, giving a side 
view of the cell in position upon the Pultrich prism, will 
explain the reason for this at once. 


Of the full aperture AB of the Pulfrich prism it will be 


414 Mr. R. W. Cheshire on a New Method of Measuring 


seen that the section represented by BC only is effective. 
It would of course be possible to utilize the full aperture AB 
of the refractometer prism by making the width of the cell 
appreciably greater than the full aperture which it is desired 
to fll, but, on the other hand, this change would increase 
the thickness of the layer of immersion fluid through whieh 
light has to pass. Unfortunately all the suitable immersion 
fluids absorb very strongly light at the blue end of the 
spectrum, and it is therefore of importance to keep the width 
of the cell as smalias possible. Lhe refractive index of the 
glass forming the base-plate must also be greater than that 
of any liquid which it is proposed to examine in the cell. 
Practical constructional difficulties rendered it inadvisable 
to diminish the thickness of the base-plate of the cell below 
about 2 mm., and this was therefore the dimension finally 
adopted. With this thickness and a base-plate of extra 
dense flint for which n,=1'74, no trouble was experienced 
in obtaining sufficient light and aperture to give good 
settings w ith the refractometer up toa value tip =I 66 for 
the fluid contained within the cell. The deposition of dirt 
in the right angle formed by the base-plate and the sides, 
or the presence there of a slight trace of the cement used in 
building up the cell, would ruin the sharp definition at the 
edge of a line seen through the telescope of the refracte- 
meter and it was therefore considered desirable to chamfer 
the inner edges of the base-plate over a small fraction of a 
millimetre, as shown in fig. 2. The upper and lower faces 
of the base-plate must be as nearly as possible plane parallel, 
and in the cell built up for the purposes of this investigation 
by Messrs. Adam Hilger, the plane parallelism of these faces 

was guaranteed to an accuracy of 5". 

The side faces of the cell through which passes the light 

used to determine the equality or otherwise of the indices of 
the fluid and the glass under test should be plane polished, 
but it is not necessary that they should also be parallel: and 
in order to secure good optical contact between the lower 
surface of the base-plate and the Pulfrich prism, it is 
desirable that the former should project very slightly below 
the level of the side faces of the cell, as shown in 1 the figure. 
The inside dimensions of the cell as actually constructed 
were 9X 2x 1°5 cm. 

The immersion fluid finally selected for use in the cell 

was an aqueous solution of mercury potassium iodide, more 
commonly known as Thoulet’s solution. An excellent 
account of the preparation and preperties of this solution is 
given in Johannsen’s ‘Manual of Petrographic Methods,’ 
where also will be found a very comprehensive table of 


Percentage transmission 


the Refractive Index and Dispersion of Glass. 415 


other useful immersion fiuids. In its most concentrated 
form the refractive index of the solution for the D line is 
1-72,and the index may be varied continuously down to 1°33 
by admixture with increasing proportions of water. The 
solution possesses the great advantage from the practical 
point of view in that the excess of water may be driven off 
by heat and the original solution with a high index of 
refraction recovered again. Many of the mixtures that 
have been suggested for the purpose of obtaining a solution 
of variable refractive index involve the employment of two 
liquids other than water~which cannot be separated and 
recovered in this way after use. The double iodide possesses 
the further useful property that it does not attack the 

Canada balsam used for cementing together the walls of the 
eell. The main drawback to the use of the solution for the 
present purpose lies in the fact that it exercises marked 
absorption at the more refrangible end of the spectrum. 


Rigo: 


to) 


a ee 


| 
= oe 


S50 yu S00 up S550 pu 600 yu 650 uy 
Wave length 


The transmission curve tor light of various wave-lengths 
passing through the cell filled with Thoulet solution of 
refractive index p= 1°51 was obtained by means of a spectro- 
photometer, with the result shown in fie. De 

It will be seen that the percentage transmissions for the 
four customary lines C, D, F, and G' are approximately 70, 


416 Mr. R. W. Cheshire on a New Method of Measuring 


63, 16, and 2 respectively. The transmission curve shown 
in the figure relates to the Thoulet solution actually employed 
in the cell for these refractive index determinations. Owing 
to a tendency of the iodine to separate out from the solution 
the tint of a solution of given index is apt to vary and to 
pass from a yellowish-green to a brown colour as this 
separation increases. The corresponding transmission curves 
will therefore change slightly, but will retain the same 
general characteristics as regards the spectrum distribution 
of intensity in the transmitted light. The original yellow- 
green solution may always be recovered again by warming 
the brown solution in the presence of a little clean mercury. 
In common with all liquids the temperature coefficient 
of the refractive index of Thoulet’s solution is very high, 
and is of the order of —0-0006 per degree centigrade. lt 
might have been expected that this circumstance would, 

itself, preclude the possibility of obtaining any s ioninel 
result to five figures, but in practice no difficulty is 
experienced if the apparatus be shielded from direct currents 
of air. After a balance has been obtained through the 
observing telescope for the equality of the indices, an 
interval of not more than a minute need elapse before two 
or three readings have been taken for the refractive index 
of the solution, and throughout this interval the solution 
will not be found to have chap nged in refractive index to an 
extent which is appreciable on the Pulfrich refractometer, 
provided that the solution has previously been allowed to 
attain the room temperature. In any case two independent 
observers could, if desired, take simultaneous readings, one 
on the refractometer, whilst the other views the liquid in 
the cell through the observing telescope. A much more 
serious difficulty to be met arises from the unequal evapor- 
ation at the surface of the liquid and from the walls of the 
cell, producing local changes in concentration and giving 
rise to the presence of striee, which eventually permeate the 
whole mass of the fluid and assume most fantastic forms 
when seen through the observing telescope with the rest of 
the field dark. Topler in his” original paper describes 
these striz and gives an excellent ‘reproduction of their 
appearance. The growth of these beautiful but very un- 
desirable strize within the fluid may be prevented by pro- 
viding the open rectangular cell with a glass cover-plate on 
the under side of which has been smeared a little vaseline. 
This has the effect of excluding the air, and after a few 
moments equilibrium is restored “between the vapours in the 
cell and no further local changes of concentration take place. 


the Refractive Index and Dispersion of Glass. 417 


The ordinary soda-fHame produced by holding a piece of 
common salt in a Bunsen flame will be found to give 
sufficient light to enable accurate observations to be taken 
on the sodium line. If more light should be required for 
any purpose, an apparatus designed to produce a very 
intense source of sodium light and described by H. EK. 
Armstrong in the Proc. Roy. Soc Vols lxxocen (IOS) as 
recommended. It will be found convenient to provide two 
sources of light, one to be used for obtaining the equality of 
the indices, and the other to be used in conjunction with the 
prism-condenser of the refractometer, which is of course 
swung out of action whilst the first source is being used, to 
determine the refractive index of the liquid as soon as the 
equality has been obtained. Itis a matter for regret that it 
has not hitherto been found possible to determine the 
refractive indices of the glass for the other standard refracto- 
metric lines C, F, and G! to the same satisfactory order of 
accuracy that is possible with the D line. The intensity 
of the G' line in the hy drogen spectrum given by the usual 
H-tubes used in refractometry is not great, and the line is by 
no means an ideal one to use at the best of ime. Moreover, 
as the curve shown in fig. 3 indicates, the immersion fluid 
itself transmits only a negligible percentage of light of this 
wave-length. This line therefore must be regarded as quite 
useless for our present purposes. The position as regards 

the C and F lines is not quite so unsatisfactory, but even in 
this case the intensity of the light given by the hydrogen 
tube is relatively feeble and is not calculated to give results 
with any certainty. It should be Rermenmn dened that the 
adjustment for equality of indices is really a photometric 
one, and consequently far more light is required than in the 
ease of the usual refractometer readings where the cross- 
lines in the eyepiece of the refractometer telescope are 
brought into coincidence with the edge of the line under 
examination, and the setting is rather of the nature of a 
geometrical or positional one. The separation of the C ana 
F lines is effected by means of coloured filters, the red filter 
employed transmitting about 89 per cent. of the © line, and 
the blue filter about 56 per cent. of the F line. The most 
satisfactory sources to be used for measurements on the 
partial dispersions would appear to be the mercury-vapour 
iamp for a green line X=546pum, and the recently de- 
seribed cadmium are (H. J. 8. Sand, Phys. Soc. Proc. 
Xxviil., 1916) for the red cadmium line A=644uy. The 


iz hides serG, Vol, 32. Now l90. OGi holo. 2¥ 


418 Mr. R. W. Cheshire on a New Method of Measuring 


mercury-vapour lamp used in conjunction with the Wratten 
mercury monochromatic filter provides all the light that is 
required for the purpose. 

Experimental readings have been taken on a small prism 
with a view to determining the sensitivity of the method 
for the D line. The glass cell was first cemented on to the 
Pultrich prism in the usual fashion with a solution of barium 
mercuric iodide (n»=1'79) and adjusted so that the fringes 
observed in monochromatic light ran parallel to the direction 
of the incident grazing light. The prism under test was 
then temporarily cemented to the under surface of the cover- 
plate with its refracting edge vertical, and the cell was filled 
with Thoulet solution to such a height that the prism was 
immersed in the liquid to a depth of about 5 mm. The 
vaselined cover-plate being appreciably larger than the open 
top of the cell could then be moved to and fro, and the 
mixture within the cell well stirred by the prism without 


admitting air and thereby disturbing the equilibrium and 


rap) 
concentration of the immersion solution. A glance through 
the observing telescope was sufficient to discomen whether 


the stirring had been carried tar enough and whether the 
solution was perfectly homogeneous. The cover- plate was 
then cautiously withdrawn 2a little distance to allow a few 
drops of the concentrated Thoulet solution to be dropped 
into the mixture contained in the cell, the mixing process 
was repeated, and observations were then taken through the 
observing telescope to determine whether the difference 
between the indices had been increased or decreased. It 
was found that after four or five trials the indices could be 
brought so nearly to an equality that one drop of Thoulet 
solution, or one drop of water, was sufficient to change the 
sign of the diference. The final adjustment was then made 
with two solutions, one with an index slightly above, and 
the other with an index slightly below that of the mixture 
within the cell. As soon as the equality had been obtained 
a reading was taken at once on the refractometer for the 
refractive index of the solution, using the second sodium 
flame mentioned above in the ordinary 1 manner. A series of 
determinations of the refractive index of the equalizing 
Thoulet solution was made in this way, and then the cell 
was removed from the Pulfrich prism, and the glass prism 
examined on the refractometer by the usual process and its. 
index determined directly. In this way it was possible to. 
investigate the sensitivity of the apparatus and to check the. 
mean yalue obtained by a direct reading on the prism. 


the Refractive Index and Dispersion of Glass. 419 


Hight separate determinations were made by the immersion- 
fluid method with the followi ing results :— 


| 
| Nuinber of | Observed ity | for ) Difference in fifth 
| experiment. | Thoulet solution. |Plave from mean value.| 
| | 
Tay teat | 151493 | uel 
DOM ane sey era sped) 0 
Beet | 151495 413 
2 RANA GMa | 151488 | wa 
Deven 151491 : aay 
Gime | 1-51489 we 
TONE BS iesisaeroe un nye Hil) 
SiN ey ion | 151491 ET 


Mean value 1:51492. 


Average difference from mean value = ‘00002. 
Value obtained for the index of the prism by a direct 
reading on the refractometer =1°51490. 


An inspection of the resuits obtained shows that the 
indices of the glass prism and the surrounding liquid can be 
equalized to an average accuracy of +°00002. The accuracy 
with which the indices may be equalized is clearly pro- 
portional to the angle of tle prism, or to the angle between 
the cathetal surfaces if the glass examined is not in the form 
ot a regular prism. In the present instance this angle was 
about 10°, and an even higher order of repetition accuracy is 
to be expected in the case of an irregularly shaped piece of 
glass in which the surfaces of the refracting edge chosen 
meet at an angle greater than this. It is evident, then, 
that the ‘¢ Schlieren-methode ” is well adapted for the 
equalizing of the indices to one or two units in the fifth 
place, and is therefore all that is required in commercial 
practice. 

It was found that when the straight edge F had been 
moved into coincidence with the image of the edge B, and 
even after it had overlapped it slightly, the field could not 
be obtained absolutely dark, and a small image of the 
extended light source A persisted in remaining “visible in 
the field of view of the observing telescope. This was 
eventually traced down to an out-of-focus flare-spot image 


2H 2 


420 Measuring Refractive Index and Dispersion of Glass. 


of the source of light formed by the objective C about 
3 metres away from the observing telescope. Suchan image 
will, of course, always send light into the uncovered portion 
of the obj ect-glass of the observing telescope in whatever 
position ihe edve F may happen to be placed. The intensity 
of the light that goes to form a flare-spot image, however, 
is not great and exerts no disturbing effect upon the readies 
It is wells however, to shield off the major portion of the 
extended source of light from the apparatus and, instead of 
a simple straight edge backed by a large source of light, to 
use one of the straig “ht edges in a slit About 1 mm. in wie 
It will be found that fe reduces the general illumination 
to almost negligible proportions. If it be considered worth 
while the intensity of the flare image can be reduced further 
by tarnishing the glass-air sur faces of the lens C along the 
lines pointed out by H. Dennis Taylor in a patent speci- 
fication No. 29561 (1904), entitled ‘*‘ A method of increasing 
the brilliancy of the imaves formed by lenses.” A paper on 
ns same subject by F Kollmorgen in the Transactions of 

the Illuminating Hueineering Society, vol. xi. (1916), might 
also be consulted. ‘The onter zones o a thin positive lens 
may be regarded as built up of a number of small prisms 
oriented with their refracting edges tangentially to the 
circular boundary of the lens, ooh no difticulty is therefore 
to be expected in dealing with such a lens in accordance 
with the method of this paper. Owing to the pressure of 
more urgent We however, it has not been found possible 
Sele te dare. elop the sated with a view to ascertaining 
its accuracy when applied to high-power negative lenses with 
appreciable edge-thickness. It was thought, nevertheless, 
that an account of what has already been accomplished 
might prove of some service to others who might be 
interested in determining all the optical constants of a given 
system. 


The author desires, in conclusion, to place upon record 
his indebtedness to Dr. Glazebrook, the Director of the 
National Physical Laboratory, for the facilities placed at 
his disposal for carrying out this investigation. 


ee aoe 


XLVI. The Photoelectric Effect on Thin Films of Platinum. 
By J. Rosinson, M.Sc., PhD., Lecturer in Physics at 
Kast London College”. 


HE photoelectric current from thin films of platinum 
does not increase uniformly with the thickness, but 
varies f as shown in Curve A, fig. 2. This applies lnecher 
the incident or emergent effect is considered ; 2. e., whether 
the light falls on the platinum film before or ator passing 
through the quartz plate-on which the film is deposited. 
For thicknesses below about 1077 cm., 2. e. from where the 
current is a maximum, the emergent ettect is larger than 
the incident effect both as regards current and velocities f. 
An explanation of the difference in magnitude of the 
incident and emergent currents was offered by Partzsch and 
Hallwachs §, who showed that in the emergent case more 
light is absorbed than in the incident case. This would offer 
a full enough explanation if the magnitude of the effects 
were exactly the same, but it would not explain the difference 
in velocities. Hence Clone was cast on the result that the 
velocities do differ. It has since been shown by Stuhlmann |, 
who used a method similar to the author’s, that the emergent 
velocity is larger than the incident ve locity. 

Partzsch and Hallwachs’ result, that more light is absorbed 
in the emergent case, shows the necessity for Sorter work to 
find :—in the first place, what the asymmetry of the currents 
is when unit quantity of light is absorbed in each case, and in 
the second place, whether for very thin films the absorption 
of light varies with the thickness in a way similar to that 
of the current. 

In the experiments to be described, an attempt was made 
to get some information on these problems. It was con- 
sidered advisable to make the measurements of the absorption 
of light on the same films on which the photoelectric 
measurements were made. 

A DEK is a nearly closed metallic cylinder standing 


en a base DE. There is a small inlet of ‘light aia se he 


* Communicated by Prof. C. H. Lees, F.R.S. 

+ Robinson, Phil. Mag. xxv. p. 122 (1918); Werner, Ark. Mat. Ast. 
Frys. viii. no. 27, p. 1 (1913); Stuhlmann & Compton, Phys. Rey. ser. 2 
ii. pp. 189 & 327 (1913). 

t Robinson, Joc. cit. 

§ Partzsch & Hallwachs, Ann. des Phys. xii. p. 247 (1913). 

| Phys. Rev. iv. p. 195 (1914). 


A492 Dr. J. Robinson on the Photoelectric 


vessel thus acts nearly as a black body, and very little light 
which enters at B can escape again. Thus it is all absorbed 
somewhere in the vessel. © is a platinum film on quartz 
arranged so that a parallel beam of light entering at B falls 


DB) 


on it. Let Ip be the intensity of light entering at B, and 


\ Sei, 

C3 
Wig 
tp) 

gees 00) 
Po 


hence falling on C. A certain proportion of this, alg, is 
absorbed by fae film, Bho is reflected, and yIp is transmitted, 
where «+8+y= 1. The amount of light (8+y)Ip 1s 
absorbed by the ve ee The amount of light eI) is that 
which produces the phote current in the film. We need to 
find 7, the photo current from the film, and elo, the light 
absorbed by the film. The photo current 7, can be meaeunel 
by charging the walls of the vessel to a potential to drive 
the electrons away from the film. 
Suppose that I, is kept constant, and with a mercury- 
vapour lamp it can be kept practically constant. Then 
a=1—(8+y), where (B+y)Io is the amount of light ab- 
sorbed by the walls of the vessel. These walls remain constant 
throughout a series of experiments, and if the photo current 
from the walls is measured by applying a potential to drive 
electrons to the film, this current is proportional to the 
amount of light absor bea by the walls, and hence to (8 +¥). 
Thus « can be obtained in terms of the photo enrrent of the 
walls of the vessel. a@ itself cannot be absolutely determined 
in each case unless some means is found for measuring the 
intensity of the light, or unless another set of experiments 
is made, so that Te can be eliminated from the equations. 


= <= si Ne 


{ee ee ee 


ee 


Effect on Thin Films of Platinum. 423 


This was at first attempted by arranging a screen at 8 of the 
same Seas as the walls of the vessel, so that the relative 
amounts of light ae on the film and the vessel could 
be altered. This was ab bandoned, as it was quite sufficient 
for the present purpose to know how @ varies with the 

thickness. 

A quartz tube was eeneuted to the side tube to let ultra- 
violet light into the vessel. Another side piece, F, had a 
ground joint for the purpose of rotating the film C. The 
rod carrying the quartz plate for the tilm was insulated at 
G by an amber piug. The contact between the film and the 
rod was mide as in a former paper, by getting a thick deposit 
of platinum at an edge of the quartz-plate, laying tinfoil 
over this so as to allow part of the thick platinum film to be 
uncovered, and clamping this prepared edge of the quartz to 
the rod. The films to be tested were deposited over the 
whole of one side of the quartz plate, and only the centre 
of the plate was exposed to the light. The plate was of 
dimensions 10 mm. x 8 mm. x 1 mm. 

The inside of the vessel was coated with soot from burning 
camphor in one series of experiments, and in another series 
consisted of zine. The platinum film was joined to a pair 
of quadrants of a Dolezalek electrometer whose sensitiveness 
was 2000 divisions per volt, when the needle was charged to 
100 volts. ‘The current from the film was measured by 
charging the walls of the reese to a high positive potential, 
and that from the walls by charging them toa high negative 
potential, so that the maximum effects were obtained in each 
ease, 

The following results were obtained :— 


TABLE I. 
| Thickness of Current from Current from the walls 
| filin. film. of the vessel. 
| 00 seconds. 8 51 
hie BO Thy, 22 46:5 
| 130 wy. 235 | 4] 
| DAD) 150 16 


The thickness of the film was measured in terms of the 
length of time to deposit it. The quartz plate was taken 
out of the vessel in each case and placed in a discharge-tube 
with a platinum cathode to have the film formed oz thickened. 


424 Dr. J. Robinson on the Photoelectric 


These results are shown graphically in fig. 2. 


ie 42: 


250 


oO g 
4 = 
9 200 '& 
SL 

E 
gE 9 
3 (50S 

~~ 

& 
w 

CY) 
o 100 $ 
< 5 
5 <5 
© 

SO 
100 200 300 


Time of Deposit of Film ia Seconds 


Curve A.—Pkoto current for various thicknesses of film. 
Curve B.—Photo current from the walls of the vessel. 

It is seen that the current from the film varies with 
the thickness in the way already demonstrated ™ ; 7. e., it 
increases gradually with the thickness at first, then it 
suddenly increases rapidly, after which it falls again. The 
current from the walls of the vessel diminishes uniformly, 
and it is proportional to the light absorbed by the vessel. 
Hence (@8+y)Iy) diminishes uniformly as the thickness of 
film increases, and hence aI, the light absorbed by the film 
increases uniformly with the thickness. Producing the 
curve for the current from the vessel backwards, we find 
the current that would have been obtained had no light been 
absorbed by the film. This value is 57 in the units employed, 
and this gives an arbitrary measure of the intensity of light 
I, on entering the vessel. Curve A (fig. 3) shows how the 
light absorbed by the film varies with the thickness. It 
was found by subtracting the ordinates of the curve B (fig. 2) 
from 57. 

It is at once obvious that the large photo current at the 
thickness 10°’ cm.is not produced by an abnormal absorption 
of light. 

In Table IT. values for the current from the film and light 


* Robinson, loc. et. 


Effect on Thin Films of Platinum. 425 


absorbed by the film as obtained from the curve A (fig. 2) 
and curve A (fig. 3) are given for various thicknesses, as. 
also are the photo currents for unit absorption of light. 


ig. 3. 


60 1150 


IN 
(Os 


= 
O 
Film per Unit 


Absorkbtion. 


from 


Ol 
O 
ié 
Light 


atte) 


oe) 
% 


Light absorbed by Film. 


e 
E, 


10.0  2Oo) 300 


Time of Deposit of Files in Seconds 


Curve A.—Intensity of light absorbed by the film. 
Curve B.—Photo current for unit intensity of ight absorbed. 


Cur 


TaBLe II. 
Thickness of film} Current from Light. Current per unit 
in seconds. film. absorbed. light absorbed. 
50 8 13°3 

80 22 10°5 21 

100 100 13 Ut 

130 235 16 147 

150 220 PA 105 

200 178 31 58 

240 150 41 | 37 


In curve B (fig. 3) are plotted the photo currents per unit 
absorption of light, against the thickness of film, and the 
maximum value of the current appears still more pronounced. 
The photoelectric sensitiveness of platinum is thus very 
large for films of the order of 1077 cm. 

June 19, 1916. 


[ 426 ] 


XLVI. Experiments with Electron Currents in Different 
Gases. (1) Mercury Vapour. By O. W. RicHarpson, 
F.RS.. Wheatstone Professor of Physics, University of 
London—King’s College, and Cuares B. Bazzont, Ph.D.. 
Instructor in Physics, University of Pennsylvania”. 


| these experiments electron currents from an incan- 
descent tungsten filament AB (fig. 1) flowed under 
various potential differences to a cold anode ( of nickel 
wire about 1°5 mm. in diameter. The filament AB was 
about 1°5 ecm. in length and -08 mm. in diameter. The 
current across the gap AB to C has been measured under 


various conditions. At the same time the nickel electrode E 


Bree 


ce--o 


‘ - 
Cl ete 


fpr ter re nre x 


at the centre of the bulb H, which is covered on the inside 
with a conducting layer of platinum, enabled the photo- 
electric effect of the radiation from the discharge to be 
examined. The connecting tube contained the tubular sheath 
F and the central wire G. An electric field was applied 
between F and G to stop the diffusion of ions from A BC to 
the bulb EH. The apparatus shown in the figure was made 
of transparent quartz, and couid be maintained at various 
temperatures by means of a specially constructed electric 


* Communicated by the Authors. 


Experiments with Electron Currents in Different Gases. 427 


furnace. Fig. 1 exhibits the relative size and position of 
the metal parts which are of electrical importance. The 
mercury was supplied from the bottom of the tube LD, 
which could be heated by a current through a metal s spiral 
outside it. Disa piece of tube 7 em. long perpendicular to 
the plane of the rest of the section, and is orn the furnace 
already referred to whose boundaries are indicated by the 
dotted lines. In this way a little mercury could be distilled 
trom D into the bulb A BC, and the pressure of the mercury 
vapour could then be regulated by means of the surrounding 
electric furnace. The currents across the gap A B—>C were 
measured by a ee and the photo- oe. eurrents 
to EH by means of a quadrant electrometer. Beyond K the 
quartz tube was ground to fit a glass cone, the joint being 

made tight with a minimum quantity of involatile grease, 
o precautions were taken to prevent it from becoming 
warm under any circumstances. Beyond the joint were 
d) a side tube leading toa Geissler tube to test the spectrum 
of the gas present, é ) a U-tube which could be immersed 
in liquid air, (3) a mercury cut-off, (4) a bulb containing 
coconut-charecoal in a liquid-air bath, (5) a tube with glass 
tap leading to Gaede pump, McLeod gauge, &. So far as 
we have been able to as scertain, the experimental tube was 
absolutely air-tight. Before beginning the observations the 
quartz tube was heated in the furnace, and maintained at 
a red heat for some time by means ‘of al blowpipe; the 
filament A B glowed with the charcoal in liquid air, and the 
pump was running, so as to get all the occluded vases out of 
the apparatus. This procedure was found satisfactory; but 
it appeared that continued heating of the filament AB 


liberated traces of hydroyen for some time after the expe- 


riments were begun. After this evolution of gas had ceased 
it was found that on heating the tube LD lines due to 


hydrogen and carbon menoxide could be detected. The 


experiments to be described refer to conditions in which 
tbese contaminants were not present, at any rate in sufficient 
amount for spectroscopic detection, unless the contrary is 
specifically stated. For the construction of the quartz appa- 
ratus we are indebted to the skill of Mr. Reynolds of the 
Silica Svndicate, Ltd. 

The relation hetmen current and potential difference across 
the gap A B—->C depends on the pressure and nature of the 
gas andon the temperature of the filament, which controls 
the maximum possible electron emission. It may possibly 
depend on the temperature of the gas as well. The effect of 
gradually raising the applied potential difference is shown 


428 Prof. Richardson and Dr. Bazzoni: Experiments 


for a series of different filament temperatures by the curves. 
in fig. 2. These refer to discharges in mercury vapeur at 
pressures comparable with 0-01 mm. At a low filament 
temperature (1:0 amp. heating current) the current increases. 


Fig. 2. 


3170 at 3O0V 


Ds!7 H 


Nn 
N 
@) 
Oo 


iN) 
(6) 
DB 
Oo 


2300 ) : 


«<— Micro amps. 
fy 
63] 
(o) 
oO 


O ice) 20 30 « 4a¥O) 50 
<—— Volés G29 


more rapidly with the voltage than the first power at low 
voltages. At higher voltages the rate of increase falls off 
until finally saturation ensues (curve H, fig. 2). The current 
only behaves in this way when the temperature of the filament 
islow. The behaviour at higher temperatures is indicated by 
curves A, B, C, and Din fig. 2. These curves are all very 
much alike, and the behaviour below about 20 volts is very 
similar to that of curve EH. With higher voltaves it is, 
however, quite different from that shown by H. Thus with 


with Electron Currents in Different Gases. 429 


a heating current of 1:05 amps. (OAA,; A, A;) when the 
driving potential reaches 22 volts (A) the current suddenly 
jumps “from 130 micro amps. to (A) 1100 microamps. At the 
same time the resistance of the gap falls ; so that a potential 
difference of only 21 volts is required to maintain the larger 
current. On increasing the potential after this condition has 
been established, the current increases uniformly along the 
line A,Ap. Above 35 volts (A,) another kind of instability 
appeared, the final reading (A;) being considerably under 
that observed when the potential (45 volts) was first applied. 
This instability is probably due to the discharge wandering 
inside the tube, but we have not paid much attention to it, as 
it does not seem to be so interesting as that observed at the 
lower voltages. On lowering the potential from 45 volts the 
uniform part of the curve A,A, was found to extend to the 
left beyond 21 volts; but when the applied potential fell to 
17 volts (A,) the current suddenly dropped on to the lower 
curve again, and the potential difference rose to 19 volts 
owing to the increased resistance of the gap AB->C. 

The potential differences were tapped off a sliding rheostat 
in series with a battery of 200 volts with its middle-point 
earthed. The line ran to the negative end of the filament 
through a 5000 ohm resistance to prevent the development 
of an ordinary high-current arc. The potentials were mea- 
sured by a voltmeter between the negative end of the filament 
and the earth, to which the anode C was also connected 
(through the microamperemeter 

The other curves B, C, D hardly eall for individual comment. 
The chief point to note is that ne potential difference at which 
the current jumps from one curve to the other drops uni- 
formly to lower values as the filament temperature is raised 
and the number of electrons available becomes greater. 
This is true whether the jump is from the low- to the high- 
current curve, or vice versa. 

The magnitude of the “ kick ” corresponding to the passage 
from O A to A,A, and vice versa, as well as the potential 
differences under which these changes take place, depends a 
good deal on the pressure of the vapour as well as on the 
temperature of the filament. At very low pressures the kick, 
as measured by the ratio of the currents at A, and at A, is 
small, but sets in at relatively high potential differences ; 
as the pressure increases the ratio of the enrrents increases 
to a maximum and subsequently falls off until at pressures 
comparable with a centimetre of mercury the kick is scarcely 
noticeable. The potentials at which the kicks take place 
‘appear to drop uniformly towards a limit as the vapour 


430 


pressure of the mercury rises. These statements are illus— 
trated by the following numbers, in which the values of the 
vapour pressure of the mercury present are very reugh 
guesses {rom the temperatures as indicated by a thermometer 
in contact with the bulb :— 


Prof. Richardson and Dr. Bazzoni: Heperiments 


] 
| 


Tt will be seen from the numbers in the 


| 


Mareen ene: | Potential | Current Potential | Current | Heating 
| SMe Waren AS aul Weegee at As. | at A,. | Current: | 
re Vapour sass Cass Bp RAEN NR Hla be a adgee 
| volts. jmicroumps.)° volts. /microaimps.} amps. 
0-0001em.| 40 406 36 ease 105 
ODI ee 120 21 792 1-04 
0-001 Se 13:5 3432 112 
0:05 Loy 36 IES) 180 1-05 
| 
10 Le alias Hah 4) 228 1:07 


second and third 


rows that increasing the filament temperature tends to diminish 
the minimum potential necessary to maintain the currents on 


the high-level range. 


This is shown also by the following 


numbers :— 
| 
Vapour Pressure. /Minimum Voltage | PIP Carne | Current across | 
emplof Elio. 7 for A,. ; oe ‘| AB-+>C. 
| about volts. amps. | niicroaips. 
| 
A ‘U0] 185 1-05 | 1584 
ks 14 1-09 2700 
13°5 1:10 27600 
(B) +001 15 1:05 1089 
13:2 109 2838 
13 110 3036 
(CO) 205 12 | 1-05 192 
12:0 1:10 Dz 
(D) $05 120 1:05 210 
Lg 1:10 200 
1:0 7 1-07 230 


The observations under A and © were obtained imme- 


diately after taking B and D respectively, and then opening 
the connexion to the charcoal tube. The pressures in A and 
C are therefore believed to be respectively somewhat less than 
those for B and D. 

These and other observations indicate that increasing the 
pressure of the mercury vapour or the temperature of the 
filament does not lower the potential necessary to maintain 


with Electron Currents in Different Gases. 431 


the discharge in the condition corresponding to «A, inde- 
finitely, but that a limiting minimum value of this potential 
difference is gradually reached. The lowest potential dif- 
ference under which we have been able to maintain the 
discharge in the state A,A, is 11°5 volts. The potential 
differences are measured from the negatige end of the hot 
filament, so that this is the greatest potential difference 
between any part of the hot filament and the anode. As 
about one-tenth of the filament at each end is too cold to emit 
electrons in appreciable numbers, the greatest potential 
difference due to the field which is effective in driving the 
electrons across the gap would be some 0°3 volt less than 
this. On the other hand, the electrons are emitted with 
kinetic energy ot thermal agitation in accordance with 
Maxwell’s law. The allowance for this is somewhat inde- 
finite, but would probably wipe out the 0°3 volt under con- 
sideration. ‘Io maintain the discharge in the state A,A, 
therefore it appears to be necessary that the electrons should 
be able to acquire an amount of energy equal to that given 
by falling through a potential difference very close to 
11°5 volts. This is nearly equal to the sum of the cathode 
and anode potential drops in the ordinary luminous high- 
current mercury arc. According to Aron’s observations the 
former is 5°4 and the latter 7:4 volts. 

The emission of light from the discharges under consi- 
deration will be dealt with below. 


The Low Potential Discharge. 


The currents with small potential differences (O A, fig. 2) 
exhibit characteristics similar to those shown in fig. 2 over a 
wide range of pressure of mercury vapour and of the tempe- 
rature of the filament. The behaviour at a pressure of 
mercury vapour of the order of 1 mm. and with a heating 
current of 1:06 amp. in the filament is shown in oreater 
detail by the following numbers:— 


MOltS. ......... Onvall 2 3) 4 5 6 a 8 9 LOR ble Ne Ss} 14 
Currents 


ee sams). }o 13 18 45 66 $8 108 138 153 180 21 24 283 338 478 
Currents . 3 0) 1a ‘ : 2.2 PR GQUMDINT TE Oa 9 6 Q4. é 
ae he 08 21 39 60 88 109 138 169 20:1 23:6 27231 349 391 
The currents are roughly proportional to the potential 
difference raised to the power 1:5. This is shown by the 
, numbers in the last row, which are caleulated by assuming 

that the currents vary as V®2, and that the value at 5 volts 
is correct. Below 7 volts the agreement with this formula 


432 Prof. Richardson and Dr. Bazzoni: Experiments 


is within the limits set by the errors of measurement, but 
between 7 volts and 13 volts the observed currents are 
uniformly below those given by the V?? law. Above 
14 volts the observed currents exceed the values caleulated 
in this way. 

It is known that electron currents between a hot anda 
cold electrode in a very good vacuum vary as V?? for small 
values of V, and the phenomenon has been accounted for 
satisfactorily by Langmuir”. *. No doubta similar explanation 
will apply to those of the present experiments in which the 
pressures were very low, but it will scarcely cover a case 
such as that of the niin betes in the last table, since an essential 
feature of Langmuir’s explanation lies in the assumption 
that the electrons move freely between the electrodes under 
the acceleration due to the electric field without interference 
or collisions with gas molecules. This condition will certainly 
be far from being “satisfied with 9 gas pressures of the order of 
a millimetre. 

The following considerations show that with monatomic 
gases a similar law for low potential differences is to be 
-expected even at high pressures. According to the researches 
of Franck and Hertz, if the potential difference is iess than 
a critical value V, ohame acne for each monatomic gas, the 
-electrons undergo collisions with the gas molecules witloun 
loss of energy; so that if they are assumed to start from the 
hot electrode (V =()) with nevligible velocity, their kinetic 
energy on eens a point in “the field where the potential 
is V (V<Y,) will be given by 


$m ene ° ° ° . ° : : (1) 


Owing to the presence of electrons in the space between 
: oer Scere 
‘the electrodes, V will have to satisfy Poisson’s equation 


V?Vs4np.. ok 


The average velocity of the electrons in the direction of 
‘the electric intensity will be governed by the acceleration 
between successive collisions, so that the current density at 


any point wiil be given by the expression f 


UW nz in r OV 
ae = ea — ne 
2 mv 2mv Qa’ 


where X is the mean free path of the electrons in the gas. 


* Phys. Review, vol. il. p. 457 (1918). 
+ Cf.‘ Electron Theory of Matter,’ 2nd Ed. p. 410. 


with Electron Currents in Different Gases. 433 


If we confine our attention at present to the discharge 
between two parallel plane electrodes, (2) reduces to 


2VV 
or =4ep. Sey Py ate Aer sioe (Op) 


Eliminating p and v from (1), (3), and (2’), 
hero 


a on AW (4) 
~ where 
87r Qm. ; 
a= ye) 6 C 6 J (5) 
A first integral of (4) is 
. adV\3 
2aV82 += (a) (6) 


If the currents are a long way from saturation, o =0, 


when V and z=0; so that C=0, and d 
Cae Os2V" .) ee yh Cr 
Since V=0 when x=0, C’=0, and if V, (V:<V)) is the 


potential difference between the electrodes, whose distance 
apart is , | 


3\3 
2a=(277). (8) 
Thus 
Mae ei’ ean 
y= Onis! Mace s t8\\3 GO) 


and the current varies as the potential difference raised to the 
power 1-5. 

So far we have only considered parallel plane electrodes. 
Other cases lead to differential equations which, up to the 
present, have proved intractable. The relation 2 « V?? can, 
however, be shown to be independent of the shape, size, 
and relative position of the electrodes by a general argument. 
In general, the relations between v, V, p, and the coordinates 
are governed by equations (1), (2), and (3). Consider any, 
the same, geometrical system under two different potential 
differences. Let v, V, and p be the variables at any given 
point in the one case, and v’, V’, and p’ in the other. Then 
the dashed and undashed variables respectively will sepa- 
rately satisfy equations (1), (2), and (3). If possible let 

Phil. Mag. 8. 6, Voi. 82. No. 190. Oct. 1916. 2G 


434 Prof. Richardson and Dr. Bazzoni: Haperiments 


V'=nV at any point and i! =nsi at the same point, where 
n is constant and s is to be determined. From (1) 


ALO. . ae 


From (3) 
ee © x(q) & — gr d V! 
and 
i= eg he i) is \ orad V! 
Zoom NN \ Ze) © 
Thus 


p=n3p. . . . 
But from (2) 


Amp = Vi =n VN =—Annp, 


and 
=p. cds... ne 
For (11) and (12) to be compatible 
sS=3i20 rn 


Corresponding to any solution V=/(z, y, z) of the equations 
there will be a solution for which V’=nV, p'’=np, and 
z’=n?7, If the total potential difference is changed then 
the current will vary as the potential difference raised to the 
power 1°5. 

When the maximum potential difference V; exceeds the 
critical value V,, the problem becomes more complicated. 
Since the electrons lose all their kinetic energy by collisions 
at the critical value eV, the equation (1), for values of V 
lying between V, and 2V,, will have to be replaced by 


lim =e(V—V)=eU.. 2 ees 


In equations (2) and (3) the variable V can be replaced by 
the new variable U. Thus the differential equation for the 
space between the critical equipotential surface and the anode 
is the same as before, the only change being that the potential 
differences are now measured from the critical value. Frem 


(2), (3), and (14), 


| oU\3 
ae ——2 aren | bas Sa ik 
2a + C= (S-) (15) 

The further treatment of the problem depends very much 
on what one supposes to take place at the critical impacts. 
The work of Franck and Hertz has shown that at this stage 
the electrons lose all their energy, and that some positive 


with Electron Currents in Different Gases. 435 


ions are apparently liberated. According to Bohr’s theory 
the last named effect is either illusory or is a secondary 
phenomenon. For the present we shall assume for simplicity 
that the only effect influencing the motion of the electrons 
at the critical equipotential surface is the loss of kinetic 
energy. We propose to defer the consideration of the more 
complicated case, in which positive ions are liberated, until 
later, when we hope to be able to submit further experimental 
results bearing on the question at issue. 
Turning to equation (3), and keeping to the case of parallel 
‘ 1 dV dV 


planes, since 2 is the same everywhere, we see that — Eee 
v dx dx 


must have the same value on each side of the critical equi- 
potential surface. Since v vanishes at the critical surface, 


ate <d2Nr : 

either — or —, must also vanish. But — is not zero for 
Ao da dx 

V<'V,, and if ove were not continuous, there would be a 


da 

_ finite charge on a surface in the gas, and this would not be 

‘in equilibrium under the forces to which it is subjected. 
dV we d?V 

Thus ae must be continuous at the critical surface and Te 


must vanish for V=V,. Considering the solution for 


V < Vi, we see from (6) that the value of aM at the critical 


dx 
surface (since C=0 in this region) is given by 
dV 
Ns = (2a)tV 4 
(= =e) Vit en Cag) 


Since U vanishes at this surface C in (15) is equal to 
2aV,7?, and 


: dU 
Thus 
F : a U 1 
Qa)sz+C =ViF(y.) =ViF(, Pn CLS) 
where 
ld NCAR Dees 11/2 
eles ne (tis 0 e919) & 
vghdaTe.on.(8n-2) 2 — Smt9 
eta eee a OU) ME | (19) 


3” n 3n+2 


1 


‘a ral 
167 


436 Prof. Richardson and Dr. Bazzoni: Eaperiments 
Since U and z vanish when V=V, and «=2, we have 
(= —(2ata=—2Vi, . - a ee 


from the solution for the space between the cathode and the 
critical surface. Thus if the distance between the cathode 
and the anode is / and the potential difference between 


them V,, 
ab — — 
2a)i= 4) 248 ( 
eS 1 iE 


and 


e \$V2? Vi a 9 V; i a Vv; -1)'- 35, e -1) 
x) —- {1+ v, at eas ay, 891\V, 


ao) 3n+2 
bee aay bette Gn ye 2 (“ae wee 


3” n 3n+2\V, 


This formula makes the currents for values of V between 
Vand 2V, increase with V less rapidly than the requirements 


of the V?? formula which holds for V< V,. The calculated . 


falling off is indicated roughly by the following numbers:— 
NE NG > lez 38) 1°8 2°0 
Woe rer ti) 257 “4 “4 


The values of i/i;’ are the currents calculated by (21) 
divided by the corresponding currents calculated on the 
assumption that the V3? law which holds for V< V, is valid 
also for V>V,. The calculated falling off is much greater 
than that shown by the results on p. 431. This might be 
expected, as the theoretical conditions are very imperfectly 
realized in the experiments. Apart from the quite different 
system of electrodes used, there was a drop of potential down 
the cathode due to the heating current of about 2°5 volts. 
This makes the etfective critical potential about 1°5 volts 
too high. The sharpness of the effects arising from the loss 
of energy at the critical value also tends to be obliterated, 
owing to the occurrence of the initial distribution of velocity 
which will prevent the average kinetic energy from actually 
vanishing at any point. The larger observed currents might 
also be attributed to the liberation of positive ions at the 
critical collisions; since we should expect any considerable 
development of positive ions to permit a much closer 


approach to saturation, owing to the consequent reduction 
of p in equation (2). 


" (2D 


11/2 


+ 


| 


with Electron Currents in Different Gases. 437 


Emission of Radiation. 

At low pressures of the order of 0:001 mm. when the kick 
from OA to A,A, (fig. 2) does not set in until potential 
differences exceeding 30 volts have been applied, we have 
not been able to detect any evidence of the presence of a 
visible discharge either on the low range O A or on the high 
range A,A,. The tests were carried out by visual observations, 
both directly and through blue glass, with appropriate dia- 
phragms to eliminate the glare from the filament, and by 
examination of the tube with a large direct-vision spectro- 
scope by Hofmann. It may be that there isa faint luminous 
discharge which is swamped by the diffused light coming 
from the hot filament; but, if so, its intensity must be 
very low. Under these conditions there is, however, an 
emission of ultra-violet radiation, whose presence can be 
detected by its effect on the electrode E (fig. 2). This is 
shown by the following examples :— 

(a) At p=about 0°001 mm. and a heating current of 
1:05 amps. through the filament the thermionic current at 
40 volts was 429 microamps. At this potential (corresponding 
to A, fig. 2) the current suddenly jumped to 1056 microamps. 
at 39°5 volts on the higher range. The point corresponding 
to A, was located at 36 volts. No effect on the electrometer 
connected to H could be detected on the low-current range 
corresponding to O A below 30 volts, but at 36 volts 5 divi- 
sions in 30 seconds were obtained. On the range corre- 
sponding to A,A, a deflexion of 50 divs. in 30 sees. at 40 volts 
was observed. Evidently a large increase in the ultra-violet 
emission sets in with the kick on to the higher range. 

(b) At a pressure similar to that in the experiment just 
described, and with a heating current of 1:07 amps., the 
current was found to Jump up at 39 volts and down at 
31 volts. At 39 volts the current was about 280 microamps., 
and it jumped to 1848 microamps. at 36 volts. The following 
observations up to 30 volts were taken on the low-current 
range, and that at 40 volts on the high-current range: — 


Volts across gap A B —~> O (fig. 1)... 15 20 25 30 40 
Electrometer deflexions in 30 secs.... 0 0 5 13 200 


In this case also there is a big increase in the ultra-violet 
emission when the high-current discharge sets in, but there 
was a distinct effect which set in between 20 and 25 volts 
on the low-current range. This effect may have been present 
in experiment (a): but under the conditions of that expe- 
riment it would have been too small to detect with any 
certainty. It is difficult to be sure that these small effects 


438 Prof. Richardson and Dr. Bazzoni: Hxzperiments 


are not due to the presence of traces of contaminants such as 
hydrogen, to which the spectroscopic test is not very sensitive 
at these low pressures. No hydrogen lines could be detected in 
the mercury spectrum under the conditions of experiment (0). 

At pressures greater than about 0:01 mm. the jump to the 
higher range was always accompanied by the establishment 
of a visible glow discharge. The spectrum of the glow is 
that of the mercury are. The behaviour of the luminosity 
when the potential across the gap is changed presents very 
interesting features. We have not been able to observe any 
luminosity except that due to the light coming from the hot 
filament when the discharge is on the low-current range. 
On the high-current range at the lowest potential which 
would maintain the discharge a faint glow appeared close to 
the surface of the anode. On raising the potential slightly 
the glow grew out from the anode into the surrounding space. 
A further slight increase of the potential difference, in some 
of the experiments at least, caused the glow to spring away 
from the anode. In this condition there was a bright patch 
in the space between the electrodes with a dark space round 
both the anode and the cathode. A further small increase of 
potential difference made the glow settle round the cathode. 
After this stage was reached the luminosity was greatest 
near the cathode. Further increase of potential difterence 
caused the luminosity to spread out further from the cathode 
and to increase in intensity. In this condition there was a 
gradual falling off in intensity towards the anode, and the 
luminosity did not appear to have a sharp boundary. 

At high pressures the changes in the potential difference 
which are necessary to cause the glow to move over from the 
anode to the cathode are quite small, being comparable with 
1 volt. Observations similar to the following were frequently 
made with pressures of mercury vapour of the order of 1 em. 
With 0:98 amp. heating current and the quartz tube at 
225° C. the following observations of the luminosity and 
discharge current at different potentials were taken :— 


Wioltgre eveees sores. << tat 12 14 25 50 

Luminosity ......... None. On anode. Crossed Filled whole Filled 
to cathode. space. wholespace. 

Current (microamps.) 16 84 184 Jette 315 


With a larger heating current (1:05 amps.), but under 
conditions otherwise similar to the foregoing, the following 
observations were taken :— 


Woltistseeee renee sou/-- 1] 12 13°2 25 50 
Luminosity ......... None. Onanode. Orossed Filled whole Filled 
to eathode. space. wholespace. 


