Skip to main content

Full text of "The London, Edinburgh and Dublin philosophical magazine and journal of science"

See other formats 


retrny feat 


Sh eerneve AS thers thy: ne 
OUP Lp f PR ere ee Mal et woe nahaned eh uy 
Or a Tah peat al were ein 

D: Hit capa! vit ewes 

vor ye MNT ott ae 

edt 4 

UNA a 7th wrheradereih WH ecpsantlane 
wre eel tale a ' ‘ by tists vane el) 

| | 


m9 PG wie y 
wie le ly tet rey te te 

ONY were nal he ten 
POPE Pee ET Ora rte waked ar dh ae eR pr a8 OF 
Te el eg) ag Vie taint edie) a uihedsy evi bots 
AS ie a Mh HNP ad ath beads a, 
i Aah Sh, Ni a ehaeeh bf 0 


aft 


¢ 
pb wegt see eth wale 
Pee bs the ald ont 5 hk 
; Paced Seep Pe melee been ¢ 
fae hale ay Wad We dF ee ee be me 
Ty ore ' Cty bee ete Palle eererri er: 
4 ee eid hah el ene 
Vie f tht me) 
bahia ba EDS 
pi ek pet hee wee 4) 
Ths sha PL) 
Leone TOLL Uta TC ELE 
Ute rt ered y wit R a tosye Teor 
18 uf S0@ ah ated of Monee » 


PR ydtrver! ate’ 


vy yeu ada 


; Veter ; 
Pert erat ATE PASE 


fe Maisity 
bi) Pec kesh 
erste 


haath Ay hei 


ag Md 
. Héshiqiduy 
HL Be a Lh oe te 
aie di Midis opedad Yyhihehs bral PP aadiceess) 
st it ay HiFi 4b Mate Sorel Pee 

hi Gad yh 7 ? 


eld 


4tedalie 


Cody 
Peevey us a 


; Wider cabl ie 
i! bea yO 
eat) ve 4 i) 
“ PAD EH BET hjsaal en 
‘ ity a dis 


HAS abt 
a . ' 
Pa ant a | 

nas) 8) ¢ 


t a fate 
+ \ wle ht taons iy ect 
ED aes f f : its WEEP ates {Anoeus 


Int OTe y 9/emianye Rees oa 


bay lay 


Ber te rita) iy 
hdl Mell le wt 


% 
*“ }" uy J 
te ae als 
‘ ahi why * Wie 
Abt eee i lea? heey 
eh ‘ \ f 1 ee Web atey 
Miah ve a1) a ‘ ‘ Sy) phphe Wi 
Osi) fab i! Hui ’ TT oo ahi ha 
i { iit 
i an Le | i fh} 


ta 


ee 
GAO PS ap 
1 rie thi titenah pt belie ‘ 
ie shee lh 
sWaePegeattas, Da if Hue 
Let bb dat ry 
ld wae Mini i Hat t 1) 
sina The euaal 
hey Meeig Nt Tate Ye saan tabre 
i AS itES La HAs tube 
Ys Sp pe Whebad) eke 
WUE Bags ea vree 


a a hits i 


Unda, oe snld 
hte M4 
hau te Derby t8 (it Lideetnhye Porria Le Re 
it) ra tan pu a tna d ye 
darnuitn dab ve y MRA ddd re 
, p Mth Hn iS it} tee 
pip. we ebaieaad sper heed 
nabbed e 


i 
see oub hie 
Be Tare Rey) 
ot wiett 4 ‘ 
He 


Q 


& 
S 


SRENORIQLINRINS 


% SCIENTIFIC LIBRARY 


RYIRINIOIR NINO OO CAIRO ONIONIN 


E 


MMamammasawMa~Ma~waKw 


UNITED STATES PATENT OFFIC 


») 


Samar 
11—8625 


ca 
GOVERNISENT PRINTING OFFICE 


wa 


4 


THE 
LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 
Fg 


CONDUCTED BY 


JOHN JOLY, M.A. D.Sc. F.R.S. F.G.S. 
AND 


WILLIAM FRANCIS, F.L.S. 


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


VOL. XV.—SIXTH SERIES. 
JANUARY—JUNE 1908. 


55393 


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


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


S80 { 


perspicua .... Admiratio generat queestionem, questio investig 
or investigatio inventionem.”—Hugo de S. Victore. a 


“ 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 celo, 
Quo micet igne lis, superos quis conciat orbes 
Tam vario motu.” | we 
dB. Pinelli ad Me 


CONTENTS OF VOL. XV. 


(SIXTH SERIES). 


NUMBER LXXXV.—JANUARY 1908. 


Dr. D. F. Comstock on the Relation of Mass to Energy .... 
Messrs. A. C. and A. E. Jessup on the Evolution and Devo- 


lution of the Elements. (Plate VII.):................. 21 
Mr. L. V. Meadowcroft on the Curvature and Torsion of a 
Helix on any Cylinder, and on a Surface of Revolution .. 55 
Prof. D. N. Mallik on a Potential Problem .............. 63 
Prof. D. N. Mallik: Experimental Determination of Magnetic 
Induction in an Elongated Spheroid .................. 65 
Prof. E. H. Barton on Spherical Radiation and Vibrations in 
Seeremerserere oe... TR ts. ait 69 
Mr. W. Ellis Wiliams on the Rupture of Materials under 
Combined Stress :—Tension and Hydrostatic Pressure.... 81 
Messrs. K. Honda, T. Terada, and D. Isitani on the Secondary 
Undulations of Oceanic Tides. (Plates 1-VL.) ........ 8&8 
Mr. J. D. Hamilton Dickson on the Joule-Kelvin Inversion 
Temperature, and Olszewski’s Experiment. (Plate VIII.) 126 
Dr. C. VY. Burton on the Thermally excited Vibrations of an 
Penne Eye casly. ei SU 08 20 ee 4g. Oi 48 147 
Mr. A. Campbell on the Use of Variable Mutual Inductances. 
epee ee he ean alk Ue ee le kD), 155 
Mr. G. A. Schott on the Electron Theory of Matter and the 
Explanation of Magnetic Properties .................. 172 
Mr. J. A. Tomkins on the Focometry of a Concave Lens .. 198 
Notices respecting New Books :— 
M. C. Chéneveau’s Recherches sur les Propriétés Optiques 
des Solutions et des Corps Dissous................ 200 
res Ol Retetiacth aos Soc! fas gga ee aeiesisree a2 ok 200 
Dr. A. Seidell’s Solubilities of Inorganic and Organic 
SULT SESE RRR Se aR Pe trae cae Iv ee va 201 


1V CONTENTS OF VOL. XV.—SIXTH SERIES. 
Page 
Proceedings of the Geological Society :— 
Prof. F. Dawson Adams on the Structure and Relations 
of the Laurentian System of Canada .............. 201 
Rev. W. Howchin on Glacial Beds of Cambrian Age in 
pouth Australia... cie<s4........05 202 


Messrs. H. Basedow and J. D. Iliffe on a Formation 
known as ‘Glacial Beds of Cambrian Age ” in South 
Australia ¢.......-.onee ies... ss. 203 

Intelligence and Miscellaneous Articles :— 

A Theory of the Displacement of Spectral Lines pro- 
duced by Pressure—A Correction ................ 204 

doord ‘Kelvans 6... eens oy... 204 


NUMBER LXXXVI.—FEBRUARY. 


Messrs. C. E. Mendenhall and L. R. Ingersoll on certain Phe- 
nomena exhibited by Small Particles on a Nernst Glower.. 205 
Prof. EK. L. Hancock on the Effect of Combined Stresses on 


the Elastic Properties of Steel. (Plates X. & XI.)....... 214 
Mr. G. A. Miller on the Groups of Isomorphisms of the 

Groups whose Degree is less than Hight .............. 223 
Mr. Andrew Stephenson on Induced Stability ....... . 233 


Mr. L. F. Richardson on a Freehand Graphic way of deter- 
mining Stream Lines and Equipotentials (Plate XII.) .. 237 
Prof. R. W. Wood on Anomalous Magnetic Rotatory Dis- 


persion of Neodymium. (Plate'X@I)E yy.) )... 2. coe 270 
Prof. R. W. Wood on the Existence of Positive Electrons in 

the Sodium Atom... (Plate XIV) 32.5.>...... .- see 274 
Dr. H. Wilde on the Atomic Weight of Radium .......... 280 
Dr. R. A. Houstoun on a New Spectrophotometer of the 

Hiifner Type. .s6 cigs ce eee ap lo 282 
Dr. C. G. Barkla on X-Rays and Scattered X-Rays........ 288 
Prof. W. McF. Orr on the Mixing of Gases.—Remarks on 

Mr, Burbary’s recent Papers.) d-e see eee ele ee eee 297 


Intelligence and Miscellaneous Articles :— 


On the Focometry of a Concave Lens, by J. A. Tomkins. 300 — 


NUMBER LXXXVII.—MARCH. 
Mr. J. Rose-Innes on the Practical Attainment of the Thermo- 


dynamic Sceale:orTemperature 205000... ee 30 
Prof. A. H. Bucherer on the Principle of Relativity and on 
the Electromagnetic Mass of the Electron.............. 316 


Mr. T. J. Bowlker on the Factors serving to determine the 
Direction of Sound « 2. 02.0 ccee A re ts os A 318 


>" eeaisie ‘ 


CONTENTS OF VOL. XV.——-SIXTH SERIES. V 


: Page 

Dr. T. H. Havelock on certain Bessel Integrals and the Co- 
efficients of Mutual Induction of Coaxial Coils... ...... 332 
Lord Rayleigh on the Effect of a Prism on Newton’s Rings. 345 
Mr. A. Stephenson on Mechanical Phosphorescence........ 302 
Prof. D. N. Mallik on Mutual Induction ..... sige Nasa ke 364 
Prof. E. Goldstein on the Canal-Ray Group.............. 372 
Prof. J. Joly on the Radioactivity of Sea-Water .......... 385 

Notices respecting New Books :-— 

Lerhbuch der Gerichtlichen Chemie ................ 393 
MeN. Clark's Mhemrolarity of Matter... 5.0... 05... 394 

M. EH. Fischer’s Guide de Préparations Organiques a 
WUIsaeerecmGMOUIANUS |. 1... Lic epee cto keys oe ae Hye es 394 
Moritz von Rohr’s Die Binokularen Instrumente ...... 394 
marmcnmempomr FAT VIOS8 0. oe pe ni ee wee a 395 
The Science Year Book and Diary for 1908 .......... 395 

Prof. M. Brillouin’s Lecons sur la Viscosité des Liquides 
5S OE ARI em BEES ee lee, 2.000 a me ma me NEC 395 
Golletin of the Bureau of Standards .:.............. 396 

- Intelligence and Miscellaneous Articles :— 

On the Evolution and Devolution of the Elements, by 

MPO Nee se olay MRE EEL cao Sag away rane) au oy 4 2th 396 


NUMBER LXXXVIII.—APRIL. 


Lord Kelvin on the Formation of Concrete Matter from 
LASDLIG (QS 0 10) Nae aoe ae dn PRP eae 397 

Dr. G. Bakker on the Theory of Surface Forces.—III. The 
Physical Meaning of the Unstable Part of the Isotherm of 


SRM EET OSRS ON esa. eas aw chl= ape’ < ales abn ops ado gn emp cree 413 
Prof. H. A. Bumstead on the Heating Effects produced by 
feaareew hays in Lead and Zine.......0 40. .-...--. 065. 432 


Mr. G. A. Schott on the Frequencies of the Free Vibrations 
of Quasi-permanent Systems of Electrons, and on the 


Explanation of Spectrum Lines.—Part I. .............. 438 
Prof. Percival Lowell on the Tores of Saturn ............ 468 
Messrs. 8. W. J. Smith and H. Moss on the Contact Potential 

Differences determined by means of Null Solutions ...... 478 
Dr. J. Morrow on the Lateral Vibration of Bars supported 

at two Points with One End overhanging .............. 497 


Mr. W. C. M. Lewis on an Experimental Examination of 
Gibbs’s Theory of Surface-Concentration, regarded as the 
Basis of — with an Application to the Theory of 
MOT... RE RAMA AE I ah OSS BEARS Nae 499 

Mr. E. Buckingham on the Thermodynamic Corrections of — 
the Nitrogen Scale. (Plates XV._XVII.).............. 526 


vi CONTENTS OF VOL. XV.—SIXTH SERIES. 
Page 
Mr. G.J. Elias on Anomalous Magnetic Rotatory Dispersion 
of Rare Earths. Remarks on Prof. R. W. Wood’s recent 


PRBOT coisas o> 64 4k oI > > - in er 538 
Dr. Hans Geiger on the Irregularities in the Radiation from 
Tiadioactive Bodies .°.. chien... ...  .,.. -bele Se ee Sern 539 


Lord Rayleigh on Further Measurements of Wave-lengths, 
and Miscellaneous Notes on Fabry and Perot’s Apparatus. 548 
Rev. P. J. Kirkby on the Positive Column in Oxygen...... 559 
Notices respecting New Books :— 
Prof. Carl Barus’s Condensation of Vapor as induced 


by Nuclei and Tons iecmes:---....-...——n 569 
H. 8. Carslaw’s Fourier’s Series and Integrals ........ 569 
J. J. van Laar’s Sechs Vortriige iiber das Thermo- 

dynamische Potential = 3 oeeere 30+ ...... nn 570 
The Scientific Papers of J. Willard Gibbs........2.2% 57 
Prof. 8. Newcomb’s Compendium of Spherical Astronomy 570 
J. M. Pernter’s Meteorologische Optik .............. 571 
EK. E. Fournier d’Albe’s Two New Worlds .......... 57 


Proceedings of the Geological Society :— 
Mr. W. Hill on a Deep Channel of Drift at Hitchin 


(Hertfordshire)... . <4.) Gee eee ee 573 
Prof. G. F. Wright: Chronology of the Glacial Epoch 
in North America. ....: .. eae ee eee ee 573 


Dr. H. C. Sorby on the Application of Quantitative 
Methods to the Study of the Structure and History 


of Rocks ois. ass oe oie ace 574 
Messrs. C. Reid and H. Dewey on the Origin of the 

Pillow-Lava near Port Isaac in Cornwall .......... 76 
Mr. R. M. Brydone on the Subdivision of the Chalk at 

Trimmingbam (Norfolk) . (aeeaeeeeren 1-1: See 577 


Prof. T. G. Bonney on Antigorite and the Val Antigorio, 
with notes on other Serpentines containing that 


Mineral... os... a 2 578 
Mr. J. V. Elsden on the St. David’s-Head “ Rock- 

Series ” (Pembrokeshire) .. ee eceepie: b)-4)--asee 578 
Mr. W. B. Wright on the Two Earth-Movements of 

Colonsay. .).... .-..-/ss -(: 6 ee ee eee ae 579 
Mr. H. Bury : Notes onthe Biver W ey). «4a Sie 


NUMBER LXXXIX.—MAY. 


Prof. R. W. Wood on the Resonance Spectra of Sodium 
Vapoar;. (Plates XVIIL. .& XL) eee. . 2. . ee 581 
Mr. Sidney Russ on the Distribution in Electric Fields of 
the Active Deposits of Radium, Thorium, and Actinium.. 601 
Dr. R. A. Lehfeldt on the Electrochemical Equivalents of 
Oxygen aad Hydrogen! |)... ..) 4. sage ieee eee 614 


CONTENTS OF VOL. XV.—SIXTH SERIES. vil 


Page 
Mr. ©. E. Van Orstrand on Inverse Interpolation by Means ; 
Meet, ICTICS. epee che ces ete ences 628 
Mr. R. D. Kleeman on the Different Kinds of y Rays of 
Radium, and the Secondary y Rays which they produce .. 638 
Prof. W. H. Bragg and Dr. J. P. V. Madsen on an Experi- 
mental Investigation of the nature of the y Rays ...... 663 
Notices respecting New Books :— 
Dr. A. N. Whitehead’s The Axioms of Projective 
Geometry and the Axioms of Descriptive Geometry.. 676 
Drs. W. H. Young and G. C. Young’s Theory of Sets 
Ree OMELGE, | nace gee, ee = Bere tice aoe ea hy me eran 676 


NUMBER XC.—JUNE. 


Lord Rayleigh on Hamilton’s Principle and the Five Aberra- 
pine Seidel 2. sw. i a ce eee le ll). 677 
Lord Kelvin on the Problem of a Spherical Gaseous Nebula. 687 
Prof. A. Morley on Laterally Loaded Struts and Tie-rods _. 711 
Prof. A. 8S. Eve on the Changes in Velocity, in an Electric 
Field, of the e, @ and Secondary Rays from Radioactive 
ee oa wo) PER Yn ol cian die teyn Cigar > 9 ace Pye.) 
Mr. 8. Russ on the Electrical Charge of the Active Deposit 
TNE gcc 5-412, PAGE «wd We SHH Sale eos 737 
Messrs. G. Owen and A. Ll. Hughes on Molecular F ABET 
gations produced in Gases by Sudden Cooling . 746 
Prof. A. W. Porter on the Effect of the Position of the 
Grating (or Prism) upon the Resolving Power of the 


(5 0 15G0 0 « cil ee aa 0) A ee ee 762 
Prof. G. H. Bryan on certain Dynamical Analogues of Tem- 
PemePeqnihbrum 4.0) 2 hPL ae eee. ok. 765 
Notices respecting New Books :— 
Bulletin of the Bureau of Standards, Washington .... 768 
Intelligence and Miscellaneous Articles :— 
On the Mixing of Gases, by. 8S. H. Burbury ......:... 768 


I oc cs a dala wm chetn cw Hae dws ceewceeecews 788 


I-VI. 
VIE: 


VILE 


IX. 
X. & XI. 


XII. 


XIII. 
XIV. 
XV.-XVII. 


XVIII. & 


XIX. 


SO) 


PLATES. 


Illustrative of Messrs. K. Honda, T. Terada, and D. Isitani’s 
Paper on the Secondary Undulations of Oceanic Tides. 

Illustrative of Messrs. A.C. and A. E. Jessup’s Paper on the 
Evolution and Devolution of the Elements. 

Illustrative of Mr. J. D. Hamilton Dickson’s Papen on 
the Joule-Kelvin Inversion Temperature, and Olszewski’s 
Experiment. 

Illustrative of Mr. A. Campbell’s 
Variable Mutual Inductances. 

Illustrative of Prof. E. L. Hancock’s Paper on the Effect of 
Combined Stresses on the Elastic Properties of Steel. 
Illustrative of Mr. L. F. Richardson’s Paper on a Freehand 
Graphic way of determining Stream Lines and Equi- 
potentials. 

Illustrative of Prof. R. W. Wood’s Paper on Anomalous 
Magnetic Rotatory Dispersion of Neodymium. 

Illustrative of Prof. R. W. Wood’s Paper on the Existence 
of Positive Electrons in the Sodium Atom. 

Illustrative of Mr. E. Buckingham’s Paper on the Thermo- 
dynamic Corrections of the Nitrogen Scale. 

Illustrative of Prof. R. W. Wood’s Paper on the Resonance 
Spectra of Sodium Vapour. 


Paper on the Use of 


ad VLE, \ L}J 


THE 


LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES.] 


y 
Yd JANUARY 1908. 


I. The Relation of Mass to Energy. By Dantew F. Comstock, 
Ph.D., Instructor in Theoretical Physics, Mass. Institute 
of Technology, Boston”. 


A: HETHER the imertia of matter has or has nota 
complete electromagnetic explanation is a question 
that it will perhaps take many years to answer with any 
degree of certainty. The experiments of Kaufmann seem to 
prove that in the case of a single electron the mass is entirely 
of this origin ; and it is impossible therefore to avoid the con- 
clusion that at least a fraction of ordinary material inertia is 
also electromagnetic. Doubtless there is a psychological 
cause for our reluctance to accept the electromagnetic expla- 
nation as complete, constant familiarity with ponderable 
bodies having blinded us to the possibility of anything being 
more fundamental ; but certain itis, that if we free ourselves 
from prejudice as much as possible and adopt the well-tried 
policy of choosing the simplest theory which adequately 
represents the phenomena,—the theory that is, which involves 
the least number of variables,—we must decide in favour of 
the complete electromagnetic explanation, which involves only 
the ether and its properties. 
2. The complexity of the Zeeman effect and the relations 


* Communicated by the Author. 


Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. B 


i 


2 Dr. D. F. Comstock on the 


between the wave-lengths of the spectral lines, make it seem 
probable that if matter is to be considered as an electrical 
system, it must be much more complex than a system com- 
posed entirely of electrons separated by distances great in 
comparison to their size. It becomes therefore of interest to 
see whether any relations can be found between the mass of 
an electric system in general, and any of its other properties. 
It will be found that a general relation does exist, which is not 
only of considerable interest in itself, but also suggests other 
relations. 


3. The straightforward calculation of the mass of an electric 


system possessing any distribution of charge and any internal 
velocities below that of light presents considerable difficulty ; 
for such calculation involves the use of the scalar and vector 
potential, and these are not effective instantaneously at all parts 
of the system. Any expression for the mass of the system 
calculated in this way will therefore involve terms which vary 
in an extremely complicated way with the internal velocities 
when these are not very small. The same is true with respect 
to the velocity of the system asa whole. In the following 
discussion the problem is attacked in an entirely different way, 
which is not open to this objection. 

As the constraints of the system are intimately involved, 
it will be well first to consider them. 

4, The position of internal constraints in general electrical 
theory isa very fundamental one. By “ constraints” are meant 
rigid connexions of any kind. These act merely as reactions 
to the electrical forces, and do not contribute to the virtual 
work. If the electrical laws are to hoid universally, 2. e, for 
minute distances as well as for greater ones, itis obvious that 
no electrical system can exist as such unless there are such 
constraints to balance the electrical forces. ven a single 
electron would dissipate itself through the mutual repulsion 
of its elements, were it not for some form of internal constraint. 
Besides holding the system together, as it were, these con- 
straints also act in another important way. They may become, 
in common with all geometrical constraints, paths of energy 
flow. We are accustomed to think of the Poynting vector as 
representing completely the energy flow in a purely electrical 
system, but of course this is not in general true. 

Take as a simple example the case of a large plane air- 
condenser moving in a direction perpendicular to the plane 
of its plates. If the condenser is charged there is obviously 
a transference of energy at a rate equal to the internal 
energy multiplied by the velocity of movement. The Poynting 


ee 
+e 


Relation of Mass to Energy. a 


vector is, however, zero. The energy transfer is not through 
space in the ordinary sense, but is along the constraint which 
holds the condenser-plates apart. ‘The plate in the rear picks 
up, as we may say, the energy of the field, and after it has 
been transmitted to the forward plate by means of the constraint 
itis there set down again. On the other hand, when the 
condenser is moving parallel to the plane of the plates, there 
is no energy flow along the constraint and the Poynting 
vector adequately represents the transfer of energy. So also 


In the case of a single moving electron, the ‘rate of transfer 


of energy is not given by the integration of the Poynting 
vector through all space, but differs from this by an amount 
corresponding to the energy-flow along the constraints in the 
body of the electron. This does not mean that there is any 
energy associated with the constraints, for of course rigid 
constraints can neither absorb nor give out energy; there is 
no storing up, but merely a transfer. 

d. It is not difficult to find an expression for this rate of 
transfer. Ifthe constraint is a simple linear one the transfer 
of energy along its direction is evidently 


—ITv’, 


where (/) isthe length of the constraint, (v') the velocity with 
which it is moving along its length, and (T) the tension 
along it. The amount of energy (Tv’) per sec. is put at the 
forward end, and is instantly available at the rear end ata 
distance (J). It the velocity (v) makes an angle (f) with the 


constraint, v’=v cos 9, and the transfer in the direction of v is 


—/Tv cos? §. 


Another type of transfer enters when there is shearing 
stress in the constraint, a transfer that is which is in a 
direction perpendicular to that of velocity. It must be 
remembered that the constraints~are described in no way 
except geometrically. | 

If we consider therefore the general case where the stress 
in the body of the constraint is represented mathematically by 
the nine stresses commonly used in the theory of elasticity, 
meamcly, X:, Y2, 4, X,, Y,, Z,, Xz, Y3, Z,, then there is a 


rate of transfer of energy in the z-direction through unit 


volume given by 


fc = — (z+ X,v,Y,+1-Z,). 
Be 


4 Dr. D. F. Comstock on the 


This can be readily shown by a consideration of the figure. 
When the velocity of the element is along (w) there is an 


A 
AY’ 


= Q’ 


0 


Y 
amount of work (v,.X,) dy.dz done per second on the 
element by the tension (X,) applied at the surface (O/A’), 
and this energy is instantly available at the surface (OA), 
where it is given out. The distance over which the energy 
is transmitted being dw (the thickness of the ‘element), the 
rate of energy-flow is 
. —v,Nadydzdx= —v,X,dr, 

where (dr) is the element of volume. 

In like manner the velocity (v,) and the shearing stress. 
(Yx.dy.dz) cause energy to be taken up at the surface (O'A’) 
and given out at the surface (OA), and we have the rate of 
flow along the «-axis 

—vyV xT 3 
and finally the velocity (vz) and the shearing stress (Ze. dy . dz) 
give 

—v,L,dT. 

Hence adding we have, if we call (f,’) the density of flow 
along «, 

f (dt = — (vyX_y+ ty a+ vzLy) at. 

Obtaining the corresponding equations in:similar way we 
have finally for the three components of the density of energy- 
flow along the constraints in any system 


fee’ aa a (Wr Xx i vyY x a: vzLe), 
fy = —(vwXy + tyVy t+ vzLy), a as 
j=— (ve Xz as vyYe2 a vzLz). 


Relation of Mass to Energy. D 

For the total density of energy-flow (/2, /,, f:) we must 

of course add to the above the components of the Poynting 

vector. Writing as usual X, Y, Z and a, 8, y for the electric 

and magnetic force intensities and calling (V) the velocity of 
light, we have 


ji 2 af oR —B ZL) Sas (vr X,, = ty Ye +L), 


a 
i 
> 


a= qq (al —yX)—(veXyt ty¥ytvely), >. (2) 


These equations give the density of the total energy-flow 
through any purely electrical system, in which the ordinary 
electrical laws hold universally. 

6. Consider an isolated electrical system moving as a whole 
through space with the constant velocity (v,). A constant 


velocity will be possible if the system retains on the average 


iets es Bes 
fe= y-(BX—2Y)—(teXz + V2 + vee), 


- the same internal structure. The total average rate of transfer 


of energy corresponding to the movement of such a system 
is evidently (v,. W), where W is the total contained energy. 
Another expression for the same thing is to be obtained by 


' integrating throughout the system the components along (v,) 


of (fe, fy, fz) given in equations (2). In order that the 
velocity (v;) may appear explicitly, however, it is necessary 
that the velocity (v), which was used in equations (2), be 
written as the sum of (v,) and another velocity (v.). Then 
(v2) is the velocity with respect to axes moving with the 
system. 

Tf 1, m, n are the direction cosines of the constant velocity 
(v1), we have for the total energy-flow (F) in the direction 
of (%), 


‘U= P=| Ufe+ mfy + nfz)dr 


a ’ ; 
=7 (0 —BZ)dr—lv, (IX2+mY_+nZpjdt—l) (voeXp + voyVe+ 22L,)dt 


mV 4y air ty 4 i 
ig Te {(%—yX) dr—me,| (IXy+mYy+nZ,)dtT—m ( (ex2X, + v2yV y+ v2zLy)dt 


V wv 
a ee (BX—2Y)dt—nv, fs. +mYe+ iDs)de—n (003 zt VoyY2+v2L,)dt 


(3) 


6 Dr. D. F. Comstock on the 


Since the proof of equations (1) is equally valid for relative 
‘motion, the integrals involving (v,) in the above correspond 
to flow of energy with respect to axes moving with the 
system. There is also an implicitly involved internal term in 
each of the Poynting vector integrals. Since the system is 
isolated, the sum of such “internal” terms must on the 
average vanish. There remains therefore to represent the 
actual average rate of transfer of energy through space only 
the explicit Poynting terms and the terms involving (v). 

The electromagnetic momentum corresponding to any 
electrical system is given by the components 


Ss ae 
Me= oy |@¥—82)ar 


1 : 
M,= wy | 2 —yX)dr 


(4) 


which, except for the factor V?, are the same as the integrals 
of the components of the Poynting vector throughout the 
system. Hence equation (3) may be written 


Wr, =/Mz+mM,+2M,—lv, (ix. +mYz+nZz)dr 


—mv,\ (IX,+mYy+nZ,)dr—nv, | ((X,+ Ee + nZ,)dt 
(34) 


Also if the electrical system here dealt with is to represent 
a material body, we may assume that the resultant momentum 
(M) is in the direction of the velocity, and hence 


M=/M,+mM,+7M;. 


This may be considered as due to the fact that the lack of 
symmetry necessarily involved in the intimate structure of any 
electromagnetic system has become a symmetrical average in 
particles large enough to be dealt with. This symmetrical 
point may of course have been reached in the case of single 
atoms. We may now write (34) in the form 


Wr,=VM- vf {(PXz +m?Y,+77Zz) 
+lm(Xy+ Yx)+ln(X24+ Zr) +mn(Yz+Z,)}dt. . (9) 


7. To reduce this expression further requires some relation 
to be established between the stresses and the electric and 
magnetic force intensities. This process is closely analogous 
to the derivation of the Maxwell stress in the free zther 


Relation of Mass to Energy. 7 


except that we here have to deal with, besides the forces in 
the constraint, only the electromagnetic force on electricity 
embedded in the constraint, and we have nothing to do with 
hypothetical stresses in the free ether. 

If (p) represents the electric density and (4x) the x-com- 
ponent of the total electromagnetic force on unit charge 
embedded in the constraint, we have 

OXe OXy OX: Leia Choxx 
On en | beam ee 


=—pX—£ (ryy—v-8) © 
=—pX—(kyy—keB) ) 


where (4) is the density of convection current caused by the 
movement with velocity (v) of the electricity of density (p). 
Making use temporarily of the vector terminology for the 
sake of brevity and calling the electric force (I), the mag- 
as (H), and the sign [ | denoting the vector product, we 
ave 


(6) 


—pA.= —pHy—[kH]. 
noe 
V of 
1 ey hy th Ob 
—p%=— 7,4 B,div B+ | cwl H— 5 Yi HI, 


1 1 OH 
= — (4 Endiv B+ [curl WH, H],— es H| }. (7) 


Since div E=47p and curl H— =Ark, 


Now it is an easily verifiable identity that 


EH, div H—[E, curl E],+H.div H+[curlH, H], 


= 2. {4(E,?—E,?—B,”) +4(H.”—H,? — H.)} ; 
3 a @ 
+ 5) (Bey + HoHy)+ 5, (HoE.+H.Hz); - - (8) 


and hence, remembering that div H=0, equation (7) becomes 


1 : 
— #,=— me p(B.’ — E,’—H,?) + 3(H2?— H,?+ H,”) 


a . 2 


+ [H, curl E],— a H| \. bia te eyeby mets: GOD 


$8 Dr. D. EF. Comstock on the 


Since 
hy 1 ‘em 
[ E, curl B]e=—| E, v Se Hi 
the last two terms in the bracket of (9) become together 
a, 
— 5 [ EH fe, 


which is minus the time rate of the density of momentum at 
the point. The time rate, however, refers to a point fixed in 
space, and to change to a point moving with the system we 
make use of the usual expression and write 


/ a 
—2 me= — ome ~ 0 (1s +m one + nom), (10) 


where m, is the 2-component of the density of momentum 
and (1, m, n) are, as formerly, the direction cosines of the 
/ 
constant velocity (v,). The operator . now refers to the 
rate of change at a point moving with the system. 
Substituting (10) for the two last terms in (9) and noticing 
that 


P 
0 mz : : j 
may be written — =) m,dz, 


ot Ot Ot Jr 


where the integration is to be taken from R (meaning merely 
from a point outside the system where m, is zero) up to the 
point P in question, account being taken of any discontinuity 
at the bounding surface, we have in place of (9) 


oe. fs) i 9 5 5 i 2 5 ‘ 30’ hg } 
5a) ~ 3p E-BY-BD)- 3, (Hi El — HH: )—yln,+ ue max 
tae 
6 e{- (GE, +H,H,)—vymm, + 4 mally } 
R 


+242 EE +H,H-)—vynm a gt - i» Ae 
Oz ie nie Ue 3) a 


This equation gives us what we were seeking, namely, the 
values of X,, X,,and X, in terms of the electric and magnetic 
forces and the density of the momentum. 


Relation of Mass to Energy. a 


Thus we may take 


X,=—- “ aH 2 Ey ay SE 2H P—H?)—v,ln oo r dz 
a ee y 2 0 te a ¢ 3 AGA ae aoe 


a 
Hy) —vymm, + val m dy 


wP 


/ 
x= — = ! ELE, es WIAs. vey Meta) os. at) LZ) 


R 


Similar values are of course to be found for the other six 
components of stress in the constraint Y,, Y,, Y., Z,, Z,, Zz. 

5. The values for these nine stress-components are now to 
be substituted in equation (5). In doing this it is to be 
noticed that the last term in each of equations (12) will, after 
substitution, furnish a term of the type 


ct Cal ae i 
fer ry, ( onde & la ( vPmd2, . . (13) 


and this, being a time derivative, gives an average value of 
ZeLo pent the’ time is allowed to increase indefinitely, since 
all quantities in the system remain finite. Also 


?4+m?+n?=1. 


Making the substitution in o and simplifying, we have as 
the value for (v,W) 


aW=VM—o, { { .(B ot Ele) ce “PE eee ) bar 


+r | {, +mH,+nH,)? + (0H, +mH, + nH.) far 
+v,") (im,+mm,+nm,)dt. . . (14) 


Now the first integral represents the total included energy, 
the two parts of the second integral represent the squares of 
the components of the electric “and magnetic forces in the 
direction of the motion of the system, and the last integral 
represents the momentum in the direction of motion, which 
in this case is the whole momentum M, since we have 
assumed that M and », are in the same direction. 


Calling 


cae ‘ pee 
| 4 ( ADe + mE, e nH)? +(1 A +mH,+ nd.) drt= Wy, 


Mass = 


10 Dr. D. F. Comstock on the 
the longitudinal energy of the system, and 
W-W,=Wr 


the transverse energy of the system, we may rewrite equation 
(14) as 


v,W === VM — v,W + 27,Wr + v°M, + . (15) 
and hence ; 


SSS SS = 7) 4 
FE 
2 af! 
Vv (1+ at ) 
This gives the total momentum of any isolated, moving, 
purely electrical system, which has on the average the same 
internal structure, in terms of its transverse energy, 2. é., the 
energy represented by the components of the electric and 


magnetic forces which are perpendicular to the velocity of 
the system. The mass of the system is then 
a 
(y) 


M_2 Ws 
dv, V? {1 


If, as we have assumed in deriving this expression, the 
system possesses the same momentum for uniform translation 
in any direction, this formula for the mass can contain terms 
of even powers only in the ratio of the velocity of the system 
to the velocity of light. If we neglect terms of the second 
and higher orders Wy has the same value as for v= 0, which 
from symmetry of the system must be two-thirds the total 
energy W. ‘Therefore 


Mass = 5 yo. ae 6s) = 


if second order terms be neglected. This formula would 
apply with extreme accuracy for the electromagnetic mass 
of ponderable bodies, for no such bodies have in nature a 
OFA 
(\) 

It should be noticed that in equation (17) second order 
terms may enter in either (W-) or its derivative with respect 
to v2. In fact such terms do enter for two reasons. In the 
first place, the setting of the body in motion requires work 
and hence adds new energy, through a second order term; 


co) . 
and secondly there is an effect due to the change which 


velocity large enough to make appreciable. 


aWy 


dv? 


(2) ayy BGO): 


(17) 


Relation of Mass to Energy. UE 


motion causes in the velocity of propagation through the 
system of electrical disturbances. This is seen in the simple 
ease of a moving electron where the crowding of the lines 
towards the equator with increase of velocity is only partly 
due to the added energy. It is evident, therefore, that for 
velocities so great that the second order terms cannot be 
neglected, the mass depends on complicated terms which vary 
with the internal structure and motions of the system, and 
it does not appear as if a general expression for the mass of 
a system for such high velocities could be found. 

Tre second order terms may in the future make themselves 
experimentally manifest through an increase of mass of 
rapidly meving a-particles. 

9. Expression (18) may readily be verified for simple 
symmetrical systems. For a single charged conducting 
sphere of radius (a) the mass for slow velocities is well- 
known to be 

pune 26 
Bee 3 Viat —— s(e. Potential) = — 

An interesting verification of equation (16) for the special 
ease of a general, rigid, electrostatic system in translatory 
motion has been furnished me privately by Mr. G. F. C. 
Searle. He obtains for such a system (Phil. Mag. Jan. 1907, 
p. 129) the expression 


W. 


ATE ed Oe aaa ada!) 


where M, is the momentum of the entire system along the 
direction (a) of motion, (T) is the total magnetic energy due 
to this motion, and (v,) is the common translatory velocity 
possessed by all parts of the system. 
_ Now it is well known that where the Faraday tubes move 
through space uniformly, as in the present case, the magnetic 
force (H) is given in terms of the electric force (4) by the 
expression 

H= ale sin 6, 
(@) representing the angle between (E) and the velocity of 
motion (v4), and (H) being i in a direction perpendicular both 
to (EH) and (v). In the present notation 


RU Chan ot es ae 
-and hence we have 


ROTEL 2 ee 240? ) 
= ee OE eee 


12 Dr. D. F. Comstock on the 
Combining (19) and (20) we can obtain 
1 9 9 
v,-2 4 5 (BY +5)+ v} 


M, ei 2 “ = 21) 
v2(1+ 3) ( 


V2 


and remembering that 
it ? $ 
ig == 3,, Hy +H: os 


we have finally 


a 
) 


which, since (v;) is along (x), is identical with (16). Thus 
(16) is verified for the case where the moving system 
possesses no internal motion. 

Perhaps the simplest symmetrical system containing 
magnetic as well as electric energy is that formed by a great 
number of charged spheres moving in straight lines out 
from a common centre, with velocities small enough so that 
the fourth and higher powers may be neglected. They are 
to be at distances from each other great in comparison with 
their size, and at equal distances from the common centre. 
If the system be now given the slow velocity (v,) as a whole 
the total momentum accompanying this motion may be deter- 
mined. Because of limited space the calculation will not be 
here given, but if it be carried out along established lines it 


: : r times the sum of the 
pot 
electric and magnetic energies, thus verifying equation (18). 
10. We conclude that, if ordinary material mass has an elec- 
tromagnetic basis, such mass for slow velocities is proportional 
to the.total electromagnetic energy-content of the body, and 
the laws of conservation of mass and energy become closely 
related if not identical. In any case the expression given 
represents the electromagnetic part of the total mass whatever 
that may be. 


will be found that the mass is 


Considerations suggested by the Foregoing. 
The Atomic Weights. 
11. If the conclusion of the last article is correct a dimi- 


nution in mass should follow a loss of energy in material 
transformations. Calculation shows, however, that in the 


Relation of Mass to Energy. 13 


case of the powerful reaction between hydrogen and oxygen 
forming water, the change of mass would only be of the order 
10- gram. In the case of radioactivity, however, the 
energy change is very much greater and an appreciable 
effect is to be expected. Thus if a radium atom gives off 
an a-particle of mass (m) with velocity (yw), then there should 
be a diminution in the sum of the masses of the a-particle 
and the remaining atom equal to 


rat § 
3 Gn’), 


since 4m” represents the energy lost, and this, calling 
m=4 (using gram-atomic weight) and #=2°5 . 10°, gives 


A (Mass)= —1°7.10-? gram; 


an amount large enough to cause discrepancies in calculating 
the atomic weights of radioactive substances from the number 
of a-particles lost. Since A(Mass) is proportional to the 
square of the velocity of the a-particle, its value would be 
greatly increased by a slight error in the determination of 
(w) and the effect could easily be much larger. 

12. A consideration of some interest is the following. If 
we adopt the disintegration theory, we are obliged to think 
of the various atoms as combinations or groups, more or less 
modified, of the lighter atoms. If there were perfect con- 
servation of mass this would introduce a certain uniformity 
in the relations between the atomic weights, a uniformity 
which apparently does not exist. On the other hand, if we 
take into consideration the inevitable change of mass when 
the electromagnetic energy of the system is modified, the 
atomic weights will involve a correction term depending upon 
the change in this energy, and hence they will no longer 
bear simple, exact relations to each other. In a highly im- 
portant paper (Zeitschrift fiir Anorg. Chemie, xiv. p. 66, 1897) 
Rydberg has shown that the atomic weights of the first 
twenty-seven elements of the periodic system approximate to 
whole numbers very much more closely than chance could 
bring about. He has also shown that the atomic weights of 
these elements are best considered as the sum of two parts 
(N+ D) where N is an integer and D is a fraction, in general 
positive and smaller than unity. If M is the number of the 
element in the system (called by Rydberg the “ Ordnungs- 
zahl”’), then N is equal to 2M for the elements of even 
valence and 2M +1 for the elements of odd valence. Below 
is given a table showing the various quantities. I have used, 


14 Dr. D. F. Comstock on the 


however, the International Atomic Weight values for 1907 
instead of those Rydberg used. 


| i | : 
| N. ) Atomic | | | N. | Atomic 
Sign. | M.| ————, Weight.| D. || Sign. M.| ——+— |Weight.| D. 
/OMJ2M+41 ismaie 2M.)2M+1 
| autos pe ES a a oo ee es ee be ee 
He 2] 4 ep as 31 | 310 | 0 
Li 3 7) os 08 8.2... 16 ee 82:06 | -06 
Be Aa 84 Oa) Gh. 2.) 17 | 35 | 85-45 | -45 
nee 5" | 14) 12:00? 0A... 18 | 36 | 399 «13-9 
Te ee 12:00 | -0 | K 19 — 89 | 89:15 | -15 
N 7 | 15 | 1401 |--99]| Ca ...| 20 | 40 | 140-1 | 1 
Be: 8 16 11600 | ‘0 |iSe ....21| | 43°) 2 
BE Oo. | 10 Le ae 44 | a on 
Ne 10205 | 20:0 oO | — | 23 | 47 —_ — 
Na ...| 11 23 | 23:05 | -05|Ti ...| 24 | 48 | 481 | -1 
Me ...| 12 | 24 | 24°36 | -36|V ...| 25 | 51 | 512 
We 18 27 | Syd) eh Vim ney aie 52. | 521 
Si 14 | 28 23:4 | 4 || Mn...| 27 55 | 550 | 0 
ees Fe ...| 28 | 56 559 | cl 
| 


The orderly arrangement of the series is striking. It will 
be noticed that in three cases only are the D’s greater than 
unity and only in two cases are they negative. 

Rydberg points out that although the heavier elements do 
not conform well to this scheme, 2. e., do not in general give 
the small fractional values of (D) noticed above, yet this is 
in reality no valid objection, for the numerical values of the 
weights of heavier elements depend much more on the value 
of the arbitrary unit chosen than do those of the lighter 
weight elements, and hence they can have litile influence - 
one way or the other in estimating the validity of the curious 
relations he sets forth. 

The whole question is of course whether these ditterences 
represent real physical deviations from something or whether 
they are merely mathematical remainders. Rydberg certainly 
believes them to represent physical realities, and considering 
the before-mentioned overwhelming improbability that the 
approximation of the atomic weights to whole numbers is due 
to chance, we can hardly doubt that he is right. 

13. Now itis to be noticed that these deviations find a ready 
explanation when the conclusions of the present paper are 
combined with the theory, so much favoured recently, that 
one element breaks down into two or more others with an 
accompanying expulsion of energy. The deviations are then 
to be explained as resulting from loss of mass accompanying 
the dissipation of energy. On the other hand, if no such loss 
of mass takes place, the existence of these deviations in the 


Relation of Mass to Energy. 15 


table of atomic weights becomes a well-nigh insuperable 
difficulty in the path of the evolutionary theory of the 
elements. 

If we follow the present suggestion, we must search for 
the components of an element, not by comparing atomic 
weights, but by comparing the corresponding values of N, 
for the atomic weights deviate because of the lost mass 
accompanying the dissipation of internal energy. 

Very recently Sir W. Ramsay has announced several 
striking discoveries which seem to add much weight to the dis- 
integration theory, and, indirectly, to the views here set forth. 


He found helium, neon, or argon appearing as a product of 


radium emanation according to the exterior conditions 
imposed, and he found lithium appearing when a copper- 
sulphate solution was left in the presence of the emanation. 
Prof. Ramsay states, I believe, that every source of error 
was eliminated and that the results were obtained many 
times. ? 

14. It should be noticed that this theory of loss of mass 
and its consequences does not require that the whole material 
mass should be of electromagnetic nature. It only requires 
that the energy lost in the transformations, explosive or 
otherwise, should be at the expense of internal electromagnetic 
energy, 7. ¢., that the forces which expel the a-particles should 
be electric or magnetic. 

Respecting Gravitation. 

15. The experiments of many investigators have shown 
that up to a high degree of accuracy the ratio of mass to 
weight for different substances is the same. Now if the 
mass 1s proportional to the internal energy as here suggested, 
instead of being proportional to the number of electric nuclei 
as might be supposed, the conclusion is apparently forced 
upon us that gravitational attraction is between quantities of 
confined energy, and not between quantities of “‘ matter” in 
any other sense. 

On this basis, the weight of a calorie at the earth’s surface 
would be of order 10—" dyne. ‘This is apparently too small 
to explain the temperature gradient in the earth although the 
calculation, depending as it does on the mechanical force on 
confined energy due to a temperature gradient, would certainly 
depend to a large degree on the medium. 

If we assume this gravitational effect, it is interesting to 
ask whether free energy would also show an attraction for 
itself. If so, the energy radiated from a gravitational centre 
like the sun would leave some of itself behind along its path 


16 Dr. D. F. Comstock on the 


as it moved through space, and it might be possible to 
account in this way for some of the energy which is ordinarily 
thought of as totally dissipated. 

Another conclusion which is suggested by the foregoing is 
that, assuming the loss of mass accompanying dissipation of 
energy, the sun’s mass must have decreased steadily through 
millions of years. If too, our conclusion respecting the 
gravitating quality of contined energy be correct, the gravita- 
tion constant of the sun has also decreased and the distances 
of the planets must have increased accordingly. This last 
increase of planetary distance can be calculated by making 
the angular momentum of the planet about the sun a 
constant, and allowing the mass of the planet, together 
with the gravities of both sun and planet, to grow less with 
time. 

So little is known as to the former radiating power of the 
sun that no even approximate calculation can be made, but 
it is not difficult to show that the order of magnitude is such 
as might make the increase in the planetary distances not 
altogether negligible during great lapses of time. 


A Proof from a different Point of View. 


16. The proof of expression (17) which has been given has 
the advantage of entering intimately into the structure of 
the general system and showing the part that non-electrical 
forces in the form of constraints must play if the fundamental 
laws of electrical action are to hold for every infinitesimal 
element of the finite volume occupied by any electrical 
system. Although this is assumed in every mathematical 
derivation of the mass of an electron, and in fact in all 
problems of a similar nature, many will doubtless object to 
this assumption on the ground that probably the ordinary 
electrical laws do not apply when the distance between 
“elements of charge,” so called, is comparable with the 
diameter of an electron. 

Although it is difficult io see how a coherent mathematical 
theory of electricity can at present be formed without this 
assumption, yet it was thought best to add a more general 
proof of (17). The following is therefore given as avoiding 
the explicit use of constraints. 

17. The statement of the law of the conservation of energy 
for an element of volume in any electrical system possessing 
electrical charges in motion, is the well-known expression 

S S. 
= (S24 oe 3 So) = (ep Met op Fe + 00H) . (23) 


Relation of Mass to Energy. 17 


Here (w) is the density of the total electromagnetic energy, 
Sz, S,, S: are the components of the Poynting vector, 7,, Ay, 
#. are the components of the total electromagnetic force on 
unit charge, (p) is the density of electrification at the given 
point, and z,, vy, vz represent the velocity through space of 
this electrification. Thus 


w= 2b (REVEL) + (P+ e+ y)}, 


where X, Y, Z, and «, 8, y, are the electric and magnetic 
force intensities respectively, and 


F,=X + (vyy—v-8), 
Ay=V + (v.a—rny), 
A.=L + (v2.8 —vyx). 


Equation (23) states merely that the rate of increase of 
energy in an elementary volume is equal to the activity of any 
foreign (z.e., non-electrical) forces which may act therein 
minus the outward flow of energy. 

Now suppose we consider an electromagnetic system 
bounded by a rigid surface (AB), which moves uniformly 
! 


iC 


z i 

. Ci 
ae 

: | x D: 


D’ 


through space with the velocity (v,) along the axis of (2) ; 
and further suppose that the volume inside this closed surface 
is divided into two parts by the plane partition (CD) which 
is perpendicular to the z-axis and which, although fixed in 
the moving system, coincides at a given instant with the 
plane (C’D") fixed in space. If this system be considered as 
isolated, then no disturbance passes through the bounding 
surface (AB). 

In equation (23) the time derivative of the energy density 
Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. C 


18 Dr. D. F. Comstock on the 


refers to a point fixed in space, and if we wish it to refer 
to a point moving with the system we must write as usual _ 


ou _ 84 _ 98%, mh cys ane iy 
Ow 
Ot 
moving point. Likewise, if we wish the velocities which 


enter into (23) to be expressed in terms of velocities relative 
to axes moving with the system, we must write 


where now means the rate of change measured from the 


ich Waa | 
Vy — V2y . ° e e . . (25) 
Vz = Vaz 


where Vez, V2y, and V2, are the components of these relative 
velocities. 

Substituting (24) and (25) in (23), and remembering the 
simple proportionality between 8,, 8,, and S, and the 
density of momentum m,, my, and mz, we easily obtain 


Om: Ow Ow (OS, , OS- 
2 eb Ne A Ce : cee y 
V e 192 +1pA,= Dt te a) 
— (vopA,+voypAyt vopf.). . . (26) 
Now (p,) may be expressed in terms of the electric and 
magnetic force intensities, together with the density of the 
momentum. This involves only the fundamental equations 


of electromagnetic theory and has been done in paragraph 7, 
reference to which will show that with the present notation 


ae) 1 2 Wo. ge 1 2 2 2 
p= 24 WY B+ 5 (ep y) } 


* £ { £ (RY +48) } + oq (XZ+ay) } | 


O'm, Om, 
— 3: +? oa . . . . e = e ° ° . (27) 


Substituting this for the (pz) which occurs on the left- 
hand side of (26), rearranging the latter, and putting 


a SIEVE te + ety} 


pan pees {Y¥? 4+ Z? + BP +97}, 
Op OT 


Relation of Mass to Energy. Wy 
we get 
2 {(V? + vy?)inz — 2vywe} 


—_ 0M, Oise (Oy , dS- 
ae i, v1 acl a 


1 a 
= 7 (XY +48) | -2. { dg (XZ+ay) | 
— (02,072 + Vay Ay + vazpe2). » » » + « (28) 


Now if this expression is integrated through the part of 
the volume (AB) which lies on the side (A) of the partition 
(CD), the terms on the left of (28) become equal to 


fav? + v,")mz— 2v,we dS, 


where the integral is taken over the part of the plane (C’D’) 
which is included in the surface. This follows from the fact 
that the rest of the surface of part A belongs to the surface 
(AB) and outside of (AB) there is no disturbance whatever. 
The terms on the right (of 28) give an average value of zero. 
This last will be evident if they are considered separately. 
The first two give directly a time average of zero, after great 
elapse of time, since neither (w) nor (vymz) ever becomes 
infinite. The terms involving the (y) and (z) derivatives when 
integrated may be written as surface integrals over the 
bounding surface (of part A), and they then represent the 
flux of energy through this surface in a direction perpendicular 
to (v), 2. é., in a direction perpendicular to the x-axis, This 
flux being everywhere zero over the surface, these terms 
vanish. 7 

The terms involving FP. Py, , also give a time average of 
zero because they represent what might be called the “ internal 
activity “’ of the forces which act on the charges in the A- 
part of the volume (AB), and since this part A is isolated 
on all sides except on the side (CD), the activity of these 
forces really means the rate at which the part (B) by means 
of them is doing work on A. In the long run B’s work on 
A must be equal to A’s work on B, if the system is to be 
conservative and the internal motions are to be stationary. 

Thus we learn finally from equation (28) that the average 
value of 


i) { GV? aT v1"), — 20,0; dS = Oy 


where the integral is taken over the enclosed part of the plane 
(CD). Since this is true for any position of the plane (CD) 
so long as it is perpendicular to (v,) we evidently have on 


Y! 


20 Relation of Mass to Energy. 
the average, integrating along (#) throughout the entire 
system, 

\{ (V2 +0), — 200; }dr=0, 7, ae 
This gives, using the former notation, and remembering that 


on the average the internal structure is assumed to remain 
the same, 


psi 2u,Wr lea 2u,Wr 
a ee 


which, since (v,) is here along (x), is precisely the result of 
equation (16), and becomes (17) on differentiation. 


(30) 


Conclusion. 


It has been shown in the foregoing that the electromagnetie 
mass of an isolated, symmetrical, purely electric system 
possessing any structure which on the average remains the 
same, and any internal motions or constraints, is expressible 
in terms of its velocity as a whole through space together 
with its “ transverse energy” and. the derivative of the latter 
with respect to the velocity. If second-order terms in the 
velocity be neglected, the mass is a simple constant multiplied 
by the total included electromagnetic energy. 

If the mass of ponderable bodies has an electromagnetic 
origin, then the inertia of matter is to be considered merely 
as a manifestation of confined energy. From this point of 
view, matter and energy are thus very closely related and the 
laws of the conservation of mass and energy become practically 
identical. 

It has been pointed out that the loss of mass, inevitable 
on this view, which takes place when energy is lost to the 
system, is large enough to be detected in the case of radio- 
active changes. If we assume the disintegration theory of 
the elements, this loss of mass affords a ready explanation of 
the general, small irregularities to be found in the list of 
atomic weights, and thus removes a serious difficulty from the 
path of the disintegration theory. For this loss of mass to. 
take place however, it is not necessary that the whole of the 
mass be electromagnetic. 

It has been shown that if material mass be electromagnetic 
and if lighter elements are formed from heavier ones through 
violent energy changes, it follows that gravity acts between 
quantities of confined energy and not between masses in any 
other sense. Several speculations are indulged in as to the 
results of assuming gravitation between quantities of energy. 


Evolution and Devolution of the Elements. 21 


Finally, the fundamental proposition is dealt with mathe- 
matically from an entirely different point of view and the 
same result obtained. 


In conclusion I wish to express my thanks to Prof. J. J. 
Thomson and to Mr. G. F. C. Searle for several valuable 
criticisms and suggestions. 

Cambridge, England, 

August 14th, 1907. 


IL. The Evolution and Devolution of the Elements. 
By A. C. and A. KE. Jessup *. 


[Plate VIL] 


i eee hypothesis that the elements are different forms of 

one original substance was first formulated in modern 
times by Prout, and though his idea that hydrogen was that 
substance has since been shown to be incorrect, yet modern 
theories have given us, in the corpuscle, a body which may 
well be the root basis of all matter. 

The recent researches of M. and Mme. Curie, Sir William 
Ramsay and Mr. Soddy, Professor Rutherford and others, 
have brought to light the fact that some of the elements are 
undoubtedly degrading into simpler forms of matter. But 
when we look for a reversal of this process on the earth, it is 
not apparent. In other words, we have as yet found no 
indications that elements with low atomic weight are changing 
into other elements with a higher atomic weight, that is, we 
have no proof of inorganic evolution. But when we turn 
our attention to the heavens, the case is altered, and it is 
entirely upon astrophysical observations that the ideas of 
evolution we are about to bring forward are based. 

It was originally our intention to give these observations 
in full, but it has appeared advisable to give in the present 
paper only such of them as are essential for an understanding 
of what follows. 

Spectroscopic evidence shows us that the nebule contain 
but few elements, all of which are in a highly attenuated 
form. The only two which have been recognized on the 
earth are hydrogen and helium, and the atomic weights of 
these are less than those of any other elements with which 
we are acquainted. 

As the nebula becomes more compact, and assumes the 
form of a star, more and more complex elements appear, such 


* Communicated by Sir William Ramsay, K.C.B., F.RS. 


B820| 


22 Messrs. A. C. and A. H. Jessup on the 


as iron, carbon, calcium, silicon, and magnesium, whose 
presence is first evidenced by their enhanced lines, as shown 
by Sir Norman Lockyer. The enhanced lines of the elements 
are known to be due to the elements being submitted to very 
high temperatures or very great electric stresses. Inasmuch — 
as we are justified in assuming the existence of stresses in the 
stellar systems, it is justifiable to assume the existence of 
enhanced lines as first evidences of evolution. Moreover, it 
may be pointed out that the appearance of new lines in the 
nebulze is generally accompanied by a faint continuous 
spectrum, usually in the green. Hence, in the heavens, it 
would seem that we have distinct evidence of evolution. 

Now Mendeléeff has arranged the elements in order of 
atomic weight, and in such a way that they fall into groups, 
and we may regard the evolution as taking place in one of 
two ways. J irstly, that the elements evolve in order of 
atomic weight, that is along horizontal lines of Mendeléeff’s 
table ; secondly, that they evolve in groups, that is, down its 
vertical columns. We have strong reasons for believing that 
the second method has most evidence in its favour. 

In the first place, the elements which appear after hydrogen 
and helium are carbon, silicon, magnesium, calcium, and iron, 
but these are in no way in order of their atomic weights, and 
moreover sodium, although much smaller quantities of it can 
be detected, and with a much lower atomic weight than 
calcium or iron, does not appear until long after these 
elements. The first positive evidence of sodium is in the 
case of the stars of group 7. 7 

Secondly, the whole of the nitrogen group is missing in 
the sun. On the supposition that the elements grow in 
groups, the fact that nitrogen did not form in the sun would 
account for the absence of the other members of its family. 
The other method of growth would imply, that each of the 
five elements belonging to the group was missed out in the 
ordinary process. 

Our view of evolution will at once explain two great 
difficulties connected with the Periodic Table, namely, the 
cases of tellurium and argon, with atomic weights greater 
than those of iodine and potassium. It is natural to assume, that 
if an element A has a greater atomic weight than an element B, 
the element formed from A will also have a greater atomic 
weight than that formed from B ; but this is not necessary, 
and the two cases cited above are examples of this fact. 
Sodium, with a greater atomic weight than neon, forms 
potassium, which has a less atomic weight than argon. 

However, to return to the first stage of evolution. The 


i. 


Evolution and Devolution of the Elements. 23 


nebulous stage of matter is the first of which we have any 
knowledge. The spectra of the very earliest nebulze consist 
of three lines only, their wave-lengths being 5007, 4959. and 
4862 t.m., and corresponding to hydrogen and two unknown 
elements. As a nebula grows more compact, two more lines 
appear with wave-lengths 4340 and 5876 t.m. These are 
due to hydrogen and helium respectively. Consequently, we 
might well suppose that in the nebulous stage of matter there 
are four substances, the first two being unknown upon earth, 
the third being hydrogen, and the fourth, which apparently 
only exists in small quantities, being helium. It also seems 
probable that except these four, no other elements exist in the 
early nebule ; and if this is the case, we are justified in 
assuming that hydrogen, the two unknown elements, and 
helium are the four original elements from which all the 
other elements form. To distinguish them from the others 
we will term them protons. 

Before specifying which elements these four protons pro- 
duce, it will be as well to indicate as briefly as possible the 
manner by which we consider that one element gives rise to 
another. 

We may assume that all matter at some period was in the 
form of corpuscles. By certain processes, a description of 
which we hope to give in a future paper, some of these 
corpuscles arranged themselves into stable integral systems, 
these systems being the four protons. These protons in turn 
developed into other atoms. Lach of these protons begins to 
collect round it more corpuscles, this condensation continuing 
until a stable system is formed, capable of separate existence. 

The assemblage round the central atom may be regarded as 
having certain properties, and a more or less definite shape. 
We do not propose to speculate at all on the nature of this 
assemblage, beyond saying that the particles are probably in 
rapid motion, and that the whole atom forms a stable system. 
We may, however, for convenience denote this assemblage by 
the term “ring.” If the atoms were really built up of con- 
centric rings, all that we have to say would apply equally 
well, so that in order to imagine the appearance of an atom, 
we may actually look upon it as composed of a series of rings 
of varying shape and size, and in all that follows we shall 
speak as though this were its actual structure, for the sake of 
simplicity ; butit must be distinctly understood that we make 
no assumption as to its true form. We may here mention that 
the formation of successive elements is attended by the escape 
of a large quantity of energy, and that the systems are steadily 
progressing from a higher to a lower energy content. In 


24 Messrs. A. C. and A. E. Jessup on the 


fact, we may go so far as to say that the whole process of 
evolution is entirely governed by the possibility of energy 
leaving the systems. 

The process of evolution may be considered to be a per- 
fectly smooth and continuous one, and the elements as we 
know them only to represent positions of maximum stability, 
which are arrived at during the process of evolution. Since 
the elements may be looked upon as denoting the positions of 
maximum stability or, so to speak, stopping places in the 
evolution process, we can readily comprehend the simul- 
taneous existence of all the elements. Of course the rate of 
evolution along different groups may vary, and therefore we 
are not surprised at the fact that calcium and iron, with 
atomic weights of 40 and 56 respectively, are found in stars 
which do not contain sodium with an atomic weight 
of 23. | 

We will now turn our attention to the periodic table as 
given by Mendeléeff and see how far our ideas of evolution 
agree with the form of the table as given by him. Several 
attempts have been made to remodel this table, of which 
perhaps the most notable are the harmonic periods of Professor 
Kmerson Reynolds, the vibrating pendulum of Sir William 
Crookes, and the logarithmic spiral of Dr. Johnstone Stoney. 
It is, however, admitted that there are defects in all of these, 
and we hope to be able to remove some of them. We may 
perhaps here mention that the idea of evolution in groups was 
first suggested by Mendeléeff’s table, and that without it our 
original observations would not have developed. It was onl 
when we came to extend our ideas that we found that they 
would not entirely agree with the table as it originally stood. 
Consequently, we have in our turn ventured to present the 
table in a modified form, which we hope may smooth over 
some of the difficulties at present connected with the classical 
researches of the great Russian. 

In the table as it now stands, it is customary to divide the 
main groups into two sub-groups. For instance, group I. 
is in Mendeléeff’s table divided into group I A. hydrogen, 
lithium, sodium, potassium, rubidium, and cesium; and 
group I[ B. copper, silver, and gold. 

Now, it will at once be seen that although all the elements 
of the same sub-group resemble each other greatly, yet 
members of different sub-groups have apparently very little 
connexion between them. Members of the same sub-group 
occupy corresponding positions on the atomic volume and 
melting-point curves, while the positions of members of 
different sub-groups differ very widely. 


a ma 
e 


Evolution and Devolution of the Elements. 25 


The same considerations apply to group II. if it be sub- 
divided as follows:—A. Beryllium, magnesium, calcium, 
strontium, barium, radium ; and B. zinc, cadmium, mercury. 
However, in chemical properties there is a certain relationship 
between the two sub-groups. 

In group III., though the members of the different sub- 
groups occupy different positions on the two curves mentioned 
above, yet their chemical properties are more closely connected 
than those of group II. If we proceed in this manner through 
the groups, we shall in all cases find the members of corre- 
sponding sub-groups differing with regard to their positions 
on the curves, and varying the closeness of their chemical 
relations as they pass from group to group. In groups 38, 4, 
and 5 the members are most alike, while those in groups 1 
and 7 are most dissimilar. 

To account for these facts we venture to put forward the 
following explanation. We would suggest that members of 
each sub-group evolve in turn from the first member of that 
group, and resemble each other in structure. All the 
members of a single Mendeléeff group do not evolve from 
each other in order of atomic weight, but only each sub- 
group, the two sub-groups of each column have an entirely 
different internal atomic structure, their apparent resemblance 
being wholly due to a similarity in the external form of their 
atoms. ‘This question will be more fully discussed later, but 
at present we must investigate the result this assumption will 
have on the form of the periodic table. 

In the first place, it is evident that if each sub-group be 
considered as being the result of a separate and distinct 
process of evolution, an increase in the number of sub-groups 
must be made. To determine what the number shall be, it is 
best to consult the atomic volume curve. It is generally 
admitted that lithium, sodium, potassium, and rubidium are 
typical members of one family, as is clearly shown by their 
positions on the above-mentioned curve. Between lithium 
and sodium are seven elements, and the same number is found 
between sodium and potassium, but between potassium and 
rubidium there are seventeen. This means that in the hori- 
zontal series, starting with lithium and sodium, there are 
eight groups, but in the next series this number has increased. 
An explanation of this is that one or more of the elements in 
the second series of eight has developed in more than one way. 

Lest this seem improbable, we may mention that it is 
exceedingly unlikely that there are more than four original 
elements or protons, while it is universally acknowledged 
that the series starting with lithium and sodium both contain 


26 Messrs. A. C. and A. E. Jessup on the 


eight members. Consequently, if our idea of evolution is to 
hold good, one at least of these protons must have given rise 
to more than one direct product of evolution. | 

It it be admitted that some double growth has occurred, 
the next step is to find the place of its occurrence. Examining 
the atomic volume curve, we find that sodium and potassium, 
magnesium and calcium occupy similar positions on the curve, 
and also that the same relations hold good between phosphorus 
and arsenic, sulphur and selenium, chlorine and bromine : and, 
consequently, these pairs must be directly connected together 
in the process of evolution. Now sodium and potassium, 
magnesium and calcium, are corresponding members of 
consecutive horizontal series of Mendeléeft’s table ; but the 
three next pairs, phosphorus and arsenic, sulphur and selenium, 
chlorine and bromine, do not belong to two consecutive hori- 
zontal series, for the elements vanadium, chromium, and 
manganese, respectively, lie between the members of these 
pairs. We may conclude, therefore, that the evolution must 
have sub-divided at some position between magnesium and 
phosphorus ; in other words, that either aluminium or silicon, 
or both, must have given rise to more than one series of 
evolution products. 

Now potassium and sodium, calcium and magnesium, 
arsenic and phosphorus, selenium and sulphur, bromine and 
chlorine must be consecutive products of evolution in their 
respective families, and we are driven to the conclusion that 
vanadium cannot be derived from phosphorus at all. Similar 
arguments apply to chromium, manganese, &. We are 
therefore justified in concluding that vanadium, chromium, 
manganese, &c., must be derived by indirect evolution from 
aluminium or silicon. 

If we compare the chemical characteristic properties of 
boron, aluminium, and scandium, with those of lithium, 
sodium, and potassium, and of beryllium, magnesium, and 
calcium, the analogy between the three groups is sufficiently 
strong to justify our considering scandium as the direct 
derivative of aluminium, and, for the same reasons, we must 
consider titanium to be the direct derivative of silicon. Now, 
silicon and phosphorus undoubtedly follow one another in 
order of atomic weight, 7. e., there are no elements of atomic 
weight lying between 284 and 31:0; and inasmuch as 
titanium and arsenic have been shown to be direct evolution 
products of silicon and phosphorus, it is quite evident that 
all the elements possessing atomic weights lying between that 
of titanium (48°1) and arsenic (75) must be derived from 
silicon by indirect processes of evolution. 


[To face page 27. 


p.Be. '1:33/p.B. 2. 

3iBe. | 95 |Boemimee,- 19. 

Mg. 24:36/Al. 27-1/Si. 28-4. 

5[Ca. 401 |Sc, 44-1/Ti. 48-1/V. 51 -2)¢r. 
Sr. 876 /Y¥. 89-0lZr. 90-6.Nb. 94-0\Mo. ¢ 
|Ba. 137-4 |La. 138-9/Ce. 140:3 

Tm.?171 |Yb.? 173 Ta. 183 W. 1$ 
Ra. 225 Th. 232°5 U.| 2 


‘ 
— 
2 


a 


ear Sl 


Evolution and Devolution of the Elements. 24 


There are now ten elements lying between these limits, 
viz., vanadium, chromium, manganese, iron, nickel, cobalt, 
copper, zinc, gallium, and germanium ; and we must consider 
all these to be derived from silicon by some indirect process 
of evolution. It may be pointed out here that by the term 
“A direct process of evolution,’ we mean one by which an 
evolution product is obtained of the same valency as its parent 
or immediate antecedent element in the scale of evolution. 
Conversely, “ indirect evolution” produces elements of 
valencies differing from that of the parent element. 

Now the next highest elements to germanium in order of 
atomic weight are arsenic, selenium, bromine, krypton, 
rubidium, and strontium ; but these undoubtedly. belong to 
the families of nitrogen, oxygen, fluorine, helium, lithium, 
and beryllium respectively ; and so may be considered as the 
direct evolution products of these families. The next four 
elements again are yttrium, zirconium, niobium, and molyb- 
denum, and these will fall natur ally under scandium, titanium, 
vanadium, and chromium respectively, a fact w hich is in 
absolute agreement with the relative chemical properties of 
the families concerned. 

We are now in a position to draw up a Periodic Table 
modified from that of Mendeleéff so far as to embody our 
conclusions, and such a table is shown opposite. 

Leaving, for the moment, the question of the four protons, 
hydrogen, proto-beryllium, proto-boron, and helium, we have 
clearly eight elements standing at the head of eight very 
strongly characterised families, viz., lithium, beryllium, boron, 
carbon, nitrogen, oxygen, fluorine, and neon. By direct 
evolution these eight ‘elements give rise to the next eight 
elements in order of atomic w eight—sodium, magnesium, 
aluminium, silicon, phosphorus, sulphur, chlorine, and argon. 
The question of argon having an atomic weight greater than 
that of potassium has already been mentioned. Following on, 
we have potassium, calcium, scandium, and titanium, which 
belong to the first four groups of the table. After titanium, 
we have the ten indirect derivatives of silicon which we have 
dealt with above, and we therefore place these in a hori- 
zontal row between the carbon and the nitrogen families. 
The next ten elements in order of ascending atomic weight 
following germanium, viz., arsenic, selenium, bromine, 
krypton, rubidium, strontium, yttrium, zirconium, niobium, 
and molybdenum, fall perfectly naturally in the consecutive 
families as already pointed out. The next vertical column 
after molybdenum is headed by manganese ; from a con- 
sideration of Mendeléefi’s table, it is quite evident that 


\ ae 
[To fuce page 27. Phil Mag., Ser. 6, vol. xv.] 


THE PERIODIC TABLE. 


Le) 
LAO. Ip Be. 1:33)p.B. 2. He. 4 
Li. 7:03)/Be. 0:1 1B, Wi jc 2. x 14:01)0. 16:0 |F. 19 Ne. 20 
Na. 23:05 Mg. 24:36) Al, 27-1)/Si, 28-4. B 31-0 |S. 382-0601. 35:45 )Ar, 39-9 
nee | i aan es, are 
K. 3975/Ca. 40:1 |Sc. 44:J/Ti, 48:1/V. d1-2/Cr. 62-1/Mn. 55:0 Fe. 55°9) Ni. )8°7| Co. 59°0 Cu. 63-6 Zn. 65-4\Ga, 70 |Ge. 72:5)As. 75:0 | Se. 79-2 |Br. 80:0 ||Kr. 81:8 
Rb. 85:5 Sy. 876 }¥. 89:0)Zr. 90:6/Nb. 94-0/Mo. 96:0) Ru. 103:0/Rh, 1030) Pd. 1065 Ag. 107-9Cd. 1124/Im. 115 |Sn. 119:0]Sb, 120-2 |e. 127-6 I. 127-0 ||xe. 128 
| pee L : | |_ 
Cs, 132'9 Ba. 187-4 |La. 138-9/Ce. 140:3 | Pr.? 140 5Nd.? 143:6/Sm.? 150°3 Bu.? 151-7 Gd.? 157 Tr.2160 | Er.?166 
| | 


l¥b.? 173 


232:5 


Ta. 183 


10s. 191 ‘Tr. 193-0 


| 
194:8 Au 
| 


Pt 


197-2 Hg. 200-0 


Tb. 


204-1 


206'9] Bi. 208:5 


28 Messrs. A. C. and A. E. Jessup on the 


manganese has no direct evolution products, that is to say, 
there are no elements of higher atomic weight than man- 
ganese with properties sufficiently closely related to manganese 
to justify our considering them as direct evolution products 
of this element. We have therefore placed manganese at the 
head of its own column, and would point out that the blank 
spaces below this element do not necessarily mean that there 
is room for elements to be discovered to fill these, for, as 
will be presently shown (page 36), it is in the highest degree 
improbable that manganese should be able to give rise to any 
direct derivatives at all. 

Following molybdenum, we have now the three elements 
ruthenium, rhodium, palladium, which clearly may be placed 
under iron, nickel, cobalt. After these three come in ascending 
order twelve elements, silver to cerium, which fall naturally 
into consecutive families as shown in our table. Between 
cerium and tantalum we have all the rare earths, and indeed, 
so little is known of their chemical properties that it is 
impossible to assign any of these to its particular family with 
any degree of confidence. We have, however, ventured to 
assign positions to those rare earths which seem to be known 
definitely, and have followed Emerson Reynolds’ example in 
considering praseodymium, neodymium, and samarium as 
belonging to the iron group. We are convinced that the 
rare earths are very abnormal: they are all characterized 
by being trivalent, and this fact alone, which means that the 
external atomic structure is that of a trivalent element, is 
sufficient to differentiate these from the other normally 
developed elements, for in most cases they cannot be evolved 
simply from trivalent antecedents. However, in the section 
of this paper which deals with the question of radioactivity, 
we put forward a suggestion which may account for their 
extraordinary characteristics. The remaining elements after 
tantalum up to uranium may, without question, be assigned 
to the positions shown in our table. 

We may now turn our attention to the four protons, that 
is to say, the four earliest forms of matter existing in nebule, 
viz., those with wave-lengths 4862, 4959, 5007, and 5876 t.m., 
and endeavour to trace how the process of evolution from 
these gave rise to the eight elements lithium, beryllium, 
boron, carbon, nitrogen, oxygen, fluorine, and neon. We feel 
that we are justified, in dealing with processes of evolution, 
in saying that the most natural tendency will be for the 
process to take place along a direct line of development. It 
is reasonable to assume that the probable effort of any element 
will be to give rise to an element of its own type. We would 


Evolution and Devolution of the Elements. 29 


urge that in elementary evolution, the principle of heredity 
is an all-important one, and that the normally derived elements 
would be characterized by the same properties as the ante- 
cedent elements in each case. Now, in the case of the 
lightest proton, hydrogen, its chemical behavour is that of a 
monovalent element ; and from our standpoint of the existence 
of the principle of heredity (see page 51), we consider that 
the direct descendants of hydrogen must be monovalent 
elements, or, in other words, that hydrogen must be the 
antecedent element of the alkali metals, and therefore we have 
placed hydrogen at the head of this family *. An exactly 
similar argument is applicable to the fourth proton helium, 
and we therefore place helium in the last group of our Table, 
thereby establishing it as the progenitor of the inactive 
gases. 

We now have left two protons, viz., those with wave- 
length 4959 and 5007 respectively. Clearly these two must 
in some way have given rise to all the elements except the 
alkali metals and the monovalent gases. In our table, 
we have placed the proton with wave-length 4959 at 
the head of the alkaline earth metals and the remaining 
proton at the head of the trivalent third group. Now 
inasmuch as we know at present nothing of the chemical 
properties of these two unknown elements, it is clearly 
impossible for us to make any definite statement as to how 
the evolution proceeded from them. Our reasons, however, 
for placing these two elements in the positions assigned to 
them are as follows :—(1) Hydrogen has an atomic weight 
of one, helium of four. The wave-length of the hydrogen 
line is 4862, and that of the helium line is 5876. We there- 
fore assume that the element whose wave-length is 4959 has 
a smaller atomic weight than the element with wave-length 
5007. (2) Silicon gives rise to a great number of elements, 
many of which are to be found in considerable quantities, 
both in the earth and in the stars. It is only reasonable 
therefore to conclude from this that the original proton from 


* At first sight, it might be pointed out that hydrogen can be con- 
sidered as the progenitor of the halogen family, inasmuch as these 
elements are usually spoken of as being monovalent. Against this, how- 
ever, we point out that the monovalency of the alkali metals is equal and 
opposite to that of the halogens. The former are monovalent in the fact 
that they have one free electron to be given up when entering into com- 
bination, while the latter are monovalent in the fact that they take up 
one electron in forming compounds. Hydrogen and the alkali metals 
are absolutely analogous in their compounds with the halogens. If 
hydrogen belonged to the halogen group, hydrogen should be more 
electro-negative than fluorine, and should form mono-atomic compounds 
with the alkali metals, more stable than the fluorides, 


30 Messrs. A. OC. and A. EH. Jessup on the 


which silicon was evolved should be found in very large 
quantities in the nebulee. Now the element whose wave-length 
is 5007 is far the most abundant of the four protons formed in 
the nebulze, and consequently, we consider that this element 
gives rise to carbon and silicon, &c. (3) Starting from lithium, 
in the whole scheme of evolution there is only one point where 
the evolution sub-divides into both direct and indirect series 
of derivatives. This occurs, as we have shown, in the case of 
the element silicon, which gives rise to one direct and ten 
indirect derivatives, and applying our hereditary principle, 
we would say that this tendency to sub-divide is due to the 
fact that such a sub-division had previously taken place, and 
would therefore argue that carbon and silicon are themselves 
indirect derivatives from one of the protons. For this reason, 
therefore, we think that the position of the element 5007 
cannot be at the head of the carbon and silicon groups. 
(4) Moreover, except in the case of the six central groups, 
the discussion of which, owing to their exceptional character, 
we for the moment defer, there is no case in the whole table 
of an element giving an evolution product with a valency less 
than its own*. For these reasons we put the proton with 
wave-length 5007 at the head of the trivalent groups of 
elements boron, aluminium, &c., and we are only left with 
the position at the head of the alkaline earth metals for the 
remaining proton with wave-length 4959. Since this element 
has without doubt a smaller atomic weight than that of the 
element with wave-length 5007, this position seems perfectly 
justified. For convenience we venture to suggest the 
names of proto-beryllium and proto-boron for these two 
elements. | 

Since drawing up this Periodic Table, a paper has appeare 
by Messrs. Cuthbertson and Metcalfe (Phil. Trans. 1907, 
Ser. A, vol. ceviil. p. 135), in which they give a Periodic 
Table of the Elements, which, so far as it is carried, is 
practically the same as the one in this paper. These authors 
have deduced their Table from observations upon the 
refractivities of the elements, and we venture to think that 
this gives considerable support to this particular form of the 
Periodic Table. 

In our new form of the Periodic Table it will be observed, 
as already pointed out, that a large number of elements 
appear as having been formed from silicon. Now, to explain 
their position, and in fact, the actual structure of all the 


* By our theory of the indirect evolution process, the indirect products 
‘must have a greater valency than the parent element (see below p. 33). 


{joe 


0. Sh i a a 


Evolution and Devolution of the Elements. au 


atoms, it will be necessary to enter more fully into the 
question of rings, to which we have already alluded. 

As was stated in our preliminary description of the process 
of evolution, we consider that originally all matter was in the 
form of corpuscles, and that some of these corpuscles arranged 
themselves into stable integral systems, these systems being 
the four elements: hydrogen, proto-beryllium, proto-boron, and 
helium. Hach of these elements began to collect round it 
more corpuscles, and this continued until another stable 
system was formed, capable of separate existence. The 
system or assemblage of corpuscles around the central atom 
may be regarded as having certain properties, and a more or 
less definite shape. We denoted this assemblage by the term 
“ring,” it being understood that we make no assumption as 
to its true form. We, however, consider each ring to have a 
definite structure, and certain properties depending on that 
structure : in fact we conclude that all the principal chenucal 


properties of each element are entirely determined by the 


structure of the ring. 

Now each ring may be supposed to consist of two parts : 
(a) the main assemblage of corpuscles, to which the ring owes 
its mass, size, and shape, and (6) a certain number of cor- 
puseles which are attached to the rest in a slightly different 
manner. They may be perhaps regarded as satellites to the 
main ring, although their connexion is probably much closer 
than that suggested by this illustration. For the sake of 
distinction from the other corpuscles we will apply the term 
“electron ”’ to them. ! | 

From a general study of these electrons there is undoubtedly 
a tendency possessed by them to form into systems of eight, 
or multiples of eight, during the process of evolution. These 
sets of electrons constitute, in some way or another, electrically 
neutral systems. or example, we feel convinced that all 
phenomena of chemical combination are due to this tendency. 
Thus, if an atom with two electrons is brought into 
immediate contact with one possessing six, the two electrons 
will leave the first atom, and form a very stable system of 
eight electrons around a second atom ; and, as we say, upon 
this fact depends the combining power and the valency of 
each element. A fuller discussion of this may well be deferred 
until we have entered into the process of evolution in greater 
detail. 

It will be advantageous in doing this to take a specific instance, 
and for this purpose we will take the element hydrogen. As 
our Table shows, we consider that hydrogen is the progenitor 
of all the alkali metals. Now the hydrogen atom is a 


32 Messrs. A. C. and A. E. Jessup on the 


monovalent element, that is to say the outer ring of corpuscles 
belonging to hydrogen contains one free electron, the whole 
constituting a stable system. We have already stated that 
the tendency of the electron is to form into stable groups of 
eight, consequently we may view the single free electrons of 
hydrogen as being possessed with a definite and measurable 
desire to complete its system of eight. It appears that in 
order to do this it adds on first a single group of eight 
electrons. The result of this would be that the free electron 
on the hydrogen atom completes its system of eight, and that 
one electron will still be left over in the free state. This - 
addition of systems of eight electrons goes on continuously — 
until the next position of maximum stability is reached. This, 
however, as we have before shown, represents the element 
lithium. Now inasmuch as the addition of each group of 


eight electrons in adding itself on, forms with the free _ 


electrons already existing a stable group of eight, and leaves 
still one free electron on the outside, so now we see that each 
evolution product thus normally produced must have the same 
valency as the antecedent element. We therefore find that 
lithium and all the elements normally produced from hydrogen 
must be monovalent. Now without binding ourselves down 
in any way to the actual shape of these rings, we do assume 
that there is some intimate connexion between the valency of 
a ring and its shape. We therefore point out that in the 
process of evolution from hydrogen each ring and corpuscle 
takes on the same shape as the outer ring of the hydrogen 
atom; that is to say, that the rings of all the alkali metals 
have that shape which is typical of the ring of the hydrogen 
atom. Now this addition of rings of the same type as the 
previously existing rings is what we have already spoken of 
as the principle of heredity, that is to say, in all normal 
processes of evolution there is definite evidence of an hereditary 
principle, inasmuch as every valent element possesses the 
same shape and valency as that of its antecedent element. 
Now this process of evolution, which we have described in 
detail in the case of the alkali metals, we consider to be per- 
fectly general throughout the whole of the Periodic Table, in 
so far as what we have called “ direct evolution ” is concerned. 
This normal process of evolution takes place undoubtedly in 
the first three families, and in the last family of our Table. 
The next point we have to deal with, is what we have 
previously defined as “indirect evolution” ; that is to say, 
when series of elements of valency differing from that of the 
parent family are produced. In the case of the first group 
of alkali metals, the members of this family are amongst the 


Evolution and Devolution of the Elements. 33 


most electro-positive of all the known elements ; that is to say, 
they possess a very great tendency to enter into combination. 
In the second group, which possesses two free electrons and 
contains the metals of the alkaline earths, this tendency is 
great, but considerably less than in the group just mentioned. 
In the third group this tendency is still further reduced, with 
the result that the tendency of the three electrons on the 
outer ring of these elements to complete themselves into 
systems of eight is very materially smaller than in the case 
of the first two groups of the Table. Consequently, although 
we have a normally evolved series of elements from proto- 
boron, we also have an indirectly evolved element in carbon. 
We venture to suggest that this may be explained as 
follows :— 

The element proto-boron in its normal process of develop- 
ment, produces the element boron. It, however, is capable 
of proceeding along the lines of indirect evolution also, and 
thus gives rise to the element carbon. Instead of proceeding 
directly along the normal lines from boron to aluminium it 
does noi add on complete rings of electrons, but only takes up 
an insufficient number of sets of eight electrons. These sets of 
eight electrons are absorbed into the outer ring, with the- 
result that a new outer ring is formed of different shape, viz., 
a tetravalent ring. The three electrons in the outer ring of 
boron are not sufficiently desirous of forming complete 


_ Systems of eight to compel the evolution to proceed as far as 


aluminium, but they will allow the incorporation of a small 
number of sets of eight electrons, with the result that instead 
of abstracting five electrons to complete their own system 
they only abstract four, so that the outer ring of the new 
element becomes tetravalent. The acceptance of a smaller 
number of the sets of eight electrons than is necessary to 
produce a complete ring is possible by the incorporation of 
these sets in the previously existing ring, so that a new ring 
of an altered type is produced. This new type of ring we 
define as a “distorted” ring, and we may go so far as to say 
it has a greater valency than the original ring. Now we 
would point out that when the process of indirect evolution 
takes place, and the so-called distorted ring is produced, the 
first step is the establishment of the new ring which then 
proceeds to develop normally until the first position of 
maximum stability is reached. Thus, in the indirect evolution 
from proto-boron, the first stage is the incorporation of the 
necessary sets of eight corpuscles with the establishment of 
the tetravalent ring belonging to the first family of indirect 
derivatives (carbon, silicon, &c.). This tetravalent ring then 


Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. D 


34 Messrs. A. C. and A. E. Jessup on the 


ean evolve normally until the first position of maximum 
stability is reached—carbon. It follows from this, therefore, 
that the formation of these distorted rings (or in other words 
“indirect evolution”) only first becomes possible with 
trivalent elements, and that it is very likely to take place 
with tetravalent elements. We press this conclusion on the 
argument previously developed, that a tetravalent element 
will be still less desirous of completing the system of eight 
electrons than even a trivalent element. Asa natural sequence 
of this, we should expect to find that the element silicon would 
show an exceptional tendency towards this phenomenon of 
indirect evolution, and, as a matter of fact, as we have already 
shown, the element silicon gives rise to a great number of 
indirect derivatives. It is important to notice that neither 
of the elements proto-boron or silicon gives only one series of 
indirect derivatives, for in the former case we have in addition 
to carbon the elements nitrogen, oxygen, and fluorine; and 
in the latter case we have vanadium, chromium, and man- 
ganese. We are convinced that there is some fundamental 
reason underlying the production in this way of several 
indirect derivatives. Taking the case of the indirect evolution 
from the element proto-boron, it will be seen that we have 
derived in this way the elements carbon, nitrogen, oxygen, 
and fluorine ; that is to say, a tetravalent, a pentavalent, a 
hexavalent, and a heptavalent element. 

In speaking of the valency of an element we define this as | 

the number of electrons which that element gives up in 
entering into combination: thus lithium is monovalent, 
beryllium is divalent; nitrogen is pentavalent, because it 
gives up five electrons; so that we speak of nitrogen, 
oxygen, and fluorine as being penta-, hexa-, and hepta-valent 
respectively. 
_ Now we have already shown that in the case of the element 
carbon, the formation of this element is caused by the 
absorption of a number of the sets of eight electrons which 
in themselves are insufficient to form a complete ring. The 
result is that the distorted tetravalent ring which gives rise 
to carbon is formed ; and furthermore, the three free electrons 
on the outer ring of the antecedent element have not taken 
up five electrons to form a set of eight, but only four. In the 
case of the element nitrogen an almost identical procedure 
takes place, only in this case three electrons have been 
extracted in place of the normal five, with the result that a 
second type of distorted ring is produced with a valency 
of five. i 

In the case of the element oxygen a similar absorption of 


Evolution and Devolution of the Elements. 30 


sets of eight electrons takes place, but only two electrons are 
taken up by the outer ring of the antecedent element, with 
the result that a distorted ring is produced of valency six. 

In the case of the element fluorine we have a distorted 
ring of valency seven produced in exactly analogous fashion. 
We would point out that from this point of view a trivalent 
element can only give rise to four indirect derivatives, and 
all of these possible derivatives are accounted for in our 
Table. 

The next case we have to consider is that of the element 
silicon. This is a tetravalent element, and consequently can 
only give rise to three indirect derivatives, viz., a penta- 
valent, a hexavalent, and a heptavalent element. Thus from 
silicon we have the direct derivative titanium, this being the 
case when the superadded ring gives up four electrons to 
complete the system with the four already existing electrons 
on the four-membered ring of silicon. 

We then have the first indirect derivative in which a 
distorted ring is produced of valency 5, when only three 
electrons are transferred. This is the case with the element 
vanadium. We have the third and fourth indirect derivatives 
chromium and manganese which possess distorted rings of 
six and seven valencies respectively. 

Now each of those elements which has been produced by 
a process of indirect evolution such as we have described, 
may itself become the source of new elements and families 
of elements produced from it by normal or direct processes 
ot evolution. Thus the elements nitrogen, oxygen, and 
fluorine will give rise to families of pentavalent, hexavalent, 
and heptavalent elements respectively. 

The three indirect derivatives of silicon, viz., vanadium, 
chromium, and manganese, are penta-, hexa-, and heptavalent, 
but they differ very considerably from the three elements 
nitrogen, oxygen, and fluorine, which have the same valencies. 
The three first mentioned elements are electro-positive, that 
is to say, the electrons in each case exhibit comparatively 
little desire to complete their systems of eight. This is an 
abnormal condition, and one that it is difficult to find any 
reason for. It may be owing io the fact that these elements 
have been produced by two indirect processes of evolution, 
or it may be owing to the size of the outer rings. That the 
three elements, vanadium, chromium, and manganese have 
practically no desire to complete their sets of eight is 
evidenced by the fact that they do not form any compounds 
with hydrogen or with any other electro-positive element. 
Although they are pentavalent, hexavalent, and heptavalent 

9 


= 


36 Messrs. A. C. and A. H. Jessup on the 


respectively, yet these valencies are not usually exerted, the 
salts of these metals being due to the existence of secondary 
valencies of some form or other; in fact, the principal 
valencies are hardly exerted at all, consequently the direct 
derivatives in the process of evolution from these elements 
should be produced in much smaller quantities than in the 
case of the typically pentavalent, hexavalent, and heptavalent 
elements, nitrogen, oxygen, and fluorine. If we take the 
case of the elements vanadium and chromium, it will be seen 
that their direct derivatives which are placed under them in 
our Periodic Table occur much less frequently upon the earth 
than the two elements themselves. 

In the case of manganese, we have a heptavalent element, 
which is the last possible indirect derivative of silicon, and 
we might point out that the heptavalency of this element is 
only exerted in the case of the heptoxide, which itself is only 
known in combination. Conversely we can argue that the 
tendency of the seven electrons of manganese to complete 
their system of eight is extraordinarily small. So small is 
this tendency that manganese is incapable of giving rise to 
any direct derivatives, that is to say, it has not the capability 
of adding on a complete ring and producing a heptavalent 
element more electro-positive than itself. Now the evolution 
process when it has arrived as far as the element manganese 
cannot stop, for manganese is a perfectly stable element, 
showing no sign of radio-activity, and consequently we must 
assume that the evolution process can and does proceed 
further. On the other hand, manganese cannot give rise to 
any indirect products of evolution such as we have described 
above, for when an indirect process of evolution takes place, 
an outer ring is produced with the transference of electrons 
numbering one less than those transferred in the direct process 
of evolution; inasmuch as the direct process of evolution from 
manganese would require the transference of one electron only, 
it is clearly impossible for a truly indirect evolution to take 
place, because this would mean the transference of no electron. 

The oxides and halogen derivatives of manganese show, as 
before stated, that this element nearly always acts by secondary 
valencies and very rarely by its principal valency. We 
should therefore expect that any evolution derivatives of 
manganese should be peculiar and quite different from any 
which have previously occurred. The only thing that man- 
ganese can do is to add on very small quantities of matter, 
and give rise to substances with quite abnormal rings. 
Theoretically these rings should have a maximum valency of 
eight, that is to say, rings which could give rise to tetroxides. 


Evolution and Devolution of the Hlements. 37 


under certain conditions; at the same time, these rings 
should have extraordinary properties inasmuch as we can 
only view the evolution from manganese as being in the 
nature of plastering up its ring, so to speak. For these 
reasons, as well as from the facts of stellar evolution, we 
consider that iron, nickel, and cobalt are the products of this 
operation. Now the properties of the external rings of these 
elements are certainly very abnormal. These three elements 
are strongly magnetic and so we may look upon their rings 
as being to a certain extent plastic ; that is to say, they are 
readily deformed under the influence of a magnetic field. 
It is perfectly true that none of these three elements shows 
the expected valency of eight, but in the elements ruthenium 
and osmium, which are the direct derivatives of iron, we do 
find evidences of this octo-valency. We would argue from 
this that the octovalent character is latent in the case of iron 
&c., that it becomes more pronounccd in the case of ruthenium, 
which gives a tetroxide known in combination, and still more 
pronounced in the case of osmium, which gives a perfectly 
stable tetroxide. 

It must be pointed out here that there is a very considerable 


difference between the octovalent ring of these elements and 


the saturated eight-membered ring of the elements belonging 


to the helium family. In the first case, we have a ring which 


ean under suitable conditions give up eight electrons, but 
which prefers to act by secondary valencies, especially the 
valencies of two, three, and four. On the other hand, the 
rings of the elements of the helium family must be looked 
upon as saturated rings which have no power to act by 
secondary valencies at all. 

There is no doubt that the elements iron, nickel, cobalt, and 
other direct evolution products represent a somewhat unsatis- 
factory condition of affairs, and that each one of them will 
tend in the natural course of evolution to recover itself 
and form more or less ordinary derivatives. We think that 
the elements copper, zinc, gallium, and germanium, and the 
elements directly below them, represent these recovery 
products. That is to say, copper, zinc, gallium, and ger- 
manium are the successive recovery products of iron, nickel, 
and cobalt, while silver, cadmium, indium, and tin are 
the successive recovery products of ruthenium, rhodium, 
and palladium, and that gold, mercury, thallium, and 
lead are the successive recovery products from osmium, 
iridium, and platinum. We will return to this question of 
recovery again when dealing with the electropotential series 


of Professor Abege. 


38 Messrs. A. C. and A. EH. Jessup on the 


In discussing the position of the four protons, viz., hydrogen, 
proto-beryllium, proto-boron, and helium, we placed helium: 
at the head of the family of rare gases, saying that it acts as 
the progenitor of this family. Ofcourse it is usually assumed 
from ordinary chemical evidence that these elements have no 
powers of entering into combination, and it might be argued 
from this that it would be impossible for helium to produce 
evolution products of its own type. Unless, however, we 
make the somewhat unjustifiable assumption that the elements 
neon, argon, krypton, and xenon are entirely devolution 
products of some radioactive processes, we must assume that 
they possess some form of residual valency. It is quite true 
that no definite evidence of this residual valency has yet been 
proved by our somewhat crude methods of chemical obser- 
vation. There is no doubt in our own minds that this residual 
valency does exist, and we are in great hopes that before very 
long some physical method may be found of determining its. 
existence and its amount. | 

In connexion with the general scheme of evolution as set 
forth, a most interesting point arises which is well worth 
noticing. We have already stated that the tendency of the 
four electrons on the outer ring of the element silicon to 
complete the system of eight is very small, and that, conse- 
quently, the tendency to form direct derivatives will be 
relatively small. The tendency therefore of the element 
silicon to give rise to indirect processes of evolution will be 
relatively greater; and in this fact we have an explanation 
of why the indirect derivatives of silicon occur in so much 
greater quantities than the direct derivatives. The indirect 
derivatives to which we especially refer are manganese, iron, 
nickel, cobalt, copper, and zinc. 

There is one most important aspect of any evolution: 
process which must not be forgotten. As stated in our pre- 
liminary account on page 23, the whole process of evolution 
must of necessity be governed by the fact that evolution is 
attended by decrease of energy content, for it is perfectly 
clear that this is the fundamental law of physical or chemical 
change. If we consider any particular series in the Periodic 
Table, we should expect to find some evidence of this decrease 
of energy content. It is naturally impossible that we should 
be able to express in any way the actual energy content of 
any atom, but we can gain considerable information by 
making comparative measurements along any series. For 
example, if we take the series of the halogens, we find that the 
first member, fluorine, has very powerful chemical properties ; 
that is to say, the desire to pick up electrons to complete the 


Evolution and Devolution of the Elements. 39 


set of eight is extraordinarily strong in this element. In other 
words, it is exceptionally “ electro-negative.” 

If we take the next element chlorine, we find the same 
properties, but to a less degree, and following through bromine 
to iodine, this desire to enter into combination decreases. 
This fall in electro-negative behaviour may be considered as 
being due to the fact that as the evolution progresses a more 


_ and more dense system is formed : that is to say, more and 


more energy escapes. 


Now Professor Abegg* has published a most important 
Table of the Values of the Hlectro-potentials of the Elements, 
and from this we can see at once that the tendency of the 
elements to take up electrons always falls as the direct process 
of evolution progresses, that is to say, the electro-positivity rises _ 
precisely as we should expect, if our theory is to hold good: 
We give a diagram showing Abegg’s values, which is modified 


a 
< SS Soe 
BSSer a 
CNIUND Nel PAARL 
TAS NP 
EAN S/N 
PUSS 


ol RRR SRE 
SE ee eS 


slightly so as to show more conveniently the relations between 

the families in our Periodic Table. It can at once be seen that. 

the values of the potentials as given by Abegg do not fit in 

with the Periodic Table as usually adopted at the present time. 

They do, however, agree much better with the groupings of the 
* Zeit. Anorg. Chem. 1904, xxxix. p. 330. 


40 Messrs. A. C. and A. HE. Jessup on the 


elements set forth in our Table. A most interesting point 
arises in connexion with the potentials of those elements 
which arise from manganese. We have shown above that the 
element manganese first gives rise to iron, nickel, and cobalt, 
whose outer rings theoretically are octovalent. This is 
accompanied by a considerable fall in the value of the | 
potentials, This is a perfectly natural sequence, because it is 
quite evident that elements possessing this type of octovalent 
ring should show considerably decreased affinity, or, in other 
words, that their electro-potentials should be small. 

Now we pointed out that this octovalent ring must not be 
looked upon as being completely established in the case of 
iron, nickel, and cobalt, and that it becomes more established 
as the normal evolution proceeds through ruthenium €c., to 
osmium &c. This establishment of the octovalent character 
is naturally accompanied by a still further fall in value of 
the potential. When the recovery occurs from iron, nickel, 
and cobalt, to copper, the value of the potential falls to. a 
minimum and then rises again to copper, and a still further 
rise occurs when the evolution proceeds to zinc, and again to 
gallium. A similar rise in the potential takes place when 
the recovery passes from ruthenium, rhodium, and palladium 
through silver and cadmium to indium, and again from 
osmium, iridium and platinum, gold, and mercury, to 
thallium. 

It will be noticed that the recovery from ruthenium, 
rhodium, and palladium is steeper than the recovery from 
iron, nickel, and cobalt, and further, that the recovery from 
osmium, iridium, and platinum is even steeper still. The 
result is that the three recovery curves cross one another 
after passing through gallium, indium, and thallium before 
they arrive at lead, tin, and germanium. At first sight it 
might be expected that the three elements germanium, lead, 
and tin should have a higher value of potential than gallium, 
indium, and thallium respectively ; but it must be remembered 
that the three former elements are tetravalent, that is to say, 
the tendency of the four electrons to enter into combination 
is distinctly less than in the case of the gallium, indium, and 
thallium triads. The values of the electric potentials there- 
fore of lead, tin, and germanium should be less than those of 
gallium, indium, and thallium. 

We may point out that this accounts for the fact that the 
electric potentials of the three groups, copper, silver and 
gold, zinc, cadmium and mercury, gallium, indium and 
thallium are inverted. It is for this very reason that we 
do not consider these three groups to consist of normally 
developed elements, that silver and gold, for example, are 


ee 


Evolution and Devolution of the Elements. 41 


not direct evolution products of copper. In support of our 
view that the individual members of these three groups are 
not connected by a process of direct evolution, we would also 
mention that the three elements zinc, cadmium, and mercury 
have been shown by Cuthbertson and Metcalfe * not to be 
connected in their refractivities. 

_A remarkable result has recently been obtained by J. J. 
Thomson +, which is entirely in agreement with the theory 
of evolution as set forth above. He has found that the number 
of corpuscles in an atom approximates to the atomic weight. 
Thus in the atom of hydrogen there is 1 corpuscle, and in a 
molecule of air there are 25-28. Now, with our hydrogen 
is associated 1 electron, with nitrogen 8+5=13 and oxygen 
8+6=14, which would give about 27 to 1 molecule of air. 
Can it be these free electrons which have been measured ? 
The objection to the result is that there are many more lines 
in the are spectrum of iron which give the Zeeman effect, 
than there would be electrons. However, when subjected to 
the great strain of the electric are, it is highly probable that 
some of the corpuscles from the outer ring of the atom are 
set into vibration, as well as the electrons, and this would 
afford an explanation of the number of lines. 

However, to return to our main subject. We have traced 
the growth of all the elements, and will now proceed to 
discuss the relations of their properties. 

Firstly, members of the same family are formed from the 
same protons, and have the same shaped rings. Hence there 
must be a very close relationship in all their properties, as is 
the case in groups 1, 2, 3, and 8, where the structure is 


perfectly regular. Groups 5, 6, and 7 are also very similar ; 


but the first members differ somewhat from the others, because 
the first ring is situated round one of different valency, or shape. 

Again, on the outer ring will depend a certain number of 
properties. Two elements with the same outer ring will show 
several similarities. Their spectra will probably be of the 
same type, and their valencies being the same, they will form 
similar compounds. 

Now Rudorf f obtained certain relations between frequency 
differences of pairs of lines in the spectrum of an element 
and its atomic weight. On plotting the values of d/A? against 
those of A, where d is the frequency difference for an element 
of atomic weight A, he obtained characteristic curves for each 
family ; moreover, the point representing sodium was at the 
intersection of the curves through K, Cs, Rb, and Cu, Ag, Au. 

* Loc. cit. 
t+ Phil. Mag. vol. xi. p. 769 (June 1906). . 
} Zeits. Phys. Chem. vol. 50. p. 100 (1904). 


42 _ Messrs. A. C. and A. E. Jessup on the 


Consequently, spectroscopically, sodium belongs to both these 
families, a fact which confirms our supposition. Precisely 
the same holds in the case of magnesium. 

Copper affords an example of special interest. As above, 
its spectrum is seen to resemble that of sodium, but there are 
also lines which are similar in type to those of tin, lead, and 
some other heavy metals. The explanation of this fact is that 
the outer copper ring is of the same shape as the outer sodium 
ring, but yet the internal structure of the two is different, 
that of copper being similar to tin and lead. 

The properties of an element will, however, not depend 
solely on its outer ring, but also upon two other main causes. 
These are (a) the actual number of rings in an atom, and (6) 
the way in which the rings are superimposed on each other. 

The effect of the first cause upon the properties of the atom 
is simply that the larger the number of rings, the more 
unstable the atom, or we may perhaps say, the greater its 
tendency to disintegration. The second cause has been 
already touched on; it may be mentioned, however, that 
wherever a whole atom is composed of rings of the same kind, 
the outer ring will behave normally ; but where the rings are 
not of the same kind, there will be what may be looked upon 
as distortion, and consequently abnormal properties are to be 
expected, and the greater the valency of a ring the more 
marked the distortion will be when such a ring is super- 
ee on one dissimilar to itself. 

e are now in a position to discuss the influence of the 
constitution of the elements on their chemical reactions. We 
will, for the sake of clearness, limit ourselves mostly to simple 
combinations of two elements. 

Chemical compounds are formed entirely by an exchange 
of electrons between the combining atoms; but the number of 
electrons taking part in the exchange, and the ease with 
which they do so, depends on the structure of the atom. 

As before stated, the chief tendency of the electrons is to 
form into systems of eight, and when this occurs, what may 
be called the principal valency is exerted. As an example, 
calcium with two electrons will give them up to oxygen 
which has six; when this occurs the calcium will become 
positively, and the oxygen negatively charged with “elec- 
tricity,’ and combination results. To secure uniformity we 
may say that in this change calcium is divalent and oxygen 
hexavalent, the principal valency being the number of 
electrons the atom can give, or eight minus the number it 
can receive. Thus nitrogen will take three electrons from 
hydrogen, or give five to oxygen, being in both cases penta- 
valent. Simple changes such as the above always take place 


Evolution and Devolution of the Elements. 43, 


in such a way that the number of elevtrons transferred is a 
minimum ; so, with sodium and chlorine, the sodium will give 
one electron to the chlorine, and not receive seven from it. 
That this is so, is evident from the following considerations. 
Given that a system of either eight or zero electrons round an 
element is the most stable, an element with seven electrons, 
or with one only, is very near the stable position. In 
the first case, the element with seven electrons will very 
readily acquire another ; and in the second case, the element 
with only one will very readily give it up. So also an element 
with two electrons will readily lose them, but not so readily 
as an element with only one ; three ae will be lost or 
gained still less readily ; and finally, in the case of an element 
with four electrons, it is a matter of indifference whether 
others are gained or lost. In other words, monovalent 
elements may be said to be, in general, the most electro- 
positive, heptavalent the most electronegative, and tetravalent 
ones to be electrically neutral. 

Loss or gain of electrons, according to the above ee 
constitutes the principal valency of an element, and it must 
be noted that more than four electrons never ieave or approach 
one element, and that, moreover, unless the outer ring is 
highly distorted, there is only one ‘principal valency. 

Now in addition an element may have secondary valencies. 
This idea is due to Abegg, but he assumes that each element 
has one principal and one secondary valency, and that the 
sum of these is eight. Also in the first three groups the 
secondary valency is latent. It seems more reasonable to 
suppose that an element may have more than one secondary 
valency, and apparently these secondary valencies are either 
all even or all odd. If this be so, we may tabulate the 
number of electrons which any given ring will lose or take, 
bearing in mind that there is only one principal valency. Thus: 


A monovalent ring will lose if electron or acquire 1, 3, 5, or 7. 
A divalent * 4 2 E i, 2, 4, or 6. 

A trivalent 6 (3 3 e 3 3 or 5. 

A tetravalent ., ei 2 or 4 a Ms 2 or 4. 

A pentavalent _,, fe 3 or 5 A BY 3. 

A hexavalent __,, eS 2, 4, or 6 & * 2. 

A heptavalent _,, A 1 Sno OEE a ie an 

An octovalent _,, 0 Q), 


3) ” 


In this it will 3 noticed that the tetravalent ring has no 
principal valency, and this would be expected, since it is the 
connecting link between elements with positive and negative 
principal valencies. 

In accordance with the principle of minimum movement of 
electrons, it will be seen that when an element has several 
secondary valencies,1it will for preference exert that which 


44 Messrs. A. C. and A. E. Jessup on the 


involves the least interchange of electrons. An excellent 
example of this is afforded by the halogens. These elements 
form the following compounds among themselves :—IF;, 
ICl;, and IBr; BrF; and BrCl. Fluorine is electronegative 
enough to make.iodine give up five electrons, chlorine will 
only make it give three, and bromine can extract one only. So, 
bromine will give three to fluorine, but only one to chlorine. 

Now it has been already pointed out that if the atom is 
built up of rings of different valency, or, in other words, is 
deranged, the properties will be affected, and consequently 
the above statements as to valency will not rigidly hold in 
such cases. A typical example of this is afforded by nitrogen, 
where there is a pentad ring on a triad. Here the principal 
valency is unaltered, being five, and five only, as nitrogen 
will take up only three electrons from hydrogen to form 
NH;. NH and NH, are not known. The secondary 
valencies are, however, affected, with the result that nitrogen 
can lose either two or four electrons in NO and NO, A 
similar state of affairs exists with vanadium, chromium, and 
manganese, where both even and odd valencies are. found. 

The next thing is to consider the valency of the elements 
normally derived from the above. It would naturally be 
supposed that, as these elements add on rings of the same 
valency as the outer one of the elements from which they are 
derived, their structure would tend to become more regular, 
and this is found to be so with nitrogen, oxygen, and fluorine 
derivatives. We should expect this influence to be best 
evidenced in the cases of antimony, tellurium, and iodine; 
and it is undoubtedly shown by the compounds of these three 
elements, Sb.03, Sb,0O;; TeO, ‘'eO,, TeO;; IBr, ICl;, 1,0;, and 
I,0; Gn compounds). Here, antimony exerts the secondary 
valencies of 3 and 5, tellurium of 2, 4, and 6, and iodine of 1, 
3, 5, and 7,so that they have become perfectly normal. The 
intermediate elements are in a state of transition so to speak, 
and therefore nothing can be definitely argued from them. 

In vanadium and chromium derivatives, however, there is 
practically a double derangement, since in the first place 
carbon is deranged (tetrad on triad), and then either a penta- 
or hexavalent ring is added to the tetravalent ring of silicon. 
The result is that these elements do possess even and odd 
valencies ; for example, molybdenum forms MoCl,, MoCls, 
MoCl,, MoCl;, and probably MoCl,. 

In the series carbon, silicon, titanium, zirconium, there is 
only one derangement, and consequently, only even valencies 
are found. 

Before concluding this section of our paper, it might be as 
well to make a few remarks on the rare earth elements. It 


Evolution and Devolution of the Elements. 45 


will be seen that we have assigned positions between cerium 
and tantalum to those which are known definitely. According 
to our theory, their resemblance is due to a special charac- 
teristic trivalent ring, which largely masks any other 
properties resulting from internal structure. Moreover, 
since so little is known of them chemically, it is unsafe to 
say whether we are justified or not. There are, however, 
three of them about which a little more is known, viz. praseo- 
dymium, neodymium, and samarium. Emerson Reynolds * 
has given several good reasons for supposing them to belong 
to the iron group. One of these is the high paramagnetic 
susceptibility of their salts. Now we have already stated 
that the iron group is especially characterized by the mag- 
netic properties of its first three elements, which arise from 
the presence of an abnormal ring. As evolution proceeds, 
this ring is not destroyed, but still exists inside the atom, 
and hence by our theory we should expect the derivatives 
of the first three elements to exhibit this particular property, 
but to a less degree. So, if it is granted that these three 
elements may be placed among the platinum metals from 
which they at first sight differ so greatly, it is to be hoped 
that no objection will be taken to the more or less temporary 
positions we have assigned to the others. 


We have now discussed our theory of the method of growth 
of the atoms, and if our surmise is correct, would it not be 
reasonable to suppose that the table of atomic weights as 
rearranged would exhibit some quantitative relations between 
the weights of the atoms of the elements? If the atomic 
weights of the elements of group I. are written down in order, 
and each subtracted from the one that comes after it, the 
following numbers result :— 

6°02 16-02 16°01 46°3 AT-4, 
and these numbers may be regarded as the weights of the 
outer rings of the elements from lithium to cesium. 

Examining the other groups, it is found that in general 
the differences are again 6, 16, and some number between 40 
and 50 ; but whereas all the numbers belonging to the first 
class are very nearly 6 or 16, and that though the other 
numbers vary considerably, they approximate to a mean value 
of 46, showing a definite repetition and relationship between 
them. If we again consider the numbers. 6, 16, resulting 
from group I., we note that they may be written : 


(14+24+3), and (14+2+3)+(1+243+44) 
respectively. : 
* Chem. Soc. Journ. vol. Ixxxii. p. 619 (1902). 


46 Messrs. A. C. and A. E. Jessup on the 


The next term of a series of natural numbers arranged in 
this way which approximates most closely to the average 
number 46 is— 


(14+243)4+ (1#24+344)+ (E72 
+(14+24+34+445+46) = 52. 

Now the average value of the differences is, as we have seen, 
approximately 46, a considerable diminution from the figure 
as given by the above series. In connexion with this, it is 
worthy of note that Runge and Precht* determined the 
atomic weight of radium spectroscopically, and found it to 
be 258, while the value found by chemical methods was 225. 
Rudorf +, in seeking to explain the discrepancy, made a 
series of observations upon the relation between the frequency 
differences of spectral lines of families of elements. He found 
that if d is the frequency difference of pairs of lines in the 
spectrum of an element atomic weight A, d/A? is approxi- 
mately constant for low values of A, but increases with A 
for higher values of A. This means that the elements 
towards the end of the series have a lower atomic weight 
than would be the case if the formula d/A?=constant were 
rigid. 

The phenomenon appeared in all the series of elements 
which were examined, and not only in group II. 

Thus we have another example of a formula yielding 
higher values for the atomic weights than those actually 
found. May it not be that these formule give the atomic 
weights which the elements would naturally assume if not 
subject to some disturbing influence? In other words, is it 
unreasonable to suppose that we are here face to face with 
the process of devolution ? 

We know that this process exists in a marked form in the 
later stages of each group, in the form of radioactivity. Is 
it illogical to postulate that its origin occurred at an earlier 
stage in the growth of the element ? 

If we may regard the formation of the elements as a com- 
bination of the processes of evolution and devolution, the 
former is most effective among the first elements of a group, 
and the latter among the last. 

Turning to devolution, and again examining the table of 
atomic weights, it is evident that there is no simple law 
connecting the various elements of high atomic weight, such 
as exists for the evolution. Moreover, since any term of the 
series of devolution is only obtainable when the difference of 
the atomic weights of two consecutive elements in any group 


* Phys. Zeitschr. vol. iv. p. 285 (1903). 
+ Zeit. Phys. Chem. vol. 50. p. 100 (1904). 


Evolution and Devolution of the Elements. AT 


is known, and owing to the many uncertainties which exist in 
the atomic weights of the elements, and the number of them 
which are not yet discovered, it is exceedingly difficult to 
obtain even an approximation to a law for devolution, 

However, there is one group which is fairly complete, viz. 
group II., and if any law is to be derived, it is naturally to 
this group that we should look for information. 

The atomic weights of the members of this family are as 


follows :— 

Proto-beryllium ......... between 1 and 4 
teria LUPIR RI 2334s Do een oh as < a ye 9-1 
MErermesttany PS: 222. oes PEAS. a4°3 
OS a eee AO*1 
nMeeEEGLUAIEN: fs. oe oc dete ee a 87°6 
ert e028. wey eee 137-4 
Peogue-radwnr. 222... 0. 22. Ji 169-172? 
3" ee eS ee 


The differences between these weights taken in order, 
omitting proto-beryllium for the present, are :-— 


Beryllium to magnesium ......... 15°2 
Magnesium to calcium ............ 15°8 
Caleium to strontium ...........- 47-6 
Strontium to barium ............. 49°8 
Barium to proto-radium, about... 34 
Proto-radium to radium, about... 52 


If we subtract the numbers given by the series, which 
correspond to the above values (namely 16 and 52), from 
these values, we obtain the following numbers :— 


Beryllium to magnesium ......... —0°8 
Magnesium to calcium ............ —0-2 
Sxlemm to strontium | .......-:.< —4-4 
Strontium to barium ............ —2°2 
Barium to proto-radium, about... —18 
Proto-radium to radium, about... 0 


It will be observed in the foregoing numbers, that alter- 
nate members —0°8, —4°4, — 18, form a descending series ; 
and if we treat other groups of elements in the same way, 
their alternate numbers also invariably form descending 
series, but, as before mentioned, the number of members in 
these groups being incomplete, we can only observe the 
above-mentioned fact, and are unable to deduce any 
mathematical formula in their case. 

Returning to the series —0°8, —4°4, —18, it is evident that 
this represents the devolution from the standard series 6, 16, 
52. Let us now consider the remaining terms of the series, 
namely —0Q°2, —2°2. and 0. 


ie 


48 Messrs. A. C. and A. EH. Jessup on the 


These may be expressed in the form —0°8+0°6, —4:44+2:2, 
—18+18,and may thus be regarded as a growth, or recovery 
from the values given above for the devolution of the elements 
of group II. To be more explicit, we may say that alternate 
members of group II. are subject first to a period of evolution, 
and then of devolution, and that the remaining elements of 
this group are subject to both these influences and, in addition, 
a further return of evolution. 

A close approximation for the two series —0°8, —4-4, —18, 
and 0:6, 2°2, 18, may be written as follows :— 


Loss. Recovery. 
Be 4(d-1) 7 a bg, 
eet i) agua ail 
Pele) 2 


The general expression for the loss if written as above is 


11 ee 
jo 2?-” where p=2, 4, or 6; 


or, in other words, is the position of the corresponding element 
in the original series. 
The general expression for the loss and recovery of the 
odd members is 


Use oC) 5; 
Ties 5 2h where—p = 1, 3,.5, or 7, 


these numbers denoting position in the series as above. 

Returning to the original series for the evolution, viz., 6, 
16, 52, the general term may be written in the form 

ek n—I a—1 
+ 2)(2 + 3)(2 + 4) — 22. 
31 

where n = 1 for the first row of the table, 2 for the next two 
rows, and 3 for the last four. 

Thus the complete formula for the evolution and devolution 
of group II. is 


n—1 ma—1 na—1 


(2+ 224 Qe ce, ped id i 


for the even pA and 
na—1 n—1 n—l 
@+2)2+32@+4 -4l Lie», Woy 
3! IDF 10 


for the odd. 
If we employ this formula, reckoning differences from 
calcium as standard, we obtain the following values for the 


Evolution and Devolution of the Elements. 49 


atomic weights of group II., and may compare them with 
those given in the international table :— 


Value from International 

Element. formula. value. 
Prote-beryllium.......... 1:3 — 
Ieewingm 2... 2-22. 9-2 9-1 
Maenesm ........2.:.. 24:1 24°36 
Beet i 40°1 40-1 
2 Ligh rr 87°7 87°6 
eevee |. Bort eee 137°5 137°4 
Proto-radium (Thulium ?).. 171°9 Lis 
MRAM hss). ss 223°9 225 


It is noteworthy that the value obtained by the foregoing 
formula for the element which precedes radium is 171°9. 
This is in close agreement with the usually accepted value 
for thulium, and thus substantiates the position in which we 
have placed it. In addition, putting p=1 and n=1, we 
obtain the value 1°3 for the atomic weight of the element 
preceding beryllium. This is in agreement with the fact that 
this weight should lie between J and 4, and the actual value 
is probable, when viewed in the light of considerations which 
are to follow. 

Before proceeding we may draw attention to some 
interesting facts concerning our original series for evolution, 
viz., 6, 16, 52, or as we wrote it :— 


6=(1+2+4+3) 
16=(14+243)4+(14+243+4) 
52=(14+24+3)4+(14+2+844)+(14+24+844+4+5) 
+(14+24874+5+46). 

There is one term of 6, two of 16, and four of 52, but in 
the growth of the elements as shown by the table, there is 
one growth of 6, two of 16, and four of 52. 

Moreover, in term 1 there are three digits, and though in. 
the first row of our table there are four elements, yet of these 
only three are electrically valent. 

Again in term 2 there are seven digits, and seven 
electrically valent elements in the next two rows of 
the table. 

In term 3 there are eighteen digits, and in the last rows of 
the table there are seventeen electrically valent elements, and 
this perhaps points to a relationship between the number of 
the digits in the above series, and number of electrically 
valent elements, in the corresponding rows of the table. 

Stating this in another manner, the increase in the number 


Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. iE 


i be od 
i") 
> 


50 Messrs. A. C. and A. E. Jessup on the 


of digits from the second term to the third is eleven, which 
is the number of elements to which silicon gives rise. 

With reference to the rise and fall which we have developed 
in our mathematical equation as shown above, we would point 
out that if there is any real basis for this phenomenon, it must 
be exhibited in the properties of the elements concerned. 
That there is indeed a very striking connexion is clearly 
shown in the following Table of oxides and chlorides which 
are known to exist for the elements of the nitrogen, oxygen, 
and fluorine groups :— 


NCI, FE Cl, As Cl, Sb Cl, Bi Cl, 
— le a Sb Cl, — 
S Cl, _ Te Cl, 

SCL, Se Cl, Te Cl, 

SO, Se O, Te O, 

S 0, — Te O, 
Cl OH Br OH ITOH 
C10 0H — Cl. 
ClO,0H BrO,OH I10,0H 
Cl 0,0H — 1 0,0H 


Moreover, since the members of the first three groups do 
not commonly exert more than one valency, we cannot expect 


to find any phenomenon analogous to the above in their ease. . 


Also in the silicon family the sub-groups are too small to 
show the same, and consequently any rise and fall in affinity 
as shown by variable valency should only be looked for in the 
nitrogen, oxygen, and fluorine groups, and it is precisely here 
it does occur. From this it appears that the alternate members 
of these groups, viz., nitrogen, arsenic, bismuth, selenium, and 
bromine, do not exert their valencies to such an extent as the 
remaining elements. The pentachlorides of nitrogen, arsenic, 
and bismuth do not exist, and though As,O; is known, it is 
much less stable than P,O; or Sb,O;. Again, selenium is 
only tetravalent in its simple compounds with oxygen and 
chlorine, whereas the other members of its family are divalent, 
tetravalent, and hexavalent as well. The selenates with 
hexavalent selenium exist, but are comparatively unstable. 
Finally, in the halogen family, no compounds of bromine are 
known iv which this element is tri- or heptavalent. 

While offering no explanation for this pheuomenon, we 
think that it is at least highly interesting, especially when 
viewed in the light of the rise and fall in atomic weight given 
by our formula. 

We would further point out that the maximum fall occurs 
in the case of the rare earth thulium in the second group. 


Evolution and Devolution of the Elements. 51 


That is to say, this element has been formed with a con- 
siderable fall in atomic weight from its theoretical value. 
We have endeavoured to derive expressions for the atomic 
weights of all the other groups of elements, but we have not 
been entirely successful owing to the want of knowledge 
of the atomic weights of the highest members. We find, 
however, that the same periodic phenomenon of rise and fall 
takes place in all the groups, and moreover that a rare earth 
element always appears at the position of the maximum fall. 
This fact is an experimentally suggestive one in connexion with 
the phenomenon of radioactivity. Now the chemical pro- 
perties of the rare earths are apparently extremely complex. 
The fact that all these substances are trivalent we attribute 
to the fact that they possess an outer trivalent ring. Of the 
internal structure of the atom in each case it is impossible 
to speak with any certainty: in fact there is little doubt 
that these substances are quite abnormal. 

Now inasmuch as these elements are produced at a period 
of maximum fall in the evolution of each family, it would 
appear that they are the result of the degradation of substances 
of higher atomic weight. It does not seem far from im- 
probable that the final degradation products of all radio- 
active elements should be a rare earth. 

In the last section of our paper we introduced the idea of 
a process of devolution, and the natural result of such an 
idea is an inguiry as to whether the structure which we 
assigned to the atoms to account for their evolution will also 
account for what is known of their devolution. As before 
mentioned, we have undoubted cases of devolution in the 
radioactive bodies, and a large number of most interesting 
facts have recently been observed in connexion with them 
by many workers. 

Of all these facts, perhaps the most interesting, from our 
point of view, is that the mass of the Alpha particles of radium, 
uranium, thorium, and actinium have been found to lie 
betweeen the masses of the atoms of hydrogen and helium. 
Now in developing our theory of evolution we laid great 
stress upon what we described as the principle of “‘ heredity ” 
in normal direct evolution ; that is to say, we consider for 
example that the outer rings of the elements, with the 
exception of thulium, in the second group, are of the same 
shape, viz., that shape corresponding to the valency of two. 
Perhaps our meaning in this connexion might be rendered 
clearer by taking an analogy with crystallization. When a 
solution of a salt is sown with the proper form of crystal, 
the salt proceeds to crystallize out along the lines which 
have been as it were laid down for it. Although perhaps 

2 


52 Messrs. A. C. and A. E. Jessup on the 


this analogy is not very good, yet it does more or less 
express our meaning, for we might consider that the 
element proto-beryllium acts as it were as a seed, and its 
evolution products grow from it, preserving very much the 
same shape and characteristics. We do not necessarily mean 
that the successive products of evolution are merely magnified 
products of the original antecedent, but rather that each 
evolution product consists of groups of nuclei, each nucleus 
having the shape of the original antecedent element, proto- 
beryllium. 


On these grounds, it is not unreasonable to suppose that if 


in the process of evolution an unstable element were produced 
it would break down and give off one or more of these nuclei. 
This nucleus would have a mass equal to that of the original 
antecedent element. It is these nuclei which in our opinion 
determine the particles. As mentioned previously, we believe 
that the atomic weight of proto-beryllium and proto-boron 


lies between 1 and 4, and this does not differ from the results. ; 


so far obtained for the mass of the particles. Ifit be granted 
that an « particle is a proton, we can trace the change through 
which an atom formed according to cur theory would go, 
For example, let us consider the case of radium. The atom 
is approximately 224 times the weight of a hydrogen atom, 
and has been evolved from an element haying an atomic 
weight not far from 1°3. That is to say, the outer ring of the 
element radium could be looked upon as being composed of 
nuclei, each nucleus being, by our principle of “heredity,” 
of the same type and shape as that of the original element 
proto-beryllium. Now this element, radium, is unstable. 
it proceeds to lose one of its nuclei. The whole of the 
remaining atom will then seek to find the next position of 
stability. It might take the position of the last element in 
group L., but this would be no more stable than its original posi- 
tion; and so it seeks to establish itself in group VIIL., forming 
the emanation. Now when the e particie, which we believe 
becomes an atom of proto-beryllium, leaves the atom of 
radium, the outer ring of radium is left with groups of eight 
electrons. That is to say, in so far as its outer ring only is 
concerned, the emanation belongs to the helium family. Now 
this emanation again proves to be an unstable atom, possibly 
owing to the fact that its outer ring is of quite a different 
type to the inner ones. It again breaks down giving up 
more a particles. Now inasmuch as the outer ring of the 
emanation is of the helium type, so the particles are 
helium atoms. 

So, when radium turns to the emanation the proto-beryllium 
protons must turn into helium protons. The single corpuscles 


a 


Evolution and Devolution of the Elements. 53 


are probably not deranged during the transformation, since 
none are given out; hence we may assume that the only 
change which takes place is in the protons, and that this is 
of the nature of polymerization. Consequently, an integral 
number of radium protons must form the atom of helium, 
and if this be so, the atomic weight of the proto-beryllium 
proton must be 1°33 or 2, so that either three, or two of them, 
unite to form helium. 

Now the value given by our formula in the last section, 
for the atomic weight of proto-beryllium, was 1°3 ; moreover, 
we are about to show that there are reasons for believing 
that the atomic weight of proto-boron is 2, and consequently 
we must assume that the atomic weight of proto-beryllium 
is 1°33. We will now consider the behaviour of thorium 
and actinium. These elements resemble each other in their 
method of disintegration, and consequently they may be 
treated together. 

By thorium, we mean the element which gives rise to 
thorium X, its emanation, and its active deposit. 

In the cases of thorium and actinium, we have two heavy 
atems, in all probability heavier even than radium, and com- 
posed of proto-boron protons. When one of these particles 
is expelled, the atom, as in the case of radium, will try to 
assume a more stable form. Now since the rare earths are 
all formed by the preceding elements adding a trivalent ring, 
it appears that such a ring is at any rate a configuration of 
great stability, and it should not surprise us if an element 
when disintegrating should form a stable derivative with a 
trivalent ring, and that this is the X derivative. In other 
words, thorium and actinium will have been transformed into | 
thorium X, and actinium X, with trivalent outer rings, but 
they will have the same sized a2 particles, as their protons 
have as yet undergone no change. If this is the case, we can 
see why thorium, actinium, and uranium should form 
thorium X, actinium X,and uranium X, while radium should 
be immediately transformed into its emanation. For the 
weight of radium being below that of the element which 
would occupy the last position of group III., radium in the 
process of devolution cannot turn into this element, but goes 
at once into the helium family. The other elements have a 
higher atomic weight, and do form this element as they decay. 
This involves the fact that all the X’s are the same substance, 
and we hope at a future date to give reasons for believing 
that this assumption is correct. 

The X elements when formed will give off an « particle, 
and, like radium, become members of the helium family. By 
a method of reasoning similar to that in the case of radium, 


54 Evolution and Devolution of the Elements. 


it is highly probable that an exact number of proto-boron 
atoms will form one helium atom. Consequently, the atomic 
weight of proto-boron must be 1:33 or 2. But proto-boron 
is of necessity heavier than proto-beryllium, and consequently 
its atomic weight must be 2. Therefore, we see that the 
atomic weights ‘of the four primary atoms form an harmonic 
series, namely 4/4, 4/3, 4/2, Al, which is interesting in con- 
nexion with Newland’s original idea of octaves. 

So little is known of the substances to which the various 
emanations give rise, that we will not hazard any suggestions 
regarding them. 

The last case before us is that of uranium, the simplest of 
all the radioactive elements. In this case, as with thorium 
and actinium, we have a heavy atom, heavier than radium, 
belonging to the carbon family. When one a particle has 
been expelled, uranium, like thorium, changes into uranium X, 
but uranium X is very different to thorium X. Instead of 
giving out a particles it gives out Bandy particles, and does 
not decay into any Seniesa! 

The mention of 8 and y particles opens out an entirely fresh 
phase of radioactivity, and to explain it fully we should have to 
enter into many new details, which in themselves form sufficient 
material for a separate paper. However, as we mentioned 
at the commencement of our paper, the space at our 
. disposal necessitated the omission of many astrophysical facts 
bearing directly upon the subject with which we have dealt, 
and now again we are constrained to pass over a fuller treat- 
ment of this important branch of radioactivity. 

To recapitulate, our conclusions concerning the periodic 
law and radioactivity are direct deductions from stellar 
observations, and many other little understood facts connected 
with matter find an explanation in the same source. We 
chose the periodic law and an outline of radioactivity as 
being perhaps the most typical examples of this theory, but 
we hope to give others at no distant date, and also a more 
detailed account of the astrophysical observations which gave 
rise to this paper. 

In conclusion, we wish to express our most hearty thanks 
to Professor E. C. C. Baly and Mr. H. E. Watson for their 
valuable assistance, and the kind interest they have taken in 
this paper. 


EXxpLaNaTion OF Dracram (Plate VIL.). 


The diagram on Plate VII. represents what we believe to be the 
structure of elements of typical groups. Circles drawn at wide 
distances from each other (2 mm.) indicate regularity of structure, 
2.e. successive elements have the same valency. Circles drawn 


i 


Curvature and Torsion of a Helix on any Cylinder. 55 


more closely show where the structure is irregular. No meaning 
is attached to the absolute size of the rings. | 
The figures indicate the valency of each ring, and consequently, 
the outermost figure of each element shows the number of 
electrons associated with it. 3' represents a rare earth ring, 
which, although drawn close to the ring next it, for the sake of 
distinction, produces no distortion in the element following it. 
Group 3 only differs from group 2 by the fact that the rings 
are trivalent imstead of divalent. The structure of the chromium 
group is the same as that of the vanadium group, but the penta- 
valent rings in the latter are replaced by hexavalent ones. 
Manganese is the same as vanadium with a heptavalent outer 
ring. 
The only difference between iron, nickel, and cobalt, is in the 
mass of the incomplete dotted ring. The same holds for the 
remaining members of the iron group. ad 
The three groups following copper have the same structure, but 
their outer rings are di-, tri-, and tetravalent respectively, instead 
of monovalent. ef 
Similarly, the structure of the oxygen and fluorine groups is 
the same as that of the nitrogen group, but the rings are hexa- 
and heptavalent instead of pentavalent. 


III. On the Curvature and Torsion of a Helix on any 
Cylinder, and on a Surface of Revolution. By L. V. 
Mrapowcrort, B.A., M.Sc., Scholar of Trinty College, 
Cambridge *. 


N the following short paper I propose to discuss the 
curvature and torsion of a helix on any cylinder, and on - 

a surface of revolution, by a new method. The method is of 
general application, and will apply to a helix on any surface. 
In this latter case, however, the formule are somewhat long 
and unwieldy ; and so I have contented myself by simply 
indicating the method of procedure, without giving the 
analytical details in full, As far as I know the results 
are not new, but the method presents several points of 
novelty and interest. 3 

I define a helix as a curve for which the ratio - is 
constant, p and o being the radii of curvature and torsion 
of the curve at any point. It follows, by a theorem given 
below, that the tangent and binormal at anv point of the 
curve make fixed angles with a fixed straight line in space. 

An equivalent definition is that a helix is a curve such 
that the tangent at any point of it makes a fixed angle with 
a fixed straight line in space. 


* Communicated by the Author. 


56 Mr. L. V. Meadowcroft on the Curvature and 


These two definitions may be proved to be equivalent by 
means of the following well-known theorems :— 


Theorem 1. 


If the tangent to a curve makes a constant angle a with a — 


fixed straight line in space, then o=ptana. Also the 


binormal makes the constant angle 774 with the same 
straight line. 
This is best proved by elementary geometry. 


Theorem 2. 


If the ratio p/o be constant, the tangent-line and binormal 
make constant angles with a fixed direction. 