Current (microamps.) 20 124 aa ae eae 884 


with Electron Currents in Different Gases. 439 


The smallest potential difference which we have recorded 
between the stage when the glow starts on the anode and that 
at which it crosses to the cathode is 0:5 volt, the largest about 
6 volts. These observations were made with a high pressure 
of mercury vapour comparable with ] cm. The smallest 
potential differences were required with the smallest filament 
temperatures and correspondingly small discharge currents, 
and vice versa. In this respect the behaviour of the glow is 
similar to that of the positive column in the ordinary low- 
pressure discharge-tube, which is gradually forced back on 
to the anode as the strength of the discharge current is 
increased. 

We have examined the glow carefully with the spectro- 
scope in the initial stages to see if there is any difference in 
the potentials necessary to excite the different lines of the 
mercury spectrum. We have not been able to detect any 
such effect, either when the glow just starts or when it is 
made to die out. The strong lines of wave-lengths 4358 
and 5460 have been examined with especial care in this 
respect. So far as we are able to observe, the potentials 
required to excite these two lines are identical, certainly to 
within one-tenth of avolt. The equivalent voltage difference 
of the two lines by the quantum relation is about eight-tenths 
of a volt. This indicates that the lines of the mercury arc 
spectrum are not excited separately by single electron 
impacts, one for each line, but suggests that they are the 
result of a complex change set up in the mercury atom, and 
affecting a large number of electrons simultaneously. 

The value 11°5 volts of the smallest potential difference 
which will start the glow is less than the value 12°5 volts 
given by McLennan and Henderson* for this quantity. The 
precise interpretation of this voltage is a difficult matter 
without a knowledge of the distribution of potential along 
the discharge. Ifit is interpreted as the potential difference 
equivalent to the energy necessary for an electron to disrupt 
the mercury atom or the mercury ion in a definite way, we 
have to face the difficulty that, according to the experiments 
of Franck and Hertz, itis impossible for an electron travelling 
through mercury vapour to acquire kinetic energy in excess 
of that equivalent to 4°9 volts. This might be overcome if 
there is a sharp drop of potential sufficiently close to the 
cathode to carry a considerable number of electrons through 
the critical region without atomic impact, on the further 
assumption that the collisions become elastic again after 
passing the critical velocity. 


* Roy. Soc. Proc. A. vol. xei. p. 485 (1915). 


440 Experiments with Electron Currents in Different Gases. 


On testing with the electrode H (fig. 1) a vigorous emission 
of ultra-violet light was found to set in simultaneously with 
the appearance of the glow. With pressures sufficiently high 
to give a visible discharge under the comparatively low 
potential differences, we have not been able to detect any 
ultra-violet emission at potentials under that at which the 
glow sets in. In fact, we have never been able to discover 
any effect on the nickel electrode E with potentials across 
AB— > C of less than 11°5 volts. Presumably this is due to 
the absorption of the single line 2536 of the low-voltage 
spectrum by the mercury vapour in the connecting tube ; 
but it may be that the frequency of this line is not high 
enough to excite the photoelectric emission from the nickel 
electrode used. We have not tested this point. 

With high pressures there is no discontinuity in the 
current when the luminosity sets in, such as is observed at 
lower pressures. This is indicated by the numbers in the last 
row of the table on p. 430. - | 

A number of observations giving results of interest were 
taken in the early stages of the experiment before the 
tungsten filament had been thoroughly glowed out. Under 
these conditions the discharge in the Geissler tube showed 
the spectrum of hydrogen quite strong, but no other lines 
except those due to mercury. It was then found that with 
a low pressure of mercury vapour a considerable emission of 
ultra-violet light set in at a potential difference considerably 
under that necessary later, when the hydrogen had been got 
rid of. This effect set in quite consistently between 16 and 
17 volts, and at first increased rapidly with rising potential 
difference. The rate of increase then fell off until, finally, the 
effect varied little, if at all, with the potential difference. This 
is illustrated by the following numbers taken with a mercury 
pressure of about 0:001 mm. and an uncertain amount of 
hydrogen :— 

Volts across A B->C...... 0 5 10 15 16 17 20 25 30 35 40 5b 6b 
MicroampsacrossAB->C. 0 13 31 45 ... ... 52 60 68 70 70 71 ... 


Photoelectric current 

from H (scale divs.in$ 0 0 0 O O 5 I1 25 33 50 58 75 76 

30 sec.). 

It will be observed that as the potential rises the discharge 
current becomes constant sooner than the photoelectric 
emission. The thermionic currents were too small for the 
kick to develop under the conditions of these observations. 


Wheatstone Laboratory, 
King’s College. 


Woop. 


2 


| 
y 
Z 
4 
; 
A 
j 
4 
: 
Z 


Phil. Mag. Ser. 6, Vol. 32, Pl. VIL. 


"ee oye g atydetoojoyd '9g 04k G o1ydvasojoyg 


Phil. Mag. Ser. 6, Vol. 32, Pl. VILL 


OS 0781 oTydBasoyoyd ‘9G Tq orydetsozoyq 


: 


Kyrer & WELO. 


RAMAN. Phil. Mag. Ser. 6. Vol. 32, Pl. IX. 


String. 


Belly. 


String, 


WT Agana AK ee 


| Simultaneous Vibration-Curves of Belly and String of Violoncello 
at the “ Wolf-note ” pitch. 


\ 
| 
{ 
| 


‘Kinenon. Phil. Mag. Ser. 6, Vol. 32, Pl. X. 


IRTtEs Ik 


THE 
LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES.] 


NOVEMBER 1916. 


XLVI. The Mobility of the Negative Jon. 
By 8. Ratner, of the University of Petrograd *. 


T is well known that in general the mobility (k) of a 
gaseous ion varies inversely as the pressure (p), so that 
the product kp is constant for a given gas, but at low 
pressures the mobility of the negative ion becomes abnormally 
great, and kp increases rapidly with diminution of pressure. 
The phenomenon of the abnormal mobility of the negative 
ion has been studied by a large number of experimenters. 
Langevin f, Kovarik ft, Lattey §, and others could observe 
the abnormal increase of the mobility only in the case when 
the pressure was reduced below a certain value (75-200 mm.) 
and therefore assumed the existence of a “critical”? pressure 
at which the abnormality setsin. Frank ||, and later Haines 4, 
have shown that in carefully purified nitrogen, argon, 
helium, and hydrogen, the mobility of the negative ion is 
abnormally great, even at atmospheric pressure. Kovarik 
and Lattey (loc. cit.) drew attention to the fact that at low 
pressure the mobility of the negative ion varies with the 
electric force. J.S. Townsend** has shown that the mobility 
of the negative ion may in general be expressed as a function 
* Communicated by Sir J. J. Thomson, O.M., F.R.S. 
Tt P. Langevin, Annales de Ch. et de Physique, xxviii. (1903). 
t Kovarik, ‘The Physical Review,’ xxx. p. 415 (1910). 
§ R. T. Lattey, Proc. Roy. Soc. A, Ixxxiv. p. 178 (1910). 
|| Frank, Verh. d. D. Phys. Ges. xii. pp. 221 & 618 (1910). 
q Phil. Mag. xxx. p. 503 (1915). 
** J.S. Townsend, ‘ Electricity in Gases.’ Oxford, 1915. 


Piet. Mag... 6. Vol. 32. No. 191. Nov. 1916. 2H 


44? Mr. S. Ratner on the 


of 2 where X is the electric force and p is the pressure, 


and that for large values of = the mobility begins to 


diminish with increase of the force. Experiments were 
carried out in order to determine whether the mobility of the 
positive ion also departs from the inverse pressure law. 
Todd * has found that at pressures below 1 mm. the mobility 
of the positive ion also becomes abnormally great ; a com- 
paratively small increase of the mobility of the positive ion 
was observed also by J. 8S. Townsend f, at higher pressures, 
but these results are contradicted by other experimenters. 
The study of the mobility of ions leads us to results which 
may throw considerable light upon the nature of an ion and 
should be thoroughly carried out in all possible directions. 
Unfortunately the methods usually employed are but little 
adapted for this purpose when the mobility becomes ab- 
normally great. Rutherford’s, Langevin’s, and Zeleny’s 
methods and their modifications serve only in the case when 


the velocity of the ion does not exceed the order of ee 


and therefore can hardly be used for a complete study of 
the mobility of the negativeion. Chattock’s method involves 
other difficulties, as in this case the mobility has to be 
investigated under very unfavourable conditions, 7. e. when 
the electric force acting on the ion is very strong and not 
uniform. ‘The only suitable method for measuring large 
ionic velocities is that given by J. S. Townsend, depending 
on the action of a magnetic force on the motion of a stream 
of ions. 

In the present paper a new method of measuring ionic 
mobilities is described and some results are given. 


Method and Apparatus. 


The method may be considered as a modification of that 
given by Chattock{, and is based on the production of a 
wind in an ionized gas when acted on by an electric field. 
In the case of the discharge from a point toa plate, Chattock 
deduced the following expression for the mobility (4) of an 
ion in terms of the discharge current (c), the corresponding 
wind-pressure (P), and the distance between the plates (d) : 

C 
k=dy- 
: Todd, Phil. Mag. xxii. p. 791 (1911). 


. 8. Townsend, Phil. Mag. July, 1914. 


* 
+ 
t Chattock, Phil. Mag. (5) xlviii. p. 401 (1899). 


Mobility of the Negative Ion. 443 


In a previous note the author * has shown that if two 
parallel plates are immersed in an ionized gas, a small 
potential difference established between the plates produces 
a wind-pressure large enough to be measured by a special 
gauge t, which will be described later on. When a surface 
ionization of one sign close to one of the plates is produced, 
the density of electrification, as well as the strength of the 
field, is uniform throughout the volume between the plates 
(if the disturbing effect of the ions is neglected), and in this 
case Chattock’s formula may be deduced in a very elementary 
way. Let P be the total surface electrification on the plate, 
X the strength of the field; w the velocity of the ions, ¢ the 
total ionization current, P the corresponding wind-pressure, 
and d the distance between the plates. Then Pw is the 
quantity of electricity streaming through a plane parallel to 
the plates in unit time and c=Pw. When the ions move 
with a uniform velocity their drag on the gas is equal to the 
product of the total electrification between the plates and 
the strength of the field t. Hence P=dpX. 

These two equations give 


W C 
x =¢p> ° ° ° ° ° ° e (1) 


where = is the mobility of an ion. In the case when the 


velocity w of an ion is not proportional to the electric force X, 
w 


xX 


is what is called the abnormal mobility of an ion. 


The experiments may be carried out at any pressure, the 
potential difference between the plates varying over a very 
wide range, viz. from zero up to that required for a spark- 
discharge between parallel-plate electrodes, and, as will be 
shown later, by this method ionic velocities may be measured 


of the order of from 10 to i 


The apparatus used is shown in figs. 1, 2, and 3. Two 
brass plates A and B (fig. 1) are mounted on a wooden 
base ST. The circular plate A, which consists of two 
connected metal sheets 8 cm. in diam., is supported by a 
rod f sliding in a brass collar e and is insulated by an 


* Comptes Rendus, clviii. p. 565 (1914). 

+ I wish to thank Prof. Debierne for suggesting to me the idea of 
this gauge. 

} This is experimentally proved by the fact, that using the above 
formula Chattock deduced for the mobility of an ion the same value as 
given by other methods. 

Pah 


444 Mr. 8S. Ratner on the 


ebonite column IL fixed in an earthed guard-ring R. The 
plate is cut out in the middle in the way shown in figs. 1 
and 3, and in this gap a strip of thin platinum foil mn is 
stretched in the plane of the plate. One of the extremities 
of the strip is insulated from the plate by the ebonite plug p, 


ies AL 


— ewe ee eH KK =| —_—_——-- = 


——_— —- &— = ee es ee 


\ by 


Ls =e 
SCUNUMMOMMnhy 


while the other extremity is brought into contact with the 
plate and is fixed on a sliding metal support n pulled by a 
weak spring s. The strip can be heated to any desired 
temperature by passing a current through it from an insu- 
lated battery of accumulators kept at the same potential as 
the plate A. The plate B, 8 cm. wide and 14 cm. high, has 


Mobility of the Negative Ton. 445 


eut out of its centre a circular hole 3 cm. in diam. opposite 
to the strip of platinum and covered with metal gauze gh. 
The gauge employed is shown in figs. 1 and 2. A vane 
abed is cut out in the form shown in fig. 2 from a sheet of 
aluminium 0:1 mm. thick, and is suspended by a fine brass 
ribbon z attached to a torsion head X which may slide on a 
rod v projecting from the plate B. On its lower part the 
Fig. : 


o mde 


o, 9 
to} 


vane is provided with a mirror 7, a small weight g, and an 
arrangement for damping. The earthed cage k& forms an 
electrostatic protection for the vane. The apparatus is 
placed on a brass plate pierced by insulated electrodes and 
is covered with a bell-jar provided with a window of plane 
glass, so that the image formed by the mirror may be 
undistorted. Before being admitted into the bell-jar the 


446 Mr. S. Ratner on the 


air passes through calcium chloride, concentrated sulphuric 
acid, and glass-wool. The plate A may be earthed or charged 
to a desired potential by means of a battery of small cells, 
one terminal of which is earthed, while the plate B is con- 
nected to a galvanometer provided with a series of shunts, 
by means ot which any current between 10~° and 107° 
ampere could be measured. The strip of platinum, coated 
with a mixture of barium oxide and aluminium phosphate, 
emits when heated a copious supply of negative or positive 
ions, according to the direction of the electric field between 
the plates. The motion of these ions, as shown above, 
produces between the plates a stream of gas which passes 
by inertia through the grating gh and a small tube ¢, and 
imparts to the vane a deviation measured in the ordinary 
way by means of the mirror 7. 

When the platinum strip is heated, no electric force being 
applied between the plates, a convection current due to the 
heating of the strip is produced in the gas, and the image 
from the mirror changes its zero position on the scale. For 
each temperature of the strip a correction of the zero-point 
of the apparatus is therefore necessary, and precautions must 
be taken in order to diminish, as far as possible, these dis- 
placements of the zero. The suspension of the vane should 
not be too sensitive, the platinum strip has to be very narrow 
and placed in a vertical line, as shown in the figure, and the 
vane should not be placed close to the edge of the tube é, 
but about 5 millimetres from it. The ionization current, as 
well as the corresponding wind-pressure, may be varied 
over a wide range by changing the temperature of the strip, 
or the electric force between the plates. 

It is seen from equation (1) that for a given distance d 
between the plates the mobility of an ion x is proportional 


to a: the ratio between the ionization current and the 


corresponding wind-pressure. A great number of preliminary 
experiments were carried out in order to verify the above 
equation. As at high pressures the mobility of an ion is 
known to be constant over a wide range of the strength of 


r) 


the field, the ratio — according to equation (1), must also 


is 
be constant under the same conditions. Further, supposing 
= = _) for negative ions and a ,) for positive, ae 
(+) 


must be the ratio between the mobilities of the negative and 
the positive ion. 


Mobility of the Negatwe Lon. 4A7 


A sample of the results of these experiments is shown in 
Table I., where p is the pressure of air under the bell-jar, 
v the potential in volts applied to the plate A, and ¢ and P 
are given in arbitrary units. 


TABLE I. 


p=600 mm., d=20 mm. 


v c v c 
negative.| °% P. Pp =e positive.| °% P. Pp =F 4). 
280 18 11 1°64 320 26 22 1°18 
400 45 27 1°66 400 34 28 1:22 
- 20 12 1-66 600 75 65 1°16 
600 66 41 161 800 71 60 1:18 
1000 | 102 63 1°62 1000 66 56 1:18 
aA 28 17 1°65 1 31 26 1:19 
Mean value of ky 1°64 Mean value of Keay 1-19 
k 1°64 
lee) = eee [3 8. 
een Lae 


Tt is seen from the Table that 5 is constant within the 


limits of experimental error and that the value of “— =1-38 


is in good agreement with the well-known ratio enna the 
mobilities of the negative and the positive ion in air. 
Experiments of this kind, together with some other control 
experiments, which will be described later, prove the efficiency 
of the method and show that the results are reliable. 

It is easy to see that the wind-pressure produced by the 
stream of gas can be measured by the gauge only in arbitrary 
units and, consequently, the absolute value of the mobility 
cannot be deduced from equation (1). This, however, is not 
necessary for the purpose of this work, as in the results given 
below the abnormal mobility of the negative ion is expressed 
in terms of its normal mobility—a constant determined to a 


high degree of accuracy by other methods. 


Results. 


1. The mobility of an ion as measured in arbitrary 


Ww 

= 
° e Cc . e Pe 

units by the ratio p? constant over a certain range of the 


448 Mr. S. Ratner on the 


applied electric force X, which is in agreement with the 
well-known experimental facts. But in the case of negative 
ionization, wnen X is gradually increased the experiments 
show that at a certain value of X=X, depending upon the 
pressure of the gas a lack of proportionality between c and P 
C 
le: 
Equation (1) shows that in this case the velocity of an ion 
increases more rapidly than the electric force, 2. e., that its 
mobility becomes abnormally great. The value of this 
critical electric force, X,, decreases rapidly with diminution 
of the pressure, so that at pressures below 200 mm. the same 
effect is reached with comparatively small forces when the 
ionic velocities are small enough to be measured by 
Langevin’s or Rutherford’s method. 


In the curves I., II., IIL, IV., V., and VI. (fig. 4), the 


suddenly comes in in such a way that = begins to increase. 


Fig. 4. 
& S 
& & oe 
a & Cy poe 
Px stig 500-m.m: 
6 
14 
cS 4 
x 
& 
5|5 
ry 
S 
3S 
3 Tr 606mmos 
.— 
4 
3 
74 : 
I 8r.m 
WI (+)200mm. 
400 800: 1200 1600 2000 2400’ 2800 3200 3600 4000 


Fiq 4 Electric Force in “ES 


mobilities x of the negative ion at pressures of 748, 600, 
500, 400, 300, and 200 mm. respectively, are plotted against 


the electric force X, and the curve VI(,), which isa straight 
line parallel to the axis of X, shows the mobility of the 


Mobility of the Negative Lon. 449 


positive ion at 200 mm. pressure. The mobilities are given 

in arbitrary units, the mobility of the negative ion at 

atmospheric pressure being equal to 0°72 in these units. 
2 Soren: 

The ‘lable Il., corresponding to curve I., is given to show 

the procedure of the experiments. 


TABLE II. 


p=748 mm., d=10 mm. 


v. G: 1Py pe | v. C. 12. PP: 


SOR 1e 14 ait 2000 | 52 60 “88 
200 | 32 44 ‘73 || 2120 | 24 21 | 1:14 
600 | 46 638 73 || 2200 | 48 39 | 1:28 

1000 | 38 53 G2) 2400) | 71 47 | lol 

1400 | 51 69 ‘74 || 2600 | 65 34 | 1-91 

1600 | 26 36 72 3000 | 89 40 | 2:23 

1720 | 31 43 ‘73 3400 | 76 30 | 2°53 

1800 | 35 46 “76 || 4000} 92 34 | 2°77 

1880 | 43 54 ‘80 


The curves show that when the critical value of the electric 
force is reached the mobility of the negative ion begins to 
increase, at first rapidly. At atmospheric pressure the 
mobility is doubled when the electric force is increased from 


volt 


1800 to 2400 aa at 400 mm. pressure the mobilities 
at 450 and 2000 ee are as 1 to 5. With larger forces 


the mobility increases less rapidly, and it may be noticed from 
the curves that it apparently tends to reach a constant value. 
The curves show also that at high pressures the velocity of 


the negative ion cannot be expressed as a function of >: 


The determination of the critical electric force X, at a 
given pressure involves great difficulties. At high pressures 
the result of experiments carried out under the same 
conditions varies from day to day to the extent of 20 per cent. 
The amount of moisture present in air does not apparently 
affect the results, although in these experiments the air was 
never dried with special care. The effect may be due to 


450 Mr. S. Ratner on the 


some other impurities in the air, and this supposition is 
supported by the fact that, as shown below, even imperceptible 
traces of chloroform vapour change completely the aspect 
of the mobility curves. At lower pressures, when the 
critical force X, is small, another difficulty arises, as the 
jonization-current and the corresponding wind-pressure de- 
crease with the potential difference between the plates and 
become, in this case, too small to be measured with accuracy. 


In the Table III. the values of X, and = at different 


pressures are given, X, being the mean approximate value 
from a great number of experiments. 


TasLeE III. 
x Xx 
Pp 1 p- 
750 1800 2°4 
600 1000 17 
500 700 1-4 
400 450 Ter 
300 250 8 
200 100 5 
150 60 4 
100 30 3 
75 2U 26 


It is seen from the Table that a decreases rapidly with 


diminution of pressure, so that at high pressures the negative 
ion may attain comparatively large velocities before it begins 
to assume an electronic state, and that is the reason why 
the abnormal mobility could not be observed at high pressures 
by other methods. It is, however, worth noticing that even 
at atmospheric pressure the largest velocity attained by the 


normal negative ion, viz. 3200 ~~ seni is small compared with 


its velocity of thermal agitation. 
2. Interesting mobility curves are obtained with larger 


values of = It has been already noticed from the curves 


(fig. 4) fee at large electric forces the mobility tends to 
remain constant. When the force is still further increased 


104) 


Mobility ia Air. 
iS 


(ay) 


Mobility of the Negative Ion. A51 


the mobility of the negative ion attains a maximum value 
and begins to diminish. This phenomenon could be pre- 
dicted from Langevin’s formula for the mobility of a charged 
particle whose mass is small compared with that of a mole- 


l 
cule: =~. xe where e and m are the charge and the 


Xm 

mass of the particle, / its free path, and u its velocity of 
agitation. The mass associated with the negative ion 
gradually diminishes with increase of electric force, and at 
this stage of evolution of an ion its mobility increases with 
the force. When, finally, the electronic state is reached by 
the ion, its mobility follows the above equation, and since 
the velocity of agitation of an electron increases with the 
electric force *, its mobility begins to decrease f. 


Fig. 5. 
if 
ie) 
0 
2m 
JIL 
ns 
oF 
Ww 100 p 
7 
50 m.m- 
ty WV+ I + 
20Q@ «=: 400, 600 800 1000 1200 1400 1600 i800 2000: 


Electric Force in Volts 
ci. 


This decrease in the mobility of the negative ion is shown 
in curves I., I1., I1I., and IV. (fig. 5), for pressures of 200, 
100, 50, and 10 mm. respectively. The curves III...) and 
IV...) show the mobility of the positive ions at 50 and 10 mm. 


* J. S. Townsend and H. T. Tizard, Proc. Roy. Soc. A, Ixxxvit. 
p 336 (1913). 

+ A full discussion of this question will be found in J, S. Townsend’s 
book, ‘ Electricity in Gases.’ Oxford, 1916. 


452 Mr. S. Ratner on the 


pressure. In this figure the mobilities at different pressures 
are given in different units in order to avoid using too small 
a scale. In the curves I. and II. the mobility at any point 
may be given in terms of the normal mobility represented 
as straight lines in the beginning of the curves, while in the 
curves III. and LV. it may be calculated from the mobility 
of the positive ion at the same pressures. The curves show 
that the velocity of the ions always increases with increasing 
forces, even in the case when their mobility diminishes. 

By their maximum points the curves are divided into two 
parts, the first part corresponding to the gradual dissociation 
of the ion and the second to the pure electronic state of an 


ion. The value of = corresponding to the maximum 


mobility is constant at pressures up to 200 mm. and is equal 
to 6°2, which is very large compared with 0°2—the value 
given by J. S. Townsend in case of carefully dried air. 

The rate of decrease of the mobility, as seen from the 
curves, diminishes with the increase of the force, and at 


sufficiently large values of x the mobility tends to reach a 


minimum value. At low pressures this bend in the curve is 
well marked, us shown by curve [V. Experiments with 


still larger values of x are made impossible by the luminous 


discharge which takes place under these conditions between 
the heated strip and the grating. It is possible that the 
luminous discharge is preceded by a feeble ionization by 
collision, which might be responsible for this bend of the 
curves. Further experiments in this direction are now in 
progress. 

With regard to positive ions, the experiments show that 
at pressures down to 5 mm. and with very large forces the 
mobility remains constant. 

3. Haperiments in hydrogen.—The same experiments were 
carried out in hydrogen—a gas in which the negative ion 
is known to exist in the electronic state under ordinary 
conditions. The hydrogen was generated in a Kipp appar- 
atus from hydrochloric acid and zine and purified by passing 
slowly through NaOH, KMnO,, and H,SO,. It was always 
contaminated with small quantities of impurities, since the 
brass-plate and the bell-jar enclosing the apparatus contained 
a great number of different joints which allowed a constant 
small leakage of air. The mobility curves in hydrogen show 
the same characteristic features as in the case of air. The 


Mobility of the Negative Ion. 453 


critical value X, of the electric force, at which the mobility 
of the negative ion becomes abnormal, is very small in 
hydrogen and increases with the amount of impurities to 
such an extent that it may serve as a good criterion for 
testing the purity of hydrogen. 


The curves P Wivand, TEP. (fig. 6) represent the mobility 


Fig. 6. 
7 
6 
Ir 
Su 
Qo, 

S 
a 
x= 
x 
eS , 
= 
= 
~Q 
9 
S 3 " 375mm, 

DL 100 mm. 
2] 
\ 
200 400. 600 600 1000 41200 #4211400 ° #2451600 1800 2000 


2 , Volt. 
Electric force in sams 


of the negative ion at pressures 750, 375, and 100 mm. 
respectively, in the purest hydrogen ebtained for these 
experiments. The critical value X, of the electric force is 


~ about 40 = at atmospheric pressure, and at lower pressures 
it becomes too small to be measured by this method. The 
value of és corresponding to the maximum point of the 


mobility curves is very large, varying from 1°8 at atmo- 
spheric pressure to 11 at 100 mm., and with further increase 
of the electric force the mobility tends, as in air, to attain a 


minimum value. These high values of ae are probably due 


to the impurities contained in the hydrogen. Experiments 


454 Mr. S. Ratner on the 


in purer hydrogen and nitrogen, as well as in carefully dried 
air, are now in progress. 

4, Experiments in heavy gases—Hxperiments were also 
made in order to test the mobility of the negative ion in 
heavy gases, such as the vapours of chloroform, carbon 
tetrachloride, and methyl iodide*. The mobility in these 
gases was found to be constant for the negative as well as 
for the positive ion, the mobility curves being straight lines 
parallel to the X axis even at low pressures (down to 5 mm.) 
and with large electric forces. 

A striking effect illustrating the influence of impurities on 
the mobility of the negative ion was observed during these 
experiments. Even after the chloroform was removed and 
the apparatus, as well as the bell-jar and the pump, was 
thoroughly cleaned and freed from traces of this gas, no 
further experiments in other gases were possible for many 
days. It was found that chloroform vapour present in air 
to an amount as small as 1 part in 1,000,000 changes com- 
pletely the aspect of the mobility curves. 


Control Experiments and Sources of Error. 


1. It is important for the accuracy of the experiments, 
that the total wind-pressure produced by the ions should be 
given by the gauge, and this is secured if all the ions reach 
the plate B within the area of the grating. In order to 
ascertain this the grating was insulated from the plate by a 
narrow air-gap, and the galvanometer connected by means 
of a suitable key, either with the plate or with the grating. 
It was found that even at the largest distance between the 
plates, and under conditions in which the lateral diffusion of 
ions is abnormally great, the total current is received by the 
grating, the charge picked up by the plate being imper- 
ceptible. 

2. The temperature of the gas close to the heated strip 
must be very high, and therefore the mobility of the ion at 
the moment when it leaves the strip must be large. In order 
to ascertain the magnitude of this source of error, curves 
representing the mobility of an ion as a function of the 
electric force were drawn for distances between the plates 
varying from 5 to 45 mm. No difference between them 
could be observed, which shows that the path traversed by 


* Tn these experiments the silvered surface of the mirror was attacked 
by the halogens. I wish to thank Mr. E. Everett for suggesting to me 
the idea of platinizing the mirror and for setting up the cathode-ray 
apparatus necessary for this operation. 


Mobility of the Negative Ion. 455 


the ion with a larger velocity is small compared with the 
distance between the plates and may be neglected. 

3. It was supposed at first that the electrie field between 
the plates was uniform. This, however, is not the case, the 
electrification between the plates being large enough in these 
experiments to disturb considerably the uniformity of the 
field. Unfortunately, this source of error cannot be elimin- 
ated or even diminished, since the ionization-current must 
be considerable in order to produce a perceptible wind-effect. 
It is useless to increase the sensitiveness of the gauge 
beyond certain limits, as the electric wind produced must be 
large compared with the convection current due to the heated 
strip. There is, however, sufficient evidence to show that 
the results are but little affected by this source of error. In 
almost all the experiments described the ionic velocities were 
very large, so that the density of electrification between the 


plates was considerably reduced. ‘The equation ose 


as shown above, was found to be true over a wide range of 
X,c,and P. The mobility of the negative ion was shown 
to be a function of the electric force only, and not to change 
with the distance between the plates, which would be the 
case if this source of error was great. 

4. The platinum strip freshly coated with salts gives off, 
when heated for the first time, a considerable amount of 
smoke (consisting probably of charged particles), which 
make the results inconsistent. In this case the strip has to 
be strongly heated at reduced pressure and in a strong 
electric field before measurements are taken. 


Electric Wind in case of Ionization of both signs. 


It seemed to be of interest to study the pressure of the 
electric wind in the case of ionization of both signs, when 
the wind is produced by positive and negative ions moving 
in opposite directions. For this purpose a small cell con- 
taining 25 mer. of radium bromide and provided with a thin 
mica window was placed in the centre of the plate A in 
place of the platinum strip, and the gas between the plates 
strongly ionized by the a rays. ‘The plate A could be 
positively or negatively charged, the plate B being earthed, 
and the wind-pressure measured in the usual way by the 
gauge. In this case the pressure of the wind at any point 
between the plates is a resultant of two opposite forces 
produced by the motion of positive and negative ions. 


; Pressure of Wind in Arbit: Units. 


A456 Mr. S. Ratner on the 


In the curves (+) and (—) (fig. 7) the wind-pressure is 
plotted against the electric force at 450 mm. pressure, in 
the case when the ions moving towards the grating are 
positive and negative respectively. These curves are in full 


400 600 800 1000 1200 1400 1600 1800 2000 


Vo/t.s: 
cm. 


Electric Force in. 


Fico. 


agreement with the results relating to the mobility of the 
negative ion given above. For small forces the wind- 
pressure is small, as the ions in this case recombine on their 
way and traverse only a small part of the distance between 
the plates. The wind-pressure increases with the force 
until the saturation current is reached, and with further 
increase of the force a striking difference between the two 
curves is observed. Under these conditions the mobility ot 


the negative ions, as measured by the ratio oF begins rapidly 


to increase, and since ¢ is constant in these experiments, the 
wind-pressure P produced by them must decrease at the same 
rate; while the mobility of the positive ions, and conse- 
quently the wind-pressure due to them, is not affected by the 
increase of the force. When the plate A is negatively 
charged and the negative ions are moving towards the 
grating, the wind-pressure produced by them diminishes 


Mobility of the Negative Lon. 457 


with the strength of the field, and when the opposite force 
due to the positive ions begins to prevail, the pressure is 
reversed and assumes a negative value. In the case when 
the positive ions are moving toward the grating the wind- 
pressure produced by them remains constant, but since the 
opposite force due to the negative ions gradually diminishes, 
the resultant pressure measured by the gauge increases with 
the electric force. At atmospheric pressure the negative 
direction of the electric wind can be produced only with 


large electric forces exceeding 3000 ae at 50 mm. 


lt. ’ 
pressure a force of 80 — is required for the same effect. 


It is of interest to note that with small electric forces 
applied within the plates the wind-pressure increases with 
diminution of pressure of the gas down to a certain limit. 
This is due to the fact that the ionization-current, which is 
in this case far from saturation at atmospheric pressure, 
gradually approaches it when the pressure is reduced. 

It would not be out of place to consider these results in 
connexion with the observations made by Joly™ on the 
motion of radium inanelectric field. Almostall the peculiar 
properties of radium described in that paper are easily 
explained, in the light of these results, by the electric wind 
produced by radium. 


On the Nature of the Negative Carrer. 


In the discussion of the results given above the existence 
was assumed of a transition stage between a negative ion 
and an electron, at which the average mass associated with 
the ion gradually diminishes with increasing electric forces. 
Whilst these experiments. were in progress a new paper on 
the mobility of the negative ion appeared}, in which a 
radically different point of view as to the nature of the ion 
is put forward. According to H. M. Wellisch there are two 
distinctly different kinds of negative carriers in a gas: 
normal ions and free electrons, the proportion of the latter 
increasing with diminution of pressure; the conception of 
an intermediate stage between an ion and an electron is 
erroneous and is due merely to the attempt at averaging the 
different properties of these two kinds of ions. 


* Joly, Phil. Mag. vii. p. 303 (1904). 
+ E. M. Wellisch, Amer. Jour. of Science, xxxix. p. 583 (1915 


2 
Phil. Mag. 8. 6. Vol. 32. No. 191. Nov. 1916. Ag | 


458 Mr. 8S. Ratner on the 
Hquation (1) may be written in the form = ade, which 


shows that the wind-effect produced by the ions is inversely 
proportional to their mobility. According to Wellisch’s 
theory the wind-pressure measured by the gauge in these 
experiments is mostly due to the slow normal ions, the effect 
produced by the free electrons being very small. The 


e ie e e e e e e e 
ratio —, as shown above, diminishes with increasing electric 
¢ 


forces, and reaches in some experiments 5 per cent. of its 
initial value. It is necessary therefore to suppose that 
the proportion of free electrons increases with the electric 
force, so that, finally, the amount of normal ions becomes 
vanishingly small. Wellisch, however, states that the pro- 
portion of free electrons is independent of the strength of the 
field, and is always less than 5() per cent. of the total number 
of the negative carriers. 

In order to verify the results of Wellisch, experiments were 
carried out which were based on the principle of separating, 
by means of an alternating electric field, the normal ions 
from the free electrons, and measuring the wind-effect due 
to one of these two kinds of ions separately. It appeared 
necessary for this purpose to study at first the wind-effect 
produced by the negative ions in an alternating field under 
conditions in which the mobility is normal, 2. e. when, 
according to Wellisch, all the ions are normal ions. The 
alternating field between the plates A and B was established 
by means of a high-speed commutator which could pro- 
duce up to 2000 alternations per second*, the potential 
difference between the plates being kept constant. The 
experiments show that when the number of alternations is 
gradually increased, the wind-pressure at first slowly dimin- 
ishes and then rapidly falls down to a very small value. If 
the wind-pressure is plotted against the number of alterna- 
tions per unit time, the curves obtained are all of the type 
of curve (+) (fig. 8), and simple calculations show that the 
sudden drop in the wind-pressure corresponds to the number 
of alternations at which the ions emitted by the heated strip 
begin not to reach the grating. 

Now, if we repeat the same experiments under the con- 
ditions, in which the mobility of the negative icn is abnormal, 
2.€. when, according to Wellisch, the wind-pressure is mostly 
produced by the normal ions and only to a small extent by 
the free electrons, the following observations are to be 


* The commutator was that used by G. Todd, Phil. Mag. xxil. p. 791 
(1911). 


Pressure of winds 


Mobility of the Negative Ion. 4.59 


expected : The wind-pressure should at first slowly diminish 
with increasing rate of alternations, and when the number of 
alternations per second is large enough to prevent the normal 
ions from reaching the grating, a sudden drop in the wind- 
pressure should be observed. 


Fig. 8. 


300 600 900 1200 
Alternations per Second, 


The experimental results are, however, altogether different 
from what might be expected if Wellisch’s theory were true. 
No drop can be observed in the wind-pressure when the 
frequency of alternations reaches the critical value, which 
decisively proves that the wind is produced only by one kind 


of ions whose mobility ( as measured by the ratio Ee) is 


intermediate between that of a normal ion and of a free 
electron. ‘The curve (—) (fig. 8) was drawn at 100 mm. 
pressure, the distance between the plates being 3 cm. and 
the p.d. between them 200 v.; 900 alternations per second 
is enough under these conditions to separate the normal ions 
from the electrons; the curve, however, shows that the 
wind-pressure continues to diminish slowly with further 
increase of the rate of alternation. The curve (+) shows 
the wind-pressure under the same conditions, but in the 
case when it is produced by positive ions. 
yan ls 


sal 


. 


1500 


460 On the Mobility of the Negative Ion. 


The experiments of Wellisch were carried out on the 
supposition that the mobility of the negative ion is indepen- 
dent of the electric force applied; this, however, is shown 
to be true only for comparatively small electric forces. The 
study of Wellisch’s paper leaves but little doubt that the 
sudden increase in the mobility of the negative ion with 
increasing forces is responsible for the characteristic bend 
in the curves, which have led him to the erroneous conclusion 
of the existence of two different kinds of negative ions. 

Thus, for a satisfactory explanation of the results given 
above, we must assume the gradual diminution of the average 
mass of the negative ion with increasing electric forces. 
The conception of “the average mass associated with an 
electron” is, however, complicated, owing to the uncertainty 
as to the nature of an ion, and may be interpreted in a way 
different from that adopted in this paper. According to the 
interpretation recently given by Sir J. J. Thomson* the 
electron may easily escape from the system of molecules 
constituting the ion and travel a certain distance in a free 
state, until it is once more attached to a molecule, and so 
on, so that the negative carrier makes its way through the 
gas, partly as a free electron and partly as a normal ion. 
From this point of view there is no “transition stage” 
between an ion and an electron, and the increase in the 
mobility of the negative ion, as well as the apparent dimin- 
ution of the mass associated with it, show oniy that under 
certain conditions the proportion of time during which the 
ion 1s moving as a free electron increases with the electric 
force. 


Summary. 


1. A new method of measuring ionic mobilities in gases 
is described. 7 

2. At a given pressure the mobility of the negative ion is 
shown to be constant only over a certain range of electric 
forces applied. With increasing electric forces X a certain 
value of X=X, is reached when the mobility of the ion 
begins to increase rapidly. The value of the critical force 
X, increases rapidly with the pressure, being equal to about 
1800 = at atmospheric pressure. 

3. The mobility of free electrons was measured and shown 
to diminish with increase of the electric force. 

4, The mobility of the negative ion in hydrogen and in 
some heavy gases was measured. 

# Phil, Mag. xxx. p. 821 (1916). 


Equilibrium of the Magnetic Compass in Aeroplanes. 461 


5. No abnormality in the mobility of the positive ion 
could be observed at pressures down to 5 mm. and with very 
large electric forces. 

6. The pressure of the electric wind in case of ionization 
of both signs was studied. 

7. The recent theory of E. M. Wellisch with regard to the 
nature of the negative ion was investigated and shown to be 
erroneous. 


These experiments were to a great extent carried out in 
the Physical Laboratory of the Polytechnic Institute in 
Petrograd. Iam indebted to Prof. A. F. Joffe for allowing 
me to work in this laboratory and to Mr. J. 8. Shcheglaeff 
for putting the necessary apparatus at my disposal. 

In its final stages the work was completed at the Cavendish 
Laboratory, Cambridge, and I wish to take this opportunity 
of thanking Prof. Sir J. J. Thomson for permission to use 
the laboratory and for his interest of the progress of this 
work. 


Cavendish Laboratory, 
August 1916. 


XLIX. The Equilibrium of the Magnetic Compass in Aero- 
planes. By 8S. G. Staruine, B.Sc., West Ham Municipal 
Technical Institute *. 


| Maer following investigation of the question of the 

behaviour of the aeroplane compass was undertaken 
at the suggestion of Mr. A. J. Hughes. It appears that the 
deviation of the compass when under acceleration causes 
grave errors in estimating the course, and that these differ 
considerably from one course to another. The greatest 
errors occur when the turn is made, as the course is changed, 
and as these errors may arise from a variety of causes, it 
becomes necessary to find the true position of equilibrium of 
the compass-card, due to the magnetic effect of the earth’s 
field upon the needle. 

The calculation of the couples acting on the needle presents 
great difficulties, as these depend upon the position of the 
needle at every instant, but if the equilibrium position at 
each instant is found, it can then be seen to what extent it 
would be disturbed, and this has led to suggestions which 
would cause the equilibrium position to be maintained true 
magnetic N. and 8. 

There is, of course, a tilting of the compass-card due to 


* Communicated by Prof. A, W. Porter, F.R.S. 


462 Mr. 8. G. Starling on the Equilibrium of 


acceleration when the machine is increasing or decreasing in 
speed. This causes the card to dip towards the fore part 
during acceleration and to the aft during retardation. The 
resulting disturbance of the card will be a maximum when 
flight is magnetic Hast or West, and this would only occur 
during straightforward flight and chiefly near the ground, 
when the compass readings are not particularly important. 
By far the most serious errors of the compass are due to 
the acceleration towards the centre of the circle when a turn 
is made, and it is with these errors that the present paper 1s 
concerned. 

As the aeroplane turns it is tilted towards the centre of 
the turn exactly as any other vehicle, and the resultant of 
gravitational and centrifugal forces may still be in the median 
plane of the machine, so that a plumb-line on it would still 
be perpendicular to the base-board, or, on the other hand, 
this resultant may be inclined to it. In the latter case, it 
would be possible to devise a compensator in the nature of a 
magnet or electric coil under gravitational control which 
would produce a magnetic field opposite to that causing the 
disturbance of the needle. This was tried and found to 
produce no effect, either of correction or disturbance, and it 
is therefore concluded that, as the aeroplane turns, the 
banking is such that the resultant weight of every part of it 
is still perpendicular to the base-board. Thus the card of 
the compass, which is free to take its new position under 
gravitation and centrifugal force, will still remain parallel 
to the normally horizontal glass of the compass-bowl. The 
machine then goes round a bend just as a large conical 
pendulum would do. 

Any disturbances due to motion of the liquid in the bowl 
and friction at the pivot are now, for the moment, put on 
one side, and the equilibrium of the needle under magnetic 
forces alone considered. 

Let H be the horizontal component of the earth’s magnetic 


field, and V its vertical component. Then, tand= Ng where 
d is the magnetic dip. “ 

In fig. 1, AB is a truly horizontal plane shown in elevation 
and KF the compass-card inclined at angle 6 to the horizontal. 
OP'Q is the plan of the compass-card, and OPQ the card if 
imagined to be rotated through 6° into the horizontal plane. 
NS is the magnetic meridian, so that NRQ is the true 
compass-course (¢'), and S’RQ is the angle between the 
magnetic meridian and the course RQ, as measured on the 
card, and is called @. 


the Magnetic Compass in Aeroplanes. 463 


For simplicity the pole of the needle is supposed to be at 
the edge of the card and the strength m, and the radius of 
the card 1, but this assumption does not affect the result. 


Fig. 1. 


Then TW=RW.tand’, 
and SW=RW .tan ¢. 
TW tang’. Bee i 
SW tang ; but cos 0= Sw: 
tan d' 
cos 0 = aia’ 


a result which will be required shortly. 

Vertical component of earth’s field.—There will be a vertical 
force Vm acting downwards upon the N. pole of the needle 
and, of course, an equal and opposite upward force upon the 
S. pole. 

This has a component H!Z perpendicular to the plane of 
the card, which has no turning effect upon the needle, and a 
component E/Y=Vmsin @ in the plane of the card, and 
perpendicular to OQ. 

Horizontal component of earth’s field—The horizontal force 
Hm on the pole E’ may be resolved into two horizontal 


464 Mr. 8. G. Starling on the Equilibrium of 


components (fig. 2), E'a=Hmcosd' in the plane of the 
compass-card, and parallel to OQ and E'b=Hmsin @’, not 
in the plane of the card. The latter is resolved into H’e 
normal to the card, and E’d=Hmsin ¢' cos @ in the plane of 
the card and perpendicular te OQ. 

Equilibrium of the needle—There are now three forces 
acting on each pole of the needle, namely, Vm sin 0, Hm cos¢’, 
and Hm sin ¢’ cos 6. 

In fig. 2, x’ is the angle between the magnetic meridian 


Fig. 2. 


and the vertical plane through the axis of the magnet, 
while 2 is the corresponding angle in the plane of the card, 
that is, S’RE’ (fig. 1). 2 is therefore the deviation of the 
compass from magnetic North, as seen by an observer in the 
aeroplane. 

If 21 be the length of the magnetic needle, the magnetic 
moment is 2/.m, and the couples due to the three components 
of field acting on the needle are 


21.m.Vsin@cos(¢—2z), 21.m.H cos ¢' sin (6—2), 
and 21.m.Hsin ¢' cos 6 cos(d—2z), 
and the needle will be in equilibrium if 
H sin ¢' cos 6 cos (@—x) = V sin 6 cos (6—2) 
+ H cos ¢! sin (6—2). 


But . = beth. 
cos g tan (6—.) = sing’ cosO — tand.sin 0, 
or tand sin @ - 


tan (6—2) = cos@ tan gd! — a een (1) 


the Magnetic Compass in Aeroplanes. 465 


For any angle of dip (d), compass course (¢’), and tilt 0, 
the value of (6—2#) can then be found, and since ¢ is 


tan 
tan @'” 

In the northern hemisphere, when the aeroplane makes a 
turn from N. to EK. and continues from Hi. to S., the tilt of 
the machine is downwards towards the south, and the plane 
of the card approaches the position of being perpendicular to 
the earth’s resultant magnetic field, but the two turns N. to 
Hi. and E. to 8. are symmetrical, the deviations being in 
opposite directions in these two quarter turns. 

In going from 8. to W. and W. to N. to complete the 
circle, the card is tilted towards the north, that is, its plane 
comes more nearly into the direction of the earth’s resultant 
magnetic field, and the control of the needle is always 
increased by the tiit, and the deviations due to tilt are again 
symmetrical in the turns 8. to W. and W. to N., and are in 
opposite directions in the turn. 

The equation of equilibrium for this southerly half of the 
complete circle is slightly different from the former, as the N. 


a then becomes known. 


calculated from cos @= 


Fig. 3. Fig. 4. 


Hm Sn Q Con © 


polar half of the card is dipping. In fig. 4 the turn S. and 
W. has just begun and the equilibrium equation is : 


H cos ¢' sin (6+) = V sin 0 cos (6 +2) 
+ Hsin qd’ cos @ cos (6+ 2), 


or tand.sin 8 
tan (b+ 2) = eee ts + cos@tang’. . . (2) 


A466 Mr. 8. G, Starling on the Equilibrium of 


It is easier for these courses to reckon ¢ and ¢’ from the 
south, although in plotting the results they are considered 
to be 180° up to 360° E. of N. 

Calculation of results—The following Tables are obtained 
for a position where the dip is 67°, so that 


tan d = tan 67° = 2°36. 


For values of @ less than 15° it is taken that ¢ and q’ are 
equal, but for values of @ above 15°, the value of @ is 
calculated from 


tan 
cos? = -——_, or tand= z 


tan ® 


Tt must also be noticed that on the turns N. to E. or H.toS., 
that is, when the north side of the card is upwards, there 
may be a position for which «>d, and the N. pole of the 
magnet is below the horizontal line OQ (fig. 1). In this 
case (b—wz) becomes (x—@). This is, in fact, generally the 
case ; for even when 0=20°, we obtain by putting d=a 
in (1), that 

tan d sin @ 
cos ¢’ 
sin ¢'=tand . tan 6, 
=P rao tam, 
= 0°860, 


or d =60° approx. 