I shall give a proof of this theorem by means of Serret’s 
Formule. Let (1, m,n) and (A, », v) be the direction cosines 
of the tangent and binormal at any point. Then by Serret’s 
formule we have : 


dl dn a ae | vdn (dy 
Patea, —” Pas? ante eee 
+, dividing each by o and integrating, we get 
Flt. =constant=A, | 


= m-+ “#=constant= B, + 


Paty =constant=C. | 
oO 
Multiply by 4, uw, vand add: .. AXN+ Bu+Cv=1, 
pe 


Multiply by J, m,n and add: .. Al+Bm+Cn=" | 


Co 
‘, the tangent line and binormal make fixed angles with 


at eee A, B, C 
the fixed direction VA? BEC” 
Having proved the equivalence of the two definitions by 
means of these two theorems, it is permissable to use the 
properties which follow from both in the following work. 
I take the fixed straight line as axis of <. Let a be the 
constant angle made by the tangent line with the axis of z. 


which proves the theorem. 


Then = = cosa = constant. .°. ga = 0. .. the principal 


normal is perpendicular to the axis of z, and so parallel to the 
plane of xy. Now the tangent line is perpendicular to the 


Torsion of a Helix on any Cylinder. 57 


principal normal, and the osculating plane contains the tangent 
and the principal normal. Hence it follows that the osculating 
plane at any point of a helix on any surface makes an angle 


te 574 with the plane of wy. 


The simplest helix is that described on a circular cylinder. 
This is known as the common helix or spiral staircase. Its 
curvature and torsion can easily be found by several methods. 
I shall not consider this curve at all. 

The first type of helix I shall discuss is that described on 
any cylinder, the axis of the cylinder being taken as axis of z. 
In this case the tangent line makes an 
angle « with the axis of z, and therefore 
with any generator. Hence the curve cuts 
the generators at the constant angle «. 

Then angle between the tangent-line and 
the axis =a, and the angle between the 
binormal and the axis rs 

Let P and Q be two neighbouring points 
on the curve, and let PQ=ds. Draw the 
generators through P and Q, and let the 
generator through Q cut the normal section 
of the cylinder through P in N. 

Let p’ be the radius of curvature of this 
normal section at P,and let d@ be the angle 
between the tangent planes to the cylinder at P and Q, 
z.e. at P and N. 

Then PN =p'ddé=ds sin a, from the figure. 

oe Draw a unit sphere with any 
point O as centre. Draw OZ 
parallel to the axis of the 
cylinder. Draw OB parallel 
to the binormal at P and OT 
parallel to the tangent line. 
Let B’, T' be the points cor- 
responding to the binormal and 
tangent line at Q. 


pe pes Lae 
1, er 


Then 
ZB=ZB'="—a, ZT=ZV=a, BI=B'l= 


bol 3 


Now TT’=d0=the angle between the tangents to the 
curve at P and Q. 

Also BB’=dn=the angle between the binormals at P 
and (). 


598 Mr. L. V. Meadowcroft on the Curvature and 


Now d@ = sinadd, \ 
dyn = cos add, 


since TZT'=BZB! =the angle between planes at P and Q, 
each. containing the generator and 

the tangent line 
=the angle between the tangent planes at 


P and Q, 2. e. at P and WW, 


i. 
1 eet oe 
a ee 
a at since pee oe : 
p sin a 
p=p cosec? « 
Again, ae Pik: 
a ds ds 
__cosasina 
a ia, 
“. @=p' cosec a sec a. 
Hence p=p! cosec? a, 
o =p! cosec «sec a. } 
Also o=ptana, as it should do. 


Verification. 
The usual method of procedure in such a case as the above 
is to project the helix onto the plane of xy, and find p’ 
the radius of curvature of the plane curve. Let vy be the 
angle which the osculating plane at P makes with the plane 
of xy, 8 the angle made by the tangent at P with the same 


plane. 
Then, by a general theorem, 
p _ cosy 
pcos’ 6 
In our case y=; 3% B=; 7 


pi =p’ cosec’ z. 
Also c=p tana, and 
o =p’ Cosec @ Sec a. 


This verifies the above formule for p and oc. 


Torsion of a Helix on any Cylinder. a9 


EHxamples. 
1. In the common helix, p ‘=a=the radius of the circular 
section of the cylinder. " 


p = acosec? a, li 
o = acosec asec z. | 


These are well-known formule. 


24,2 
2. In an elliptic cylinder p'=— : , where p is the per- 


pendicular from the axis onto the tangent plane at P, a and 
being the semi-axes of the principal elliptic section of the 
cylinder. 


a*b? 
= cosec? a, 
P Pp 
ab? 
a ee COSeG a SeC @. 
p 


The next step is to apply the above method to find the 
curvature and torsion of a helix on any surface of revolution. 
As before, I shall take « as the angle which the tangent at 

-any point makes with the axis of. z, and I shall further 
Beppase that the axis of z is the axis of the surface. Let 

=/(7) be the equation of the generating curve, r being the 
distarice of any point from the axis. 

Let P and Q be two neighbouring: points on 1 the helix, and 
fot PQ=—ds. 

Let the cylindrical coordinates of P and Q be (<, r, 6) and 
(2+dz, r+dr, 6+d¢) respectively. 

Tnen 

ds? = dz? + dr’? +d. 


dz* sec? a = dz?+dr?+r'dd’, since dz=ds cos a. 
ae a. (r)?+ll=rdo*, since dz=/' (r)dr. 
Heaths = 
dd < V tan? a. f/(r)?— 


the + or — sign being taken according as r increases or 
dass with @. 


Draw the spherical diagram as before and let TZT’=do. 
It will not now be equal to d@. 

dw=the angle between a vertical plane at P containing 
the tangent line and the corresponding plane at Q. 

Now the “tangent plane at P makes an angle W with a 


60 Mr. L. V. Meadowcroft on the Curvature and 


vertical plane perpendicular to the meridian plane at P, 
vr being given by tan VE =a" 

The tangent line at P lies in the tangent plane at P and 
makes an angle a with the vertical. It easily follows from 
this that cosy=tan cota, x being the angle which the 
vertical plane containing the tangent line makes with the 
meridian plane through P. 


At Q x will have the value y+dy, where 
—sin y dy = sec’? cot a dy. 
dw = dp+ (y+dxy)—x 


bis sec? yr cot a 
= dp— Go Se 
Now i 
tan fp = ——- 
Y= FO 
sec? dy = mcr ; 
r 
cota f/"(") , 
TT Sb Sg AOI 
mm, [ Vtan’ a. f'(r)?—1 jek So VD F_(?) an 
a r /1—tan? cot? a f'(7)?- 
Di hye ne _ fll 
_ tant ef =H) 70) gy, 
rf'(r)/tan? a. f(r? —1 
ow 
bid ag 
an sin a 
se gi eg 
eae” art de as 
.  tarPraf (ry —f\r)trf"(r) 1 
=-+ : Wf NL 5 C08 Be 
=A «Taal ey— be) ae 
awn, rf'(r)y tan? a fi(ryr-l P raee 
ie ~tan2a. f{r)3— f(r) +77" (r) "sin a cos a 
Again, 
iy 
sg COS & get 
rf'(r)? Vtan? a. f'(r)?—1 1 
ees 


~ “tan? a. f/ (r)®§—f'(r) + 2f"(r) * cos® a 


Torsion of a Helix on any Cylinder. 61 


Verification. 


These formule for p and o may be verified by the same 
method as was used in the verification for helices on 
cylindrical surfaces. 


We have 
ds? = dr? +dz?+7°dd’. 


This reduces, as above, to 
dr | taney (ea Lh) = dg’. 
2 1 ws dp 
tan? a. f' (x)? L= (Gr). 
Hence the p, r equation of the projection of the helix on 
the plane of ay 1s 
2 Hh. NF 2p 
‘tan’ a./ Co ial ee 
ip 
9 9 i a 
PT tanta. f(r 
Ole. 
Par tan*e . f(r} 


: — a. f(r)’ —f"(r) ae 
tan? a’. f(r) ; 


ee es | BY ah. 
f dp, tanta. f'(r) pti 

_ Oe kOe L 
P=sin?'a  —~tan?a. f(r) —f" Se sin «cosa 


Also c=ptana; and so we find o. 
These results verify the values of p and o obtained by my 


method. 
Examples. 
ie ga oLe paraboloid of revolution. 


Here a J = if 1 is the semi-latus rectum of the generating 


parabola. 


fO)=5 FO=; 


Also r increases with @; and so we take the plus signs in 


62 Curvature and Torsion of a Helix on any Cylinder. 


the expressions for p and o. 


pA) tanta. Fel 
iia P i? 1 Vr —E cotta 
ee — 3 ae 
like gees sIn a cosa 2% 
tan? Ae oe ial a Su 


ae Sie o” sin? acos*a = 7°—/? cot? « 


This is a known result. 
2. Helix on a sphere. 
Here g= V a2—?”. 
2 


(OS FQ ==— 


If we take < to be positive TE is negative. 


tan? a .~, 
a l 


ee ici 3 » ra? *sine cosa 


az ae 


ue 7? — a? cos” : 
, on reduction. 
sin a 


\/ 72 — a? COS? a 
sagt sin & 


os J/ 7? — a? cos? @ 
Similarly c= ae : 


This again is a known result. 
3. Helix on a cone of semivertical angle 9. 


Here c=r cot 6. 
f(r) = cot B, f" (7) = 0. 
Also a is positive if the vertex be pointing downwards. 
r cot? BY tan? a cot? B—1 1 
tan? a.cot?B—cotB ‘sinacose 


I 


pP 
2 
Ve a jan - oe De sin? @ Gosia i : ; (tan? «—tan’ 8) sin? a COs? a. 


Prof. D. N. Mallik on a Potential Problem. 63 


Now it can easily be shown that a helix on a cone cuts all 


COS a 
the generators at a constant angle #, where cosa = : 
cos 8 
1 sin?acos?a cos? B—cos? « 
pa ” " cos? « cos’ B 
“sin? «sin? @ 
= 2 


Pp = Pr COsec a COSEC @. 


Similarly c=r sec e cosec ow. 


This, again, is a known result. 


Apparently the above method would apply to a helix on 
any form of surface, but the results would be too complicated 
to be of any use, except in very special cases. The only 
difficulty is to find dw. The tangent line at any point P 
makes an angle a with OZ and lies in a known plane, 2. e. 
the tangent plane at P. It might be possible to calculate 
from this the angle made by the vertical plane containing 
the tangent at P with (say) the plane r=0. Say this angle 
was found to be yw. Then dw=dw ; but unless the surface 
was of a very simple nature, the resuits would be useless 
because of their length and want of simplicity. - This is not 
of much importance, as the most interesting helices are those 
described on surfaces of revolution. These may very con- 
veniently be dealt with by the method given above. 


IV. A Potential Problem. By D. N. Maur, 
F V is the potential of an ellipsoid and X Y Z the com- 


ponent forces, then of course 
X=+Az &e., and V=( Az du+... 
At an internal point, A B © are constants and then 
V=V,+34 (Aa?+ By? + Cz’). 
(Minchin’s Statics, vol. ii. p. 326.) 


The same method is also applicable for an external point, 
although I have not seen it given in the text-books. 


* Communicated by Prof. A. W. Porter. 


64 Prof. D. N. Mallik on a Potential Problem. 


In this case 


A= 2m pabe { (a2 an ar) 3( 6? ae wy )e(c? +a) ’ 
2 2 ae 


where Ente ae t¢ 
atpw P+y +p.’ 


.. V=—27p ciel ( watt ay 
where Q=J/(C+WC+y) (C+). 


We have now to change the order of integration. 
For this, let 


Flog he ene 55 
Oty P+y C+p 


be a curve in « and mw, mw being the abscissa (y and z being 
regarded as constants). 

Then the integral depending on 2 extends evidently over 
the region included between this curve and a line parallel 
to the p-axis, 7. e. from mw given by 


x ye 2 
le Ola eae 0 N=, 
and from «=2, to z= where 
7 Yy Car 1 
a + pb + Bap C+ 


(with corresponding values of y and z). 


- n? ya re) 2 ee a av » 
ce fa Gps 2mpabe | O@ry } yritts 


=—T7p ce | oy ( a cote i) 
» Q \att+y  aftp 


‘id a“? “ a 
dw 4 y 1), 


J Q ery" C4W 7 PT 


since ay hia te setae =k 


Patna College, Bankipore. 
6th June, 1907. 


Goes J 


V. Experimental Determination of Magnetic Induction in an 
Elongated Spheroid. By Prof. D. N. Matrtx*. 


L. ik a paper on the magnetic induction in a spheroid 
due to acoil carrying current (Phil. Mag. Oct. 1907), 


I proved that if M be the induction at any point of such a 
spheroid, aes a6 
F 2etelog (1+ ™) : 
M = a (1+47k)2mm E + ea | 
) : | 37 ( 1+ a ) ir 
= A), say, pe 


‘. if B = total magnetic induction over the semi-ellipsoid, 
B=(MdS=A{pdS=Aa{da, 
: 4 
=(1+ dark) 2armi( 1— aa), nearly, 
where dS is an element of surface, and dz an element 
of the major axis ; 


: dB 
jg CONSt. 


ag. 
In this form it is suitable for experimental verification, 


and I have made a series of experiments to test its accuracy. 


A 


Hey, 


Length of 70d....50 Ci 
Diameter .......... 2-6 Ci. 


2. For this purpose, a rod AB (fig. 1) was prepared 
* Communicated by the Author. 
Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. EF 


66 Prof. D. N. Mallik: Experimental Determination 


approximately spheroidal in form of very mild Bessemer steel 
well annealed, and the coil DC was wound round a thin wooden 
cap which just fitted it to half its length. 

A current, the constancy of which was indicated by an 
ammeter, being passed through the coil, the soft steel rod was 
ma onetized, and the object of the experiments was to explore 
the magnetic induction over the projecting portion of the rod. 

3. In order to do this, a thin coil 4 inch in thickness 
having 36 turns (six turns, six deep), in series with a ballistic 
galvanometer, was slipped over the rod and placed at different 
positions beginning with middle of rod (the zero position), 
and corresponding throws on the reversal of current were 
noted. In the preliminary experiments (I. and II.) the thin 
coil was placed at random at different positions which were read 


eS) 


Throws (in scale readings). 
S 


| x Lee I | 
© LxeLl | 


| 
| 


| 
2 
Agel Distances Fors Zero eT > Cenmed Boston of Thin Coil ae cms. ). 
off by means of a telescope. The results are represented by 
Curve I., fig. 2, in which abscisse represent distances from 


of Magnetic Induction in an Elongated Spheroid. 67 


‘the zero position, and ordinates the throws obtained at 


corresponding positions. 

4, In the final experiments the thin coil was moved up, a 
centimetre at a time, by means of a mechanical arrangement 
allowing of the actual displacement being read off on a 


vertical scale parallel to the axis of the rod. The following 


table gives the results obtained after a slight correction on 
account of the direct action of the magnetizing coil. 


| 


| } 
| 


Distance | 4Xthrow. | 4xthrow. | Mean | Throw | 
from zero. | lxp. III. | Exp.IV. __ throw. per em. of rod.) 
| 0 | 376 | 36-7 | 9-29 
) 1 Peano) - sera 3°56 43 
2 ee a eee 79 | 66 
3 mee! | | Ded moe 68 
4 | 266 26'1 6°59 ‘63 
5 24-1 oe Pe ea | 62 
6 | 21-4 pee a ee? ed 
| me 194 1925 | 483 5t 
| 8 | zi), | fein 495 | ‘55 
) 9 or) | ae eee ‘DD 
| 10 13 [> (eiseereee na22 51 
| 11 10°73 1063°%..) «. 267 | 55 
| 12 so | ok? ee 48 


[In Exp. III. the exvloring coil was moved from the zero position, and 
in Exp. IV. Zo the zero position. | 


The results are also plotted in the Curve II. It will be 
seen that both sets of experiments give practically identical 
results, viz., that the curve is straight with a slight upward 
trend for the first three cms. from the zero. In order to 
explain this, we note that the rod is not altogether spheroidal 
in form, but is practically a cylinder with a rounded off end, 
so that the small coil for the first few centimetres from the 
zero position embraces more lines of force than if the rod 
were accurately spheroidal. We conclude therefore that the 
formula (§ 1) is experimentally verified for an elongated 
spheroid. It will therefore be necessarily also true of a cylin- 
drical rod with rounded off ends, if the rod is sufficiently thin. 

5. Further it is seen that in the latter case the lines of force 
are radial and uniformly distributed. 

Moreover, if m be the total amount of induced magnetism 
over such a cylinder, 


Arm = B, 
and p=linear density of this magnetism, 
Amr\pd c=B, 
or dB 
Pais p=const. ; 


68 - Magnetic Induction in an Elongated Spheroid. 


or linear density of induced magnetism in a long thin cylinder 
due to a coil embracing half its length is constant. 

6. The formula being thus experimentally verified, it may 
obviously be used to find how the value of k corresponding 
to different values of current, varies. 

For this the small coil was placed in the central position 


and the throws of the galvanometer for different values of 


the current were noted. The following table gives the 
results of experiment. . 


Current in amperes. Throw. 
"2 1:05 
°4. 21 
"6 a5 
zo) Ard. 
i! 6 
1°2 704 
1-4 8°07 
1°6 972 
1°8 10°02 
2 1155 
These throws are plotted on a curve (fig. 3) in which these 
Fig. 8. 


—> THROWS 


CURRENTS /N AMPERES. i 2 
are abscissee and the currents are ordinates. We see that 
up to current of 8 ampere & is absolutely constant, and then 
begins to vary slightly with the current, the variation being at 
firstinappreciable. Unfortunately, however, the effect of the 
current in heating the coil as well as the rod was so great 
when the current was over 2 amperes, that it was not thought 
desirable to proceed with the experiments for stronger 
currents without considerable modification in the apparatus. 

I have to thank Prof. Trouton for permission to work at 
the University College Laboratory for the purposes of the 
above investigation. 


[Ghee 


VI. On Spherical Radiation and Vibrations in Conical Pipes. 
By Hh. H. Barton, D.Sc., F.RSE., Professor of Haperi- 
mental Physics, University College, Nottingham™. 


T is well known that the vibrations in parallel pipes may 
be treated by plane waves and elementary methods. 
When, however, a change is made from parallel to conical 
pipes the waves cease to be plane, and the method hitherto 
available is powerless to deal with the phenomena. Yet the 
importance of the subject is at least equal to that of parallel 
pipes, since the brass instruments in musical use are 
conical or quasi-conical, and also the oboe, bassoons, and 
English horn. Thus, apart from a knowledge of spherical 
radiation and its application to such pipes, the student is left 
without a clue to the phenomena occurring. He is accord- 
ingly somewhat at a loss to understand why a conical pipe, 
closed at the vertex and open at the base, should have the 
same pitch and the same complete series of harmonic tones as 
a parallel pipe open at both ends. Whereas, a parallel pipe 
if closed at one end falls in pitch about an octave and loses 
all the evenly-numbered partials. 

The mathematical aspect of the matter is of course feel 
with great generality and elegance in the classical Hah iran 
(see, for example, Rayleigh’s ‘Theory of Sound,’ vol. 
chaps. xi., xii., & xiv.). But the use in such treatises of the 
velocity potential as the dependent variable, slight as this 
obstacle is, may prove sufficient to prevent some readers from 
assimilating the articles in question. If, however, following 
Riemann +, we take for the dependent variable the so-called 
condensation, which isa more familiar conception, the analysis 
is somewhat. simplified and the whole problem is solved by 
methods within the range of every one familiar with the 
elements of the calculus. It seems desirable, therefore, that 
the physical student should be provided with a treatment 
intermediate between the recondite mathematical treatises on 
the one hand, and the mere statement of the musical facts 
respecting conical pipes on the other hand. This plan, already 
found useful in dealing with such students, is here given in the 
hope it may thus prove of service to others. Another matter 
which seems very puzzling, in the simple statement without 
proof, is the fact that in a conical pipe the anti-nodes remain 
in equidistant positions as for a parallel pipe, but that the 
nodes are all shifted, some considerably and others slightly. 


- * Communicated by the Author. 
+ Partielle Differentialgleichungen und deren Anwendung pri physi- 
kalische Fragen. 1882. Sixth part. 


70 Prof. E. H. Barton on Spherical 
This point is here illustrated graphically, the reason for the 


shifts which occur and their relative amounts being made 


clear in a single diagram. 

Fundamental Assumptions.—In attacking the problem eS 
the spherical radiation of sound in air we simplify the analysis 
by making at the outset the following assumptions :— 

(1) That the action of gravity is negligible. 

(2) That the effect of viscosity is negligible. 

(3) That the motion is vibratory and devoid of rotation. 

(4) That the vibrations are small, so that writing the 

density 


p=po(1 + s) 


the “condensation” s is to be regarded as a small 


quantity often negligible in comparison with unity. 
(5) That the velocities and accelerations of the air are 
small quantities whose squares and products are 


negligible. 


It is easily seen that the above assumptions simply narrow, 


down the discussion to the case in question, and do not 
involve the loss of any generality we wish to retain. 


In estimating the acceleration of the air two methods are 
open to us: (a) We may follow in thought an individual 


particle and note how much is velocity is increased per 
second; or: (b} We may fix attention on a spot in space and 
note how the speed changes of that particle (whichever it is) 
which is found there at the time in question. In other words, 
we may note the increase of speed of an individual in the 
procession, or the increase of speed of the procession as it 
passes a fived point on the route. The relation between the 
two accelerations is given in hydrodynamical treatises. In 
our use of acceleration the first form should in strictness be 
taken, but, with the limitation (5), the distinction drops, as 
the difference is only of the second order of small quantities. 
Thus the second form, which is simpler, may always be used. 

We have now to derive the differential equation for aerial 
vibrations in space of three dimensions, solve it and simplify 
- to the case of spherical radiation, then apply the solution to 
the various cases of conical pipes. 

The differential equation is based upon (i.) the so-called 
equation of continuity, and (ii.) the equations of motion. 

These we now take in the above order. 
. Equation of Continuity.— Consider aninfinitesimal parallele- 
piped of edges dx, dy, and dz, and let the velocities of the 
air parallel to the axes of w, y, and z be denoted by u, v, and w 
respectively. 


Radiation and Vibrations in Conical Pipes. 71 


Then the mass of air entering the parallelepiped at the 
face dy dz in the time dt is pu (dy dz) dt, and that leaving by 
the opposite face in the same time is 


| put Ae) ae | ay dz dt. 
Hence, the mass lost by this pair of faces is 
d(pu) ; 
We te dy dz dt. 


Therefore, considering the other four faces in like manner, 
we have as the total mass lost by the parallelepiped in time 
dt the quantity 

flor) , dpe) 


L dz dy 


+ OY) } tw dy dz dt. 


But this quantity can also be expressed in terms of the 
decrease of density, viz. : 


dp : 
ag oe dy dz dt. 
Whence, equating the two forms, we have 


dp _d(pu) | dev) | d(pw) _ 
a i es ee 


Now, since p=p)(1+s), the first term of (1) becomes pp ds/dt. 
Again, 


d(pu) ,du , dp_ du ‘ 
eae po(l+s) ds * "Ga Por. nearly, 


the product of the two small quantities wand dp/de being 
negligible. Thus for our case (1) becomes 
ds du’ dv . dw 
a = 2 
di * dx dy + Or mee? i 


and this is the form of the equation of continuity for small 
oscillations of a light elastic fluid. 

Equations of Motion——We have now to express, for the 
fluid in our parallelepiped, the condition that the product, 
mass into acceleration, equals the moving force to which it is 
subjected. The mass into acceleration is (p dx dy dz) du/dt. 
The moving force is the excess of that due to pressure, p, 


72 Prof. E. H. Barton on Spherical 
behind over that due to pressure, »+dp, in front, or 


d, d 
pdy dze—(p+ Pda) dy dz =—F de dy dz. 
Hence, equating, we find —dp/dx =p du/dt, or 


~ de a (3) 


But p is some function of p, and, whatever the form of 
the curve coordinating p and p, the small portion with which 
we are concerned may be considered straight. Hence, for 
our small vibrations, we can write 


dp/dp=a’, some constant. . . . . (4) 
We can easily see, by the method of dimensions, that a in (4) 
is of the nature of a velocity. For the left side of (4) is 
dimensionally 

i be ed Pi 

ML-* 
in which M, L, and T represent mass, length, and time re- 
spectively. From (4) we get 


= 17T-*=(velocity)?, 


dp a?dp_ a®p,ds a 
Lf = o—. =a ds nearly. <a 
p p—-*Po(l +s) : ) 


It should be noted here that we are not entitled to inte- 
grate (4) and draw from such result any conclusions about 
the general relation between p and p. On the contrary, the 
relation between them must be determined independently, 
and the general value of dp/dp =a? derived from it. 

Substituting (5) in (3), we obtain 


a* dsida=—dufdit.' . . \. an 


Whence, for the other axes, we have by symmetry of 
notation 


a’ ds|dy=—dor/dit, .. . . ee 
and @ds/dz——dw/dt: . . |: ioe 


These three constitute the required equations of motion. 

The Differential Equation—We have now to derive from 
equations (2), (6), (7), and (8) the partial differential 
equation. We see that in addition to s, 2, y, z, and ¢, the 
only variables we wish to retain, the above equations involve 
also u, v, and w. These last three must accordingly be 


Radiation and Vibrations in Conical Pipes. (3; 
eliminated. On differentiating equation (2) with respect to 
time, we obtain 

as | du d?v d?w 


ue Myaaeeeetesi desde: 4 


Again, if we differentiate equation (6) to x, (7) to y, and 
(8) to z, and then add the results, we have j 
Bes. O'S. 3 ES) Dau Pou Pw 

—a( 72+ We Mae) ee dicks dede io 2°) 
But since d?u/dx dt=d?u/dt da, because the order of partial 


differentiation is indifferent, the right sides of (9) and (10) 
are identical. Thus, equating their left sides, we have 


ds ge d2s = 
ge: 


oe 
dt? dx? dy? dz? (11) 
or d?s 


1 =0V*s, 
a a ae i a? 
TE aaa ee 


where for brevity’s sake y? denotes 


This is the general form of the differential equation appli- 
cable to small vibratory disturbances of a light compressible 
medium in space of three dimensions. 

Plane Waves.—As a check upon this result, let us reduce 
it to the case of plane waves in the plane of yz, and proceeding 
therefore in the direction «. Thus s is a function of ¢ and x 
only and is independent of y and z. Hence V’s=d?s/dz” 
and (11) becomes 

ad? sy yeas 


=a 


ot Oa ete he OS LD) 


which is the well-known form of equation for plane waves. 
To examine if a has the right value here let the specific 
volume be U, then p=1/U and dp=—dU/U?. 


—P _( = 

aA 2— f(T = —F = 4 

@ dp (a7) U=EU [p= P/e: (13) 
in which E denotes ihe volume elasticity and y the ratio of 
the two specific heats, Thus, the ordinary value is seen to 
hold. 

Solution for Spherical Waves.—Let us now transform to 
polar coordinates defined by 


z=rsin@cosd, y=rsin@sing, and z=rcos§. (14 


) 


74 ‘Prof. E. H. Barton on Spherical 


Then it is shown in text-books on the calculus (e. g. William- 
son’s ‘ Differential Calculus’) that 
os. Bas ae Bree ee cr 
Ca eS LAA Ne Weeks OAD ES eae: eA eg oe 
eae dnt ane waksin 70) + sn @ dg?” co, 
Now suppose we have symmetry about the centre of the 


system, so that s is a function of r onlv. Then ds/d@=0 and 
ds/db=0. Hence, for this case, (15) reduces to a 


Ps 2d _1&(r8) 


7s ES =- — ( 
Ve dr dr 4+ dr - 
Equation (11) accordingly becomes 
o@(rs)  d*(rs) 
Oe = eee) nn (17) 
The general solution of this may be written 
rs=fi(r—at)+f(rtat) - . . - (18) 


where 7, and /, denote arbitrary functions. This solution 
obviously consists of both diverging and converging waves 
of spherical form, of any periodic or non-periodic type and 
travelling with radial speed a. For diverging waves of 
simple harmonic type (18) becomes 


s=“ cos k(r—at). SS a 


To denote the speed wu of the air along the radius 7, we 
derive from (6) and (19) 


—du a ds a’e 


: 1 
BEY oo ee as { — k sin k(r—at) — cos k(v— at) | - (20) 
Thus 
at es ca ee \ 
Uu=—a ptt = — | cos k(r—at)— Zain k(r —at) f- (21) 


Again, if displacements along 7 be denoted by &, we have 
by another integration 


J udt, or 
a pe 1 : 
—=— es {sin k(v —at) + Fp oo k(n at). . (22) 
We may thus see, from equations (19), (21), and (22), that 


it is only the condensation s whose change of phase is 
restricted to the ordinary one inseparably associated with the 


Radiation and Vibrations in Conical Pipes. 75 


advance of a progressive wave. Its amplitude, however, 
suffers diminution by varying inversely as 7. But, owing to 
this diminution or attenuation with advance, we have in the 
other equations the factors 1/r and also 1/7*, one applying to 
a sine and the other to a cosine function. Hence the speed 
wand the displacement € each exhibit, during advance, an 
additional slight change of phase (as the sine and cosine 
terms are differently diminished), beyond that always pre- 
sent in a progressive wave. But, where 7 is great enough 
to make I/kr negligibly small, (21) and (22) become 
approximately 


We cos k(r—at), Sg) 250-45 AT 
and os os STR ieee tert. frum ile) 


Refleaion at Pole—Let us now regard the two spherical 
waves of (18) as a converging one and a diverging one 
to which the other gives rise by reflexion at the pole or 
centre of the system. And let it be required to determine 
the relation between /; and /, so resulting. The total 
current across the surface of a sphere of radius ris 4ar?u 
and, for r=0, must vanish, since all is symmetrical about 
the pole or origin of coordinates. That is, w cannot be 
infinite and so make 7’u finite for r=0. But, if 4rru 
vanishes at r=0 for al! values of ¢, so also will 4ar?du/dt 
vanish. And this condition is easier to fulfil analytically. 


Thus from (6) and (18) we have 
Anr dufdt = —Ana@rds/dr 
=4ma*} fi(r—at) + fo(r + at) }—40a'r ffi’ (r—at) +f,'(r+at)}, 
where /’ denotes the first derived function of /. 
Hence, putting du/dt=0 for r=0, we find 
O70 —@) po ee ee ol Coe 


as the relation between /,; and f,. But we see from (18) that 
the right side of (25) is the value of (rs) for r=0. Hence, 


we may write as the condition at the pole 
TS) FOG Be ee a eee) 
or, 7s must vanish with r. 

Thus, at the pole, a condensation is reflected as a rare- 
faction and vice versa, somewhat as in the case of reflexion at 
the open end of a parallel pipe. 

Conical Pipes.—To apply equation (17) to conical pipes 


76 Prof. E. H. Barton on Spherical 


we must choose a form of solution corresponding to stationary 
waves. Thus, let 7s be everywhere proportional to cos kat. 
Then d?(rs)/dt?= —/°a?(rs), and equation (17) transforms into 


EOS) 4 12(r8) =0. pho a . ae 


The general solution of this may be written 
rs=(Acoskr+Bsinkr) coskat, . . . (28) 
where A and B are arbitrary constants. These are to be 
determined for each case by the position and nature of the 
ends of the pipe. There are accordingly a number of 
separate cases to consider. 

Open Ends.—First let both ends of the conical pipe be 
open. Then obviously the condition at the end is approxi- 
mately s=0. TF or,at the ideal open end there can be neither 
condensation nor rarefaction. Let the coordinates of the 
ends of the pipe be 7, and 7., measured from the vertex. of 


the cone if completed. Then, we have from (28) for the 
terminal conditions, | 
A coskr;+Bsinkr;=0 and Acoskr,+B sin kr,=0; 
whence, by the elimination of A/B, we obtain 
sin k(rg—7,)=0 or k(re—r1) =n. 
This may be written 
tA 2 ole ai 


=a ony | ee 


where n is an integer; for since s is proportional to cos xat, 
k=2aN/a=27/dr, N being the frequency and X the wave- 
length of the motion. Thus, for a conical pipe with open 
ends, the pitch of the prime tone and the form of the series 
of other natural tones are like those for an open-ended 
parallel pipe. This might have been anticipated from the 
similarity of the differential equations and the conditions for 
the open ends in each case, There is, however, this slight 
difference that r,—7, is the slant iength of the conical pipe 
and not its axial length. As to the segments into which the 
pipe is divided when emitting its higher natural tones, it 
follows from equation (29) that the antinodes are equidistant. 
When dealing with the next case, it will be seen that this 
simplicity does not extend to the nodes. 

Closed Ends.—The condition at closed ends is obviously 
u=0; consequently du/dt=0 there also. But by equation (6) 
du/dt=a?ds/dr, if u denotes the velocity along 7. We may 
thus write as our condition for a closed end ds/dr=0. 


Radiation and Vibrations in Conical Pipes. EG 
Applying this to equation (28), we find, after a little trans- 
formation, 

A(cos kr, + kr, sin kr,;) = B (kr, cos kr; —sin kr), 
and A(coskr.+ krzsin krz) = B(krz cos krz—sin frp). 
On dividing out by the cosines and writing tan @, for kr; 


and tan @, for kr,, we may eliminate A/B between the above 
equations. Thus 


tan 6,—tan kr, 
. 1+tan 6, tan kr, 
or kr ,—tan7! kro =kry—tan kr.» 6) ss 17 (80) 


=tan (6, — hr) =tan (6,—kr.), 


The transcendental form of this equation shows that the 
nodes are not equidistant ina conical pipe. We will pre- 
sently find where they are in the important case of a complete 
cone with open end. 

Closed Cone.—To treat the case of a cone continued to the 
vertex and with base closed, we have simply to write 7,=0 
in equation (30) and R as the slant length of the cone for 7.. 
This gives 

Galt cig eb eee ee ewe Con} 


To solve this equation, which we may regard as tanv=a, 
we may proceed graphically. Thus plot the two graphs y= 
and y=tanz. Then their intersections will give the roots 
required. See fig. 1, p. 80 as an illustration of this. The 
equation may also be solved by successive approximations by 
which (in another connexion) Lord Rayleigh finds (‘ Theory 
of Sound, vol. i. p. 334), 


OF = n/m =0, 14303, 2°4590, 3-4709, 44747, 54818, 64844, &e. ; oo 
= 6, 0,, 03, 0, 0, O6, 0, Say, 


Thus these quantities, denoted by the 6s, each multiplied 
by 7, give the first seven values of AR in equation (31). 
Now since 0,=0, we may write (k,R)=76@,41. But we 
also have as the general relation k,=27N,/a. Hence, we 
may write for the frequency of the nth tone natural to the 
closed cone 
a 


A acc a a  ) 


Thus the frequencies are directly proportional to the speed 
of sound, inversely proportional to the. slant length of the 
cone and the relation of the various possible tones in the 
series is defined by equation (32) giving the roots of (31). 


78 Prof. E. H. Barton on Spherical 


Open Cone.—We now consider the case of a complete cone 
with base open. At the open base, we have, as before for an 
open end, the condition rs=0. And at the vertex, the origin 
of coordinates, we have from equation (26) that rs=0 there 
also. So that although one end is open and the other 
closed, we have the apparent anomaly that the same con- 
dition applies to each. Hence if R is the coordinate of 
the base, 7. e«. R is the slant length of the cone, we have 
from (29) 

R=m,,,/2. or N,,=ma/2h,") Gee 


where N,, is the frequency of the mth natural tone and Am its 
wave-length, a being the speed of sound. This then shows, 
what has before been remarked upon as strange, that a cone 
open at the base and closed at the vertex gives practically 
the same fundamental and the same /ul/ harmonic series of 
other natural tones as are obtainable from a parallel pipe 
of the same length and open at both ends. Of course, when 
the corrections for open ends are taken into account, the 
statement as to pitch and length suffers a slight modification. 
For the double open-ended parallel pipe has two ends needing 
correction and the cone only one; moreover their diameters 
may differ. But if a cone and an open-ended parallel pipe 
are prepared of slightly different lengths so that their funda- 
mentals are in unison, then their other partials will be in 
accord also. This may easily be verified by pipes of zine 
tested with a set of tuning-forks forming the harmonic 
series of relative frequencies 1, 2, 3, 4, &e. 

We see from the first form of (34) that the wave-length is 
inversely as the order of the tone produced ; hence the 
antinodes are all equidistant. This, however, does not apply 
to the nodes. To determine the positions of the nodes we 
must refer to equations (31) to (34). Now equation (33) 
gives in terms of @ the various values of N for a closed 
cone of jiwed slant length R. Let us, however, substitute 
the variable r for the constant hk and, dropping the subscript 
of N, rewrite this equation as follows :— 


2Nr/a=6,, On. Os, G,, Os, G., or G7, &e. ee (35) 


We may now regard both N and ¢ as variables which must 
satisfy (35), 7 being the slant length of a closed cone. Again, 
equation (34) gives the frequencies of the various tones 
natural to the open cone. Let us rewrite it, dropping from 
N its subscript and writing for m on the right side the series 
of natural numbers which it represents. We thus obtain 


Radiation and Vibrations in Conical Pipes. (1) 


Here we consider the slant length R of our open cone as a 
constant and N to vary in accordance with the numbers on 
the right side of the equation. Now, on dividing equation 
(35) by (36) we eliminate the pemenoy! N and obtain the 


required relations between v and R, viz. 


Lat 0, 0, or 6, 0, 0, or é:, G5, ee @. or 6, 0, 0., 63, Ga or 0: = 
a alld Re cat 4 5 > | 
01, O>, 03, 64, 0; or 0, Gi, G., ae 0, G3. 0, or 0, ( (37) 
6 3 or Fi ) &e. 


—_ 


it is necessary to cross combine the right sides of (35) and 
(36) in this way to obtain all the values sought. The deno- 
minator of any one of the fractions on the right of (37) 
shows the order of the tone being emitted by the pipe. The 
various values of 7/R obtained by taking the various 6’s in 
the numerator of that fraction locate the nodes for the tone in 
question. The series of @’s in each numerator is finite, being 
limited by the obvious fact that r/R cannot exceed unity, 
The first few nodal positions are given in Table I. They are 
also exhibited graphically together with the positions of the 


Tasie I.—Nodal Positions of an Open Cone. 


Order im of . 
Natural Tone ; 
i. e, denominator Nodal Positions ; 


Pik dae ae 4. é., values of 7/R=0/m in equation (37). 


equation (37). 
wr ee a 
CT ee O 7152 
2) Ya ae 0 0-4768 | 0°8197 
LE ee 2A 0 0°3576 | 0°6148 | 0:8677 
5 ean te 0 02861 | 0-4918 | 06942 | 08949 
| eee ie 0 0:2384 | 04098 | 0:°57385 | 0°7458 | 09136 
ce saat 0 


02043 | 0°3513 | 04958 | 0:°6392 | 0°7831 | 0-9263 


Values of 0’s in 
numerators of 0, 6, 6, 0, 6. 0, 6, 


ra) 


fractions on right |f g | 41-4303 | 24590 | 3.4709 | 44747 | 5-4818 | 6-4844 
side of equation (37) 


80 Spherical Radiation and Vibrations on Conical Pipes. 


In this diagram the graph y=wz is shown 
by a full line and the branches of y=tan x by broken lines. 
The abscissee of their intersections show the values of the 


antidodes in fig. 1. 


roots of the equation, tanw=2. 


Fie. 1.— Vibrating Segments of an Open Cone. 


The circles show the equidistant antinodes and the crosses show the 
nodes located by the intersections of r=y and y=tan 2. 


n *. 

VALUES OF ¥ 

ov /N GRAPHS... 
-< 


x 
x 
g 
» 
Fit 


20-0 


6YTONE s Z = 5 a 


17-5 


| 
( 
S270ME { 
i 
15-0 — 
; 
{ 
QTATONE. 
[25 { +4 
{ 1 
{ t 
tl 
‘ 
{ | 
= 10-0 es a! : 
STONE ; 
{ un 
| on 
! _ 
Ha) \ 6 ies 
{ i! BS 
2”PTONE ; S 
( 1 o 
5 i 
50 pease 
j 1| 
i 1 & 
1 1} wy 
12° 70NE i i o 
1) | 
2-5 1 
it ! 
tt I 
4 f 
a 4 2 i | / | 
as / Z / 
0 : / : / — | — x 
Q ei i ‘en ‘ST OTT “OT ‘671 71 RADIANS. 
6/ G2 6; Oz Os Os Oz (THE 05 SHOW THE VALUES 
OF THE FPOOTS OF EQUATION 


To show the segments of one pipe of fixed len 


TAN H=x00 


oth when 


co) 


emitting its various tones, a number of such diagrams would 
beneeded. Toavoid this repetition, a series of pipes is shown 


On the Rupture of Materials under Combined Stress. 81 


in the upper part of the diagram. These are of various 
length so that the longest at the top is shown with the 
segments corresponding to its seventh natural tone. Passing 
down the series, we reach the last pipe, the shortest of all, 
with one segment only as when emitting its fundamental. 
The antinodes are indicated by small circles and the nodes by 
crosses. The equidistant positions midway between the anti- 
nodes are shown by dots. 

Experiments by Mr. D. J. Blaikley have shown the 
existence of these nodes in positions slightly displaced from 
equidistances, each node being moved towards the vertex 
from the corresponding equidistant position. Further, it 
was found by Blaikley that the displacement was the greater 
for those nodes which are nearer the vertex. Or, in other 
words, “‘the nodes in the cone are at increasing distances 
apart, reckoning from the open end, and at the apex of the 
cone is a node common to all the notes.” These nodal 
positions were established by Blaikley with an experimental 
bugle made in sections. On taking this to pieces, thin 
metal diaphragms were inserted at the positions of the nodes 
for a certain note, and their presence was found not to 
prevent the production of the note in question. The dia- 
phragms had each a few small holes to admit the passage of 
the player’s breath, but prevented all free vibration at the 
place. 

University College, Nottingham, ‘ 

June 7, 1907. 


VII. On the Rupture of Materials under Combined Stress :— 
Tension and Hydrostatic Pressure. By W. EL.uis 
Wiurams, B.Sc., Research Fellow of the University of 
Wales*. 

HE connexion between the rupture of a solid body and 
the stress causing it is at present but vaguely under- 
stood, and a number of different theories have been put 
forward to enable the tendency to rupture to be calculated 
from the stresses acting on the body by the methods of 
elastic theory. The principal of these are f :-— 


(1) Lamé.—Rupture occurs when the greatest tension 
has reached a certain limit. 

(2) PoncELeEr and St. VENANtT.—Rupture occurs when 
the greatest extension has reached a certain limit. 


* Communicated by Prof. E. Taylor Jones. 
+ Love, ‘ Elasticity,’ vol. i. p. 106. 


Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. & 


82 Mr. W. Ellis Williams on the Rupture of 


(3) TrEsca.—Rupture occurs when the difference be- 
tween the greatest and least principal’ stresses 
exceeds a certain limit. . 

(4) CouLoms.—Rupture occurs when the greatest shear 
has reached a certain value. 


The theory which has been most widely adopted is that 
attributed to Poncelet, which agrees closely with experi- 
mental results for a few substances, notably for cast iron. 
In this theory the extension is to be calculated from the 
applied stresses by means of the ordinary elastic constants, 
and the theory requires for each body only one other constant, 
the limiting extension. 

Some years ago Voigt* made a series of experiments on 
the effect of hydrostatic pressure on the breaking strength ot 
rocksalt and of wax, and obtained results in direct opposition 
to Poncelet’s theory. Voigt measured the breaking-stress 
of rocksalt specimens enclosed in a cylinder containing 
carbonic acid gas under a pressure of 50 atmospheres, and 
found that the tension necessary to break the specimen was 
the same as when the experiment was performed under 
atmospheric pressure, while according to Poncelet’s theory 
it should be twice as great. Voigt concludes from his 
experiments that the third theory due to Tresca is the most 
satisfactory. The results could, however, be equally well 
explained by Coulomb’s theory, as both give nearly the same 
result for this kind of stress combination. The following 
experiments were carried out to confirm and extend Voigt’s 
results, and especially to repeat them under the highest 
possible hydrostatic pressure. The range of pressure obtained 
in the present experiments is 900 atmospheres or 9 kg. per 
sq. mm., which is twenty times the breaking-stress of rock- 
salt, the material experimented upon. 


Description of Apparatus. 


The hydrostatic pressure was obtained by means of a 
Schaffner and Budenberg screw-pump furnished with a 
Bourdon gauge, reading to 1500 atms. 

The specimens were tested in a steel cylinder of the 
following dimensions :—Inner diameter 4 cms., outer diameter 
10 ems., length 80 cms. 

The cylinder was fixed vertically in an iron stand built 
into the stone floor of the room, so as to avoid all shocks and 
vibrations which might tend to break the test-pieces. The 
specimens were ruptured by means of the electromagnetic 


* Voigt, Annalen der Physik, lvii. p. 452 (1899). 


Materials under Combined Stress. 83 


pull on an iron rod placed partly inside a coil carrying an 
electric current. The actual arrangement of the apparatus 


COLLIIA YL IPUT, 


7 


is shown in the accompanying figure. In order 
that the force might be as great as possible two 
coils were used, wound on the same brass tube 
and about 15 ems.apart. This tube served also 
to guide the iron cores, which were connected 
together so that they moved as one rigid piece, 
the distance between them being the same as 
that between the coils. In order to diminish 
the friction between the cores and the guiding- 
tube, a small frame carrying three rollers was 
fixed to each core. In the part of the brass 
tube above the upper coil a long slit was cut 
reaching to within 2 ems. of the top of the coil. 
A piece of thick steel wire was soldered to the 
upper end of the iron core and bent so as to 
travel in this slit, it thus prevented the iron core 
from turning, and at the same time served as a 
hook by which the core could be hung from the 
piece to be tested. The other end of the test- 
plece was supported on a ring-nut travelling on 
the brass tube on which a corresponding screw 
was cut. By adjusting this ring the core could 
always be suspended in the same position in the 
tube although the test-pieces might vary in 
length. In order to show when the test-piece 
was broken, it was arranged that the iron core 
in falling should break the electric circuit. 

For this purpose a ring carrying a small 
ebonite mercury cup connected by a piece of 
flexible wire to the upper coil was made to slide 
on the tube below the hook mentioned above. 
A wire coming from the cylinder cover dipped 
into this cup and closed the electric circuit of 
the two coils. When the test-piece was broken 
the steel hook in falling carried the ring with 
it and so broke the connexion. In the experi- 
ments the current was gradually increased until 
the connexion was broken and the maximum 
reading of the amperemeter noted. 

The connexion between the force exerted by 
the iron cores and the current in the coils was 
measured in a preliminary experiment. The 
brass tube carrying the coils was placed in the 


cylinder in the position which it occupied in the breaking 


G 2? 


84 Mr. W. Ellis Williams on the Rupture of 


tests. The cylinder being left open, the iron core was hung 
from one pan of a balance placed directly above the cylinder, 
and the force for each current measured by placing weights 
in the other pan. The values obtained are given in the 
following table :-— 


| 
Current | Force in | Friction in ahd | “lee | . 
in amps. | grammes. | grammes. | Current. | Force. | Friction. 
p Pata fevasercs Dey 
0 | 14553 2 1125 oh Re 5 
43 70 2 130 | 3120 | We 
yas. | 1049 3 148) | 3570") ae 
Gari ole) 1282 5 157 |") S7OOk a aaeem 
816 | 1670 5 160, | 3920) | ieam 
9-03 | 1905 5 | 


It will be seen that the maximum force obtained was 
nearly 4 kilos, and that the friction never exceeded 1 per 
cent. of the total force. 


Preparation of the Rocksalt Pieces. 


Owing to the extreme brittleness of the rocksalt it was. 
found necessary to suspend the pieces in such a way that there 
should be no bending moment on the piece, that is to say 
the line of resultant stress must pass through the centre of 
each cross section of the rocksalt rod. For this purpose the 
apparatus shown in fig. 2 was made. It consists of two. 
cylindrical pieces made in aluminium, with holes Fic. 2 
of the proper size into which the rocksalt rods ete 
could be fastened with wax. Lach piece has a 
short steel rod pointed at each end screwed in. 
The inner two of these points serve to suspend the 
piece in the testing apparatus, while the other two 
serve to mount the piece in the lathe during the 
preparation of the rocksalt rod. These steel rods 
were carefully fitted in on the lathe so that they 
were exactly parallel to the axes of the cylindrical 
pieces. The rocksalt was first split with a knife 
into prisms about 3 cms. long, and was then 
worked approximately cylindrical with a fine file. 

The aluminium end-pieces were then fixed on, 
and while the wax was still soft the whole was 
pushed into a brass tube into which they accurately 
titted, and left in until the wax hardened. This ensured that 
the axes of the two end-pieces sheuld be in the same straight 


l 


= 
Ss 
= 
= 
=! 


nuit 
[rmentf 


SS 


ns 


Materials under Combined Stress. 85 


line, and consequently the axes of the four steel points also 
lay in this line. 

The whole was then placed on a watchmaker’s pair of turns, 
and by means of a pulley on one of the pieces the rocksalt 
was turned with a file and then polished with a piece of moist 
wood. The pieces were turned down a little thinner in the 
middle as shown in the figure, the diameter of the thinnest 
part being about 2 or 3 mms. This process of preparing the 
rocksalt ensured that the two points from which it was sus- 
pended were exactly in the same straight line with the axis 
of the rod. 

As the fluid used in the pump and cylinder was water, it 
was necessary to protect the rocksalt from its action. For 
this purpose a small brass tube was screwed on the lower of 
the two aluminium pieces, and the vessel thus formed was 
filled with nitrobenzol, which completely covered the rock- 
salt. It was previously ascertained that this fluid had no 
dissolving power for rocksalt and did not affect its breaking 
strength. 

RESULTS. 

It was found from the beginning that the value of the 
breaking tension varied very much for different specimens. 
At first it was thought that this variation was due to imper- 
fections in the apparatus and methods of preparation of the 
rocksalt rods. All improvements in the apparatus did not, 
however, alter the results, and the conclusion was finally 
come to that the differences were due to variations occurring 
in the rocksalt itself. 

Two different crystals were used in the experiments—one 
about 6 x 4x 2 cms., and the other a large square block about 
10 ems. each way. 

The first piece was all used up in preliminary experiments 
in determining the breaking stress of rocksalt with weights, 
and the results obtained were much better than the results 
afterwards obtained in exactly the same way with rods cut 
from the iarger block. It has been shown by Voigt that the 
breaking stress of crystal specimens depends on the direction 
in which they are cut relative to the crystalline axes, and 
that in rocksalt this variation is very marked. I found that 
in specimens from the large block cut in a direction making 
an angle of 45° with the crystalline axes, the breaking stress 
was 4000 grammes per sq. mm., or more than four times the 
maximum result with pieces cut parallel to the axes. It is 
possible therefore that the variations observed are due to 
small irregularities in the crystalline structure of the pieces 


86 Mr. W. Ellis Williams on the Rupture of 


which cause the direction of the axes to vary from point to 


point of the crystal. Although owing to these variations ‘it! 
was impossible to make any exact comparison between the 
breaking strength with and without external pressure, yet 
as the hydrostatic pressure employed was many times greater 


than the breaking stress of rocksalt, the results show clearly 


that the phenomena are not in accordance with Poncelet’s 
theory. Thus the strength of rocksalt under atmospheric 
pressure varied from 400 to 900 grammes per sq. mm. with 
a mean of 600 grammes. According to Poncelet’s theory 
the strength under 1000 atmospheres should be 10,000 
grammes, while the values actually obtained range between 
900 and 1000 grammes. 


The actual results obtained are shown in the following | 


table :— 
Tests without external pressure. 


Specimens from first block, broken with weights :-— 
449, 372, 390, 391, 497, 466, 530, 510, 407, 364, 
485, grms./sq. mm. 
Mean :—447 erms./sq. mm. 
Specimens from second block of salt, broken with weights :— 
503, 600, 901, 946, 570, 381, grms./sq. mm. 
Mean :—649 grms./sq. mm. 


Specimens from second block, broken in the cylinder wita 
electromagnetic apparatus, but without pressure :— 


630, 700, 700, 400, 720, 380, 300, 427, grms./sq. mm. 
Mean :—537 grms./sq. mm. 
Tests under pressure. 


All specimens from second block. 


Pressure. Breaking Stress. 
150 atms. 932 grms./sq. mm. 
ZOO ie, 502 i 
DOO. 3 o01 it 
960", 800 ue 
OGa'* 840 49 
SFOs Pee 5940 aS 
SOD 587 . 


180, 853 » 


Materials under Combined Stress. 87 


Two specimens were not broken with highest current 
employed, giving strength greater than 


820, 762 grms./sq. mm. respectively. 


The five specimens tested at a pressure from 900-1000 atms. 
give a mean of 724 grms/sq. mm. 


Haperrments on Aluminium Wires. 


It was thought desirable to extend the experiments to 
ductile materials, which might possibly behave in a different 
manner to a brittle substance like rocksa't. Lead and tin 
were first tried, but it was found difficult to obtain satisfactory 
results owing to the fact that rupture takes place gradually, 
and the result depends on the time during which the 
stress is applied. Finally, aluminium was chosen as having 
a comparatively low breaking strength, and giving consistent 
results when tested with weights. A length of hard-drawn 
aluminium wire, ‘48 mm. diameter, was cut into pieces a 
few centimetres long, and alternate lengths tested with and 
without pressure. The results were :— 


Breaking stress without external pressure. 
Mae 13'S, §13°6,. V450943-2 Ialos/sq. mm: 
Mean :—13°6 kilos/sq. mm. 


Breaking stress under 700 atms. pressure. 
T42, 14°2, 12-7, 12°38, 15:2 kilos/sq. mm. 
Mean :—13°8 kilos./sq. mm. 


There is thus no appreciable difference in the breaking 
stress with and without external pressure. 


The above experiments were carried out in the Physical 
Laboratory of the University of Munich, and my best thanks 
are due to Professor Rontgen for his help in carrying out the 
work, and especially for suggesting the methods employed 
to overcome the difficulties connected with the preparation 
of the rocksalt pieces, and the electromagnetic apparatus for 
obtaining the breaking tension. 


University College, Bangor, 1907. 


r 88 7 


VIII. On the Secondary Undulations of Oceanic Tides. (An 
Abstract.) By K. Honpa, T. Terapa, and D. IstTAnt, 
Tokio Imperial University, Japan*. 


[Plates I.-VI.] 


CONTENTS. 


. Introduction. 

. General Conclusions. 

. Results in Detail. 

. Experiments with Models. 

. Formule for Calculating the Periods of the Oscillations in Bays. 

. Method and Results of Determining the Period of Oscillation by 
Calculation. 

. Sea-Waves and Secondary Undulations. 

. Oscillations of Large Bays and Anomaly of Tides. 


COLOQD2  CORNOKXORNONONOA 
Oo NI > OU CO DD 


§ 1. Introduction. 


TPXHE tidal curves obtained by self-recording tide-gauges 
are often accompanied by a peculiar zigzag or secondary 
undulation of considerable amplitude, which becomes per- 
ceptible at a station situated in a bay or an estuary. ‘This 
fact seems to have been noticed by many earlier observers. 
Without discussing all these earlier contributions to 
the subject, we will briefly mention some comparatively 
recent valuable investigations. David Milnef discussed the 
remarkable undulations of July 1843 which were noticeable 
on the coasts of Great Britain, and ascribed their origin to 
the storm then prevailing in that district. Admiral Smytht 
referred to the same phenomenon observed at Mazzara, where 
it has long been termed Mrabia or Marrobbio. Sir George 
Airy§ believes that the secondary undulation is the seiches 
between the Sicilian and African coasts. This phenomenon || 
has also been observed on the coast of Italy and the northern 


* Communicated by the Authors. 

* D. Milne: “On a remarkable Oscillation of the Sea observed a 
various Coasts of Great Britain,” Trans. Roy. Soc. Edinb. 1844. 

t Smyth: Memoir descriptive of the Resources, Inhabitants, and 
Hydrography of Sicily ; London, 1824. 

§ G. Airy: “On Tides in Malta,” Phil. Trans. Roy. Soc. London, 
clxix. 1878. 

|| G. Grablovitz: “ Ricerche sulle maree d’Ischia,” Rend. Ace. Lincet, 
vi. 1890. “Sulle osservazioni mareogratiche in Italia e specialmente su 
quelle fatte ad Ischia,” Atti del I. Congr. Ital., Genova, 1893. R. Sieger : 
“* Niveauveranderungen an Skandinavischen Seen fund Kiisten,” Verh. 
Qten Deutsch. Geogr., Wien. Is. Briickner : “ Ueber Schwankungen der 
See und Meere,” Verh. 9ten Deutsch. Geogr., Wien, 1892. 


Secondary Undulations of Oceanic Tides. 89 


coasts of Europe. In 1895 W. Bel] Dawson™ read a paper 
before the Royal Society of Canada, in which he demonstrated 
the existence of secondary undulations of considerable 
amplitude. 

Professor Duffft concluded, from observations at many 
stations along the coast of the Bay of Fundy and the Gulf of 
St. Lawrence, that the periods of the secondary undulation 
were peculiar to each station, and that the phenomenon here 
partook of the nature of seiches excited by the low barometric 
conditions. H.C. Russell} states that at Sydney the undu- 
lations are in most’ cases due to atmospheric disturbances. 
Napier Denison§, of Toronto Observatory, made a systematic 
study of the subject in connexion with the barometric changes, 
and attributed the phenomenon to long waves generated by 
air waves of considerable wave-length, which accompanied 
the low barometric pressure. For several years Professor 
Giovanni Platania|| has been investigating the secondary 
undulations in the Gulf of Catania, as well as on other ceasts 
of Italy, and has noticed that the oscillations of conspicuous 
amplitude. occur in connexion with barometric disturbances. 

In the bays of Japan on the Pacific coast, the phenomenon 
is sometimes so remarkable that it is commonly known as 
Yota. In the harbour of Nagasaki it is called Abski; and 
here the amplitude of undulation frequently exceeds 60 cms. 
This yota or abiki is trequently observed during the calm 
weather which immediately precedes an approaching low 
barometric pressure. 

Professor F. Omori] studied the secondary undulations of 
Ayukawa, Misaki, and Hososhima mareograms in connexion 
with his discussion of the several destructive sea-waves, and 
found that the periods of the waves are the same as those 
observed in ordinary cases. The records** taken from the 
tide-gauges at Indian coasts, of the waves which accompanied 


* Dawson: “Notes on Secondary Undulations,”’ Proc. Roy. Soe. 
Canada, May 1895. 

+ Duff: ‘“Seiches on the Bay of Fundy,” Amer. Journ. Sci. iii. 1897. 
“ Periodic Tides,” ‘ Nature,’ lix. 1899. 

{ Russell: “The Source of Periodic Waves,” ‘ Nature,’ lvii. 1898. 

§ Denison: “ Secondary Undulations of Tide-gauges,”’ Proc. Can. Inst. 
i. 1898. “ The Origin of Tidal Secondary Undulations,” zb7d. i. 1898. 

|| Platania: “ Le librazioni del mare con particolare riguardo al Golfo 
di Catania,” Att: del V. Congr. Geogr. Ital., Napoli, 1904. “I fenomeni 
in Mare durante il ferremoto di Galabria del 1905,” 1907. <“Nuove 
ricerche sulle librazioni del mare,’ 1907. 

4 Omori : Publications of Earthquake Investigation Committee, xxxiv. 
1900. 

** Omori: Publications of Earth. Inves. Comm. lvi. 1906; Proc. Tukio 

Math.-Phys. Soe. 11. 1905. 


90 Messrs. K. Honda, T. Terada, and D. Isitani on the 


the great eruption of Krakatoa 1888, were also found to show 
the same periods as were frequently found in ordinary cases. 
He explained this interesting fact by supposing that a bay, 
or a certain portion of the sea, makes some fluid pendulum 
motion with its own period. Professor H. Nagaoka™*, in his 
paper on the hydrodynamical investigations of sea-waves, 
expressed the desirability of & special inquiry looking towards 
an explanation of this phenomenon. The suggestion was 
taken up by the Earthquake Investigation Committee, with 
the result that the task of making a series of systematic 
observations was imposed upon us. 

The observations were carried out during the years 1903 
to 1906; the number of bays and coasts where work was 
done amounts to about sixty in all, extending from Hokkaido 
to Kiushiu. The work consisted jirstly in finding the periods 
of the undulations peculiar to each bay or coast; and 
secondly in comparing the phases of the different portions of 
a bay by simultaneous observations—which latter, however, 
was accomplished only for a number of the most typical 
bays. 

‘a most cases portable, self-recording tide-gauges (PI. L., 
figs. 1,2) were used, the construction and theory of which 
has been already published in the Phil Mag. vol. x. p. 253 
(1905). 


§ 2. General Conclusions. 


The general conclusions f, which have been drawn from the 
thorough study of the numerous records obtained, are given 
in the following propositions. In discussing our results we 
have availed ourselves of the valuable records made by Lord 
Kelvin’s tide-gauges at ten different stations. These instru- 
ments have been set up at various places on the coasts of the 
Pacific and of the Japan Sea, and some of them have been 
in working operation twenty years. 


1. On the Pacitic coasts, free from any inlet in the coast- 
line, the secondary undulation is quite unnoticeable, 
and of a very irregular nature. 

2. On the coasts of the Japan Sea the secondary undula- 
tion on the open coast is observable, though the periods 
of the undulations are not regular. 


* Nagaoka, Proc. Tokio Math-Phys. Soe. i. 1903. 
+ In Pl. Il. a map of Japan is given. The locations of the principal 
observation stations are marked by points. 


Secondary Undulations of Oceanic Tides. 91 


3. Ina bay of considerable area, or in a shallow bay with 
a narrow opening towards the ocean, the secondary 
undulation is in ordinary cases imperceptible. 

4, In a deep bay or estuary, the breadth of which is not 
large in comparison with its length, the secondary 
undulations are most pronounced. 

5. In bays or open coasts, which are not far from each 
other, a common undulation is observed. 

6. The secondary undulations in many bays change their 
periods continuously and through certain ranges. 

7. In some bays the periods of the undulation are fairly 
constant. 

8. In many cases the same trains of secondary undulations 
appear in the same phase with respect to the tidal 
wave, on consecutive days of ordinary weather. 

9. The phases of the prominent, fundamental_undulation 
at different parts of a bay are equal. 

10. The periods T of the most pronounced undulations are 
fairly given by the relation 


where / is the length of the bay measured along its 
depth, / the mean depth of the bay, and g the force 
of gravity. 

11. Just outside a bay the undulation, which inside has 
been observed to be of considerable amplitude, may 
also be traced, but its amplitude is very small. 

' 12. In a bay, the periods of the conspicuous undulation 
observable in the case of a storm, or in that of a sea- 
wave of distant origin, are the same as those ordinarily 
observable in the bay. 


It has long been believed that the secondary undulation in 
a bay is the ‘seiche between two opposite sides of the bay ; 
but according to our observations, the phases of the most 
conspicuous undulation are the same throughout the bay, so 
that this view cannot be universally true. ~ Napier Deniscn 
considers the undulation to be long waves continued from 
the ocean into the bay, on which supposition all the con- 
clusions above enumerated, except the 9th, 10th, and 12th, 
can easily be explained. But the fact that there is a pro- 
minent undulation peculiar to each bay, cannot be explained 
by merely considering progressive waves. 
_ This undulation, however, can be explained in the following 
way. As a representative example, take a rectangular bay 


92 Messrs. K. Honda, T. Terada, and D. Isitani on the 


of constant depth. Suppose a regular series of long waves 
are continuously propagated in the direction of the length of 
the bay and are reflected at its end. Through the interference 
of the incident and reflected waves, a standing wave is formed 
having its loop of the vertical motion at the end of the bay. 
If the wave-length be such as to form the node of the vertical 
motion at the mouth of the bay, the period is the same as 
that of the fundamental oscillation of a tank having double 
the length of the bay; and therefore the amplitude of oscil- 
lation must necessarily be magnified by the successive 
occurrences of the long waves. The period of the oscillation 
is then, neglecting the mouth correction, expressed by the 
relation | 
4l 
WV gh? 
where /gh is the velocity of the long wave. This action of 
the bay may then be suggestively compared to the experiment 
in which a column of air in a resonance tube is made to 
vibrate in unison with a tuning-fork placed over its mouth. 
If the waves of different periods proceed from the ocean 
toward the shore, the one whose period coincides with that 
of the oscillation having its node at the mouth of the bay 
will excite the most energetic oscillation of the bay water. 
Thus, bays on the coast-line may be compared with a series 
of resonators, each of which takes up selectively from the 
chaos of very complicated sounds or noises and resonates to 
the note of its proper period. The plausibility of such a 
conception seems to be heightened to a rather unexpected 
degree by the present investigation. Moreover, the fact that 
the rising and falling motion of the level of the bay in respect 
to the principal undulation is in the same phase for several 
stations, stands in favour of the above view. Darwin and 
Otto Kriimmel* seem to have entertained an analogous idea. 
In a bay, besides the uninodal oscillation above referred 


i pe 


to, oscillations with two nodes, three nodes, &c., are also’ 


possible ; the periods of these oscillations are respectively 
4,1, &. of the period of the fundamenta! oscillation. In 
some cases, a lateral oscillation of the bay excited by incident 
waves is also possible, the period of which is principally 
determined by the oscillating water in the bay. These addi- 
tional modes of oscillation were actually found to exist in 
some bays such as Hososhima, Ofunato, Hakodate, &e. 


* Darwin: ‘The Tide, ch. x. p. 169. Otto Kriimmel: “ Ueber 
Gezeitenwellen,” Rede bei Antritt d. Rectorates d. Konig]. Christ-AIbr.- 
Univ. zu Kiel, Marz 5, 1897. 


Secondary Undulations of Oceanic Tides. 93 


In the oscillation of the bay-water just referred to the 
period of the forcing wave, which corresponds to the maximum 
resonance, is not sharply defined, but within a small range 
the oscillation remains fairly conspicuous, as we have often 
proved. 

In the bays of regular shape, such as Ofunato and Hoso- 
shima, the position of the mouth line is determinate; but in 
the bays of complex shape, such as Shimoda and Susaki, 
several mouth lines are conceivable. By the choice of the 
mouth lines, the length and mean depth of the bay vary 
within a considerable range, so that the period of the proper 
oscillation changes within a certain range. Hence such a 
bay may resonate to any one of the incident waves whose 
period falls within the same range. In the two bays above 
mentioned, the period of the conspicuous undulation was 
actually found to vary within a wide range. ; 

As to the cause of the long waves, which manifest them- 
selves as secondary undulations, we may mention the wind, 
the cyclone, the earthquake, &. It isa matter of fact that 
the seiche in many lakes, which is the result of interference 
of direct and reflected waves of long wave-length, is often 


excited by a strong wind. In the same way the wind blowing 
on the surface of the ocean may cause long waves of several 
kilometres. Such waves, too, are often caused by a deep 
cyclonic centre. Near such a centre fluctuations of pressure 
and of wind velocity go on incessantly, and these varying in- 
fluences, acting in an impulsive way, may cause waves of long 
periods. An upheaval or depression of the sea-bottom, due 
to an earthquake or to a submarine eruption, may also be a 
cause producing the same result. 


§ 3. Results in Detail, 


In the present abstract, it is not proposed to give the 
details of observations made at all the stations; we will 
confine ourselves to mentioning some of the remarkable 
findings in the bays. in which the secondary undulations 
appeared most markedly, and out of the sixty stations at which 


operations were carried on, we have chosen the following :— 
(i.) Hakodate. 


Hakodate. situated on the middle coast of the strait of 
Tsugaru, which separates Hokkaidé from Honshiu, is the 
best anchorage in Hokkaid6. The bay is approximately 
semi-circular in shape. Here Kelvin’s tide-gauge has been 
continuously working during the last twenty years, and has 


94 Messrs. K. Honda, T. Terada, and D. Isitani on the 


recorded several important sea-waves which had their origin 
near the American coasts of the Pacific, as well as near our 
coasts. | 


Mig. 1. 


TACHIMACH/ZAHT 


X 
HEA ODATE 


BAY OF HAKODATE 


x 
AAl171S0 


x 
JOMIKAWA 


At Hakodate the secondary undulations are very promi- 
nent, sometimes exceeding 30 cms. in amplitude, and of 
fairly regular periods. The periods* of the most conspicuous 
range from 45™°5-57™5. Sometimes its octave 2179-245 
is found superposed on the undulation of the above period. 
In the undulation accompanying the sea-waves the octave 
generally appears in a marked degree. 

When observations at Hakodate and Kamiiso were carried 
on simultaneously, it was found that only the fundamental 
oscillation appeared on the records. The comparison of our 
records showed that the phase of the undulations was the 
same for these two stations. When like observations were 
made simultaneously at Hakodate and Tachimachizaki, it 
was noticed that although the conspicuous undulation appeared 
in the bay, yet just outside it was very insignificant. These 
observations agree well with our view regarding the secondary 
undulations, which we have propounded in the foregoing 
Section. 

If we consider the longer period to correspond to the 
fundamental oscillation, and the shorter period to the lateral 
oscillation of the bay, the calculated periods for these oscilla- 
tions are respectively 45™3 and 23™6, which coincide well 
with the observed values. As we shall see hereafter, the 
periods corresponding to these two modes of oscillation, as 
given by our model of the bay, are also 47™0 and 23™°6. 


* Figures in heavy type always represent conspicuous undulations. 


Secondary Undulations of Oceanic Tides. 95 


These results confirm our view regarding the nature of the 
secondary undulation. 

The amplitude of the secondary undulation is usually in- 
creased by a low barometric pressure, which is approaching 
the bay. As a good example, we may cite the cyclone of 
Sept. 21-22, 1904, which was approaching from the Pacific 
side of Honshiu toward Hakodate. The undulations in the bay 
continued over a whole day, with a considerable amplitude 
the maximum exceeding 40 cms. The periods of conspicuous 
undulation were 47™1-56™-9 and their octaves. 

The bay is especially sensitive to sea-waves; and it is 
remarkable that waves orivinating on the American coasts 
have often been recorded by the tide-gauge of the bay. The 
periods of the Ecuador wave, 1906, in the bay were 24™-2- 
267-0 and 48™-3-51™°8; while those of the Valparaiso one 
were 23™-5-26™0 and 47™-0-52™:0. The periods of the great 
sea-wave of Sanriku, 1896, observed in the bay, were 23™6 
and 47™°3-52™-1; those of the small wave, 1897, in the same 
bay were 22™1 and 45™5. 


Gi.) Aomori. 

The large bay of Aomori has the form of a dumb-bell, and 
is connected with the strait of Tsugaru by a wide neck. The 
observations were made at Aomori, wien the centre of a deep, 


low pressure was approaching the district from the Japan- 
sea side. On this occasion a regular undulation of 103™ 


Fig. 2. 


JS8SUCARU STR. OMINATO 


Bar oF Aomor/ 


x 
AOMORI 


appeared on the record and continued for a day and a half. 
Upon this an undulation of a period ranging from 234 to 
26™°3 was superposed. Besides, a period of 295™ could also 
be traced. 

The undulation of 103™ is probably the lateral oscillation 
of the bay, and that of 295™ the fundamental oscillation 
having its node at the mouth. ‘The calculated periods 108™ 
and 284™ are thus in a fair agreement with these observed 
periods. ‘The corresponding periods as given by our model 


96 Messrs. K. Honda, T. Terada, and D. Isitani on the 


are respectively 108™ and 303™. The undulation of the 
shortest period may be the higher mode of the lateral 
oscillation. 


(iii.) Ofunato. 


The bay of Ofunato has an elongated form, and is some- 
what crooked near its mouth, so that at Ofunato, which is 
situated at the end of the bay, the sea is extremely calm. 


“A 
OFUNATO 


Hosoura xV¥ OFUNATO 


This affords a good example for illustrating the effect of 
shadow, 2. ¢. a wave whose wave-length is large compared 
with the size of the obstacle goes round it, but a wave whose 
wave-length is comparatively small has its influence nullified 
by the obstacle, and the sea bebind it remains quite calm. 

The form of the bay is specially adapted to permit of 
comparison of the phases of the secondary undulations at 
different stations along its length; those chosen were Ofunato, 
Sunagosaki, Takonoura, and Hosoura. The periods observed 
are 5™5, 12™-8-16™'8, 36"-0-39"1, and 41™5—43™5.) /1 ae 
phase of the significant undulation 36™-0-39™1 is found to 
be the same for these four stations; which undulation is 
probably the fundamental oscillation of the bay. The caleu- 
lated period is 36™:-4—in close agreement with the observed. 

At Ofunato and Hosoura, w here an undulation of the period 
12™-8 168 was sometimes obser ved, its phases were opposite 
to each other. The undulation was quite imperceptible at 
Sunagosaki, which is situated about midway between Ofunato 
and Hosoura, so that this may probably be a binodal oscilla- 
tion of the bay. Waves of the short period 5™5 appeared at 
Hosoura and Sunagosaki, but not at Takonoura and Ofunato; 
the absence of the wave at the latter stations is possibly due 
to the effect of shadow. 


Secondary Undulations of Oceanic Tides. Sf) 


(iv.) Moroiso. 

About 4 km. north of Misaki, in the province of Sagami, 
‘there lies a small forked bay, the one branch of which is 
called Moroiso, the other Aburatsubo. In Aburatsubo a 
Kelvin’s tide-gauge is constantly working, the records of 


Fig. 4. 


x Observed station. 


which Professor F. Omori has frequently referred to as 
Misaki mareogram. The undulation is very regular and 
marked, having the period 18"-8-15™6; the calculation of the 
period for the fundamental oscillation gives a fairly coincident. 
value 13™4, | 

The record shows an appearance of the beat of two waves 
of nearly the same wave-length. So it was suspected that 
the phenomenon might be due to. the interference of the two 
distinct modes of oscillations of the two branches of the bay, 
which constitutes a vibrating system with two degrees of 
freedom. But this was found not to be the case, since the 
simultaneous observations at different parts of the bay showed 
the identity of the undulations with respect to their forms 
and phases. By comparing the records of the simultaneous 
observations at the inside and outside of the bay, we could 
distinctly trace corresponding undulations in the two records. 
The amplitude of the wave outside the bay is, however, very 
small as compared with that of the undulation inside the 


bay. 
(v.) Susaka. 


Susaki is a deep bay on the middle coast of Tosa in 
Shikoku. Init the observations were made at four stations— 
Yamasakibana, Otani, Heshima,and Kure. The diagrams of 
Yamasakibana are very conspicuous, and characterized by 
the simplicity of the undulations; the periods observed are 
3079, 3574-3875, 40™-0-46"'8, and 500-5470. The 
periods observed at Otani are 17™6-18™'2, 35™°4 and 53™3 ; 

Piil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. H 


98 Messrs. K. Honda, T. Terada, and D. Isitani on the 


those at Heshima 24™°6-27""6, 39™°7, and 551. At Kure 
secondary undulations of the periods 15"-0-16"'3 and 61a°3 


Io. 5. 
lig. 


YaMASAKI BANA 
xX 5 


X WESHIMA 


BAY OF 


SUSAA/ 


. 


were noticeable. Here it is to be observed that at the last 
three stations the records are not so rich in results as those 
at Yamasakibana. 

Comparing the records of Kure and Yamasakibana taken 
on the same days, it will be seen that the waves with the 
periods 35™-40™ appeared only at the latter station, and those 
of 16™ only at the former. Again, comparing the records of 
Yamasakibana and Otani of the same days, the waves of 
39™-40" are found common in both stations, while waves of 
18™ are peculiar only to the latter. At Heshima, which is 
situated at the mouth of the minor inlet of Otani, waves of 
16™ are noticeable by their absence, while waves of 35™— 
40 are apparently traced, though not of so great amplitude 
as at Yamasakibana. 

Thus, we may infer that the undulation of 35"-40™ is the 
fundamental oscillation of the bay, having its node near 
Kure, and that the undulation of 16™ at Kure is the oscilla- 
tion of the minor inlet. The undulation of 18™ at Otani is 


Secondary Undulations of Oceanic Tides. Shy 


probably the seiches between Awa and Otani. Calculated 
periods, corresponding to those supposed modes of oscillation, 
show a fair agreement with the actual periods. The experi- 
ments with the model also lead to the same conclusion. 


(vi.) Hososhima. 


Hososhima is an elongated bay on the eastern coast of 
_Hiuga in Kiushiu; here Kelvin’s tide-gauge is constantly 
working. It recorded several sea-waves, which were origi- 
nated on the American coasts and which had travelled across 
the Pacific. Simultaneous observations were recorded at 
Hososhima and Isegahama, situated inside and outside the 
bay respectively. In the bay, extremely regular undulations 


Fig. 6. 


BAY OF 
FIOSOSHIMA 


x 


a 
HOSOSHIMA. 


/SECAHAMA 


S 


appeared superposed on the tidal wave, whose periods varied 
trom 17™°8 to 20-3 according to the tidal phase. In calm 
weather the amplitude of undulation amounted even to 25 em. 
The period of the undulation slightly decreased as the tide 
passed from low water to high. The fundamental period, 
calculated to be 19™-0, agrees well with the observed one. 
_The change of the period caused by a change of depth due to 
tidal influence, has also the range which is to be expected 
from the theory. Besides, longer periods such as 34™-0- 
38™°7 and 43™4-49™-]1 are sometimes observed. 

Outside the bay, the undulation of the period 17™-8-20™3 
is very faint, while the undulations of the longer periods are 
barely observable. By placing the diagram of a record taken 
in the bay upon the corresponding one taken on the open 
coast, we can distinctly trace undulations in the two records, 
which correspond to each other. If we bring the records 
of any two consecutive days into coincidence as regards 
the tidal phase, we observe the same succession of undu- 
lations. 


H 2 


100 Messrs. K. Honda, T. Terada, and D. Isitani on the 
(vil.) Nagasaki. | 
Nagasaki is a well-known harbour on the western coast of 


Kiushiu; the observations were made near the end of the 
bay. Since March of 1905 a tide-gauge of our system has. 


Fig. 7. 


F UH AHORI 
4 


been set up in the same place by the Office, and many 
significant records obtained. 

In the bay, the secondary undulation is so conspicuous that 
it is usually known as abiki. The observed periods are 
22™- 5-252, 32™-0, 34™°5-37™6, 40™ 1, 44™-5-45™2, 53™-6, and 
69™-0-72™-0; the amplitude of the conspicuous undulations. 
often exceeds half a metre. On one occasion, about ten 
years ago, the amplitude of the aliki was over 2 metres, and 
a large number of boats and steamers are said to have been 
damaged. The largest amplitude since the beginning of the 
tide-gauge observation was 1°54 m. observed on May 2, 1905. 
P]. ILL. contains the record of the famous abzki. 

The conspicuous abiki is generally associated with weathers 
in which the isobars in the neighbourhood trace a devious 
course, because of the co-existing low barometric centres. 
Since it is well known that a tornado is frequently associated 
with such a distribution of isobars, it seems then very pro- 
bable that in such a weather a sudden local disturbance of 


Secondary Undulations of Oceanic Tides. 101 


pressure may often occur, and that this barometric dis- 
turbance, giving rise as it does to waves of considerable 
length, may be a cause of the abiki. A deep barometric 
centre with regular concentric isobars, which is approaching 
the district, excites short waves of considerable amplitude, 
but does not cause an abiki of a marked amplitude. 

As to the modes of oscillation of the bay, two are con- 
ceivable. The one is the seiches between the Fukahori 
side and the end of the bay; the other the fundamental 
oscillation having its node at the mouth. The periods 
calculated on this supposition are 22™°6 and 37™°5 respectively, 
in good agreement with the observed periods: experiments 
with the model gave also fairly coincident values. 

Near the western mouth of the bay of Nagasaki lies 
Fukahori, where Kelvin’s tide-gauge incessantly operates. 
The undulation is generally imperceptible ; the same periods 
as those at Nagasaki are also traceable. It is interesting to 
note that, though the uninodal oscillation of the bay is very 
prominent at Nagasaki, it is not especially so at Fukahori, 
the latter being situated near the node of the oscillation. 
Hiven the great abiki of Nagasaki on May 2, 1905, was only 
30 cm. in amplitude at Fukahori. On the other hand, the 
seiches between Nagasaki and Fukahori sides are conspicuous 
even at the latter station, where the oscillation nearly forms 
the loop. 


§ 4. Experiments with Models. 


To confirm our theory, we decided to experiment with 
models, in order to endeavour to find the actual mode of 
oscillation of the bay. Several models of bays were made 
exactly proportionate to the original, and the periods of the 
oscillating water in these models were compared with the 
observed ones. In reducing the period of oscillation in the 
model to the actual one, it was assumed that the period is 
proportional to its length, and inversely proportional to the 
square root of its depth, provided the latter is a small fraction 
of the former. 

The construction of a model proceeded thus : first, contour- 
lines of the bed of the bay to be modelled were drawn on 
separate zinc plates, and these plates cut along these lines. 
They were then placed one above the other, the space between 
each two being kept by blocks of wood of such thickness 
that the ratio to the actual depth was in each case maintained. 
The interspace between the plates was then filled with cement. 
The model thus constructed was immersed in a large rectan- 


102. Messrs. K. Honda, T. Terada, and D. Isitani on the 


gular tank (150 x 76 x 19 cm.?) filled with water to the water- 
line of the model. | 

The waves were then excited by a pendulum-bob oscillating 
in the water. This was a lead ball, 7 cm. in diameter, 
suspended at the end of two pieces of cord, which from the 
point of suspension above passed downward through two 
holes in a movable horizontal bar, the latter being used to 
regulate the effective length of the pendulum. ‘The part of 
the pendulum which oscillated with the bob was thus re- 
stricted to that part of the strings below the horizontal rod, 
the length of which could be varied at will by moving the 
rod upward and downward. With the pendulum arrange- 
ment, it was easy to obtain a period less than three seconds, 
but if one wished to obtain a longer period it would be 
necessary to use a pendulum of a considerable length. To 
avoid this inconvenience a horizontal pendulum was utilized. 
This was constructed by taking a horizontal brass bar, the 
one end of which rested in a steel cup embedded in a vertical 
framework, while the other was held by a cord which sloped 
from a point at the top of this framework. The cord was 
here attached by means of a steel ring, which rested on a 
knife-edge. A heavy lead ball was then suspended on the 
end of three strings from a frame attached to the movable 
end of the horizontal rod. By properly inclining this vertical 
framework, periods larger than three seconds could easily 
be obtained. 

When the penduium was made to oscillate in front of the 
model with its bob under the surface of the water, the water 
in the model oscillated smoothly, with no appreciable surface 
waves. In order to keep the amplitude of pendulum oscil- 
lation constant, a slight force by hand was applied near the 
upper end of the pendulum at suitable intervals. To avoid 
the reflexion of the excited wave from the walls of the tank, 
a thick layer of a damping material, such as wood-shavings, 
was laid in front of the reflecting walls. 

By simply exciting waves with the above arrangement, 
the water in the model made a standing oscillation, whose 
amplitude was generally small; but as the period of the 
pendulum approached to the proper period of the bay, the 
amplitude of oscillation gradually increased ; and when 
the period of the penduium coincided with this proper period, 
its amplitude was a maximum. In this case the mode of 
oscillation was the same as that conceived by us, that is, the 
end of the bay was a loop for the vertical motion, and a 
node for horizontal motion; while its mouth was a node 
for vertical motion, and a loop for horizontal motion. ‘The 


Secondary Undulations of Oceanic Tides. 103 


phase of the water particles in the bay was the same for all 
_ parts of the bay when the oscillation was fundamental. In 
an elongated bay, a binodal or trinodal oscillation was easily 
produced. 

It was convenient for the observation of the mode of oscil- 
lation to follow the motion of the fine cork powder, or better, 
fine aluminium powder which had been scattered over the 
surface of the water. To diminish the effect of surface 
tension of water as much as possible, a few drops of oil were 
poured on the water in the tank ; but before the fine powder 
was scattered over it, the water was well stirred. In this 
way the path along which the water particles moved could 
be easily traced. We also took the photograph of the model 
in the tank, when the bay water was oscillating. By placing 
a camera in a vertical position over the model and giving an 
exposure equal to about half the time of a period of oscillation, 
a photograph was made on which could be traced the path 
of each moving aluminium particle ; the aggregate of these 
paths then showed convincingly the actual mode of the 
horizontal motion. 

To determine the proper period of oscillation of a bay, 
the period of the pendulum was so adjusted as to give a 
nearly maximum amplitude of oscillation. The pendulum 
was then stopped, and the period of the subsequent oscil- 
lations was determined by means of a stop-watch. Though 
the period of the pendulum varied slightly from the above 
value, the period of the subsequent oscillation was quite 
constant. If the period of the exciter differed considerably 
from the proper period of the bay, the oscillation after the 
stopping of the pendulum was rapidly damped, and this 
gave us a good means of detecting whether the period of 
the exciter was near to the proper one, or not. 

We experimented with models of seven bays, in which 
regular and conspicuous undulations were observed: the 
results for three of these are given below :— 


(i.) Bay of Hakodate. 


The dimensions of the model were as follows :—Length 
1: 20200, and depth 1:548, so that the factor r, by which 
the observed period in the model must be multiplied in order 
to obtain the period of oscillation of the actual bay, was 863. 
The bay had two modes of oscillation : the fundamental, with 
its node at the mouth of the bay, and the lateral, which 
oscillated between Hakodate and Tomikawa with its node 
midway between. The periods of these oscillations were 
3°27 and 1°64 respectively ; multiplying by 7 we get 47™-0 


104 Messrs. K. Honda, T. Terada, and D. Isitani on the 


and 23™°6 in good agreement with the observed values. 
These two modes of oscillation were clearly seen from the 
photographs taken in the way before described. Figs. 8 
and 9 of the annexed cut are the stream-lines traced on the 


photographs. It is exceedingly interesting to trace the 
stream-lines in the case of the lateral oscillation. Certain 
of these lines extend from Hakodate to Tomikawa gradually 
diverting towards the middle, while other lines run toward the 
mouth of the bay from the Tomikawa side. Fig. 1in Pl. IV. 
is a photograph of the fundamental oscillation of the bay. 

In experimenting with models, it was observed that the 
period of the forcing wave, which corresponded to the maxi- 
mum resonance, was not well defined; within a certain 
range of the period, which did not much differ from the 
period of free oscillation, the oscillation remained fairly con- 
spicuous. In the actual case such a phenomenon was also 
observed: conspicuous undulations of 45™5-57™°5 were 
frequently observed, though the period of free oscillation of 
the bay is 47™0. 


Gi.) Bay of Aomori. 

The dimensions of the model were as follows :—Length 
1:110700, and depth 1: 731, so that the factor r was 4090. 

This bay had also two modes of oscillation, as in the Bay 
of Hakodate: the fundamental and the lateral oscillation. 
The periods of these oscillations were 4°45 and 1°60 respec- 
tively ; multiplying by r we get 303" and 108". During 
our observations, the oscillations of periods 295™ and 103™ 
were observed, which agree well with the above values. 

Fig. 2 in PI. IV.is a photograph of the lateral oscillation of 
the bay. Here the greater part of the stream-lines extends 
from Aomori to Ominato, while the other part runs from 
Aomori toward the mouth of the bay; the case is Just analogous 
to the corresponding oscillation in the Bay of Hakodate. 


Secondary Undulations of Oceanic Tides. 105 
(ii.) Bay of San Francisco. 

During the last fifty years the tide-gauge at San Francisco 
has recorded several sea-waves originated at different coasts 
of the Pacific, the periods of which are 17™-3-19™2, 24™-3- 
278, 34™°3-41"-2, 47™-4 and 116™, of which the first is an 
octave of the third. 

Now the bay is so irregular in shape that it is very difficult to 
find out by calculation what modes of oscillation correspond 
to the actual periods; hence a model* was constructed and 
experiments made. The scale of proportion used was as 
follows :—Length 1: 40000, and depth 1: 366, so that the 
factor r was 2076. The model was too large to go in the 
tank, so it was placed in a small pool in the court of the 
University. 

Since the greater part of the model was very shallow, the 
oscillation rapidly subsided when the exciting wave was 
stopped, so that the period was always determined by ob- 
serving the maximum resonance of the bay. For the wave 
incident upon Golden Gate, the principal modes of oscillation 
of the water were those between West Berkeley and Sausalito 
sides. The remaining portion of the bay, including both 
ends, seems to have but little influence on these modes of 
oscillation. | 

By exciting waves of the periods ranging from 3%1 to 
3°5, the water in the bay oscillated with the fundamental 
mode of oscillation, having its node near Golden Gate and 
its loop at West Berkeley side. The mode of oscillation most 
easily excited was a binodal seiche between the narrowest 
mouth-line and West Berkeley side. The period of the wave, 
which gave a maximum resonance to the binodal seiche of 
the bay, ranged from 1°1 to 1*4. By slightly changing 
the period of the wave, the corresponding displacement of 
nodal line was observed. We could also produce a trinodal 
seiche of the bay, the period of which was nearly 0*8. 
Multiplying these periods by 7, we get 1077-122", 38™-48”, 
and 28". The period 116™, which in all probability corre- 
sponds to the fundamental oscillation of the bay, was actually 
observed in the bay in the case of the sea-wave of South 
America, 1868. The periods corresponding to the binodal 
and trinodal oscillations above «described have often been 
observed in the bay. 

In concluding the present section, it may not be out of 


* We used the chart published by the Washington Coast and Geodetic 
Survey and presented to Professor Omori by Dr. O. H. Tittmann, Super- 
intendent of the Office. 


106 Messrs. K. Honda, T. Terada, and D. Isitani on the 


place to mention some experiments in oscillations made with 
tanks. We tried many forms of vessels, but will venture to 
describe only one of them. 

A circular vessel of constant depth (a x 15? x 8 cm.®) made 
of sheet zinc was partially filled with water. On a table two 
indiarubber tubes, each about 20 cm. long, were laid some 
30 cm. apart, parallel to each other. A wooden plate. on 
which the circular vessel rested, was set upon these tubes in 
a horizontal position. By moving the plate to and fro with 
different periods, we could produce any desired mode of 
oscillation. The aluminium powder was scattered over 
the surface of the water, and then the oscillations were 
photographed as usual. PI. V. fig. 1, shows the stream-lines 
in the fundamental oscillation; Pl. V. fig. 2 and Pl. VI. 
fig. 1, those of the second and the third harmonics respectively. 
They clearly indicate how each water particle moves. The 
theoretical treatment of this tank motion is given in Lamb’s 
‘ Hydrodynamics.’ 

Next, instead of periodically moving the wooden plate, 
two diametrically opposite points on the wall of the vessel 
were held by the fingers and the zine walls simultaneously 
pressed, both inward and outward, with a proper period. 
Pl. VI. fig. 2 indicates the stream-lines of the oscillating 
water thus started; a, a the points touched by the fingers. 
These stream-lines formed a system of hyperbolas. Here it 
is to be remarked that in such a complex motion of water, 
it was impossible to judge by the naked eye what form these 
stream-lines actually have. 

Thus the above investigation affords a good method of 
experimentally solving some difficult problems in tank 
motion, for which mathematics so far has failed to be of 
avail. 


§5. Formule for Calculaiing the Periods of the Oscillation 
wn Bays. 


The oscillation in a bay is nearly the same as the seiches 


in a symmetrical lake, each half of which has exactly the 
same form as the bay under consideration. The hydro- 
dynamical condition at the mouth of the bay is, however, 
slightly different from that at the middle part of the lake. 
Hence the period of oscillation in the bay is not exactly the 
same as that in the symmetrical lake ; this difference is here 
called the mouth correction. If this correction be known, 
the problem of finding the period of oscillation of the bay 
water reduces itself to that of finding the period of oscillation 
in the lake. 


Secondary Undulations of Oceanie Tides. 107 


Gi.) Rectangular bay of constant depth. 


Let / and h be the length and the depth respectively of a 
rectangular bay of constant depth; then the period T of the 
free oscillation of the bay, which has its node at the mouth 
and its loop at the end, will be given by the formula 


A] 
T= —= ae wick bura ty: + js i} 

V gh’ @) 
provided the correction due to the mouth be neglected. 
This correction may be approximately found in the following 
way. 
Take the origin of the rectangular coordinates at the 
middle point on the mouth of the bay ; z-axis in the direction 
of length, positive inwards, and y-axis upwards. Assume 
the vertical displacement 7 inside the bay to be given by 


TH Qt 


N=asin-,, COS => 
il 


21 

If we neglect the vertical acceleration, we have 
PN 

Ow 


where € is the horizontal displacement ; hence 


n=—h 


ee ee 
ch De 7? 


and 
i — ane COs Fy sin a 


If 6 be the breadth of the bay, the kinetic and potential 
energies inside the bay are given by 


K.E. = tphb\ Pda and P.H. = 4qbp \ ada : 


assume also the kinetic energy outside the bay to be 
Phb*p&°, where & is the value of £ at e=0, p the density of 
water, and P the dimension of a number. Neglect the 
potential energy outside the bay, which is very small, and 
write down the condition of the constancy of energy. From 
the ovo, which are obtained by putting ¢=0 
and >| in this equality, we get the expression for the 
period of oscillation :-— 


Al 
eam 
Vv gh 


(1+4P?). aimee 1 EZ) 


108 Messrs. K. Honda, T. Terada, and D. Isitani on the 


Lord Rayleigh * found the reaction of air upon a vibrating 
rectangular piston, whose length y 1s very great compared 
with its breadth b, to be equal to the addition of a mass 


oe log) 
(5 a4 a) e 


6 


where y==0°5772 and ==, » being the wave-length. If 


the reaction be uniform over the piston, we have for y=h 


hb? (3 Kb 
NDT, ee a 

Now, ina problem of long waves, we usually neglect vertical 
acceleration and consider horizontal acceleration nearly con- 
stant for different depths. Vertical planes, which are 
parallel to wave ridges and fixed relative to water, make a 
to-and-fro motion similar to the case of aerial vibration. 
The node of an aerial stationary wave corresponds to the 
loop of the water wave and vice versa. If we use the analogy 
for the expression of the kinetic energy, we have 


ee be ah 
P= me —y—log =) 


inal 73 loo 
eae ey) 


This relation seems to be sufficient for the estimation of the 
order of magnitude of the mouth correction. 


Gi) Irregularly shaped bay. 


Professor Chrystal, in his hydrodynamical theory of 
seiches, has satisfactorily worked out the problem of seiches 
for irregularly shaped lakes. When the shape of a lake 
does not considerably differ from that of a rectangular tank, — 
the following method of calculating the period may be of 
some practical importance, though not very rigorous. 

Consider a nearly rectangular lake of the length J and 
the mean transverse section Sj. The section S=S)+A8S 
varies slightly such that square of AS/S, may be neglected 
in comparison with unity. If the variation of S be every- 
where gradual, we may assume that the horizontal displace- 
ment of water in every section is in the direction of the 


* Lord Rayleigh, Phil. Mag. vol. viii. 1904. 
+ Chrystal, Trans. Roy. Soc. Edin. vol. xli. 1905. 


Secondary Undulations of Oceanic Tides. 109 


length of the lake and also uniform in each section. The 
vertical acceleration is neglected in comparison with the 
horizontal. 

Take the origin of rectangular coordinates at one fendiof 
the lake, w axis being i in the direction of Lop E, n, } have 
the same meaning as before. Then the kinetic and the 
potential energy are given by 

K.E. = 4) pSéde and P.E. = $\gbpy?d 


2 


respectively. Again, from the condition of ¢ uy bp 


ox 
by — ied Oa’ 


where X=Sé. By the condition of the constancy of energy, 


we have 
Bx? mer 
ii Ss dar an (& dx=const. 


Assuming for the first approximation 


Dad . Te 
X=a sin—- cos nt, 


l 
substituting this value in the above equality, and putting 
4=0 and t= — 7 two relations are obtained ; whence the ex- 
pression for the period of oscillation follows at once :— 


~ 7m (7 AS 
l= aah +53), 8 i. a +e) b, net) 


where Jb) =surface-area and hyly=So. Here the expression 


re Qe = _AS 
Al= 34 cos j ~ + )da 


may be considered as the correction to the length. It shows 
that any contraction or expansion towards the central part 
of the lake prolongs or shortens its natural period respec- 
tively, and that a contraction or expansion towards the ends 
shortens or prolongs it respectively. 

To apply the above expression in the case of a bay, we 
need only to consider a lake whose shape is symmetrical 
with respect to the vertical plane through the mouth line, 
and to find the period of the seiches in the lake by the above 
formula. This period, if it be corrected for the mouth, is 
the required period of oscillation in the bay. 


110 Messrs. K. Honda, T. Terada, and D. Isitani on the 
(ii1.) Dumbbell-shaped bay. 

The above formula does not hold when a portion of the 
lake is very much contracted. In this case, then, we may 
treat the problem in a quite different way. 

When two basins communicate with each other by a 
narrow canal, the mode of oscillation of the longest period 
takes place when the levels of the two basins rise and fall 
alternately. If the breadth of the canal be very narrow 
compared with the dimensions of the two basins, we may 
assume that the rise and fall of the level are uniform for 
each basin, and that in the canal the level is invariable, the 
motion of the water being chiefly horizontal. Then, denoting 
the areas of the basins by S and S/, the breadth, the depth, 
and the length of the canal by b, h, and / respectively, the 
displacernent of water in the canal in its direction by &, and 
the vertical displacement of the surface of S and 8’ by y 
and 7’ respectively, the potential and the kinetic energy are 


given by 
P.B= PY Sq? 48q), and KE.=P 2. 


Again, the correction to the kinetic energy on each end of 
the canal is nearly 


in which » is the wave-length, if the basins be infinitely 
wide, and may be considered nearly equal to four times the 
length of the basin in the direction of oscillation. 


Since Sn=—S’7/ and Sy=)hé, 


we obtain in the usual manner for the period of oscillation 
i Wey  2b73 mb 
T= 20 o ws ie -y—-log 7) \. (4) 
X 9 co) 
bl 1+ =) te duals 


Special interest is due to the case when one of the basins 
becomes infinitely large ; in which case the problem reduces 
itself to that of a bay communicating with the open sea 
through a narrow neck. Taking X\=A’=41L, where L is the 
length of the bay measured along the probable direction of 
propagation of waves, we obtain from the above equation, 


Ta27, [Se as 1993 os) Bie: 
7 Ro 0-923 + log = (5) 


Secondary Undulations of Oceane Tides. Lid 


$6. Method and Results of Determining the Period 
of Oscillation by Calculation. 


In the calculation of the period of oscillation in a bay by the 
simplest formula, it was necessary to estimate the length and 
the mean depth of the bay from the charts ; the charts used 
were those published by the hydrographical section of the 
Naval Department. As the mean depth, we took the ratio of 
the total volume of the water in a bay to the area of the 
surface. The length of the bay was measured along a line 
drawn lengthwise and perpendicular to the contour-lines of 
the bed of the bay. 

To find the mean depth of a bay, we began by drawing 
contours on the chart, where the depths at a number of points 
referring to low-water springs are given. After drawing 
as many contours as the case requires, we measure with a 
planimeter the areas between them. ‘These areas multiplied 
by the corresponding depths, increased if necessary by the 
half range, give the partial volumes of water. Dividing the 
sum of these partial volumes by the area of the free surface, 
we get the mean depth. 

We calculated by formula (1) the periods for all observed 
bays. For several typical bays we also calculated the cor- 
rection due to the change of the section, as well as that due 
to the mouth, and compared the corrected values with the 
observed. 

The calculation of the correction due to the variation of 
the section was carried out in the following way. We drew 
on the chart several lines at suitable positions normal to the 
line of length, and then taking the length as abscissa and 
the corresponding breadth as ordinate, we got a breadth- 
diagram. We next drew the mean breadth-line at a distance 
equal to the whole surface of the bay divided by the length. 
Taking now this line as the new axis of coordinates, we could 
27x 


easily draw the diagram for Ad cos- i 


; whence by mechan- 


ical integration, we get the value of fas COs ae dx. Pro- 
ceeding in a similar way for the sectional area, we got the 
value of fas cos de From these two the required 


correction was obtained by simple operations. 


112 Messrs. K. Honda, T. Terada, and D. Isitani on the 


The results of calculation of two corrections for three bays 
are given below :— 


Bay 4 Ngh,. | Corr. for section.| Corr. for mouth. 7. 


AIOINOET ).c..4e. oe 213™ +24:8m +46:5™ 284™ 
Br th PR), ar / 97°5 + 85 0 106 
Ofunato......... 39°5 —107 + 86 36:4 
ESUGUIEA, ccc 0a. 60°0 | —168 15°8 590 


According to our investigation, the bay of Aomori oscil- 
lates in two different modes with the periods 295™ and 103™. 
The longer period corresponds to the fundamental oscillation ; 
while the shorter one corresponds to the lateral oscillation, 
for which the mouth correction is naturally zero. The last 
two bays in the above table are good examples, showing that 
the correction due to the section and that due to the mouth 
nearly cancel each other. Similar remarks apply for many 
other bays. . 

When the mouth of a bay is decidedly contracted, formula 
(1) or (3) fails to give the period of oscillation, in which 
case formula (5) is to be used. The bay of Osaka, which is 
almost surrounded by land and which communicates with 
the external sea through two necks, Akashiseto and Yura- 
seto, may be taken as an example of such bays. 

For the resultant conductivity of necks, the sum of 
separate conductivities was duly taken. The data for the 
calculation of the period estimated from the chart is :— 


Akashiseto |).0°6,;=3°9 kms, “G=6°2 km. -hj;=41ae 
Yuraseto ,=50 kms ei =l2 km: (hy=2eaee 


Sd AT S10? homes A150 im, = 


whence by formula (5), T=270™. This agrees nearly with 
the largest period observed, 260"—310™. 

In the following table are given the periods calculated by 
formula (1) for all observed bays*. The periods corrected 
for the variation of the section and for the mouth are given 
in heavy type. ‘The letter S indicates that the periods which 
it follows are those for the lateral or seiche-like oscillation 
in the bays. In the table only the observed periods, that we 

* In the table, 1-5 are the bays along the coast of Hokkaidé, 6-9 
those along the coast of Japan Sea, 10-25 those along the southern 
coast of Honshiu, 26 that in Shikoku, and 27-29 those along the coast of 
Kiushiu. Futami is a small bay in Bonin Island (Ogasawara), and Kiirun 
the sole harbour in Formosa. 


113 


considered to correspond to the calculated ones, are given 
side by side. It must be understood that for each bay 
there are many other periods observed, though not so con- 
spicuous as those here quoted. 


Secondary Undulations of Oceanic Tides. 


Bay. Observed Period. 


Calculated Period. 
. Of er eS '  18°8™—16°5™ 17-3™ 
PONEMIULO .....:...1.c000 0 10°9 9:0 
SEPHMANAKA ........cc00 0 49°5 48°2 
PPE TPRLOROLAM ..c00:--cecesesle- 51:1—54:0 48°9 
5. E00 Cr 45°5—57-5 45°3 
SO 21°9—24°5 23:6 8 
Gnd 295 284 
\. Be ere ere eee 103 106 S 
FOS DE rl se ne 541 52-7 
Sp URES eee 56°7—62°9 590 
Oy COM OMONIEA s.....-s0c.0---<-- 11°9—12'9 EEE 
TU LV 05 ee nO ee 45°4 
ee ee 21:3—22:0 24:08 
Dipeel@amaishh 3. -............ 24-8 —26:0 24:8 
. ) (aa 20°3 22-5 
i) ‘Kojrrohama............... 24°6 26'0 
1S Ch ———————a 18°3—20°1 EL 
if. Oe 27:°5—29°9 26°4 
PRYCIEE, 2-2. 222.-..~ 2.2200 18°3 18°4 
Lo.) 006 36°0—39°1 364 
Uy. Oye 6-4— 76 T5 
EPA VUGAWH ......0.:2...0.006 68— 89 8-9 
SMeeMGraiso: ©..5.............- 13°8—156 13°4 
2s pmimoda .....:..... tbe 13°8—18°2 too 1b9 
Li. Tes $e 390 363 
22. Mikawawan............... 208 217 
Pe eushimnOte ..............- 16:5—186 18°3 
. 11-6—13:0 12°38 
Ces ONC so oi 260—310 27 
. 2 tie a 106—150 126 
oh. 61—66 63 
Ape lain): 60:0 61°6 
mat. Sic ape eheeee a 39-1 
jt be Bes a 17:6—18:2 1608 
5 oe Beane 15:0—16°3 17:0 
ai Gal) ROSOSAIIVE «215. . 40500250. 17-8 —20°3 19-0 
<n 65— 8-7 63 
Deen AEAGSEE oe co acess oo es's 15:0—19-0 15k 
Boe) Wagasakie.. 20) 00...56.. 34:5—376 37'5 
a Bates hot J) 225 — 25-2 2268 
{algal rise pray WP ae: Oe ee 16:0—20:0 136 
2) Si osha ieee eee 25°3—29°6 25°8 
It will be seen from the table that in most cases, the 


4] 


T=—= 


/ gh 


agree fairly well with those actually observed. 


Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. 


periods calculated by the simplest formula (1), that is, 


Thus for 
I 


114 Messrs. K. Honda, T. Terada, and D. Isitani on the 


many cases the corrections due to the mouth and the varia- 
tion of the section seem to be superfluous. This probably 
arises from the fact that in many bays the correction due to 
the section nearly cancels the correction due to the mouth ; 
for such bays gradually contract and the depth decreases 
as we approach the end, therefore the correction due to the 
variation of the section is negative, and the mouth correction 
being always positive, the two tend to annul each other. As 
a good example, we may refer to the bays of Ofunato and 
Tsuruga. If, however, the mouth of a bay be contracted 
and shallower than the inside, the correction due to the 
section is positive, and the total correction may, In some 
cases, amount to a considerable quantity. In such cases, 
the value calculated by the simplest formula must be de- 
cidedly less than the observed value, and can be brought into 
agreement only by taking the two corrections into consider- 
ation ; a good example of this is furnished in the case of the 
bay of Aomori. That the calculated periods for the bays of 
Mororan and Okirai, both of which have rather narrow 
mouths, are a little less than the observed values, is also 
explained on the same view. 


§ 7. Sea- Waves and Secondary Undulations. 


As already remarked, Professor F. Omori found that the 
periods of sea-waves observed in a bay are the same as those 
of the usual secondary undulation. We have made a like 
investigation for different bays, and our findings confirm his 
conclusion, especially in the case of sea-waves of distant 
origin. This fact now may be explained in the following way. 

Sea-waves are probably of such a complex nature, as to be 
represented by the sum of a series of long waves of different 
periods and amplitudes. If a group of these waves proceed 
towards a bay, the bay takes up and resonates to the undu- 
lation whose period coincides with that of the free oscillation 
of the bay. Thus the prominent undulation in a bay, what-- 
ever the component waves of a sea-wave may be, is the 
undulation whose period is nearest to that of the free oscillation. 

The above consideration applies chiefly to sea-waves that 
are of distant origin, and consequently of small amplitude. 
If, however, the origin is not very far from a bay or an open 
coast, progressive waves of long wave-length, irrespective 
of their periods, are sufficient to cause a disastrous effect on 
the coast; for by Green’s law of amplitudes, long waves con- 
siderably increase their amplitude the nearer they approach a 
shallow shore. Thus it is that the report on actually de- 
structive sea-waves almost always speaks of high wave-fronts 


Secondary Undulations of Oceanic Tides. 115 


approaching towards the shore, indicating that the waves are 
of the nature of progressive, but not of stationary oscillations. 
When that disastrous wave of 1896 visited our coast of 
Sanriku, there were instances in which the periods of the 
wave did not coincide with those observed in ordinary 
cases. 

In the investigation of the nature of sea-waves, the tide- 
gauge is at present the sole instrument available ; and in 
order to obtain the best results it should be set up where 
there is an open coast, or better in the neighbourhood of a 
small isolated island. If the instrument be placed in a calm 
bay, as is usually done, the waves are much modified by the 
proper oscillation of the bay. 

In the following pages discussions relative to sea-waves, 
originated from three different causes, will be given. 


(i.) Sea-waves of the Krakatoa Hruption, 1883. 


Great sea-waves caused by the eruption of Krakatoa, 
August 27, 1883, swept over the entire area of the Indian 
‘Ocean, and even forced their way as far as the northern 
parts of the Atlantic and the Pacific, leaving as they went 
evidence of their visit on all intervening tide-gauges. These 
records have been published in the Report of the Royal 
Society of 1888, and described by Captain W. J. H. Wharton. 

As to the cause of the periods of these great Krakatoa 
waves, few theories have been proposed. Captain Wharton * 
attempted to explain the period of two hours by assuming 
that the sea-bottom was upheaved for about an _ hour. 
According to Professor H. Nagaoka J, the earth is continually 
vibrating to the period 67™, which is the period of the funda- 
mental oscillation; and this vibration was what actually 
determined the periods of the Krakatoa waves. Our theory 
differs from the above by not assuming the slow up and down 
motion of the sea-bottom. Now any portion of the sea partly 
bound by land, if suddenly excited by some great disturbing 
cause, oscillates with its own stationary mode and may con- 
tinue for some time after the cause of the excitement has dis- 
appeared. Sunda Strait, from the character of its boundaries, 
may be taken as a good example. The south-west end of the 
channel opens widely into the Indian Ocean, while the north- 
east end, which is very narrow and shallow, leads to the Java 
Sea. The strait, asa whole, may be compared, by an acoustical 
analogy, to a conical open pipe. The loop of the gravest 


* Wharton ‘The Report of the Krakatoa Eruption,’ p. 97. 
+ H. Nagaoka, Proc, Tokyo Math.-Phys. Soe. iv. No, 2 (1907). 
12 


116 Messrs. K. Honda, T. Terada, and D. Isitani on the 


mode of oscillation possible in such a channel must lie midway 
through or perhaps somewhat nearer the narrower end. 


Fig. 10. 


N SUMATRA WA v9 See 
rs ; 
ar 
fe gt 
au 
S ay SAVA 
Ze 


“ 
QI, 
on 
Cc: 
Say 


Hence the eruption of Krakatoa which burst forth at the 
loop of this oscillation would be very favourable to excite 


natural, stationary oscillation of the strait as a whole. The 


initial disturbance would soon settle into a regular oscillation 
natural to the system, and this oscillation would be propa- 
gated into the external ocean asa train of regular waves, 
whose period is determined by that of the source. 

Taking the length of the strait as 160 km. and its mean 
depth as 183 m., we obtain from our formula * T=126™, 
which was actually recorded by the tide-gauge of Batavia. 
Besides the mode of oscillation above described, a binodal 
oscillation between the two sides of the strait, the Javan and 
Sumatra sides, might possibly be generated by the eruption, 
which occurred at the loop of this mode of oscillation also. 
The period of this oscillation is calculated to be about one hour, 
which nearly coincides with the periods recorded at many 
stations along the Indian coast. In addition, the modes of 
oscillation higher than the above two, with comparatively 
small amplitudes, might possibly have been in co-existence. 

Beyond the north-east end of the strait the sea becomes. 
shallow and the bed abounds in irregularities, which may 
cause a scattering of the waves that have been propagated 
from the end of the strait, through complicated reflexion and 
refraction. Besides, the sectional area of the north-east end 
of the strait is estimated to be about 1, that of the south-west 
end; so that the energy propagated from the former mouth 
must have beena small fraction of that from the latter. These 
considerations probably account for the smallness of the waves. 
propagated in the north-eastern direction. 


21 
* In this case, our formula evidently becomes T= Vane because both 
ends are open. gh 


Secondary Undulations of Oceanic Tides. 117 


The tide-gauge nearest Krakatoa at the time of the 
eruption was that of Batavia. It recorded two-hour waves, 
but not a trace of one-hour waves appeared. The absence 
of these latter, however, raises no serious objection against 
our supposition, because the narrow opening to the north-east 
of the strait is very unfavourable for the propagation of the 
energy of the lateral oscillation, much more so than for that 
of the longitudinal. 

Thus the major part of the energy of oscillations was pro- 
pagated into the Indian Ocean, and left its record on the 
tide-gauges even so far distant as the ports of Southern Africa. 
Examining the records given in the above cited report, we 
may in general distinguish two types of undulations—the 
one those propagated directly from Krakatoa, and the other 
those of the stationary oscillation of bays or estuaries excited 
by the incident waves. Prominent undulations recorded 
along the coast of India belong to the former type. Com- 
paring the records at Madras and Vizagapatam or Negapatam 
and Port Blair, an identity of waves may easily be recognized. 
We see also the trace of Vizagapatam waves in Nagapatam 
records, and vice versa. Most of these stations are not situated 
in either a bay or an estuary where an oscillation of such a long 
period as possible. — 

For remoter stations, we see in general that the disturb- 
ances are chiefly due to the second type, 2. ¢., the proper 
oscillation excited by the synchronizing components of the 
incident waves. Hence for such bays the periods of oscillation 
for Krakatoa waves must coincide with those calculated from 
the dimensions of the bays. ‘To verify this point we require 
reliable charts giving all necessary data; but having no such 
charts at hand, we had to be satisfied with deducing the mean 
depths of the different bays, quoted in the Krakatoa Report, 
from their periods of oscillation and their lengths estimated 
from charts* available. The results of our calculation for 
Port Elizabeth, Table Bay, Port Adelaide, Port Phillip, 
Lyttleton, and Honolulu all gave reasonably mean depths, 
when judged from the charts at hand. 


(.) Sea-waves accompanying Earthquakes. 


A number of great sea-waves caused by earthquakes have 
been recorded by the tide-gauges at different stations. We 
have studied these records, and found that the periods of the 
waves as given by the tide-gauge at a distant station are the 


* Berghaus, Physikalischer Atlas, Shuter’s Atlas, and Encyclopedia 
Britannea. 


118 Messrs. K. Honda, T. Terada, and D. Isitani on the 


same as those of the ordinary secondary undulations. In this 
connexion we quote a few of the most accurate records from 
modern occurrences. 

On June 15, 1896, the most destructive and disastrous. 
sea-wave of modern times visited the coasts of Sanriku in 
Japan. It originated in the sea about 150 km. off the coast 
of Sanriku, the product of a submarine earthquake, and soon 
assumed such proportions that at Yoshihama the height re-. 
corded was 24m. It swept away many towns and villages 
along the coast of Sanriku covering a distance of about 320 km. 
In all 22,000 lives were sacrificed. The sea-wave also crossed 
the Pacific, reaching the western coast of America. It 
left its records on the tide-gauges of Honolulu and 
San Francisco. 

In the following table the observed periods, together with 
those observable in ordinary cases, are given. 


' 


| Station. | Sea-Wave. Ordinary Undulation. 
| Hanasaki .........| ee | (Coast) 

Hakodate ......... 23"; 47m-3—H2m-]) 21-9245; 45m5—57MG 
Revato ow | 8, 16 | 12; 213220 
Ayukawa ......... | fae | 68--8-9 

Obosti ses 7 | (Coast) 
Mof0IS0..5-2024..2- | 15 | 13°8—15°6 
Honoluls \....,..5: | 26°0—23°4 | 

San Francisco dy 6:2, 34:3 


Now, according to the results of our investigation, the 
periods of the principal sea-waves on the coast of Sanriku are 
7-8" and 15"-17™. The records of two coast-stations, 
Hanasaki in Hokkaido and Choéshi in Shimésa, though they 
lie on opposite sides of Sanriku and are a considerable 
distance from each other, showed the existence of the same > 
wave. At Miyako, which is the nearest station at which the 
sea-wave was scientifically observed, the periods of the wave 
were different from those in ordinary cases. Though actual 
observations are wanting, the same remark seems to apply to 
the places where the sea-wave produced its most disastrous 
effect. In distant bays, Hakodate and Moroiso, the periods 
of the wave are the same as those usually observed. The 
periods observed at Honolulu and San Francisco are also 
the same as those observable in the case of other sea-waves. 

On February 1, 1906, in our time, a strong earthquake was 


- Secondary Undulations of Oceanic Tides. 119 


felt on the Pacific coast of Ecuador; it was accompanied by 
a sea-wave, which traversed the Pacific from America to 
Japan. The wave was recorded by the tide-gauges of 
Hakodate, Ayukawa, Kushimoto, and Hososhima. The 
earthquake of Valparaiso, which was also followed by a 
sea-wave, occurred on July 17 of the same year. The wave 
arrived at Hakodate, Ayukawa, and Kushimato. The above 
two waves also slightly affected the tide-gauges* at 
San Francisco, San Diego, and Honolulu. The periods 
observed in these bays are :— 


Bay. | Ecuador Wave. | Valparaiso Wave. Ordinary Case. 

ee | ¥ : 
Hakodate ......... | 51-8 —48m-3 | 47™-0—52™-0 45™:5—57™°5 

4s es RE a 24:2 26-0 23°5—260 | 21°9—24°5 
Ayukawa ......... 73-83; 20 Tos De 6°8—8'9; 22 
Rushimete.<....... 210 > 13-2 12:3; 21:4 116—13°0; 21:5—23°7 
Hososhima......... DOA a0 02. 0 PM ERRE ZS 17-8—20°3 
San Braneiseo. 22.) = - «= «+». 25°3 
See iceo, ....| 16-7; 335 322; 43 
Honolalu) 22.5062. | 24-8—26°8 26:2 


Thus in the bays on our coast the periods of the sea-waves 
are the same as those observed in ordinary cases. The periods 
observed at San Francisco and San Diego are those frequently 
observed in the case of other sea-waves; they correspond to 
the multinodal oscillations of these bays. The periods observed 
at Honolulu are quite constant; they possibly correspond to 
the fundamental oscillation in the inlet of Honolulu. 

In passing, it is interesting to calculate the velocity of 
propagation of sea-waves through the Pacific. If the path of 
the sea-waves which travel across the Pacific be known, the 
velocity of propagation can be estimated. Now the sea-wave 
which has usually a long wave-length compared with the 
depth of the ocean, must be refracted according to the con- 
dition of the bottom, so that it is very difficult to know the 
actual path by which these waves travelled through the ocean. 
We theretore conceived several paths between the place where 
the waves originated and the observed station in question, 
and then we measured the lengths, and also the mean depths, 
of each by mechanical integration tf. The velocity of propa- 


* The tide-gauge records utilized are those furnished by Dr. Tittmenn 
to Prof. Omori. 


+ The chart published by ‘‘ Deutsche Seewarte ” was used. 


120 Messrs. K. Honda, T. Terada, and D. Isitani on the 


gation of the long waves over each path was then calculated 
from the various mean depths. The times of transmission of 
the waves over each path were compared, and the path with 
the minimum time was taken to be the actual one. From 
the path thus found and the actual time of transmission, we 
calculated the mean velocity of propagation of the sea-wave, 
as given in the following tables. The time of the occurrence 
of the earthquake we took as the starting-time of the 


sea-wave. 
Sanriku Wave. 
| | he 
| Station. Distance. | Mean Depth *. | N gh. Time Interval.| Velocity. 
| Honolulu ...... | 6000 km. 4°92 km. 220 m./sec. 7° 446 216 m/sec. 
San Francisco | 7970 5°51 234 108 34™ 209 
| 
. 
| Ecuador Wave. 
| 
i Hakodate ...... 14330 km. 4°92 km 220 m./sec. 205 16™ 195 m./see. 
Ayukawa ...... 3 4s . 20 [2m 200 
Kushimoto ...| 15280 4-81 17 20h 44m 208 
| Hososhima | 15610 5 i 20 38m 211 
| Valparaiso Wave. 
| = —— 
i j 
| Hakodate ...... 17080 km. 466km. | 214 m./sec. 23 48m 200 m./sec. 
| Ayukawa ...... ze - | = | 23h 17m 204 
Kushimoto ...| 17600 4-58 | 212 23h 3]™ 208 


Hitherto it has been customary to measure the path of the 
sea-waves as supposedly lying along the great circle of the 
earth ; but the actual distribution of the depth being com- 
plicated, this is not a proper method. 


The value /gh of the fourth column in the above tables 
represents the theoretical velocity of long waves. This 
theoretical velocity is always greater than the actual mean 
velocity, but here the difference is not constant, while in one 
case it is very small, in another considerable. This fact has 


* One of us (K. Honda) calculated the mean depths of the ocean by the 
same method; but having used ‘* Berghaus Physikalischer Atlas,” the 
values were considerably greater than in the present case. See Proc. 
Tokyo Math.-Phys. Soc. 1. No. 9 (1906). 


Secondary Undulations of Ocearre Tides. 121 


been noticed by several earlier writers, such as Milne, 
W. J. H. Wharton, E. Geinitz, C. Davison, &e. The point * 
noticed by C. Davison as regards the mean velocity, if it be 
taken into account, does not diminish the discrepancy more 


_ than one per cent. 


iii.) Sea-waves accompanying Cyclonic Storms. 


Disturbances of the sea-level caused by cyclonic storms 
produce phenomena which may be conveniently classified 
under three heads, namely, short waves; long waves ; and the 
abnormal rise of sea-level. 

The violent, short waves, commonly called in Japan Gekiro, 
have periods usually of a few minutes, and are superposed by 
waves of still shorter periods but of considerable amplitude. 
These waves are always associated with strong gales, and have 
probably the same origin as ordinary wind-waves f. 

As to the long waves, examples are very numerous in 
which secondary undulations of remarkable amplitude occur, 
accompanying the cyclonic centre which was passing the 
vicinity t. In some cases remarkable undulations occur for 
a few hours, but soon abate into usual waves with the rapidly 
retreating cyclonic centres ; while in other cases they last for 
one or two days with considerable amplitude. In any ease, 
the periods of remarkable waves in a bay are generally the 
same as those observed on ordinary days. The period of the 
most prominent undulation, however, is different in different 
cases: in some the fundamental period is most pronounced, 
while in others one of the higher harmonics is conspicuous. 
Generally speaking, the duration, as also the period of the 
prominent undulation, seems to depend on the width, the 
velocity, and the distance of the cyclonic centre. Again, in 
many cases, especially for the Japan Sea coast, the periods of 
conspicuous waves accompanying cyclones in different stations 
and for different occasions are nearly similar. Onthe Japan Sea 
coast 11 is frequently found that a series of waves recorded at 
one station is quite similar in form to one obtained at another, 
distant station on a different occasion. Similar cases also 
appear with regard to the Pacific coast, though often masked 
by the undulation proper to the bays. In connexion with 
this fact, it will be noticed with interest that the wave-lengths 
of the most prevalent, cyclonic waves are generally com- 
parable with the dimensions of the area of the cyclonic 


* Davison, Phil. Mae. vol. 1. (1900). 
+ Wheeler, ‘ Tides and Waves, Ch. X. 
t See also ibid. Ch. XI. 


122 Messrs. K. Honda, T. Terada, and D. Isitani on the 


depression prevalent in our sea. It is in all probability true 
that the barometric fluctuations constantly going on at a 
cyclonic centre, acting in an impulsive way, may give rise to 
a train of long waves whose wave-lengths are comparable 
with the dimensions of the centre. 

Abnormal upheavals of sea-level have also been frequently 
observed on our coast. In some cases the level gradually 
rose and fell with the passing cyclonic centre, while in others 
the sea was abruptly upheaved, apparently by the action of 
strong gales. On one occasion a considerable rise of the level 
was observed, when neither cyclone nor earthquake was 
reported. 

It may be added here that in some bays remarkable secondary 
undulations occurred in ordinary calm weather, although 
there was no cyclone in the neighbouring sea. These undu- 
lations seem to be associated with the unstable distribution 
of atmospheric pressure. For example, the abnormal undu- 
lations in Nagasaki known as “abeki”’ are usuaily accom- 
panied by devious isobars and twin-centres of low pressure. 


§ 8. Oscillation of large Bays and Anomaly of Tides. 


Thus far we have generally discussed secondary undulations, 
the periods of which are much shorter than those of the prin- 
cipal tidal components, viz. the diurnal and semi-diurnal. 
We will now proceed 1 to consider those undulations of much 
longer periods, that commonly exist in the tides of very large 
inlets or estuaries. 

Hxaggeration of oceanic tides, which takes place in shallow 
seas and in estuaries, has often been explained merely by 
Green’s law of amplitude*. Airy? attempted to explain 
anomalies of tides observed in some rivers, by the considera- 
tion that for a wave of finite amplitude different parts of the 
wave-profile travel with different velocities ; but his argument 
has been proved untenable{. Again, inferior tidal components, 
known as compound tides or over-tides, which become con- 
spicuous only in shallow basins, have been explained on the 
analogy of combination tones in acoustics§. Itseems to us, 
however, that the theory alone is not sufficient to account for 
the facts thatin some gulfs or bays the amplitudes of superior 
tides are often comparable with those of the proper tidal 
components, and also that most pronounced, compound tides 

* Green, Camb. Trans. vi. 1857; Math. Papers, p. 225. 

Tt Airy, ‘Tide and Waves,’ Art. 198. 

t McCowan, Phil. Mag. [5] xxxv., 1892. 

§ W. Thomson, Proc. Roy. Soc. vii. G. H. Darwin, Brit. Assoc. 
Report, 188. 


Secondary Undulations of Oceanic Tides. 123 


are different for different bays and gulfs. TFerrel™ attempted 
to explain some irregularities of oceanic tides, by considering 
oceans as making stationary oscillations like the seiche in 
lakes. Recently R. A. Harris+, acting on a similar theory, 
constructed a co-tidal chart of the world. According to his 
view, all the water on the globe is divided into several 
distinct portions, each of which has the period proper to its 
own stationary oscillation. This point has been subjected to 
eriticism by G. H. Darwin. Harris applied his theory also 
for the explanation of tidal phenomena in many bays and 
straits, the standing oscillations of which are forced by tidal 
waves incident on their mouths. He has considered, however, 
exclusively the forced oscillation with diurnal and the semi- 
diurnal periods, and has not considered those oscillations 
peculiar to each bay. 

Now, according to our view, any bay or gulf, either small 
or large, may be put into a standing oscillation, if it be excited 
by the incident waves of a proper period. If the proper 
period of the bay happens to coincide nearly with one of 
the tidal components, that component will become more or 
less prominent, according to the degree of proximity of 
the proper and the exciting period. In this way, tides of 
the superior orders or indefinite components arising from 
meteorological causes, may sometimes become prominent 
in a bay, though almost insensibly small in the open sea. 
Moreover, the case may occur in which a solitary wave of 
wide extent, caused by some disturbances either meteorological 
or geotectonic, excites the oscillation of a Jong period proper 
toa bay. These oscillations in a bay will more or less deform 
the tidal curve, and cause an anomaly of the tides peculiar to 
the bay. Believing this view correct, the proper periods as 
given by our formula were calculated for different bays or 
gulfs, which are notorious for abnormal range of tide, and 
also those for which some remarkable irregularities of the 
tide were observed in the mareograms given in the Report of 
the Krakatoa eruption. The results of these calculations 
were found to confirm the above views. Some of the results 
will be given below. 

(i.) Bay of Fundy, Canada.—Near the end of this bay, 
spring-tides range 15 m., while near its entrance the rise is 
only 2°5 m. to 3°D m. an the calculation, the mouth-line 
was taken from Cape Cod to Cape Sable, and the end of the 
bay was taken at Port Greville. Then /=460km. h=141m.; 

* Ferrel, “ Tidal Researches,” Rep. Coast and Geodetic Survey, 
Washineton, 1874. 

+ R. A. Harris, * Manual of Tides.’ 


, = em - 


124 Messrs. K. Honda, T. Terada, and D. Isitani on the 


hence T= 13-0. If the mouth-line be taken between Yarmouth 
and Machias, the calculated period is 11"6. In any case, the 
proper period of this bay would be very near 12 hours. The 
abnormally high tide may then in part be explained by the 
coincidence of the proper period with one of the semi-diurnal 
tides, though the tidal phase is slightly retarded toward the 
end of the bay: and therefore the phenomenon cannot be 
wholly attributed to the standing oscillation. 

Gi.) Bay of Bengal.—Near the mouth of this bay the tidal 
range is small, being less than half a metre at the southern 
coast of Ceylon, while in the bay the range is 1-2°7 m., and 
this increases rapidly near the end of the bay. Since the tidal 
phase is nearly the same for the greater part of the bay, the 
principal part of the tide is then probably due to the standing 
oscillation of it. Taking the mouth-line from the eastern 
coast of Ceylon to the northern end of Sumatra, and the end 
of the bay at Akyab in Burma, we obtain /=1500 km. and 
h=1950 m., which gives T=12-0, which coincides with the 
period of semi-diurnal solar tide. 

Gil.) Madura Strait, Java.— Mareograms of Ujong, 
Sourabaya, and Karang Kleta reproduced in the Krakatoa 
Report show very marked irregularities of tide in the narrow 
Strait of Sourabaya, which connects the end of the wide Strait 
of Madura with the Java Sea. Since the width of the Strait of 
Sourabaya is extremely small in comparison with that of the 
Madura Strait, we may consider the latter strait as a rect- 
angular bay, ending at the former strait. The length of the 
Strait of Madura is about 160 km., and the mean depth of 
the basin is estimated to be about 30 m., so that the calculated 
period becomes 8"°8. 

In order to see whether any long wave corresponding to the 
calculated period actually exists in records, the mareogram 
of Karang Kleta was analysed by means of the tide- 
rectifier*, and an evident trace of waves with the mean 
period of 8-0 was found. The exciting cause of this wave 
may be the compound tide usually denoted by MK, the 
period of which is about 8"2, which nearly coincides with 
the proper period of the Madura Strait. 

(iv.) Port Adelaide, Australia.—A mareogrampot this Port, 
given in the Krakatoa Report, shows a remarkable variety of 
diurnal inequalities on successive days. On eliminating the 
principal parts of diurnal tides, the resulting curve shows an 


* Terada, Publications of Earthquake Investigation Committee in 
Foreign Languages, xviii. 1904. 


ee 


Secondary Undulations of Oceanic Tides. 125. 


apparent beat of semi-diurnal tides. The period of the cha- 
racteristic component, which forms a beat with the usual 
semi-diurnal tide, is about 10°°9, as estimated from the 
rectified curve. Itis probably to be attributed to the standing 
oscillation of the St. Vincent Gulf. Taking the mouth-line 
from Troubridge Point to Cape Jevis, and the end of the gulf 
at Wakefield, we obtain /=140 km., and the estimated mean 
depth is 21°55 m. Hence T=10*8, which coincides very well 
with the observed period. 

(v.) Port Phillip, Australia—Mareogram of Williamstown 
given in the above quoted Report shows some irregularities of 
tide, which suggests the existence of very long undulations pe- 
culiar to the bay. The curve was rectified and a period of 83 
was detected. Now the period of seiches in this nearly enclosed 
basin can by no means become so long as 85, unless the mean 
depth of the Port be less than one metre. This period is 
probably due to the undulation of the whole basin with its 
narrow neck communicating with the open sea. For the calcu- 
lation of the period corresponding to this mode, the necessary 
data were estimated from the chart given in Harris’s Report : 
ter o*, f= 2-95 km.; b= 4-08 km., h=18-4m. and 
L=200 km. The period calculated from these data is 839 
by formula (5), § 4, which almost coincides with the observed 
period. 

Besides the above enumerated examples, there are many 
bays, or straits, the forms of which seem favourable for their 
own standing oscillation of long periods, and in which the 
ranges of tides are comparatively large. The periods of these 
oscillations were estimated as follows :— 


Bay or Strait. T in hours. 


IS eee | 15 
| Mozambique Channel 7 
| Bristol Bay, Alaska 
| Hecate Strait, British Columbia 
| Hudson SLO oO) i | 11 
_ Bristol Channel, England | 
| Gulf of St. Malo, France 


= 


iy 


As we have no tidal records of these bays and channels at 
hand, discussion must be omitted. We conclude this section 


$3201 


126 ir. J. D. Hamilton Dickson on the 


by calling the attention of oceanographers to the proper 
oscillation of large bays. 


In conclusion we wish to express our best thanks to 
Professor H. Nagaoka, under whose supervision the present 
work has been carried ont. Equal thanks are due to Professor 
F. Omori, who favoured us with many records of sea-waves. 
Lastly, our cordial thanks are due to Dr. Y. Yoshida, 
Dr. N. Watanabe, Dr. S. Iwamoto, Dr. Y. Inouye, Dr. J. 
Fukuda, and Dr. Hirata, who have been zealous cooperators 
in the course of the observations. 