= cos 0 tan ¢’. 


For values of @ greater than 23°, a is always greater 
than ¢' for 
sin d’ = 2°36 tan 23°, 


=O Or 24 — nO: 
> == 00% 


Thus if @ is more than 23°, tand tan 6 must be greater 
than unity, and there is no possible value for ¢’, such that 
¢'=x', which means that 2’ must always be greater than ¢’, 
and the North polar end of the needle is below the horizontal 
OQ from the start of the turn. This appears also from the 
curves. As an example, the calculation for a tilt of 2=70° 
is given, and only a summary of the results for the other 
angles of tilt from 10° to 85°. : 


the Magnetic Compass in Aeroplanes. A67 
N. to E., @=70°. 
| 
_ | 2°36 sin 70° & ; Se ale ei Gat : 
ra FY - cos 70° tan g’.| tan (v7 —@). |(aw—).| tan = cos 70° Gennes: 
0° 2-22 0 2-22 66° 0 0°] 66° 
10 2:25 0-060 9°19 65 0°514 27) 92 
20 2:36 0:129 2-93 66 1:06 47 | 113 
30 2-56 0-197 2:36 67 1°69 59 | 126 
40 2-90 0-287 2-61 69 9-45 68 | 137 
50 3-46 0-407 3:05 72 3:49 74 | 146 
60 4-43 0-592 3:84. 75 5:06 79 | 154 
70 6:48 0-94 5-54 80 8-08 83 | 163 
80 12°8 1:93 10-9 85 16°6 86 | 171 
85 25:5 3:92 21:6 87 33:4 88 | 175 
Seu Nos Ges 10s: 
9:36 sin 70° hs f 
ler carnt | co (0° tang’.) tan(@+2z). | (6+). | 9. i 
0° 2-22 0 2:22 66° 0° | 66° 
10 9-25 0-060 2-31 67 27 | 40 
20 2°36 0-129 2-49 68 Al 
30 2°56 0°197 2°76 70 59 11 
40 2-90 0-287 3:19 "3 68 5 
50 3-46 0-407 3°87 76 74 2 
60 4:43 0-592 5-02 "9 79 0 
70 6:48 0:94 7-42 82 Soha aia 
80 12°8 1:93 14-7 86 86 0 
85 O55 3:92 99:4 88 88 0 
Grieg Loe: 15° 20°. 23°. 302; 40°, 
q'. | p. d. Q. d. 
N./s. ||n./ 8 v.| gs N.| § Neca enone Ils: 
0 | 22/22//31| 31 | 0/39! 39 || 0/43| 43 | 0, 50/50) o| 58 58 
10 | 24/21 |/34| 28 | 11/44! 34 |/11/ 49! 37 | 11| 57/43/13! 69 |47 
20 | 24/19/37] 25 | 21149] 29 || 211/54! 32 | 23| 66/35/125| 792/38 
30 | 26 | 17 || 88 | 22 | 31) 53} 25 |) 31 | 59] 27 || 384) 75) 28) 387) 912) 29 
40 |23|14||/39| 18 | 42/58] 1932/42/65] 21 144) 81/221/ 47/102 |23 
50/18} 11]| 38] 14 ||52)60| 15 52/71] 16 54} 93/17] 58/1152) 15 
60117] 81/36] 11 ||61/62) 12 ||61)75| 13 | 64/104] 11]/ 66/128 |11 
70/13! 6|| 29 71\59) 72/71/81| 8 72/119] 8\l741152 | 7 
80| 7| 3||17 81/47| 3/81/86] 3481/1438} 4|/82/161 | 4 
g5/ 4/ 1/| 9I- 85 | 30 2185186} 2186] 161| 1]/86/170 | 2 


468 Mr. 8. G. Starling on the Equilibrium of 


é. 60°. (02: 80°. 


] Fagen a a 
Ne Sse N.| 8. | NE Sem N. | 8. 


o| o | 64 | 64 | o| 66| 6 | o|. 67] 67 | o | 67 | 67 
10 | 193| 824] 452]] 27 | 92] 40 |] 45 | 112] 23 ||64 |181 3 
0 | 36 |100 | 31 || 47 | 113 | 21 || 64| 132] 5 \l77 Vian eee 
30 | 49 |113 | 20 || 59 | 126) 11 || 73] 142 )—3 |\Sa) 150 Mle 
40 | 59 |125 | 13 || 68 | 137! 5 || 78] 150|—6 |/84 | 1552] —12 
| 82 | 156 |—7 ||86 |1603|—11 
| 84 | 161 |—5 | 87 |1643|/— 9 
| 86 | 167 |—4 |88 |169 |— 6 
| 88: 173 | —2 189 ag eeenes 
| 89 | 177 | —1 1893) 1773)— 1 
Hi! 


The values of z, the deviation of the compass from true 
magnetic north as seen upon the card, are now plotted against 
o’ the true compass-course of the aeroplane, the values of x 
when @¢’ is greater than 90° being derived from the Tables, 
as before described. 

Discussion of Curves.—It will be seen (fig. 5) that, as the 
machine turns from N. to H., the equilibrium position of 
the compass may be anything from true magnetic N. to true 
magnetic 8., according to the angle of tilt at which the machine 
takes the curve. For tilt below 23° (that is, the complement 
of the angle of dip) the error never reaches 90°, but at a tilt 
of 23° the compass-card is perpendicular to the total earth’s 
magnetic field when the machine is flying H. Consequently, 
there is no controlling field whatever, and the card will then 
set anywhere. This vanishing of control is shown by the 
instability of the compass, as indicated by the fact that the 
23° curve bends sharply up (or down) to reach the 180° or 
(0°) point, when the magnetic course reaches 90°. Itisa 
general rule that the steeper the curve at any point, the less 
will be the stability of the compass-card at that point. 

For tilts above 23°, it is obvious that the compass 
becomes exactly reversed when flying E., for the direction 
of the resultant magnetic field of the earth is now pulling 
the needle into the reverse of its usual direction with respect 
to the bowl (fig. 6). This is indicated by the fact that the 
curves for tilts above 23° run to the 180° point on the course 
90°. It may also be noticed that as the tilt approaches 90° 
the error on the U° course approaches 67°, in fact, at this 
point the card approaches in action to a dip circle. 

On the courses 180° to 360° the deviation due to tilt is 
very much less than for the 0° to 180° courses, the reason 


\ 

j 

\ 

5 
- 
q 

) 
W 
i] 
"i 
O, 
FY 
ie 
ety 
iy a 
ey 
tae 


the Magnetic Compass in Aeroplanes. 469 


Deviation from Magnetic North in card degrees 
Soonrnomorn wows DON A ® On 
2S) tsi ykte) 4S On SS roo) foo) 


ae otis 
Bits 

EEE HEEHE 

mae KRONE 
EL oe AANA 
TICS 
aes CNY 
“Saas np ye A 
ACHE 
NSN if 
ANOS 


AN 5 
RAIN AAG 
AAR VY 

NC Nit oa 


‘g ‘SI 


t i 

fe ee 

wae Poe inl 

nee Soames 

ry ETON Os 
| an [a 
eee Re 

sea CS 

Sh IS 

segues? S\ Qeeuues 
maT | 


being that the card comes more nearly into parallelism with 
the earth’s resultant magnetic field. The card is more stable 
and the period of oscillation should be diminished. The 


470 Mr. 8. G. Starling on the Equilibrium of 


deviation becomes reversed at B and B (fig. 5), for a reason 
to be explained later. 


Fig. 6. 


In actual flight the positions indicated by the curve wiil 
never be actually reached because of the changing course of 
the machine, but they give an indication of the movement, 
since they show the point towards which the card tends to 
move at each course. 

For example, with a tilt of 80° or 85°, there is a powerful 
couple acting on the needle, tending to reverse its direction, 
and this acts for a large part of the whole turn. 

Liffect of liquid in Bowl.—In the case chosen for a complete 
circular turn in the direction N.-E.-S.-W.-N.., the deviations 
alternate in direction, being first H., then W., again W., and 
finally E. Ifthe rotation of the machine tends to drag the card 
after 11 by reason of the viscosity of the liquid in the bowl, this 
will sometimes increase the error and sometimes diminish it. 
Thus on the N. to E. (0° to 90°) turn, the effect of the liquid 
will be to increase the error, and it might be expected that 
this is the worst of all the turns—in fact, the card would pro- 
bably go round faster than the machine if the tilt is great. 

On the E. to S. (90° to 180°) course, the action of viscosity 
would be to check the motion of the card due to magnetic 
effects, and this course should not be so bad as the first. Of 
the last two, the liquid tends to correct the card from 180° 
to 270° and to assist in disturbing it from 270° to 360°. 

Method of correcting error.—lIt is clear from the preceding, 
that the vertical component of the earth’s magnetic field (V) 
is responsible for a large part of the disturbance. If, 
therefore, this could be balanced, an improvement would be 
effected. Now, the component of V in the plane of the 
compass-card is V sin @, and therefore an arrangement which 
will produce a field at the card equal and opposite to this is 
required. A pair of bar magnets NS (fig. 7) attached toa 
frame, which can rotate about an axis passing through the 
compass and in the median line of the aeroplane, produce a 
vertical field, say F, at the compass. On rotating these 
about the axis through an angle equal to the angles of tilt, a 
field F sin@ will be produced in the plane of the card. If 


the Magnetic Compass in Aeroplanes. ATI 


this is adjusted to be equal and opposite to V sin @, it will 
be so for all angles of tilt. It is suggested that this rotation 
of the pair of magnets might be produced by attachment 
with the steering-gear or with a stabilizer. 


Fig. 7. Fig. 8 


As there are certain mechanical difficulties in the way of 
this, it may be preferable to produce the correcting field by 
raising one magnet and lowering the other, the magnets 
being connected together by a cord passing over pulleys(fig.8). 

In order to ascertain whether this correction (fig. 7) is 
suitable for allangles of tilt, the field in the plane of the card 


4'magnets 


le 
-2 
ww 
YQ 
ez 
8 
Ie 
4 
eoecee ae) 
18cm {2 
18 Y 
oF = 
66 
3 
2 
-} 
30 


for a travel of 12 cm. on either side of the zero position for a 
pair of magnets 20 cm. long at a distance of 23 cm., one on 
either side of the compass, is drawn (fig. 9). A sine curve is 


472 Mr. 8. G. Starling on the Equilibrium of 


shown in dotted line alongside of it. From the parallelisin of 
the two, it is apparent that a fair approximation to a proper 
correction might be obtained, if the movement of the 
magnets could be arranged to be proportional to the tilt of 
the machine. 

Effect of earth's horizontal component alone.—If the earth’s 
vertical component of magnetic field could be entirely elim- 
inated, the errors of the compass due to tilt would be 
reduced, but not entirely eliminated. On putting V=0 in 
the equation from which equations (1) and (2) are derived, 
we have 

cos g’ sin (@—z) = sin qd! cos 8 cos (P—2), 
or tan (¢6—=2z) ='cos OP tan ¢d’. .-. eee 


It is now possible to find in a manner somewhat similar 
to the former, for the values of ¢ and of cos@tan®@’ are 
already known, so that (6—w«x) and w can be tabulated. 
These are now done for the values of tilt 20°, 40°, 60°, 80°, 
aud 87. 


Bal 20°: 40°. 60°. 80% 85°. 

res dl Ee ee Re ee | 

¢'-\(p—2).| p. | 2. ||(P—2).| o. | 2. (G—a).) g. | 2. Ke 2), g. | £. (P—2).| o. 

0| 0 | 0,0] 0 | Oo) 0 | 0) 0] © | 6) 0s 
10} 9 |11/2| 8 |13] 5]| 5 |19g]i4a|) 2 |45]43] 1 [64/63 
20) 19 |21)2]| 16 |25| 9} 10 |36 |26 || 4 |64/60|) 2 |77|75 
30| 28 |31| 3 || 24/37/13) 16 |49 [33 || 6 |73le7] 3 |81)7 
40| 38 |42/ 4 || 33 |47|14}) 23 |59 136 || 8 |78/70]) 4 | 84/80 
50| 48 |52|4 || 42 |58/16]) 31 |67 |36 || 11 |82/71]) 6 | 86/80 
60| 58 |61) 3 || 53 |66|13]) 41 |74 |33 || 17 |84/67|| 9 |87|79 
70| 69 [71/2] 65 |74| 9] 54 |80 [26 | 25 |86/61]) 13 |88)75 
30| 79 |81|2]| 77 |82| 5] 71 |85 |14 || 45 |88|43}) 27 | 89/62 


These curves (fig. 10) are now plotted to the same scale 
as the previous set. They then make it clear that whereas 
the total disturbance is semicircular, being of the same sign 
(positive) from 270° to 90° and negative from 90° to 270°, the 
disturbance due to the tilt for horizontal component of the 
earth’s field acting alone is quadrantal. Itis the combination 
of the semicircular disturbance due to the earth’s vertical 
component and the quadrantal disturbance due to the 
horizontal component that gives the want of symmetry 
about the 0° and 180° ordinates in the first set of curves. 
It is also seen why the curves at B become negative, for 
here the quadrantal effect predominates, at the large tilts 
the disturbing effects of the earth’s vertical field being 
small, since the card approaches to parallelism with the 
resultant field. | 


the Magnetic Compass in Aeroplanes. 473 


Deviation fromMagnetic North in card degrees. 


‘OL “SUT 


i | 


eo 


— 
eam 
(a Ea aes 


Ce 
E+ 


EZ 


Suggested quadrantal compensation.—To correct perfectly 
for the quadrantal error, soft iron is necessary. It must 
also be possible to vary the intensity of its effect as the tilt 
changes. ‘These conditions would be fulfilled by a pair of 

Pals Mag. 8. 6. Vol. 32. No. 191. Nov. 1916. 2K 


A74 Mr. 8. G. Starling on the Equilibrium oj 


soft iron bars, placed under or over the compass-bowl and 
hinged together (fig. 11), and so arranged that a cord would 


Bie. 1. 


PG 


> 


raise the ends of the bars N S as the tilt increases. As no 
measurements are available, it is not possible to say that the 
raising and lowering could be made to give a correcting 
field which would vary suitably with the tilt; but the 
correction would be of the right sign and would approximate 
to a sine curve as the course changes. As the curves of 
quadrantal deviation are somewhat similar to sine curves, it 
might be possible to obtain an approximate correction in 
this way : 

In the event of a quadrantal correction of this kind being 
too small to be effective, it is also suggested that a pair of 
soft iron blocks placed one on either side of the compass-bowl 
would, by the attraction between the needle and them, 
produce a deviation of the right character. It would thus 
be necessary to move them nearer to the bowl as the machine 
tilts, the distance away when the machine is level being so 
great that they do not appreciably affect the position of the 
compass-card. 


Measurements on corrector of type fig. 7.—Two vertical 
magnets, one on either side of the compass-bowl, are carried 
by an aluminium bar (fig. 12). The whole can rotate about 


the Magnetic Compass in Aeroplanes. 457 


the compass-bowl of an aero-compass. ‘Two sets of magnets 
are used, one, 2 pair of 8-inch magnets, and the other, 4-inch 
magnets. The lubber line being set N. and 8., the magnets 
are placed in position one at a time and their level adjusted 
with the arm horizontal, until the card is in equilibrium in 
the magnetic meridian. 

For various tilts of the arm @, the deflexions are cbserved. 
The tangents of the deflexions are plotted against 0, thus 
giving a number proportional to the deflecting couple. The 
values of 6 and of —@ are observed, but the deflexions for 
the latter are so near the former that, after the first obser- 
vations, only those for one direction of rotation of the arm 
(+6) were observed. 

For values of @ near 90° the deflexion approaches 90°, and 
the readings of the deflexion become difficult to make with 
sufficient precision to give the tangent of the deflexion 
accurately. 


8-inch magnets. 


d=16 cm. * d=18 cm. ad=20 em. 

Compass | tan tan tan 

0. nae Defl. go. || 9 | Card.| Defl.| goq || 9. | Card. | Def.) 27 
0 180° 0 10 0! 180 0 | 0 0 | 180 0 i 
5 206 26 | 0-48 d | 199 19 | 0°34 +3) A aA! ue 

10 228 AS tel 10 | 215 35 | 0O'70)} 10 | 210 30 | 0°58 
20 244 64 | 2-1 20 | 236 56 | 1:5 || 20 | 2380 OI lee 
30 252 Te Waal 30 | 245 65 | 21 30 | 240 60 | 1:7 
40 254 74 | 35 40 | 250 70 | 2:7 40 | 244 64 | 21 
50 258 TS 4-7 50 | 252 (OA Beal 50 | 248 68 | 2°5 
60 260 80 | 5:7 60 | 254 74 | 3:5 60 | 249 69 | 26 
70 260 SON org 70 | 254 74 | 35 70 | 250 70 | 2:7 
80 260 80°), 557 1 80 (255175) | 3°71) 80 1250) K-70)" 2:7 
90 260 80) a:% W090 | 255')) 75e.| 3:7 |) 901 250) | 70) ) 2:7 

4-inch magnets. 

10 211 ol | 06 10 | 205 25 | 0-41|| 10 | 200 | 20 | 0:36 

20 233 Doe lecs 20 | 225 45 | 10 20°) 213 33 | 0°65 

30 241 61 | 1:8 380 | 234 54 | 14 || 80 | 224 44 | 0:97 
40 246 66 | 2:2 40 | 240 60 | 17 40 | 230 10 id a 
50 250 70 | 2:7 50 | 243 63 | 2:0 || 50 | 235 55 | 1-4 
60 250 70 | 2:7 || 60 | 246 66 | 2:2 || 60 | 288 58 | 16 
70 251 71 | 29 70 | 246 68 | 2:2 70 | 240 60 | 1:7 
380 253 73 | 33 || 80 | 247 67 | 2.4 || 80} 241 61 | 18 
90 253 73 |.3°3 || 90 | 247 OF QO 242 62 | 1:9 


476 ~—=-Prof. H. 8. Carslaw on Napier’s Logarithms: 


With a limiting tilt of the aeroplane of 90°, the vertical 
component of the earth’s magnetic field produces its maximum 
effect, the magnetic intensity being 0°18 tan 67°=0°18 x 2°3. 
For 6=90°, the deflecting couple should equal this 
(0°18 x 2°3) ; but the field due to the compensating magnets 
is then 0°18 x tan (deflexion), therefore for proper compen- 
sation tan (deflexion) =2°3. 

This condition is seen from the curves (fig. 9, p. 471) to 
be very nearly fulfilled with the 4-inch magnets ata distance 
d=18 cm. The sine curve having maximum value 2°5 is 
plotted as a line of dots, and is shown to be very nearly 
coincident with the corrector curve; showing that the com- 
pensation for vertical component would be very nearly perfect 
with 4-inch magnetsat18em. Any discrepancies are due to 
the fact that the field produced by the compensating card is 
not uniform, and the card magnets are of considerable size. 


L. Napier’s Logarithms: the Development of his T heory. 
By Professor H. 8. Carstaw, Sydney, N.S. W.* 


§1. Iyrropuctory. 
4 Dates paper deals with Napier’s idea of a logarithm. 


In my view there are three distinct stages in the 
development of this idea in his work. In the first he is 
concerned with a one-one correspondence between the terms 
of a Geometrical Progression and the terms of an Arith- 
metical Progression. There are traces of this in the 
Constructio t in his use of the series 

1 Ly? 
107, 107 (i- i): 10" (1- = ke, 
and in the word logarithm itself, derived from Aoyos aptO pos, 
and generally taken to mean ‘‘the number of the ratios.” 
In the second he has passed from this correspondence, and 
his logarithms are given by the well-known kinematical 
definition, which forms the foundation of the theory of the 


* Communicated by the Author. Read to the Roy. Soc. New South 
Wales, Aug. 2, 1916. 

Tt The Mirifict Logarithmorum Canonis Constructio was published in 
1619, two years after Napier’s death, but had been written several years 
before his Mirifict Logarithmorum Canonis Descriptio, published in 
1614. I shall refer to these works as the Constructio and the Descriptio. 
The Descriptio was translated into English by Wright (1616), and 
Filipowski (1857), the Constructio by Macdonald (1889). The former is 
a rare book, both in the original and in translation. Several of the more 
important pages of the latter are reproduced in the ‘ Napier Tercen- 
tenary Memorial Volume,’ Plates I-VI. (London, 1915). 


the Development of his Theory. 477 


Constructio. In the third, referred to in the Appendix to 
the Constructio, he has reached the idea of a logarithm as 
defined by the property :— 

The logarithms of proportional numbers have equal differences, 
with the additional condition that the values of the logarithms 
of two numbers are given. 

In the second and third stages he has obtained, what we 
would now call, a function of the independent variable—the 
number—, but the function of the third stage is more general 
than that of the second, which it includes as a special case. 

If this view is correct, the statement that “‘ Napier’s theory 
rests on the establishment of a one-one correspondence 
between the terms of a geometric series and the terms of an 
arithmetic series” * should not be taken too literally. 
Further, the custom of employing the term ‘“ Napier’s 
logarithms ” to describe only the logarithms of his Canon is 
unfortunate. It will be seen in the course of this paper that 
logarithms to the base 10—as we now know them—are 
Napier’s logarithms just as much as the logarithms of his 
Canon. 


THe First STAGE. 


§ 2. The idea that multiplication and division could be 
reduced to addition and subtraction by the correlation of a 
geometrical series and an arithmetical series was not a new 
one. Aristotle was familiar with it, and since his time many 
mathematicians had returned to it. If we take the series 


Wy ee) Os TO dy By” as 15. 
2, A, on 16, 32, 64, 128, 256, Leak 32768, 


the product of 128 and 256 in the geometrical series can be 
read off as 32768, which corresponds to 15, the sum of 7 
and 8, in the arithmetical series. 

The Swiss Biirgi in his Arithmetische und Geometrische 
Progress Tabulen{, constructed some time between 1603 
and 1611, but first published in 1620, used the series 


Oi<0:.-10'x 1, 10 x 2, LOM es. 


108 10°(14 55 108 “asa at 10°(14 eae 
104}? 108 108 


- * Cajori, ‘The American Mathematical Monthly,’ vol. xxii. p. 71 
1916). 

+ A facsimile of the title-page of Burgi’s work, and of one of the 
pages of his Tables, will be found in the ‘ Ni apler Ter centenary Memorial 
Volume,’ Plates XII. and XIII. Comparison with the references in 
Cantor’s Geschichte der Mathematik, Tropfke’s Geschichte der Elementar- 
Mathematik, and Braunmihl’s Geschichte der Tr igonometrie will show 
that in none of these is the title quoted correctly. 


478 Prof. H. S. Carslaw on Napier’s Logarithms : 


His tables cover the range 10° to 10°, and for all practical 
purposes are as satisfactory as Napier’s Table of Logarithms 
of 1614. If Napier had simply used the idea of the cor- 
respondence between the terms of a geometrical series 
and the terms of an arithmetical series, his work could 
not be regarded as so great an advance upon Biirgi’s as it 
really is. 

But it is clear that at the beginning of his labours, which 
extended over a period of 20 years, Napier’s mind was 
working on the same lines as Biirgi’s, and that he used the 
series 


0, 1, pd ahh 


L 1 _ i \4 
7 t REL ( Bia Le sss! 
OR 10 (1 aio (1 \ a 


The geometrical series occurs in the Constructio. He 
employed it in the calculation of his logarithms, but neither 
then, nor later, are his logarithms the terms of the corre- 
sponding arithmetical series. His word logarithm (see 
above, § 1) is evidently a survival of this stage of his work. 

Napier meant his Tables to be used in calculations 
involving the trigonometrical ratios. In his time, the sine, 
cosine, &c., were lines—or, more exactly, the measures of 
lines—in a circle of given radius. Napier took the radius 
as 10’. It may be that Burgi chose 10° in his Tables for a 
similar reason. With our notation, Napier’s numbers would 
correspond to 7-figure Tables of Natural Sines, &. If 
greater accuracy were required, the radius was taken as 10”, 
and sometimes even a higher power of 10 was used. These 
sines, &c., following Glaisher*, we shall refer to as line- 
sines, Xe. 


THE SECOND STAGE. 


§ 3. Napier opened out entirely fresh ground when he 
passed to his kinematical definition of the logarithm of a 
sine or number. By this definition he associated with the 
sine, as it continually diminished from 10’ for 90° to zero 
for 0°, a number which he called its logarithm; and the 
logarithm continually increased from 0 for the sine of 90 to 
infinity for the sine of 0°. 

The fundamental proposition in Napier’s theory in the 


* The Quarterly Journal of Pure and Applied Mathematics, vol. xlvi. 
p. 125 (1915). To this paper I am indebted, not only for a most con- 
venient notation for the different systems of logarithms, but also for 
an account of Speidell’s work, hitherto inaccessible to me. 


the Development of his Theory. 479 


Descriptio (1614) and the Constructio (1619) is to be found 
in Prop. 1 of the Descriptio :— 

‘The logarithmes of proportionall numbers and quantities 
are equally differing” *. 

And in Section 36 of the Constructio it appears as :— 

“The logarithms of similarly proportioned sines are equi- 
different.” 

Glaisher has introduced a convenient notation nl, for 
Napier’s logarithm, in this system, when the radius is 10". 
He also uses Sin, # for the line-sine of the angle x, when the 
radius is 10", and he keeps the symbol sin w for the sine in 
the modern sense of the term. With this notation we have 
Sin, 2 

Oi 

In this paper I follow his notation, and log, x is used in 
its modern sense for the logarithm of w to the base e, the 
system commonly called hyperbolic logarithms. 

The fundamental theorem, referred to above, can now be 
stated as follows :— 

five: b=c:d, then 

nia = nlsb— nlc — lady iawn yay. (CL) 
Also we are given that 
TUL OR 0) 1S) ead 


Napier’s Canon consists of a Table of Logarithms in which 
(1) and (2) are satisfied. His definition of the logarithm 
by means of the velocities of two points moving in two 
different lines leads us to the formula 


He nO logy (=) : 


But, of course, neither this, nor the fact that his function 
nl,z has —1 for its differential coefficient, when «=10’, 
could be known in his time. 


sin v= 


THE THirp STAGs. 
§ 4. Since wo: uw=v:1, we have 
nl, (wv) —nl,u=nl, v—nl, 1. 
Thus nl, (uv) =nl,u +n), v—nl, 1, 
and it must be remembered that nl, 1 is not zero. 
When r=7, nl, 1=161180896°38 (cf. Constructio, Section 
Sr ecleay. nl, (u/v) =nl, u—nl,v+nl, 1. 


* In quoting the Descriptio I follow Wright’s version, and for the 
Constructio I adopt Macdonald’s, 


480 Prof. H. 8. Carslaw on Napier’s Logarithms: 


Thus multiplication and division are changed into addition 
and subtraction. But the logarithms of numbers with the 
same figures in the same order cannot be read off from one 
another, since in this system, 


nl, (10"a) =nl,,a—m/(nl, 1—nl, 10), 


and nl, 1—nl, 10=23025842°34 (cf. Constructio, Section 53). 

It is obvious that if a system of logarithms could be 
devised in which the logarithm of unity is zero and the 
logarithm of 10 is unity, the calculations would be immensely 
simplified, and the table curtailed; because one of the chief 
defects of Napier’s Canon, as well as of Biirgi’s Tables, was 
that, if the numbers did not come within the range covered 
by it, more or less awkward calculations were needed to 
overcome this difficulty. 

Napier’s Canon was first printed in the Descriptzo (1614). 
After his death in 1617 the Constructio was published by 
the care of his son. It had been written several years before 
the Descriptio. To this work was added an Appendix, by 
the hand of Napier himself, ‘ On the Construction of another 
and better kind of Logarithms, namely one in which the 
Logarithm of unity is 0.” This Appendix begins with the 
words :— 

“ Among the various improvements of Logarithms, the more 
amportant is that which adopts a cypher as the Logarithm of 
unity, and 10,000,000,000 as the Logarithm of either one tenth 
of unity or ten times unity. Then, these being once fixed, the 
Logarithms of all other numbers necessarily follow.” 

It is clear from Napier’s words that, when he wrote the 
Appendix, not only did he see the advantage of such a 
system, but he was in a position to draw up a Table of 
Logarithms in which these conditions would be satisfied. 
Indeed, he gives three distinct methods of finding these 
logarithms. The kinematical definition of the logarithm was 
superseded, and the correspondence between the terms of a 
geometrical series and the terms of an arithmetical series 
was left far behind. This is the third and final stage of his 
work. 


BriGGs AND NAPIER. 


§ 5. In the change from the logarithms of the Canon to 
this “ better kind of logarithms” Briggs was associated with 
Napier; but, chiefly because of the unsatisfactory account of 
the matter given by Hutton in his ‘ History of Logarithms’ *, 


* Hutton’s ‘Tracts on Mathematical and Philosophical subjeota? 
vol. i. Tract 20. 


the Development of his Theory. 481 


‘the share of the former in the discovery has been exaggerated. 
The fault is not due to Briggs; and though his reference 
to the question in the preface to the Arithmetica Logarithmica 
(1624) is familiar, I reproduce it again here :— 

‘‘T myself, when expounding publicly in London their 
doctrine to my auditors in Gresham College, remarked that 
it would be much more convenient that 0 should stand for 
the logarithm of the whole sine, as in the Canon Mirificus, 
but that the logarithm of the tenth part of the whole sine, 
that is to say, 5 degrees 44 minutes 21 seconds, should be 
10,000,000,000. Concerning that matter I wrote immediately 
to the author himself; and as soon as the season of the year 
and the vacation time of my public duties of instruction 
permitted, I took journey to Edinburgh, where, being most 
hospitably received by him, I lingered for a whole month. 
But as we held discourse concerning this change in the 
system of logarithms, he said that for a long time he had been 
sensible of the same thing, and had been anxious to accom- 
plish it, but that he had published those he had already 
prepared, until he could construct tables more convenient, if 
other weighty matters and his frail health would permit him 
so to do. But he conceived that the change ought to be 
effected in this manner, that 0 should become the logarithm 
of unity, and 10,000,000,000 that vf the whole sine ; which 
I could not but admit was by far the most convenient of all. 
So, rejecting those which I had already prepared, I com- 
menced, under his encouraging counsel, to ponder seriously 
about the calculation of these tables.” 

Napier also mentions his discovery of the new system in 
the dedication of his Rabdologia (1617) in a passage quoted 
in my previous paper *. 

It will be seen from Briggs’s own words that the modi- 
fication which he suggested to Napier was to keep the 
logarithm of the radius as zero, but to take the logarithm of 
one-tenth of the radius as 10,000,000,000. His reference 
to the Canon is sufficient to show that he does not look upon 
the radius as unity. In the construction of the Tables of 
Logarithms, after Napier’s death, he takes it as 101°, and it 


* See also Macdonald’s English translation of the Constructio, p. 88. 
This paper may be regarded as a supplement to a paper entitled “ The 
Discovery of Logarithms by Napier of Merchiston,” Journ. of Proc. Roy. 
Soc. N.S.W. vol. xlviii. p. 48 (1914), which deals chiefly with the 
construction of Napier’s Canon. I take this opportunity of amplifying, 
and to some extent correcting, the references in that paper to Briggs’s 
share in the discovery of the “better kind of logarithms.” <A paper 
covering much the same ground as the above will"be found in ‘The 
Mathematical Gazette,’ vol. viii. (1915). 


482 Prof. H. S. Carslaw on Napier’s Logarithms : 


is for this reason that the characteristics 9, 8, &c., are to be 
found in the logarithms of the sines, &e. 

Using the notation bl, « for the logarithm of x in the system 
suggested by Briggs when the radius is 10”, we have 


bly a—bl, b=bl, e—bl, d, 
when a: b=e:d. 
Also bl, 10° OF amadhiol,. 107-2 11Gte. 
In this system we have 
bl, (uv) =bl,uw+ bl, v—bl, 1, 
bl, (u/v) =bl, u—Dbl, v + bl, 1. 
Also bl4910?° = 10b14910—9bl,)1=0. 
blo 10°= 9bl,910 — 8bl,91=10". 
Thus by lO ole amd: bli lie 10 x TO 


The advantage of the new system consists in the fact that 
the logarithms of numbers with the same figures in the same 
order could be read off from each other, since we have 


bl, (10"a) =bl,.a—m x 107°. 
§ 6. The change upon which Napier had resolved, previous 


to Briggs’s visit, was a much more important one. He 
“conceived that the change ought to be effected in this 
manner, that 0 should be the logarithm of unity, and 
10,000,000,000 the logarithm of the whole sine.” And in 
the Appendix we see that he often passes from logarithms of 
sines, and drops all reference to the radius. In the new 
system, logarithms were to be dejined by the relations :-— 
If a: b=c:d, then 


nla—nlb=nle—nld 
with nl1=0 and nl10=10”, 


It need hardly be added that 10!° was taken for the 
logarithm of 10 instead of unity, for the same reason that 
10’ (or 10") was taken for the radius in dealing with the 
trigonometrical ratios. 

Later, Briggs takes the logarithm of 10 as unity, and 
itnedmees the notation of decimal fractions in his Tables, a 
notation employed, probably for the first time, by Napier 
himself. 

If this account of the growth of the idea of a logarithm in 
Napier’s work is correct *, it seems unfortunate that the 


* See also Gibson’s paper in the ‘Napier Tercentenary 1 Memorial 
Volume,’ pp. 111-187. 


the Development of his Theory. 483 


term Napier’s logarithms is usually confined to the logarithms 
of his Canon. His “better kind of logarithms” actually 
consists of the logarithms now in daily use—the logarithms 
which we call logarithms to the base 10. In some text-books 
they receive the awkward name of Briggsian logarithms. 
Certainly Briggs calculated them, and the rapidity and 
industry with which he performed this immense work in 
computation will always be the admiration of mathematicians. 
But the discovery of the system was Napier’s, and the 
logarithms are as much Napier’s logarithms as those of his 
Canon. 


SPEIDELL’s New LogarirHMes (1619). 


§ 7. In most accounts of the discovery of logarithms 
reference is made to Speidell’s ‘ New Logarithmes’ (London, 
1619), and it is stated that they contain the first table of 
logarithms to the base e*. Attention is also usually called 
to the fact that, while logarithms to the base e are frequently 
spoken of as Napierean logarithms, they are quite different 
from the logarithms of Napier’s Canon; and it is pointed 
out that the place of the number e in the theory of logarithms 
and the possibility of defining logarithms as exponents were 
discoveries of a much later day. These two statements, at 
first sight, seem inconsistent. A word or two regarding 
Speidell’s system will make the matter clearer, and will also 
confirm the view I have taken above as to Napier’s final 
conception of the logarithm. 

Speidell’s ‘ New Logarithmes,’ like Napier’s: Canon, refer to 
the trigonometrical ratios. Using Glaisher’s notation sl, 
for Speidell’s logarithm of x when the radius is 10”, we have 


sl, e=107t!— nl, a. 
Tt follows that 


sl, (uv) =sl,u+sl,v—sl,1, 
sl, (u/v)=sl,u—sl,v+sl, 1, 


and sl,1 is not zero. 

The sole advantage of this system was that it avoided the 
use of negative quantities in calculation with logarithms. 
Such quantities were then outside the range of the “ vulgar 
and common arithmetic.” 


* In Glaisher’s paper already referred to, he published the interesting 
discovery that an Appendix (1618) to Wright’s English translation of 
the Descriptio contains a table of hyperbolic logarithms by an anonymous 
author, whom Glaisher identifies with Oughtred. 


484 Prof. H. 8. Carslaw oa Napier’s Logarithms : 


Since nl, c=10" log, (= : 
we have x 
ch e=10" 419" log.( 7). 
Thus Sin, @ 


sl, Sin, #=10"*14 107 log, . ? 
=107(10+ log, sin x). 


In a sense Speidell’s ‘ New Logarithmes’ may be said to 
be hyperbolic logarithms, but the sense is the same as that 
in which the logarithms of Napier’s Canon are sometimes 
said to be logarithms to the base e-!. But this is a misuse 
of the term *. Still Speidell’s logarithms of sines, from the 
accident that the sine is now used in a different sense, have 
actually the same figures as our hyperbolic logarithms of 
sines. 

In the ‘New Logarithmes’ (1619) he takes the radius 
as 10°, so that these tables give 


sl, Sin; 7=10°(10+ log, sin 2). 
$8. But subsequently Speidell did publish a table of 


hyperbolic logarithms of numbers, which gives the values 
of 10° log. x for the numbers 1 to 1000. This table probably 
appeared either separately, or attached to an impression of 
the ‘New Logarithmes,’ in 1622 or 1623. In this system 
he takes 


sl,v=nl,1—nl, 2. 
It follows that 
sl, (uv) =sl,u+sl,v, 


sl, (u/v) =sl,u—sl, v ; 
E 10” 
nlp—10" los, (=) 5 


we have Sia — Ml looea.: 


and since 


But it is clear that in both Speidell’s systems of logarithms 
the connexion with hyperbolic logarithms is accidental, and 
the same is true of the logarithms discovered by Glaisher, 
to which reference is made at the beginning of this section. 

Like Napier and Briggs, Speidell sees that the fundamental 
property, that the logarithms of proportional numbers have 


* Cf. Glaisher, loc. cit. p. 146, footnote. 


the Development of his Theory. 485 


equal differences, can be taken as the starting-point of the 
theory ; and that, if the logarithm of unity is zero, the 
logarithms of the product and the quotient of two numbers 
are, respectively, the sum and difference of their separate 
logarithms. 


THE DIFFERENTIAL HQUATION SATISFIED BY THE 
LOGARITHM OF 2. 


§ 9. We have seen that the theory of the different systems 
of logarithms described in the previous pages rests upon the 
fundamental property :— 

Bia 7b—c sd* then 


da) — (6) =A(c) —A(d), 


where A(x) stands for the logarithm of «. 
The function X(#), therefore, satisfies the equation 


M.0-+h) —A(w) = (14 =) —r4). 


ah 
Math)—da) 1 M(1+ 5) =>) 
h Nia: h 


Proceeding to the limit h-0, of course: keeping « fixed 
we have 


\'(a)=—, where A=n'(1). 
Therefore (wv) =A log «+ B, 


and the system is made definite by adding two other 
conditions. 
In Napier’s Canon, writing p for the radius, we have 


nl «=A log «+B, 
with nl p=0, and nl’ p=—1. 


Therefore nee ee (*) 
In Briggs’s modification of the system, we have 
bla=A log «+B, 


with vi are oe sci, 
bl p=0 and bly =10 


486 On Napter’s Logarithms. 


Thus ee, ( p. ) 


110 ee 10 1 
blav=10 a0 = logw(2). 


And Napier’s final form is, of course, 


nl =10! logy z. 


Biirgi’s Arithmetische und Geometrische Progress Tabulen 
also come under the same law. If the terms in the Arith- 
metical Progression are taken as the logarithms of the terms 
in the Geometrical Progression, and Bl & stands for what I 
may call Biirgi’s logarithm of w, we have 


L 
log. (=z) 


Lv 
sau (ces =10leg,, + (55): 
oe i0') 
for Hee 


s being any positive integer. 
Finally, treating Napier’s series 


0, i, 2, 


he earn 1\ 
10", 10 (1-55): 101 Ta) aa 


in the same way, and denoting this logarithm by Nl az, 
we have 


1 Tale 
loge(1— Tor 
for ANS 
o=107(1— fe 2 


s being any positive integer. 


Sydney, August 1916. 


P 487] 


LI. On Multiple Reflexion. By L. StrBerstzin, Ph.D., 
Scientjiic Adviser to Adam Hilger, Lid.* 


| purpose of the present paper is to give a very simple 
method of dealing with reflexions from any number of 
plane mirrors. The subject has been taken up in connexion 
with some technical problems concerning the construction of 
the kind ef triple mirrors known as central mirrors. 
Consider first a single plane mirror. Let the unit vector 
n, represent its normal, drawn away from the reflecting side. 
Let the direction of the incident ray be given by the vector r, 
and that of the reflected ray by r,. The tensors 7, 7, of 
these vectors are irrelevant. It will be convenient, however, 
to make them equal. If both are taken as unit vectors, then 
their scalar product rr, will give at once the cosine of the 
angle included between the incident and the reflected rays. 


Now, by the fundamental law of reflexion, r;— r=AB has 


“OA 
the direction n, and the size — 2uyr, that is, 


ee ty (LT) et ee (CAC) 
or, using the dot as separator, 
r,=[1—2n, . ny |r=Qyr. 
Thus, the linear vector operator or the dyadic, 
0,=1—2n,.n,, 


when applied tf to the incident ray r, gives the reflected 
ray 1. In view of this property the operator 0, can be 
called the reflector belonging to the mirror in question. It 
is a pure versor, 1. é. it leaves intact the tensor of the operand. 


* Communicated by the Author. 

+ It may be applied to the operand either as prefactor, r,=Q,r, or as 
a postfactor, r,=rQ,, the operator for a simple mirror being self-conjugate 
or symmetrical. The operator for a multiple mirror, however, is not 
symmetrical, and to avoid confusion we shall use it always as a prefactor. 


488 Dr. L. Silberstein on 
In fact, squaring (A) and remembering that n,;?=1, we have 
rP=r+A4(nyr)?—A(nyr)?=7", 


identically. Thus, r being a unit vector, so is also ry. 
Notice in passing that 


0,0,=27?=1—4n,.n, +4(n,.n,)?=1—4n, .n,+4n,. mj, 


2. e. Q,7=1, which is an obvious property. 

Let, now, the ray r,, reflected from the first mirror, 
impinge upon a second mirror whose normal is ng, again a 
unit vector. ‘Then the ray r,, reflected from the second 


mirror, will be 
i) = O.¥; = OQ, 


where Q,=1—2n,.n,; and since Q, applied to the vector 
Qir gives the same result as 0,0, applied to r (associative 
property), no separating signsare needed. ‘Thus the reflector 
of the double mirror 1, 2 is simply Q=0,0,. Similarly, 
if rz 1mpinges upon a third mirror, the reflected ray is 
r3= Or, where 2 =0;0.0,, and so on. 

Thus the reflector of a multiple mirror consisting of any 
number « of plane mirrors, taken in the prescribed succession 
Ih ican ora ts 

1=1 
0=0,,.. 0,050; = 10, -. <a 


1=K 


where 
O,=1—2n;.1;,-. . ~. +. nner 


n; being the unit normal of the ith mirror*. If x is the 
incident and r’ the finally reflected ray, we have 


r=Or, << 4... = See 
and, by what has been said before, 
f=... |. 2 Ee 


If, therefore, r is a unit vector, so is r’, and if @ be the 
angle between the incident and the finally reflected ray, we 


have simply 
cos0=rOr.’ . .. . 2 en 


* Notice that, for a given multiple mirror, © will, in general, be 
different according to the order of succession of the component reflexions, 
that is, of the suffixes 7. In short, the “products” 030.9, etc., are 
associative, but, in general, not commutative. They become so only for 
certain multiple mirrors, z. e. for some particular arrangements of the 
component mirrors. If a finite beam of parallel rays r impinges upon, 
say, a triple mirror, some pencils may be reflected as 23Q.Q,r, others as 
39,0, and so on. Thus the several reflected pencils will, in general, 
diverge from one another. | 


Multiple Reflexion. A89: 


In particular, when the arrangement of the component 
mirrors is such that 


O=—1, 


then every incident ray will be sent back parallel to its own 
path. Such multiple mirrors are called central mirrors. 

By a well-known theorem of vector algebra, the general 
reflector (1), being a pure versor, could always be expressed 
by Q=a.it+b.j+e.k, where both a, b,c, and i, j, k are 
some normal systems of unit vectors, both right-handed, 
or both left-handed. No use, however, will be made here 
of this fundamental property, since obviously the most 
natural entities to represent the properties of any multiple 
mirror are the normals ny, Ng, etc. themselves. These appear 
in © as dyads, such as n,.n; or ny.ny, ete., or as scalar 
products n,’?=1, nyn»= cos (mj), ng), and so on. In certain 
cases it may be advantageous to employ the unit edges of 
consecutive mirrors, 2. e. apart from the scalar factors 
sin (nj, 02), etc., the vector products Vunj;n,, and so on. 


The utility of the above method of treatment, in which the 
clumsy and often unmanageable formule of spherical 
trigonometry are replaced by the simple operator (1), needing 
no drawings whatever, will best be exhibited on a number of 
examples. Weshall begin with the simplest case of a double 
mirror and then proceed to more complicated ones. In 
each case the procedure will consist in simply “ multiplying ” 
out the dyads n.n contained in the component reflectors. 
And in doing so we have only to remember that juxtaposed 
vectors, not separated by dots, are fused into ordinary scalar 
products. Thus, n,.nyn,.n,=n,(N;N,) .n,=a@).n, .ny, where 
djg=NjNn,= cos (nj, n,). In short, the “product” of any 
number of dyads is always a dyad, including an ordinary 
scalar factor. In what follows we shall employ the general 
notation 


MF Cos.(1;, 1) — 0 ,,— Geen) 1) 


The incident ray or operand r need not be written out in 
each case; it is enough to deal with the operators Q; and 
with their resultants, 2. e. with the reflectors themselves. 
All of the operations involved being associative as well as 
distributive, the multiplication will be done as in ordinary 
algebra, the only precaution (owing to non-commutativity) 
being the preservation of order. 

Double mirror.—The unit normals of the component 
mirrors being ny, Ny, and nyny=ay=a the cosine of their 


Phil. Mag. 8. 6. Vol. 32. No. 191. Nov. 1916. 2.1L 


490 Dr. L. Silberstein on 
included angle, we have, by (1), (2), 
O=0,0,=1—2[n).n, +n,.n,]+4ano.nj, . (6) 
or, introducing the vector p=n,—2ang, 
Q=1—2p.n,—2n,.my. . . . . (7) 


The meaning of this operational equation is seen at once 
by remembering that r'=Or ; thus 


r—r’=2(rn,)p+2(rm.)m,, . . . (7a) 


i. e. whatever the incident ray r, the vector r’—r is normal 
to the common edge of the two mirrors. In other words, 
the projections of rand r’ upon the edge are equal to one 
another. 

From (6) we see that, in general, 0,Q, differs from 0,04, 
since the last term 4an,.n; is not symmetrical. Thus, a 
beam of parallel rays r (broad enough to impinge upon both 
mirrors) is split by the double mirror into two beams r’, r’’ 
oblique to one another, e. g. such that 


r!—r"=4a| ny .ny—ny . De |Y. 


In particular, if the double mirror is orthogonal, we have 
a=n,n,=0, and, independently of the order of reflexions, 


Q=0,0, =0,0,=1—2n, - n, —2n, » To. 
But since n, | n,, we have 
t= ~1) +N» -lo+te . €, 


where e is a unit vector along the common edge of the two 
mirrors*. Therefore, for an orthogonal double mirror, 


OQ=—[1—2e.e], © 4. hy a (8), 


that is to say, the reflexion from such a double mirror is 
equivalent to the reflexion from a simple mirror whose 
normal is e, followed by a simple reversal (—1). This is 
valid for any incident rayr. More especially, if the incident 
ray is normal to the edge, or re=0, we have r'=Or=—r, 
that is to say, the ray is sent back parallel to itself. The 
latter property is familiar from ordinary geometrical con- 
structions. 