TX. On the Joule-Kelvin Inversion Temperature, and 
Olszewski's Experiment. By J. D. Hamitron Dickson, 
M.A., Fellow of Peterhouse, Cambridge; F.R.SE* 


[Plate VIL] 


YPHE condensation of gases and the means of reaching 

low temperatures have, within the last dozen years, 
brought into prominence what is called the Joule-Kelvin 
inversion-temperature. Whether Mayer’s hypothesis would 
be established as a fact, led to Joule’s experiments on the 
expansion of a gas at high pressure. These experiments 
seemed to establish this hypothesis as true; but the more 
ingenious method of experimenting suggested by Kelvin, 
in which sudden expansion through a small aperture from a 
high pressure to a low one was replaced by continued 
expansion through a porous plug between two pressures 
differing but little from each other, brought out the fact 
that in general such expansions produced cooling, and thus 
disproved Mayer’s hypothesis. 

This result was at first found to be true of all gases ; but 
further examination of hydrogen showed that heating was 
experienced in its case. The theory of the porous-plug 
experiment was given at the time by Kelvin, and the prime 
use made of it then was to determine how far “absolute 
temperature” differed from temperature given by an air- 
thermometer. ‘The most careful precautions were taken, 
by means of concomitant correction experiments, to determine 
exactly the conditions of the porous-plug experiment ; and it 
may be said today, that after a lapse of nearly 60 years 
no improvement has been made in them. 


* Communicated by the Author. 


Joule-Kelvin Inversion Temperature. ADy 


Kelvin’s formula, governed by the conditions of the Joule- 
Kelvin experiment, is, in the usual notation, 


av! wT 
Q=| part pomp'd—t) 7 a “yk tak Meas AL 


where Q is the quantity of heat lost by the gas through 
traversing the porous plug froma higher to a lower pressure. 
There are two other forms in which this may be written, 
namely, 


a=("(G-san, | ee, el 
Q = (U'+p'r’)—(U+pv), - - - -. @) 


each being, on occasion, more suitable than the other two 
for a particular discussion. In the practical application of 
the results of their experiments, Joule and Kelvin employed 
a formula sent them by Rankine*, which accorded with 
their experimental results, showing that the fall of tempera- 
ture varied inversely as the square of the absolute temperature. 
After this followed Andrews’s epoch-making experiments on 
the continuity of state; anda little later, van der Waals’s 
Thesis, which, leaving the Boyle-Charles laws as a first 
approximation to the equation of state of a gas, gave distinct 
and permanent advance to our theoretical knowledge of the 
properties of a gas by means of the well-known equation 


(p+ %)(—0) = Be. Soha eee 


In his thesis he disuussed Andrews’s experiments, and showed 
that this equation produced the form of the isothermal which 
James Thomson had suggested as probable. He also dis- 
cussed the Joule-Kelvin porous-plug experiment, and by 
means of the same equation deduced for it a theoretical 
relation, when the pressure is not too high, which he 
expressed (in his notation) by the equation 
— 10334 2a Pi—p 

) = 7293 dy 1+ at ‘ Py ee (9) 
The accuracy of this equation for air was remarkable ; for, 
by calculation, he found that the fall of temperature per 


atmosphere was 0°265, while the average of seven results 
quoted from Joule and Kelvin’s work, and ranging from 


and 


A424 c(t, 


* Kelvin, Collected Papers, vol. i. pp. 376, 391. 
+ English translation, p. 444. 


—_—\ z= ae 
1) 
: a 


128 Mr. J. D. Hamilton Dickson on the 


"2429 to 2881, was *2593. There was some discrepancy in a 
similar calculation for carbonic acid, but not enough to 
reject the claim that his theory was substantiated by’ its 
coincidence with experiment. Shortly after van der Waals’s 
thesis was published Clausius proposed a modification of it 
in two directions, the principal one being that the molecular 
attraction constant should vary inversely as the temperature. 
On this assumption, for small pressures, we get the relation 


9 
JC,—— = pear . . . ° 2 5 . (6) 


thus introducing the inverse square of the temperature, 
in accordance with Joule and Kelvin’s experiments and 
Rankine’s theoretical equation. 

It will not be a matter of surprise that two formule 
so different should both approximately satisfy the experi- 
mental results, when we consider that on the p, ¢ plane 
these results would be plotted within a range of some 
4 or 5 atmospheres and some 80 degrees of temperature. 
Two such curves, threading their way tor the short distance 
necessary among the experimental points plotted on this 
area, would practically coincide. 

Great developments of the experimental methods of Cail- 
letet and Pictet have been made in the last 25 years, notably 
by Dewar, Olszewski, and Linde. To the first of these, 
by the invention of the vacuum-jacketed flask and by his 
subsequent masterly use of it, we owe the production of 
liquid oxygen and hydrogen in bulk under atmospheric 
pressure, and also solid oxygen, air, and hydrogen, thus 
reaching the lowest temperature yet attained ; while, on the 
other hand, the last-named physicist has devised means to 
produce liquid air and oxygen commercially. Olszewski, 
in particular, has studied the lowering of temperature by 
expansion and has, for his researches, modified in some 
important points both the Joule and the Joule-Kelvin 
experiments. In the Olszewski experiment a finite time is 
occupied in the release of the gas under pressure, thus 
differing from the method followed by Joule. There is 
also a departure from the Joule-Thomson porous-plug expe- 
riment in employing a great difference of pressures, and 
(at least as far as can be made out from published papers) 
in restoring the small aperture in place of the porous plug. 
The questions arise: Is the theory of the Olszewski expe- 
riment the same as that of the Joule-Kelvin experiment ? 
If not, what is the theory ? And with these questions there 


Joule-Kelvin Inversion Temperature. 129 


arises another, What is meant by the Joule-Kelvin inversion- 
temperature ? 2 

For some time the idea has held ground that by merely 
dealing with the Joule-Kelvin experiment at higher or lower 
temperature, one temperature would be found at which the 
escaping gas would show neither heating nor cooling. This 
is correct provided the temperature does not exceed a certain 
limit. But the further extension has been made, that the 
same would still hold true if a great difference of pressures 
was employed. Here, however, the conditions of the Joule- 
Kelvin experiment may be intringed, and without formal 
provf we are not at liberty to make this assumption. The 
properties of a gas depend in general on two of the three 
quantities, pressure, volume, and temperature, because we 
have an equation of state connecting the three. If a certain 
property depends on the alteration of one of these quantities 
only, some condition must in general have been assumed, 
even although tacitly. In the first idea of the inversion- 
temperature; the tacit assumption was (at least) that the 
difference of pressures was small; perhaps there was also 
the idea that the pressures themselves were small. But in 
any case, Kelvin’s formula fully provided for any pressures 
so long as the conditions of the experiment were maintained. 
The Boyle-Charles laws were originally employed in the 
numerical reduction of the Joule-Kelvin observations ; later, 
a formula due to Rankine was used ; today we may employ 
van der Waals’s relation as a step nearer the actual condition 
of a gas. 

I shall use equation (1) for my purpose, and substitute in 
it from equation (4), noting that in general undashed letters 
refer to the initial state (the state of higher pressure) and 
dashed letters to the final state. Thus we get 


Q = Re(—> yee lar 2u(=,--) i bein £9) 
whence, on putting Q=0, the inversion-temperature ¢; is 


given by 
2a b b 
— D (1-2) gen . . e ° (8) 


or, if ¢, is the critical temperature, 


27 b b 
sere haar ee \ 
i= Ze (0 a} Ba ie ue 


Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. K 


130 Mr. J. D. Hamilton Dickson on the 


If the initial pressure is great while. the final one is small, 
we may neglect the fraction b/v', and take the simpler form 
of the equation, 


27 b | 
€; =F t.(1-2); ad tae: ‘ioe 
and if both pressures are small, we get the limiting equation 
27 
c= tah) cise) hea ee (11) 


To show the differences between these equations, let us 
take Olszewski’s * highest values for air, namely, p=160, 
t;=532° abs. Inserting these two values in van der Waals’s 


equation, 
1682 
(y+ 23) (e—1-598) = 28352, 2. (2) 


in which the values of a, 6, R for one atmosphere as unit 
pressure, and a cubic centimetre as unit volume, have been 
deduced from the values a=:0037, 6="0026 given by him f 
for the pressure of a metre of mercury as unit pressure, 
and the volume of the gas at 0° C. and under this pressure 
as unit volume, we find the value of wv for one gram 
of air to be approximately 10°07 c.c. Hence equation (9) 
gives ¢-=93°, while the limiting equation (11) would give 
78°°8. Neither of these values is near the true value, 133°; 
but we see that the effect of the value of 6, although it 
is only about 14 c.c. in comparison with some 773 c.c. as the 
specific volume of air, when associated with a certain high 
pressure, is to give values for the critical temperature which 
differ in the ratio of 6 to 7. Otherwise, equations (9) (for 
two nearly equal pressures at 160 atmos.) (10), (11) make the 
ratio of the temperature of inversion to the critical tempe- 


~yature 4°86, 5°72, and 6°75 respectively. From van der Waals’s- 


constants for air, and his own formula, equation (5), the tem- 
perature of inversion is about 770° abs., or 5°84 x 133°, 133°. 
being the critical temperature quoted by Olszewski, Linde, 
and Berthelot. It may be noted that these temperatures 
increase as the initial pressures diminish. 

In 1887 E. Natanson {, by an elaborate series of experi- 
ments on carbonic acid, conducted on Joule and Kelvin’s 
lines, showed that the cooling of this gas on passing the 
porous plug was not independent of the pressure. 

In 1898 Witkowski, after a most careful experimental 

* Phil. Mag. June 1907, vol. xiii. p. 723. 


+ English translation, p. 400. 
t Wied. Ann. xxxi. p. 518 (1887). 


a ae 
a be 7 | 
pe 


eo 
. 


Joule-Kelvin Inversion Temperature. 131 


study of the properties of air, published a memoir * on the 
behaviour of this gas in the Joule-Kelvin experiment ; but, 


instead of basing his investigation on any theoretical equation, 


he employed his own experimental results, involving them in 
the thermodynamic equation by the method of quadratures. 
Taking U for the internal energy, he calculated the values of 
the function U + pv for the pressures and temperatures within 
the range of his experiments. With these he plotted a 
series of curves of constant pressure, the coordinates being 
temperature and the excess of the function U+pv for 
pressure p above its value for one atmosphere. From these 
curves he constructed curves for constant values of U+ pv, 
with pressure and temperature as coordinates. The range of 
these curves being from about 130° to 273° (abs.), and 
therefore not including the (so-called) inversion-temperature, 
he pointed out that it was possible, if they had extended 
farther, to have found the slope of the U+pv curves 
changing sign, and hence to have determined this tem- 
perature. Thus to Witkowski is due the proof of the 
fact, based on accurate experiment, that for the deter- 
mination of any such temperature it is necessary to take 
the pressure also into consideration. 

In 1906, after the publication of Olszewski’s results on the 
release of hydrogen from high pressure, Porter t examined 
the question, using also the function U+pv. He plotted 
curves of constant pressure (such curves may be called 
isopiestics), using as coordinates the temperature and the 
variable part, independent of temperature, of the indefinite 
function U+pv. But instead of employing experimental 
results, he used various theoretical equations, and in par- 
ticular van der Waals’s equation of state. [quation (7) may 


be written 
Riv’ 2a ci hehe. > Bahk 
Q=(52 mia) =“) 
v—b v w= bh *s 


Br: Q— (yd —5)-(n—2); | 


and we have also 


U+ po = $(t)—“ + pe, oe) 1 ones dane) 
where ¢(¢) is an unknown function of the temperature. 


a . . . 
Porter took pv—~ for his abscissa (the minus appears as 


* Anzewer d. Akad. d. Wiss. in Krakau, 1898, pp. 282-295. 
+ Phil. Mag. April 1906, vol. xi. pp. 554-568. 
K2 


132 | Mr. J. D. Hamilton Dickson on the 


plus in his paper) and temperature for the ordinate, both 
in reduced coordinates. 

In discussing the problem we may proceed along two 
lines, and it will be instructive to follow both. We may 
eliminate v and v’ from equation (8) by means of an 
equation (4) for each; or, we may examine the properties 


of the isopiestics plotted to the reduced values of pe—< 


and ¢ as coordinates. There is important information to be 
got from both methods ; meanwhile, equations (13) and (14) 
show that the two methods are equivalent. 

The function U+pv being of the nature of a potential, 
I shall, in what follows, use this name for it; and in this 
connexion zsopotential curves will also be employed. We 
have to find the equation connecting the inversion-temperature 
with the initial and final pressures. As the rest of the 
investigation deals only with this temperature, we may drop 
the suffix 2, and understand that ¢ refers only to it. 

The result of eliminating v and 2! between the equations 


re =(1-=)(1-5), - ee 


v 
Pst eng (v6) =Ri, rn 
@ + = (i —}) = Rt. 
may be written in the simple form 
(P+ortir+u)=8ru, ... . (1d) 


where 
: a Rt ) 
P=ioty), T= t= b=p-P,| 
| EI: 16 
No ote o 
2 ; J 


and “u= 


If the pressures differ only slightly, as in the Joule-Kelvin 
experiment, we may in the limit take them as equal, and 
consequently put 6=0, P=p, u=r, and equation (15) 
becomes 

(p+r+ir) —Sar. . 28. ae 


This equation agrees with equation (10) in Dewar’s paper * 


* Proc. Roy. Soc. March 1904, vol. Ixxiii. p. 260. 


Joule-Kelvin Inversion Temperature. 133 


on “ Physical Constants at Low Temperatures,” namely, 


Ha). Vd pV 
;= B(1+5' ie) 


We can also verify this result directly. For putting 
v=v’' in equation (8), it becomes 


2a (v—b\ 
je 7Ce) Breil wr edit ey) 
f bRt . , , 
whence, if ?= 5g? Van der Waals’s equation (4) becomes 
a A ee ae sa 
p+ p-fy=Re he =2750(1—%), 
or 
a 3 Ret uz wee: Ré 
ee TG 85 


which is equation (17). But equation (15) is not in a suit- 
able form for the determination of t from given values of p 
and p’. Expanding, and arranging it in powers of 1, it takes 
the form 
974 +16 (7 +3P)r°—2{4(r4+ P)(S7—11P) +387} 0? 
2°16 {4(a + P)?(r7—P)—(o7- P)& hr 
+ 16(a + P)*—8{2(32 — P)?—(a+ P)7$8?+ &=0; (19) 


which shows at once that the inversion-temperature is a 
complicated function of the initial and final pressures. It 
is interesting. however, to note that it depends, not explicitly 
upon them, but only upon their difference 6, and their 
arithmetic mean P. Further, since 6 occurs only under the 
square and the fourth power, the phenomenon, but for other 
reasons, would be reversible. Putting 6=0, we ought to 
recover the equation for the Joule-Kelvin experiment ; and 
this is the case, for the terms without 6 are 


9r* + 16 (7 +3P)73—8 (7+ P)(S7r—11P)?? 
—64(7+P)*(7—P)r+16(7+P)4, 

which can be written as the product 
tot oe OE rae EP)? } (ar + PF er)?) © (20) 
the first of whose factors equated to zero, and with p written 
for P, is equation (17). The second factor cannot vanish, 
since each of its terms is positive. 
We can now answer the question, What is meant by a 


Joule-Kelvin inversion-temperature ? Gas is allowed to | 
expand through a porous plug from a high pressure of any 


~~ - 


a a 


134 Mr. J. D. Hamilton Dickson on the 


magnitude to a lower pressure of any magnitude subject to 
certain conditions, namely, it must enter and leave the 
neighbourhood of the plug at the same temperature ; some 
considerable time must elapse between the starting of the 
experiment and the commencement of observations so that 
the flow of gas may be steady both as regards pressure and 
temperature ; the gas must leave the plug with kinetic energy 
differing so little from that with which it approaches the plug 
that this difference may be neglected ; then, the temperature 
in question is an inversion-temperature for the gas concerned 
and the pressures employed. | 
For the discussion of the general question it is convenient 
to express equation (19) in reduced coordinates. Let a, a’ 
be reduced pressures, and put 
eo a! = a—o! 


a 
ce pS Gia ot 
Fe a a ee 


. (21) 


the equation then becomes homogeneous and of the fourth 
power in a/b*. On dividing this factor out, it takes the form 
IP* + 16(1 +380)? — 2{4(1+¢)(5—11e) + 3}? 
—16{4(1+¢)?(lL—o) —(5—a)eiB | 
+16(14+0)*—8{2(3—c)—(1l+o)}e+e=0, . (22) 
whence the reduced temperature corresponding to any pair 
of pressures «, a’ is si feP 
As numerical examples of equation (22), I have taken the 
case of a gas with a critical pressure of 40 atmospheres, and 
have supposed expansions, on the Joule-Kelvin process of 
experiment, to take place from 160, 120, 80 and 40 atmo- 
spheres in each case to one atmosphere. Thus the values of 
a are 4, 3, 2, 1 respectively, and that of ' is i Linde and 
Berthelot quote 39 atmospheres as the critical pressure of air, 
so that the following results will approximate to those for 
air, on the suppositions made. 
The four equations are:-— 
for a=4, ge! oo 7 
98* + 19°5758°— 36-068°—66°668 +1856=0; 
hor ha = 3: ae 
98* + 18-6856? —37:106?— 66-3738 + 18-31=0; | 
f +. . (28) 


; > — * Beh 
for eae a =m 


98! + 17°7956® —38°108?—65°868 +17-82=0; 
for a=, gle 


984 + 16-9086" —39-0536°— 65-068 +17-07=0. ] 


Joule-Kelvin Inversion Temperature. 135 
Of these the required roots are 
1°845, 1°886, 1-923; 1-9605, 


respectively: hence, on the hypothesis of van der Waals’s 
equation of state, a porous-plug experiment conducted with 
the limitations of the Joule-Kelvin experiment, gives an 
increasing temperature of inversion for a falling initial 
pressure, the expansion taking place against a pressure of one 
atmosphere. The following Table collects these results :-— 


Reduced | Initial 
temp. | pressure. 


Inversion-temp \Olszewski’s temps. 
i: a for Air 


a. B. 


Abs. Cent. Abs. Cent. 


(o) 
4 | 1-845 | 6227 | 160atm.| 8984 | 5534 | 539 259 
3 


1886 6°366 | 120 ,, 8468 5738 
2 19238 6-492 80 ,, 863-4 590°4 515 240 


1 19603} 6-618 40 ,, 880°5 607°5 471 198 


and as air has a critical pressure of nearly 40 atmospheres, 
I have assumed it (approximately) as being treated by these 
equations, and have therefore used its critical temperature of 
133° abs. to get the temperatures in the fifth and sixth 
columns of the table. The seventh and eighth columns 
contain temperatures from Olszewski’s published numbers *. 
Hence it would appear that the Olszewski experiment differs 
fundamentally from that of Joule and Kelvin, for his results 
show a falling inversion-temperature with a falling initial 
pressure, which is contrary to the results of the theory of the 
Joule-Kelvin experiment. 

The conclusion thus reached will be verified by the second 
mode of consideration of the problem, namely, by direct 
examination of the isopiestics drawn on the temperature- 
potential plane, and based on van der Waals’s equation. 
Taking « to represent potential, and y to represent tem- 
perature, in reduced ea we have 


Dae Sy) 2 
v=ah— . 8y=(36— (a+ ss) SU AAW) 
a and ¢ being reduced pressures and volumes respectively. 


* Phil. Mag. June 1907, vol. xiii. p. 723 


136 Mr. J. D. Hamilton Dickson on the 


It will be necessary to get the equations to these isopiestics 
Hliminating ¢ from (24), after a few steps we have 


7294? — 216.027 + (729 — 27a) a2— 64y? 
+ 108ax—32ay +(108a—4a2)=0. . . (25) 


We shall also require the envelope of these isopiestics ; 
its equation, following the usual process, is immediately 


16(7292? — 2162°y + 7292? — 64y’) 
+ (27x? — 1082 + 32y—108)?=0 


or (+2) (@+2)?— sry b =o . + eae 


showing that the envelope is a parabola. This envelope is 
the locus of points representing an expansion between 
pressures differing only infinitesimally in a Joule-Kelvin 
experiment, and (for the present) appears to be independent 
of the magnitude of the pressures, but we shall see later that 
only a finite portion of the curve is involved in the question, 
and also a limited range of pressures. We ought to arrive 
at the same result from Porter’s equations (pp. 555, 556 of 
his paper). Writing them in the present notation they are 


Qo= i ee SSS Ee ee 
y=4(3 , )»and ess Ber in 5? 
whence 3 \3 

ence y ( ) (a +2)’, 


thus verifying equation (26). 

These isopiestics and their relation to the envelope are 
shown in fig. 1, Pl. VIII. The zero-isopiestic consists of the 
straight line Ow, and the parabolic are Ow, whose equations 
are 

27 a= Sy =O 0 en bs se Er 


and 


27a?4+27a+8y=0. . . . . . (28). 


The envelope V MLN is touched by this isopiestic twice, namely, 
at the points M, N. As @ increases from zero the successive 
isopiestics each touch the envelope at two points, the one 
leaving M, the other N, and approaching each other towards 
L. These points of contact are given, for the a-isopiestic, 
by the equations 
“+2=+ 


: 
9 


(a6), y= a ee 


where u2=9—a. Hence we see again that a cannot exceed 9, 


Joule-Kelvin Inversion Temperature. 137. 


and that the 9-isopiestic touches the envelope in two co- 
incident points at L, which we may call the final point of 
contact, its coordinates being e=2/3, y=3. Thus the part 
of the parabola which constitutes the envelope is contained 
between M and N, and deals only with pressures within the 
range a=0...9. Further, a Joule-Kelvin experiment, in 
which the pressures are nearly equal, and as governed by 
van der Waals’s equation, cannot involve pressures higher 
than nine times the critical pressure. This limitation has been 
noticed by Dewar, Berthelot, and others ; and it agrees with 
Porter’s full-line curve (p. 556 of his paper), for its equation 
may be put in the form 


(£+12n—4577=144(9—£), . . . (80) 


where & is reduced pressure, and 7 is reduced temperature, 
and real solutions of this equation require & not to exceed 9. 
Hquation (30) is that of a parabola, cutting the axis of tem- 
perature at the reduced temperatures 3/4 and 27/4, the latter 
result agreeing with equation (11). 

From equations (29) we get the positions of the points M, 
N, namely, M is (—2, 2), N is (2, 62). The following 
Table will help to show how these points of contact follow 
each other, (2, y,) being the lower, and (a, y2) the higher 
point of contact for any one isopiestic. For values of 2 


a vy } Yi- V5 Yo 

0...) 067 0-75 2-00 675 

Te Hebd: / —0-59 O84 1°92 6-49 | 

= a ae Bi : ae on 

eee soacte ate is ef. ae 

ee a8 8 ae 3 a Loe 

yeasts. —0-22 1°33 1°56 5°33 
ee 0: 1-69 1-33 469 | 

Fi een bee x ae: ae 

Soar aie 2-08 vil 4:08 

| 
heed | r=067, y=3-00. 


greater than 9, no isopiestic touches the envelope; that is, 
there is no inversion-point for a Joule-Kelvin experiment, 
the pressures differing by only a small amount, when these 
pressures exceed nine times the critical pressure. But there 
are inversion-points for finite differences of pressure within 
certain limits, as we shall find later. 

If we seek for the general circumstances in which a given 
temperature may be an inversion-temperature, we find that 


138 Mr. J. D. Hamilton Dickson on the 


for such a temperature two isopiestics must intersect. I 
distinguish between an inversion-point and an inversion- 
temperature ; and it will be convenient to refer to the two 
isopiestics passing through any inversion-point as the higher 
and the lower according as the corresponding @ has a greater 
or less value. An examination of fig. 1 will lead to the 
following conclusions, which have been verified by calcu- 
lation. All points of intersection of any two isopiestics lie 
within the curvilinear triangle MONLM ; hence inversion 
points only exist within, or on the contour, of this triangle. 
The highest isopiestic which can take part in determining 
an inversion-temperature is a=27, for this isopiestic cuts 
the triangle only at the point O. Any isopiestic between 
this and «=9 may be the higher of the two isopiestics of 
any inversion-point. Any isopiestic from a=9 to a=0, 
inclusive, may be the lower of the two required; and no 
isopiestic above «=9 can be the lower one. Hvery lower 
isoplestic (2. e. one whose a< 9) can only have associated with 
it as the higher isopiestic, one from a limited range of iso- 
piestics extending from itself to a particular value of a, 
depending upon the lower isopiestic, and less than 27. Thus, 
in fig. 1, it is easy to see that the range for a=1 extends to 
about 22—the last being that isopiestic which touches a=1 
and lies not quite half-way from 20 to 25; similarly for 
a==5, the range gets to about 14. No temperature can be 
an inversion-temperature which exceeds 62 times the critical 
temperature. for every temperature below this value, any 
point on the horizontal line representing that temperature, 
and lying within the curvilinear triangle, gives a pair of 
pressures which have this temperature for inversion-tempera- 
ture. Thus in order that three times the critical temperature 
may be an inversion-temperature, we may begin with a 
pressure of about 214 times the critical pressure and expand 
into a vacuum ; or we may use an initial pressure of 20 times 
the critical pressure and expand down to the critical pressure ; 
or finally we may employ two nearly equal pressures about 
© times the critical pressure. 

The equation (25) for the isopiestics may be regarded from 
other points of view, each of which will, on occasion, be 
preferable, as helping us to see more clearly the limitations 
of the question. It expressesa relation between «, y, and a— 
potential, temperature, and pressure. By making a constant, 
we have already got the equations of the isopiestics in the 
temperature-potential plane. If we make 2 constant, we 
shall have the equations of the isopotentials in the temperature- 
pressure plane ; and if we make y constant, we shall have the 


7 


Joule-Kelvin Inversion Temperature. 139 


equations of the inversion-isothermals in the potential-pressure 
plane. The isopotentials correspond to Witkowski’s* second 
(derived) curves. Equation (25) rewritten in a more suitable 
form for an isopotential is 
(8y + 2a)? + 21627y + 27(a2?—40 —4)a—7292?(a#+1)=0, (31) 
in which # is to be given any constant value. For different 
constant values of 2, this is the equation of a series of para- 
bolas whose axes are parallel and make with the axis of 
temperature an acute angle of 14° 2/ [the scales in fig. 2 
reduce this angle to 1° 26’]. In fig. 2 these parabolas are 
shown ; they have there been derived directly from measure- 
ments made on fig. 1, and not by means of equation (31). 
The curve ABCD 1s the curve corresponding to the envelope 
VMULN in fig. 1. Like it, it extends from y=°75 to y=6°75, 
the greatest pressure associated with any point on it being 
9, when y=3 (corresponding to the point L). This curve 
is the parabola whose equation has already been given in 
(30), and of which another useful form is 
(iy aa pe lige f(b) 
it touches the axis of pressure at the negative pressure 
—27. The axis of temperature in fig. 2 corresponds to the 
zero-isopiestic in fig. 1. The advantage of the new figure 
is now apparent. The inversion-points in fig. 1 were all 
confined within a rather confusing curvilinear triangle ; 
they are now contained within the more open curvilinear 
quadrilateral BDEB’ bounded by the portions of the axes. 
BD, B’H, and the parabolas BB’, DE whose equations are, 
respectively, 
(4y+a)?+24y—6a—27=0 . . . (83) 
and 
pee EN a Marge age 5 


The circumstances in which inversion will take place are 
now to be found from a line of veraahaah temperature. Thus 
if we look along the line for which y =2 we see that inversion 
will take place for a gas expanding from an initial pressure 
of about 244 times the critical pressure into a vacuum, the 
potential being 6; or between pressures 22 and 1, the 
potential being -5; or between pressures 15 and 32; the 
potential being °3; or finally between two nearly equal 
pressures of about 8 times the critical pressure. Witkowski’s 
isopotentials lie within the rectangle bounded by y=0°9, 
y—2°1, and e=U) e=4. 

Instead of representing the isopotentials we may obtain 
perhaps a better display of the circumstances of inversion 


* Loc. cit. 


140 Mr. J. D. Hamilton Dickson on the 


by means of the inversion-isothermals,—curves of constant 
inversion-temperature drawn with potential and pressure 
as coordinates. These are given in fig. 3, and have been 
drawn from readings taken directly from fig. 2. If we 
want to arrange for 1:4 times the critical temperature 
as an inversion temperature, we may expand the gas from 
about 25 times the critical pressure down to a vacuum ; or 
from about 194 down to about 14 times the critical pressure ; 
or from 133 to 24; or from 10 to 32; or, finally, a Joule- 
Kelvin experiment may be made for this temperature as an 
inversion-temperature if we expand the gas from a little 
above to a little below 52 times the critical pressure. Joule- 
Kelvin experiments with small differences between the two 
pressures are represented by points on the parabolic are 
FGH...KJ, whose equation is 


9(3a7—2)?=16(9—2).  . .9 ee 


The area of inversion is here again a curvilinear quadri- 
lateral. It is bounded by the axis of x, and the isopotential 


«= —°6, by the zero-isothermal, and by a parabolic are, part 
of which is the curve ST whose equation is 
2727 +4e—108. . 2... 


These inversion-isothermals will afford the means of clearly 
testing the reduction of Olszewski’s observations made 
recently by Porter. The basis of his calculation is that 
from Oiszewski's experiments the inversion-temperature is * 
“about 5°8 times the critical temperature.” Putting y=5°8 
in equation (25), the quadratic for @ is 


Ao? + (2702 —108a + 77'6) a— (7292 — 523-82 — 2152-96) =0, 
ea 


from which the following Table has been calculated. 


Xe. a). } as. 


| 2 
| 1:70785 35119 | 
| 17105 598 1:76. | 
1-715 6-41 0-64 
1:7185 7-065 0 
1-725 8:03 —0'93 | 
| 1-750 10°69 Met Tos 4 | 
| 18 14-33 00a 
19 19-68 —1215 | 
|. 20 24-06 | —16-46 | 
: | 


* Loe. cit. p. 063. 


Joule-Kelvin Inversion Temperature. 141 


In fig. 4 (a continuation of fig. 3, but with the scale of 
potential doubled, to bring the circumstances more clearly in 
evidence) the inversion-isothermal is drawn from this table. 
The available portion PQR of this isothermal lies to the left 
of the vertical through P, where it cuts the axis of potential. 
The highest available pressure for this isothermal is 7-065 
times the critical pressure, and the associated low pressure 
must be that of avacuum. Porter notes* that “in Olszewski’s 
experiment with hydrogen the initial pressure was eight 
times the critical pressure, and the final pressure was atmo- 
spheric, 7. e. roughly 3}, the critical value.’ These circum- 
stances do not coincide with the present more complete 
determination of the point in question based on van der 
Waals’s theory, which Porter also employed ; nor, in fact, 
with the data given by Olszewski. In the paper com- 
municated to the Philosophical Magazine “by the author,’ 
Olszewski states T that he began with “about 170 atmo- 
spheres”? and got “considerable” cooling at “about 
—190°”; that he next employed 150 atmospheres at 
—103° and again got a fall of temperature ; that he only 
succeeded finally in getting constancy of temperature when 
the initial pressures during a series of 25 experiments “ fell 
from 117 to 110 atmospheres.”’ If then we still take 
20 atmospheres as the critical pressure, these pressures cor- 
respond to values of « lying between 5°85 and 5-2 (and not 
to a=8), and the 5°8 inversion-isothermal shows that the 
final pressures (reading from the curve) would require to be 
about 1°2 to 1°9 times the critical, or 24 to 38 atmospheres, 
values which cannot be considered as approximations to the 
experimental value of one atmosphere. If we take a lower 
value of the critical pressure (in accordance with Olszewski’s 
later experiments), say 15 atmospheres, then @ ranges from 
7:8 to 7-2; but the verticals through these points cut the 
isothermal in the region of negative pressure, which is in- 
admissible. To put the critical pressure lower would only 
enhance this divergence from admissibility. The conclusion 
seems clear, therefore, that on the data, and employing 
van der Waals’s equation of state, the physical nature of 
Olszewski’s experiment has still to be determined. 

In this conclusion I believe I am confirmed by the recent 
observations published by Olszewskif{ on Air and Nitrogen. 
These experiments show a diminishing temperature of in- 
version with a falling initial pressure, the final pressure in 
each case being one atmosphere. For an experiment following 
exactly the conditions of the Joule-Kelvin porous-plug 


* Loc. cit. p. 563. “t Phil. Mag. May 1902, vol. iii. p. 539. 
t Phil. Mag. June 1907, vol. xiii. p. 723. 


ht Baas iA 7 
pi 
ly 
‘. ‘ 
' 


142 Mr. J. D. Hamilton Dickson on the 


experiment, the constancy of the potential function U+ pv is 
accepted. The inversion-points concerned are (fig. 1) the 
points of intersection of the isopiestic «=, (approximating 
to the case of air) with isopiestics ranging from «=0'5 to 
a=4:0. These points lie near N*, within the sharp apex of 
the curvilinear triangle. A somewhat expanded portion of 
this part of fig. 1 is given in fig. 5, which shows the inter- 
sections of the isopiestics «=1, 2, 3 with e=j,. These 


Fig. 5. 
fs 
4-0 
\ 
2 
3 
| N 
SN 
S 
K 
& 
q 
S 
K 
he 
SOTENTIAL 


four isopiestics touch the envelope at increasing distances 
from N (the point of contact of the zero-isopiestic, at the 
reduced temperature 27/4), and it is clear that the points of 
section of a=), by 2=1, 2, 3 must be at successively lower 
points in the figure; that is, at successively lower tempera- 
tures. In other words, as « increases in this neighbourhood, — 
the lower pressure being always constant, the successive 
inversion-temperatures must fall. This verifies the result of 
the direct calculation earlier in the paper. In fact the locus 
of actual Joule- Kelvin inversion-points in this neighbourhood 
* These isopiestics intersect again within the inversion triangle near M, 
but the temperatures at M are only slightly above the boiling-point, and 
differ greatly from those in the Olszewski experiments. It may be noted 
that this is an inversion region deserving attention. All lower isopiestics, 
for which «< 1, have three inversion-points for each inversion-temperature 
lying within certain limits, less than the critical temperature ; otherwise, 
for a given isopiestic, there is only one inversion-point for each inversion- 
temperature. It may, however, not be possible to obtain these experi- 
mentally, for other reasons. ° ie 


Joule-Kelvin Inversion Temperature. 143 


for air is the isopiestic for one atmosphere, when the expan- 
sions deal with pressures such as those given by Olszewski. 
The same thing is brought out even more clearly by a little 
consideration of the apex of the curvilinear figure BDEB' 
in fig. 2, say, the portion lying between the ordinates y=9'7 
and y=6'75. The parabola DE gives the maximum initial 
pressures possible for expansion into a vacuum as the tem- 
peratures fallfrom 6°75. The coresponding curve of maximum 
pressures for expansion against the pressure of one atmosphere 
will be very slightly below DE. This shows clearly the 
fall of initial pressure with a rising inversion-temperature. 
Olszewski’s air results are approximately represented by the 
hyperbola 

ear he Sy j= O:01)46 aie) 6138) 


whereas the theoretical line is very near DH, whose equa- 
tion is 


fa (a OF 34 
y TAG ial ie . . . . . ( ) 


The conclusion seems to be that the Olszewski experiment, 
although an expansion experiment like the Joule-Kelvin 
experiment, differs essentially from it. We may ask where 
the differences appear. They seem to embrace at least the 
following:—We may amplify the analytical condition of the 
Joule-Kelvin experiment and consider it to involve the 
constancy of the function U + K+ pv, where K is the kinetic 
energy of the gas. In the Joule-Kelvin experiment, K is so 
small, both before and after the gas passes the porous plug, 
that it may be neglected. The same result might be attained 
if the values of K before and after passing the plug were 
approximately equal. In the Olszewski experiment it is 
clear that the value of K before the gas reaches the nozzle is 
small; but it is by no means small after passing the nozzle. 
From the numerical data so far published for the hydrogen 
experiment, the value of K after passing the plug must be 
comparable with the value of pv at the critical point, on the 
basis of van der Waals’s equation*. Again, Olszewski in the 
hydrogen paper already referred to, states t that “the ex- 
pansion took place slowly and lasted from 4 to 5 seconds.” 
But in the experiments of Joule and Kelvin associated with 
their main porous-plug experiments, they found{ that on 
either opening or shutting a certain stop-cock, after the 
temperature of the air had become pretty constant, fluctua- 
tions of temperature continued to be very perceptible in 
different cases for periods of from 3 or 4 minutes up to 

* See Note at ‘end. + Loe. ert. p. 539. 
$ Kelvin, Coll. Works, vol. i. p. 359; and generally pp. 357-362. 


144 Mr. J. D. Hamilton Dickson on the 


nearly half an hour after the pressure had become sensibly 
uniform. They add that having discovered this fact, they 
found it necessary to delay recording the observations until 
13 or 2 hours after the pumps began working. Further, 
Joule and Kelvin commenced by using an aperture in the 
nozzle, but gave this up in favour of a porous plug. Their 
purpose* was “in order that the work done by the expanding 
fluid may be immediately spent in friction without any 
appreciable portion of it being even temporarily employed 
to generate wis viva, or being devoted to produce sound.” 
Olszewski apparently continues to use the small aperture f. 
These appear to be essential differences in the conduct of the 
two experiments. In any case Professor Olszewski’s further 
detailed description of the experimental arrangements and of 
the apparatus will be awaited with interest. 

It is worthy of note that if Joule-Kelvin experiments were 
to be carried out for points on the envelope MLN, the kinetic 
energy would in every case be so small that it might be 
neglected. Such experiments, by the actual determination 
of the envelope, would further our insight into the nature of 
gases as subject, or not, to van der Waals’s law. It would 
also be possible to conduct them with comparative ease, for 
the ranges of pressure and temperature are neither too great, 
nor do they extend into regions too difficult of access. For 
air, B is about 100° abs. and D is about 850° abs., while the 
highest pressure need not exceed some 350 atmospheres. 
For hydrogen, B is a little above its boiling-point, D is not 
far from the melting-point of carbonic acid, and the highest 
pressure necessary would not exceed 135 atmospheres. 

In 1902, on the appearance of Professor Olszewski’s paper 
on the inversion-temperature of hydrogen, I calculated in- 


version-temperatures (although they were not published) on | 


the basis of van der Waals’s, Clausius’s, and Reinganum’s 
equations. The results gave the connexion between this 
temperature and the associated volumes of the gas at high 
and at low pressure. The van der Waals’s formula has been 
the subject of the present paper, and is given in equation (8) 
or (9). From Clausius’s equation 


Rt ¢ ‘ 
p= me (w+ BY’ . 1) ) DOT eae 
we get 
poy? 2iat+p (v—2)(v—2« )idov +28(v+u) +8 (40) 
Saas 8 a (v+B)? (oe + 8)? 5) 


* Kelvin, Coll. Works, vol. i. p. 346. 
t+ See note at end. 


= ee 
ars 
; 


Ri= 


Joule-Kelvin Inversion Temperature. 145 


which, when the pressures differ only by a small amount, 
becomes 


ae Da a+B (v—a)(3vt+B) mS TAT) 


Sar aire. 


and, when the pressures are themselves small, reduces to 
et) aya 
ee a se Lae aaa) Geo (42) 


From Reinganum’s equation 
Prints 1G 
jh (v—b)! tee 
where a and > are constants, we get in like manner the 
equations 
2a (v—b) (v'—b) (v—b)3(v' — 6)? 
eB ve! {2cv'—b(v+ ov’) {Qv*v"? —2bov! (vu + 0') +(e? +07)? 
(44) 


(43) 


giving, for nearly equal pressures, 


R= a ("="), EMAL) cect) (665) 


26\ v 
and when the pressures also are small 
a 
Rt= 9). Dae Wh teos, oe) 
The critical temperature, on siti h Ck equation, is 
2+2,/6 1 itt 
Hee ees) Bry 6 p 093835. + GD) 


Berthelot * in two great memoirs has discussed the theory 
of the equation of state, especially for low pressures, in a 
thoroughly exhaustive manner. His comparison of theory 
with experiment has led him to formulate a modified form of 
van der Waals’s equation for moderate pressures, namely, 

Rt a 
eae heat ee Se ee ee) 
but with the understanding that instead of the critical con- 
stants being derived from a, 6, and R, the constants to be 
chosen are ¢,, p,, and R, with the relations 
1 Peles Late? Oi He, 
v= 40 a= 64. “pi? and V.— 39 pul . (49) 

* ‘Tavre Jubilaire dedié a H. A. Lorentz,’ 1900, p. 417. “Sur les 
thermométres a Gaz,’ Trav. et Mém. du Bur. Intern. des Poids et 
Mesures, 1903. 

Pil. Mag. 5. 6. Voll. No. 85. Jan. 1908: L 


146 On the Joule-Kelvin Inversion Temperature, 


These four equations of state give for the maximum value 
of the temperature of inversion 


6°75t. 3-182 / +n 5°36t,, and 4-24t,, 


respectively. For carbonic acid a and B are nearly equal 
‘in Clausius’s equation, so that for this gas the limiting 
temperature of inversion is 4°5t,. 


In conclusion, the subject of the above discussion has been 
the possibility of coincidence between Professor Olszewski’s — 
experiment and that of Dr. Joule and Lord Kelvin with the 
porous-plug, guided by van der Waals’s equation of state. 
That this equation requires some modification, or at least 
entails a Jimitation in its applicability to all gases and in all 
conditions, is admitted ; and it may even be that the Olszewski 
experiment and the van der Waals’s equation may each assist 
the other in the progress towards truth; but, today, on the 
conditions assumed, there is apparent incompatibility between 
them. This does not at all affect the purpose, as I read it, of 
Professor Olszewski’s experiment, nor its success, namely, the 
determination of a temperature suitable for moderate pres- 
sures, such that if expansion of a gas be made below that 
temperature, cooling will take place. Incidentally, the full 
meaning of the term “a Joule-Kelvin inversion-temperature ” 
has been developed; and whatever form the fundamental 
equation of state may ultimately take, this meaning will 
probably not require alteration. Although the theory is 
simple, the calculations involved have been long and heavy, 
but it is hoped that, as nearly every one was repeated, no 

serious errors have crept in. 


Note.—The texts at one point in the description of Professor 
Olszewski’s experiment are confusing. The German text of the 
Krakau Bulletin (1901, p. 455) speaks of “die im Innern mit 
Saimischleder ausgefiitterte Blechbuchse.” ‘The translation from 
this in ‘ Nature’ (Ixv. p. 577, April 1902) is “a thin metal box 
which is lined with chamois leather.” The translation in the Phil. 
Mag. iii. p. 537, May 1902) is “box ... which is stuffed with 
chamois leather.” The difference between stuffing and lining the 
box materially alters the question. A German friend, who is a 
well-known authority on the German language, tells me that 
ausgefiutterte means ‘lined, as a glove lines the hand.” The 
detailed description of the experiment promised by Professor 
‘Olszewski will, however, clear up the matter. 


Peterhouse, Cambridge, 
October 1907. 


te 


X. On the Thermally excited Vibrations of an Atmosphere. 
By C..V. Burton, D.Se.* 


i Lord Rayleigh’s paper “ On the Vibrations of an Atmo- 

sphere,” ¢ the free vibrations of a gaseous mass in certain 
ideal circumstances are investigated. In the present note an 
attempt is made to carry the investigation a step further by 
considering the vibrations which in “like circumstances arise 
from fluctuating thermal conditions ; the problem presented 
in each case being the determination of the pressure as a 
function of time and coordinates when the thermal conditions 
are specified. 


Equations of a gas subject to small fluctuations of thermal 
state, but otherwise at rest. 


It will be convenient to prefix the symbel 6 to indicate the 
disturbances of the various quantities involved, as arising 
from the given small thermal fluctuations. Thus 


p is the density at any point when there is no disturbance; 

p+6p the density when disturbances are present ; 

T, T+6T the undisturbed and disturbed (absolute) tem- 
perature at any point ; 

Pp; p+6p pressure ; 

u, v, w are the velocity-components at any point in the 
disturbed condition, and vanish when disturbances are 
absent ; 

R as usual stands for p/pT, and is an absolute constant. 


Thus RT=p/p=(velocity of sound)? when Boyle’s law is 
applicable 
=a SENOS 0 Pa 
The equation of continuity is 
O(p+ op) , diet Sp)us , Oi(p+ Spe} _, Ot(e+ Sp)w} 
ot or Oy 0< 


—0,. (2) 


where 0/0¢ is time-differentiation at a point fixed in space. 
That is, to a first order, 


Dap, N(on) 
Aron 02 


O(pr) , A(pw) _ ie 
ay ony © Vo si(X) 


-- 


* Communicated by the Author. 
J ‘Collected Papers, vol. iti. p. 335; Phil. Mag, vol. xxix. 1890, 
pp. 175-180. 
. 2 


148 Dr. C. V. Burton on the Thermally 
Again, 
dsp _ 1, dT , nn dde 
Gee as Oe 
Set Cran eae 
2 pees ORT Olpr) | few) 
pss; —B1| St 


To a first order also, when V is the gravitation potential *, 


Ou Ov Ow OV , O(p+ép) 
e( ) + OP), |, 


ae ar ar =~ + 8) 3 
that is (since p is not a function of ¢) 


O(pu) A(ev) O(pw)\ _ OV | ddp / 
“aE? ae fe) \ os =| 8p oe ee Ae (5’) 


From (4), (5’), and (1) we obtain 


pas, spree 2 +9 + OP OT + bov?V4 vp | 
OT | sas, 2(08 OV O80 BV | 08o| Ova 
52 +a°V*op+a (+ a5 ct Sy ae + =e *), (6) 


provided that V°V =0. 

When there are no impressed forces capable of influencing 
the motion, the last principal term on the right hand of (6) 
disappears. When in addition there are no changes of tem- 
perature, the first term on the right hand also vanishes and 
(6) reduces to the well-known form appropriate to small 
disturbances in a gas obeying Boyle’s law. 

When no forces act on the gas we have from (5), to a first 


order, 
& Ov Ow\_ _1 (dep dép Se 
Of’ Ot’ OF is p\ du’ oy’ dz 


so that, if Aw, ... are increments corresponding to a finite 
lapse of time, 
te 30): 10 

Au, Av, Aw)=— (= ~,~ |] \opdt 3a 

( ? ) ) p Ou’ Oy’ Oz J is 3 ( ) 
p being by definition independent of ¢ If we are dealing 
with gas which is uniform in the undisturbed state, p is every- 
where and always constant, and (7) shows that, in this case 
to our order of approximation, if uv, v, w at any time satisfy 


3°5p_ 


Or =—R 


s 


* Defined, forrany given point, as the work to be done against grayi- 
tative forces in bringing unit mass from infinity up to that point. 


078) 


io 


a 


O’H 9 9 9 
i bag, 1 V8 V Opt ya” 


Eacited Vibrations of an Atmosphere. 149 


the irrotational condition (for example, if O=w=v=w ini- 

tially) they will continue to do so throughout the motion. 
Hquation (6) may also be put into another form, which we 

shall find convenient. For if y is the ratio of the two specific 

heats of the gas, it can be shown that 

OoT y—1/d6p , 0H 

= a <= NpaNagS 

Sy, agar on) ss 

where OH/Ot is the value at any point of the rate at which 

heat is being received per unit mass. 


Using (8), we have in place of (6) 
Bs OV dap OV 


Rp 


Consider next an atmosphere such as Lord Rayleigh has 
discussed in his paper already referred to. This atmosphere 
is bounded below by a plane, which we take as the plane of 
xy, while in all other directions it is unlimited. The acceler- 
ation g due to gravity acts everywhere perpendicularly to 
this plane (that is parallel to the axis of z) and is of uniforne 
intensity. 3 

In the undisturbed state the density p is given by 

I. 


Ea, ray. 
p=me * , while p=pew s...°: - (10) 


the “ undisturbed” temperature T being everywhere uniform. 
Let the given thermal conditions which cause the disturbance 

be expressed by 
l= Boe) SOHO Ie AO! CRI 


so that at each instant every plane parallel to yz is an iso- 
thermal surface ; the additional temperature-distribution 6T 
being made to travel through the atmosphere in the direction 
of increasing «, like a progressive wave-train. The disturb- 
ance of pressure due to 6T will evidently be of the form 

ap a ee ns Ca) 
@ being a function of <. 

From (6), remembering that 


ce =+g, while eee =- op 


~ ~ ~ 


(because T and 6T are independent of z), we obtain 


y Dddp_10%p__ Rp oer 


28 
MgO? ch oo, a Or ange (13) 


nies ; 8p, ON 
Oz Ov Oy, | OY Oz Oz 


).@) 


150 Dr. C. V. Burton on the Thermally 
Substituting in this from (11) and (12), and using (10), 


0a ga ny iahee RB ios 
ga t ge t (ae )e= are os. 


a solution of which is 


nt — ka?” 

or ; 
RBp yn? = — 25 +i (ke— nt) oy 

6p = ae NeW : (15) 

This solution corresponds to the forced vibrations, arising 
-from the temperature-changes expressed by 6T. The most — 
general solution of (14) is found by adding to (15) the solution 
of (14) with its right-hand member replaced by zero. Such 
an addition would represent free vibrations, and it may be 
remarked in passing that the equation formed by equating to 
zero the left-hand member of (14) is equivalent to (27) of 
Lord Rayleigh’s paper. It should, however, be remembered 
that the quantity there denoted by p would in our notation: 
be dp/p. 

By means of (5), bearing in mind the periodic character 
of the motion, it is easily verified that 

RBkn i(ka—n 
ree a eo o=0, 0=0. aa 

Since w vanishes thronghout, it follows that the motion 
would not be modified (fluid friction apart) if any number of 
constraints were introduced in the form of fixed rigid hori- 
zontal planes. For these thermally excited vibrations, then, 
the solution has the same form whether we suppose the 
atmosphere unlimited upward, or whether we consider a thin 
lamina of air between two neighbouring horizontal planes. 

As Lord Rayleigh has done in the case of free vibrations™, 
and with much the same degree of plausibility, we may pass 
from the consideration of a thin flat lamina of air to that 
of a thin spherical sheet. In view of what has just been 
said concerning the simpler problem, it seems that the 
thermally-excited vibrations of such a sheet of air should 
not differ widely from those of an outwardly unlimited 
atmosphere. The object here is to obtain some analogy to 
the barometric variations which arise from diurnal heating 
and cooling. The problem considered is not that of the- 
earth’s atmosphere, but it may be hoped that it is not too 
remote from actual conditions to afford-some light thereon. 


* Loe. cit. 


Ezeiied Vibrations of an Atmosphere. 151 


Bidlosins Lord Rayleigh, all amplitudes concerned are treated 
as small, the mean ‘temperature i is taken as uniform over the 
whole of the spherical sheet considered, and of course the’ 
limitations imposed by the thinness of the sheet preclude any 
systematic motion of the air corresponding to trade-winds or | 
the like. 

It will be assumed that in the free vibrations of the spherical 
sheet, the relation between density and pressure follows the 
adiabatic law; so that in the case of thermally excited 
vibrations the only transfer of heat of any account is that 
arising trom the heating-effect of the sun and from radiation 
into space, noappr eciable effect being g produced by the passage 
of heat horizontally from one part of the atmosphere to another. 
Accordingly (9) is the form of equation appropriate to the 
problem, gravity, however, being evidently without percep- 
tible effect on the motion, so that the last principal term on- 
the right hand may be omitted. Thus 


Oop oH nis : 
= =(y—-lp B= yay op: =<. CE 


To represent roughly the heating-effect of the sun, it is 
assumed that, during the day, heat is being gained per unit 
mass of air at a rate proportional to the sine of the sun’s 
altitude; while, day and night, heat is being lost per unit 
mass at a rate proportional to the fourth power r of the absolute 
temperature of the air. As the fluctuations above and below 
the mean temperature in any latitude are supposed to be- 
small, the rate of loss referred to may be treated as constant. 

Taking, for simplicity, the time of an ae let @=co- 
latitude, « #=longitude. 

Then sin (altitude of sun)=sin (4t—@) sin 6, — * being al 


solar day, and the origin of time being sunrise at ae meridian 
a=0. 
Our assumptions are thus expressed by 


“2 =. ee) ce ay 

ne when 0<ht—w<7; : , | (1d) 
oH __pr ; 

ot when 7<hi—o<27. 


To our order of approximation, \dH= 0 for any whole day, 
which leads to 


A sin @ - , ia 
DSS ae ve 
BT em = . % . . . . (19) 


10 y) we —_ 


15z* Dr. C. V. Burton on the Thermally 


This corresponds to absolute zero of temperature at the 
poles ; we are more concerned, however, with the diurnal 
changes, which probably furnish a better representation of 
actual conditions than do the mean temperatures for different 
latitudes deduced from our assumptions. From (18) and 


(19), 


a8 = Asin a sin (kt—w) -=}, when 0< kt—a <7; 
7 


Dis. us A sin @ 
aba: 
Also 


0°H : | 
ae > Ak sin @ cos (kt—@), when 0<kti—w<7; 
07H 

va =, when 7 <“ki—w<t0m 2) 2) ge 3 


It is now proposed to examine the disturbances of pressure 
arising from the temperature-disturbances (21) imagined as 
affecting a spherical sheet of air whose mean temperature is 
uniform. 

Expanding 0’H/0¢° in a Fourier series we find 


oH 
OF 


\ 
2 


, When 7 < kt—ow< 27. “a 


(21) 


ee 


=+tAksin @ cos (kt—o) — = Ak sin @ sin 2 (kt—@) 
+ terms of higher frequency. (22) 


The right hand of this equation must now be expanded in 
such a form that @, w appear only in a series of surface har- 
monies. It can be shown that (22) is equivalent to 


07H 1D. 
nya =4Ak sin @ cos (kt—@) —Ak{ 55 sin?6 sin 2 (kt—@) 
D fe 3 sha 
+ eG sin? @ (7 cos? @— 1) sin 2 (kt~@) 
643). eae : 
+ 357304 Sim” 6 (33 cost 2—18 cos?6+1) sin 2 (At —w) + eS 
+ terms of higher frequency; . . . .. 2) 20gee 


in which the first term, considered as a function of 6, @ is a 
surface harmonic of degree 1, and the terms given within the 
bracket of degrees 2, 4, 6 respectively. For convenience let 


excited Vibrations of an Atmosphere. 153 


us write 

R.H. member of (23) = 8, cos kt +8)! sin kt 
+(S.+8,+5,+...) cos 2ht+ (S./+8)'+8,'...) sin 2k 

eee rl we a E24) 

where each S is a surface harmonic (of degree indicated by 

the suffix) multiplied by a constant physical factor. For the 

determination of the disturbance of pressure dp, we may 

assume provisionally a solution of (17) in the form 


dp=Q, cos kt + Q,' sin kt +- (Ql+ Qi + Qo +...) cos 2ht 
+ (Qo! + Qa’ + Qe! +...) sin 2kt + ...3 (25) 


the ()’s being also surface harmonics multiplied by constant 
physical factors. Now 
; Oop . 2 d6p Lie Ouk ie 2oep 1; pare ee 
*Sp= a etc — sin 0&2) = r> (26 
ee Or ‘-r Or he sin 6 sa(sin ° foley iat sin? 8 Qa?’ @5) 
and if q is the outward radial velocity of the air at any point 
within the spherical sheet, we have from (5), taking the axis 
(say) of < vertically upward, 


pot =— op (p being independent of ¢). . (27) 


But q¢ vanishes at both spherical boundaries of the sheet, 
the radii of which may be called » and 7++Ar: hence when 
Ar is very small, ¢ is evidently insignificant throughout the 
notion; so that 

09 


Ot 

Also 0=(qg at outer boundary) — (g at inner boundary) 
=0g/drAr, so that 0g/dr=0 always. 

eee 57/07=0,.°. 5 23) 


which justifies our assumption that dp is independent of +. 


Using (28) and (29), (26) becomes 
Le il. Oy eG Bi!) OF toe 
eo RED PRE ss = he 10s §: ht 
| Vv op= 7 tan sa(sin 0<,)+ in? O93 {Qi cos kt + Q,’ sin 


@ 
+ (Qo+ Q,+Qe+.-..) cos 2ht +6 Qe’ + Qu’ + Qe' +...) sin 2hkt} 
— (Q, cos kt + Qy’ sin kt) — = (Q, cos 2ht + Q,’ sin 2kt) 
4.5 : GNF Penge 
bere (Q, cos 2kt + Q,’ sin 2kt) — = (Qg cos 2ht + Qe’ sin 24t) 
eee es i hath add ina i ') ) 


by a well known property of surface harmonics. 


=0, and ue ale peter Vag) 


154. Thermally excited Vibrations of an Atmosphere. 
By means of (24), (25), and (30), (17) may be written 


(2-1 pe YQ cos kt + Q,' sin At) 
m (18-2 ie Gs cos Jew OM eos 


+( 4024.5 \(Q, cos 2ht-+Q, sin 2kt) 


=F (4e—6 a tla eV cos 2ht + Q,! sin 2kt) 


=, cos kt +8;'sin dt + (S.+8.4+8, +...) cos 2hé 
+(8,°+8,/ +8, +...) sin 2kt+...... (31) 


By equating terms in cos kt, sin kt, ... we obtain 


| (Q; cos kt + Q,' sin ) Gee aon 2) —(y—1)p(S,coskt +8 Si, "sin kt) 
—4(y—1)Apk sin @ cos (kt—@) 


by (24) ; with corresponding values for Q, cos 2kt + Qo! sin 2ét, 
&e. 

Substituting in (25) the values thus obtained, we have 
finally for the solution corresponding to (thermally) forced 


vibrations, 

1 (y—1 ie 

op=5- “HES " pAk sin 8 cos (ki—@) 

15) (y—- 1)pr’ 3 ; Z 

29° ae eye 2k: sin?@ sin 2(it —@) 

5 (y= 1)pr" Ob win? 2 Sue 

64 De 20 22 k sin?6(7 cos?@ — 1) sin 2(kt —@) 

6439 (y pr? NO} <in?O(33 costO ~18 ‘ ai 

+. S54504 "TP doyat ie” HE O88 cost? — 18 cos’ 


at He ee SN Re 


It is unfortunately very difficult to base any sort of nume- 
rical estimate on these tentative results. In agreement with 
Lord Rayleigh’s investigation the denominator of the first 
term of dp ee cr k= Vy V2/r 3 ya? being here 
the quantity which Lord Rayleigh Aone by a. Simi- 
larly the denominator of the second term vanishes when 
th= ya ya? /6/r. It appears that both the denominators are 
small, corresponding to well-marked resonance-effects ; but 
we cannot even say with certainty what is the sign of each. 


The Use of Variable Mutual Inductances. 155: 


We are dealing in fact with just that degree of isochronism ° 


between the Pee periods of the spherical sheet of air and the 


periods of the (thermal) disturbing agency for which our 
assumption of no dissipation ceases to represent even approxi-: 


mately the actual conditions. | 

It may be remarked, however, that the value of dp given 
by (32) indicates a semi-diurnal barometric amplitude not 
widely different from the diurnal, provided equally favourable 
resonance conditions may be assumed. If the agreement of 
period between free vibrations and disturbing agency were 
somewhat better for the second term of 6p than for the first, 


the greater prominence of the semi-diurnal period might’ 


perhaps be accounted for. 


A less remote representation of atmospheric conditions ° 


might have been realized by introducing into the value 
of QH/dt an additional factor, a function of co-latitude and 
longitude, so as to take some account of the fact that the 
diurnal variation of temperature is greater over a continent 
than over anocean. Sucha factor, however, is not competent 
to introduce ir:to the barometric fluctuations at a given station 
terms whose periods are submultiples of that of the thermal 
excitation. 


XI. On the Use of Variable Mutual Inductances. By 


ALBERT CAMPBELL, B.A.* (From the National Physical: 


Laboratory.) 
[Plate IX. | 


CONTENTS. 
1. Introductory. 
2. Mutual versus Self Inductance. 
3 & 4. Construction and Adjustment of Variable Mutual Inductances. 


5, 6,7 & 8. Uses of the same for measuring unknown Mutual 


Inductance, Self Inductance, and Capacity. 
9. Appendix. The Scale of the Excentric-Coil Mutual Inductance. 


1. Introductory. 
HE accurate measurement of small self inductances and 
capacities has of Jate years assumed an_ increased 
importance owing to its direct practical application in. con- 


nexion with wireless telegraphy ; and much important and, 
interesting work has been done on the subject especially as. 
regards the calculation and measurement ot seif inductance.. 


For some time past I have found that suitably designed 
mutual inductances are of the greatest assistance in such 


measurements, and I incline to think that by their use some 


* Communicated by the Physical Society : read October 25, 1907. 


156 Mr. A. Campbell on the Use of 


of the sources of error inherent in other methods are avoided. 
The chief method (B) here described was designed for the 
measurement of the small self inductances (1 to 200 micro- 
henries) used in wave telegraphy, and it has proved most 
useful in such work. Before I proceed to describe the 
designs and the methods of use which I have found valuable, 
let us consider and compare mutual and self inductances in 
relation to their capabilities in practical use. 


2. Mutual versus Self Inductance. 


For a number of reasons mutual inductances are more 
easily dealt with than self inductances. Those of invariable 
value, when properly designed, have the following 
advantages :— 

(a) The absolute values can be calculated with much more 
certainty from the geometrical dimensions, since the formulas 
for mutual inductance are of high thecretical accuracy while 
those for self inductance are much less satisfactory *. 

(b) Unless the conductors are highly stranded, the current 
distribution varies with frequency and in general the self 
inductance will also vary. By keeping the two circuits at a 
relatively large distance from one another the mutual induc- 
tance is practically free from this effect. 

(c) The effects of distributed capacity are probably less in 
mutual than in self inductances. In all cases the distributed 
capacity of one of the two coils can be made very small by 
sufficiently decreasing the number of turns in it (or opening 
them apart) while increasing the number in the other coil to 
keep the M constant. | 

When the mutual inductance is of the variable type, it can 
always be designed so that its value can be varied continuously 
from negative to positive through zero. This is a very great 
advantage, for with variable self inductance standards the 
incapability of reaching a zero value is a distinct drawback. 


3. Construction of Variable Mutual Inductances. 


For short range standards the construction used by Lord 
Rayleigh may be employed{. In this the primary coil is 
inside the secondary and can be rotated round a common 
diameter as in the Ayrton-Perry variable self inductances, 
and the whole positive range is comprised within 90° of 
turning, which gives a scale not nearly open enough for 


* For example see Rosa, Bull. Bur. Stands. vol. ii. p. 161 (1906) ; 
and Strasser, Ann. der Phys. vol. xviii. p. 765 (1905). 

+ Phil. Mag. vol. xxii. pp. 469-500 (1886); Collected Papers, vol. ii. 
p. 155. 


Variable Mutual Inductances. BT 


high accuracy. I have therefore designed a different and 
more elaborate arrangement which gives a much more 
extended scale. I proceed to describe the first model, which 
has been in use for a considerable time with most satisfactory 
results. 

The general arrangement of the apparatus is shown 


diagrammatically in figs. 1 and 2, which are plan and side 


ja 
Rea et» Pe 


—a 


view respectively. The primary ~* circuit consists of two. 
equal coaxial coils C and C’, which are connected in series 
their windings being in the same direction of turning. The 
secondary consists of the coils D and F in series. Of these 
coils, D is movable, being mounted on an excentric axis Q 
so as to be free to turn in a plane parallel to those of C and 
C’ and midway between them. Rigidly connected with the 
movabie coil is a pointer H which moves over a circular 
scale of about 180° in extent and graduated to read directly. 
The coil F is subdivided into ten sections, which are in 
series, each of them being 0:1 millihenry, and their junctions 
are brought to a set of separate terminals or studs with a 
turning head. The range of the moving coil extends from 
—0-002 to +0°11 millihenry. This gives a continuous range 
from 9 up to 1 millihenry, readable near zero to 0:02 micro- 
henry, to 1 part in 500 at 0-1 millihenry, and to 1 in 5000 
at 1 millihenry. The subdivision of the coil F is easily 
earried out by the following artifice. The coil is wound 
with uniformly stranded wire of ten insulated strands, all 
the strands are connected in series, and the. whole adjusted 
to give 1 millihenry f. If the stranding has been properly 


* Itis only for convenience of description that the circuits are here 
distinguished as primary and secondary. ‘hey are really quite inter- 
changeable. ; | 

+ Mutual Inductances with stranded wires were used many years ago 
ae Brillouin (Theses presentées a la Faculté des Sciences de Paris, 

2). 


158 Mr. A. Campbell on the Use of 


done, it will be found that no one of the sections differs from 
its neighbours by more than 1 part in 1000, and each is 
01 millihenry. The placing of the movable secondary coil 
midway between the planes of the two primary ones ensures 
that small axial displacements shall have very little effect on 
the mutual inductance. 


4, Adjustment and Calibration of Variable Mutual 
Inductances. 


The equality of any pair of sections can be tested by con- 
necting them in series with their windings in opposition in 
circuit with a ballistic galvanometer and reversing the current 
in the primary. It should be noticed here that, if a primary 
coil have any number of secondary circuits, the mutual 
inductance to all the secondaries in series is equal to the 
algebraic sum of their separate mutual inductances (+ or — 
according to the direction of the winding). Owing to this 
very important property we can build up and step down in the 
values as easily as if we were dealing with resistances, and 

there is the further simplification that we can subtract as 
well as add the values. The marking of the scale and the 
setting of the coil F are done by comparison with a fixed 
standard mutual inductance such as I have described else- 
_ where*, The comparison may be made by Maxwell’s method f, 
using a sensitive ballistic galvanometer or a vibration galva- 
nometer as detector. When a vibration galvanometer is 
used as in fig. 3, it shouid be remembered that, for a 


Fig. 3. 


balance, two conditions must be satisfied, viz., 
M,/M.=R,/R, 
and L,/L, =R,/R,, 


* Phys. Soc. May 1907 ; Phil. Mag. [6] vol. xiv. p. 494, Oct. 1907. Also 
see Proc. Roy. Soc. A. vol. lxxix. p. 428, June 5, 1907. 
- + Maxwell, Elect. and Mag. 2nd edition, ii. § 755. 


Variable Mutual Inductances. ; 159 


where R, and R, include the resistances of the secondary 
coils. In order that the second condition may hold, it is 
necessary to introduce into one of the secondary branches a 
coil a whose self inductance can be continuously varied ;_ by 
alternate adjustments of R,/R, and the self inductance of this 
coil, a balance is easily obtained. The fact that R, and R, 
are partly of copper coils is apt to introduce some inaccuracy. 
The copper resistance, however, can usually be largely 
swamped without losing too much of the sensitivity. 

In connexion with this method I may mention that it is 
perfectly applicable to the case of comparing the mutual 
inductances between one primary circuit and two separate 
secondaries. This case is shown in fig. 4. 


Fig. 4. 


By this means a thorough intercomparison (or adjustment) 
ean be made between the various sections of a subdivided 
mutual inductance. By further sets of secondary coils for 
higher multiples (10, 100, and so on) any desired range may 


be obtained. By stranding (or otherwise subdividing) the 


primary coils as well as the secondary ones a multiple-range 
inductance is the result. In all cases the subdivision can 
be effected by other means than stranding the wires, which 
is merely used to avoid the separate adjustment of the 
sections. The main objection to the subdivision by stranding 
is that it increases the distributed capacity of the coils. 
With coils of many turns this might be a serious drawback, 
but with the relatively few turns required for 1 millihenry 
the effect can only be very small and [ have not yet noticed it in 
practice. It should be kept in mind that, as in all standard 
inductances, eddy currents should be prevented by using only 
highly stranded wire for the winding of every section and by 
avoiding the use of any metal parts near the coils. 

By varying the relative dimensions and positions of the 
two fixed primary coils CC’ (fig. 2) and the movable 


160 Mr. A. Campbell on the Use of 


secondary D, the scale can be considerably modified, and can 
. be designed to suit special purposes. The most generally 
useful type of scale becomes more and more open as the 


readings approach zero, for this gives good accuracy of 


reading even at very low values. Fortunately in the present 
instance this type of scale can readily be attained. Further 
discussion of this point will be found in an Appendix. 


5. Employment of Variable Mutual Inductanees. 


Variable inductances of this type may be used for several 
purposes. Their most obvious application is to the calibration 
of ballistic galvanometers for magnetic testing; and I have 
found them of the greatest value for quickly adjusting a 
galvanometer circuit to read B directly for any given iron ring. 
I need not dwell on this application at present, but proceed 
to describe several methods to which they are applicable. 


6. Measurement of Mutual Inductance (Method A). 


Any unknown mutual inductance, whose value lies within 
the range of the variable standard, can be at once determined 
by connecting the primaries of the unknown and the variable 
in series to B (fig. 5), a source of alternating or intermittent 

Fig. 5. 


° 


current, while the secondaries, with their windings in oppo- 
sition, are connected in series to a vibration galvanometer G. 
The variable M is then adjusted to bring the deflexion to 
zero, and the reading gives directly the value of the 
unknown M. This is an extremely simple method as it 
involves no knowledge of any resistances. <A_ ballistic 
galvanometer and a commutated current may be used. This 
method does not apply to mutual inductances higher than 
the maximum value of the variable standard; for these 
another method will be described below (see § 8). 


7. Measurement of Self Inductance (Method B). 


In fig. 6 let a variable mutual inductance M whose primary 
includes the subdivided coil be connected into a Wheatstone’s 
network, as shown, along witha self inductance L,. Let 
the resistances of the arms be P, Q, R, and 8 respectively, 


Variable Mutual Inductances. 161 


the self inductance of the arm P being L, (the secondary 
coils of M) and that of Q being L,. Let H be asource of 
periodic current and G a vibration galvanometer tuned to 
resonance with it, so that we may take the wave form of the 


Fig. 6. 


currents to be asine curve. Let the instantaneous potentials 
of the three upper corners be 1, 0, and v2 respectively, and 
the instantaneous values of the currents into the upper corner 
be 7,, %, and 2 as marked. Let p=2an where n is the 
frequency, and for convenience of writing let pV¥—1 be 
denoted by a, so that 22 =—p?. The mutual inductance M 
may be made positive or negative according to the way in 
which the coils are connected; and in all that follows we 
might write + M for M throughout. When the galvanometer 
shows a balance, v;=v,, and the instantaneous valne of the 
current through G is zero. 


Also eae ie 
— Fa al = 13. 


Accordingly we may write 


(P+ L,2)i,— Mai =(Q + Lpa)iz ; 
[P+ (Ly +M)e]i;=[(Q4 (L2—-M)a]iz 


also liye 
Hence 


Fherofore 


S[P+(h,+ M)z]=R[Q+ (L,—M)zl. 
Equating the real and imaginary parts each to zero we have 
PP Gila He wbne 4) eidtnspsfl) 
and pela M)=-R(Po— MM). FQ) 


* (Nov. 22, 1907) I find that a case slightly more complicated than 
that of Equation (2) has been worked out and applied in an ingenious 
‘manner by Gratz (Wied. Ann. vol. 50. p. 766, 1893). 


Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1968. M 


162 Mr. A. Campbell on the Use of 


Exactly the;same equations hold when the positions of the 
source and the galvanometer are interchanged. 

The most useful case is when the noninductive arms are 
made equal, 7. e. S=R; then (1) and (2) become 


P=Q, 
and L,—L,=2M. 2 2 


This case gives an extremely convenient way of measuring 
small self inductances, which is done as follows. 


The arrangement is shown in fig. 7. ‘The noninductive 
arms are equal (R, R). In thearm AB there is the secondary 
coil a of self inductance Lin series with a practically non- 
inductive rheostat 7. Inthe arm AC is placed a “ balancing” 
coil b also of self inductance L and of resistance equal to or 
slightly greater than that of 6. By adjusting r the bridge 
will balance when M=0. ‘The small self inductance N to be 
measured is now inserted in series with coil } in arm AC, 
and a balance obtained by altering r and M. Then, by (3), 
N=2M. Thus N is found directly from the reading of M, 
and the range of values that can be measured runs from 0 up 
to twice the highest reading of the variable mutual inductance. 
[For values of N above this range the more general case 
(equations (1) and (2)) may be used.} The L’s of the coils 
a and 6 should be adjusted to equality once for all by putting 
M at zero and setting one of the coils. An exact setting is 
convenient, but not necessary, for if L, and L, differ slightly, 
they can be balanced (without N) by a small reading M,. If 


Variable Mutual Inductances. 163 


M be the reading for balance when N is inserted, then 
N=2(M—M,)*. 

fiven if a and bare well matched, it is always well to begin 
by reading their difference if any. 

It will be noticed that the method is really a differential 
one; when N is introduced into the arm AC no alteration has 
to be made in the other arms except to increase the resistance 
of AB by an amount equal to the resistance of the coil N. 
But although it has all the advantages of differential mea- 
surement, the reading can be made to give N directly without 
having to take a difference at all. This is due to the use 
of the inductive balancing coil 6 f. 

The method has the advantage that it does not require the 
knowledge of the absolute value of any resistance. The non- 
inductive bridge arms must be equal; to check the equality 
they can be interchanged. For the noninductive adjustable 
resistance 7 I usually employ a special rheostat consisting of 
two slightly flattened thin wires running parallel to one 
another at a few millimetres distance, with a sliding contact 
piece across them to complete the circuit. The inductance 
of such a rheostat can be approximately calculated, and 
may thus be allowed for when measuring very small self 
inductances. The inductance of the part added to com- 
pensate for the introduction of N has merely to be subtracted 
from the result. 

In practice the method proves very convenient; with the 
variable mutual inductance described above, self inductances 
of any value from 0-1 microhenry up to 2000 microhenries can 
be measured directly withont the bridge being altered in any 
way except in the rheostat7. In a later model shown in the 
accompanying Plate 1X. the whole scale of the movable coil 
corresponds to 20 microhenries, and at this value it can be read 
to l or 2 in 1000—at 200 microhenries to 1 or 2 in 10,000. 
All the resistances of the coils are very low, and the sensitivity 
can be considerably increased by using M. Wien’s method 
of connecting the vibration galvanometer to the bridge by 


MImeans of a transformer of suitable ratio (n/n, small). The 


* (Oct. Ist, 1907.) Lf, instead of the coil N, a condenser K with 

series (absorption) resistance g be added, then 
1/p?K = 2M, 
and g=change in r. 

+ If coil 6 be made noninductive we revert to Maxwell’s method of 
comparing the M of a pair of coils with the L of one of them. 
Equation (2) then reduces to L/M=~—(1+R/S). Maxwell, ‘ Eleet. & 
Mag.’ 2nd edit. vol. 11. § 756. 

M2 


164 Mr. A. Campbell on the Use of 


method will also give the difference between two unknown 
self inductances introduced into ABand AC. | 
The addition of a condenser to Method (B) gives an inter- 


esting corroborative method (Method C). If (fig. 8) the 


condenser K is in shunt with 8, 


it may be shown that, for a balance, 


SP—RQ=p*RSK(L,—M), .-. . . @ 


and SCL,—-M)—RG,+M)=QRSK. . . . () 

When R=S: and pl, —1o— 1, 
P—Q=,RK(i—M), . : . ae 

and 2M = QE fe) egies, yl 


Kiquation (7) gives a check of K against M without having 
to measure the frequency. The adjustment is dependent on p, 
however, and therefore a steady frequency is necessary, which 
is not required in Carey Foster’s Method. 

From (6) and (7) we get 


Q(P—Q)=2p'M(L—M). . . . . @} 


8. Measurement of Mutual Inductance greater than 


Standard (Method D). 


When the unknown M is beyond the range of the variable 
standard the following method is applicable. 

Let the primaries of the variable standard and the unknown 
M be connected up as in fig. 9, adding to the latter if 


Variable Mutual Inductanees. 165 


necessary a coil ¢ of sufficient self inductance to make N 

considerably greater than L. Let a balance be obtained 

{by Maxwell’s Method) and hence 
Peeper es 6 ee =. (9) 


and Neste 1, G0) 


_ Now, without altering any other branches, let the secondary 
coils be introduced (in opposition) into the galvanometer 
circuit as shown in fig. 10. 

If a balance be now obtained by adjusting the variable M, 
then it is easy to show that 


(P+R)M,=(Q+8)M, .. . . (1) 
and» PS—p'M,(L—M,)=QR—p2M,(N—M,). . (12) 


- Since (9) and (10) still hold, these equations both reduce 
to. 


PO NE 2 te 3S 


which gives M, immediately in terms of the reading M, and 
_ the resistances R and S (neither of which include copper 
coils). The self inductance of the added coil ¢ need not be 
known, but if R and S have not a slide wire between them, 
¢ ought to be slightly adjustable, which is easily arranged by 
having part of it a small coil of a few turns that can be 
shifted about over the remainder. 


166 Mr. A. Campbell on the Use of 


9. Measurement of Capacities (Method E). 
As in fig. 11, let the primary P of the variable standard’ 
be connected in series with a condenser K to a periodic 


source H, while the secondary Q is connected to a vibration 
galvanometer as shown. Let M be adjusted until the galyano- 
meter gives no deflexion, in which case we have 


ah re 


or +peMK=—1. 


The secondary musi be connected in such a direction that 


v7 


the lower sign is taken, and thus 
pM Bed) oe 3! aoe 


where M is in henries and K in farads. 

It is interesting to notice that this formula is similar in form 
to that for resonance with a condenser and self inductance, 
viz. pp>LK=1. With self inductance we usually tune to get 
a maximum current through the galvanometer; with mutual 
inductance, on the other hand, we tune for zero current. 
Tf M is in henries and K in microfarads, equation (15) gives 

 Tso"2 
a. a (16) 


and if M and K are known, the frequency can be determined. 

If the frequency is not fairly high, the product MK 
becomes inconveniently high, e. 9. M=1 henry, K=10 mfds. 
gives n== 50~per sec. In contrast, n= 10,000 for 256 
microhenries and 1 mfd. 

The following is a practical example of the method :—The 
source was a microphone hummer for 2000 ~ per sec. and 
the detecting instrument a tuned telephone. A balance was 


wl 


Variable Mutual Inductances. 167 


obtained for K=10°06 mfds. and M=0-626, millihenry, 
whence the frequency was found to be 2004~ per sec. 
With another condenser (a carefully calibrated subdivided 
one), capacities of 1:000, 0-900, 0°800, 0-701, 0°601, and 
0°551 gave 


n=1999, 2005, 2009, 2001, 2004, and 2001 ~ per sec. 


respectively; the variable M was not, however, so accurately 
readable in this case. 

In this method it is very important that the insulation 
resistance S of the condenser shall be high. If the leakage 
is not negligible, equation (14) becomes 


. Ss . 
se Kies 
S(1—SKa). 


Putting the real part equal to zero we have 
1/(1 + p?S?K?) =0, 


showing that an exact balance can only occur when 
p’S?K?=a0. In practice the insulation resistance of a good 
mica condenser is quite high enough to give a good balance ; 
even a good paraffined paper condenser will work fairly well, 
but an ordinary one gives only a minimum deflexion on the 
galvanometer *. 

The application of this null method to the high frequencies 
used in wave telegraphy is made difficult by reason of the 
coexistence of two frequencies in the waves used. 


APPENDIX. 


Scale of Hacentric-Coil Variable Mutual Inductance. 


The problem of finding the mutual inductance between 
two circles (thin circular coils) not in the same plane and 
whose axes are parallel has not yet been solved, as far as I 
know. For any particular case the result may be calculated 
by quadratures as follows. 


* This may, however, be due more to absorption than leakage. - 


168 Mr. A. Campbell on the Use of 


In fig. 12 let the two circles be H and G, of radii a and A 
respectively, 6 being the distance between their planes, and 


Fig. 12. 


g the distance between ‘their axes. Let the circle H’ be the 
projection of H on the plane of G, and let BCD be an 
elementary strip concentric with H/ of radial width 07, where 
O’/B=r. Let CO'B=6@. Then the M between circle H and 
strip BCD 


. (M of circle H and annulus TBD) 


fag c > =. o 
Z 2 | — 47 nine vary (k— *) F,+ (Ea b 10-° |r, 


@ 2arnn10-° 


Sia 
oM= = Vatry ie { 2a(F\—E;)—(r—«)(%) E, }or, 


or 


where (18) 
k=2 Var] /(at+r)?+0= sin y, 
k'= cos y, wien 
and F’, and E, are complete elliptic integrals to modulus &. 
_ Also 7 a balls 
| @= cos-! Girls 


he 2rgq 
Thus when a, A, 6, and g are given, by calculating 9M 
by (18) for a series of equally spaced successive values of 7, 
from r=g~ A to g+A, and adding in the usual way, the 
value of M can be approximaiely found. By this process 


= 


Variable Mutual Inductances. 169 


which is rather laborious, the following approximate 
values were found, when A=20, a=10, b=5cm., and 


nN, No= 100,000 :— . 
q: NE. 


em. millihenries. 
0. 9-3 
5 a7 

10 79 

FS 6°7 


It is interesting to notice that the maximum value of M 
does not occur when the axes of the coils are coincident 
(g=0). To avoid the labour of the calculation by quadra- 
tures, some rough experiments were made with coils of finite 
section (about 0°6 x 0°6 sq. cm.). From one set of these the 
series of curves shown in fig. 13 was obtained. The coils 


Fig. is; 


S Sasa ( Jered 
t 
2-6 b Si s> 
aS 
tv 

4 A = {0, aa=aS om 
5 M2 = 100000 

Si 

Zz 
78 — 

t 
ISS 
m 

oO 


were of mean radii A=10, a=5 ecm. respectively, n, ine 
being 100,000, and g and 6 were given various values. It 
will be seen that for 6=2°6 the maximum M does not occur 
at the symmetrical position (just as in the case calculated 
above). For equal coils, however, the coaxial position gives 


170 Mr. A. Campbell on these of 


the maximum M. The curves also show that the value of ¢ 
for which M vanishes and changes sign becomes greater and 
greater as the mean planes of the coils are set at greater 
distance from one another. The ratio of A/a has been made 
2 in this example, as that isa suitable proportion for use in a 
practical instrument. Having now the curves connecting M 
with g, the distance between the axes of the two coils, it is 
easy to investigate the variety of angular scales obtainable 
by turning one of the coils round an axis parallel to its own 
axis. 

Let a definite value of 6 be taken, say 6=5, which is 
convenient in practice, and in fig. 14 let the smaller coil be 


Fig. 14. 


movable about an axis perpendicular to the plane of the 
paper at Z, and let the radius ZQ=p. Let the distance ZO’ 
of the axis Z from O/ (the point where the axis of the other 
coil cuts the plane of the paper) be A. Let the angular scale 
reading YZQ=d¢. 

Then by choosing various values for p and h we can get a 
wide variety of movement of the centre Q (of the movable 
coil) relatively to Q’, and hence a variety of scale calibra- 

Bs. 
tions. Since cos¢= ree , the angle ¢@ corresponding 
to a reading M can be got from q, f, and p._ The scale can . 
be most easily set off by the following geometrical construc- 
tion. Draw a circle with centre Z and radius p, and make 
ZO' equal toh. Then find on the (6>=5) curve in fig. 13 
the g corresponding to a particular value of M. With this 
value of g as radius, a circle with centre O/ will cut the 
circle X at the point of the scale required. By taking suc- 
cessive values of M the direct-reading scale can be set oft 
round the circumference of the circle X. 


NO Da it = 


Variable Mutual Inductances. LTE 


As an example, in fig. 15 are given three scales for h=5‘5, 
p=9'9; h=7, p=T2; and h=10, p=4:2 respectively, when 
A—10,0—9, 6=5, and n,-r_ = 33,000. 

Although these scales give very different ranges, the 
general characteristic of all three is an openness at both 
ends; no proportions that I have tried tend much to make 
the scale uniform throughout. The openness at the lower 
values is a distinct advantage, however, when dealing with 


Fig. 15. 


125, 


100° 
small inductances ; but care must be taken, when marking 
that part of the scale, that the subdivisions shall be accurately 
marked and not put in merely by interpolation. 


Scale (2) in fig. 15 is the best of the three and is nearly 
that which I have used in practice. 


In conclusion I desire to express my best thanks to 
Dr. Glazebrook, for his kind encouragement and valued 
advice. 


Sept 21, 1907. 


4 O 


XII. The Electron Theory of Matter and the Explanation of 
Magnetic Properties. By G. A. Scuort, University College 
of Wales, Aberystwyth *. 
vA: \* a previous series of communications to this 

Journal it has been shown that spectrum-lines 
cannot be ascribed to the free vibrations of systems of electrons 
in orbital motion of atomic dimensions, partly because the 
intensities of most of the waves emitted are far too small to 
produce observable effects, partly because the wave-lengths 
of those waves, which are of sufficient intensity, are generally 
far too short. When we consider the experimental evidence 
in favour of the view that the atoms of a gas, which absorb 
and emit light, are only a small fraction of the whole number— 
for instance the researches of Stark T and his associates in 
the canal-ray spectrum, the recent work of Pfliiger { on the 
absorption in a Geissler tube, and of Wick § on the fluores- 
cence of resorufin,—we are led to the conclusion that spectrum 
lines are due to some special mechanism in the atom, which 
is set in operation during the act of ionization and operates 
only as long as the ion lasts, but is unaffected by disturbances 
which are sufficient only to produce free vibrations of the 
electrons of the atom without altering its structure. 

Be this as it may, spectroscopic phenomena throw much 
less light on the structure of the average atom than has 
hitherto been supposed; we are thus driven to seek information 
from other sources, such as the phenomena of magnetism. 

§ 2. It has been shown by Richarz || and R. Lang@ that 
an electrolytic ion, revolving in an orbit of atomic dimensions 
with a period either greater or less than those of light-waves, 
is equivalent to a molecular magnet, whose moment is of the 
order of one twenty-fifth of the average moment of an atom 
of iron magnetized to saturation. We conclude that an atom 
built up of revolving electrons is capable of accounting for 
magnetic phenomena, provided a sufficiently large proportion 
of the atoms of the substance are effective. 

More detailed investigations. by Voigt ** and by J. Jd. 
Thomson tf, however, indicate a peculiar difficulty, in so far 
as they lead to the conclusion that an assemblage of mutually 
independent electrons, revolving in elliptic orbits, give on 

* Communicated by the Author. 

+ Stark, Drude Ann. xxi. p. 401. 

J Pfliiger, Ann. d. Phys. (4) xxiv. p. 515. 

§ Wick, Phys. Zitschf. viii. p. 181. 

|| Richarz, Wied. Ann. lii. p. 410. 

q R. Lang, Drude Ann. ii. p. 483. 

** Voigt, Drude Ann. ix. p. 116. 
tt J. J. Thomson, Phil. Mag. [6] vi. p. 673. 


aie ak 


The Electron Theory of Matter. ie 


the average no magnetic moment at all. If, however, the 
electrons are subjected to impulses irregularly distributed at 

comparatively long intervals, magnetic effects may be pro- 
duced (Voigt); or if the electrons are subject to sutticiently 
rapid dissipation of energy, so that their orbits continually 
diminish in area, an average magnetic moment results and 
the system is paramagnetic ( Ae Thomson). Neither 
hypothesis is quite satisfactory: for Voigt’s hypothesis 
requires either very large numbers of independent electrons 
in the atom, or very powerful impulses, to produce an appre- 
ciable effect (§ 22), while that of J.J. Thomson requires very 
great dissipation of energy, and besides leaves little scope for 
accounting for diamagnetism (§ 22). As a matter of fact 
we shall see that neither hypothesis is necessary; for a 
system of electrons in orbital motion must group themselves 
in rings, if their radiation is to be sufficiently small for per- 
manence ; and a ring, as opposed to a system of independent 
electrons, possesses magnetic properties (§ 21). 

§ 3. The magnetic effects to be explained may be con- 
veniently grouped as follows :— 

(1) Effects due to the mutual interaction of the molecules 
of the substance; these are usually taken to include hysteresis 
and coercive force, and the numerous interactions between 
strain and magnetization. It should, however, be borne in 
mind that some of these effects, such as the variation of 
magnetic susceptibility with magnetic force, can be explained 
without intermolecular action (Theory of Stefan* and 
W. Siemens 7). 

(2) Effects due to the actions between the atoms of the 
molecule. The smallness of the magnetism of oxides and 
salts of iron and cobalt in proportion to the amounts of iron, 
or cobalt, contained in them, is an effect of this kind. 

(3) Effects due to the constitution of the atom itself. The 
fact that the amalgams of iron and cobalt studied by Nagaoka ¢ 
are as strongly magnetic, in proportion to their concentrat tion, 
as the pure metals , coupled with the usual supposition, made 
on chemical srounds, that the molecule of a metal is mon- 
atomic, can hardly be explained except by supposing that 
magnetism is, in the last resort, an atomic property ; and that 
the largeness of the magnetism of iron and its congeners is 
due to some peculiarity of atomic structure. This conclusion 
is supported by the experiments of St. Meyer§ and others, 
from which it’ results that atomic magnetism is a periodic 

ers Wien. Ber. \xix. (2) p. 165; Wiedemann, Elektrizitdt, iii. 

D210 
a + W. Siemens, Wied. Ann. xxiv. p. 115. 


{ Nagaoka, Wied. age. lix. -p. 665 .., 
§ St. “Meyer, Wred. Ann. \xyii. p.'325; Ixix. p. 236 


174 Mr. G. A. Schott on the 


function of the atomic weight and is in some very close 
connexion with the atomic volume. In fact iron and its 
congeners do not occupy a quite exceptional position amongst 
the elements; for instance, Meyer finds that erbium in 
Eb,O3 is four times as magnetic as iron in Fe,Qs. 

In view of these facts we shall in what follows restrict 
ourselves to the explanation of atomic magnetism; never-_ 
theless the results have an immediate bearing on the magnetic 
properties of amalgams and powders of iron and its congeners 
for small concentrations, and generally of elementary sub- 
stances, which are weakly paramagnetic or diamagnetic. 
The problem then is this :—-How far can an atom, built up of 
coaxal rings of revolving electrons, be made to account for 
the magnetic properties of elementary substances, without 
the use of additional hypotheses, such as that of external 
impulses (Voigt), or that of great dissipation of energy 
(J. J. Thomson) ? 

- § 4. Let us consider a ring of n electrons, which, when 
undisturbed, revolve in a circle of radius p with uniform 
angular velocity . A controlling field is presupposed of 
such a kind that the steady motion 1s stable; and the velocity 
is supposed to be such that it is practically permanent in 
spite of radiation. Let ¢ be the velocity of light; write 
B=op/c. We can assign a limit to 8 for any given value of 
n, Which must not be exceeded if the ring is to have the 
necessary degree of permanence. 

The equations of steady motion are 


cp? d(mB . 
oe he8) a: .. 0 Si 


c 


3 me = SERRA: . ) oe 
where m is the transverse mass of the electron, e its charge 
in electrostatic units, P the central force of the controlling 
field, and K, U, V functions of 2 and 8 given by the 


equations 


t=rn—lL 5 
K= > : cosec —, 
t=1 mt 
s=20 8 
U —=2> | s°S an (2n8) — ene. 6°) | Jon (2sne)de], 
4—3 0 
s=@a 2s+ 1 
V= = (s+ 3) cot! om i | BAA") Saraaf (s+ 1)B} 


Bs 
+(1 +8°)| Josii{(2s+ La}de | te 
0 
* Schott, Phil. Mag. [6] xii. pp. 22/23. 


Electron Theory of Matter. 175 


Equation (1) enables us to assign the upper limit to @ 
(vide table*): for instance, we might admit a secular change 
in the wave-length of the D lines, and therefore of @, of not 
more than, let us say, six ten-thousandths of an Angstrém 
unit in thirty years, that is, 10~1° of its value per second ; or 
an equivalent change in m, both leading to value of 8 of the 
same order. 

§ 5. The controlling field may be partly due to other rings 
of moving negative electrons, but is mainly due to positive 
charges of such amount as to make the whole system neutral 
except when ionized. The positive charge may be distributed 
throughout the atom (J.J. Thomson’s model), or may be 
concentrated in discrete charges (Nagaoka’s model). In any 
case the steady part of the field must be symmetrical about 
the axis of the ring, and derivable from a potential ®; and 
the part of ® due to any ring can be calculated as if all the 
charges of the ring were uniformly distributed along it and 
revolving with it. In addition the disturbing ring produces 
periodic forces, the fundamental frequency of which is the 
difference between the angular velocities of that ring and the 
one under investigation. These forces excite perturbations 
and cause additional radiation ; they must be small enough 
not to upset the permanence and stability of the system. The 
“necessary conditions being supposed satisfied, the periodic 
forces need trouble us no further. Thus we may write 


Se? | 
Daca ail te Waban Nawcouts anes 


§6. Ata large distance the ring is equivalent to :— 
(1) a charge ne at the centre ; 
(2) a polarized element of electric moment p = Ser; 
(3) a magnetized element of magnetic moment 


i 
i= {Le E = 
9 ¢ 
r is the vector from the centre to the zth electron of the ring, 
v is its velocity relative to the centre, and } denotes sum- 
mation for 2 from 0 to n—1. 


When the centre of the ring is in motion with velocity w 
the polarized element (2) is itself equivalent to a magnet of 


I: é 
moment - E <| t+. The translatory motion of the atom as 


a whole may be neglected, even in the case of a gas at high 


* Loc. cit. p. 24. 
t H.A. Lorentz, Math. Encyclopaedie, V. 14, §§ 12-15. 


— '—_— 
a 
é J , 
» 


176 Mr. G. A. Schott on the 


temperature, since its velocity is exceedingly small in com- 
parison with c, the velocity of light. Hence w may be taken 
to mean the velocity of the centre of the ring relative to its 
position in the undisturbed atom. 

§7. Let the axis of the undisturbed ring be Oz; the 
azimuth of the zth electron may be written in the form 


21 : ; : . 

wt ++ —, where 6 is an arbitrary angle; its coordinates 
VL é 

(x, y, 2), referred to axes fixed in the plane of the ring, are 


Qar1 


Dan) 
e=pcos(ot ++ rus yY=p sin(wt+8 + ), z=0. (4) 


n 
When the ring is disturbed these coordinates are increased 
by terms of the form 
Qari Qari 
de= —é sin (or+3+ =m) —7 cos( wt +84 ) ) 


7 | 
Da7 Qari 

éy= Ecos (ot +84 =") <n sin (or +84 7 P : (5) 

eae eer J 


where &, 7, € are the components of the displacement, sup- 
posed small and measured from the position that the electron 
would have occupied at the same time, if it had not been 
disturbed from steady motion. They may be expressed as 
the sums of functions of the type 


eis 
Ae—* cos (qth +2), 


where / is an integer between + ” such that the number of 


Estes 
nodes, and also of loops, in the disturbed ring is 2k. q is the 
frequency relative to the rotating ring, +o that relative to 
an outside observer. ; 

§ 8. The particular type of disturbance we have to consider 
is that due to a constant external magnetic force h, which 
we shall suppose makes an angle @ with the axis of the 
undisturbed ring, the axis being drawn to correspond toa 
right-handed rotation. We shall choose the arbitrary angle a 
so that h lies in the plane of wz; then the components of h 
are (2 sin 8, 0, hcos @). Assuming the usual expression for 
the mechanical force on a moving charge as given by the 
Maxwell-Lorentz theory, we find for the components of the 
force on the ith electron in the directions of the tangent, 


’ 
“¢ 


Electron Theory of Matter. 177 
inward radius and axis, the values 


: Qari : 
(0. —Beh cos 0, — Beh sin @ cos (ot +8 - —)} » ¢€G) 


7 

Since the disturbances produced are of the same types as 
the disturbing forces, with the same frequencies and damping, 
but, it may be, displaced in phase, we have only to consider 
two types of disturbance :— 

(1) A disturbance due to the radial foree —Behcos@. q, k 
are both zero; & 7, € are constant and the same for 
each electron of the ring. 

(2) A disturbance due to the axial force 

— Beh sin @ cos (or +34 — Ga kf. 

In each case we have g+/w=0, that is, the disturbance, 
like the force, is stationary for an outside observer. Never- 
theless it is periodic relative to the electrons of the ring, and 
like any other forced vibration may exhibit phenomena of 
resonance. This is a very important fact; from it follows as 
a consequence that a ring of revolving electrons may, and 
actually does, possess magnetic properties much greater than 
would be the case if its electrons were all independent. 

Further, it follows from the stationary character of the 
disturbance with reference to space, expressed by the equa- 
tion g+kw=0, that the radiation of energy into space is 
exceedingly small, in fact only of the same order as the 
radiation due to the steady motion ; hence the displacement 
in phase is almost zero, which means that the maximum of 
the disturbance lies very nearly in the plane through the axis 
and the magnetic force. 

9. By means of the expressions of § 6 and the equations 
(4)-(6) we find the following expressions for the electric 
and magnetic moments of the whole ring, on the supposition 
that squares and products of the displacements may be 
neglected :— 


a eee ee —_ 
w . 


ee 


| 
: 
: 
F 
7 
{ 


(a) Steady motion. 


ov p=O. wy, - 
Be grate 2 @ 
m=(0, 0, dneBp) . . 
(6) Disturbance due to the axial magnetic force, and 
therefore radial mechanical force (— Beh cos 0). 


8p, =ne(0, 0,8), S=ne(0, 0, )=0 . 9 e 
; f RN 5 
em, = Sne(On rote) 2S PID 
It must not be forgotten that this force not merely 
displaces the ring but also changes its velocity. 


Phil. Mag. 8. 6. Vol. 15. No. 85. Jan. 1908. N 


178 Mr. G. A. Schott on the 


(c) Disturbance due to the radial magnetic force, and 
therefore axial mechanical force 


(—peh sin 8 cos (wt +6+ = 


ne 


dp. =4ne(A’—B, A+B!, 0), dpo=dnew(A, A’, 0) \ (9) 
SAN, (: 


om, =1neB(—C, —C, 0) e e e e 9 
where.we have written 
QT 


(E, n, )=(A, B, C) cos (wt-+3+ a 


: 2ar1 

+(A!, BI, C) sin (ot+5+—= . 

§ 10. The varying electric moment dp, in (¢) gives rise to 

a magnetic force rot “Es) ; its components are linear func- 

tions of the direction cosines of the radius vector r with 

respect to axes fixed in the ring, and therefore their values 

vanish on the average for a large number of rings, whose 

axes are distributed equally in all directions, and for a 
distant point. 

The electric moment ép gives rise to a magnetic moment 


ac: "|, where for w we may take the velocity of the centre 


of the ring relative to its undisturbed position (§ 6); but 
this velocity, as well as dp itself, is small of the order of the 
disturbance, so that the resulting magnetic moment is of 
the second order and may be neglected. 

Thus we need not consider the polarization in finding the 
magnetic field due to the disturbing ring. 

The magnetic moment in the steady motion by (a) gives a 
component in the direction h equal to gne8p cos @; since this 
must vanish on the average the mean value of cos @ vanishes, 
as it must do for rings equally distributed. 


It only remains to consider the moments 6m given by (8) a 
and (9); by the principle of superposition of small disturbances. 


we may find the eftect of each separately, and add them 
together to find the total effect. 

§ 11. To find the effect of (8) it is necessary to consider 
the motion of the ring during the variable period accom- 
panying the establishment of the magnetic field h. Since 
the motion of the ring itself is periodic, the simplest method 
of proceeding is to resolve the mechanical force into simple 
harmonic components, either by Fourier’s series, or integrals, 
and to determine the corresponding forced vibrations, and 
thence, by summation, the resulting motion. 

Suppose the variable period to extend from t=0 to t=T; 


— So 


Electron Theory of Matter. 179 
then we may write, between these limiis, 
: s=0 t 

h= > Hs exp. seg Bee NG lh bO) 


where the coefficients are small for large values of s, and the 
series may be supposed differentiable. The axial magnetic 
force, which alone concerns us here, is got by multiplying by 
cos @, and gives rise to a mechanical force to the centre 
— Beh cos @ (§ 8, (6)). Since however the magnetic force is 
changing, an induced electric force E is produced along the 
circle, which causes a tangential mechanical force cH. To 
find it we apply Faraday’s Law of Induction to the circuit 
made up of the orbit of the electron from t=0 to t=r, 
together with so much of a circle and radius as are necessary 
to make a closed circuit. This circuit differs from the 
original circular orbit, described an integral number of times, 
by quantities of the order of the disturbance ; since it occurs 
on both sides of the equation with h, or E, as a factor we 
may substitute the circle for it without making an error of 
more than the second order. Thus we get, from the general 
equation of Maxwell’s theory, 


Oe BP ook 
H=—5 —(heos@).. . . . . (il) 


§ 12. A great simplification ensues because we may gene- 
rally suppose t very large compared with the time of 
revolution of an electron. In fact 7 will rarely be as small 
as 10~® second—the value for a coil of radius 1 cm., length 


100 em., with 1000 turns and of resistance 10 ohm would be 


about 4.10—? see.—while the period for an electron describing 
an orbit of atomic dimensions with a velocity one thousandth 
of that of light is about 2.10~” sec. Since such a velocity 
is already very smail to be consistent with stability for a ring 
of more than 3 electrons (in J. J. Thomson’s well-known 
model the lower limit is in general several hundredths of 


_ the velocity of light), our assumption appears to be amply 


justifiable on experimental and theoretical grounds for any 
but the smallest rings. 
Hence if g be the frequency of any one of the harmonic 


e — Gg AX . . . g se 
components of the disturbance, = and @ fortiori *°, is an 


exceedingly small quantity. ra 

We shall now develope the equations of the disturbed 
motion ; as the full investigation is long it will not be pos- 
sible to do more than indicate the method. We first use the 
equations of motion of an electron under any forces * to get 


* Abraham, Elektrizitit, vol. ii. p. 123. 
N2 


180 Mr. G. A. Schott on the 


the equations for an electron slightly disturbed from uniform 
circular motion. Next we calculate the changes in the forces 
exerted by the controlling field, which are due to the 
displacement of the given electron from its position in steady 
motion, and, when necessary, those due to displacements of 
neighbour ing rings. Lastly, we calculate the changes in the 
forces exerted by the remaining electrons of the ring. The 
last problem is the most troublesome when the velocity of 
the ring is not small. 

§ 13. With the notation of §7 the equations for the 
disturbance (&, 7, €), whether periodic or not, are 


ees Ee : fae are ai 
stm wt t+ Lama ee 

m is the transverse mass as before, in which a small ceil 

change is admitted as possible for the sake of generality ; 


d(mpB) _- 
73 oT, 


= tt 10k (EOP) sue | 
* aiaay et etme Uppity) I= eT 

an ead ENG thot 

| 


pis the longitudinal mass and is given by p= 


oP, 58 are the components of the disturbing force due to all 
causes. The electron is supposed to be slightly disturbed from 
steady motion in a circle of radius p with angular velocity o. 

§ 14. If the controlling field be due to fixed charges, the 
changes in the forces P= sS=—- oe (this being zero 
in the steady motion) are due to the displacement (&, , &) 
of the electron, which is also its displacement relative to the 
controlling field. Since the field is symmetrical about the 
axis cf the ring, the component displacement along the 
rng, f, produces no change, and for the same reason no 
force is produced along the ring itself. A force of this 
latter type would be produced by induction if the charges 
producing the field themselves moved. 

For shoriness write 


(12) 


‘ - Bs 
2 


Electron Theory of Matter. 181 


Then the variations of the component forces due to the 
displacement relative to the controlling field are 


SVE OW. 
Pipes eo a (13) 
oS=Sy— TE. 


| The new symbols R, 8, T, which are usual ones for second 

Bi, differential coefficients, all cause no confusion, for in the 

| steady motion the force components, which ald be denoted 
by S, T, both vanish. 

When the field is due to a central positive charge (Nagaoka), 
or to a sphere of uniform positive electrification (J. J. “Thom- 
son), S vanishes, but in general this is not the case. 

€ 15. For periodic diatarbances the forces on the zth 
electron due to the rest of the ring are given by the following 
Eipantons: = 


y 
5 ol=~é ae )N—4Me@? 2 @K—aH eS + A- A, 


{8 eS ace 


tafe { (1+6°)M+ (2K —2H+ we { Bt 


2/6 legs) eae Me SIR 
5 ey — apy | - 


Pp a + 14) ace + Wet 2} 


+B+B,—2U ~ 


é 
1 : 
sof EO By Lig ied Foye aoy -OFG 


—2M(1—6°)@?4 4+ (K—2H)(1 +eye! 


+04 + SQ + +aq 5 ot) 


ees | (1) — 6)? 1+30? 
a Ne ieee 


+(K-—2H 24 26 re 28° 
JOE Bye © +f - D+3q—p5: PPot 3a — 2) 


(14) 


182 Mr. G. A. Schott on the 


where K, U, V have the same meanings as before (§ 4), 


and 
i=n—1] gkmri Pace, TT . il 
ae = jeosee™ sin or R= 2 * cot cosec — 2 
i=—n—1 1 3 7 if é =) gk 
N= 2 (5 cosee cara Coote sin eh: 
s=m 2s+1 B 
We s(s = cot FT Bda.1{(2s+1)8}—| Tossa 4 2st Do} de |, 
zh 2 2n i 


A, A,... are functions of 8, qg, and k defined by the 
alae 


j=- —-@ 
D) 
A= > FN aly ol 
Sri Zn @ 


with similar expressions for B, B, C, C, D, D, where 


pepe eases oil; Pe 
a (m, i= a3 J (IB) = Br 1 Jan (lx)de. 
0 


— 18? +18? (8 
b(m, 18) =— : S 1g3,,(I8) + : BP i} Jin (led. 


c(m, 18) = pera (8) + “TF iad a8) 
aie (15) 
eae voll * J 4(n)do a 


0 


d(m, 18) = - =F rest (ig) + “2? 1854 (18) 


2 2 2 & # 2 B 
4. a4 =. tnig aes a if Jo (la\ele 


Ay, Ap; .-. are the values of A, A,... in the particular case 
when £=0 and g=0. 

The expressions (14) and (15) remain true for a complex 
A emagel g, corresponding to finite damping. 

§ 16. The functions a,... all vanish when the argument /@ | 
aah When /@ is not zer o, they diminish rapidly as-the e 
order m increases. 

In the particular case of vibrations stationary in space, for 
which ¢g+w vanishes, as in our present problem, we have 


Electron Theory of Matter. 183 


m=1+2k, while in A... we have /=2jn, and in A... 
=2s+1. The terms in A... corresponding to j=0 vanish, 
and the largest terms left are those corresponding to 7= +1, 
which are of the orders m= +2n+2kh, and very small. We 
may neglect A ..., as well as the quantities By, U ; this only 
amounts to neglecting the radiation, which in the present 
case is of the same order of smallness as the steady motion 
radiation from the ring. For the same reason we shall 
neglect the small terms involving the variation of mass, m, of 
the electron in equations (12). 
A second simplification occurs in any case of periodic 
motion, whether damped or not, and therefore also in the 
present problem. The terms due to radiation on the left- 


2 
hand sides of equations (12), that is, those involving = 


as a factor, are easily seen to cancel, term by term, corre- 
sponding terms on the right-hand sides of equations (14). 


Hence in (12) we have only to take into account the 
following terms : 


pE—(w+m)on, (M+ m)o£+ mn—pern, me. (16) 


on the left-hand sides ; while on the right-hand sides of (14) 
we need only take into account the real parts of the factors 
of £ and ¢7 in the first, of o£ and 7 in the second, and of ¢ 
in the third equation. 

§ 17. We shall now apply our results to our first problem, 
the calculation of the part of dm due to induction, that is, to 
the axial magnetic force A cos 0, which produces the radial 
mechanical force —fehcos @ (6), and while it is changing, 


the tangential induced force eH=— = 3 (heos@) (11). 


When everything has become steady, 6m,;= (0, 0, }ned(@p)) 
(§ 9. (6) (8)). Now we have ; 


oo = 2 dp=—y, - whence 6 (8p) =a(® -n). 


Bence Sm =Jnea(~ =) along Ox. S88 OCELT ) 
This equation shows that, although the variations of the 


field are comparatively slow, nevertheless é is of the same 
order of magnitude as 7 and ¢. < 


In the present case we have k=0; hence by 5 af 
H=M=N=0. 


184 | Mr. G. A. Schott on the 

Further, ae is a very small quantity (§ 12). It follows 
from equations (15) that A—A,, C—C), D—Dp, can be 
expanded in ascending even powers of ; beginning with 


the second, while B can be expanded in odd powers, beginning 
with the first. Bearing this in mind, we see that on the 
right-hand sides of (14) the coefficient of & is of the order 


‘a B\2 
(2) and that of » of the order 6 in the first equation, 


g | 
that of € of order = and those of 7 and € of order zero in 


the remaining equations. Now @&=—&, wqn=H, wE=E; 
hence we retain the terms of the first order, that is, of order 
£ and 1 in the first equation, and those of zero order, that is, 
of order &, 7 and @ in the others. . 
2 ‘2qBY 2 

Thus we write A—A,=}<A,” ote ) ss aes 2) , and 
neglect C— C,, D—LD,, where the dash denotes differentiation 
with respect to the argument in A, and B,. By help of (15) 


we easily find 
2B Ry 78 i oe + BW, i 


where V, W are the series already defined (§§ 4, 15). 
§ 18. Collecting our results together and using (6), (11), 
(12) ... (18), we get the following equations of vibration :-— 


} a+ a (2ke-~5,) | —17 J (utm)o-+t J(2K8 85m) } 


aE 
Taig. 7 (h cos @). 


dV dV 


! 2 teal tee oe oe 
E{ Gtm)o+ 5 (2K eQ) | +\R—po?+ al 2K-2V+8 55 \ 


dB / 
—oS = — Beh cos @. 


7S—¢T =... 


We integrate the first equation, substitute from it in the 
second, and eliminate ¢ by means of the third. In the 
result we use the equation of steady motion (2) and obtain 
tke equations 


(E —7) | eo? + :(2Ke—2%3) t —mo’n= —4 Beh cos 6 


i: n)me? +n(R— Rs nt = = —1 eh cos 0. 


@ 


= 
f 


Electron Theory of Matter. 185 


These give 
3 Bak «2k 
mo” +R-—~+— 
7 Np 


. ai 
= Beh cos 6 - 


S? 4) e AGT 
24 pas Me react er Cf eee OU L 
ma +(R Taare a a + 5 B Bae) 
The component of the resulting increase in moment in the 
direction of h is &m,cos €@=4neB cos a(é —), bye (27) 
Substituting the value of > —7, and averaging for all rings 
@ 


of an element of the substance, we find for the mean effective 
moment per ring due to axial magnetic force 


F Shee 
Pe neo? mo? +R—- T oe ‘as 
6m,= hoe : R_ SS ) bs e (2K- i dV i ° (20) 
id ( ng p m + cmp B dp J 


§ 19. We shall now use our equations to determine the 
part of the moment due to the radial magnetic force, which 
produces the axial mechanical force 


— Beh sin @ cos (0 +6+ =n") (§ 8 (6)). 
/ 
The increase in the moment is given in § 9 (¢) (9), and is 


equal to 


2G 360 G, 


| 
bole 
OS 
Keo) 


dm, 


ZT 


) 271 
where f=C cos (wt+54 Tee — cu sin (ot + O — =). 


Thus we need caleulate only the axial displacement €. 
In the present case we have k= —1, g=o, so that as before 
g+kow=0. We find by § 15 
M=H~—K, N=2K—H. . (21) 


1 T 
H= 4 col ss 


It follows from equations (15), by using the continuation 
formulze for Bessel Functions, that in the present case 
1dV 2V_ 2(1—£6?)W 
So ee OW) pela 4. VW. 
eda BB Paeat3) A h D 
dv (22) 


C—CQy=A—4y—BGg +3V, D—D=V. 


A— Ay= 


186 Mr. G. A. Schott on the 
Remembering that the radiation terms disappear as before, 
we must take account of the following terms on the rien 
hand sides of (14), because of (21) and (22) :-— 
—=—&[2K—(1—6”?)H + A— A, | —im[ (1-8?) K-—H) + V4 A—Ap}, 
5 2—56?— 
e[(1—8)(K-H)+V+.A—4)]—[ 0-2) K— ORB 
+V+A Me As | 
ais 2 
c[ (1+ colina Se ea : 
respectively. 
Substituting these results in (12), together with (13) and 


(16), we get the equations of vibration in the form 
— fo? © (2K (1B) + AA} 
2 ’ 
—un[ (w+ m)o? — 3 {(1—6?)(K—H)+V+A—Ao}] =0 
3 
EL (w+ me? — 2 {(1-8)(K—-H) + V+ A—Ag}] 
2 _ Bee ee 
+n | R(t m)eo* + ih { (1-p)K-= "Fy 
p | 
+V+4~ Ay b | en 
2 i 2/2 2 
—n8+ ¢[T— ma? - ‘ {4G +2°)K- Se avh] 


202 
= — Beh sin @ exp. se ze) 


since the disturbing force is axial and equal to the real 
part of — Beh sin 6 exp. i( ot +8+ =) § 8 (6). 
Write 
pO OD + (23) 


and use the equation (2), mw?= a — sil )K Vs 


Electron Theory of Matter. 187 


we get 
(€—Q40~ =0, | 
(é—n)(Q+ =) + n {R4 —— | — i =0, L (24 


a8+e{ t- = ie a < GEE) ue H| = — Be hsin O exp. s(@t +6+ <7") 


From these equations it follows at once that 


Beh sin 6 
Bee? 6346") Ss? 
== 2 H— = Bay: 
a fil pera ae 
BU nea =” 


Since the corresponding moment is equal to —4ne@8C and 
is along the axis of w, we get the effective moment of the 
ring by multiplying by sin@. Averaging the result for all 
values of @, we get for the mean effective moment due to the 
radial magnetic force the expression 


1 ne? p? 
Me = h wr we BB+") 4 : S? " ; 7a (25) 
p 


; es: 6 a{r+t +55 goelengell Ora} +! Ly 


§ 20. The total mean effective moment per ring is equal to 
ém=6m,+ 6m, ; since all these quantities are proportional 
to h, at any rate for small displacements, we introduce the 


specific moments, defined by equations such as p= = 

If N be the number of rings per unit volume, the magnetic 
susceptibility is given by k= =Np, if the several rings do not 
act upon each other ; this is at any rate an approximation for 
small concentrations, such as we find in amalgams of iron and 
cobalt. 

In order to form some idea as to the performance of our 
expressions (20) and (25) we shall consider some special 
arrangements of rings, without enquiring into their stability 
and permanence. 


188 Mr. G. A. Schott on the 

(1) Single ring with a fixed positive charge at the centre 
(type of Nagaoka). 

Let the positive charge be numerically equal to ve. Then 


Steady motion requires oF =(, 1. e. z=0;\ hence 


ve" 20C-. ve" 
Ze zy. aS a b=. T= =e 
Pp Pp Pp 


Since H=j cot in (25) by (21), we get, from (20) 
and (25), 


oe ne"p* . 

a 12cm (26) 
Snp° 

Do= > ie al 

ein Cra a: 


(2) Single ring inside a fiwed positive sphere of uniform 
density (type of J. J. Thomson). 

Let the positive charge be numerically equal to ve, and the 
radius.of the sphere b. - Then 


ae ee 0. T= 


These give = 


eve mo” + ad 7 F 
ne p> b° 
Pi 12cm Sve 2 1 dV\) ° | 
ig de) Meee ye eG a 
Pe aes Da Ge enp\2K B il (21) 
snp? 7 ; ! 
= Pe Le he Soe t a i = . ° . . . . | 
P= 3848) 77 On’ act 


§ 21. We shall now calculate the value of & given by (26) 
or (27) for the weakest iron amalgam (0°19 per cent.) studied 
by Nagaoka *, on the assumption that each atom contains a 
single ring of 7 electrons of the type considered in § 20 (1) 


* Nagaoka, Wied. Ann. lix. p. 66. 


: 
: 


Electron Theory of Matter, 189 


and (2), the atoms being independent. For this particular 
amalgam the magnetization curve is practically a straight 
line up to H=3200 C.G.S. so that the condition of inde- 
pendence is satisfied; hence we may put h=H. 

Let p=x.10-8 cm., where 10-§ em. is the conventional 
radius of the atom. We at once find from the second 
equation of (26), or (27) that 

owl. sip 0be 10=%) a 


? 
for when n=2, tan-- =1, while B? is very 
enn=2, tang =1, while 8? is very small, and when 
od 
° . Ths TT e ) . . 
n= , limit n tan > = 5, While # is nearly unity. 


The ratio p/p. is of the oe where a is the radius of 


the electron, given by a= Since 27p/n is the distance 


5e7m 
between consecutive electrons, 5 
negligible. ; 2 

A°19 per cent. iron amalgam contains ‘025 gr. iron per ©. és. 
taking the mass of the hydrogen atom to be 171. 10-*4 gr. we 
tind N=4 107°. Hence since k= Np 


0007 . a? >k> :0004. 2°, 


is small, and p,/p. 


Nagaoka’s curve gives k=:0013. 

Wesee that an atom containing only a single ring of radius 
10-° cm., for which «=1, gives a value of from one half to 
one quarter of the experimental value of 4, An atom con- 
taining from two to four independent rings, or one slightly 
larger ring, could account for the actual atomic magnetism 
of iren ; since, however, the rings could not be independent, 
the above investigation will not apply. Before proceeding 
to the study of systems of rings, it will be convenient to 
consider the performance of the formule of Voigt and 
Thomson already referred to. 

§ 22. Voigt considers a medium containing a large number 
of independent electrons describing elliptic orbits and subject 
to damping, the irregular stationary state being maintained 
by subjecting each electron to periodic shocks which renew 
its energy. Let wbethe mean kinetic energy of an electron, 
and ¢, its mean potential energy, yr, its mean kinetic energy 
just after a shock. Voigt’s formula (51) (/. ¢. p. 128) is in 
our notation equivalent to the following :— 


me P Pu Be 
P~ 128m VA, ARE EAS YA RUNS (AS) 
for a single electron. 


190 Mr. G. A. Schott on the 


We notice at once that np is precisely of the order 
of p;, as given by (26) or (27), inless eee be very large. 
On account of damping yy, is probably many times y, but o; 
is of the order vy, unless we suppose the potential energy to 
be much increased by the shock. This, however, can hardly 
be the case, since a shock does not instantaneously change 
the position of the electron, but merely its velocity. It follows 
that while (28) can account for a much larger diamagnetic 
moment than is actually found, it can hardly account for a 
sufficiently large paramagnetic moment. 

Voigt gives a formula (134) (1. ¢. p. 134) for the magnetic 
moment due to rotating electrons, which is equivalent. to 


__ ak } 
DP hem G5: 
for a single electron, where & is the radius of gyration, and 
for simplicity the densities of both electric charge and mass 
have been supposed uniform. If we assume the mass to be 


2 2 
wholly electromagnetic, so that m= a while /?= | we get 
a? ; 
= 74. 


The moment due to a negative electron is thus far too small 
to account even for diamagnetism ; but that due toa rotating 
positive sphere of atomic radius would, with the same assump- 
tion as to its mass, give a diamagnetic moment comparable 
with the paramagnetic moment due to a ring as given by (26) 
or (27). This will be of importance to us in § 23. 

A third formula (103) (/. ¢. p. 143) may be written 


LN) i eA 0 
m= F-(To— gon) - . 1 1a 


where & is the radius of gyration of a rotating electron and 
hy its mean initial angular velocity of rotation. The formula 
gives the mean magnetic moment due to an assemblage of 
rotating electrons, whose rotation round the axis of symmetry 
is unresisted, whilst that about other axes is strongly resisted, 
the stationary state being supposed to have been reached. 
Apart from theoretical difficulties due to the term ho, this 
term is not sufficiently well defined to render the formula 
amenable to calculation, so that a comparison with experiment 
is impossible. 


Electron Theory of Matter. 191 


Lastly, J. J. Thomson gives a formula (1) (J. ¢. p. 689), 
equivalent to 


“ae ef eae See 2 
a. Lem \. a P') «tts 


where ihe motion of the electron in its orbit is supposed to be 
resisted by a force equal to mx times the velocity. In con- 
sequence p is supposed to diminish. To obtain definite results 
integrate by parts, and let P, be the initial value of p ; we get 


Aé : e a 
: = { Bes —p bee ( “p'at } ae (Po —p*), (31) 


since p?<p,” in the integral. 

This result shows that the paramagnetism due to dissi- 
pation of energy from the moving electron is at most of the 
order of magnitude of p,, as given by (26) or (27). 

From this discussion we conclude that the formule (28) 
and (30), although from their form capable of accounting for 
the paramagnetism of iron, can be regarded as little more 
than empirical ; because the very terms, which make them 


capable of doing so, the factor me in (28), and the term ho 


in (30), are too ill-defined to serve as the basis for numerical 
calculations. On the other hand, the formule (29) and (381), 
which are free from theoretical objections, will only account 
for diamagnetism. The formule (26) and (27) again can 
account for the paramagnetism of iron, but cannot explain 
diamagnetism, because the term p,, which gives diamagnetism, 
is far too small to counteract the term ps, which gives para- 
magnetism ; thus these formule, although successful to the 
extent of accounting for the paramagnetism of iron, and 
therefore superior to the others hitherto proposed, are 
incomplete. The remedy has been already indicated in the 
remarks made respecting (29) : the paramagnetic moment pz 
ean be balanced, in whole or part, by the diamagnetic moment 
due to a charged sphere of either sign and of atomic dimen- 
sions, set in motion by the induction set up in starting the 
magnetic field. 

§ 23. We are thus led to consider the following system:— 

(3) A single ring of negative electrons inside a rigid, but free, 
positive sphere of uniform density (type of J. J. Thomson). 

Itis not difficult to prove that the tilting of the ring due to 
the radial magnetic force produces no couple tending to rotate 
the sphere; this couple is due entirely to the inductive effect 


192 Mr. G. A. Schott on the 


of the changing field at starting, which is partly external, 
partly due to the changes in radius and velocity of the ring 
studied in § 18 and represented by equations of the type (19). 

It is not difficult to see that the part of the couple due to 
the ring bears to the part directly due to the external field a 
ratio, which is of the same order of smallness as the ratio 
pit, In § 20. The corresponding parts of the magnetic 
field due to the sphere bear to each other a ratio of the same 
degree of smallness, and therefore also do the resulting parts 
of the magnetic moment of the sphere. The large part we 
shall find to be of the order js, the small part of the order 
p,; the latter we shall neglect. 

For the same reason we neglect the moment p,, and the 
. change in it due to the reaction of the sphere; and, in 
estimating the change produced in p, by the sphere, we take 
into account only the rotation due to the external field. This 
we shall now consider. 

Let L be the resulting couple, and © the corresponding 
angular velocity ; then 


L 
— rE 


where I is the moment of inertia’of the sphere. 

Divide the sphere into elementary rings with the diameter 
parallel to the external magnetic force h as axis ; let dV be 
the volume of a ring, p’ its radius, e the uniform electric 
density, and e’ the charge of the sphere. The induced 
electric force EH acts along the ring and is by Faraday’s Law 


co) 


: Pha) y p> 
given by 27p'H=— = ie (mph), so that H= — 5, Hence 


also 
eb? - 


5 iw a 
Le atigves 2d — 
{ da 2c | ie DC bs 


and el? bi 

oO =—_ Bel h, ° : 2 4 A : (82) 
the sphere being supposed at rest in the absence of an 
external field. 

§ 24. We shall now study the field due to the sphere. In 
the first place, the external field is the same as that due to an 
infinitely small magnet at the centre, whose moment is 
given by 

e’b? e'2h* 
ee a 6 


Electron Theory of Matter. 193 
Hence the sphere has a specific moment given by 
e*h* 


Para aaa ge Aare IN ° ° (33) 


which is diamagnetic and in general of the order otf 
magnitude of po. 
§ 25. In the diagram E is the position of the ith electron of 


the ring at time ¢; its azimuth is 6=ot+6+ ae The 


direction of the external field, h, which by (32) is fs that 
of Q, lies in the plane «Oz, and makes the angle 6 with Oz. 


Fic. i: 


The component angular velocities of the sphere therefore are 
QO, cos 8 along Oz, and Osin @ along Ow. Since we require 
only the change in p, we require only that component of the 
magnetic force due to the sphere, which gives an axial 
mechanical force, that is, we require only the radial magnetic 
force. Since the angle between the radius OE and the angular 
velocity sin @ is 2 while © cos @ is perpendicular to OH, 
the outward radial magnetic force is entirely due to Q sin 9 
and is equal to 


Co! 2) 2 rig 
a (1 AE O sin @ cos ¢ *. 


Therefore the axial mechanical force is the real part of 
eb 


sf. ZT 1— is )B el sin @ exp. i(ot+d+ =) (34) 


It must be added to the right-hand side of the third of 
equations (24) and is of the opposite sign, as is to be expected 
from Lenz’s Law. 


* A. H. Bucherer, Phys. Ztsch. vol. vi. p. 269. 
Phil. Mag. 8. 6. Vol. 15: No. 85. Jan. 1908. Q 


194 Mr. G. A. Schott on the 


2 2 
§ 26. Proceeding as in §19, we put P= “f, T= as 
S=0, and e?=v’e?, and by (34) obtain 


veb 3p”\ 2 np? Gog 


Pe Veby. Bp Val ene ee 
p= pi Bel | pe J 3(3 + 8?) tans e (35) 


The resulting moment of the system is p=p2+p 33 hence by 
(33) and (35) we get 


p=—Srar+ { 1 Sor(1— 5} aera OO 
The greatest value of p is given by putting I=, when 
it reduces to p, as given by (27); as I diminishes p 
diminishes also ; when IJ takes its least value, we corre- 
sponding to purely electromagnetic inertia, we have 


Aa ap \p on vg } 


When p=0 the expression (36) is obviously negative ; 
when p=d it reduces to 


v*e7h* 2v7eb 8nb? T 
- ar + | 1- Spay lata ay: + G8) 


which is certainly positive because ach tan = > : dl 
RENO dp 


and sr ay < F- Similarly a is positive between these 


limits ; hence p vanishes once, and only once, as p increases 
from zero to its greatest value 6. This means that for a sphere 
of given mass and radius, and for a ring of a given number 
of electrons and given velocity, there is a critical radius, such 
that the system is paramagnetic when the radius of its ring is 
larger, and diamagnetic when smaller. Thus the system of a 
sphere and a single ring can account qualitatively for both 
paramagnetism and diamagnetism. 

§ 27. The magnetic moment is represented by the curves 
in fig. 2, which correspond to the four limiting cases :— 


I. A sphere of infinite mass, and a ring of two, practically 


few elect c v7 e"b oe: Sn 7 16 
very few electrons 5 jag”: 5g: gee 


Electron Theory of Matter. 195 
Fig. 2.—Curve For SPECIFIC MOMENT OF SpHERE AND Rina. 


I. Fixed sphere, few electrons. Mass of sphere infinitel 
If. Fixed sphere, many electrons. great, 5 i 


III. Movable sphere, few electrons Mass of sphere purel 
IV. Movable sphere, many electrons. Oe tiene 


| JRE ee 
) (TSR 
ee abi ter sei Hd 


196 Mr. G. A. Schott on the 


II. A sphere of infinite mass, and a ring of an infinite, 
practically a large, number of electrons ; 


8n T 
tan — = 7. 


III. A sphere whose mass is purely electromagnetic, that is 
ave” : 
equal to 52h? and a ring of few electrons ; 
web: ak 
rete Nek, 
IV. A sphere whose mass is purely electromagnetic, and a 
ring of many electrons. 


The following conclusions can be drawn from these 
eurves :— 


(1) In similar sytems both paramagnetism and dia- 
magnetism, as measured by the specific moment, are 
proportional to the volume. 

(2) Increasing the mass of the sphere increases para- 
magnetism, and diminishes the critical radius. 

(3) Increasing the radius of the ring increases para- 
magnetism. 

(4) Increasing the number of electrons in the ring 
diminishes paramagnetism and increases the critical 
radius. 


§ 28. We must now study the numerical values of the 
moments given by our formulz and compare them with the 
values found by experiment. For this purpose it is most 
convenient to use the atomic susceptibilities of the elements; 
St. Meyer* gives a table referred to one gram atom per 
litre with a unit for & equal to10-® If we take 10~™ gr. to 
be the mass of the atom of hydrogen, we find the number of 
atoms in one ¢.c. of a solution or powder containing one gram 
per litre to be 1071; hence we must multiply Meyer’s numbers 
bye 

‘ahugeins the weak iron amalgam studied by Nagaoka (§ 21) 
contained 4.10?° atoms per c.c. and had a susceptibility 0013. 
This gives p=3'2.10-% for iron. Meyer’s table gives 
p=—2:0.10-*% for bismuth ; all other elements have inter- 
mediate values with the possible exception of erbium, which 
Meyer estimates to be four times as magnetic as iron when 
pure, corresponding to p=1:3.10—*%. The greatest para- 


* 0 ¢, ee p20. 


Electron Theory of ALatter. 17 


magnetic moment thus is 16,000, perhaps 65,000, times 
as great as the greatest diamagnetic moment, ‘The curves 
give the extreme values 1°8. fee Or 1, b?, in the ratio 
10:1. 

By taking 6 a few times the conventional atomic radius, 
10-8 em. (or by choosing several rings) the greatest para- 
magnetic moment can easily be brought within the range of 
the system, which so far is quite satisfactory. We cannot, 
however explain diamagnetism so easily, although the margin 
available is very large, for the diagram shows that our single 
ring system cannot possibly be diamagnetic unless the radius 


of the ring is less then the critical radius, the greatest value 
of which is 0°72. 6 


Now 2 7 eannot be chosen at will, because the ‘equation of 


steady motion (2) must be satisfied. In the present case of 
2 
ne 


a neutral system we have v=n, P= Fs P 3 thus (2) becomes 


> bmp (14+8?)K—V 


nv Vv 


The last term is positive; in fact K is siightly greater 
n 
than 9 logn, and: in the series V Ja..,{@st+ DP} 
(2s+1)7 
an 
in this value for the cotangent, we get a series whose value 


is equal to = {g?—(1 4-8”) log ¥1—6?}. Hence 


anne 7 4 
positive, because B<1, and (s+4) cot a5 putting 


14+67)K-V_ 1+/f" iw E 
ies ana a {log (n,/1—?) —B?} >0 


for all values of 8 which are actually possible with radiation 
small enough for permanence. 


Substituting the value : = 0°72 in (39), we find 


logn < 20h, < 2:0, n<8, 
on the assumption that 6? is negligibly small, which is 
certainly true for such small values of n. Hence we conclude 
that no single ring system can be diamagnetic unless the 
number of electrons is less than 8. 
This number obviously does not give sufficient margin to 
account for the large number of diamagnetic elements know n, 


198 MNS OX Toraking oa Why 


even if there were no other objections. Hence it becomes 
necessary to examine systems with more than one ring. 

§ 29. The effect of the presence of other rings besides the 
one considered is twofold :—(1) Hach ring contributes a term 
to the resultant specific moment of the system, so that the 
second terms in (36) and (38) must be replaced by sums of 
terms, one for each ring. (2) Hach ring modifies the con- , 
trolling field of every other ring, partly on account of its 

i 
| 


own steady motion, partly on account of its disturbance ; the 
result is that the quantities T— a S and R in equations (24) 
and (25) are altered. In particular T——, which is zero in 


our models, becomes negative, and the rings become linked 
together. In consequence the total moment due to all the 

rings is by no means the sum of the moments they would | 
contribute if independent of each other. When the linking - 
is very loose the total moment is paramagnetic, but less than 
the sum of the separate moments, possibly less than that of a 
single ring; so that it may very well happen that the resulting 
moment of the system of sphere and rings is negative for | 
certain arrangements of rings. The problem is, however, too | 
complicated to be pursued further here, but it does suggest 

that magnetic properties are functions of the structure of the 

atom, depending on the number and arrangement of its 
electrons, just as directly as do the atomic weight and atomic 

volume. 


XIII. Note on the Focometry of a Concave Lens. / 
By J. A. Toxins, A.2t.C.Se. (Lond.)*. 7 Jf 


N a recent number of the Philosophical Magazine / 
Prof. Anderson described a simple method for the de- 
termination of the focal length of a concave lens in which f 
is calculated from observed values of u and »v. 
The following methods, in which the focal length of a 
concave lens is measured directly, occurred to the writer 
after reading Prof. Anderson’s article. They correspond to 
two well-known methods which are employed in the case of a 
convex lens. They do not appear, however, to have been 
previously described. 
I. Light from the cross-wires C (fig. 1) of an optical bench 
is brought to a focus at F by means of a convex lens L,. The 
concave lens L is then placed in the position shown and 


* Communicated by the Author. 
Tt Phil, Mag. June 1906. 


Focometry of a Concave Lens. 199 


adjusted until the rays reflected by the plane mirror M, passing 
through the lenses again, form an image of the cross-wires on 
the screen carrying them. 

Big: 1: 


WV ° 


F is then at the principal focus of L and the distance 
between L and I is the focal length. 


II. As in the first method, light from the cross-wires C 
(fig. 2) is brought toa focus at F. The concave lens L is 


Fig. 2. 


then inserted and adjusted until an image of the cross-wires 
is formed in the focal plane of the telescope T, which has 
been previously focussed for parallel rays. The distance 
between L and F is the focal length as before. 