* If i, j, k be any normal system of unit vectors, the dyadic 
i.i+j.j+k.k is equivalent to 1, or is, in Gibbs’s nomenclature, an 
idemfactor: {i.i+j.j+k.k]r=r, for any r. Notice that, since e 
appears only through the dyad e.e, the sense of e is, obviously, a matter 
of indifference. | 


Multiple Reflexion. 491 


Returning to the general formula (6) we have for the 
angle 6 between the rays r and r’, cos0=rOr, i. e. 


4(1— cos 0) = 1+ 7.’ —2aryre, 


where 7;=ru,, %2=Yrn, are the projections of the incident 
ray upon the mirror normals. On the other hand, since 
r=ryn,+7rM,+ (re)e and r is a unit vector, we have 


ry? + Po? + 207%, = I — (re)?, 
so that the last equation can be written 
4(1— COS 0) = 1—4ar,r,—(re)’. ° . ° (9) 


This gives the angle @ for any double mirror and for any r 

From this general formula we see at once that there is no 
such double mirror which would send back (parallel to its 
own path) every incident ray, in short, that there are no 
central double mirrors. In fact, cos@=—1 would mean 
Aarr.+ (re)?=0, and this cannot he fulfilled for all directions 
of r. 

Triple mirror.—The reflector in this ease is O=202;,0,.Q0,, 
that is, the product of 0;=1—2n;.n; into the operator (6) 
of the preceding section. Writing, therefore, NjNo2=Gy5, Cve., 
we have, for any triple mirror (the order of reflexions being 


pe eo) 
= O07..= 1—2/n, ° ny +N, -No+Ng 13 | 
= A] ayoNg - ny -~ Ag2N3 . No+ 3 N3 . ny | — 89Ao3N3 Ny. (10) 


Here again, the third and fourth terms being non- 
symmetrical, an incident beam of parallel rays will give rise 
to six reflected beams QOyo,r, Q,3,r, etc., which will, in 
general, not be parallel to one another. ‘These reflected 
beams become parallel to one another, 7. e. Q becomes 
independent of the order of reflexions 1, 2,3, when, and only 
when, 

Ay2= Ao3 = U3 = 0, 


2. e. when the three component mirrors are perpendicular to 
one another. In that case, nj), ny, n; being a triad of normal 
unit vectors, the dyadic n, .nj+no. Ny + Ns. n; becomes an 
idemfactor or 1, and therefore, 


OS hs ae oc . ) CLD) 


That is to say, every incident ray is sent back parallel to 
itself. The orthogonal triple mirror is a central mirror. 
Returning to the general triple mirror, let 7, 72, 7; be the 


212 


499 Dr. L. Silberstein on 


direction cosines of the incident ray, 2. e. the projections of r 
upon 04, Dy, 3. Then the direction cosines 7’, 72’, 73 of the 
reflected ray r’ will be, by (10), 


=, (l= 27 eee sk eee 
=9,(1— 27, Agere Slate ; [ 
3 =73(1—2r3 + 409379 + 449171 — 84420937). \ 
These scalar formule can at once be used for numerical 
calculation. 
The angle 0=(r, r’) is given by cos 0=rOr, 7. e. by (10), 
1— cosd 


— Aa 2 2 2 a 2 © qa 7 ? a a a 7 
5 =r treet 73? —2| Aye? + Ao3?973 + A373? | + 40y20037371- 


On the other hand, we have, by squaring 


Y=1N,; +roN,+4rsNs *, 
rs Lary tre? +79? + 2[ ayer 72 + 4937073 + A317 37) |, 


and therefore, for any incident ray, whose order of reflexions 
is 123, 
3— cos 6 
L 


From this general formula, which enables us to calculate 
at once the angle @ for any incident ray, we can see also 
that the orthogonal mirror is the only possible central mirror. 
In fact, the right-hand member of (13) becomes equal to 1 
(Oh 2 cos paaied, for any Fe ton of r, when, and only 
when, 0, M2, n; are normal to one another. 

If the three mirrors constitute a regular pyramid, i.e. if 


S=9 i tet 1s + 2ajoG3731- = 2 ee 


1g =0o3—a; = COS @, SAY, .. .. = sae 
then the reflector (10) becomes 
O=1—2[n,.n, +n, .ny+n; . n; | 
+4a[no.0y+N3.Ny+n3.n)|—S8a'n3.n,, {10a) 
and the formula (13) for the angle @=(r, r’), 
a =r+r?+7+2cos*aryrs, . (134) 


for any incident ray, the order of succession of the reflexions 
being 123, for (10a), and either 123 or 321 for (13a). 
If the order is 231 (or 132) or 312 (or 213), we have only 
to write, in the last term of ae a), 72”, and 73rz respectively. 


* We assume, of course, that mj, Me, N, are not coplanar, z. e. that the 
three reflecting planes constitute a pyramid, not a prism. 


Multiple Reflexion. 493 


More especially, if the incident ray is equally inclined to 
the three reflecting planes of the regular pyramid, we have 


i — Tp a cot = 
1 2 3 /3 9° 
and (13 a) becomes 
3— cos @ PN thie 9 @ 
p= (i+ 3 COS «@) ot 3° (13 6) 


In this case the angle @ is independent of the order of 
reflexions, as was to be expected. The reflected beams, 
although not parallel to one another, are equally inclined to, 
and symmetrically disposed around, the direction of the 
incident beamr. ‘These reflected beams coincide in direction 
when, and only when, o=90°%, 7. e. when the mirror becomes 
an orthogonal and, therefore, a central mirror. 

Further discussion of the above formule and the con- 
struction of similar ones for quadruple and more complicated 
mirrors are left to the reader. Here but two further 
remarks on the general reflector Q, :— 

Reversal of the order of reflexions.—Let r be the incident 
ray, r’ the finally reflected ray when the order of reflexions 
is 123...«, and s’ the finally reflected ray when the order of 
reflexions is x...321. Then, 1f Q=O©,...0,0,0), as in (1), 


Ge aOres— 10. 
whence 
ere =esese, py em Ob. 


and therefore, for any multiple mirror, 
RSG | leeameuerme cic. (IA?) 


while r’s’'=rOQ’r. That is, the reflected rays r’, s', although 
not parallel to one another, are always equally inclined to the 
incident ray r. The equality 0)53=639;, exhibited by (13 a) 
is but a special instance of this general property. 

Images of gwen objects.—Hitherto we have considered r 
and r’ as determining the directions of the incident and the 
reflected rays. In order to obtain the image of a given 
point-object, let the end-point of the vector r, drawn from a 
fixed origin O, determine the position of the object, and the 
end-point of r’, drawn from the same origin, the position of 
the image. Then, in the case of a simple mirror, we have 
again 

I ) 
r’=Q,r=[1—2n, . n, |r, 


provided that O is a point of the reflecting plane itself. 


494 Dr. Manne Siegbahn and Dr. Hinar Friman on 
Similarly, in the case of a double mirror, 
r=0,Our, 


if O is taken on the common edge of the component mirrors. 
Thus, in the case of any multiple mirror, we have, as before, 


=I 
= OP Opes TUT (Oe 

tK 
where Q;=1—2n;.n;, provided that all the component 
mirrors have a common point of intersection, and that this 
point is taken as the origin of the vectors r, 1’. 

Under these circumstances, therefore, the treatment of 

point-objects and their images is formally the same as that 
of incident and reflected rays. 


September 5, 1916. 


LIT. On an X-Ray Vacuum Spectrograph. By MANNE 
SIEGBAHN, Dr. phil., and Hinar Friman, Dr. phal.* 


i order to examine the high frequency spectra of the 
elements by long wave-lengths the authors have had 
a vacuum spectrograph built. Hereby, as our former 
measurements T have shown, the following conditions must 
be satisfied. First, the crystal must be movable, as otherwise 
irregularities in the structure of the crystal may be of great 
influence (comp. Rutherford and Andrade and EH. Wagner). 
Secondly, in order to get a good resolving power, besides 
using a fine slit it is of great importance to focus the rays. 
The apparatus built on this principle is shown in fig. 1, in 
both horizontal and vertical section. The spectrograph 
consists of a round metallic box of 6 mm. thickness, a height 
of 8 cm. and an inner diameter of 30 cm. The upper part 
BB, 3°5 cm. broad, is carefully plane-ground, as well as the 
corresponding part CC of the cover D. This as well as the 
bottom is furnished with radial reinforcements in order to 
resist the pressure better. The cover has a handle anda 
screw, the latter being used to lift it after the air has been 
admitted. The box is supported by three set-screws. Inthe 
middle of the bottom there is a conical hole with a metallic 
cone H, fitting well in it. This cone, being kept in the hole 
by a ring screwed into the box. after lubricating can be turned 
without the air passing through. The crystal table, placed 


* Communicated by the Authors. . 
+ Phil. Mag. vol. xxxi. p. 403 (1916) ; vol. xxxii. p. 39 (1916). 


an X-Ray Vacuum Spectrograph. 495, 


on the cone, consists of a slide F, that can be moved by 
means of the screw H. On the slide the table N is placed, 
that can be turned round the axis a with the aid of the 


SS WINN NSSSS EASE RANE EE ANN 


ni A a) ~--| ae Za 
es iy [ ial i} ® Ij 10 G R 
Serer an | = j 


screw h. This arrangement allows of a careful adjustment 
of the crystal K. This is attached to the table by means of 
a bow and two screws. The lower part of the cone is joined 
to a metallic arm M, about 20 cm. long, with an index Z. 


496 On an X-Ray Vacuum Spectrograph. 


This moves along a scale S, that indicates the turning angle 
of the crystal directly in degrees. The arm M can be moved 
to and fro by a slowly rotating eccentric-pulley. Through 
the side of the box, placed opposite to one another, two 
tubes R and U pass. The former is in communication 
with the pump by means of an indiarubber tube. In the 
latter there is another tube V, the end of which is covered 
with a plane-ground plate p, having a slit o 0-1 mm. wide 
and furnished with edges of gold. Between this slit and 
the crystal a screen I of lead is placed, with a slit s of 
2mm. width. In the tube V isa cylindrical shield of lead 
1°5 cm. thick, with an opening of about 1mm. The X-ray 
tube (of the construction previously * described) is attached 
to the spectrograph in such a way, that the extended part of 
it is put into the tube U, and then an air-tight elastic 
connexion obtained by means of picein. 

The photographic plate is placed in the casket G. This 
is attached to the sledge L by screws passing through the 
two feet, which are furnished with longilateral openings by 
means of which the casket can be turned or removed. 
Through this arrangement the plate (in position 0) can 
be adjusted perpendicularly to the line first slit—crystal 
rotation axis, and at the same distance as the slit from the 
crystal. The sledge is supported by three steel bullets (comp. 
Moseley), two of which rest in a circular groove, the third 
in one of the radial cavities. In this way the sledge can 
take up perfectly fixed positions, the number of which is 
thirteen, situated at about the same angular distance from 
each other. These angles have been very carefully deter- 
mined. In the investigations described in the next paper, only 
five positions have been used, which are denoted by 0, I-IV. 
The aperture angle of the casket is about 37°, and thus the 
plates in two adjacent positions have an angle of 4°-5° 
common. 

The X-ray tube is evacuated with a molecular air-pump. 
The fore-pump is employed simultaneously for the molecular 
air-pump and the spectrograph. If the latter is separated 
from the X-ray tube through a foil placed on the greased 
plate p, a suitable vacuum can be obtained ina few minutes. 


Physical Laboratory, University of Lund, 
June 1916. 


* Phil, Mag. vol. xxxii. p. 39 (1916). 


Fe 407 7 


LIT. On the High-Frequency Spectra (L-Series) of the 
Elements. Lutetium—Zinc. By K1nar Frman, Dr. phil.* 


(Plate XI] 
| a preceding paper t Dr. Siegbahn and the author have 


given the results of an investigation of the high-frequency 
spectra (L-series) of the elements tantalum—uranium. In 
this domain we found at least 11 line-groups. With the 
vacuum spectrograph described in the previous paper (p. 494) 
I have been able to continue these examinations down to 
zinc (34 elements). Here I have reached a wave-length 
of 12°346.1078 cm. (Le,-line of zinc), while the greatest 
wave-length measured by Moseley was 8°364.1078 cm. (Kz- 
line of aluminium). 

Most of the elements examined as compounds were finely 
pulverized and rubbed on the anticathode, which was scratched 
beforehand with a file, or still better with a knife. By this 
method, also used in the investigation above-mentioned, a 
small quantity of the substance (some mgr.) is enough for 
an experiment. In some cases, however, the elements in 
metallic form were soldered on the anticathode (Cd, Ag, Pd, 
Zn) or pressed in tongs of copper soldered on it (In, Ru). 

With the elements lutetium-silver very sharp photo- 
grammes { were obtained with a rocksalt crystal as reflector 
and an exposure time of an hour. In the later experiments 
a gypsum crystal was used. This also gave several rather 
sharp lines, but the lines of weak intensity did not appear. 
As window between the spectrograph and the X-ray tube 
in the beginning a thin foil of aluminium (0:001 mm.) was 
used. From tellurium it was replaced by a thin foil of gold- 
beater’s skin that very well transmits the soft rays (Moseley). 
It has a disadvantage, however, in also transmitting the lumi- 
nescence light from the anticathode, which often was very 
strong and blackened the plate. The spectral lines generally 
were referred to the Ke,-line of copper, the wave-length of 
which has been very carefully determined by Siegbahn and 
Stenstrom §. The reflexion angles calculated have an 
uncertainty of about 0°3 per cent. As grating-constant of 
rocksalt the value 2°814.107% cm., given by Moseley, was 


* Communicated by the Author. 

+ Phil. Mag. vol. xxxii. p. 39 (1916). 

{ Some of these photogrammes are reproduced in my doctor-dissertation, 
Lunds Univ. Arsskr. N. F. Avd. ii. Bd. xii. Nr. 9 (1916). Here a 
more detailed account of my investigations is given. 

§ Phys. Zettschr, xvii. p. 48 (1916). 


ne. 


2 


Zi 


wuUmMe= 


gh-Frequency Spectra of the Elements Lutet 


2 


val 


498 


Boi GOT Te | SSlele =| aoa = \ G6GtL- | SOG ler Ll savor. e at 619: | 629-1 “ey Ty 
‘- | o77.. | 823-1 | LOGL | GPL] IShL | PIPL | PLP-L | OGFT | 819-1 Es OLO-L | 1891 | 268-1 | IA 04 
te | SOIGE 66ers 2 LOST + | 99G-T | BbIG-L | 98G-1 | 664-1 | GGT ne e8L-1 | F6L-1 AE BO 
“| @O¢-L | 698-1 | SIFT | 029-1 | 89G-1 | 9F9-1 | L¢9-T ee ae erPsL | FesT sa eg 0)0 [ay 
ae OTE - | col | Olek 1 “| eoOn alsaaom =| GOleL.. | lewL a see 106-1 | 916-1 SO AGS) 
1ep-L | OLFL | L261 | 1e¢-1 | 699-1] GPL1_ | G89-T | GLL1 | F841 | 986-1 eS 16-1 | S861 oe eC g9 
“| (9@G-1) | (G9G-1) | 2469-1 28 a PPS-L 198-1 wg at E403 | FG0-% Ss feo 9 118 
"| 069-1 | 66¢-T | 399-1 eee | Ole | SIG |. CcoL ed aS T2ke | TeEe eee ener GO 
es 699-1 GL-T “ec oe | peel |-000-c ay ee S 00Ge | Olas Pape (SSSURt EO 
| AGE lat e{0) 215] heme If stp 2ora Sed aeleoe| Cedi | LONG alae orG cae ve 60E-3 | BLE “= 1" DAT OQ 
P| OCG: Leateeeoul | Sao. [ “el aieeG | OGEG =) OCGe = oe ca ZORG | GLP PS OSes 
So (G00 @) ZG0- Tusa i er Oren | «Gree a OG et Gan Laas G ae 292-3 | G12: SETH 
Be : un OFT I ple | LOSG J Lobo le Saba ms eo 99-6 | FL9-G Pera 6 LG 
ae GES “* | 0¢¢-4 | LOF-% | 69¢-% | 999.2 oa ee 941-6 | 981-6 Be SESE Va) 
uu PSG: 08-2 me s| G60 | PIGiG | FSO:c 9 | S09-% mS i 168-3 | 668-3 Sate S| eG 
pe E82. ORE) OG Paice a eet Gece lel sty ot es OniEen cee pee eed (0) 
nee GWG “| 100-8 | 188-3 | PL0-€ | FP0-8 ee as 06a-E | 662-8 Alea eg 
ee CBL:G 6FS:S | GELS |e leO Geel 26G.cmlepelea ee re GP-S | ShPS SE Se 
1g8-3 | 688-3 | S063 | 666-2 eee: GLice | = 1[S¢.G. 3) siee-¢ oon me #69: | F09-8 Susie, 
Ash le 10: OoLe se He deem aqaate 1 re <? Gone Ie pie “eles tT ap 
AY ae ee 188-8 “* | 669-6 | FIG-E | S646 | 929-8 ye -S BF6-8 | 696-8 S| SPORT. 
os a a GTG-¢ “* | @Z8-6 | 869-€ | 666-6 | G98 “| SLb | OPL-F | OTF Pe Nei Nig 
go 169-8 OGLE | 080-F | G06:2 | GFL | 1Z07 = COG ee COG te Beene |e DanOP 
Gh Ao) ar he — hice | 96Gb e" oe ee unr GF 
ry iat ae a “a = ee nee = a appro crea een nea 4s 
ne ie ee ei 50 a ae ote ee eos | core 4 Giro rong ep 
aa ce >a ite * ae JIS-G | §6F-G aa -* 60-4 | #oLG | TSL¢ me GUN TE 
ea a a bade a or 1 aoe = naa meniaecleacats iat mee cae 
pee eae ae ee at x aA aaa) wa ie rho. | oRpn 2 v0) lees go 
ae <i ae, ae, a a ie ae9.0 a ; a ioe a ws |e dgron 
te a a oa a - La 160.2 Se a age ee licecel ep Toe Pea 
a ae =~ os ep eee a PLS es ee ooee | Teese a Pe uote 
ce ees - i As Tone - [ee gy eg 
ee ae e oe ~ = ae i ae “ie Fer s "toed ne ie 


rh ay ZL Th sof Eg ey To tg) “lt fp Sh a "7 = 

ee ee Ee ee ee ee eee 
COE SA | 

5S a ee ee ee ee a aE a a EE eB 


‘[ WIaVv], 


Diffusion Cells in Lonized Gases. 499 


used. With gypsum this constant was tound to be 7°621 . 107* 
em., being the mean of two determinations only differing 
by 0°16 per cent. 

The values of the wave-lengths are put together in 
Table I., and in Plate XI. a oraphical representation of the 
results (from zirconium to uranium) after the relation of 
Moseley is given. As seen, 14 line-groups seem to be sure 
in the L-series. Of these the /-group was discovered by 
Siegbahn *, who has examined these lines in the elements 
tungsten—uranium. The graphical representation shows 
further, that only in some cases (a, #1, 8.) a Moseley relation 
holds good for the whole range, while the other groups 
generally form curves slightly convex to the X-axis. The 
order of the rare earths here found agrees with that generally 
adopted. The order Te-[ found by Siegbahn + for the 


K-series is also verified here. 


I have much pleasure in recording my thanks to Dr. M. 
Siegbahn for suggesting this work to me. 


Physical Laboratory, University of Lund, 
June 1916. 


LIV. Diffusion Cells in Ionized Gases. By E. B. Woop, 
M.A., O. A. bE Lone, B.S., and K. T. Compron, PhD., 
Assistant Professor of Physics, Princeton University t. 


i les a recent paper§ Mr. W. H. Jenkinson has shown 
very conclusively that an electromotive force is set up 
between two similar electrodes in a gas if the degree of 
ionization of the gas surrounding the two electrodes is 
unequal. The effect of this phenomenon on the relative 
potentials of an electrode and the surrounding gas had 
already been investigated by Professor John Zeleny||. The 
interpretation given by Mr. Jenkinson to his results, however, 
appears to us deserviny of further examination. 

Following the analogy of Nernst’s treatment of electro- 
lytic cells by the concept of solution tension and solution 


* Verh. d. Phys. Ges. xviii. p. 150 (1916). 

+ Verh. d. Phys. Ges. xviii. p. 39 (1916). 

t Communicated by the Authors. 

§ “Concentration Cells in Ionized Gases,’ Phil. Mag. xxviii. p. 685 
(1914). | 

|| Phil. Mag. xlvi. p. 184 (1898). 


200 Messrs. Wood, de Long, and Compton on 


pressure, Mr. Jenkinson interpreted his results by the 
equation 


RT mn 3 
= (pa pele 
EK e (22 iB) log po e e se e e (1) 


where p; and py, are the partial pressures of the ions of either 
kind just outside the electrodes, x is the proportion of the 
total current carried by the positive ions, and E is the electro- 
motive force resulting from the unequal ionic concentrations 
at the two electrodes. According to this equation the sign 
of the electromotive force should depend on the value of «, 
7. e. on the kind of ion, positive or negative, which effects 
the greater transfer of electricity. 

From the sign of the electromotive force observed in his 
experiments (the electrode exposed to the ionizing radiations 
always acquiring a positive charge), Mr. Jenkinson concluded 
that it must be the positive ions which determine the potential 
step between a metal and the surrounding gas. He takes 
from his experiments support of the view that a layer of 
occluded gas at the surface of a metal is essential to the 
production of a contact potential effect, and that positive ions 
only pass from this layer to the surrounding gas. 

In view of the increasing importance of discovering the 
conditions which exist at the surface of a metal, we feel 
justified in pointing out amuch more probable interpretation 
of Mr. Jenkinson’s experiments which involves neither the 
formation of an occluded gas layer nor the passage of 
electricity to and from a metal by the sole agency of 
positive ions. 

Theory.—Consider the simplest case, that of two infinite 
parallel metal plates and a source of ionizing radiations 
which ionizes the intervening gas more strongly near one 
plate than the other. If the two plates are originally at the 
same potential, the ions will move by diffusion from the 
regions of greater to those of less concentration ; but this 
process will never produce uniform ionic concentration in all 
parts of the gas because of recombinations which take place 
simultaneously with diffusion. In this respect the present 
case differs from the similar case of an electrolyte. 

If the positive and negative ions diffuse at the same rate 
the plates will obviously remain at equal potentials ; but if 
the negative ions diffuse at a more rapid rate than the 
positive ions, as is usually the case, the plate in the region 
of less ionic concentration will acquire a negative charge 
and a negative potential with respect to the other plate. If 
the plates are insulated, this process will continue until the 


Diffusion Cells in Ionized Gases. 501 


electric field is just sufficient to cause the positive and 
negative ions to drift at equal average rates toward the 
negative plate, numbers of them recombining as they drift. 

Such an explanation of the electromotive force of con- 
centration-cells in ionized gases accounts for the sign of 
the electromotive force observed by Mr. Jenkinson without 
introducing any hypothesis regarding the phenomena at the 
surface of the metal. That it is the correct explanation is 
indicated by the following experiments. 

The process just described may be put in a form for quan- 
titative experimental test by applying Prof. J. J. Thomson’s 
well-known equations for the time rate of increase of the 
number n of ions per unit volume at any point in the gas: 


dn dn, d 

a =g—anyno+ Dive — Uy De (Xn), 
ding d?n. aie 

— =J—ANy Ng + D, ae aa bors (Xing); 


where subscripts 1 and 2 refer to the positive and negative 
ions, respectively. gq represents the rate of production of 
ions by the ionizing agent; «and D are the coefficients of 
recombination and diffusion, respectively ; uw is the ionic 
velocity in a field of unit intensity; X is the electric intensity, 
and w is the coordinate normal to the plates. 

We may eliminate the terms involving g and « by sub- 
traction and integrate the resulting equation. The inte- 
gration constant vanishes, since the current through the gas 
vanishes in the final steady state. By Poisson’s equation it 
follows that in the steady state n,=n, at any point in the gas. 
Making this substitution and integrating again, we find 


ID =1)) n 
1 Ee 
a Ui eR loo —, 


Ujtu, ~ No 


Xd= 


where d is the distance between the plates and mp and n, are 
the ionic concentrations in the immediate vicinities of the 
plates. Xd=EH, where E is the electromotive force, and 
D=RTu/e*. Thus we have 


(2) 


This equation is given in Winkelmann’s Handbuch der 
Physik, 2 ed. vol. iv. p. 616, being there derived by a con- 
sideration of partial pressures asin the case of an electrolyte. 


€ Ut 1% 


* J, J. Thomson, ‘Conduction of Electricity through Gases,’ p. 45. 


502 Messrs. Wood, de Long, and Compton on 
When a current flows through the gas 


Ly 4 Ug 
U, + Ug yu 


give the proportions of the electricity carried by the positive 
and negative ions, respectively. Furthermore, the partial 
ionic pressures p are proportional to the ionic concentrations n. 
Thus we may write equation (2) in the form of equation (1) 
used by Mr. Jenkinson. The interpretation of x here sug- 
gested, however, is typical of a diffusion-cell rather than an 
ordinary concentration-cell such as Mr. Jenkinson considered. 

Experimental Tests.— We tested the validity of equation (2) 
by an apparatus whose essential features are shown in the 
diagram. Brass plates A and B were surrounded by guard- 
rings G, mounted on ebonite blocks and enclosed in the 


. 
7 


Khdhdd 
WEEE 


g 


Y) 
ss 


thick-walled lead box L L, which was lined with brass to avoid 
disturbances due to contact difference of potential. A tube 
of radium at R was also enclosed in a lead box provided with 
a fan-shaped opening. ‘This box could be set against stops 8 
so as to expose the gas near either plate to the ionizing 
y rays. The potential divider P enabled the potential of the 
plate A to be adjusted until the electrometer H connected 
with the plate B showed no deflexion. The difference 


Digjusion Cells in Ionized Gases. 503 


between the voltmeter readings with the radium in the two 
positions gave twice the electromotive force of the diffusion- 
cell. An insulated switch, not shown in the figure, enabled 
the connexions to the two sets of plates to be interchanged. 
By means of two switches k, and ky, actuated by an 
electromagnet pendulum-bob M, a field of two volts from 
the battery B, could be suddenly applied to the plate A for a 
definite small interval of time, usually about 0°05 second. 
During this interval ions of one sign within a fraction of a 
millimetre of the plate B were swept into the plate, pro- 
ducing a deflexion ot the electrometer. This could be 
repeated with the ionizing source acting near the other 
plate, or with the connexions to A and B reversed. The 
ratio of these deflexions gave the ratio m,q/7. All other 
quantities in equation (2) being known, these measurements 
enabled a quantitative test of the equation to be made. 
Unfortunately our tube of radium did not produce sufii- 
ciently strong ionization to permit an accurate measure of 
the ratio n,/o, the electrometer deflexions being too small in 
comparison with a somewhat irregular shift due to an un- 
balanced induction effect while the field was changed by the 
switches k; andk,. We were consequently forced to use an 
X-ray tube as the ionizing source in this part of the expe- 
riment, and the measurements were subject to the somewhat 
erratic variations inits action. However, with this arrange- 
ment, we obtained results of which the following are 
examples. n, and mn, are in terms of the electrometer 
deflexions noted just after the fall of the pendulum M. The 


TABLE [. 
No. Na. E (cale.). E (obs.). 
24 | 7 0:00497 0:0042 
18°5 | 5 0:00528 0:0034 
24 5 0:00633 0:0067 


electrometer sensitiveness, about 5000 divisions per volt, 
was such as to make the apparatus very sensitive to dis- 
turbing influences, so that we feel that our measurements 
may be taken asa fair support of the theory. Doubtless, by 
taking greater precautions to insure steady strong ionization 
and by increasing the distance between the plates A and B 
so as to increase the amount of recombination, and therefore 
the ratio n,/, the experimental test could have been made 
more accurate. 


D04 Diffusion Cells in Ionized Gases. 


We tested equation (2) in another manner, by substituting 
dry carbon dioxide gas for air between the plates. In this 
case, the difference between the ionic velocities u, and wu, is 
much less than in air, from which we should expect a smaller 
electromotive force. In order to avoid large changes in the 
strength of the ionizing source during the substitution of 
one gas for the other, we used the radium for the ionizing 
source in this test and increased the electrometer sensitiveness 
to about 15,000 divisions per volt. The averages of a number 
of consistent measurements gave H=0-00090 volt for air 
and E=0-:00035 volt for carbon dioxide. This difference is 
in the right direction and of the general size predicted by 
equation (2), though an exact prediction cannot be made 
because of the differenee in the degrees of ionization and 
recombination in the two gases. 

Finally we tried the effect of filling the apparatus with 
moist carbon dioxide, in which case the sign of the electro- 
motive force should be reversed, owing to the fact that the 
positive ions diffuse more rapidly than the negative ions in 
this gas. This test, however, was unsuccessful because the in- 
sulation broke down before the measurements could be made. 

In conclusion we might mention an attempt to test 
equation (2) without the necessity for measuring the ionic 
concentrations at the plates by the aid of the equation 

d’n dn ‘ 

da? + A dz or ete le 
which follows from Thomson’s equations given earlier in the 
paper if we put A=u,X/D, and B=—a/D,. This equation 
is of the type of the equation of motion of a particle moving, 
subject to friction, under the action of a force proportional 
to the square of the distance from a fixed point, and has not 
been solved. Approximate solutions indicated that the ratio 
n,|% is not independent of 7, as we might have anticipated. 
Thus even with an exact solution of this differential equation 
it would still be necessary to make some sort of measurement 
of the ionic concentration, so that this method was not 
pursued further. 


Weare glad to acknowledge our indebtedness to Professor 
HE. P. Adams for suggesting approximate solutions of the 
differential equation mentioned above, and to Professor 
H. L. Cooke for suggesting the method which we employed 
for measuring the relative values of the ionic concentrations 
at the plates. 


Palmer Physical Laboratory, 
Princeton, N. J., U.S.A. 


[ 505 ] 


LV. Results of Crystal Analysis.—III. 
By L. Vegarn, Dr. phil., University of Christiania *. 


[Plate XII.] 
Sih. A S an addition to the results of Crystal Analysis 


given in two previous papers f, I shall give an 
account of the determination of the structure of the following 
crystals: 


Xenotime (YPO,), 
Anatase (‘TiO,). 


Both crystals belong to the ditetragonal bipyramidal class 
of the tetragonal system. 


Xenotime. 


§2. This mineral has a chemical formula analogous to that 
of zircon (ZrSiO,); but although the formula may have a 
similar appearance, there are probably considerable differences 
with regard to chemical constitution. Thus the two elements 
associated with oxygen in the case of zircon have both a 
valency of four, while in the case of xenotime the one 
element (Y) has a valency of three, the other one (P) a 
valency of five. 

Now, in spite of their chemical differences, the two 
substances have crystals which, both as regards form and 
physical properties, are very similar. Both belong to the 
ditetragonal bipyramidal class, and the ratios of the axes are 
very nearly the same. 


Hor) Zircon) a\ic= 1 : 0;630N% 
» mdenotime a :c=1 : 06177. 


In a previous communication it was found that in zircon 
the Zr and Si atoms took up a similar position in the lattice 
as regards their relation to the oxygen atoms. The arrange- 
ment corresponds to a formula ZrO,S8i0, and not Zr(SiO,). 
In view of the crystallographic similarity, the question arises 
whether also the atoms in xenotime might be arranged in a 
zircon lattice, which would mean that Y and P—in spite of 
their difference of valency—would have the same relation to 


* Communicated by the Author. 

+ L. Vegard, Phil. Mag. January 1916, Paper I.; ibid. July 1916, 
Paper IT. 

Phil. Mag. 8.6. Vol. 32. No. 191. Nov. 1916. 2M 


506 Dr. L. Vegard en 


the oxygen atoms. Thus the determination of the structure 
of xenotime is of considerable importance for the question 
of the part played by the valency in the constitution of 
solids. 

The experimental determination of the X-ray spectra of 
the different faces is somewhat difficult on account of the 
fact, that most crystals are more or less transformed into 
amorphous mass, are partly ‘‘metamict,”’ and as a conse- 
quence, it was impossible to get reflexion from a number of 
specimens tested. 

At last I got from the Geological Laboratory some fresh 
and pure samples which gave fairly good reflexion for some 
of the faces. 

The spectra of the faces (100) and (110) were obtained 
from a sample from the Langesundsfjord, Norway; those of 
the faces (111) and (101) from a crystal from Raade, Norway. 

Several attempts to get reflexion from the base (001) 
were not successful. 

The glancing-angles (@,) and intensities (I,) of the 
observed spectra for the four faces are given in Table I. and 


in fig. 1. 


TABLE I. 
YEO: 
(100). (110). (101). (111). 
N. a 

On . I Ne One Ibn . An. ie On. ie 
ee 72 16'| 1:4 | 5° 9'| 60 | 6°58 | 35 m 0 
Oe 4 89 2) 1-0. OMI So | 14 QA OG Ue 0 
Sen te Loe ST al 15 3! 0 e ) 
Aas Aes Pa Ee $e oi ie 15° 42’ 1:5 


In the elementary cell dy o?doo = dio? , we find from 


observations a number of molecules equal to 0°127 or 3, 
which is the same as was found for zircon. 

In a similar way, as in the case of zircon, we are led to 
a similar space-lattice for the Y and P atoms, as found for 
the Zr and Si atoms in zircon. If so, both the P and 
Y atoms are arranged in a lattice of the diamond type. 

The lattice of P can be made to cover that of Y by a trans- 
lation along the tetragonal axis a distance ¢/2. 


Results of Crystal Analysis. 507 
The spacings of the four faces are in the relation : 
doo Gheg ° Aso1 ° Oni 6 AKO) € 1:043 : 0°468, 
Fig. 1. 


Xenotime 


or approximately 


Ae ] 
y/o sees mac 5° = eee (1a) 
f+ (2) 4/24 (2) 
C C 
If for a moment we do not take into consideration the 


oxygen atoms, but only the Y and P atoms, the lattice would 
give the following relation : 


i. 9 
A091: dii0 5 doy : dyy,= 1 B/D er 


WF CINING) (18) 


We see that the relations (1a) and (10) are the same 
except for the face (111), where the spacing observed is 
only + of the value to be expected from the lattice. 

The question is now, whether the oxygen atoms might 
be arranged in such a way as to make the three first 
maxima of the (111) face disappear or become vanishingly 
small. 


2M 2 


508 Dr. L. Vegard on 
Arrangement of Oxygen Atoms. 


§ 3. As far as the Y and P atoms are concerned, we have 
assumed a lattice of the Zircon type. The first question we 
shall put is: Can such an arrangement of the oxygen atoms 
as that found for zircon explain the spectra of xenotime ? 

The spectra of the faces (100), (110), (101), might be 
accounted for by a proper choice of the parameters, and the 
test will be whether the zircon lattice can explain the 
peculiar spectrum of the face (111), in other words, explain 
the fact that the first three spectra are too weak to he 
detected. With regard to the determination of the intensities 
I shall only refer to my previous paper (II. p. 84). 

The functions 7,(n) and f,(n), which determine the strength 
of the spectrum of order n, are for the (111) face of an 
arbitrary lattice of the Zircon type given by the expressions 
(II. eq. 12) 


A(n)=N, 4+ (—1)N.+(Ni 4+ N2+4N3) cosn 5 


+2N; (cos na, + (—1)" cos nag), 
f(r) =(Ni—Nz) sin ns ; 


N; is the atomic number of oxygen; N,, N, those of the 
two other elements. In our case we have to put N;=39, 
Eloy Neo. 

We can, however, prove that there is no value of the 
parameters which can give a distribution of intensities in 
accordance with observations. Thus it is impossible to make 
the first order spectrum sufficiently small as compared with 
the fourth. 

Remembering that the amplitude A,=,/f;(n)?+)9(n)?, 
we shall find that the smallest value of A, is 24, and the 
largest value of A, is 172. This would mean that the first 
order spectrum should be at last half as strong as that of 
the fourth order, and the spectra of the second and third 
orders would not be sufficiently weak at the same time. 

Now the maximum observed was not very strong, but we 
should easily have detected a peak equal to 4+ of that observed 
for the fourth order spectrum. We can then safely conclude 
that the oxygen atoms in xwenotime cannot be arranged in a 
lattice of the Zircon type. 

If in the lattice proposed for the Y and P atoms of 
xenotime, both sorts of atoms were replaced by atoms of the 
same reflecting power, the first three orders of the face (111) 
would vanish. Then we must arrange the oxygen atoms so 


Results of Crystal Analysis. 509 


as to increase the weight of the P planes so that the group of 
P and O atoms can balance the heavier Y atoms, and we are 
led to try to arrange the oxygen atoms tetragonally round 
the P atoms. 

There are mainly two different arrangements possible :— 

(1) The oxygen atoms are arranged on a line through the 
P atoms parallel to the tetragonal axis. This would make 
the spectra of the faces (110) and (100) normal; but in the 
case of the face (100) the second order spectrum is nearly 
as strong as that of the first order, consequently such an 
arrangement is excluded. 

(2) Four oxygen atoms are arranged in a plane through 
the P atoms perpendicular to the tetragonal axis. 

In order to keep up the highest symmetry class there will 
only be two arrangements of this description possible : 

(a) The oxygen atoms are placed on lines through the P 
atoms which are parallel to the sides of the base ; (b) they 
are placed on lines through the P atoms which are parallel 
to the diagonals of the square base of the lattice. 

Let us first consider the first possibility (a), and calculate 
the intensities for the faces (111) and (110). 

After some reduction we find for the amplitudes, 


‘ 
(111) A, =(0°47+(—1)"1-22 + cos na) cos 
(110) A,=1°69+ cos 2na, 


where e is a parameter determiningi the distance between 
P and O. 

In order to explain the disappearance of the three first 
spectra of (111) we should have to choose a value of a 
smaller than 25°, because a greater value would make the 
third order spectrum too strong not to be detected ; and 
even for #<25° the first order spectrum would probably be 
too strong to escape detection. Further, we cannot assume 
a very small value of «, because it would bring the centres 
of the P and O atoms of the group too close tovether. 

Finally, a value of «<25° would produce a distribution 
of intensities of the face (110) which is not in accordance 
with observations. The observations give for the intensities 
of the first three orders the relative intensities 100, 58, 25, 
showing that the intensities of the second and third orders 
are larger than in the normal spectrum. In other words, 
the oxygen atoms must be arranged in such a way that 

A,<A,< As, 
while for #< 25° the arrangement considered would give 


A,>A,>A3. 


(2) 


510 Dr. L. Vegard on 


Hence we conclude that the arrangement (a) must be 
excluded. ta 
We shall then consider the second possibility (6), which 
gives the spacings represented in fig. 2 for the four faces 
Fig. 2. 
Xenatime 


%, 
! i 
4100) 20 —>, 2k 
hj h 
ii It 


YP Oz o> YZ 


! it 

TIO) ba ——>} 2a 4 
| 

ii! 


| 
VP O> 


YP Oz 02 Ale 


examined. The amplitudes are determined by the following 
expressions : 


(111) A,=[2°444+ (—1)"(1'94+4 cos ne) | cos, | 


(110) A,=4°37+4 cos 2ne, . 
(100) A, =1°69 >> cosine, - 
(101) A,=1°'69+ cos ne. 


If / is the distance from an oxygen atom to the central 
P atom: 


(3) 


Gag ae 


Ar | 

a= — =. 

V2 4 
The equation for the amplitude of (111) shows that the 
second order spectrum will disappear for all values of a. 
The first order spectrum will vanish for 2=60°, and for this 
value the ratio of the intensity of the third to that of the 
fourth order spectrum will be 1:7 and the third order 

spectrum might well be so weak as not to be detected. 

In order to get the best possible agreement between 


Results of Crystal Analysis. 511 


calculated and observed values for the other three faces, we 
put z=6o-. 

In the following Table II. are given the calculated 
intensities corresponding to this value of a, as also the 
observed intensities. 


TABLE II. 


(111). (110). (100). (101). 


Teal. Tobs. Teal. Tobs. Teal. Tobs. Teai. Lobs. 


0 <20 45 58 74 70 9 17 
12 <20 20 25 1 0 
100 100 


The agreement between calculated and observed values is 
quite satisfactory. 

§ 4. Comparing the lattices found for the xenotime with 
that of zircon, we see that the lattice at the same time 
accounts for the crystallographic similarity and the difference 
in chemical constitution. 

Both lattices have the symmetry properties which are 
necessary to give a crystal of the ditetragonal bipyramidal 
class. 

If we would imagine the oxygen atoms removed the two 
lattices would be exactly of the same type. 

In both lattices the same relative number of oxygen atoms 
are arranged with tetragonal symmetry in a plane perpen- 
dicular to the tetragonal axis, which accounts for the fact 
that the ratio c/a is almost the same for both minerals. 

The difference found for the arrangement of the oxygen 
atoms accounts for the different chemical constitution to be 
expected for the two minerals. In zircon both Zr and Si 
take up essentially the same position towards the oxygen 
atoms; in xenotime, however, the oxygen atoms are 
attached in a singular way to the P atoms and we get groups 
(radicles) of the composition POQ,. Thus the Rontgen-ray 
analysis of the space-lattice gives as the constitution formule: 


ZrO,S8iO, for Zircon and Y(PO,) for Xenotime. 


512 Dr. L. Vegard on 
Anatase (Ti0.). 


§ 5. This mineral is isomeric with rutile, but although both 
crystals belong to the ditetragonal bipyramidal class of the 
tetragonal system they are not isomorphous. for rutile 
the ratio c:a@ is equal to 0°644, for anatase it is equal to 
ATE 

Also, in the case of anatase, we had some difficulty in 
obtaining reflexion from a sufficient number of faces ; but 
finally we succeeded in finding a number of maxima for the 
following five : (100), (001), (110), (111), and (112). 

The positions and intensities of the observed maxima are 


given in Table III. and in fig. 3. 


TABLE ITT. 

(001). (100). (110). | (111). (112). 
nh. ee ee ee ee 

) [1 0 I 6. il f) i 0 I 
elie 7° 30'| 10 118° 19'15:8 | 9°21/13-2 | 5° 0'/46 |11°48'| 1:5 
Dives ee 15 8/06/27 26/14/18 58|;08/)10 2/0 La: 0 
i ies Ve eo Opal irene aNg is eae oy PQ Os 
Abas: BO 150 Ord | nee Pee Ree ines [20 DAL Os 


Tf we would write the formula (TiO,), as in the case of 
rutile, we should find that only ;', of a molecule is associated 
with the elementary cell dyo9’dqo;, or just half the number 
found for the rutile cell. 

For the ratios of the spacings we get: 


dioo 5 dio g diy : door 2 Ono :1°418 : 2°643 3 1:765 5 1-126, 


or approximately 
ah 4 1 

=1; V2i—G=—— =: ————— 

f2+() 2 /() 45 

c C 2 
These are exactly the ratios which correspond to a lattice 
of the diamond type which is drawn out in the direction of 
the tetragonal axis. For the sake of comparison we 
remember that in the case of the isomeric substance, rutile, 
the lattice of the Ti atoms is composed of two lattices of the 


diamond type put inside one another in such a way that 
the elementary cell is not altered. The lattices give the 


Results of Crystal Analysis. 513 


right number of atoms in the elementary cell and explain 
the fact that the numbers in anatase are just half the number 
found for rutile. We should probably state this fact in the 


Fig. 3. 
Anatase 


0 0.1 COE ik mora 0.4 0s 


best way by writing the molecular formula of anatase TiOz, 
and that of rutile (TiO,)., then we should have in both 
crystals 3 molecule associated with the elementary cell. 


The Position of the Oxygen Atoms. 


§ 6. A fact of the first importance for the determination of 
the positions of the oxygen atoms is, that the faces (100) and 
(001) give spectra with very different relative intensities. 
This fact excludes any pseudocubic arrangement, or the 
tetragonal structure is also in the present case to be ascribed 
to a tetragonal arrangement of the oxygen atoms. 

The oxygen atoms cannot be arranged with tetragonal 
symmetry in a plane perpendicular to the c-axis, as in the 
case of zircon and xenotime. Such an arrangement would 


514 Dr. L. Vegard on 


make the (001) spectrum normal, contrary to the obser- 
vations, which give an abnormally weak second order 
spectrum. 

We are then naturally led to assume the following 
arrangement : 

Two oxygen atoms are associated with each Ti atom with 
the latter in a central position forming a kind of moleeular 
element. In the case of anatase all the molecular axes are 
parallel to the c-axis. 

The spacings corresponding to this arrangement are shown 
in fig. 4. 


Fig. 4. 
Anatase 
t i ' j 
' ! t { 
(iD i 1 j«—g —sI Pe 
i ! j j : 
: 
77 O 07; 6) rd) TA: 


(100) 
I { 
iH ii 
Ti Oz Ti Op 
I 
(110) 
t t 
H f 
Ti Op Ti Oz 
t I 
! t 
HZ) ce ie 
; ; 
' { 
TiO, Of 


We see that the faces (100) and (110) should give a 
normal spectrum, and that the relative intensities of the 
faces (001) and (112) should be equal. Both consequences 
are in agreement with observations; it is only to be remarked 
that in the case of the face (112) the reflexion is so weak 
that the very weak spectra of second and third order could 
not be detected at all. 

The arrangement has one parameter (the length of the 
molecular axis), and it remains to be seen whether a proper 


Results of Crystal Analysis. 515 


choice of the parameter can give the right intensities for the 
spectra (001) and (111). 
The amplitudes are determined by the following expressions: 


(OOP Ae eavio-E Cos na,’ ~)).2) en 


(111) A,=(1°375 + cos n8) cos, re ae) 
ce l 
ee 4 = Ua = aap 
where / is the distance from the Ti atom to one of its oxygen 
atoms. 

It follows from the lattice that J/e<4 or B<m, and 
a<4m. Jet the smallest value of « which gives the right 
intensities of (001) be a, then the amplitudes of (001) will 
be unaltered if we interchange a by the values: 


27—oa, 2r+ao, 4—ay. 


Which of these is the right one must be determined from 
the (111) spectrum. 

The right value of a is very nearly equal to 60°, and from 
(111) we find «=27—a)=300° and B=75°. 

Table IV. gives the calculated and observed intensities 
for the two faces. 


TaBLeE LV. 
B=T5°. 
(001). (111). 

Teal. Tobs. Teal. Tobs, 

100 100 100 100 

T6 6 0 8) 
0'5 15 23 66 

0-8 14 9 11 


The calculated and observed values agree as well as can 
be expected, and we have no doubt found the right lattice 
for anatase with the right symmetry properties. 

Also in the case of anatase the tetragonal structure is 
produced by the oxygen atoms. 

The molecular element must require much more space in 
the direction of its axis than in a direction perpendicular toe 
it. As all molecular axes are parallel to the tetragonal axis, 


516 Dr. I. Vegard on 


the lattice must be drawn out in this direction, and also in 
this case the position of the oxygen atoms accounts for the 
value of c/a, which in this case is greater than unity. 

To make clear the relation between the lattices of anatase 
and rutile, we imagine that in the latter substance we 
remove one of its two Ti lattices of the diamond type and 
the oxygen atoms associated with the Ti atom. The lattice 
left with the molecular axes in tetragonal arrangement in 
planes perpendicular to the tetragonal axis does 1:0t seem to 
be stable. If, however, all molecular elements are turned 
through an angle of 90°, so as to have their axes all parallel 
with the tetragonal axis, the configuration becomes stable 
and forms the mineral anatase. 