III. In this method a distant object is viewed through the 
concave lens L by means of the telescope T (fig. 3). The 


Fig. 3. 


—— 
—_ oo 


— 


— — 
_—— < 
_— 


—— 


eh ete es ee, 


latter will therefore be focussed as for an object at F, the 
principal focus of L. 


or 
f 7 
q 


200 Notices respecting New Books. 


To find the position of F, the lens L is removed and a 
screen § is adjusted until an image of it is formed in the 
focal plane of the telescope T. As in the previous methods, 
the focal length required is the distance between L and F. 

In this experiment an auxiliary convex lens L, may be 
employed, if necessary, so that F may be brought to within a 
short distance of the telescope. 

Of these methods II. and III. appear to be more accurate 
than I. 


Technical College, Bradford. 
Oct. 1907. 


XIV. Notices respecting New Books. 


Recherches sur les Propriétés Optiques des Solutions et des Corps 
Dissous. Par M. C. Catneveav. Paris: Gauthier-Villars. 
O07. Pp. 190. 


(eee memoir forms the thesis presented by the author for his 

Doctorate to the University of Paris. It contains the results 
of numerous determinations of the refractive index and dispersive 
power of various solutions (aqueous and others), and the conclu- 
sions deduced by the author from the results of his experiments. 
The apparatus employed by him consisted of a modified form of 
Feéry refractometer. It is interesting to note that according to 
the author’s researches there is little to choose between the law 
of Gladstene and that of Lorentz, although the latter appears to 
give rather more accurate results when dispersion is taken into 
account. Students of physical optics will find a great deal to 
interest them in this valuable contribution to our knowledge of 
the optical properties of solutions. 


Rivrsta di Scienza. Anno I (1907), No. 1. Bologna: Nicola 
Zanichelli. 1907. Pp. 192. 


Tue first issue of this new periodical, which is described as an 
“ International Review of Scientific Synthesis,” contains the 
following articles :—‘ La mécanique classique et ses approxima- 
tions successives,” by E. Picard; “Zur modernen Energetik,” 
by W. Ostwald ; ‘‘ Problemi di chimica organica,” by G. Ciami- 
cian; “ 1] concetto di specie in biologia,” by F. Raffaele; “ Die 
natiirliche Zuchtwahl,’ by H. E. Ziegler; “Il carattere delle 
legei economiche,” by C. Supino; “ Impartiality in History,” by 
W. Cunningham; ‘“ Questions pédagogiques,” by J. Tannery. 
In addition to these, there are reviews (in Italian) of various 
scientific works, and of recent progress in various departments 
of science. The scope of the new periodical, it will be seen, is 


Geological Society. 201 


very wide, and from the list of future contributors it appears that 
the editors have been successful in securing the cooperation of 
many distinguished men of science from many lands, each of 
whom writes in his own language. We wish this high-class 
international publication every success, and hope that it will be 
instrumental in counteracting some of the evils which arise from 
over-specialisation. 


Solubilities of Inorganic and Organic Substances. By A. SEIDELL, 
Ph.D. New York: D. Van Nostrand Co. London: Crosby 
Lockwood & Son. 1907. Pp. 368, 


Tu1s important work of reference may be regarded as taking the 
place of Professor A. M. Comey’s ‘ Dictionary of Chemical 
Solubilities,’ published in 1894. Its compilation must have 
involved a very large amount of labour, and chemists will feel 
grateful to the author for the care and devotion which he has 
brought to bear on his somewhat formidable task. The substances 
are arranged in alphabetical order, and in addition an excellent 
index is provided at the end of the book. Great care has evidently 
been taken by the author to select the most reliable data in cases 
where a number of determinations was available. In warmly 
commending this useful work to the attention of chemists and 
physicists, we should like to draw their attention to the following 
extract from the author’s Preface :—“A glance through the pages 
of this book wiil show the incompleteness of the data for many of 
the most common chemical compounds. Furthermore, many of 
the results given are of doubtful accuracy, although the best 
available. Itis hoped, therefore, that a realisation of the present 
incomplete state of our information concerning solubilities as 
evidenced in these pages will stimulate investigations of many of 
those substances which have hitherto been studied iucompletely or 
not at all.” 


XV. Proceedings of Learned Societies. 


GEOLOGICAL SOCIETY. 
[Continued from vol. xiv. p. 787. ] 
November 6th, 1907.—Sir Archibald Geikie, K.C.B., D.C.L., 8e.D., 
Sec.R.S., President, in the Chair. 
ae following communications were read :— 
1. ‘On a Collection of Fossil Plants from South Africa.’ By 
Prof. Albert Charles Seward, M.A., F.R.S., F.G.S. 
2. ‘ Permo-Carboniferous Plants from Vereeniging (South Africa).’ 
By Prof. Albert Charles Seward, M.A., F.R.S., F.G.8., and Thomas 
Nicholas Leslie, F.G.S. 


3. ‘On the Structure and Relations of the Laurentian System of 
Canada.’ By Prof. Frank Dawson Adams, D.Sc., F.R.S., F.G.S. 

This paper contains an outline of the results of the examination 
by Dr. Barlow and the author of an area of 4200 square miles, 


Phil. Mag. Sr. 6. Vol. 15. No. 85. Jan. 1908. E 


202 Geological Soctety :— 


comprised within the Haliburton and Bancroft sheets of the Ontario 
& Quebec series of maps. The paper opens with a short account 
of Logan’s work in the original Laurentian area. The main con- 
clusions reached by the author may be thus summarized :—(1) The 
Laurentian System of Sir William Logan consists of a very ancient 
series of sedimentary strata, largely limestones, invaded by great 
volumes of granite in the form of bathyliths ; (2) This sedimentary 
series is one of the most important developments of the pre- 
Cambrian rocks in North America, it presents the greatest body — 
of pre-Cambrian limestones on the continent, and it is best desig- 
nated as the Grenville Series; (3) The invading masses of 
granite are of enormous extent: they possess a more or less 
distinct gneissose structure, due to the movements of the magma, 
which developed a fiuidal and, in the later stages of intrusion, a 
protoclastic structure in the rock; (4) The granite-gneiss of the 
bathyliths not only arched up the invaded strata into a series of 
domes, but ‘stoped’ out portions of the sides and lower surface 
of the arches, the fragments torn off from walls and roof by the 
invading granite being found scattered throughout the mass of the 
invading rock: this ‘stoping,’ however, probably developed only a 
small part of the space which the granite now occupies; (5) The 
invading granite not only exerted a mechanical action upon the 
invaded strata, but also gave rise to a variety of metamorphic 
products, among others amphibolite produced by its action in the 
limestone, which accounts for the fact that while the invaded strata 
are chiefly limestone, the fragments of the latter, where found in 
the granite, consist of amphibolite; (6) The invading bathyliths 
and allied intrusions of granite occupy the greater part of the great 
Northern Protaxis of Canada, which has an area of approximately 
2,000,000 square miles. It has, therefore, been considered advisable 
to restrict the name Laurentian to this great development 
of the ‘Fundamental Gneiss,’ which, although intrusive into the 
Grenville Series, nevertheless underlies and supports it; (7) The 
relation of the Grenville Series, which forms the base of the 
sedimentary portion of the geological column in Eastern Canada, to 
the Huronian and Keewatin Series, which are the oldest stratified 
rocks in the western part of the Protaxis, has yet to be determined, 
the two not having so far been found in contact; nowhere, more- 
over, either east or west, has the original basement on which the 
first sediments were laid down been discovered ; these are every- 
where torn to pieces by the granite-intrusions of the Laurentian. 


November 20th.—Sir Archibald Geikie, K.C.B., D.C.L., Se.D., 
Sec. R.S., President, in the Chair, 

The following communications were read :— 

1. ‘Glacial Beds of Cambrian Age in South Australia.’ By 
the Rev. Walter Howchin, F.G.8., Lecturer in the University of 
Adelaide. 

The known extension of the beds in question is 460 miles from 
north to south (Onkaparenga River to Willouran Range). The 


Glacial Beds of Cambrian Age in Australia. 203 


greatest width across the strata between Port Augusta, at the 
head of Spencer’s Gulf, in an easterly direction to the Barrier 
Ranges of New South Wales, is about 250 miles. The beds occur 
as part of a great conformable series, in the upper part of which 
Cambrian fossils have been found. ‘The rocks above the Glacial 
Beds are mainly purple slates and limestones; below they are 
quartzites, clay-slates, and phyllites, passing into basal grits and 
conglomerates, resting on a pre-Cambrian complex. ‘The beds 
consist of a groundmass of unstratified indurated mudstone, more 
or less gritty, carrying angular, subangular, and rounded boulders, 
up to 11 feet in diameter. In most sections there are more or less 
regularly-stratified bands. The thickness of the glacial series has 
been proved up to 1500 feet. The commonest rock-type among 
the boulders is a close-grained quartzite; but gneiss, porphyry, 
granite, schistose quartz, basic rocks, graphic granite, mica-schist, 
and siliceous limestone occur. The discovery of ice-scratched 
boulders has placed the origin of the beds, according to the author, 
beyond doubt. The strie are often as distinct and fresh-looking as 
those occurring in a Pleistocene boulder-clay. Up to the present, 
eighty definitely-glaciated boulders have been secured, besides the 
known occurrence of other erratics too large for removal. Under 
strong pressure and movement in their bed, some of the boulders 
exhibit evidences of abrasion; but this produces features which 
eannot well be confounded with those due to glaciation. The 
pressure that has induced cleavage has caused the elongated 
boulders to revolve partly in their bed and place their long 
axes parallel to the cleavage-planes. In this movement, some of 
the stones have become slightly distorted, and many show the 
effect of fracture in the form of pseudo-striation on exposed 
surfaces. The lines, however, are of equal size and depth, and 
parallel to each other over wide surfaces ; while the glacial strie 
are generally patchy in their occurrence, of varying intensity, 
and divergent in direction. A series of illustrative sections are 
described. It is considered that Mr. H. P. Woodward’s suggestion, 
that the ‘boulder-clay’ had its origin from ‘floating ice,’ appears to 
be most in accordance with facts. The interbedded slates and lime- 
stones may possibly indicate the occurrence of interglacial conditions. 


2. ‘On a Formation known as “ Glacial Beds of Cambrian Age ” 
in South Australia.’ By H. Basedow and J. D. Iliffe. 


Some 8 miles south of Adelaide a typical exposure of the con- 
glomerate is bounded to the east by a series of alternating quartzitic 
and argillaceous bands of rock, comprising the central and western 
portions of a fan-fold, partly cut off by a fault. Further evidence of 
stress in this margin is given in the fissility, pseudo-ripple-marks, 
contortion and fracture, and obliteration of bedding in the quartzite- 
bands, and in the pinching-out of them into lenticles and false 
pebbles. On the west side the conglomerate is bounded by the 
‘Tapley’s Hill Clay-Slates,’ and there is evidence from the nature 
of the junction-beds that the conglomerate itself is isoclinally folded. 


204 Intelligence and Miscellaneous Articles. 


In that portion of the conglomerate which is adjacent to its 
confines, ‘boulders’ of quartzite are apparently disrupted portions 
of quartzite-bands, since these are in alignment with the truncated 
portions of bands still existing, and are of similar composition. 
The authors are not at present in a position to account for the 
presence in the conglomerate of boulders of rocks foreign to the beds 
that border the conglomerate, or of such as possess markings 
comparable to glacial strie, by their theory of differential earth- 
movements; but they consider that a boulder-bed subjected to 
lateral pressure would probably lend itself to the production of 
‘false pebbles,’ through the disruption of intercalated hard bands 
within itself or on its boundaries. 


XVI. Intelligence and Miscellaneous Articles. 


A THEORY OF THE DISPLACEMENT OF SPECTRAL LINES 
PRODUCED BY PRESSURE.—A CORRECTION. 


Y attention has been drawn by Professor W. J. Humphreys to 


a slip on page 576 of my paper under the above title in volume | 


14 of this Journal. Since the formula for 6A towards the bottom of 
the page is in electrostatic units, the datum e/mc= 1-8 x 107 should 
be replaced by e/m=1:8x10"% The value of 6A/d\, is 8x 10-" 
per atmosphere as stated. 
O. W. RicHARDson. 
Princeton, N. J., 
Dec. 9, 1907. 


Lorp KELVIN. 


Wira the beginning of a new volume we have to mourn the Joss 
of Lord Kelvin. Ever since the year 1871, when Sir William 
Thomson’s name first appeared as one of the Editors of the 
Philosophical Magazine, this journal has been very deeply indebted 
to him for his kindly interest and invaluable advice, as well as for 
many contributions of the highest value. 

Notwithstanding the incessant calls on his time and energy, 
arising from his numerous public duties as well as from his uninter- 
mittent scientific labours, he was never too busy to give his 
ungrudging help to the Magazine whenever it was sought. 

Our readers will doubtless join with us in an expression of 
profound admiration of Lord Kelvin’s intellectual greatness and 
of affectionate regard for the simple kindliness of his nature. 


HONDA, TERADA, & ISITANI. 


Ere. 1. 


Phil. Mag. Ser. 6, Vol. 15, Pl. I. 


Pre 2, 


a 


Phil. Mag. Ser. 6, Vol. 15, Pl. IT. 


. ONDA, TERADA, & ISITANI. 


f 


MAP OF JAPAN Hoxnncé 3 a 
1- 10000000, 
eee FRINCIPAL OBSERVED 
STATIONS. YODATE 
R) 
sr 
43 Aomori 
~ ya Mirano 

s 

s 

= : 

% AYUKAWA 
@) 
Q B S: 
\ 
hg Bs 
a = 
J at W CHOsHi 
MoRorso 
S oO 
‘ an 
% 
G 


g - 


SUSAA1 
Fg US. ‘ 
ef 


2 HOSOSHIMA 


HONDA, TERADA, & ISITANT. Phil. Mag. per. 6, Vol. 15, Pl. EL. 


=) g 5 a 
= oO © 
= : fe} 
1 B 
5 
7 
Paty 
cr) 
IS) 
mT” 
rt) 
a 
re 
Ss 
= 
=~ 
Ss. 
° 
oT € 
E 
= ‘a 
a LS 
vA = 
ide) 
So 
Ou 


HONDA, TERADA, & Phil Mag. Ser. 6, Vol: 15, Pl. IV: 


| 
j 
\ 


rl 
A 
«| 
H 
— 
m 
H 


HONDA, TERADA, & 


Wor ioe. Wl 


6, 


Mag. Ser. 


il 


Ss 
AY 


HONDA, TERADA. & ISITANI. 


Phil. Mag. Ser. 6. Vol. 15. Pl. VIL. 


A.C, & A. EB, Jussvur. 


THE STRUCTURE OF THE ELEMENTs, 


2 
a 
~ 
- 

= 


wey Te 


sXe. 


\ 


Je}. Ru Rh, Pa. 


© 
\ Z 


4 


Ce al a 


2 Ba 


© 


Oy} 


é 
ia 


3\5\5 @ 


24}3\2 Ra. 


’ 


Phil. Mag. Ser. 6, Vol. 15, Pl. VIII. 


i 
N 


\ 
N 


IN RRARN AE a 
AYA 
SANA 
‘a 


LN. 


_——— TT 
os : 
j i 
ea 


ees | 
° 4 
ae ey Ss 30 


“Oo 


Phil. Mag. Ser. 6, Vol. 15, Pl. VIII. 


DioKson, 


6 «PSP bo) R 


10. 
2» 9 
8 
7 
6 


We Ir 1 Tpyoraduay | 
| | 


otent ia bs 


zs! 


rem perakure 


Potential 
Y D 8 9 
_ 
rl 
eee | 3 fees | gs «1 /e 9 onsing 3 Ls B 9 a0 | 201 | 
pas = 
— + 
y) 
yy 
4 y/ 
N 
|| 
| 
ties 


oO 
-10 
-20 
PAS 
~30 


CAMPBELL. Phil. Mag. Ser. 6, Vol. 15, Pl. IX. 


CAMPBELL’S VARIABLE MUTUAL INDUCTANCE. 


Nie 
iW. 7 


THE 
LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES.] 
¢ FEBRUARY 1908. 


XVII. On certain Phenomena exhibited by Small Particles on 
a Nernst Glower. By C. HE. MeEnpDENHALL and L. R. 
INGERSOLL *. 


N the course of some recent work on the determination 
of high temperature melting-points (see Phys. Review, 
July 1907), the present writers noted several rather curious 
phenomena attending the heating of minute particles of 
various metals placed upon the surface of a Nernst glower— 
whose temperature could be controlled with a suitable rheostat 
—and examined with a microscope of low power. These 
phenomena seemed sufficiently novel and interesting to merit 
separate study, and the results of our observations are con- 
tained in the present paper. 


Phenomena of Undercooling of Molten Globules. 


The first phenomena are associated with an undercooling of 
molten globules which is so great as to be in itself worth noting. 
If a small particle of, say, platinum be placed on the glower, 
melted into a globule about 1/10 mm. or 1/20 mm. in diameter, 
and then allowed to cool either slowly or rapidly, it will not 
solidify at the melting temperature, but will remain very 
evidently fluid until a temperature is reached from 50° to 
300° lower than the melting-point, when solidification will 
suddenly occur, accompanied by a quick “ flash ” or brighten- 
ing of the drop. Upon reheating there is no flash and the 
drop melts at its proper melting temperature. The same 


* Communicated by the Authors. 
Phil. Mag. 8. 6. Vol. 15. No. 86. Feb. 1908. Q 


206 Messrs. Mendenhall § Ingersoll on Phenomena 


general effect is observed with gold, palladium, platinum, 
silicon, rhodium, and iridium. 

The only record we can find of the previous observation 
of such an effect is that given in Winkelmann’s Handbuch 
(II. 1, p. 487, 1st. ed.), where the flash is considered to be 
a luminescent phenomenon associated with the giving off of 
absorbed hydrogen by silver, no mention being made of 
undercooling. The article refers to a paper by Dr. T. L. 
Phipson (B. A. Report, 1859, p. 76), who noted such a flash 
in cooling a globule of silver, just at its melting-point. 

It seemed to us worth while to determine more closely 
the conditions which made possible the retention of the fluid 
state at temperatures as much as say 300° below the normal 
melting-point—especially to see if the occlusion of gases 
played any great part in it, and also to see whether the 
brightening of the drop was a luminescent effect connected 
directly with molecular rearrangement—or simply a tem- 
perature effect due to the liberation of the latent heat of 
fusion. Microscopic examination of the particle shows con- 
clusively that the flash accompanies a change of state, for in 
general there is a marked change froma smooth surface often 
showing decided motion, to a rough, corrugated, and very 
evidently solid surface. 

Bearing upon the problem as stated above we have noted 
the following :— 

The temperature at which “ flashing ” occurs is extremely 
variable. In general the smaller the drop and the higher 
its melting-point, the more it can be undercooled before 
flashing occurs ; the difference in this respect between a drop 
*1 mm. in diameter and one *2 mm. in diameter is quite 
noticeable. The more the globule works into and imbeds 
itself in the glower material, 2. e. the more it departs from a 
spherical form, the less it can be undercooled. The possible 
undercooling is also slightly less if the cooling occurs rapidly. 
The melting must be thorough and complete or else no under- 
cooling whatever will occur. Granted thorough melting, 
further heating above the melting-point seems to be without 
influence *. 

The behaviour is essentially the same if the glower is 
placed inan atmosphere of CO,. Flashing also occurs in the 
same way if the globule is heated on a non-conducting clay 
surface, 2.e.a Nernst-glower heater. The phenomenon was 
readily obtained with gold particles on such a heater, and 

* Tapping the support of the glower did not affect the undercooling 


which, considering the minute particles involved, is only what might be 
expected. 


shown by Small Particles on a Nernst Gilower. 207 


also with silver if care was taken to protect the silver from’ 
direct contact with the clay surface. For in the latter case 
it would not form a globule but simply flux into and be 
absorbed by the clay surface, just as was the case when tried 
on the Nernst glower itself. 

The brightness of the flash increases with the extent of 
undercooling. Flashing appears to precede the solidification ; 
with a large drop the flash is noticeably more prolonged than 
with a small one, and very evidently occurs before the 
solidification. 

From these facts we conclude that the phenomenon is 
simply one of exaggerated undercooling made possible because 
we are dealing with a very small mass of metal which assumes 
an approximately spherical shape. Furthermore, that the 
flashing is not a direct luminescent phenomenon but is due 
to the sudden increase in temperature resulting from the 
liberation of the latent heat of fusion ; the longer duration 
of the flash in the larger globules corresponding to the greater 
amount of heat which must be dissipated. Probably, then, 
the previous observations referred to are erroneously classified 
as “luminescent.” 

We have made a number of measurements of the tem- 
perature at which flashing occurs, but these seem without 
interest except to note the maximum degree of undercooling 
which has been observed, namely, about 370°C. for both 
rhodium and platinum. That is to say, rhodium, which 
melts at about 1910°C., has remained molten as low as 
1540°C. ; and platinum, melting at 1745°C., as low as 
1370° C. 

In the case of rhodium we have observed a second anomaly 
at about 1050° C., consisting of an easily reversible change 
in the radiating power of the surface of the drop. Asthe tem- 
perature is lowered the drop becomes at about 1050° C. rather 
suddenly brighter—the change being seen to spread rapidly 
over the globule—and remains so (relative to the glower). 
If the temperature is again raised the reverse change takes 
place. The effect may be due to a molecular rearrangement, 
or to the absorption of some gas, but in either case the 
energy changes invoived must be small, otherwise flashing 
or some similar temporary recalescent change would be 
observable. 


Phenomena of Motion of Particles. 
Quite distinct from these cases of undercooling are a 
number of cases of motion of various metals on the surface 
of the glower parallel to the current flow. The most striking 


Q 2 


208 Messrs. Mendenhall & Ingersoll on Phenomena 


of these is observed when a small particle of cobalt, say, is 
placed on the surface of a glower carrying a direct current. 
When the temperature is raised to about 1200°C. it will 
commence a slow and regular end-over-end rolling motion 
along the glower, which changes in direction when the 
current is reversed, and which is not affected by holding the 
glower vertical, or, indeed, by turning it completely upside 
down. It is regretted that a somewhat careful study of this 
and the allied phenomena has not led as yet to as definite 
and satisfactory an explanation of the cause as might be 
desired ; although of course it may be said at once that, as” 
the glower is an electrolytic conductor, such motions are 
probably related to the other, and little understood, motions 
of matter in an electrolyte which have been observed many 
times (see discussion in Winkelmann) although the connexion 
is not very obvious. Our observations and such conclusions — 
as we have been able to draw may not, however, be without 
interest, and will be discussed under two heads corresponding 
to two distinct, although evidently related, effects which have 
been observed, namely :— 

(a) A rolling or sliding of molten globules of metal on the 
surface of the glower. 

(6) A motion of solid particles of metals and their oxides. 
This is the curious end-over-end rolling above-mentioned 
which will take place with particles of the most irregular 
shape. 

Case a.—A rolling or sliding of molten globules of 
metal has been observed with most of the metals which will 
melt into a globule on the glower ; rhodium perhaps showing 
it the best. The motion may be slow and accompanied by a 
plowing up of the surface of the glower as if the globule were 
trying to embed itself, or—and this is especially true just 
when the substance is first melted—it may be as fast as the 
eye can follow. In direction* it may be either with or 
against the current, according to the metal used, and in every 
case where both can be observed agrees as to direction with 
the motion in the solid state described below (Case 6). With 
gold and platinum the motion is with the current. This 
rolling is very possibly similar, as to cause, to an effect noted, 
we believe by Quincke, in which a globule of mercury placed 
in a tube of acidulated water is observed to move rapidly, 
only in this case against the current. 


* This rolling has also been observed, in the same direction, with 
rhodium on special glowers composed of MgO and Al,O,, instead of the 
oxides of ziroonium and yttrium as is the case with commercial glowers ; 
so that it does not depend on the special constitution of the glower. 


shown by Small Particles on a Nernst Glower. 209 


Case b.—Motions of solid particles of metals and their 
oxides. These effects are so striking that we have been led 
to collect considerable data concerning them. To describe 
in detail a typical case which has already been mentioned :— 
If a sliver of cobalt, say 2 mm. long and ‘2 mm. thick, is 
placed lengthwise on a glower carrying direct current, at a 
temperature below the cobalt melting-point, it will slowly 
rise on end and continue turning over till it again lies flat, 
the process being repeated so that the metal as a whole moves 
along the glower against the current flow ; and the motion 
will continue till the particle stops on the cool glower terminal 
(see fig. 1). A more exaggerated case—and one which 


Fig. 1. 


: i 
<— Motion. VY Fis 
¥ | ¢? 


shows that the forces, such as they are, must be comparatively 
large, if they are assumed to act only on the small area of 
the particle in actual contact with the glower—is that of a 
horseshoe-shaped piece of metal, say 4 mm. in diam., hung 
over the glower, with its plane at right angles to it. This 
will slowly turn (in the same sense as before) until its prongs 
point upward when it usually becomes unstable and falls off. 

In an attempt to explain these actions we have noted the 
following facts :— 

(1) The motion always reverses with reversal of current, 
and this is true no matter what the position of the particle, 
whether it is just rising from a horizontal position, or standing 
vertical. For some metals the motion is with, for others 
against, the direction of the current, as ordinarily defined. 
In Table I., column headed “direction,” a + sign means 
motion wrth the current, a — sign motion against the current. 

(2) The rapidity of the.motion under similar conditions 
varies greatly in different metals. In the column headed 
** magnitude,” the numbers are roughly proportional to the 
rapidity of the motion, cobalt and copper being by far the 
best in this respect. The actual velocity of progression along 
the glower of a small piece of either of these metals would in a 
favourable case be a matter of perhaps a centimetre per 
minute. With most metals, however, the motion is so slow 
that it is practically impossible to get a progressive end-over- 
end translation of the particle along the glower ; so in taking 


210 Messrs. Mendenhall & Ingersoll 0 on Phenomena 


the observations which form Table I. the effect was considered 
to exist when a particle showed a definite tendency to tip 
over one way or the other, the direction reversing with 
reversal of current. 

_ (8) The size and shape of the particles i is not important, 
most irregular shaped particles rolling as readily as nearly 
round ones; in general small particles have a more rapid 
angular motion, though they may not cover as much ground. 
Long flat pieces usually move very readily even when touching 
the glower only at the tip. The forces involved are not at 
all proportional to the size of the particle, but are sufficient 
to move small particles up a vertical glower against gravity. 
The fact that particles will also move along the under side of 
the glower, and that they are found to be more or less firmly 
stuck to the glower when it is allowed to cool, indicates that 
there is incipient fluxing or sticking of the metal to the 
glower even at temperatures below the melting-point. 

(4) The rapidity of the motion increases both with the 
temperature of the glower and the current density in it. By 
superposing a direct on an alternating current through the 
glower (the latter doing most of the heating, and the former 
serving to direct the motion), we have inia measure separated 
current and temperature influence, and, find that the 
current is the more important of the two. If the current 
density in the glower is sufficient, the motion will take’place 
-at a temperature a good deal below the melting-point of the 
metal: for example, cobalt, which melts at say 1500° C., has 
been observed to move at 1150°C. In general the higher the 
melting-point of the metal, the higher the inp 
necessary to produce noticeable motion. 

(5) We have attempted to influence the motion by pro- 
ducing electrostatic and electromagnetic fields near the 
particle on the glower, but without success. Nor could the 
motion be produced on a hot piece of glower material not 
carrying a current, but having electrostatic and magnetic 
fields along and at right angles to it in such a way as to 
simulate the effects of a current. Negative results were also 
obtained when particles supported on mica were brought up 
underneath a running glower to within microscopic distances 
of its surface. 

(6) We have not been able to produce the effect on any 
metallic conductor, nor have we succeeded in observing ‘it 
on any electrolytic conductor which differed much in com- 
position from the ordinary Nernst glower, the trouble being 
either that these became molten and sticky at too low a 
temperature and with too small a current density (for 


shown by Smali Particles on a Nernst Glower. 211- 


example glass or common salt), or require too high a tempe- 
rature (for example MgO). The phenomena were, however, 
readily produced on a glower made of pure zirconium oxide. 
No motion will occur over a thin layer of MgO coated on a 
glower surface,—probably because under these circumstances 
the MgO carries practically no current. 

(7) A particle of a given metal moves with difficulty if 
at all over a part of the glower which has been fluxed* with 
the same metal. The motion of another metal over this part 
of the surface is in general not so much altered by this 
fluxing of the first. Heating for a short time at a high 
temperature will remove these effects produced by the 
fluxing in of any metal, probably because the metal is 
vaporized off in this process. | 
_ (8) Placing the glower in an atmosphere of CO, or ina one- 
millimetre vacuum does not perceptibly alter the effect. Nor | 
does a jet of oxygen blown at the glower or particle make 
any difference, unless it be strong enough to cool the particle 
considerably, especially at its point of contact with the 
olower. | ! : ea Ge | 

(9) Oxides iof many metals (see Table, p. 212) show a 
similar motion, foe | r 

(10) In the table the approximate atomic weights. of 
zirconium and yttrium (whose oxides are the principal con- 
stituents of the glower) are inserted, and it will be noted 
that, with two exceptions (boron and ruthenium), all metals 
having an atomic weight greater than these metals move with 
the current, and those with lower atomic weights move against 
the current. In order to test further this rather striking, but 
probably accidental arrangement, we have made many attempts 
to observe the motion on glowers made of oxides of metals 
having much smaller atomic weights—especially MgO and 
Al,O,—in order to see whether the motion of cobalt, for 
example, would be reversed on such a glower. So far we 
have been entirely unsuccessful in observing anything but 
the motion of molten globules (above referred to) on such a. 
glower, and the metals which make molten globules have 
almost all higher atomic weights than the zirconium of the 
ordinary glower, so no reversal would be expected in this 
case. But this negative result is by no means conclusive, 
for, in the first place, it is extremely difficult to make a 
glower of Mg and Al oxides which will run on direct current 

* With most metals (the less oxidizable metals such as gold, platinum, 
rhodium, and iridium are exceptions) heating to a sufficiently high 


temperature results in the metal soaking into and fluxing witb the 
glower material. 


212 Messrs. Mendenhall & Ingersoll on Phenomena 


TABLE I, 
Motion when Motion when 
bee aes solid. mw oS solid. 
° © i) ® 
es das 
Substance. | .2 = 8 = 2 2 S Substance, = = i = g 5 3 
x) , 25) 5a o Ss =: 
SP : Ag 8 a 62 z FS) 8 Es 
A A= SJ =< = S 
Boron ...... 109 + 3 |) Rhodium ...| 103] + | + | 3 
Megnesium | 24 | Palladium...| 106 2 
Oxide...... ® 2 | Sibvers.t53614: 108 +(?) 
on ianiie Mmmm Pinay 8 | Seth 8. 137 
Titanium ...| 48 _ = | 5) Oxide. ses + 1 
Chromium... 52 — | 1 | Tantalum | 183 | 
Manganese .| 55 - 1 | aa + . 
heeded fi 5G : Iridium..;,.:.| 193} ... 1 
Oxide...... —~ | 5 |! Platinum ...) 195 te 1 
| Cobalt ...... 587 — | 10 | Gold <i. .s2- 197 | + | +(?) 
Nickel ...... 58:6 hy fis (eS 1 | | 
Copper ...... 63°2 de | = 10 | | 
Yttrium...... 89 | 
Materials of | 
*Zirconium...| 90... Glower. 
105 | 
| Ruthenium. | 102 = | 2 | 


t 


* Zirconium not being, properly speaking, an element, the atomic weight of 
the particular “ fraction” contained in the glower is uncertain to about the 
extent indicated in the table. 


for more than a minute or two, and then only very un- | 
steadily, especially at low temperatures; and, secondly, the 
resistance of these oxides is so high that the current cor- 
responding to a given temperature is very much less than 
with the ordinary glower. The circumstances of observation, 
then, are much less favorable, and the effect to be expected 
much less in amount than with ordinary glowers. 

Besides these cases of mass motion we have noted two 
cases of real or apparent diffusion of metal along the glower. 
The first occurs when a particle of metal (gold, platinum, 
and the less oxidizable metals excepted) is heated until it 
melts and fluxes with the glower. In some cases this means 


shown by Small Particles on a Nernst Glower. 213 


simply a sinking of the metal into the material of the glower 
without apparent motion in either direction, but with some 
metals the process involves vigorous action, and shows a very 
definite tendency to proceed in one direction, depending on 
the current (see Table I.). Fluxing may be carried to a point 
where the conduction in the glower becomes largely metallic 
instead of electrolytic, and that part of the glower in con- 
sequence becomes relatively cool. Continued heating ata high 
temperature will vaporize most of such metals out of the 
glower. | 

The second case, which is possibly one of only apparent 
diffusion, has been observed with platinum, rhodium, and 
palladium, and is probably characteristic of the less oxidizable 
metals. It consists in the formation of a discoloured streak 
or ‘‘ tail” extending out from the drop along the surface of 
the glower and ending in a point. In the cases noted it has 
been directed against the current flow, when direct current 
was used (2. e. opposite to the direction of mass motion), while 
with alternating current it is found on both sides of the 
globule. Inthe case of platinum, it will occur at temperatures 
a hundred degrees or more below the melting-point, although 
the formation is more rapid at higher temperatures. Micro- 
scopic examination of such a platinum trace shows the surface 
of the glower covered with what is apparently a very thin 
layer of platinum for perhaps a millimetre from the glebule. 
Nevertheless, chemical tests kindly made by Professor Lenher 
have so far failed to prove positively that the “ tail ” 7s metallic 
platinum; but the minute amount of material present makes 
chemical examination extremely difficult. 

Undoubtedly these cases of diffusion, inasmuch as they are 
unidirectional and depend on the current, imply the existence 
of forces related to those which must cause the mass motions 
above described, but it is impossible to say at present just 
what this relation is; for there seems to be no fixed relation 
between the directions of diffusion and of mass motion in the 
various cases. 


Conclusion. 


It seems probable that the motions of both solid and molten 
particles are due to the same causes; furthermore, that the 
forces involved, such as they are, are localized at the point 
of contact of the particle and glower. If we consider this 
fact in connexion with that before noted of the sticking of 
the particie to the glower, involving as it does considerable 
forces of cohesion between the metal and the glower, we 
may perhaps suggest a general explanation of the phenomena. 


214 Prof. E. L. Hancock: Effect of Combined 


For some of the glower current may be supposed to pass 
around through the particle, and there would then be, on 
account of the electrolytic character of the glower conduction, 
oxygen given off where the current enters, and absorbed 
where it leaves, the metal, and thus the forces of cohesion 
on the two sides be rendered unequal. The result would be 
a couple tending to turn the particle or globule over in one 
direction. Somewhat similar conditions would exist in the 
case of a particle of oxide placed on the glower, as ali oxides 
conduct more or less at these temperatures. It is not evident 
what determines the direction in which a particle will move, 
nor is there any apparent reason for the curious grouping 
with respect to atomic weights, of the metals moving with, 
and moving against the current, which grouping may indeed 
have little or nothing to do with the fundamental cause of 
the motion. A possible explanation of this grouping, however, 
might be based on the fact that the metals of higher atomic 
weights which can be tested in this way, are the less readily 
oxidizable ones which will form globules when molten, while 
those of lower atomic weights are more readily oxidized and 
flux with the glower when melted. That different metals 
should, under similar conditions, move oppositely is, from 
the view just given, not surprising. There are, however, 
several exceptions (for example, tantalum and barium, which 
are readily oxidizable) which are against this latter way of 
looking at the matter. ni 


Physical Laboratory, University of Wisconsin, 
June 1907. 


XVIII. Lyfect of Combined Stresses on the Elastic Properties 
of Steel. By WH. L. Hancock, Assistant Professor of Applied 
Mechanics, Purdue University, La Fayette, Ind.* | 

[Plates X. & XI.] 
Le the past year tests have been continued under 
the direction of the writer to determine the effect of 
combined stresses on the elastic properties of ‘steel. These 
tests form a part of the general plan for such tests carried 
on in the laboratory for testing materials of Purdue University 

(see Proceedings Amer. Soc. Testing Materials, vol. v. p. 179, 

and vol. vi. p. 295). The tests already made and reported to 

this Society have included tests in tension and compression 
while the material was under torsion. The tests reported in 
this paper were made by first subjecting the material to 
tension or compression, and then while under such stress 
applying certain increments of torsion. In the tension-torsion 


* Communicated by the Author, having been read at the meeting of 
the American Society for Testing Materials on June 20, 1907. 


Stresses on the Elastie Properties of Steel. ras 


tests tensile stresses of 0, 33, 50, 69, 81, and 100 per cent. of 
the normal elastic limit in tension were first applied, and 
while the material was under such tension it was tested in 
torsion. In the compression-torsion tests compressive stresses 
of 0, 33, 50, 83, and 100 per cent. of the normal elastic limit 
in compression were applied, and while the material was under 
such stress it was tested in torsion. A third series of tests 
were made on full-sized steel shafting, by first subjecting 
the material to a certain torque, and then, while under such 
torque, testing it in flexure. 

_ The writer hopes, at an early date, to be able to make a 
general review of all tests made, and to accompany it with a 
proper analysis of all data thus far obtained. In the mean- 
time the tests themselves are reported. 

Materials——The material used in the tension-torsion and 
the compression-torsion tests was a grade of steel tubing 
furnished by the Shelby Steel Tube Co. The tension-torsion 
test-pieces were 32 inches long, one inch outside diameter; 
and 28/32 inch inside diameter. The compression-torsion 
test-pieces were 8 inches long and of the same size and 
thickness as the tension-torsion test-pieces. Simple tension 
tests showed the material to have the following physical 
properties:—Maximum strength 41,000 lbs. per sq. inch; 
elastic limit 21,000 Ibs. per sq. inch and per cent. of elonga- 
tion in eight inches of 32. It was thoroughly annealed and 
of uniform thickness. The material used in the torsion- 
flexure tests consisted of solid nickel and mild carbon-steel 
shafting furnished by the Carnegie Steel Co. The nickel- 
steel was of the same chemical composition as the carbon- 
steel, except that it had about 3 per cent. of nickel. The 
pieces of shafting tested were 5 feet long and turned down 
to 1°5 inches in diameter. These pieces were squared slightly 
at the ends to provide for the application of the torque. 

Method of Test—The method of testing pieces in tension 
while already under torsion has been explained (see Proc. 
Amer. Soc. Test. Mat. vol. v. p. 179). The same apparatus 
and arrangement of apparatus was made use of in testing in 
torsion while under tension. After the desired tensional load 
was applied, sufficient torsional load was put on to overcome | 
the friction of the ball-bearing heads, and then the piece was 
tested in torsion. Elongations were measured with a Johnson 
extensometer and torsional deformations by means of an 
Olsen troptometer. Deformations in both tension and torsion 
were measured on an eight-inch gauge length. To determine 
the torque necessary to overcome the friction of the ball- 
bearing heads, due to the tensional load, the troptometer was 
set at zero, and sufficient sand added to the pails at the ends 


216 Prof, E. L. Hancock: Effect of Combined 


of the arms to cause a perceptible motion. This method 
proved accurate and satisfactory. 

In making the compression-torsion tests, the upper ball- 
bearing head was placed upon the platform of the testing- 
machine with the same side up as when it was on top of the 
machine. The other ball-bearing head was left on the under 
side of the moving head of the testing-machine. In this 
position the compression specimen was inserted, and the same 
set of jaws that applied tension and torsion in the tests just 
described, applied compression when the moving head of the 
machine was lowered, and afterward applied torsion. The 
compression specimens were 8 inches long, and this length, 
after allowing 2°5 inches on each end for insertion in the 
jaws, gave a compression length of 3 inches. Compression 
and torsion were both measured on a two-inch gauge length. 
Deformations were measured with an Olsen compressometer 
and troptometer. The disposition of the specimen in the 
machine and the arrangement of the compressometer are 
shown in Plate X., although this is not a photograph of one 
of the test-pieces and the troptometer is not shown. Torsion 
loads were obtained by adding the required amount of sand 
to the pails attached to the ends of the arms, in a manner 
similar to that used in the tension-torsion tests. After the 
required compressional load was applied, sufficient sand was 
a in the pails to overcome the friction of the ball-bearing 

eads. 

This was indicated by a perceptible movement of the 
troptometer index. In making these compression-torsion 
tests great care was used to avoid any possible column action of 
the specimen. Itis believed by the writer that no such action 
took place in the case of any tests reported in this paper. 

The torsion-flexure tests of nickel- and carbon-steel shafting 
were made by means of specially devised apparatus. The 
same arms used in the preceding tests were attached to the - 
ends of the specimen. The attachment of these arms and the 
general disposition of the specimen with relation to the testing 
machine is shown schematically in Pl. XI. fig.1. The testing- 
machine, upon which the apparatus was mounted, and which 
was used in the flexure part of test, was an ordinary tension 
machine of 20,000 lbs. capacity. The torque was applied by 
means of weights, indicated in the drawing by the arrows P,. 
The plan throughout was, first, the application of a certain 
torque, and then while the shaft was under this torque sub- 
jecting it to a flexure test. While the torque was being 
applied the knife-edge A was removed, allowing the shaft to 
turn freely over the knife-edge B. When the desired torque 
had been applied, the knife-edge B was removed and the 


Stresses on the Elastic Properties of Steel. 217 


knife-edge A inserted. The downward force P, at the end 
of the shaft indicated by A was obtained by allowing the 
desired weight to hang from the end of the arm. The up- 
ward force P,, at the same end A, was obtained by allowing 
the desired weight to hang from a bicycle chain which ran 
over a suitably mounted bicycle wheel and fastened to the other 
end of the arm, in such a way as to give the required vertical 
upward pull, The friction of the wheel and chain were 
negligible. The arm at the end C was the fixed arm. The 
specimens were tested in flexure by applying the loads P at 
the centre by means of the testing-machine. Torsional 
deformations were measured on a gauge-length of 4 feet 
9 inches by means of an Olsen troptometer. Flexural deforma- 
tions were measured with an Olsen deflectometer. After the 
torque had been applied the beam of the testing-machine was 
balanced, eliminating from consideration the weight of the 
cross-beam and the weights used in applying the torsion. 

Results.—The lowering of the torsional elastic limit of steel 
tubing due to the various tensional loads is shown by Pl. XI. 
fig. 2, Curve 1 shows the results of simple torsion tests, Curve 2 
the results of torsion tests while the material was under tension 
to 2/6 the elastic limit in tension. Curves 3, 4,5, and 6 
show the results of torsion tests while the material was under 
tension of 3/6, 4/6, 5/6, and 6/6 respectively the elastic limit 
in tension. Hach curve represents, at least, an average of 
two tests. The results of the compression-torsion tests 
are shown by fig. 3. Curve 1 shows the results of simple 
torsion tests of the material. Curve 2 shows the results of 
torsion tests while the material was under a compression of 
2/6 the elastic limit in compression. Curves 3, 4, and 5 
show the results of torsion tests while the material was under 
compression to 3/6, 5/6, and 6/6 respectively the elastic limit 
in compression. Hach curve represents an average of two or 
more tests. The values of 2/6, 3/6, &. of the elastic limit were 
not in every case exactly noted, but the results used in the tables 
and curves show a discrepancy in only one or two cases. 

The results of the torsion-flexure tests of nickel- and carbon- 
steel are shown by figs. 4 & 5 (Pl. XI.). Fig. 4 shows the 
results of the tests of mild carbon-steel shafting. Curve 1 
shows the result of a flexure test of a piece of the shafting 
when no torsion is applied. Curve 2 shows the results of a 
flexure test of a similar piece of shafting while it was under 
a fibre stress of 22,800 lbs. per sq. inch on the outer fibre. 
Curves 3 and 4 show the results of flexure tests of shafting 
while under a stress on the outer fibre, due to torsion, of 
30,400 and 38,000 lbs. per sq. inch respectively. Fig. 5 
shows the results of tests of nickel-steel shafting. These 


218 Prof. E. L. Hancock: Effect of Combined 


pieces of shafting were the same size as those of carbon- 
steel. Curve 1 shows the result of a simple flexure test. 
Curves 2, 3, 4, and 5 show the results of flexure tests while 
the material was under a torque sufficient to produce a fibre 
stress in the outer fibre of 15,200, 22,800, 30,400, and 
38,000 lbs. per sq. inch, respectively. | 

The change in the elastic limit in tension, torsion, and 
flexure due to the presence of another stress—torsion, tension, 
compression, and torsion—is shown in fig. 6. The abscisse 
represent the amount of the particular stress initially applied, 
and the ordinates the portion of the elastic limit (normal 
elastic limit) obtained in tension, torsion, and flexure as a 
result of the initial stress. All the results that have been 
obtained by the writer are shown in the diagram, so that, not 
only are the tests made during the past year represented, but 
also all those that have been reported heretofore. These points 
are averaged by lines 1, 2, 3, and 4 in fig. 7. The figure 
needs no explanation. ; 

The change in the unit stress at the elastic limit, part of 
deformation at elastic limit, and the modulus of elasticity 
are shown by fig. 8. The amount of stress applied initially 
is represented on the horizontal axis, and the corresponding 
change produced by this initial stress is shown on the vertical 
axis. It is seen that the unit stress at the elastic limit, 
deformation, and modulus of elasticity are all lowered by the 
stress initially applied. 

The results obtained from the tests of steel tubing, in 
torsion while under tension, are shown in Table I. The table 
shows that an amount of tension applied, equal to 0, 33, 50, 
69, 81, and 100 per cent. of the elastic limit in tension, pro- 
duces an elastic limit in torsion of 100, 68, 60, 48, 31, and 25 
per cent. of the normal elastic limit in torsion respectively. 
The same table also gives the results of the compression- 
torsion tests, showing that an amount of compression applied, 
equal to 0, 33, 50, 83, and 100 per cent. of the normal elastic ~ 
limit in compression, produces an elastic limit in torsion of 
100, 73, 42, 36, and 27 per cent. of the normal elastic limit in 
torsion respectively. Itis seen from this table, as well as from 
fig. 8, that the unit stress and unit strain at the elastic 
limit are lowered considerably due to the combined stresses, 
and that the modulus of elasticity is also lowered, but to less 
degree. That is, the strength of the material suffers most 
when combined stresses are acting. 

The results of the flexure tests on steel shafting are shown 
in Table II. Here the nickel-steel seems to withstand the 
combined stresses better than the mild carbon-steel. ‘This is 
seen by comparing the percentage of normal elastic limit, for 


Stresses on the Elastic Properties of Steel. 219 


the carbon-steel 100, 87, and 67, and for the nickel-steel 100, 
90, 83, 77, and 70. The percentage of normal deflexion at 
the centre for carbon-steel is seen to be 100, 92, and 79, while 
for nickel-steel it is 100, 98, 100, 99, and 89. The modulus 
of elasticity is changed but little in either case, that is, the 
combined stresses have little effect upon the stiffness of the 
shafting, 

Table III. gives the bending moments in both torsion and 
flexure for both carbon- and nickel-steel shafting. Table IV. 
gives the results of computing unit stresses by the various 
formule for combined stresses. The first and second columns 
give the greatest tension and shear respectively, on any internal 
plane, when no account is taken of the change of form due to 
the acting stresses. The third and fourth columns give the 
greatest tension and shear respectively, on any internal plane, 
when it is assumed that Poisson’s ratio is 1/4, The fifth and 
sixth columns give the greatest tension and shear respectively, 
on any internal plane, when it is assumed that Poisson’s ratio 
is 1/3. The last two columns give the unit tension and unit 
shear on the outer fibre applied to the specimen during the 
test. In the case of the compression-torsion tests the first, 
third, fifth, and seventh columns give unit compression instead 
of unit tension. (For the formule from which these results 
have been computed, reference is made to a former com- 
munication of the author, see Proceedings of Amer. Soc. for 
Test. Mat. vol. vi. p. 295; Phil. Mag. vol. xii. p. 418, 1906.) 


Conclusions. 


The results of tests reported in this paper show :— 

(1) That combined tension and torsion lowers the elastic 
limit in torsion, as shown in fig. 7. 

(2) That combined compression and torsion lowers the 
elastic limit in torsion in about the same way as in the case 
of tension-torsion. 

(3) That combined torsion and flexure lowers the elastic 
limit of the materials in flexure, as shown in figs. 6 and 
7, and that this lowering seems to be less than for any 
other case of combined stresses thus far investigated. This 
lowering is slightly less for the nickel-steel shafting than for 
the carbon-steel shafting. 

(4) That the unit deformation at the elastic limit of the 
tubing in tension-torsion and compression-torsion is lowered, 
as shown in fig. 8. | 

(5) That the deflexion of the steel shafting is made less 
when the torsion is increased, the change being greater in the 
ease of the carbon-steel than in the case of the nickel-steel. 
(Deflexion here means deflexion at the elastic limit.) A 


ee 


220 Prof. E. L. Hancock: Effect of Combined 


comparison of figs. 4 and 5 shows that, within the limits 
of elasticity, the amount of deflexion for any given load is 
about the same for the carbon-steel as for the nickel-steel. 

(6) That the modulus of elasticity, both in torsion and 
flexure, is lowered slightly by the presence of combined 
stresses (fig. 8). | 

(7) That the maximum shear on any internal plane should 
control in design of parts subjected to combined stresses. In 
Table IV. the computed maximum tension and maximum 
shear that is greater than the tensile or shearing strength of 
the material has been underlined. It is seen that in only a 
few cases does the maximum tension exceed the tensile 
strength of the material, while the computed maximum shear 
is, in very many cases, greater than the shearing strength of 
the material. These results are generally in accord with 
results previously obtained by the writer. 


TABLE [, 


Results of Tests of Steel Tubing, Tension-Torsion, and 
Compression-Torsion. 


Compression-Torsion Tests. 


Norre.—For these results the word Zension in the first two columns above 


should be changed to compression, otherwise the headings are the same. 


8000 | oO | 10500} 100 , -0021 100 | 10000000} 100 
8000 | 33 7700 | 73 | -0022 104 | 7000000 70 
12000 | 50 4500 | 42 0012 57 | 7500000 15 
20000 | 83 3800 | 36 | -0018 85 | 4200000 42 
24000 | 100 | 2900 | -27 | ‘0012 57 | 5000000 50 


Tension | . Angular | Percent.|Modulus of} Per cent. | 
stress, | Percent) “llasae | Eereon. deformation} of elasticity | of normal 
| of elastic} limit in | of elastic : 
lbs. per limit j : --. + | at elastic | normal | of shear, | modulus 
; imit in} torsion | limitin}] 3. - 

BU ARED. |) aan Tabtaaaed |. kavcien limit per | defor- Ibs. per of 
applied. | “°?*!0"- ; ‘junit length.) mation. | sq. in. | elasticity. 
0 0 10500 100 0021 100 10350000 100 
7000 33 7200 68 ‘0017 82 9075000 87 
10500 50 6300 60 00138 64 10350000 100 
14000 89- 4500 43 ‘0011 53 9075000 87 
17000 81 3300 31 ‘0007 35 9680000 93 
21000 | 100 2700 25 ‘0007 35 8086000 77 


Stresses on the Elastic Properties of Steel. 


TABLE II. 


221 


Results of Tests on Mild Carbon- and Nickel-Steel Shaftings 


in Flexure while under Torsion. 


MILD CARBON-STEEL. 


: : Per cent 
Elastic Modulus of | Per cent.| Deflexion ; . 
Speci mest ol limit in Se elasticity |ofnormal| at elastic Se on 
peci- jappee,) sexure, |. tor | in flexure, | modulus| limit, Sona, 
men. | lbs. per |i}, ner elastic iS aes Ib Of CIRA ca normal | elastic 
sq. In. | 4 Ss limit. Ae eit mee deflexion.| limit 
q- sq. in. icity. : applied. 
Teves. 0 47000 100 32900000 | 100 530 100 0 
iasas'¢= 22800 | 41000 87 28400000 86 “490 92 75 
5 eee 30400 | 31000 67 32400000 88 420 79 100 
BN eos 38000 | 22500 | 48 31500000 95 280 53 125 
NICKEL-STEEL. 
| Soa 0 78500 100 33300000 100 840 100 0 
2......| 15200 | 68000} 90 | 27700000} 83 | -820 98 | 40 
SP seks 22800 | 84000 83 31500000 94 "840 100 60 
ree 30400 | 59000 17 32300000 Sf) :8a0 99 80 
i ae 38000 | 34000 70 32300000 97 "750 89 100 
Tas E IIT. 


Relation between the Bending Moment in Torsion and the 


Bending Moment in Flexure in the Torsion-Flexure 


Tests of Steel Shafting. 


CARBON-STEEL. NICKEL-STEEL. 


Torsional Bending Torsional Bending 
ee ae ees || nent 
in inch lbs. | in inch lbs. in inch Ibs. | in inch lbs. 
ree eee FR eine Gy <| saz 
7 A 14400 12950 9800 21800 
Diannmay eee 19200 9800 14400 20300 
AEN cionacoerapeede 24000 7100 18200 18650 
3 ERROR Dr End) A. Ake a aa eee 24000 17000 
Phil. Mag. 8. 6. Vol. 15. No. 86. Feb. 1908. R 


= ae 
/ 
: 


of Steel. 


és O 


/ 222 Effect of Stresses on the Elastic Propert 


ooroe |  000e¢ 
008zz |  000F9 
oozet | 00068 
00000 : 


00F0S OO0TE 
00822 OOOTF 
00000 OOOLF 
006 Q00FG 
008 000026 
00¢F 00021 
OOLL 0008 
00SOT 00000 
00000 000% 
0012 000T8 
00es OOOLT. 
00°F O00FT 
0089 00¢0T 
00GL 0002 
00SOT 00000 
00000 000TZ 
“sq d 


"sesseqg ofduiig 


wore ra 

oorse | oros | onszo | n0cL 
ocszo? | ove, | 09066 | 09084 
00z0e | oozeL | OOTLb | 00084 
ooore |- o0ess O08LF 00s8L 
00664 | ‘OoFZe | OOBFL | O0EEB 
Orr | OOLGg | OO8sr | 00S 
0080r | o0ctg | oosse | O0LE9 
OOSTS | O00LF | 00863 | 000Lb 
00zFI | 00602 | OOsEt | 00802 
QO00T | ooorE | O0e6 | 008ET 
OO¢TT O0GFT 00801 00881 
O00FT | COOKE | OOTET | OOIST 
oooot | coors | 00st | 000 


‘SOBSOTIG OAT, 


fh je = I 
"sossedqg ont, 


—- 


8h 


‘sossoryg quoreddy 


OOSTLZ |’ eanxe_yT "T'q-uotss0 7, o/p 

goatee OADXOLT "T'T-UOssoy, g/g 

ooze, |" * @INXepL TT-Wows10 7, ¢/z 

ooco, ft steereewenses eanxe],q efdutig 
*18998-199IN 

OOZTL | eanxepy ‘TW-Uorss0 7, F/G 

oossr ft oINXOPY T'G[-Uolwssoy, F/F 


OSIIS eeeveoe ee OANXOT TT’ -UOws10 7, $/¢ 

OOOLF eoeeee Pee ereecerescce OANXO] AT ojdaig 
"P9998" UOgABH PITAL 

*oumuapt-wors4oy, U2 Burg fnyg pea7g 


OOkEZ [rere UOIsIOT, "T'A-"dw0yH OQ. 
00l0e |v’ uowsoy *T'q-"dm0g ¢g. 
gover fon ** uolsdoy, "T'q-"dur0g Z/T 
oogst | uoIsdOy, "T'q-dm0g ¢/| 
oocor fen odie sia uorssoy, efduig 
ooorg [eet uorssa1dmoy eydmig 

"u0rso 7, -worssaudmoy we burqny 79990 
OORIS [ttt  wowsAOY, "T-"8U97, OOOT 
OOSLI [reese ' UOISIOT, ‘T'-"suay, Tg. 
ooest fee UOISIOT, ‘T'W-"sUey, G9. 
OOFRT [ree WOOT, "T'A-"suay, Z/T 
5 ae hae + nomssoy, “TA-suay, 8/1 


00cOT deodersece eoseveveres uoIssOT, [dug 


O00TZ eee ceeererenvencee eee UOIsUaT, ejduiig 
‘uosdo,y -worsua 7 us burqny, ja0gg) 


—- 


—— | —__————_——— — 


AB 
"s891]9 JO PUT 


‘S[NWUIOT JUSIOHIGG wWo1z poyndwuoyj—sassauLG LINQ—'AT WAV, 


XIX. The Groups of Isomorphisms of the Groups whose 
Degree is less than Eight. By G. A. MILLER *. 


A Sees main objects of the present paper are to develop 
some fundamental theorems relating to the group 
of isomorphisms of a known group, especially when it is 
‘represented as a substitution group, and to give a complete 
‘list of the groups of isomorphisms of all the abstract groups 
which may be represented on seven or a smaller number of 
‘letters. As the groups of low degrees present themselves 
most frequently, it is believed that such a list, if it is entirely 
reliable, will render good service. Fifty-four different ab- 
stract groups can be represented on seven or a smaller number 
of letters, while the number of distinct substitution groups 
on these letters is 95. A complete list of these groups is 
found in the ‘ American Journal of Mathematics,’ vol. xxi. 
(1899) p. 326. In this list the distinct abstract groups are 
denoted by Greek letters. 

In accord with common usage the group of isomorphisms 
will be denoted by I, while the group under consideration 
and its holomorph will generally be represented by G and K 
respectively. The following known theorems are frequently 
used :—Jf a group is generated by two characteristic subgroups 
which have only the identity in common, its I is the direct 
product of the I’s of these two characteristic subgroups, and its 
K is the direct product of ther K’st. In particular, the K 
of an abelian group of order pi, pf?, py? - - - (Pi Pay Pay ++ + 
being different prime numbers) is the direct product of the 
K’s of the subgroups of orders pl, p$?, pf, ---- The 
symmetric group of degree nu, n # 2 or 6, is simply isomorphic 
with its 1, and the alternating group of degree n, n=&3, has the 
same group of isomorphisms as the symmetric group of the 
same degree. When n=2 the | is identity, and when n=6 
it is a well known imprimitive group of degree 12 and 
order 1440¢t. It should also be remembered that the I of 
the cyelic group of order p*, p being an odd prime, is the 
cyclic group of order p*-! (p—1), and that the I of the 
cyclic group of order 2*,2>1, is the direct product of the 
group of order 2 and the cyclic group of order 2*-. 


* Communicated by the Author. 
+ Transactions of the American Mathematical Society, vol. i. (1900) 
. 396. 
¢ Bulletin of the American Mathematical Society, vol. i. (1895) 
p. 258 ; Holder, Mathematische Annalen, vol. xlvi. (1895) p: 345, 


R2 


224 Mr.G. A. Miller on the Groups of Isomorphisms 


The theory of the groups of isomorphisms of cyclic groups 
includes the theory of primitive roots in number theory in 
view of the known theorem that the necessary and sufficient 
condition that a number (m) has primitive roots is that the I 
of the cyclic group of order m is cyclic. It has also been 
observed that some of the most useful theorems of number 
‘theory, such as Fermat’s and Wilson’s, are included in 
elementary theorems relating to the group of isomorphisms *, 
‘Moreover, the groups of isomorphisms furnish one of the 
best means to construct groups having known subgroups, 
and in many other group-theory considerations they play 
a fundamental role which is continually receiving more 
attention. 


§ 1. General Theorems. 


THEOREM I.—ZJfanabelian group G which involves operators 
whose orders exceed 2 is extended by means of an operator of 
order 2 which transforms each operator of G into its inverse, 
then the group of isomorplisms of this extended group is the 
holomorph of G. 

As the non-invariant operators of order 2 in the extended 
group correspond to themselves in every holomorphism of 
this group, its I may be represented as a transitive substi- 
tution group of degree n, where n is the order of G. After 
‘any holomorphism of G has been established, these operators 
of order 2 may be arranged in n different ways. Hence the 
order of I is n times the order of the I of G. Moreover, I 
contains an invariant subgroup which is simply isomorphic 
with G since we may obtain a holomorphism of the extended 
group by multiplying each operator of G@ by the identity 
and the remaining operators of the extended group by an 
arbitrary operator of G. It may be observed that this theorem 
includes the known theorem that the I of the dihedral group 
of order 2m, m>2, is the holomorph of the cyclic group of 
order m. 


THEoREM II.—The square { of a complete group which is 
not a direct product has the double holomorph of this group for 
its group of isomorphisms. 

Let H and H’ be two identical complete groups which are 
not direct products, and let G be the direct product of H 


* Cf, Transactions of the American Mathematical Society, vol. iv. 
1903) p. 158; ‘ Annals of Mathematics,’ vol. iv. (1903) p. 188. 

+ The direct product of two identical groups is called the square of 
one of them. 


a ae 
“ 


of the Groups whose Degree is less than Exght. . 225 


and H', We proceed to prove that H and H’ are charac- 
teristic subgroups of G. Suppose that H” were another 
invariant subgroup of G which could correspond to H ina 
holomorphism. Hence H” could be constructed by esta- 
blishing some isomorphism between H and H’ or between 
invariant subgroups. This isomorphism could not be (a, 1), 
since H does not contain any invariant operator besides the 
identity. If it were (a, 8), a, 8>1, the quotient group with 
respect to the subgroup of order e8 would be composed of 
invariant operators under G. Hence H” could not be a 
complete group, as its constituents would admit outer iso- 
morphisms under H and H! separately, since these groups do 
not contain any complete group invariantly. From this it 
follows that H must correspond either to itself or to H! in 
every holomorphism of G. As the holomorph of H is the 
square of H, it follows that this holomorph contains all the 
operators of the I of G which transform H into itself. Hence 
this I is the double holomorph * of G. The preceding proof 
clearly also establishes the theorem: The direct product of 
two distinct complete groups neither of which is a direct product 
as a complete group. 


TaEeorEM III.—The group of isomorphisms of the group 
obtained by extending a cyclic group of order 2m, m>2, by 
means of an operator of order 4 which transforms each operator 
of the cyclic group into its inverse is the holomorph of the cyclic 
group. 

The proof of this theorem is similar to that of Theorem I. 


TaEorEM IV.—Jf a complete group has only one subgroup 
of index 2, the direct product formed with it and the group of 
order 2 1s simply isomorphic with its group of isomorphisms. 

Corollary I. The direct product of the symmetric group of 
order n, n> 6, and the group of order 2 is simply isomorphic 
with ats I. 

Corollary I. The direct product of the metacyclic group of 
order p (p—1), p being any odd prime, and the group of 
order 2 1s simply isomorphic with its I. 

The proof of this theorem follows from the fact that such 
a direct product contains as a characteristic subgroup the 
subgroup of index 2 under the complete group, for if a group © 
contains only one subgroup of half its order this subgroup 
is generated by its operators which are squares; and, vice 
versa, if a subgroup of index 2 is generated by operators 
which are squares, it is the only subgroup of this index. This 


Sea eee of the American Mathematical Society, vol. iv. (1903) 
p. 104 


226. Mr.G. A. Miller on the Groups of Isomorphisms 


subgroup corresponds to itself in every holomorphism of GG 


The invariant operator of order 2 also corresponds to itself. 
Hence G contains a characteristic subgroup of index 2, and 
each of the remaining operators may correspond to either of 
two operators after any holomorphism of this characteristic 
subgroup has been established. As the order of I is equal to 


that of G and as I contains a complete group which is simply 
isomorphic with the complete subgroup of index 2 in G, it 


follows that I is simply isomorphic with G. 


-Turorem V.—The necessary and sufficient condition hates 


operator s of G transforms the operators of G according to an 


invariant operator under its group of cogredient isomorphisms 


is that the conjugates of s under G may be obtamed by multi-- 


plying s by mvariant operators under G. The highest order 


of such an invariant operator is the order of the corresponding. 


operator in the group of cogredient isomorphisms.. 


This theorem follows directly from the isomorphism be- 


tween G and its group of cogredient isomorphisms. It is 


clear that a holomorphism corresponds to an invariant 
operator in the group of cogredient isomorphisms whenever. 


the operators which correspond to themselves in the holo- 
morphism constitute an invariant subgroup, the corresponding 


divisions are invariant, and the remaining operators corre- 


spond to themselves multiplied by invariant operators. 


Similarly, it may be observed that a holomorphism corre- 
sponds to an invariant operator of I when the operators. 


which correspond to themselves in this holomorphism consti- 
tute a characteristic subgroup, the corresponding divisions 
are characteristic, and the operators of G which are not in 
the given characteristic subgroup correspond to themselves 
multiplied by characteristic operators of G. 


THEorEM VI.—Jf a substitution-group of degree n contains 
a subgroup of degree n—1 and involves no subgroup which is 


both of degree n and also of index n, then its group of 1s0- 


morphisms is simply isomorpluc with a substitution-group of 


degree n which contains the given group invariantly. 
No two substitutions of G could transform the n subgroups 


of degree n—1 in the same manner, since each substitution ° 
transtorms these sae in the same manner as it trans— 


forms its elements*, If an operator of I were commu- 
tative with each one of these n subgroups it would also be 


commutative with every operator of G. Ass this is impossible, 


p- 145 


* Bulletin of the American Mathematical. Society, vol. ii. (1896) 


of the Groups whose Degree is less than Eight. . 227 


it follows that I is simply isomorphic with a substitution 
group of degree n and that the group of cogredient isomor- 
phisms is simply isomorphic with G. This proves the theorem 
in question. 

If a substitution group G of degree n contains a subgroup 
of the same degree which leads to a representation which is 
conjugate to G, then G admits outer isomorphisms. In 
particular, if a simple group appears only once among the 
total number of substitution groups of degree n, and if it 
involves a subgroup which is both of degree and of index n, 
then it admits outer isomorphisms. For instance, the simple 
group of order 168 presents itself only once among the 
substitution groups of degree 7 and contains a subgroup of 
degree 7 and order 24. It must therefore admit outer iso- 
morphisms, as is also known from other considerations. As 
it contains only one set of 7 subgroup of order 24 and 
degree 7, and as it is not invariant under a larger group of 
degree 7, its I can be represented as an imprimitive group of 
degree 14 which involves two systems of imprimitivity and 
is of order 336. Such considerations apply toa large number 
of substitution groups and are frequently useful to obtain I. 

The group of isomorphisms of every finite group is finite. 
In fact, it is always easy to find an upper limit of the order 
of I for any known finite group, as this order cannot exceed 
the number of different ways in which a set of generating 
operators may be selected, and hence it can certainly never 
exceed (g—1)!, g being the order of the group. It is easy 
to see that there are only three groups for which I has this 
maximal order, viz. the four-group and the groups of order 2 
and 3. Assuming that 0! = 1, the identity might also be 
classed among these groups. In most cases it is easy to find 
much lower upper limits for the order of I. | 


§2. The Groups whose Degree is less than Six. 


There is only one group of degree 2, and its group of 
isomorphisms (1) is the identity. One of the two groups of 
degree three is the cyclic group of order 3 and hence has the 
group of order 2 for its 1, while the other is symmetric and 
therefore is simply isomorphic with its I. Two of the groups 
of degree 4 are cyclic. As their orders are 2 and 4, their I’s 
are the identity and the group of order 2 respectively. Two 
others are simply isomorphic with the four-group, and hence 
have the symmetric group of order 6 for their I. A third 
set of two groups of degree 4 is composed of the alternating 
and the symmetric groups. These have the symmetric group 


228 Mr. G. A. Miller on the Groups of Isomorphisms | 


of order 24 for their I, according to the well-known theorem 
quoted above. The seventh and last group of degree 4 is the 
octic group which is known to be its own I group. The 
orders of the groups of isomorphisms of the 7 substitution 
groups of degree 4 are therefore 1, 2, 6, 8, 24. 

The orders of the two cyclic groups of degree 5 are 5 and 
6 respectively, and hence their I’s are the cyclic groups of 
orders 4 and 2. Since the group of order 12 and degree 5 
is the direct product of the symmetric group of order 6 and 
_ the group of order 2 it is simply isomorphic with its I, 
according to theorem IV. This is also true of the non- cyclic 
group of order 6, as this is simply isomorphic with the 
symmetric group of this order. The metacyclic and the 
semi-metacyclic groups of orders 20 and 10 repectively are 
known to have the former for their I, while the alternating 
and the symmetric groups have the latter for their common I. 
Hence the orders of the groups of isomorphisms of the 8 sub- 
stitution groups of degree 5 are 2, 4, 6, 12, 20, 120. 


§ 3. The Groups of Degree Six. 


In the preceding section we considered all the possible 
abstract groups whose order is less than 7, and hence we 
know the groups of isomorphisms of the 9 substitution 
groups of degree 6 whose orders do not exceed their degree. 
There are ten other groups of this degree which are simply 
isomorphic with groups of lower degree, viz. three of order 8, 
two of order 12, three of order 24, and one of each of the 
orders 60 and 120. Hence only 18 of the 37 groups of 
degree 6 are distinct, as abstract groups, from those of lower 


degrees. Two of these 18 are simply isomorphic with the - 


group of order 8 which contains 7 operators of order 2, and 
hence have the simple group of order 168 for their I. The 
remaining group of order 8, (abcd)cyc.(e)), is of type (2, 1), 
and hence has the octic group for its 1. The group of order 
9 is of type (1, 1), and hence has the transitive group of 
degree 8 and order 48 which involves operators of order 8 as 
its I. Since the group of order 16 is the direct product of 
the octic group and the group of order 2, its I is known™ to 
be the transitive group of degree 8 and order 64 which 
Cayley denoted by 


(ae. bf. cg .dh)N $ (abcd) s(efgh)s} dim. 
The group of order 18, which is the direct product of the 


* Quarterly Journal of Mathematics, vol. xxviii. (1896) p. 252. 


of the Groups whose Degree is less than Enght. 229 


symmetric group of order 6 and the group of order 3, has for 
its I the direct product of the symmetric group of order 6 
and the group of order 2, according to the theorem ; if a 
group is generated by two characteristic subgroups which 
have only the identity in common, its I is the direct product 
of the I’s of these characteristic subgroups. Since the other 
group of order 18 is obtained by extending the group of 
order 9 and of type (1, 1) by means of an operator of order 
2 which transforms each of its operators into its inverse, its 
I is the holomorph of this group of order 9 and hence is the 
doubly transitive group of degree 9 and order 432, according 
to the theorem given above. The two groups of order 24 
which are not simply isomorphic with the groups of lower 
degrees are the direct products of the alternating group of 
order 12 and the group of order 2. As these factors are 
characteristic subgroups of this direct product, its I is the 
symmetric group of order 24. 

Two of the groups of order 36 are simply isomorphic with 
the square of the symmetric group of order 6, and hence 
have the double holomorph of this symmetric group for 
their I. This double holomorph is the group of order 72 
and degree 6. The other group of order 36 is a characteristic 
subgroup of this double holomorph and the order of its I 
cannot be less than 144, which is the order of the I of this 
double holomorph, as we shall soon prove. Since this group 
of order 36 is generated by two operators of orders 3 and 4 
respectively, and as the former of these operators could not 
be selected in more than 8 ways while the latter may be 
selected in no more than 18 ways, it follows that the order 
of this I is 144 if the order of the I of the given double holo- 
morph is 144. When we prove the latter fact we shall also 
prove that this group of order 36 and the group of order 72 
and of degree 6 have the same I. 

Since the groups of order 48 are the direct product of the 
symmetric group of order 24 and the group of order 2, they 
are simply isomorphic with their I. The group of order 72 
contains subgroups of order 12 and degree 6 which give rise 
to a transitive representation, and hence it admits outer 
isomorphisms according to the given theorem. Moreover, 
it contains a characteristic subgroup of order 36 which does 
not admit more than 144 holomorphisms. The holomorphism 
of the group of order 72 is fixed by that of this subgroup, 
since the former contains only one operator of order 2 which 
is commutative with every operator of a subgroup of order 6 
contained in the latter. Hence each of these groups has the 
same I and its order is 144. If this group of order 144 is 


230 Mr. G. A. Miller on the Groups of Isomorphisms — 


represented as a transitive substitution group of degree 12 
with respect to a subgroup in the given group of order 72, 
the two sets of 6 subgroups of order 12 will correspond to 
two systems of imprimitivity. Hence there are two transitive 
groups of degree 12 and order 144 whose heads are obtained 
by making the group of degree 6 and order 72 simply iso- 
morphic with itself, while the published list of these groups 
gives only one*. As it is well known that the I of the 
alternating and the symmetric group of degree 6 may be repre- 
sented as an imprimitive group of degree 12 and order 14407, 
it has been proved that the substitution groups of degree 6 
which cannot be represented as substitution groups of lower 
degrees have groups of isomorphisms of the following orders : 
168, 8, 48, 64, 12, 432, 24, 72, 144, 1440. The two groups 
of order 48 are clearly distinct, as one involves operators of 
order 8 while the other does not have this property. The 
13 distinct abstract groups which may be represented as 
substitution groups of degree 6, but of no lower degree, have 
therefore 11 distinct groups of isomorphisms. | 


§4. The Groups of Degree Seven. 


Only eleven of the forty groups of degree ’7 are simply 
isomorphic with groups of lower degrees. Three others are 
simply isomorphic with the cyclic groups of orders 7, 10, 12 
respectively, and hence they have for their I’s the cyclic 
groups of orders 6 and 4, and the four-group. Two of the 
remaining three groups of order 12 are simply isomorphic. 
with the direct product of the four-group and the group of 
order 3, and hence their I is the direct product of the 
synimetric group of order 6 and the group of order 2. This. 
is also the I of the other non-cyclic group of order 12 which 
is not simply isomorphic with groups of lower degrees, since: 
the holomorph of the cyclic group of order 6 is the direct. 
product of the symmetric group of order 6 and the group of 
order 2. We have now considered the [’s of the 5 possible 
abstract. groups of order 12, and found that their orders 
are 4, 12, and 24 respectively. Three of these groups: 
have the holomorph of the cyclic group of order 6 for 
their 1. | ) | : i 
. The three groups of orders 14, 21, and 42 have the last of 


* Quarterly Journal of Mathematics, vol. xxviii. (1896) p. 223. * 
+ Bulletin of the American Mathematical Society, vol. i. (1895): 
p. 258. | ie 


of the Groups whose Degree is less than Eight. 231 


these for their I, since the metacyclic group of order p (p- 1) 
and degree p, p being any odd prime, is the group of iso- 
morphisms of all its subgroups whose orders are divisible by p 
and exceed p. The group of order 20 has the direct product 
of the group of this order and of degree 5 and the group of 
order 2 for its I, according to theorem I. The two groups 
of order 24 which are the direct product of the symmetric 
group of order 6 and the four-group may be obtained by 
extending the direct product of the four-group and the cyclic 
group of order 3 by means of an operator of order 2 which 
transforms each operator of this direct product into its 
inverse. Hence by the theorem given above, these groups 
have the holomorph of this direct product for their I. The 
order of this holomorph is 144. The group which is the 
direct product of the cyclic group of order 4 and the 
symmetric group of order 6 is invariant under the holomorph 
of the cyclic group of order 12, and has the group of cogre- 
dient isomorphisms of this holomorph for its I. Hence the ~ 
order of this I is 24, and it is the direct product of the 
symmetric group of order 6 and the four-group. 

The direct product of the octic group and the group of 
order 3 has these two groups for characteristic subgroups, 
and hence its I is the direct product of the octic and the 
group of order 2, while the I of the dihedral group of order 
24 is the holomorph of the cyclic group of order 12.. The 
orders of these two I’s are 14 and 48 respectively. It 
remains to consider the I of the group denoted by : 


§(abed)s com. (efg) all} dim.* 


As this is invariant under the direct product of the octic and 
the symmetric group of order 6, and only two operators of 
this product are commutative with each of its operators, its I 
is the group of cogredient isomorphisms of this direct 
product. This I is the direct product of the dihedral group 
of order 12 and the group of order 2. We have now con- 
sidered the five types of groups of order 24 which can be 
represented as substitution groups of degree 7 but not of a 
lower degree. The orders of ‘their I’s are 144, 24, 48, 16, 
The two I’s of order 24 are simply isomorphic. i 
The group of order 36 contains three invariant tetra- 
hedral groups. As each of these may be transformed into 
itself by the symmetric group of order 24, the I of the group 
of order 36 contains this symmetric group invariantly and is 


* American Journal of Mathematics, vol. xxi. (1899) p. 326. 


232 Isomorphisms of Groups whose Degree is less than Eight. 


the direct product of the symmetric groups of orders 6 and 
24 respectively. The group of order 40 is simply isomorphic 
with its I, since it is the direct product of the group of order 
2 and a complete group which involves only one subgroup 
of half its order. The group of order 48 contains a charac- 
teristic subgroup of order 8 and two invariant subgroups of 
order 6. From these subgroups it follows that the order of 


its I is 96, and this I is the direct product of this group of - 


order 48 and-the group of order 2. 

Since two of the groups of order 72 are direct products of 
characteristic subgroups, their I’s are the direct products 
of the groups of isomorphisms of these subgroups. The 
orders of these I’s are 48 and 144 respectively. The re- 
maining group of order 72 involves three invariant tetra- 
hedral groups and contains all the operators of its I which 
transform each of these invariant subgroups into itself. As 
these subgroups are transformed according to the symmetric 
group of order 6 by I, it follows that the order of this group 
of isomorphisms is 432. With respect to the characteristic 
abelian subgroup of order 12, it is isomorphic to the direct 
product of two symmetric groups of order 6, since the 
group of cogredient isomorphisms is isomorphic with the 
symmetric group of order 6 with respect to the same 
subgroup, and does not permute the given three tetrahedral 
groups. 

The group of order 120 is the direct product of two 
characteristic subgroups, and its I is the symmetric group of 
order 120. Since the group of order 144 is the direct 
product of two characteristic subgroups which are also com- 
plete groups, it is simply isomorphic with its I. The simple 
group of order 168 is known to have the group of degree 8 
and order 336 for its I, while the group of order 240 is 
simply isomorphic with its I, according to the given theorem. 


The remaining two groups are the alternating and the sym- 


metric, and hence have the latter for their common I. Hence 
the groups of isomorphisms of the 29 groups, which may be 
represented on seven but on no smaller number of letters, have 
the following orders: 4, 6, 12, 42, 40, 144, 24, 48, 16, 96, 
432, 120, 336, 240, 5040. 


_ University of Illinois. 


a 


fi 23d: | 
XX. On Induced Stability. By ANDREW STEPHENSON.” 
| Nae conditions under which an imposed periodic change 


in the spring of an oscillation exerts a cumulative 
effect in magnifying the motion, have already been investi- 
gated +. We shall now examine the influence of such a 
variation on instability of equilibrium and, in certain cases, 
of steady motion. 
The equation of motion about statically unstable equilibrium 
under variable spring is 


2— (uw? —2an? cos nt)x=0, 
and the complete solution is given by 
pes A,.sin {(e—rn)t+e}, 
where | 
—A, fu? +(c—rn)*} +an?(A,_1+Ar41)=0. © . (r) 


In the limit when 4, taken positive, is infinite 
a 
Ap pA 


so that the series is convergent in both directions. If the 
eliminant of the equations (7) gives a real value for c, the 
elementary oscillations are of constant amplitude and the 
equilibrium is stable. 

An approximation t for ¢ is readily obtained when «@ is 
small, for in that case, r being positive, 


an? 


Air= we+(c+ faye eed 


approximately, and on substituting for A, and A_, in (0) we 
find that for a real ¢ when « is small n must be large compared 
with «~: then 


c2 = Qa2n? — p?, 
and for stability x must be greater than p/eWV/2. 
Thus the system can always be maintained about a position 
of otherwise unstable equilibrium by a periodic variation in 


spring of sufficiently large frequency. The inverted pen- 
dulum, for example, is rendered stable by rapid vertical 


* Communicated by the Author. 

+t “On a class of forced oscillations,” Quarterly Journal of Mathe- 
matics, no. 168, 1906. “On the forcing of oscillations by disturbances 
of different frequencies,” Phil. Mag. July 1907. 


{ Similar to that employed in the former of the two papers referred 
to above. 


234 Mr. A. Stephenson on 


vibration of the point of support. The above analysis shows 
that quite apart from gravity, an imposed motion of small 
amplitude and high frequency produces a comparatively 
slow simple oscillation about its own direction ; the impressed 
action must therefore exert a restoring moment on the body 
proportional io the displacement of the mass centre from the 
line of the applied motion, and if this moment numerically 
exceeds the outward moment due to gravity stability is 
ensured *. 

If the pivot is given a simple vibration of amplitude a with 
frequency x per 27 seconds, the equation of motion about 
the unstable position is 


cae 
&— 7 (g—an? cos nt) e=0. 


For a simple pendulum 1 metre long n?=10 approximately, 
and if a=10 em. for stability n>44°7; thus a frequency of 
7°2 per second is sufficient to maintain relative equilibrium. 
If the pendulum is 20 cm. in length w?=50, and if a=10 
the condition would give n>20, a frequency of over 3:2 
per second. In this case, however, «/n is not sufficiently small 
for the approximate formula to be applicable. 

There is no difficulty in verifying the stability under the 
imposed motion by experiment. It is found furthermore that 
the pendulum may be rendered approximately steady in a 
position sensibly oblique by a comparatively small inclination 
of the direction of vibration, and it is of interest to enquire 
under what circumstances this occurs. 

If the path of the pivot makes a small angle @ with the 
vertical the equation of motion is 


ae | I 
L— a (g—an? cos nt) z= Bazan? cos nt. 


For the particular solution giving the forced motion due 
to the disturbance represented by the term on the right, we 
have 


wo 
x = >B, cos rnt, 
0 
where 


* The stability of this particular system is worked out from first 
principles for a special case in a paper “On anew type of dynamical 
stability,’ read before the Manchester Literary and Philosophical Society 
in January 1908. ' 


Induced Stability. 235 


except for r=0 and 1, for which 


—By e+ 72 ss ty (OY 


k2 
and 
B(% Lhas. B 1 
+n w) + saan (2B) + ) = Ban ha 
The series is evidently convergent. Since ue = pw’ is small 
compared with n?, 
1 ah 
Be= a ja Br-1 


approximately if r>1. 
Also B, = 7 » Bo, and therefore a 9 ce Tia: 
B, is chevetore negligible i in (1)’, and finally 


ein eis 
Bb, ~~ fo Ight” 
nah 


As before for the stability of the “free” motion n? must 
exceed y?/2a, which is equal to 2gk?/a*h: if, therefore, n is 
in the vicinity of this limit Bo, the mean oan of the 
pendulum, is large compared with 8, the inclination of the 
path of the pivot. Since 2g/an? is small, being less than 


ah/k’, it is clear that By is large compared with 5 (Bets te. 


the forced oscillation about the inclined Sees is compara 
tively small. This fact is very evident experimentally. 
As n increases By decreases to the limit @, and the rod 
approaches the line of vibration. 
2. A case of steady motion for which the stability ga: 
is of the type 
r+rAx=0 


is furnished by a solid of revolution rolling on a horizontal 
plane. Consider the rectilinear motion of such a body sym- 
metrical about the middle plane normal to its axis. The 
small oscillations are determined by 


(A+ Mb?) + {Cp?(C-+ Mb*)/A — (b—p)Mg}e=0, 


where p is the angular velocity, b the radius of the mid- 
section, p the other principal radius at any point on it, and 
the other constants are in the usual notation. 


When, therefore, p*<g eens the motion is unstable. 


236 On Induced Stability. 


If now the plane is given a simple oscillation vertically, 
g must be replaced in the above equation by g+an? cos nt, 
and the motion is rendered stable if - 
eee 2 as C(C+MB) [g(b—p)MA _ 7 
2 A+ ML? A(A+M0*) LC(C+ MB)? JS’ 
where, as before, in making the approximation it is assumed 
that — a issmall. The range of action is limited by 
the conditions of the problem which require that an? must 
not exceed g. | 
3. In general the single equation determining the oscil- 
lations of a system about steady motion is of order higher 
than the second, as for example in the case of the spinning 
top, which we now examine in the present connexion. The 
unstable position of equilibrium of a symmetrical top may be 
rendered stable either by an axial spin or by an imposed 
vertical vibration of the point of support. The two actions 
together might therefore be expected to reinforce one another, 
if either singly is not of sufficient intensity. 
For the small oscillations about the position of equilibrium 


&—py—(pw?—2an? cos nt)a = 0 
y +pi— (pw? —2an? cos nt)y = 0, 
and a solution is 
2 =A, sin {(e—rn)t+e}, 


= 3 A,.cos {(c—rn)t+e}, 
where 
—{p?+(e—rn)?—p(e—rn) A, +an?(A,_1 + Ars) =9,. 


This set of conditional equations is similar to the system (7) 
in § 1, and admits of corresponding approximate treatment. 
If « is small and n is large compared with yu and p, we find 


w+c?—pe—2en? = 0, 
and therefore for stability in this case 
2a?n? > w?—ip?. 
Thus wien the imposed motion is comparatively rapid the 
two actions are simply cumulative. 


Manchester, October 1907. 


b> 337 J 


XXI. A Freehand Graphic way of determining Stream 
Lines and Equipotentials. By L. F. RicHarpson *. 


[Plate XIT.] 


SCHEME OF PAPER. 


I. On the need for new methods. 
Il. The first idea of freehand solution and confirmation of its 
accuracy. ; 


III. The conditions which the solution of V7V=0 must satisfy in 
order that it may be determinable by a single graph. | 
(a) When the guiding lines are normal to a family of surfaces. 
Possible types—test cases. 
(6) Thin shells. 
(c) Screw symmetry—example. 
IV. Points and lines of equilibrium. 
V. Equations other than Laplace’s—variable conductivity. 
VI. Boundary conditions. 
VII. Miscellaneous notes on draughtsmanship. 
VIII. Estimation of errors. 


L Dhe Need for New Methods. 
i Laplacian differential equation 


vv vv, ov 

02? i oid a oer | 
has received an extraordinary amount of attention during the 
last century owing to the great number of physical quantities, 
the space distribution of which can be determined from its 
integrals. The analytical integrals hitherto obtained by such 
means as Fourier series, Bessel functions, spherical and other 
harmonics make it possible to determine the distribution when 
the boundary conditions bear relation to certain simple types 
of surface, such as parallelipipeds, cylinders, spheres, ellipsoids, 
anchor rings, &e. 

Now for physical research this is well enough. It is 
usually possible to arrange the instruments so that the parts 
involved are of these simple forms. The wires may be wound 
in circular rings of small cross-section, as in Helmholtz’s 
galvanometer. The pieces of substance for the measurement 
of specific properties may be shaped into square bars, as in 
Forbes’s experiments on the flow of heat. Or, as in Kelvin’s 


* Communicated by the Physical Society ; read November 8, 1907. | 
Phil. Mag. 8. 6. Vol. 15. No. 86. Feb. 1908. S 


238 Mr. I. F. Richardson on a Freehand Graphic way 


attracted disk electrometer, parallel plates may be made 
practically infinite by his device of the guard-ring, 

But for the purposes of the engineer this is of very limited 
application. If he is to handle partial differential equations 
freely, they must be applicable to bodies of most various shapes, 
such, for example, as the toothed core-plates of dynamos, the 
water surrounding ship shells and screw propellers, the space 
between turbine blades, and a host of other forms, too irregular 
to be readily described. | 

Further than this, the method of solution must be easier 
to become skilled in than are the usual methods with harmonic 
functions. Few have time to spend in learning their 
mysteries. And the results must be easy to verify—much 
easier than is the case with a complicated piece of algebra. 
Moreover, the time required to arrive at the desired result 
by analytical methods cannot be foreseen with any certainty. 
It may come out in a morning, it may be unfinished at the 
end of a month. It is no wonder that the practical engineer 
is shy of anything so risky. , 

Harmonic functions have, however, one very strong point 
in comparison to the methods put forward in this paper, and 
that is their accuracy. Once we have determined V as an 
infinite series of harmonic functions, it is usually not much 
more labour to obtain an accuracy of 1 in a million than of 
1 in ten. 

Now it is true that in the determination of absolute electric 
standards measurements are made to 1 in 100,000 or to an 
even greater refinement. But for most chemical and physical 
work 1 in 1000 is more like the limit attained. And in any 
new branch of research, two, five, or even ten per cent. are 
very welcome. The root of the matter is that the greatest 
stimulus of scientific discovery are its practical applications. 
And here, in the design of machinery for example, cost rules 
everything, and this can seldom be foreseen as near as 
1 per cent. 

To sum up. The existing methods of solving Laplace’s 
equation are susceptible of great accuracy, but they are slow 
and uncertain in time and, most serious of all, they can only 
be applied to very special boundary conditions. There is 
obviously a demand for a method of solving that group of 
partial differential equations—of which we may regard 
Laplace’s as the simplest type—which shall, if necessary, 
sacrifice accuracy above 1 per cent., to rapidity, freedom 
from the danger of large blunders, and applicability to more 
various forms of boundary surface. 


al 
ia 


of determining Stream Lines and Equipotentials. 239 


Il. The First Idea of Freehand Solution. 


. The real simplicity of the space distributions of electric 
and magnetic phenomena,—so much disguised in the algebraic 
integrals of the differential equations, but rescued from con- 
fusion and clearly set forth by the vector analysts, Heaviside, 
Walker and others,—leads one to hope for equally simple 
methods of calculating their numerical values with reference 
to any boundary whatever. 

The beautiful figures of stream and equipotential surfaces 
published by Maxwell, Lamb and others as the result of 
harmonic analysis, and by Hele Shaw as the result of experi- 
ment, suggest that by imitating their characteristic properties 
freehand we may, in some small part, attain the result 
desired. 

Maxwell in § 92 of his ‘Elementary Treatise on Elec- 
tricity and Magnetism’ speaks of tentative methods of 
altering known solutions of the Laplacian equation by drawing 
diagrams on paper and selecting the least improbable. The 
object of the present thesis is to point out that this method 
can do far more than merely alter known results, and that it 
may be so far from tentative as to yield an accuracy of 
one per cent of the range. 

This method of treating potentials, although still far from 
combining all desirable qualities, and suffering from the 
restriction to certain types of symmetry, yet from its great 
freedom within those types may, it is hoped, supply to a 
certain extent the demand we have indicated. 

On turning to Maxwell’s figures and picking out those in 
which V is independent of z so that we have 


it will be seen that while the curves are of the most various 
shapes yet the chequerwork of all the diagrams possesses 
these two properties in common :—(1) the corners are 
orthogonal, (2) when the chequers are small enough the 
ratio of their iength to breadth is the same in all parts of 
the field. 

The proof of this follows most conveniently from the con- 
sideration of the motion of a liquid when the lines of flow 
lie in parallel planes and the motion is the same at all points 
of any normal to these planes. Draw three adjacent stream 
lines defining two adjacent tubes of flow. 


8 2 


240 Mr. L. F, Richardson on a Freehand Graphic way 


Take two points A and B on the mid line of one tube, and 
from A and B draw normals to the direction of flow cutting 
the mid line of the other tube in D and C respectively. 


Fig. 1. 


Halfway between AD and BC draw a line PQS normal to 
the direction of flow so that PQ is the width of one tube and 
QS of the other. Now if the fluid is incompressible and we 
have drawn the tubes so that the flow in each is the same, 
then the respective velocities are to one another inversely as 


a k k 
PQ and QS. Let the velocities be PQ and Qs: Next let 


us take the line integral of the velocity round the small 
rectangle ABCD. The sides AD and CB are normal to the 
flow and so contribute nothing. The sides AB and CD 


contribute nies i 
ae Xk { ia DG 
2 ee ef a 
ABO Ge PQ OS 
AB xe : 
Now PQ is the ratio of the length along the flow to the 


breadth across the flow of the small chequer which has A, Q, 
B, P, at the mid points of its four sides. It will be convenient 
to have a special name for this quantity, and I propose to 
call it the “chequer ratio”? with the understanding that 
‘length along the flow is always in the numerator, and that 
the chequer is so small that its size no longer causes an 
appreciable deviation from the accuracy obtained by using 
infinitesimals. Then we have :— 

Difference between successive 

Line integral of the velocity around ABCD=kx chequer ratios in a direction 

oil perpendicularly across flow. 


07 determining Stream Lines and Equipotentials. 241 


Now the curl of the vector velocity is defined as the line 
integral round a small circuit divided by the area of that 
circuit—that is in this case by the area ABCD which will in 
the limit be equal to the mean of the areas of the two 
adjacent chequers. So that we have :— 


difference of successive chequer ratios in a direction 
perpendicular to velocity 


curl of the velocity =k 
mean chequer area 


If the velocity has no curl the chequer ratio must not vary 
along any line normal to the flow. It may vary from one 
normal to another, but if on the other hand we prefer to make 
it constant all over the field, then at any point the distance 
between successive normals will be inversely as the flow, so 
that these normals will be contours drawn at equal intervals 
of a velocity potential. 

To return: since the fluid is incompressible the condition 
for the existence of a stream function is satisfied, and since 
the stream-lines are drawn so that the flow between each 
successive pair is the same, it follows that these stream-lines 
are the contours drawn at equal intervals of a stream 
function . Now it is proved in works on Hydrodynamics 


that ae + ov 


is equal to the curl of the vector velocity. 


on? Oy 
Therefore :— 
‘ ditference of successive chequer ratios in a direction 
Ov rf ov aE perpendicular to the contours of J 
a oy mean chequer area 


And since may be any one-valued function of position on 
the plane, it is seen that all hydrokinetical considerations have 
been eliminated from the above equation, which is purely a 
proposition in differential geometry. The only implication 
being that contours are drawn at equal intervals of y what- 
ever be its physical meaning. 

To draw chequers freehand so as to satisfy a difference 
relation of this sort between the chequer ratios is likely to be 
toilsome, and we will here consider only the case when 
V7yr=0. 

_ Supposing then that a chequerwork has been obtained in 
which the chequer ratio is everywhere the same and in 
which the given boundary conditions are satisfied, then by the 

Mi AV 
; oe 7 Oy 


uniqueness of the solution o =( this chequer- 


work gives us what we want. 


aa 24, “a 


242 Mr. L. F. Richardson on a Freehand Graphic way 


_ It remains to be shown what accuracy may be expected 
from the freehand method. This is of course largely a 
personal matter. I exhibit my own handiwork. Others will 
doubtless obtain greater precision. Throughout this paper I 
have chosen test cases in which the analytical verification 
should not be too difficult. Hence the diagrams look rather 
stiff and formal and do not in any way do justice to the 

_ freedom of this graphic method. 


Fig. 2. 


RDS 
He PX 


2 2 

Example of the solution of cad + 57 =0-—Along one 
pair of opposite edges of a square V =1, along the other pair 
V=0. Find V at all points inside. By symmetry V will be 
equal to } along the diagonals. And again by symmetry, 
the lines joining mid-points of opposite sides will be normal 
to the contours of V. So that it is only necessary to find V 
in half one-quarter of the square. Further, by symmetry, ata 


of determining Stream Lines and Equipotentials. 243 


corner of the square the contours drawn at equal intervals of 
V must make equal angles with one another. One starts 
then by ruling out an accurate square, putting in the 
diagonals, joining the mid-points of its sides and setting off 
the equal angles with a protractor. It is convenient to 
divide the range of V into ten equal parts. Having thus 
prepared the paper, lines were sketched and amended until 
further improvement became very slow. The pencil-lines 
were then firmly fixed in ink. Coordinate lines were drawn 
in and the values of V at six points were read from the 
diagram and are given in parenthesis in the accompanying 
table. The whole work from the beginning of the drawing 
took two or three hours. 


; 
(47) 
5 
466 
(-40) | (:865) 
1-0 ‘5 
396 | -364 
(307) | (23) | (20) 
1:0 
300 293 202 
5 0 0 0 0 


Not until this had been done did I look up the correct 
values which had been computed from the analytical solution 


. m—1 
v= n= (- 1) eS sech met cos ma .cosh mz. 

These are given in the table beside the numbers read from 
the graph. From these we find the errors +007, +°007, 
—-002, +°004, +:001, +:004. Treating these as all of the 
same sign, their mean is ‘0042. 

The error of a graph may well be compared with the total 
range of V within which the determination was made 
freehand. In this case the range was 0°5 so that the mean 


244 Mr. L. F.-Richardson on a Freehand Graphic way 


error was 0°84 per cent. of the range, a degree of accuracy 
which would be sufficient for many purposes. 

_ Having shown that the freehand method is a practical one 
in a plane, it will be well next to enquire to what types of 
symmetry it may be extended. 


III. Possible Types of Symmetry. 


_ Inasmuch as a single chequerwork is to determine the dis- 
tribution in the whole of the space considered, we are confined 
to two coordinates. The freehand method at present offers 
nothing to compete with the analytic forms in which three 
coordinates appear, such as: 7 


v2 (A, sin ma + B,, cos mx)(C,, sin ny + D_cosny)(Re~ /m?+n2. 2+ Get Vmitn? 2 


V=2 e*”(A,, cos m+ Bn sin md)J,,(kr), 
mk i 


(Whittaker, ‘Modern Analysis,’ p. 318) 


V=> Cnr”(Ax cos np + B, sinng) (sin 6)”. Tae 
(Byerly, ‘ Fourier’s Series,’ p. 196) 
or others like them. | | 

The expressibility of V in terms of two coordinates implies 
that V is constant along a certain family of lines in space, 
namely, the intersections of the surfaces over which the said 
two coordinates are respectively constant. Any particular 
type of symmetry is most conveniently distinguished by 
specifying the family of lines along each of which V must be 
constant. 

As it will frequently be necessary torefer to these lines 
and to distinguish them from the normals to the surfaces 
V=constant, I propose to call them the “ guiding lines,” 

It is indeed conceivable that by adding together several 
space distributions in each of which V is constant along a 
different family of lines, we might attempt the solution of 
problems which it may be impossible to treat by two co- 
ordinates directly, such for example asithe motion of a perfect 
fluid past a three-bladed screw-propeller, or the electrostatic 
field due to a ring of electrons. It may even be possible to 
treat the most general distributions by means of sections of 
the potential surfaces drawn on the leaves of a bock of tracing- 
paper. But these extensions must be left to those who desire 
the results. This paper deals only with two coordinates. 


J" 1) eed 
a Re 


of determining Stream Lines and Equipotentials. 245 


Case (a). The guiding lines are everywhere orthogonal to a 
family of surfaces.—Let these be the surfaces over each of 
which y=F;(2, y, z) is constant. 

Then choosing a particular surface, say y=, we wish to 
draw thereon a chequerwork of orthogonal lines, and we wish 
this chequerwork, by the motion of each point of it along the 
guiding line at that point, to sweep out two families of surfaces 
in space, in such a way that one family may be equipotentials 
and the other stream-surfaces. This requires that these two 
families, which we may denote by | 


a=F(2, y, z)=const., B=F(a, y, z)=const., 


should be everywhere orthogonal. Therefore the surfaces 
a, 8, y are mutual orthogonal, and consequently the surfaces y 
must satisfy the condition necessary in any member of a triply 
orthogonal system (Salmon, ‘ Geometry of Three Dimensions,’ 
Ath ed. §§ 476 to 486). 

But more than this. For we wish to be unrestricted as to 
the direction of the orthogonal traces of « and 8 drawn upon 
the surfaces y. Therefore, since three mutually orthogonal 
surfaces necessarily intersect in their lines of curvature 
(loc. cit. § 304), it follows that at every point of the surfaces y 
there are lines of curvature in every direction. The only 
form which possesses this property is the sphere or its limit 
the plane. Therefore the surfaces y are either spheres or 
planes. This is necessary. We have not proved that it is 
sufficient. As frequent reference will be made to the theorems 
proved in Lamé’s Lecons sur les coordonnées curvilignes, it will 
be convenient to employ expressions such as (Lamé, § xi. 15) 
to indicate equation 15 of § x1. of this treatise. The relation 
of our notation to Lamé’s is that his p p; p2 are replaced by 
a 8B y and that a2 B y are used as subscripts respectively 
instead of absence of subscript, 1 and 2. Otherwise the 
notations are the same. 

In particular, if F is any function of position, we will 
denote by Hy the quantity 


1 
V(32) +50) * (32) 


which is the reciprocal of the space-rate of F along the normal 
to the surface F = constant. 

Consider the lamina bounded by the two spheres y and 
yt6y. The thickness of the lamina is H,dy. If z«=V=the 


246 Mr. L. F. Richardson on a Freehand Graphic way 


potential, then the surfaces 8= constant are surfaces of flow. 
Denoting in like manner the distances between two adjacent 
surfaces of these families by H,da, Hg58, we see that H,ée 
and H,68 are the length and breadth of a chequer traced on 
the surface y. 

A tube of flow is bounded by the four surfaces 8, 8+ 68, 
y, y+6y. And its cross section is therefore H,. H,. 88. dy. 

Now if the flux has no divergence, then along a tube of 
flow magnitude of flux multiplied by cross-section =constant. 
But the magnitude of flux is equal to the negative space-rate 


of the potential 2 along the line of flow, and this is se 
He bert - 68 . dy must be con- 


Therefore along a line of flow 


stant in order that the vector space-rate of the scalar a—the 
Hamiltonian Ya—shall have no divergence. Or equivalently 
the condition that 


‘ .. Od awe 
Wa 0s 2. (#857) == || 


Now we may by freehand trial and amendment so arrange 
the orthogonal lines on the surface yp that the above relation 
shall hold true on the surface yo; but we must further 
enquire what conditions the spheres y must satisfy in order 
that 2 (4 =0 shall be true for all values of y when 

a a 
it is true for one Y); and this moreover when « and # are 
otherwise undetermined. 

At this stage the fact that the surfaces Y are spheres makes 
a remarkable simplification. For supposing for a moment 
that they did not possess this property and that r¢ and 7 were 
their principal radii of curvature at any point, then by 
(Lamé, § xxx. 24) 


aie Sees 
fe he OY. te ey 
where h,= ee and similarly for 6 and y. 


Equating the two curvatures, 


wi 


of determining Stream Lines and Equipotentials. 247 


Now es a "ok 
Syl) = eels 2, i 


and therefore vanishes; so that the chequer ratio H. is a 
function of « and 8 only. Hs 
As has already been stated, to make (7?a=0 on the surface 
Yo we must have 
2 (= Hy, )=0 
daw EM? 


which may conveniently be arranged by making 


A 
fia ko Hy, 


where kp is a constant, so that the chequer ratio is given as a 
function of position on yp. This is more than sufficient in 


that it makes 
2. H B Hs) 
OB\ H. 


vanish as well, but the loss of generality involved is found 
not to matter, while the simplicity gained is a great con- 


venience. Next, because ee is independent of vy it follows 


Hg 


that on any other surface 7, we still have 


H, 

He — ea H,,. 
But if (7?«2=0 isto be satisfied on this second surface we 
must there have 


where &, is a second constant. Therefore regarding yp as 
fixed and y,=y¥ as movable we have Hy= Hy, xa function of 
y on 

But Hy, j is a function of « and 8 only. 

This relation is equivalently — by the two equations 


a (log be =0; (log H,)= 0, 


aya8 


248 Mr. L. F. Richardson on a Freehand Graphic way 
whence by (Lamé, § xxx. 24) 


But if the surfaces y are planes, then by the equations 
(Lamé, § xxx. 24) already quoted 


2 (-)=0= 2(k) 


and consequently = and = are independent of y. But it 
a. 8 


is shown by Lamé (§ xxxviii.) that the curvature of the are 
of intersection of the surfaces « and 8 is equal to 


Mi ae: ie: 

(3) re =): 

So that if the radius of curvature of this are be p then p is 
independent of y. But p is equal to the length of the normal 
from the point considered onto the line of ultimate inter- 
section of two consecutive planes of the family y which pass 
one on either side of the point considered. As the plane 
moves this length must remain constant. And as this is to be 
true for every point in space, it is easy to see that if the 
surfaces y are planes they must intersect in a common axis. 
We have in this case symmetry about an axis. Or if the 
axis be at an infinite distance, the planes are parallei, and we 
have V independent of one of the Cartesian coordinates 
x,y, z. But if, quite generally, the surfaces y are spheres 


yy 
we have only 2 independent of y and therefore S inde- 
B , 
pendent of y. If the centres of the spheres y lie in a strane 
line, then since the orthogonal traces of the surfaces « and 8 
on a sphere y may turn round anyhow, we may choose for 8 
the planes intersecting in the line of the centres of the 
spheres y. Then r¥=0, and consequently 7% is independent 
of y so that the traces of «=const. on the planes B are 
circles. This is the system of toroidal coordinates which has 
been treated by Professor Hicks in Phil. Trans. 1881, 
Part II. Now the above reasoning would lead us to expect 
in these a type of symmetry which can be dealt with by two 
coordinates—other than symmetry about an axis. But on 
referring to Hicks’s formule it is easy to show that this is not 
possible for if V be made independent of either of those two 
of his coordinates which determine position in a plane passing 
through the axis, then the other of these two will not divide 


of determining Stream Lines and Equipotentials. 249 


out of the equation V7? V=0, so that all three coordinates 
must still be present in the integrals. Clearly then, our 
deductions, though necessary, are not sufficient. I have little 
doubt that the omission lies in this: that to leave us unre- 
stricted as to the direction of the orthogonal traces of « and 
8 upon the surfaces y, it is not sufficient that the surfaces v 
should be spheres. For the curves normal to y which we 
have called the “ guiding lines”? must be such that they form 
one set of lines of curvature of any surface whatever passing 
through them. To satisfy this condition it seems likely that 
except when the radius of the spheres y is infinite, the 
guiding lines will have to be straight and the spheres con- 
centric. This is the symmetry when V is independent of the 
radius in spherical coordinates, but may vary anyhow with the 
latitude and longitude. 

The only case remaining uninvestigated is that in which 
the surfaces y are spheres with centres which do not lie on a 
straight line. | 

By this application of Lamé’s formule, aided by those due 
to Hicks, we have discovered no new type of symmetry 
which allows two coordinates to be used instead of three. 
We have proved that within the stated limits the well- 
known types are the only possible ones. A summary of 
these may be useful. | 


Summary of Types of Symmetry when the guiding lines are 
orthogonal to a family of surfaces. 


If y°V is made equal 
to f(V, a, 8) over one Analytical 


Guiding lines. Ee anor mina. surface y, its value methods. 
on the others will be 


functions. 


Parallel straights. Constant. v2V=/(V, a, 8). | Conjugate 
| Zonal harmonies 


Circles with their of the 
centresonacom-| Proportional to . cylindrical, 
‘mon axis and distance V VE, ea). | spherical, 
their planes nor-| _— from axis. | spheroidal, 
mal thereto. aud toroidal 
! systems. 
A rae a, Bp | 
Radii from a com- Wi= Pe / 


3 Constant. . . 
mon point. where 7 is the distance. 


from the radiant point. 


t 
EEE 


ee ee “a 
! hs 


250 Mr. L. F. Richardson on a Freehand Graphic way 


* ‘Example of Symmetry about an Aais.—Byerly in his 
‘Fourier’s Series and Spherical Harmonies,’ p. 230, sets 
the following problem :—‘“ A cylinder of radius one metre 
and altitude one metre has its upper surface kept at tem- 
perature 100°, and its base and convex surface at the 
temperature 15°, until the stationary temperature is set up. 
Find the temperature at points on the axis 25 cm., 50 em., 
and 75 cm. from the base. and also at a point 25 cm. from 
the base and 50 cm. from the axis.”” To solve this the first 
thing necessary is to prepare a chart bearing chequers of the 
appropriate shape for each distance from the axis. The 
graph of any solution of V?V=0 symmetrical about an axis 
would serve this purpose. or example several of the figures 
out of Maxwell’s ‘ Electricity and Magnetism’ would do. But 
I preferred to prepare a standard chart by ruling equidistant 
parallel equipotentials normal to the axis of revolution, and 
then stream-lines parallel to the axis at distances from it 
proportional to the square roots of the natural numbers 
0, 1, 2, 3, 4, 5, &. The cross section of the cylindrical 
shell enclosed between successive stream-lines is then the 
same for every pair, aad the chequer ratio proportional to 
the distance from the axis. This having been done in red 
ink, a sheet of tracing-paper was pinned over it, the section 
of our given cylinder was drawn in black and equipotentials 
and lines of flow were drawn in pencil. These were then 
rubbed out and amended with the aim of making the pencil 
chequers everywhere very similar to the red rectangles 
underneath. When improvement becameslow, the blurredlines 
were made firm and definite with ink and the chequers con- 
sidered individually and marked as to whether they were too 
square or too thin. The lines were then drawn on a clean sheet 
of tracing-paper, the chequers again examined individually, 
and finally the lines fixed in ink (see fig. 3). Coordinate lines 
were then ruled and the values of V at their intersections were 
read from the graph. This process, from the ruling in of 
the given contour to the determination of V in numbers, 
took me four hours. The analytical method would perhaps 
have been more rapid im this case; but for an irregular 
shaped contour with an irregular boundary distribution 
the freehand solution would still take about the same 
time, while analytical methods may be almost indefinitely 


tedious, 


251 


of determining Stream Lines and Equipotentials, 
Fig. 3. 


The results are tabulated below :— 


Distance from base of cylinder .... 25 50 75 25 cms. 
: axis a pas 0 0 0 50; %, 
Potential read from the graph as | 175 88 =~ 64 14 


decimal of unit range. 
Last multiplied by the actual range 149 333 544 119 


(100°— 15°) 
Last +15°=actual temperature .. 29°99 473 69-4 269 


Not until these numbers had been written down did I look 
at the correct values found by Byerly from the Bessel function 


series, namely :— 
oe AG 712 25'8 


Errors of freehand determination., +°3 —-—°3 -—‘8 +11 


Disregarding signs the mean of these errors is ‘63 
which is equal to 0°74 per cent. of the range of 85°. 


ae eS 


252 Mr. L. F. Richardson on a Freehand Graphic way 
Example of the type of symmetry when V is independent of 


the radius in spherical coordinates.—“ On a uniform spherical 
shell there are equal sources at the north and south poles 
and equal sinks at the extremities of a diameter lying in the 
equatorial plane. ‘The sources and sinks send out and receive 
uniformly in all directions. The flux has no divergence except 
at the sources and sinks and no curl anywhere. Find the dis- 
tribution of potential onthe surface.” Todo this we might draw 
orthogonal lines on the surface of a globe so as to make the 
chequers ratio constant. Or because, in Mercator’s projection, 
any small part on the globe transforms into a small part of 
the same shape on the map, we may transform the boundary 
conditions and obtain the required solution by drawing 
chequers of constant chequer ratio on the map. Blank 
Mercator projections suitable for this work may be obtained 
from George Philip & Son, Fleet Street. In the present 
example the lines of flow radiating from the pole become lines 
straight, parallel and equidistant at infinity. And as the 
graph progressed it was found that by their symmetry 
with the sinks on the equator, the foregoing condition must 
be very nearly satisfied at 10° from the poles, a region which 
is within the confines of the map. Again, in this case it is 
only necessary to determine V in one octant of the sphere, 
and symmetry helps us in other ways. The accompanying 
graph (fig. 4) is the best of four or five separate attempts. 
The time taken to make these was collectively four hours. 
Special attention was given to the equipotential curve which 
passes mid-way between the two equatorial sinks, and as the 
result of the aforesaid trials it was found to pass through a 
point 44° due north of the sink on the equator. This suggested 
that the true value should be 45°, and on looking at a sphere 
this is seen to follow from symmetry although it was not 
obvious on the map. Thus again we have a confirmation of 
the passable accuracy of the graphic method—the error her 
is 1 degree in 90 or 11 per cent. of the range. j 
So far we have only treated the problem as relating to a 
spherical shell. But we may next suppose the sphere solid 
and V to be independent of the radius. We will then have 
a solution of Laplace’s equation in space. Since the chequer 
ratio is constant, the magnitude of the flux is inversely pro- 
portional to the linear dimensions of the chequer (on the 
sphere not on the map) and is consequently proportional to 


i on : . : 
— along any guiding line, r beg the radius. But if we draw 
r ; 


any small cone enclosing the polar axis—which is now a line 
source—we see that the outflow between two spheres 7 and 


of determining Stream Lines and EHquipotentials. 253 


r+6r is proportional to 6r x (magnitude of flux) x (peri- 
meter of the trace of the cone), and by the above this is pro- 
portional simply to 6r. Therefore the polar axis must be a 


60 Vi 


ae 
ih 
m 
— 
a 
G 
2 
ae 
28 


Bema eee 2 Lie 


[> A 
THA 


line source of uniform strength and similarly for any other 
source or sink when the guiding lines are straights passing 
through a common point—the strength must be independent 
of the radius. 

_ Case (6).—General method for conduction in a thin shell 
of any shape, the thickness and conductivity being any given 


Phil. Mag. 8. 6. Vol. 15. No. 86. Feb. 1908. T 


254 Mr. L. F. Richardson on a Freehand Graphic way 


functions of position on its surface, and all conditions being 
constant throughout the thickness of the shell at any point of 
its surface. ‘Take a solid bounded by a surface of the shape 
of the shell and draw small rectangles at numerous points of 
length along flow 
breadth across flow 
is directly proportional to the product of the thickness and 
conductivity at each point of the shell. For then the flow 
through each chequer will be the same. Suppose that these 
standard chequers are in some distinctive colour, say red. 
Now lay off in black the boundary conditions of the special 
problem and draw a black chequerwork to have the same 
chequer ratio as the red at each point, much as was done for 
- symmetry about an axis. The standard red chequers need 
not be connected so as to form two systems of orthogonal 
lines but may be scattered anyhow over the surface, all that 
is necessary is that they should be sufficiently small and 
numerous. 

Or it may be convenient to use a projection of the surface 
as was done in the case of the spherical shell above. 

Case (c).— When there exists no family of surfaces normal 
to the guiding lines. Without pausing for generalities we 
will proceed at once to :— 

Screw symmetry about an axis.—Let us discuss this with 
the aid of cylindrical coordinates r, ¢, 2. Ata point P on 
the axis OZ let a perpendicular be drawn extending to 
infinity. This perpendicular, which isto project only on one 
side of the axis, is imagined to revolve round the axis and 
slide along the same with proportional velocities. In one 
rotation round the axis let it move / along the axis. Then 
the line sweeps out a surface, at all points of which the 


the surface, so that their chequer ratio= 


expression z — Far ¢ is constant. Let us put z— jb =o. 


Then as @ varies we pass from one of these screw surfaces 
to another formed by shifting the first parallel to z. The 
range of the coordinate » is from 0 to/. The intersections 
of w=const. with the cylinders r=const. are a family of 
screw-threads. ; 

Let dn be an element of distance measured along any 

Lae dyn _ length of turn of screw 
screw guiding line, so that a a 
afunction of r only. And let us make V a linear function of 
distance along each screw-thread so that OV =a function 
of r only, o” 


of determining Stream Lines and Equipotentials. 255 


Then as dy, dw, and dr are in perpendicular directions 
they are independent and 
BV _ dV Bo , BV dn, BV a 
@ 0° G2. me oc or 02’ 
OV ivav 


which reduces to S2 Fae + a function of r only. 
Therefore ; 
OO OV Qo 3 on 0\0V 
32 ~3:(30) = (3230 T3230) 30 
_3'V 7 8 @V)_ BV 
J dwt 0- de \dn/ Oe?” 
Again, 
wy OV oO OV ops (b0V¥ 
d¢ do 06 9 db 27 do ae ie 
Since . 
a emnet. of = pe Screw a function of x only. 
And so 
OV _1 9 V)_ 1 (Quo , dx By—v 
O¢? 27 d¢\da@/” 27\0d' do OP On/ de 
ee. ow 
ae ee 


ov 


Now substitute these values of oo ae a ond se in the ex- 


pression for VV in cylindrical a and we have 


eo Yeo ‘ala 1 chs 
i Or ae Or +(1+ dor?) Bw?’ 
which contains only two coordinates r and w. So that if we 


make V7V=/(V, 7, @) over any surface the same will be 
true throughout fin whole region filled by the screw-threads 


passing through the surface, provided that ov is such as to 
2V 

make a - + a constant along every guiding screw. 

One way of phos this is to make V increase by the same 

amount per turn of ee screw, along every screw-thread and 


o7V 
so that ved and —— ot are both constant along every guiding 


SCrew,. 


T 2 


. ee 
- i: 
' sad @ 
<4 


256 Mr. L. F. Richardson on a Freehand Graphic way 
. In the following pages, except where specially indicated, 
we wil] consider only the case s = 

As this result does-not appear to~be given in the text- 
books, it may be well to confirm it by a slightly different line 
of reasoning, as follows. The tangent of the angle between 
the tangent to any screw-threads and a plane normal to the 
axis of symmetry is oe 


Therefore the first space rate of any function of position 
V along the tangent to a screw-thread is 


Fests Cody ih} 


af oF =a function of » only, eee every screw-thread 
then we have a function of r only +29 = ‘ —19% 


throughout the whole region. Therefore this last equation 
will still remain true after differentiation by ¢ or by g, thus 


6 20" Saye 
T Og? OG02” 25 
Ol Weaits, + aa 


Ofoz 24a” 
| 0’-V : 
Hquating the two values of bes thus obtained we have 


pee 2 92V 
f O¢” Ag? a2 7. 


which on substitution in the expression for V?V gives 


o7V 1S A fo. 
MONE Get r Or +(1 eS 


But it is now to be observed that if the distribution of V 
on any plane passing through the axis of symmetry is known, 
then V is determined ev erywhere. And on such a fixed 
plane the contours of <z are identical with those of w. So 
that we may replace z by @ in the last equation, and the 
previous result is confirmed. 

We have shown that if we make V?V= TCV, 7, @) over 


any surface intersecting all the screw-threads, the same will 


i 


of determining Stream Lines and Equipotentials. 257 
be true throughout the whole region, with the stated pro- 


. . V . e 
visions as to the value of oN . The geometrical meaning of 


0” 
this result is that if we draw any infinitesimal rectangle 
normal to one of the screw-threads and draw screw-threads 
through each of its four corners, then the infinitesimal tube 
thus formed will be everywhere rectangular in normal cross 
section, and more than this, the rectangle will have the same 
ratio of length to breadth and will be of the same size at all 
points along the tube. For if we consider one pair of opposite 
faces of the tube as equipotentials and the other pair as 
lines of flow, then these properties are seen to follow 
from the fact that YV?V is constant along a screw-thread 
when V is constant along the same. And indeed these pro- 


perties are immediately obvious from the appearance of the | 
system. : iw eo 

Consequently, if we take any family of surfaces « passing © 
through the guiding screw-threads, there will always ‘be an | 


orthogonal family of surfaces 8, also passing through the 


screw-threads. If the surfaces « are the contours drawn at — 
equal intervals of the potential V the surfaces 6 are stream- | 


surfaces. And =* may be named the “Chequer Ratio” 


H 


consistently with what has gone before. If V7« is to vanish 


we must have 


. ; ‘H, 
acl, x (length of portions of successive screw-threads =e 4 


between two stream-lines lying on the same stream-surface) 


Since the screw system is uniform the length of the - 


portions of successive screw-threads intercepted between two 
stream-lines lying on the same surface 8 can be proportional 


to nothing else than the length of one turn of the screw- — 
thread at the radius considered. For the two stream-lines’ | 


on the same screw-thread. So that the projection of the 
distance between the said pair of points onto the axis of 
the screw will be always the same fraction of / as the points 
move from one screw-thread to another. 

Now the length of an are ds of a screw-thread being ~ 


Vdz?- +72d6=d04 / Ue Be 
Aor? 


‘in question must by symmetry make equal angles with planes | 
normal to the axis of the screw, at each pair of points lying” 


258 Mr. L. F. Richardson on a Freehand Graphie way 


the length of one turn is 


Q=27 
{aman /F, +72 = n/ 14 a ; 
@=0 


Therefore we may satisfy the equation V22=0 by making 


fi. é Ary? ; ; | 
i, proportional to / jae —aT as the radius varies. A 


oe Ary? 
table giving the values of / 1+ 7B for a number of 
gs. 4 
values of zs annexed. 
“ 40°72 For standard chart. Z 
ar at Ss: 
c i 2ur 
Z times this equals arc | equals tangent of angle 
of one turn of thread. | H,. Hg. between helix and cirele. 
0 10000 | -5507 | 18157 © 
05 10482 _ 5639 17735 371831 
10 11811 | 5985 1:6707 15916 
4 *to 13741 | 6456 1:5490 1:0610 
-20 16060 _ 6979 1°43828 "7958 
30 21338 | "8045 1:2430 53052 
“40 2°7049 | 9058 1:1040 39789 
50 3°2969 — 10000 10000 31831 
60 3°9003 . |. 10877 9194 26526 
‘70 45105 | 11697 8550 22736 
80]. 5°1251 1:2468 “8021 "19894 
‘90 57426 13198 “T5717 ‘17684 
1:00 6°3623 | 1:3892 ‘7199 "15916 
15 9478 16955 D898 ‘10610 
2:0 12°606 19554 | “5114 07958 
25 15740 | 2°1850 4577 06366 
30 18876 | 2:3928 ‘4179 05305 
oo 22°014 | 2°5840 3870 04547 
00 00 a 0 00 


It is hoped that these values are correct to less than 
half a unit in the last place. 


As there is no surface normal to the screw-threads, it is 
not possible to draw standard rectangles of the appropriate 
chequer ratio for each distance from the axis. But as the 
whole distribution of V is determined when the section of it 
by a plane passing through the axis of the screw is known, 


of determining Stream Lines and Equipotentials. 259 


we may draw on this plane the sections of tubes formed by 


Tee : 
the surfaces « and @ in such a way that “7 is proportional to. 


B 
ry, Buea 


The sections of these tubes will in general not be rectangles; 
in fact, the angles and ratio of sides of the chequers formed 
by the traces of a and 8 on the plane $=constant will both 
now depend on the orientation of the chequer as well as on 
its distance from the axis. It will therefore be necessary 
to make a chart of standard chequers in various orientations 
at a number of distances from the axis. Plate XII. is such 
a chart. The rectangles in the right-hand margin represent 
normal cross sections of the tubes formed by the surfaces 
aand 8. Ina line with each of these are five sections of a 
tube of the same size and shape by the plane of the paper, 
when the angle between one face of the tube and the normal 
to the axis of the screw is successively 0°, 224°, 45°, 674°, 
90°. In order to be clearly visible these parallelograms are. 
drawn quite large. What each really represents is the 
shape of an infinitesimal chequer situated at the central 
point of the large one. Practically the difference will not be. 
important. 

Now this standard diagram can be covered by a sheet of 
tracing-paper, and two intersecting families of lines drawn on 
the tracing-paper in such a way that the parallelograms formed 
by them are everywhere similar to the chequers underneath, 
which have the same distance from the axis and the same 
orientation on the paper. Then if this tracing-paper plane. 
rotate round the axis and slide along it so as to follow the 
guiding lines, the equipotential lines on the paper will 
sweep out the contours at equal intervals of V in space in 
such a way that V?V =0 and the other family of lines will 
sweep out stream-surfaces. — 

A quantity which it is frequently necessary to determine 
is the magnitude of the flux 


V5.) + (8) + Ge) =a; 


Since bal al An 
Suge Oh 


260 Mr. L. F. Richardson on a Freehand Graphic way 


where A is an absolute constant, we must have 
| Aqr?y? 
iu / Taig 


So that @ is a stream function analogous to the forms in use 

when the guiding lines are parallel straights or circles with 

their centres on, and their planes normal to, a common axis. 
In types previously studied, when the graph was drawn 


on a surface normal to the guiding lines, H, and H, were 


proportional to the length and breadth of a chequer and 
could be measured directly. But here we must first compare 
the linear dimensions of a freehand chequer with those of 
the standard oblique section of the tube bounded by two 
stream-surfaces and two equipotentials, and then refer to the 
normal section of the same tube in the right-hand margin of 
the chart. 

The standard chequers were obtained in the following 
manner :— ) 
seat Lt tant 4 Le | 
a elng equal to constant X ike P , some other re- 
lation is necessary to determine H, and H separately. The 
relation H, x H, = 1 was chosen for this purpose, as this 


gives a neat appearance to the standard chart. It was also 


H 
found convenient to make the constant such that —= =1 


H, 
when 5 =0°5. The values of H, and H, were calculated 


and are given in the accompanying table. The sides of the 
rectangles in the right-hand margin of the standard chart 
were drawn proportional to 2H, and 2H,. 

To obtain the slant section, the tangent of the angle 
between the tangent a guiding-line and the plane normal to 
the axis of the screw, was first calculated. It is equal to 
= and is given in the table under that head. The rect- 
angles were then projected with ruler and compasses in a 
manner which is perhaps sufficiently indicated by fig. 5, 


which shows the construction when 7 =0-05 and the angle 


between a radius from the axis of symmetry and the tangent 
plane to the surface «=constant meeting at the point con- 
sidered is 45°. 


= : is 
a 
- 


of determining Stream Lines and Equipotentials. 261 


Of the innumerable solutions of V?V =0 possessing screw 
symmetry of the sort described, which may be obtained by 
the aid of this standard chart, perhaps the simplest is the 
field due to a helical line source, such for example as the 
distribution of temperature ina mass of electrically insulating 
material which encloses a helical copper wire carrying an 
electric current. ‘lo avoid the introduction of a difficulty 
not characteristic of screw symmetry. I have assumed a core 
of non-conducting material in the form of a circular cylinder 
surrounding the axis. This relieves us of the necessity of 
considering the axial line of equilibrium, which would other- 
wise have to be treated by an extension of the method in 
Section IV. The external surface of the medium is also 
taken as a circular cylinder and is assumed to be at constant 
temperature. Consistently with our boundary conditions 


we may suppose that a =0. Now symmetry will help us 


in several ways, for since the chequer ratio on the standard 
chart is the same whether any particular half-turn of the 
screw passes over or under the chart, one sees on beginning 


262 Mr. L. F. Richardson on a Freehand Graphic way 


to make the drawing, that the two surfaces w = const., which 
pass respectively through the electric current and half-way 
between two adjacent turns of the current, must be surfaces 
of flow. Again, very close to the electric current the flow of 
heat will be nearly the same as that due toa straight current 
tangential to the helix, that is to say, the lines of flow will 
be normals to the helix and the isothermals will approximate 
to circular cylinders concentric about the tangent. 

The particular dimensions chosen were “= 0°05 for the 


core, 0°3 for the source, and 0°5 for the outer cylinder. 
Fie. 6. 


5 
A 


Beh 


oF THE SCREWS 


Owing to orientation of the chequer affecting its shape 
this graph took twice or thrice as long to adjust as did the 
others in this paper. Its errors are discussed in Section VIII. 
hereafter. 

The magnetic field due to the helical current may doubt- 


ae sas Vie 
less be determined in a very similar manner. Here will 


be a constant other than zero and the cyclical properties of 
the field will add a further complication. 


of determining Stream Lines and Equipotentials. 263 


IV. 


We have hitherto passed over without mention the pecu- 
liarities relating to points of equilibrium—these are points 


at which the first space-rate of the potential vanishes in 


all directions. In the neighbourhood of these the chequers 
become unusually large, and if any chequer goes right up to 
an equilibrium point it will not have the shape characteristic 


of its neighbours, but will take a peculiar form of its own. 


There are several diagrams of this in Maxwell’s ‘ Electricity 
and Magnetism.’ See, for example, vol. ii. fig. xvii. 

Now if V be expressed in terms of rectangular coordinates 
u and v lying in the plane of the graph with their origin at 
the equilibrium point, then linear terms in V must vanish, 
and we have 


V=Ar? + Bo + Co? + Ev? + Fu?v + Guv? + terms of higher 
degree. 
Now let us make \/?V vanish. 
For guiding lines parallel straight and normal to the plane 


of the graph 


OV OV 
g + Pg 


When the graph is on a plane passing through an axis about 
which there is symmetry of revolution and ~ is normal to 
this axis, we must add to the above value of VV the term 


V7V= 


= 2(A+C) +u(6E +2G)+2(2F+6H). 


Ou 


where 7 is distance from the axis. 

Now when the point considered is not on or close to the 
axis, it will be possible to put in so many chequers that the 
first two chequers in any direction from the equilibrium 
point require for their measurement so small a range of u 


and v that the fractions = <, &c., will be small, and therefore 


the additional terms which come in for symmetry about an 
axis may be neglected, and we have the same form for V?V 
in both cases. 

Further, since Mercator’s projection does not alter the 
shape of any small pieces, V/?V will have the same form in 
the neighbourhood of an equilibrinm point on the Mercator’s 
plan of the distribution on a sphere. 


264 Mr. L. F. Richardson on a Freehand Graphic way 
This being so, the general form of V in all three cases is 
V=a(w—v?) +b. uvt+g(ui —3uv?) + h(v?—3u?v) + higher terms, 


where a, 6, g, and h are arbitrary constants. When the 
ratios of a and b to the succeeding coefficients do not vanish, 
then the first two terms are all that we need consider. Now 
it may easily be shown that by a proper rotation of the axes 
of reference, so that wv tranform to w,v,, the sum of these 
two terms may be transformed into either of them separately. 
‘We need therefore only consider one, say bw,v;. The con- 
tours of this function are hyperbolas and are orthogonal to 
2 

b 
ratio in the neighbourhood of the equilibrium point. 

A graph of this function for the special case of unit 
chequer ratio is given in Webster’s ‘ Dynamics,’ p. 525, and 
shows that two equipotentials meet at right angles at the 
equilibrium point, and that two stream-lines also pass through 
the same point and bisect the angles between the equi- 
potentials. The eight curved chequers which meet in the point 
each have consequently three corners of 90° and one of 45°. 
A graph of this function may be used as a “ standard equili- 
brium point” to keep the eye informed of the necessary 
proportions of the first and second ring of chequers sur- 
rounding the point. ) | 

If, however, the coefficients a and } vanish, while g and h 
do not, then the terms of the 3rd degree become all 
important. 

By rotating the axes the sum of the two terms of the 3rd 
degree may be reduced to either separately. A rough graph 
of the contours of these functions is given by Fiske in 
Merriman & Woodward’s ‘ Higher Mathematics,’ p. 248. 
Here three equipotentials intersect in the equilibrium point. 
And three stream-lines bisect the angles of 60° which are 
formed in this way. | 
~ Now when a graph has to be drawn and is found to contain 
an equilibrium point, the general arrangement of the potential 
will give us the clue as to whether two, three, or more equi- 
potentials intersect in the equilibrium point. And this being 
known, we have only to draw in the standard type at the 
proper dimensions and chequer ratio. 

When the graph is drawn on a plane passing through an 
axis about which there is screw symmetry of the sort described 
in Section IIIc, then the appearance is different, for we 
have to add to the value of \/?V for circular symmetry about 


those of a(u,?—v,°). The ratio > is determined by the chequer 


_of determining Stream Lines and Equipotentials. 


bo 
for) 
eu 


an axis the term 


[2 3°V 2 
ee = ree, vH).. 
Tee 32 ine C+ 2uG+6vH) 


And therefore writing 
[? _p 


Agr)? 


Lee 
we have 
— V=a(R2P— 0?) +b uv t9( R28 —3ur) +h —3 Rv). 


A simpler way of looking at the matter is to consider a tiny 
plane element normal to the guiding screw which forms the 
line of equilibrium. The normals to the surfaces w=const. 
lie in this plane. If d§,, distance along such a normal, then 


——— eed 


do=d Wp Oo) (198) =as.y/ ML 
$55 4Ra ee ie r Od a 


Substituting this in the expression of VV in terms of 7 and 
@ we have | 
oo have id? V 
INT aS tes Se as = 
i — Or ah r Or OS,” 


just as if 8, was 2 in circular symmetry about an axis. 
From this we see that the appearance of the equilibrium 
point on a small plane element normal to the guiding screw 
will be exactly similar to the forms already dealt with. Its 
appearance on a plane which passes through the axis of the 
screw inay be sketched without much difficulty by comparing 
the chequers in the right-hand margin of the standard chart 
with their projections as drawn in the middle of the chart. 


V. Hquations other than Laplace’s. 


It has been shown above that in order to solve the equation 


27 = 
ot = a = any given function of V, a, y, 


a relation between differences of chequer ratios has to be 
satisfied. And the same will be found to be true for the 
other forms of the equation V*V = a function (of V and of 
position) which can be treated by two coordinates. <A 
difference relation of the sort referred to would involve the 
comparison of each chequer with a standard set having 
graded chequer ratios, followed by the calculation of V*V by 


266 Mr. L. F. Richardson on a Freehand Graphie way 


arithmetic. And although it would doubtless be possible to 
carry out the necessary operations, yet it would almost cer- 
tainly be quicker and more accurate to use-arithmetical finite 
differences altogether, writing in the numerical values of V 
at a set of points on the paper and adjusting these numbers 
until the finite difference equation is satisfied,—in a manner 
which may be described in a future paper. In view of this 
I will not attempt to elaborate freehand methods for V7V= 
a given function of V and of position. 

There are, however, certain common space distributions 
which may be treated graphically with simplicity although 
they do not satisfy V*V =0. 

Firstly, when the conductivity is a continuous function of 
position, and the direction of the flux is normal to the contours 
of a potential, and the magnitude of the flux is the maximum 
space-rate of the potential multiplied by the conductivity, 
and the flux has no divergence. For example: the flow of 
heat and electricity in isotropic but non-homogeneous bodies, 
or the soakage of water in a saturated subsoil the upper 
layers of which are more porous than those below. Let K 
be the conductivity and suppose that it is constant along each 
guiding line but varies from one such line to another. 
Then, when the lines have a family of surfaces normal to 
them we must have 


0 Hg. Hy. Ky, . 
a4) ani )=0 


in order that the flux shall not diverge. This is very easily 
assured by preparing the paper with standard chequers 


having their chequer ratio at proportional to H,.K. In 
fact, we have an example of this in Section [Ila above ; 
for circular symmetry about an axis may be regarded for 
this purpose as flow between parallel planes in a medium 
having conductivity directly proportional to the distance 
from the axis. And reciprocally. 

Similarly in the case of screw symmetry, standard chequers - 


are to be prepared having 
= proportional to Ky / 1 — 
Two other cases can probably be treated freehand, namely, 


the flow of heat in bodies where the conductivity varies wtth the 
temperature, and, of great practical importance, the distribution 


~ of determining Stream Lines and Equipotentials. 267 


of. magnetic induction in soft iron, taking into account the 
variation of the permeability with the force. But these again 
will be left to those who need the results. 


VI. Note on Boundary Conditions, 


It may be convenient to the reader if we bring together 
certain well-known facts concerning boundary conditions. 

Let us regard V simply as a function of position, not 
necessarily satisfying (77V=0 or any other equation ; and, 
as always, let contours be drawn at small intervals of V each 
equal to k. Then the first space-rate of V in any direction 
at a point is inversely as the intercept cut off from a line in 
that direction by two contours of V one on each side of the 
point, and is directly as K. Suppose, further, that the whole 
distribution of V can be represented by a single graph. 

1. If we have to make V such that the magnitude and 
direction of its maximum first space-rate, the Hamiltonian 
vector \/V, satisfies given values over a boundary of a given 
shape. ‘Then it is easy to set off the ends of the contours of 
V with a ruler and scale, for their directions are known and 
also the distance apart of successive pairs. 

2. If we are not given V/V over the boundary but only 
the first space-rate of V ina given direction. Then there are 
an indefinite number of ways in which the contours of V may 
cut the boundary ; and as it will not generally be possible to 
say which of these is consistent with the internal conditions, 
they must be drawn and modified freehand as the approxi- 
mation to the internal conditions proceeds. This is usually 
not difficult. | 

3. To make V continuous at any surface cutting the distri- 
bution, all that is necessary is that the ends of the contours 
of V approaching from the two sides should meet one 

another at this surface. Whether they meet at an angle or 
not does not matter. 

4. To make the jirst space-rates of V in every direction con- 
tinuous at any surface where V is continuous, not only must 
the contours of V meet one another, but they must pass 
smoothly into one another without making an angle. For if 
they made an angle and a straight line were drawn tangent 
to one branch of the contours at the angle, then the ratio of 
successive intercepts of this line by the contours of V would 
not become unity when the contours were drawn at indefi- 
nitely small intervals of V,so that the second space-rate along 
this straight would be indefinite at the angle. 

5. Suppose next that a non-divergent vector is normal to 


268 Mr. L. F. Richardson on a Freehand Graphic way 


the surfaces V=const., and that the magnitude of the vector 
is equal to the space-rate of V along the said normal, 
multiplied by a scalar function of position ; which according 
to the particular application will be the conductivity, perme- 
ability or some other specific constant. Then we may require 
the conditions which must hold at a boundary where the 
specific constant has a discontinuity while V is continuous—as, 
for example, where magnetic flux passes from air into mild 
steel. These conditions, which I take from Prof. J. J. 
Thomson’s ‘ Elements of Hlectricity and Magnetism,’ may be 
stated thus :—If K,, K, are the aforesaid specific constants 
on the two sides of the boundary, and 6, and @, are the 
corresponding angles which the direction of the vector makes 
with the bounding surface, then : . 

is i 

K fan @, = K tan 05. 

Now if the graph be drawn on a surface which is normal 
to the guiding lines, the direction of the vector lies in the 
graph, and 6; and @, are the actual angles which one sees. 
The same is true of the Mercator’s map of a spherical surface 
distribution, since the angles are unchanged by projection. 
But with our method for screw symmetry, the angles 6; and 
8, do not immediately appear, and comparison must be made 
with the angles of the slant sections of the rectangular tubes 
given on the standard chart. 


VIL. Ascellaneous Notes on Draughtsmanship. 


(a) Since with the exception of given boundaries and lines 
deduced from symmetry no part of the field can be said to be 
correct until the whole field is correct, it is advisable to begin 
by covering the whole field with intersecting lines, however 
erroneous they may be, and then to carry out amendments 
over wide areas at one time. 

(>) In the tinal stages of a drawing intended improvements 
often overshoot the mark or cause unforeseen disturbances in 
the surrounding chequers. It seems well, therefore, to lay 
aside the indiarubber after a certain accuracy has been 
reached, and, placing a sheet of tracing-paper over the rough 
diagram, to draw the intended improvements upon this. 
And so with all later stages. The tracing-paper diagrams 
are then compared with one another and the best selected. 

(c) The graphic addition of two scalar functions of position 
is conveniently performed in the way described by Maxwell 
(‘ Elementary Treatise on Electricity °) by laying the contours 
drawn on a sheet of tracing-paper at equal intervals of the 
one over those of the other, covering the two with a clean 


of determining Stream Lines and Equipotentials. 269 


sheet of tracing-paper, and drawing the diagonals of the 
chequers formed by the intersecting contours. 


VIII. Estimation of Errors. 


To one reading an account of this freehand method without 
having worked an example, it might seem as if there were no 
way of setting a limit to the errors of any particular graph. 
This, if it were true, would be a serious fault. But, happily, 
it is not so: for it is commonly necessary to make several 
drawings and then select the best of them: so that by the 
time the draughtsman has reached a drawing which he can 
scarcely improve upon, he has before him deviations from it 
in divers directions. The difference, then, between the 
selected graph and the second best graphs is a measure of 
the errors of the latter and an outside limit to the errors of 
the former. The actual errors of the selected graph will be 
less than this limit, and may be estimated by comparing the 
errors in the shape of the individual chequers in the best and 
second best graphs, and taking a fraction, thus :— 


individual chequer error in best graph Nev eeaeee between best al 
same in second best graph second best graphs H 

This is the true measure of the errors of the best graph. 
It depends, of course, on a general mental estimate or appreci- 
ation, and is consequently not susceptible of exact definition. 
But this does not much matter, for if the value of an error 
be known within two times either way it is usually sufficient. 

The difference between the best and second best graphs is 
less dependent on a mental estimate, and consequently sets a 
firmer limit to the possible error. 

Taking, for example, the graph of the field round a helical 
line source given in section III. c¢, and laying over it the 
tracing of the unpublished second-best graph, one sees that 
the difference in position of the lines in the two graphs 
nowhere exceeds 4 the linear dimension of the chequer, at the 
point and in the direction considered. Now I should estimate 
that the error of the shape of individual chequers in the 
published graph averaged 4 of the same quantity in the other ; 
so that 4 of the linear dimensions of the chequer may be taken 
as the error of position of the lines in the published graph. 
Now the graph exhibits ten tubes of flow ; so that + of one 
tube is 24 per cent of the range. This is in the worst parts 
of the field. Elsewhere the error will be less, but it may 
still be expected to exceed the errors found when the graph 
is drawn on a surface normal to the guiding lines, because in 
the case of screw symmetry we have the added difficulty that 
the shape of the chequers depend upon its orientation. 


Phil. Mag. 8. 6. Vol. 15. No. 86. Feb. 1968. U 


[ 270 | 


XXIT. Anomalous Magnetic Rotatory Dispersion of Neo- 
dymium. By R. W. Woon, Professor of Experimental 
Physics in the Johns Hopkins University *. 


[Plate XIII.] 


HERE has been a good deal of discussion as to whether 
anomalous dispersion of the rotatory polarization oceurs 
in the vicinity of the absorption-bands of the rare earths. 
The work of Bates, as well as my own, appears to have proven 
beyond much question that the aniline dyes do not exhibit 
the phenomenon, as was claimed by Schmauss, who published 
curves which apparently showed very pronounced anomalies 
at the absorption-bands. Schmauss also investigated the 
magnetic rotation of solutions of the rare earths (Annalen 
der Physik, x. p. 853, 1903), and found that there was, 
in every case, an abnormal increase of the rotation on 
the red side of the bands, and an abnormal decrease on the 
blue side, as in the case of ordinary anomalous dispersion. 
Bates was unable to find any anomalies whatever, but an 
examination which I made of a solution of praseodymium 
appeared to confirm, at least in a measure, the results of 
Schmauss (Phil. Mag. May 1905; Phys. Zeit. vi. p. 416, 
1905). 

i I stated in my earlier paper, it was difficult to get 
results in which one could feel absolute confidence, owing to 
the great experimental difficulties. I had no polarimeter of 
precision, and was obliged to work with a pair of nicols only, 
and determine the rotation of the plane by the method of 
extinction. 

If one could deal with rotations of sufficient magnitude to 
enable-one to obtain a dark band in the spectrum of the 
transmitted light, one could feel sure of the results, and this 
I have at last succeeded in doing. To my great surprise, 
however, in the case of the one band which has been carefully 


studied, the anomaly is the same as with sodium vapour, that — 


is the rotation increases very rapidly as the absorption-band 
is approached from the short wave-length side. It has been 
found possible to set the nicols for extinction and cause a 
restoration of the light in the vicinity of the absorption-band 
by the excitation of the magnet, as can be so beautifully done 
with the non-luminous vapour of metallic sodiumf. This is 
proof positive that the phenomenon exists, and I have even 


* Communicated by the Author. 

+ Macaluso and Corbino, Rend. Real. Ac. det Lincei (5) vii. p. 293. 
Wood, “Magnetic Rotation of Sodium Vapour,” Phil. Mag. Oct. 1905; 
and July 1907. 


a 


Anomalous Rotatory Dispersion of Neodymium. 271 


ania in photographing the spectrum of the restored 
ight. 

The neodymium nitrate was prepared in the form of a 
solid film, amorphous and isotropic, pressed between two very 
thin cover-glasses. The nitrate was prepared from the 
double ammonium salt by precipitation with oxalic acid, 
ignition, and treatment with nitric acid. The nitrate, freed 


from acid, was boiled down to the consistency of syrup, in 


which state a drop placed on a strip of glass will immediately 
solidify into a clear glass bead without crystallization, a fact 
found by Anderson in the course of his investigations upon 


the absorption spectra of the rare earths. By forming a 


prism of about 40 degrees of this substance between plates 


of heated glass, I have detected anomalous dispersion at the 


absorption-band at wave-length 5790, something which I 
have never been able to do with solutions, even with com- 
pensated prisms. 

The films are not at all difficult to prepare. The cover- 
glasses are heated and a small drop of the molten substance 
pressed out between them. If crystallization occurs, it in- 
dicates that the solution has not been sufficiently boiled down. 
Films varying in thickness from 0:1 to 0°5 mm. were used, 


the best results being obtained with a moderate thickness. 


The light from an arc-lamp was passed through a nicol 
and the cores of a large Ruhmkorff magnet, between the 
poles of which the film was mounted, then through a second 
nicol, after which it was concentrated on the slit of a spectro- 
scope. The nicols were set for extinction, which ‘could be 
done to within a tenth of a degree, after which the magnet 
was excited. A restoration of the light immediately occurred 
throughout the whole range of the spectrum in the case of 
the thickest films, the intensity being greatest however in 
the vicinity of the absorption-band. With thinner films the 
restoration was confined to the immediate vicinity of the 
band, a bright and very narrow line shining out exactly in 
coincidence with the centre of the absorption-band, with 


fainter bands on each side (PI. XIII. fig.1). The direction of 


the rotation was the opposite of that produced by a plate of glass 
placed between the poles of the magnet, and was about four 
times as great. It immediately occurred to me that what 
we may term the rotation of the substance as a whole (that 
is the rotation due to infra-red and ultra-violet electrons) 
could be practically compensated by employing glass plates 
of the requisite thickness, leaving an outstanding effect due 
solely to the *absorption-bands which fell within the limits of 
the visible spectrum; and this method was successfully 
U2 


(272 ~~ Prof. R. W. Wood on the Anomalous Magnetic 


employed in some of the work, though there does not appear 
to be any especial benefit derived from its nse. 

As is well known, when the nicols are set for extinction, 
and the light is restored in the spectroscope by the excitation 
of the magnet, a dark band enters the red end of the spectrum 
and moves towards the violet, when the analysing nicol is 
turned in the direction in which the rotation of the plane of 
polarization has occurred. ‘This means of course that the 
rotation increases as the wave-length decreases. In the case 
of the neodymium film, the reverse was found to be true, the 
band moving up from the blue towards the yellow, until it 
was driven into the absorption-band and lost, indicating that 
the rotation is abnormally great on the short wave-length 
side of the band, as is the case with the vapour of sodium. 
No very marked effect could be detected on the red side. 

We will now examine in detail some of the effects which 
are seen with films of varying thickness. 

The phenomena which are to be discussed occur at the 
absorption-band at wave-length 5790, and as the appearance 
of the band is a little different from that exhibited by solu- 
tions of the salt it seemed best to photograph it. A film was 
pressed between two plates of glass in the form of a very 
acute prism, which when placed in front of the spectrograph 
slit, and illuminated with white light, gave a photograph 
which showed the actual form of the absorption curve. The 
absorption-band is seen to be double, a strong one with its 
centre at 5790, and a fainter one close to it and on the blue 
side (fig. 1). A barium-iron arc was impressed on the same 
plate as a comparison spectrum. 

When a very thin film is placed between the poles of the 
magnet and the current turned on, the restoration of the 
light is only noticeable in the immediate vicinity of the ab- 
sorption-band. A bright and rather narrow band (about 
12 A.E. in width) appears exactly at the centre of the 
absorption-band, that is at wave-length 5790. On the blue 
side of this there appears a fainter and broader band midway 
between the two absorption-bands, while still further along 
there is another narrow band, in coincidence with the fainter 
of the two absorption-bands. 

A photograph of this “magnetic spectrum” was made. 
but it is on too small a scale to reproduce well, and I have 
accordingly made an accurate drawing from it on the same 
scale as the absorption spectrum. The state of polarization 
in this magnetic spectrum appears to be most peculiar, If 
the analysing nicol is turned a degree or two ig the direction 
in which the rotation has occurred, the bright band at wave- 


Rotatory Dispersion of Neodymium. 273 


length 5790 moves distinctly towards the blue, while rotating 
the nicol in the opposite direction causes a motion towards 
the red. 

If we attempt to explain this behaviour of the band by 
assuming a certain distribution of plane-polarized light in it, 
we find that we have a discontinuity in the position of the 
plane of polarization at the centre of the band. It appears 
to me that the more probable state of things is as indicated 
in fig. 2, the polarization at the centre of the band being 
circular, passing to the state of plane polarization (oppositely 
inclined) on each side of the band through intermediate 
ellipses. This means of course that the two circular com- 
ponents of the plane vibration, which by their unequal 
velocity in the magnetized medium give rise to the rotation 
of the plane of polarization, are unequally absorbed, a phe- 
nomenon which has been observed by Cotton in the case of 
naturally active bodies. 

From a very careful study of the spectrum and the 
direction of the shift of the dark and light regions in it, ] am 
inclined to regard the rotations as -indicated in fig. 2, the 
incident vibrations being vertical. 

With a thicker film very strong anomalous rotation can 
be observed in the green. After the restoration of the light 
by the magnetic field, if the analysing nicol be turned in the 
direction in which the plane of polarization has been rotated, 
a broad dark band moves from the blue-green region towards 
the absorption-band in the yellow, becoming narrower as it 
moves along, until at the moment when it is at the edge of 
the absorption-band it is so narrow as to be barely visible. 
This indicates of course that the curve of the magnetic rota- 
tion becomes very steep as we approach the absorption-band 
from the blue side, precisely as in the case of sodium vapour 
(fig. 3). On the other side of the absorption-band I am 
unable to determine how the curve runs. The rotation is 
apparently about the same as in the blue-green region, and 
of the same sign, but no certain trace of any moving band 
could be discerned as the nicol was rotated. The curve is 
evidently not as steep on the red side, or the band of ex- 
tinction could certainly be detected in the neighbourhood of 
the absorption-band. So far as could be determined, rotation 
of the nicol extinguished the red and orange portions of the 
spectrum simultaneously, the faint suggestion of a dark band 
coming in from the red side being in all probability an 
illusion, for as is well known extinction appears to begin 
at the faintly visible end of a spectrum, even when it is 
extinguished uniformly. 


274. Prof. R. W. Wood on the Existence of 


We must remember that we are observing rotations which 
result from a pair of absorption-bands, which may behave in 
different ways, that is they may conspire for wave-lengths 
on the blue side and oppose each other for the longer waves 
on the red side. If, for example, the stronger band at wave- 
length 5790 gave positive rotations for the red waves, and 
negative rotations tor the green, and the fainter band gave 
negative rotations for both red and green, the absence of a 
sharply marked extinction-band on the red side is at once 
explained. Both types of rotation curves are theoretically 
possible according to Drude, according to the fundamental 
hypothesis adopted. The second type is characteristic of 
sodium vapour, as is well known. 

I am now preparing a set of these films for Dr. Bates, of 
the Bureau of Standards, who plans to investigate them with 
a large and very accurate polarimeter. 


XXIII. On the Existence of Positive Electrons in the Sodium 
Atom. -By R. W. Woon, Professor of Le«perimental 
Physics in the Johns Hopkins University *. 


[Plate XIV.] 


lear? greater part of the evidence which we have obtained 

thus far regarding the structure of the atom, indicates 
that the centres of vibration which emit the spectral lines 
are negatively charged corpuscles. The positive charges 
appear to be associated with the atom as a whole, and the 
assumption is often made that the positive electrification is 
— of uniform distribution. | 

The Zeeman effect shows us that the D lines of sodium are 
due to vibrators carrying negative charges, a fact which is 
true of all other lines which show the effect. That a negative 
charge is associated with the centres of vibration which emit 
the J) lines is also shown by the direction (positive) of the 
magnetic rotation of the plane of polarization, for waves of 
very nearly the same frequency as that of the D lines. As 
is weil known band spectra do not show the Zeeman effect at 
all, consequently we are unable to apply this test to the 
investigation of the nature of the charge associated with the 
centres of emission of the lines of which the bands are 
made up. 

Some of the lines which make up the complicated channelled 


* Communicated by the Author. 


Positive Electrons in the Sodium Atom. 275 


absorption spectrum of sodium vapour, have, as I have shown 
in previous papers (Phil. Mag. Oct. 1905, Nov. 1906), the 
power of rotating the plane of polarization when the light is 
passed through the magnetized vapour in the direction of 
the lines of force. 

White light is passed through a nicol prism and a steel 
tube which passes through the pole-pieces of a large electro- 
magnet. An analysing nicol, condensing-lens, and spectro- 
scope follow in succession. The tube contains metallic 
sodium and is highly exhausted, for the vapour loses its 
rotating power when mixed with an inert gas. If we set 
the second nicol for extinction, the spectrum of the crater of 
the arc-lamp disappears, but on heating the tube and exciting 
the magnet, a vast number of bright lines appear in the red 
and green-blue regions of the spectrum. Spectra obtained 
in this way, since they are radically different from spectra 
of other types, I have named “ magnetic-rotation spectra.” 
Macaluso and Corbino observed the effect at the D lines, 
employing a sodium flame between the poles of a magnet, 
but they missed the complicated bright-line spectra which 
only appear when very dense sodium vapour is formed in 
vacuo. 

In the case of the rotation for wave-lengths in the vicinity 
of the D lines, there is no difficulty in determining the. 
direction, i.e., whether positive or negative, for the broad 
bands of rotated light which border the absorption-lines can 
be moved from side to side by slight rotations of the analysing 
nicol ; or we may employ the device so frequently used, the 
Fresnel double prism of right- and left-handed quartz, which 
tells us at a glance the direction of the rotation. In the 
ease of the narrow lines of the channelled spectra, no in- 
formation can be gathered as to whether the rotation is 
positive or negative by rotating the analysing nicol, for the 
smallest possible turn from the position of extinction causes 
the continuous spectrum to brighten up, obliterating the 
rotation lines. It is, however, of the utmost importance to 
determine the nature of the rotation in this case, as it will 
furnish many additional clues to the structure of the atom. 
An attempt was first made to employ metallic arcs in place 
of the white-hot crater, as the source of the light, on the 
chance that some of the lines might be of the right wave- 
length to suffer rotation in the region of the channelled 
spectra. If any of the lines were found to be rotated by the 
vapour, the direction of the rotation could be easily determined 
by rotating the analysing nicol until they were extinguished. 


276 Prof. R. W. Wood on the Existence of 


No lines were found, however, which had just the right 
wave-length. It then occurred to me that the selective 
rotatory power of the vapour could be utilized to furnish a 
source of light made up of just the right wave-lengths ; in 
other words, magnetized sodium vapour between crossed 
nicols could be used as a light filter. The light passed by 
the crossed nicols when the magnetic field was excited was 
accordingly sent through another magnetized tube of vapour ~ 
and examined with a third nicol and spectroscope. It was 
hoped that by setting the third nicol for extinction, and 
causing the bright-line spectrum to appear again by excitation 
of the second magnet, it would be possible to determine the 
direction of rotation of the lines by observing in which 
* direction it was necessary to rotate the third nicol in order 
to blot them out. The first magnet, with its sodium tube 
and polarizing prisms, delivers plane-polarized light of exactly 
the wave-lengths of the bright lines of the magnetic-rotation 
spectrum. ‘This light is then passed through a second 
magnetized tube of sodium vapour, a nicol prism, and a 
spectroscope. The nicol having been set for extinction the 
bright-line spectrum disappeared, reappearing again as soon 
as the magnet was excited. It was found, however, that 
rotation of the third nicol was wholly without effect on the 
appearance of the lines, notwithstanding the fact that the 
light was originally plane-polarized. The magnetized sudium 
vapour appeared to have completely depolarized the light. 
The cause of this phenomenon is not difficult to explain. 
The lines which make up the magnetic-rotation spectrum, 
though they appeared as narrow as the iron arc-lines in a 
photograph which I made two years ago with a concave 
erating of 12 feet radius, are not in reality monochromatic. 
The action of an absorption-line is to rotate the plane of 
polarization of waves of nearly the same wave-length through 
various angles depending on their proximity to the absorption- 
line. It is these waves which are transmitted by the nicol. 
The line therefore has a finite, though narrow, width, and 
the second tube of magnetized vapour rotates the mono- 
chromatic constituents, of which the lineis made up, through 
various angles. Some of the light in the line is therefore 
passed by the third nicol in every position. 

From their analogy to the bright rotated lines which 
border the D lines when examined under similar conditions, 
we should expect all of the lines of the magnetic-rotation 
spectrum to be double, and I have spent a good deal of time 
in attempts to show their duplicity, using an échelon grating. 


_- 


Positive Electrons in the Sodium Atom. pobre 


No very definite results were obtained, however, and more 
recent experiments show pretty conclusively that the rotatory 
power of most of the absorption lines is confined to wave- 
lengths on one side of the line only. This same action is 
observed in an exaggerated degree by the ultra-violet 
absorption-line of mercury (\= 2536) which, as I have shown 
in a previous paper (Astrophys. Journ. July 1907), broadens 
very unsymmetrically. The form of the absorption-curve, and 
the magnetic rotation as shown with the Fresnel rotating 
quartz prisms, is shown in Pl. XIV. fig. 1,a@and 6. The 
spectrum obtained by passing white light through the vapour 
placed between crossed polarizing prisms is shown in fig. 1, 
the fainter line being rotated 270°. 

The behaviour of mercury vapour will be fully treated in 
a subsequent paper, and for the present we need only remark 
that an absorption-band is possible which only gives an 
appreciable magnetic-rotation for wave-lengths bordering it 
one side. 

This shows us that the lines of the magnetic-rotation 
spectrum would not necessarily appear double, even with the 
highest resolving powers (neglecting rotations larger than 
90°). Though the lines appear as narrow as are-lines even 
with a large grating, the magnetized sodium vapour and 
polarizing prism show us that in reality each line embraces 
a narrow range of the spectrum, the individual components 
of which are rotated through very different angles by the 
vapour. 

The experiment which finally showed clearly the nature of 
the rotation was made with a pair of Fresnel quartz-prisms. 
They were much thinner than those usually employed, as it 
was felt that it would be better to work with a single broad 
band of extinction, than a large number of parallel bands. 
The magnetically rotated lines are faint in comparison with 
the continuous spectrum from which they are derived, and it 
is consequently important to have the background upon which 
they are to show up as dark as possible. With a thick 
Fresnel prism we have the continuous spectrum at its full 
intensity traversed by a number of parallel dark bands, 
which correspond to the points on the slit at which the plane 
of polarization is parallel to the plane of extinction (long 
diagonal) of the analysing nicol, which is placed immediately 
behind the slit. There isin consequence more or less diffused 
light from the grating, which renders the background (the 
dark bands), upon which the rotated lines are to appear, 
much too luminous. To get rid of this effect, the best 


278 Prof. R. W. Wood on the Existence of 


method is to use a thin prism, and cover the slit except for 
a small portion immediately above and below the single 
dark band of extinction. 

With this arrangement of the apparatus the magnetically 
rotated line should penetrate the dark band from above or 
below, according to whether the rotation is positive or nega- 
tive. If we excite the magnet and gradually heat the sodium 
tube, we see sharp needles of light shoot down from the 
continuous spectrum into the dark region immediately to 
the right and left of the D lines, as has been described by 
Macaluso and Corbino, Zeeman, and others. If we reverse 
the magnetic field the needles of light shoot up from below. 
The direction in which the plane of polarization is rotated 
by the D lines indicates that they are caused by vibrations of 
negative electrons. The important question to be answered 
is whether the absorption-lines of the band-spectra rotate the 
plane of polarization in the same or in the opposite direction, 
and whether they all behave alike. 

The magnetic-rotation spectrum being much brighter in 
the red and orange than in the green and blue region, the 
first observations were made in this part of the spectrum. 
The spectroscope was a medium-sized instrument, consisting 
of a telescope and collimator of about 180 cms. focus, 
furnished with a plane grating. 

The sodium tube was heated until the fine black absorption- 
lines in the red appeared distinctly in the continuous spectrum 
above and below the horizontal dark band due to the Fresnel 
prism. The current was then thrown into the magnet, the 
self-induction of which is so great that the field does not 
rise to its full intensity for several seconds, so that there was 
plenty of time to see exactly what happened. As soon as 
the switch was closed numerous needles of light commenced 
to penetrate the dark region, some of them shooting down 
from above, others shooting up from below. Of these, some 
only extended halfway or less across the dark band, while 
others crossed it completely. On opening the switch the 
luminous needles slowly withdrew from the dark background 
into the bright region from which they came, reminding one 
of the tentacles of an alarmed hydroid. The phenomenon is 
one of the most beautitul that I have seen for some time, 
for it shows us at once that some of the absorption-lines 
rotate the plane of polarization in the positive direction, 
while others rotate it negatively. 

A very satisfactory photograph of the phenomenon was 
obtained on a Wratten and Wainwright panchromatic plate 


Positive Electrons tn the Sodium Atom. 279 


with an exposure of one hour. An enlargement was made 
of the plate, which is reproduced in fig. 2a. Fig. 2bis a 
contact print from the original negative, and fig. 2¢ is a 
print from a plate made with a small two-prism spectroscope, 
showing the entire visible spectrum and the phenomenon at 
the D lines. It will be observed that in the case of some of 
the lines the bright needles of light have withdrawn almost 
entirely from the absorption-spectrum, leaving a dark line. 
(Compare the upper and lower spectra at the points indicated 
by the arrows.) 

If the positive rotation at the D lines can be used as an 
argument that they are due to negatively charged electrons, 
it appears to me that the two types of rotation in the 
channelled spectrum is an evidence that we have both 
positive and negative electrons in the atom. It is perhaps 
unwise to speak of a positive electron, since electron has 
come to mean the disembodied negative charge, after it has 
been expelled from the atom. 

Whether the two types of magnetic rotation proves the 
existence within the atom of both positively and negatively 
charged discrete particles is for the theoretical physicists to 
answer. ‘The observations recorded in this paper merely 
prove that some of the absorption-lines give a rotation 
opposite to that given by the D lines. 

Becquerel has inferred the existence of positive electrons 
in certain crystalline minerals, from the change in the 
appearance of the absorption-band when the crystal is placed 
in a magnetic field. The conditions in this case are, how- 
ever, much more complicated than in the case of sodium 
vapour, for he is dealing with molecules of the rare earths 
in combination with or imbedded in other substances. 

It will be extremely interesting to determine the direction 
of the rotation of the lines in the green and blue region, for 
these have been found to coincide with the regularly spaced 
series of lines in the fluorescence spectra excited by mono- 
chromatic radiations *. Iam now investigating this subject 
in collaboration with Mr. Felix Hackett, Fellow of the 


Royal University of Ireland. 


* See previous paper on “The Fluorescence and Magnetic Rotation 
Spectra of Sodium Vapour, and their Analysis.” Phil. Mag. Nov. 1906. 


[ 280 ] 


XXIV. On the Atomic Weight of Radium. 
By Hunry Wiipt, D.Se., D.C.L., FBS. 


[* my paper read before the Society last year 7, it was 

shown from the relations of the specific gravities of the 
alkaline-earth metals to their atomic weights, and also from 
the similar relations of the series of alkaline metals, that 
radium would have a proximate specific gravity of 5, and an 
atomic weight of 184, notwithstanding the assertions that 
have been made that the new element would be a heavy 
metal comparable with thorium (sp. gr. 11) and uranium 
(sp. gr. 18), and that its atomic weight ranged between 225 
and 258. It was also shown that the atomic weights of the 
two series Hn and H2n of my tables, are definite multiple 
differences and not intermediate numbers. 

In the several accounts which have been given of the 
atomic weight of radium, it is stated that the experimental 
determinations were made with radium chloride. Now it is 
well known to chemists that the series of alkaline metals and 
alkaline-earth metals, magnesium, calcium, strontium, and 
barium, unite with chlorine in one proportion only. 

An important gain to chemical science which the multiple 
differences of the atomic weights have led up to, is the 
quantitative determination of the combining proportions of 
new elements in anticipation of the experimental results. 

Taking the instance of radium chloride, I have estimated 
its combining weight after the classical method of Marignac{ 
and Dumas § in their experimental determination of the 
atomic weight of barium as follows :— 


1. One part of silver corresponds to 1°176 parts of 
1-000 
ive: 

2. The atomic weight of silver being 108, we have radium 
chloride=1°176 x 108=127 minus Cl (35)=92 the 
combining weight of radium with chlorine. 

3. Now 2x 92=184 is the bivalent atomic weight of 
radium with bivalent oxygen in the positive and 
negative series H2n, as shown in my general Table, 
with a possible increase of one unit in the experi- 
mental determination, as in the instance of barium 
(136-137). 

* Reprinted from the Memoirs and Proceedings of the Manchester 
Literary and Philosophical Society, vol. lii. pt. 1 (1907). From a 
separate copy communicated by the Author. 

+ Manchester Memoirs, vol. li. No. 2 (1906). 

t Bibl. Univ. Archives, 1858, p. 81. 

§ Ann. Chim. Phys. vol. lv. p. 189 (1859). 


radium chloride, or 


13°6 200 


9x 24—8=Pb=208 


11°44 207 


Dr. H. Wilde on the Atomic Weight of Radium. 281 
Hn. H2n. 

Sy 1, Dik, He=, 2 Diff: 

— 6 — 6 
oe ein 7 0.0.8=Gl=. 8 
o59¢ 7* 164? 9-2 

| —16 —16 
1x23. 0=Na= 23 1x 24—0=Mg= 24 
098 23 174 24 

—16 —16 
2x23—7=Ka= 39 2x24—8=Ca= 40 
0°86 39 158 40 

— 23 — 24 
3x23-—7=Cu= 62 3x 24—8=Zn= 64 
89 633 72° 65 

— 23 — 24 
4x23—7=Rb= 85 4x24—8=Sr= 88 
Cy ee 254 87° 

— 23 — 24 
5 x 283—7 = Ag =108 5 x 24—8=Cd =112 
106 108 | 869 112 

— 23 — 24 
6 x 23—7=Cs =131 6 x 24—8= Ba =136 
1:38 132 aia) a 

—23 — 24 
7x 23—7= —=154 7 x 24—8= — =160 

12°2t 10°13 f 

— 23 —24 

8x 23—T= —=177 8 x 24—8= Ra = 184 
221 50t 
— 23 — 24 


* Accepted Atomic Weights. + Specific Gravities. { Estimated. 


I have previously shown that the positive series of elements 
H2n closes with lead (208), and that if any higher member 
_ of the series of alkaline-earth metals exist, it would have an 


982 Dr. R. A. Houstoun on a New 


atomic weight of 232, and an approximate specific gravity 
ag. 

Assuming this hypothetical member to be radium, the 
combining equivalent of its chloride with silver (Cl 35 and 
Ag 108) would be 1°399 in accordance with the determi- 
nations arrived at with the other members of the same series, 
and not 1°371 as determined experimentally for the inter- 
mediate atomic weight 226, recently assigned to radium f. 


XXV. A New Spectrophotometer of the Hifner Type. By 
Rozsert A. Houstoun, PA.D., D.Sc., Assistant to the 
Professor of Natural .Philosophy in the University of 
. Glasgow}. 

pee? the past forty years a great number of different 

spectrophotometers have been worked out. This will 
readily be seen by a reterence to Kayser’s Spectroscopie, 
vol. ii. chap. 1, where a complete list is given§. But 
although so much attention has been given to designing and 
testing new instruments, few spectrophotometers have been 
made, and most laboratories are without one. This must in 
part be ascribed to their cost, and also to their construction, 
which usually prevents them from being used for any other 
purpose. The object of the present paper is to describe an 
attachment which may be fitted to any spectrometer or 
spectroscope, converting it into a spectrophotometer, and 
which may be removed at once when the ordinary use of the 
instrument is desired. A method for applying the instru- 
ment to the measurement of absorption in the ultra-violet 
is also described. 


The Hiifner Spectrophotometer || consists (fig. 1) of a 
spectroscope, before the slit of which a Glan-Thompson prism 


* Manchester Memoirs, vol. li. No. 2 (1906). 

+ Comptes Rendus, vol. cxlv. p. 422 (1907). 

t Communicated by Professor A. Gray, F.R.S. 

§ This list does not mention the new form of the Konig spectro- 
photometer designed by Dr. Martens (“ Ueber eine Neukonstruktion des 
K6nigschen Spektralphotometers,” F. F. Martens u. F. Grunbaum, Ann. 
der Phys. xii. p. 984, 1903), and the spectrophotometer of Dr. J. Milne 
(‘ Nature,’ Ixxii. p. 891, 1905). 

|| ‘“‘ Ueber ein neues Spectrophotometer ” von G. Hiifner, Zs. f. physic. 
Chemie, iii. pp. 562-571 (1889). 


Spectrophotometer of the Hiifner Type. 283 


N and a glass rhomb AC—the Hiifner rhomb—is brought. 
The edge A is ground very sharp and bisects the slit gh. 
If we disregard the polarizing effect of the rhomb, the upper 
beam bounded by the rays sAg and thr, which illuminates 
the lower half of the slit, consists of natural light, and the 
beam bounded. by the rays ugp, vAg, which illuminates the 
upper half of the slit, consists of light polarized in either a 
vertical or horizontal plane. Another nicol is inserted in 
the path of both beams. If we look into the eyepiece, we 
see two spectra, one above the other, and by rotating this 
nicol we can alter their relative intensity. 

The light which comes from the upper half of the slit 
must be plane-polarized before entering the second nicol, 
and the light which comes from the lower half of the slit 
must be unpolarized natural light. But it is partially 
polarized by both the Hiifner rhomb and dispersion-prism. 
Hiifner was perfectly aware of this, and in his improved 
instrument made by Albrecht he got over the difficulty by 
choosing the glass and angle of the dispersion-prism so that 
it removed the polarization produced by the rhomb (c/. article 
cited, p. 564). Messrs. Hilger have apparently rediscovered 
the same method of removing the polarization produced by 
the rhomb, and it is described at length by Mr. F. Twyman* 
under the impression that it is new. 

Any dispersion-prism will not do then. Its angle must be 
chosen so as to remove the polarization produced by the 
rhomb. 

I have recently designed a prism which performs the 
functions of both the rhomb AC and the Glan-Thompson 
prism N, but which polarizes both beams in planes at right 
angles to one another, and thus can be used with any dis- 
persion-prism whatever (fig. 2). 


Fig. 2. 


ABC is made of glass w»=1'526, the sides AB, BC, CA 


* “Tmprovements in the Hiufner type of Spectrophotometer.” 
F, Twyman, Phil, Mag. April 1907. 


284 Dr. R. A. Houstoun on a New 


being each 2cms.long. It is cemented to a prism of Iceland 
spar, BDEC, cut with its axis perpendicular to the plane 
of the paper. The angle D is 127° 12’, H is 115° 49’, and 
BCH is 36° 44’. 

The action of the prism may be better understood by 
considering the beams of light to go in the reverse direction 
—from the object-glass of the collimator to the slit. The 
beam gr is broken into two by the Iceland spar prism, ed 
being the ordinary beam and ab the extraordinary. The beam 
pq is broken into two, but only the extraordinary ef emerges, 
the ordinary being totally reflected at the surface CE. The 
beams ef, cd meet 15 cms. out in an elliptical spot of light 
measuring 2°0 by 2°4 cms., the long axis being vertical. 
The beam abd is quite 2 ems. clear. 

If now we have as source of light an incandescent mantle 
behind a screen, with an aperture at the proper place not 
much larger than 2-0 by 2:4 cms., and if we look into the 
eyepiece, we see two spectra one above the other and polarized 
at right angles to one another. ‘The ordinary component of 
the Jower beam misses the slit entirely, the extraordinary 
component of the upper beam misses the object-glass of the 
collimator. 

The prism was fixed in a rectangular brass cell open at 
the ends with the edge A protruding. The brass cell was 
mounted in a short piece of brass tubing with a milled head, 
which turned inside another piece of brass tubing fixed in 
front of the slit. The prism could thus be rotated about the 
axis of the collimator—the only adjustment necessary. The 
inner tube containing the cell could be removed when re- 
quired, the outer tube remaining fixed to the slit. When it 
was returned to its plane the milled head prevented it sliding 


in too far and damaging the edge A. The prism was attached 


to one of W. Wilson’s well-known College spectrometers. 
All that is then required to complete the equipment of the 
instrument as a spectrophotometer is one of the nicol-prism 
polarizers with a divided circle, which is supplied by the 
makers to fit over the object end of the collimator. This is 
preferable to the object end of the telescope as it avoids the 
depolarizing effect of the dispersion-prism. The eyepiece 
supplied with the instrument for ordinary use magnifies, how- 
ever, too strongly, when it is used as a spectrophotometer. 
Much the better place for the nicol, however, is the eye- 
piece. In spectrophotometry it is necessary to waste as little 
light as possible. The object-glasses of collimator and 
telescope had a clear aperture of one inch. A nicol with an 
aperture of one inch would be too expensive, and makers 


Spectrophotometer of the Hiifner Type. 285 


usually fit a much smaller nicol, thus causing a large loss of 
light. In the eyepiece the rays come closer together, and a 
smaller nicol may be used. An ocular nicol with divided 
circle, reading telescope, and screens for cutting out the 
regions of the spectrum not under observation was therefore 
made by F. Schmidt and Haensch, Berlin, to repiace the 
ordinary eyepiece. The prism for attaching to the slit was 
made by R. & J. Beck, Ltd. Its angles do not require to 
be made accurately, the only point necessary being that the 
axis of the Iceland spar should be parallel to the edge A. 
It is extremely important that the slit prism be set with the 
axis of the Iceland spar perpendicular to the refracting edge 
of the dispersion-prism. If that is not the case, the two 
beams are not polarized in and perpendicular to the plane of 
incidence when they enter the dispersion-prism. Their 
planes of polarization will then be rotated, and the positions 
in which the ocular nicol extinguishes the upper and lower 
spectra will not be at right angles to one another. The 
cea position of the slit prism may thus be found by 
trial. 

If it is preferred to keep the one beam unpolarized, as in 
the ordinary Hiifner spectrophotometer, then a prism of the 
form shown in fig. 3 might be used. ABC and BFD are 


Fie. 3. 


glass, FDEC is Iceland spar. There is now no light lost by 
reflexion in the nicol. Owing to the space saved, wider cells 
might be used and the absorption of more dilute solutions 
measured. 

The limitation of the size of the aperture causes no difficulty 
in practice. 

With reference to the accuracy obtained, it is not possible 
with a Hiifner rhomb to get the two spectra under comparison 
to touch one another so sharply as in the case of a Lummer- 
Brodhun cube or a biprism such as is used in the 
Kcenig-Martens spectrophotometer, as my experience of the 

Phil. Mag. 8. 6. Vol. 15. No. 86. Feb. 1908. Xx 


286 Dr. R. A. Houstoun on a New 


latter instrument has shown me*. The latter instrument 
also has the advantage that the polarizing prism is placed 
after the dispersion-prism and the error due to depolarized 
light removed. In practice, however, since twice as much 
light is lost in the latter instrument, the Hiifner spectro- 
photometer is as accurate, and is of course simpler. The 
University of Glasgow possesses a Wild spectrophotometer, 
but so much light is lost in the latter instrument that in 
accuracy it is very much inferior to the others. | 

To test the possibilities of the instrument for spectrophoto- 
metry in the ultra-violet, I attempted to measure the absorption 
of copper sulphate photographically. Copper sulphate was 
chosen because it is a salt for which accurate values are 
given in Landolt and Bornstein’s tables. It absorbs the red 
end of the spectrum, but the results show that rough 
quantitative work in the ultra-violet could be done very 
easily. 

PW sia was attached to the end of the collimator. The 
diameter of its aperture was less than one quarter the diameter 
of the lens, and hence there was a great loss of light. In 
place of the telescope there was placed a camera, the focal 
length of its lens being about 51 cms. Wratten and Wain- 
wright’s panchromatic plates were used. The spectrum from 
6700 to 3500, where the absorption of the glass begins, was 
taken at one exposure and measured more than 5 cms. in 
length. 

First of all both beams were left free and the nicol set so 
that its plane of polarization made angles of 55°, 52°, and 
50° with the plane of polarization of the lower beam. At 
52° the one spectrum was brighter, at 50° the other, so 51° 
was taken to be the angle at which the nicol would be set 
for equal brightness. A photograph was then taken of the 
flame spectrum of Li, K, Na, and Th, and the wave-length 
of a point of the plate determined in terms of its distance 
from the D lines. 

Tken the cell with the copper sulphate was placed in 


position, and exposures made for different positions of the 


nicol. A bunsen flame with a sodium bead was placed 
between the incandescent mantle and the cell. The sodium 
line appeared therefore on all the plates. The distance of 
the point on the plate where the two spectra touching one 
another were equally black was measured from the D line, 
and the wave-length of the point could thus be determined. 


* “Untersuchungen iiber die Absorption des Lichtes in isotropen 
Kérpern,” R. A. Houstoun, Ann. der Physik, xxi. p. 535 (1906). 


Spectrophotometer of the Hiifner Type. 287 


If a gives the position of the nicol, the fraction of the 
intensity of the incident beam transmitted by the solution 
is given by 

a ean 

y Stameai” 


Dial 9 ¢ denote the concentration of the solution in gram- 
molecules per litre, and d the thickness of solution passed 
through, then A the “molecular extinction coefficient” is 


defined by 
_ The following table gives the results :— 
| Time of x A | A from the 
a exposure, 3 ° tables. 
49:8 10 mins. | 4750 0-053  ~—-0-009 
47-4 a 5640 0-187 0-29 
45:0 ay 5710 0°322 0-37 
41°6 = 5850 0-514 | 0:57 
38:1 4 5980 0714 | 0-83 
30°0 15 mins. 6130 LFF 1:24 
23°1 20 mins. 6200 1:69 | B50 
14:9 Py 6410 | 2-45 2-42, | 


3 ; c=0°5424 om.-molecules/litre. 
d=1cm. | 
The width of the D line on the plates is 0°6 mm. 


_ One exposure gives A for only one wave-length, but if the 
absorption had more than one maximum the exposure might 
give A for several values of X. The values of A taken from 
the tables are of course more accurate. I attempted to 
determine the wave-length by putting a piece of didymium 
glass before the slit, but this method was not so accurate as 
the one used. 

The cost of the instrument has been borne by the Carnegie 
Trust for the Universities of Scotland. 


X 2 


[ 288 J 


XXVI. Note on X-Rays and Scattered X-Rays *. 
By Cuarzes G. Bargwa, IA., D.Se.t 


[* a paper on Polarized Rontgen Radiation ¢ I showed 
that on the zther-pulse theory of Réntgen rays, we 
should expect the primary beam proceeding from an X-ray 
tube in a direction perpendicular to that of propagation of 
the cathode stream to be partially polarized, and on the 
theory of scattering by electrons in light atoms the secondary 


rays proceeding in a direction perpendicular to that of pro- — 


pagation of the primary, from substances of low atomic 
weight, ought to be almost completely plane polarized. 
Experiments were described which in both cases demonstrated 
the polarity $§, and the effects were shown to be of the order 
of magnitude expected on these theories. The evidence in 
favour of the theories appeared so strong, that a more com- 
plete study of the distribution of intensity of the secondary 
radiation was considered unnecessary and unimportant. 
Prof. Bragg, however, in arecent paper || regards some of 
the evidence in favour of the zther-pulse theory as a little 
over-rated, and proposes in its place the hypothesis that an 
X-ray beam consists mainly of “neutral pairs” (each consisting 
of a positive and a negative particle rotating in a plane con- 
taining the direction of propagation of the “ pair”). This he 
considers affords an easier explanation of the phenomena of 
X-rays and is not improbable a priori. 

It appears altogether unnecessary to fully discuss such an 
hypothesis, for the study of the distribution of intensity of 
secondary radiation from light atoms affords a simple and 
apparently conclusive test between the ether pulse and the 
‘‘neutral pair” theories. At the same time this makes the 
investigation on polarization more complete, and furnishes a 
delicate method of testing the presence of scattered rays in 
a complex radiation. 

According to the eether-pulse theory of Rontgen rays, when 
a primary pulse passes over an electron in a light atom, the 
electron is accelerated in a direction opposite to (being 
charged negatively) the direction of electric intensity in the 


* The expenses of this Research have been partially covered by a 
Government Grant through the Royal Society. 

+ Communicated by the Author. 

{ Phil. Trans. A. vol. ccexiv. 1905, pp. 467-479. 

§ Royal Soc. Proc. A. vol. lxxvii. 1906, pp. 247-255. 

i| Phil. Mag. [6] vol. xiv. Oct. 1907, pp. 429-449. 


On X-Rays and Scattered X-Rays. 289 


pulse. During the passage of an unpolarized primary beam, 

these accelerations are uniformly distributed in direction in 

a plane perpendicular to that of propagation of the beam. 
Now the electric intensity at a point P in one of the 


ef sin 6 


resulting secondary pulses is expressed by * , where 


e is the charge and / the acceleration of the electron, 
» the distance from the electron to the point P, and @ the 
angle which the line joining the electron to the point P makes 
with the direction of acceleration. 

If P is on the line of propagation of the primary pulse 
over the electron 0= = , and the intensity = L 

If P is in the plane through the electron perpendicular to 
the direction of primary propagation, @ varies uniformly 
from 0 to a for the primary pulses. 

The intensities of radiation then in these two directions 
are proportional to 


9 (>) 979 . 9 
Pay ef? sin? 6 
Se and pp a respectively. 
I (od fhe 


When the primary beam is unpolarized, the average value of 
sin* @=4, consequently the intensity of secondary radiation 
in the direction of propagation of the primary is double that 
in a direction at right angles. 


On the neutral pair hypothesis it is supposed that a pair 
which strikes a substance consisting of light atoms is liable 
to be taken up only by an atom revolving in the same plane. 
It is sometimes ejected again, and its subsequent rotation and 
translation continue to take place in the one plane; there- 
fore all pairs ejected in any direction at right angles to that 
of propagation of the primary rotate in the plane of primary 
and secondary propagations. A tertiary beam is therefore 
strongest when in the same plane. This is Bragg’s explana- 
tion of the polarization of the secondary beam as shown by 
the tertiary beams. 

The results of experiments described below, however, show 
that this cannot be the true explanation of the polarization 
effect ; neither is the theory capable of explaining the partial 
polarization of the primary. It is easily seen that on 
this hypothesis all the possible planes of rotation of pairs in 
2 primary beam contain the direction of propagation of that 
beam, and therefore that any pair has a chance of being 


290 | Dr. C. G. Barkla on 


ejected again along that axis. If we consider the radiating 
mass at the centre of a sphere with the direction of primary 


propagation as the axis, then the different possible planes of 


ejection are the planes through this axis, and as in each plane 
we may assume there is equal chance of ejection in all 
directions, the number of pairs received by any element of 
surface of the sphere is proportional to the density of the 
lines of longitude on that element. This varies as the secant 
of the latitude from 1 at the equator to infinity at the poles. 
The total energy of radiation received by a small area near 
the poles is therefore many times as great as that received by 
a similar small area near the equator ; that is, a small beam 
of secondary rays proceeding in a direction near that of 
propagation of the primary, is many times as intense as one 
in a direction approximately at right angles to this. 

To test between these theories, the simple apparatus was 
arranged as shown in the diagram. 


we 


A narrow pencil of X-rays emerged through a small 
circular aperture of °85 cm. radius in a lead box, and at a 
distance of 38 centimetres in this pencil a square: slab of 
carbon (8x8 em.) was placed in a vertical plane with its 
face equally inclined to the lines joining its centre to the 
centres of two equally distant rectangular apertures A, and 
A, in lead screens. Behind these were situated electroscopes 


® 


X-Rays and Scattered X-Rays. 291 


of the type described in previous papers—modifications of 
those devised by C. T. R. Wilson. Their thin paper and 
aluminium faces were against the apertures, which were of 
dimensions 3°2x5 cm. and 5x5 cm., and were placed 
symmetrically with regard to the carbon at a distance of 
27 centimetres. 

With this arrangement the radiation from each atom before 
entering one electroscope had to travel approximately the 
same distance in carbon and air as that entering the other. 
The intensities of the secondary beams entering the electro- 
scope were then proportional to the intensities of radiation 
from the atoms in the two directions RA, and RA,. One of 
these RA, made only a small angle with the axis of the 
primary beam, while the other RA, was perpendicular to 
that axis. 7 

By first placing a sheet of copper in the position R, it 
was found, by alternately placing one electroscope in the two 
positions A, and A,, that there was no appreciable difference 
between the intensities of secondary radiation from copper in 
the two directions. 


This is what was expected from the results of previous 
experiments on copper, for it was found that when a polarized 
primary beam fell on copper the intensities in the two 
principal directions at right angles were equal, indicating 
that the copper radiation was equally intense in all directions 
whatever the position of the plane of polarization and direction 
of incidence of the exciting primary beam. The character 
of the secondary radiation from copper had also been shown 
to be approximately independent of that of the primary 
producing it. The equally intense copper radiations were 
therefore used to standardize the two electroscopes afterwards 
used in place of the one. 

Thus in one experiment after correction for the normal 
ionization in the electroscopes, for the effects of the secondary 
rays from air and for the ionization that may have been pro- 
duced by very penetrating primary rays in the electroscopes, 
the deflexions of electroscopes A, and A, were 12°5 and 
18°05 respectively when copper was the radiating substance. 
Now as these were due to equally intense beams, the 
deflexion of 12°5 in electroscope A, was equivalent to 18°05 
in A, When the copper was replaced by carbon the 
deflexions were 7 and 5:2 respectively, and by repeated 
observations 7°9 and 5°95. Taking the average of these we 
have 7°45 and 5°575. If the electroscopes had been equally 


292 . Dr. GC. G. Barkla on 
sensitive the ratio of deflexions would therefore have been 
7°45 x 18:05 | 


19:5 > ood. 
2. e. 10°76 : 5°D75 
or 1-95.28, 


That is, the average intensities of the secondary beams from 
R received by the electroscopes A, and A, were in the ratio 
a et 

As this ratio appeared to vary considerably in different 
experiments by an amount much greater than the possible 
experimental error would account for, it was apparent that 
it depended to a certain extent on the character of the primary 
radiation. 

Gas was therefore admitted into a regulating X-ray tube, 
until it was brought to that state when a discharge through 
it did not produce X-rays which could be detected by the 
electroscopes at all. A discharge was then passed through 
until rays were emitted in sufficient intensity to be experi- 
mented upon. The ratio of intensities was then found to be 
2-0: 1. When the bulb was somewhat “‘ harder ” this became 
1:93: 1, and when “ harder” still the ratio became 1°80: 1. 
In order to experiment on a beam of much greater average 
penetrating power, the more absorbable portion of this beam 
was then cut off by a plate of aluminium °04 cm. in thickness. 
This diminished the ionizing power of the primary to about 
30 per cent. of its original value, while the transmitted beam 
was on the average about twice as penetrating. Using this 
penetrating primary beam, it was found that the ratio of 
intensities had dropped to 1°51 : 1. 

These experiments were repeated with other X-ray tubes 
and the results were confirmed, the ratio never exceeding 


2:1, and decreasing with increasing penetrating power of. 


the primary rays used. 

It must, however, be observed that the primary beam used 
in these experiments was partially polarized, the variation in 
intensity of secondary beams from carbon in the two principal 
directions at right angles to the direction of primary propa- 
gation amounting to about 20 per cent. when the beam was 
of the very absorbable type, but only to 6 or 7 per cent.* 
when the rays were much more penetrating. Thus the 


* There is, of course, probably no lower limit to the variation, but 
these were approximately the limits found for the different beams of the 
type used in these experiments. 


Lase  é 
ss 
»G : 


—— a 


~~; 


X-Rays and Seattered X-Rays. 293 


intensity of the beam RA, depends to this extent on ifs 
position with regard to the cathode stream in the X-ray 
tube. The above ratios were obtained when the beam RA, 
was parallel to the cathode stream, and therefore have the 
maximum values. The intensities of secondary radiation in 
directions RA, and RA, when easily absorbed primary rays 
were used, were therefore in ratios varying from 2:1 to 
2: 1:2, i.e. from 2:1 to 1°66: 1, according to the position of 
the plane of polarization of the partially polarized primary 
beam. 

For an unpolarized primary beam, the ratio of intensities 
in the two directions RA, and RA, would therefore be 
approximately 1°85: 1. 

The results of these experiments in this way showed that 
the ratio of the intensities of secondary rays from carbon in 
the two directions indicated varied from about 1°85: 1 with 
very soft rays to about 1:45: 1 for penetrating rays. Now 
the first ratio is of the order—probably a little less—that 
might have been expected if we assumed perfect scattering 
on the ether-pulse theory, the rays entering electroscope A, 
making a small angle with the direction opposite to that of 
primary propagation, and most of those entering A, being 
not quite perpendicular to the direction of propagation of 
the primary. 

On the other hand, on the neutral pair hypothesis the 
ratio should be roughly 7 or 8: 1. This, however, is on the 
assumption that a pair rotating in one plane continues even 
after ejection from an atom which it strikes to rotate accurately 
in the same plane. If we admit any deviation from this, 
then the ratio of intensities becomes less than the above. 
But the polarization which exists in a secondary beam from 
carbon is sufficiently complete to show that if the effects are 
to be explained on such an hypothesis, the behaviour of the 
pairs must approximate very closely to this ideal, and con- 
sequently the ratio of intensities in the two directions RA, 
and RA, could not possibly have been lower than 4:1 or 5:1. 

We thus have what appears to be most conclusive evidence 
against the “neutral pair” hypothesis, and the most con- 
vineing verification of the ether-pulse theory that could be 
given by such experiments. 

Not only is there perfect agreement between theory and 
experiment here, but experiments showed that where the 
secondary rays were otherwise found to be less purely a 
scattered radiation judged by their penetrating power, and 
by polarization experiments, there was also a reduction of 
the ratio of intensities discussed in this paper—the result to 
be expected on the pulse theory. 


294. Dr. C. G. Barkla on 


Thus :— | 

Using the rays which are emitted by an X-ray tube in 
which the pressure is as high as possible when producing 
X-rays of appreciable ionizing power, the absorbability of 
the secondary rays from carbon differs exceedingly little if 
at all from the primary. Direct experiments did not indicate 
the slightest difference. 

Using such a beam as the primary to be tested for polari- 
zation by study of the intensities of secondary radiation from 
carbon, the maximum amount of polarization of a primary 
was observed—about 20 per cent. variation. When such a 
primary was used to produce a secondary beam from carbon, 
the secondary beam was found by a study of the tertiary 
rays to be fairly completely polarized. 

When such a primary beam was incident on carbon, the 
ratio of the intensities of secondary radiation in the direction 
opposite to that of propagation of the primary and in a 
direction at right angles was very close to the theoretical 
value for perfect scattering, 7. e. 2 : 1 (within 5 per cent.). 

As the primary rays became more penetrating, the dif- 
ference in the ionizing powers of the primary and secondary 
beam from carbon became more marked; the difference in 
their penetrating powers increased ; the evidence of polariza- 
tion of the primary became less pronounced*; and the ratio 
of intensities of secondary radiation opposite to and perpen- 
dicular to the direction of primary propagation decreased. 


According to this theory, then, all those substances of low 
atomic weight—up to and including sulphur—when subject 
to “soft” X-rays should emit secondary rays varying in 
intensity in the two directions approximately in the ratio 
2:1. Itis possible, I think probable, that the behaviour of 
sulphur varies much more rapidly with a variation of the 
penetrating power of the primary than any of the others, 
and that for some primary rays this ratio of intensities may 
be very different from that found by experiments on the 
lower part of this group. 

Paper, when tested in the same way as carbon for “ soft” 
rays, gave off radiation whose intensity was found to vary 
in the ratio 2°01: 1, which again after correction shows the 
true ratio of intensities not appreciably different from that 
for the “softest”’ rays from carbon. | 

* This might possibly be due to the penetrating primaries actually 
being less polarized, but when considered with other changes it appears 
probable that much of the decrease of observed polarization is due to the 
secondary beams not consisting of purely scattered rays, and so not 
showing the polarization. The best results showing polarization in the 
secondary beam were always obtained by using easily absorbed rays. _ 


| 
| 
: 
: 


X-Rays and Scattered X-Rays. 295 


Though accurate experiments were not made on aluminium, 
it was found to behave in the same way. 

The penetrating power of the principal portion of the 
secondary radiation from Cr, Fe, Ni, Co, Cu, Zn has been 
found to be entirely different, being almost if not quite com- 
pletely independent of the penetrating power of the primary 
producing it. The origin of the radiation will be discussed 
elsewhere. ‘The intensity from one of these we should expect 
not to depend on the direction of propagation of the primary. 
This is what was found for the radiation from copper. 
When, however, the behaviour of iron was compared with 
that of copper, ‘it was invariably found that the ratio for 
iron was greater than that for copper. This is shown below. 
The three comparisons were made at different times. It 
appears to indicate a small amount of scattered radiation, 
though possibly not more than 2 per cent. of the total radia- 
tion from each atom. 


| | | | ; 
Corrected | Corrected Ratio of Ratio of 


Radiator.| Time.  deflexion | deflexion | deflexiors intensities of 
OF Ay.;|. |i, COAgae is Mis Ae RA, and RA,. 
5 ie eae 40 secs.| 22:3 298 | 75:100 ru 
Bos... ee 14:25 | 84: 100 11221 
Gi se... Seiad |) 27-5 42-45 | 65:100 1:1 
eek... Pau | 27°7 3715 | 74:100 1:13:1 
Mic...) 75°), 34:2°° (| ABBR h70 2100 1:1 
ee. Thy: 26°7 |p s84 Bie oh 77-2: 100 1:10:1 
| | 


It seems quite possible from other considerations that a 
certain amount of scattered radiation does appear, as the 
atomic weight is getting near that for which a scattered 
radiation was found by much cruder methods. 

Again, Ag, Sn, Sb, I, &e. belong to quite another class of 
substances, tor the radiation from these depends to an extra- 
ordinary extent upon the character of the primary. It has 
been shown that when primary rays of moderate penetrating 
power are used, this secondary is not a scattered radiation, 
though from tin its average penetrating power may be very 
like that of the primary producing it ; it is not polarized, 
and it does not give evidence of polarization in a primary 
where it exists. Though very different from the radiation 
from one of the elements in the Cr—Zn group, we should for 
these reasons expect the beam to show very little or no 
variation in intensity in the beams RA, and RA,. The results 
of comparisons with copper are given. 


eS ea 


296 On ne and Scattered X-Rays. 

: oy [ . | : 

. | Created] Corrected | Ratio of | Ratio 

) Radiator. ‘Time, deflexion deflexion  deflexions of of 

of A. | of A, | A,andA,. | intensities. 
ie citulic:. 60 secs. 332.4 606Kaa 54:9: 100 FFD I 
Se 180 , 6:27 12-15 4 54100 . 95-1 
| 

Monte e . 50 ,, 236 3888 | 608:100 | 1:1 

| ae eh al 180 ,, 6°65 10°5 | 63°3 : 100 | 104:1 

bi gd 40 ,, 975 | 49-45 | 65: 100 | 1:1 

PRM be cece 240 ,, 13°6 Y 305 66°3 : 100 | 1:02:1 


| 


From these experiments there was no indication of a 
decided variation ; we conclude that there was equality of 
intensity in the two directions. Another group of substances 
classified according to their behaviour under X-ray trans- 
mission includes W, Pt, Pb, and Bi. These emit a radiation 
which appears to differ from that from any of the other 
groups referredto. When the radiation from lead was tested 
in a similar way, no evidence of scattering was obtained. 

Experiments were also made to determine the behaviour 
of a number of these elements under more penetrating rays, 
but the conditions of experiments have not yet been varied 
sufficiently to lead to definite conclusions. 


These results thus afford strong confirmation of the 
electromagnetic pulse theory of X-rays s, and of the theories 
of scattering and of polarization based on this. 

The method of comparing the intensities of radiation in 
the two directions indicated is a delicate one for detecting 
the presence of scattered rays, or of measuring the control 
over radiating electrons, exer cised by the primary “pulses while 
producing or stimulating the production of secondary rays. 

The scattering of “ soft ” X-rays by substances of low 
atomic weight has been shown to be almost perfect, and the 
appearance - of a radiation which is very far from a purely 
scattered radiation, when more penetrating rays are used, has 
been shown in a very marked way. The small proportion 
of scattered rays in other secondary radiations has been 
demonstrated. 

These experiments, however, do not show that under 
suitable conditions scattered rays may not be obtained from 
elements hitherto found to emit only an uncontrolled secondary 
radiation. 

I wish to thank Mr. F. J. Thorpe, B.Se., for assistance in 
conducting some of these experiments. 


George Holt Physics Laboratory, 
Liverpool. 


iy 


i ane ae) 


XXVII. Lhe Mixing of Gases. Ro maaar. Wee. Burbury’s 
regent Papers. By Prof. W. McF. Orr, M.A.* 


| BEG to make some remarks on the two papers by 
Mr. Burbury on the Diffusion of Gases which have 
recently appeared in this Magazine +. 

Mr. Burbury appears to dispute the propositions: (1) that 
when two gases mix by diffusion the process is an irreversible 
one ; (2) that every irreversible process is attended by a loss. 
of available energy. 

As regards the former proposition, Mr. Burbury, in the 
first paper, after discussing some points in the kinetic theory 
of gases, suggests ft that the diffusion would be reversible, 
provided the system were completely isolated, a condition 
which it is impossible to realize in nature, but admits that it 
is irreversible in any experiments that can practically be 
made. On this view, the point at issue may be said by one. 
unfamiliar with the kinetic theory of gases to be an abstract 
question affecting a system under conditions so unnatural 
that, whichever way it is decided, it nevertheless allows the 
natural process to be, as a matter of fact, irreversible. 

Mr. Burbury does not discuss the latter of the two 
propositions directly, but turns his attention instead to an 
equivalent theorem relating to increase of entropy, and 
merely for the particular case in hand, 2. ¢., the diffusion of 
gases. On this point he urges$ that there is no gain of 
entropy in any one collision between molecules, and that 
therefore no increase can result from all the collisions taken 
together. It may, I think, be fairly replied, again from a 
standpoint outside the kinetic theory, that it is quite un- 
necessary to consider such questions as this and others 
which he raises. The usual definition of entropy makes no. 
reference to single molecules but deals with matter in bulk ;. 
the Kinetic Theory of Gases is an interesting annexe of 
Thermodynamics, but the latter in no way depends on the 
former ; ifthe molecular theory gives results which contradict 
our experience, necessarily limited to that of gross matter, 
then, as happens from time to time with other branches of 
science, it requires revision, as regards logical development, 
or premisses, or both. Moreover, a being who could have 
experience of individual molecules would, in all probability, 
not understand the notions of pressure, temperature, heat, 


* Communicated by the Author. 
+ Phil. Mag. July and Sept. 1907. 
£ 2 ¢. play: 

§ ZL. c. p. 124. 


298 Prof. W. McF. Orr on 


work, energy, &c., in the sense in which we adopt them ; 
our distinction between heat and other forms of: kinetic: 
energy might seem to him specially artificial, and in this 
connexion it may be remarked that, although it is impossible 
to separate a mixture of (perfect) gases into constituents at 
the same pressure as the mixture, if the temperature is kept 
constant, without doing work on them, yet the gases give 
out in the process an equivalent quantity of heat, so that 
their energy remains unaltered. 

.The second proposition in all its generality, or the equi- 
valent theorem concerning entropy, does not, indeed, appear 
to be formally deducible from any usually accepted state- 
ment of Thermodynamic laws; each may form his own 
opinion as to how far experience and experiment warrant 
its acceptance. J imagine, however, that it is admitted 
almost universally that every process occurring in nature 
is irreversible and also that it involves a loss of available 
energy. | 

But in the particular case of the diffusion of gases, or at 
least of some gases, it is a fact established by direct experi- 
ment, and, I had supposed, beyond all dispute whatever, 
that there is a loss of available energy, a statement which 
Mr. Burbury appears to challenge * (for the case of oxygen 
and nitrogen). Lord Rayleigh, in the opening paragraph of 
the paper to which Mr. Burbury refers in his second paper, 
called attention to this in 1875 ; and the rest of Lord Ray- 
leigh’s paper is devoted, not so much to a proof that there is 
such a loss, but to calculations of its amount. In the simple 
and common experiment there described, a tube containing 
hydrogen and closed at the upper end stands over water ; 
the hvdrogen escapes through the pores more rapidly than 
the air enters, thus creating a partial vacuum, and the water 
accordingly rises. The available energy is thus actually 
seen to be used, to some extent, in pumping water up a 
glass tube. And, if we were able to separate the hydrogen 
and the atmospheric air, in their original states, from the 
mixture, keeping the temperature constant, without doing 
work on the system—a process the possibility of which 
Mr. Burbury appears to suggest in his second paper t—it 
would be easy to devise a heat engine by which we could 
continue to pump water from a lower to a higher level, and 
in which the working substance would go through a closed 
cycle taking in heat at a constant temperature (atmospheric), 
but requiring no condenser,—thus obtaining a perpetual 

: * L.c. p. 128. 
Tt Z. c. p. 423. 


the Mixing of Gases. 299 


motion of the second kind, as Ostwald and Planck call it ; 
it seems unnecessary to describe a suitable arrangement in 
detail. There would, of course, be a loophole of escape if 
there were any reason to think that the energy is actually 
drawn from the plug. 

Mr. Burbury makes no allusion to this paragraph of Lord 
Rayleigh’s, but proceeds to refer to one of the possible 
processes of separation of gases which Lord Rayleigh 
employed in calculating the amount of work necessary for 
the operation. In this it is supposed that a tall narrow 
vertical tube is mounted on a large reservoir containing a 
mixture of two gases at sensibly constant pressure: the 
composition varies in different parts of the tube, owing to 
the effect of gravity, there being a greater percentage of the 
lighter gas at the top of the tube than at the bottom. 
Lord Rayleigh supposes that a small quantity of gas is 
(1) removed from the top of the tube (an equivalent amount 
being, I take it, automatically supplied from the reservoir so 
that the composition of the gas at every point of the tube 
remains unaltered) ; (2) compressed until it attains the 
pressure of the gas in the reservoir ; (3) allowed to fall to 
the level of the reservoir ; and (4) forced into the reservoir, 
_ (but, as I understand it, kept separated from the gas in the 
reservoir by an impermeable membrane) ; the temperature 
is supposed to be constant throughout. Thus, from the 
original contents of the reservoir there has been separated a 
small portion of gas which is of a different composition from 
that which remains. Mr. Burbury, however, apparently 
ascribes to Lord Rayleigh a somewhat different process ; he 
supposes each element of the vertical column, in succession, 
brought down to the level of the reservoir (but without being 
replaced by other similar elements and without being forced 
into the reservoir). This he describes as a partial separation 
of the gases, which appears to be a misdescription ; it is 
merely an alteration in the position and density of gases 
without any change in the proportions of their constituents. 
He then regards these operations as the second half of a 
complete cycle of which he proceeds to supply the first 
half : evidently he does not use the phrase “* complete cycle” 
in its usual sense. In this first half he supposes a tube, 
which is a facsimile of the vertical one, but placed hori- 
zontally at the level of the reservoir, to be filled with the 
reservoir gases, interchanges the molecules of the two gases 
in it until each element has the same constitution as the 
corresponding element of the vertical column, expands each 
element until it has the same pressure and density as the 


300 Intelligence and Miscellaneous Aveaes 


corresponding element of the vertical column, and sets the 
tube vertical in a position corresponding to that‘of the column. 

When these operations are prefixed to those which he aseribes 
to Lord Rayleigh, a series of processes is obtained which, of 
course, might as well be terminated after the first, 2. @., the 
partial separation of the gases in the horizontal tube ; ; and 
Mr. Burbury contends that Lord Rayleigh ~ has proved. 
nothmg-inconsistent with the supposition that this' separation’ 
—at constant pressure ‘ ‘and ' constarit’ ‘temperature—can be 
performed without the expenditure of work. ‘In one ‘sense 
this. contention’ may be admitted, ‘for, ‘if I apprehend the 
situation’ correctly, Lord Ray leigh’ s’ investigation is not 
directly relevant to the series of processes‘which Mr. Burbury 
describes. If, however, in the reservoir, a small portion of 
gas of the same constitution as that at the top of Lord Ray- 
 Teigh’s tube be separated from the remaining contents, and 
if this’ partial’ separation can be performed at. constant 
temperature and‘ constant pressure without doing any work 
on the: gases, then, ‘by’ reversing” ‘the processes which Lord 
Ray leigh, pe understand correctly, actually does employ, it 
follows from’ his invéstigation that the’ gas system would 


again constitute the working substance in a heat. engine 


which contradicts the Second: Law of: Thermodynamics. . — 


RN Intelligence and Miscellaneous Articles. 
ON THE FOCOMETRY OF A CONCAVE LENS. 


To the Editors of the Philosophical Magazine. 


GENTLEMEN,— 


ig has just been pointed out to me that the first method I de- 
scribed in the “‘ Note on the Focometry of a Concave Lens ” 
in the Philosophical Magazine for January, is given in the Syllabus: 
of the Course of Practical Instruction in Physics at the Royal 
College of Science, London (1902). 
I was, unfortunately, unaware of this at the time of writing, as 
it is not given in my copy, which is ‘an earlier edition (1892). 
The second method is, of course, similar in principle to the first ; 
while the third is derived from the second by reversing the direction 
of the rays. 
Yours faithfully, 
J. A. TOMKINS. 


pi: 
ta 
ta 
Pe 
1 
‘ 


Ny > 


Hancock. Phil. Mag. Ser. 6, Vol. 15, Pi. 


Fic, 1. 


PLATFORM OF TESTING 
MACHINE 


Uv 


Fic. 5.—Torsion-Flexure Curves for Nickel-Steel Shafting. 


7 


90000 


0000 


70000 


60000 


oO o1 oz os OF O05 06 07 08 809 10 


Ka) STRESS 


( 


Tewsrowar (\) 
Torsiowac (2) 
FLETURAL 


AMOUNT OF f 


Fic. 2,—Ouryes showing Results of Tests of Steel Tubing 
in Tension—Torsion. 


b 
26 
¥e | 
£3 Simece Jonson 
Re (2)|7onsioW WHLE UN OR ee ALIN TENSION | 
ay ) = -|%bL . 2 
a (5) = Seat 
NR - | 56 | 3 
gs (6) . -| % 
soo 
B00 
700 
600 
500 — 
400 
300 
200 
100 }-@ i. 
Pike OF TWIST FORS/NCH LANGTH 
Toh 02-03) 04 05« 0607) :0B, 09 PADIANS 
057 11% -I7| BEB 285-342 «1399 «456 «513 DEGREES 


° 
& 


oO? 


on 


Tic. 6.—Curves showing Lowering of Dlastic Limit of Steel 
due to Combined Stresses. 


OF ASTIC LIMIT 1M FLEXURE OUE TO 7afsion 
ER ING OF EX ASTIC LIMIT Ii TENSION DUE TO Torsion 
ST/e LIMIT iy TORS/ON DUR TO TENS/ON AND COMPRESS/ON 


Nore 


70 flere ne-Torslon Newz STEEL SHAFTING 
seus - Tors: IN STEEL SHRETING 
NAEKERS TO 70RS/OV-TEWSVON NibyEe STEFL 


~Tewslon Steet Tusing 
=ToRshon Steke Tuaiye 
To Co, 


oF Exasric Limir 


en nee ns 


TORSIONAL a) 
NOUN OF eee (2) 


STRESS 


Compasssiowaz (3) 
Tomsiont @) 


18000) 


5000 


6000 


Fiaeh SrREss. ourer FIBER, Pounds PER se iA. 


Tic. 3.—Ouryes showing Results of Tests of Steel Tubing in 
Compression-Torsion. 


Symece ToRsyon 
J, COMPRESSJON —T0F 
Ys - - 
he ~ : 


0 *00} 


RESSANTORS ION SteeL TUcine 


Ge : . 


ANGLE OF | TWIST) PER UNIT LGNGTH. 


7005 “006 <007, | .00B) AAD/AWS 
285 352 399 456 OEGrees 


“002 4003 
WA 171 


004 


057 228 


Fig. 7.—Curves showing Lowering of Blastic Limit of Steel 
in Tension, Torsion, Wlexure, and Conrpression, due to 
Combined Stresses. 


& 
= 
re inzowel ING OF THE ELASTIC Lid v FLEXURE fue 70 7pRsion 
HO) of Sly ~| ~ Comeress = 
8 
Bie - cs | TEWSION | = 
$14 e's Torsion | - ENS/OM| 
to (Arto COMPRESSION, 
03 > a 
sy 
lc 
On) . 
g Re, 
w x, TN 
S 
K =, 0) 
G 07 =] = : 
Aj a 
Sane ia 
io — = 
s (2) 
Sa 
an 
See $05 ; 
Sess 
Gree 
SRK 04 
SS 
re 
C08 | — 
§ 
3 
= 07 


: 


eee 
% &% ¥ % % 


Uinsunr on Srasss FP iIED INrincey 


fe OF ELASTICLIM/T 


— 
lin SIMPLE ALEXURE Tesh I, 
(2) FLgxURE WHILE|UNDEM TORSHON. SH. PER SQ, 11. 
70000 : —_ 
an = 5 = 
a |. z : a | aa 
60000 
50000 
40000 
30000 
9 
re 
20000 
5 
& 
{0000} 
hy 
DEFLECTION IN| (CHES 
i SENS (cee Se 


Phil Mag. Ser. 6, Vol. 15, Pl. XT. 


Fic. 4.—Torsion in Flexuve Curves for Carbon-Steel Shafting, 


035 04 G5 06 07 O8 09 10 


Fis. 8 


DiACKAMS SHOWING THE EFFECT 
OF COMBINED STRESSES ON - 


o 


N 


C) 


()Moouius or Liasriciry 
@ (2)\DEFonmaA tion ar Lzasric Limir 


} 
| 
| 
D(3)UmT Frac STRESS ar Ecastic Lin/T | 


») Nore -— Aexune 
W-S=WICHEL STEEL SHAFTINE | wateecuroes! 
C-S= CrpsonSreet SHAFTING) TORSION | 
\ N= Nicene Sree r 
C= Carson Steet EEA 
85-9050 Steet TUs/NG ToRSiON. 


UNSIDE DIAMETER, 
6590-50 Hesrecrivery 


7-7 = STEEL TUBING { ToRs/OoNW WHKE 
Unoer Tension | 
C-T= Sreet Tuaincs TORSION WHE | 
UNDER COMPAESS/0N | 


A 


A 


“nT OF DEFORNAT/O 


Le 


(3) 
4% % ae eer ee 3 


Amounr oF STRESS APPLIED FRACTION oF ELASTIC LIMIT. 


Ss lf 

S HESS} a 
: < a < Lt ES Bo iil 
oe ee 


Phil. Ma; 
memart =| 


ie 
tt 


3 


Segoe SS 


} 


STANDARD CHART FOR SCREW SYMMETRY. 


te 


AXIS oF SCREW 
a Se il —--------------------- = - 


if rid 
yee 


*) 


Phil. Mag. Ser. 6. Vol. 15, Pl. XTIT. 


Woop. 


BI BEE PSD: ae OPE LOR Oe eR Pee foe Boge 8 kl eel epe Bee moun 


EE) | 


hy 


l 


{ 


\ 


\ 


pice Put 
v\ 

" 

( | 

0) 


1 


pealbap w1 
UOTJDION ~<— 


O 
“LOY OVW 


T ‘OI 


Woop. Phil. Mag. Ser. 6, Vol. 15, Pl. XIV 


FPN YM 


Pe 


- 7 j 7 rs J 
Ad} a eS eo 


FHE 
LONDON, EDINBURGH, axn DUBLIN 
PHILOSOPHICAL MAGAZINE 
AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIES.] 


/ MARCH 1903. 


XXIX. On the Practical Attainment of the Thermodynamic 
Scale of Temperature—Part II]. By J. Rose-[yyes, 


meee. B.Sc.* 
HE first part of this paper, which was published in the 


Philosophical Magazine some years ago, contained a 
method whereby the thermodynamic correction to a gas- 
thermometer could be calculated from a knowledge of the 
Joule-Thomson effect for the gas, and of its isothermal com- 
pressibility at some one temperature (Phil. Mag. July 1901, 
pp. 131-133). The theory of the method given in that 
paper | still believe to be sound; but the accuracy of the 
final numerical results must depend not merely on the 
soundness of the theory followed, but also on the accuracy 
of the experimental “data employed in the calculations. 
Several physicists have lately seen reason to entertain doubts 
concerning the trustworthiness of the measurements of the 
cooling-ettect carried out by Joule and Kelvin. I have 
therefore thought it might be of some interest to develop a 
theory of the gas-thermometer in which less reliance is placed 
than formerly on the Joule-Thomson measurements. Not 
that we can afford to neglect the Joule-Thomson measure- 
ments altogether; every theory of the gas-thermometer is 
obliged to rely on them to some extent. But we are not 
obliged to go the length of accepting the absolute values of 


* Communicated by Prof. F. T. Trouton. 
Phil. Mag. S. 6. Vol. 15. No, 87. March 1908. Y 


302 Mr. J. Rose-Innes on the Practical Attainment of 


the results obtained by Joule and Kelvin; we can form a 
consistent theory if we accept the Joule-Thomson figures as 
giving us simply the relative values of the cooling-effect for 
various temperatures and pressures of the streaming gas, and 
if we use these relative values in conjunction with other 
experimental data. Proceeding in this way, I have succeeded 
in diminishing some of the difficulties which were encountered 
in the first part of the paper. 


Integration of the fundamental Differential Equation. 


The differential equation for a gas streaming through a 
porous plug is 


(See Kelvin’s Reprinted Papers, vol. i. p. 2485; see also 
vol iis p..179.) 

The notion underlying the present plan is that in certain 
cases the Joule-Thomson measurements might be, as regards 
their absolute value, less trustworthy than observations made 
on some other physical quantity, and yet might be trusted to 
give us the relative values of the Joule-Thomson effect with 
sufficient accuracy; thatis to say, the ratio of the values of the 
Joule-Thomson effect for two different sets of conditions might 
be given sufficiently well by means of the measurements. We 
may express this algebraically by supposing that the Joule- 
Thomson effect is equal to the experimentally determined 
value multiplied by A, where X is a constant factor, which 
must subsequently be either calculated or eliminated from 
the equations. 

- But even though complete confidence is not placed in the 
Joule-Thomson measurements as regards their absolnte value, 
they may be supposed sufficiently accurate to tell us the 
order of magnitude of JK e ; for such gases as are actually 
used for thermometric purposes J is always small, and 


will be treated as such in what follows. Further, the experi- 
dt 
ments of Joule and Kelvin show us that yess may be 
treated as a function of the temperature only, if we are 
neglecting squares of small quantities. _ 
Let us denote by the letter y the experimentally deter- 


mined values of 3 then the above differential equation 


the Thermodynamie Scale of Temperature. 303 
may be written , | 


t ee —v= TK yy. 


Divide by ¢? and we have 


1 dv v  JKyry 
(a), Ta ee 

Suppose that the suffixes 1 and 0 applied to v and ¢ refer 
to the boiling and freezing points respectively ; then let us 
integrate the last equation with regard to ¢ from ¢, to t, 
at constant pressure p ; we shall obtain 


2 | 4JKp .. 
Rory rf) re dt = XI (say). 


a5 ty : 
Multiply by ‘ and transpose ; we obtain 
. (any ; 
v t t 
ia a 
% to an) 
Similarly by integrating at constant pressure Pp we may 
obtain 


where dashe:l letters refer to the pressure p’. Subtracting 
and dividing by ¢,—t), we have 


t es 2 
al—-a= oa Gre ee 


any 


where a denotes the coefficient of expansion at constant 


pressure. Hence 
r =(2'-a)+ 4 4 (a- = oz 
ht) ol % 


lf we calculate X from this formula we shall be able to 
turn the numbers found by Joule and Kelvin into absolute 
values. We notice thatasmall percentage error in the value 
of X will involve only the same percentage error in the finally 
accepted values for the Joule-Thomson effect,—itself a small 
quantity. Hence it is permissible to employ approximate 
methods to some extent in calculating >. Thus we see that 
we require to know the values of ¢) and ¢,; if we employ 
yalues for these quantities obtained from an uncorrected 


¥ 2 


304 Mr. J. Rose-Innes on the Practical Attainment of 


hydrogen or nitrogen thermometer, we shall thereby intro- 
duce an error involving squares of small quantities into the 
finally accepted values of the Joule-Thomson effect : such an 
error may be safely disregarded. For much the same 
reason it is permissible to employ approximate methods in 
evaluating the integral I. We can effect the evaluation most 
conveniently by a graphical method,—plotting a suitable 
curve and finding its area by means of a planimeter in the 
usual way. 

Having secured a sufficiently accurate value of 2, we may 
proceed to calculate the absolute values of the Joule-Thomson 
effect from the experimental results of Joule and Kelvin. 


We will suppose that J RS throughout the field of observa- 


tion may be fairly well reproduced by means of a series in 
an 


descending powers of i, say > ~ where n is either positive 


or zero. The fundamental differential equation may then be 


written 
dv an 
t (Zi), —-v= = rs 
Divide by 2, | 
= (5, eee hat 
t\di/, ie ed 
Integrate with respect to ¢ along an isopiestic, and we 
obtain 
a 


n 


i) 
(n+1) et” 


en P is a function of p only. Multiply by pt, and we 
ave 


eK) Ss 


Qn 
eal CEN 


The product pP is a function of p only and may be 
denoted by f(p): we thus have 


— ait An 
eam A ak a 


Denote pv by the single symbol ~y, and differentiate with 
respect to p, keeping ¢ constant, 


a3 tht _ a&n 
7), =f @) ICES DY. 


the Thermodynamic Scale of Temperature. 305 


l aie 
For several gases the quantity (< ) is difficult to deter- 
= dp t 
mine experimentally at temperatures where the Joule- 
Thomson effect has been measured ; for such gases, however, 
it is always found possible to select some one isothermal as 
being fairly well determined ; the temperature of this iso- 
thermal is never far removed from ordinary atmospheric 
i 
temperatures. The value of (**) along the selected iso- 
t 

thermal is found in all cases to be independent of the pres- 
sure to the degree of accuracy to which we are at present 
working. It tollows that 


7 (5) $5 tnt 
t dp/, (n+l1l)e 


mnst be constant along the selected isothermal ; hence /’(p) 
will also be a constant along the same isothermal, say e. 
Further, the value of /'(p) from its form must remain 
unaltered for any isopiestic change; hence it will remain 
equal to e for any condition of the gas which can be reached 
from the above-mentioned isothermal by means of an iso- 
piestic change. In other words, /’(p) must be equal to the 
constant € to our present degree of accuracy. 


Since f'(p) =e, it follows that 
T(p) = R+ep, 
where R is an arbitrary constant introduced by the integra- 
tion. Employing this value of /(p) we obtain 
An 
(n+1)t” 


If v is kept constant while p and ¢ are both made to 
increase together, the term ept will ultimately become more 
important than Rt. As it seems improbable that this can 
represent the true state of things at high temperatures we 
ought to try to make e vanish. We can secure this result 


if we can put | , 
) Ui gs 4 Se Oe oe 
( ap hs (n+ 1) 


along the isothermal we have selected for measuring the 


w= Rt+ept—p2 


. Ai ae a 
deviation from Boyle’s law. Choosing the series & a so as to 


306 Mr. J. Rose-Innes on the Practical Attainment of | 


fulfil the last condition, we obtain the characteristic equation 


We shall next enquire what is the change in the coefficient 
of expansion corresponding to the above equation. We have 


oe ee 
Aree (n+ 1) t)” 
nee On 


P (n+ 1) a 


Multiply the last equation by 2 the last but one by 
9 


t : 
!_ and subtract, we obtain 


Yo “9 
i 4o F say 3 dn mat}. 
Co to Vo to (n + ip) ty” (n+1)é" + 1) t,” 


Similarly for pressure p! we should have 


? 
Ui Eat 1 64 An Zn } 
bed hatgrd 2g! ae  CEMIDTT (a+ (n+1)t,” a (n+ 1) ty") ° 


These last two ie after a little reduction yield 


a! -2a= (-- t ee Qn an ’ 
mae t—to 2 (n+1) ttt Tp ly art f° 
We see nies that the series > 2 must be chosen so as to 


satisfy three conditions :—(i.) The observed Joule-Thomson 
effect at different temperatures must be proportional to the 


values of the expression ee ei ; (i1.) the value of Ge) must be 


represented by —S 7a for some single isothermal ; 
n 


Gii.) the observed change in the coefficient of expansion 
must agree with the calculated value. 


Variation of the above Method. 


In the investigation given above I have assumed that 
observations of the Joule-Thomson eftect had been made at 
a sufficient number of different temperatures to fix fairly 


the Thermodynamic Scale of Temperature. 307 


definitely the form of the curve oS plotted against ¢ 


between the limits ¢, and 4. It may eon however, that 
the observations are not sufficient in number to fix the curve 
with-as much definiteness as could be desired, and in such a 
case the resulting uncertainty in the value of I may lead to 
errors in the calculated Joule-Thomson effect too large to 
be fairly comparable with the squares of small quantities. 
In order to avoid errors from such a cause the most con- 
venient plan is to take some account of the observed value of 


(F) in settling the form of the curve. This plan may 
t 

be carried out in substance by means of the following 
device :— 


Suppose we find a series in descending powers of ¢, say 


is —where n is either positive or zero—such that So is ; 
equal to the observed Joule-Thomson effect, and — Ph Gras z 


is equal to the observed value of i along the selected 
t 


is an approximation to the series we 


C 
isothermal. Thus zon 


are seeking, but it needs correction. Let us find a second 


BF : e Poe: 
series in descending powers of t, say = Fo where n is either 


positive or zero—such that ps s is equal to the observed 
Cn 

(n+1) 

Cn + Ken 
ie 


"Tl . : 
Joule-Thomson effect, and & is zero along the 


selected isothermal. Then the series = , where « is a 


constant at our disposal, is equal to 1+. times the observed 


Joule-Thomson effect, while 1 eS ee, is equal to the 


A ie 
d 
observed value of A along the selected isothermal. 
t 


Cnt Ke 
a elk 
ditions mentioned above, and we can make it fulfil the third 
by properly choosing «. In fact, when we employ this new 
series, we find that the difference between the coefficients of 


Hence the series > ~—— fulfils the first two of the con- 


308 Mr. J. Rose-Innes on the Practical Attainment of 


Pe ed ; a) ae 
expansion at constant pressures p and p’ is given by 


t LEY gE ¢, t iL ) 
pl , L sid n ? Gamat tee ks 
eo0t 7 i —t Seat ~) 13, n+1 (sc5 1 get 
en 1} ] )\ 
$08 (7 Ferrie ae ae tj", % 


This equation may be considered as fixing the value of « 5 


‘sl ° (G +k é, . 
once «is known we know also the series er which 


fulfils the three conditions Jaid down at the end of last 
section. Itis elear that L+« is equivalent to the A of last 
sectlon. 


alpplicution of the above Theory to Hydrogen and to Nitrogen. 


We have next to consider the application of the above 
theory to the gases actually used for thermometric purposes. 
Tt is found that for hy drogen and for nitrogen we ean fulfil 
the three conditions, specified at the end of ‘the first section, 
with sufheient aceuracy by putting 


an 2s 
te ma iste? a Ty 


It is worth remarking that the te algebraic form 
which we choose for the series does not perceptibly affect 
the final numerical results for the thermodynamic correc- 
tions, so long as we employ the same experimental data to 
calculate the constants, and so long as we keep within the 
limits of temperature and pressure over which the Joule- 
Thomson effect has been observed. We may express this by 
saying that the three condifions determine the thermo- 
dynamic corrections with arithmetical uniqueness within the 
field of observation over which they lold good. Hence, if 
we do not attempt to extrapolate, we may choose the form of | 
our series solely with a view to ease in arithmetical caleula- 
tion, and the form suggested above is on the whole the most 
convenient for such a purpose. 

The experimental data relating to hydrogen and to nitrogen 
will be considered separately. 

Hydrogen. —This gas was subjected to the porous-plug 
experiment by Joule and Kelvin ; the results are given in 
Kelvin’s Reprinted Papers, vol. iii. Pp. Lae There was a 
heating eect which amounted, per 100 inches of mercury 


the Thermodynamie Scale of Temperature. 309 


of differential pressure, to 0°:100 C. at temperatures of 4° C. 
or 5° ©., and to 0° 155 C. at temperatures of from 89° C. to 
93°C. The change of a with the pressure was examined by 
M. Amagat (see Annales de Chimie et de Physique, 6° série, 
t. xxix. p. 127) ; it is unfortunate that his experiments apply 
to high pressures only, but since the value of « appears to 
vary ‘steadily with the pressure over a considerable range of 
pressure, we may probably make use of his results w ithout 
introducing any sensible error. The decrease in @ corre- 
sponding to an increase of 100 atmospheres pressure is 


‘00018. The value of a ean also be obtained from 


M. Amagat’s experiments (see his paper, Ann. de Chimie 
et de Physique, 5° série, vol. xxii.) In the first part of the 
present paper I practically accepted M. Amagat’s statement 
that the isothermals of hydrogen,—y plotted against p— 
form a set of parallel straight lines with a slope correspond- 


ie = 00078. Though M.Amagat warns his 


readers Sif this law is only approximate, I eame to the 
conclusion that it was sufficiently accurate for the purpose I 


had in view ; accordingly I took (=) to be ‘00078, and 
t 


ing to 


attached this value to the temperature 50° C. as being mid- 
way between the boiling- and freezing- points. But a more 
careful examination of M. Amagat’s results has shown me 
that the rough method 1 employ ed was somewhat rougher 
than I supposed. The curvature of the isothermals—though 
not very apparent to the eve in a diagram—indicates a 
smaller deviation from Boyle’s law for those portions of the 
isothermal waich are nearer the axis of zero pressure ; and 
I find that the isothermal ter 17° 7 C. yields a value of 
‘dw 
(iy 
nearly with that given by MM. Leduc & Sacerdote, viz. 00064 
at 16° C. (see Comptes Rendus, t. CXxWe beOT, p. 299), and is 
ere ay aa from the) value viven by Prof. K. Onnes, 
viz. *000623 at 20° C. (Schalwijk, Académie des Sciences 
ad’ Amsterdam, June 1901). 

Nitrogen. —'Phe isothermal compressibility of this gas has 
been studied by M. Amagat (Comptes Rendus, t. xcix. p. " 1153); 
the gas was subjected to the porous-plug pale by 
Joule & Kelvin (Kelvin’s Reprinted Papers, vol. 1. p. 42 1); 


and the coetticient of expansion at various constant pressures 


)= 000645 at low pressures. This value agrees very 
t 


310 Mr. J. Rose-Innes on the Practical Attainment of 


has been determined by M. Chappuis (Zvav. et Mém. du Bur. 
Int. xiii, pp. 21-25). 

The experiments of M. Amagat on the compressibility of 
nitrogen were conducted chiefly at high pressures; but 
there is one measurement of pv at atmospheric pressure on 
the isothermal of 16° C., so that we may obtain the desired 


value of (s) at this temperature by means of an inter- 


It 
polation. In the earlier part of this paper I gave —0-00035 
as the value founded on M. Amagat’s experiments: and this 
figure is very nearly in accord with that subsequently 
found by Lord Rayleigh from his own experiments, viz., 
—0:00034 (Proc. Roy. Soc. Ixxiii. no. 490, p. 153). The 
small difference between these two estimates is not sufficient 
to cause any appreciable change in the final numerical 
results, so that I have simply retained my former value of 
dy 
dp ) 

M. Chappuis’s most trustworthy determinations of the 
coefficient of expansion were made with a thermometer 
having a reservoir of iridio-platinum. His observations with 
this instrument may be divided into two groups: the first 
group having a mean pressure of 1:001855 m..of mercury 
gave a value of the coettficient 0°00367315, the second group 
having a mean pressure of 1°386787 m. of mercury gave a 
value of the coefficient 0°00367775. 

The value of the Joule-Thomson effect was determined b 
Joule & Kelvin for three initial temperatures, viz.: 7°204C., 
91°-415 C., and 91°965 C. ‘These last two temperatures are 
so close that we should not expect to find any marked 
difference in the Joule-Thomson effect as we pass from one 
to the other; and, as a matter of fact, the cooling-effect per 
100 inches of mercury, for the actual gas employed, is much 
the same in both cases. But the figures which are given for 
the Joule-Thomson effect of pure nitrogen, as the result of 
an extrapolation, differ considerably in the two cases ; so 
much so that it is clear the difference cannot be due to the 
small change in the initial temperature, but that we are in 
presence of a serious experimental error. I have, as in the 
earlier part of this paper, altogether rejected the value at 
91°-965 C. on the ground that the experimental gas in this 
case contained a large percentage of oxygen, and that the 


figure given for pure nitrogen is the result of a considerable 
extrapolation. 


A] 


—— 
wo 


the Thermodynamie Scale of Temperature. 311 


Making use of the data referred to above, we obtain the 
following values for the constants ag, a;, a2 :-— 


Hydrogen. Nitrogen. 
ec a —°00060654 Vy ......... —°0018319 V, 
hx SE oe ean 10105 V) 
MGA: GHEY git ai Le: 108-73 Vo. 


Here V, is the volume of 1 gramme of the gas at standard 
temperature and pressure. 


Thermodynamic correction to the Gas- Thermometer. 


In the earlier part of this paper I considered the thermo- 
dynamic correction to the gas-thermometer on the assumption 
that the Joule-Thomson effect could be reproduced with 
suflicient accuracy by an expression of the form 


ay ag 
a = ae a 
a tik 


The algebraic work there given is still applicable, since I 
propose to adhere to the same algebraic formula for the 
Joule-Thomson effect, though with altered values of the 
constants. © 

Alydrogen—tThe coefficient of increase of pressure at 
constant volume has been carefully investigated by M. Chap- 
puis. His measurements are probably the best that have 
yet been published ; unfortunately, owing to experimental 
difficulties, there is still an appreciable discordance between 
several of his determinations. As the result of his experi- 
ments conducted at three different times, he gives the 
following numbers for the coefficient :— 


00366254 with initial pressure 998-9 mm. of mercury. 
"00366296 __,, + saixy LODO Ds rs, ‘3 
"00366217 ,, > ev, LOOT Et ya 

(See Trav. et Mém. du Bur. Int. xiii. p. 61.) 


Of the three determinations here given the two first were 
made with a reservoir of iridio-platinum, and the third with 
a reservoir of hard glass. The variations in the value of the 
initial pressure are small and may be disregarded for our 
present purpose. Iam inclined to place the greatest reliance 
on the last determination, which was made with a reservoir 
of hard glass; since M. Chappuis himself, in another publi- 
cation, refers to the inconvenicace caused by the permeability 
of metals to hydrogen at high temperatures, and it is quite 


312 


possible that even at lower temperatures some slight inter- 
action may take place between the walls of a metal vessel 
and the enclosed gas (“ L’échelle thermomeétrique normale et 
les échelles pratiques,” pp. 8-4). For these reasons! prefer to 
take the figure ‘00366217 as the most probable value for the 
coefficient of the increase of pressure at constant volume ; 
we thus obtain 273°062 as the “uncorrected estimate ”’ of 
the freezing-point of water. By employing the values of the 
constants given at the end of last Section, we may readily 
calculate that the proper correction to be applied is 0°-064, 
so that we obtain 273°-131 as the estimate of the freezing- 
point. 

The thermodynamic corrections to the readings of the 
hydrogen thermometer are given in the following Table :— 


Mr. J. Rose-Innes on the Practical Attainment of 


Tasxe f. 

/ | Correction. | Correction. 

(2 emmemenber 

san | Constant | Constant eR | Constant Constant 

| | pressure | volume | pressure volume 

thermometer. | thermometer. | thermometer.) thermometer. 

Op: Abin 0%) ¥ yiprktye | 60° 4 —0023 —-0015 

| 10 —-001L | —007 |, 70 —-0019 —-0013 
20 ORB IG «t= 0012 ||: «80 — 0014 — 0010 

| 30 | —0023 | —0015 || 90 —0008 | —-0005 

| 40 | —0025 | —-0016 100 0 0 

| 50 | 0025 | 0016 || | | | 

| ! 


The constant-pressure thermometer is supposed to be under 
the pressure of 1 atmosphere, and the constant-volume 
thermometer is supposed to be under the pressure of 
] atmosphere at the freezing-point. 


Nitrogen.—We saw that M. Chappuis’s value for the — 


coefficient of expansion at a constant pressure of 1-002 m. 
was ‘00367315 ; froin this figure we obtain 272°246 as the 
‘uncorrected estimate ” of the freezing-point of water. We 
may employ the values of the constants given at the end 
of last Section to calculate the necessary correction: we 
easily find that it is equal to 0° 890. Hence we obtain 
273° 136 as the * corrected estimate ” of the freezing-point. 

The following Table gives the thermodynamic corrections 
for the constant-pressure thermometer under 1 atmosphere 
pressure, and for -theconstant-volume thermometer haying 
the pressure of 1 atmosphere at the freezing-point :— 


the Thermodynamic Seale of Temperature. 313 


TaBueE II. 

} 
Correction. \ Correction. 
| Temp. Constant | Constant | Temp | Constant Constant 
) pressure | volume 1 | pressure voluine 
/ thermometer. thermometer.| -thermometer. thermometer. 

0°. 0 Gn aie snc @OiOe lett —=-OPGS41) jos 0026 
10 = D074) ih yt OOR al eFOdaI ie orgs.) |) —0022 
| 20 —0126  —0019 || 89 | 0102 | —016 
— 80 —0159 | — 0025 1°90 | —-0056 | —-0009 
| 40 —0175 | —-0027 | 100 0 0 
| 50 0176 | —-0028 || | 


The numerical results given in the tables of last Section 
show that the thermodynamic corrections to the constant- 
volume thermometer are small quantities, when we employ 
hydrogen or nitrogen at standard density ; and it becomes 
of importance to examine what sort of relation these 
corrections bear to the mistakes of reading due to the un- 
avoidable errors of experiment. It has been already 
remarked that in the case of hydrogen the nature of the 
walls of the containing vessel appears to exert a perceptible 
influence upon the pressure of the ges. Great interest 
therefore attaches to a careful comparison, carried out by 
M. Chappuis, between the readings of two hydrogen thermo- 
meters, both used at constant-volume, but differing in the mate- 
rial of their bulbs. The following figures were obtained :— 


TABLE III. 
Excess of reading of mercury-in-glass 
thermometer over constant-volume 
Temperature. hydrogen thermometer with : 

| Glass reservoir. | Metai reservoir. 

10° C. 051 | 052 
20 ‘076 "085 

30 "095 "102 

40 "108 ‘107 


(Trav. et Mém. du Bur. Int. xii. p. 39). The pressure at 


314 Mr. J. Rose-Innes on the Practical Attainment of 


treezing-point of the hydrogen thermometers was in both 
cases 1 metre of mercury. 

An inspection of the above Table shows us that the 
difference of reading between two hydrogen thermometers is 
quite appreciable ; it is as high as 0°-009 C., for instance, at 
20° C. Hence, at this temperature, one at least of the two 
thermometers must be wrong in its-reading by a fairly large 
figure in the third place of decimals. This suggests that 
even where the two thermometers agree, as in the neighbour- 
hood of 40° C., we cannot be certain there is no error: it is 
just as likely that both thermometers happen to exhibit at 
this place the same divergence from the thermodynamic 
scale, the divergence being of the same order of magnitude 
as that actually proved to exist at 20° C. 

We readily find from the tables of last Section that the 
thermodynamic correction at 20° C. for the hydrogen thermo- 
meter, under the conditions of pressure observed by M. 
Chappuis, is less than 0°-002 C. Such a quantity is con- 
siderably smaller than the error proved to exist in at least 
one of the two hydrogen thermometers. A similar remark 
applies to the thermodynamic correction at 30° C. ; and it is 
quite likely, though not actually proved, that the correction 
is smaller than the experimental error at other temperatures 
between the freezing- and boiling-points. Hence the thermo- 
dynamic corrections to the readings of a constant-volume 
hydrogen thermometer are usually neglected. 

As the standard instrument for thermometric purposes it 
is usual to take just such an instrument as those examined 
by M. Chappuis, viz., a constant-volume hydrogen thermo- 
meter with a pressure ‘of 1 metre of mercury at the freezing- 
point. But since we do not really know ‘which of his two 
thermemeters is the more correct in its indications, we may 
inadvertently have let our choice fall upon the more faulty 
one for the standard instrument. Hence we cannot trust 
our standard thermometer in the third place of decimals for | 
anything more than thermoscopic purposes. And since the 
difference of reading of the two hydrogen thermometers 
at 20°C. is greater than the thermodynamic correction 
to the constant-volume nitrogen thermometer at the same 
temperature, we cannot be certain that the usual standard 
thermometer is everywhere superior in the accuracy of 
its actual readings, to the constant-volume nitrogen ther- 
mometer, 


the Thermodynamic Scale of Temperature. 315 
Conclusion. 


We have found that there is a good agreement between 
the two estimates of the freezing-point which are given in 
the present part of this paper, and are derived from the 


_ experimental data concerning hydrogen and nitrogen re- 


spectively ; the closeness of this agreement entitles us to 
consider the present method as superior in accuracy to those 
previously employed. We may also place considerable con- 
tidence in the tables of thermodynamic corrections given in 
this paper ; because, as has been already pointed out, while 
we confine ourselves within the limits of temperature and 
pressure, within which the Joule-Thomson effect has been 
observed, the precise algebraic form of the expression chosen 
to represent the effect has practically no influence on the final 
numerical results. But outside these limits the algebraic 
form of the expression chosen to represent the Joule-Thomson 
effect may, and in general will, exercise considerable influence 
on the final numerical results. For this reason J] have not 
given the thermodynamic correction for temperatures lying 
either below the freezing-point or above the boiling-point. 
And until we know the true form which ought to be employed 
for the Joule-Thomson effect, the results of calculating the 
thermodynamic corrections for such temperatures must be 
largely speculative. 

The above considerations point to a large gap in our 
knowledge, and some physicists have endeavoured to meet 
the difficultv by employing a rational formula for the Joule- 
Thomson effect based upon some molecular hypothesis 
regarding the constitution of gases. The drawback to such 
a method of procedure is that the trustworthiness of the final 
numerical results is made to depend not merely on the two 
laws of Thermodynamics and on data derived from experi- 
ment, but also on the truth or falsehood of a speculative 
hypothesis. The introduction of such speculations into the 
treatment of a purely thermodynamic question must be 
considered as being to a large extent a step backwards. On 
the other hand, it is not possible to remain contented with the 
present state of our knowledge of the thermodynamic 
correction to a gas-thermometer ; we have to confine our 
theory to temperatures which lie between the boiling- and 
freezing-points, or which are but slightly removed from 
those limits. 

The most simple and at the same time the most effective 
method of meeting the difficulty would be to repeat the 
Joule-[Thomson experiments over a much wider range of 


316 Prof. A. H. Bucherer on the 


temperature. If an empirical formula were found to fit the 
values of the Joule-Thomson effect over a wide range of 
temperature, then we might very well conclude—apart from 
any molecular hypothesis—that this formula was the proper 
one to employ. It is true that measurements of the Joule- 
Thomson effect are far from easy to carry out satisfactorily ; : 
still, the difficulties are not insuperable , and there is no 
reason why the success of Joule and Kelvin in this line should 
not be repeated. The measurements of these last-named 
experimenters appear to have been confined within narrow 
limits of temperature, not so much because observations were 
impossible at temperatures outside these limits, as because 
Lord Kelvin imagined he had already discovered the true 
formula for the Joule-Thomson effect. 

The plan here advocated of repeating the Joule-Thomson 
experiments over a wider range of temperature is all the 
more feasible since we have shown that we require only the 
relative values of the Joule-Thomson effect. Thus any 
source of error which multiplies all the Joule-Thomson effects 
by the same factor would be eliminated The errors of ex- 
periment which give rise to ordinary “‘ wobbling” would 
also be eliminated by the present method. Indeed, the only 
sources of error which are liable to affect the final numerical 
results to a sensible degree are those which tend persistently 
to increase or diminish the Joule-Thomson effect at higher 
temperatures as compared with that at lower. Provided 
that such sources of error were either abolished or properly 
allowed for, we could place considerable contidence in the 
final numerical results, and probably sneceed in throwing 
great light on a fundamental problem of thermodynamics. 


XXX. On the Principle of Relativity and on the Electro- — 
magnetic Muss of the Electron. A Reply to Mr. KE. Cun- 
ningham. By A. H. Bucuersr, D.Sc., Professor in the 
Bonn University *. 


N the October number of this Magazine Mr. E. Conaaee 
raises some objections to the theory of relativity as de- 
fined by me in the April number. 1 wish to say a few words 
in reply in order to show that Mr. Cunningham’s remarks are 
due to a misconception on his part of the real meaning and 
bearing of the principle as used by me. 
As appears from my paper, my object has been to find 
a purely phenomenological method of calculating electro- 
magnetic effects, which should harmonize with all the facts 


* Communicated by the Author. 


Principle of Relativity. 317 


o£ observation, leaving it to future endeavours to find a 
physical interpretation of thismethod. No doubt this way of 
proceeding implies a certain resignation. But in view of the 
failure of the electromagnetic theories advanced as yet, it 
seems the safest road to follow. 

My method rests on ths following principles and de- 
finitions :— 

(1) The validity of the Maxwellian “ differential ” equations 
associated with ordinary kinematics. 

(2) The distinction, for the mere sake of calculation, between 
active and passive systems, whenever forces are concerned 
which two electromagnetic systems in uniform relative motion 
exert on each other. 

(3) The prescription: Whenever the force on one of the 
two systems due to the other is required, choose the former 
as the passive one and calculate the force on it exactly as in 
the original Maxwellian theory, as though it were “ at rest in 
the ether” and the other “‘ moving through the ether.” 

I have proved (J. ¢.) that this prescription is consistent, 
2. e. that the forces thus calculated are identical whichever of 
the two systems is chosen as passive. Z'his implies the prin- 
ciple of relativity of motion for the systems considered. 

Mr. Cunningham will admit that this method of calculating 
is perfectly definite, and by a careful comparison with the 
Lorentz - Einstein principle he will convince himself that 
the two principles are essentially different. The remark of 
Mr. Cunningham that my principle is identical with that of 
Einstein except that [omit the transformation of time and 
space coordinates, appears untenable also from the following 
consideration. HWvidently, according to Mr. Cunningham, a 
transformation of the forces experienced by a moving electron 
in a condenser field and in the field of an electromagnet as 
calculated by me (/.c.) should yield the expressions given 
by Hinstein and Lorentz. Butan inspection of my equations 
proves the impossibility of such a transformation. In fact 
no other known theory of electromagnetism leads to these 
forces. 

As I employ the ordinary kinematics, only a spherical 
electron wiil fit in my theory. Mr. Cunningham has overlooked 
this circumstance. 

However, it does not follow that the same formula as 
Abraham’s should be applied, as this formula is connected 
with the expression of the field energy, and the latter is 
introduced by me as a special hypothesis (J. ¢. p. 418). 

Whereas it will be impossible to point to any discrepancy 
Phil. Mag. 8. 6. Vol. 15. No. 87. March 1908. Z 


— ay: an 


318 Mr. T. J. Bowlker on the Factors serving 


between my theory and experimental facts in the electro- 
dynamics of the relative uniform metion of electric and 
magnetic masses, the Lorentz theory finds unsurmountable 
difficulties on theoretical grounds. As was first conclusively 
shown by Abraham, the Lorentz deformation excludes a 
purely electromagnetic basis of mechanics, The work of 
the external electric forces acting on the electron does not 
have its exact equivalent in the increase of the electro- 
magnetic energy of the electron. Therefore a certain inner 
energy of non-electromagnetic character must be ascribed to 
the electron. The same conclusion must be drawn from 
Mr. Cunningham’s calculations, if they are properly interpreted. 

Mr. Cunningham says the Lorentz-Hinstein theory is the 
only theory that can account for certain optical phenomena. 
In fact, he asserts that it is required “to explain how a 
light-wave travelling outwards in all directions with velocity 
C relative to an observer A may at the same time be travelling 
outwards in all directions with the same velocity relative to 
an observer B moving relative to A with velocity v.” 
Mr. Cunningham then proceeds to show that this require- 
ment is satisfied by the Lorentz-Hinstein transformation. 

I am not aware that such a “ requirement ” is necessary to. 
explain any known fact of observation. 


XXXI. On the Factors serving to determine the Direction of 
Sound. By T. J. BOwLKER*. 


N the summer of 1906, while on a steamship off the coast 
of Maine, U.S.A., I was roused about midnight by the 
blowing of foghorns, and presently followed the shock and 
grinding of a collision. It appeared to me that the accident 
could only be explained by a mistake in judging of the 
direction of the foghorns of the colliding vessels. This accident 
suggested a study of the factors determining the direction 
from which sound appears to come. y 
During the winter of 1906-1907 I made some experi- 
ments. In one of them I placed the ends of two rubber tubes 
of equal lengths at the ears and moved the end of one towards 
or away from the source of sound. With equal lengths of 
tube I thought that the friction and resonance effects would 
be the same. The sound, as heard through the tubes, did 
/ appear to move somewhat to one side or other of the 
head, but the movements did not appear to have any relation 

to the wave-length, and the movements were very irregular. 


* Communicated by the Author. 


to determine the Direction of Sound. — alg 


I realized how hopeless it was to try such experiments in a 
small laboratory where perhaps 90 per cent. or more of the 
sound had already suffered one or more reflexions ; the 
experiment merely suggested a possible method of studying 
the position of nodes and antinodes in such a closed space. 
Later an apparatus to determine the direction of sound by 
magnifying the intensity effects by means of two sound- 
- receivers on opposite sides of a large flat board was surpris- 
ingly inadequate except in the case of the higher notes, and 
led to the conclusion that it was very probable that phase had 
a good deal to do with our judgment of the direction of 
sound. 
I postponed further experiments on the direction of sound 
until the summer of 1907, when I should have an opportunity 
of trying them in the open air. | 
In the meantime I saw Lord Rayleigh’s paper in the 
Philosophical Magazine proving conclusively, in an ingenious 
manner in the laboratory, that phase in some cases gave a 
sense of direction. It only remained to show to what extent 
and within what limits these effects were produced. In 
June 1907 therefore I proceeded to try experiments in the 
open air, the apparatus being a set of adjustable organ- 
pipes, made for me by Hutchings & Votey of Cambridge,/ ne 
Mass., and cylindrical tubes of various lengths applied to) ~~“ ~ 
the ears. 1\ eas 
The tubes were of sheet aluminium, and they could have 
their length altered by sliding one within another. 24+inches | 
was chosen as the diameter, because then the tube could be 
readily fitted quite closely round its whole circumference 
against the head. | 
The source of sound was placed at a distance of about 
30 feet from the observer in an open field. It was at once | 
found that with two unequal tubes applied to the ears, and , . 
with the observer facing the source of sound, the source/ | > ae 
appeared to move to the opposite side to that on which the’ | 
longer tube was applied. (This, as will be seen later, is only | 
true within certain limits.) Be ust 
After certain preliminary experiments, the ground was 
pegged out in an arc of a circle 28} feet in radius. The 
source of sound was placed at a point of this arc, and pegs 
were placed at intervals of 4° along the circumference up to 
60° on each side of the source. The observer was situated at) 4 4 
the centre of the circle. As the image of the source moved || 4,” 
the observer continually faced the image and noted its ; 
apparent position on the arc. This position could be fixed 
with an error of not more than one or two degrees. 


Z 2 


320 Mr. T. J. Bowlker on the Factors serving 


Some of the results obtained are given in the table below. 
In order to avoid psychological errors of imagination and 
expectation, no guesses at what would happen were made, nor 
any theory formed as to the probable position of the image 
until two or three complete sets of experiments had been 
made. It was found, however, that after a little practice the 
location of the image could usually be obtained with such 
certainty that these precautions were unnecessary. 

In the following tables 7. indicates that the tube was over 
the right ear or that the movement of the sound-image was 
to the right, whilst /. indicates the same for the left. 

L is the length from the extremity of one tube to the 
extremity of the other when both are applied to the head. 

@ isthe angular displacement of the image when the longer 
tube is nearer the source. 

@’ is the angular displacement when the shorter tube is 
nearer the source. 


Note of Wave-length = 19 inches. 


Length of Diff. of | Displacement L sin 0. Lsin@'+D. 
tubes over length of of 6=angle to | 0'=angle to 
ears. ear-tubes=D.| image=0@. the left. the right. 
Inches. Inches. ~ Inches. Inches. 
7 over r 
4 1 Dea. | 6 
6 over } | 
ete aap 2 6° 1. 21 
9r. ; 
ao eee 3 8°, 2-9 
ae he 4 10° 1, 37 
| 12 r. ; 
i @L pee 6 16°1. 67 
ee es 8 10°, 4:4 
eS ie} F 
ea dw 93 Sota 9-9 187 | 
18 r. (24°, Es = 
"61 | he 12 fee 123 16-2 
19 r. J 26° 1. ‘ j 
| aL eae 13 Horn 13:6 16-1 


Now if an image, apparently in front, is produced when 
the phases arriving at the ear are in agresment, L sin @should 
be equal to the difference of length of the ear-tubes, and 
Lsin @’+D should be approximately equal to the wave- 
length. We find that this is the case. There are, however, 
some anomalies, notably when the angle suddenly changes 
from 16°]. with a tube 12 inches long over the right ear. to 


” 


to determine the Direction of Sound. 321 


10°]. when the tube is made 14 inches long. This is doubt- 
less due to resonance in the 14-inch tube increasing the 
intensity at the rightear. Subsequent experiments confirmed 
this view. In later experiments large loose wads of cotton- 
wool were inserted at the outer ends of the tubes. This 
proves a very effective means of checking resonance, whilst 
not interfering very much with the intensity. In this manner, 
using sounds of various waye-lengths, it was proved that 
an image appeared in front of the observer when the 
phases of the waves arriving at the ears were approximately 
in agreement. A better arrangement was devised, however, 
later, and an account of these later experiments appears 
further on. 

The next point was to determine, to what extent the image 
of the source was displaced whilst the observer continually 
faced the source, and how this displacement depended on the 
wave-length. For this purpose two tubes were taken, and 
the observer continually faced the source and tried to note 
the direction in which the image appeared: in some cases 
only one tuhe was applied to the ear. 

One example is given below. 


Note of Wave-length 26°8 inches. 


Angular dis- | Angular dis- | Angular dis-| Angular | 


Length of ses Tila 
tube placed ; engthsof| placement | displace- | 
es Gise ee aie enhok tubes placed| with the | isasie with | 
ao with tube over with tube over ee ie longer ae be | Faq 
other ea¥ | the right ear. | the left ear. || ‘7° ©7°- oh be re TG Wk eth alee 
free. 2 right ear. the left ear. 
Inches. Inches. | | 
| =----- 4° 1, AE 9 &6 | $G°E4 69! aS 
re >... 8° 1. 4° y, fa SG tee? LP es 
ame! 25.07. Perab @ 1h A 18 ee Ao | | ) 
| BRL 8>: 2401. |) 16? x. 1 && a 40° LL po fio rAd 
ES i inn-2o- a tht.) 24° r. 17. &6 (66°91. &60°r| 20°r | 
ee 3-3. 40° 1. & 46° Sele r.& 30°1) 19 &6 Mr, ol 5 eee 
i yaieee 40° r, =40° r. & 40°91) 232 & 6 = x 30° r, | 
> ne 30° r. 30° 1. loci he te iheaesl 
a. 90° r. 94°). || 25 &6 4° yr. 1 ae 
ie. 10° r. 16° 1. | ia sn 
SL ae 0° 8°] ) 
= eee 0° UV | 28 &6 | AP ro 12} 6? L 
=< 5 Q/e ae can 0 2S, GA | 2 gel De PW 
a WP i 4°r | 
BES o2---- Ay 1. o1t ) 


It would appear from the above that the image crosses 
over from left to right, or from right to left, when there is a 


322 Mr. T. J. Bowlker on the Factors serving 


difference of about half a wave-length in the arrival of the 
sound at the two ears—the image being on that side_at which 
| the sound-wave arrives first. 

It will be noted that in the earlier stages a displacement 
was more readily produced when the phase was advanced on 
the left ear. I was thus asymmetric with regard to my 
hearing, and JI have found the same to be true of others, 

I also found that the hearing mechanism connected 
with the formation of the position of the image is capable of 
fatigue. After listening intently for some time with one ear 
towards the source, the position of the image does not return 
to exactly the same place as before. I tried to determine 
whether fatigue for one note resulted in a lessened displace- 
ment of the sound-image of another note, but | was unable to 
settle this point. These phenomena may be of interest to 
the experimental psychologist. 

Similar experiments were tried with notes of other wave- 
lengths, the result always indicating that there was a sudden 
appearance of an image on the other side of the field of 
sound-view when the difference of phase at the two ears was 
approximately half a wave-length. The maximum angle of dis- 
placement increased up to 90° as the wave-length of the sound 
increased ; a wave-length of about 36 inches being the first 
that gave an image displaced 90°. Sounds with longer waye- 
length gave a displacement of 90° before a phase-difference 
of half a wave-length was reached, and the sound-image then 
seemed to spread over a continually increasing length of are 
on each side of 90°. 

y It is hard to determine the actual position of a sound- 
4, | image when the observer is facing the source, and the sound- 
image is displaced through a largeangle. Up to 20° I think 
I can tel] the position to within 2°, from 20° to 30° to within 
4°, from 30° to 40° to within 6°, from 40° to 50° to within 
8°, after 55° I may be 10° or more out. When possible I 
\ find it best to point with the arm outstretched in the direction 
\from which the sound appears to come, but when holding 
two tubes to the ears one cannot do this. 
~ In the above experiments resonance effects were still 
rather disturbing, and, asthe general behaviour of the sound- 
images had been ascertained, a new arrangement was 
introduced. 

In this the ear-tubes were of equal length, and a right- 
angled bend was inserted in each, so that the outer portions 
of the tubes were approximately vertical. The plane through 
the outer edges of these tubes was horizontal. i 


= 


~ 


to determine the Direction of Sound. 323 


With this arrangement the resonance was the same in both 
tubes, and the intensity was also the same for both. 

The distance apart of the centres of the tubes was 31 inches, 
and their diameter was 2 inches. The tubes were luted into 
the righi-angled bend with stiff modelling-clay. 

With this arrangement, when facing the source, the image 
did not always coincide exactly with the source, perhaps 
owing to the tubes being slightly unsymmetrical with regard 
to the head when closeiy pressed against it to make a tight 


_joint at the junction of head and tube. 


by applying the ear-tubes and facing the source, then turning 
the head until the image appeared to be straight in front 
and then noting the reading. (The error here would pro- 
bably not be more than 1°.) Then the head was turned 
slowly to the left, this caused the image to move more or 
less to the right, at a certain position in the rotation of the 
head an image appeared suddenly to the left—this new image 
made an angle with the symmetric vertical head plane 
approximately equal to the angle the original image now 
made with it. This point is called the “‘two-image”’ point, 
and the angle that the image makes with the vertical head 
plane at this point is called the “cross-over” angle. It is 
near the maximum displacement that phase will produce 
with the particular wave-length under observation. 

After this “‘ two-image”’ point has been noted, the head is 
still turned towards the lett until the new image appears 
directly in front. This point can usually be found within 
1° or 2°. The procedure is repeated to the right. 

The results obtained are as follows :— 


With Note of Wave-length 51 inches. 
Zero at 4° left. 


“ Two-image ” point at 57° left, no such point to the 
right. 

“ Cross-over angle”’ from 90° right (wide image) to 90° 
left (wide image). These images were over several degrees 
of are to right and left of 90°. 

The ‘ ‘two-image ” point 57° left, if we take it as occurring 
at a difference ‘of phase equal to half a wave- length, and 
reckon from the zero at 4° left, gives us 2 (31 sin 53°) — 49-6 
inches as the wave-length. 


394 Mr. T. J. Bowlker on the Factors serving 


With Note of Wave-length 40 inches. 
Zero at 2° left. 


“* Two-image point ”’ at 40° left and 38° right. 

“ Cross-over angle”? from 90° right (wide image) to 90° 
left (wide image). 

Taking 39° for the mean position of the two-image point, 
we get 2 (31 sin 39°) =39 inches as the wave- length, 


Note of Waye-length 27:6 inches. 
Zero at 4° left. 


“Two-image point”... 22° r. and 30° 1. (mean 26°). 
New imames VA O'S .2 7. 51° r. and 55° 1. (mean 53°). 


Cross-over angle from 50° right to 40° left and from 40° 


left to 40° right. 
Wave-length from “two-image point ” 
=2 31 sin 26° ==27-3 inches. 
Wave-length from new images 
=31 sin 538°=24°'8 inches. 


Note of Wave-length 19-4 inches. 


Zero at 0°. 
“'Two-image point... 19° 1. and 16° r. (mean 174°). 
New images ......... 42° 1. and 46° r. (mean 44°). 
Cross-over angle’ .!) 30° % te soor | 


and 36° 1. to 36°r. 
Wave-length from “ two-image point” 
= 2x31 sin 174°=18'6 inches. 
Wave-length from new images 
= 31 sin 44°=21°4 inches. 


Note of Wave-length 15:8 inches. 
Zero at 0°. 

“Two-image point”... 17° 1. 17° x. 
New images <........... 20° 1, G2 deg2o des aene amt. 
Cross-over angle ...... 18° to, 137. 
Wave-length from “ two-image point ” 

=2 x 31 sin 17°18 1nehes: 
Wave-length from new images 

=31 sin 29°=14°9 inches. 

31 sin 65° 


and a, = 14°1 inches. 


to deternune the Direction of Sound. 329 


Note of Wave-length 10°4 inches. 
Zero at’ 3° rt. 
Pel wo-imace pot”... 13° ly de? r., 41° 1. 
New images ............ ZAM ae 0” Li 
Cross-over angle ...... 1S coal a: 
Wave-length from first pair of two-image points 
=2(3i sin 155-)—16"7 inches. 
From first pair of new images 
=31 sin 234°=12'4 inches. 
From next image 
D2 iq Oo 
= ei = 12°4 inches. 


Below this wave-length the images were too near together 
to have their position clearly determined, the sound of one 
image confusing the apparent position of the other image. 

Sometimes it appeared as if three images were in the field 
together, then the one nearest the source would appear the 
loudest and draw off the attention from, and apparently 
obscure the image directly in front of, the observer. 

The tubes were shortened so that the distance between the 
centres of their ends was reduced to 15 inches. 

With this pair of shortened tubes the following results 
were obtained :— 

Wave-length 10:4 inches. 
Zero at 0°. 

“‘Two-image point”... 22° |. 

New images at.......... 41°]. & 40° r. 

Cross-over angle ...... 20° tro 20am 

Here the two images were heard with the head facing any- 
where over the arc from 12° r. to 30° r.and from 12° ].to 30° 1. 

Wave-length by two-image point 

= ol sim Uo leamehes, 
Wave-length by new images 
=15 sin 404°=9°8 inches. 
Wave-length 8 inches. 

“'T-vo-image point’-—two images were evident during 
nearly the whole range—there were practically always two 
and sometimes three images evident, though I had some 
doubts whether the central image was always real or a resuit 
of attention to the two side images. 

New images...... 24° 1. & 21° r. 

Cross-over angle 12°, 2. e. the maximum angle of displace- 

ment was 12°. 

Wave-length by new images =15 sin 224°=5'7 inches. 


ro 
. 


326 Mr.T. J. Bowlker on the Factors serving 


So far the tubes had been used with the full aperture—a 
circle of 2 inches diameter. Now they were closed by disks 
luted on with modelling-clay—the disks being perforated by 
holes 5; of an inch in diameter. 


Note of Wave-length 6:4 inches. 
Zero 4° 1. 


Images at 24°1. and 22° r. ; but now three images could 
always apparently be heard together when facing one of 
them. When blowing the note hard the central image could 
be heard most strongly, but as the sound gradually ceased 
the central image died away, first leaving the impression that 
the two side images only existed. 

Cross-over angle—or maximum angle of displacement of 
image about 9°. 

Wave-length by new images = 15 sin 23°=5°8 inches. 


Note of Wave-length 5:8 inches. 
Zero at 0°. 


Images at 16° |, and 14° r. 

When facing the image at 16° |. the image further to the 
left seemed stronger and tended to draw off one’s attention. 
When facing 14° r. the image at the source, which now 
appears only about 8° to the left, tends to obscure the image 
in front at 14° r. 

Maximum displacement 6° or &°. 

Wave-length by images =15 sin 15°=3°9 inches. 


Note of Wave-length 4:4 inches. 


Here it was very difficult to determine even approximately 
the position of an image, the one to the left of the two or 
three in the field of view seeming the loudest as a rule. | 
I thought I had images in front, however, at 8° r., 10° L., 
22° 1., 36° 1., but it was impossible to distinguish between 
having an image directly in front and having a two-image 
point in front. I only felt sure that phase was still playing 
a part in fixing the maxima and minima which gave rise to 
the centres of the sound images. 


From the measurements of the maximum angle of dis- 
placement that I had taken in the above experiments it would 
seem that for wave-lengths below 20 inches the angle of 


to determine the Direction of Sound. 327 


displacement * was very roughly proportional to the wave- 
length—for a wave-length of 4°4 inches the displacement 
would only be 5° or 6°. 

Now, to produce a difference of phase of half a wave-length 
the head, without the tubes, would have to be turned through 
about 23°—so if the head faced the source and was then 
turned to one side or the other, the image would appear to 
following the turning of the head if the intensity remained 
the same at the two ears, and when the head had turned 
through 23° the image of the source would have turned 
through 17°—this 17° must be compensated for by change 
of intensity at the two ears. As a matter of fact I am 
inclined to think that in the case of the higher notes—perhaps 
in the case of all notes—the zone or arc in which the sound- 
mage appears is settled by the relative intensity at the two ears ; 
the actual position of the images within this zone being produced 
by the maxima and minima within it produced by phase- 
difference at the ears. 

In order to explain the existence of a movable image of the 
sound within this zone, we may suppose that the transmission 
of the sound impulse through some specialized part of the 
auditory apparatus or brain takes a definite time from each 
ear, and that the point where the impulses meet is the focus 
that gives rise to the sensation of a sound-image. 

To explain the existence of two images, and perhaps three, 
we may suppose « and @ to be the crests of two successive 
waves ; then, if the observer is facing the source, the crests 
ot @ arriving at the ear simultaneously produce an image at 
the centre of the sound-zone ; @ at the right ear and B at the 
left give an image to the right of the centre, and « at the 
left ear and @ at the right give an image to the left of the 
centre. 

If two equal tubes be applied to the ears, one with aperture 
2 inches in diameter, and the other of 5°; of an inch, and both 
apertures be turned square on to the source, I find a 
deflexion of 20° to be produced in the sound-zone, and within 
this the image moves about as before. 

In order to determine whether intensity affected the 
position of the image when notes of medium pitch were 
concerned, one ear-tube was closed by a disk of aluminium 
luted on with clay. A hole was made through the centre of 
this disk, through which hole a cylinder of paper, 2 inches 


* Further experiments with wave-lengths down to 2 inches would, 
however, seem to indicate that this law does not hold even roughly for 
short wave-lengths, the displacement being much larger than this law 
would give. 


328 


long and 3 inch i 4 diameter, was inserted ; the area of this 
tube was about sx of the area of the pas and it pro- 
jected 3 inch bey. ond the disk. The outer 24 inches of the 
tube was very loosely plugged with cotton-wool to suppress 
resonance. 

A series of experiments was made. In one set the ear 
tube without the perforated disk was used, in the other the 
tube was used with the disk attached. 

The tube was placed over the one ear, the other ear being 
free; the head faced the source of sound, and the apparent 
angular displacement of the image was noted. 

The results are given in the following Table :— 


Mr. T. J. Bowlker on the Factors serving 


Note of Wave-length 27°6 inches. 


/ 
| Tube applied to the Right ear.|) Tube applied to the Left ear. 


Length of tube | / 


applied to Angular dis- | Angular dis- | Angular dis- | Angular dis- 
the ear. placement | placement placement placement 
without per- | with per- without per- with per- 
| forated disk. | forated disk. | forated disk. | forated disk. 
Inches. | | | 
see aes ) 12°. 30° 1. 22 r- 102 
oh eee oe 24°), 36° 1. | 16° r. ie = 
5 ae 452] 80° 1, & 45° r. | 29°m, 30° 
A sath es. 0 60° 1. & 50° r. PAP wl] 52° me. 12° 1 
re eee 32° r. 20° r. || 34° r. 36° 1, 
eet oaiee 98° x 14° r. | 145° r. & 50° 1. Reed 
Ee sssee 10° r 0° 40° 1. 0° 
ie Hate h ee 0° |10°1 ) 26° Ty BAe 
26 16°1 40° 1 | 0° 1.] 18° 


From this it is evident that intensity plays an important 
part in fixing the position of the sound-image when there is 
a great difference of intensity at the two ears. 


Another experiment which shows the part played by 
intensity, and yet how largely the apparent position is affected 
by phase at the same time, was tried. Im this case the 
observer's left ear was towards the source of sound, the 
vertical plane through the ears passing through the source. 
Ear-tubes of various lengths were applied to the right ear, 
and the head was kept fixed, the angle of displacement being 
noted. 

The results are as follow :—- 

The source is considered zero—the point in front of the 


to determine the Direction of Sound. 