Photographs of models of the lattices of xenotime and 
anatase are shown on Plate XII. 

§ 7. The absolute dimensions of the lattices are given in 
Table V. For the sake of comparison I have also given the 
dimensions of the Zircon group. 


{ONE AY 
Substance. Qe C. cla. V. f 

. 10-* em. | 10-* em. 10-22 em.3 Pa 
ZrO,Si0, s0c000 9:20 5°87 0:639 4:97 (Si) 1-08 
GnON aoe oa85 629 | 0673 5-50 2-08 
(TiO,), (Butile)| 9-05 5°83 0-644 4°77 1-99 
TiO, (Anatase).| 5:27 9°37 STONE 2°60 1:95 
To, eae 9-60 5:94 0-618 5:49 1-23 


V is the volume of the elementary lattice, J is the distance 
from an oxygen atom to the central atom to which it belongs. 

We see that also with regard to the absolute dimensions 
of the lattice xenotime comes very close to the Zircon group. 

The minerals of the Zircon group showa small, but regular 
increase of dimensions with increase of atomic number of 
the central atom of the molecular group. Comparing the 
dimensions of rutile and anatase, we notice that in the 
direction which is perpendicular to all molecular axes the 
linear dimensions are nearly equal, and the value of a of 
rutile is nearly equal to the value of ¢ of anatase. This 
may be due to the fact that in both cases there is the same 
number of molecular elements inside the lattice which have 
their molecular axis directed along the axis considered. 

We may notice that the absolute dimensions are calculated 


Results of Crystal Analysis. 517 


from the measured glancing-angle and the known wave- 
length. If V’ is the volume of the elementary cell, M and p 
molecular weight and density, x the number of molecules in 
the cell, N the number of molecules in a gramme-molecule 
(61°5 x 10”), we have the relation : 


M 
I 
Vip=nx- 

If we knew p and n and had measured V’, we might 
find M. As the glancing-angle can be found with a very 
great accuracy, the reflexion of Réntgen rays from crystals 
might furnish us with a valuable and accurate method of 
determining the atomic weight of an element which had 
entered into combination with elements of known atomic 
weight. 

If we only want ordinary relative values of the atomic 
weight, the accuracy mainly depends on the accuracy of the 
glancing-angle, and not on the number N. To fix the 
idea, let us suppose that we have two cubic crystals, one 
of which we take as a standard. Let the glancing-angle of 
the cube faces be @ and 6, then 


See A y 
pi ph ae Wein J 


Summary. 


(1) The crystalline structure of xenotime Y(PO,) and 
anatase (TiO,) has been determined. 

(2) The lattice of xenotime is not of the zireon type ; the 
difference, however, is only due to a different arrangement 
of the oxygen atoms. In xenotime they are arranged in 
groups of four round each P atom, while in zircon both Zr 
and Si have essentially the same relation to the oxygen 
atoms. The lattice gives the constitution formule Y(PO,) 
and ZrO,S8iO, for xenotime and zircon respectively. 

(3) In spite of the different arrangement of the oxygen 
atoms, which accounts for the chemical differences, the 
lattice also accounts for the crystallographic similarity of 
the two substances. Both possess the right symmetry, and 
in both cases the same relative number of oxygen atoms are 
arranged in planes perpendicular to the tetragonal axis, and 
will expand the lattice perpendicular to this axis to the same 
extent and make the ratio ¢/a smaller than unity and almost 
equal for both crystals. 

(4) We can imagine the lattice of anatase to be derived 


518 Prof. A. Ogg and Mr. Lloyd Hopwood: Critical Test 


from the zircon lattice by removing the Zr atoms and the 
oxygen atoms associated with them, and substitute the Si 
with Ti atoms. The oxygen atoms seem no longer to be 
able to remain in the position they had in zircon, the 
molecular axes all turn through an angle of 90° so as to 
become parallel to the tetragonal axis, and so form the stable 
lattice of anatase. 

(5) Also, in the case of anatase, the deformation of the 
lattice from the cubic form is due to the oxygen atoms. 
The molecular axes being all parallel, the lattice most 
expands in the direction of the tetragonal axis to make room 
for the oxygen atoms, and thus make the ratio c/a greater 
than unity. 

(6) The absolute dimensions of the lattices of zircon and 
xenotime are very nearly the same. The volume of the 
lattices of the Zircon group shows a small but regular 
increase with increasing atomic number. 

Also, the distance from a central atom to one of the 
oxygen atoms associated with it increases with increase of 
atomic number of elements belonging to the same family. 
Generally the distance is smaller for a greater affinity to 
oxygen. 


In conclusion, I wish to express my indebtedness to 
Professor W. C. Broégger, for supplying me with the crystals 
necessary for the research, and to Mr. H. Schjelderup for 
valuable assistance in making the observations. 


LVI. A Critical Test of the Crystallographic Law of Valency 
Volumes; a Note on the Crystalline Structure of the 
Alkalt Sulphates. By A. Uae, M.A., Professor of 
Physics, University College, Grahamstown, South Africa, 
and EF. Luoyp Hopwoop, M.Se.(Lond.), A.&.C.Sce., 
Assistant in Physics, University of London, University 
College *. 


Qa complete investigation of the structure of crystalline 

substances involves two distinct inquiries. The first is 
concerned with the various ways in which the structural units 
may be arranged to form a homogeneous structure, and the 
second with the nature of these units. 

While it is generally believed that the geometrical theory 
of crystal structure is now complete, the laws governing 
the relations between crystalline structure and chemical 


* Communicated by Prof. W. H. Bragg, F.R.S. 


of the Crystallographic Law of Valency Volumes. 519 


constitution are but imperfectly understood. One of the 
generalizations in this connexion which has gained wide 
support, and is commonly accepted, is the theory due to 
Pope and Barlow, known as the Law of Valency Volumes. 
According to this theory *, the entire space occupied by a 
crystal can be regarded as a close-packed assemblage of 
approximately spherical cells of various sizes, representing 
by their relative volumes the spheres of influence of the 
component atoms of any particular crystalline structure. 
The volumes of the cellular domains allotted as above 
are further supposed to be approximately in the ratio of 
the integers which respectively express the fundamental 
chemical valencies of their contained atoms. 

There is a certain amount of indirect evidence { against 
the truth of this theory; but the application of the X-ray 
method of investigating crystal structure now furnishes us, 
for the first time, with incontrovertible evidence of its failure 
in a typical case. 

In the present paper the results of an investigation of 
the crystalline structure of the sulphates of potassium, 
rubidium, ammonium, and cesium are set forth and dis- 
cussed in reference to the Law of Valency Volumes. 

These erystals all belong to the orthorhombic system, 
and form one of the best-known examples of an isomorphous 
series. 

The crystals were examined in a Bragg X-ray spectro- 
meter, using an X-ray bulb with a palladium anticathode. 

It was soon evident that the elementary cell, or unit 
rhomb, of each of these crystals contained four molecules. 

Thus it was found that the glancing angle in the second- 
order spectrum reflected from the (100) face of potassium 
sulphate was 5° 51’. Substituting this value, and that of 
the appropriate wave-length of the X-radiation used, in the 
usual formula 

nmrX = 2d sin @, 
we find 
digg = Or fa xX LOn? em: 


Now the molecular weight of K,SO, is 173-04, its density 


is 2°666 gms./cm.3, and its axial ratios 
Goce Gs Ona 


* Vide * Annual Reports on the Progress of Chemistry,’ vol. v. (1908) 
pp: 268 e¢ segg.; also ‘Mineralogical Magazine,’ vol. xvii. (April, 1916) 
pp. 314-328. 

+ Tutton, ‘ Crystalline Structure and Chemical Constitution ’ 
(Macmillan & Co., 1910), pp. 123 et seqg. 

{ Tutton, ‘ Crystalline Structure and Chemical Constitution,’ p. 119. 


520 Prof. A. Ogg and Mr. Lloyd Hopwood: Critical Test 


Taking the mass of the hydrogen atomas 1°64 x 10-4 grm., 
the mass of unit rhomb, if it contains 4 molecules, is 


4x 173°04x164x10™* or 113°51x10-” orm, 
and Gone X 20 bbe ilo. x 10a 
and hence, from the known ratios of a:6:c, we find 
O= Oo oles 10-8 eme 
6 = 10:008x 1078 cm., 
¢= 7424x107 cm. 


it 


The equality in the values of dio) and a shows that the 
assumption made in our galculation, that there are 4 molecules 
in the elementary cell, is correct. 


TABLE I. 
Great Molecular | Density at | Molecular Axial ratios. 
eee weight. 2Q°/4°: volume. ae ee 8 
KESO) i bunks. 173°04 2666 64-91 0:5727 :1:0°7418 
(NH,),S0,...... 131-20 1-72 74-04 0:5635 : 1: 0°7319 
RESO, a!) 265714 3615 73°34 0°5723 : 1: 0°7485 
WSO, cee 359°14 4-246 84:58 0:5712:1:0°7531 
Length of sides of unit rhomb. 
Cara Volume of 
Se unit rhomb. 
a b. c 
cms. cms. _, emsa Je cm oa 
RESO Ae. ..| 573110 10-008 x 10 7424x10 | 425°78x10 
(NH,),80,...... 5951x10~° | 10560x10~° | 7°729x107° | 485°71x107~ 
BESO) v....... 5949x107° | 10394107" | 7:°780x10~° | 481-14x107~ 
Cs SOnise.: 6-218x10~° | 10°884x10~° | 8:198x10~° | 55488x107> 


In Table I. the volumes of the elementary cells, and the 
relative and actual values of the lengths of their sides, are 
shown for the whole series of crystals. 

The most striking fact disclosed by the above table is the 
very close agreement, almost amounting to identity, in the 
volumes of the unit rhombs of rubidium sulphate and 


of the Crystallographic Law of Valency Volumes. 521 


ammonium sulphate*. This is of fundamental importance. 
It shows us that the replacement in the elementary cell 
of eight atoms of potassium by forty atoms of the four 
radicle groups NH,, causes no more distension of structure 
than if eight atoms of rubidium had replaced the eight 
atoms of potassium. [urthermore, the substitution of eight 
atoms of cesium for ezght atoms of potassium causes double 
the distension of structure that the substitution of forty 
atoms of the four radicle groups NH, does. 

To the authors this appears to be conclusive evidence 
against the general truth of the theory of crystal structure 
based on the closest packing of the constituent atoms or 
their spheres of influence. 

Further evidence in support of the fact that the elementary 
cells of each of the above-mentioned crystals contain four 
molecules, and that the space-lattices of ammonium sulphate 
and rubidium sulphate are strictly comparable with each 
other, is exhibited in Table II. 

This table shows the agreement between the values of the 
glancing angles calculated on the above assumption, and the 
observed values, for the whole series. The wave-length of 
the particular line in the X-ray spectrum of palladium used 
was *084x 107% cm.f, except in the case of the rubidium 
and cesium crystals. The reflexions from some of the faces 
of these two crystals were rather faint, and were taken 
with slits too wide to entirely cut out the effects due to 
M— oli x 107° cm. 

; In general, there was no difficulty in separating these two 
ines. 

The complete structure of this series of crystals has not 
yet been worked out. With the object of doing this, the 
authors contemplated investigating the related series of 
crystals—the alkali selenates; but this work has _ been 
interrupted. Some information, however, may be deduced 
from Table II. 

The absence of the first- and third-order spectra from the 
(100) planes shows that the spacings are equal to one-half 
of the sides of the elementary cell. For the (010) planes 
the fourth order is strongest, the second being found only in 
the case of ammonium sulphate. The fourth order is also 


* Since the sum of the fundamental valencies of (NH,),SO, and Rb.SO, 
are respectively 24 and 12, the volume of the elementary cell of the former 
should, according to Pope and Barlow, be double that of the latter 
crystal. _8 a 

{ Really the two lines \='5828X10 and r="6872x10 . 


Phil. Mag. 8. 6. Vol. 32. No. 191. Nov. 1916. 2N 


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523 


of the Crystallographic Law of Valency Volumes. 


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524 Prof. A. Ogg and Mr. Lloyd Hopwood: Critical Test 


strongest in the reflexions from the (001) planes, except in 
the case of ammonium sulphate, which gives a stronger 
second order. These general considerations lead us to think 
that there are planes of sulphur atoms with spacings equal 
to half the side of the unit rhomb, with planes of metal 
atoms lying between them. The presence of a very strong 
first-order spectrum from the (111) planes of ammonium 
sulphate would seem to indicate a tace-centred lattice for the 
sulphur atoms. 

It appears, then, that a lattice with representative points 
at the corners and at the middle points of the faces (see 
fig. 1) would explain the spectra and give four molecules per 
unit rhomb, 


Fig. 1. 


The dimensions of the crystals are such that the tri- 
angles OHA and BHG are nearly equilateral. It has 
therefore been suggested by von Federow that the space- 
lattice of the rhombic sulphates is pseudo-hexagonal. Tutton 
considers the evidence so strong for such a structure that he 
has adopted it. 

The arrangement we have suggested would give the 
representative points in the (001) planes at the corners 
of hexagonal prisms. But the plane KLMN is derived from 


of the Crystallographic Law of Valeney Volumes. 525 
OAGB by sliding, and therefore the lattice would not be 


hexagonal. To make the system hexagonal, the repre- 
sentative points on the plane KLMN would have to be 
at the middle points of the sides and the centre of the 
rhomb. The middle plane must, however, be different in 
some way from the end planes. This might, of course, be 
brought about by the oxygen atoms. 

Without entering further into a detailed analysis of the 
conditions which will satisfy the symmetry and explain 
the intensities of the various orders of the spectra, we at this 
stage merely wish to indicate the probability of the sulphur 
atoms being located at the corners and centres of the faces 
of the unit rhomb. This would give a pseudo-hexagonal 
structure for the sulphur atoms in planes parallel to the 
(001) face. 

The metals are also probably arranged in hexagons. 


Summary. 


An account is given of an X-ray investigation into the 
crystalline structure of the Isomorphous Alkali Sulphates. It 
is shown that for each of these crystals there are four molecules 
in the elementary cell. It follows from this that the dimen- 
sions of the crystal units of ammonium sulphate and rubidium 
sulphate are almost identical. This furnishes a critical test 
of the Valency Volume Theory of Pope and Barlow, which 
is shown to break down in this case. Further evidence in 
support of this contention is provided by the agreement 
between the observed and calculated values of the angles of re- 
flexion from these crystals. Although the complete structure 
of the series is not solved, the probability of the hexagonal 
arrangement of the metal and sulphur atoms is pointed out. 


Most of the crystals used in this investigation were lent by 
Dr. A. HK. H. Tutton, F.R.S. They had been prepared with 
optically worked faces and used by him for his classical work 
on the thermal expansion of crystals. We have to thank him 
for his very great kindness in this matter, and also for the 
keen interest which he took in the work. 

It is also a pleasure to record our thanks to Professor 
W. H. Bragg for the valuable assistance and advice which 
he was ever ready to give during the progress of the 
measurements. 


University College, London, 
April 1916. 


a 


LVII. Notices respecting New Books. 


A Theory of Time and Space. By Atrrep A. Ross. 
1914. Cambridge University Press. 


INKOWSKYS mathematical treatment of the theory of Rela- 
tivity reduces time to imaginary space, and presents the 
inter-relations of space and time in terms of the properties of a 
four-dimensional space. Mr. Robb in this truly remarkable book 
approaches a similar outlook by an altogether different route. The 
presentation is geometrical and of the approved modern type. That 
is to say, there are laid down certain postulates from which, with 
the aid of a series of closely connected theorems, there is con- 
structed a geometry in which every element is determined by four 
coordinates. The argument is difficult to follow; and the full 
bearing of the conclusions reached is not easy to appreciate. The 
author is meanwhile content with having established a consistent 
geometry, and leaves developments and applications for a further 
volume. The fundamental notion which is the basis of the whole 
is the recognition of what Mr. Robb calls conical order. According 
to this conception, an event which is neither before nor after 
another is not necessarily simultaneous with it. The only events 
which are reallv simultaneous are those which occur also at the 
same place. When events occur at different places we may be 
able to say that one is neither before nor after the other, but we 
cannot say that they are simultaneous. In the four-dimensional 
space which is thus imagined a point represents a state of a 
particle at a given time, and “the theory of space becomes 
absorbed in the theory of time.” 


LVIII. Proceedings of Learned Societies. 
GEOLOGICAL SOCIETY. 
[Continued from p. 176.] 


June 28th, 1916.—Dr. Alfred Harker, F.R.S., President, 
in the Chair. 


oheat following communications were read :— 


1. ‘On a New Species of Hdestus from the Upper Carboni- 
ferous of Yorkshire. By A. Smith Woodward, LL.D. F.R.S., 
V.P.G.S. With a Geological Appendix by John Pringle, F.G.S. 


2. ‘The Tertiary Voleanic Rocks of Mozambique.’ By Arthur 
iFiolmes; 3 :Se.,-A. R-C. Se. DAL C2 #-G.s- 


Until recently, the district of Mozambique—geographically as 
well as geologically—was one of the least known of the East 
African coast-lands. During the seasons 1910-11, a prospecting 


Geological Society. 527 


expedition was organized by the Memba Minerals Ltd., and during 
the second season, Mr. HE. J. Wayland, Mr. D. A. Wray, and the 
author visited the country as geologists to the Company. 

With the exception of a coastal belt of Cretaceous and Tertiary 
sediments, flanked on the west by later Tertiary volcanic rocks, the 
whole territory consists of a complex of gneisses and other foliated 
rocks, intruded upon by granites belonging to at least two different 
periods. From Fernao Vellosa Harbour to Mokambo Bay the 
junction of the sedimentary formations with the crystalline com- 
_ plex is a faulted one, and the volcanic rocks, now greatly dissected 
by erosion, are distributed on each side of the fault. The lavas 
are of post-Oligocene age, and are clearly the result of fissure- 
eruptions, the feeding channels being exposed as numerous small 
dykes that penetrate the underlying rocks. 

Throughout the area the prevailing lavas are amygdaloidal 
basalts, in which the chief amygdale minerals are chlorite, heulan- 
dite, and forms of silica. An andesite dyke of later date occurs 
near the Monapo River. In the north, near the Sanhuti River, 
picrite-basalt, basalt, phonolite, and sélvsbergite have been found, 
and related lavas occurring elsewhere in the area are tephritic 
pumice and egirine-trachyte. Thus, within the limits of a 
small volcanic field, series of both ‘alkali’ and ‘cale-alkali’ 
types of lava occur together. The ‘alkali’ series can be closely 
matched by the lavas of Abyssinia, British Hast Africa, Réunion, 
and Tenerife. The amygdaloidal basalts of the ‘calc-alkali’ series 
are similar to those of the Deccan, Arabia, and East Africa, and 
also to those (of late Karroo age) occurring in South Africa and 
Central Africa. 

In all, ten analyses have been made, and the variation-diagrams 
constructed from them support the view that each of the series 
was evolved by a process of differentiation acting on a parent 
magma. From the composition of the amygdale minerals (which 
are referred to the closing phase of lava-consolidation), it is 
deduced that the parent magma of the ‘alkali’ series was rich in 
carbon dioxide, and undersaturated in silica; whereas that of the 
‘cale-alkali’ series was rich in water, and oversaturated in silica. 
The radioactivity of the lavas indicates that the depth from which 
the parent magma came was probably between 383 and 44 miles 
from the earth’s surface. The boundary-fault along which the 
lavas are aligned seems to mark a zone where pressure was relieved 
to an extent and depth sufficient to promote fusion. 


Dr. A. Srraman, F.R.S., exhibited cores from borings 
in Kent, showing pebbles of coal embedded in Coal- 
Measure sandstones. With the coal-pebbles occurred a few 
partly-rounded fragments of chert, and in one of these radiolaria 
had been identified by Dr. G. J. Hinde. The chert resembled 
that which had been described from Lower Carboniferous rocks 
elsewhere. Its occurrence suggested that the sequence of strata had 
been similar m South Wales and Kent, and, taken in connexion 


528 Geological Society. 


with the piping of the limestone-surface at Ebbsfleet and the 
absence of Millstone Grit in Kent, tended to confirm the view 
that there is unconformity between the Coal Measures and the 
Carboniferous Limestone in that county. 


Mr. F. P. Menyext exhibited a geological sketch-map 
of. the northern margin of Dartmoor. 

He said that the central part of Devon was toa great extent 
a terra incognita; but,-as regarded the fringe of altered Carbon- 
iferous rocks along the northern border of the Dartmoor granite, 
he had been led, in the course of observations originally concerned 
with the petrology alone, to the conclusion that it might prove 
possible to establish a definite order of succession. This was 
rendered feasible by the occurrence of some well-characterized 
bands of rock, especially limestones and tufts, which were exposed 
in every good river-section. It was true that almost everywhere 
overtolds, sometimes accompanied by thrusts, were to be de- 
tected, and tended to make the observer somewhat doubtful of 
his ground. Nevertheless, it seemed impossible to escape the 
conclusion that, as one approached the granite from the north, 
continuously older rocks were met with, and the extremely 
continuous character of some of the beds seemed to show that, 
despite all minor disturbances, the general sequence could be 
trusted. 'The comparison of the different lines of section leading 
up to Dartmoor showed them to be strikingly similar. The 
granite was, moreover, intruded all along at precisely the same 
horizon, and its direct offshoots never reached into the lower of 
the two important bands of limestone, but were confined to the 
altered shales at the bottom of the series, which afforded, where 
fresh, good examples of andalusite-hornfels. The series, which 
extends from south of Sourton to Drewsteignton and perhaps right 
round to Doddiscombeleigh, appears clearly older than the shales 
which have been so carefully searched for fossils in the Exeter 
region by Mr. F. J. Collins. These last are considered to be of 
Pendleside age, and nowhere contain any traces of limestone. 
The probability is thus indicated that the distinctly calcareous 
series under consideration may represent part of the Carboniferous 
Limestone. 

It may be noted that, although a number of bands of epidio- 
rite representing intrusions of dolerite occur roughly parallel to the 
strike of the sediments, the contemporaneous rocks are never of 
such basic character. The main band of tuff stretches from Lake, 
near Bridestowe, to beyond Sticklepath, and, of the numerous 
well-preserved rock-fragments that it contains, most are of 
rhyolitic or trachytic character, with some which represent altered 
andesites. 


IX ‘Id ‘ZE 1A ‘9 “10g ‘’M Tg ; NVI] | 


oe ‘2 
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Phil. Mag. Ser. 6, Vol. 32, Pl. Xt. 


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THE 
LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


- AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES ] 


DECEMBER 1916. 


ea 

LIX. On Convection Currents in a Horizontal Layer of 

Fluid, when the Higher Temperature is on the Under Side. 
By Lord Rayuries, O.M., £RS.* 


HE present is an attempt to examine how far the inter- 
esting results obtained by Bénard f in his careful and 
skilful experiments can be explained theoretically. Bénard 
worked with very thin layers, only about 1 mm. deep, standing 
on a levelled metallic plate which was maintained at a uni- 
form temperature. The upper surface was usually free, and 
being in contact with the air was ata lower temperature. 
Various liquids were employed—some, indeed, which would 
be solids under ordinary conditions. 

The layer rapidly resolves itself into a number of cells, the 
motion being an ascension in the middle of a cell and a 
descension at the common boundary between a cell and its 
neighbours. ‘'wo phases are distinguished, of unequal dura- 
tion, the first being relatively very short. The limit of 
the first phase is described as the “semi-regular cellular 
regime”; in this state all the cells have already acquired 
surfaces nearly identical, their forms being nearly regular 
convex polygons of, in general, 4 to 7 sides. The bounduries 


* Communicated by the Author. 

+ Revue générale des Sciences, vol. xii. pp. 1261, 1809 (1900); Ann. da. 
Chimie et de Physique, t. xxiii. p. 62 (1901). M. Bénard does not appear 
to be acquainted with James ‘Thomson’s paper “ On a Changing Tesselated 
Structure in certain Liquids” (Proc. Glasgow Phil. Soc. 1881-2), where 
a like structure is described in much thicker layers of soapy water cooling 
from the surface. 


Phil. Mag. 8. 6. Vol. 32. No. 192, Dee. 1916. 20 


530 Lord Rayleigh on Convection Currents in 


are vertical, and the circulation in each cell approximates to 
that already indicated. This phase is brief (1 or 2 seconds) 
for the less viscous liquids (alcohol, benzine, &c.) at ordinary 
temperatures. Even for paraffin or spermacetti, melted at 
100° C., 10 seconds suffice ; but in the case of very viscous 
liguids (oils, &e.), if the flux of heat is small, the deforma- 
tions are extremely slow and the first phase may last several 
minutes or more. 

The second phase has for its limit a permanent regime of 
regular hexagons. During this period the cells become equal 
and regular and allign themselves. It is extremely pro- 
tracted, if the limit is regarded as the complete attainment 
of regular hexagons. And, indeed, such perfection is barely 
attainable even with the most careful arrangements. The 
tendency, however, seems sufficiently established. 

The theoretical consideration of the problem here arising 
is of interest for more than one reason. In general, when a 
system falls away from unstable equilibrium it may do so in 
several principal modes, in each of which the departure at 
time ¢ is proportional to the small displacement or velecity 
supposed to be present initially, and to an exponential factor 
ev’, where g is positive. If the initial disturbances are small 
enough, that mode (or modes) of falling away will become 
predominant for which g is a maximum. The simplest 
example for which the number of degrees of freedom is 
infinite is presented by a cylindrical rod of elastic material 
under a longitudinal compression sufficient to overbalance 
its stiffness. But perhaps the most interesting hitherto 
treated is that of a cylinder of fluid disintegrating under 
the operation of capillary force as in the beautiful experi- 
ments of Savart and Plateau upon jets. In this case the 
surface remains one of revolution about the original axis, 
but it becomes varicose, and the question is to compare the 
effects of different wave-lengths of varicosity, for upon this 
depends the number of detached masses into which the 
column is eventually resolved. It was proved by Plateau 
that there is no instability if the wave-length be less than 
the circumference of the column. For all wave-lengths 
greater than this there is instability, and the corresponding 
modes of disintegration may establish themselves if the 
initial disturbances are suitable. But if the general dis-’ 
turbance is very small, those components only will have 
opportunity to develop themselves for which the wave- 
length lies near to that of maximum instability. 

It has been shown * that the wave-length of maximum 


* Proc. Lond. Math. Soc. vol. x. p. 4 (1879); Scientific Papers, vol. i. 
p- 861. Also ‘Theory of Sound,’ 2nd ed. §§ 357, &e. 


a Horizontal Layer of Fluid. 531 


instability is 4:508 times the diameter of the jet, exceeding 
the wave-length at which instability first enters in the ratio 
ef about 3:2. Accordingly this is the sort of disintegration 
to be expected when the jet is shielded as far as possible from 
external disturbance. 

It will be observed that there is nothing in this theory 
which could fix the phase of the predominant disturbance, 
or the particular particles of the fluid which will ultimately 
form the centres of the detached drops. There remains a 
certain indeterminateness, and this is connected with the 
circumstance that absolute regularity is not to be expected. 
In addition to the wave-length of maximum instability we 
must include all those which lie sufficiently near to it, and 
the superposition of the corresponding modes will allow of 
a slow variation of phase as we pass along the column. ‘The 
phase in any particular region depends upon the initial cir- 
cumstances in and near that region, and these are supposed 
to be matters of chance*. The superposition of infinite 
trains of waves whose wave-lengths ciuster round a given 
value raises the same questions as we are concerned with 
4 considering the character of approximately homogeneous 
ight. 

“ln the present problem the case is much more compli- 
cated, un!ess we arbitrarily limit it to two dimensions. The 
cells of Bénard are then reduced to infinitely long strips, 
and when there is instability we may ask for what wave- 
length (width of strip) the instability is greatest. The 
answer can be given under certain restrictions, and the 
manner in which equilibrium breaks down is then approxi- 
mately determined. So long as the two-dimensional cha- 
racter is retained, there seems to be no reason to expect the 
wave-length to alter afterwards. But even if we assume a 
natural disposition to a two-dimensional motion, the direc- 
tion of the length of the cells as well as the phase could 
only be determined by initial circumstances, and could not 
be expected to be uniform over the whole of the infinite 
plane. | 

According to the observations of Bénard, something of this 
sort actually occurs when the layer of liquid has a general 
motion in its own plane at the moment when instability 
commences, the length of the cellular strips being parallel 
to the general velocity. But a little later, when the general 
motion has decayed, division-lines running in the perpen- 
dicular direction present themselves. 


* ‘When a jet of liquid is acted on by an external vibrator, the reso- 
lution into drops may be regularized in a much higher degree. 


202 


532 Lord Rayleigh on Convection Currents in 


In general, it is easy to recognize that the question is 
much more complex. By Fourier’s theorem the motion 
in its earlier stages may be analysed into components, each 
of which corresponds to rectangular cells whose sides are 
parallel to fixed axes arbitrarily chosen. The solution for 
maximum instability yields one relation between the sides of 
the rectangle, but no indication of their ratio. It covers the 
two-dimensional case of infinitely long rectangles already 
referred to, and the contrasted case of squares for which 
the length of the side is thus determined. I do not see that 
any plausible hypothesis as to the origin of the initial dis- 
turbances leads us to expect one particular ratio of sides in 
preference to another. 

On a more general view it appears that the function 
expressing the disturbance which develops most rapidly 
may be assimilated to that which represents the free 
vibration of an infinite stretched membrane vibrating with 
given frequency. 

The calculations which follow are based upon equations 
given by Boussinesq, who has applied them to one or two 
particular problems. The special limitation which charac- 
terizes them is the neglect of variations of density, except in 
so far as they modify the action of gravity. Of course, such 
neglect can be justified only under certain conditions, which 
Boussinesq has discussed. They are not so restrictive as to 
exclude the approximate treatment of many problems of 
interest, 

When the fluid is inviscid and the higher temperature is 
below, all modes of disturbance are instable, even when we 
include the conduction of heat during the disturbance. But 
there is one class of disturbances for which the instability is 
a maximum. 

When viscosity is included as well as conduction, the 
problem is more complicated, and we have to consider 
boundary conditions. Those have been chosen which are 
simplest from the mathematical point of view, and they 
deviate from those obtaining in Bénard’s experiments, 
where, indeed, the conditions are different at the two 
boundaries. It appears, a little unexpectedly, that the equi- 
librium may be thoroughly stable (with higher temperature 
below), if the coefficients of conductivity and viscosity are 
not too small. As the temperature gradient increases, in- 
stability enters, and at first only for a particular kind of 
disturbance. | 

The second phase of Bénard, where a tendency reveals 
itself for a slow transformation into regular hexagons, is not 


a Horizontal Layer of Fluid. 533 


touched. It would seem to demand the inclusion of the 
squares of quantities here treated as small. But the size of 
the hexagons (under the boundary conditions postulated) is 
determinate, at any rate when they assert themselves early 
enough. 

An appendix deals with arelated analytical problem having 
various physical interpretations, such as the symmetrical 
vibration in two dimensions of a layer of air enclosed by 
a nearly circular wall. 


The general Eulerian equations of fluid motion are in the 
usual notation :— 


Du_y_1ldp De_y_ 1d Dw _7_ 1 dp (1 
ae a) pide, DE pin Wie wy ands.) 
where 


Deiat) d d d 

SS == === = Tans SAL a Raa (9 
eden oy ae oy 
and X, Y, Zare the components of extraneous force reckoned 
per unit of mass. If, neglecting viscosity, we suppose that 
gravity is the only impressed force, 


N=) Y=0, Zoi Genco) 


z being measured upwards. In equations (1) p is variable 
in consequence of variable temperature and variable pres- 
sure. But, as Boussinesq* has shown, in the class of 
problems under consideration the influence of pressure is 
unimportant and even the variation with temperature may 
be disregarded except in so far as it modifies the operation 
of gravity. If we write p=po+ dp, we have 


9p =9po(1 + 8p / po.) =9pa— gor, 


where @ is the temperature reckoned from the point where 
p=po and @ is the coefficient of expansion. We may now 
identity p in (1) with po, and our equations become 


Du__1dP Dv__1dP Dw__1aP ig ay 
Dt ~~ p de’ DEE p dy’ DiLw ae to (4 


where p is a constant, y is written for ga, and P for p+gpz. 


* Théorie Analytique de la Chaleur, t. ii. p. 172 (1905). 


D384 Lord Rayleigh on Convection Currents in 


Also, since the fluid is now treated as incompressible, 


du dv dw 
Rpt ae el . 2) Ti ee 


The equation for the conduction of heat is, 


De & WA allel 
Dt ~~ \da? dy? = 


: ae 


in which « is the diffusibility for temperature. These are 
the equations employed by Boussinesq. 

In the particular problems to which we proceed the fluid 
is supposed to be bounded by two infinite fixed planes at 
e=(0 and z=€, where also the temperatures are maintained 
constant. In the equilibrium condition u, v, w vanish and 
@ being a function of z only is subject to d*@/dz?=0, or 
d@/dz=, where 8 is a constant representing the tempera- 
ture gradient. If the equilibrium is stable, @ is positive; 
and if unstable with the higher temperature below, @ is 
negative. It will be convenient, however, to reckon @ as 
the departure from the equilibrium temperature ©. The 
only change required in equations (4) is to write @ for P, 
where 


a=P—py\ @dz. J 3 oo. ps 
In equation (6) D@/ Dt is to be replaced by D@/Dt+w§. 


The question with which we are principally concerned is 
the effect of a small departure from the condition of equi- 
librium, whether stable or unstable. For this purpose it 
suffices to suppose u,v, w, and @ to be small. When we 
neglect the squares of the small quantities, D/D¢ identifies 
itself with d/dt and we get 


fe ide dee. 2 le 
et Sp dae? dt me meomaa: cat) pd ae 


dé dD) e102 Gao 
q the = "(Fat Tat aaa) + cane (9) 


which with (5) and the initial and boundary conditions 
suffice for the solution of the problem. The boundary 
conditions are that w=0, 02=0, when z=0 or €. 

We now assume in the usual manner that the small 
quantities are proportional to 


CMe eRe OE allt ae: 


a Horizontal Layer of Flud. D390 
so that (8), (5), (9) become 


lax Imes lda 
nu=—-—, nv=— , nw=—~—-+978,. (11 
p p pie = oe 


tuum -dw/dez=0,. «\. ~~) 2) 
nO + Bu=x«(@idz?—P—m’?)O, . . . (13) 
from which by elimination of u, v, w, we derive 


mee ew 
P+ dz =nw—yé. @ ° ° ° (14) 

Having regard to the boundary conditions to be satisfied 
by w and 6, we now assume that these quantities are pro- 
portional to sinsz, where s=q7/f, and gq is an integer. 
Hence 


ate ie te teaatadiia tS) } C=, ) oe aus! (115) 
WMP +m? + s)\w—y(?+m)d=0, . . (16) 

and the equation determining 7 is the quadratic 
(P+ m* +s”) +nk(? +m? + s”)? + By(? + m?)=0. (17) 


When «=0, there is no conduction, so that each element of 
the fluid retains its temperature and density. If 8 be 
positive, the equilibrium is stable, and 


na tiv iByl? +m’) 
a J/{Ptmi+st ? 


indicating vibrations about the condition of equilibrium. 
If, on the other hand, 6 be negative, say —’, 


i EV {B'y(E+m?)} 
mars art (19) 


When x has the positive value, the corresponding disturbance 
increases exponentially with the time. 

For a given value of [?+m?, the numerical values of x 
diminish without limit as s increases—that is, the more sub- 
divisions there are along z. The greatest value corresponds 
with g=1 or-s=7/¢. On the other hand, if s be given, 
|n| increases from zero as 1?+m? increases from zero (great 
wave-lengths along z and y) up to a finite limit when /? +m? 
is large (small wave-lengths along « and y). This case of 


(18) 


536 Lord Rayleigh on Convection Currents in 


no conductivity falls within the scope of a former investi- 
gation where the fluid was supposed from the beginning to 
be incompressible but of variable density * 

Returning to the consideration of a wars conductivity, we 
have again to distinguish the cases where # is positive and 
negative. When 6 is negative (higher temperature below) 
both values of n in (17) are real and one is positive. The 
equilibrium is unstable for all values of (?+m? and of s 
If 8 be positive, m may be real or complex. In either case 
the real part of m is negative, so that the equilibrium is 
stable whatever /?+ m? and s may be. 

When # is negative (—’), it is important to inquire for 
what values of +m? the instability is greatest, for these 
are the modes which more and more assert themselves as 
time elapses, even though initially they may be quite 
subordinate. That the positive value of n must have a 
maximum appears when we observe it tends to vanish both 
when /?+m? is small and also when [?+m? is large. Setting 
for shortness /?+ m?+s?=a, we may write (17) 

nao +nko*—B'y(o—s’?)=0, . . . (20) 
and the question is to find the value of o for which n is 
greatest, s being supposed given. Making dnjdo=0, we 
get on differentiation 


W+2nko-— PB y=0;. 0. « 2. 
and on elimination of n? between (20), (21) 


ee La 


Ko" 


J. . 
Using this value of n in (21), we find as the equation for 
9) 2 be! 

ey rn 
o Ka" 
When « is relatively great, o=2s”, or 
Pam ss. 4.4.) 
A second approximation gives 
By 


SK? 


24 m?= 57+ 


The corresponding value of n is 


= F4{1- oat ios 


AKS 
* Proc. Lond, Math. Soc. vol]. xiv. p. 170 (1883) ; Scientific Papers, 
vol, i. p. 200, 


a Horizontal Layer of Flucd. 537 


The modes of greatest instability are those for which s is 
smallest, that is equal to m/f, and 


2 7 
T B Y, \ 
(pei == See Ba SAT Aa 
Ge One mie Ge ( : 

For a two-dimensional] disturbance we may make m=0O 
and /=27/x, where X is the wave-length along a The dr of 
maximum instability is thus approxunately 


DYER GE RU NIN SARE GLY GaAM TOR ame NH MeCN (Gren) 
Again, if l=m=27/>, as-for syuare cells, 


Ne ens ech wo VEN Caen) 
greater than before in the ratio ,/2: 1. 

We have considered especially the cases where « is 
relatively small and relatively large. Intermediate cases 
would need to be dealt with by a numerical solution of (23). 

When w is known in the form 


w= W ezemy sin SZ. gee, ‘ ; i $ (30) 


n being now a known function of /, m, s, u and v are at once 
derived by means of (11) and (12). ‘Thus 
dw im dw 
cs pr a a I= ss... . Cl 
e= Bam dz? ° 24m? dz Oy) 
The connexion between w and @ is given by (15) or (16). 
When § is negative and n positive, @ and w are of the same 
sign. 
As an example in two dimensions of (30), (31), we might 
have in real form 
Wi VR COST. SIT 2 CUR Mmr mre Coa.) 


Maa VSI. COS N24) v= Oumyue 0). (38) 


Hitherto we have supposed the fluid to be destitute 
of viscosity. When we include viscosity, we must add 
v(V7u, V/7v, Vw) on the right of equations (1), (8), and 
(11), v being the kinematic coefficient. Equations (12) and 
(13) remain unaitected. And in (11) 


Vi @jde— Pm a eh. 6). (84) 


We have also to reconsider the boundary conditions at -=0 
and z=¢€. We may still suppose 02=0 and w=0; but fora 
further condition we should probably prefer dw/dz=0, 
corresponding to a fixed solid wall. But this entails much 
complication, and we may content ourselves with the 


938 Lord Rayleigh on Convection Currents in 


supposition d?w/dz?=0, which (with w=0) is satisfied by 
taking as before w proportional to sin sz with s=gn/f. This 
1s equivalent to the annulment of lateral forces at the wall. 
For (Lamb’s ‘ Hydrodynamics,’ ¢§ 323, 326) these forces are 
expressed in general by 


dw du dw dv 
Lae ae ae ae dy ila e o. % (35) 


while here w=0 at the boundaries requires also dw/dx=0, 
dwidy=0. Hence, at the boundaries, d?u/dzx dz, d’v/dy dz 
vanish, and therefore by (5), d?w/dz*. 

Equation (15) remains unaltered :— 


Bw+in+K(P +m? + s*)}0=0, -- eae 
and (16) becomes 
{n+v(? +m? +s?) }(P +m? +s?)w—y(? +m?)0=0. (86) 
Writing as before c=/?+4+m’+ 5’, we get the equation in n 
(n+ Ko)(n+va)o+ PBy(?+m?)=0, . . (37) 


which takes the place of (17). 

If y=0 (mo expansion with heat) the equations degrade, 
and we have two simple alternatives. In the first n+ x«c=0 
with w=0, signifying conduction of heat with no motion. 
In the second n+va=0, when the relation between w and 8 
becomes 


Bw+o(e—v)d=0.. . . . . (38) 


In both cases, since n is real and negative, the disturbance is 
stable. 

If we neglect « in (37), the equation takes the same 
form (20) as that already considered when v=0. Hence 
the results expressed in (22), (23), (24), (25), (26), (27) 
are applicable with simple substitution of v for x. 

In the general equation (37) if 8 be positive, as y is 
supposed always to be, the values of n may be real or 
complex. If real they are both negative, and if complex 
the real part is negative. In either case the disturbance 
dies down. As was to be expected, when the temperature 
is higher above, the equilibrium is stable. 

In the contrary case when £ is negative (—{’) the roots 
of the quadratic are always real, and one at least is negative. 
There is a positive root only when 


By? 4m?) > Kv0%s + sci ley el 


a Horizontal Layer of Fluid. 539 


If «, or v, vanish there is instability ; but if « and pv are 
finite and large enough, the equilibrium for this disturbance 
is stable, although the higher temperature is underneath. 
Inequality (39) gives the condition of instability for the 
particular disturbance (/, m,s). Itis of interest to inquire 
at what point the equilibrium becomes unstable when there 
is no restriction upon the value of /?+m*. In the equation 


B'y(F +m?) —kve*?= B'y(o—s?)—Kva®=0, . (40) 


we see that the left-hand member is negative when 7? +m? 


is small and also when it is large. When the conditions 
are such that the equation can only just be satisfied with 
some value of /?+m?, or a, tle derived equation 


hop morya 7". Wie nvei sah. (Addy) 

must also hold good, so that 
G57) ee teeter) ot oe eel hone (EZ) 
and Bigeeo Tere ios Nay eaiyal sa. (43) 


Unless B'y exceeds the value given in (43) there is no 
instability, however / and mare chosen. But tle equation 
still contains s, which may be as large as we please. The 
smallest value of s is 7/f. The condition of instability 
when J, m, and s are all unrestricted is accordingly 


271 a Ky 


COU ap a en Sieh | a aS (44) 


If B'y falls below this amount, the equilibrium is altogether 

stable. I am not aware that the possibility of complete 

stability under such cireumstances has been contemplated. 
To interpret (44) more conveniently, we may replace £9’ 


by (O,—®,)/¢ and y by 9(p2—p3)/p1(O.—®), so that 
! Gg P2—P1 
—s 9 e e ° ° ° ° 45 
B'y BN Ae (45) 


where @,, ©, po, and p; are the extreme temperatures and 
densities in equilibrium. Thus (44) becomes 


Po— pi — 27 arKv pis. 
(oe gen <i) (46) 


In the case of air at atmospheric conditions we may take 
in C.G.S. measure 


v="14, and «=8v (Maxwell’s Theory). 


540 Lord Rayleigh on Convection Currents in 
Also g= 980, and thus 
ear ONS) 


> 


woe Sn 


For example, if €=1cm., instability requires that the 
density at the top exceed that at the bottom by one-thirtieth 
part, corresponding to about 9° C. of temperature. We 
should not forget that our method postulates a small value 
of (p2—p)/pi1. Thus if «v be given, the application of (46) 
may cease to be legitimate unless be large enough. 

It may be remarked that the influence of viscosity would 
be increased were we to suppose the horizontal velocities 
(instead of the horizontal forces) to be annulled at the 
boundaries. 

The problem of determining for what value of 1?+’, 
or o, the instability, when finite, is a maximum is more 
complicated. The differentiation of (37) with respect to o 
gives 


(47) 


n+ 2no(k+v)+3xv07—B'y=0,. . . (48) 
whence 
Au rit At aneee 
(IEEE |. 
o*(K == v) 
expressing nin terms of o. To find a we have to eliminate 
n between (44) and (45). The result is 


o°kv(K—v)? + 0°B'y(K+7)?—a*. 2B'ys? (x? +07) —Bry’s* =0, 
(50) 


from which, in particular cases, o could be found by 
1umerical computation. From (50) we fall back on (23) 
by supposing v=0, and again on a similar equation if we 
suppose c=0. 

But the case of a nearly evanescent n is probably the 
more practical. In an experiment the temperature gradient 
could not be established all at once and we may suppose the 
progress to be very slow. In the earlier stages the equi- 
librium would be stable, so that no disturbance of importance 
would occur until n passed through zero to the positive side, 
corresponding to (44) or (46). The breakdown thus oceurs 
for s=/¢, and by (42) ?4+m?=7'/2¢?. And since the 
evanescence of n is equivalent to the omission of d/dt in the 
original equations, the motion thus determined has the 
character of a steady motion. The constant multiplier is, 
however, arbitrary ; and there is nothing to determine it so 
long as the squares of u, v, w, 6 are neglected. 


a Horizontal Layer of Fluid. 5AL 


In a particular solution where w as a function of w and y 
has the simplest form, say 


Me GOSS COSI Yye alan ae auras 1 (OL) 


the particular coefficients of « and y which enter have 
relation to the particular axes of reference employed. If 
we rotate these axes through an angle ¢, we have 


w=2cos fa' cos¢—y' sin pb}. cos {z' sin 6+y’ cos b} 
= cos {z'(cos f— sin g)} . cos {y'(cos¢+ sin d)} 
+ sin {2'(cos é— sin d)} .sin {y/(cos $+ sin d) } 
+ cos {z'(cosg+ sin d)} . cos {y'(cos ¢— sin d)} 
— sin {2 (cos¢+ sin d)}. sin {y'(cosP— sin g)}. (52) 


For example, if 6=i7, (52) becomes 
w= cos (y',/2)4 cos (2/2), | (53) 


It is to be observed that with the general value of 4, if 
we call the coeificients of 2’, y', | and m respectively, we 
have in every part ?+m?’=2, unaltered from the original 
value in (51). 

The character of w, under the condition that all the 
elementary terms of which it is composed are subject to 
?-+m?=constant (£*), is the same as for the transverse dis- 
placement of an infinite stretched membrane, vibrating with 
one definite frequency. The limitation upon w is, in fact, 
merely that it satisfies 


(Pda? + P/dy?+h)w=0.. . . . (5d) 


The character of w in particular solutions of the mem- 
brane problem is naturally associated with the nodal system 
(w=0), where the membrane may be regarded as held fast : 
and we may suppose the nodal system to divide the plane 
into similar parts or cells, such as squares, equilateral 
triangles, or regular hexagons. But in the present problem 
it is perhaps more appropriate to consider divisions of the 
plane with respect to which w is symmetrical, so that dw/dn 
is zero on the straight lines forming the divisions of the 
cells. The more natural analogy is then with the two- 
dimensional vibration of air, where w represents velocity- 
potential and the divisions may be regarded as fixed walls. 