329 


observer is +90°, the point directly behind is —90° (where 
not marked negative the angle is to be taken as positive). 


| 


Note of wave-length 13°8 in. Note of wave-length 19-2in. || Note of wave-length 40 in, 
l \ 
Length of | Angular dis- Length of | Angular dis- | Length of | Angular dis- 
ear-tube. | placement. ear-tube. placement. ear-tube. placement, 
Inches. | Inches | Inches 
- (Se Nee EOS O° | eae! dake one ie 
Li ta Sal +60 2 ade d Be het 45 ORGS BE hy: 8 
ee 55 Di fasacloaks 80 Balreact tic 8 
laces a 45 Gi Aueeesac 75 Sy eee 90 
BE ce ccicc ws | 36 Me i ees 60 15 eee ie 90 
0 Sh Wosaceis- 50 FR Le saetascat 80 
1). ee | 30 SE Fock ee: 45 Ly a APs 70 
1D Seen ) 80 ORO: ame 36 Tig Fei cc cect 60 
ere 70 eel #33 25. 16 WG 55.5534 60 
Bee Se nswoan 60 | 0b 2 ere ne 0 (0 aa 50 
Be giveonas 36 Hpac | ce ekoaipee 20 |S oye ae 30 
LG eee eeee 0 Begotten ak 40 WO sfaga 3 12 
i 10 tn pees 60 e-Souet 0 
5 Sear 15 ROP ae 60 
Lr Sac 10 | ong 7 Sree 45 | After this the angle of 
| Se ES a reseee 38 | displacement is small. 
And it now always remains) 19 ......... 20 
within 20° of 0. cg. toeteee eee 0 
|) 2B cere. 50 1 


I simply record these results without attempting any ex- 
planation, beyond suggesting that sound-conduction through 
the bone and some portion of the brain-substance may perhaps 
play a part in fixing the sound-zone when the Image is 
perceived. 

It is to be remarked that as no special care was taken to 
suppress resonance in the above experiments, it is probable 
that some of the results are affected by it. 

In order to determine the amount of displacement pro- 
duced by a difference of phase in the arrival of sound at the 
ears, a pair of tubes 6 inches long and 2,2, inches in diameter 
were taken. A pair of flat rectangular plates 44 inches by 
24 inches were also taken, and in each of them a slit 
1} inches long and 5 inch wide was cut across symme- 
trically near the centre. These plates were luted on to the 
outer ends of the ear-tubes with modelling-clay, and were so 
arranged that the central line of the slit was =; of an inch 
distant from the axis of the tube. 

The observer faced the source of sound and applied these 
tubes to the ears, the slit of one tube being vertical and 


330 Mr. T. J. Bowlker on the Pantene serving 


nearer the source, the other vertical and further from the 
source. 

The position of the image was noted, and the tubes were 
then rotated through 180° about their axis, and the position — 
of the image again noted. 

This arrangement made the conditions at both ears pre- 
cisely the same as regards resonance and intensity for both 
observations, the only difference being that there was a 
difference of 24 inches in the length of the path of sound to 
the two ears in the two cases. 

With this apparatus the results with two observers, myself 
and Mr. O. B. Clarke, were as follows :— 


| Angular displacement of image. 
Wave-length | 
of sound. | 4 | 
/ Observer O. B.C. | Observer T. J. B. 
— oo | Steed eed 
Inches / 
By eh ree 1S 0! . Les 12° 
Zhe Sheet eee ! 18 14 
VE oe oe eee 20 14 
ease iam ieee tee pe 18 14 
hi 1) Se tp haa Pts 24 16 
| aaah | cere cee 18 14 
2k eee | 18 12 


It is remarkable that the angle of displacement should 
remain nearly the same with such a wide variation in wave- 
length. 

Now if we take the diameter of the head through the 
ears to be 54 inches, a movement of the head of 25° 
would make a difference of 2,4 inches in the distances of 
the ears from the source of sound. 

Assuming that when the head is turned through 25° the 
direction of the source is indicated exactly by the sensa- 
tion of the image, this experiment shows that in the case | 
of one observer an average of 6°, and in the case of the 
other observer of 11°, had to be made up by the displace- 
ment of the sound-image due to intensity or other cause. 
As regards myself, 1 find that when my right ear is 
turned partly towards the source of sound and with eyes. 
shut I point to it, T underestimate the angle from the 
symmetrical head-plane for sounds of 24 inches wave-length 
and over, and overestimate the angle when the left ear is. 
turned towards the source. 

With regard to the general intensity of the sound, I 
find that when the sound is fairly loud the apparent. 


to determine the Direction of Sound. 331 


direction, when using ear-tubes and facing the source, 
remains almost the same when the sound becomes louder, 
the angle of displacement becoming, if anything, somewhat 
smaller. In the case of the notes of greater frequency, 
however, the sound when dying away ” will, in my ease, 
move through a considerable angle to the left, ‘if its original 
position is near the source to start with—this is probably 
due to my left ear being more sensitive for feeble shrill oe 
than the right ear. 

In connexion with these methods of examining sound- 
images, an interesting experiment is to take two “tubes of 
2 inches or more in ‘diameter, one say 12 inches long and 
the other 4 inches, and fen to a band of three or four 
instruments played in the open—the notes will be found to 
be scattered over a wide range, most being to the side of the 
short tube, some being in front and some being to the side 
of the long tube. . 

In listening with such a pair of tubes to two dogs furiously 
barking, the effect is at first quite alarming—one seems to 
be in the middle of a pack of dogs some of which are rushing 
viciously at one’s throat, Preferably the tubes should be of 
metal sheet, and it is best, though not necessary, to surround 
them with a sheet of some rubber cloth or composition. 


It remains to say something about the interference of 
trains of waves after reflexion in producing a change in the 
position of the image. 

In one experiment, an organ-pipe was placed 5 feet from 
a flat wall and the observer stationed on a line, making an 
angle of 30° with the wall, through the foot of the perpen- 
dicular from the organ-pipe on the wall. There was more 
difficulty in settling “the direction than when there was no 

wall, and the apparent position seemed to be at the foot of 
the perpendicular on the wall: with other positions of the 
observer, the apparent image of the source varied between 
the source and this point. 

When the positions of the observer and the source were 
interchanged, [ obtained these results when working with a 
note of wave- ‘length 26°8 inches. 


Source of sound 40 feet distant. 

Har 30 inches from the wall—the only image lay on a 
line from the ear parallel to the wall. 

(The image of the first harmonic of the stopped pipe was 
displaced 10°.) 


As the head was moved ‘nearer to the wall, the image 


332 Dr. T. H. Havelock on 


moved gradually nearer to the source and arrived at the 
source when the ear was 16 inches from the wall. 

A similar experiment with a note of wave-length 19-4 
inches gave the image :— 


on line parallel to the wall, with ear distant 17, 39, 61 inches 
at the source, with ear distant 24, 48, 58 inches 
40° from the wall, with ear distant 32, 52 inches. 


From these experiments it is clear that in noting the 
direction of a fog-horn at sea the observer should be well 
away from any reflecting surfaces of any kind. (In one 
experiment an umbrella held to one side of the head ata 
distance of 2 feet displaced the sound-image 20°.) I find 
that it adds to correctness in fixing the direction to have a 
flat board slung on the shoulders vertical and parallel to the 
axis of the ears. This increases the intensity in front and 
shuts off sound from the rear. I think it would also be 
better to have two short blasts of 3 seconds each, every half 
minute, at sea, rather than a long blast every minute. 

Also fog-horns should be placed well above any reflecting 
surfaces, but it might add to their carrying power if a large 
i or sounding-board was placed horizontally directly over 
them, 


XXXII. On certain Bessel Integrals and the Coefficients of 
Mutual Induction of Coaxial Coils. By T. H. Havetock, 
M.A., D.Sc.; Fellow of St. John’s College, Cambridge ; 
Lecturer in Applied Mathematics, Armstrong College, 


Newcastle-on- Tyne ™. 


HE calculation of coefficients of mutual induction has 
been discussed by several writers from the time of 
Maxwell to the present, more accurate expressions being 
required as experimental methods have become more refined. 
The expressions are generally in one of two forms: they are. 
either given in elliptic integrals, in which case numerical 
calculations are tedious, or given by a certain number of 
terms of a series. 

The present paper brings forward another method of 
expressing the coefficients, namely in terms of integrals 
involving Bessel functions; series are obtained from these 
integrals, their general terms found, and their convergence 
tested. In certain cases series are obtained which seem to be 
simpler and better adapted for numerical calculation than 


* Communicated by the Author. 


certain Bessel Integrals. 333 


those in use at present ; the cases examined are those involving 
two coaxial coils treated simply as cylindrical current-sheets. 
The Bessel integrals used here occur frequently in other 
physical problems ; ; it has been thought better to give the 
investigation of these separately in the first section, where 
the series are obtained and examined independently of the 
applications made later. 


§ 1. Some Integrals involving Bessel Functions. 


The integrals which we have to evaluate are of the type 


(ee) 


I = e™Ti(m) Sip) “du, 
0 


and we require their expression in series suitable for large 
and for smali values of p. 
When p is zero we have the following known integrals :— 


x Cie | 

5 RAR iy iY GE 
{ am =; (1) 
‘Twas, a7, 
s, Jy (Ho = Be Sat ix . : : : : . ° alte : . (2) 

dw la y. 
Bue hee) aiid 2h b Zien : - ~ = C . - (3) 

f Ji(ua)S (pd) f= ee oS, J (e—e pe nt he b>a 

= 5 "(2 4.@)E— (?—a? *) FS, 


where E, F are complete elliptic functions of modulus a/0. 
Also, putting k for a/b, we have in the last case 


" d Le RBS F 5 
{ J,(p) J(u) 2 — i = 16 k? — 128 — 9048" ¢, « dieters ) 
0 
Sh he. Bt Df «aye Care SB 2( 2y—asl. aes 
he ae ) HC de (4) 
r= ri(r+ 0)! 


Phil. Mag. 8. 6. Vol. 15. No. 87. March 1908. 2A 


334 Dr. T. H. Havelock on 


This series is convergent for / less than unity ; it converges. 
rapidly, and in most cases we shall find the first three terms 
sufficient. 

Consider now the integral 


ie.8) 


r=| e "32 (dp. 


If p is large we can easily find a suitable series for I in 
ascending inverse powers of p; we substitute for the Bessel 
function its equivalent series in ascending powers of mw and 
then integrate each term separately. Then, since we have 


a « ' (2s +2)! per 2s+2 
J(u) = i ee f 


and 


ee) 


—plH 2s (2s)! 
e ad dp = ose 
0 ie 


we obtain the series 


io 2 A Qs 2)!}? 1 
ay, 2 d as Ss ai = {( ee - 
> é 1 (“) ie = ) 9” eh) (s 2s Divo 


(6) 


ee) 


“ee oj) dus 6 ue Os +1) EN a 
Oe 2/2 de ee Seal wes je iS ean 
{ rw = 2-0 peepee? © 


ages dp = : (2s)!(2s+2)! 1 
Dey 2 Ne Se - ‘ Nestea 
i) ar are me ie) 


fla) Io) 5 1 ays! 
de eg Loe ieee) ae i 


The series are convergent for p>2. 
Further we have * 


Ee oo o eal eS es Se) ee 
Ja(m)Sa(awe) = rE ¢ 1) Tico (4) ee 


a 


* Nielsen, Cylinderfunctionen, p. 20. 


certain Bessel Integrals. 335 
where X.< 1, and F is the hypergeometric series given by 


F (a, b, ¢; “)= 1 + & a. © ~e f 


a(a+1)(a+2) \O(O+1)(64+ 2) 34 
5!e(e+1)(¢+2) rin 


Then, using the same method, we obtain series for more 
general integrals suitable for large values of p. We obtain 
thus 


s(2s+2)tF(—s—I, —s, 2,7) 1. 
Foal) oe oe j 


(10) 


{ scone de=r5(—1 
0 s—0) 


s(2s+1)!F(—s—1, —s, 2,27) 1 


i} eS, (4) Sin) = =n 3 (=1) (11) 


sits cba] pe ; 
ee . du _. 2, 4.#(2s)!F(—s—1, —s, 2,¥)_1 
{ é TM)IiOn) Ts ae eset £) s!(s+ | ia ae es 
i | oe | APs JE 1 
uae = =e ae ex 2-5 ie ee 
alt Te(l+> ast 3p (1+30?+ es 
= (L+6N+ 60+ +0) 5 + ay. dt . (12) 


To obtain series suitable for small values of p we have 


a0 


y= | eM I2H)de 
0 


Tv 


— ae eos ata | ey Joe cos da)dp 
aE 
0 0 


2 (? cos20d0 
om Vp? +4 cos?0 


0 


The summation in this integral is now divided into two 
parts, one between the limits 0 and ae and the other 


between 5—é and e; ¢ may be taken indefinitely small 
2A 2 


336 Dr. T. H. Havelock on 


ultimately, but at present it is regarded as indefinitely — 
than p. 


Then we have 


2 2° eos 26d0 2 * eos 20d0 


oo in +4 cos 20 Ve +4sin”9 
iL 2 
—fy,47 aE o ge i fte “Sekt ES RE (13) 
In y; we write, as far as terms in p’, 
2cos20 _—cos 24 es ip sy ee 
V/p?+4cos’?@  cosé L 8 cos?@ * 128 <0]: 


Then we integrate the terms separately, substitute the 


ad e Mi = e 
imits 0 and = —e, and assuming e small we expand as far as 


2 


necessary ; we obtain thus 


1 ee 
in=2+ logze— lis gee ee") 


133 i 1 =) 
Tae 


+ op?'(- ia tiga Ge 5°85 (14) 


In the second integral y:, p is small compared with @ 
throughout the range; we substitute for cos 20 and sin @ 
their expansions in powers of @ and expand by the binomial 
theorem. We obtain 


mel (1264 agg > Se +3 s pe 
ee al 3 3p? +402 45 p? +40? © 3 (p? +40")? VJ p? +46?’ 


including all parts which will give terms containing p* on 
integration. Integrating the parts separately and expanding 
as far as p*/e* we find 


3 15 ‘)( oun Weta, OP: |, = Lip’ 
ee ver hs Ge oon im(! Se? + 1786) 
seta aot) ee 
ak = += rh pe ie == 

e(uate 


—: NG +e). yt eee Se 


> oe 
| certain Bessel Integrals. 337 
Substituting these values of y; and y, in (13) we find that 


the terms in loge and inverse powers of € cancel; then 
making e indefinitely small the terms involving positive 
powers of e vanish and we obtain as far as terms in p* 


y= {mse dy 


15 8 1 31 
ee 8 Te) ee Rg Selene peter: 
=54(14 + igh 1024” *) log. ari hice agus i 


Further, if we integrate (11) with respect to p and put 
the constant of integration equal to 4 on account of (1), we 
obtain 


. I 
{ ey? te ae 
0 


=F : ee 5 ee 5 63 , y Eee 
ee {let io?” — rom?" )!o6.5—F —P+ goag?* fi) 
Integrating again with respect to p and taking account of 


(2) we find 


i Ss” fie i! uf 8 

mae —— — 2 ss Tp Ep SOE 4}j pias 

i) e™J, : {50 ioe i024? pies 
1 


moe ae st 1 


4 

2 
L. SE ON AN eS ue : 
+ eq RP (Jog 85, * +4) —aong? (lox > - 5) }- (18) 


In the same way, if p? + (1—))? is small compared with 2, 
and if © >1, we obtain from (11) the more general 
expansion 


mri i} eS ,(w) Ty (rp) dp 


3 a 3 p?+(1—a)? cite re i es8/ aah 


ik shop a(n)? op’ sinh Bes 
SaeeAy Ae reat sie CES 


Finally, integrating (19) with respect to p and taking 
account of (3) we can obtain a similar series for the integral 


{ “e-PHS (uw) Ty(dps) oe. jis eee 


338 Dr. T. H. Havelock on 
§ 2. Mutual Induction of Single-layer Coil and 


Coaxial Circle. 


If we have two coaxial circles of radii a and 6, with a 
distance ¢ between their planes, we have their coefficient of 


mutual induction given by 


am (coe tb cos (b—¢’) dd dd! 
M = $(2%!as as = ( a ae 
4| ee Bu Vc? + a?+b?—2ab cos (6—¢') 


= (ne cos (6—¢’) db dd i ere (A /a2 +b%—2ab cos (pb—¢') ) dr 


= 4ntab | oJ, (ray diab) da. oo ot 
0 


From the integrals (10) and (19) in the previous section 
series for M could be obtained suitable for c large or small 
compared with a or 6. Further, if we have a single-layer 
solenoid of length 2h, radius 6, and nm turns of wire per unit 
length together with a concentric coaxial circle of radius a, 
less than 6, we obtain their coefficient of mutual induction 
by integrating (21) with respect to ¢ between the limits —h 
and +h. Hence we obtain 


M = 87?abn ( (1—e-*) Ji(Aa) Jy (Ab) a . 22 
e 0 


Using then the integrals given in (11) and (20) we have 
series suitable both for long and for short coils. However, in 
the latter case the difference between the radii of the coil 
and the circle must be small compared with one of them, and 
unless this holds series already in use probably give a better 
approximation than those obtained from (22)*. Finally, if 


Ke Cf. E. B, Rosa, Bulletin of the Bureau of Standards, vol. iii. p. 209, 
1907. 


certain Bessel Integrals. dd9 


we have a solenoid of length 2h and radius a, and a circle 
of the same radius ata eee ce from the central section of 


the coil, we obtain from (16) 


e h 
M=47n7a’n ( drJP7(Aa) J. oa dae | ord | 
vo. h e 


2 id = c aa ay a in 
Aetna \2—-¢ ene a JP(a) ns) 


a) ab Trey "ny 9 du 
=4nta'n| 1— | r hy ag: 5 Tu EI. (24) 


Substituting for these integrals from (7) and (17) according 
as h-te and h—c are large or small compared with a, we can 
obtain series giving the ‘induction through any section of a 
solenoid, whether it isa long or a short coil. 


§ 3. Two Coaxial Solenoids of Equal Length. 


Suppose we have two single-layer coils of equal length 
2h and of radii a and 4, placed as in the figure ; let n, and 


340 Dr. T. H. Havelock on 


ng be the number of turns of wire per unit length on the two 
cylinders. Then if we write |2—.'| for the absolute value 
of «—w', we have from (21) 


ie h pie 
M= Antabnyn, | J1(Aa)J,(Ab)dr a acl. dale ht | 
0 


ro) ] : 
= 16n°abnyn, | - a= ne ee 2 a} 0 J(Aa)J (Ab) dr 


With 6>a and h>b, we use the series (3). (4) and (12) ; 
thus we find for the coefficient of mutual inductien of the 
two coils: 


ae et l fay 1 va\= 5 a\® 
ae 2 es es ee re Paes mas 
1 aes E at ia($) - TCs, * 3048 (7) q 
1b 1 a®*\ (63 1 a? aN (6S 
ai ct — a+) (7) + ort 3et ED) 


b 4 
= ss(1+65 5 nee +2) 7) 


This gives an expression for M which is easy of calculation 
and rapidly convergent; moreover, from (4) and (12), 
additional terms in the two series within the brackets in (25) 
can be calculated if required from the general terms 


$1.3.5....(2r—3)}?(2Qr—1) (¢)" 
2rtlye! (r+1)! b 


and 
(2s)! F(—s-1, —s, 2, ae i 


(—) gers ea DI h, 


We consider as a numerical illustration a case which has 
been used in comparing other similar series, namely :— 


a=) cem.; b=10 cn.F .2—100 em: =” >, —7.— ae 
Then we have 


h 


b a 1 
5 10s eee 


| (25) 


i 


certain Bessel Integrals. d41 
and from (25) 
| 1 1 5 1 enon) Mig 
M= = 200072n? (10-5 — $+ — 26 + ul + giz a es 80 = 99 103 


Using the terms shown, we find that this gives 


Mima = l0or2o.... 2 2 Kee 


Further, by calculating a few more terms, we easily see 
that the result in (26) is correct as far as the figures shown. 
Other series for this case are those of Maxwell* and 
Heaviside +, while a complete expression in elliptic functions 
has been given by Cohen{. In the last case, although an 
exact theoretical expression is found, yet in practice the 
accuracy depends upon tables of elliptic functions and upon 
the resuit of long and complicated calculations. These three 
expressions have been compared numerically by Rosa and 
Cohen § for the case used above; they give the following 
results for M/s?n? :— 


Miarrelicceseries 62. ce... ccie cee LGOS 7-25 
Hen VISEUE;S) SETIOS «conics saedecde we 19067°08 
Cohen’s elliptic-function formula... 19057°36 


Comparing these with the result given in (26) we inter 
that the two latter formule do not give better results than 
Maxwell’s, at least when the ratio of length to diameter is 
large ; Cohen’s formula is applicable to all values of this 
ratio, but it is not suitable for calculation. The series given 
in (25) appears somewhat simpler than Maxwell’s ; it is con- 
vergent for all coils with the length greater than the radius 
of the outer coil, and as one knows the general term of the 
series the result can be calculated to any required degree of 
accuracy ; it can easily be verified that the series converges 
quite rapidly even for coils whose length is not much greater 
than their breadth. 


* Maxwell, ‘ Electricity and Magnetism,’ vol. i. § 678. 
T Heaviside, ‘ Electrical Papers,’ vol. ii. p. 277 

as Cohen, Bulletin of the Bureau of se iilerds, vol, ili, p. 295 
907). 

a § Rosa and Cohen, Bulletin of the Bureau of Standards, vol. iii. p. 316 
907). 


’ i 
9 =" 


342 Dr. T. H. Havelock on 


§ 4. Short Coil inside a Long Coil. 


Another case is that of a coil of length 2h, within 
concentric coaxial coil of length 2 h,, as in the figure. 


Fig. 4. 
i 
| 
be 
a 
a 
a | 
ey, eee 
a ? ae 
y 
che» 
gS es A, eh ees > 


Then if n, and n, are the turns per unit length, we have 
1 he 
M=47’ abn io i J (Aa) J {(Ad) yar al o-Ala—2 | da! 
—hy —hy 


to ho—hy pe hethy 
= 877" “abn [Fa—B | Gina: Db —e" 8 sa, (F4) 4], . (27) 


Hence, if h,—h, is large compared with 8, a suitable series 
can be obtained by using (12) in these two integrals. We 


find 
{ b b 
abs n= 2h’ on 
[82?abnyng 7 |aNig—hy heh, 


tates oh h )-( b y+ 
16 ( b? hy—hy ho thy 
b = b ) ee. ] 
+35(1+ +35; +5 )4 (goa) hag t+hy 6 va ee 


The general term can be obtained from (12) if required : 


certain Bessel Integrals. 343 


for purposes of calculation the series can be written in the 
following way : 


a 1 
M/8777a?n,no=h, —L? | Z (u—v)— —d(w—v’) 
L4 16 


me. Mee eae 
+ 39 (+E) 0?) — 555 (BE + a) Ca— e) + eo By 
where 

1 1 


¢=a0: d=a?+b?; w= -— >; v=; 
: ; hg—hy? hy +h, 


We shall test this formula by the following numerical 
example which has been used to compare other series : 
fy o.em. 5 ky=2°d cnr... b-=5-em.,.; o= 4 em. ; 


m=10, and n,=40 turns per cm. 


Then we substitute in (28) the following values : 


2 


Z 2 
GA: d=4); OS ores ean: 


Calculating the terms shown, we find to. the order indicated 
M=-0012000 henry. 


From the form of the series we see that this is larger than 
the true value ; and in fact, by taking an extra term we find 
to the same order 


M=-0011999 henry. 


Rosa and Cohen * have calculated the same example by 
three different series using a similar number of terms and 
give the results :— 


M. Series. 
OOLIG ISIC. Roiti. 
SUCRE RO Ne Searle and Airey. 
OOPS Sor. Russell. 


§ 5. Short Coil outside a Long Coil. 


With the same notation, suppose hz is small and /, large. 
Tt has been thought that in this case the formula for M is 


* Loe. ctt. 


d44 On certain Bessel Integrals. 


different from before and more complicated. But we have 


-.) hy hs 
M=47’abnyng} J,(Aa)J COLA aol e Alen? lal 
0 —hy a 


2 hy — hy mth 
= 87°abnyn, E I—b| (Ca be I w)d, (F#)<4]. 
0 


Comparing this with (27) we see that A, and hg are merel 
interchanged ; so that there is a similar series to (28) for this 
case also. 


§ 6. Self-induction of a Cylindrical Coil. 


As a final example we consider the self-inauction of a 
single-layer coil ; then if we have 


.2h=length of coil ; a=radius ; 
N=2nh=total number of turns of wire; 


we can easily deduce from the integrals in $4 an expression 
for the self-induction of a coil in the form 


L=4n7°¢ i fe oe sinh (an) | Jy (Aa)dr 


aN? Atti GA ce eae 
— i) at a 2 WE 
i h aire Br h 7 ‘\ - se (m) A 
For the integral in (30) we can now use one of the 
series (8) or (18). 
If h>a, we have from (8) the series 


a*N? Ava Vpayt alraie 
Deen tts eta) — mG) 


B/G.” ko ae , 
as; ) — 91 =| — ose eS , . . 2 (31) 


where the general term is given by 
(2s!) (2s+2)! ul 
s! (s+2)! {(o+1)!s?28*7\h 
The first four terms of this series have been obtained by a 
different method by Russell *. We see that the series in 


general is simple and rapidly convergent for coils whose 
length j is greater than their width. 


* Russell, Philosophical Magazine, vol, xiii, p. 445 (1907). 


(29) 


Lifect of a Prism on Newton’s Rings. 345 


For short coils we have also the alternative series given 


in (18). We find then 


] wg te a Aa 
Bog N?| 2 gal (ag) ges a | ie ghee 
L=27a} | hee gil) } les. is 1 


Oe 


By using an expression for L in terms of elliptic integrals 
and expanding, Coffin* has obtained a series for L with 
which (32) agrees ; Coffin’s series was evaluated up to terms 
in (h/a)*. Instead of using one such complicated series, it 
seems that the two series (31) and (32) sheuld cover between 
them all the cases that occur in practice. 


XXXII. Lect of a Prism on Newton’s Rings. 
By Lord Rayueien, O.W., Pres. RS. 


THEN Newton’s rings are regarded through a prism 

(or grating) several interesting features present them- 

selves, and are described in the “ Opticks.” Not only are 
rings or arcs seen at unusual thicknesses, but a much 
enhanced number of them are visible, owing to approximate 
achromatism—at least on one side of the centre. The first 
part of the phenomenon was understood by Newton, and the 
explanation easily follows from the consideration of the case 
of a true wedge, viz. a plate bounded by plane and flat 
surfaces slightly inclined to one another. Without the prism, 
the systems of bands, each straight parallel and equidistant, 
corresponding to the various wave-lengths (A) coincide at 
the black bar of zero order, formed where the thickness is 
zero at the line of intersection of the planes. Regarded 
through a prism of small angle whose retracting edge is 
parallel to tue bands, the various systems no longer coincide 
at zero order, but by drawing back the prism, it will 
always be possible so to adjust the effective dispersive power 
as to bring the nth bars to coincidence for any two assigned 
colours, and therefore approximately for the entire spectrum. 
‘In this example the formation of visible rings at unusual 
thicknesses is easily understood ; but it gives no explanation 
of the increased numbers observed by Newton. The width 
of the bands for any colour is proportional to A, as well after 
the displacement by the prism as betore. The manner of 


* Coffin, Bulletin of Bureau of Standards, vol. ii. p. 118 (1906). 
+ Communicated by the Author. 


346 Lord Rayleigh: Effect of a 


overlapping of two systems whose nth bars have been brought 
to coincidence is unaltered ; so that the succession of colours 
in white light, and the number of perceptible * bands, is 
much as usual. 

‘Tn order that there may be an achromatic system of bands, 
it is necessary that the width of the bands near the centre 
be the same for the various colours. As we have seen, this 
condition cannot be satisfied when the plate is a true wedge ; 
for then the width for each colour is proportional tod. If, 
however, the surfaces bounding the plate be curved, the 
width for each colour varies at different parts of the plate, 
and it is possible that the blue bands trom one part, when 
seen through the prism, may fit the red bands from another 
part of the plate. Of course, when no prism is used, the 
sequence of colours is the same whether the boundaries of 
the plate be straight or curved.” 

In the paper t from which the above extracts are taken, 
the question was further discussed, and it appeared that the 
bands formed by cylindrical or spherical surfaces could be 
made achromatic, so far as small variations of X are con- 
cerned, but only under the condition that there be a finite 
separation of the surfaces at the place of nearest approach. 
If a denote the smallest distance, the region of the nth band 
may form an achromatic system if 


== a Ned 35 y(or Ne (1) 


At the time pressure of other work prevented my examining 
the guestion experimentally. Recently I have returned to 
it and I propose now to record some observations and also 
to put the theory into a slightly different form more con- 
venient for comparison with observation. 

For the present purpose it suffices to treat the surfaces as 
cylindrical, so that the thickness is a function of but one 
coordinate #, measured along the surfaces in the direction 
of the refraction. The investigation applies also to spherical 
surfaces if we limit ourselves to to points lying upon that 
diameter of the circular rings which is parallel to the re- 
fraction {. If we choose the point of nearest approach as 
the origin of w, the thickness may be taken to be 


= 6 HOR, iy xs Wh sou ae er 


* Strictly speaking the number of visible bands is doubled, inasmuch 
as they are now formed on doth sides of the achromatic band. 

+ “On Achromatic Interference Bands,” Phil. Mag. xxvii. pp. 77, 189, 
1889 ; ‘Scientific Papers,’ 11. p. 313. 

t In the paper referred to the general theory of curved achromatic 
bands is considered at length. 


Prism on Newton’s Rings. 347 


where 6 depends upon the curvatures. The black of the’ 
nth order for wave-length » occurs when 


i’ NRO ge we ss (CO) 
or 

PN OO. «spo . (4) 
so that 

ae in 


EI 


The nth band, formed actually at wv, is seen displaced under 
the action of the prism. The amount of the linear displace- 
ment (£) is proportional to the distance D at which the 
prism is held, so that we may take approximately 


me ee ° . ° ° . . (6) 


8 representing the dispersive power of the prism, or grating. 
The condition that the nth band may be achromatic (for 
small variations of X) is accordingly 


sera © 
or é 
x REDE =tindt—a, . shits t (has (8) 
a quadratic in n. The roots of the quadratic are real, if 
BOOED SON iar eal ol cidingiac ue wale) 


If a be zero, the condition (9) is satisfied for all values of D, 
so that at whatever distance the prism be held there is 
always an achromatic band. And if a be finite, the con- 
dition can still always be satisfied if the prism be drawn 
back far enough. 

From (8) if 7, nz be the roots, 


9g Ale ales 
z as A TS. <a (10) 
Again, if a=0, that is if the plates be in contact, n, =0, and 
Hi, pe am up eek ore s) 


The order of the achromatic band increases with the dis- 
persive power of the prism and with the distance at which 
it is held. The corresponding value of « from (4) is 


B= AhED, Musee. | 2h wat LD 


348 Lord Rayleigh: Effect of a 


If a be finite, there is no achromatic band so long as D is 
less than the value given in (9). When D acquires this 
value, the roots of the quadratic are equal, and 


or 


Ny = MN = 4a/r, we) ee (13) 


This is the condition formerly found for an achromatic 
system of bands. If D be appreciably greater than this, 
two values of m satisfy the condition, viz. there are two 
separated achromatic bands, though no achromatic system. 
From (8) 

mn, = l6abk? D’,. . 7 


Thus if D be great, one of the roots, say n,, becomes great, 
while the other, see (10), approximates to 2a/A, that is to 
half the value appropriate to the achromatic system (13). 

There is no particular difficulty in following these pheno- 
mena experimentally, though perhaps they are not quite so 
sharply defined as might be expected from the theoretical 
discussion, probably for a reason which will be alluded to 
presently. It is desirable to work with rather large and 
but very slightly curved surfaces. In my experiments the 
lower plate was an optical “ flat’? by Dr. Common, about 
six inches in diameter and blackened behind. The upper 
plate was wedge-shaped with surfaces which had been in- 
tended to be flat but were in fact markedly convex. In 
order to see the bands well, it is necessary that the luminous 
background, whether from daylight or lamp-lght, be uni- 
form through a certain angle, and yet this angle must not 
be too large. Otherwise it is impossible to eliminate the light 
reflected from the upper surface of the upper plate, which 
to a great extent spoils the effects. In my case it sufficed 
to use gas-light diffused through a ground-glass plate whose 
angular area was not so great but that the false light could 
be thrown to one side in virtue of the angle between the 
upper and lower surfaces of the wedge*. It will be under- 
stood that these precautions are needed only in order to see 
the effects at their best. The most ordinary observation and 
appliances suffice to exhibit the main features. 

Another question which I was desirous of taking the 
opportunity to examine was one often propounded to me by 
my lamented friend Lord Kelvin, viz. the nature of the 


* Compare ‘Interference Bands and their Applications,” Scientific 
Papers, lv. p. 54. 


Prism on Newton's Rings. 349 


obstruction usually encountered in trying to bring two 
surfaces nearly enough together to exhibit the rings of low 
order. In favour of the view that the obstacle is merely 
dust and fibres, | remember instancing the ease with which 
a photographic print, enameled by being allowed to dry in 
contact witha suitably prepared glass plate, could be brought 
back into optical contact after partial separation therefrom. 
_ My recent observations with the glass plates point entirely 
in the same direction. However carefully the surfaces are 
cleaned by washing and wiping—finally with a dry hand, 
the rings of low order can usually be attained only at 
certain parts of the surface*. If we attempt to shift them 
to another place chosen at random, they usually pass into 
rings of higher order or disappear altogether. On the other 
hand, when rings of low order have once been seen at a 
particular place, it is usually possible to lift the upper glass 
carefully and to replace it without losing the rings at the 
place in question. I have repeatedly lifted the glass when 
the centre of the system was showing the white of the first 
order or even the darkening (I do not say black) corre- 
sponding toa still closer approximation, and found the colour 
recovered under no greater force than the weight of the 
glass. Some #me is required, doubtless in order that the 
air may escape, for the complete recovery of the original 
closeness ; but in the absence of foreign matter it appears 
that there is no other obstacle to an approximation of 
say 2. 
In making the observations it is convenient to introduce 
a not too small magnifying lens of perhaps 8 inches focus 
and to throw an image of the source of light upon the pupil 
of the eye. With the glasses in contact it is easy to trace 
the rise in the order of the achromatic band as the eye 
and prism are drawn back. As regards the latter a direct- 
vision instrument of moderate power (three prisms in all) is 
the most suitable. An interval between the glasses may be 
introduced by stages. When the approximation is such as 
to show colours of the 3rd or 4th orders at the centre, it 
becomes apparent that the best achromatic effects are attained 
when the prism is at a certain distance, and that when this 
distance is exceeded the more achromatic places are separated 


* The plates are here supposed to be brought together without sliding. 
By a careful sliding together of two surfaces, the foreign matter may be 
extruded, as in Hilger’s echelon gratings, where optical contact is 
attained over considerable areas. 


Phil. Mag. 8. 6. Vol. 15. No. 87. March 1908. 2B 


ee ae 


390 Lord Rayleigh: Effect of a 


by a region where the bands are fringed with colour. This 
feature becomes more distinct as the interval is still further 
increased, so that without the prism only faint rings or none 
at all can be perceived. Tor the greater intervals the inter- 
position of a piece of mica at one edge is convenient. In. 
judging of the degree of achromatism, I found that narrow 
coloured borders could be recognized as such much more 
easily by one of my eyes than by the other, and the difference ~ 
did not seem to depend on any matter of focussing. 

In observing bands of rather high order, the question 
obtruded itself as to whether the achromatism was anywhere 
complete. It will have been remarked that the theoretical 
discussion, as hitherto given, relates only to a small range 
of wave-length and that no account is taken of what in the 
telescope is called secondary colour. So long as this limita- 
tion is observed, the character of the dispersive insirument 
does not come into play. It appeared, however, not at all 
unlikely that even with gaslight the range of wave-length 
included might be too great to allow of this treatment bemg 
adequate ; and with daylight, of course, the case would be 
aggravated. It is thus of interest to examine what law of 
dispersion is best adapted to secure compensation and in 
particular to compare the operation of a prism and a 
grating. 

As to the law of dispersion to be aimed at, we have from 
(A), if N=Ap+ OA, 


- _ £4nry—a) 2 in dd. by 4n6r >? 
aE A eae Gos) 
If & be the displacement due to the instrument, & should be 
a similar function of 6A. In this matter the constant terms 
(independent of 6) are of no account, and the terms in 6X 
may be adjusted to one anotlier, as already explained, by 
suitably choosing the distance D. In pursuing the approxi- 
mation, what we are concerned with is the ratio of the term 
in (6)? to that in 6x. And in (15) this ratio is 


1 7dr 


Rie 3nd —4 (16) 
thus in the particular cases 
89 depo’ (17) 
C= We Xoo ikl ee 
1 or 


bs ice 


eee | : 
a = snr —— A 
gate 2 Ao 


Prism on Newton's Rings. 9. 
Corresponding expressions are required for the dispersive 
instruments. In any particular case they could of course be 
determined ; but no very simple rules are available in 
general. If the intrinsic dispersion be small—the necessary 
effect being arrived at by increasing D, we may make the 
comparison more easily. Thus in the case of the grating 
the variable part of & is proportional to 6a simply, so that the 
ratio of the second and third terms, corresponding to (16), is 
zero. And in the case of the prism if we assume Cauchy’s 
law of dispersion, viz. ~=A+BaA~?, we get in correspondence 
with (16) 
3 ON 
Res (19) 
So far as these expressions apply, it appears that the dis- 
persion required is between that of a grating and of a prism, 
and that especially when a = 0 the grating gives the better 
approximation. It would be possible to combine a grating 
and a prism in such a way as to secure an intermediate 
law, the dispersions cooperating although the deviations 
(in the case of a simple prism) would be in opposite 
directions. 

I have made observations with a grating. using for the 
purpose a photographic reproduction upon bitumen*. This 
contains lines at the rate of 6000 to the inch and gives very 
brilliant spectra of the first order. I thought that I could 
observe the superior achromatism of the most nearly achro- 
matic bands as compared with those given by the prism, 
but the conditions were not very favourable. The dispersive 
power was so high that the grating had to be held very close, 
and the multiplicity of spectra was an embarrassment. If it 
were possible to prepare a grating with not more than 3000 
lines to the inch, and yet of such a character that most of 
the light was thrown into one of the spectra of the first 
order, it might be worth while to resume the experiment 
and, as suggested, to try for a more complete achromatism 
by combining with the grating a suitable prism. 


Terling Place, Witham, 
Jan. 30, 1908. 


* ‘Nature,’ liv. p. 382, 1896; ‘Scientific Papers,” iv. p. 226. 


[ 352 7 


XXXIV. On Mechanical Phosphorescence. 
By ANDREW STEPHENSON®. 


a y iy explanation of the phenomena of phosphorescence it 

is generally held that the energy communicated by 
the incident light is stored as potential energy of the mole- 
cules, and that on the removal of the light the change of 
configuration ceases to be stable and the system gradually 
returns to its initial state, the potential energy setting up. 
vibratory motion, and thereby becoming dissipated as emitted 
light. The change occurring in the substance during this. 
process is considered to be of such a radical nature as is 
designated by chemical change. 

We propose here to show that a simple type of mechanical 
system exhibits the chief phenomena of phosphorescence, and 
furthermore that the agreement between the properties of 
the phosphorescing substances and those of the system is of 
a quantitative nature. 

The energy stored by the system under the incident dis-. 
turbance is kinetic, so that the explanation offered has nothing 
in common with that stated above. 

It will appear that the argument is general in character,. 
applying to any mechanical system that can be treated by 
the method of normal coordinates. On account of the gain 
in vividness, however, it is convenient to make use of a: 
particular system; the main facts forming the basis of the 
theory are then capable of simple exposition apart from the 
analytical development. 

2. Consider a particle suspended from a fixed point by a 
light elastic string; if we regard the system as confined to. 
a vertical plane it is fixed by two coordinates, and the normal. 
motions are the vertical vibration and the horizontal (pen-. 
dulum) swing. If the system is frictionless—a case which 
may be considered by way of introduction—the normal — 
motions are in general practically independent; the vertical - 
vibration creates a small periodic change in the spring of 
the pendulum motion, while the latter exerts a second order 
forcing disturbance on the former, but neither of these 
actions is cumulative in effect unless the periods are 
properly adjusted. When the periods vertically and 
horizontally are approximately in the ratio 1/2 there is. 
mutual interaction: if the system is initially in vertical 
motion, any small horizontal swing is gradually magnified 
through the periodic change of approximately double. 


* Communicated by the Author. 


On Mechanical Phosphorescence. 30% 
frequency in its spring*, while if, on the other hand, the 
motion is initially horizontal, a vertical vibration is forced by 
the isochronous variation in the tension. Some notion of the 
marked nature of this interaction may best be gained from 
experimental observation. 

Now let us suppose that the vertical coordinate is subject 
to kinetic friction, while the other is frictionless. Then, 
whatever the relation between the periods, the pendulum 
motion forces a small oscillation vertically, in general of 
second order amplitude, maintaining it against friction; thus 
the energy is gradually dissipated and the frequency of the 
emission is double that of the pendulum. 

Consider the effect of a periodic disturbance applied radially. 
In general it will merely produce an isochronous forced 
vibration vertically, but if the frequency is approximately 
double that of the pendulum motion, this oscillation will in 
turn magnify any existing small pendulum swing; thus the 
system stores energy absorbed from incident disturbance of 
the particular frequency. 

For a given intensity of disturbance the amount of energy 
obtained in this way will depend upon the time of action, but 
evidently it must have some definite upper limit. 

When the exciting force is removed the pendulum motion 
generates a comparatively small forced oscillation vertically 
of double its own frequency, which continues until the energy 
is dissipated against the kinetic friction. Thus the system 
gives out the stored energy in a definite frequency during an 
interval which is comparable with, and may indeed be large 
compared with, the time of action of the incident disturbance. 
That is, the system phosphoresces. 

It is to be noted that in the special case when the frequencies 
of the normal motions vertically and horizontally are in the 
ratio 2/1, the forced oscillation vertically under the exciting 
disturbance, the energy stored, and the subsequent rate of 
energy emission have their greatest values. 

A disturbance acting on a coordinate through periodic 
change of spring has a continually cumulative effect when 
the ratio of its frequency to that of the free motion of the 
coordinate lies anywhere within a certain range. In the 
above, therefore, energy is stored under incident force of 
any period within a range, but the resulting emission is 
always of one definite frequency, 2. e. double that of the free 
pendulum motion. 


* “On a Class of Forced Oscillations,” § 2, Quarterly Journal of 
Mathematics, no. 168 (1906). 


304 Mr. A. Stephenson on 


The class of phosphorescent substances exhibits exactly 
similar phenomena. 

The qualitative agreement being thus far established, we 
proceed to examine the quantitative properties of our typical 
system ™. 

3. If ris the length of the string and @ the inclination to the 
vertical at time ¢, the equations of motion of our system under 
the radial disturbance are 


rO+2% O490=0.  « =). 0 
r+2e7r —r 04+? (r—l)=an? cos2nt . . . (il) 


In the case of a system which receives energy from, or 
gives up energy to, a surrounding medium the “Frictional ’ 
coefficient, «, may be large during emission and negligible 
during absorption. Thusin the present case if the suspended 
body exposes a large surface the effect of periodic disturbance 
communicated through the air is accounted for by the term 
on the right of (ii.), and « represents merely the internal 
frictional force which may be small: when, on the other 
hand, the body is giving up energy by the generation of 
periodic motion in the air, « is large. We shall therefore 
assume that « is zero while the x ery is storing energy, 
but is large during emission. 

Assuming the amplitude of 8 to be small initially, we have 


yp=l—2ea cos 2nt. 


where J is the length of the string in equilibrium, and 


Putting 0=¢/r we have from (i.) 
rp t(g— 7) b=0, 
| — 2% ee ie Ns oe eae t)p=0 — 
( 7 COs 2n b+(e— >. n” cos 2nt jo=0, (iii.) 
where p?=g/l. 


* The results of the analysis in § 3-§ 5 are summarised in § 6. 


Mechanical Phosphorescence. 309 


The solution of this equation is of form * 


d= > | ars cos { (p—2r= Ln)t+ e} + A~(2r—1) COS { (P +2r—In)t+ e} | ) 
where ¢ is arbitrary. (iv.) 


On substitution in (iii.) we find 


Mh 2r—1 { a? (p—2r—1. nye} ~ [4 ae (p—2r—3 2)? b airs 


oa { 4n?—(p —2r+1. n)* baa | =0...(@@r—1) 


a4 we any} _ i[ {ae -(o—m? fa 


ar { 41°=(p—5n)* | a, i axes. COs) 


ay ja oo ny} = aL 4 an —(pt ny} (ia 


ae { dn? —(p—any? b as |=0 ae (1) 


aa we —(ptny b Ff dnt —(p—n)* ban 


+ { dn?— (p+ Bu)? bas =i. (—1) 


as {at (p+ an)? | r 7 { 4? (p+ nye an 


Hr { 4? (p+ in? Las | =O. (23) 


— 


_ (2-1) {ut (pt+2r—1. ny h — = | { dn? — (p +2r— Bn)? baron 
+ { An*—(p+ Or + in)’ } aerts| =0...(—2r+1), 


* The method is similar to that employed in the paper already 
quoted; a change of notation is introduced which makes the solution 
easier to handle for our present purpose, although the simple idea 
underlying the method is thereby rendered less readily apparent. For 
the general principle reference may be made to the previous work. The 
solution is here carried to a higher approximation. 


356 } Mr. A. Stephenson on 


In the limit v= 


G+ (Qr—1) 


ast a | numerically, 
7+ (Q@r—3) he: 


so that the series is convergent. The above equations are 
sufficient to determine p and the a’s. Our special object is 
to find the range of n near w for which p is imaginary; such 
a value of p indicates a continually increasing amplitude of 
o, and therefore of 6, due to the influence of 7. In the 
general case it is not a practicable matter to obtain the 
solution of this problem in finite terms, and we shall there- 
fore assume that a2 is small. The coefficients then diminish 
rapidly, and we shall neglect those lying beyond a, and a_s. 
Then to the required degree of approximation 


3 
and on substituting in (1) and (—1) we have 


Be Es : 
a, {we (p—n)— Soe = * {4n?—(ptnp fa 


eg .0) ee eee Zan, rey 
[w—(p—nyr— a {w—(ptnyi—n BE! 


a { 4n?—(p+n)?| { 4n?—(p—n)?} + 
2 


w*?—n* being of order «/l this equation is correct to the 
order («/l)* and determines p to the order (a/l)?. Putting 


w= w(.+k7) and neglecting powers of «/l above the third 
we find 


we ee 

Sree 5 
‘ae s 
4+ 2k7 


D1 a 
Bei 


and therefore for a cumulative influence on the 6 motion n? 


Thus p is imaginary if & lies between the limits +3+ 


Mechanical Phosphorescence. 307 


must lie within the range between 


and gradually diminishes to zero on either side towards the 
ends of the range. 

If an?, the amplitude of the forcing disturbance, is kept 
constant, the magnitude of « depends upon the position of n 
within the range. By taking ¢ greater than 2n we ensure 
that | « | increases along with n, the variation being marked 
if e—2n is small. 

It appears, then, that the system stores energy under 
incident disturbance of any frequency within a certain range 
the central value of which is greater than double the frequency 
of the pendulum motion; and as the intensity of the dis- 
turbance increases the range becomes wider and the central 
value slowly greater. 

4, Expressed in real form the equation (iv.) becomes 


70 = Ae!” |*sin (nt +8)+ Be! ?!* sin (nt—B)+ ..., 


where 3—k 
tan 9= + pee aes 


the terms of the first approximation only being retained. 

A and B depend upon the initial conditions, but if A is 
not zero initially the motion tends to the steady phase given 
Py 

r0= Ae! |* sin (né+ 8), 


in which the passage of energy to the @ coordinate is maximum. 
If the initial @ is sufficiently small this state may be closely 
approximated to before much of the total energy is stored. 
To find what happens after the initial stage we must 
now consider the reaction of @ upon r. Through the term 
6? in (ii.) a direct disturbance is applied to the r motion, 
and this gradually reduces 2a, the amplitude of r, until 


melt 


gat SAE 
nap 137 = 3 ) i.€., until n is on the verge of the 


range of frequency within which magnification takes place. 


358 Mr. A. Stephenson on 


The effect of the » motion on @ is then merely to keep the 
latter in a forced vibration of constant amplitude, and the 
system is therefore in the steady state, no further energy 
being absorbed. It may be noted that when n=yp for the 
steady motion «=0: in this case of exactly double frequency 
it is evident without analysis that at a certain amplitude of 
swing, if the phase is properly adjusted, the periodic varia- 
tion in tension is exactly balanced by the radial incident 
disturbance, so that + remains constant and all the incident 
_ energy is reflected. 

It is of interest to inquire how the amount of energy 
required for saturation depends upon the intensity of the 
disturbance and its frequency within the necessary range. 

When the system is saturated, 


and B=90° or 0° respectively, so that 


a eater 
sin 

The signs and the phase of @ depend upon the sign of a, 
the initial value of a. There is no loss of generality in the 
choice of a so that a is positive. Then as the @ swing 
increases a is gradually diminished from a to the steady 
value, which is therefore positive also. Hence in the above 
equations for a//, k, and @ the upper or lower values are to 
be taken according as n? is less or greater than p’. 

By substitution in (ii.) 


248 2 n a) i Bo hh wae , 
Fen (“5 —1) Leos 2nt — 5 1b®n?(1F cos Qnt) 


+24 + : (= — 1) l cos 2nt +n} =an’ cos 2nt, 
where 2X is the change in the mean value of r due to the 
@ motion. The terms in (5 -1) being of the second 
order are neglected. We have then 

A=4 1b’ n?/c’, 


mie , 
+6? = #| 2an?+=- (e— n')(~, =} } a. aikarete © 


The effect of the increase, X, in the mean value of 7 is to 


and 


Mechanical Phosphorescence. 309 


diminish uz”; and if this is appreciable the frequency of 
emission ple the removal of the incident disturbance must 
gradually become greater as the energy is given out. 

Again, n? 
ah gee , where o’?<3. 
be l 


9 
pe uErpm, (V-) +=—(1—3o")a, Was te AR ay 


according as n is less or greater than uw, Now if ¢>2n, a is 
negative when «, is positive, and therefore the above equation 
gives 6° positive if n greater than yw, but negative if n less 
an mw. It follows that in the latter case there cannot be 
steady motion and the amplitude of the @ coordinate oscil- 
lates periodically. In the steady motion which holds for the 
upper half of the range it is evident-from (vi.) that the 
total store of energynecessary for saturation is proportional 
to the square root of the incident intensity, and gradually 
decreases to zero as the applied frequency is taken greater 
within the range. It is to be noted that in the deduction of 
(vi.) only the terms of the first approximation were retained. 

d). When the exciting disturbance is removed the @ swing 
directly forces a small oscillation vertically of exactly half 
its own period, the energy being thereby dissipated against 
the kinetic resistance of the vertical coordinate. It is neces- 
sary to obtain the conditions under which the emission is of 
constant frequency throughout its decay. Evidently from 
the preceding % must be small compared with «; and on 
substitution for 6? in the value of X we find that this is 
brought about by taking ¢ sufficiently close to 2n. Further- 
more, during emission there is motional resistance to the r 
coordinate, and by making this large we bring 8 towards the 
limit 7/4, and therefore cause the frequencies of 0 and r to 
approach the limits w and 2w respectively to any required 
degree of De desselinticlon Thus the variation in the phos- 
phorescence frequency during decay is brought within any 
assigned range however small. 

We now seek to express the rate of emission of energy as 
a function of the time that has elapsed since the removal of 
the applied force. 

The forced motion of + due to @ is of amplitude propor- 
tional to the square of the @ amplitude, and the work done 
per oscillation against the motional resistance is therefore 


; ee ary hs 5 


360 Mr. A. Stephenson on 


proportional to the fourth power of the swing. Hence, if HE 
is the energy of the 4 motion at time ¢, 


di : 
Wis ——s — gH 4 
where ga constant. This gives 
= ie 
C+4qt 
where C=1/H,; therefore 
dE q 


di. (C+#qt)" 
Now I, the intensity of the phosphorescence, is equal to 
the rate of energy emission; thus 


tone (C+ gt), 


2.é., the reciprocal of the square root of the emission intensity 
has a uniform time gradient. 

The initial intensity is proportional to the square of the 
total store, and therefore increases proportionally with the 
incident intensity until the latter becomes so large that our 
approximation (vi.) does not apply. 

6. It may be well now to summarise the properties of the 
system as regards phosphorescence in the probable order of 
their appearance to an experimental observer :— 


1. The emission frequency is independent of the exciting 
disturbance. 

2. The system stores energy under incident disturbance 
of any frequency within a certain range which 
includes the emission frequency ; the greater the 
intensity of disturbance the greater the range and the 
greater the excess of its central value above the emission 
frequency. : 

3. For different frequencies within the range the initial 
rate of energy storage is maximum near the centre, 
and decreases gradually towards each end. If the 
frequency of the storing coordinate is less than half the 
frequency of the other, the system ultimately reaches 
a steady state only for that part of the range which 
lies above the emission frequency. In the steady state 
the energy stored is proportional to the square root of 
the incident intensity, and it also varies with the 
applied frequency, gradually decreasing to zero as 
the latter is taken greater within the range. 


Mechanical Phosphorescence. d61 


4, The reciprocal of the square root of the emission in- 
tensity has a uniform time gradient. ; 
. When the system is saturated the initial emission in- 
tensity is proportional to the intensity of the incident 
disturbance when the latter is small. 


7. Now when a substance phosphoresces :— 


1. The emission spectrum in any neighbourhood is inde- 
pendent of the exciting light. 

2. Phosphorescence is pr oduced by light of any frequency 
within a range which includes the emission fr equency 
near its ae end. 

3. The intensity of emission is maximum for an exciting 
frequency greater than that of the emission. 

4, The reciprocal of the square root of the emission in- 
tensity when plotted against the time, gives a straight 


2S 
line during an aierval in which the greater portion 


(over 90 per cent.) of the stored energy is dissipated ; 
there is then a rapid bend in the graph which quickly 
becomes straight again at a smaller inclination, 

5. The energy stored under given disturbance increases 
asymptotically to a definite limit, and when the system 
is saturated the initial emission intensity is propor- 
tional to the intensity of the exciting light, provided 


the latter is not too large * 


There is thus a remarkable agreement between the pro- 
perties of the phosphorescent substance and those of the 
system—an agreement which is all the more striking in view 
of a very noteworthy difference, namely, that phosphorescent 
matter gives a band spectrum while the system gives a line. 
A secondary point of contrast arises in the rate of Hae of 
emission ; as we have seen for the system the I->—¢ graph 
is simply a straight line, while in the case of the anes ito 
escent matter some amie change takes place when only a 
small ee of the energy remains, and the rate of 
change of I~? is diminished. Connected with this there is a 
hysteresis effect + which has no analogue in our mechanical 


The phosphorescent bands differ very much in brilliance 


* “Qn the Decay of Phosphorescence in Sidot-blende and certain 
other Substances,” by Messrs, E. L. Nichols and E. Merritt, Physical 
Review, vol. xxill. (1906) p. 37. 

+ Nichols and Merritt, loc. ect. 


362 Mr. A. Stephenson on 


and duration; in the case of the system these qualities depend 
on the natural frequency of the coordinate directly acted on 
by the incident disturbance. If the frequency of this co- 
ordinate is equal to that of the emission, it is evident that 
maximum intensity in emission is obtained. : 

8. The simple vapours that give line spectra in fluorescence 
show no trace of phosphorescence, and it would appear that 
the band spectrum of phosphorescence is, perhaps, an essential 
property of the phenomenon. The difference between the 
system and the phosphorescent substance may thus be funda- 
mental, but the agreement in other respects is of such a 
nature as to indicate the possibility that in essence there is 
mechanical identity between the two; that the incident light 
forces a vibration in a coordinate, which in turn communi- 

eates kinetic energy in intensifying vibration in another 
coordinate until the saturation limit is reached; the reverse 
process being set up on the removal of the exciting light, 
with the emission of ener gyi in a frequency double that of 
the storing coordinate. 

The analog ue affords marked economy of thought in con- 
necting a number of the chief facts of phosphorescence. It 
may also be of service in suggesting new directions of 
experimental examination. We e chalt conclude with a 
reference to two of these. 

9. If the spring of a simple oscillation is subject to periodic 
variation, the amplitude is magnified not only in the case of 
double frequency, but more generally when the ratio of the 
natural frequency to that of the disturbance lies anywhere 
within a range in the vicinity of 47, where 7 is any integer”. 
As r becomes greater the magnifying influence “diminishes 
very rapidly, but when r=2 it is easily observable, although 
not nearly so pronounced as in the ease of double frequency, 
r=1. Thus if our system is subject to a radial disturbance 
of frequency nearly equal to that of the pendulum motion, 
energy will be absorbed, and at once emitted again during 
the action of the disturbance in appr oximately double 
frequency. 

It would be of interest to inquire whether phosphorescent 
substances exhibit a similar phenomenon. As the effect is 
of the second order, it would be necessary to have the exciting 
light of great intensity in order to give the matter a fair 
trial. Experiments to determine the various effects of the 
infra-red rays have been carried out by Nichols and Merritt 


* See the paper already referred to in § 2. 


Mechanical Phosphorescence. 363 


with Sidot blende as phosphorescent substance *. The waves 
in this region have the effect of greatly retarding the storage 
of energy when incident along with the exciting disturbance; 
and in general they also diminish the total store and rate of 
emission if applied during phosphorescence. ‘There is an 
exception to be noted, however, in the case of Sidot blende ; 
one particular band of the phosphorescence spectrum is rendered 
more intense throughout its decay by application of the 
infra-red during emission. This effect is consistent with the 
property enunciated above for the mechanical system, but 
further experiment is necessary to determine the range of 
Gnfra-red) frequency which produces the brightening. If 
the range were found to contain, or lie near, half the phos- 
phorescence frequency, a further point of resemblance would 
he established between the mechanical system and the phos- 
phorescent substance. 

The general infra-red effects, on the other hand, do not 
appear to come within the analogue, possibly on account of 
the difference referred to in the preceding section. 

Another point which is worthy of experimental investi- 
gation is the variation in the rate at which energy is stored 
under given intensity of excitation for different trequencies 
within the effective range. It has been shown in $§ 3 that 
the initial rate of energy absorption in the mechanical system 
is maximum for a frequency rather greater than the central 
value of the range, and gradually diminishes towards zero at 
the ends of the range. 

If the excitation intensity is not too great it is found that 
an appreciable timef is required for the saturation of some 
phosphorescent substances, and there does not appear to be 
any insuperable difficulty in the way of the experiment, 
The inquiry would be of special interest in determining the 
saturation limit for different frequencies throughout the 
range. 

Manchester, November 1907, 


* “The Influence of the Red and Infra-red Rays upon the Photo- 
luminescence of Sidot Blende.” Physical Review, vol. xxv. (1907) 
p- 362. 

+ About five minutes in the only case recorded by Nichols and Merritt, 
p. 45, in the paper first mentioned. For trustworthy results it would be 
necessary to make measurements of the short duration phosphorescence 
before the bend of the 1~2—¢ eraph. 


ee i 


es eae 


XXXV. Mutual Induction. By Prof. D. N. Matix *. 


i es M be the potential energy between any two shells, 

vr’ the magnetic strengths of these shells, and w the 
solid an subtended at any point of one of the shells by the. 
other, and dn an element of outward drawn normal and ds 
an element of surface of the first, then we know that 


M =p | 22 d 


Again, let ds’ be the element of surface at any point of the 
second surface and 
p = distance between the points, 
dn' = element of outward drawn normal of the second 
surface. 


v= far (5 ) 
raw fe AC )ow 


«x. 1. Two circular wir es in any relative position to each 
other, carrying currents 2, 2’. 

Let C1, Co be the radii of the spheres of which they are 
circular sections, the origin bemg the centre of both 


spheres. 
Then 
hi (pei) i ! 
Mee ee ee ds 
‘i | 06.. 06s fe + C9? —2¢1Cz Cos 7 gens 
= —i' {2 onl. — S (2). P,,(cos7) . ePdwes? dy | dpdd' 
1 2 


= 71'> n(n +1) r PR cosrduendb dee 


where ¢ > Cg, 


r= £ between any two radii of the two circles, and 
#, ® have their usual meanings. 


But 
P,,(cos7)=P,,(u) P, cos (7) +> Tr (u) Tx (cos 7’) cos m(d— $"’) 
and 
P,,(cos r’) =Px(u’)P,, (cos 0) + >, Tx (u') Tr (cos 0) cos m(p'— 9"), 
* Communicated by the Author. 


Prot. D. N. Mallik on Mutual Induction. 365 


where = Zbetween the perpendiculars to the two circles, 
¢'" defining the plane containing these lines, 
= Zbetween any radius of the and circle and the 
axis of the first circle $” defining the plane 
containing these lines. 


integrating for @ between 0 and 27 


6! we have 


2 n+l a 
M=47ii'Snun4)). an P,, (cos @) | P,(u)du{ P,(w!) dp! 


Oe dP 


ae ‘q(1—w)(1— —n')3(4 Ly" P,(cos 8) — : dul? 


which is a well-known result. 

Ex. 2. Mutual induction between two parallel rect- 
angular wires symmetrically situated and carrying unit 
currents. * 

Let x, y, h, and x’, y', h', be the coordinates of any two 
points in the rectangles of w ich the wires form the contours. 


oe Oo» da dy da' dy’ 
Oh Oh) /(a—2' P+ (y—y' P+ hh’ 


This is directly integrable, for 


22) ay) 
is — wu!) +2 ripe log VAC PHV+(a—2')f, (1) 


and flog fie +7 +u du 
=ulog(Vi4+W+u)— Vl+wW.. 2 © (2) 


But we may also proceed as follows :— 


le dy da dy 
= can) \ part ey pista ey. 
; ane oh Ja ty2t (hn? 


Then M=— 


POMS: Fr ‘ 

where ee poe, 
" oe Pe Fae erat dal df ee 
OS BRS (ily eB Ja? +y?+ hl 


if 2a, 2b are the sides of the first rectangle. , 
Phil. Mag. 8. 6. Vol. 15. No. 87. March 1 gOS, | 2° 


366 Prof. D. N. Mallik on 


Me 2. & Aue is — (2) +. . .) 
(1+ S(ay7) ++) ya 


e Pa) Pa) n=O M=—O Lay: 
=—Ilba—. avi, > (2n+1)!(2m+1) i 


O 2n—1 Pe) hee 1 
x ae NJ! ee 
(30) (31 J Na? 4074 (hh? 


where 


(2 0 “a 1 da ke 
Ja®aete ory 7 , PP Mh 
which can be written down from (1) and (2). 
If the distance between the planes is small = d,. 


foe a nearly 
Oh * Ol! Va? +b? +(h—h'y? ~ (a® 4624 +d*)3 o* 
and 


a2" e h2m : 


Qn+1)!Qm+1)! 


«(ax) (30) ree 


which is derdctly’ suitable for calculation if the terms 
decrease rapidly (the first three terms involving integrals 
which can be easily evaluated). 

The final result can, however, be written down :-— 

For we have > 


(2,) Fe) =a’ Feat) +72) aly* Bg ae 


M=-—I16ab { 33 


Put 1 1 
ie SS eS SS Pig 
ce. Va?+b?+(h—h') p wei 
then 
: 1°3..€2r—1 
Fr(a) = as py eee 
and 


anata 04 
= (—1)"s Fai P (F)- 


ees eee 
ap ) ls iat 
ak : ; 
, 


Mutual Induction. 367 
Again, if F(b") eR =P hod 
4P 
F’(b?) =—= we Fal @+DP-+2 —~.Pr ‘| 


but it can be shown that 


aber (r+ 1)P-=Phat. 


Fo?) = — ; ; ore 


and F" (52) i oe ores Hier + — Peas 


Mee) ae 


== 22 . ores bette 


.. / 
since S PL! +(r+1)Pr=P741. 


5 Pe + (r+ 2)Pe=Priis 


a . PY. + (r+3)Plas= Pre 
f 1 1 s 
F(b le Geld eer ° p r4+2s+1 ors 3 
where 0 \S 
P = (<.) P e 
KB =\5 2 (u) 
But 


Biel / sf ],/2 1 !\ s—2 s—1 (f,/2 
—” y+ =) (2p o Wee ty + 


str 8 s(s—1) b's—2 ree 
(2, =) (3,) ( yG)= Ge ~ ‘ ee ryaetl *r# 7 Dp * prt2s—1* Rie 
OS 7 a [(=) Pp oe a po 


private p rts ite 


Finally if h—A’ is small 


So (a)= a8 
DAM Ob: te, 


+ 


£ 


: : 


368 Prof. D. N. Mallik on 
and 
2 ae 
Oh” OAT” \prtesti rte 
1 s+ 
= prreets ° rae 


and accordingly, 


1 argu’) (s7/) ath (h—h'y? 


tid r! b'\$s+1 s(s—1) (0'\3 5 
rd) u ge prtsts bere 2.5 1.2 (=) “ Pie r 


Ez. 3. Two concentric and coplanar elliptic wires. 

Describe coaxial spheroids of which the ellipses are parallel 
sections. 

Then if p is the distance between any two points of the 
spheroidal shells of which the given ellipses are plane 
sections, 


= = 7 ¥(2n-+1)Pa(Hs) Pal!) Qu(2) Qul?") 
+ 3E2 SEIT TE WIL (QE (#)e08m(G—9) 


where hr, hi! are major axes of the elliptic areas, 


and i=-(\22 = 2 ale )asas'. 


Remembering that pdn=h’rdr, | 
and pds=h’r(r?—1) sin @ dé dd 
= —h’*r(7?—1)dyud?d ; 


we get 


Sete 5 


=—_ h>(2Qn 5 vO) Ca a 1)(* of yee . iy year (1) Pin(p’ ) dp dp’ dp dg' 


2 19 d n d eg m m ' ! 
—2h>> “ oat (1) ee) W! j eu {zr (w)T, (uw) dp du 


x cos m(d—¢') dp dd’ ; 


Mutual Induction. 369 


and the limits of ¢, ¢! are 0 to z, and those of p, ry are 1 to 0 
taken twice over. 
2n+1 ‘ is a dP» dP! 
|, M=— 49h 1), “ea (Gr) 
n(n - NG Pape du. ae dp! MEY 
_ghSS (n—m) (2 —=1)(r? 7) re Bane AQ 


(n+m)! me) ee dr’ 
x fz (4)Tr (we!) du dy’. 
0 m 
We proceed now to evaluate i) Tr (m) dp. 
1 
We have 
29 ayy a"P, 1 doe Cie 23 
(_—-z ae m f= ae pe 
BPS TIN. nbs 
- ae qe 
Gea alae | 
*. differentiating m—2 times 
dm™-1p, m—2P d™=—-|P_ ye nee (s 
pepe t on —2)4 iu ducuam apace 
Now writing | TS | 
, dpm — m—|? 
(2) becomes 
pD,_,= D+ (n—m+2)D" 4. . |. (8) 


*. from (1) and (3) we have 
fa-w9 2 Di du=(1—p?)? D,,_, 


+m\(1—p?)2 ot pies : +(n—~m—2)D),_ |e - (4) 


Again, 
ein <2 =nbe ee 
"Lae m—1 times : 
A= W2)D! (m1) 4602, — 2AM IDs 
=nD" | — nD, _ ae 23 
1. e. 


(1—p?)D" =nD" + u(2m—2—n)D" _ i+(m—1)(m—2—n)D" _ oC 


370 Prof. D. N. Mallik on 
.. from (3) and (5) we get 
fa-w)FDEqe 
=a—n)?> nD, _, + (2m—2—n) | Dr + (n—m+2)D"_, 
# (m—1)(m—2—n)Dp,_ | . . a 


Multiplying (4) by (2m—2) and (6) by m and subtracting, 
we get 
(m— 2) 1— p’) 2 D’ du=2(m— 1)\(1— 2 Des 


+m(n+m—1)(n—m+ Xe —p”) Sei Die du. 


1.é. m __ 2(m—1) 2\ = n 
(rt d.=2—D a ie 


a (n+m i soe ; “ai 


(ray = [Paty ua 


: Ps gg GPa 
(ra =(a 1 ies 
it ee = pln (Phil. Mag. Oct. 1907). 


We accordingly have 


and 


a SAeara (nt+m—1)(n—m+2) m—3 
\ LU Ee Wee Eee l (0) + 2m To aap — a Uae 3 (0) 
7 eke: 
In order to evaluate D,_,(0) &e., we proceed as follows :— 
Let 1 n 
1 > ft ee 
Phen ! 20 
4 
or dy __) 
and a’ a Py 
dp” dpe 
2r+l1 . 


= 3.5...(2r—J1)k" . (1—2uk+kh) 2 


Mutual Induction. 371 
(522) = coefficient of k” in the expansion of 


= a 


2r+1 


5...(2r—1)hA+h?)7 2, 
as | chanel id +1)(2r+3)...r7+n—1) 


n—? wT ' 
pay: 2 ° ») F 


1 (@+a—1)! (dad) 2 
~Sfh=1 /ptn—? 


2 rtn—2\, ap 
( 9 2 JF 
2. @. 
nm—im+1 
Dies aS es a ee (n+m—2)! 
: 2 


m—2 (e*): | 


2 
etm —1) (n—m+2) m—3 


(n+m—A)! 


m—2 "m—4n+m—5, n— m+3, 
= = 1 ee v4 ‘ 2 
— e 
ae 1) (n+m—2)! + Pe 
Shy ee nm+m— 3,2 m+1, (m 1) +m— In? —(m— 1)*]- a 
Se j 2 


= t'"(say) : 5 ale if 


where'n—m-+1 must be even ; otherwise the integral is equa) 
to zero. 


Also 
(5) =(), if nis even 
du =o 
Pais Lia we til) Teo 
==r( = Qa-1 5 a ae res ri nis o ° 
cy 
Hence, ultimately, 


7 4p+3 AQsp41 AQ’ap Ti { (2p + 1) !}? 
ey ea on ‘ u2p p+1 4p 
M= —47?h(1?—1)(1 >, pil Qpt2e a de! BG 
(=m)! (P17 1) AOE AVE a im 
16h 22 em)!” = 2 a Bias ae t 
where m must be odd, and » must be even 
4. If the ellipses are parallel, with the line joining the 
centres perpendicular to their planes, the same investigation 


will apply, only the limits of integration for ¢, ¢’, 4, a will 
be different. 


XXXVI. On the Canal-Ray Group. By EH. GOLpstEIn* 


T is permissible to speak of a ‘canal-ray group,” in so 
far as there exist—as will be shown—seyveral forms of 
radiation which have certain characteristic features in common 
with each other and with the canal rays properly so called, ’ 
while in other respects they exhibit differences of behaviour. 
By true canal rays we mean those which arise on the side 
away from the ‘anode, at a cathode completely filling the 
opening of the tube and provided with small perforations or 
narrow slits, and which in the case of holes form narrow 
cones, andin the case of slits beams of slight divergence. 
Although it is to be surmised that eventually it will be found 
possible to consider the various members of this group of 
radiations from a common standpoint, and to reduce their 
differences to a quantitative gradation, I am nevertheless of 
opiion that, in the interests of further investigation, it is 
advantageous s, in describing the various members, not to 
group them together as yet, but to bear in mind their dif- 
ferences as well as their similarities. 

The true canal rays on leaving the cathode openings 
proceed in straight lines, and, in the case of a cathode com- 
pletely filling the tube, tov vards the side away from the 
anode; they are not appreciably deflected by weak magnetic 
or electrostatic fields. In the ease ofa plane parallel plate, 
perforated with a number of holes or slits, the various pencils 
in general converge towards each other and towards the axis 
of the plate. Their colour is rosy in hy drogen, bluish in air 
or nitrogent. At the glass wall, and in general in any 


oo) 


compounds containing sodium, they excite golden-yellow 


light; in compounds containing lithium the excited light is 
red, and in magnesium compounds green. In all these ae 
the line spectrum of the metal is observed. 

Closely allied to the canal rays proper are the rays whigt 


* Communicated by the Author from the Verhundl. d. Deutsch. Physik. 
Gesellsch. iv. p. 228 (1902). ; 

[Norr, Jan. 30th, 1908.]—I am greatly indebted to the Editors of the 
Philosophical 1 Magazine for reprinting the above paper which was pub- 
lished first in the Verhandl. d. Deutsch. Ph y sik. Gesellsch. nearly six years 
ago. It contains a condensed summery of many years’ observations con-- 
cerning canal rays since my first publication (1886) on this subject. As 
far as way be judged from recent literature, my observations do not 
appear to have been of much use hitherto. I would consider it an 
advantage if the above reproduction would prove to the reader that 
canal rays are a much more complicated phenomenon than generally 
admitted.—E. G. 

+ The common impression that canal rays are golden-yellow in air 19 
based on the effect produced by a form of radiation discussed below. 


On the Canal-Ray Group. 313 


may be observed in the case of perforated or entire cathodes 
when the ordinary cathode rays are deflected to one side by 
a magnet. They spread, in the case of a cathode which com- 
pletely fills the opening of the tube, not backwards, but 
towards the anode. Towards magnets saad electrostatic fields 
they are not more sensitive than the true canal rays. In 
hydrogen they appear rosy, in air golden-yellow (¢f. below). 

They excite luminescence in sodium , ithium, and magnesium. 
Like the canal rays, they are propagated in straight lines, 

but in the case of a plane plate form a moderately divergent 
pencil, and in general closely follow the direction of the 
ordinary cathode rays which start from the same surface 
and are not influenced by external forces. Their direction 
relatively to the cathode, as w ell as the side towards which 
they are propagated, is indicated by the shape and position 
-of the shadows cast by solids placed in their path*. In 
what follows they are briefly termed K,-rays fF. ‘ 

The first layer of the cathode oie whose behaviour I 
have recently described in detail f, vexhibits similarities and 
close relation iship with the canal rays, so much so that one 
might be led to suppose that the canal rays are identical with 
the first layer of the cathode glow, and are merely that 
portion of it which passes unaliered through the openings in 
the cathode. 

Contrary to this supposition, it has been found § that the 
first layer consists of rays which are extraordinarily suscep- 
tible to influences that affect canal rays but slightly. This 
seems to exclude the possibility of a complete identity of the 
two. ‘The rays of the first layer will be referred to in what 
follows as S,-rays. 

* E. Goldstein, Verhandl. d. Deutschen Physik. Gesellsch. 11. p. 207 
1901 
T in ote, Jan. 30th, 1908.|—The KX,-rays have first been described in 
my earliest paper on canal rays (Sttzungsber. d. K. Akad. d. Wissensch. zu 
Berlin, 1886), where on p. 698 it is shown that rays of the same colour 
and the same magnetic behaviour as canal rays proper are emitted also 
by non-perforated cathode-disks standing in the free gas-space, and are 
found in front of the cathode. In 1901 [ described the following expe- 
riment (Verhandl. d. Deutsch. Phystk. Gesellsch. iii. p. 207) : :—“ As 
cathode in a tube of about 4 cms. width a nickel disk of 5-10 mm. is 
used and a wire or a glass rod of some millimetres thickness is fixed for 
casting a shadow. Evacuating the ,hydrogen) gas to a proper degree 
and bending the ordinary cathodic rays to the side by a magnet, one will 
observe that the rosy ¢c canal-rays form a continuous mass of light filling 
the width of the tube, while behind the object islyinga sharply-bounded 
weakly divergent shadow-space. Undoubtedly we see here rays with 
qualities of canal rays, extending away from the cathode.” 

+ E. Goldstein, /. ¢. iv. p. 64 (1902). 

§ E. Goldstein, /. c. iv. p. 64 et seg. (1902). 


374 Prof. E. Goldstein on 


The §S,-rays are also closely related to the K,-rays. For 
if the luminosity of the S,-rays is different over different 
portions of the cathode—a result which may, for example, 
be produced by the approach of a magnet—the luminosity of 


the K,-rays emitted from the corresponding portions of the. 


cathode surface varies in a similar manner. 
_ There arises the problem of the true origin of the canal 
rays. 

That the canal rays produced at the back surface of 
the cathode have their origin at the front surface is rendered 


very probable by the already established fact* that the 


direction of the individual pencils is determined very largely 
by the shape of the front surface of the cathode, but appears 
independent of the shape of the back surface. If the latter 
is plane, e. g., the canal rays are convergent if the front 
surface is convex, and become divergent if the front surface 
is made concave. 

As regards the origin of the canal rays, it appears to me 
that the part played by the openings has hitherto not received 
sufficient attention. | 

In using slotted plates of not too great a thickness for the 
production of canal rays, we must regard not so much the 
entire plates as the walls of the openings—e. g., the opposite 
sides of a slit—as forming the cathodes ; these, along with 
the edges bounding the openings, are effective in the pro- 
duction of canal rays, and it is incorrect to regard the 
openings in general as merely forming so many neutral 
channels of escape. 

There further seems to me to be a close connexion between 
the canal rays and a form of radiation which is also obtainable 
with plane, not perforated cathodes, and which forms rays 
both tangential to the cathode plane and moderately inclined 
to it, which pass inwards over the cathode from one of its 
edges to the opposing edge and then into free space. In the 
case of a circular plate, these rays obviously only produce a 
uniform glow around the plate. But if the plate be cut 

Fig. 1. 


away in several places along chords (fig. 1), the rays which 
start from these chords may be recognized—in hydrogen, 


* EK. Goldstein, Sitzwngsber. d. k. Akad. d. Wissensch. zu Berlin, 1886, 
p- 695; Wied. Ann. lxiv. p. 43 (1898). 


the Canal-Ray Group. 375 


é. g., aS narrow rosy pencils which pass from the edges 
lying opposite the chords in a radiai direction into space ™*. 
The luminosity of these pencils is comparatively feeble, and 
hence not easily noticeable; it may, however, be greatly 
increased by arranging two such congruent cathode plates 
coaxially at a distance of a few millimetres apart. The 
luminosity is then not merely doubied, but increased many 
times. ‘lhe reasons for this increase of luminosity will be 
investigated elsewhere. If in a spherical tube (8-11 ems. in 
diameter) there be fixed facing each other two small con- 
gruent squares (side=8 mm.) with their sides parallel, then 
in (hydrogen) gas at a suitabie pressure there may be seen 
a bright cross (fig. 2) whose luminous arms, of a rosy colour, 


Fig. 2. Fig. 3.. 


proceed from those portions of the space between the two 
squares corresponding to the middle portions of the sides. 

If the cathode consists of a pair of regular pentagons, there 
is produced a five-rayed star, whose five arms, however, do 
not arise from the middle points of the sides, but from the 
angular points of the space between the plates (fig. 3). 

In the case of a pair of regular hexagons, there appears a 
six-rayed star whose rays again appear to arise from the 
middle points of the sidest. In general, in the case of even- 
sided regular polygons, the star-shaped figure is formed by 


* EK. Goldstein, Physik. Zeitschr. i. p. 133 (1899). 

+ In order to carry out these experiments, we may place two polygonal 
plates facing each other, and either provide them with separate leadine- 
in wires (attached to their back surfaces or edges), or else make use of a 
single leading-in wire, the connexion between the two plates beine 
established by means of a thin rod fixed to suitable points. 4 


376 ; Prof. E. Goldstein on 


rays which form continuations of the shortest radii of the 
polygon, while in the case of odd-sided polygons the rays 
appear to form continuations of the longest radi. It will 
be seen that this arrangement is consistent with the suppo- 
sition that the rays originate from the opposite sides. If the 
polygons have unequal angles, then in the case of odd-sided 
figures the pencils may appear at points other than the 
angular points, and*in the case of even-sided figures at 
points other than the middle points of the sides, but always 
in a manner consistent with the edge that they start 
from the opposite sides. 

The rays of these star-shaped figures possess all the 
properties which may be observed in connexion with true 
canal rays. 

If the double cathode consists of two congruent rectangles, 
then a luminous pattern is produced like that shown in fig. 4, 


which corresponds to a certain pressure of the gas. There 
arise in the space between the cathodes, from their longer 
sides, two wide rosy luminous bands, whose initially almost 
pare allel boundaries conv erge more and more with decreasing 
pressure of the gas. From the smaller sides there arise two 
thin pencils of much feebler luminosity, not represented in 
the figure. The appearance of the two wide bands does not 
change appreciably if the shorter sides of the plates be 
connected by plane walls, the cathode then forming a hollow 
parallelepiped. 

If the line of sight be parallel to the long sides of the 
cathode, the pencils appear in the form of feebly divergent 
bands, which proceed from the entire width of the space 


the Canal-Ray Group. 377 


between the plates (fig. 5). <A precisely similar appearance 
is presented by the radial arms of the stars formed when 
using the various double polygons mentioned above, 
when the line of sight is parallel to the planes of the 
polygons. 

A more exact knowledge of the path followed by 
the rays in the space between the plates may be derived 
from the fact that the rays themselves map out this 
path on the plates. Ifa double square has been used 
for some time as cathode ina gas at suitable pressure, 
then on stopping the discharge the two opposed plates 
show a pattern which at first sight appears compli- 
cated, but which on closer inspection is seen to consist 
simply of the superposition of four bands, each arising 
from a side of the square (fig. 6a). The bands have 
a boundary which is concave outwards, and therefore 
represent.a system of rays which gradually narrows 
as it approaches the opposite side of the square. The 
rosy arm of the cross shown in fig. 2, which issues 
from the opposite side, is the direct continuation of 
the concave. band. While the discharge is still in 
progress it is possible to observe directly that the rosy bands 
between the plates have the same direction and dimensions 
as the traces left on the plates when the discharge is arrested. 


TS ee eee 


Fig. 6a. Pig. 6b. Ries t 


On the plates of a double rectangle there appears a pattern 
which is the result of the superposition of two pairs of con- 
cave bands, one pair of which is shown in fig. 60. Corre- 
sponding patterns are obtained with cathode plates of other 
shapes. The patterns result partly on account of differences 
in the reflecting power connected with the formation of thin 
layers of oxide, and partly from shallow grooves produced in 
the originally plane cathode surface. : 

From the concave boundary of the traces and of the pencils 
themselves, it may be inferred that the rays proceeding from 
any side are subject to an attraction due to the neighbouring 
sides. We may, for example, imagine that in the case of 
the square the rays are originally convergent along straight 
lines, somewhat after the fashion indicated by the dotted 
lines in fig. 7, and that it is in consequence of the attraction 
exerted on them by the adjacent sides that they assume the 


378 Prof. E. Goldstein on 


actually observed form (shown by the full lines). Similar 
remarks apply to rectangles Kc. 

It may be asked in how far all these observations sérve to 
explain the genesis of the canal rays proper. In this con- 
nexion it is only necessary to remark that a straight slit in a 
plane parallel plate represents nothing else than a double 
rectangle or a hollow parallelepiped, and that a plate with a 
number of slits forms a system of such double rectangles. 
Hach double rectangle acts like an independent cathode of 
the same shape, and thus there arises a number of luminous 
bands similar to those shown for the case of a single double 
rectangle in fig. 4. If the plate containing the slit completely 
fills the opening of the tube, only a single luminous band can 
be produced—viz., that which issues from the edges of the 
‘slit on the side towards the anode. 

If the perforations of the cathode consist of holes and not 
of slits, each hole acts like a small hollow cylinder. In the 
case of a single hollow cylindrical cathode there is produced 
a rosy feebly divergent luminous cone. 

In the small double rectangles of plates provided with 
slits, and, mutatis mutandis, in the short hollow cylinders of 
cathodes perforated with holes, the canal rays pass both 
tangentially along, and at a moderate inclination to, the inner 
‘walls, from the front to the back surface, and into free space. 
The fact that the rays do not exclusively follow tangential 
paths explains why even slits a few millimetres in width 
appear to be completely filled with canal rays, and why each 
luminous band when viewed in a direction parallel to the 
shorter side always appears divergent: those rays which are 
inclined to the walls of the slit and which proceed from 
opposite walls crossing each other while still in the slit and 
then passing out in a divergent form. In order to explain 
details, especially the relatively feeble divergence of the 
Iuminous bands, it must be borne in mind that just as the 
tangential rays are subject to the attraction of the adjacent 
sides, so also the inclined rays are attracted by the surface of 
the emitting cathode, and become concave towards it. The 
rays therefore emerge not along their original directions, but 
along the tangents to the last elements of the curves which 
they follow between the walls of the slit. 

The fact, on the other hand, that with decreasing pressure 
of the gas the boundaries of the bands proceeding from the 
slits become more and more strongly convergent when viewed 
broadside-on, may be explained by taking into account the 
relation between the canal. rays and the first layer. The 
canal rays start on the front walls of the narrew double 


the Canal-Ray Group. 379 


rectangles not only at the places in contact with the first 
layer, but those rays which pass in a grazing direction over 
the walls enter the slit along the directions which the rays 
of the first layer have at the corresponding portions of the 
edge. Now there appears to apply to the S,-rays of a plane 
cathode a law entirely analogous to that which has long ago* 
been established in connexion with the ordinary cathode rays 
easily deflected by a magnet—viz., that if the pressure be 
maintained constant, the rays are the more strongly inclined 
towards the axis of the plate the nearer the starting-point of 
the rays is to the edge of the plate, and that the less the 
pressure the more does this inclination of the rays increase. 
This not only explains why with decreasing pressure the 
boundaries of the band arising from a slit converge more 
and more (corresponding to the i increasing divergence of the 
S,-rays), but also accounts for the fact ‘that the luminous 
bands of a system of parallel slits in a plane plate converge 
towards each other and towards the axis of the plate. 

In the case of cathodes which either entirely or nearly fill 
the opening of the tube, we also have to consider, when 
working at low pressures, ihe recently investigated + obliquity 
of the boundaries of the first layer due, on the one hand, to 
the repellent anodic action exerted on its base at the cathode, 
and, on the other, to an attraction of the rays by a cathodic 
zone of the wall which lies in the neigh- 
bourhood of the outer limit of Crookes’s 
space. If under this influence the ex- 
treme §,-rays assume the form shown 
in fig. 8, there results a canal-ray band 
with very strongly convergent boun- 
daries (shown dotted in fig. 8). 

In this connexion it may not be out 
of place to refer briefly to the effect of 
very weak magnets on the canal rays. 
In the case represented in fig. 8, it is 
possible, by means of a weak and small 
horseshoe magnet held horizontally—the 
slit having an equatorial direction—to 
produce great changes in the directions 
of the boundaries of the luminous band, 
and to render one or other of the boun- 
daries very much more oblique. 


Fig. 8. 
* E. Goldstein, Sitzungsber. d. k. Akad. d. Wissensch. zu Berlin, 1881, 


p- 799; Wied. Ann. xv. p- 274 (1882). 
+ E. Goldstein, Verhand!. d. Deutsch. Physth. Gesellsch.iv. p. 64 (1902). 


380 Prof. E. Goldstein on 


This apparently so powerful effect of a magnet on the 
direction of the canal rays is primarily conditioned by the 
change in the direction of the S,-rays which results from 
the approach of the magnet. Yet even this strong effect on 
the §,-rays is not primarily due to the magnet, the primary 
effect being the magnetic deflexion and distortion of the 
common cathode rays which excite phosphorescence, and the 
approach of their extremities towards the cathode. If, for 
instance, the magnet deflects the rays towards the right hand, 
thena certain portion of the right-hand wall on which the 
condensed rays impinge becomes a secondary cathode, which 
attracts the rays of the first layer, and, by causing them to 
be attracted towards a region lying nearer the cathode than 
before, inclines them more towards the right. The con- 
sequence is that the right-hand boundary of the convergent 

canal-ray band becomes inclined even more obliquely upwards 

and towards the jeft—Let it be expressly stated that the 
possibility, in addition, of a direct effect of the magnet on 
the S,-rays is not fesoedl 

If a gold wire be ar ranged as cathode along the axis of a 
eylindrical tube, then, as is well known, a deposit of gold is 
produced on the ‘wall around the cathode (in an atmosphere 
free of oxygen). On replacing the wire by a double square 
of material (aluminium) which does not disintegrate, with 
the planes of the squares normal to the axis of the tube, and 
on producing the cross-shaped system of canal-rays depicted 
in tig. 2, it is found that after a comparatively short time the 
thickness of the gold deposit is very materially reduced in 
the four places where the arms of the cross meet the tube, 
and ultimately the wall of the tube is in those regions denuded 
of gold. This property of canal rays may be exhibited in a still 
more striking and technically more convenient manner by in- 
troducing into a tube a double cathode of material subject to 
disintegration (silver, goid, nickel), in which case those places 
where the arms of the cross strike the tube remain entirely 
free from deposit. ‘These regions are in general of a rhombic 
shape. When they have been mapped out, it is found that 
on rotating the cathodes about their axis through, say, 45°, 
the regions formerly free from deposit become covered, while 
the already covered regions where the rays impinge against 
the tube become denuded. In general, the particles deposited 
are transferred from the places where the canal rays have the 
greatest density to places of lower density *. 


* [Nors, Jan. 30th, 1908.]—Later on Prof. J. J. Thomson (Proc. 
Cambridge. Phil. Soc. xiii. p. 212, 1905) has also observed that canal rays 
disintegrate metallic surfaces, and the same statement has been made by 

Mr. Kohlschiitter (Zettschr. f. Litektrochemie, xii. p. 869, 1906). 


the Canal-Ray Group. 381 


To a smaller extent, the ordinary cathode rays also possess 
the property of disintegrating metal surfaces against which 
they impinge, as I have pr eviously shown *. Grandquist has 
pointed out in this connexion that the minimum thickness of 
deposit occurs where there is a maximum of phosphorescence fF. 
In comparative experiments carried out by myself, the action 
of the canal rays was markedly stronger. A separation of 
the two kinds of rays may be etfected by means of the str ong 
magnetic deflexion exerted on the ordinary rays which excite 
phosphorescence. 

Connected with this transport of metallic particles are the 
traces which are left in the canal rays on the cathode surfaces 
atter continued use. Mere differences of colour and reflecting 
power may already be obtained in a very much shorter time, 
and, as mentioned above, depend on the formation of layers 
of oxide. 

The bands and narrow cones of the canal rays are—even 
in the case of cathodes completely filling the opening of the 
tube—surrounded by a duller, but very easily noticeable, 
nebulous light, which fills the entire space even in the case 


Se 
of cylindrical tubes of large cross-section, or spheres 10 cms. 


in diameter. In order to investigate this light more closely, 
itis convenient to make use of plate screens completely filling 
the opening of the tube and provided with only a single slit 
or hole. It is thus possible to establish the fact that this 
light contains two forms of radiation. One of these consists 
—as may be ascertained from the shadows cast by suitable 
objects—of regular, rectilinear, divergent rays, which proceed 
from the slit (or hole) in the form of a wide cone, enveloping 
the canal rays proper. The aperture of the cone exceeded 
120° in my experiments. If the 8,-rays which proceed from 
a slit viewed end-on be deflected - the right by a magnet, 
the nebulous rays in the left-hand portion of the tube become 
brighter. The luminescence of sodium &c. may be excited 
by these rays just as well as by the canal rays proper. At 
low gaseous pressures, these regular nebulous rays cast deep 
shadows, which appear quite dark. But with increasing 
pressure the space corresponding to the shadow becomes 
gradually filled with the second form of radiation, which 
consists of diffuse rays. These are produced in a much 
weaker form in hydrogen than in air (or nitrogen), in which 
they exhibit a golden-yellow colour. 


* E. Goldstein, Thdtegkeitsbericht der Phys.-Tech. Reichsanstalt fiir 
1894, p. 70; Zettschr.f. Instrumentenk. xvi. p. 211 (1897). 
aa G. Grandquist, Oefver's. ES , Svensk. Akad. 1897, p. 


575. 
Phil. Mag. 8. 6. Vol. 15. No. 87. March 1908. 2k) 


382 Prof. E. Goldstein on 


It seems permissible to suppose that these diffuse rays stand 
in the same relation to the canal rays proper and to the regular 
nebulous rays, as does the light of the third layer of the 
ordinary cathode glow to the regular, ordinary cathode rays— 
2.é., that the diffuse rays are due to the impact of the canal rays 
proper Sc. against the particles of the gas*. - 

The diffuse golden-yellow rays play an important part in 
investigations relating to the canal-ray group. To their 
existence is due the impression that the canal rays proper 
themselves emit golden-yellow light. In reality, the light 
of the canal rays proper is bluish, and it is only the stron 
diffusion that envelops them in a golden-yellow light which 
to a large extent obscures the bluish one. To the same cause 
is to be ascribed the apparently intense chamois to golden- 
yellow colour of the similarly bluish rays of the first layer in 
air. It is only at very low pressures, at which diffusion 
becomes small, that the bluish colour appears more clearly. 
If by means of an anode fixed in the neighbourhood of the 
cathode a shadow be produced in the S,-rays, this shadow is 
also, at moderate pressures, not entirely destitute of light, 
being filled with the diffuse golden-yellow light. This same 
light also fills the shadow cast in Kj-rays when the pres- 
sure is not very low. It seems that the K,-rays, and also 
the nebulous rays enveloping the canal rays, are bluish in air, 
and only owe their golden-yellow appearance to the admixture 
of the diffuse rays. : 

Evenin the case of double cathodes (squares, pentagons, &c.) 
arranged in free space, a closer investigation shows that 
the (in hydrogen) rosy arms of the cross- and star-shaped 
figures form the most prominent, it is true, but not the only 
kind of rosy light that is related to the canal rays. The 
bright narrow arms (when viewed in the direction of the axis 
of the plates) are, as a matter of fact, seen to lie between 
very feebly illuminated wide fields of rosy light. Shadow- 
casting objects show that this light is also propagated (away 
from the cathode) strictly along straight lines, and that it arises 
from every part of the space between two parallel sides (of the 
polygonal cathodes), in the form of wide fans—strongly 


* (Note, Jan. 30th, 1908.|—In a lately published paper Prof. J. J. 
Thomson (Phil. Mag. [6] vol. xiv. p. 359, 1907) declares to have “found 
that positive rays are to be found in all parts of the tube which have an 
uninterrupted view of the ordinary ‘ Canalstrahlen,’ or of that luminous 
patch next the cathode of which the Canalstrahlen are the prolongations.”’ 
Rays of a peculiar intensity are observed by Prof. Thomson “right in 
front of the cathode.’—It seems that my above statements published 
previously on the diffuse rays and on the K,-rays have escaped Prof. 
Thomson’s attention. 


the Canal-Ray Group. 383 


divergent in planes parallel to the cathodes—whose rays are 
also slightly divergent in a plane normal to the cathodes, in 
a manner similar to that of the star-rays themselves, as shown 
in fig. 5. The objective existence of the dull fan-shaped 
pencils of rays which fill up the gaps between the rosy arms 
of the star may also be shown by using double cathodes of 
material which undergoes disintegration. It is then observed 
that, in addition to the regions entirely free from deposit 
which correspond to the intersections of the bright arms with 
the walls of the tube, there also appears around the wall a 
moderately wide zone in which the deposit, though present, 
is markedly reduced in thickness, as may be seen by trans- 
mitted light. In accordance with this, it is also found that 
in planes which are normal both to the planes and to the 
sides of the double cathodes, there proceed from the spaces 
between the plates dull, rectilinear rays, in the form of very 
wide, divergent, very feeble, wide-angled fans of rays which 
accompany the moderately divergent and easily seen rays 
represented in fig. 5. These strongly divergent rays, whose 
planes of divergence are normal to the plane of symmetry of 
the double plate, are to be regarded as analogous to the 
rectilinear nebulous rays mentioned above. 

In the formation of the moderately divergent pencils shown 
in fig. 5 there may also take part—in the case of double 
cathodes—rays which, like the S,-rays, appear to proceed 
from the inner surfaces of the plates, more especially from 
prominent lines and symmetrically situated points of the 
polygonal surfaces ; in the case of rectangles, e. g., from a 
portion of the bisecting line parallel to the longer sides. The 
rays traverse a short distance in a direction nearly normal to 
the plate, and are then deflected towards it through a large 
angie. According to the amount of this deflexion, which 
changes with the pressure, they enclose, together with the rays 
proceeding from the other plate, spaces of varying shape. 
Thus at low pressures and with a relatively small deflexion 
they converge with the corresponding rays of the other plate 
and form a convergent middle sheet in the outwardly divergent 
beam of fig. 5. A more detailed account of these very com- 
plicated relationships of the rays cannot, however, be given 
within the limits of a general sketch. 

Up to the present, it has been possible to identify (at least) 
the tollowing members of the canal-ray group :—1. The rays 
of the first layer. 2. The canalrays proper. 3. The K,-rays, 
which also occur at the front surfaces of continuous (not 
perforated) cathodes. 4. Regular nebulous rays, which 
envelop the canal rays proper. 5. Diffuse rays. 

The above list might be extended. 

2D2 


384 On the Canal-Ray Group. + 


There arises the question as to whether the available material 
is sufficient to enable us to introduce a guiding and simplifying 
principle into the study of these complicated phenomena. In 
the literature of the subject, two views have been expressed 
regarding the nature of the discharge. According to one, the 
discharge and the radiation effects by which it is accompanied 
are regarded as a process taking place in the ether, the nature 
of which is not further particularized. According to the other, 
the discharge and radiation are identified with the motion of 
charged particles. At the present time the two views would 
appear to be capable of a certain amount of reconciliation, 
owing tc the fact that many investigators are prepared to 
abandon the association of ordinary ponderable matter with 
“ electrons,” and to regard the masses in question as only 
apparent. In so far, therefore, as purely electrical processes 
may be regarded as actions taking place in the ether, a fusion 
of the two points of view does not appear improbable. The 
champions of the electron theory have gained heuristic aids 
by expressing the consequences of their hypothesis in mathe- 
matical form and attempting to verify the formule by 
experiment. In this connexion it has appeared that in the 
case of the ordinary cathode rays the following may be re- 
garded as the determining variables :—the potential at which 
the rays are produced and propagated, the mass of the pro- 
jected particles, and the magnitude of their charges; the 
deflectibility of the rays under the action of electric or mag- 
netic fields increasing with the amount of charge, and de- 
creasing with increase of mass and potential. 

Possibly from the point of view of the electron theory the 
required general principle, primarily of heuristic importance, 
might be established were it possible to determine definitely 
the as yet doubtful direction of propagation of the S,-rays. 
Should this direction turn out to be—as is often assumed, 
though without proof—towards the cathode, it would afford a 
general view-point for certain actions of the canal-ray group. 
The disintegration of the cathode itself on the one hand, and 
the removal of the disintegration deposits by the canal rays 
already described on the other, might then be brought 
together under the general point of view according to which 
the S,-rays and the canal rays cause removal of the metal 
wherever they terminate, while in connexion with the 
spreading of the S,-rays in a direction away from the 
cathode, the rays would produce the same effect at their 
places of origin as do the canal rays at their terminations. 
Similar considerations may be applied to the fact that the 
canal rays excite luminescence in sodium, lithium, and 


The Radioactivity of Sea- Water. 385 


magnesium on encountering these at their terminations, and 
that a similar luminescence occurs at the surface which forms 
the base of the S;-rays on the cathode. Should the propaga- 
tion cathode-wards of the S,-rays be established, then on the 
assumption of the already existing hypothesis that the first 
layer of the cathode glow consists of carriers of positive 
electricity, we might attempt to deduce the origin of the 
other members of the canal-ray group of the S,-rays. Thus, 
in order to account for the K,-rays, we might suppose that 
during the impact of the positive S)-rays on the cathode there 
occurs a partial rebound of the particles, which are then 
characterized by a strongly diminished charge, and must 
therefore in accordance with theory be much less sensitive.— 
A nearly parallel beam of S,-rays arriving in a nearly 
tangential direction at the walls of a slit would suffer various 
changes of charge, according to the distance of its individual 
rays from the cathodic walls of the slit, and wouid, on account 
of its variously charged particles being at various distances 
from, and hence suffering varying deflexions by the negative 
surfaces, be transformed intv a more strongly conical beam, 
whose components would, in the space between the walls, 
proceed partly ina tangential, partly in oblique directions. 
The concavity, towards the neighbouring sides, of the positive 
beams which sweep across the polygonal plates would thus be 
accounted for, as well as the fact that outside the space 
between the plates the rays again proceed along straight 
lines, forming tangents to the last elements of their curvilinear 
path between the plates. The decrease of charge taking 
place during the motion across the plates would explain why 
the canal rays emerging on the opposite side are much less 
sensitive than the S,-rays. The nebulous rays, which emerge 
in a highly divergent form, would also be accounted for. 


XXXVIL. The Radioactivity of Sea-Water. By J. JOLy, | 
Sc.D., F.R.S., Professor of Geology and Mineralogy in the 
Oniversity of Dublin, Hon. Sec. Royal Dublin Society™. 


AM acquainted with only one determination of the 
radium in sea-water taken directly from the ocean— 
that recorded by A. 8. Eve in his paper “On the Ionization 
of the Atmosphere over the Ocean” f. Strutt had previously 
determined the radium in sea-saltt, and obtained the value 
* Communicated by the Author. In part reprinted from the Scientific 
Proc. Royal Dublin Sce. xi. No, xxi, ; 
+ Phil. Mag. ser. 6, vol. xiii. p. 248. 
t Proc. Roy. Soe. vol. Ixxvini. p. 151. 


386 Prof. J. Joly on the 


0:15 x 10-7? in grams per gram of salt. This must be reduced, 
however, to about one-half, seeing that, at the time of this 
determination, the quantity ‘of radium associated with a gram 
of uranium was believed to be about double as great as it 
has since been shown to be—this quantity entering into the 
calibration of the electroscope. It is necessary to add that 
Strutt gives his determination as ‘‘ approximate only.” 

The sea-water dealt with by Eve was collected in mid- 
Atlantic, between Montreal and Glasgow. The amount of 
radium found was 0-0003 x 10-2 gram per gram of sea-water. 
If we reckon sea-salts as composing 3 per cent. by weight of 
sea-water, Strutt’s determination would give 0:0023 x 10-¥ 
gram per gram of sea-water. 

Hive also made an experiment on sea-salt, using 400 grams 
of Omaha sea-salt guaranteed pure by the importers. He 
finds 0:02 x 10- gram per gram, or for the equivalent sea- 
water, 0:0006 x 10-*”. 

We thus have the three divergent results :— 


From radium in sea-salt (Strutt), 0°0023 x 10-®. 
From radium in sea-salt (Eve), 00006 x 1052. 
From mid-Atlantic sea-water (Eve), 0:0003 x 10-, 


In reviewing these figures Eve states that it is safe to con- 
clude that sea-water does not contain more than 0:0006 x 10—” 
gram radium per gram. 

In April of the present year a friend sent me from 
Valencia, Co. Kerry, a sample of sea-water collected in the 
harbour at flood-tide. A very large volume of water enters 
this extensive harbour at each flood; and as there are no 
large rivers along the coast, water taken at the harbour mouth 
may be regarded as free from impurities derived from the 
shore. The quantity sent to me (in a stoppered glass bottle 
which I had transmitted, specially cleansed, for the purpose) 
was 2450 c.cs. This was evaporated on the water-bath to 
about 1200 ¢.es., with every precaution against contamination, 
and in a part of the laboratory in which radioactive prepara- 
tions had never come. ‘The radium determined in this 
Valencia sea-water was 0:0356 x 10—- gram per gram. The 
apparatus and mode of extraction I was using at this time 
were closely similar to those developed by Strutt as described 
in his paper (loc. cit.). The readings of the electroscope 
were standardized by the emanation from Joachimsthal 
pitchblende, a portion of which I had analysed for the 
uranium. 

This determination, showing more than ten times the 


Radioactivity of Sea- Water. 387 


amount of radium given in any evaluation known to me, I 
naturally regarded as requiring confirmation. 

In the month of May I received from Mr. 8. W. Kemp, 
of the Irish Fishery Department, 2800 c.cs. of sea-water 
taken in the Irish Channei a few miles to west of the Isle of 
Man, lat. 53° 53’ N., long. 5° 16’ W. This also was trans- 
mitted in bottles which I had supplied for the purpose. The 
water was evaporated down to about 800 c.cs., and when 
investigated yielded only 0°0038 x10~” gram per gram. 
This figure seemed to support sea-salt determinations ; but 
I was so convinced of the genuineness of the first result that 
I provisionally concluded a real difference to exist between 
near-shore and more open-sea samples of water. I had, 
indeed, changed the mode of extraction of the emanation, 
substituting a method in which ebullition proceeds in a 
partial vacuum, and at its conclusion the space above the 
liquid in the flask is completely filled with distilled water, 
thereby securing that every trace of emanation is displaced 
into a receiver, from which it is transferred into the electro- 
scope. I had found by comparative experiments on known 
quantities of uraninite in solution, that this mode of extraction 
gives a somewhat lower constant for the electroscope, showing 
that it is capable of effecting a more complete extraction. 
The change in manipulation was therefore in favour of giving 
a higher, and not lower, result. 

Shortly after this measurement, Mr. Kemp sent me three 
other samples of sea-water. Calling the two samples already 
referred to (a) and (>) respectively— 

Sample (c) is from lat. 51° 26’ N., long. 1295’ W. A 
point about 65 miles due west of Valencia. Quantity, 
2665 c.cs. 

Sample (d) is from 1°5 miles south of Crow Head, Co. 
Kerry. Quantity, 2740 c.cs. 

Sample (¢) is from lat. 51° 35’ N., long. 10° 43’ W.—that 
is, about twenty miles west of Bantry Bay. Quantity, 
2764 c.cs. The last two samples were collected on the same 
day, June 21st. 

These were evaporated down as before, each to about 900 
or 1000 c.cs. The results were as follows :-— 


(c) 0°0126 x 10-”. 
(2) OO fF! 10: 
(e) 0°0268 x 10-®. 
It will be seen that all these figures are of the same order 


ot magnitude as the result on the Valencia water. The dis- 
crepancy with the Isle of Man sample is considerable. While 


388 Prof. J. Joly on the 


very sure that errors arising from contamination did not enter 
these results, it seemed probable that some other source of 
error existed. If this arose from any cause residing in the 
nature of the material dealt with, it could only be that there 
was in some way concealment of the emanation in the lower 
results, and that the higher results were the more nearly 
approximate to the truth. Sea-water is rich in sulphates ; 
and the possibility of the process of concentration resulting 
in a precipitation of the radium in non-emanating form 
occurred to me. ‘There isa small quantity of barium in sea- 
water. A very minute precipitate of this might bring with 
it the radium ; and, again, the calcium sulphate, which is 
one of the earliest substances to fall, might also be concerned. 
The emanation, which never increases beyond a certain 
amount, might be imprisoned in very fine precipitated 
particles. The flasks were examined with this possibility in 
view. ‘None of them were perfectly clear of precipitate. In 
(a) it was least, perhaps ; but the dulling of the glass on the 
inside of the flasks showed that in every case some solid 
matter had come down. It is true that, in so far as this is 
redissolved on ebullition, it can do no harm ; but the risk of 
error here was evident. 

Accordingly all five were somewhat diluted and trans- 
ferred to larger flasks, and a few ec.cs. of very pure HCl 
(which I had distilled from a nearly pure acid) added to each. 
This instantly cleared the precipitates away. They were then 
boiled and closed. When in due time the whole five came 
to be redetermined, the results came out as follows :— 


(a) Valencia, taken in March, 00400 x 1074 
(b) Isle of Man, taken in April, 0:0386 x 10-8. 
(c) 65 miles W. of Valencia, June, 0:0314 x 10-—= 
(d) 1:5 miles 8. of Crow Head, June, 0°0226 x 10-22. 
(e) 20 miles W. of Bantry Bay, June, 0°0393 x 10-™. 


These, with the exception of (d), are in close agreement ; 
and I find it hard to believe that such a degree of uniformity 
could arise without a real basis for the results. The deficiency 
of (d) might originate in a loss of radium in the process of 
evaporation, the radium being in part left as a precipitate 
clinging to the evaporating dish. The higher results appear 
to be the more reliable ; indeed, there is the possibility that, 
in a liquid so rich in sulphates, even these are not a full 
revelation. I do not regard this, however, as probable. 

These results, of course, do not warrant the conclusion that 
mid-ocean sea-water is equally rich in radium. It might be 


Radioactivity of Sea- Water. 389 


that the acid-treatment is effective in bringing into solution 
suspended radioactive particles of coastal origin. 

It will be evident that measurements made on sea-salts 
must be unreliable ; for there is no assurance that the sample 
used will contain its due share of the radium which would be 
carried down among the first constituents precipitated in the 
preparation of the salt. 


Shortly after the above results were communicated to the 
Royal Dublin Society, five samples of oceanic water collected 
in the course of a voyage from Madeira to the Bay of Biscay 
by Dr. F. Stevenson reached me. I will call these 7, g, h, 2, /, 
for convenience of reference; giving their place of derivation 
later on. 

Fully realizing the necessity of every precaution, Dr. 
Stevenson had himself cleansed the vessels in which these 
waters were transmitted, not only with fresh-water but several 
times with sea-water, before finally filling up. ; 

As there was sufficient of each sample to enable two 
determinations tu be made, I resolved to vary the procedure 
in the several observations. 


yj. (1) A quantity amounting to 2825 c.cs. was evaporated 
on the water-bath in my laboratory to 15V0 c.es. and 
tested without addition of acid. The result was 
0°0029 x 10-17. This was now acidified with 12 c.cs. 
of pure radium-free HCl, the subsequent examination 
now gave 0°0144x 10-”. 

(2) 1650 c.cs. concentrated by evaporation in my labora- 
tory, acidified with 80 ¢.cs. HCI, closed and examined: 
result O0°0150x10—*. Tested a second time the 
result was 0°0213 x 10—. 

g. (1) 2826 ¢.cs. concentrated in my laboratory to 1500 c.es. 
and acidified with 30 ¢.cs. HCl gave 0°0193 x 10-”. 
Tested a second ‘time, adding 50 e.cs. HCl, result 
0273 x 10-". 

(2) 1500 c.cs. not concentrated but directly transferred 
to the flask, adding 80 c.es. HCl, afforded 0°0044 x 10-2, 
A second experiment on this gave 0°0080 x 10-®. 

h. (1) 2500 c.cs. concentrated in the Botanical Laboratory, 
T.C.D. (a new building in which no radioactive work 
had been done), with 10 ¢.cs. HCl gave 00020 x LO-»., 
Tested again with addition of 50 ¢e.cs. HCI, the reading 
was 00146 x 10—-. 

(2) 1626 ccs. with 70 c.cs. HCl concentrated in Botanical 
Laboratory: 0°0084 x 10-®. 


390 Prot. J. Joly en the 


2. (1) 1200 c.es. not concentrated, no acid : 0:0031 x 10—-”. 
Same with 35 c.cs. HCl: 0°0260 x 10-”. 
(2) 1200 ec.es. not concentrated, with 20 c.es. HCl: 
0:0304 x 10-?". 
j. (1) 1400 c.cs. not concentrated, no acid: 0°0146 x 10-”. 
The second lot of 7 was lost by an accident before its 
examination. 


I give these results with some detail in order that the 
variations obtained in the readings may be brought out. 
It is apparent that acid is not always requisite to enable-the 
emanation to be liberated ; thus in 7 the reading is in the 
second decimal place, although acid was not added. But if 
there is concentration then acid appears to be requisite, and 
in some cases a small amount of acid increased to a larger 
at once raised the amount of emanation obtained. It will be 
noticed, also, that the second test of any sample invariably 
affords a higher reading. This is probably due to some 
effect attending the first boiling, whereby a more perfect 
solution is obtained. Second tests are on a slightly more 
dilute solution than first tests, owing to the addition of dis- 
tilled water in the process of displacing the emanation and 
gases into the receiver. In the cases of g (2) and h (2) the 
sea-water, although sufficiently acidified, did not read into 
the second decimal place, although nearly approaching it. 
It is to be observed about these two results that the samples 
of sea-water bad stood for nearly two months in the bottles in 
which they were received, and it is possible that radioactive 
matter may have settled out or become adherent to the vessels. 

Taking the means of those determinations where there had 
been sufficient acidification, the results stand as follows :— 


jf. Taken Nov.- 4th, 1907; lat. 32° 58’ N:; long: seo Ne 
(About 70 miles N. of Madeira and 430 from coast of 
Morocco.) 
Radium per c.c. 0°0169 xX 10-” gram. . 
g. Taken Nov. 5th; lat. 84° 50’ N., long. Vos eee 
(About 530 miles west of Strait of Gibraltar and 240 
miles from Madeira.) 
Radium 0:0147 x 10—” gram per c.e. 
h. Taken Nov. 6th; lat. 37°:34'\ N., lono. 13° bee 
(About 300 miles west of Cape St. Vincent.) 
Radium 0:0115 x 10—” gram per c.c. 
2. Taken. Nov. 7th; lat..40° 397 N.. Jono 12 ae ee 
(About 180 miles west of coast of Portugal.) 
Radium 0:0282 x 10—! gram per c.c. 


— iy 


Radioactivity of Sea- Water. 391 


j. Taken Nov. 8th; lat. 44° 03’ N., long. 16° 21’ W. 
(About 80 miles north-west of nearest land, at entry of 
Bay of Biscay.) 

Radium 0:0146 x 10~* gram per e.c. (no acid used). 


The mean of these results is 0°0172 x 10— and the mean 
of the maximum results on the five sea-waters is 0°0217 x 10—®. 

More recently still a sample of sea-water reached me from 
the Arabian Sea. This was collected by my former pupil 
Mr. R. Friel, of the Indian Civil Service. The bottle used 
was supplied by me. I concentrated this sea-water imme- 
diately on its receipt from its original bulk of 1550 c.es. to 
1000 c.cs., acidifying it with 50 c.es. of re-distilled HCl, part 
of which was used in washing out the bottle in which the 
sea-water travelled. The entire operation was effected in 
the Botanical Laboratory. The result was :— 


k. Sea-water from Arabian Sea, taken Nov. 25th, 1907 ; 
lat. 10° 40’ N., long. 58° 0’ E. (About 400 miles 
west of Cape Guardafui and 500 miles south of the 
coast of Arabia) :— 


Radium per c.c. 0°0243 x 10—-” first experiment. 
Or03T4 > 3. second ms 
FOZ TS” 53 mean result. 


Selecting the result c from the first experiments as derived 
from a true oceanic water, and accepting the general mean 
of the five samples of water taken between Madeira and the 
Bay of Biscay, and finally the mean result on the water from 
the Arabian Sea, we get a general mean of 0:0255x10-¥. 
Although it is my intention to add to these determinations, and 
more especially to examine water brought up from ocean depths, 
it seems allowable to conclude that results so sustained on 
waters derived from points so far separated, must represent 
approximately the radium content of oceanic water. Any 
other assumption seems inconsistent with the fairly uniform 
composition of sea-water, as regards other constituents, over 
the world, where special conditions do not intervene to 
dilute or concentrate it. Hve’s result points, indeed, to 
possible deficiency of radioactive matter in central ocean. 
My own to some increase near the land. These indications 
call for further investigation. 

As to the validity of the determinations the general agree- 
ment among results arrived at, some by concentration, some 
by direct bottiing of the water as it arrived, and again 
between samples evaporated in the Geological Laboratory 
and those evaporated in a building only just opened, leaves, 


392 The Radioactivity of Sea- Water. 


I think, no room to suppose that an unsuspected source of 
radium contamination can exist. This being so, the principal 
question arising is whether such measurements truly _re- 
present the whole of the radium present. It is indeed 
remarkable that the quantities of emanation dealt with are 
by any process capable of extraction from the large volumes 
of highly saline liquid involvec. Thus Ramsay and Soddy* 
estimate that the volume of emanation in equilibrium with one 
gram of radium is but one cubic millimetre; a number closely” 
agreeing with Rutherford’s calculated value f. It follows 
that in about a litre of sea-water there is a maximum of 
about twenty billionths of a cubic millimetre. That such a 
minute quantity can be extracted with such an approximate 
agreement among observations is remarkable. On this point 
il ‘thought it of interest to see if I could extract with accur acy 
a known small quantity of emanation generated in the sea- 
water. I accordingly introduced into one of the flasks con- 
taining 1400 c.es. of bea ater, after the determinations were 
concluded, one milligram of uraninite in solution. A few 
hours afterwards the sea-» ater was boiled in the usual manner 
and the emanation transferred to the electroscope. The yield 
was almost exactly correct ; the small discrepancy (on the 
side of excess) being no more than a small experimental 
error would account for. This experiment, indeed, does 
not prove that after prolonged ebullition there may not 
remain over a residuum of emanation in sea-water, but it 1s 
an assurance as to the reality and meaning of the positive 
values observed. 

It appears to me that the above results go some way 
towards explaining the difficulty which Eve experienced in 
accounting for the amount of ionization observed over the 
ocean (loc. cit.). It must be remembered that over a fluid 
medium emanation may be derived from considerable distances 
below the surface, not only by convection currents bringing 
fresh portions of the medium to the surface, but by the extrac- 
tion of emanation whenever air becomes mixed up with the 
water. The process, in the latter case, being in fact the same 
as that frequently used in extracting emanation from liquids. 
It seems as if the atmosphere over the ocean might draw upon 
considerable depths of water for supplies of emanation, 
whereas over the land the emanating part of the radium 
must be practically only those atoms at or very near the 
surface of exposed solids. 

Oceanic radioactivity is, most probably, in part referable 

* Proc. Roy. Soe. Ixxiii. no. 494, p. 546. 
+ ‘Nature,’ Aug. 20, 1903. 


Notices respecting New Books. . 393 


to the denudation of rocks and the transport of the dissolved 
and suspended radioactive materials to the ocean. The con- 
siderable amounts of radium contained in deep-sea sediments, 
even from the most central parts of the ocean, suggests that 
much of the uranium reaches the sea in solution or in very 
fine suspension, and in part ultimately finds a resting-place 
among the oozes everywhere slowly collecting. On this 
point it hope shortly to complete my determinations. If this 
is so, 1t 1s improbable that we will ever be in a position to 
form an equation between the river supply of radium and 
the oceanic Cee ee 


XXXVIIL. Notices respecting New Books. 


Lehrbuch der Gterichtlichen Chemie. Bearbeitet von Dr. Grore 
Baumert, Dr. M. Denngrept, und Dr. F. VoreTLanpEr. Erster 
Band. Braunschweig: F. Vieweg u. Sohn, 1907. Pp. xvi 
+ 490. 

BRIEF notice of the second volume of the new edition of this 
authoritative work of reference appeared in these columns a 

few months ago, and the first volume, now before us, completes the 
work. The nature and classification of poisons are considered in 

a brief introduction. The ve is divided into two parts, a 

“general” and a “special” one. In the “general” part, an 


exposition is given of the fecal aspect of the subject, and some 


generalities connected with toxicological analyses are dealt with ; 

the investigation of food-stuffs, utensils, toys, &c. for the detection 
of poisons is next considered, and finally useful and complete 
directions are given for the testing and purification of the more 
important reagents and apparatus used in analyses of this kind. 
The * special Z part, which fills by far the greater part of the 
volume, deals in detail with the detection of the various poisons. 
It is divided into three sections. In Section I. are considered 
inorganic poisons, which are divided into metallic poisons, poisons 
belonging to the group of the alkaline earths and alkalies, those 
belonging to the group of acids and the halogens, and, lastly, 
poisonous gases. Arsenic receives exceptionally full treatment. 
The section on organic poisons deals with cyanides, carbolic acid, 
chloroform, alcohol, &c., alkaloids, pfomaines, organic acids and dye- 
stuffs. Section ILI., which is very brief, deals with the conduct of 
analyses in general. An Appendix contains a brief digest of the 
various laws relating to poisons. Notwithstanding the fact that 
the work is primarily addressed to chemists, and that it deals with 
a subject surrounded with great difficulties, the style is so simple 
and attractive that a member of the legal profession should have 
no difficulty in reading it, and in deriving from it information which 
would prove of great assistance to him in the investigation of 
criminal cases. 


a94 Notices respecting New Books. 


The Polarity of Matter. By ALEX. CuarK, M.A. 
London: Gall & Inglis. 1907. Pp. viii+134. 


“THE atmospheric air is a magnetisable substance.” ....“ When 
iron filings are placed on a sheet of paper over the poles of a 
magnet, each fragment forms a small portion of an elliptic curve. 
These curves do not represent lines of force, as sometimes sup- 
posed, but merely the posture in which the particles are carried 
towards the magnet.” .... ‘‘ Electricity, therefore, is merely a 
modified form of the force inherent in the particles of matter, and 
is the same force which in other circumstances we call gravita- 
tion.” .... “ Having arrived at the parting of the ways, Hertz 
unfortunately took the wrong turn. Instead of accepting the 
conclusion that they (Hertzian waves) are heat waves, and 
endeavouring to account for their magnetising effect, he assumed 
the identity of electricity and light, which is impossible.” Thus 
our author; and we are left wondering how this book ever came to 


be published. 


Guide de Préparations Organiques a PUsage des Etudiants. Par 
E. Fiscuer. Traduction autorisée d’aprés la septieme édition 
allemande par H. Decker et G. Dunant. Paris: Gauthier- 
Villars. 1907. Pp. xvin+110. 


Tuts small Jaboratory guide should prove of great assistance to all 
students entering on a course of laboratory work in organic 
chemistry. In the selection of the preparations, the author was 
guided partly by the price of the necessary materials and appa- 
ratus, partly by the time required to carry out the various operations 
described, and partly by considerations of safety. The author 
wisely devotes the introductory portion of the work to the precau- 
tions to be observed by the experimenter in order that accidents 
may be avoided. The book is divided into two parts, Part I. being 
devoted to preparations of importance to professional chemists, 
and Part II. to compounds which are of special interest to the 
biologist or medical man. An index of the various compounds 
dealt with completes this exceilent little work. 


Die Binokularen Instrumente. Nach Quellen bearbeitet von Moritz 
von Rone. Berlin: Julius Springer, 1907. Pp. vii+224. 


Tus monograph deals in a thorough manner with the theory and 
history of binocular instruments. Its preparation must have 
involved a great deal of labour and patient study, and the author 
is to be congratulated on having rendered a very valuable service 
to manufacturers of binocular instruments and others interested 
in the subject, by bringing together in a comparatively small 
volume so much useful information, and more especially by placing 
before his readers so thorough a survey of the history of the 
subject. Some idea of the amount of historical research 
embodied in this volume may be gathered from the bibliography of 


Notices respecting New Books. 395 


the subject given at the end of the book, which extends over 
nearly 20 pages and contains hundreds of references. A thorough 
knowledge of the history of a subject such as that aealt with in 
the present volume is essential to all would-be discoverers and 
inventors, as it prevents waste of time and—what is perhaps more 
trying—bitter disappointment at finding that an apparently new 
invention was well known to an older generation. We can give 
no higher praise to the volume than by saying that it is fully 
worthy of its author’s reputation, and that the arduous task which 
he has attempted has been accomplished with all the thoroughness 
which might have been expected of him. 


Annuaire pour [An 1908. Publié par le Bureau des Longitudes. 
Avec des Notices scientifiques. Paris: Gauthier-Villars. 

THis year’s issue of the Annuaire contains various physical and 

chemical tables, and the following special articles :—La distance - 

des astres et en particulier des étoiles fixes, by M. G. Bigourdan ; 

and L’Heole d’ Astronomie pratique de U Observatoire de Montsouris, 


by M. E. Guyon. 


The Science Year Book and Diary for 1908, Edited by Major 
B. F. S. BapEn-PowWELL. 


Tse fourth issue of this year-book differs but little from its 
predecessor. The diary part of the book is as before of paper of 
excellent quality. The ‘ Biographies” section is still very 
imperfect. We note that Professor Rutherford’s address is given 
as ** McGill University, Montreal.” 


Lecons sur la Viscosité des Liquides et des Gaz. Par Marcen 
Britiovury, Professeur au Collége de France. Seconde Partie. 
Viscosité des Gaz. Caracteres Généraux des Théories 
Moléculaires. Paris: Gauthier-Villars. 1907. Pp. 142. 


THE author commences this concluding volume of a treatise on 
viscosity by a description of the earlier researches on the viscosity 
of gases, in which the pendulum and oscillating disk methods were 
employed. The experiments of O. E. Meyer, Bessei, and Girault 
are dealt with and criticised. The next chapter is devoted to an 
account of the researches of Maxwell, Kundt and Warburg; the 
various forms of apparatus employed by these physicists are 
illustrated and described very fully. The flow of gas through a 
cylindrical tube is next dealt with, and an excellent resumé is 
given of the work of Graham, Meyer, and Warburg. Absolute 
determinations, the viscosity of gases at high temperatures, and 
the behaviour of vapours are then considered, while various 
comparatively recent determinations are described in the last 
chapter of the first Section of this volume. 

The remaining Section is devoted to molecular theories. After 
a brief historical survey of the subject, the author deals first with 
the kinetic theory of gases, and then sketches the outlines of the 


= —s_ ” ieee nee Dims of 8 ks. | 


396 Intelligence and Miscellaneous Articles. 


kinetic theory of liquids. Some general considerations based on 
the available experimental data conclude this Section. ie 

The work is one which will prove of very great interest to all 
students of molecular physics, and its value is greatly enhanced by 
_the copious references to original memoirs. 


Bulletin of the Bureau of Standards. Vol. 3, Nos. 1-4. 
Washington : Government Printing Office. 1907. Pp. 728. 


A MERE cursory glance through these volumes is sufficient to es- 
tablish the great interest and importance of the investigations now 
being carried out by the Washington Bureau of Standards. Here 
we have papers on such subjects as the calculation of mutual in- — 
ductances in various cases of practical importance, a comparison 
of the units of luminous intensity of different countries, the es- 
tablishment of the thermo-dynamic scale of temperature by means 
of the constant-pressure thermometer, the production of high- 
frequency oscillations from the electric arc, incandescent lamp 
photometry, the measurement of the capacity and power-factor of 
condensers, a determination of v, and others. No one who is 
anxious to keep abreast of modern progress in either pure or 
applied science can afford to neglect this standard publication. 


XXXIX. Intelligence and Miscellaneous Articles. 


THE EVOLUTION AND DEVOLUTION OF THE ELEMENTS. 
BY HUGH RAMAGE. 


Messrs. A. C. and A. E. JEssup, in a very suggestive paper 
in the Philosophical Magazine for January, put forward as new 
the view that the chemical elements have evelved in groups, 
that is, down the vertical columns of Mendeléeff’s table. This 
view was suggested by me in a paper on “ A comparative study 
of the spectra, densities and melting-points of some groups of 
elements and of the relation of properties to atomic mass,” 
published in the Proc. Roy. Soe. vol. Ixx. p. 1, 1901, and the 
cases of argon and tellurium were also discussed. This paper 
indicates the order of evolution as :—Li-Na-K—Rb-Cs and 
Li-Na-—Cu-Ag-—Au ; also Be-Mg—Ca-Sr-Ba and Be-Mg—Zn—Cd- 
Hg, &c. The suggestion that those sub-groups beginning with 
copper, zinc, and gallium have been indirectly evolved from 
carbon, and not directly from sodium, magnesium, and aluminium 
respectively, receives no support from the spectra of the metals. 
In fact, the evidence given by the spectra, and also by certain 
properties of the metals, is strongly opposed to it. 

The view that the formation of successive elements is attended 
by the escape of a large quantity of energy is contrary to that 
which has impressed itself upon me. The question, however, is 
too complex to be dealt with in a short note. 

Carrow Hill, Norwich, 

January 31st, 1908. 


», *o8 
\ 


’ be 


THE Ns 


LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[SIXTH SERIBS.] — 


ALP Fel E L908 


Tue following paper was written by Lord Kelvin some months 
before his death, and the subject with which it is concerned was 
occupying his attention down to the last few days of his life, in 
fact, till the commencement of his last illness. Very shortly before 
his death he wrote out in detail a paper on “ Homer Lane’s Problem,” 
which he had communicated to the Royal Society of Edinburgh in 
January 1907, under the title “The Problem of a Spherical Gaseous 
Nebula”; and it is proposed to reprint this paper, along with an 
appendix, written at Lord Kelvin’s desire by his Private Secretary, 
Mr. George Green, in the next number of the Philosophical 
Magazine. 

The present paper has been carefully corrected by Mr. Green 
and myself, but it was left in a perfectly finished state, and little 
or no editing was necessary.—J. T, Borromiey. 


> 


XL. On the Formation of Concrete Matter from Atomic ~ 
Origins. By the late Lord KEtvin*. 


§ 1. eo arae te due to gravitational attraction between 

materials given originally in small parts widely 
distributed through space, is probably the most ancient 
history of all the bodies in the universe. What the primitive 
forms or magnitudes of those pieces of matter may have been 
can never be made known to us by historical evidence. If 
they had been all globes, or irregular broken solids, of 
diameters a few kilometres, or a few thousandths of a milli- 
metre, and if among them all there existed all the atoms 


* Communicated by Dr. J. T. Bottomley, F.R.S. 
Phil. Mag. Ser. 6. Vol. 15. No. 88. April 1908. 2H 


398 Lord Kelvin on the Formation of 


which at present exist, the present condition of the universe. 
might be very much the same as itis. Towards a speculative 
answer we might be guided, and perhaps wrongly guided, by 
what we see of meteorites, stony or iron. We can hardl 
regard it as probable that those broken looking lumps of solid 
matter, with their corners and edges rounded off by melting 
in their final rush through our atmosphere before arriving at 
the Earth’s surface, were primitive forms in which matter 
either was created, or existed through all infinity of past time. 
On the contrary, it may seem to us quite probable that the 
primitive condition of matter was atomic ; ; perhaps every 
primitive particle was a separate indivisible atom ; or perhaps. 
some of the primitive particles were atoms, and some of them 
doublets such as the O., No, H,, which we know as the molecular 
constituents of gaseous oxygen, gaseous nitrogen, gaseous 
hydrogen, accor ding to modern chemical doctrine. Or perhaps 
some of the primitive particles may have been given in groups 
consisting of a moderate number of atoms ready for building 
into er ystals ; or they may have been given as very small com- 
plete crystals each consisting of a very y large number of atoms. 

§ 2. To illustrate the dynamics of the real conglomeration, 
which we believe to be an event of ancient history, consider 
an ideal case of 1083 million million million cubic metres of 
solid matter ; the sum of their volumes being equal to the 
Earth’s volume. Let the density of the material of each cube 
be equal to the Harth’s mean density, 5°67. The sum of their 
masses will be 6°14 thousand million million million metric 
tons, being equal to the Harth’s mass. Place them at rest in 
cubic or der, equally distributed through a vast spherical space, 
of radius one thousand times the Earth’s radius, and therefore 
of volume equal to a thousand million times the Harth’s 
volume. Let every one of the cubes be oriented with its. 
faces and edges parallel to the planes and lines of the cubic 
order. In this order, the lines of shortest distance between 
the centres of constituents are perpendicular to the three 
pairs of parallel faces of the cubes. The distance from centre 
to centre would be one metre, if the cubes were given in 
contact, occupying a sphere equal i in bulk to the Earth. The 
distance between the centres of nearest neighbours is therefore 
a kilometre, when they are given in their wide spread initial 
arrangement. 

N 3. Leave now the cubes all free to fall inwards in virtue 
of mutual gravitation. Hach one of those on the bounding 
surface of the whole group will commence falling towards 
the centre of the sphere, with acceleration one millionth of 
the acceleration of a body falling freely near the Harth’s 
surface: that is to say 9°8 millionths of a metre per second 


(/. tel 
, a 7 
7 i 
Jae 
. 


Concrete Matter from Atomic Origins. 399 


per second. The centreward acceleration of all the others 
will be in simple proportion to their distances from the centre 
of the assemblage. This is easily seen by remarking that, 
according to Newtonian principles, the resultant force on 
each cube is the same as if all the cubes at less distances than 
its own from the centre were condensed in a point there, and 
all the cubes at greater distances than its own were annulled. 
Hence as long as the cubes all fall freely inwards their dis- 
tribution remains uniform, and their boundary spherical. 

§ 4. Hence the cubes, if all similarly oriented as in 
§ 2, will, at one instant of time, all come into contact, each 
with all its six nearest neighbours; and for an instant they 
will be fitted together making a great globe of the same size 
as the Earth, but with angular projections at the spherical 
boundary according to their cubic arrangement. All the 
parts of this composite globe are, at the instant of first contact, 
moving inwards at speeds simply proportional to their dis- 
tances from the centre. If the cubes were perfectly elastic, 
they would rebound outwards with velocities equal to those 
they had at the instant of first contact; and a periodic 
falling inwards and rebounding outwards would follow; 
which would continue for ever, if there were no ether to resist 
this gigantic oscillatory movement. But what would be done 
by cubes of real matter, with its true molecular constitution 
and imperfect elasticity, in and after such a prodigious, purely 
pressural, collision, is not a subject for profitable conjecture. 

§ 5. Let now the cubes be placed initially at rest, with 
their centres arranged as in § 2, but with orientations given 
at random, and let them fall freely. Contacts between some 
neighbours will begin to occur when the shortest distances 


between centres are equal to 3°, or 1°732, metres, being the 
length of the body diagonal of each cube. At this instant 


the whole assemblage occupies 3°, or 5°196, times the Earth’s 
bulk: and the average density is somewhat greater than the 
density of water. 

§ 6. The velocity of the outermost cubes at the instant of 
impact, in the case of the similar orientations of § 2, is 11,175 
metres per second. If the cubes were reduced to infinitely 
small material points at their centres, and so could continue 
to fall freely to the centre of the sphere, they would all reach 
that point in °897 of a year from the time of our ideally 
assumed initial state of rest. As the material points fall 
towards the centre of the sphere, the mean density of the 
assemblage continuously increases. The following table 
shows the number of minutes during which the system must 

2H 2 


400 Lord Kelvin on the Formation of 


continue to fall to reach the centre, after it has attained the 
densities indicated :— 


Time before reaching the 


. Mean Density. 
centre *, 
47°7 minutes O-2 
Dias! a 0-5 
1 a L*O 
Oo phate 3°67 


If the beginning is an assemblage of cubes oriented at 
random, as in § 5, and with their centres in cubic order, as in 
§ 2, contacts would commence at a time about 8 minutes 
earlier than the time of coming to fit exactly on the supposi- 
tion of uniform orientation. 

§ 7. In the case of cubes initially oriented at random, 
collisions will not begin simultaneously as in § 4; but will begin 
with a crushing to powder at colliding edges or corners. 
The stupendous system of collisions which follows the com- 
mencement, would, if the material of the cubes is of any 
known substance, metallic or rocky, cause, in the course of a 
few minutes, melting of the whole mass: unless the pro- 
digious pressure in the central parts should have the effect of 
preventing fluidity in those parts, which does not seem probable. 

§ 8. The same general description is applicable to the 
ideal case of a vast number of large and small fragments of 
any shapes, instead of our equal cubic metres of homogeneous 
matter ; provided only that the initial distribution through the 
great spherical space is of uniform average density all through. 

§ 9. Let us now, instead of masses large or small of 
concrete matter, begin with a vast number of atoms; or of 
atoms, and doublets such as Oo, N», H., given at rest dis- 
tributed uniformly in respect to average density through a 
sphere of a thousand million times the Harth’s bulk: and 
having the sum of their masses equal to the Harth’s mass. 
Every particle (atom or doublet) will have the same centre- 
ward velocity at the same time, as that found for the ideal 
cubes in § 6; untilsome of the atoms or doublets get into 
touch with neighbours, that is, come so near one another that 
mutual molecular forces become effective. This must be the 
case when the mean density is considerably less than one- 
tenth of the density of water, as we see by considering the 
known properties of gases and vapours, and of liquids of 
small density. Hence the time during which the atoms 
will continue to fall freely without jostling one another must 
be a few minutes less than the time of getting into touch at 


* Calculated by means of a formula given on page 538 of Lord 
Kelyin’s “ Baltimore Lectures,” Appendix D. 


Concrete Matter from Atomic Origins. 401 


mean density somewhat greater than that of water, on the 
supposition of cubes randomly oriented at rest, as defined 
in § 5. 

§ 10. When the density becomes so great that the atoms 
begin to seriously jostle one another, we have the first step 
to the formation of concrete matter from atomic origins. 
Our present knowledge of the properties of matter does not 
suffice to allow us to follow, with definite and complete 
understanding, the progress after this step. We can see a 
prodigious cloud of atoms crowding turbulently around the 
centre. Atoms coming from all directions meet and collide. 
The energy of the relative motions of the atoms is still, let 
us suppose, sufficient to prevent them from ever remaining 
in contact except during very short times when any two of 
them are in collision. And let us suppose the central mean 
density to be still considerably iess than -1 of the density of 
water. The whole assemblage now constitutes the gaseous 
fluid mass which forms the subject of Homer Lane’s cele- 
brated problem*. As long as the whole mass remains thus 
in gaseous condition, loss of heat to the surrounding ether 
allows mutual gravitation to condense the whole assemblage, 
doing its work by increasing the kinetic energy of the relative 
motions. A result, as found by the mathematical solution of 
Homer Lane’s problem, is that all of the assemblage outside 
a certain distance from the centre sinks in temperature, 
while all within that distance rises in temperature. I may 
here explain that the temperature of a “ perfect gas” means 
the kinetic energy per unit mass of all the relative trans- 
latory motions of its molecules in free paths from collision to 
collision. 

§ 11. During the whole time of the gaseous stage which 
we have been considering, the crowding becomes denser and 
denser in the central regions, until there every atom comes 
to be always in collision with all its nearest neighbours. 
Throughout all the space around the centre in which this 
condition has been reached the crowd constitutes a fluid of 
the species called liquid. In the outlying parts the crowd 
is much less dense ; each atom, instead of being always in 
collision, is in collision only during comparatively short 
intervals of time ; and, for the rest of its time, is moving in 
approximately straight lines, not perceptibly disturbed by 

* J. Homer Lane, American Journal of Science, July 1870; A. Ritter, 
Wiedemann’s Annalen, 1878 ...1882; Prof. A. Schuster, Brit. Assoc. 
Report, 1883: Sir William Thomson, Phil. Mag. vol. xxiii. p. 287, 1887 ; 
Prof. J. Perry, ‘Nature,’ vol. lx. 1899 ; T. J. J. See, Astr. Nachr. No. 4053, 
Bd. 169, 1905; T.J. J. See, Astr. Nachr. No. 4104, Bd. 171, 1906; 
Lord Kelvin, Royal Society of Edinburgh, Jan. 21, 1907; Lord Kelvin, 
‘Nature,’ Feb. 14, 1907. 


402 Lord Kelvin on the Formation of 


other atoms far or near. The less dense crowd of atoms in 
those parts constitutes a fluid of the species called gas or 
vapour. LHvery collision between two atoms in those outlying 
parts, and in the central region every change of speed.and of 
direction of atomic motions, due to mutual forces between 
neighbouring atoms, sets ether locallyin motion. The motion 
thus produced in ether gives rise to etherial waves which 
travel outwards through the outlying parts of the assemblage ; 
and continue their outward course into void ether all around 
the assemblage. These etherial waves carry away gradually 
into infinite space the kinetic energy of the atoms, originally 
given to them by gravitation between all parts of the con- 
tracting assemblage. A first effect of this loss of energy 
would be to continue the raising of temperature in the central 
region, which in § 10 was said to take place during the whole 
of the gaseous stage of the evolution. As time advances, the 
dense gas or liquid in the central parts comes to a maximum 
of temperature. After this there is a general diminution of | 
temperature, by the conduction and radiation of heat out- 
wards: the whole mass goes on cooling, and is automatically 
kept largely stirred by irregular convection-currents, of cooled 
liquid flowing downwards from the surface, and of hotter and 
less dense liquid rising from below. 

§ 12. If, as would be the case were the liquid melted iron, 
or water, the solidified material is less dense than the liquid 
at the same temperature and pressure, a continuous crust 
would form all over the surface of the globe ; which would 
grow thicker and thicker inwards, by freezing of the interior 
hquid in contact with it, in virtue of conduction of heat 
outwards through the crust. This solid crust, completely 
enclosing a liquid, would be burst by the liquid expanding - 
as it freezes (just as a closed water-pipe is burst by water 
freezing inside it). If the spherical shell bursts into many 
fragments these would all float. The freezing of the liquid 
exposed in the openings thus produced in the crust would 
quickly fill up the gaps; and thus the process of freezing 
all around would spread inwards to the centre. Such may 
possibly be the history of the earliest solidification of part 
of the Karth’s mass, forming a metallic central nucleus by 
coalition of primeval atoms of iron, nickel, gold, platinum, 
or other dense metals. 

§ 13. But the Earth, while not improbably metallic in its 
central parts, is probably in the main of “earthy ” or rocky 
materials. It seems highly probable that, unlike the materials 
mentioned in § 12 which expand in freezing, all the “ earthy” 
materials of the Harth contract in freezing. Bischoff*, in 


* Bulletin de la Soc. Géol. 2nd series, vol. iv. p, 1312. 


Concrete Matter from Atomic Origins. 403 


experiments made about eighty years ago, found that melted 
granite, slate, and trachyte, all contract by more than ten 
per cent. in freezing ; and sixteen years ago, Carl Barus * 
found that diabase (a partially crystalline basaltic rock) is 
fourteen per cent. denser than melted diabase. He found the 
melting temperature of diabase to be about 1170°C. ; and the 
late Professor Roberts-Austen, by experiments which he 
kindly made at my request, found the melting temperatures 
of several different rocks under ordinary atmospheric pressure t 
to be as follows :-— 


Melting-point. Error. 
_! 21.3) 2 anaes aaa 1520°C. gu 
Hornblende ... about 1400° ,, 
Pe 1440° ,, + 30° 
EG el Altea Lito tae + 15° 
Lb 71S 7k ieee about 880° 


ward thermal conduction; and the amount of material in 
the thoroughly liquid condition will increase until the whole 
globe becomes liquid, except a vaporous or gaseous atmo- 
sphere of comparatively small total mass, resting on the liquid 
all around its spherical surface. The density of this liquid 
increases from surface to centre, in its earlier stages probably 
only because of the greater pressure ; but ultimately also in 
consequence of subsidence of the denser chemical ingredients, 
after the convective currents have become too slack for 
thorough mixing up of all the materials. 

§ 15. Crystalline freezing may begin at the surface, 
because of the rapid loss of heat by radiation outwards, but 
each solid crystal sinks because its density is greater than 
that of the fluid in contact with it. In sinking, it is melted 
and redissolved by the hotter fluid below. This process goes 
on until the temperature at every part of the liquid is reduced 
to that at which some of the ingredients such as quartz, felspar, 
hornblende, mica, crystallize out of the liquid, under the 
hydrostatic pressure at the depth of the portion considered. 
This formation of crystals leaves a mother liquor consisting 
of ingredients which freeze at a lower temperature. At this 
stage the portions frozen at the surface do not melt in sinking 
through the liquid, and they fall down to the centre, or to the 
central nucleus if there is one. Thus the main rigidification 

* Phil. Mag, 1893, first half-year, pp. 173-175. 


+ See Addendum to “‘ The Age of the Earth as an Abode fitted for 
Life,” Phil. Mag. 1899, first half-year, p. 89. 


404 Lord Kelvin on the Formation of 


commences in the central region, by conglomeration into a 
granite of crystals descending through a vast surrounding 
Java ocean. The solid granite thus formed extends outwards 
till it comes to the surface. 

§ 16. The views regarding the solidification of the Harth, 
described in § 15, were first published in the Proceedings of 
the Victoria Institute, for 1897, in an article on “ The Age 
of the Earth as an Abode fitted for Life,” republished in the 
Phil. Mag., Jan., 1899. From this article the following 
passage is quoted :—“ If the shoaling of the lava ocean up 
“ to the surface had taken place everywhere at the same time, 
‘the whole surface of the consistent solid would be the dead 
“level of the liquid lava all round, just before its depth 
“ became zero. On this supposition there seems no possibility 
“that our present-day continents could have risen to their 
“present heights, and that the surface of the solid in its 
“ other parts could have sunk down ito their present ocean 
“ depths.” 

§ 17. Our question is:—How can we explain why the 
Earth is not at present a mass of solid granite of approxi- 
mately spherical surface, deviating from sphericity just so 
much that, if it were covered with water, the water would 
be at the same depth in every part, when in equilibrium 
under the combined influence of gravitation and centrifugal 
force due to its diurnal rotation? A possible, but it seems 
to me an almost infinitely improbable, explanation is that 
ocean depths are scars due to collisions with outside bodies, 
and mountain heights are due to matter left on the Earth 
by such collisions. When we look at the scarred surface 
of the Moon, we cannot but feel that it would be pushing 
possibilities beyond the verge of absurdity to attribute the 
geographical features of the Harth, and the corresponding 
features of the Moon, all to blows received by them, or 
matter shot down on them, from without. 

§ 18. After solidification, as described in § 15, contraction 
by loss of heat would almost certainly produce abundance of 
vertical cracks, proceeding inwards from all parts of the 
spherical surface (on the same dynamical principle as that 
which explains the well-known “ crackling” seen on the glaze 
of many kinds of pottery). But there’seems no possibility 
that the wide-spread hollows of the Antarctic, Pacific, 
Atlantic, and Indian Oceans, and the great areas of elevation 
in the continents, Europe and Asia, Africa, America, and 
Australia, and the seven to ten kilometre heights in the Andes. 
and Himalayas, can have followed, by any natural causes, 
merely from the condition described in § 15. I have come 
to this conclusion after careful consideration of the dynamics 


Concrete Matter from Atomic Origins. 405 


of cooling and shrinkage, and possible cavitations, through 
two hundred kilometres below the surface all round the 
globe. Itseems indeed quite certain that when the Harth came 
to be almost wholly solid, it must have had in itself some great 
heterogeneousness of constitution or figure, from which its 
present geographical condition had its origin. This hetero- 
geneousness must have had z¢ts origin in some heterogeneous- 
ness of the primordial distributions of atoms: and we must 
abandon the uniform distribution which we chose in § 9 
merely as an illustration, But we have much more than 
geography to account for. We have to account for :—(1) 
the diurnal rotation of the Harth; (2) the Earth’s motion 
through space, at about thirty kilometres per second, rela- 
tively to the Sun; (8) the Sun’s motion through space 
towards a point in the constellation Hercules, first indicated 
by Sir William Herschel, and more recently estimated at 
about nineteen kilometres per second, relatively to the average 
of sufficiently well observed stars. These three deviations 
from the spherical and irrotational conditions of §§ 2...16 are, 
it seems to me, essentially connected with the explanation of 
merely geographical heterogeneousness demanded in §§ 16, 17. 

§ 19. Any system of bodies large or small, or of atoms, 
given at rest, and left subject only to mutual gravitational 
and collisional forces, fulfils throughout all time two laws :— 

Law (1). The centre of inertia of the whole system remains 
at rest. 3 

Law (2). The sum of moments of momentum™* of the 
motions of all the parts, relatively to any axis through the 
centre of inertia, is zero. 

The corresponding laws for a system set in motion in any 
manner, and left to move under the action of mutual forces 
only, are as follows :— 

Law (1’). The centre of inertia of the whole system moves 
uniformly in a straight line. 

Law (2'). The sum of moments of momentum of the motions 
of all the parts, relatively to any axis through the centre of 
inertia, parallel to any fixed line in space, is constant. 

* (1) The momentum (a name first given in the seventeenth century 
when mathematicians wrote in Latin, and retained in the nineteenth and 
twentieth centuries) of a moving particle is the product of its mass into 
its velocity. 

(2) The moment (a nineteenth century name) of momentum of a 
particle round any axis is the product of its momentum into the shortest 
distance of its line of motion from that axis, into the sine of the inclina- 
tion of its line of motion to that axis. 

(8) The moment of momentum round any axis of any number of 
moving particles is the name given to the sum of their moments of 
momentum round that axis. 


It makes no difference to this definition if any set or sets of the particles 
are rigidly connected to make a rigid body or rigid bodies. 


406 Lord Kelvin on the Formation of 


Law (3). There is a certain definite line, fixed in direction, 
through the centre of inertia of the system, such that the 
sum of moments of momentum round it is greater than that 
round any other axis through the centre of inertia ; and the 
moment of momentum round every axis perpendicular to it, 
through the centre of inertia, is zero. 

That maximum moment of momentum is called the resultant 
moment of momentum of the system. Its axis may be called 
the rotational axis of the system. 

§ 20. Consider a vast assemblage of atoms, or of small 
bodies, given at rest at any time, distributed in any manner, 
uniformly or non-uniformly, through any finite volume of 
space. According to $19, Law (1), the centre of inertia 
of the whole assemblage would remain at rest, and the total 
moment of momentum round every axis through it would 
remain zero, whatever motions the atoms receive in virtue of 
mutual gravitational attractions, and mutual repulsions in 
collision. 

§ 21. Consider now separately some part of the whole 
assemblage, which to avoid cireumlocution I shall call part 8, 
including all the primitive atoms or particles which at present 
form the Solar System, but not including any other great 
quantity of matter. Part § has, at each instant, a definite 
resultant moment of momentum round a definite axis through 
its centre of inertia ; and its centre of inertia is, at each instant, 
moving with a definite velocity in a definite direction. Ina 
vast assemblage such as we were considering in $ 20, which 
may be the whole matter of the universe (finite * in quantity 
as it may with all probability be supposed to be), let there 
be denser parts and less dense parts. In the denser parts, 
there will be gravitational coalition ; in the less dense parts, 
there will consequently be rarefaction. The present existence 
of the Sun is undoubtedly due to gravitational coalition in 
some of the denser parts. The velocity of the centre of 
inertia of the Solar System is due to the gravitational attrac- 
tions of matter outside S, so also is the moment of momentum 
of the Solar System round any axis through its centre of 
inertia. The rarefaction of the distribution of particles, 
large or small, around 8, leaves the matter belonging to 8 
more and more nearly free from force acting on it from 
without; and it becomes more and more nearly subject to 
Laws (1’) and (2’) of § 19. 

§ 22. The approximately constant momentum of the Solar 
System in its motion through space is chiefly the momentum 
ot the Sun’s motion, because his mass is much greater than 


* See Lord Kelvin’s “ Baltimore Lectures,’ Lec. XVI. § 15. 


Concrete Matter from Atomic Origins. 407 


the sum of the masses of Jupiter and all the other planets and 
satellites. On the other hand, the resultant moment of 
momentum of the motions of all parts of the Solar System, 
relatively to the fixed line of greatest moment of momentum 
through the inertial centre of all, is chiefly due to the orbital 
motion of Jupiter and Saturn, and but in small part (about 
gz of the whole) due to the Sun’s axial rotation ; as we see 
by the following table of moments of momentum, given by 
Mr. See * in his paper of 1905, “‘ Researches on the physical 
constitution of the heavenly bodies.” 
M. of m. of orbital motion, 


that of Sun's axial rotation 
being unity. 


SC ere we! eee ene 1-0000000 
Wirearyo i). os 28h ABS 0:00069654 
Picante 2.2 POF) 26 AS 0°035444 
ranma PR UN? AIR ah 0:0517385 
WOMSreP as et Lene. oe 0:00676526 
“UTES IG ae lg NA, lel ee 36°98288 
SLUT ee ee ete epee 14:98374 
(SESE ey oe nn ee Ls See 3°26959 

PN oPIREINC a opt « in seeping 4°83260 


We have not now the simple and direct gravitational 
coalition, by motions towards the centre of a spherical 
assemblage, which we hadin §$ 1 ...16,and which gave us 
Homer Lane’s beautiful problem of a spherical gaseous 
nebula. Our vast assemblage S has moment of momentum ; 
and its main condensation has led to the formation of our 
rotating Sun. Local condensations of smaller portions of §, 
each having some share of the moment of momentum of the 
whole, have produced the planets, all revolving round the 
Sun, and rotating round their axes, in the same general 
direction ; anti-clockwise, when viewed from the northern 
side of the general plane of their orbits. 

§ 23. In Kant’s and Laplace’s Nebular Theory the local 
condensations, from which have been evolved the planets 
moving in their orbits round the Sun, and the satellites 
moving in their orbits round the planets, were, according to 
the suggestion presented to us by Saturn’s rings, supposed 
to begin as rings of detached particles, which later became 
gravitationally drawn together into spheroidal groups, and 
formed ultimately liquid or solid approximately spherical 
bodies. This may probably be the true history of many of 


* Astr. Nachr. Bd. 169, Nov. 1905. 


408 Lord Kelvin on the Formation of 


the planets and satellites; but Sir George Darwin * has given 
the very important suggestion that the separation from a 
planet of material to become a satellite may in some cases 
have been a single portion of the mass, breaking away from 
what was earlier a rotating mass of liquid, in the shape of a 
figure of revolution, contracting by loss of heat, and there- 
fore rotating with greater and greater angular velocity as it 
became denser. 

§ 24. Suppose for example the mass which is now Harth and 
Moon to have been at one time a single oblate spheroid of 
revolution. Its figure would then have been exactly elliptic, 
if its rotational angular velocity, and its density, were each 
equal from surface to centre. If it was denser in the central 
regions, its figure would have been an oblate figure of revo- 
lution, but not in general exactly elliptic in its meridional 
sections. While the spheroid shrinks in cooling it becomes 
more and more oblate, till, all round the equator, gravity is 
exactly balanced by centrifugal force ; or till the spheroid 
becomes lopsided, as suggested in § 25 below. Farther 
continued shrinkage cannot give a stable oblate figure of 
revolution. It might cause an equatorial belt to be detached 
from the main body ; or the result might be as suggested in 
§§ 25, 26, below. 

§ 25. Poineairé’s “ pear-shaped ” figure of equilibrium of 
a rotating liquid suggests the idea that the first instability 
produced by cooling and shrinkage, with constant moment of 
momentum, may possibly give rise to a stable figure with a 
protrusion on one side of the centre of what was the equa- 
torial circle, and a flattening of the surface on the other side 
of the centre of inertia. This idea is to some degree sup- 
ported by the elaborate and powerful mathematical investi- 
gations of Poincaré t and Darwin { on “ pear-shaped ” figures 
of liquids rotating in stable equilibrium, under the influence 
of gravity and centrifugal force. | 

§ 26. Continued cooling and shrinkage would produce more 
and more protrusion on one side of the centre of inertia, 


* Phil. Trans. 1879, p. 447, “On the Precession of a Viscous Liquid 
aud on the remote History of the Earth ”; Phil. Trans. 1880, p. 713, “ On 
the Secular Changes in the Elements of the Orbit of a Satellite revolving 
about a tidally distorted Planet”; British Association Report, 1905, 
p. 3, Presidential Address. 

+ H. Poincaré, “Sur la Stabilité de l’Equilibre des Figures Pyri- 
formes,” Phil. Trans. 1902. 

{ Sir George Darwin, “On the Pear-shaped form of Equilibrium of 
a rotating Mass of Liquid,” Phil. Trans. 1902; “On the Stability of the 
Pear-shaped figure of Equilibrium of a rotating Mass of Liquid,” Phil. 
Trans. 1903; “On the Figure and Stability of a Liquid Satellite,” 
Phil. Trans. 1906. 


Concrete Matter from Atomic Origins. 409 


until the protrusion becomes unstable, and a comparatively 
small portion of the whole liquid breaks away from the main 
mass, at the thin end of the “ pear.” 

That separation must have been a sudden and violent 
catastrophe, however gradual may have been the changes of 
figure and distribution of matter which led to it. If at the 
time when it took place, the whole material was perfectly 
liquid, the act of separation would leave no permanent marks 
on either of the two bodies. After some moderate time of 
subsidence from the violent oscillations suddenly produced 
by the catastrophe, the Earth and Moon would have subsided 
into the comparatively tranquil conditions of rotating liquid 
spheroids, revolving round the centre of inertia of the two ; 
disturbed from hydrostatic equilibrium only by the interior 
convective currents due to cooling at the surfaces ; and with 
no prospect of ever freezing into the largely unsymmetrical 
shapes which we now see on the visible half of the Moon’s 
surface, and over the whole surface of the Earth. 

§ 27. To account for the evolution of present configurations 
and conditions, it seems to me that we must suppose the 
material of Moon and Earth, at the time of the separational 
catastrophe, to have reached some such condition as that 
described in § 15 :—a conglomeration of crystals with still 
liquid lava filling all the interstices between them. Sucha 
conglomeration would have plasticity enough to pass through 
the changes of figure which, according to Darwin’s theory, the 
material of Moon and Harth must have experienced before the 
separational catastrophe : and yet may have possessed suf- 
ficient subpermanent or permanent resistance against change 
of shape to allow them to keep permanent traces of the wounds 
left on the two bodies by the convulsive separation. 

§ 28. The scar, and subsequent surgings, left on the 
semi-plastic Harth by the tearing away of the Moon from it 
might account for persisting deviation from rotational sym- 
metry, and from equilibrium of gravity and centrifugal force, 
as great as that which is presented by the elevations of Africa, 
Asia, Europe, America, and the depths of the Atlantic, Pacific, 
and Indian Oceans. If, at the time when the Moon left the 
Earth, the material was all in the semi-solid semi-plastic con- 
dition of granite conglomerate, with a mother liquor of melted 
basalt in the interstices among the crystals, this quasi unset 
Portland cement, constituting the two bodies, might well in 
the course of fifty or one hundred million years become as 
nearly solid as we know both the Harth and Moon to be at 
present. It must however be quite understood that the 
present features of the Earth, with mountains and ravines, 


410 Lord Kelvin on the Formation of 


and ocean depths, have been produced by long continued 
geological actions of upheavals and erosion. 

§ 29. Immediately after the separation, the Moon, about 
gy of the Harth’s mass, would begin moving from the 
perigee of a somewhat approximately elliptic orbit round 
the centre of inertia of Earth and Moon, much disturbed on 
account of the great and violently changing deviations from 
sphericity of the two masses. The period of this orbital 
motion of the two bodies round their centre of inertia would 
be longer than the rotational period of the Earth, which 
would be but little changed by the catastrophe. In becoming 
rounded into a spheroidal form, the Moon would come to 
rotate round its own axis in a somewhat shorter period than 
that of the whole mass before the separation. Thus, in the 
beginning of the new regime, we have three different 
periods ; the shortest being the rotational period of the Moon 
round an axis through her centre of inertia ; somewhat longer 
than this the rotational period of the Earth ; and considerably 
longer than it, the orbital period of the two bodies round their 
centre of inertia. 

§ 30. The changes of shape of the two semiplastic spheroids 
in their subsequent motions under the influence of mutual 
gravitation between all their parts, would give rise to a loss 
of energy: while the total moment of momentum would 
remain unchanged. The main action would be loss of kinetic 
energy of the Harth and Moon, by transformation into heat 
of quasi-tidal work within the two bodies. ssentially con- 
comitant features would be augmentation of the distance 
between them, involving work done against mutual gravity, 
and gradual transformation of the moment of momentum of 
their rotations into augmentation of the moment of momentum 
of their orbital motions round the centre of inertia of the 
two. The energy of the initial rotation of the Moon would 
be small compared with that of the Harth. The whole kinetic 
energy of the rotations, and the motions of centres of inertia, 
of the two bodies, at the present time exceeds by a relativel 
small quantity the present kinetic energy of the Earth’s 
rotation. The work done in separating the Moon to its 
present distance from the Earth, and in giving it the kinetic 
energy of its orbital motion, has been wholly drawn from the 
Karth’s rotational kinetic energy at the time of the disruption, 
with the exception of a small contribution derived from the 
Moon’s initial kinetic energy of rotation. A comparatively 
early result of the motions of the two bodies must have been 
to bring the Moon to keep always the same face to the Harth 
as she does at the present time. 


—_ > 


Conerete Matter from Atomic Origins. 411 


§ 31. Sir George Darwin, with comprehensively pene- 
trating dynamical insight, has traced the course of events 
following the stage reached in §¢ 29. He has given a rea- 
sonable account of the evolution of the present eccentricity 
of the Moon’s orbit; and he has made the remarkable and 
important discovery that the axis of the Harth’s rotation 
could not remain as it is at the stage of § 29, perpendicular 
to the plane of the orbital motion of Harth and Moon. 
We might readily enough work out the general character of 
the motions and transformations that would follow the stage 
of § 29, if the Harth’s and Moon’s rotational axes did in 
reality remain perpendicular to the plane of their orbital 
motions. But Darwin finds that this possible and easily 
understood association of motions would be unstable; and 
that the slightest deviation from exact perpendicularity of 
the Harth’s axis to the orbital plane would become, not 
diminished but, augmented by the Harth’s viscous resistance 
against change of shape. With this hint it is almost as easy 
for us to see, by dynamical reasoning, that the Earth’s axis 
must, through millions of years, have become more and more 
oblique to the orbital plane, as it is, for us, with Archibald 
Smith’s hint*, to see that a “teetotum” or a boy’s spinning- 
top, having a well rounded bearing point, and set to spin at a 
sufficiently high speed round an axis oblique to the vertical, 
and dropped on a hard horizontal plane, will in a short time, 
perhaps less than a minute, be found spinning round a fixed 
exactly vertical axis (“sleeping”), and will go on so for 
ever, if the materials of the top and plane are perfectly hard, 
and if there is no resistance of the air. 

§ 32. Darwin’s theory of the birth of the Moon might 
seem improbable ; might seem even an extravagant attempt in 
evolutionary philosophy, insufficiently founded on knowledge. 
In reality it is rendered highly probable, it is indeed forced 
upon us, by tracing backwards to earlier and earlier times 
the dynamical antecedents of the present conditions of Earth, 
Moon, and Sun. 

§ 33. A hundred years before the doctrine of energy 
thoroughly entered the minds of mathematicians and natu- 
ralists, Kant made known the truth that the Harth’s rotational 
velocity is diminished by tidal friction. When we consider 
the dynamics on which this statement is founded, we see 
that it implies reactive forces gravitationally exerted on the 
Sun and Moon by terrestrial waters. Ignoring the Sun, as 
less influential than the Moon in respect to the tides, we 


* “Note on the Theory of the Spinning Top,” Camb. Math. Journal, 
1839, vol.i.p. 42. 


412 Formation of Concrete Matter from Atomic Origins. 


“see that the mutual action between the Moon and the Earth 
‘must tend, in virtue of the tides, to diminish the rapidity 
“of the Earth’s rotation, and increase the moment of the 
“* Moon’s motion round the Harth.”*.... “ The tidal defor- 
““mation of the water exercises the same influence on the 
““Moon as if she were attracted not precisely in the line 
“towards the Earth’s centre, but in a line slanting very 
“slightly, relatively to her motion, in the direction forwards. 
“The Moon, then, continually experiences a force forward in 
“her orbit by reaction from the waters of the sea. Now, it 
““might be supposed for a moment that a force acting 
“forwards would quicken the Moon’s motion; but, on the 
“‘contrary, the action of that force is to retard her motion. 
“Tt is a curious fact easily explained, that a force continually 
‘acting forward with the Moon’s motion will tend, in the 
“Jong run, to make the Moon’s motion slower, and increase 
“her distance from the Earth.” * 

§ 34. Thus we see that in the present regime the Moon is 
getting farther and farther from the Earth, and the Harth’s 
rotational velocity is becoming less and less; the sum of 
moments of momentum being thus kept constant. Hence in 
more and more ancient times, the Moon must have been 
nearer and nearer to the Earth, and the Earth’s rotational 
velocity must have been greater and greater. Trace then 
the course of motions backwards for a sufficient number of 
millions of years, and we find the Earth much hotter than at 
present, and in the semi-solid semi-plastic condition described 
in § 28 above. The distance of the Moon from the Earth must 
then as now have been increasing, and the Earth’s rotational 
velocity then as now diminishing. But these two changes 
must have been much more rapid then than now, because of the 
viscosity of the semi-solid material of the Earth, and because 
of the Moon’s shorter distance from the Earth and therefore 
greater gravitational influence on the Earth, then than now. 
Sir George Darwin had perfect right to trace the regime 
backwards, until there was contact and continuity between 
the Earthand Moon. The continuity of the whole mass must 
have come to an end with the sudden and violent catastrophe 
described in § 28 above, to which Sir George Darwin boldly 
went back with sober truthfulness. It is conceivable that 
meteorites large or small may have at various times pro- 
duced disturbances ; but I cannot see any probability for any 


* Quoted from Sections 7 and 14 respectively of an address by 
Sir William Thomson, to the Geological Society of Glasgow, “ On Geo- 
logical Time,” Feb. 27, 1868; republished in Lord Kelvin’s ‘ Popular 
Lectures and Addresses,’ vol. ii., see pp. 21 and 33, 


oa ati 
: i 
i] 


The Theory of Surface Forces. 413, 


other history of Earth and Moon, differing materially from 
that which Darwin has given us. 

§ 35. Returning now to § 1, an unanswerable question 
occurs’:—Were the primordial atoms relatively at rest in the 
most ancient time, or were they moving with velocities, 
relative to fixed axes through the centre of inertia of the 
whole, sufficiently great to give any considerable contribution 
to the present kinetic energy of the universe? It is con- 
ceivable that all the atoms were relatively at rest in the most 
ancient time, and that “the potential energy of gravitation 
may be in reality the ultimate created antecedent of all the 
motion, heat, and light at present in the universe’’*. 


XULI. On the Theory of Surface Forces—Il1l. The Physical } v 
Meaning of the Onstable Part of the Isotherm of James / 
Thomson. By G. BaKKer f. 


§ 1. The Equations of Lord Kelvin jor the Pressure of the 
Vapour in the neighbourhood of a Curved Capillary 
Layer. 


[* this periodical {| I have demonstrated, that in a plane 

capillary layer of a liquid, which is in contact only with 
its vapour, the hydrostatic pressure (p,) perpendicular to its 
surface is equal to the vapour-pressure, but that on the con- 
trary the hydrostatic pressure (p2) parallel to the surface of 
the capillary layer has a gradient in the direction of the 
normal to the surface. The relation between the pressure pe 


: 
) 


and £ where p denotes the density in a point of the capillary 
p 


layer, was given by the curve HU WVK of fig. 1 (p. 414) ; the 
curve HRGFPK presents the theoretical isotherm of James 
Thomson. I will now extend these considerations to a 
capillary. layer which has the form of a spherical shell. 
Therefore I consider firstly a spherical bulb of vapour in 
eqailibrium in the interior of a liquid. The capillary layer 
round about the vapour is in this case a spherical shell, and 


* Quoted from “On Mechanical Antecedents of Motion, Heat, and 
Light,” Brit. Assoc. Rep., Part IL., 1854; Edin. New Phil. Journal, vol.i., 
1855; Comptes Rendus, vol. xl., 1855; republished in Su William 
Thomson’s ‘ Math. and Phys. Papers,’ vol. ii. p. 34. 

+ Communicated by the Author. 

t “On the Theory of Surface Forces—II.”: Phil. Mag. Oct. 1907, 
pp. 515 & 526. 


Phil. Mag. 8. 6. Vol. 15. No. 88. April 1908. 2F 


414 Dr. G. Bakker on the 


the surface of the capillary layer is therefore concave to the 
side of the vapour. Because the forces of attraction between 


Fie. 1. 


‘ 
' 
‘ 
See? 
L?% 
¢ 
‘ 
’ 
4 
4 
a 
é 
s 


 HRGFPK-theor fsoth. 

HU WV K-p,-r Curve 

SurfaceN H RGM N-S.LFGUL 
SF-FW 


v-AIds 


the elements of a liquid or vapour are perceivable through 
only a very small range, the potential V of these forces at 
some internal point depends only on the elements embraced 
in the sphere of action of which it is the centre, and should 
be thus for a homogeneous phase proportionate to the 
density p. or the potential function of the forces of attrac- 


tion we have thus: 
pee 


Ce af 


which gives for the potential V the following difterential 
equation : 


> 
L 


ON CaO ae . 


For our case, and in many others, it is necessary to intro- 
duce instead of the coordinates z, y, z more general coordi- 
nates and to transform the left of the equation (1) into a 
more general expression. For this purpose, we divide the 
whole space into elements of volume by a set of orthogonal 
surfaces. Such a system of surfaces cut one another, as 
every one knows, according to the theorem of Dupin, into 
lines of curvature. The differentials of these curves we shall 


* “On the Theory of Surface Forces—I.”: Phil Mag. Dec, 1906, 
p- 558. 


Theory of Surface Forces. 415 


denote by du, dv, anddn. For the potential energy of the 
agent, which presents mathematically the fluid, we have 


found : 


i Gg 
W=— sep ||) Bea dy dz— ral Widadydz*,. (2) 


where R denotes the force acting on unit of mass. 
This equation can be written : 


ie — sa MN {(2) al Gol ‘d S ) yan ayer 
- All) V2du dv dn. 


The variation of W gives: 


allt OV s 3° or or ONO dudvdn 


an ov =n 
ey VON OATC IS Gta) one oe By te (3) 


We put: dudv=d8,, dudn=d8,, dvdn=db; ; 
ae 


{st oY a8, aie Sag ON aes =|\2*as, SV 
Ou 
res ic Bets) ) 
Tet lu. 


The integral refers to the whole space, and the surface- 
integral, being therefore null, contributes nothing to the 


value of SW. 
By the development of all the terms of the first volume- 


integral of (3) we me : 
(jor ales ae (55) 2(5,4) (5. “%)} 
= oa WGN RAC ICOMe ee te en ie Cameras C2) 


We will now consider the “parallelepiped curvilinear ”’ 
constructed upon the differentials du, dv, and dn, which meet 
* “On the Theory of Surface Forces—I.”’: Phil. Mag. Dec. 1906, 


p. 569. 
2 2 


A16 Dr. G. Bakker on the 


in the same point. The base of this parallelepiped is : 
dS,=dudv. lf we denote the element of the surface opposite 
to dS; by d§8,' we have, neglecting the infinitely small of 
higher order, 


ote. _ d8//—d8, 
an (d8,) = ange 
We find therefore 
V V 
(50 3 5 av 
ae oe Te —— dS,\dn+ aA (dS8,/—d8). . (3) 


du and dv are the elements of the curves of curvature in 
the considered point of the surface 8,. Therefore 


a dn dn = i iz 1 AS 
pune ie (1+ os raat (x Zs Rana dn. (6) 


If the surface-elements, denoted by d§,, be specially the 
elements of the equipotential surfaces and if dn denotes the 
differentials of the lines of force, we have 


| Ou * Bel oa ee Fhe pane 
and the equation (4) at with the aid of ©) and (6) : 


avs oe ae is a 
where dt denotes C element of volume. 


For the potential energy of the forces of attraction we have 
also the general expression: 


W =4\Vpdr. 
Varying only the density, we have 


JV8p dr={poVar, 


SW =3)Vepdr+t{p8V dr=(pdVdr.. . . (8) 
The equalization of (7) and (8) gives: 
OV 


Oo 
fo) (on Law P 
ea (7 ne PV hea ee 


and thus 


Theory of Surface Forces. 417 


For a homogeneous agent we have: 


joe 
on Ca 


The differential equation passes therefore for a homogeneous 
agent into the equation : 

V+477r’p =(). 

According to the theory of Gauss and van der Waals the 
potential energy of the forces of attraction must be in this 
case 

V=-—2ap, 
where a denotes the coefficient cf the expression for the so- 
called molecular pressure of Laplace. Hence 


2a f N= 4. 


For a capillary layer, having the form of a spherical shell, 

concave towards the vapour, we have therefore : 
eM 2A diy & 

wR I <i. ain ta Aa os 
where R denotes the radius of the sphere of equal density, 
which passes through the considered point, while dh indicates 
the differential of the normal (radius) to the surface of the 
capillary layer. This differential is reckoned positive in tue 
direction : liquid —~ vapour. 

If p denotes the pressure for a point of the theoretical 
isotherm, which corresponds to the density of the considered 
point in the capillary layer* we call u=le dp the thermo- 
dynamical potential in the considered point of the capillary 
layer. If yw, indicates the value of w in the homogeneous 
phase of the liquid we have : 


VP 2ap=py— pt. 2) 2. OS EE) 
For the homogeneous liquid- and vapour-phases, which 
limit the capillary layer, the potential V may be expressed by 
the formula of Gauss and van der Waals : 
V=-— 2ap, 
and therefore : 
Hi= Fy 
when yw; and yw, indicate resp. the value of the thermo- 
dynamical potential in the liquid and in the vapour. In 
* We say that a point of the theoretical isotherm corresponds with a 
point in the capillary iayer, when the specific volume of the first has the 


reciprocal value of the density of the latter. 
+ Phil. Mag. October 1907, p. 516, equation (4). 


418 Dr. G. Bakker on the 


fig. 2 HGPK presents the theoretical isotherm, and surface 
NHADN gives the absolute value of the integral w=\vdp. 
Fig. 2. 


Surface QSAD*+St HECS 
ABC = p,-v Cwre 


If the points A and C correspond with the points of the 
spherical surfaces, which limit the capillary layer, the 
equality of the thermodynamical potentials requires there- 
fore : 


Surface NHADN=Surface NKCQN. 


We put: NH=v,, KN=»v,, DA=v,’ and QC=v»,’, and we 
consider approximately NHAD as a trapezium. We have 
then : 
(v1. + e1')(p, — pi) = (te + 02')(P,— po). » . (12) 
In this equation v, and v, indicate respectively the specific 
volume of the liquid and vapour in contact with a plane 
capillary layer; v,/ and v,' on the contrary denote respectively 
the specific volume of the liquid and vapour in contact with 
a capillary layer, which limits a spherical bubble of vapour. 
In the same manner 7, indicates the pressure of the vapour 
in contact with a plane capillary layer, while »; and p, are 
respectively the pressures in the liquid and in the vapour, 
when these homogeneous phases are separated by a capillary 
layer, which has the form of a spherical shell. 
From the equations (10) and (11) we may deduce : 
92° WS nan 
ier: — 2 a =fy—pP, - = «' & (10 a) 


where #,=,; indicates the value of the thermodynamical 
potential in the liquid. | 
By differentiating (10 a), we obtain : 
a gt Vg se ae 
Md ie OR ee 


a i 


Theory of Surface Forces. 419 


and on substituting the value of p in (10) we find by 
integrating, 


Por Pi= | cha dh. 


T£é R denotes a value hae een the minimum and maximum 
values of the radii of the spheres, which limit the capillary 
layer, we may write : 


2 dV \2 
Sees Be =f (ar ) Solin Yor yb) 


Now we have for the cohesion respectively in the direction 
of the lines of force and perpendicular to this direction* : 


A Gebyt, (haves Ve 
sls = te a } 


| Lea Ee 
and = Pear) +5}. 


When further p,; and p, denote respectively the hydr ostatic 
pressures in the same directions we haye also f : 


Se 


—S8)=pi-—p2- 


aie a ) 
Pi Pa aaah J 
The total departure from the law of Pascal being the 
surface-tension H of Laplace, we have thus : 


B= aap |, (ae) = Ba), (Ge) a 


and equation (13) becomes 


Hence: 


2H 
ROMP S 92: a a ee (13a) 


By the aid of (12) we can write therefore the equations of 
Lord Kelvin in the form : 


V+ 01' PEL 
Vat vy’ — (vj +7)” BR | 


Po=Pi— 
and 


(14) 


V2 + Vo! yA a e 
05-4 0s — (0) Bad 
* Phil. Mag. Dec. 1906, p. 564. 


+ Phil. Mag. Dec. 1906, p.564. The hydrostatic pressure p, is therefore 
in the direction of the radius of the capillary layer. 


Pitti 


420) Dr. G. Bakker on the 


§ 2. The curves, which present the relations between the hydro- 
static pressure p, tn the direction of the normal to the 
surface of the capillary layer and the reciprocal value 


v= —of the density in the considered point. 


If @ denotes the thermic pressure* in a point of the 
capillary layer, we may consider the hydrostatic pressure in 
every direction as the difference between @ and the cohesion. 
In the direction normal to the surface of the capillary layer, 
we have therefore : 


pi=O—-S, 
and (see above) 
a ih ie ; 
17 Aa iyla Van ) 


Hence 
V2 r2 sdV \2 
ae ra 
Differentiating in the direction of the normal h, we find 
thus : 
im _d0_VaV x av ev 
dh ~~ dh -2a.dh.” Qadh. dh. 
Further : d@=— pdV 


and | 
aN BAF ON 
72 —V—2ap= Rah {see above, equation (10) J. 

Hence | 
Pn (TY =D) -h LB 
dh ~~ a \dh )-R =i dh, } Uae Gs 

Now we have found above for the departure from the law 
of Pascal . 


*  aeeel oe ) 
PLE ioe Ant arn }* 
Hquation (15) gives therefore : 
dpi _ 2(p1— Po) ? 
reignite eet. (16) 
We have thus the following theorem : 


The gradient of the hydrostatic pressure p, in the direction 
of the normal to the surface ina point of a capillary layer, 
which has the form of a spherical shell, is the product of the 
departure from the law of Pascal and the curvature. 


* The thermic pressure is the power of repulsion in the theory of 
Young. 


Theory of Surface-Forces. 421 


This theorem may also be demonstrated without the aid of 
a special potential function in the following manner :— 

We divide the capillary layer (that may be limited by a 
arbitrary surface) into volume-elements by a system of 
orthogonal surfaces, of which one is presented by a surface 


of equal density. Every element, such as ABCD A'B'C'D! 


Fig. 3. 
DD 


must be in equilibrium under the action of the hydrostatic 
pressures round about this element*. Let R, and R, 
denote respectively the radii of curvature of the elements AB 
and BC of the lines of curvature ona surface of equal density, 
while 2 and @ be the angles AMB and BNC. Further, we 
indicate the hydrostatic pressures respectively in a direction 
perpendicular to the surfaces of equal densities and parallel 
to these surfaces by p, and py, while the differential AA’ of 
the normal MA is denoted by dh. The pressures on the faces 
BCC’B’ and ADD’A‘ are thus p, x surface BCC’B'’=p,8R.dh 
and the pressures on the faces ABB’A' and DCC’D': p,aRidh. 
The components of these hydrostatic pressures in the direction 


of the normal through the middle of ABCD are thus: 
poBRedh singe and poeRydhsin $8. 


For the pressures on the elements of surface ABCD and 


A'B'C’D’, we have 


paB-BRy and (p,+ Pr ath \ a(Ry+dh) . B(Ry-+dh). 


* Gravitation is not considered. 


‘7 
{ rs, 
a 


422 Dr. G. Bakker on the 


When we substitute $a for sinda and 38 for sin 48, the 
equation of equilibrium may be written : 


a@Rypdh+ aBRipodh— (r.+5 ci os dh Ja(Ry + dh) (Ry +dh) —pyaQR,Ro. 


Neglecting the infinitely small of higher order and omitting 
the common factors dh, a and £, we find : 


ay 
p2Ry+poRy =p, Rk, +p, R.— ae RR»... 


or — Pt = (p, =p) (+ = . 


For a capillary layer which has the form of a spherical 
shell, we have thus : 


Bh = —- Te ° ° e ° ° . (17) 


If we consider the positive in the direction : liquid—vapour, 
equation (17) corresponds to a spherical drop of liquid, sur- 
rounded by vapour ; for, in fig. 3, we have considered dh 
positive in a direction opposite to the direction of the radius 
of curvature. The equation (16a) may be expressed as the 
following general theorem : 

The gradient of the hydrostatic pressure py in a direction 
normal to the surfaces of equal density ina point of an arbitrary 
capillary layer is equal to the product of the departure from the 
law of Pascal and the curvature of the surface of equal 
density, that passes through the considered point. 

If the curvature is null and the capillary layer therefore 
plane, we have properly : 

ch ean 2 (pi—P») 44) 
ai Sie 
or pi1=constant= pressure of the vapour. 


Fora capillary layer, which limits a spherical bubble of 


vapour, we have already found : 
dp, _ 2(pi— po) 
dh cc UT A oe (16) 


Integrating the equation (16), we find immediately the 
theorem of Lord Kelvin: 


2 {2 2H 
-p= py) Capod= 


— aaa 
Fine. 


Theory of Surface Forces. 423 


Now we have (see above) for every point of the capillary 
layer : 


r.—m=75( 4): 


Py >pP>2, and thus always 8 SAU 

For a capillary layer, which limits a bubble of vapour, the 
hydrostatic pressure p, increases therefore continually in the 
direction : liquid—vapour. ‘The value y, of p, in the homo- 
geneous phase of the liquid is presented in fig. (4) by the 
ordinate of the point A, and the value p, in the homogeneous 
phase of the vapour by ‘the ordinate of the point C. If 
PB is the tangent at a point B of the curve ABC, we have 


PW=BW cot e=(p1— pr) : ap =iR. 


Hence 


OP 
dh 


Fig 4. 


In the same manner, as I have demonstrated for a plane 
capillary layer *, we may prove that the curve,which represents 
the potential V of the attractive forces between the volume- 


elements as a function of h, has only one point of inflexion, 
a j dpy 
ov that ah has only one maximum value. Because Th 


2 
is proportionate to nar) fsee equation (15)}, the curve, 


which presents p,; as a function of h, must therefore have 
likewise a point of inflexion and the p,—/-curve has thus 
a form as the curve ABC in fig. 4. The equation (15) 
may be written : 


Epiias 2 (| ) aigh 
dv  4nf\dh } Rav’ 
* Phil. Mag. Oct. 1907, p. 517. 


Ra i 


424 Dr. G. Bakker on the 


If therefore the gradient = 


density p(v= = has the same sign for every point of the 


of the reciprocal value of the 


capillary layer*, the curve, which presents p, as a function of 
v, has a form as the curve "ABC in fig. 2, where HGPK pre- 
sents the theoretical isotherm. 


§ 3. The curves, which represent the relations between the hydro- 
static pressure po in a direction normal to the radius and 


the reciprocal value v= — of the density in the considered 
point of the capillary layer. 


The radius of the capillary layer, which has the form of 
a spherical shell, gives the direction of the lines of force. 
The hydrostatic pressure p, is therefore in a direction normal 
to the lines of force. Jf @ denotes the thermic pressure and 
S, the cohesion in the designed direction, we have 


po=O—Ry. 


Now the cohesion S, in a direction normal to the lines of 
force is given by the formula: 


ee 1 a ae 


~ 8rft ii ) 

Hence a gc; liens iy iE 

Pa’ Bahr? 8x (Fr 
In the direction of the lines of force the cohesion §, is 

given by 

ater 

rua day L\dh )- p ede Bs 
Therefore : 


vz 1 (dV\? 
nat sim seit 


Saft | Bafldh 
Further we have: a=2fd*. (See above p. 417.) 
Hence 


+ v? ; 
oi =0- ir ee 


* In the Ann. der Phys. xvii. p. 478 (1905) I have demonstrated the 
great probability of this supposition, 

+ Phil. Mag. Dec. 1906, p. 560. 

t The hy drostatic pressures p, and p, may be also considered respec- 
tively as the maximum and minimum yalue of the pressures in the 
considered point. 


Theory of Surface Forces. 425 


We will denote half the sum of the max. and min. value 
of the hydrostatic pressure by p, and write : 


Se AS feHintiony be (18a) 


In differentiating (18a) we have: 
dp_ dO 2VdV wioey =~ 


2a 


dv dv 4adv dv 
because dd=—pdV. 
Further we have: 
V+2ap=m44—p™. 
Thus dp dO y,— 
du du 2ap ~ 


wis the thermodynamic potential of the homogeneous phase, 
which corresponds with the density of the considered point 
in the capillary layer. When the index 1 corresponds to 
the homogeneous phase of the liquid, we have 


| ial 5 \* vdp. 
Fig. 5 


N 
Q 
D 
v3 
7 
~ AEC = PitP2 ~ ~y Curve 
G HRGEPK = théor Isoth. 
= wy ABC=Pi-V Curve 
= AUWVG=P2-v Curve 
> Surf=OAGM=Surf=LEGHM 
* Surf =NHAD =Surf=NKCQ 
= HK= empirical fsoth 


a4 
tl 
tt 
i) 


V-Azs 


*, If we construct, therefore, in fig. 5 the curve AEC, which 
presents half the sum of the pressures p, and pz (p) as a 


* Phil. Mag. October 1907, p. 516, equation (4). See also above. 


i re ee 
aa @ 


4260 Dr. G. Bakker on the 
function of v= 2 so is the point Ef of the curve, where the 


ordinate has its maximum value, exactly the point of the 
theoretical isotherm for which w=, or : 


surface DAGM=surface LEGM. 
When, namely, »=,;, the equation 
V+2ap=1— pb 
gives V=—2ap, 
and the equation (18 a) becomes 
p=90—ap’. 
Now, we have for the theoretical isotherm: 
p=O—ap’. 
Hence p=p. 


§4. The capillary layer, which envelops a spherical 
drop of liquid. 


Hitherto we have considered a bubble of vapour, which 
was enveloped by liquid. Let us now consider a spherical 
drop of liquid enveloped by the vapour. The capillary layer, 
which envelops the liquid, is in this case convew on the side of 
the vapour, and the differential equation for the potential of 
the forces of cohesion is, instead of the equation (10a) 
above: 
so V eee, 


ee ae 


while the difference between p, and p; is found by the aid 
of the equation (13) in changing R into —R. Hence 


2 r»2(4/dvV\2 2H 
Oey ie R en (7) dh= ap . e ° (195 


In the same manner as above, we find that the thermodyn. 
potential has the same value in the homogeneous phase of 
the liquid as in that of the vapour. If A and C in fig. 6 
are the points of the theoretical isotherm, which correspond 
respectively with the volumes of the liquid and vapour, 
which limits the capillary layer, we must have thus: 


surface DASQ=surface SCKH. 


Theory of Surface Forces. 427 


The gradient of the hydrostatic pressure p, normal to the. 
surfaces of equal density may be deduced from the equation 
(16), when we change R into —R. Hence 


apy 2d 2 (pi— po) \* 
ti. oe (20) 


Volum -axrg 


The departure from the law of Laplace or the difference 
1 dV 2 5 Spi : dp, C 
Pi-po= lee, being positive, the gradient apa 
always negative. That the p,;—v-curve has a point of in- 
flexion may be demonstrated in the same manner as above 
in the case of a capillary layer, which envelops a spherical 
bubble of vapour. Integrating (20) we have the well-known. 
equation of Lord Kelvin: 
2H 
Pe 


The p,—v-curve may he studied in the same manner as. 
above. 


§5. Lhe physical meaning of the unstable part of the 
Isotherm of James Thomson. 


We will now firstly resume the considerations with regard 
to the p,;—v-curves in one figure. In the case that the 
vapour envelops the capillary layer, and that therefore the 
liquid is in the interior of the capillary layer, the maximum 
value of the pressure p, in the vapour corresponds with the 


* This equation is the same as the (relation (17). Above we have- 
found this equation as a particular case of the more general relation (16a). 


428 Dr. G. Bakker on the 


point P in fig. 2 or fig. 6; while in the case that the 
capillary layer is enveloped by the liquid, the minimum value 
of the pressure p, in the liquid is given by the point G in 
figs. 2 and 6. We get, therefore, for the complete set 
of p,—v-curves the curves of fig. 7. The curve Ay C, cor- 
responds to a vapour-bubble, for which the pressure 77 the 


Fig. 7. 


‘BIRISISS IYZ JOSIMT 


Volum -axts 


liquid, round about it, has a minimum value. The corre- 
sponding radius of the spherical vapour-bubble is given by 
the formula: 


Eesti © cs! 9 Seine 2 cis (21) 
Vo+ V2 — GE v') Pim Pin.’ 
where Pmin.=pz corresponds to the point A;. Because the 
total departure of the law of Pascal has not necessarily 
reached already its maximum value, I have written H’ instead ~ 
of H, H presenting the maximum value of the surface- 
tension*. In the following manner we may show that 
nevertheless H’ is of the same order of greatness as H. For 
ihe temperature T=0°844 T, I have found f that the thickness 


* When, namely, the radius of the spherical vapour-bubble has a value 
of the same order as the sphere of action, or smaller, the surface-tension 
is a function of the radius. 

+ Zeitschr. f. phys. Chem. li. pp. 858 & 361 (1905), In the Phil. Mag. 
for October 1907, p. 522, I have given for a temperature T=0°82T; : 

ald surface-tension 
vapour-pressure * 


v 


Theory of Surface Forces. 429 
h of a plane capillary layer may be expressed by 


where H is the ordinary constant vf Laplace, and p, the 
pressure of the vapour. If we accept the equation of state 
of van der Waals for the homogeneous phase, we have for 
this temperature * 

Pmin. =(), 


while the proportion between the specific volume v, of the 
vapour and that of the liquid is about 19. Hence (see 
equation (19)): 


R, =circa 


‘ ! 9! ! 
ee i 5.) 
2h 

Now the radius of the bubble of vapour must be at feast 
the average distance between two molecules, and because 
this distance has in the vapour a value of the order of the 
radius of the sphere of action, the cohesion of the vapour 
being practically null, we see that in the formula (22) the 
minimum value of Re is of the same order as h, and H’ must 
therefore be also of the same order of greatness as H, then 
h, the thickness of the capillary layer, is of the order of the 
sphere of action. 

The curve A,©, corresponds to the other limit. The 
radius of the corresponding spherical drop of liquid is given 
by the formula: 


pee 


+ ay! eA sy 
i mle Aa)’ & 
Ug+ Uo —(%j4+%, ) Pmax.— P1 


(23) 


H” denotes the departure from the law of Pascal, v, and 
¥. are respectively the abscissa of H and K in fig. 7, and 

and v,'' the abscissa of A, and P. If the radius of the 
spherical drop of liquid has a measurable value we have: 
is Meda 

Every point of the part PK (fig. 7) of the theoretical 
isotherm we have thus brought in connexion with a spherical 
drop of liquid. I£ C; is the considered point, the ordinate 
and the abscissa of this point give respectively the pressure 
p» of the vapour which REE 3 the drop and the recip- 
rocal value of the density of the vapour. The pressure p; of 
the liquid in the interior of the drop is given by the ordinate 
of A;, for which the thermodynamical potential has the same 


* J.D. van der Waals, Kontinuitit &c. p. 105 (1899). 
Phil. Mag. 8. 6. Vol. 15. No. 88. April 1908. 2G 


430 7 Dr. G. Bakker on the 


value as for the point C;. The reciprocal value of the 
density of the drop is given by the abscissa of A;. A cone 
formable consideration we may make for each pair of points 
below the part HK of the empiric isotherm. If, for instance, 
in the points A, and C, the thermodynamic potential has the 
same value, these points determine the state of a spherical 
bubble of vapour and of the liquid which surrounds the bubble. 
In the same manner we have found above, that a pair of points 
above the part HK of the empiric isotherm determine the state. 
of a spherical drop of liquid with its vapour. 

A part of the theoretical isotherm that we, however, have 
not considered hitherto is the part A,P (fig. 7). This part 
of the isotherm corresponds, as everyone knows, with the 
phases, which cannot be realized under uniform pressure, 
because the pressure would be increased with the increase of 
the volume. Therefore Maxwell says in his text-book ‘ Theory 
of Heat’: “ We cannot, therefore, expect any experimental 
evidence of the existence of this part of the curve, unless, as 
Prof. J. Thomson suggests, this state of things may exist in 
some part of the thin superficial stratum of transition from a 
liquid to its own gas, in which the phenomena of capillarity take 

lace.” 
, Now the curve AEC in fig. 5 presents the relation between: 
half the sum of the maximum and minimum value of the 


pressure | Pek =? \ in every point of the capillary layer 


and the reciprocal value of the density in this point, and, as 
I have demonstrated above, the point E on the curve AEG, 
where the ordinate has its minimum value, and where the 
thermodynamical potential has the same value as in the 
homogeneous phases of the liquid and the vapour (the cor- 
responding points are A and C) lies exactly on the theoretical 
isotherm. Therefore we have the following theorem: 

Every pair of points of the isotherm, for which the thermo- 
dynamical potential has the same value (fig. 8) (as Ag and Cy, 
A, and C,; &e.), corresponds above the rectilinear part HK of 
the empiric isotherm to a spherical drop of liquid, such, that 
the state in the interior of the drop and the state of the vapour, 
which surrounds it, is determined in a singular manner by the 
situation of this pair of points. In the same manner, every 
pair of points below the rectilinear part HK of the empiric 
isotherm (As and C3, A, and Cy, &e.), for which the thermo- 
dynamical potential has the same value, corresponds to a 
spherical bubble of vapour. If we now construct the curves, 
such as A,B,C,, AsBsC,, &e., which present the relation between 


Theory of Surface-Forces. 431 
half the sum p a a of the maximum and minimum pressure 


and the reciprocal value of the density for every point of the 


Fig. 8. 


.o) 


capillary layers, which envelops the spherical drops and the 
spherical vapour-bubbles, the minima of these curves present 
exactly the unstable part AyEB,Cg of the theoretical isotherm 
of James Thomson. 

The densities of the unstable phases of the isotherm 
are thus present in the considered capillary layers under 
a system of pressures, which are dependent on the direction, 
the maximum and minimum values of these pressures being 
respectively zm the direction of the lines of forces and 
perpendicular to the latter. The theorem above gives thus 
the physical meaning of the unstable part of the theoretical 
isotherm of James Thomson. 


2642 


FBO. 


XLII. On the Heating Effects produced by Réntgen Rays in 
Lead and Zinc. By H. A. Bumsteap, Ph.D., Professor 
of Physics, Yale University *. 


N an earlier number of this Journal f the writer described 

a series of experiments, from which it appeared that, 
when Roéntgen rays were equally absorbed in lead and in 
zinc, approximately twice as much heat was generated in the 
lead as in the zine. These experiments were carried out 
in the Cavendish Laboratory of the University of Cambridge, 
and the further prosecution of the investigation was in- 
terrupted by the writer’s return to America. An unusual 
pressure of other duties prevented the resumption of the ~ 
work until Jast summer and autumn, when a considerable 
series of observations was made with such variations of the 
conditions as might be expected to reveal certain possible 
errors in the original experiments. It soon became apparent 
that errors had been present, and that the difference in the 
quantities of heat generated in the two metals was much less 
than had appeared from the earlier experiments. The source 
of the original mistake was inherent in the apparatus used 
(a special form of radiometer), and although it could be 
diminished it was not easy to eliminate it altogether: there 
still remained an uncertainty of from 5 to 10 per cent. as to 
the equality of the heat in the two metals. I had accordingly 
planned, before publishing this correction, to attempt to 
bring the result within narrower limits, by substituting for 
the radiometer, a thermopile, with which the principal 
source of difficulty could be easily avoided. In the mean- 
while, however, a paper by H. Angerer has appeared f, in 
which a series of very careful experiments of this kind are 
recorded. Angerer’s results leave no doubt, I think, that 
the heating effects in lead and zine are equal to within a few 
per cent. ; the total effect is so small and the experimental 
difficulties are so considerable, that it does not seem practical 
at present to seek for a possible small difference within these 
limits. Certain facts in connexion with the emission of 
electrons by metals make it not improbable that there may 
be some liberation of atomic energy when ultra-violet light 
or Rontgen rays fall on a heavy metal. Such considerations 


have been advanced by Lenard §, by W. Wien ||, and by 


* Communicated by the Author. 

+ Phil. Mag. February 1906, p. 292. 
t Ann. der Phys. xxiv. p. 370 (1907). 
§ Ann. der Phys. viii. p. 169 (1902). 
|| Zéed. xviii. p. 991 (1905). 


Heat produced by Réntgen Rays in Lead and Zinc. 433 


J.J. Thomson *; and they are in a measure supported by 
the recent work of Bestelmeyer f, Cooksey t, and Innes §. 
But so little is as yet known about the mechanism of this 
emission of electrons, that it is by no means certain that 
the facts observed by these investigators necessarily involve 
the liberation of atomic energy. And in any event, the 
results of Angerer indicate that this energy, if it is set free, 
forms only a small part of the total produced by the ab- 
sorption of Réntgen rays. 

The source of my own erroneous results was found in the 
greater rate of loss of heat by the zinc, for a given tem- 
perature above its surroundings, than by the lead. In the 
original experiments small strips of the two metals (of different 
thickness so as to produce equal absorption of the rays) 
were held by an ebonite support so that each strip was 
opposite one vane of a radiometer made of thin aluminium- 
foil and suspended by a quartz-fibre. The whole was enclosed 
in a heavy metal case from which the air could be exhausted 
to the point of maximum radiometric sensitiveness. An 
aluminium window, with a movable lead screen outside, 
permitted either or both of the strips to be subjected to the 
action of Réntgen rays; and through a glass window the 
deflexions of the radiometer could be read by telescope and 
scale. The position of the strips could be reversed and the 
balance of the two vanes tested by a device which is described 
in the former paper. The repulsion of one of the vanes was 
of course primarily dependent on the temperature of the 
surface of the strip to which it was exposed ; to make the 
radiometer deflexions a measure of the quantities of heat 
developed in the two metals, it was necessary that the rate 
at which the two metals lost heat, per degree excess of 
temperature above their surroundings, should be the same. 
If then the steady state was observed when the heat lost 
was equal to the heat generated, the rise in temperature 
of either strip would be proportional to the heat developed 
init. I sought to realize this condition by covering both 
metals with thin aluminium-foil which was stuck to the 
metal by a very thin layer of wax. It was recognized that, 
if any considerable part of the total heat were lost over the 
supports to which the ends of the strips were attached, the 
zinc would be at a disadvantage in comparison with the lead 
owing to its greater conductivity and thickness. This 


* ‘Conduction of Electricity through Gases,’ p. 319. 
+ Ann. der Phys. xxii. p. 429 (1907). 
J} Am. Jour. Sci. xxiv. p. 285 (1907). 
§ Proc. R. 8. A. lxxix. p. 442 (1907). 


a ee 
a) ae y 


434 Prof, H. A. Bumstead on the Heating Effects 


possibility appeared to be excluded (as well as any sensible 
difference in the emissivity of the surfaces) by a control 
experiment in which the strips were heated by exposure to 
the light of an incandescent lamp, instead of Rontgen rays. 
The deflexions of the radiometer were almost exactly equal 
in this case, and the whole behaviour was such as to indicate 
that the conditions specified above were fulfilled; the 
question was discussed in the previous paper and the ex- 
periments with light were taken as excluding the possibility 
of the result being due to the more rapid loss of heat by the 
zinc. My recent experiments, however, force me to the 
conclusion that the zine did lose heat more rapidly (for a 
given temperature) mainly over the supports, and that, in 
the control experiments, this was accidentally compensated 
by a greater absorption of the incident light by the zine, 
possibly owing to a thinner layer of wax between it and 
its covering of aluminium-foil. The experiments with light 
were made at the very end of my stay in Cambridge and 
could not be repeated on account of lack of time: if they 
had been repeated under slightly varied conditions, the error 
would doubtless have been discovered. The agreement 
between the two metals, however, was so good, and the 
improbability of two unrelated large errors which exactly 
compensated each other was so great, that I was led to put 
more confidence in this result than it deserved. 

When the experiments were again taken up, however, 
I did consider the possibility that the emissivity of a surface 
and its absorption of light might vary together—thus 
destroying the force of the control experiment. It was not 
likely that the emissive power of a surface for low tempera- 
ture radiations would be proportional to its absorption of 
the high temperature radiation from an incandescent fila- 
ment, but I nevertheless made some experiments to test the 
matter. For this purpose two lead strips were used, one 
of which was left with its original dull surface and the other 
covered with aluminium-foil. Réntgen rays and light were 
both used: the rays gave deflexions agreeing to about 
5 per cent. ; with light, on the other hand, the dull lead strip 
gave four times the deflexion of the other. It thus appeared 
(as was expected) that the rate at which heat was lost varied 
very little with the state of the surface, while the absorption 
of light varied greatly. Thus there appeared to be no reason 
for distrusting the control experiment. 

A series of experiments was next made, by means of an 
electroscope, on the amount of secondary radiation from lead 
and zinc, in order to test the possibility that a considerable 


cl 


produced by Réntgen Rays in Lead and Zinc. 435 


fraction of the energy in the case of the zinc might escape in 
this form. It appeared from these that the total intensity of 
the secondary rays escaping from both surfaces of the zinc 
strip (as measured by the ionization produced) was less than 
1/15 of the primary rays absorbed. 

The energy measurements were again taken up, and a 
change was made in the method which would render it 
independent of the rate of loss of heat from the metals. The 
strips of lead and zine were held at the ends by massive brass 
clamps, connected to binding screws outside the case by means 
of rods insulated from the case. In this way a known current 
of electricity could be sent through either strip ; the resistances 
of the strips were measured, and thus a known quantity of 
the energy could be developed in either strip and the 
corresponding deflexion of the radiometer obtained. The 
loss of heat through the electrodes was so rapid, however 
(especially in the case of the zinc), that measurable deflexions 
could not be got with the Roéntgen rays. I accordingly 
substituted for each strip a five-barred grid, carefully cut 
from the same materials, each bar being one millimetre wide ; 
one electrode-clamp held the beginning of the first bar, 
the other the end of the fifth bar, so that the current passed 
through the five bars in series. This served the purpose 
although the deflexions produced by the rays falling on the 
zinc were still too small for very accurate measurement. 
The resistance of the zinc grid was 6°5x10—> ohm and of 
the lead 64x 10-° ohm; currents of 10 to 12 milliamperes 
were used in the lead and from 35 to 50 milliamperes in the 
zine. ‘The deflexions produced by either grid were found to 
be proportional to the quantity of heat developed in it, but 
the sensitiveness of the two grids was very different. Thus 
In one experiment which may serve as an example of many 
which were made, the energy necessary to produce a deflexion 
of 1 cm. was :—with the zinc grid, 26°8 ergs per second ; 
with the lead grid, 7°82 ergs per second. When the 
Roéntgen rays fell upon the zinc, the deflexion of the 
radiometer was 2°6 cm.; when on the lead, 10°8 em. These 
measurements give 70 ergs per second in the zinc and 84 
ergs per second in the lead; the ratio of the two is 12. 
This is a much smaller difference than was obtained in the 
Cambridge experiments; and, what is more significant, 
any errors (due, for example, to a gain of heat by both strips 
from other portions of the apparatus struck by the rays, or 


to imperfect screening of one strip) would favour the lead 


on account of its greater sensitiveness. So that the difference 
would be less than 20 per cent. rather than more. 


v Ace i. 


436 Prof. H. A. Bumstead on the Heating Effects 


A more careful consideration of the conditions of the 
original experiments showed that it was not impossible that 
the result obtained was due to escape of heat by conduction 
through the ebonite disk which supported the strips, the 
aluminium leaf which covered the disk (to prevent electrical 
effects), and the copper wires by which the strips were 
earthed. The lead strips had also between them and the disk 
pieces of cardboard to bring their front surfaces into the 
same plane with those of the thicker zinc strips. I accordingly 
repeated the experiments with the following modifications :— 
the cardboard was left out and the zinc strips sunk into 
recesses cut in the disk; the aluminium-foil was scraped 
away from the.vicinity of the ends of the strips; and the 
strips were earthed by manganin wires 0:05 mm. in diameter 
and 3 cm. long ; also the strips were covered with aluminium 
paint instead of foil to make the coefficient of absorption for 
light less uncertain. 

Two series of observations with this arrangement were 
made (one with Rontgen rays, the other with light), during 
which the balance of the radiometer vanes was tested, the 
position of the metals reversed and various corrections applied 
as detailed in the former paper. ‘The ratio of the lead effect 
to the zine effect was :— 


With Rontgen rays ........- 1:47 +04, 
Wachtel... ascence eo. Meee 1:43 +°06. 


These results show that the heating effects of Rontgen rays 
in the two metals are equal, with an uncertainty of between 
) and 10 per cent. 


Some time before the above results were obtained, two 
experiments of another kind were made with the view of 
testing certain aspects of the hypothesis that Rontgen rays 
caused atomic disintegration. The first was an attempt to 
find out whether any rays similar to a-rays existed among 
the secondary radiations given off when a heavy metal is 
exposed to Réntgen rays. An iron tube was provided with 
an aluminium window at the side, through which a beam of 
Rontgen rays could be sent ; this beam fell upon a lead plate 
at an angle of 45°. To the upper end of the tube, 3 em. 
above the centre of the lead plate, was cemented a glass 
plate, the inner side of which was coated with powdered zinc- 
blende. The tube was exhausted to 0'l mm. and the zine- 
blende screen examined by means of a lens in the ordinary 
manner, while the rays fell on the lead plate. No scintil- 
lations were seen ; the sensitiveness of the eye was tested by 
alternate observations of another similar screen placed above 


produced by Réntgen Rays in Lead and Zine. 437 


a weak radium preparation which gave a few scattered 
scintillations ; this was placed near the Rontgen tube and 
observed while the tube was excited. A large Miiller water- 
cooled tube was used (20 cm. in diameter) and a heavy 
discharge sent through it: the focus was only 28 cm. from 
the lead plate, so that the latter was exposed to very intense 
rays. The air between the lead and the screen would 
have formed at atmospheric pressure, a layer only 0:004 mm. 
thick. 

I also tried to find out whether Roéntgen rays had an 
accelerating effect upon the disintegration of a radioactive 
substance. The active deposit of thorium was used in 
preference to one of the more permanent radioactive sub- 
stances. An aluminium plate, one side of which had been 
exposed to thorium emanation for some hours, was placed over 
a hole in the wall of an electroscope with the exposed side 
inward. A number of measurements of the activity were 
made and then the plate (without being removed) was exposed 
for ten minutes to strong Roéntgen rays from the large bulb 
with its focus 20 cm. from the plate. The ionization was 
then measured again several times. The following is an 
example of the results obtained :— 

Activity before exposure ...... 7°56 +°056 
Activity after exposure ...... 754+ °076 
Repetitions of the experiment gave similar results. 


Conclusions. 


1. The result previously obtained by the writer that the 
heat generated in lead by Rontgen rays was twice that in 
zine is not confirmed by further experiments, which show 
that the quantities of heat are equal, with an uncertainty of 
from 5 to 10 per cent. The source of the error was imper- 
fect heat-insulation of the metals; this escaped the control 
experiment on account of a difference in the coefficients of 
absorption of the surfaces for light, which, by an unfortunate 
accident, was just sufficient to compensate the other 
inequality. 

2. No rays capable of producing scintillations on a zinc- 
blende screen are present among the secondary radiations 
from lead when exposed to Réntgen rays. 

3. The disintegration of the active deposit from thorium 
emauation is not hastened by exposure to Rontgen rays. 

Sloane Laboratory, 


Yale University, New Haven, Conn., 
Dec. 6, 1907. 


J é 


Pea38 4 


XLII. On the Frequencies of the Free Vibrations of Quasi- 
permanent Systems of Electrons, and on the Explanation of 
Spectrum Lines. Part I. By G. A. Scuort, B.A., B.Sc, 
University College of Wales, Aberystwyth *. 


$1. fi a previous communication ¢ I have shown that a 
ring of electrons, rotating in a field due to electrons, 
all of which exert only electromagnetic forces, has a prac- 
cally determinate radius and velocity, provided only each 
electron be expanding at a very slow rate, uniform or 
not. With the aid of expanding electrons we can build up 
a purely electromagnetic system possessing a determinate 
structure, and therefore also determinate free periods. 

Secondiy f I have examined the waves emitted by a 
rotating ring of equidistant electrons, when disturbed from 
steady motion in any way, and have shown that of all the 
free vibrations, which can be excited in such a ring even by 
violent disturbances, only a few can produce waves sufficiently 
powerful to give observable spectrum-lines. It follows that 
only a small proportion of the free vibrations of a system of 
electrons can be used to account for spectrum-lines. In all 
probability a similar limitation exists for other vibrating 
systems; but as they do not so readily lend themselves to a 
calculation of relative intensities, it has hitherto escaped 
notice. It is indeed hardly conceivable that this difficulty is 
peculiar to systems of electrons. 

§ 2. In order to account for known spectra we must study 
more complex systems of electrons; but if, for the present, 
we confine ourselves to quasi-permanent systems, that is to 
systems which can last for very many periods of vibration 
without appreciable change of structure, the problem is very 
much simplified. In fact such systems are necessarily built 
up of circular rings of equidistant electrons, relatively far 
apart, and each rotating with its own determinate velocity 
about a common axis. This may be seen as follows. : 

The electromagnetic field due to an electron which describes 
a closed orbit with period T can be expressed in the usual 
way in terms of a scalar potential ¢ and vector potential A 
by the equations 


E=—yo— “. H=curl A; 


* Communicated by the Author. 
t Schott, Phil. Mag. [6] vol. xii. p. 21. 
{ Phil. Mag. [6] vol. xi. p. 189. 


Frequencies of Free Vibrations of Electrons. 439 


Ata distance from the electron large compared with its 


radius 
Fre peek Qa} R ) 
i ye pis =e ewe <a e saage ti / 
d pee a) Rose (: P t Lat 


Fag tf Eeak iy eee ae oe 
A = > “| Ot = = )dt, 


j=-@ 0 


(1) 


where R is measured from the position of the electron at 
time t’, and u is its velocity at that time. Thus 


R=V 8 P+ P+ 


where the coordinates (v’, y', z’) of the electron are assigned 
functions of ¢’ of period T. 

The electric and magnetic forces due to a single electron 
thus consist of series of harmonic terms, no member of which 
in general is missing. ‘The energy radiated away is therefore 
very considerable, and the motion can only be permanent for 
a very small velocity. We know from Earnshaw’s Theorem 
that systems of discrete electric charges at rest are unstable: 
in general a minimum velocity is necessary for stability, but 
this is usually inconsistent with permanence. Hence we 
conclude that a system including stray electrons, each de- 
scribing its own orbit independently of the others, is not 
permanent and stable. Ultimately these electrons either 
escape from the system or combine into groups, describing 
the same orbits. 

§ 3. In order that a group may be quasi-permanent a 
number of the harmonic terms in the forces due to its several 
electrons must annul each other by interference. The con- 
dition for this is easily seen to be, that the coordinates of the 
ith electron of the group be given by equations of the form 


The potentials then reduce to series of the form 


pe ksh by. 2mns eect av eceacy 
=> 7) ROOF (e— at) at’ AR) 


where R is now measured from any one of the » electrons. 
The first n—1 periodic terms are now missing from the 
potentials ; the largest periodic term leftis that given by s=1, 
or by j=n. 

§ 4. A similar interference occurs when the field is that 


s=——D 


440 Mr. G. A. Schott on Frequencies of Free Vibrations 


due to waves of small amplitude travelling round the ring. 

When the displacements of the ith electron, measured from 

its position in the quasi-permanent motion, are proportional 
271 


- to the real part of exp.| —at+/—1, (g¢—2=@)], the 


T 
forces of the field consist of an infinite series of harmonic 
Darl Ie 
ee where s 
takes all integral values from —o« to +; of these the 
most intense are given by s=O, and to these the radiation of 
energy is almost entirely due. For the case of a circular 
orbit the rate at which energy is lost by radiation through 


terms, the frequencies of which are q+ 


the sth harmonic is relatively of order J an( =) where p is 


the radius of the orbit, > the wave-length of the harmonic, 
and m=k-+sn. For the harmonics m=0, m= +1, the order 
is about the same; for greater values of m it is much less, 
diminishing, as m increases, faster than a geometrical pro- 


2 ) bi feme : 
gression whose ratio is (ae , that is, about 1: 1,000,000 
For the general case the intensities have not been worked 
out, but there is every reason for expecting essentially the same 
result. We conclude that in general any disturbance of 
allowable amplitude (not greater than the distance between 
neighbouring electrons), for which k=0, +1, gives rise to 
appreciable radiation. 

§ 5. Let us now consider a system of rings or groups of 
electrons. As we have seen, each ring, on account of its 
permanent motion, emits waves which disturb the other rings; 
and similarly is disturbed by them. It executes forced 
vibrations and radiates energy. If the system be a solitary 
one, the whole of this energy is lost to it, and hence the 
system cannot be permanent; but if the structure is such 
that the forced vibrations of each ring correspond to values 
k other than the values 0, +1,..., the loss of energy is small, 
and the system is nearly permanent. We must bear in mind 
that for spectroscopic purposes an absolutely permanent 
system is not necessary; it is sufficient if we account for the 
degree of homogeneity and the fineness of spectrum-lines 
as we know them. Lummer and Gehrcke estimate that 
interference with one million wave-lengths difference of path 
can be realized with the red cadmium line. Further, the 
width of this line when at its finest is of the order 01 A.U., 
of which the greater part is perhaps due to Doppler effect ; 


of Quasi-permanent Systems of Electrons. 441 


let us assume a width of one millionth of a wave-length in 
round numbers. Then we may say, that if the red cadmium 
line be due to one of the free vibrations of a system of rings 
of electrons, the structure of the system, as determined by 
the velocities of its rings, can alter by as much as one 
millionth during the emission of the line, that is during 
one million periods; otherwise the line would be broader than 
is really is. Thus the greatest change of velocity consistent 
with the observed homogeneity and fineness of spectrum-lines 
is of the order 10-! in one period of the red cadmium line, 
that is in 2.10- second; in all probability it is much 
smaller. The perturbations admissible in our system of rings 
must be consistent with this upper limit, that is, the rate of 
loss o£ energy caused by them must be less than 10-¥ of the 
energy of the system in one period. 

§ 6. It is to be noted that the rate of change of structure 
just calculated is the greatest possible consistent with ob- 
servation. If we assume that it actually exists, we thereby 
make a special hypothesis as to the emission of spectrum- 
lines. For the loss of energy implies a diminution of the 
orbital velocities of the rings of electrons, and a corresponding 
diminution of the free periods of the system. When the 
system begins to radiate it emits the most refrangible part of 
the line, the light emitted becomes gradually less and less 
refrangible, and after one million vibrations or so the emission 
ceases entirely. The system cannot again emit red cadmium 
light until by some external agency its internal energy has 
been restored to its original value. Thus we are compelled 
to suppose (1) that only a fraction of all the atoms in a 
radiating gas at any instant emit any one spectrum-line ; 
(2) that the atoms begin to emit red cadmium light, and only 
begin to do so, when their internal energy reaches a perfectly 
determinate upper limit, and cease to do so when if falls to a 
certain lower limit ; while they emit no energy whatever far 
a considerable range outside these limits. There is sufficient 
evidence in favour of the first supposition to make it appear 
reasonable ; but it is so difficult to construct a mechanism in 
accordance with the second, that for the present we shall 
assume the radiation from our system to be much smaller 
than the limit just calculated. 

§ 7. The question now arises: What types of groupings 
of electrons, and what arrangements of groups are consistent 
with small radiation, implying the absence of all perturbations 
for which k=0, +1? 

Let us consider the field due toa circular ring, which is of 
course given by the general expression (2). In this case, 


im FS clea a Sh iP 


442 Mr. G. A. Schott on Frequencies of Free Vibrations 
using polar coordinates, we may write 
R=/7r? +7r?—2r7' sin 6 cos (¢’/—4), 


with 7’=constant, ¢’=ot’+6, and easily find on changing 
the variable of integration from t' to y=¢'—4@, 


cade, a Wage t R 
d aa ei. R cos ns (wt +3-g—° —x)dx 
Ja) cae » “cos (nswR/c) 
Fah — C08 ns(wt +6 $){ hi Wee mene ydy 
See) ™ 31 7 : 
+ = sin ns (wt + 5-9) seri lL cos x dx, § (3) 
s= =— 


where R=V 72 +7?—2rr’ sin 6 cos XV 


Thus ¢ occurs only in the circular functions, so long as we 
are dealing with a fixed point (7, 6, 6). To find the mecha- 
nical force on an electron we differentiate as usual with 
respect tc the coordinates (7, 6, d) and the time (é), as the 
case may be. When the electron is moving, the coordinates 
are given functions of ¢; but these values of the coordinates 
as functions of ¢ are only to be substituted after all the 
differentiations necessary in deriving the forces have been 
performed. 

§ 8. When the electron in question belongs to a second 
circular ring of 7 electrons, rotating with angular velocity o! 


about the same axis as the first, we substitute d=o't + 6'+ ae 


and thus find for the potentia!s and forces series of harmonic 
terms of the form a nS { oly 22} The pertur- 
bations produced are of the same type, which is obviously 
that of § 4 with k=0, g=ns(@—o’), k=ns ; and the emitted 
waves are given by m=ns+w7's', where s' is any positive or 
negative integer. The radiation is small provided ns+n's! 
cannot take the values 0, +1, zero values of s, s’ being 
omitted, since they imply zero frequency and no wave at all. 

Tt is at once obvious that the values 0, +1 will occur ; if 


, iE n 
n, n' be incommensurable, and — be the last convergent to —> 
n 


we need only make s=n!, s‘=—n to get m=0, and s=+Q, 
s‘'= FP to get m=+1. Thus we cannot entirely avoid the 
dangerous harmonics in question. 

We can, however, ensure that their amplitudes are small. 


of Quasi-permanent Systems of Electrons. 443 


For generally the coefficients of high harmonics in (3) are 
small provided the rings be not close together; and it is 
sufficient if either s or s’ be a large number when the rings 
are fairly near together, smaller when far apart. This con- 
dition can always be satisfied, except for special values of n, n’. 

§ 9. When the electron belongs to a ring which has not 
the same axis as the first, the coordinates r and 0, as well 
as d, involve ¢, and so also do the coefficients of the circular 
functions in (3). Expanding them in Fourier’s series we see 
that the mechanical force on the moving electron now involves 
harmonics of all integral orders, and not merely of the orders 
n, 2n,3n,.... Consequently it is no longer possible by a 
suitable choice of the values of n, n’, to ensure that the 
amplitudes are very small for the dangerous vibrations for 
which m=0, +1. Hence considerable radiation occurs, and 
the system is not permanent. <A fortiori the same thing 
occurs when the rings are not circular and are oriented in an 
arbitrary manner. 

We conclude that no system of electrons can be permanent 
unless its electrons be grouped in circular rings of equidistant 
electrons, all rotating about the same axis. 

§ 10. This conclusion can be at once extended to a system, 
which is not solitary, as we have hitherto supposed, but is 
surrounded by a large number of other systems, just as an 
atom in a radiating gas is surrounded by a very large number 
of other atoms. It is true that the energy emitted by the 
system, or atom, in this case is, at any rate in part, replaced 
by energy absorbed by it from the field due to the surrounding 
systems, or atoms, and that to an amount depending on the 
absorptive index of the gas for its own radiations. But the 
observations on the degree of homogeneity and the fineness 
of spectrum-lines, on which the conclusion of § 9 has been 
based, themselves apply to a complex of atoms, and not to a 
solitary atom. 

In the same way measurements, such as those of H. Wiede- 
mann, of the amount of energy radiated per second from a 
flame or other source, enable us to calculate the net loss of 
energy of an atom due to radiation, that is, the excess of the 
amount emitted above the amount absorbed. If the absorption 
be considerable and be neglected, the net loss is still correctly 
estimated on the average; but since in the case of large 
absorption the energy actually radiated by the flame comes 
from a thin surface layer only, the amount of energy emitted 
from a single atom, and the net loss of each surface atom, 
are both underestimated. Since in this case the observed 
spectrum-lines are due to the surface layer alone, and the 


444 Mr. G. A. Schott on Frequencies of Free Vibrations 


radiation from its atoms is underestimated, the argument of 
§ 9 is rendered all the more cogent. 

§ 11. We shall now examine the frequency equations of a 
ring of the quasi-permanent system of § 9, in order to form 
an estimate of the frequencies to be expected from a system 
whose scale is comparable with that of an atom. Estimates 
of this kind are frequently omitted on the ground that they 
are illusory, because the dimensions of an atom are not 
sufficiently well known. However, in the present state of 
knowledge this criticisin is not quite justifiable; and besides 
we merely require superior limits for our purpose. Such 
quantities as the number of molecules in 1 c.c. of gas under 
normal conditions (4.101%), the mass of the atom of hydrogen 
(10-*4 gr.), and the diameter of a molecule (at most from 
10-® to 10-7 em., probably nearer the lower limit), have been 
calculated by various methods with consistent results, and 
are sufficiently well established to make estimates of fre- 
quencies and wave-lengths of considerable use in judging the 
merits of a proposed model of an atom. We should have 
considerable hesitation in accepting as a working model a 
system whose wave-lengths were very different from those of 
light-waves. 

§ 12. Thisis the proper place to notice a fundamental differ- 
ence between systems of electrons in orbital motion and systems 
built up of Hertzian vibrators or of elastic bodies. Systems of 
the latter types of different linear dimensions, but otherwise 
similar, have their wave-lengths in the ratio of their linear 
dimensions. Now, speaking roughly, wave-lengths of light- 
waves are a thousand times as great as the linear dimensions 
of the atom; therefore the wave-lengths of the free vibrations 
of the Hertzian, or of the elastic, system must be roughly one 
thousand times its linear dimensions, in order to furnish 
a satisfactory atomic model. Obviously simple vibrators, 
spheres, ellipsoids, rods and the like, do not satisfy this con- 
dition, so that special assumptions are necessary; for example, 
a suitable Hertzian vibrator might consist of two conductors 
so close together as to have a capacity one million times the 
linear dimensions of the system. 

On the other hand, a ring of electrons involves three linear 
quantities in its specification, namely the radius of the ring, 
and the radii of the negative and positive electrons, of which 
the two latter remain the same when the linear dimensions of 
the system are altered. For this reason the wave-lengths 
of the free vibrations of the ring are in no direct relation to 
the radius of the ring; in fact it may happen that a very 
small ring emits longer waves than a large one. The size of 


of Quasi-permanent Systems of Electrons. AA5 


the ring must be considered in every calculation of the ratio 
of the wave-length to the radius. 

§ 13. We shall now study the frequency vibrations of a 
circular ring of equidistant electrons, which forms part oi 

a system of rings of the type specified i in § 9, with a view 
to estimating the wave-lengths, and comparing them with 
those of light-waves. For this purpose it is not necessary to 
prove that the system is stable ; it is sufficient to assume its 
stability and permanence. If it be found that a sufficient 
number of wave-lengths are comparable with those of light- 
waves to account for a reasonably large number of spectrum- 
lines, we may accept the system provisionally as a satisfactory 
model and examine its stability; if not, it must be rejected, 
whether stable or not, as by itself incapable of accounting for 
spectra by means of its free vibrations. If no system be 
found which satisfies all these conditions by itself, we need 
not necessarily reject the electron models as useless ; for we 
never have to deal with isolated atoms of any element, but 
only with the element in bulk. It is quite possible that the 
emission of spectra, as we know them, is a property of 
complexes of atoms, and not of the individual atom. 

§ 14. We shall for the sake of simplicity begin with 
the study of the models proposed by Nagaoka and by 
J.J. Thomson, in which no account is taken of those terms 
in the frequency equations, which are due to radiation, an 
omission implying velocities small compared with that of 
light. We shall then extend our investigations to the general 
case, where the effects of radiation are taken into account, 
and no special hypothesis is made as to the nature of the field 
in which the ring moves. 

Before proceeding to the study of special systems we shall 
mention a few points which concern all systems equally. 


Notation. 


Number of electrons in the ring =n. 

Radius of the ring =p. 

Angular velocity =o. 

Velocity of light =C. 

Velocity of electron/velocity of light =wpe/C=8. 


The azimuth at time ¢ of the zth electron is given by 


aes 


d=oat +6+ — 
its coordinates in steady motion ete 


Z= ees OP, Y—p sind. 2-0, 
Phil. Mag. 8. 6. Vol. 15. No. 88. April 1908. 2H. 


446 Mr. G. A. Schott on Frequencies of Free Vibrations 


When the ring is slightly disturbed from steady motion 
the displacements of the 7th electron, in the direction of 
motion, towards the centre and parallel to the axis, are denoted 
by (&, 7, 8); they are measured from the position which the 
electron would have occupied at the same time in the steady 
motion, not from a point fixed in space. Thus when disturbed 
the coordinates of the electron are 


w=(p—m) cos(P+E/p), y=(p—n)sin (b+E/p), 2=b 
Squares and products of the displacement are neglected in 
the equations of vibration (not of course in the energy). 

This is substantially the method of representation used by 
Maxwellin his paper on Saturn’s Rings, and by Nagaoka 
and J. J. Thomson in their investigations. 

§ 15. We may resolve the disturbance (&, , €) into a 
series of harmonic components proportional to terms of the 


271 ‘ : : 
form exp. 4 pi—-k—~), where & is an integer, and p is a 


complex constant of the form g+ux. 

This harmonic represents a wave with 2é nodes and 2k 
loops, travelling round the ring with angular velocity g/k 
relative to the rotating ring; for, apart from the damping 
factor, the displacements are unaltered when we increase 7 
by n, and t by 27rk/gq. The angular velocity of the wave 
relative to fixed space is w+q/k; accordingly q is the fre- 
quency relative to the rotating ring, as it appears to an observer 
revolving withit; g+ kw is the frequency relative to jixed space, 
as it appears to a stationary observer. We may speak of ¢ as 
the relative frequency, of g+kqw as the absolute frequency, 
or the frequency simply. 

Accordingly we notice that the waves emitted by the 
disturbed ring into the surrounding medium on account of 
the disturbance (g, k), consist of a series of simple harmonic 
waves with frequencies given by the formula ¢+ (k+sn)o, 
s any integer (§ 4). . 

It is obvious that we can obtain all the different types of 
disturbance possible, either by giving & every integral value 
and making s=0, or more conyeniently, by giving k n inde- 
pendent integral values, and s all values in turn. We shall 
select the values 


=o) ae = 7 
= - a O, +1,.. — + <, when n is even, 
n—] n—3 n—3 n—Il 


»--. —l, 0, +1,... + —,{ + , when 2 is odd. 


po) tate 2 


of Quasi-permanent Systems of Electrons. AAT 


In this case the harmonic s=0 is, for each value of k, by 
far the most important (§ 4). 

The distinction drawn: between relative and absolute fre- 
quencies is vital, and is particularly insisted on by Maxwell ; 
it is necessary again to insist on it since it is often overlooked. 
In fact the results of Nagaoka as to the arrangement of the 
relative frequencies of his ring in bands and series, for this 
reason do not apply to the frequencies of the waves emitted 
by his ring, and thus have no direct application to spectrum 
series. : 

§ 16. We must now consider the velocity of the ring, 
remembering that the ring cannot have a determinate struc- 
ture and determinate free periods unless its velocity is 
determinate. In order to account for the determinateness of 
the wave-lengths of spectrum-lines, we must make one or 


other of two hypotheses ; either :— 


(A) Hach system emitting lines has a definite structure ; or 
(B) The determinateness of wave-length is due to some 


action between the several systems, which constitute the 


radiating gas, in virtue of which only those waves which 
have definite frequencies ever become intense enough to 
produce observable lines. In this case each constituent 
system can be continually changing its structure within 
certain limits. All that is necessary is that the conditions 
for homogeneity and fineness of the lines satisfied ($ 9), so 
long as the system happens to be one of those which is 
producing a spectrum-line. 

Although for obvious reasons the first hypothesis seems 
the more probable, the possibility that the second may be 
true cannot be left out of account entirely. All that it 
requires is that the system, during the emission of a spectrum- 
line, be not altering its velocity by more than one 10—th 
part in one second (§ 5). 

§ 17. The determinateness of structure necessary for 
hypothesis (A) can for a single ring only be obtained by 
means of expanding electrons ($ 1). Fora system of rings 
it may be thought that the condition of permanence in spite 
of mutual perturbations of the rings, might alone suffice to 
fix the velocity of each ring within limits narrow enough 
to account for the observed fineness of spectrum-lines. But 
a closer examination of this question shows that in this way 
we can under no circumstances obtain conditions more than 
enough to fix the ratios of the velocities ; and even then it is 
doubtful whether the limits can be drawn sufficiently close 
together to give sufficiently fine lines, assuming that the 
remaining condition can be otherwise obtained. 


2H 2 


448 Mr. G. A. Schott on Frequencies of Free Vibrations 


Jeans, it is true, obtains the required condition by intro- 
ducing a hypothetical non-electromagnetic force between the 
electrons; this procedure amounts to giving up the sim- 
plicity which is the greatest advantage of the electron theory. 
But it has another disadvantage ; it is of no use for a system 
in which the electrons are in orbital motion, and is in fact 
employed to avoid the necessity of orbital motions. In con- 
sequence it can only explain spectrum-lines by means of 
dynamical considerations, and gives the relation between the 
frequencies (N) of the lines in the form 


N?=/(m), m an integer. .\'. i)" 5 ee 
To account for spectrum series we require the form 
N=/(m), man integer. . . .) 1). ae 


This fact led Lord Rayleigh to suggest that the numerical 
relation between the frequencies of a spectrum series is of a 


kinematical rather than a dynamical character ; for it is by 


no means evident that the equation (4) can, by simple 
extraction of the square root, always be reduced to the form 
(4) in such a way as to agree with observed series to the 
accuracy required by experiment. The reduction has in fact 
been accomplished only in very special cases (c/. the model 
Of WV tee 

Any system, on the other hand, which admits of the exist- 
ence of orbital motions, for that very reason admits the 
possibility of a kinematical explanation of spectrum series, 
an advantage which ought not lightly to be given up. But 
every such system also requires the hypothesis of expanding 
electrons in order to completely fix its structure. 

§ 18. I have stated elsewhere f that the rate of loss of 
energy due to radiation from a ring of electrons in steady 
motion is approximately equal to 


e287 Worn l+y —— 
R =o A / el exp. 2n(y—4logy=*), y=V/ 1-8}, 
provided n be large; and to J. J. Thomson’s value 


20 n(n+ 1)(ng)*"*? 
R= —- ——_, > + 
Bhieitagy: eet 


for small values of n, for which 6 alsois small. 


* Ritz, Ann. Phys. (4) xii. p. 264. 
+ Schott, Phil. Mag. [6] vol. xii. p. 22; vol. xiii. p. 194. One or two 
mistakes have corrected been here. 


of Quasi-permanent Systems of Electrons. 449 


On the hypothesis of expanding electrons, the intrinsic 
electromagnetic energy of each electron of the ring diminishes 


at the rate C’6?m 7 where a is the radius of the electron. 


In consequence a tangential pull is exerted on each electron 
by the ether, due to the expansion, and a drag, due to the 
radiation ; the velocity of the ring increases or decreases until 
the pull and drag balance. When this state is reached the 
radiation from the ring takes place at the expense of the 
intrinsic energy of its electrons ; and we have 


a 
R=nlB?m-- 
a 
Hence we get 


sas } 1 kek 6) i 
Vey . eXp. 2n(y—3 log ne cc ata. 0) 


when v is large ; and 


— 


ee ss ar a eae, 


alos 9) 1. Cmp? a a (7) 
) ‘ i i 


when 7 is small. 

§ 19. If we do not accept the hypothesis of an expanding 
electron, but adopt hypothesis (B), all we can assert is that the 
radiation R is at most at the rate of one 10—!th of the instan- 
taneous kinetic energy, or thereabouts (§ 16). 

The kinetic energy of the ring is of the order 


3nC?m6?. 
Hence we get 
R107? 2 ame’, 
as 7 Cmp? 
VJ n'y. exp. 2n (y-3 log = < be One. a <a (8) 
when n is large ; and 


n (n a 1) Cmp? a! 
she SA Tio’, ! 12 ware 1 
acai is Rens AUN ioe, ? 


when n is small 
We notice that the conditions (6), (7) and (8), (9) differ 
only by having a sign of equality replaced by one of inequality. 


450 Mr. G. A. Schott on Frequencies of Free Vibrations 


The quantity ae : changes as 8 changes, because both m 


and p change; its order is, however, easily assigned. In 


fact we may take e=3°5 .107-° (H.8.U.), e/Om=1° 9108 


p=10-° cm. or thereabouts; so that oe is of order 


1°5.10-'*. Hence 2’ is of the order 2. 10-5. 

As to d/a, we cannot well admit as possiblea change in 
mass of the negative electron of as much as 4 per cent. 
per annum: this makes @ of the order 4. 10-*. 

But a continual change in mass of the electron implies, on 
hypothesis (A), a secular chan ge in wave-length of all 
spectrum-lines. A change of as much as 1/100 A.U. per 
annum could hardly escape detection ; this is 1/600,000 of 
the wave-length for the D-lines. The corresponding relative 
rate per second is 5. 10- 13; taking this for the largest pos- 
sible value of a/a we find « at most of order 2. 10-%. 

We shall adopt 2. 10-6 as the upper limit for both wand al. 
It is considerably higher than the values used in my first 
paper. The following table gives the corresponding values 
of 8. On the hypothesis of the expanding electron they are 
actual values for each ring, otherwise they are merely 

maximum values, in each case for the assigned value of a. 


=| Lal Ge live “i 4. | 5. | 6. | ae 8. 9. 10. 
[2 


os 3 0 12 AO d 4. 10° i 10% ‘16. 10—3/3-9.10- sie 10-3] -014 "022 031 


| ———— | ff ff 


| 


200. as 1000. 


150. 


660) 


175. 


a 


| 11.| 12. las] 20.| 30.) 40.| 50. | 60. 5. |r fa 80. |90. 0. |100.| 125. 
“ool Tt 838 |-897. 


—_—— OBB Loo (ay lone Gen (en ew a eee 
‘041 -053 |-090 147 |-252 -333 |:398 es ag 18 "550 | 576 "622 
| 


ie 
ees 


——— 


We must particularly notice that, regarded as maximum 
values of 8 deduced from the observ ed degree of homogeneity 
and fineness of spectrum-lines (§ 5), these velocities have 
been calculated from the radiation R alone. Since the ex- 
pression for R is calculated from the field at a great distance 
from the ring, the only assumptions made are those required 
by Maxw ell’s ‘theory of the electromagnetic field. The table 
therefore gives maximum values quite independently of the 
forces assumed toact between the electrons at atomic distances ; 
and is true for every model of the atom, which assumes the 
radiation to be electromagnetic, whether the forces between 
the electrons be purely electromagnetic or not. 


of Quasi-permanent Systems of Elactrons. AdL 


Since £, y are less than unity and y—3 log ae 
the form of equations (6) to (9) shows that when we multiply 
a, or «', by a power of 10, for a given value of 8 we di- 
minish x by an amount nearly proportional to the exponent. 
For instance, if in § 5 we assume the breadth of a spectrum- 
line to be 1 A.U., in place of -01 A.U., and the number of 
waves in atrain to be 100,000 instead of 1,000,000, we must 
multiply «' by 10? and diminish n by 10 per cent. 

Physically speaking we see that a ring with a given velocity 
gives finer lines and longer wave-trains the greater the 
number of electrons (of course only on hypothesis B). 

§ 20. We shall here collect together certain results which 
occur in all the theories which we shall discuss. The 
notation is that of Maxwell in his paper on Saturn’s Rings ; 
we shall use it throughout for the sake of uniformity. 

Neglecting 8, the force exerted on any one electron of a 
ring of n equidistant electrons by the rest is a repulsion along 
the radius equal to Ke?/p?, where 


is negative, 


t=n—1 1 
= = 4 sin (7i/n)* 
{= 


For values of n equal to 10 or more K= ~ a zs nearly. . 


When the ring is subject to a disturbance (q, &) parallel to 
the axis, to the same approximation the force on any one 
electron whose displacement is ¢ is in the direction ¢ and 
equal to Je?¢/p*, where 


t="—1 sin? (kari/n) 


J=2 sind (a2/n) 


— 


When it is necessary to specify & we write Jz for J. 
Obviously Jo=0, and Jj;=K; the maximum value of J 
occurs for k= a or = , and for moderately large values 
of n is equal to 0:017 . n? nearly. 
When the ring is subject to a disturbance (gq, &) in the plane 
of the orbit, with components (&, 7), to the same approxi- 
mation the force in the direction & is 


—Ne&/p* + uMe*n/p’, 
and in the direction 7, 
— i Me?E/p? + Le?n/?, 


452 Mr. G. A.-Schott on Frequencies of Free Vibrations 
where as usual real parts alone are to be taken, and 
‘—n-W fain’ (kat/n) _, sin® (emia f ! 
LS = Esin® Gre/n) sin (ifn) ~ 2sin Cri/n) 
a 


4=n— 
> 1 in sin (2kmi/n) . cos (71/7) 
soe 8 sin? (zri/n) ‘ 

orl sin? ery n) _ sin” (kn2/n) 
a cae pee sin® (77i/n) ~ #sin (i/n) } 


edn: another constant H, where 
j= i 2 = 
sin? (k7i/n) 
H= = 4 sin (7i/n)’ 
we may also write 
L=J+H—-2K, N=2J—H. 


With the same notation as before we have 


7/8 
Ho =0; Hy =4 cots 
M,=0 Me ia 
Pane 1 4 2Qn 
2 in 
L)p=—2K, L,=1 cots —K, 
N,=0 N,=2K—+4 eo me 
’ 0 ) i 4 Qn 


n—l1 
H, L, N have their greatest values for k= 5? or 73 


for{ moderately large values of n they are respectively ;K, 
0:017 .n3, 0°034 . n?. ans el 


M has a minimum for 4=” 9 or 


The constants H, J, L, N are even functions of K, while M 
is an odd function. 
We note the useful series 


a (2s+1)x 
To 
i ; 2n 4 
al 


M=kK—Z 4(b—s—) cot 28D, 
SANG 2n 


s=k—1 
J =PK 3402900520 oh vae 
nr 


a 
. 
fe 


of Quasi-permanent Systems of Electrons. 453 


§ 21. Nagaoka’s model *. 

The model consists of three parts : 

(1) The ring of n equidistant negative electrons each of 
vharge 2 and mass m. 

(2) A central positive charge ve of mass M. To ensure 
the limited stability required by § 5, v must be very large 
compared with n. 

(3) A swarm of negative electrons sufficient to make the 
system neutral on the whole. According to § 9 perma- 
nence requires them to be grouped in rings coaxial with the 
ring (1). Stray electrons will either escape from the system, 
to be shortly replaced by others, or will owing to radiation 
lose their kinetic energy and coalesce to form rings, or join 
rings already present. Since the system cannot long 
remain charged without attracting negative electrons, when 
it is positive, and repelling them, when it is negative, it 
must on the average during its existence be very nearly 
neutral. 

Nagaoka considers it likely that his system forms a flat 
disk or ring, all the negative electrons crowding towards the 
invariable plane. He appears to assume that the system as a 
whole can be permanent; but there appears to be some 
doubt, whether a flat disk or ring of many electrons moving 
with such small velocities as are necessary for permanence, 
under the influence of forces acting inversely as the distance, 
ean be stable at allf. 

§ 22. It must be noted that in deriving his results Nagaoka 
neglects the field due to the swarm of negative electrons (3). 
This is strictly correct when the swarm forms an elliptic 
homeeoid completely enclosing the ring; but Nagaoka con- 
siders it more probable that it forms a flat ring, approxi- 
mately in the plane of the ring (1). The total charge of the 
swarm is that of v=n négative electrons, the whole system 
being neutral, and is herefore comparable in magnitude with 
the central positive charge. Hence its action cannot be 
neglected without further investigation. In order to form 
same idea of its effect we may treat it as an elliptic cylinder 
of the same cross-section, the long axis of the section lying 
in the plane of the ring. 

Let the long axis of the section be 2a, the mean radius of 
the swarm ¢, the short axis negligibly small. We easily find 
for the radial force on a negative electron, at a distance p 
from the axis in the plane of the ring, a repulsion from the 
mean line (¢c) of the swarm, which inside the swarm is equal to 

* Nagaoka, Phil. Mag. [6] vol. vii. p. 445; Tokyo Proc. vol. ii. 
nos. 17-21. Schott, Phil. Mag. [6] vol. viii. p. 384. 

+ Pellat, Comptes Rendus, March 4 and April 8, 1907. 


454 Mr. G. A. Schott on Frequencies of Free Vibrations 
2(v—n)e* 


2(v—n)e? c~p 
a cle~p + V (c~p)?—a?) 


Borel ee and outside is equal to 


2 
The radial force due to the central charge being —, thie 


action of the swarm is only negligible when its mean radius 
(c) is large compared with the “radius (p) of the ring. 

pees On the assumption that the velocities are small 
enough to allow us to neglect the effects of radiation, we 


find :— 
(1) for the equation of steady motion, 


y—K)é 
a 
(2) for the frequency equation for axial vibrations, 
ame bg eh 
pigs 1 Ph AU) <4 2 


(3) for the frequency eae for orbital vibrations, 


i ae Me 
ae. fom mp? 


Ne? ¢ | 42K re - 
mp , {B08 mp” } ail =O; 


By means of (10) equations (11) and (12) may be written 
in the forms 


2 a 
Foo: altri. ooh 
Pusher Nad =2K). op) 2M 
ai sy les y—K ee es 
BN 6 UN (ie 2K ie 
—K Ky =0. . wee a 


These equations no longer involve p explicitly, and the 
a p : 2 
coefficients of 2 are functions of vy and n alone. This cir- 
@ 


cumstance makes Nagaoka’s system extremely simple; but it 
must be distinctly noticed that the simplification is only 
obtained at the expense of neglecting the eftect of the negative 
swarm (3), and therefore applies only to rings which are small 
compared with the whole system (§ 22). 

§ 24. In order to calculate the roots of these equations 
Nagaoka assumes that n/vis so small that the roots can be 
expanded in ascending powers of the quantities J/v,... 


p< 
— 


of Quasi-permanent Systems of Electrons. 455 


Neglecting higher powers he finds, for each value of k, the 
following frequencies (relative): 
Axial vibrations—undamped («=0). 


my JaK pei) 


@ 2y @ 2y 


Orbital vibrations. 
Two undamped vibrations («=0). 


N-L-2K-4M gq, ,_ 4N-L—-2K+4M 


2y @ 2v 


7) 
One damped vibration. 


qs _ 2M ae 
ae kgs Pi, 


One vibration of instability of the same frequency. 


fe 2M. Kellie oo /BN. 


3 
@ V @ V 


The first four vibrations have relative frequencies very 
nearly equal to +. It must be noted that a negative fre- 
quency is to be interpreted as equivalent to an equal positive 
frequency. Hence in future negative frequencies will not 
be specially distinguished from equal positive ones. 

§ 25. These vibrations give rise to the following waves. 

Axial vibrations—two sets of undamped waves for which 
the absolute frequencies are given by 

1" T 
Nope IK Moy ISK 


| ae @ Zury 


where & takes m values between + : (§ 15). 


These frequencies are not all different. Since J—K is an 
even function of 4, the frequencies (N,) for k= —1, —2,... 
are the same as the frequencies (N,) for k= +1, +2,... and 
vice versa. Further we have Jj,=0, J;=K. Thus omitting 
all zero frequercies we get finally a set of frequencies given 
by the scheme 


N 1+ ae aaa ee 
2 2v heey 


1t+—> MES ; d bas hal 


Only those frequencies have been written down which can 


456 Mr. G. A. Schott on Frequencies of Free Vibrations 
correspond to observable spectrum-lines, namely those for :=0, 
+1, and exceptionally k= +2. 

Orbital vibrations. 


In the same way as above, remembering that N, L are even 
and M is odd, and that No, Ig +2K, My all vanish, we obtain 
the set of undamped waves 


a, 94 4N, 1, =2K—4M, MS be 
Z Zyv 
|, 9 4N,—I,-2K-—4M, . 4N,—1, oe 
L242 ee SS 2 2 | Oe 
N _ 4N,—L,—2K+4M, | 2y 2y a 
w Dy 4 AN b. 2K 44M, 
| 2y oh se? 
|, 4N,-L,—2K 44M, | 
- ee 


(16) 
Lastly we obtain a set of damped waves, and a set of waves 
of instability of the same frequency, given by 


Fig My ene tlatiy a 
DD had] Von ais | (17) 
okie of / ¢ es e 


‘a v 3 Po ae y Pie aR 


§ 26. Remembering that the suffixes 0, 1 correspond to 
relatively strong, and 2 to weak lines, we see that a single 
ring can give rise to the following lines : 
4N,—L,—2K4+4M,, 

2v | 
(6) An octet of 5 strong lines and 3 very weak lines, all o 
frequency w very nearly. 
(c) A quintet of 3 strong lines and 2 very weak lines, all 
of frequency 2m very nearly. 
(d) A triplet of 3 very weak lines, of frequency 3 very 
nearly. 


(a) A strong line of small frequency » 


In general the very weak lines will not be observable. 
If all the frequencies are such that all the lines fall within 
the limits of the visible spectrum, we get three strong lines, 
simple or complex, and occasionally a fourth very weak line. 

If Nagaoka’s assumption that vis very large compared with 
n be not true, the results are altered to a certain extent, but 
not very materially until the quantity K becomes comparable 
with v; this requires that » be comparable with vy, a con- 
tingency which can hardly occur unless v be small. 


of Quasi-permanent Systems of Electrons. 457 


These results show clearly that a single ring cannot pos- 
sibly account for spectrum series, and that for two reasons : 


(1) The absolute frequencies (N) are approximately in 
arithmetical progression. 

(2) The number of lines observable is far too small, because, 
as I have proved elsewhere, the intensities of successive lines 
after the first two or three diminish far too rapidly. 


§ 27. We must now examine whether a system of ring 
can account for spectra, each ring contributing one or more 
lines. This view it is true has difficulties; for the similarity 
in the structure of lines of the same series, and in their 
behaviour in a magnetic field and under pressure, is frequently 
assumed to indicate that they are produced by the same 
vibrator. But seeing that all rings of the same system are 
necessarily linked together, perhaps all that we need assume 
is that they are produced by vibrations of the same type, 
though of different rings. Thus it becomes necessary to 
investigate under what conditions the frequencies (N) just 
found fall within the limits of the spectrum. For this 
purpose it is necessary to estimate the value of o. 

@ is given by equation (10) of § 23. Introducing the 
quantity B=ap/C, we get 


mC? 6? : 
(2) = (v—K)e . ° ° ° ° ° (1 5) 
and 
(ye? 


(19) 


We must remember that the table of § 19 gives an upper 
limit to the value of 8 for each value of xn. Thus when n, v 
are given, equations (18), (19) give respectively upper and 
lower limits for w, p, which must not be transgressed if the 
system is to give fine spectrum-lines. 

Again, an upper limit for p is given by the condition that 
the ring must be small compared with the whole system (§ 22) ; 
thus p cannot be as large as the radius of the atom which the 
system represents. With this upper limit for p equation (19) 
gives a lower limit for 8, and equation (18) a lower limit 
for w, quite independently of the condition for fineness of the 
spectrum-lines. 

This condition, however, with the table of § 19 gives a 
lower limit (7) for n, This is obvious on hypothesis (A), 
where x is determined by 8 by the table. On hypothesis (B) 
we saw that to produce lines of a given fineness and degree of 
homogeneity, the value of n given by the table is the least 


SR ass ia i 


458 Mr. G. A. Schott on Frequencies of Free Vibrations 


possible for the assigned value of 8; any smaller value would 
give broader lines and shorter wave-trains (§ 19). 

Since v is the total number of negative electrons in the 
system, whether all be grouped in rings or not, the quotient 
v/ng gives the greatest possible number of rings of the system 
which can produce lines of the requisite fineness. This 
number is probably very considerably overestimated, for it 
assumes that nearly all the rings have the same number of 
electrons, while it is nearly certain that many of them have 
many more than the lower limit (7%) for n just found. 
Multiplying v/n) by the number of lines given by a single 
ring, we get an upper limit for the number of lines given by 
the system. 

§ 28. In order to see what kind of results can be obtained 
in this way with different assumptions, we shall make the 
calculation for the values v=10, 100, 1000, 10,000, 100,000, 
and for p=10-*, 10-8, 10-* cm. The constants required are 
é/mC?, and mC?/e?. For small velocities m=6.10-% gr., 
e=3'5.10- (E.8.U.), C=3.10% cm. Hence 


2 (mCP =2°2). 107 an) 2 1-4 10 


For larger velocities m is greater but a sufficient correction is 
easily applied. 

The method of calculation is best understood from a parti- 
cular case, say v=1000, p=1079. 


Neglecting K we get by (19) 6?=0°22. Thus to a first. 


approximation B=0°47, np>=66. 

This is too large for us to neglect the change in mass, 
which amounts to an increase of about 7 per cent. according 
to Kaufmann. The value of K for n,=66 is about 48, practi- 
cally 5 per cent. of v. Thus on the whole we must decrease 
8? by 12 per cent., and 8 by 6 per cent., getting B=0°44, 
Ny== IY: 

The following table gives the results of the calculation. 


| 10-° 036 110) 11 |-14 | 400! 19 |-44 1300 59 >1)... |... |>1 


| 
| 


pean 4 10. 100. 1000. 10,000. 100,000. 


1078-013 | 4-2 | 
Times 


044] Woes eee Ny. | (Be | aos) | Mp. |) (Bs | w. ii, Lefa |». 2. | B- | 
| wisi eet eaten 


———— — 


8 |-046] 14] 11 |-15 | 44 | 19 |-45 | 140] 61 |>1 


| 
| 
| 


| 


0043, 0:13) 6 O14 0-45 3-047 14 11-15 |44 | 19 |-46 | 15 | 62 


ee 
4 


w. | Ng» 


* 


of Quasi-permanent Systems of Electrons. 459 


The unit for w is 10’ revolutions per sec. 

The frequency, measured by 27C/A, in the same unit is 1:9 
for the extreme ultraviolet, 0°24 for the extreme red, 0°48 for 
the violet. 

§ 29. In studying the frequencies of the lines of the four 
types given in § 25 we must bear in mind that Nagaoka’s 
expressions presuppose that v is large; for values of v less 
than 100 they give only rough approximations to the truth. 
As a matter of fact we are interested mainly in systems 
for which v is large, on account of the difficulty as to 
stability; the small values are merely added for the sake of 
completeness. 


Ni Leek San, 
se ee 


fal 


Type (a).—N=o 


By § 20 we find 
2 
AN, —L|—2K + 4M, =403n(Log n—O0°300 + el 
where Log denotes the common logarithm. 


For n=6, 4N,—L,—2K 4+4M,=11°7. 
For r= 60, AN, —L,—2K+4M,=357. 
For n=450 4AN,—L,-—2K + AM, =4275. 


“= If we assume thai all the rings in each system have dif- 
ferent numbers of electrons, as for instance is the case in 
J. J. Thomson’s model, but that the larger rings differ only by 
one electron at a time, then the number of electrons in the 
r(r+1) 


largest ring, 7, is given by =v. The values are, for 


v=10, 100, 1000, 10,000, 100,000, 
eile yO 141, 447, 


where the values for 10, 100 can only be regarded as rough 
approximations, since for small rings the difference from ring 
to ring may exceed one unit. 

These numbers show that the largest ring in every system 
gives a line of type (a) in the observable spectrum for some 
value of p between 10-* cm. and 10-7cm. Thus every 
system may give as many lines of this type as it has rings 
with a number of electrons at least equal to ny for the given 


460 Mr. G. A. Schott on F requencies of Free Vibrations 


system. Hence ihe largest possible number of lines of type (a) 
are, for 


v=10, 100, 1000, 10,000, 100,000, 
ee i yee 123, 386, 


on the supposition (1) that the number of electrons is different, 
but only by one unit, from ring to ring. 

On the very much less likely supposition (2) that the rings 
all have the same number of electrons, and that the least 
possible (np), we get absolutely the largest possible number 
of lines, namely, for 


v=10, 100, 1000, 10,000, 100,000, 
1, ye BO: 526, 1612. 


§ 30. Type (b).—N =o nearly. 

The table (§ 28) shows that these lines only fall within the 
observable spectrum for systems for which v < 1800, for larger 
systems they lie beyond the extreme ultraviolet. 

Type (c).—N=2o nearly. 

These lines fall within the observable spectram when v< 470, 
otherwise they lie beyond the ultraviolet. 

Type (d).—N = 3 nearly. 

These lines fall within the observable spectrum when v< 210, 
otherwise they lie beyond the ultraviolet. 

We must also remember that lines of type (d) are very 
weak and can only be seen in exceptional circumstances. 

§ 31. Collecting our results together, we conclude that the 
number of lines to be expected is probably less than the value 
(1), and certainly less than the value (2), given in the following 
table :— 


gb == (0) 100, 1000, 10,000, 100,000. 
No. of strong lines (1)= 3, 21, 68, 123, 386. 
yy Weak) L52 (CD a Ti 0, 0, 0. 
> etrong.~ ,, S(2)= 36, 180, 526, 1612. 
i Weak ‘45, Ke—wde 12, 0, 0, 0. 


All these are estimated on the assumption that 10-7 em. is 
the maximum allowable radius, even for the largest rings. 
A smaller maximum materially reduces the values given ; in 
fact a maximum 10-§ em. gives no lines whatever for any 
system (§ 29). 

§ 32. In order to identify these systems with atoms we must 
calculate their masses and compare them with the atomic 
weights, bearing in mind that the mass of the hydrogen atom - 
is equal.to that of 1700 negative electrons. 


ae 


of Quasi-permanent Systems of Electrons. 461 


If M be the mass of the positive charge and A the atomic 
weight of the atom, we find at once 


Vv M 
= epee | 9 
A= 1700 1700m’ e . es s . (20) 


so that 1700 A is an upper limit to the value of v. 

Thus we find that a system, representing the H atom in 
mass, cannot give more than 92 lines (1), or 130 lines (2), 
in all. 

Similarly, a system representing the Fe atom, for which 
v<95,200, cannot give more than 376 lines (1), or 1560 
lines (2). 

These numbers include all the lines that can be given by 
the system under all possible conditions, whether they occur 
in series, or in bands, or as stray lines. When we consider 
that they are upper limits for the selected values of p and y, 
while these selected values are themselves in all probability 
chosen too high, we are driven to the conclusion that a single 
system, constructed on Nagaoka’s model, cannot account for 
spectra as we know them; but we cannot on these grounds 
reject the possibility that such a system may account for a 
part of a spectrum, e. g. a band or series, or even for a single 
spectrum of the element. 

To decide this question we must consider the stability of the 
system. 

§ 33. In § 24 we saw Bas each of the n sets of orbital 
vibrations includes a vibration of instability, that isa Lau 
whose component vibrations in the plane of the orbit (&, 7) 


are proportional to e*‘ cos ( gi— pom = 2), with «= or / 2X. ea 


The frequency of the emitted wave is g= o( bt am =) prac- 


tically Aw; its period (IT!) is 27/kw. Hence c= ae 
which means that during each period the amplitude is 
multiplied e”3N/”» times. 


By § 20 we have 


, 2st oe) 
2n 


Njea2K = CoB, 
s=0 ke? 


for k>0, zero for k=0. The greatest value of N/k? obviously 
Phil. Mag. 8. 6. Vol. 15. No. 88." April 1908. 21 


462 Mr. G. A. Schott on Frequencies of Free Vibrations 


occurs for k=1, and is equal to 2K —4 cot - that is, approxi- 
0°12 


2 
nv 
mininium values of x given in the table of § 28 we find, for 


y=10, 100, 1000, 10,000, —_ 100,000, 
ay BN 5.90, +38, “15, 07, 05. 
ral ae 


The method of reckoning here adopted is the one which 
is most convenient for our purpose ; we compare the increases 
in amplitude of the waves, not after one revolution of the 
ring, but after one period of each wave, the period being 
of course less for the waves with the greater number of 
loops. With the former method of reckoning the waves 
with the greatest value of & are the most dangerous, with 
our present method those for which k=1. The latter is 
obviously more favourable to the ring than the former. 

The physical interpretation of the numbers given is obvious; 
in a system of 10 electrons an initial disturbance of the type 
considered is at least doubled in 77 period, in a system of 
100,000 electrons in 13°9 periods. In the usual sense we 
should say that the ring is unstable. 

§ 34. To this conclusion Nagaoka™ raises two objections : 


mately equal to 0°733 . n(Log n—0'162+4 Using the 


2M\ . Me - 
(1) Since the frequency o-*) is small, the vibration in 


question does not enter into the system in general. 
(2) The analysis only holds for small oscillations. 


2M 


As to the first objection, we must remember that oe 


measures the frequency relateve to the ring; the frequency 
of the corresponding emitted wave is the absolute frequency 


oy 
7) (b+ at) ; the external waves which most strongly excite 


the dangerous vibration have the same absolute frequency. 
Any external wave of nearly the same absolute frequency 
will excite it to a certain extent, the more strongly the more 
nearly coincidence of frequencies is approached. Now the 
system emits orbital waves of frequency more or less nearly 
equal to kw (§ 25), which are in fact assumed by Nagaoka 
to account for spectrum-lines. Hence when the system is 
not isolated, but surrounded by a large number of similar 
systems all emitting these waves, it is necessarily subject 


* Nagaoka, ‘lokyo Proc. vol. ii. No. 17, p. 4. 


— _* 
> 
. 


of Quasi-permanent Systems of Electrons. 463 


to waves which are capable of exciting the dangerous vibra- 
tions in question to a greater or less extent. 

As to the second objection, it is of course a valid one ; 
but it can be surmounted by taking higher powers of the 
disturbance into account. We shall find that for our pur- 
pose it is quite unnecessary to pursue the investigation so 
far as to decide whether, after a greater or less excursion 
from the circular shape, the ring returns or does not return 
to its original condition; that is to say, whether it is stable 
or not in the usual sense. We can apply the condition of 
§ 5, that the ring must give trains of a minimum number 
of waves, whose period does not change during emission by 
more than an assigned small fraction of its value. 

§ 35. The equations of motion of an electron of the ring 
are, for. motions in the plane of the orbit, which alone concern 
us here, 

Dheo\ sar. P 5 

i =) i AWE yh. yx Shee ca), 
me dada — 

SF aoe i ae emg (22) 


where P, T are the mechanical forces towards the centre and 
along the tangent. 
When the ring is disturbed we have 


r=p—7, OG=at+&/p, 
where n=pAe sin (gt+a), 

&=pBe™ cos (gt+2), 
for free vibrations on the supposition that linear terms only 
need be taken into account. 


For another electron of the ring, the 7th from the selected 
one, we have 


gg 
O=ot+&/p+—, ps 


. 2 2 * 
where &, 7; involve gé+ Peay pelle place of gt+ a. 
2 
d ve 
The force P involves a term —~—, due to the attraction 


of the central positive charge, together with a small term due 

to the electrons of the ring from 7=1 to i=n—1; the force 

Tinvolves the latter term alone. Nagaoka’s analysis depends 

on the fact that the latter terms are small in comparison with 

the first ; we may assume that this remains true, provided 

the disturbance never becomes very nearly equal to the 
ye 


ee iF ih, 

ok a 
Pr rsh . ao 
ice ae 


‘ 


464 Mr. G. A. Schott on Frequencies of Free Vibrations 


distance between neighbouring electrons of the undisturbed 
ring. Even for values of vas great as 100,000 we saw in 
§ 29that rings of more than 450 electrons can hardly occur ; 
we must suppose the radius to be comparable with 10—~ em. if 
they are to give observable spectrum-lines. This makes the 
distance between neighbouring electrons equal to 10-9 em. at 
least. Hence the disturbance must be supposed to be appre- 
ciably less than this value. 

§ 36. On this assumption we may neglect the effect of the 
squares of the small terms, £,... in the small terms of P, 
and altogether in T. Taking the averages of the equations of 
motion for one period we find 


seed we Ne 
@ 20 — 0 + p — We ° e ° e (23) 


where the cross-bar denotes average values. 

In averaging wemay obviously neglect the variation of the 
factor e** during one period, particularly if we suppose ¢ to 
refer to the middle of that period. Hence 


Er — Fi = 0: 
so that (24), and therefore (22), is still anes 


Further 
En =—hgpABe,  P=g’p Be 


Qe La dt ai ve? 
aoa >» Pp {l—Ae* sin (gt +a) yo p2(1 — Arent)? 


so far as it depends on the central charge. 


2 
”- The part due to the ring is — sd as before. Thus the 


equation (23) becomes ty 


| okt 1 7 po, 2K¢ ver __ Ke’ 
w.(1+4 +2 Bers 5 BY sii ix tL a ee 
Now equation (22) gives to a first approximation 
2g, _M 
By eae 
A ie 
wo v—K 


of Quasi-permanent Systems of Electrons. 465 
With this value we find that the factor of w? becomes 


2N+M 


es 2(v—K) 


AB e2kt 


very nearly, the second term of which is very small compared 


2 

with A%e?"*, since one is small. Hence we get very 
nearly 2(v— BK) 

a 8 =e | 

i ip’ Cl Bee ya ie (25) 


2 
for in the small term =5 we may without appreciable error 


put 1—A’e?* in place of unity. 

Comparing (25) with (10) of § 23, we see that the effect of 
the second and higher order terms is the same as if the area 
of the ring underwent a progressive diminution, its value 
after a time ¢ from the beginning of the disturbance being 
less by the fraction A’e*. The effect on @ is to produce an 
increase of nearly # of this amount; and since every frequency 
is proportional to @, the frequency of every line increases by | 
the same fraction of its initial value. This change is obviously 
not one which can be counteracted by the expansion of the 
electron on hypothesis (A), because it is a progressive 
change while the latter is ‘secular. 

N 37. At this stage we introduce condition of fineness 
and homogeneity of spectrum-lines (§ 5). In order to be on 
the safe side let us only assume trains of 100,000 waves, and 
lines of 1 A.U. width, and let us neglect the part of the width 
due to Doppler effect. In other words, we assume that the 
frequency of the lines does not change by more than one 
ten thousandth of itsvalue during one hundred thousand 
periods. Is this consistent with equation (25) ? 


By § 33 the value of « is fobs = which for a system of 


1000 electrons is at least equal to yes Hence for t=10°.T 
iat am we a we aera Ne 
This must not exceed 107*. Hence we find 

Meo’ 


Obviously the value of A is not appreciably affected by the 
fineness of the lines, but only by the value of «, and especially 


466 Mr. G. A. Schott on Frequenctes of Free Vibrations 


by the number of waves in a train. If this be supposed to be 
only ten thousand in place of one hundred thousand, the limit 
for A is increased to 107°”. 7 

§ 38. By the ordinary theory of resonance we know that 
a periodic disturbing force excites every free vibration of the 
system on which it acts to a greater or less extent, unless it 
be localized at the nodes of one of the free vibrations, which 
in that case is not excited at all. But the waves which act 
on our system are not so localized, and necessarily excite the 
vibrations of instability as well as ail the others; so that 
amplitudes greater than 10°”, or for that matter greater 
than 10~ °°, are certain to occur. It follows that Nagaoka’s 
model cannot give trains of as many as 10° or even 104 waves; 
and that interference phenomena with large phase differences 
are impossible for the waves emitted by it. Hence it cannot 
account for optical phenomena by means of its free vibra- 
tions, quite apart from the difficulty of accounting for fine 
spectrum- lines in sufficient number. Nevertheless its study 
is extremely instructive, for on account of its simple struc- 
ture it is possible to obtain numerical results and deduce 
definite conclusions. Besides, the interesting properties 
studied by Nagaoka in his later papers for the most part are 
not due to his particular assumption of a central positive 
charge, but to the arrangement of the negative electrons in 
rings, and so may be expected to belong to other ring systems 
also. 

It should be particularly noted that the arguments of 
§§ 33-37 apply toall systems possessing frequency equations 
with complex roots. We may therefore conclude that in 
general no system can account for long wave-trains unless the 
imaginary part of each of its frequencies is vanishingly small 
(of order 107° or so) in comparison with the real part. 

§ 39. It will be convenient here to summarize the con- 
clusions to which we have been led during the course of this 
investigation. 

In the first place, we have been led to assign new meanings 
to the terms “ permanence” and “ stability.” No system 
which includes electric charges in orbital motion can be 
absolutely permanent, for orbital motions always imply 
radiation of energy. By the introduction of the hypothesis 
of an expanding electron we can, it is true, supply the loss of 
energy at the expense of the intrinsic energy of the electrons, 
and so give the system, as it were, a permanence of a higher 
order, which we may call secular; but obviously sooner or 
later the structure of the system must be changed. Therefore 


' ia 
a 


{ 


of Quasi-permanent Systems of Electrons. 467 


we call a system “ permanent” when it lasts sufficiently long 
to satisfy the observed conditions as to constancy of mass and 
of other properties. 

Such a system may be stable or not in the ordinary sense ; 
that is to say when disturbed in any way, the disturbance 
may diminish and the system come back to its original con- 
figuration, except in so far as that configuration has been 
altered by radiation ; or the disturbance may increase until 
the system falls away altogether from that configuration. 
Whichever happens, we call the system “stable,” when the 
progressive change in it, produced by terms of the second 
order in the disturbance, is so slow as to permit of the emission 
of trains of many waves and of the production of spectrum- 
lines of the fineness actually observed. 

When the structure of the undisturbed system is fixed by 
the hypothesis (A) of the expanding electron, its free periods 
of vibration are determinate, although subject to secular 
changes. If it be stableit can produce fine spectrum-lines 
when disturbed, and the observed width of the lines must be 
accounted for by means of secondary causes, Doppler effect 
and the like. But ifthe hypothesis (A) be not adopted, then 
a part of the width must be ascribed to progressive changes 
of period due to unbalanced radiation. We have seen that in 
any case a degree of permanence sufficient to allow of the 
production of fine spectrum-lines can only be expected from 
systems built up of coaxal circular rings of equidistant 
electrons, because other systems involve radiation, on account 
of mutual perturbations of their electrons, which cannot be 
balanced by the energy set free by the expansion of these 
electrons. 

As regards the particular model of Nagaoka, we have seen 
that it can give a large number of spectrum-lines, but not 
sufficient to account for a whole spectrum by means of the 
free vibrations of a single system, however numerous its 
electrons may be. Further, this model is far too unstable to 
permit of the production of trains of waves long enough to 
account for the observed phenomena of interference with 
great path differences. 


Physical Institute, Bonn, 
April 29th, 1907. 