The simplest case is, of course, that in which the cells are 


5AQ Lord Rayleigh on Convection Currents in 


squares. If the sides of the squares be 27, we may take 
with axes parallel to the sides and origin at centre 


w=COS#+COsy, .° . | eae 


being thus composed by superposition of two parts for each 
ef which #?=1. This makes dw/dv=—sinz, vanishing 
when a=+7. Similarly, dw/dy vanishes when y= +7, so 
that the sides of the square behave as fixed walls. To find 
the places where w changes sign, we write it in the form 


a “ry (ey) 
w= 2 cos >. cos g > Re (56) 


giving «+y=+7, w—y=+7, lines which constitute the 
inscribed square (fig. 1). Within this square w has one sign 
(say +) and in the four right- 


angled triangles left over the Fig. 1. 
— sign. When the whole plane nye 
is considered, there is no want j 


of symmetry between the + and 
the — regions. 

The principle is the same 
when the elementary cells are — 
equilateral triangles or hexa- 
gons ; but I am not aware that 
an analytical solution has been 
obtained fur these cases. An 
experimental determination of 
k? might be made by observing 
the time of vibration under gravity of water contained 
in a trough with vertical sides and of corresponding 
section, which depends upon the same differential equation 
and boundary conditions*. The particular vibration in 
question is not the slowest possible, but that where there 
is a simultaneous rise at the centre and fall at the walls all 
round, with but one curve of zero elevation between. 

In the case of the hexagon, we may regard it as deviating 
comparatively little from the circular form and employ the 
approximate methods then applicable. By an argument 
analogous to that formerly developed t for the boundary 
condition w=0, we may convince ourselves that the value 
of k? for the hexagon cannot differ much from that appro- 
priate to a circle of the same area. Thus if a be the radius 

* See Phil. Mag. vol. i. p. 257 (1876); Scientific Papers, vol. i. 
pp. 268, 271. 

+ Theory of Sound, § 209; compare also § 317. See Appendix. 


a Horizontal Layer of Fluid. 543 


of this circle, is given by J, (ka)=0, J, being the Bessel’s 
function of zero order, or ka=3°832. If 6 be the side of 
the hexagon, a? =3/3. 6/27. 


APPENDIX. 


On the nearly symmetrical solution for a nearly circular 
area, when w satisfies (d?/dx? + d?/dy? +k*)w=0 and makes 
dw/dn=0 on the boundary. 

Starting with the true circle of radius a, we have w a 
function of r (the radius vector) only, and the solution is 
w=J,(kr) with the condition Jy (ka) =0, yielding ka=3°832, 
which determines k if a be given, or a if & be given. In 
the problem proposed the boundary is only approximately 
circular, so that we write r=a+p, where a is the mean 
value and 


p=%,cos@+6,sind+...+a,cosnO+,sinnd. . (57) 


In (57) @ is the vectorial angle and a, &ec. are quantities 
small relatively to a. The general solution of the differ- 
ential equation being 


w=ApJ (kr) +J,(kr){ A, cos 8+ B, sin 9} 
+...¢d,(kr){A, cosnO6+B, sin}, . . (58) 


we are to suppose now that A,, &., are small relatively 
to Ay. It remains to consider the boundary condition. 

If @ denote the small angle between r and the normal dn 
measured outwards, 


dw _ ees 


nae os d— 79st Di gee aie (Oo) 
and 
tam g= “= a "(—a,sinnO +8,cosnd) (60) 


with sufficient approximation, only the general term being 
written. In formulating the boundary condition dw/dn=0 
correct to the second order of small quantities, we require 
dw/dr to the second order, but dw/dé to the first order only. 
We have 


— =Ao{Jo! (ka) + kp Jo! (ka) + 4h? 0°S9/"(ha) } 


+{J,'(ka) +kpJn!'(ka)}{ An cos n8+ Basin n6}, 


dw 


Paha Ona) { — —A, sin nO+B,, cos nt 


544 Lord Rayleigh on Convection Currents in 
and for the boundary condition, setting ka=z and omitting 
the argument in the Bessel’s functions, 

Ao{Jo.cosptkp Jy +4 p7J 9/4 

a ve + kp Jn''}{An cos né + Bz sin nO } 

— — —— In {—A, sin n@ + B, cos nO} { —a, sin 6 


+8, cos nO} =02 2 aa 


If for the moment we omit the terms of the second order, 
we have 


Agdo’ thkAod, fa, cos nO + 8, sin nO} 
+J,{A,cosn6+B,sinnOt=0;. (62) 


so that Jo (2) =0, and 
kA o «tn +Jn-An=0,° “Ago”. Ga t+Jn -Be= One) 


To this order of approximation z, =ka, has the same value 
as when p=0; that is to say, the equivalent radius is equal 
to the mean radius, or (as we may also express it) k may be 
regarded as dependent upon the area only. Hquations (63) 
determine A,, B, in terms of the known quantities «,, Bn. 

Since J) is a small quantity, cos in (61) may now be 
omitted. To obtain a corrected evaluation of <¢, it suffices 
to take the mean of (61) for all values of 6. Thus 


Ao{2Jo’ + LhPS 9 (cen? -- a) } 
1 ARI, 1d ,faclla,A, 4 6,8) 


or on substitution of the approximate values of A,, B, 


from (63), 
ee , 9 my 


fey, 
wo 


This expression may, however, be much simplified. In 
virtue of the general equation for J, 


ee 


and since here’J,’=0 approximately, 


Jo =—J,, Me = — 28 Ny se de 
Thus 


Jo/(z) = 3h J ° > (6,” == Baw {5 a x} b) 7 (65) 


the sign of summation with respect to n being introduced. 


a Horizontal Layer of Fluid. 545 


Let us now suppose that a+da is the equivalent radius, 
so that Jo/(ka + kda) =0, that is the radius of the exact circle 
which corresponds to the value of & appropriate to the 
approximate circle. Then 


Jo (z) tkdaJo"(z)=0, 


and ( 
wigs Jo’ ae 2 2 Aneraseea 1 
da=— pry = hE (ay +Be)1 a7 + ae 66) 


Again, if a+da’ be the radius of the true circle which has 
the same area as the approximate circle 


iif 
al eat Be) wis) 4 COO) 


and 
baie BY G Mdute) () 
MPN (68) 


where z is the first root (after zero) of Jy (z)=0, viz. 3°832. 

The question with which we are mainly concerned is the 
sign of da’—da for the various values of n. When n=1, 
J,(z) = —J,/(z)=0, so that da=da’, a result which was to be 
expected, since the terms in a, 8; represent approximately 
a displacement merely of the circle, without alteration of size 
or shape. We will now examine the sign of J,/J,' when 
we 2,and 3. 

For this purpose we may employ the sequence equations 


da' —da= —> 


on 
Bed 
Oe Sa —In-1 Jn =4dn-1—4) nat, 


mach allow J, and J,’ to be expressed in terms of J, and Jo, 
of which the former is here zero. We find 


Jom=—do, Jpg =—4e dy, Ja =(1—24277) Jo; 
dy =) 03 ae cine: J3=(12277—1)Jp. 
Thus 
LE A ae 
Z 4 aly ht 2 bh? Y 
whence on introduction of the actual value of 2, viz. 3°832, 
we see that J>/J_’ is negative, and that J3/J3’ is positive. 
When n>z, it is a general proposition that J,(z) and 
Jn(z) are both positive*. Hence for n=4 and onwards, 
Jn/Jn is positive when <=3°832. We thus arrive at the 
* See, for example, Theory of Sound, § 210. 


Phil. Mag. 8. 6. Vol. 32. No. 192. Dec. 1916. 2P 


546 Mr. C. W. Raffety on some Investigations of 


curious conclusion that when n=2, da’>da, as happens for 
all values of n (exceeding unity) when the boundary con- 
dition is w=0, but that when n>2, da’<da. The existence 
of the exceptional case n=2 precludes a completely general 
statement of the effect of a departure from the truly 
circular form ; but if the terms for which n=2 are absent, 
as they would be in the case of any regular polygon with 
an even number of sides, regarded as a deformed circle, we 
may say that da’<da. In the physical problems the effect 
of a departure from the circular form is then to depress the 
pitch when the area is maintained constant (da’=0). But 
for an elliptic deformation the reverse is the case. 

At first sight it may appear strange that an elliptic 
deformation should be capable of raising the pitch. But 
we must remember that we are here dealing with a vibration 
such that the phase at both ends of the minor axis is the 
opposite of that at the centre. A parallel case which admits 
of complete calculation is that of the rectangle regarded as a 
deformed square, and vibrating in the gravest symmetrical 
mode *. It is easily shown that a departure from the square 
form raises the pitch. Of course, the one-dimensional vibra- 
tion parallel to the longer side has its pitch depressed. 


LX. On some Investigations of the Spectra of Carbon and 
Hydrocarbon. By Cuarues W. Rarrety, /.R.A.S.F 


[Plate XIII] 


HE spectrum of the Bunsen flame (the “Swan” 

spectrum) has been made the object of a vast amount 

of experimental work directed towards the determination 

of its chemical origin, a résumé of which is given by 

Dr. Marshall Watts in a paper in the ‘ Philosophical 
Magazine’ of July 1914}. 

The present communication gives the results of a photo- 
graphic study of the “Swan” spectrum obtained from 
different sources, and is intended to direct attention to 
certain features which appear to be worthy of further 
research—namely, the physical significance of the changes 
produced in the spectrum by a change in the conditions 


* Theory of Sound, § 267 (p=q=2). 

+ Communicated by the Author. 

t “On the Spectra given by Carbon and some of its Compounds; 
and, in particular, the ‘Swan’ Spectrum,” W. Marshall Watts, Phil. 
Mag. July 1914. 


the Spectra of Carbon and Hydrocarbon. DAT 


of the source, and the origin of certain new lines and bands 
which have been recorded photographically. 

The ‘“‘Swan” spectrum, characteristic of the flame of 
hydrocarbons, consists of the well-known groups of flutings 
in the yellow, green, and blue attributed to the element 
carbon, as well as some bands of more complicated structure 
in the violet which are ascribed to hydrocarbon. This 
spectrum is given brightly and with purity by the base of 
the flame of a Mecker gas-burner (coal-gas burning in air). 

Other flutings of similar structure and obviously be- 
longing to the same system of bands can be distinguished 
in the red-orange region of the flame spectrum ; but these 
are very faint and difficult to photograph. In the spectrum 
of the electric arc between carbon poles in air, the yellow, 
green, and blue “Swan” bands are reproduced brightly 
with the addition of the intense fluted bands in the violet 
and ultra-violet which are ascribed to cyanogen since 
nitrogen must be present for their production. In the 
absence of hydrogen, however, the hydrocarbon bands do 
not appear. 

In the red and orange regions of the carbon are spectrum 
we find very complex series of lines in which cyanogen is 
represented by a number of flutings”*, but the bands of 
carbon, faintly visible in the flame spectrum, are nevertheless 
obscured and inconspicuous, their heads being traceable only 
with some difficulty. 

In the “Swan” spectrum, carbon is represent:d by the 


following groups of flutings + :— 


Red-orange Yellow Green Blue Violet 
Group. Group. Group. Group. Group. 
Rane OL 9) I. 9635°4 9165°3 4737-2 4381-9 
6121°3 5985'D 9129°4 4715°3 43713 
6059°9 5940°9 9101°0 4697-6 4365:0 
6005°1 5901°9 5081-9 4684-9 (“the three”) 
5958-2 oL72°7 


An attempt was made to apply the electrical conditions of 
the arc discharge to the products of the flame in which 
carbon was in the process of changing its chemical com- 
bination. 


* “The Less Refrangible Spectrum of Cyanogen, and its Occurrence 
in the Carbon Arc,” Fowler and Shaw, Proc. Roy. Soc. A, vol. ]xxxvi. 
(1912). 

+ See also ‘Index of Spectra,’ W. Marshall Watts, Appendix V., 


pp. 67, 68. 
2B 2 


548 Mr. C. W. Raffety on some Investigations of 


The arc between soft-cored carbon rods 8 mm. in dia- 
meter, connected to an alternating current supply of about 
200 volts, was struck within a flame of coal-gas in air, 
the current being regulated by an inductance and a variable 
resistance in circuit. It was thus hoped to intensify the 
carbon bands and also to discover whether any modifi- 
cations occurred in the hydrocarbon bands which are 
produced under these conditions. 

An image of a horizontal arc was projected with a lens 
on the slit of the spectrograph in the usual way, the are 
having been struck in a flame of coal-gas which completely 
enveloped it. After careful adjustment it was found that 
the normal blue region at the centre of the are became 
replaced by one of a pronounced green colour, and at 
the same time the arc emitted a characteristic sound by 
which the change could always be recognized. 

A visual examination of the spectrum revealed the carbon 
flutings with especial brilliancy, the series of lines forming 
them appearing much extended so as to fill the spaces sepa- 
rating the groups. At the same time the red-orange bands 
of the normal flame spectrum emerged from their obscurity, 
being exalted in brightness until their “ heads” became 
quite conspicuous. The red line of hydrogen (Ha, » 6563-0) 
also appeared. 

In the more refrangible part of the spectrum the cyanogen 
bands still possessed great brilliancy, there being, apya- 
rently, sufficient access for atmospheric nitrogen in the 
flame ; but together with them, the hydrocarbon bands were 
well developed. 

Fig. 1, Pl. XIII., is an enlargement from a spectrogram 
of the normal carbon are and the arc-in-flame, taken with a 
grating spectrograph, showing the yellow and green bands 
of carbon. 

The difference in intensity between the two spectra, how- 
ever, is actually much greater than is here indicated as, in 
order to produce clear photographs and avoid extreme con- 
trast, longer exposure was given to the one of the normal are. 

Fig. 2, Pl. XIII., shows the corresponding intensification 
of the red-orange bands as compared with the normal are. 

From experiment it appeared that the flame most favour- 
able for the production of the carbon bands was the ordinary 
luminous jet rather than the air-fed Bunsen or Mecker 
flames ; and the arc was moved about from one part of 


the flame to another, but was finally located about the 
centre of the luminous region. | 


— the Spectra of Carbon and Hydrocarbon. 549 


. Probably the chief interest of the arc-in-flame spectrum 
lies in the region about 24382 to 4364, and in the hydro- 
earbon bands. In the former region are the three lines 
which appear to form part of the ‘‘ Swan ”’ spectrum, and to 
be attributable to carbon. These three lines occur in the 
flame spectrum, where they are superposed on the first 
portion of the hydrocarbon “ /”’ group, but are frequently 
so faint as to be hardly visible amorgst the lines forming 
the band. Their brightness, however, changes considerably 
with the conditions of combustion. In a Mecker flame with 
a rapid flow of gas, the region immediately above the grid 
is blue and the flame roars, but when the flow is sufficiently 
reduced it becomes almost silent and the colour changes to 
yellowish green. 

The spectrum then shows an increase in brilliancy of the 
“Swan” bands, and simultaneously ‘“‘the three” appear 
clearly, standing out on the background of the hydrocarbon 
band. 

In the normal carbon are in air these lines are not con- 
spicuous, but are recorded with suitable conditions and 
exposure. 

In the spectrum of the arc-in-flame we find “ the three ” 
present with very considerable intensity, and their sym- 
pathetic variation with the carbon bands lends support to 
the conclusion that they have the same origin. 

The sources from which the carbon and hydrocarbon 
spectra were obtained were as follows :-— 


1. Flame of coal-gas in air from a Mecker burner. 

2. Carbon arc in air. 

3. Carbon arc in flame of coal-gas in air. 

4. High-tension transformer spark in air between 
carbon electrodes (capacity and _ self-induction 
in circuit). 

5. Transformer spark between carbon electrodes in an 
atmosphere of coal-gas (capacity and self-induction 
in circuit). | 

6. The same as No. 5, but in a flame of coal-gas in 
alr. ) 

7. Transformer spark in absolute alcohol, and also 
in glycerine (no capacity or self-inductance in 
eircuit)): 


Visual observations of the spectra were made with various 
spectroscopes, and the enlargements which accompany this 


950 Mr. C. W. Raffety on some Investigations of 


paper are reproduced from negatives obtained with the 
following instruments :— 


1. Grating spectrograph of 22-inches focus. 

2. Grating spectrograph of 6-feet focus. 

3. Prismatic spectrograph with compound prism and 
camera of about 9-inches focus. 


Fig. 3, Pl. XIII., shows the hydrocarbon bands as they 
appear.in the spectrum of the Mecker gas-flame, the com- 
parison spectrum being that of iron in the carbon are. The 
line-series are seen to be complex, and the whole system of 
bands is both beautiful and intricate. 

The region of “the three,” and the less refrangible part 
of the hydrocarbon spectrum as given by the are-in-fame, 
is shown in fig. 4, Pl. XIII. The former are strong and 
widened, and a close examination indicates a multiple 
structure. 

In the case of the hydrocarbon bands the most striking 
features, apart from the question of variation of relative 
intensities, appear to be the new companion lines to the 
isolated line of the flame spectrom about 24324, which 
appears single with small dispersion. 

The intricacy of structure of the bands, however, espe- 
cially in the “head” (X 4314), requires instruments of 
considerable power, and only a partial resolution is shown 
in these spectrograms. 

The ionization within the flame is so great, that to 
maintain the disruptive discharge of the spark a source 
not only of sufficient potential, but also of considerable 
output is required to work with a condenser of the requisite 
capacity, and a high-tension transformer was employed for 
this purpose. 

In addition to an inductance in the primary circuit of the 
transformer, another high-tension inductance was inserted 
in the secondary cireuit to control the current and prevent 
arcing at the electrodes within the flame. 

In the oscillatory discharge circuit formed by the con- 
denser and spark-gap, a variable inductance was introduced 
so as to control the oscillation period of the spark. With 
the spark between carbon electrodes in air, with capacity 
but without self-induction, the line spectra of nitrogen and 
carbon are seen. In the case of the same spark within 
a flame of coal-gas there appear, in addition, the “ Swan ” 
bands of carbon, the flutings of cyanogen, and, less strongly, 
the hydrocarbon bands. 

The two spectra are thus superposed ; but their relative 


the Spectra of Carbon and Hydrocarbon. 5D E 


intensities can be changed considerably by an alteration of 
the length of the spark, and by other variations of the con- 
ditions of discharge. 

When sufficient self-induction is introduced into the 
oscillatory circuit, the line spectra of nitrogen and carbon 
disappear, leaving a bright ‘“‘Swan ” spectrum with “the 
three,” the hydrocarbon bands, and those of cyanogen. 

The spectrum then becomes practically identical with that 
of the arc-in-flame. 

A spectrum very closely similar is given also by the 
spark with self-induction in an atmosphere of coal-gas, 
the survival of the cyanogen bands being attributable, 
presumably, to an admixture of some nitrogen with the 
gas. 

The spectrum was obtained from the transformer spark 
between carbon electrodes in a glass cylinder through which 
a gentle stream of coal-gas was maintained, the gas being 
brought into the sparking cylinder through the electrodes 
themselves, which were drilled so as to form tubes. 

Here again, with the introduction of self-induction into 
the discharge circuit, the line spectrum of carbon gives 
place to the ““Swan” bands. The spectrum from about 
4550 to A 3800 is reproduced in fig. 5, Pl. XIII. 

The modification of these spectra from that given by the 
Mecker flaine suggested the investigation of the spark under 
alcohol, in order to ascertain whether or not the same new 
lines were present in this case. The transformer discharge 
was accordingly taken between electrodes of platinum under 
absolute alcohol. With condenser in circuit the discharge 
was intermittent and violently disruptive, giving a bright 
continuous spectrum; but when the capacity was discon- 
nected, the spark became much easier to maintain and 
appeared of a pronounced greenish colour. 

An examination of the spectrum revealed the “‘ Swan” 
bands brightly. Owing to the rapid liberation of solid 
earbon in the liquid, photographic exposures were attended 
with considerable difficulty, as the aleohol very soon became 
turbid and semi-opaque. By using a fairly large quantity, 
however, and by bringing the electrodes close up against the 
side of the glass containing vessel in order to reduce the 
absorption by the liquid, it became possible to give the long 
exposures necessary. 

The spark needed constant adjustment which added very 
considerably to the difficulty of obtaining good spectrograms. 
The image on the slit of the spectrograph was apt to wander, 
and required to be brought back on the same spot repeatedly, 


502 Mr. C. W. Raffety on some Investigations of 


Fig. 6, Pl. XIIL., is reproduced from a spectrogram enlarged 
with a cylindrical lens. 

Examination reveals a close agreement with the spectra 
of the arc and the spark in the coal-gas flame, and with the 
spark in an atmosphere of coal-gas. 

In addition, certain conspicuous new bands occur, some of 
which appear to be identical with members of Deslandres’ 
negative groups *. 

‘Their chemical origin seems as yet to be unknown. The 
approximate wave-lengths of the heads of the bands shown 
in the spectrogram are as follows :— 


AX 4102°5 
4068-0 
38540 
3827-5 (?) 
The bands of cyanogen are completely absent. 


[In connexion with the spectrum of the arc-in-flame, there is a 
very interesting point to be noticed with regard to the behaviour 
of lines due to impurities in the carbon rods. Various elements 
oceur as impurities: calcium, iron, sodium, aluminium, silicon, 
and magnesium are common, and often their spectra are strongly 
represented. The action of the flame surrounding the are is to 
quench out these lines more or less completely, and a good 
illustration of this is given in fig. 7, Pl. XIII. 

The two spectra here shown were obtained from the same 
carbon rods—the upper one being that of the normal open are, 
and the lower one that of the are in a coal-gas flame. In the 
first, the lines of caleium—Ad 4226-9, 3968-6, 3933°8—are very 
strong; lines of silicon, aluminium, &c. also appear. In the 
spectrum of the arc-in-flame these lines are either absent or very 
faint, the most striking case being the practically complete 
disappearance of the strongest calcium lines, H and K.] 


It was realized that to study in detail the modifications of 
the hydroearbon bands, and the other features of these 
spectra, greater instrumental power was desirable; for 
although the optical performance of the smaller grating 
spectrograph was excellent, an instrument of greater focal 
length would have given better resolution on the photo- 
graphic film. 

The spectra were accordingly photographed with a grating 
spectrograph of 6-feet focus. The mounting of this instru- 
ment leaves much to be desired when long exposures are 
concerned, as its situation is subject to very considerable 


* See ‘Index of Spectra,’ W. Marshall Watts, Appendix V., p. 77; 
Deslandres and D’Azambuja, C. &. 1905, exl. p. 917. 


the Speetra of Carbon and Hydrocarbon. 553 


vibration with corresponding detriment to the best definition, 
and it is also liabie to temperature changes to some extent. 
Under these disadvantageous conditions the recording of 
the Mecker flame spectram, necessitating exposures of many 
hours’ duration, was attended with considerable difficulty ; 
and even greater difficulties were experienced in the case of 
the spark under alcohol. 

The results obtained, however, were fairly satisfactory, espe- 
cially in the case of the arc-in-flame, where the intensity of the 
light rendered short exposures possible, and the detrimental 
etfects of intermittent vibration and temperature changes 
could be avoided. 

The spectrum of iron in the carbon arc was employed as a 
reference spectrum for wave-length determinations. 

The Mecker burner was placed in a horizontal position as 
before, the luminous region immediately above the grid being 
viewed from one side. The image thus formed on the slit by 
the condensing lens was a narrow line of light, and fairly 
uniform illumination was obtained over the entire width of 
the spectrum. 

Fig. 8, Pl. XIII., is reproduced from a spectrogram taken 
with the larger instrument, and shows the main portion of 
the hydrocarbon spectrum as given by the Mecker coal-gus 
flame, with Fe: comparison. The very fine and complicated 
structure of the head of the main band (about A 4314) is 
here revealed, although the densest region is still beyond the 
resolving-power of the instrument. 

It is to be noted that in this case “the three” (AA 4382, 
4371, 4365) are inconspicuous. 

he same region of the spectrum of. the arc-in-flame is 
shown in fig. 9, Pl. XIII. Seen on the scale given by 
the increased instrumental power, the wealth of detail is 
remarkable and the differences from the flame spectrum 
become the more apparent. 

The greatly increased intensity of “the three” is seen to 
be due in part to the appearance of new components ; while 
the flame-line of 14324 has now become a group with 
a new isolated line on the more refrangible side. Photo- 
graphs of this regiun were obtained also in the 2nd order 
spectrum of the grating ; one enlargement is reproduced in 
meg LO. tle XT. 

Examination reveals that, in addition to the appearance of 
new lines in the are-in-flame spectrum, there are differences 
in relative intensities as compared with the flame spectrum. 

The flame shows two strong lines—)A 4292°2, 4291°3 ; 
but in the spectrum of the arc-in-flame, the less refrangible 


dod Mr. (. W. Raffety on some Investigations of 


line is much fainter, while a third line—r 4293°26—is 
strong. This line appears to coincide with a faint line 
in the flame spectrum, but its intensification in the case 
of the arc-in-flame is as remarkable as is the reduction of 
intensity cf the line 4292-2. 

The recent announcement by Prof. Newall of the identifi- 
cation of the hydrocarbon lines with those of the “G” group 
in the solar spectrum has given a greatly increased interest 
to the hydrocarbon spectrum ; and a further study of the 
variations mentioned in this paper may yield important 
information for the interpretation of the solar absorption 
lines. If it should be possible to trace variations in the | 
lines of the “* G” group depending on the particular region 
of the sun from which the light is received, a comparison 
with different laboratery spectra such as those herein de- 
scribed might lead to definite conclusions as to temperature 
and the probable location of the absorption. 

In the flame spectrum there are no lines between the 
head of the band (about 24315) and the isolated line 
X 4324; but the arc-in-flame shows many lines in this 
region. 

The author is indebted to Dr. W. Marshall Watts for his 
kindness in making some preliminary measurements of the 
spectrograms. The wave-lengths of the components of 
‘“‘the three,” as they occur in the normal arc, the arc-in- 
flame, and the flame, and of the group of lines about A 4324 
typical of the arc-in-flame, are as follows :— 


Normal Are. Are-in-flame. Flame. 
Be 4d01 6D 2h 2 ee 4381°70 
SOLO on hoes 4380°57 


35 HSB as 4372°33 
ASi0j4) a 10 Te one 70°77 

| 69-09 
3; 4364-84 4364-84 4364-03 

4324-03 

23°30 

23-19 

23-08 

22-20 

4321-6 


(Wave-lengths on the International scale.) 


the Spectra of Carbon and Hydrocarbon. 555 


The spectrograms of the spark in alcohol obtained with 
the larger spectrograph were insufficiently exposed for 
successful reproduction; but the negatives show clearly 
a close agreement of the main features with those of the 
arc-in-flame, although the fainter lines are not recorded. 

The normal carbon arc shows only faint lines in the region 
between about X4380 and the head of the cyanogen band 
X 4216, and a comparatively long exposure is required to 
photograph them. 

For this reason lines due to impurities in the carbon rods 
are very strongly recorded on the plate, even though their 
actual intensities are not great. The region in question is 


shown in fig. 11, Pl. XIII. 


An Investigation of the Fainter Regions of the 
Hydrocarbon Band. 


In some spectrograms of the hydrocarbon hands obtained 
with the smaller grating instrument, faint traces of a new 
series of lines were detected in the region between » 4107 
and 24025. The discovery of radiations in this apparently 
dark interval at the centre, between the main head and tail 
serles, was considered to be of sufficient interest to justify an 
attempt to explore this region as thoroughly as possible in 
the hope of connecting the two main parts of the hydro- 
carbon spectrum. 

The first negatives with the grating spectrograph showed 
that the main difficulty was the presence of a faint con- 
tinuous spectrum extending throughout the band, which 
tended to mask the lines. The latter were so very little brighter 
than the continuous background on which they appeared 
that increasing the length of the exposure beyond a certain 
point did not serve to increase their visibility, but merely made 
the whole image darker. Fortunately, however, the intensity 
of the continuous spectrum seemed to be variable with the 
precise state of the gas-flame, becoming fainter as the supply 
of gas was reduced and the flame became more silent. 

In order to obtain a spectrum of greater luminosity, and 
decrease the length of the exposures, it was decided to 
employ a prismatic spectrograph. The instrument used 
was provided with a compound prism of high dispersion 
and a camera of about 9-inches focus. The effective working 
aperture was fairly large, and the angular dispersion of the 
prism for the region in question was very considerable. 

Several successful spectrograms were obtained, which 
showed the new series more distinctly. 

The flame of the Mecker burner was adjusted and carefully 


956 Mr. C. W. Raffety on some Investigations of 


focussed upon the slit of the spectrograph, and a visual 
examination made with an eyepiece. ‘The flow of gas was 
varied and the precise region of the flame image selected, 
until the hydrocarbon band appeared at its brightest. Only 
the main head at about 14314 was at all distinct ; but the 
visibility of the fainter doublets on the more refrangible side 
was taken as an approximate guide to the best conditions. 

A reproduction of one of the spectrograms is given in 
fiz. 12, Pl. XIII. The main series of lines forming the 
‘tail’? is well shown ; for not only are the lines com- 
paratively bright, but they are also free from the masking 
effect of the continuous spectrum, since the latter does not 
extend with appreciable intensity quite so far. 

The faint new series can be discerned converging to a 
point situated between the first two (least refrangible) 
members of the main “ tail” series. 

In the case of lines so faint showing against a veil of con- 
tinuous light, measurements of wave-lengths are attended 
with considerable difficulty and some little uncertainty. 
The following approximate values were obtained. ‘The first 
two columns are from photographs obtained with the 
grating instrument, on which the lines appear very faintly, 
and the third is from a spectrogram with the prism. 


Spectrogram 1. Spectrogram 2. Spectrogram 8. 
eae Grating — Prism 
ae (determined graphically). (determined graphically). 
4106-7 i 4108 
4095°0 A094°8 4095°3 
AO84°2 4084°6 A084°8 
4075°1 4074:0 A0N7T4:0 
40659 4065-0 4066-0 
4059°3 4059-0 A0D60°7 
40526 4053-0 4053-0 
4047-2 ae 4047-8 
4043-4 4044-0 A043°2 
A0 40-1 4039°5 4040°0 
hi 4037°6 : 4037-0 
ti) 1038" : 40313 
40252 — : | 


The horizontal dotted lines in column 1 indicate that there 
occurred lines in these intervals too faint to be measured. 


the Spectra of Carbon and Hydrocarbon. 557 


In the attempts to photograph these lines, the fainter 
members of the other series starting from the head of the 
main band were being simultaneously recorded on the 
plates. The enlargement reproduced reveals the pairs of 
lines proceeding with gradually increasing separation up 
to what appear to be two nebulous pairs or narrow bands. 
Between these and the new faint series only a short space 
intervenes, so that the seemingly void region has been 
greatly reduced. 

In structure the faint series closely resembles the main 
series of the tail, but its scale is smaller. 

As this paper is not intended to be a detailed communi- 
cation, no discussion of these line series is attempted and 
their investigation awaits more exact data. 


Some Bands of Unknown Origin associated with the 
“ Swan” Spectrum. 


In the course of some work on the Mecker flame spectrum, 
some small-scale spectrograms were taken with a grating 
camera. 

On close examination of these, it was noticed that certain 
faint bands were visible in the region between the green 
and the blue carbon flutings. These bands were too faintly 
recorded to enable anything more than rough wave-lengths 
to be obtained, nor was any structure visible in them; but 
during the investigation of the lines of the hydrocarbon 
bands with the prismatic spectrograpb, the new bands 
appeared on the negatives with much greater clearness. It 
became important, then, to measure them more accurately 
and endeavour to ascertain something of their structure. 

At best the bands are very feeble, being only just visible 
on the faint continuous spectrum which extends between the 
“Swan” bands. Ordinary plates were used, as the range 
of wave-length under investigation fell within their limits 
of sensitivity ; ; but long exposures were necessary owing to 
the faintness of thelight. As in the case of the hydrocarbon 
bands, a limit to successful photography was imposed by the 
fogging produced by the feeble continuous radiation; but 
in spite of this difficulty, some useful spectrograms were 
obtained. 

For the purpose of wave-length determinations the spark 
spectrum of iron was employed, the discharge being passed 
between iron electrodes with capacity and _ self-induction 
in circuit. It fortunately happens that iron gives excellent 
reference lines in the region under investigation, and a 


558 On the Spectra of Carbon and Hydrocarbon. 


narrower Fe: spectrum was superposed on that of the Mecker 
flame. 

Fig. 13, Pl. XIII., is an enlargement from a negative 
obtained with the prismatic spectrograph on which the 
unknown bands are recorded. 

The least refrangible edge falls about midway between 
the two strongest iron lines and appears to consist of a 
double line. The second and stronger head is nearly coin- 
cident with the iron line X4891. A third head is also 
somewhat vaguely indicated, but the most remarkable 
feature is the ribbed or fluted structure of the band, the 
close and regular sequence of which can be traced nearly 
up to the carbon fluting (A 4737-2) in the blue. 

The faintness of the detail, however, makes the components 
somewhat difficult to recognize. 

In addition to these bands, a well-marked single line 
occurs close to the blue carbon band. Owing to the long 
exposures necessary the “Swan” spectrum is greatly over- 
exposed; and this accounts for the appearance of the bright 
green carbon band (AX 5165), for an ordinary plate has little 
sensitivity to this region. 

The approximate wave-length measurements are as 
follows :— 


Fe: Comparison Spectrum. Unknown Bands. 
r. A. 

4957-6" (1) 4942°0 
4939-0 4937°5 
4924°1 
4920'7 (2) 4390°0 
4919-1 
ABO (3) 4853 (?) 
A878:°4 
4871-9!” (4) line 4743 
4860°0 


In order to show the line X 4743 well, the exposure must 
be suitable. It is seen more clearly in fig. 14, Pl. XIIL., 
where it is recorded at the expense of the bands. 

These features appear to be of constant occurrence in 
association with the “Swan” spectrum as given by the 
Mecker coal-gas flame. There does not seem to be any 
reason for attributing them to the presence of some chance 
impurity, and the inference is that they are probably due to 
carbon or one of its compounds. The fainter details are 
liable to variations and changes in relative intensities, but 


On the Collapse of Short Thin Tubes. 559 


until they are more fully investigated, no definite conclusions 
as to their origin and significance are possible. 

Similarly, the variations in the hydrocarbon spectrum 
with the changes in the conditions of its source and the 
appearance of new lines with “the three” await further 
research. 

The discovery of such variations and of new bands asso- 
ciated with the spectra of carbon and its compounds may 
be of importance in connexion with astrophysics, and 
particularly in the study of comets. 


September 1, 1916. 


LXI. The Collapse of Short Thin Tubes. 
By AvBeRT P. CaRMAn™®. 


[Plate XIV.] 
ee problem of the collapse of tubes is to find an 


equation to express the relation between the collapsing 
pressure and the dimensions and elastic properties of the tube. 
In this general form no solution has been found, and it is 
probable that no simple solution is possible. We have 
reached, however, from theory and from experiments, fairly 
simple equations (1) for long thin tubes, and (2) for long 
tubes of moderate thickness. The term “long” is used for 
a tube the length of which is beyond a certain ‘critical 
length,” which is generally taken as about six diameters. 
The thickness of the tube is expressed in terms of the ratio 
of the wall thickness ¢ to the mean diameter d. If t/d is 
not more than about °025, the tube is ordinarily called 
“thin.” By “moderately thick’? we mean tubes for which 
t/d has a value between ‘03 and -07. The equation for long 
_tbin tubes was deduced first by G. H. Bryanf, from 
theoretical considerations, and has the form p=c(é/d)*, 
2K 

ear (E is 
Young’s modulus, and s is Poisson’s ratio for the material 
of the tube). The form of this formula has been confirmed 
by the experiments of Carman and Carr { on seamless steel, 
lap-welded steel, and brass tubes, and by Stewart § for lap- 
welded tubes. The values of the constants found by these 


where c is a constant and equal theoretically to 


* Communicated by the Author. 

+ Proc. Camb. Phil. Soc. (1888). 

t Univ. of Illinois Eng. Experiment Station Bulletin (1906). 
§ Trans. Am, Soc. Mech, Eng. (1906). 


560 Mr. A. P. Carman on the 


experiments are, however, about 25 per cent. less than those 
indicated by theory, probably in consequence of the material 
being beyond the elastic limits. The formula for moderately 
thick tubes has the form p= a, — b, where a and 6 are 
constants. This formula is purely empirical both in its form 
and its constants. 

Carman concluded from his early experiments on small 
brass tubes (Phys. Rev. vol. xxi., 1905) “that there is a 
minimum length for each tube, beyond which the collapsing 
pressure is constant, and further, that this minimum length 
is quite definite. Again, for lengths less than this critical 
minimum length, the collapsing pressures rise rapidly. As 
definitely as can be determined from these small tubes, the 
collapsing pressure varies inversely as the length for lengths 
less than the critical length.” ‘This last law is stated by 
Gilbert Cook in what he has called ‘‘Carman’s equation ” 


(Pil. Mac. July 1914) p2703). ip = ip, where p is the 


collapsing pressure of an infinitely long tube, L is the 
critical length, / the length of the given tube, and p’ the 
corresponding collapsing pressure. A curve drawn with 
lengths as abscisse and collapsing pressures as ordinates, 
would thus consist, as R. V. Southwell has noted (Phil. Mag. 
Jan. 1915) of two discontinuous branches, a straight line 
parallel to the axis of abscissee and a rectangular hyperbola 
intersecting the straight line at the point corresponding to 
the critical length. For both theoretical and practical 
reasons, the form of this pressure-length curve at and within 
the critical length has recently aroused much interest and 
discussion. ‘he practical interest came first from the 
problem of spacing “ collapse rings ” in boiler-flues. Another 
practical problem comes trom the collapse of steel flumes by 
atmospheric pressure, when accidents have suddenly let out 
the water and reduced the pressure almost to the zero on 
the inside. The theoretical interest comes from a formula 
deduced by R. V. Southwell in a very important paper on 
“‘ Hlastic Stability,” read in 1912 before the Royal Society 
of London. In this paper Professor Southwell has deduced 
the formula 
t z ep eid me i 

p- INI —1) Bt aml ae 


where p is the collapsing pressure, E the Young’s modulus, 


Lee : t 
is Poisson’s ratio, z is a constant depending upon the 


Collapse of Short Thin Tubes. 561 


end constraints, /,d, and ¢ are the length, diameter, and wall 
thickness, and & represents the number of lobes into which 
the tube collapses. As is known, long tubes collapse into 
two lobes, shorter tubes into three lobes, and still shorter 
tubes into four lobes (see Phys. Rev. vol. xxi. p. 396, figs. 3, 4, 
1905). This formula is represented by a family of curves, 
corresponding to the values of 2, 3, 4,etc., for k. Southwell 
points out that the envelope of this family of curves is very 
nearly a rectangular hyperbola. For longer tubes for which 

4 
k=2, and - is very small, Southwell’s formula reduces 
directly to Bryan’s formula. Southwell at first used the 
term ‘critical lengths” to designate the lengths at which 
the tube may collapse into either 2 or 3 lobes, or into either 
3 or + lobes, that is, the points of intersection of the branches 
of this curve. He has, however, also deduced theoretically 
an expression for the critical length in the sense in which 
Love and Carman used the term*. In this expression 
L=k Va?/t, k is a constant. Mr. Cook, in the Phil. Mag. 
for July, 1914, gets a value of 1°74 for k from the discussion 
of a series of careful experiments made by him on short 
steel tubes of three inches mean diameter and different 
thicknesses. Cook concluded that the critical length is 
apparently “from about 13 to 18 times the diameter” 
instead of about 6 diameters as originally suggested by 
Carman. Unfortunately, Cook’s apparatus limited him to 
tubes less than 13 inches in length; that is, to lengths of 
about 4 diameters, and so his curves do not reach the 
important bends or “critical” points. Mr. Cook himself 
says: “the tests cannot be regarded as sufficient in number 
or covering enough range of dimensions to confirm definitely 
the equation L=k /d*/t.” 

The following experiments have been made to obtain more 
data on the collapsing pressures of tubes, particularly near 
and inside of the ‘‘ critical” bend in the curve for the 
relation between pressure and length. The work to be 
described is part of work which is being done under the 
auspices of the Hngineering Experiment Station of the 
University of Illinois, and with the help of Mr. 8. Tanabe, 
Research Fellow. The apparatus was mostly that which 
was used by the author a number of years ago, and the 
methods of the experiments used were similart. The tube 

* Southwell, Phil. Mag. vol. xxix. p. 69. 
nm Bulletin No. 5, Engineering Experiment Station, University of 

1nois. 


Pint. Mag. S. 6. Vol. 32. No. 192. Dec. 1916. 2Q 


562 Mr. A. P. Carman on the 


to be collapsed was closed by end-plugs, and these were 
supported by an internal red so as to eliminate end pressures 
on the tube when the tube was placed under external water 
pressure. Leakage to the inside of the tube was prevented 
by wrapping the joints with “friction” insulation tape, and 
covering the whole joint with asphaltum. Hnd constraints 
were thus avoided. “The eee described here were 
on seamless-drawn steel tubes of 1, 2, and 3 inches diameter. 
‘The tubes were carefully miuchined to different thicknesses, 
so results have been obtained for several values of t/d tor 
each kind of tube. The nickel-steel receptacle was 40 inches 
long, so that results well beyond the “critical length” could 
be obtained in most cases. Asa matter of interest and of 
possible importance, experiments were made on the collapse 
of a series of small tubes of brass, aluminium, hard-rubber, 
and glass. These results are not given and discussed in 
this paper, but it may be stated that the curves obtained 
have similar forms to those for steel. The collapse of the 
glass tubing was striking, as the tube was reduced to a 
fairly fine powder upon collapse. The strain was apparently 
uniform throughout, and all parts seemed to give way at the 
same time. 

The results of the experiments with steel tubes are given 
in Tables I, II., and III. These results are also shown by 
the curves of figs. 1, 2,and 3 (Pl. XIV.). The number of 
lobes of the collapsed tube is given in the fourth column of the 
Tables, and is also indicated by a small numeral at the point on 
the curve. Thisis made a matter of record on account of its 
importance in Southwell’s theory, but it is not discussed in 
the present paper. The data given here are the results of the 
collapse of over 125 tubes. There were in practically each 
case two tubes of the given size, and thus an average could 
be made for each point. This also gave an immediate check 
on freak ecllapses. Very few freak collapses, however, occur- 
red, and these were easily explained by irregularities that 
appeared upon the inspection of the collapsed tube. While the 
machine work on these tubes was done with great care by the 
mechanician of the department, it was impossible always to 
get the value of ¢/d with the desired exactness. It was thus 
necessary to make a correction in the observed collapsing 
pressure, so as to have sets of results for a curve between 
pressures and lengths with t/d constant. These corrections 
were made by interpolation, assuming that the collapsing 


3 ! 
pressure varies as (5) . Since the total variations of (5) 


in these corrections were small, there was little assumption 


AP 


Collapse of Short Thin Tubes. 563 
TABLE I. 
*One-inch’”? Steel Tubes. Inside diameter ‘942 inch. 


Length td Pressure of Collapse. Number of 
in inches. /d. lb. per sq. in. Lobes. 
1-72 "0245 2490 4 
Me 0190 1600 3 or 4 
1-78 0150 1030 3 or 4 
1°78 01 420 4 
2°63 0245 2060 3 
2°63 0190 1180 3 
2°63 0150 670 3 
2°63 ‘010 200 3 
3°69 0245 1390 2 or 3 
3°69 ‘0190 800 3 
3°69 ‘0150 550 3 
3°68 ‘010 120 3 
o72 0225 1170 3 
5°63 "0245 920 2 
5°63 0225 730 2 
£63 ‘0190 520 2 
5°63 0150 320 2 
5°63 010 115 2 
5:9 0227 675 3 
75 0245 900 2 
7:59 0225 710 2 
7:56 ‘010 50 2 
744 015 210 2 
7-62 ‘015 190 2 
762 ‘019 420 2 
9°6 0245 900 2 
9°56 "0225 625 2 
9°6 ‘019 440 2 
9°56 015 150 2 
116 0245 750 2 
11°63 "0225 600 2 
mw G3 019 375 2 
116 015 145 2 


(if any) in the above interpolation and corrections. We 
note : 

(1) The curves for different thicknesses and diameters are 
all similar in shape. Thus the series of curves for the 1-inch 
tubes and that for the 2-inch tubes show the same character- 
istics. The very thin tubes show the greater variations, but 
this is to be expected owing to the difficulty in getting 
uniformity in thickness and material for thin tubes. 

(2) By taking the point for the length of six diameters, 
and drawing an hyperbola through this point, using the 


equation p’= Pas and taking p as the collapsing pressure at 


the length of six diameters, there is, in most cases, a satisfactory 


2Q 2 


are Mr. A. P. Carman on the 
TABLE Of, 


‘Two-ineh ’ Steel Tubes. Inside diameter 1:873 inches. 


Leneth t/d Pressure of Collapse. Number of 
in inches. { lb. per sq. in. Lobes. 
3°91 0245 2870 3 
3°88 0197 1770 3 
4: ‘O15 920 3 
3°88 ‘010 300 2 
5°94 0245 2050 3 
5°94 0197 1280 3 
5°94 015 540 3 
5°90 ‘010 150 z 
73 0245 1630 2 
7-63 ‘0197 980 2 
763 ‘015 515 2 
7°66 ‘O10 135 2 
11-34 “0245 1080 2 
11°34 ‘0197 590 2 
11°34 015 260 2 
11°34 ‘C10 70 2 
15°13 "0245 960 2 
15:13 0197 510 2 
15°13 ‘0150 220 2 
15°13 0099 55 2 
18-94 "0245 794 2 
23 0245 780 2 
27°83 0245 780 2 

TaBxe III. 


‘“Three-inch’”’ Steel Tubes. Inside diameter 2°87 inches. 


| 
Length | t/a Pressure of Collapse. Number of 
in inches. ; lb. per sq. in. , Lobes. 
5&8 0227 2640 + 
9°12 0227 1960 3or 4 
12°14 “0227 1390 2 
Lal df (227 1040 2 
12:19 "0227 1010 2 
15:06 0227 873 2 
15:06 0227 910 2 
18°13 0227 756 2 
18°28 “0227 895 2 
24:13 "0227 684 2 
24:25 0227 770 2 


agreement between the observed and calculated curves for 
lengths less than six diameters. In figs. 4, 5, and 6, we 
have the observed and caleulated curves for lengths less than 


Collapse of Short Thin Tubes. 565 


six diameters. In fig. 6 we have the calculated hyperbola 
extended beyond the length of six diameters. The particular 
curves of figs. 6 and 7 are for J-inch and 3-inch tubing of 
thicknesses of t/d=*0245 and ‘0227, but the tubes of other 
diameters and thicknesses show similar characteristics. It 
is seen that, for lengths greater than six diameters, the 
hyperbola thus calculated shows pressures increasingly less 
than those given by experiment. At this length of six 
diameters there thus appears to be a “critical length.” The 
curve at this length bends rapidly towards the horizontal 
[especially and particularly so] in the case of the thick tubes. 
In the case of the thinnest tubes both for the 1-inch and the 
2-inch tubes, that is, for tubes in which the ratio of t/d is :001, 
the agreement with the hyperbola is not so good and, indeed, 
the maximum bend seems to occur at a much shorter length. 
While the curves show much uniformity and the same 
characteristics are found in the 1-inch and the 2-inch tubes 
for this thickness, yet tue percentages of error for such low 
pressures of collapse are necessarily greater. 

(3) The experimental curves of this investigation are not 
in agreement with the Southwell formula L=k v/d*/t, 
which Cook has used. This is shown in typical cases in 
figs. 6 and 7, where the curves for L=1:75 4/a*/t are drawn 
for the l-inch and 3-inch tubes having a ratio t/d of °0245 
and *0227. On the same figures we have the rectangular 
hyperbolas drawn through the experimental point fcr a 
length of six diameters. If we write the formula L=k wa‘ /t 
in the form 


be ed 
vila’ 

we see that the critical length should increase directly as 
the diameter, and inversely as the square-root of t/d. It is 
evident that the length corresponding to the critical bend 
does vary as the diameter of the tube, but the curves do not 
show that the thinner tubes have the longer critical lengths. 
Indeed, the very thin tubes for both the 1-inch tubes and 
the 2-inch tubes rather indicate a shorter critical length 
than a length of six diameters. It will be remembered that 
Cook was unfortunately limited by his apparatus to lengths 
of less than four diameters, and so his curves could not show 
the above. 

The experimental curves for seamless: steel tubes, for 
which the ratio ¢/d is between ‘0245 and 015. thus show 
that there is a “critical” bend at the length of about six 
diameters, the part of the curve for the shorter lengths being 


566 Dr. C. B. Bazzoni: Heperimental Determination 


a rectangular hyperbola. Beyond this critical bend, the 
curve approaches more or less rapidly to a straight line 
parallel to the axis of abscissee. The above, in connexion 
with previous results on the collapsing pressures of long 
tubes, thus should allow us to calculate with reasonable 
approximation the collapsing pressures of short steel tubes 
of these thicknesses. 


Laboratory of Physics, University of Illinois, 
Urbana, Illinois, May 1916. 


LXIT. £ aperimental Determination of the Ionization Potential 
of Helium. Sy Cuarues B. Bazzoni, Ph.D., Harrison 
Research Fellow, University of Pennsylvania*. 


’R*HE energy of impact necessary to remove an electron 
from simple atoms can be calculated directly from 
Bohr’s theory. The application of Bohr’s formula to helium 
indicates that an energy corresponding to that of an electron 
falling through 29-3 volts is necessary to remove one elec- 
tron from the helium atom, and that an energy corresponding 
to a fall through 83°4 volts is necessary to remove both 
electrons. If the theory is correct, no assumed configuration 
of the electrons in the atom can give any other probable 
values but these, provided that impact ionization, so called, 
results directly and solely from impacts as is generally 
assumed. It follows that an experimental determination of 
the ionization potential of helium ought to serve as an 
excellent direct check on the validity of Behr’s theory itself. 
The first determination of this potential was made by 
Franck and Hertz? using a very sensitive method for 
detecting small amounts of impact ionization. The value 
obtained was 20°5 volts. The only other determination has 
been made by Pawlow ft, who used a method essentially the 
same as Franck and Hertz, and got a result, 20 volts, in 
excellent agreement with those investigators. Pawlow, 
however, found it unnecessary tv apply any correction for 
the initial velocity of the impinging electrons, while Franck 
and Hertz’s result involves such a correction. Recently 
K. Compton § has calculated the ionization potential, making 
use of a formula which he has developed and of various 
recorded data on helium ionization. He obtains 22°5 volts 


* Communicated by Prot. O. W. Richardson, F.R.S. 
+ Ber. der D. Phys. Ges. 1918, p. 34. 

t Proc. Roy. Soe. vol. xe. p. 338 (1914). 

§ Phys. Rey. 2nd ser. yol. vii. pp. 4 & 5. 


of the Ionization Potential of Helium. 067 


as the most probable value. All of these values are too low 
to fit into Bohr’s theory. Bohr has suggested that the effects 
observed in the experiments might have been due to ioniza- 
tien of impurities by radiation trom the helium under 20°5- 
volt bombardment, or to the liberation of electrons from the 
metal parts of the apparatus from the same cause. ‘The fact 
that the characteristic frequency of helium as calculated by 
Cuthbertson * from dispersion data (5:9 x 10!) corresponds 
very closely to 20'5-volt impacts, lends some support to this 
explanation. Further, certain experiments of Rau’st in 
which he finds 80 volts necessary to excite the ordinary 
many-lined spectrum of helium and 80 volts necessary to 
excite 14686 have heen taken, although without sufficient 
justification {, to imply that Bohr’s calculated value is the 
correct one. In view of the remarkable agreement with fact 
of many of the deductions from this theory of Bohr’s, it has 
seemed worth while to re-determine the ionization potential 
of helium, avoiding the suggested possibilities of error, and, 
further, in view of the extraordinary susceptibility of helium 
to disturbances due to minute traces of impurities, with an 
arrangement such that all traces of occluded gases could 
positively be got rid of in the first place and the gas sub- 
sequently be maintained absolutely without contamination. 
The apparatus used in this investigation has been designed 
with these ends in view. 


Fig. 1. 


The experimental tube, which is constructed of transparent 
quartz, is shown in fig. 1. The filament F, which is of 


* Proc. Roy. Soc. 1910. 


+ Srtz. der Phys. Med. Ges. zu Wiirzburg, Feb. 1914. 
t Richardson & Bazzoni, ‘ Nature,’ Sept. 7, 1915, 


568 Dr. C. B. Bazzoni: Experimental Determination 


tungsten about °8 cm. in length and ‘08 cm. in diameter, was 
surrounded by a coaxial cylinder of sheet copper (A), the 
lead (H) from which was connected with a quadrant electro- 
meter. As the currents dealt with were sometimes of con- 
siderable magnitude, the electrometer was provided with 
an adjustable capacity. The sensibility of the electrometer 
with no capacity was roughly 600 mm. per volt. At P isa 
ground cone for connecting the tube with the air-pump and 
with the apparatus for producing and purifying the helium. 
The joint at this cone is preferably covered with sealing-wax 
without the use of tap-grease, but in these experiments it 
was actually covered with soft wax, which seemed a satis- 
factory arrangement. The heating current was taken from 
a 50-volt storage battery, and, since it was absolutely essential 
that the temperature of the filament should be maintained 
constant over relatively long periods, the filament was put 
in one arm of a Wheatstone bridge, which was kept accu- 
rately balanced by the use of three continuously adjustable 
resistances in parallel in the main circuit. This heating 
circuit was insulated from the ground. The negative end of 
F was connected to a sliding contact on a rheostat, which 
was in serles with a 200-volt storage battery. The potentials 
were measured directly through a high-resistance Weston 
standard voltmeter between the negative end of F and the 
ground. This voltmeter was checked against a Clark cell 
and a standard resistance, and found correct within 0:1 volt 
over the range used. 

Connected to P there was first a discharge-tube for 
examining the purity of the gas with a spectroscope ; next 
a U tube to be surrounded with liquid air to get rid of 
mercury ; next a mercury cut-off; then a U tube with 
charcoal; then another mercury cut-off ; then a discharge- 
tube containing phosphorus pentoxide. There was next a 
stopcock beyond which was a quartz tube containing clevite, 
with a lateral tube containing potassium permanganate 
crystals and a final connexion with the pump. Since the 
ionization potential of helium is higher than that of any of 
the substances with which it may be expected to be con- 
taminated, it is of the highest importance to get the gas 
absolutely pure. In this arrangement the major portion of 
the hydrogen was removed by sparking for a long time with 
oxygen over phosphorus pentoxide. The residual hydrogen 
- and other impurities were then taken out by the liquid air 
and charcoal. The gas finally obtained in the experimental 
tube was of a high degree of purity, and in the work that 
follows showed no trace, spectroscopically, of contamination 


of the Ionization Potential of Helium. 569 


with anything excepting where it is specifically stated 
otherwise. As might be expected, the main difficulty was 
in getting the occluded gas out of the cylinder A and in 
properly ageing the filament. The occluded gas was finally 
eliminated by running the pump steadily while maintaining 
the entire apparatus (excepting the seals I, ],, and HE) at a 
bright red heat for two hours by the use of a blowpipe. 
Spurious effects due to the filament surface were got rid of 
by glowing the filament out at a very high temperature 
during several hours. The only impurity which appeared 
subsequent to this treatment was mercury, which could, 
of course, be kept down only by the constant use of 
liquid air. 

In addition to the points mentioned above, there is a 
difference between this apparatus and the arrangements 
used by Franck and Hertz and by Pawlow which might 
have been of considerable importance. These authors made 
use of a device by which a large number of electrons were 
subjected in a space A to an accelerating field, after which 
they passed into a retarding field in a space B where impact 
ionization occurred. The amount of impact ionization was 
measured by the number of positive ions collected by an 
electrode bounding space B, all of the negative electrons 
being returned by the field into the space A. Such an 
arrangement is very sensitive for detecting the potentials at 
which ionization sets in, but it has the disadvantage that it 
affords no opportunity for an accurate comparison between 
the extra current due to impact ionization and the electron 
current which is the cause of it. Tor this reason the results 
of such experiments may always be open to the objection 
that the impact ionization results observed are possibly due 
either to the presence of small amounts of some more easily 
ionizable impurity, or to the occurrence of multiple impacts 
under low voltages, which, although giving rise to some 
ionization, may only form an insignificant part of the impact 
lonization in the same gas when higher voltages are avail- 
able. The present arrangement, on the other hand, measures 
directly both the primary electron current and the extra 
current it gives rise to by impacts. 

A series of observations made before the last traces of 
occluded gases had been expelled is of interest in showing 
the marked effect of minute quantities of impurities. In 
fig. 2 we have the steady deflexion of the electrometer after 
a 30-second charge plotted against the negative driving 
potential from EF to A. The heating current and the 
capacity across the quadrants of the electrometer are shown 


570 Dr. C. B. Bazzoni: Experimental Determination 


beside each curve. In A we havea fresh charge of prre 
gas—ionization by collisions is seen to set in at, roughly, 
20 volts. In B, which was run with the same gas after the 
filament had been glowed for about an hour, the setting in 


Fig. 2. 
‘isd eae | ete Bias 1 = 
i) | ”, Kaen 
180 ee 
eS N “Olraf. 
S770 Mii 


He 


+ 
ec 


<—— E fectrom 


io 20 20 i 40 50 60 . re 


of ionization at 14 or 15 volts indicates the presence of 
hydrogen, although there was no certain spectroscopic evi- 
dence of this gas at this stage. In C we have the curve for 
the same gas when the spectroscope showed hydrogen faintly, 
and in D the curve when hydrogen was plain in the spectrum 
together with a faint indication of CO. In curve C there 
is evidence of the setting in of a second ionization above 


“ 


of the Ionization Potential of Helium. 571 


40 volts. This, together with the value of the primary 
ionization potential taken from A, is a preliminary con- 
firmation of the experiments of Franck and Hertz. 

After the cylinder and filament had been properly cleaned 
up, results were obtained of which the curves in fig. 3 serve 
as specimens. With the apparatus completely evacuated, 


NM ® 
ye) 


? 


<— Electrometer M90 sec. for B 


0 =e -20 -30 -@&O aa 


washed out repeatedly with air, and then reduced to a liquid- 
air vacuum, the current-voltage curve has the form shown 
in D. Saturation, which is nearly reached at —1 volt, is 
complete at —10 volts, after which the current remains 
practically constant. In B the apparatus contained pure 
helium at a pressure of about °85 mm. Liquid air had been 
on the U tube for six hours, but the green line of mercury 
was still faintly visible in the spectrum. The electron 
currents here were of such magnitude, due to the relatively 
high temperature of the filament, that °3 microfarad was 
necessary across the quadrants of the electrometer. Under 
these conditions complete saturation was not attained. The 
curve, nevertheless, shows clearly that impact ionization sets 
in when the potential applied is 19°5 volts. Curves of which 
A is a specimen were obtained after liquid air had been on 
the U tube for 24 hours or more, and the spectrum contained 


o72 ~=Dr. C. B. Bazzoni: Experimental Determination 


no trace of mercury or of any other impurity. The electron 
currents were also reduced by lowering the temperature of 
the filament until no capacity was necessary across the 
electrometer. Here saturation is complete at 10 volts, and 
ionization is seen to set in at 19°5 volts. Curve C was taken 
after the liquid air had evaporated, and a trace of mercury 
was present. The mercury, which presumably ionizes under 
4-9 volt impacts, makes it impossible to get complete satu- 
ration, but, nevertheless, the amount present is too small to 
obscure the setting in of the helium ionization, which is 
again seen to start at 19°5 volts. 

In order to deduce the correct ionization potential from 
these data, it is necessary to consider two disturbing factors— 
first, the drop of potential along the filament and, second, 
the initial velocity with which the electrons leave the wire. 
The first factor might be important because the filament is 
cool at the ends, so that no electrons are liberated exactly 
where the voltmeter is connected. The total drop along the 
filament with the currents used was found to be °75 volt. 
Allowing one-tenth of the length of the filament at each end 
as the part where the cooling effect would be appreciable— 
which, judging from a visual examination, was an ample 
allowance—the corresponding correction to be applied to 
the voltmeter readings is seen to be 0:1 volt. The distri- 
bution of velocities amongst the emitted electrons was 
studied by applying positive potentials to the filament, and 
it was found that the maximum velocity of emission at the 
temperatures used was one volt. It is seen, therefore, that 
when the voltmeter read 19°5 volts there were some electrons 
coming from the negative end of the filament with velocities 
corresponding to 20°4 volts. The results are consequently 
in general accordance with those of Franck and Hertz. If 
the true value is different from 20°4 volts, it is more likely 
to be somewhat below this value than above it, since very 
few of the electrons have initial velocities approaching one 
volt. The experimental arrangement, however, with a large 
anode completely surrounding the filament, and with the 
greater part of the drop of potential close to the filament, is 
a sensitive one, and ought to detect the ionization even when 
the number of electrons impinging with the necessary energy 
is small. 

The phenomena of ionization in gases like helium and 
mereury vapour, involving, as they apparently do, elastic 
collisions below a certain velocity, and totally inelastic ones 
above that velocity, are worthy of close study. The hypo- 
thesis of a critical velocity for a totally inelastic Impact was 


of the Ionization Potential of Helium. 573 


introduced by Franck and Hertz to explain the results of 
certain of their experiments, and seems in general to be in 
accordance with the facts. The ordinary interpretation of 
this hypothesis leads to the statement that an electron cannot 
exist in mereury vapour with a velocity above 5 volts, or 
in helium with a velocity above about 20 volts, provided that 
the dimensions of the receptacle are sufficiently large compared 
with the mean free path of an electron at the given pressure. 
A check on this hypothesis can be obtained from a con- 
sideration of the successive current maxima at multiple 
values of the critical voltage. With an apparatus like that 
used by Franck and Hertz, in which the positive ions are 
dealt with, no cenclusion can, however, be drawn from the 
successive values of the current on account of the retarding 
effect of the accumulated positive ions. In mercury the suc- 
cessive maxima appear to be, as a matter of fact, nearly equal 
one to the other. In studying this point with the apparatus 
used in the present investigation, the pressure and the 
filament temperature were necessarily much reduced in 
order to get that approach to saturation in each successive 
current which was needed to allow the current increase at 
the succeeding critical value to be apparent. The pressure 
worked with (‘2 mm.) was unfortunately one for which the 
mean free path of the electrons is not sufficiently small, 
compared with the radius of the receiving cylinder, to 
render it safe to draw definite conclusions from the 
observations. The results obtained are illustrated by the 
curves of fig. 4,in which the electron current arriving at 
the copper cylinder is plotted against the accelerating 
voltage. A is a preliminary curve in which particular 
attention was paid to the points around 40 volts. In B 
the points are carried to 85 volts. It will be observed that 
a definite increase takes place at multiple values of the 
ionization potential, but that the successive currents are not 
related to the original current in a 1.2.4.8 ratio, as 
would be the case if every electron collided inelastically 
with an atom and there were no disturbances due to positive 
layers. The only effects which can be produced by the 
positive ions in this experiment are, first, their very marked 
effect on the distribution of potential—a matter not yet 
sufficiently investigated—and, second, their action in remoy- 
ing a certain number of free electrons by recombination. 
The current ratios taken from curve B are, roughly, 
1.1:25.1°76. 2°18. On the assumption that three tenths 
of the electrons undergo collisions at the specified ranges 
these numbers would be 1.1°30.1:69 and 2:20. This 


574 On the Ionization Potential of Helium. 


proportion is about what would be expected from the dimen- 
sions of the apparatus and from the pressure of the helium 
used. It seems doubtful whether electron collisions with 
heliam under the ionizing potential are perfectly elastic, but 
it is difficult to draw definite conclusions about this question 
from the present experiments. 


Fig. 4. 


pa | 


\ 


0 10 20 20 40 50. 60 <— Volts. 


The curves of fig. 4 also indicate strongly that the true 
ionization potential of helium is not more than 20 volts, 
rather than 20°5 or 21 volts. This follows from the fact 
that the jump in ionization is seen already to have taken 
place at 40, 60, and 80 volts, at which potentials the 
correction to allow for initial velocities is relatively 
important. 
ae Me sciiadn: it is to be observed that the results obtained 
in these experiments cannot be ascribed either to the loni- 
zation of impurities—for there were no impurities present in 
the critical experiments—or to the liberation of electrons 
from the metal parts of the apparatus, for this would have 


registered a positive charge on the electrometer. It would 


The Compression of the Earth’s Crust in Cooling. 575 


seem, therefore, definitely established that ionization takes 
place in helium under an applied potential of about 20 volts. 
This fact can be reconciled with Bohr’s theory only by 
making certain new assumptions as to the mechanism of 
impact ionization in helium. These new assumptions have 
not yet been subjected to experimental investigation. 


I wish to express my obligation to Professor O. W. 
Richardson for valuable advice given throughout the course 
of these experiments. The quartz tube was constructed by 


Mr. Reynolds of the Silica Syndicate. 


Wheatstone Laboratory, 
King’s College. 
August 20, 1916. 


LXIII. The Compression of the Earth's Crust in Cooling. 
Sy Haroup Jerrruys, B.A., M.Sc., Fellow of St. John’s 
College, Cambridge *. 


I. Introduction. 
ig is generally agreed among geologists that the principal 


cause of the elevation of continents and mountains is 

that the crust of the earth must be in a state of horizontal 
compression, under which it frequently gives way, the strata 
being then folded into a shorter length in the neighbourhood 
of the point where the weakness has been shown. Such a 
compression appears to be the only mechanism that has been 
sugvested that is qualitatively capable of producing the 
observed results. The cause of the compression itself is, 
however, very uncertain. The contraction hypothesis is the 
most satisfactory of those that have been offered, but grave 
doubts have frequently been expressed about its quantitative 
adequacy. According to this hypothesis, the earth was 
originally at a very high temperature, and different parts 
have since cooled by different amounts, changing in volume 
in consequence. A state of strain is thus set up in the crust. 
The mathematical aspect of the theory is due originally to 
Dr. C. Davison? and Sir G. H. Darwin}; they showed 
that as the inner layers after a certain time are cooling more 
rapidly than the outer ones, and therefore contracting more 
rapidly, the outer layers will have to undergo compression 

* Communicated by the Author. . 

+ Phil. Trans. 178 A, pp. 231-242 (1887). 

Se Trans. 178 A, pp. 242-249 (1887); or Sci. Papers, vol. iy. 
p. vos. 


576 Mr. H. Jeffreys on the Compression 


in order to continue to fit the inner ones. At the same time 
they showed that the inner parts are being stretched. The 
boundary between the parts in process of extension and 
compression respectively is known as the “level of no 
strain.” Initially it was at the surface, and has moved 
steadily downwards. The amount of compression geologically 
available is not, however, the integral amount since the time 
when the earth became solid, for if we consider a layer at a 
definite depth, in the earlier part of the interval stretching 
was taking place there, and compression has only taken 
place since the layer of no strain was at that depth. In 
calculating the available compression we must then take the 
amount that has occurred since that date, and ignore the 
stretching that took place at first. That this is true can be 
seen from the analogy of a football which has been inflated 
to such an extent that the outer case is just taut. If itis 
partially deflated, the size of the bladder decreases, and we 
have a condition resembling that of an outer layer too large 
to fit the inner parts. ‘he case then crumples up and 
departs from the spherical form. On the other hand, if the 
ball is inflated further, the case only expands a little and 
remains spherical. Thus compression can produce folding, 
but tension cannot. In the case of the earth, the effect of 
long-continued extension is to produce permanent set, thus 
relieving the strain without departure from symmetry, and 
even a small subsequent compression can cause folding. 
Sir G. H. Darwin showed that the effect would be sufficient 
to shorten every great circle of the earth by a quantity of 
the order of 100 miles. This estimate was probably exces- 
sive for the form of the theory that he adopted, his initial 
temperature and his assumed coefficient of expansion being 
both too high. Nevertheless it appeared inadequate to 
satisfy the requirements of geology. A later paper™ by 
Davison showed that when the increase of the coefficient of 
expansion with temperature was taken into account, a 
considerable improvement was obtained, but the compression 
was still insufficient. 

Both of these estimates rested on the theory of the cooling 
of the earth given by Lord Kelvin, who treated the earth as 
a simple body, initially at a uniform temperature, and losing 
its heat from the surface. The recent discovery of radio- 
activity has, however, shown that so much heat is actually 
being generated within the crust of the earth, that if the 
same amount per unit volume were being produced through- 
out the mass, the temperature gradient at the surface would 


* Phil. Mag. xli. pp. 133-1388 (1896). 


of the Karth’s Crust in Cooling. aul 


be 300 times its actual value“, and the earth would be 
getting hotter instead of colder. There are many objections 
to this view, and it seems that the amount of radioactive 
matter per unit volume must decrease so rapidly with depth 
that the total is insufficient to supply more than about #2 of 
the present lesy of heat from the surface f. At the same 
time the ratio of the amounts of uranium and lead in 
minerals gives valuable direct evidence concerning the age 
of the mineral, and hence a minimum estimate of the age of 
the earth {, which is very much greater than that derived 
on Lord Kelvin’s hypothesis. ~ From a review of the evidence 
obtained by several different lines of investigation, Holmes 
has found that the observational data can be satisfied 
exceedingly well if the age of the earth be taken to be about 
1600 million years, and if the rate of liberation of heat 
per unit volume decrease exponentially with the depth. The 
interval of time concerned is so much greater than that 
found by Lord Kelvin that his theory of cooling requires to 
be revised so as to take into account our most recently 
acquired knowledge, and at the same time the contraction 
theory, which depends on it, needs similar revision. This is 
the principal object of the present paper. 

The level of no strain is found now to be at a considerably 
greater depth than the older determinations gave, and at 
the same time the amount of compression at the surface is 
much increased. For both reasons the volume of crumpled 
rock is increased. On the basis of the exponential distri- 
bution of radioactive matter the available compression is 
133 kilometres; in other words, enough to shorten every 
great circle of the earth by this amount. Actually, folding 
is not uniformly distributed over the earth, but nearly 
confined to certain definite lines of weakness, as one would 
naturally expect. In some places, as in the case of the valleys 
of Hast and Central Africa, a tension is actually indicated. 
Some great circles thus show verylittle crumpling, and others 
are free to be folded to a greater extent than would be pos- 
sible if the distribution of mountains were uniform. Whena 
numerical estimate is made of the amount of compression 
required to produce the known mountain ranges, that found 


* A. Holmes, ‘The Age of the Earth,’ 1913, p. 129. 

+ Holmes, “ Radioactivity and the Earth’s Thermal History,” Part I, 
Geol. Mag. March 1915, p. 109. 

t Holmes, ‘The Age of the Harth,’ p. 157; or “ Radio-activity and 
the Measurement of Geological Time,” Proc. Geol. Assoc. vol. xxvi. 
pp. 289-309 (1915). 


Phil. MagoS. 6. Vol. 32. No. 192)sBec 1916. 2R 


578 Mr. H. Jeffreys on the Compression 


to be available on the contraction hypothesis appears to be 
quite adequate. 


Il. The effect on underground temperatures of a uniform 
distribution of radioactive matter through a horizontal layer. 


Let the number of calories generated per unit time per 
cubic centimetre be A. 

Let the temperature at depth 2 at time t be V. 

Let the depth of the radioactive layer be 2X. 

Then, when 2 is between 0 and A, V satisfies the equation 


OVE reer VA 
We Tea ce’ eT 
and when z is greater than A, we have 
av _,o°V 
ot ov 
Here p is the density of the rocks and e¢ their specific 
heat. If & is their conductivity, 
Wae/ep. 2". 4.) 
The boundary conditions are that when t=0, V=mz+S 
for all values of the depth, and when #«=0, V=0 for all 
values of the time. Further, V and 0V/d must both be 
continuous at z=2. 
Evidently V=—A(«—n)*/2k satisfies the first equation. 
Substitute then 


V=u+A{r?—(@—r)?}/2k when z is less than A,. (4) 


-0... 


and 
V=u+ Anr?/2k when w« is greater than X. (5) 


Then Ou 429% 20° 
Ot 02" 


u=0 when =O; wand du/d are both continuous at =X. 


When ¢=0, 
u=me+S—A{r7—(a —2)*}/2k when @ is less than 4 (7) 


,— 0 everywhere... . 7 uae 


u=me +8 —Ad7/2k when z is greater than 2.f 


Now it has been shown by Fourier * that the solution 
of (6) that makes u=/(#) when ¢=0, the value of f(x) being 
specified for all values of 2 from —x to +0, is 


Pal age TAla+2ghV9). . TES) 


* Analytical Theory of Heat,’ p, 354. 


of the Harth’s Crust in Cooling. 579 


In the case here considered, /(#) is only specified from 
x=0 to =a, but we have the condition that u is always 0 
when #=0. This is satisfied automatically if we specify 
{(«) for negative values of x, so that f(7)=—j/(—2). Thus 
u can be determined. 


We find then 


UM 
u=me +S Erf o 


A ; 3 p_2 
+ 37 | (a + 2h?t) Erf ott 


ae | 
Zi. 2 


Be 8G 5 —> —(xr—))2/4h2 
+ Qxrht?or 29 Sg a eS 7(a—D)e (z—))?/4h7¢ 


— hin (ae + je | i) 


(e+)? + 2h} Bef 


LL 2 2 
— ~{(a—r)?+2ht} Erf Dts 


2 


and V is at once obtained on substitution in (4) and (5). 
Tt can easily be verified that it satisfies all the conditions. 

If we differentiate V and then put x=0, we find for the 
temperature gradient at the surface the value ~ 


ro =m-+ r 
ta a 2h Vt 


thy / (<) (1—e") . (10) 


In this substitute the values given by Holmes *, namely 
mt =0°:00005 C. per cm., Se 20081Ce 
g°) = (0°-00038 C, per em., 
t—) 


S A 
we a |»—» rf 


t=9°05 x 10™ secs.—1°6 x 10° years, 
A=63'9 x 10~™ calorie per cm. cube per sec., 
k=0°005 C.G.S. units, bh OS ee), (Tee) 


Put RRR) i eeu) vs, ) (12) 


* Geol. Mag. March 1915, p. 108, 

+ This value of m agrees well with that given by J. Johnston, Journ. 
Geol. xxiii. p. 736 (1915). 

t This melting-point is approximately that of basalt, representing 
material just under the acid shell and intermediate in composition 
between the acid rocks and the deep-seated peridotites, 


2R2 


580 Mr. H. Jeftreys on the Compression 


Then 
l(1— Erf /) oe eae = (063. 


This gives 1=-066, ae 
A= 2°49 10° cm. |. 


Thus the thickness of the radioactive layer on this 
hypothesis of a uniform distribution of active matter is 
25 kilometres. 

It may be remarked here that for any value of ¢ such that 


l( =/2ht*) is small, u can be expanded in powers of 7. Put 
th 
im afht=q. . . . . ne 
Then ort 92 
Erf (q+)= Erig+—7—e q? Pps ge 


to the second order in l. 
Also e— (94 = e-P(1 — 2414 2471? —I?) 
to the same order. 


Now substitute in wu and work to the second order in 1. 
Then it is found that V reduces to the form 


ae 
ena (s—4 sp) Bt yey, + Se—m + (15) 
where T=) oF 25 S9Ne 
and p=A(A—2)*/2k if a<X. 


Ill. The effect of other distributions of radioactive matter. 


So far the liberation of heat by radioactive matter has 
been assumed to be given by the special law that the distri- 
bution is uniform down to a particular depth, and zero 
below that depth. This restriction will now be removed. 

Consider a distribution compounded of an infinite number 
of such distributions ; the amount of heat liberated per unit 
volume per unit time by those extending to depths between 
» and X+dX is '(A)dA for depths less than X and zero for 
depths greater than X. If ¢’(A) is supposed specified for all 
values of X from 0 to infinity, then the total rate of evolution 
of heat at depth A is 


i ” $A)M=$(o)— b(n). 


Hence if we want to find the effect of a supply of heat 
according to the law ¢(«)—d¢(A), we must write $/(A)dr 


of the Earth’s Crust in Cooling. 581 


for A in the previous solution and integrate with regard to 
® from zero to infinity. . 

In particular, in the case of the exponential law used by 
Holmes *, ¢(0)—¢(A) must be equal to Ae~*; in this 
case f' (A) = Age, and on substituting in V and integrating 
we find 


A x A ee 
Va So ee yc alg) 


It must be noted that equation (15) is only valid when 
t is so great that 2/2Aé? is small, and similarly, (16) only 
holds when 1/2ahé? is small. On the other hand, wu is an 
even function of 2X, and therefore contains no term in A’. 
Hence (15) and (16) will hold provided the fourth power otf 
d can be neglected. 

Differentiating (16) with regard to x and then putting x 
zero, we find that the known velocity gradient at the surface 
gives a quadratic equation to determine a. The positive 
root of this equation is a=4°14x 1077/1 cm. _ 


IV. The straining of the crust in cooling. 


If it be assumed that throughout the process of cooling the 
earth preserves a state of spherical symmetry, then it is 
evident that as the changes of temperature are not the same 
at all points, a state of strain must be set up, consequent on 
the variations of volume that take place. 

Thus, consider a shell of internal radius r and external 
radius r+ or. 

Let the coefficient of linear expansion with temperature 
be n. It is supposed to be a function of the temperature. 

Let the initial density of the shell be p. 

Also let the rise in temperature in a definite time be V, 
which is supposed small. Then V isa function of r. The 
density of the shell will change at the same time from p to 
p(l—anV). Let the radius at the same time change to 
r(1+a). 

The external radius will change to 


r(l+a)-+8r4 1+ Sra } : 
Hence the mass of the shell after the change of temperature is 
Lape: {1 ey 2 Choon \ 


* Geol. Mag. March 1915, p. 108. 


582 Mr. H. Jeffreys on the Compression 


But the mass is unaltered. Hence we have the equation of 
continuity 


dat 2 (ra)—3nV=0. sii at (1) 


V being supposed known throughout the earth, this is a 
differential equation to determine «, subject to the boundary 
condition that a is zero at the centre of the earth. Thus for 
any shell # is determined by the changes of temperature 
within that shell. If the shell simply expanded independently 
of the interior, the radius would increase by rnV instead of 
by ra, so that the excess r(a—nV) is due to stretching. 


Denote a—nV by k, and substitute forain(1).. . (2) 
Then the stretching is given by the equation 
Le at EON a! V ) 
Gh a es - . 0 
whence 
a shee (nV) 
k= — at Ae Par ee ° a ° (4) 


Now consider the changes that must take place in a short 


time di. If the integral stretching be K, then k= Ok 


ot 
and for V we must write Or dl Hence 
aK __ 1" 2 (,2¥) | 
aa —3f Po SAn Seem 


If now c denote the radius of the earth, and 2 the depth’ 
of a point below the surface, then r=c—z. As the tem- 
perature changes only extend downwards through a small 
fraction of the radius of the earth, we can neglect the square 
of w/c. Finally then 


oF ( LG DE CR) 


or, integrating by parts and again neglecting (a7ieyes 
oO, 
= =-no +2( n& do. en 


Let now n=e+e'V, where e and e’ are two constants. 
Consider first the case of a distribution finite in depth. 


Put An?/2k=a, S—a=pP. Then 2=398°, B=802~ 


, NS ee ae 


of the Earth's Crustin Cooling. _ 583 


Then V=met+8 Erf (2/2hi?) +a—p, 
ov rae oh es v gt abet 
OR eww wr Qhie 


Then performing the integration we find 


ok ae ! 9! 1 / 2 B q —q2 
mp he teat 2 mht g+eB rt 9) | e-4 
— an | (e+ e'a)e-2 + e’mht2{2ge—% +72(1— Erf g)} 


: 1 
+ 6/8 jes Erf g+ FT er Erfg v2) \ | 


The terms depending on p have been omitted. 

When g is zero, this is negative, but when g is large 
enough, OK/dt is positive. Tor one value of g, OK/d¢ is 
zero, and the rock is being neither stretched nor crumpled ; 
this value of g corresponds to the “level of no strain.” 
Hvidently if then we put QAK/d¢=0, and regard (6) as an 
equation to find g, the solution will have 1/c as a factor. 
Hence in accordance with the approximations already made, 
g may be put zero inside the square brackets in making an 
approximation to a solution. 

Then g has to be found from 


O=[ete'ate'(mhr/ nt +B) 29m *|Rq/tm* 
3Bh 
~~ a (at) 


With the data already given, this gives g=0°207. In 
accordance with Fizeau’s results, e/e’ is taken to be 300° C. 

Hence the depth of the level of no strain is 79 kilometres. 
This is greater than the depth of the radioactive layer, so 
that pis zero, as has already been assumed. 

In addition to the level of no strain, we require the 
amount of compression at the surface. This is obtained 
directly by putting g=0 in OK/oé, and then integrating 
from ¢=0 to the present time. ‘lhe approximation is not 
good in the early part of the period of integration, but 
as the compression must in any case vanish with the time, 
that short interval may safely be neglected. 

Then 


| eas mhJ (mt) + €'B/V 2]. 


K=— “ee(- “tet cate mh (tr) + eB) V2}. 


584 Mr. H. Jeffreys on the Compression 


Fizeau’s results indicate that average values of e and é 
are given by 


e=7x 10°%=1°C. 3 6 = 24x10 =e eee 
Adopting these numbers, we find that at the surface 
K = —5 68 x 107°; 


from this the shortening of the circumference of the earth 
by compression is at once found to be 227 km. 

A rough approximation to the average effect of the 
compression may be made by adapting the method of 
Rev. Osmond Fisher. The volume of rock crumpled up 
is { 82re(—Kr).da, taken through the compressed layer. 
At the surface K=—5°68x107%; at a depth of 79 km. 
it is zero, as also is OK/Ou. . 

Hence, with sufficient accuracy for the present purpose 
we can interpolate for K by the formula 


lta eae =: || - ics 
K=—5-68 x10 (1 aa): 


If this crumpled rock were then spread in a uniform 
layer over the surface of the earth, the depth of it would 
be 300 metres. 


Next, consider the exponential distribution of radio- 
active matter. The temperature is in this case given by 


A £ A 
Vi ma (S— ap \Ert 57 + ( ee 


Q2hV t 
Put Alarvk =a, S-a=B; 
then Ai (Oe B= 4bes: 
Then 
K ak 1 
ee = et 2e’mhttg + €'8 Br q+ e'a(1—e-2m) | a 
3Bh 


GRE ie e’a)e—¥ + e’mht? {2qe—" + a3 (1— Erf q) t 
i; Ty ae 
$684 ef Extg+ Zo (1 Ent gv/2) } 


— 6 ets, — 8 or Eire (g+9))} |, 


where y=hat?. Its present value is 7°82, so that the 
terms depending on it are small enough to be neglected. 


of the Earth’s Crust in Cooling. 585 


Solving by successive approximation to find the level of 
no strain, we find that the value of qg corresponding to it is 
200, and hence the depth is 76 km. The depth is thus 
not much different from that found on the hypothesis of 
a homogeneous radioactive layer of uniform depth. The 
value of K at the surface is 3°32x 107%, so that the com- 
pression is adequate to shorten the circumference by 
133 kilometres. If the crumpled rock were spread uni- 
formly over the surface, it would cover it to a depth of 
170 meires. 


V. The amount of compression required to produce eaisting 
mountains. 


In the cases of certain particular ranges, geologists have 
succeeded in obtaining a direct quantitative estimate of 
the amount of compression that must have been necessary to 
produce the folding that is observed. In the Appalachians, 
for instance, the width of the rocks, measured perpendi- 
cularly to the chain, is estimated to have been shortened by 
about 40 kilometres. Similarly, the compression in the 
Rockies is 25 miles, in the Coast Range in California 
10 miles, and in the Alps 74 miles*. ‘The larger ranges, 
particularly in Asia, have not been so exhaustively treated ; 
but a rough idea of their importance can be obtained by a 
comparison with the Rockies, or in the case of narrower 
ranges, the Coast Range, whose geological age is about 
the same. The Alps are probably abnormal, and have not 
been used as a standard. ‘The elevation of a continent or a 
large tableland involves little crumpling within it, and no 
great amount at the coast as long as the slope is there gradual. 
Hence, in determining the amount of compression, we need 
consider only the steep slopes of mountains. The amount 
in any range is here supposed to be proportional to the 
mean height+, and in the following table the latter has 
been obtained roughly from the maps in Philips’ Student’s 
Atlas. From this estimated compression combined with 
the length of the range the area lost by folding is at once 
found. ‘The amount of compression found is of course 
essentially provisional and must be revised when further 
geological evidence is available. 


* These data are from Pirsson & Schuchert, ‘Textbook of Geology,’ 
p- 361. 

+ This would be exactly true if the strata were similarly folded in all 
mountains. 


586 Mr. H. Jeffreys on the Compression 


TasLE.— The Great Mountain Ranges. 


Length iipan Sos ee ed 
ome Height pression cae! 


Range. (km.). : (thousands 
S (metres). (km.). of sq. km.). 
Scandinavian) cscsseceneeee 1400 1000 10 14 
PNT Sy. Soca ne epee hos maeees 1000 3000 118 t 118 
Carpathians \2ccovene- fences 1300 1000 10 13 
PAPENIMINES ea -beseee eee eae sae 900 700 10 9 
Raralis 8, ccea enti comme eee 2200 700 10 22 
Caucasus and Armenia 8500 2000 40 140 
MWA tect osc See eee ece nee 1400 2000 40 56 
Tem al avya son's ca cee mete aap ences 4000 5000 100 400 
SMS] MIAM G5C..o heels sees 1200 2000 30 36 
Karakorum and Hindu 92400 1000 * 20 48 
BTS las Bt Py ee ee en 

MVEA TIT csi eee eee ee Aae 2300 1000 * 20 46 
Miran Sian 2. .00 sacs weeweree 2000 3000 60 120 
Ae Renee ee by Sti 24 Sh 1600 2500 50 80 
AIG YSSINIA, | 5.0.6.7 c0acaseowac 2000 2000 40 80 
Wrakensber? ......c52sce000e0 1500 1000 20 30 
PAGED S ginelie c's eh a vec eee 2000 1500 30 60 
Coast Range &e. ............ 4000 2060 16+ 40 
1 £3 (0) 12 a RE 7000 2000 40tT 280 
mIpBalachians, ....ccssuessees 800 1000 60 t 48 
_2\ EC (GSLs A aes eld “Gt 7000 2000 40 280 

Total it a ee 1920 


* It is assumed that the folding needed to produce the plateau of Tibet 
was all at the margins, and is thus included in that found for the Himalayas 
and Tian Shan. Thus, for the mountains within the area it is only necessary 
to consider the height above the general level of the plateau. 

i, These data are those that have been obtained directly from the geological 
evidence. 


It is thus found that in the formation of mountains 
the surface of the earth has been diminished by crumpling 


of the Earth's Crust in Cooling. 5387 


by about 1°8 million square kilometres=1°8 x 10!* square 
centimetres. 

Now if K be the relative compression at the surface, 
the area crumpled is 


Ane?{1—(1—K)?} = 87e?K. 


By substituting, we see that the value of K_ requisite 
to account for all the mountain ranges of the earth is 
1°8 x 107°, which is only about half of the amount (3°3 x 107°) 
that the contraction hypothesis has been shown able to 
produce. If we allow for possible folding in regions at 
present submerged and for old ranges, now almost denuded 
away, the agreement may be better, or the available com- 
pression may even be insufficient; but in any case the 
theoretical and the observed compressions are of the same 
order of magnitude, so that it seems highly probable that 
the contraction hypothesis is adequate to account for a 
very large fraction of the mountain-building that has taken 
place, and perhaps for the whole of it *. 


VI. The influence of denudation and thermal blanketing. 


Suppose that by some means not yet specified, a large 
area of the solid earth has been raised up to form a 
continent, and at the same time other parts have been 
lowered to form oceans. Then the influence of the atmo- 
sphere causes the continent to be denuded, and the solid 
outer layers of it come to be redeposited on the ocean bed. 
Then the uniform law of cooling already considered is 
disturbed in two ways. In the first place, it has been 
shown that the stretching or crumpling of rocks contains 
as a factor the quantity S—{r7g/a)dr, and the second 
term of this is a large proper fraction of the first. If, 
however, the upper layers, with their radioactive con- 
stituents, are removed, then, instead of the limits of the 
integral being 0 and «, they will be Ay and «, where 
Ay is the depth of the rocks denuded away. Thus the 
integral is much diminished in value, and consequently 
denudation increases the factor as a whole. Hence a 
greater strain will be thrown on the crust in continental 
areas. Similarly, if the ocean bottom is considered, the 
compression is seen to be diminished there. In the second 
place, although convection currents in the ocean will cause 


* Tam much indebted to Dr. Bonney for calling my attention to the 
possibility of the comparison here made. 


588 Mr. H. Jeffreys on the Compression 


heat to pass outwards in time, yet the ocean will neverthe- 
less act as a thermal blanket, reducing the cooling below it 
and hence the surface compression. . 

For both reasons, then, an oceanic area will tend to 
require round its margin a smaller horizontal pressure 
than the mean, if it is to remain spherical. Similarly, 
a continental one requires a larger pressure round its 
margin. ‘These two statements are inconsistent, since by 
the Third Law of Motion the pressures should be equal. 
Evidently, then, the earth cannot remain spherical. The 
pressure is inadequate to compress the surface layers of 
the continental areas so as to keep both these and the 
ocean areas part of the same sphere, so that the outer 
layers on the continents must expand somewhat relative 
to the inner ones, the margins remaining fixed by cohesion. 
The only way in which this can happen is by a reduction 
of the radius of curvature of the crust—in other words, the 
continents will tend to rise. Similarly, the ocean bed will 
sink. The effect capable of being thus produced is very great. 
Thus, if the level of no strain were at a depth of 70 km., 
and a surface layer, 3000 km. in length, were 3 km. too long 
to continue to fit the inner sphere, the radius of curvature 
would be increased by as much as 500 km.,and the elevation 
would be 15 km. inthe centre. The estimate of the available 
excess of length is probably conservative. ‘This differential 
compression is then probably a very important cause of 
the elevation and maintenance of continents and oceans. 
The adjustment of the figure of the earth to make the 
compression constant all over is not likely to be complete, 
however, as further stresses would be produced in the 
crust, which would cause fracture. The adjustment is 
not then likely to be much greater than would give iso- 
static compensation of the changes produced by denudation. 
The remaining part of the differential compression will be 
shown in extra folding in the continents, and diminished 
compression, or even, in extreme cases, a tension, in the 
ocean bed. 

Mr. Holmes has called my attention to the importance of 
another factor. The crumpling here considered is that which 
has occurred at any depth since the level of no strain reached 
that depth. Before it had gone so far down, there would bea 
considerable tension in the interior of the continents, which 
might be capable of causing rifts and accelerating intrusions. 
Such rifts would occur very soon after the denudation. The 
flexure of the crust here indicated would, on the other hand, 
increase steadily with the time, and is essentially subsequent 
to the denudation. 


ae 
re 


of the Earth’s Crust in Cooling. 589 


VII. On the causes of Isostasy. 

From a detailed consideration of the evidence for and 
against the theory of isostasy, Professor Joseph Barrell * 
has come to the conclusion that while the uppermost layers 
of the crust of the earth are probably very strong, the 
strength gradually diminishes with depth, so that at a 
depth of about 400 kilometres, in the middle of the layer 
of weakness, called the asthenosphere, the strength is only 
about #; of what it is at the surface. In virtue of this 
weakness a large area of excessive mass per unit area tends 
to sink, while lighter areas rise. The smaller the dense 
area is, the worse is the compensation, in consequence of 
the fact that the maximum shearing stress occurs at a 
smaller depth, and less stress is transmitted to the astheno- 
sphere. Barrell considers that the adjustment takes place 
by progressive local melting under strain, with subsequent 
recrystallization when the tangential stresses have been 
removed. Section VI. of the present paper may give an 
additional important cause of the maintenance of isostasy. 

A further problem of extreme difficulty is that of deciding 
how the continents and ocean basins ever came to be formed. 
If there were no isostatic compensation, they might be 
attributed to crumpling by compression, of a type not very 
different from that which produces mountain ranges. It 
appears, however, that continental areas, though varying 
enormously in height and shape, are fairly permanent in 
position, and that their height is reallv due to their being 
made of lighter materials than the ocean bed. The question 
is to decide how the lighter materials succeeded in being 
collected to certain points, leaving the denser rocks exposed 
in other parts. The most satisfactory explanation of any 
widespread inequality in the shape of the crust of the earth 
is the theory of Gravitational Instability, due to Jeans. 
This theory shows that if the earth be regarded as com- 
pressible, and any arbitrary displacements be given to its 
constituent particles, the alteration in the potential energy 
is of three parts :— 

(1) The elasticity of the materials causes it to be 
increased. 

(2) The normal displacements of matter at the boundary 
cause 1t to be increased. 

(3) The changes of density within the earth lead to 
changes in the gravitational potential which in 
general cause the potential energy to be diminished. 

* ‘Journal of Geology,’ vol. xxii. (1914) pp. 28, 145, 209, 289, 441, 
537, 665, 729 ; vol. xxi. (1915) pp. 27, 425, 499; or a review by the 
present writer, ‘Observatory,’ April 1916. 


590 Mr. H. Jeffreys on the Compression 


If, then, the third part is able to counterbalance the other 
two, the spherical state of the earth is unstable and a 
departure from it will occur. It has been shown by Love 
that such instability will occur first for deformations ex- 
pressed by spherical harmonics of the first order, in the 
case of a homogeneous sphere. If the outer layer of the 
sphere is of somewhat lower density than the rest, as in 
the actual earth, the problem is more complex ; if the outer 
layer is, however, thin and of small strength, the effect 
of the change can be predicted. For the same displace- 
ments inside, the changes (1) and (3) in the energy are 
the same as before; but if displacements of mass occur 
over the outer surface to such an extent as to compensate 
isostatically the displacements below it, the change (2) 
attains its minimum value: this is seen at once from the 
fact that the isostatic state, being that in which the interior 
of the earth is in hydrostatic equilibrium, must be a state 
of minimum potential energy for a given thickness of the 
surface layers. Hence one of the causes making for 
stability is reduced if heterogeneity is allowed for and 
the inequalities are supposed compensated. (rravitational 
instability will then be first manifested in the case of those 
degrees of freedom that are consistent with the isostatic 
state. Thus if inequalities can be produced at all in this 
way, it is highly probable that they will be isostatically 
compensated from the start. The explanation of the low 
density of the continents and the high density of the 
matter below the oceans may then be connected with 
gravitational instability, accelerated by the mechanism of 
isostasy. 


VIII. The effect of changes in the rotation of the Earth. 


An essential feature of the present investigation is the 
hypothesis that the earth was formerly in a very heated, 
and probably fluid, condition. This leads directly to several 
other consequences. In this liquid state the coefficient of 
viscosity must have been practically zero; but at present 
it is exceedingly high, probably greater than 10” c.g.s. 
units. Now, as the substances of which the earth is 
formed are not in general pure, they would pass through 
a pasty state in solidifying and the viscosity would thus 
increase continuously. Hence it would take in the process 
every value between its initial and final values. Now 
Sir G. H. Darwin showed that if the viscosity had a certain 


of the Earth’s Crust in Cooling. 591 


variable value, depending on the distance of the moon from 
the earth, but always between 10% and 10”, the process 
of transference of angular momentum from the earth to 
the moon in consequence of tidal friction would take place 
at the greatest rate possible; and the changes would be 
complete in about 50 million years. This optimum viscosity 
must evidently have been attained at some time during the 
process of cooling, by what has been above stated, so that 
for a considerable part of the time the conditions must 
have been suitable for evolution to take place at the 
maximum rate possible. The truth of the theory of Tidal 
Hyolution thus follows as a natural consequence of the 
present paper, and must therefore be considered with it. 
Most of the evolution must have taken place in a com- 
paratively short time, perhaps about 200 million years, 
and since then the viscosity must have been too great for 
much change to take place ; so that now the mean distance 
of the moon must be nearly constant. The most rapid 
part of the change must have occurred at a fairly high 
temperature, probably well above the boiling-point of water, 
but after solidification. It is thus to be placed before the 
oldest sedimentary rocks, and the consequent change in 
the ellipticity can have had little direct effect in the forma- 
tion of the present mountain ranges, which have for the most 
part been elevated since the Carboniferous period. In one 
way, however, it may be important. The ellipticity of the 
earth would decrease as the velocity of rotation decreased, 
and the consequent shortening of the equator would form 
pre-Cambrian mountain ranges with a north and south 
alignment. On account of the plasticity of the rocks, 
however, which then extended nearly to the outer surface, 
these might soon cause fracture below them and disappear. 
In any case their existence might, however, leave a record 
in the form of definite lines of weakness, which would 
tend to localise subsequent folding caused by contraction. 
The tendency of folding to occur repeatedly along the same 
line in different geological periods may be regarded as 
confirming this suggestion. 


In conclusion, the author wishes to express his indebtedness 
to Mr. Arthur Holmes, whose helpful criticism from the geo- 
logical point of view has been invaluable. 


F592) I 


LXIV. An Inquiry into the Possible Existence of Mutual 
Induction between Masses. By Mitts WALKER and 
W. Witcoms STAINER *, 


E have for long been familiar with the analogy in the 

behaviour of matter in motion and electricity in 

motion. The mathematical expressions which state the 

relations between mechanical force, mass, and velocity are 

identical with those which express the corresponding electrical 
relations. 

The object of the experiments described in this paper was 
to ascertain whether there is any measurable action between 
masses which would correspond to mutual induction between 
electric circuits. 


Fig. 1. 


Be 


oe 


If L is the inductance of the simple electric circuit A., of 
negligible resistance (fig. 1), we know that an electric 


dt 


will start a current, 2, flowing in the circuit. If there is no 
resistance, the current will go on flowing and the circuit will 
contain a store of energy, 412°. Thecurrent can be stopped 


by applying a back H.M.F. and the circuit made to yield up 


* Communicated by the Authors, being an abstract of a paper read 
before the British Association at the 1915 meeting. 


Possible Existence of Mutuat Induction between Masses. 593 


its energy. We shall then receive back the E.M.F. time- 
integral \ Bde. 

Similarly, if M is the moment of inertia of the flywheel, 
A in fig. 1, a rotational impulse 


(rae (a2 Hae 
dt 


will give to the flywheel an angular velocity, « (where T 
stands for the turning moment necessary to give the angular 
aeceleration d«/dt). If there is no friction, the flywheel will 
go on spinning, and it will contain a store of energy, M2’. 
The wheel can be stopped by applying an opposite turning 
moment, and can be made to yield up its energy. We shall 
then receive back the turning-moment time-integral \ Tdt. 
So far the analogy is complete. But now observe the effect 
of the electric circuit A; upon the adjacent circuit B,. 

When di/dt has some value generating the E.M.F. E in 
the circuit A, there exists also an E.M.F-. in the circuit B., 
the amount of which will depend upon the closeness of the 
magnetic coupling to the two circuits. Any change in the 
current in A, will be accompanied by a téndency for 
current to flow in B,. 

The question naturally arises, “‘Is there any action 
between masses analogous to this mutual induction between 
electric circuits ?”’? If we accelerate the flywheel A (fig. 1), 
does it produce any force upon the suspended disk B ? 
It is quite possible that a small force of the kind might pass 
unnoticed if not specially looked for, just as the gravitational 
attraction between two movable objects would ordinarily 
escape observation. 

Einstein and Grossman, in their mathematical inquiry 
into the theory * of gravitation, deduce the existence of 
such an effect, the order of magnitude of which is so small 
that it cannot be observed by any known apparatus. 

In 1912 the authors constructed at the Manchester School 
of Technology the apparatus shown in fig. 2. There were 
special facilities existing at the school for carrying out this 
work. A heavy steel flywheel and an electric motor for 
driving it had already been constructed for other work ; 
moreover, the School building afforded a very rigid support 
for the suspension, as it forms a hollow rectangle 90 metres 
by 60 metres, not very easily deflected by a torsional stress. 
The well of a lift provided a very suitable place to hang a 
long delicate suspension. 

* See Fokker, Phil. Mag. [6] xxix. pp. 77-96. 
Phil. Mag. 8. 6. Vol. 32. No. 192. Dec. 1916. 28 


594 Messrs. Miles Walker and W. Witcomb Stainer on the 


Ase FF Aah bowP 


2. 
AK KX 
N feeshinets 
Flywheel of cast steel. L. Screws for adjusting K. 
Suspended disk of porcelain. M. Wheel and wire band for turn- 
Lower windage screen of wood. ing torsion head at the top of 
Upper windage screen of wood. the building. 
Inner screen of sheet steel. N. Spring-retained ball-bearing. 
Outer screen of sheet steel. O. Over-speed cut-out. 
Spindle of fibre carrying por- P. Ball-thrust bearing taking main 
celain disk. weight. 
. German-silver rod for adjusting Q. Coupling. 
suspension to centre. R. Motor. 
. Mirror attached to disk. S. Steel stanchion fixed in ground 
Bifilar suspension. and stayed to distant walls. 
. [ron tube protecting suspension. 


Possible Existence of Mutual Induction between Masses. 595 


A steel flywheel A (fig. 2), 56 cm. in diameter and 11 cm. 
thick, is mounted on a vertical shaft, O, supported by ball- 
thrust bearing, P, and driven by an electric motor, R. Above 
the flywheel is suspended a disk, B, 51 cm. in diameter, 
made of very pure porcelain, weighing about 10 kg. The 
first suspension was made of very thin steel strip, but the 
final suspension is made of two round steel wires, each 
0:025 cm. in diameter. The length of the suspension is 
21 metres. The distance between the bifilars is about 
0-15 cm. The torsional control on B is extremely small, 
amounting to only 28 dyne-cm. for a deflexion of 1 radian. 
The angular swing of B upon its principal axis has a natural 
period of 2460 seconds. A mirror I (fig. 2) is mounted on 
a stiff wire attached to the disk B, which enables the move- 
ment to be accurately observed on a scale at 6 metres 
distance. By means of a telescope a movement of 0°01 cm. 
corresponding to a deflexion of 1/12,000 of a radian, can be 
estimated, so that one can observe the effect of acting on the 
edge of the disk with a force amounting to only 107! of the 
weight of the disk. The chief constants for this apparatus 
are given in Appendix I. 

The experiment consists in rapidly accelerating the fly- 
wheel A and observing the effect (if any) on B. In order 
to isolate B from the effects of any air currents and vibration, 
two screens, C and D, are interposed between A and B; the 
screen © is supported on a pillar of masonry to which the 
bedplate of the motor is attached, and the screen D is 
supported by a steel girder, 8, fixed in the ground and 
stayed by means of wires from distant walls, which were 
not appreciably affected by vibration. Two sheet-iron 
covers, HE and FE, are placed over B to keep off currents 
of air. The outer cover F is to protect the inner cover 
from currents of air which might cause a difference in 
temperature in parts of the inner cover. 

As might be expected, it was found to be very difficult to 
screen B from accidental disturbances when the flywheel 
was run up to speed. Many months were expended in 
finding out the source of the disturbances and reducing 
their magnitude. One cause of trouble was a very slight 
swirling action of the screen D, which communicated a slight 
swirl to the air inside, which caused B to rotate. To 
eliminate this effect in the later experiments, B was sur- 
rounded by an inner screen, shown at 8in fig.3. This inner 
screen was completely suspended from the top of the building 
by means of three wires. The torsional control upon it was 
185 times greater than the control on B, so that it was hardly 

282 


596 Messrs. Miles Walker and W. Witcomb Stainer on thé 


affected by the swirling action of the air, and it did net 
communicate much force to B, because the natural period 


time was 1/32 of the periodic time of B. 


Fig. 3. 
aL Tr 
K : 
‘wim 
- Fill. 
! % 
eau or 
il, 
i | A J 
| | 
i | 1-2 
| s 
I 


. Suspended disk of porcelain. 
Sheet-iron cover. 

. Mirror attached to disk. 
Bifilar suspension. 


Suspended screen. 
. Glass bottom of screen. 
. Trifilar suspension for screen. 1 


co ~1 90 no St 


3 
il 
4 
Clamp for bifilar suspension. 5. 
6 
0 


oo 


A oY ra 


. Mirror attached to screen. 

. Brass frame to change distance- 
between trifilars. 

. Adjusting screws for trifilar. 


Bracket enabling the screen to 
be built around the disk. 


. Limiting stops. 
. Iron tube protecting the suspen- 


sion. 


The procedure in making the experiment was as follows : 
The disk B was brought as nearly as possible to rest. The 
residual motion was usually a slow swinging motion which 


followed strictly a sine law over one or two scale-divisions. 


] 


Possible Existence of Mutual Induction between Masses. 597 


This was observed for several hours, so that it could be 
accurately plotted and the periodic time noted. 

The flywheel A was then rapidly accelerated (say anti- 
clockwise) and run up to a speed of 2700 revs. per min., so 
as to give an impulse to B, if that were possible. The 
speed was then maintained constant for one-half the natural 
period of the swing of B. ‘The flywheel was then rapidly 
slowed down and the direction of rotation reversed, and the 
speed increased to 2700 revs. per min. (in a clockwise 
direction). This process was repeated a number of times so 
as to induce resonance in B. ~ It will be seen that any action 
on B due to acceleration of A was in phase with B, while any 
action due to velocity of A was 90 deg. out of phase. As the 
time taken to reverse the flywheel occupied only 1? minutes 
(a time small in comparison with the half-period of swing), 
the time-integral of the forces acting on B (if any) might 
be regarded in the nature of an impulse acting on B when 
it was near the centre of its swing. Any change in the 
velocity of B was most easily calculated from the amplitude 
of the swing. From the change in velocity of B we can 
calculate the change in angular momentum Brom, of B. 
This was expressed as a fraction of the total change of the 
angular momentum Amom, of the flywheel A. 

In the early experiments made in 1913, it was found that 
if there was any effect of the kind looked for it was of an 
exceedingly small order, and that the observed movements 
of B were mainly due to accidental disturbing forces. At 


mom. 


this time it was possible to assert that the ratio 
certainly less than 2°3 x 10-8. Bas 

In the later experiments the chief aim was to diminish 
the disturbing forces as tar as possible, so that the negative 
result might be stated with the smallest possible limit of 
error. ‘The introduction of the suspended screen and other 
refinements greatly improved the steadiness of B when A 
was running. 

Fig. 4 gives the result of a typical experiment. The 
horizontal scale gives the deflexion in centimetres as observed 
on a scale at a distance of 6 metres. The vertical scale 
(read from top to bottom) gives the time in minutes. The 
actual deflexion in radians is obtained by multiplying the 
readings by 8°33 x 1074. 

The method of calculating the gain in angular momentum 
by Bis as follows: The amplitude of the swing of B before 
the acceleration of Ais noted. Let this be 6;. Then, after 


was 


598 Messrs. Miles Walker and W. Witcomb Stainer on the 
A has been run up to speed and reversed several times, it is 
found that B is swinging through a bigger angle. Let this 
be @.. Then the gain in angular velocity is 


2arn(O—0,). 


Fig. 4. 


dMinules. 


80 81 82 8 8&4 85 8 87 88 8&9 90 
Centimetres defiexion on a scale at 6 metres. 


This multiplied by the moment of inertia of B, 4:2 10°, 
gives the gain in angular momentum. 


In Fig. 4, 6,=0°17 x 8:33 x 10~* radian 
and G2 Ooi 35 <0 
Therefore 


Bom, = 0°53 X 8°33 x 10-4 x 4-2 x 10°x =" =4°8 rm. em,/sec. 


The flywheel A has undergone the following changes in 
angular momentum :— 


At (1) from 0 to + Mv 
(2) 9 =F Mv to —Mv 
(Bie —Mv to + Mv 
Aye +My» to —Mv 


(yee. —Mzr to + Mv 


Possible Existence of Mutual Induction between Masses. 599 
Therefore the value of Amom.=9 x 1:24 x 10° orm. cm.fec. 
So that the ratio 

Back. 


Thus, by the introduction of the suspended screen and 
other refinements, the steadiness of B while A was running 
was so much improved that we can now state that for the 


==A-8/0 x14 x i= 4:3 «10. 


apparatus shown in fig. 2 the ratio of —““* is less than 


5x 10710, Aanom. 

It will be seen from fig. 4 that the swinging of B is 
getting gradually out of phase with the impulses 3, 4, and 5. 
This is evidence that the increased motion of B is partly due 
to some cause in phase with the velocity of A, while the effect 
looked for should be in phase with the acceleration of A. 


APPENDIX I. 
Steel Flywheel (A) (shown in Fig. 2).—The fly wheel takes 


the form of a solid disk of cast steel 22 in. in diameter and 
41 in. thick. 
The speed of revolution is 2700 revs. per min. 
Angular velocity, 2700 x 27/60 radians per second 
= 2°83 x 10? radians per second. 
Radius of gyration =7r,/ 2/2. 
=11 x 2°54 1:414/2 cm. 
=e OL Cnt: 
Velocity of a point at the radius of gyration 
=19°9 x 2°33 x 10° cm. per sec. 
= 5°65 x 10? cms. per sec. 


Weight of flywheel =480 lb.=220 kg. 

Angular momentum = 5°65 x 10? x 2°20 x 10° grm. per sec. 
= 1:24 x 10°. units. 

The kinetic energy =0°5 x 220 x 10? x (5°65 x 10°)?, orm. 


cm. units. 
=m) < 10)? orm cnr units. 


Porcelain Disk (B) (Fig. 3). 
Weight =10'5 kg. (actual measurement). 


Overall diameter =51 cm. 


600 Notices respecting New Books. 


Rim section: in a radial direetion=7 cm. 
in an axial direction=4 cm. 
Volume of rim =17 x 4x [(51/2)?—(37/2)?] 
= 3°84 x 10° i cm. 
Weight of rim (taking specific gravity of porcelain at 2°4). 
= 2a) opm, 
Moment of inertia of disk about principal axis 
=4°26 x 10° grm. cm.?. 


LXV. Notices respecting New Books. 


Oliver Heaviside. Hlectromagnetic Theory. VolumeIlI. London: 
‘The Electrician’ Printing & Pub. Co., Ltd. Pp. ix+519. 


HIS is the third Volume, impatiently expected by all true 
admirers of Oliver Heaviside, of his unparalleled work on 
Electromagnetic Theory. The present volume has been delayed 
by “not favourable circumstances” for eight years (1904-12). 
Still it was published as long ago as September 1912, and the 
reviewer profoundly regrets that, again through “ circumstances,” 
the present Notice has been delayed over four years. But the 
consolation is that the reviewed volume—as, in fact, every work of 
Heaviside—is never too old. It has the freshness and the life 
of originality, and is, both as regards contents and form, stimu- 
lating beyond saying. Ina very short Preface the author explains 
that he has ‘excluded parts of the third volume and included 
parts of the fourth.” This confession is regrettable, inasmuch as 
it seems to imply that the plan of publishing a fourth volume has, 
for the time at least, been abandoned. 
The present volume is inscribed: 


IN MEMORY OF 
George Francis FitzGerald, F.R.S. 
“ ‘We needs must love the highest when we” know him. 


A large portion of the substance has been reprinted from 
‘The Electrician’ and ‘ Nature,’ one section from ‘ Ency. Brit.’ 
10th Ed., and another from Perry’s Book, 1901; but many pages 
of the volume are now first published. The whole volume consists 
of : Chapter IX., treating of Waves from Moving Sources, Appendices 
J and K, and Chapter X., entitled Waves in the Ether, but con- 
taining such (non-ethereal, but equally interesting) matter as 
“The Teaching of Mathematics” and ‘ Scientific Limitations on 
Human Knowledge,” and, on the other hand, “ Rotation of a 
Rigid Body” and “ Deep Water Waves.” In short, it is what 
one calls Miscellanea. The better so for the intelligent reader. 


Notices respecting New Books. 601 


In Chapter IX. we have, first of all, a number of remarkably 
elegant solutions of problems concerning the “waves from sour ces,” 
electrified points, lines, strips, and planes travelling at any speed. 
Then (pp. 43-61) there is a very natural digression into the 
“Drag of Matter upon Ether” and associated questions. The 
author tries several modifications of the “ circuital equations ” and 
discusses also Lorentz’s equations. Here he exclaims: “ To find 
Lorentz. After profound research I succeeded in discovering.... 
Lorentz’s Versuch einer Theorie, etc. This important application 
ot Maxwell’s theory....ought to have been done in English at 
once to save repetitional labour. Though sad, it is a fact that few 
Britons have any linguistic talent... .. Foreigners, on the other 
hand, seem to be gifted linguists quite naturally.” Heaviside 
forgets that they also have to ‘learn the foreign languages, English 
included, and that it is not an easy task in many eases. Then 
the author proceeds humorously: “ Very well; I would say, let 
them give us poor islanders the benefit of their skill by domg 
all their best work into English. And why not make English the 
international scientific language? It would be all the same to 
the foreigners.” This presupposes the absence, in all other 
nations, of any attachment to their traditional means of expression. 
It is easier to translate such classical things as Lorentz’s ‘Versuch’ 
into English—and, as we know, an English translation of this by 
ne means voluminous work is being prepared by Dr. Andrade. 
“On examination,” Heaviside finds “ that Lorentz’s equations do 
lead to the Fresnel wave-speed,” a property universally known 
these twenty years. Notwithstanding this, many readers will be 
glad to see (pp. 53-56) the way in which Fresnel’s coefficient does 
follow from Lorentz’s equations. ‘he next section is dedicated to 
criticism of Larmor’s equations, and contains some just remarks 
on the lack of clearness in ‘ Ether and Matter.’ The 1ext sections 
are dedicated to the theory of moving electrified cones and the 
force acting upon them and upon an electrified line, of a trans- 
versely moving electrified line segment, and of moving electrified 
hyperboloids. Further, the waves sent out from a growing plane 
source of induction and froma plane strip suddenly started are 
investigated. The problem of electrification moving along a 
straight line gives the author a good opportunity to develop his 
valuable ideas on operational solutions and on their algebrization. 
Here also a very interesting construction of simply periodic wave 
trains from certain electronic steady solutions is given (§ 486). 
§$ 488-500 treat chiefly of the interesting problems of sudden 
motions of an electron in connexion with Réntgen rays, and 
contain illuminating remarks on the peculiarities of the speed of 
light and an impor fant inv estigation of the energy wasted in the 
pulse from a jerked electron and the energy left behind. The 
chapter closes with an investigation of the motion of a charged 
spheroid along its axis. 
The short Appendix J contains anote on the size and inertia of 
electrons, now of historical interest only. 


602 Notices respecting New Books. 


Appendix K, on Vector Analysis, was written as a review of 
Gibbs-Wilson’s well-known treatise. 

Chapter X. contains under the title of Waves in the Ether such 
a multitude of topics, masterly treated and suggestive of new 
problems, that even a mere enumeration of them would exceed 
the limited space of a Philosophical Magazine review. We shall, 
therefore, content ourselves with pointing out that the reader will 
find in that chapter Heaviside’s most important investigations con- 
cerning the radiation from a moving electron, viz. “‘ Elliptic or 
any other Orbit” in § 513 and “ Theory of an Electric Charge in 
Variable Motion” in $514, pp. 432-491, the remainder of this. 
section (pp. 491-498) being dedicated to the interesting problems 
connected with spherical sources of energy, 2. e. seats of oscillating 
impressed forces. 


A System of Physical Chemistry. By Professor W. C. M°C. Lewis, 
Professor of Physical Chemistry in the University of Liverpool. 
Two volumes. Text Books of Physical Chemistry. Edited by 
Sir Wirt1amM Ramsay, K.C.B., F.R.S. London: Longmans, 
Green & Co. 1916. Price 9s. net each. 

THERE is probably no other subject which is so progressive at 

the present time as that of phvsical chemistry. The seed set by 

vau't Hoff and by Willard Gibbs developed and grew, and the 
plant has began to bear fruit on its many branches. The works 
of Gibbs were of rather too abstruse a character to be easily 
assimilated. Those of Nernst are alittle too lax in their treatment 
tobe quite satisfactory, though they represent the activities of a 
very brilliant investigator. Small books there are in plenty which 
deal more or less satisfactorily with the elementary side ot the 
subject. But there has up to the present been no book which 
deals with the matter in an absolutely satisfactory way. The 
present two volumes in a well-known series promise to fill the gap. 

Professor Lewis has endeavoured to give a sufficiently complete 

account both on the experimental and the thermodynamic side, 

and we do not hesitate to say that he has succeeded. 

The first volume deals with the kinetic theory with chief refer- 
ence to the laws of chemical equilibrium and reaction velocity 
including catalysis (positive and negative). This is preceded, 
amongst other things, by an exceptionally good account of the 
various characteristic equations which have been suggested for 
fluids. 

It is in the second volume that the unique value of the book 
will be perceived. The theorems of thermodynamics are not all 
simple, and in chemical books in particular they have often been 
badly slurred over. But here it is otherwise. In one or two. 
particulars we might have put the matter otherwise. For example, 
does not the first law of thermodynamics state more than that 
“‘ there is always a definite quantitative relationship between the 
heat that has disappeared as such and the work which has been 


Intelligence and Miscellaneous Articles. 603 


done”? As understood at the present day it asserts what this 
quantitative relationship is. 

Professor Lewis obviously understands the mathematics which 
he employs. His exposition of the meaning of a certain partial 
differential coefficient in the footnote on p. 67 is a model of 
lucidity, and will be of great use to those who are less familiar 
with the matter than he is himself. 

The osmotic pressure of concentrated solutions is very fully 
considered in its relation to other physical constants in Chap. VIII, 
while in Chap. X. there is a full account of chemical equilibrium 
(from the thermodynamic standpoint) when capillary or electrical 
effects are of importance; including adsorption and Donnan’s 
theory of membrane equilibrium. There is still a great deal more 
to be said about these latter subjects, but the account given is an 
excellent digest of what is at present known. 

Ja Chapter XII. Nernst’s heat theorem is considered. Part III. 
is devoted to considerations based upon thermodynamics and 
statistical mechanics, including applications of the unitary theory 
of energy (energy quanta) to physical and chemical problems. This. 
part is of very great interest. 

We have noticed no mistakes in typography. The references 
are very complete. The illustrations are good. Thereisa subject 
and an authors index which are well compiled. Professor Lewis 
is to be congratulated in all respects. 


ey Ti, Intelligence and Miscellaneous Articles. 


REFRACTION OF X-RADIATION. 


To the iditors of the Philosophical Magazine. 
GENTLEMEN ,— 


(oes experiments carried out by Professor Earkla to detect the 

refraction of X-rays (Phil. Mag. April 1916) have led to 
results which show that the refractive index of potassium 
bromide for X-radiation (A="5x10-§ cm.) is between -$99995 
and 1:000005. 

In the year 1914 I carried out a series of experiments in the 
Physical Laboratory of the University of Birmingham, in which 
it was attempted to refract a beam of X-radiation by passing 
it through a system of prisms. The refracting materials used for 
constructing the prisms were lead, aluminium, and sulphur. The 
two former were carefully shaped ona milling machine from solid 
blocks of material, while the sulphur prisms were prepared by 
casting, using the metallic prisms as moulds. No attempt was 
made to use optically worked surfaces. Owing to the acquisition 
of the University buildings by the War Office for the purposes of 
a military hospital, these experiments were discontinued on the 


604 — Intelligence and Miscellaneous Articles. 


outbreak of war. Up to that time I had not obtained any evidence 
of refraction. It may be of interest to indicate briefly the method 
of procedure adopted in these experiments, and also the con- 
clusions which were drawn regarding the order of magnitude of 
the possible refractive index. 

The general arrangement of apparatus was very similar to that 
adopted by Professor Barkla. By an arrangement of lead slits an 
exceeding narrow beam of X-radiation gave rise to a fine hori- 
zontal line about two or three centimetres long on a photographie 
plate placed at about one metre beyond the levelling-table carrying 
the prisms. In the path of the beam was placed a train of eight 
prisms each. of 90° refracting angle with their refracting edges 
horizontal and perpendicular to the beam. By the side of these 
prisms was placed a similar set with their refracting angles 
reversed. Any evidence of refraction would be recorded by a 
lack of alignment of the two halves of the line image obtained 
on the plate after several hours’ exposure. No such evidence 
was obtained. A simple calculation showed that the refraction of 
X-radiation, if any, in these materials would correspond to a 
refractive index lying between 1°000005 and :999995. | 

H. B. Kzznz, D.Sce., 


University of Birmingham, Assistant Lecturer in Physics. 
Nov. 11, 1916. 


ADDITIONS TO PROF H. 8S. CARSLAW’S PAPER ON 
NAPIER’S LOGARITHMS IN THE NOVEMBER NUMBER. 


The following footnotes should be added :— 


Page 479. ‘To the sentence forming the third and second lines 
from the bottom of the page: When 7r=7, nl, 1=161180896-38 
(cf. Constructio, Section 53): add the following footnote :— 


The error in Napier’s Second Table affects the accuracy of his Canon, 
and this number should be 161180956'51. The correction can be made 
from the corrected result given by Macdonald in his English translation 
of the Constructio, pp. 94-5, for it is not difficult to show that 
mil ==" 10°. 


Page 480. To the sixth line: and nl; 1—nl; 10=23025842°34 
(cf. Constructio, Section 53): add the following footnote :-— 


This number should be 23025850:93, since it is easy to show that 
nl, 1—nl, 10=nl, 10°, and Macdonald gives the corrected logarithm of 
10° (doe. cit. pp. 94-5). 


J. Mag. Ser. 6, Vol. 32, Pl. XIII. 


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Phil. Mag. Ser. 6, Vol. 32, Pl. XIV. 
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INDEX “ro VOL. XXX. 77 


ACETYLENE, on residual ioniza- 
tion in, 396. 

Aerial waves generated by impact, 
on, 96. 

Aeroplanes, on the equilibrium of the 
compass in, 461. 

Airey (Dr. J. R.) on the roots of 
Bessel and Neumann functions of 
hieh order, 7; on Bessel functions 
of equal order and argument, 237. 

Alkali sulphates, on the crystalline 
structure of the, 518. 

Alpha particles, on the tracks of the, 
from radium A in photographic 
films, 129; on the straggling of, 
222. 

Aluminium, on‘the Hall and Corbino 
effects in, 3038. 

Anatase, on the crystalline structure 
of, 505. ; 

Anderson (Prof. A.) on the mutual 
maenetic energy of two moving 
point charges, 190. 

Antimony, on the Halland Corbino 
effects in, 803; on the high-fre- 
quency spectrnm of, 497. 

Ayrsenic, on the high-frequency spec- 
trum of, 497. 

Atmospheric electrical quantities, on 
the diurnal variation of, 282. 

Banerji (S8.) on aerial waves gene- 
rated by impact, 96. 

Barium, on the high-frequency spec- 
trum of, 497. 

Bars, on vibrations and deflexions of, 
353. 


Bazzoni (Prof. C. B.) on experiments ri 


with electron currents in different 


gases, 426; on the ionization po- 


tential of helium, 566. 

Beam, on the streneth of the thin- 
plate, 172. 

Bessel functions, on the roots of, of 
high order, 7; on, of equal order 
and argument, 232, 237. 

Berkeley (the Earl of) on a semi- 
automatic high-pressure installa- 
tion, 153. 

Biggs (H. F.) on the decrease in the 
paramagnetism of palladium caused. 
by absorbed hydrogen, 131. 

Bismuth, on the high - frequency 
spectrum of, 39; on the Hall and 
Corbino effects in, 303. 

Boiling-points of homologous com- 
pounds, on the, 371. 

Books, new:—Barton’s Introduction 
to the Mechanics of Fluids, 261; 
Richardson and Landis’ Funda- 
mental Conceptions of Modern 
Mathematics, 262; Richardson 
and Landis’ Numbers, Variables, 
and Mr. Russell’s Philosophy, 262; 
Chree’s Studies in Terrestrial Mag- 
netism, 345; Ince’s Course in De- 
scriptive Geometry, 346; Gibb’s 
Course in Interpolation and Nume- 
rical Integration, 347; Conway’s 
Relativity, 347 ; Carse & Shearer’s 
Course in Fourier Analysis and 
Periodogram Analysis, 3847: Bell’s 
Course in the Solution of Sphe- 


- 606 


rical Triangles, 348 ; Ford’s Intro- 
duction to the Theory of Axuto- 
morphic Functions, 348; Bulletin 
of the Bureau cf Standards, 348; 
Miller's The Science of Musical 
Sounds, 350; Whittaker & Wat- 
son’s Course of Modern Analysis, 
351; Carmichael’s Diophantine 
Analysis, 852; Robb’s Theory of 
Time and Space, 526; Heaviside’s 
Electromagnetic Theory, vol. iii., 
690; Lewis's A System of Phy- 
sical Chemistry, 602. 

Brodetsky (Prof. 8.) on the absorp- 
tion of gases in vacuum-tubes, 
239. 

Bromine, on the high-frequency spec- 
trum of, 497. 

Burton (Dr. C. V.) on a semi-auto- 
matic high-pressure installation, 
153. 

Cadmium, on the Hall and Corbino 
effects in, 303; on the high-fre- 
quency spectrum of, 497. 

vapour, on the condensation 
and reflexion of, 564. 

Cesium, on the high-frequency spec- 
trum of, 497. 

Carbon, on the spectrum of, 546. 

Carman (A. P.) on the collapse of 
short thin tubes, 559. 


Carslaw (Prof. H. 8.) on Napier’s 


logarithms, 476, 604. 

Cathode-ray tube magnetometer, on 
a Wehnelt, 381. 

Cathode rays, on the velocity of 
secondary, 202. 

@erium, on the high-frequency spec- 
trum of, 497. 

Chapman (A. K.) on the Hall and 
Corbino effects, 503. 

Cheshire (R. W.) on a new method 


of measuring the refractive index - 


and dispersion of glass, 409. __ 
Cobalt, on the Hall and Corbino 
effects in, 303. 

Collapse of tubes, on the, 559. 
Compass, on the equilibrium of the, 
in aeroplanes, 461. Bc 
Compton (Prof. K. T.) on diffusion 

cells in ionized gases, 499. — 
Convection currents in a horizontal 
layer of fluid, on, 529. 
Copper, on the Hall and Corbino 
effects in, 305. 
Corbino and Hall effects, on the, 303. 


INDEX. 


Croxson (C.) on the Starr effect of 
the 4686 spectrum line, 327. 

Crystal analysis, results of, 65, 505. 

Crystalline media, on image forma- 
tion by, 248. 

Crystallographic law of valency vo- 
lumes, on a critical test of the, 
518. 

Das (A. B.) on electric discharge in a 
transverse magnetic field, 50. 

Dawes (Prof. H. F.) on image for- 
mation by crystalline media, 248. 

Dielectric, on the mechanical rela- 
tions of, and magnetic polarization, 
162. 

Dielectric constant of mica in intense 
fields, on the, 112. 

strength, on the cause of 
lowered, in high-frequency fields, 
242. 

Dieterici’s equation, on, 295. 

Diffusion cells in ionized gases, on, 
499, 

Dispersion of glass, on a new method 
of measuring the, 409. 

Dysprosium, oun the high-frequency 
spectrum of, 497. 

Earth’s crust, on the compression of, 
in cooling, 576. 

Elasticity, on two fundamental pro- 
blems in the theory of, 15. 

Electric absorption of gases. in 
vacuum-tubes, on the, 239. 

conductivity of mica, on the, 
112. 

— — discharge in a transverse mag- 
netic field, on the, 59. 

field, on the cause of lowered 

dielectric strength in a high-fre- 

quency, 242; on the effect of an, 

on the 4686 spectrum line, 327. 

quantities, on the diurnal yva- 

riation of, 282. 

resistance, on the change in 
the, of sputtered films after depo- 
sition, 141. 

Electrodynamics, on the principle of 
least action iu the theory of, 195. 
Electron currents in different gases, 

experiments with, 426. 

Erbium, on the high-frequency 
spectrum of, 497. 

Kuropium, on the high-frequency 
spectrum of, 497. 

Evans (Dr. E. J.) on the Stark effect 
of the 4686 spectrum line, 327. 


INDEX. 607 


Fluid, on the flow of compressible, 
past an obstacle, 1. 

——, on convection currents in a 
horizontal layer of, 529. 

Fluorescent vapours and their mag- 
neto-optic properties, on, 265. 

Friman (Dr. E.) on the high-fre- 
quency spectra of the elements 
tantalum—uranium, 39; on the X- 
ray vacuum spectrograph, 494 ; 
on the high-frequency spectra of 
the elements lutetium-zinc, 497. 

Gadolinium, on the high-frequency 
spectrum of, 497. 

Gases, on the discharge of, under 
high pressures, 177; on the ve- 
locity of secondary cathode rays 
emitted by, under the action of 
hich-speed cathode rays, 202; on 
the absorption of, in vacuum- 
tubes, 239; on scattering and re- 
gular reflexion of light by absorb- 
ing, 8329; on the condensation 
and reflexion of molecules of, 364 ; 
onexperiments with electron cur- 
rents in different, 426; on diffu- 
sion cells in ionized, 409; on the 
mobility of the negative ion in, 
44], 

Geological Society, proceedings of 
the, 175, 526. 

Glass, on a new method of mea- 
suring the refractive index and 
dispersion of, 409. 

Gold, on the high-frequency spec- 
trum of, 39; on the crystalline 
structure of, 66; on the Hall and 
Corbino effects in, 303. 

Green (Dr. G.) on a method of de- 
riving Planck’s law of radiation, 
229, 

Hall effect, on the, 200, 303. 

Helium, on the ionization potential 
of, 566. 

High-pressure installation, on a 
semi-automatic, 153. 

Hobbs (Miss EK. W.) on the change 
in resistance of a sputtered film 
after deposition, 141. 

Hodgson (Dr. B.) on the absorption 
of gases in vacuum-tubes, 239. 

Holmes(A.) on the tertiary volcanic 
rocks of Mozambique, 526. 

Homologous compounds, on the 
boiling-points of, 371. 

Hopwood (F. .L1.) on the crystalline 


structure of the alkali sulphates, 
518. 

Hydrocarbons, on the spectra of, 546. 

Hydrogen, on the decrease in the 
paramagnetism of palladium caused 
by absorbed, 131. 

Tkeuti (H.) on the tracks of the alpha 
particles from radium A in photo- 
oraphic films, 129. 

Image formation by crystalline 
media, on, 248. 

Indium, on the high-frequency 
spectrum of, 497. 

Induction, on the existence of mutual, 
between masses, 592. 

Iodine, on the high-frequency spec- 
trum of, 497. 

Ton, on the mobility of the negative, 
44]. 

Ionization, on residual, 396. 

—— potential of helium, on the, 566. 

Tonized gases, on diffusion cells in, 
499. 

Iridium, on the high-frequency spec- 
trum of, 39. 

Tron, on the Hall and Corbino effects 
in, 3038. 

Ishino (M.) on the velocity of secon- 
dary cathode rays emitted by a 
gas under the action of high-speed 
cathode rays, 202. 

Jeffreys (H.) on the compression of 
the earth’s crust in cooling, 575. 
Keene (Dr. H. B.) on the refraction 

of X-radiation, 603. 

Kimura (M.) on the scattering and 
regular reflexion of Lght by an 
absorbing gas, 329. 

Kingdon (K. H.) on some experi- 
ments on residual ionization, 396. 

Knipp (Prof. C. T.) on a Wehnelt 
cathode-ray tnbe magnetometer, 
381. 

Lanthanum, on the higb-frequency 
spectrum of, 497. 

Laws (B. C.) on the strength of the 
thin-plate beam, 172. 

Lead, on the high-frequency spec- 
trum of, 39; on the crystalline 
structure of, 66. 

Leest action, on the principle of, in 
the theory of electrodynamics, 
195. 

Light, on the scattering and iregular 
reflexion of, by an absorbng gas, 


329. 


608 


Liquid mixtures, on the vapour- 
pressures of binary, 295. 

Livens (G. H.) on the mechanical 
relations of dielectric and mag- 
netic polarization, 162; on the 
principle of least action in the 
theory of electrodynamics, 195; 
on the Hall effect and allied 
phenomena, 200. 

Logarithms, on Napier’s, 476, 604. 

de Long (O. A.) on diffusion cells in 
ionized gases, 499, 

Lutetium, on the high-frequency 
spectrum of, 497. 

Magnetic energy, on the mutual, of 
two moving point charges, 190. 
compass, on the equilibrium 

of the, in aeroplanes, 461. 

field, on electric discharge in a 
transverse, 50, 

—— polarization, on the mechanical 
relations of dielectric and, 162. 
Maenetometer, on a Wehnelt 

cathode-ray tube, 381. 

Magneto-optic properties of fluor- 
escent vapours, on the, 265. 

Makower (Dr. \V.) on the straggling 
of alpha particles, 222; on the 
efficiency of recoil of radium D 
from radium C, 226. 


Mallik (Prof. D. N.) on the electric. 


discharge in a transverse magnetic 
field, 50. 

Masses, on the existence of mutual 
induction between, 592. 

Membranes, on vibrations and de- 
flexions of, 358. 

Mennell (F. P.) on the geology of 
the northern margin of Dart- 
moor, 528. 

Mercury, on the high - frequency 
spectrum of, 39. 

vapour, experiments with elee- 
tron currents in, 426. 

Metallic films, on the change in the 
resistance of, after deposition, 
141. 

Mica, on the dielectric constant and 
electrical conductivity cf, 112. 
Mirrors, on the reflexion from plane, 

487. 

Molybdenum, on the high-frequency 
spectrum of, 497. 

Multiple reflexion, on, 487. 

Musical instruments, on the wolf- 
note in stringed, 391. 


INDEX. 


Napier’s logarithms, on, 476, 604. 

Negative ion, on the mobility of the, 
44], 

Neodymium, cn the high-frequency 
spectrum of, 497. 

Nichols (EK. H.) on the diurnal 
variation of atmospheric elec- 
trical quantities, 282. 

Nickel, on the Hall and Corbino 
effects in, 808. 

Niobium, on the high - frequency 
spectrum of, 497. 

Oge (Prof. A.) on the crystalline 
structure of the alkali sulphates, 
518, 

Osmium, on the high - frequency 
spectrum of, 39. 

Palladium, on the decrease in the 
pavamagnetism of, caused by ab- 
sorbed hydrogen, 1381; on the 
high-frequency spectrum of, 497. 

Paramagnetism of palladium, on 
the decrease in the, caused by 
absorbed hydrogen, 131. 

Photoelectric effect on thin films of 
platinum, on the, 421. 

Photographic films, on the tracks of 
alpha particles in, 129. 

Planck’s law of radiation, on a 
method of deriving, 229. 

Plates, on vibrations and deflexion 
of, 353. 

Platinum, on the high - frequency 
spectrum of, 39; on the Hall 
and Corbino effects iz, 303; on 
the photoelectric effect on thin 
films of, 421. 

Plummer (Prof. H. + C.) on the 
boiling-points of homologous com- 
pounds, 371. 

Point-charges, on the mutual mag- 
netic energy of two moving, 190. 
Polarization, on the mechanical re- 

lations of dielectric and, 162. 

Polonium, on the high-frequency 
spectrum of, 39. 

Poole (H. H.) on the dielectric 
constant and electrical conduc- 
tivity of mica in intense fields, 
112. 

Praseodymium, on the high-fre- 
quency spectrum of, 497. 

Radiation, on a method of deriving 
Planck’s law of, 229. 

Radium, on the high - frequency 
spectrum of, 39. 


TN DEX: 


Radium A, on the tracks of the alpha 
particles from, in photographic 
films, 129. 

D, on the efficiency of recoil 
of, from radium C, 226. 

Raffety (C. W.) on the spectra of 
carbon and hydrocarbon, 546. 

Raman (Prof. C. V.) on the wolf- 
note in bowed stringed instru- 
ments, 391. 

Ratner (S.) on the mobility of the 
negative ion, 441. 

Rayleigh (Lord) on the flow of 
compressible fluid past an ob- 
stacle, 1; on the discharge of 
gases under high pressures, 177 ; 
on the energy acquired by small 
resonators from incident waves 
of like period, 188; on vibrations 
and deflexions of membranes, 
bars, and plates, 3538; on con- 
vection currents in a horizontal 
layer of fluid, when the higher 
temperature is on the under side, 
529. 

Reflexion, on multiple, 487. 

Refraction of X-radiation, on the, 
603. 

Refractive index of glass, on a new 
method of measuring the, 409. 

Resonance radiation of mercury 
vapour, on the, 329. 

—— spectrum of fluorescent vapours, 
on the, 265. 

Resonators, on the energy acquired 
by small, from incident waves of 
like period, 188. 

Rhodium, on the high-frequency 
spectrum of, 497. 

Richardson (Prof. O. W.) on experi- 
ments with electron currents in 
different gases, 426. 

Robinson (Dr. J.) on the photo- 
electric effect on thin films of 
platinum, 421. 

Rubidium, on the high-frequency 
spectrum of, 497. 

Ruthenium, on the high-frequency 
spectrum of, 497. 

Samarium, on the high-frequency 
spectrum of, 497. 

Siegbahn (Dr. M.) on the high- 
frequency spectra of the elements 
tantalum-uranium, 39; on an X- 
ray vacuum spectrograph, 494. 

Silberstein (Dr. L.) on fluorescent 
vapours and their magneto-optic 


Phil. Mag. 8. 6. Vol. 32. No. 192. Dee. 1916. 


609 


properties, 265; on multiple re- 
flexion, 487. 

Silver, on the Hall and Corbino 
effects in, 303; on the high- 
frequency spectrum of, 497. 

Spectra, on the high-frequency, of 
the elements tantalum—uranium, 
39; on the high-frequency, of the 
elements lutetium—zinc, 497, 

Spectrograph, on an X-ray vacuum, 
494. 

Spectrum line, on the Stark effect 
of the 4686, 327. 

Stainer (W. W.) on the existence 
of mutual induction between 
masses, 992. 

Starling (S. G.) on the equilibrium 
of the magnetic compass in aero- 
planes, 461. 

Strahan (Dr. A.) on cores from 
borings in Kent, 527. 

Strontium, on the high-frequency 
spectrum of, 497. 

Swann spectrum, on the, 546. 

Tantalum, on the high-frequency 
spectrum of, 39. 

Tellurium, on the high-frequency 
spectrum of, 497. 

Terbium, on the high - frequency 
‘spectrum of, 497. 

Thallium, on the high-frequency 
spectrum of, 39. 

Thin-plate beam, on the strength 
of the, 172. 

Thorium, on the high - frequency 
spectrum of, 39. 

Thornton (Prof. W. M.) on the 
cause of lowered dielectric 
strength in high-frequency fields, 
242. 

Tin, on the high-frequency spectrum 
of, 497 ; on the Hall and Corbino 
effects in alloys of, 309. 

Tinker (F.) on the vapour-pressures 
of binary liquid mixtures, 295. 

Tubes, on the collapse of short thin, 
559. 

Tungsten, on the high-frequency 
spectrum of, 39, 

Tyrrell (G. W.) on the picrite- 
teschenite sill of Lugar, 175. 

Uranium, on the high-frequency 
spectrum of, 39. 

Vacuum-tubes, on the absorption 
of gases in, 239. 

Valency volumes, on the crystallo- 
eraphic law of, 518. 


2T 


610 


Vapour-pressures of binary liquid 
mixtures, on the, 290. 
Vapours, on the magneto-optic pro- 

perties of fluorescent, 265. 

Vegard (Prof. L.) on results of 
crystal analysis, 65, 505; on the 
electric absorption of gases in 
vacuum-tubes, 239. 

Vibrations of membranes, bars, and 
plates, on, 353. 

Violoncello, on the wolf-note in the, 
391, 

Walker (Prof. M.) on the existence 
of mutual induction between 
masses, 592. 

Watson (Prof. G. N.) on Bessel 
functions of equal order and 
argument, 232. 

Waves, on aerial, generated by 
impact, 96. 

Weatherburn (Dr. C. E.) on two 
fundamental problems in _ the 
theory of elasticity, 15. 

Wehnelt cathode-ray tube magneto- 
meter, on a, 381. 

Welo (L. A.) on a Wehnelt cathode- 
ray tube magnetometer, 381. 


END OF THE THIRTY-SECOND VOLUME. 


INDEX. 


Wolf-note in bowed string instru- 
meuts, on the, 391. 

Wood (E. 8.) on diffusion cells in 
ionized gases, 499. 

Wood (Prof. R. W.) on scattering 
and regular reflexion of light by 
an absorbing gas, 329; on the 
condensation and reflexion of gas 
molecules, 364. 

X-radiation, on the refraction of, 
608. 

X-ray vacuum spectrograph, on an, 
494. 

Xenotime, on the crystalline struc- 
ture of, 505. 

Ytterbium, on the high-frequency 
spectrum of, 497. 

Yttrium, on the high - frequency 
spectrum of, 497. 

Zinc, on the Hall and Corbino 
effects in, 303; on the high- 
frequency spectrum of, 497. 

Zircon group, on the crystalline 
structure of the, 68. 

Zirconium, on the high-frequency 
spectrum of, 497. 


Ss 


SMITHSONIAN INSTITUTION LIBRARIES 


Mt 


|—hUr 


12 


2 4980