\ A [ 468 j 


XLIV. The Tores of Saturn. By Percitvat LowEiu*. 


()* June 19th (1907) a new phenomenon disclosed itself 

in the Saturnian ring-system. On the morning of that 
day, the planet being so placed at the time as to present its 
rings almost edgewise to the earth, a curious detail was ob- 
served at Flagstaff in the shadow which then banded the 
planet’s equator. This equatorial shading, which was in 
truth the shadow of the rings upon the ball, seemed almost 
to belie its function because of the lack of density it offered to 
the eye. Farfrom being dark, it was only moderately dusky, 
and furthermore presented when first looked at a tripartite 
appearance. On more careful scrutiny, its lack of homo- 
geneity proved to be due to a narrow black line that threaded 
it medially throughout its length, the black core being perhaps. 
one-fourth as wide as the less dense background upon which 
it stood. At the same time the rings themselves could with 
attention be made out as the finest knife-edge of light cutting 
the blue of space on either side the planet’s disk. As the 
sun was at that moment 32’ north of the plane of the ring= 
system, while the earth was 2° 16’ south of it, the two were 
on opposite sides of the system, which fact combined with its. 
then visibility shows that the rings are never wholly lost in 
the Flagstaff glass. 

The planet was not looked at again at Flagstaff until 
October 31, other work occupying the observatory in the 
meantime. In November, however, it was critically studied.. 
The dusky band was evident as in June and the black line 
made core to it as before, being plainly perceptible to all 
the observers. On the 12th and 13th of the month I mea- 
sured both with the micrometer, the measures on the latter 
date being the more numerous and exact ; for the band was 
then measured between the threads, outside them and from 
centre to centre of the same, while the thread-like core was 
estimated in terms of the thread itself. The mean of the ~ 
measures with the suitable corrections applied gave ;— 


for the whole shadow ......... Af 
and: forts black cones) see seee see O10 


The band was tinged a faint cherry-red (Nov. 5); rather 
more strongly so than the planet’s own belts which could be 
seen both north and south of it. The black medial line in 
the midst of it was by no means even. It both undulated 
slightly and showed irregularities of outline, one black bead. 


* Communicated by the Author, 


EEE 
' om 


Prof. P. Lowell oa the Tores of Saturn. 469 


in especial being noticeable about halfway from the planet’s 
centre to its (the planet’s) eastern limb (Nov. 13, 145 G.M.T.). 
The line also seemed not quite central in the belt but a little 
nearer its northernedge. Thesun was now 1° 39/5 south of 
the ring-plane, while the earth was 50’ north of it. So that 
both bodies were now again on opposite sides of it, having 
respectively changed across. 

Although seen -by all the observers at Flagstaff, the black 
core was not caught by Barnard at Yerkes, nor has it been 
reported from the Lick. This, however, is in keeping with the 
definition at the first place disclosed already by its greater 
space penetration for stars. 

The rings themselves were equally visible, in fact were now 
easy objects, although, as before, only the edge of their plane 
was presented to the eye. But, in addition to the general 
line of their light, agglomerations were plainly discernible 
on them, attention being directed to that end. The agglo- 
merations were symmetrically placed, two on either side the 
ball, and continued observation showed them to be permanent 
in position. Micrometric measures were made apon them 
by both Mr. Lampland and by me from November 3rd to. 
November 9th. The most complete were those of the latter 
date, which, while agreeing in place with the earlier ones, 
gave not only the centre of the agglomerations but their 
beginning and end in ‘distances from the planet. My 
measures on that evening, with which those of Mr. Lampland 
substantially agree, were as follows :-— 


RIGHT. 


November 9, 1907. 


15° 25™ to 15" 50™ G.M.T. 


In equat. radii 
In seconds of of Saturn from 


are from the the Planet’s 

Mean of nearer limb. centre. 

3 measures. Inner edge of inner thickening.. 1'00 aa 

a Be Outer ,, of + BONS Bee 1-43 

7 Gap (most conspicuously vacant 5'':36 1:58 
spot). 

3 S Inner edge of outer thickening.. 6°92 1°75 

4 3 Outer ,, Ae - terra 1-92 

LErFr. 

2 measures. Inner edge of inner thickening.. 1°15 112 

2 a Outer ,, 33 2 2 422 1-46 

cae Gap (most conspicuously vacant 5°58 1:60 
spot). 

3 i Inner edge of outer thickening... 67°53 ish 

2 Outer ,, 2 5 7%, SE AG 1:91 


A470 Prof. P. Lowell on 


At the same time measures of the whole ring gave:—- 


Mean of 
2 measures. On the right measured from the nearest 
limb, reduced to equat. radii from 
the planet’s centre’ ........ 22 seen 2°164 
1 measure. On the left measured from the nearest 
limb, reduced to equat. radii from 


the planet’s centre, 1... . a5 sae 2°145 
4 measures. From W. to E. end visible double 
measures, reduced as above ...... 2°191 


The equatorial radius, also measured, came out :— 
Mean of 
2S MeCASUILES,...\, . ae eee 6 oe 9"-245 


Several things are deducible from these measures. First: 
Itis evident that the rings could not be followed quite to the 
outer limit of ring A, as that stretches to 2°25 or 2°30 radii 
from the centre of the planet, according to whose measures 
of the system we adopt, while the measured breadth was now 
at most 2°19. This implies that the outer part of ring A has 
less thickness than the rest; for we cannot refer the effect 
to less breadth of ring there being intersected by the line of 
sight, since the earth was on the opposite side of the ring- 
system from the sun. The average width of the thread of 
light upon which the agglomerations were strung was by 
comparison with the black core of the shadow not over 
80 miles. 

Secondly : The present measures indicate that the rings 
approach the body of the planet closer than they have been 
measured before. 

Thirdly: The measures of the positions of the agglome- 
rations show that Olbers’ explanation of them, endorsed by 
Seeliger, fails to account for the appearances. 

Fourthly : These positions point to another explanation of 
some interest. And 


Fifthly : This latter explanation proves also to account for 


the phenomena of the shadow, and incidentally to answer a 
query propounded by Seeliger on previous observations of it. 
To make this clear we will begin by quoting Seeliger upon 
the observations and deductions on the last occasion when the 
rings were presented as now edgewise to the earth. 

“The ring, according to Mr. Barnard, was completely in- 
visible at the end of October, 1891, even in the most powerful 
glass of the Lick Observatory. He caught it for the first 
time on October 30, 1891, 1» 7 G.M.T. As Mr. Oudemans 
could see nothing of the ring on October 29, 178 97 G.M.T., 


the Tores of Saturn. 47] 


its reappearance falls within the narrow interval between 
these two dates. At that time the earth had an elevation 
above the ring of 1° 56’. Mr. Oudemans further remarked 
on the same day: ‘a fine dark streak * runs across the equator,’ 
and explained this as the projection of the dark ring upon 
Saturn’s disk, because it still remained visible after the 
Sun had risen a little over the plane of the ring. Against 
this interpretation there is practically nothing to be “said ; 
only it may perhaps be remarked that here might well be the 
work of the bright ring too. For the shadow made by this 
ring does not lie exactly between two planes, and a part of it 
could constantly be found in the shadow of the other ring, 
even after the Sun had risen a little over the plane of the 
ring-system. One has thus the advantage of explaining a 
somewhat greater breadth of the dark streak; and this is to 
welcomed, as the minor part of the crape ring, as above 
remarked, is almost completely transparent, and therefore 
lets the disk of Saturn behind it be seen and itself does not 
appear quite black.” 

‘“ Before the ring begins to be visible, it must disclose itself 
as a dark stripe across Saturn’s disk. This was a fact seen 
and drawn by Mr. Barnard. He found on October 22, for 
the breadth of the dark band 0'51 and for the position of its 
middle point from the north limb (of the planet) 7'"40; from 
the south one 6°56. Mr. Barnard asserts that the measured 
breadth entirely agrees with the ephemeris data. I find, 
however, for an elevation of 1° 41’ for the earth above the 
plane of the ring, its width to be 0'"16 or 0'24, according 
as the bright ring alone is considered or the nes ring is In- 
cluded as well. On the contrary, I find very good agreement 
between the observations and the data for the position of the 
dark band of 7/64 and 6°72 from the north and south limits 
respectively, and of 7/60 and 6'°76 if the dark ring be 
also comprised. The above remarked divergence between ob- 
servation and theory demands explanation, since Mr. Barnard 
on October 29 found an even greater breadth of the dark 
band of 0'65.” 

‘From the very interesting notices of Mr. Barnard on the 
appearance of the ring when it became visible on October 29, 
1891 (October 30, 1891 ?, see above), the following points 
deserve prominence. The ring could first be seen ata distance 
of about 2" from the edge of the disk; also the two halves 
on either side were not the same; and lastly Mr. Barnard 


* Mr. Oudemans here refers to the whole shadow, as appears from 
Barnard’s measures of it cited below. Neither he nor Barnard saw its 
black core. 


SS ae 
> 


472 , Prof. P. Lowell on 


perceived two bright knots on the western one. The first 
fact can, if one pleases, be explained by the proximity of 
Saturn’s bright disk ; but another circumstance comes in here, 


as will be shown. The second fact demands no further — 


discussion ; while for the explanation of the third Mr. Barnard 
assumes that he saw two of the inner satellites. But Mimas 
only could be in question ; and it might raise difficulties to 
explain both knots by it. So the explanation needs no 
extended discussion, as it appears to me very probable that 
here another phenomenon is exhibited.” 

Seeliger then goes on to attribute it to greater showing in 
the line of sight ; and he deduces from the known dimensions 
of the ring-system the places where such ansal broadening 
would cause apparent maxima, to wit : 


i 4 ee radii of Saturn from the planet’s centre. 


We may here remark parenthetically that apparent agglo- 
merations or thickenings of the rings have been noticed by 
several observers since the time of Herschel, by Bond, Wray 
and Struve in especial, and agreeing more nearly than the 
above with the phenomena observed at Flagstaff ; but all the 
observers attributed them to causes other than what we shall 
now set forth. 

Taking up first the shadow phenomena seen at Flagstaff, 
calculation shows that the shadow of the whole ring-system 
including the crape ring, with the Sun as on November 9 

5 a : 
(1907) 1° 395 above the ring-plane, would be 026 wide 
only. The position of the Earth does not sensibly change 
this. Now the shadow was nearly twice as wide as this, 
being 0/46. Such then cannot be the cause. Nor canit be 
the penumbra of the dark core, as that would be but 0:05 in 
width, a quantity indistinguishable in fact by the eye. The 
only explanation left is that in the black core we are looking 


at the shadow of ring A, practically plane and in the dusky 


shaduw about it through particles situated above and below 
that plane lying in. the other rings. In other words, that 
ring B and ring C are for the most part not flat rings but 
tores. 

Turning now to the phenomena of the rings themselves, 
the agglomerations on the Olbers-Seeliger theory of their 
showing should, as computed in the same memoir by Seeliger, 
be found at 


1°60 and 1°98 radii of Saturn from the planet’s centre; 


for these are the points where the line of sight from the 


the Tores of Saturn. 473 


Harth traverses the greatest ansal breadth of the rings at their 
densest. 

Instead, however, of being so found, the present thickenings 
oceur in striking contrast to this, the maxima showing where 
the minima should and the minima where the maxima 
would be; since their centres are situate at 1°27 and 1°83 
with a conspicuous gap at 1°60 and another falling-off at 
1°92 outward. It is not, then, to line of sight massing from 
particles in one plane that the observed effect is due. 

But the moment we let our thought wander out. of the 
plane we light upon an explanation which satisfies the 
phenomena. For suppose portions of the rings to be not 
flat rings but tores, that is, rings after the manner of anchor- 
rings, encircling the planet. Then, viewed edgewise, such a 
tore would make its presence perceptible by humps of light 
in two patches symmetrically placed on either side the planet; 
to wit, at its ansae where the sight-line would penetrate the 
greatest amount of it. The agglomerations, then, can re- 
present tores, but cannot represent flat rings *. 

Thus we are led by the phenomena presented by the rings 
to the same explanation to which those of the shadow con- 


ducted us. Furthermore, it is to be remarked that the line 


* Norr.—It seems necessary to suppose that we see through the ring 
to its partially illuminated side; for from observations made or published 


since this article was written, it appears that the agglomerations disappear 


when either the Sun or the Earth passes through the plane of the rings. 
Thus Mr. Lampland’s observations of the rings gave :— 


Dec. 31. Agglomerations visible. 
Jan. J, Ansae too faint to detect structure. 
»  3&& 4 Ansae continuous. 
» 7%. No agglomerations. Rings easily seen.. 


Earlier observations by Aitken at the Lick, July 23—Oct. 12, show that 
no agglomerations were seen between those dates, See Barnard to the 
same effect. 

Since this was written Barnard has published his observations with 
his explanations. His explanations, however,—for he gives two—one 
that the eye sees through the underside of the rings and that such light 
is greatest where the rings are densest, the other the exact opposite, that 
the light is most where the ring is least crowded,—are self-condemning on 
several counts; one, for instance, that the inner condensation does not 
fall by his own showing on the ansal position of any part of ring A but 
wholly on the crape ring. Lach explanation might possibly account for 
one agglomeration alone, but for that very reason fails for both together. 
The presence of the gaps is another fatal objection to them. 

As seen at Flagstaff, under the same seeing that disclosed the dark 
core to the rings’ dusky shadow, the agglomerations were fairly con- 
tinuous, though uneven, for the whole length of them measured on 
November 9th. Their vertical width was about 0':20, while that of the 
continuous ring was about 0'02, giving for the width of the main plane 
of the rings some 80 miles or 180 kilometres.—P. L. 


A474. Prof. P. Lowell on 


of argument in each case is independent of the othér. For 
in the one case, in the shadow, we are reasoning on what we 
note from a transverse viewing of the tores; in the other, 
the rings themselves, from a longitudinal aspect of them 
in the bright agglomerations. As the two deductions 
lead to.the same result, each gains corroboration from the 
other. 

So much for the facts. They conduct us to a conclusion 
of interest from the point of view of celestial mechanics. To 
see this we will briefly recapitulate what has previously been 
shown of the stability of the rings. Laplace first showed 
that the rings could not be, as they appear, wide solid rings, 
inasmuch as the strains due io the differing attraction of 
Saturn for the several parts must disrupt them. Peirce then 
proved that even a series of very narrow solid rings could 
not subsist, and that the rings must be fluid. Finally, 
-Clerk-Maxwell demonstrated that even this was not enough, 
and that the rings to be stable must be made up of discrete 
particles, a swarm of meteorites, in fact. But, if my memory 
serves me right, Clerk-Maxwell himself pointed out that even 
such a system could not eternally endure, but was bound 
eventually to be forced both out and in, a part falling upon 
the surface of the planet, a part going to form a satellite 
farther away. 

Even before Clerk-Maxwell’s time, Edward Roche in 1848 
had shown that the rings must be composed of discrete 
particles,—mere dust and ashes. He drew this conclusion 
from his investigations on the minimum distance at which a 
fluid satellite could revolve around its primary without being 
disrupted by tidal strains. 

The dissolution which Clerk-Maxwell foresaw can easily 
be proved to be inevitable if the particles composing the 
swarm are not at considerable distances from one another ; 
and that they are not at such distances apart is certainly the 
case with rings A and B, as is witnessed by the light those 
rings send us, even allowing for the comminuted form of 
their constituents. Now a swarm of particles thus revolving 
round a primary are in stable equilibrium only in the absence 
of collisions. But in a crowded company collisions, due 
either to the mutual pulls of the particles or to the pertur- 
bations of the satellites, must occur. At each collision, 
although the moment of momentum of the two particles 
remains the same, energy is lost unless the bodies be per- 
fectly elastic, a condition not found in nature, the lost energy 
being converted into heat. In consequence, some particles 
will be forced in toward the planet while others are driven 


| / ae or 
i 
% 
| 


out; the greater number falling in until at last they are 
brought down upon the body of the planet. 

Now the interest of the observations at Flagstaff consists 
in their showing us this disintegration of the rings in process 
of taking place, and furthermore in a way that brings before 
us an interesting case of celestial mechanics. 

In considering the action of one body upon another revolving 
around a third, the points germane to our present inquiry 
are the perturbations in the radius vector and in the longitude 
of the second body. 

Now, by the method of variation of parameters the radius 
vector of the perturbed body—the disturbed particle in the 
present case—may be expressed, as has been done by Airy, by 


the Tores of Saturn. 475 


zs ay(1 = é;") 
} == €, COS (0,;—@,) 


La 


where the subscripts refer to the variable elements. The 
perturbed longitude may similarly be expressed by 


3 
O,=mt+e,+ (26 = + - &e.) sin (7,¢+e,;-—@,)+ &e. 


These may be expanded in an ascending series of terms 
according to powers of the excentricities and of cosines 
of multiple arcs of the mean motions of perturber and per- 
turbed by Fourier’s series. The resulting expression is 
composed of terms similar to those in the undisturbed orbit, 
and of others denoting the effect of the perturbation. The 
latter are of the typical form : 


Pere’= 


pr—gqn! 


cos (pn— qn')t—Q, 


where P is a function of a anda’, the radii vectores of the 
perturber and perturbed, and of uw the mass of Saturn and 
the perturbed body. 

The form of these terms shows that they will become con- 
siderable in proportion as pn—gn' is small, since their 
coefficients are divided by this quantity. Now as 2 and 7m’ 
are the mean motions of perturber and perturbed, if these 
are commensurate there will always be terms of the sort 

I 
which will be large, namely, those in which Z— 7 . for pand g 
p nn 
are always integers, In consequence of the method of 
expansion. 


A76 Prof. P. Lowell on 


The various terms with the argument (pn—gzn!)t will have 
coefficients of different powers of the excentricities. The 
lowest of these which can occur in the expressions will be of 
the order p—g. The term, therefore, in which «4+ 2,=p—q 
is the term least diminished by the excentricity coefficient, 
and therefore the most potent in its effect. 

From this it is evident that two bodies will mutually disturb 
each other in their revolutions about a third according as © 
their periods are : 


1st. Commensurate. 
2nd. Differ by the smallest integer. 


The most disturbing ratio is when the periods are:— 
120 ew Dis. Wee 


the next, Lic Se MO Ree aC. 
then, Le Ase | Qe bathe. « 
and so on. 


The initial ratio in each line will be the most effective in 
that line, because the cycle of the disturbance will be repeated 
in the time it takes the outer body to come again into coen- 
junction with the inner, and this for the ratio 3:5, for 
instance, will be three times as long as for that of 1: 3. 

The same thing can be seen geometrically by considering 
that the two bodies have their greatest perturbing effect on 
one another when in conjunction, and that if the periods of 
the two be commensurate, they will come to conjunction over 
and over again in the same points of the orbit, and thus the 
disturbance produced by one on the other be cumulative. If 
the periods are not commensurate the conjunctions will take 
place in ever shifting positions, and a certain compensation 
be effected in the outstanding results. In proportion as the 
ratio of periods is simple will the perturbations be potent. 
Thus with the ratio 1: 2 the two bodies will approach closest 
only at one spot, and always there, until the perturbations 
thus induced themselves destroy the commensurability of 
period. With 1:3 they will approach at two different spots 
recurrently ; with 1:4 at three, and so on. The number of 
points round the orbit at which they will meet is in fact as 
the sum of the powers of the excentricities in the lowest 
coefficient of the terms with the commensurable argument. 

We see, then, that perturbations, which in this case will 
result in collisions, must be greatest on the particles having 
periods commensurate with those of the satellites. But 


the Tores of Saturn. ATT 


inasmuch as there are many particles in any cross-section of 
the ring, there must be a component of motion in any collision 
tending to throw the colliding particles out of the plane of 
the ring, either above or belowit. Such extra-plane particles 
_ would, therefore, be most numerous just inside the points of 
commensurability, because, though the moment of momentum 
is preserved and thus particles be thrown outward from the 
point as well as in, owing to the loss of energy they must be 
more numerous on the inside, 

Considering, now, the commensurate ratios between the 
periods of particle and satellite which can enter into the 
problem, we find these in the order of their potency to be: 


With Mimas, 


With Enceladus, 
With Tethys, 


be et 
H= Oo He OO 


Such periods of commensurability as 2: 3 of Mimas and 
1: 2,2:3 of Enceladus do not come into question as they 
take place outside the ring-system. Now calculation shows 
that the distances corresponding toa period of 1 : 2 of Mimas, 
of 1: 3 of Enceladus, and of 1: 4 of Tethys fall in Cassini’s 
division, which separates ring A from ring B. The first or 
outer tore should therefore occur just inside that division 
or in the outer part of ring B. This is precisely where we 
find it ; for the inner edge of Cassini’s division is at 1°92 radii 
of Saturn from the centre of the planet, and the outer tore 
begins at 1°92, thence to stretch inward toward the disk. 
Pursuing our inquiry with the next most effective ratio, that 
of 1:3 of Mimas’ period, we note that its corresponding 
distance falls at the boundary of ring B and ring C at 1°495 
radii of Saturn from the centre. Now itis inside this, to wit, at: 
1:46 and 1°42, that the inner tore begins. Furthermore, this 
tore is much longer than the outer one. We turn, therefore. 
to the next most potent ratio, that of 1:4 of Mimas’ period, 
to find that its distance falls at 1°24. This then accounts 
for the greater length of the inner tore. 

The remarkable way in which theory thus accounts for 
observation is of interest, and the more so from involving a 
case of celestial mechanics interesting in itself. 


Phil. Mag. 8. 6. Vol. 15. No. 88: April 1908. 2K 


[> 478.4 


XLV. On the Contact Potential Differences determined by 
means of Null Solutions. By 8S. W. J. Suir, M.A., and 
H. Moss, B.Sc., Royal College of Science, London *. 

1. Introduction. 
4, Palmaer’s “null solutions.” 
7. A method of search for other null solutions. 
§ 9. Null solution of KCN. 
§ 12. Effect of Na,S upon the p.d. between Hg and KCl. 
16. Null solution of KCl. 
: 17. Effect of oxygen upon Paschen’s relation. 
§ 18. Inferences from results obtained with null solutions 
of KON and of KCl. 
§ 19. Null solution of KI. 
§ 20. Null solution of KOH. 
§ 21. Summary of conclusions. 


§ 1. TRODUCTION.—tThe processes which occur during 

the polarization of electrodes are of considerable theo- 
retical and practical importance. Being largely surface effects, 
they can be followed, in the case of mercury, by observation 
of the changes of the surface-tension which result from 
polarization. Mercury electrodes have been the subject of 
much study in this way, but while the interpretation of some 
of the results seems clear, there are others which have been 
the subject of frequent discussion. Chief amongst these is 
the significance of the maximum surface-tension between 
mercury and the electrolyte which occurs at some particular 
degree of polarization in almost every solution that has been 
examined. The maximum was for long supposed to indicate 
zero potential-difference between the mercury and the solu- 
tion; but the adequacy of this hypothesis, as a general 
interpretation of the significance of the maximum, is now 
very doubtful. 

§ 2. Evidence was given by one of us (Phil. Trans. A. 
1899, pp. 47-87), in discussing the validity of the Lippmann- 
Helmholtz theory of electrocapillarity, that the maximum 
surface-tension between mercury and certain electrolytes is 
not reached when the potential-difference is zero. It was 
shown that, for equally concentrated solutions of the parti- 
cular electrolytes KCl and KI, the potential-differences at 
the respective maxima differ to such an extent that if one is 
assumed to be zero the other must be nearly a quarter of a 
volt (J. ¢. pp. 70, 71). The assumption of zero potential- 
difference in either case is arbitrary. It is possible that the 
potential-difference at the maximum surface-tension is zero 
in some electrolytes; but the evidence from the electro- 


* Communicated by the Physical Society ; read February 28, 1908. 


Potential Differences determined by Null Solutions. 479 


capillary curves (l.c. pp. 68 & 82), while not conclusive, 
favours the view that it is not so in any of the moderately 
dilute solutions of KCl and KI (giving “‘ depressed”’ maxima) 
referred to above. 

Palmaer has, however, published experiments recently 
from which it would seem that in the case of »/10 KCl, the 
potential-difference at the maximum surface-tension is, if 
not zero, at least very small. This result, if conclusive, 
would be of great theoretical value, and we have therefore 
performed the experiments described below to test the 
validity of Palmaer’s deductions. 

§ 3. Palmaer published in 1903 (Zeitsch. f. Elektrocheme, 
ix. pp. 754-757) a summary of certain experiments with 
drop electrodes, by means of which he sought to show that 
the true contact potential-difference between mercury and 
n/10 KCI solution is about 0°57 volt, and he has quite re- 
cently (Zeitsch. f. physik. Chemie, lix. pp. 129-191, 1907) 
given a full account of these experiments and of others, from 
which his final conclusion is that the potential-difference in 
question is 0°5732 + 0003 volt at 18° C. 

One of us found that the maximum surface-tension between 
mercury and 7/10 KCl solution was produced by a polarizing 
electromotive force of 0°568 + ‘01 volt Cl. c. p. 71). It 
would thus appear that, after all, in the case of »/10 KCl, 
the potential-difference at the maximum surface-tension is 
practically zero, for it is represented by the difference 
between the two numbers just given. We shall attempt to 
show however that, although of much interest in connexion 
with electrocapillarity, Palmaer’s deduction from his experi- 
ments is not necessarily true, and that his results leave the 
knowledge of the contact potential-difference between Hg 
and 7/10 KCl substantially in the same state of uncertainty 
as before. 


4, Palmaer’s Null Solutions —When certain conditions, 
first fully discussed by Paschen, are fulfilled, a mercury 
electrode immersed in n/10 KCl shows a potential about 
0:57 volt higher than that of an electrode of mercury streaming 
into the same solution. Palmaer found that by adding cer- 
tain substances to this 7/10 KCI solution, he could not only 
reduce the observed potential-difference between the still and 
dropping mercury until it became very small, but could cause 
it, passing through the value zero, to change in sign. In 
this way he found two different “null solutions,” one con- 


taining small quantities of KCN, KOH, and Hg(CN),, and 


are a 
2K 3 


- , BAe ae = 
f ¥ . = 
Lan 6 lle’ 
is 


480 Messrs. Smith and Moss on the Contact 


the other H,S and a small quantity of acetic acid, for which 
the potential-difference between the still and dropping elec- 
trodes was zero. It is of course well known that a concen- 
trated solution of a mercury salt behaves like a Palmaer 
null solution in the respect that there is practically no p.d. . 
between still and dropping mercury electrodes immersed in 
it; but small variations in the composition do not here 
produce effects of the kind observed by Palmaer. Moreover, 
the absence of a p.d. in this case has been explained in 
a way which is not applicable when the amount of dissolved 
mercury is very small. 

§ 5. Experiments of the kind performed by Palmaer were 
suggested by Nernst (Ann. d. Physik, lvii. p. 11, 1896), 
and it is obvious that the results can be explained, either 
according to the Helmholtz theory of dropping-electrodes or 
to that of Nernst, if we assume that when the observed E.M.F. 
is zero a solution has been found which exhibits no potential- 
difference with respect to mercury. In such a case there 
would be no tendency to produce an E.M.F., whether, in the 
general case, the double-layer potential is altered by extension 
of the mercury surface (Helmholtz), or whether the double- 
layer forms practically instantaneously but is accompanied 
by concentration changes in the solution (Nernst). 

§ 6. Connecting each of his null solutions in turn to a 
solution of n/10 KCl, Palmaer measured the p.d. between 
mercury in the null solution and mercury in the ”/10 KCl. 
Allowing for small contact potential-differences between the 
electrolytes, he found practically identical values (of which 
the mean is the number already quoted) for the p.d. 
Hg | 7/10 KCl. Strictly, these measurements prove only 
that the p.d. between mercury and each of the null solutions 
was the same. 

Palmaer found only two satisfactory null solutions; but 
experiments with several others are described below. From 
these experiments it will be seen that the p.d. between 
mercury and a null solution of the type described by Palmaer 
is not always the same, and hence is never necessarily zero 
as Palmaer assumes. 


§7. A method of search for other Null Solutions—In 
searching for other null solutions we were guided by the 
relation deducible from Paschen’s experiments, that although 
the p.d. between Hg and an electrolyte when the surface- 
tension is a maximum need not be zero, it is nevertheless 
always equal to the p.d. between a dropping electrode of the 


Potential Differences determined by Null Solutions. 481 


Paschen type (Wied. Ann. xli. p. 42, 1890) and the same 
electrolyte (cf. Phil. Trans. l. c. pp. 83 et seg.). We have 
found, in the course of the present experiments, that the 
statement of this relation, which is one of the most important 
in electrocapillarity, requires qualification. The relation is 
true only when the medium through which the mercury falls 
before it enters the solution does not contain a constituent 
which interacts chemically with the mercury and the solution. 
In some cases, for example, the relation does not hold until 
the air surrounding the dropping electrode and the solution 
is replaced by hydrogen, nitrogen, or other non-oxidising 
medium. ; 

§ 8. Assuming the truth of the above relation, it was 
obvious that any solution for which the maximum of the 
electrocapillary curve lay at the origin (applied H.M.F. zero) 
would be a null solution. If the maximum for a given 
solution were slightly (a) to the right or (8) to the left of 
the origin, then the still electrode for that solution would be 
slightly (a) positive or (b) negative to the dropping electrode. 
The change in sign of the potential of the dropper with 
respect to the still electrode, with a small change in the 
composition of the electrolyte, would not correspond neces- 
sarily (as assumed by Palmaer) with a change in the sign 
of the potential of the still mercury with respect to the 
solution, but only with a change from one side of the origin 
to the other of the maximum of the electrocapillary curve. 

Since it is almost certain that this maximum does not 
always correspond with zero potential-difference, it seemed 
equally certain that a null solution of the kind studied by 
Palmaer could not always, and need not ever, be one 
exhibiting no contact p.d. with respect to mercury. 


§ 9. Null Solution of KCN.—It has long been known that 
moderately concentrated solutions of KCN give electro- 
capillary curves of which the maxima lie to the left of the 
origin (applied E.M.F. negative), while weaker solutions 
give curves of which the maxima’ lie to the right. Ifa 
solution could be found of which the maximum lay at the 
origin, this would be a null solution of the kind studied by 
Palmaer. A null solution of KCN was apparently found 
approximately by Amelung in a research which Palmaer 
describes (Zeits. f. physik. Chemie, lix. p. 164, 1907) but 
regards as unsatisfactory. This solution used according to 
Palmaer’s method would give a value of roughly 0°7 volt for 
the contact potential Hg | 7/10 KCl. It was obtained by 


ee 


482 Messrs. Smith and Moss on the Contact 


gradual dilution of a saturated solution of KCN until the 
p-d. between the drop electrode and the still electrode 
became zero. 

We have found and examined a null KCN solution in 
another way. A normal solution of KCN was made up and 
this, together with weaker solutions of definite strength, 
obtained from it by dilution, was then examined as described 
below. As it was found that the electrocapillary properties 
of solutions, apparently of equal concentration, produced from 
different samples of KCN frequently differed considerably, 
the precise constitutions of the different solutions cannot be 
specified. This, however, is of no importance to the validity 
of the experiments. For the sample of KUN used in the 
experiments, the null solution contained about 0:26 KCN; 
but another sample from another source would probably have 
given a different result. 


The following quantities were observed in the case of each 
solution :— 


1. The electrocapillary curve—the reading for maximum 
surface-tension, and the H.M.F’. required to produce it, being 
noted as carefully as possible. 


2. The electrocapillary curve of a solution of KCl of equal 
strength. 


3. The horizontal distance between the descending branches 
of tle two curves (cf. Phil. Trans. J. c. p. 69). 


4, The E.M.F. of the cell Hg | KON : KOI | Hg. 
). The E.M.F. of the dropping electrode circuit 
ig ; KCN 


Hg, the measurements being taken when the 


| 
q 
v 


end of the continuous part of the jet was in the surface of 
the solution (Paschen). : 


6. The E.M.F. of the dropping electrode circuit 
Hg | KCN : KCl | Hg. 
| : 


Some of the electrocapillary curves are shown in fig. l, 
the others being omitted to avoid confusion. The KCN 
curves are very flat near the maxima, and the E.M.F.s 
corresponding to these maxima are relatively difficult to 
determine. 


vi Bie 
‘000'0/_ iy = s0OSA ; ‘000¢ OH GRA 


F)°24| 261-0 =| o00/ STI W FT prd/y 


483 


725 


Q 
a OF 
al 
a 
Y 
ie 
a 
sersiicioey) cage 08% 
A 
& 
QR, 
:: 
Xo 
eR eet nt re. 


Potential Differences determined by Null Solutions. 


‘ee 


0% 


” i is 
9 @ 

+. ¢ 

. a ws fe’ 


484 Messrs. Smith and Moss on the Contact 


The chief results of the measurements are summarized in 
the following Table :— 


eS 


b 
| 1. Oa he! Bas 5. ae 7. | 
| | | Horizontal | | 
| jag Ten KCL Max 82, distance || KON : KOl| + | KON | +\| Kow! KOl| + 
_ solutions. (seale- | (scale- ) _ curves 
| reading). reading). volt. volt. volt. volt. | volt. 
| Rees fae Ee | =. ee 

Olin. | 3665 | 3675 | 039 645 642 03 | 679 

Be | 9650 |) 36-70'| “020 ear) ea 018 694 

24n. | 3610 | 3653 | -010 687 | ‘687 ‘012 698 | 
25n. | 86:08 | 36°35 0 691 | 690 ‘003 699 | 
26n. | 36:04 | 36-30 Ocal ogo oe 693 —-001 | 697 | 
O7n. | 36:00 |. 36-27 | —005 | 697 | 696 —-005 693 | 
50n. | 35°81 | 3617 | —-020 716 | ‘715 —-016 | ‘703 | 
100. | 3570 | 3610 | —059 | “748 | = “745 mie! 705 : 


| 


them will be very small, probably less than a millivolt. 
This is confirmed by the agreement between the numbers in 
Columns 4 and 5, since the former represents the difference 
between Hg | KCl and Hg | KCN (cf Phil. Trans. 1. ¢. 
pp. 69 et seq.). 

Each number in Column 7 should be equal to the sum of 
the corresponding numbers in Columns 5 and 6. A com- 
parison serves to indicate the limits of the uncertainty of the 
dropping-electrode measurements. The jet fell freely in 
contact with the air. By comparison of the numbers in 
Columns 3 and 6 it will be seen that Paschen’s relation held, 
within the limits of errors of experiment, in every case. 

The null solution was that for which the surface-tension 
was a maximum in the natural state (applied E.M.F. zero). 
Its strength was, in round numbers, 2/4 KCN, and assuming 
with Palmaer that, in consequence, Hg | n/4 KCN=0, it 
would give, by Column 5, Hg | »/4 KCl=0°69 approxi- 
mately. 

§ 10. As will be seen later, a solution of »/4 KCl, ex- 
amined according to either of the methods described by 
Palmaer, would have given a null solution from which he 


would have deduced the value Hg | n/4 KCl=0°565 approx. 


Potential Differences determined by Null Solutions. 485 


The reason of the difference of 0°125 volt between this and 
the value found by the KON null solution is suggested by 
inspection of the electrocapillary curves. 

The maxima for the KCN solutions are lower than those 
for the KCl solutions of corresponding strength, and the 
‘horizontal distance” between the curves after they become 
parallel is approximately 0°125 volt greater than the distance 
between the maxima. Consequently (cf. Phil. Trans. J. c. 
p. 67) the solution at the maximum surface-tension in KCN 
is 0°125 volt more positive to the mercury than it is at the 
maximum in KCl. 

§ 11. Every solution for which the maximum of the electro- 
capillary curve is at the origin is a null solution, but the 
potential-differences between mercury and different null 
solutions will be ditterent if the maxima for these solutions 
are not the same. 

If traces of other substances can be added to a 7/4 KCl 
solution in such a way as to move the maximum of the elec- 
trocapillary curve to the origin without appreciably raising 
or depressing it, then such a solution will be about 0°125 
volt less positive to mercury than the null solution of n/4 KCN. 
This null solution when measured against pure n/4 KCl will 
give an H.M.F’. of 0°565 in place of the H.M.F. of 0:69 
given by the n/4 KCN null solution. Hence, if Palmaer’s 
solutions were produced from /10 KCl without appreciable 
alteration of the maximum, the difference between his results 
and ours is immediately explained. 


§12. Hifect of NaS on the p.d. between Hg and KCl.— 
The p.d. between mercury and an electrolyte is controlled, 
according to the theory of Nernst, by the concentration p of 
the mercury ions in solution. If 7 is the potential rise from 
the solution to the mercury, then 

(3.70) Coy: ay ol 4 6 a Ree Re oe aioe eds 19 
where T is the absolute temperature and & is a constant. 
The rise can therefore be diminished by decreasing p. From 
Palmaer’s experiments and others (e. g., Behrend, Zeits /. 
phys. Chem. xi. p. 481, 1893), it can be inferred that a large 
diminution of p is produced by saturating a KCl solution in 
contact with mercury with H,S. We therefore thought it 
likely that by addition of Na,S to the KCl solution it would 
be possible to move the electrocapillary curve to the right, 
2. e. so that the maximum approached the origin. It remained 
to determine by experiment the relative amount of NaS 


* An interesting method of deducing a similar equation was given by 
Professor J. J. Thomson in the Philosophical Magazine for 1895. 


486: Messrs. Smith and Moss on the Contact 


necessary for the purpose of reaching the origin. Electro- 
capillary curves were obtained with solutions containing 
gradually increasing proportions of Na,S. The general 
composition of the solutions was 


x(n KCl) +y(7 NaS, 9H,0), 


where «+y=0'1; 7. e., considering the sum of the contents, 
the solution was always 1/10th normal, and the content in 
kations remained approximately constant. Curves were 
obtained for a number of erika: values of y. Some of the 
results are shown in fig. 2, in which the values of y repre- 
sented are as follows :— 


- oF solutions...) t> iLL. ae Iv. V. VI... VIL: VILL) 0X.) XS 


Walue of 7 —...... 0 0,2, 0,372 0,374 “0,577. 01, 70,2505 ee 03-08 +10 


SN 


HOS 
ah 


Fee: 2 
From these results it will be seen that less than ‘001m Na. 
added to °099n KCl was sufficient to move the maximum of 


—f- 


ace Jen 


Sar 


Decrease in pb Kap | njraKel 


Caused by addctrarn of Na, 


Potential Differences determined by Null Solutions. 487 


the electrocapillary curve to the left of the origin. Further, 


when the value of y was about ‘00037, an extremely small 
yariation produced a very large change in the position of the 
maximum. This is represented in fig. 3, which shows how 
the displacement of the descending branch of the electro- 
capillary curve depends on the percentage composition of 
the 7/10 solution. Since the concentration in kations was 
the same for all solutions, this curve represents (Phil. Trans. 
l. c. p. 80) how the potential of the solution with respect 
to the mercury rises with increase in the amount of the 


sulphide. 
He F 


Ae 
aN 
eS 
Ss 
© 


Uy 
o 
So 


2000 


7000 


§ 13. The extremely rapid concentration variation of the 
potential near y=*00037 was very inconvenient for our 
purpose, for it happened that the solution which we sought, 
2. €. one having its electrocapillary maximum at the origin, 
would contain an amount of sulphide corresponding approxi- 
mately to this value of y. The experiments of Behrend 
(l. c. p. 481) suggested that the rapid variation of potential 
at a critical value of y occurred near the completion of the 
conversion of feebly soluble chloride of mercury into still 


equiv, Gram molecules of NaS fer litre . 


Dy yk’ 


488 Messrs. Smith and Moss on the Contact 


less soluble sulphide. Thus, assuming ¢ to represent the 
concentration of Hg,Cl, in solution before addition of Na,S, 
if the amount of sulphide added were (c—c’), 2. e. less than 
¢, we might have (see however § 15) :— : 


cHg,Cl,+ (e—c')Na,S =c'Hg,Cl,+ (c—c’) Hg. 
+2(c—c')NaCl. 


Assuming « and £ to represent the fractions of the chloride 
and sulphide respectively, which are electromotively active, 
the ionic concentration p of the mercury in solution would 


be 
p=2ac!+2B(e—c'), and hence 9p/dc'=2(a—B£). 


Then, from equation (i.) above, we should have 
07/d¢ =kT(a—B)/(ac'+ Be—Cc). 


From this it is seen that if @ is very small compared with a, 


the value of O7/Qc’ will become very large when c’ becomes * 


very small, 7. e. when the amount of Na,S added is very 
nearly equivalent to the amount of Hg,Cl, in solution. 

§ 14. When the amount of Na,S added is in excess of the 
He,Cl, in solution we may have 


cHg.Cl, + (e+c¢') NaS =cHg.S +c’ NaS + 2cNaCl. 


From this it is seen that the potential of the liquid with 
respect to the electrode will continue to rise when, after the 
whole of the chloride is decomposed, more and more Na,S is 
added. For, assuming the law of mass action, the ionic 
concentration of the Hg obtained from the Hg.S will diminish 
when, by introduction of further quantities of Na,S, the 
concentration of the 8 ions in solution is raised. 

Suppose now that to a solution containing an excess of 
NaS a small quantity 2c” of an acid HX is added. Then 


we shall get 
cHg,Cl,+(¢+c¢')Na.S + 2c7HX 
=cHg.8 + (e’—c’')Na.S + ¢'H.S+ 2ce”NaX + 2cNaCl. 


Now if we assume the coefficients of ionization of the 
NaS and ihe H.S to be y and 6 respectively, the concen- 
tration of S ions arising from these two substances will be 
y(c'—c'') +8c''; while, before the addition of the acid, the 
concentration of S ions arising from the Na,S would exceed 
yc’. The value of 6 will in general be less than that of y, 
because, in moderately dilute solution, H,S is a very poor 
conductor compared with NaS. Therefore addition of the 


Potential Differences determined by Null Solutions. 489 


acid HX will reduce the concentration of the 8 ions in the 
solution containing cHg,S, and will in consequence cause 
the concentration of the Hg ions to rise. Thus addition of 
the acid will tend to reduce the effect of the Na,S and will 
cause the electrocapillary curve to move towards the right. 
Hence, if too much NaS has been added, so that the 
maximum is to the left of the origin, it may be possible to 
obtain a null solution by the addition of a small quantity of 
an acid. Moreover, it may be of advantage experimentally 
to proceed in this way by overshooting the mark and then 
adding acid, rather than by attempting to hit it by reducing 
the amount of Na,S. For, as can be seen, the rate at which 
the curve will move towards the right will be less when a 
small quantity of acid is added than when an equivalent 
quantity of sulphide is taken away. Thus, when the acid is 
_ absent, the concentration p, of S ions, outside the Hg,S, is 
such that Op,/d¢' exceeds y; but when the acid is added 


op./d¢' = —(y—4). 


§ 15. It is not contended for the above equations that they 
necessarily do more than indicate the course of events. The 
Na,8 used probably contained traces of higher sulphides, and 
the chemical relations between the substances concerned are 
complicated. The final product of the action of Na,S on the 
mercury in solution would no doubt be a double sulphide of 
HgS and Na,S, and it is worth notice with regard to this 
that, after the very rapid change at ‘00037n NaS, the 
potential of the electrode continued to decrease fairly rapidly 
until about ‘002n NaS had been added. The rate of decrease 
fell comparatively suddenly at this point, and for subse- 
quent additions of Na,S (until the whole ot the KCl was 
replaced) remained very small. 

The concentration of mercurous ions in a solution of KCl 
standing over mercury and calomel has been estimated 
indirectly by Behrend and others, and from the values 
obtained it would appear that, if the above interpre- 
tation of our observations is correct, only a small fraction 
of the mercury in solution can be in the ionized condition. 
It is important to note, however, that the comparatively 
large value of the solubility of Hg, Cl,, to which the results 
of § 12 seem to lead, may on account of the influence of 
dissolved atmospheric oxygen be more apparent than real. 
It iscommonly agreed that the interaction between Hg and a 
solution of a chloride, from which the existence of Hg, Cl, 
in the solution results, occurs through the intervention of 
the oxygen dissolved in the latter. This interaction will 


490 Messrs. Smith and Moss on the Contact 


continue (1) until the solution becomes saturated with 
Hg, Cl,, or (2) until the supply of oxygen near the electrode 
is used up. In the latter event, further formation of Hg,Cl, 
would cease until, by diffusion, a fresh supply of oxygen 
approached the electrode. Since, however, the solubility of 
oxygen in n/10 KCl is probably much greater than that of 
Hg, Cl,, it is probable that the interaction terminates in 
accordance with the first of the alternatives and before the 
whole of the oxygen per c.cm. near the electrode is removed. 
If now a small quantity of Na,S is added to the solution it 
will precipitate the whole or a part of the Hg» Cl, with which 
the solution is saturated. But the further solution of Hg, Cl, 
will be possible by interaction between the mercury, the KCl 
in the solution and the excess of dissolved oxygen. In fact 
the complete removal of Hg, Cl, from solution by means of 
Na,S, as represented in the equations given in §§ 13, 14, 
will not be possible until the whole of the dissolved oxygen 
is used up. 

From this point of view, what happens when an aerated 
solution of KCl containing Na,S is poured upon mercury 
may be described, figuratively, as a competition between the 
salts for the oxygen and mercury in the surface layer. Since 
any Hg,Cl, formed before the removal of the Na.S is com- 
plete will be at once decomposed by the latter and preci- 
pitated as sulphide, the net result of this competition is that 
the KCl can only interact permanently with such oxygen as 
the Na oS leaves uncombined. The critical amount of the 
latter is reached when it leaves none *. 

In any attempt to form a complete picture of the process, 
however, it would be necessary to take account of possible 
difference in the subsequent rates of approach of Na,S and O 
io the surface layer, from above, by diffusion and convection. 
This would lead us too far from our present aim. 

The exact composition of our critical NaS solution was 


uncertain, and the further study of the question presented _ 


chemical difficulties which it did not seem profitable to inves- 
tigate with the materials at our disposal. 


§ 16. Null Solution of KCl.—The argument developed 
above proved successful as a working hypothesis. An 
n/10 KCI solution containing ‘001n Na,S was first made up, 
and others containing different amounts of: acetic acid 
(suggested by Palmaer’s experiments) were then obtained 


* Some results of experiments still in progress in connexion with this 
view were indicated when the paper was read. 


Potential Differences determined by Null Solutions. 491 


from it. The results of the experiments with these solutions 
are summarized below:— 


1. 2. 3. 4. | 5. 6. ys 


ee ee 


| - Dawe Spite. ane 
| Horizontal 


Strength|Max.S8.T.|E.M.F.for| distance to 


sol po kal|+ | gol | 4 | 
*:10 a | 


of acetic} (scale- | Max. S.T. n | \ sol. a KCl|+ 
acid. |reading).| (volt.) |jgKC! sits * (volt.) (volt.) | (volt.) 

0 3682 | —-123 ‘661 | 660 —-129 564 
‘O0llz | 3681 | —-051 622 | 618 —-049 560 
0017 n | 36:42] —-020 588 582 —-018 564 
00232 | 36-80 | —-005 878 ‘571 —-009 | “564 
0028 | 36-80 010 568 | 562 ‘005 562 
0040n| 3680 | 020 | 556 | 549 016 564 


36°80= Maximum §&.T. for 7/10 KCl. 


The numbers representing the normality in acetic acid are 
only approximate. Nothing turns upon the accuracy of their 
determination. Some of the corresponding electrocapillary 
curves are shown in fig. 4. 


The measurements are less exact than in the case of the 
KCN curves. Owing no doubt to the presence of small 


492 Messrs. Smith and Moss on the Contact 


quantities of dissolved oxygen, the p.d. between the large 
Hg electrode and the solution sometimes changed perceptibly 
during the short time required to determine the electro- 
capillary curve. It will be seen, however, that a series of 
curves with practically equal maxima of surface-tension was 
now obtained. The null solution lies between IY. and V., 
and corresponds within the limits of experimental error with 
a solution of which thé maximum of the electrocapillary 
curve is at the origin. Part of the difference between the 
numbers in columns 4 and 5 was certainly due to increase 
with time of the p.d. Hg | solution. The capillary curves 
were taken first and are not exactly parallel to the deci- 
normal KCl curve. The distance decreases slightly towards 
the lower ends of the curves. The p.d. between the 
strongest of the solutions and the n/10 KCl solution was 
probably less than a millivolt (cf. Palmaer, Zeit. p. Chem. 
hes 1907, ps toon 

By examination of the data for solutions IV. and V. it is 
seen that a null solution of the present kind, interpreted 
according to the assumptions of Palmaer, would give, for 
the p.d. Hg | n/10 KCI, a value agreeing with his result. 
But, as will be seen, his assumption of zero p.d. between 
Hg and the null solution is not at all necessary. The onl 
necessary conclusion is that the p.d. between the still Hg 
and the solution is the same as that between the dropping 
Hg and the solution. 

From the data in the Table it is seen that, for reasons 
already given, the maxima being practically equal in all the 
solutions, the p.d. between the dropping electrode and the 
solution should be in every case the same. This result is 
probably of significance in the theory of . electrocapillarity. 
Its truth is shown by the practical equality of all the numbers 
in Column 7. 


S17. Eject of Oxygen upon Paschen’s relation With 
respect to the data in columns 3 and 6, $ 16, showing the 
fulfilment of the Paschen relation for the present series of 
electrolytes, it is important to state that the dropping-elec- 
trode experiments were performed in an atmosphere of 
hydrogen (cf. §7, above). A piece of wide glass tubing 
open at both ends encircled the dropping electrode. The 
lower end of this tube was immersed in the electrolyte, and 
the upper was closed by a rubber cork in which were three 
holes. The drop electrode passed through the central hole: 
the others served for the inlet and outlet of the hydrogen 
respectively. 


Potential Differences determined by Null Solutions. 493 


The following data are given as examples of the necessity 
for the exclusion of air. In one case, the maximum of the 
capillary curve being +°01 volt, a Paschen electrode against 
still mercury in the same solution showed —:05 volt, 
increasing to —‘07 when the mercury jet was partially 
immersed. The solution quickly became cloudy and the 
mercury:tarnished so rapidly that it fell as a dirty powder. 

In another experiment the jet was surrounded by hydrogen, 
the water used in preparing the solution having been recently 
boiled to decrease the amount of dissolved air. Before com- 
plete displacement of the air by hydrogen, the Paschen 
electrode H.M.F. was —-024 volt. It fell gradually to 
—°014 volt and then remained constant. The mercury fell 
quite clean. Some of the solution was now withdrawn, and 
its electrocapillary curve was determined. The maximum 
lay at —-015 volt. 

In a third experiment, the jet being surrounded by hydrogn, 
the electrode E.M.F. was —-024, agreeing with the electro- 
capillary maximum. Oxygen was now admitted and dis- 
placed the hydrogen. The Paschen electrode H.M.F. 
became -°140. With the jet partially immersed the E.M.F. 
increased to —°240. 

Various experiments with other gases and with insulating 
liquids were tried, e. g., it was found that the Paschen rela- 
tion was, under certain conditions, fulfilled when air was 
replaced by benzene; but further description of these experi- 
ments is omitted from consideration of space. 


§ 18. Inferences from results obtained with Null Solutions 
of KCN and of KCl.—Summarizing the results of the two 
series of experiments described, it is seen that, with respect 
to the p.d. Hg | electrolyte as considered by Palmaer, there 
are two kinds of null solution. One is formed by the addition 
to the electrolyte of very small quantities of certain sub- 
stances which leave the shape of the electrocapillary curve 
unaltered, with the maximum undepressed, but move it parallel 
to itself towards the left. This kind of null solution, inter- 
preted in the way described by Palmaer, will give the result 
that the p.d. is zero at the maximum of the electrocapillary 
curve. 

A second kind of null solution is obtained by using a 
different electrolyte and altering its concentration until the 
maximum of the capillary curve is at the origin. This kind 
may have a depressed maximum compared with the original 
electrolyte, KCl for example, and will in that case give a 
value, for the p.d. Hg | KCI, greater than that obtained by 


Phil. Mag. 8. 6. Vol. 15. No. 88. April 1908. 21 


494. Messrs. Smith and Moss on the Contact 


Palmaer if the measurements are interpreted in the way he 
describes. If, on the other hand, the curve for the null- 
solution electrolyte has a higher maximum than the corre- 
sponding curve for KCl, then the p.d. Hg | KCl found by 
Palmaer’s method will be less than that which he gives. 
We have not attempted to find a null solution exactly of this 
last kind, although it is known that the electrocapillary 
maximum of a saturated KOH solution is near the origin 
and is considerably higher (Phil. Trans. l. ¢, p. 68) than that 
of concentrated KCl. 

We have, however, obtained null solutions by the addition 
of small quantities of Na,S (and acetic acid) to n/10 KI 
solution, which has a very depressed maximum, and to 
n/10 KOH solution, which has a higher maximum than 
n/10 KCl. The results of these experiments exhibit the 
truth of the conclusions we have drawn. , 


§ 19. Null Solution of KI.—The attempt to obtain a null 
solution from 2/10 KI succeeded approximately when the 
composition of the solution was about ‘001n Na.S + -099nKI. 
The maximum surface-tension occurred when the applied 
H.M.F. was —-01 volt. It was unnecessary for our purpose 
to obtain a nearer approximation to a null solution. The 
E.M.F. between a dropping electrode falling through 
hydrogen into this solution and a still electrode was at first 
imperceptible. After some time it was ‘006 volt. Mercury 
in this solution measured against mercury in 7/10 KCl gave 
an E.M.F. of :793 volt. <A fresh null solution of KCl, made 
up by trial for comparison, measured against n/10 KCl in 
the same way gave ‘365 volt. The two solutions measured 
against each other gave °228 volt. The results of the 
measurements are tabulated below:— 


1. 2. 3. 4, 5. eae 7. 


er 


| Horizontal 
Max.8.T.|E.M.F. for, distance to | im | null } null 


Null Solution. (scale- | max. 8.T. |” Ol curve! 
reading). 10 
volt. volt. volt. volt, volt. 
Pat," neat oath on 
| 228 


n | 
{0 KCl+2Na,8...| 30°5 — ‘Ol | ‘569 565 | = — Ol 


Potential Differences determined by Null Solutions. 495 
These experiments show that the p.d. Hg | n/10 KCl 


cannot be 0°57 unless, interpreting them according to 
Palmaer’s method, the p.d. between n/10 KCl and n/10 KI 
is nearly a quarter of a volt. Such a potential-difference is 
impossible according to the theories at present in vogue 


(cf. Phil. Trans. 1. c. p. 62). 


§ 20. Null Solution of KOH.—In attempting to obtain a 
null solution from n/10 KOH it was found that the addition 
of NaS (‘001n to -002n) produced a gradual shift of the 
maximum to the left which continued over a long time. A 
solution with the maximum at the origin could not be obtained 
conveniently with Na,S alone on account of this time effect 
It was found however that if, after the solution had stood 
for some time over the mercury, dilute acetic acid was added 
drop by drop and the solution stirred, the maximum could be 
brought from the left to zero and the time effect was now 
negligible. The results of the experiments with this null 
solution are given below :— 


me 2 | 8: 4 ain 5. | 6. 
| | | Horizontal | 
‘Max.S.T./E.M.F. for distance to | | null.; 2 | null 
ee (scale- | max. S.T. Uae ae | | sol. } 10 KCl | + } sol [+ 
“| yeading).! ‘10 : 
| volt... | volt. volt. volt. 
—<_o  . |... or a —————— 
n | 
ae | | | 
tres 30°61 | +01 | “515 509 008 
+HC,H,O | 


| 


In considering the value 0:509 volt in column 5 it has to 
be remembered that there is now an appreciable p.d. between 
the solutions. Its value calculated in the usual way is 
n/10 KOH | x/10 KCl=-016 volt at 20° C.; and conse- 

uently we now obtain, according to Palmaer’s method, 
Hg | n/10 KCl1=°525 volt. But, according to the interpre- 
tation we have offered, this result signifies only that the 
potential reckoned from the solution to mercury at the 
maximum is about *04 volt less in the case of n/10 KOH 
than in that of n/10 KCl. A result which, as before, is in 
agreement with the forms and relative positions of the 
electrocapillary curves. 

2L2 


SS, 


8 


z 


Surfa de 7ension Re adings. 


496 Potential Differences determined by Null Solutions. 


The electrocapillary curves for the null solutions of KI 
and of KOH are shown in fig. 5. 


aa 


a 


ahh | 
Applied \E./0.F s 1000 = is IE Voll: 


O 2500 5000 7500 


Fig: 5 


The results of our experiments may be summarized as | 
follows :-— 


§ 21. Summary of Conclusions. 


1. The potential-differences between different null solutions _ 


and mercury are not the same. This is proved, allowing for ~ 
the contact p.d. between electrolytes, by measuring each 
p-d. against the p.d. Hg | KCl, and also by measuring one — 
null solution against another. The result is also deducible | 
from the electrocapillary curves alone, without introduction 
of the question of the p.d. between electrolytes. 

2. If the null-solution potential-differences are assumed to 
be zero, the values obtained for the p.d. Hg | n/10 KCl vary 
from about 0°53 volt to 0°79 volt. Assuming that the p.d. 
at the electrocapillary maximum is most likely to be zero 
when the maximum is undepressed, it is probable that the 
p-d. Hg | n/10 KCl does not exceed 0°53 volt. 

3. The polarizing H.M.F. required to produce the maximum 
surface-tension between mercury and a given electrolyte has 
been proved by Paschen to be equal, in many cases, to the 
E.M.F. of the corresponding dropping-electrode circuit. 
This relation is shown to be true in four particular cases in 
which each E.M.F. is equal to zero. It is also shown that 


Lateral Vibration of Bars supported at Two Points. 497 


in none of these cases is the p.d. Hg | electrolyte necessarily 
zero. 

4. It is shown that the Paschen relation immediately fails 
when the chemical action at the drop-electrode due to 
atmospheric oxygen becomes appreciable. 

5. The Paschen H.M.F. between mercury and a solution 
of KCl remains unchanged on the addition to the solution of 


_ small quantities of Nass, although the natural p.d. between 


mereury and the KCl solution is thereby altered by more 
than half a volt. 

6. A critical percentage of Na,S was found for which the 
natural p.d. just mentioned altered with extreme rapidity 
as in the cases studied by Behrend, in which he observed the 
variation of the E.M.F., Hg | He, (NOs). | Hg, produced 
by the gradual addition of KCl or KBr at one electrode. 
In the present case the change in the p.d. Hg | electrolyte. 
was deduced directly from the electrocapillary curves. 


XLVI. On the Lateral Vibration of Bars supported at Two 
Points with One End overhanging. By Joun Morrow, 
M.S¢., D.Eng.; Lecturer in Engineering, Armstrong College 
(University of Durham)*. 

W HEN a bar is carried on two supports and has one end 


overhanging by an amount ¢, as shown in the figure, 


ed 


ee ny) Sie BE See ee a a 
its natural period of vibration is to be determined from the 
equation 


(cosh ml sin m/—sinh ml cos ml) (cosh me sin me—sinh me cos me) 


— 2 sinh mlsin ml (1+ cosh me cos me) =0, — 
in which 


m = (2aN)} (pw/EI)?; 
E = Young’s Modulus for the material ; 


J = the moment of inertia of the section about an axis 
perpendicular to the plane of bending ; 


w = sectional area of bar (assumed uniform) ; 
p = density of material ; 
N = frequency of natural vibrations. 
* Communicated by the Physical Society : read November 8, 1907. 


oe ) 


3 


498 Lateral Vibration of Bars supported at Two Points. 


For given ratios ¢c/] we can obtain values of ml from the 
above equation. This has been done incidentally by Professor 
Dunkerley i in connexion with his work on the whirling of 
shafts *, but the calculations were not sufficiently extended to 
give very accurate results. By comparison with the table 
given below, it will be seen that the approximate formula 
used by Dunkerley cannot be relied on to give more than 
the first two significant figures. 

Having recently required more accurate solutions, I have 
found it necessary to make a more elaborate calculation, and 
as the results have been obtained to six figures it appears 
desirable to place them on record. 

If we write a for c/l and @ for ml, and expand each term 
of the equation in ascending powers of 6 and al, we get, as. 
far as the twenty-first powers, 


46? —(-044 +4 a3 +3 a)6° + (000035273369 + 0021164 a? 
+0037 at + 0021164 a7 +-00079365 a®)6~ (58730 + 
106°8890 a? + 293'945 at + 1007-811 a? + 8817834 a8 + 
106°889 at! + 26-722 a!?)10-864 + (-033 + 13'2 a3 +49 at 


+508°8 a7 +700 a8 + 509 al! +297 a? +13°2 a? 4+ 2-4 a®)10-Ne® 


=). 


Assuming values of a°and calculating @ we obtainjthe 
numbers tabulated below:— 


Ratio c/?. | Value of @. 
Unity tet ene 1°50592 
Three-quarters .. 1:90170 
Onechalf (2 oh wee 2°51895 
One-third) cps. 2-94042 
One-quarter .... 3°05881 
One=titth . sea eee 3°9975 
Onessixdis hae 3°11752 
One-seyenth .. 7 3°12647 
One-eighth ...... 3°13148 
One-ninth “'s 20. 3°18449 
Onéetenin VR. a 3°13641 
Zero 531. Fee eee 3°14159 


* Phil. Trans. A, 1894, p. 279. 


On Gibbs’s Theory of Surface-Concentration. 499 


Dr. Chree has given the following approximate formula 
for use when c// is small *:— 


ml=m(1—4n? c/I?), 


It is stated to be satisfactory so long as (c/l)s°/6 coth + 
is small compared with unity. Asa matter of fact we find 
that, when c// is one-sixth, Chree’s formula is correct to within 
0-005 per cent., and the error is less than one per cent. even 
when c// is one-half. 

The number of vibrations per. second is given by the 
formula 

4 : 
pe 
27 VV pal* 
September 1907. 


XLVI. An Experimental Examination of Gibbs’s Theory of 


Surface-Concentration, regarded as the basis of Adsorption, 
with an Application to the Theory of Dyeing. (From the 
Muspratt Laboratory of Physical and Electrochemistry, 
University of Liverpool.) By W. C. M. Lewis, M.A.+ 


CoNTENTS. 


I. Object of the investigation; the characteristics of Adsorption 
Phenomena. ; 
If. Theoretical Discussion ; Gibbs’s Theory of Surface Concentration ; 
Milner’s Calculations. 
If]. Experimental Methods and Apparatus; Results. 
IV. Application to the Theory of Dyeing. 
V. Summary. 


I. OBJECT OF THE INVESTIGATION. 


NDER the term “‘ Adsorption” are grouped phenomena 

which may be regarded as forming an intermediate 

stage between chemical combination on the one hand, and true 

absorption or solution on the other. Different types of 

Adsorption have been studied, but in general a solid substance 
(e.g. charcoal) has been utilized as the adsorbent body. 

The present paper is an account of an experimental attempt 
at measuring adsorption effects quantitatively and interpreting 
the values obtained in the light of Gibbs’s theory of surface- 
concentration. The treatment is novel in that the adsorption 
has been measured at a liquid-liquid interface ; for it is only 
when dealing with liquid interfaces (or surfaces) that it is 


* Phil. Mag. [6] vol. vii. p. 517, May 1904. 
+ Communicated by the Physical Society ; read February 28, 1908. 


500 Mr. W. C. M. Lewis: Experimental Examination 


possible to measure interfacial- or surface-tension—a funda- 
mental factor in the thermodynamic consideration of the 
question, 


The Characteristics of Adsorption Phenomena. 


The earlier investigations on adsorption dealt in general 
with the adsorption of gases on the surfaces of various solid 
materials. The case of vapours is of much more recent date, 
the most important instances being those of Travers * and 
Trouton t. Modern investigators have, however, dealt 
chiefly with the question of the adsorption of dissolved sub- 
stances at the surface of some solid body in contact with the 
solution. Among the numerous recent memoirs on the 
subject may be mentioned those of Kiister , Schmidt §, 
Walker & Appleyard ||, Biltz 1, Lagergren **, and in par- 
ticular Freundlich ++. Freundlich’s investigations contain 
the most systematic and accurate determinations yet recorded 
of the adsorption at a charcoal surface of a series of organic 
substances in aqueous solution. The general method was to 
prepare a solution of a certain substance (say a dye-stuff), and 
to a known volume of this solution to add a certain quantity 
of very finely powdered pure blood-charcoal. A certain 
amount of the dye is removed by the charcoal, the change in 
concentration being determined colorimetrically. 

Among Freundlich’s most important results may be men- 
tioned that the equilibrium state is reached exceedingly 
rapidly—in about five seconds; that, using the same solid 
body the quantity adsorbed varies with the chemical con- 
stitution of the solute, e.g. aromatic acids are adsorbed in 
greater quantity than aliphatic ; and finally, that the effects 
differed with the solvent used. 

We may sum up the characteristics of adsorption which 
differentiate it from chemical combination thus :— 

(1) The order in which a series of dissolved substances 
are adsorbed does not differ even when one sub- 
stitutes as the adsorbing material, bodies as unlike 
as charcoal, silk, clay, and cotton. This is unlikely 
on the assumption of chemical combination. 

* Travers, Proc. Roy. Soc. series A, vol. Ixxvi. p. 9 (1907). 

+ Trouton, Proc. Roy Soc. series A, vol. xxvii. p. 292 (1906). 

t Kiister, Zectschrift Phys. Chemie, vol. xiii. p. 445 (1894). 

§ Schmidt, Zeit. Phys. Chem. vol. xv. p. 60 (1894). 

|) Walker & Appleyard, Journ. Chem. Soc. vol. lxix. p. 1834 (1896). 

€ Biltz, Ber. d. deut. chem. Gesell. vol. xxxvii. p. 1706 (1904); 
vol. xxxviil. p. 2963 (1905). 

** Lagergren, Zeit. Phys. Chem. vol. xxxii. p. 174 (1900). 

+t Freundlich, Zezt. Phys. Chem. vol. lvil. p. 385 (1906); vol. lix. 
p. 284 (1907). 


fal @ 
- 
oe P 
» 
% 
fi 


of Gibbs's Theory of Surface- Concentration. 501 


(2) The reaction goes to an end almost instantaneously. 

(3) Heat effects, even in concentrated solutions, are 
undetectable. Chemical action, on the other hand 
at relatively low temperatures is usually accompanied 
by evolution of heat. 


Characteristics which distinguish Adsorption 
from Absorption. 


The difference existing here (which has been attested by 
numerous determinations) may be stated thus: There is not 
a direct proportionality between the concentration of the 
solution or partial pressure of the solute and the amount 
adsorbed. To take an example from Walker and Appleyard’s* 
paper on the adsorption of picric acid at the surface of silk. 

The quantities of picric acid (remaining after the reaction) 
in the aqueous solution and in the silk itself are estimated, 
the following being the results obtained :-— 


TABLE LI. 
milligrams Picric Acid 'milligrams Picrie Acid Raiiere 
' inice.e.solution. ‘| in 1 gram silk. a 
aR Pal Lila 
(a) (b) : | 
0-064 oye. 0-005 | 
1-98 37 0°053 
70 75 0-094 


Assuming the molecular weight of the picric acid to be the 


: : ok 
same in water and silk, the ratio — should have been constant 


b 


had true absorption taken place. 


If. Tarorerican Discussion. 
Gibbs's Theory of Surface Concentration. 


A theoretical investigation of this subject from the stand- 
point of thermodynamics forms one of the chapters in Gibbs’s F 
memoir on “Kquilibrium in Heterogeneous Systems.” He 


* Walker & Appleyard, loc. cit. 
+ ‘Scientific Papers of J. Willard Gibbs.’ English edition, vol. i. 
p- 219. 


502. Mr. W. C. M, Lewis: Experimental Examination — 


first takes up the consideration of the mass or bulk equi- 
librium of a heterogeneous system, 2. e. equilibrium in which 
any surface phenomenon is of insensible magnitude, and then 
proceeds to examine the case in which the surface area is 
relatively large and the influence of surfaces of discontinuity 
upon the equilibrium of heterogeneous masses becomes of 
importance. ‘To use his own words :— 

*“ The solution of the problems which precede may be 
regarded as a first approximation in which the peculiar state 
of thermodynamic equilibrium about surfaces of discontinuity 
is neglected. To take account of the condition of things at 
these surfaces, the following method is employed :-— 

“Yuet us suppose that two homogeneous fluid masses are 
separated by a surface of discontinuity, i.e. by a very thin 
non-homogeneous film.. Now we may imagine a state of 
things in which each of the homogeneous masses extends 
without variation of the densities of its several components, 
or of the densities of energy and entropy, quite up to a 
geometrical surface (to be called the dividing surface) at 
which the masses meet. We may suppose this surface to be 
sensibly coincident with the physical surface of discon- 
tinuity. 

‘Now if we compare the actual state of things with the 
supposed state, there will be in the former in the vicinity of 
the surface a certain (positive or negative) excess of energy, 
of entropy, and of each of the component substances. These 
quantities are denoted by e°, °, mj, m§, etc., and are treated 
as belonging to the surface. The * is simply used as a 
distinguishing mark, and must not be taken for an algebraic 
exponent. 

“Tt is shown that the conditions of equilibrium already 
obtained relating to the temperature and the potentials of the 
homogeneous masses are not affected by the surfaces of dis- 
continuity, and that the complete value of 6¢* is given by the 
equation 


de§ = C67? + ods + 10m + pdm + etc. 


in which s denotes the area of the surface considered, ¢ the 
temperature, #4; fl. ete. the potentials for the various com- 
ponents in the adjacent masses..... “ j rei eee 

“The quantity o we may regard as defined by the [above] 
equation itself or by the following: 


&=tn* + os + pymi + poms + ete. 


* Gibbs, ‘ Scitntific Papers,’ vol. i. p. 365. 


of Gibbs’s Theory 07 Surtace- Concentration. 503 


from which by differentiation and comparison with the former 
we obtain 


do= —n dt—Vydp, — Prd, — ete., 
where »,;, T',, T',, etc. are written for 


a a ae elC., 

s s s 
and denote the superficial densities of entropy and of the 
various substances. We may regard o as a function of 
t, #1, fe, etc., from which, if known, 7,, Ty, I'., may be 
determined in terms of the same variables. An equation 
between o, ¢, M1, Mz, etc. may therefore be called a jfunda- 
mental equation for the surface of discontinuity.” 

The final equation obtained above has been simplified and 
applied by Gibbs * to an actual case, viz. :— 

“Tf liquid mercury meet the mixed vapors of water and 
mercury in a plane surface, and we use w, and pz to denote the 
[chemical] potentials of mercury and water respectively and 
place the dividing surface so that [', = 0, 7. ¢., so that the 
total quantity of mercury is the same as if the liquid mercury 
reached this surface on one side and the mercury vapor on 
the other, without change of density on either side, then 
T,,, will represent the amount of water in the vicinity of the 
surface above that which there would be if the water-vapor 
just reached the surface without change of density, and this 
quantity (which we may call the quantity condensed [2. e., 
adsorbed! upon the mercury) will be determined by the 
equation 

da 


D,. = — djs 
In this equation and the following, the temperature is 
constant and the surface of discontinuity plane. 
“ Tf the pressures in the mixed vapors conform to the law 
of Dalton, we shall have for constant temperature 


dp, = Cd pry 5 


pz denotes the part of the pressure in the vapor due to the 
water-vapor, and ¢ the density of the water-vapor. Hence 


do re 
SF OPatas . e e e ° es 
dps 


ise 


* Gibbs, ‘ Scientific Papers,’ vol. 1. p. 235. 


004 Mr. W.C. M. Lewis: Experimental Examination 
Now applying the gas law P = RTc, finally we obtain 


This equation is of fundamental importance in that, if we 
assume its applicability to adsorption measurements, it con- 
tains the relationships between bulk-concentration, surface- 
tension, and quantity adsorbed in an_ experimentally 
determinable form. 

The same expression may be obtained more simply as 
follows :—* 

Consider the equilibrium at a surface (say of solid or 
liquid) in contact with a solution. 

Let o = surface-energy per unit of surface. 

s = area of surface exposed to the solution. 
m= the mass of solute adsorbed at the surface of the 
solid in excess of that normally present. 

The temperature is supposed to be constant. 

U = total energy of the heterogeneous film per unit 
of surface. 


U can be increased :— 


(1) by increasing the surface area, in which case the work 
required = ods ; 

(2) by increasing the concentration of the solute in the 
interfacial layer. This is proportional to dm* namely 
pdms, where pw is the chemical potential of the solute. 


Then dU = cds + pdms, 
d(U — pm’) = ods — midu. 


And since d(U — um’) is a complete differential, we have 


ac =($) 
ia) ds J, 


but = mass adsorbed per unit area of surface, 2. e. = I’; 
do 
f= =. 
du 
; - 
Now since du = RT, 


where c = the concentration of the solute in the 
bulk of the solution, 


* I am indebted to Prof, W. B. Morton for this deduction. 


of Gibbs’s Theory of Surface-Concentration. d05 
it follows that 


Norr.—It should be noted that the expression obtained above occurs. 
in Freundlich’s paper already referred to. From the nature of his. 
experiments it has been quite impossible to attempt a verification 
experimentally. 


We have now to prepare a solution of certain strength ¢ 
and measure the quantity adsorbed I’ as well as the value of 


a, On substitution of the values in the equation we may 


de 

obtain a verification or otherwise of the expression *. It 

may be stated at once that the results of experiments to be 

detailed do not show equality on the two sides of the ° 

equation—the discrepancy being always in the direction 
' 

of T being many times greater than eee 


RT de’ 
IJ. ExpertimentaL MertuHops. 
Use of a Liquid-liqud Interface. 

As already stated, previous investigations with the ex- 
ception of Milner’s have dealt solely with adsorption at a 
solid surface—which does not allow of its surface-extent or 
surface-tension being determined. As it is necessary to 
evaluate these quantities, the possibility presented itself of 
measuring the adsorption of a solute at the interface between 
its solution and another liquid. This latter liquid must of 
course be absolutely chemically inert towards the solute 
itself and its solution. Choice fell on a hydrocarbon oil, 
since a body such as this is characterized by its chemical. 
inertness. The next object was to obtain a _ substance, 
preferably soluble in water, which would possess the property 
of lowering the interfacial tension between water and oil. 
From their behaviour at the air-liquid interface it seemed 
likely that the soaps, saponine, and bile-salts would act in 


* While this work was in progress, the results of somewhat similar- 
determinations were published by Milner (Phil. Mag. Jan. 1907 [6] vol. xiii. 
p- 96). These referred to equilibrium at the az-liquid surface. He has 
deduced an identical expression to that already given in this paper, and 
has applied it to acetic acid in aqueous solution and aqueous sodium 
oleate. He has found that the actual quantity removed from solution: 

3 c do 
(I) is nearly ten times the calculated effect (at de): 

Reference might also be made to experiments by Zawidski (Zed. Phys. 
Chem. xxxv. p. 77 (1900)) on the formation of the foam produced by the 
addition of saponine to aqueous solutions of hydrochloric and acetic acids. 
No measurements applicable to the present case were recorded. 


i oo oe ae 
: ; : 
a3 


506 Mr. W.C. M. Lewis: Experimental Examination 


this way at the oil interface. The first experiments to be 
described were carried out with aqueous solutions of bile-salts. 

In order to make certain that no chemical or solubility 
effects existed between the oil and the solutions, the following 
tests were carried out :— 

(1) A portion of the oil was shaken up with water which 
was tested with phenolphthalein—neutral reaction, and 
therefore absence of free fatty acid as impurity. 

(2) A portion of the oil was boiled for three hours with reflux 
condenser with i0 methyl alcoholic potash—excess potash 
being titrated with standard acid. It was found that the 
acid required for neutralization was the same in amount as 
that for a blank experiment in which no oil was present. 
This proves the absence of fatty esters in the oil. 

(3) Some of the bile-salt powder was shaken up with the 
oil, the latter allowed to stand, filtered, and a ‘““drop-number” 
taken with the pipette (see later) against distilled water. 
The same drop-number was obtained as with the untreated 
oil fresh from the stock. Since the tension is a very delicate 
test for the presence of the salt, the above shows fairly 
conclusively that there was no salt present in the oil after 
filtration, 2. e., the salt is insoluble in the oil. 

(4) The oil was shaken up with a solution of the salt, the 
oil filtered off and a drop-number taken against water. The 
same resuit exactly was obtained as with the oil which had 
been simply shaken up with distilled water filtered off and a 
similar drop number taken against water. 

The above tests point conclusively to the fact that no 
effects of the nature of chemical combination or solubility 
take place between the oil and the bile-salt. 


Examination of the Bile-salt. 


A quantity of “sodium glycocholate ” was obtained from 
Merck. On close examination, however, it appeared that — 
this was far from pure. Besides sodium glycocholate there 
is also sodium taurocholate and other fatty acid alkali-salts, 
There were no inorganic substances such as sodium chloride 
or carbonate. Several determinations of the molecular 
weight (which is required, as will be seen later) by means of 
the lowering of freezing-point and rise of boiling-point of 
water gave as a result 140. Assuming complete dissociation 
in water, this would give the undissociated molecular weight 
280. This latter was confirmed by a determination by the 
rise of boiling-point of alcohol, the result being 283. The 
osmotic molecular weight in water (viz. 140) is, however, the 
quantity required in the subsequent calculations. 


of Gibbs's Theory of Surface-Concentration. 507 


Method of measuring the Interfacial Tension. 


The Drop-Pipette method was employed, the apparatus 
being of the form shown in fig. 1. This 

ye). was first filled with oil (by suction at C) 

i up to the mark HE. The tap B was then 
closed during a determination. The aper- 
ture F is carefully wiped before using, 
and the pipette placed in a fixed position 
relatively to the solution in all deter- 
minations. The bulb D is of about 45 c.c. 
capacity between Hand H,. The distance 
HF is about 8 cms. As soon as the tall 
jar containing the solution is placed in 
position, tap A is turned full on, the 
constriction above only allowing very 
slow entrance of air ; and hence the drops 
of oil rismg through the denser solu- 
tion are formed with exceeding slowness 
—at 12 to 15 secs. interval. This, as 
Lord Rayleigh * has pointed out, is of 
great importance in order to obtain 
fairly accurate determinations; the approx- 
imateness of the method being due to 
the application of a statical theory to 

what is really a dynamic phenomenon. 

The method consisted simply in count- 
ing the number of drops formed while the 


oil fell from E to Kj. 


Theory of the Drop-Pipette Method. 


It may be used in the first instance 
simply to determine the relative tensions 
of solutions of different concentrations, 
thus :— 

Let V = total volume of oil used, 
n = total number of drops formed, 


then the volume of each drop = ac) 


and if p=the density, 

Vp 

= 

oe tension o is taken to be proportional to the weight of 
a drop, 


the weight of a drop= 


r= K YP, 
n 


* Lord Rayleigh, Phil. Mag. [5] vol, xlviii. p. 321 (1899), 


As. ‘> Ly Ps 
rah 


008 Mr. W.C, M. Lewis: Experimental Examination 


Similarly for another liquid 


Oo; = K MP1, 
, ny 
Hence 
T nip 
Oo} np, 


As a result of the exhaustive work of Guye and Perrot *, 
Kohirausch +, and Lohnstein t, a determination of surface- 
tension in absolute measure by means of the drop-pipette 
can be obtained as follows :— 

The expression giving the weight of a drop in terms of 
the tension is 

g = ra®, 
where a=tension in millgrms./mm. 
g=weight of a drop in milligrams. 
y=radius of the orifice in mms. 


® is a function of (<). where 
eee 2a 
a? is given by a 
p being the density of the liquid in the pipette. 


This applies to cases of liquid-gas tension, but it may 
perhaps be extended to the present. case by taking into con- 
sideration the density of the medium in which the drop of 
oil is formed, 7. e. the water. 

Calling the density of the water p,, we have 


2 ae 
Pw Po 
We have now to evaluate ®. The ordinary method of 


using the Lohnstein-Kohlrausch formula is to take some 
approximate value for «; from this calculate a, and hence 


cal 


(“). Kohlrausch § has given a table showing the values 
of ® for different values of el and by inter- or extra- 


polation ® can be evaluated. We then compare the value 
of g given by the expression ra® with g actually obtained 


by experiment ; and finally, by a series of approximations, 


* Guye & Perrot, Archives d. Sc. Phys. 4th ser. vol. xi. p. 225 (1901) ; 
vol. xv. p. 182 (1903). 

t+ Kohlrausch, Ann. d. Physik, vol. xx. p. 798; vol. xxii. p. 191 (1906). 

{ Lohnstein, Ann. d. Physik, vol. xx. pp..237, 606; vol. xxi. p. 1030 
(1906). 

§ Kohlrausch, Joc. cit. p. 805. 


of Gibbs's Theory of Surface- Concentration. 509 


@is given sucha value as to make the calculated and ob- 
served values of g identical. 
Now a determination by Pockels* gives the value 48°3 
dynes as the tension between water “and a petroleum oil.” 
Assume in the present case 


a=48 dynes per cm. 
=4°89 milligrams per mm. 


Then C= URS = 
“m= 9°85 
joe aye 


Hence (F )=0-182. 


Unfortunately this value of (=) is a little beyond the lower 


limit of Kohlrausch’s table, viz. :— 


= p, 
0-300 445 
0-288 4-48 
0-200 4-66 


For Je 152, the value of ® by inspection of these 


numbers (assuming the values do not change in direction 


down to (<) = °152) would be 
Dies 
hence g = 34°62 milligrms. 


Now g found experimentally =35:00 milligrms. 

The value for the interfacial tension was therefore taken 
to be 48 dynes per em. To distinguish this oil from another 
hydrocarbon oil which was found to have a tension of 
33°6 dynes, they will be throughout designated oils “A” 
and “ B,” viz. :— 


Interfacial { Water-oil A ......... 33°6 dynes per cm. 
tension | Water-oil B ......... 48 dynes per cm. t 


* A, Pockels, Wied. Ann. lxvii. p. 668 (1899). 

+ Note.—It may be as well to state here that although oil A was as 
carefully tested as oil B, the writer would lay greater stress on the 
accuracy of the determinations made with oil B. 


Pint. Mag S26. Vol. 15. No. 885 April 1908... 2 M 


210 Mr. W. C. M. Lewis: Haperimental Examination 


Variation of the Interfacial Tension with the Concentration 
of Sodium Glycocholate Solutions. 


First SgRIEsS.—Orn A. 


Aqueous solutions of the sodium glycocholate were pre- 
pared of the following concentrations :— 


1°/, 0°8°/, 05°/, 0:33, 025°, 0-18, 0:125°, 
| 0-1°/, 0:05°/, 0-01°/, 0:001°/,. 


‘On being freshly prepared, the aqueous solutions of these 
concentrations were quite transparent, but with considerable 
rapidity—2 to 3 hours—in the case of the more dilute, a 
white turbidity manifested itself. All tension measurements 
were therefore made with fresh transparent solutions so that 
the values obtained might be quite comparable. 

The following table gives the results obtained with oil A. 
The first column gives the concentration of the solution ; the 
second, the corresponding drop-number ; the third gives the 
relative tension as deduced from the relation already obtained 
(pp. 507, 508), viz., that the drop-numbers are inversely as the 
tension ; the fourth column gives the absolute values of the 
tension based on the oil-water determination 33°6 dynes/em. 
and by applying the values of column 3. 


Tapne 11.-—Oil A. 


| -Per‘eent. |") Drop= | Relative Tension 
Concentration. number. Tension. | dynes/em. 
0 ig3p, | . 1-000 33°60 
0001 191 0-958 32°20 
0-01 2 | Os 18 08 
0-05 2 OW 25°21 
Ol 280 | 0654 21-96 
0:125 297 | -O'GIG 20°70 
0-18 B6D he Ol) OD 16°81 
0:25 418 | 0-438 1471 
0°33 489. | 0:974 12°57 
05 5386 | 0341 11-48 
0:8 558 0°328 11:03 
1:0 566 0323 10°86 


The above values in columns one and four are plotted in 
fig. 2. It will be noticed how exceedingly marked is the 
lowering effect of the sodium glycocholate upon the inter- 
facial tension. 

We are therefore, now, in a position to obtain an experi- 


of Gibbs’s Theory of Surface- Concentration. 511 


mental value for = by simply taking the value of the 


tangent to the curve fig. 2 at the desired concentration. 
Fie. 2. 


AGE CONCENTRATION 
sad Oo” 0.2 0.3 Ob 0.5 OG OF Os 04 ra 
OiL A. Sod.GiyeacuoLeaATE A. 


Szconp SERIES.—O1L B. 


A similar series of determinations with oil B are given in 
the following table :— 


TABLE Ll “OireR: 


Per cent. Drop- Relative Tension 
Concentration. | number. Tension. | dynes/cm. 
0 E23) Aylin HOOD: »)' 48 
0°0312 202 0633 | 30°38 
00625 [heer Oigaee ||. | 26°35 
07125 317 | 0-404 19°39 
0165 319, | , 0341 16°37 
0-200 417. |) 0807 14°73 
0°250 P MAg ee Ogee 13:05 
0°300 522 0-245 11°76 
0°330 531 0241 11°57 
0360 557 0:230 1104 | 
0400 o6e” | O:225 10°80 
0°500 | 585 0219 10°51 


bo 
bod 
_— 
UNS) 


912) Mr. W. C. M. Lewis: Lzperimental Examination 


The above absolute values of Tension and Concentration 
are plotted in fig. 3. 


PERCENTAGE 


Ol Bo Sopatycoctoteate B. 


In-order to ascertain how far the values for the Tension 
obtained directly from the drop-number (column 4, Table II.) 
agreed with those obtained by applying Kohlrausch’s formula 

g=ra® 
in each case, a comparison is made of the weights of the 
respective drops, using the values of the tensions for the two 
concentrations *25 per cent. and °5 per cent. 


| 

Per cent. Weight of drop | Weight of drop | 

) Concentration. g=ra®. actually found. | 
ie ee ee 
: 
| zero, 7.e. distilled water | 34°62 milligrms. | 385 milligrms. | 
. } 
25 per cent. | 8°89 a 8-60 tA | 


BS beduk Le Oe ve 6:93)... 


of (ribbs’s Theory of Surface- Concentration. 513 
Having in the foregoing obtained data for the experimental 


determination of the coefficient in the case of two dif- 


de 
ferent oils, the next procedure was to determine the value 
of I’, 2. e. the mass adsorbed per cm.? of oil-surface in excess 
of that which would have been there had no adsorption 
taken place at all. To express this quantity I’ more briefly, 
the term ‘Adsorption Coefficient,” following the nomenclature 
of Ostwald, has been employed. 


Measurement of the Adsorption Coefficient. 
Although the Adsorption Coefficient is defined as the 


excess mass of solute adsorbed, yet since the normal amount 
of solute per square cm. surface (when no adsorption has 
taken place) is very small in the present instance, this 
normal amount may be neglected in comparison to the total 
amount adsorbed per square cm. ‘This is justifiable, as is 
evident from the following figures from an experiment 
detailed later :-— : 

The bulk concentration of the glycocholate solution 
=*25 per cent. or ‘0025 grams per c.c. 

Assuming the thickness of the surface layer to be of 
the order of the range of molecular attraction, namely, 
13-4 x 10~° ems. (according to Parks*), while the quantity 
adsorbed per cm.” in this particular case is 5-4 x 10-® gram, 
the layer evidently possesses a concentration of ‘403 gram 
per c.c. The surface concentration is therefore about 160 
times the bulk concentration. 

It has been assumed that the thickness of the layer is of 
the dimensions of the range of molecular action. Of course 
the actual case is that there is a gradual “shading off” of 
the excess from the surface into the bulk of the solution, 
but practically all the excess is in the molecular surface- 
layer. This follows from the fact that the magnitude of the 
adsorption depends on the surface-tension, which is a pheno- 
menon whose effects do not extend beyond the range of 
molecular attraction. As Gibbs + says:—‘It is only within 
very small distances of such a surface that any mass is 
sensibly affected by its vicinity—a natural consequence of 
the exceedingly small sphere of sensible molecular action.” 


The experimental determination of the adsorption co- 
efficient of sod. glycocholate at the surface of a hydrocarbon 


* Parks, Phil. Mag. [6] vol. v. p. 517 (1903). 
+ Gibbs, ‘Scientific Papers,’ vol. 1. p. 219. 


d14° Mr. W. ©. M. Lewis: Experimental Examination 


oil was carried out, employing two distinct methods accord- 
ing as the adsorption took place under one of the two following 
conditions :— 


(1) Adsorption at a very curved surface. 
(2) Adsorption at an approximately plane surface. 


(1) Adsorption at a very curved surface. 


A certain volume of sod. glycocholate solution of known 
strength is shaken (by a motor-driven shaker) for several 
hours, with a known volume of oil so as to form a uniform 
emulsion. The bulk concentration of the solution after 
emulsification is estimated, and the fall in concentration gives 
the total quantity of glycocholate adsorbed. To measure 
the adsorbing area, the droplets of the emulsion are examined 
under a microscope having a scale of known value in the 
eyepiece, and the average diameter of a droplet taken. 
Hence we obtain the radius 7 and the volume of a droplet 

—‘rr*. Knowing the total volume of oil emulsified, we 
obtain the total number of droplets *, and since each drop 
has a surface area 47r?, we finally obtain the total adsorbing 
area. Dividing the total quantity adsorbed by this area, the 
adsorption coefficient is determined. 

The following are typical examples :— 


Determination with Oil A. 


One litre of sod. glycocholate solution approximately 
33 per cent. was made up, and of this 500 c.c. were shaken 
with °447 c.c. oil for 12 hours and allowed to stand 18 hours. 
The volume of oil was obtained by weighings—the density 
of the oil being “907. A drop of the emulsion was examined 
under the microscope. 


Average radius of a droplet ='0000425 cm. 
Hence surface area ot one dropiet=4 x 3°1416 x (:0000425)?. 
and volume of one droplet=4x 3°1416 x (:0000425)?. 
Total volume of oil emulsified =°447 c.c. 

*. Total number of drops formed=1°3 x 10”. 

*. Total adsorbing surface = 31,553 cm.? 
To estimate the quantity adsorbed :-— 
A drop-number was taken with the pipette through a 


portion of the original solution. Drop-number=483, cor- 
responding to a tension 12°80 dynes, which on fig. 2 indicates 


* Assuming that the density of an oil-droplet in the emulsion will 
sensibly coincide with the density of the oil in bulk. 


of Gibbs’s Theory of Surface-Concentration. 515 


a concentration 0°318 per cent. After adsorption had oc- 
curred, a drop-number through the emulsion gave 459 
corresponding to a tension 13°44 dynes, which again cor- 
responds on fig. 2 to a concentration *295 per. cent. 


.. Fall in concentration=*023 per cent. 


This was the general method adopted to determine change 
in concentration, it being much more delicate than any 
purely chemical means. It will be noticed that in the above 
the assumption is made that the emulsion particles them- 
selves would not affect the value given by the pipette for 
the tension of the solution in which the emulsion is suspended. 
Justification of this is afforded by the concordance between 
the results obtained for the adsorption by this method and 
by an entirely different method to be described later. 
Continuing :— 

Fall in concentration of :023 per cent.='115 gram 
for the 500 c.c. solution employed. 


Hence the mass adsorbed per cm.?, 7. e. 


bptneegl 65 
+ 5S 
P56 <10-° sram- 


Similar determination of 1, using Oil B :— 

The solution made up gave a drop-number 4531, corre- 
sponding on fig. 3 to concentration -317 per cent. 250 c.c. 
of solution were emulsified with -160 c.c. oil (density -900), 
and after emulsification the drop-number was 507, corre- 
sponding on fig. 3 to a concentration *290 per cent. 


Hence the fallin concentration ......... =°027 p. cent. 
.. Total mass adsorbed from the 250 c.c. soln.=:067 gram. 


Total adsorbing area ...... = 11058 cm.” 
"20-9 <10-° pram per: cart 


A further determination with a solution whose original 
concentration was *2 per cent. gave a value for the adsorption 
coefficient 

['=4-7 x 10~® gram per cm.” 


An estimation of the probable error of these values really 
depends on the estimation of the radius of the emulsion 
particle. As great care as possible was taken to obtain 


ee 


ee 


516 Mr. W. ©. M. Lewis: Haperimental Examination 


these very uniform in size. The radius of a drop is taken 
as being ‘0000425 cm. The maximum value would be 
represented by | 

r = ‘00005 em., 


and as a minimum, 


r = °000035 em. 


Substituting these values in the first determination with 
oil B, we obtain the following result :— 


r = 000035 cm., T= 4:4x10-° grm. per em? 
x= -000050. ,; BE = 6:0 1Of ge 3 
"0000425. ,, ee Tee be 

It is evident from these figures that the method is not 
sufficiently accurate to detect a different value of I’ cor- 
responding to solutions of strengths 317 and *2 per cent. 


respectively. The different values obtained are within the 
limits of experimental error. 


4 


Comparison of Experimental values of T with Calculated 
values. 


The calculated value for I is, as already pointed out, 
ce do 


“hide 
Let us take as a example the second determination with 
oil B :— i, 
The concentration of the solution =c="2 per cent. 
=°002 orm. per €.¢. 
Temperature T= 289 abs. 


R is the gas constant. For one gram of solute, 
_ 2x4:2x 10! ergs 
~ molecular weight 


The molecular weight determined as already described is 
140 in aqueous solution. 
oO 


The coefficient ie read from the curve fig. 3 at the point 


ae” 
where c=2 per cent., gave 
do _ 9°5 dynes Marah. 
de “QU2 "erin: perc. 


The true value of this probably lies between the limits 
4800-4700, 2. e., an error of about 2 per cent. 


—s ‘<o 
us o" 


or 
asad 
=~] 


of Gibbs’s Theory of Surface-Concentration. 


Hence 
e do — ‘002x4750x 140 
RY de 2x £210" x 289 
= 3°90 X 1Un erm. per cn.” 
I (found) = 4:7 x 10-® erm. per cm.? 


That is to say, the actual quantity adsorbed is about 
85 times the calculated amount. This is far beyond ex- 
perimental error, and there can be no doubt that a real 
discrepancy does exist, To confirm this, however, a different 
method of experimentally determining I’ was resorted to 
in which emulsions were dispensed with, as it was thought 
possible that the radius of the drops might be too near the 


value for the range of molecular forces. 


(2) Adsorption at an approximately plane surface. 
Fig. 4, In the previous emulsion method we have 
been dealing with tension and adsorption at 
very curved surfaces. Owing to the discre- 
pancy obtained in the above method, it is 
necessary to carry out determinations at a 
practically plane surface—i. ¢., one of very 
small curvature in comparison with the cur- 
vature of the sphere of molecular action. 
For this purpose the apparatus shown in 

fig. 4 was devised. It consists of a bulb A, 
of about 170 ¢. c. capacity, which is filled with 
oil. This bulb is connected to a long vertical 
narrow tube (15 metres high), having a rubber 
joint and pinchcock near the base so as to 
regulate the flow of oil. The upwardly directed 
nozzle of this tube i is inserted through a cork 
into a wider tube B (1 em. radius), which con- 
tains the solution through which the oil rises in 
the form of large drops. The tube B becomes 
constricted near the top, and then opens into 
a large cup-like vessel C. The sod.-glyco- 
cholate solution (about 1 litre) is poured into 
B and C until the constricted end of B is about 
3 ems. below the surface of the solution. The 
upwardly moving drops of oil adsorb on their 
‘surface some of the solute and carry it through 
the constriction, the drops eventually coalescing. 

The object of the constriction is to prevent the 
sod. elycocholate, which has once more been partially returned 
to solution (by the coalescence), from being carried back 


518 Mr. W. C. M. Lewis: Experimental Examination 


into the lower part of the tube. When A is empty, the 
tube B (which has a rubber joint near the top) is nipped at 
this point and the upper portion disconnected. The lower 
solution (which is 250 ¢.c. in volume) is run off from below, 
and a drop-number through it taken with the pipette. The 
fall in concentration is read off by means of fig. 3—or, 
rather, a somewhat enlarged form of fig. 3 ; the enlargement 
being necessary for accurate readings on account of the 
very small changes in concentration produced by this method, 
the fewer large drops offering a very much smaller surface 
than the emulsion particles. 


Haample: Oil B. 


One litre of -25 per cent. sod. glycocholate was prepared. 
The solution was poured into B and C as described, and after 
the drops have been passing for about 2 hours the reservoir A 
was empty and the contents of tube B run off. Through this 
solution a drop-number gave 466, 2. e.,a tension of 13°2 dynes, 
corresponding on the curve to a concentration *243 per cent. 


Hence the fa]l in concentration=*007 per cent. 


.. for 250 c.c. (the volume of B) the total quantity 
removed =°0175 grm. 


The volume of the oil-reservoir =168 c.c. 

The time of formation of 50 drops was taken at the 
beginning, in the middle, and near the end of the expe- 
riment. It was found that the middle value was the mean 
of the beginning and end, viz. 50 drops in 40 seconds. 


Total time of emptying A. . = 7940 secs. 


.. total number of drops . . = 9925. 
Hence; volume of a.drop, .). = Dib? ee 
.. radius of a drop = 71 60'ea 
and hence total adsorbing area = 3192 cm.’ 


*, Adsorption coefficient = 5:4 x 10-® gram per cm.’ 


A second determination was made in which three times 
the former quantity of oil was used, and hence a greater 
fall in concentration was observed, which is in favour of a 
more trustworthy result :— 


Total volume of oases.) . Spee: 

Total time of dropping . . =20520 secs. 
Total adsorbing area . . . = 7414 cm? 
Total mass adsorbed. .2) i. «9 =) 7022 cama 


.. Adsorption coefficient = 3°1 x 10~°. 


of Gibbs’s Theory of Surface-Concentration. 19 


It is worthy of note that these values are of quite the same 
order as those obtained at the very curved surface in the 
emulsion method. 

The calculated amount for this particular concentration 
(0°25 per cent.) gave 


Uy ES 
RT de 


We therefore find in this case also, that the experimental 
value for I’ far exceeds the calculated value. Thus over 


—- ® s. —8s he > ‘ ied . 
= 3°6 x 10-° gram per square cm. 


98 per cent. of the total observed effect is wnaccounted for in 


the theoretical calculation. 

This result is unexpected, especially when one remembers 
that in this latter method of determining [I the condition of 
planeness of surface is fulfilled to a very high degree of 
approximation ; for the drops at the surface of which 
adsorption took place were of practically the same diameter 
as those produced by the drop-pipette in carrying out the 


da": : 
measurements of Je 3 bens the drops are very large in com- 
c 


parison to the sphere of molecular action. One would have 
expected, therefore, that the calculated result (dependent as 


nt de d : ? 
it is on the value of za) would be in good agreement with 
de 


the experimental value of I’. 

One point of considerable interest lies, however, in the 
good agreement observable in the experimental values of TI 
obtained by the two methods. On the one hand it appears 
to point (at least as far as one may rely on the accuracy of 
the determinations of 1) to the fact that the tension even at 
very great curvatures does not differ appreciably from that 
at an approximately plane surface. On the other hand, since 
the portion of the adsorption which is unaccounted for is so 
much greater than the value required by theory, it is possible 
that even considerable variations in this latter would be 
undetected owing to the “‘swamping”’ effect of the excess 
values. 


DIscussIoN OF RESULTS. 


The net result of the foregoing experiments is to show | 
that between observed and calculated values there is a very 
great discrepancy—the actual quantity adsorbed being about 
eighty times the calculated amount. The most obvious 
explanation lies in the existence of some assumption in the 
theory which has been overlooked in practice. 


920) Mr. W. C. M. Lewis: Lvperimental Examination 


The following are Gibbs’s assumptions in the deduction of 
the equilibrium equation :— 

(1) The adsorbing surface is plane. 

(2) The solvent is to show no concentration at the inter- 

face. 

(3) For the particular equation used in the present paper, 
there is supposed to be only one component capable 
of being adsorbed. 

(4) This component and the solvent in which it is dissolved 
are supposed to form a single phase. 

(1) Planeness of the surface-—It may be readily assumed 
that adsorption measurements made at the surfaces of oil- 
drops of sensible magnitude approximate exceedingly closely 
to those at a plane surface. 

(2) Absence of surface-concentration of the solvent.—This 
was simply assumed to be the case, no means of testing its 
validity having as yet suggested itself. 

@) Adsorption of a single component.—This follows from 
the general experimental conditions, the solute, the sodium 
glycocholate, being the only substance whose adsorption is 
measured. 

(4) Adsorption from a monophase system.—This assumption 
is impled from the actual example of surface-concentration 
given by Gibbs as exemplifying his theory, viz.:—A mixture 
of mercury- and water-vapours meeting at a liquid mercury 
surface, the water being the component which suffers surface- 
concentration. Mixtures of vapours are essentially monophase 
systems ; and the question is, are we dealing with a mono- 
phase system in the case of an aqueous solution of sod. 
glycocholate ? The evidence given by its osmotic behaviour 
in raising the boiling-points of water and alcohol and lowering 
the freezing-point of water, is strongly in favour of its being 
a true electrolyte, and hence of iis solution being a mono- 
phase system. 

Temperature Effects accompanying Adsorption. 


All attempts at a direct determination of heat evolution or adsorption at : 


the oil surface were negative. It may be shown, however, indirectly that 
a small evolution of heat must occur, from the observed variation of the 
interfacial tension with temperature. Thus, with a solution of concen- 
tration ‘3 per cent. and oil B, the following results were obtained :— 


| 
Temperature. _ Tension in dynes per cm. | 
16° ©. | 11:76 
41° C. | 12°96 
bi. ©, | 14-4 


— 


of Gibbs's Theory of Surface-Concentration. 527 


We have here the remarkable phenomenon of the tension increasing 
with the temperature. But in similar experiments, using water alone 
instead of solution, there was a marked fall in the tension, as was to be 
expected, viz. :— 


Temperature. Tension in dynes per cm. 
| 16° ©. | 48 | 
; / 
41°C. | 40°8 | 
67° C. oT 1 


‘The increase in tension with temperature in the case of the glyco- 
cholate solution is to be accounted for, therefore, by partial “ desorption ”’ 
having taken place ; that is,an increase of temperature in the case cf the 
elycocholate solution decreases the quantity adsorbed, and hence, in 
accordance with the principle of Le Chatelier, adsorption must be accom- 
panied by heat evoluézon. 

Further experimental determination may make it possible to calculate 
the quantity of heat (Q) evolved per gram-mol. adsorbed by applying the. 
equation 

dlog K Q é 
Ged, at tues 
where K = ratio of concentration in surface layer to the bulk concen- 
tration. 


IV. APPLICATION TO THE THEORY OF DYEING. 


The various theories which have been advanced to account 
for the process of dyeing may be roughly divided into 


(1) Purety chemical combination or solid solution. 

(2) Purely physical surface effects (2. e., Adsorption). 

(3) Partially surface effects and partially chemical 
combination. 


A great mass of evidence has been brought forward in 
support of these different views, but it is not proposed in this 
place to discuss them *. Mention should be made, however, 
of the most recent contribution, namely that of Freundlich +, 
whose results point very strongly to the absence of chemical 
combination, but leave it an open question as to whether 
dyeing is an example of adsorption in the sense of Gibbs’s 
theory. 

It is evidently essential to this view that dye-stuffs in 
solution should possess the property of lowering the inter- 
facial tension. This, of course, cannot be determined in 

* For au account of these various theories one may consult the work 


of W. P. Dreaper—‘ Chemistry and Physics of Dyeing’ (1906). 
+ Freundlich, Zeitschr. Phys. Chem. vol. lix. p. 284 (1907). 


t 


522) Mr. W. ©. M. Lewis: Eaperimental Examination 


Freundlich’s experiments owing to the presence of the solid 
phase, but considerable support would be given to the ad- 
sorption theory of dyeing if it were shown that dye solutions 
actually did lower the tension at the interface between the 
solution and an inert liquid. This liquid must of course be 
such as to exclude both chemical combination and solution. 

With this object further measurements were made with 
the hydrocarbon oil B already experimented with, which was 
to function as the substance to be “* dyed.” 

Aqueous solutions of various concentrations of the follow- 
ing dyes were prepared :—Congo red (sodium salt), methyl 
orange (sodium salt). 

It is hardly to be expected that in any of these cases there 
would be solution in, or chemical combination with, the oil. 


Congo Red. 

A preliminary trial with this substance showed that there 
was a distinct lowering of interfacial tension. The following 
table gives the results obtained for solutions of different 
concentrations. 


Taste LV. 
| / 

Per cent. | Drop- Relative | Tension in | 
Coucentration. | number. , Tension.  dynes/em. 
————— — A ee Ee — — a> ae oe 

0-0 pee By gal AD 48 

0-02 | 53400) (ott 43:87 | 
0:05 | 148 +] 0-865 41:52 

0-1 165 | 0-775 37-20 
0-12 ry 163 0-775 37:20° | 
0-2 | 165 0-775 37:20 | 
0-25 165 0-775 37-20 


) 


The results of Table IV. are plotted in fig. 5 (lower 
curve). ; ‘ 

It will be noticed that after the concentration has reached 
0:1 per cent. there is no further effect on the tension. This ~ 
probably represents the maximum true solubility of the dye ; 
and the minimum value for the tension might possibly be 
made use of as a practical method of ascertaining the optimum 
concentration for the dye-bath. 

From the lower curve (fig. 5) we obtain the value of the 
do 
de* 
line. We may proceed, therefore, to apply the adsorption 


coefficient for the curve itself is practically a straight 


of Gibbs’s Theory of Surface-Concentration. 523 
equation already obtained and calculate the adsorption co- 
efficient (I°). 

Thus, taking as an example the solution of concentration 
O-1 per cent., 


o = 00 MGrmm per e.c:, 


do ___8 dynes 
de. *00) stmife.c.” 
Bi Gida ‘O01 x 8 x 690 jel of 2 
GET do 7 289x2 x £2 x 10 aol AE Bram Per em. 
where 690 = the molecular weight of the undissociated 


Congo red, assuming it to be a monosodium derivative. Of 
course there is doubt as regards the value, since we are 
dealing with the substance when dissolved. 


50 i 


ie 


| PERCENTAGE COM Sk. oe 
fi) 0.0s Of ; OS OZ 0.2 


ee ConacoReD ~——% 
of { 
fletaye OnancGe <———@— 


Thus, assuming complete dissociation in solution, the 
value of the molecular weight would be 345, and the 
resulting value for the adsorption coefficient would be 
1:15 x 10-7 grm. per cm.? On the other hand, if the dye is 
colloidal the molecular weight might possibly be much higher ; 
though the behaviour of sod. glycocholate, by analogy, is 
rather against this. 


524 Mr. W.C. M. Lewis: Euperimental Examination 


Experimental result :— 

250 c.c. of 0-1 per cent. Congo-red solution were shaken 
for fifteen hours with °175 c.c. of oil. The resulting 
emulsion particles were examined under the microscope. It 
was found that 7 = :0000435 cm. 

Hence, calculating exactly as in the sod. glycocholate 
3x 175 
| 0000435." 

The quantity adsorbed was determined as usual by taking 
a drop-number with the pipette through the emulsion. The 
drop-number was found to be 158, = that is, a tension 
38°73 dynes/cm., corresponding on fig. 5 to the concen- 
tration ‘082 per cent. 


solutions, we obtain the Total adsorbing area = 


.. Fall in concentration = 045 grm. 


.. Adsorption coefficient [ = 3:7 x 10-° grm. per em.? 


Tt will be observed that this is very much the same value 
as was obtained in the case of the sod. glycocholate, and 
shows a large discrepancy as regards the calculated value. 
The fact, however, of a lowered tension shows that adsorption 
effects in the sense of Gibbs’s theory cannot be absent. 


Methyl Orange. 


The solubility of this substance in water is small—a 
saturated solution being about 0°073 per cent. The follow- 
ing results were obtained :— 


TaBLe V. 

Per cent. Drop- | Relative Tension in | 
Concentration. number. Tension. | dynes/em. | 
LE iy 128 1-0 48 

0-018 138 ‘927 44°49 
0-036 145 “892 42°81 
0073 =| 152 842 40-41 


These values are plotted in fig. 5 (upper curve). 

It will be noted that fig. 5 has a marked difference in 
appearance to figs. 2 & 3. This is of course due to the 
solubility limit being reached in the former ease before the 
tension has fallen much in value. 


of Gibbs’s Theory of Surface-Concentration. 525 


Experimental determination of T. :-— 


250 c.c. of -07 per cent. solution were shaken with oil 
and formed an emulsion. 


The Fall in concentration = ‘032 per cent. 
.. Total mass adsorbed = ‘08 grm. 
3x °204 ‘ 
Adsorbing surface area = Danae em.” 
oF = 3'3 x, lOG Rae per cnk- 


Caleulated value = 1:°2x1077__—=é~, Bs 


We have here a further repetition of the observed 
anomaly. 


In considering the behaviour of these dyes in relation to 
the adsorption theory, it may be admitted that despite the 
discrepancy between the observed and calculated results, 
substantive dyeing, at any rate, must be more or less an 
adsorptive process. Possibly the actual mechanism of dyeing 
consists jirst in adsorption and afterwards coagulation possibly 
due to “colloidal neutralization.” There may finally be 
some degree of chemical combination, which varies from case 


to case. 


Note on Rosaniline hydrochloride and Crystal Violet. 


A saturated aqueous solution of Rosaniline hydrochloride (about 
06 per cent.) gave a drop-number of 140—7.e., a Tension of 43°87 
dynes/em. The decrease in interfacial tension is less ntarked than in 
methyl orange or Congo red. No further experiments were carried out 
with this substance. Also Crystal Violet dye concentration 1 per cent. 
gave a drop-number 230, and hence also shows a lowering of tension. 


V. SuMMARY. 


1. The object of the investigation has been to examine 
from an experimental standpoint Gibbs’s theory of surface- 
concentration, as the chief cause of the phenomenon of 
Adsorption. By modifying Gibbs’s fundamental expression 
for surface-concentration, an equation is obtained giving the 
mass of solute adsorbed per cm.” in terms of the concentration 
of the solution and the change in tension at the interface, 
aqueous solution—hydrocarbon oil. 

2. The material employed consisted of aqueous solutions of 
bile-salts, the solute being adsorbed at the hydrocarbon-oil 
surface. The mass adsorbed per cm.” oil surface was 


Phil. Mag. 8. 6. Vol. 15. No. 88. April 1908. 2N 


526 Mr. E. Buckingham on the Thermodynamic 7 


determined by two methods. The corresponding tension 
measurements were carried out by the drop-pipette method. 

3. The results show a considerable discrepancy between 
the actual amount adsorbed and that calculated on Gibbs’s 
theory—the actual amount measured being always in excess 
to the extent of twenty to eighty times the theoretical values. 

4, Experiments with dye-stuffs show similar behaviour, 
the discrepancy being of the order stated. 

5. As regards the discrepancy noted, no suggestion is as 
yet offered. Further experiments are in progress employing 
other electrolytes and non-electrolytes as material for measure- 
ments of a similar nature. 


In conclusion, [ would express my indebtedness to Professor 
Donnan for his advice and criticism during the course of this 


work. 


XLVI. The Thermodynamic Corrections of the Nitrogen 
Scale. By EpGAR BUCKINGHAM *. 


[Plates XV.-XVIL.] 


Introduction. 


laa p denote the ratio of the cooling observed to the fall 

of pressure at the porous plug in the Joule-Thomson 
experiment. Jor small and moderate pressures, experiment 
shows that this ratio is nearly or quite independent of the 
fall of pressure and of the mean value of the pressure. 
Assume that the observed value represents the limiting value 
for an infinitesimal fall of pressure. Let uw be expressed in 
centigrade degrees for a fall of pressure of 1 dyne/em.? 
Let C, be the specific heat at. constant pressure expressed 
in ergs. Let v be the specific volume of the gas. Let @ 
be the absolute thermodynamic temperature. Then the 
fundamental equation of the porous-plug experiment may be 
written : 


6 


This may easily be put into the form 


(25) =e+nC, ]. 


Ss hear oC 
oo ree a (p=7=—const.) | (eee 
0 % Jo, 


* Communicated by the Author. 


Corrections of the Nitrogen Scale. 527 


The absolute temperature by the constant-pressure scale 
satisfies the equation of definition 


il v 
Tl (o— 7G) : c e é Py - C (3) 


whence by eliminating v from equation (2) we have 
T_ 0 _ 0 (°uC,dd (4) 
To A — % 9 CNL a ee Re 


Let To, %, v refer to the ice-point. Ty) is then the 
reciprocal of 2, the mean coefficient of expansion of the gas 
at the constant pressure 7 between the ice and steam points ; 
6, is the absolute thermodynamic temperature of the ice- 
point ; vis the specific volume at the ice-point under the 
pressure 7. Kquation (4) may be used to find the value T, 
on the absolute constant-pressure scale, of any temperature 
of which the value is @ on the absolute thermodynamic scale. 
As a preliminary to this, the value of @) must be found by 
successive approximations setting T=T)+100, @=6)+100, 
and, to start with, 0¢)>=273. 

To perform the necessary integration, the value of uC, 
must be known as a function of @. The value of @ is very 
approximately 273+ ¢, where ¢ is the centigrade temperature 
by any of the common scales. Hence, since the second 
member of (4) is merely a small correction-term, it is 
sufficient if we can express wC, as a function of 273+4#, 
which will be denoted by 7. In other words, if we can 
find the form of the equation wC,=/(7), we may write 
wC,=f(@) in integrating equation (4). 


2. Use of the Law of Corresponding States. 


Our experimental knowledge of the value of w for any one 
gas is not sufficient to enable us to decide upon an equation 
#Cy=f(t) which shall inspire any confidence as a basis for 
extrapolation outside the small range of temperature within 
which the experiments have been made. We therefore have 
recourse to the law of corresponding states, and thus bring 
all the experiments on the various gases into one connected 
series. 

We have to consider experiments on carbonic acid, oxygen, 
air, nitrogen, and hydrogen. We assume that for these 
gases there exists a single reduced equation of state, 


Sis Mcteich ee ; 
BW ae a ee ame 5) 


where Pe, % Te are the critical constants. This assumption 
2N2 


528 Mr. E. Buckingham on the Thermodynamic 


is plausible for the diatomic gases and air, while we know by 
experiment * that the isothermals of air and carbonic acid 
are congruent when drawn to the proper scales. Admitting 
this assumption, we conclude that in corresponding states, 
the reduced value of wC, per gram-molecule, or 


Mp, 
y = Wp, -. « . 
is the same for all the gases, M being the molecular weight. 
But since the value of wu is, for such pressures as are used in 
gas-thermometry, independent of the pressure, and therefore, 
at any given temperature, independent of the volume, we 
may, in considering the Joule-Thomson effect, regard two 
gases as being in corresponding states if only they are at 
corresponding temperatures. It follows that there must be 
a single equation, 


mE: W0,=$(2) . ee 
YS OT), oy (8) 


which is satisfied by the experimental values for all the 
gases. ‘To discover the form of this general reduced equation 
for wC,, we plot the value of y as ordinate against the reduced 
temperature 7’ as abscissa ; draw a smooth curve to represent 
the distribution of the points as nearly as possible; decide by 
inspection upon the general type of equation to be adopted ; 
and, finally, adjust the constants by trial. If our use of the 
law of corresponding states is permissible, the points should 
all lie close to a single smooth curve ; if they do not, the 
assumptions are faulty or the experimental data erroneous. 

A similar process may be pursued with pw alone, instead of 
uC, We may plot the values of 


1 eyes 
pole) Can niall ae aa 


against 7’, and attempt to determine a curve 


Tiga Ce CARER 
Since we know that we are dealing with approximations 


and that the reasoning is not rigorous, the justification of our 
procedure depends in either case upon its success. 


* Amagat, Journ. Phys. (8) vi. p. 5 (1897}, 


Corrections of the Nitrogen Scale. 529 


3. Numerical Data. 


We have now to consider numerical values. In Table I. 
are given the values of p,, T,, Cp, and M. The specific heats 
are assumed to be constant from 0° to 100° except for 
carbonic acid. The value for nitrogen is from Holborn & 
Austin *, the others from Lussanat. For carbonic acid 
JLussana’s mean value for a pressure of 1 metre of mercury 
is used as a basis, and the variation with temperature, 
assumed to be linear, is taken from the mean of the results 
of Regnault, H. Wiedemann, and Holborn & Austin. All 
the values refer to a pressure of 1 metre of mercury. 


TABLE I, 


Critical Constants, Specific Heats, and Molecular Weights. 


Gas P.. cn Cp X 108, M. 
; (atmospheres). | (=273+4,). (ergs). (mol. wt.). 
CO cncs- 729 3044 7-90 (1+0-00107 2) 44 
Oe cssuc.- 50°8 1542 9°37 32 
) 39°3 133 9°91 28°8 
oe 33°6 128 9°87 eos 
3 13 32 142-2 2 
(1974) 


The data available on the cooling effect, w, are those of 
Joule-and Kelvin { on all five gases; of Natanson § and 
of Kester || on carbonic acid; and of Olszewski{] on the 
inversion-point for hydrogen. Of Joule and Kelvin’s. 
seventeen experiments on hydrogen, only those have been 
considered in which the impurity of the gas was small 
(Nos. 3, 5, 6, 7, 9, 13, 14, 15; 16, 17), the correction for 
impurity being very large and uncertain. The data for 
plotting the experimental curve y=/(7) are collected in 


* Phys. Rev. xxi. p. 260 (1905). 

+ Nuovo Cimento (8) xxxvi. p. 134 (1894). 

t Kelvin, Math. and Phys. Papers, 1. pp. 418--429. 
§ Wied. Ann. xxxi. p. 502 (1888). 

|| Phys. Rev. xxi. p. 260 (1905). 

q Phil. Mag. [6] ii. p. 585 (1902). 


530 Mr. E. Buckingham on the Thermodynamic 


Table II. Column 1 gives the initials of the observers to 
whom the observations on » are due; column 2, the symbol 
of the gas ; column 3, the reduced temperature of experi- 
ment; column 4, the value of “ “uC, deduced from the 
observed value of ~%; column 5, the number of separate 
experimental values of which mw is the mean; column 6, 
the average amount of impurity in the gas ; column 7, the 
value of y computed by equation (11); column 8, the 
differences y—y,,,,- The values given by Kester are extra- 
polated to zero impurity from experiments with amounts of 
impurity up to 2:2 per cent. 

The points in Plate XV. represent the values given in 
columns 3 and 4 of Table Il. They do all lie close to one 
smooth curve, including Olszewski’s inversion-point*. The 
curve drawn is a satisfactory representation of the experi- 
mental results; it is an equilateral hyperbola with the 
equation 


Mp. ie 53 =s 
Paella f= 


It will be found upon examination that the points which, 
from the number of separate experiments involved and the 
purity of the gas, appear a priort to deserve the most 
confidence, lie on the whole closest to the curve. 

Hquation (11) is an empirical equation with three 
constants. It is therefore natural that it may be fitted 
to the observations somewhat more closely than one with 
only two constants: for example, the equation given by 
Rose-Innes fT, 


10.) at a 


A 
#=——B, ° . e ° e e e (12) 


which follows, for low pressures, from. van der Waals’s 
equation of state, or the reduced equation 


A J, 

uC, = ee B, whe ov Se cou wee (13) 
deduced by D. Berthelot ¢ from his form of the equation of 
state. But aside from the fact that there is one more 
arbitrary constant available, equation (11) has an advantage 
over the two just mentioned, in that it does not make the 
curve asymptotic to the vertical axis. Inspection of 


* Compare A. W. Porter, Phil. Mag. [6] xi. p. 554 (1906). 
t+ Phil. Mag. [5] xlv. p. 227 (1898). 
t Trav, et Mém. Bur. Int. xiii. (1903). 


dL 


e 


Corrections of the Nitrogen Scale. 


TasiE IT.—Reduced Cooling Effect. 


5 is ie 11:39 — 5:04 5 4°5 —5'11 +0:07 


ih 2, 3, 4. 5. 6. if 8, 

Bi re es Number Average y 

servers. | Gas. Nive I = e I of impurit; cale, aoa 

(=7/7,). To p&p experiments. (per at, (by equ. 11). $5 ee 

Soa ea ae clo 0-896 | 118°5 8 te 1938 —53 
ie ee 3 0:964 103:4 11 rane 104-2 —0'8 
tee ‘ 1-027) 88:9 3 ~ 90°8 —19 
eel 7 1-092 83:0 2 a 79'5 +35 
ie ke Mats fs 1158 "75 fl ae. 70:5 +72 
—_— sae 7 1-214 712 2 i 64-2 a7 0 mee 
Mi teresa. . 0-963 99°3 ae a 104:5 — 52 
J.and 7. ...| f 0-921 108-3 6 68:2 1159 —7-6 

: nm . 1-010 87:1 1 18 93-9 —6'8 

i. | f 1-074 768 1 08 82:3 —5'5 

- 2 : 1-204 584 1 ot 65:3 —69 

‘ 4 1:217 58‘ 1 1-7 63:9 —5'8 
ee Or |e eee |S gia | 1 a = 2% 29-9 ape sik 

: . 2:35] 234. 1 36 18°6 +4'8 

i" a 7 2390 166 1 4-0 18-0 —14 

i 7 1-827 31:2 1 546 29'9 Bera 

ol : 2:359 169 1 29-4 18:5 oe me 

’ a i 2385 15:4 1 51-0 18:1 —2:7 
Gender al At, 2106 21-9 8 ae: 23-0 (Sea 

i & F 2:350 18-7 8 ee 186 +01 

a . 2:750 12-7 6 ies 136 —0:9 
J.andT, ...| N,. 2°189 22"1 1 7:9 21°4 +0°7 

i da . 2-847 12:3 1 2-9 126 05. | 

* i : 2852 16:4 1 12:5 12:5 +39 

tei. "i, 6-02 + 00 newer nae —0:40 +0-4 
; Jeand T. S 8:74 — 3:36 4 39 ~ 3°57 +021 


om °§ ox. 
Pa 


532 Mr. KE. Buckingham on the Thermodynamic 


Plate XV. shows that the observations on carbonic acid 
indicate approach to an asymptote at about 7r’=0°4 or 
Tt’ =0°5. Repeated trials have shown that the observations 
cannot be so well represented by any simple curve asymptotic 
to the vertical axis. If such a curve be made to fit the 


carbonic acid points at all well, the agreement is spoiled for - — 


the important group of points between t’/=2 and 1'=3, 
representing the observations on oxygen, air, and nitrogen. _ 
Nearly as good an agreement may be obtained by the ~ 


Pe 


more elementary method of plotting yp’ si. against 1’. 
This is done in Plate XVI. Curve A has the equation 
: 163-9 
108s P= 545 2°36 — 0-42) = 15:9, 


and is based on the assumption that the critical pressure of 
hydrogen is 19°4 atmospheres. Curve B, with the equation 


206°7 I 
709397 90'3: shine ° F (15) 


assumes that p= 13 atmospheres for hydrogen. Slight 
changes may, of course, be made in the constants of equa- 
tions (11), (14), and (15), but no great improvement is 
possible. 


| 
10 ae 


4. Integration of the Constant-Pressure Equation. 


The various detached observations on the Joule-Thomson 
effect may thus be coordinated and made to support one 
another. It seems probable that any one of the three 
empirical equations, (11), (14), and (15), gives the true values 
quite as closely as the observations on any one gas separately, 
except perhaps in the case of carbonic acid. Moreover, the 
existence of a general curve which represents the observa- 
tions affords a very plausible means of computing the values 
of w or of wC, for any one gas all the way from its critical 
temperature to some twelve times its critical temperature. 

Let us, for the present, confine our attention to equation 
(11). From this we may evidently obtain an unreduced 
equation of the form 


wlp= = —c . 2 | ad fe) Soe 


—b 


for each gas. If we now integrate equation (4) on the 
assumption that wC, is correctly represented by equation (16) 


Corrections of the Nitrogen Scale. ddd 


and identify 7 with @, we get 


T 6 . ge gi b) 17 
maa OL a(b- a )eBee deo} «am 


where 


? (18) 
aes 


In Table III. are given numerical values of the constants 
for use in computing, including the values of @ found by 
carrying on the successive approximations until the last 
two values of @) differed by less than 0°-001. The values 
of the coefficient of expansion « are from Chappuis’s* de- 
terminations. The figures given for each gas forma consistent 
system of values which may be substituted in equation (17) 
for finding the relation of the constant-pressure scale of that 
gas to the thermodynamic scale. After finding the value I 
corresponding to a given value of @, we form the differences 
@—@ and T—Ty. The quantity as 6, is the centigrade 
thermodynamic temperature t, while T — Ty is the centigrade 
constant-pressure temperature tp. The “scale correction,’ 
or the amount by which the temperature on the constant- 
pressure scale lags behind that on the thermodynamic scale 
when both are made to agree at 0° and 100°, is then simply 
tg — tp. 


TasueE III. 
Data for computing the Constant-Pressure Scale-corrections. 
| 
| Air. | Nitrogen. |. Hydrogen. hed _ 
Pressure, r(mm.) 1001 1002 1000 
Dy 66°5 | 64 16 
a | 336589 |’ 241595 |" 222721 2-45709 LS 
iets ........ | 450392 4-56992 | 498394 | £28553 | 
HERON 8 iss was es | 1°367282 1367315 | 1°366004 | 1:374097 
AU ndenccasacee | 272-270 272°246 273°221 | 267°310 
ag ene eee | 273°273 273°286 273°049 273°267 | 


* Trav. et Mém. Bur. Int. xiii. (1903). 


534 Mr. E. Buckingham on the Thermodynamic 


Equations (14) and (15) may be treated in a similar way. 
For equation (15) the form of the integral, assuming C, to 
be sensibly constant, is the same as for equation (11). For 
equation (14) it has one more term and is therefore slightly 
less convenient for computing. The values of the constants 
need not be given, as they follow from Table II. and equa- 
tions (14) and (15). 


5. Constant-Pressure Corrections for Nitrogen. 


If the foregoing methods be used to compute the thermo- 
dynamic corrections of the constant-pressure hydregen scale, 
the results do not agree, except qualitatively, with those of 
D. Berthelot* and Callendarf. ‘The two methods of com- 
putation used by Callendar also give very different results. 
All the computed corrections are small, and it seems probable 
that the true corrections are negligible except in work of 
the highest precision, but the absolute values cannot be 
determined with certainty from the data at present available. 

In the case of nitrogen the state of affairs is more satis- 
factory. In Table IV. are collected the results of Rose- 
Innesf, Callendar, Berthelot, and the writer for temperatures 
between the ice and steam points. The figures given by 
Rose-Innes and Callendar for atmospheric pressure have been 
multiplied by 4 to make them comparable with the others. 
The agreement of the four sets of corrections is satisfactory. 


TABDE WY, 


Thermodynamic corrections of the Constant-Pressure Nitrogen 
Scale between the ice and steam points, for 7=1000 mm. 


(The corrections are to be subtracted from the coustant-pressure readings.) 


} 


| 
1°20 ‘Rose-Innes | Callendar | Berthelot Buckingham ine 
(1901). | (1903). | (1903). | (1907). 
tae 00120 | 0-0109 0-010 | 0-0078 0-010 
ee ae 0°0205 0:0188 0:017 0:0137 0°017 
SO). besser 0:0261 0°0236 0:022 00179 0-022 
| CEE es, 0:0288 0:0260 0°024 0:0203 0°025 
EO. | ash eee 0°0289 0:0260 0'024 0:0209 0°025 
Gi cee 0:0269 0°0240 0-022 0'0198 0°023 
7, Lees 0°0228 0°0204 0'019 0:0172 0-020 
PO tect 0'0168 0-0151 0-014 0°0129 0°015 
00071 0°008 


SOS neces | 0:0092 00081 0:007 


* Trav. et Mém. Bur. Int. xiii. (1903). 
t+ Phil. Mag. [6] v. p. 48 (1903). 
{ Phil. Mag. [6] ii. p. 130 (1901). 


Corrections of the Nitrogen Scale. . 535 


The corrections at higher temperatures, where the 
hydrogen thermometer cannot be used, are of more interest. 
A comparison of these values is given in Table V., Callendar’s 
values having been reduced to 7=1000 mm. as before. The 
relation of the several sets of corrections may be seen from 
the curves in Plate XVIJ. Curve A represents Berthelot’s 
values. Curve B, represents the values computed by equation 
(14) with Holborn and Austin’s values of C,. Curve B, 
represents the results obtained by means of equation (11). 
Curve C represents Callendar’s results. The greatest dis- 
crepancy at 1000° is less than 0°4, an amount which is 
-negligible in the present state of gas-thermometry. 


TABLE V. 


Constant-Pressure corrections for Nitrogen at high 
temperatures. w=1000 mm. 


(The corrections are to be added to the constant-pressure readings.) 


| f. | ae 3. 4. 5. 6. 

} —a |e ee So ; 
a ; Buckingham | Buckingham | Buckingham | 

| z2C Callendar. | Berthelot. (equ. 11). (equ: 14). qnls): 
200 4 0-135 O13 | 0128 0-105 
400 ... oe 0457 | 0523 0-433 0-456 | 
450 ...| 0-660 | | 
600 | 0344 | 0-992 0-827 | 
800 ... =| 1248 | 1-492 1-262 | 
1000 ...} 2-047 1654 | 2-007 1-706 1-721 
1200 ... aft tis vam ei 2-157 2-170 
1600 .. | 3596 
2000 .. | | 4-672 


' 


The agreement of B, with A and of B, with C is so close 
as to suggest premeditation. In reality equation (14) was 
developed and the values represented by curve B; computed 
before the writer had succeeded in obtaining a copy of 
M. Berthelot’s paper. That paper then suggested further 
work which resulted in the development of equation (11) 
from which were computed the values represented by curve By. 
Equations (14) and (15) were deduced on the assumptions 
that the critical pressure of hydrogen was 19°4 atmospheres 
and 13 atmospheres, respectively. It is evident therefore, 
upon comparing columns 5 and 6 of Table V., that small 
errors in the critical constants, while they have some influence 
on the reduced equation for w or wC,, are almost entirely 
eliminated in the process of finding the coefficients of the 
corresponding unreduced equation. 


Eee ee 


-—" 


oF; 


536 Mr. E. Buckingham on the Thermodynamic 


6. Relation of the Constant-Pressure and Constant- Volume 
Scales. 


It is impossible to treat the theory of the constant-volume 
thermometer by means of the Joule-Thomson effect, without 
making some assumption regarding the form of the equation 
of state for the low pressures concerned in gas-thermometry. 
But if the equation of state be known, the relation of the 
constant-pressure and constant-volume scales may be found 
directly. There is then no object in integrating the general 
equation of the constant-volume thermometer, for the 


thermodynamic corrections of the constant-volume scale 


may be found from those of the constant-pressure scale 
already computed. 

Let us assume that the isothermal lines pu=f(p) are 
sensibly straight at low pressures, as experiment shows them 
to be for the more nearly ideal gases. Let the departures 
of the gas from Boyle’s law at low pressures be represented 
by the equation 


Pat2=pirr[1+K(p.—p,)], . . . . C9) 


in which K is small and nearly constant. Then if # and 8 
are the coefficients of expansion and of pressure, we may 
easily deduce the equation 


pyle ok ct Ting 
BT ye TO. 


in which ¢, and t, are the numerical values of any given 
temperature on the centigrade constant-pressure and centi- 
grade constant-volume scales, respectively, and 7, the constant 
pressure in the one case, is the same as pp, the initial pressure 
at the ice-point, in the other. 

If we assume that the behaviour of the gas at low and 
moderate pressures is represented by the equation of Clausius, 


P+ aeeay |©-D=Re, - ot eS 


we find that as the pressure approaches zero, the coefficient 
K of equation (19) approaches the limit 


K=7 (0-f- Be) bi a 


(20) 


The values of 6 and of 2 may be found from Chappuis’s * 


* Trav. et Mém. Bur. Int. xiii. (1903). 


Corrections of the Nitrogen Scale. 537 
experiments on nitrogen, and the numerical equation is 


0 147°95 

ie 9, (9 oo1410— a) 
The unit of pv is here taken to be the value that pv 
approaches at the ice point as the pressure approaches zero, 
and @) is taken equal to 273°2. 


7. Constant-Volume Corrections for Nitrogen. 


Hquations (20) and (23) have been used to compute the 
difference of the two nitrogen scales and thence the thermo- 
dynamic corrections of the constant-volume scale. In this: 
latter process the constant-pressure corrections given in 
Table V., column 5, were used. A comparison of results is 
given in Table VI. and, graphically, in Plate XVII. Curve A’ 
represents Berthelot’s values ; curve By’, those just computed ; 
eurve C’, Callendar’s values. If B, had been used as a basis 
instead of B,, we should have had another curve B,' lower 
than B,' by the same amount as B, is lower than By. 


TABLE VI. 


Constant-Volume corrections for Nitrogen at high 
temperatures. Pp=1000 mm. 


(The corrections are to be added to the constant-volume readings.) 


| | 
°C. | Callendar. | Berthelot. | yea) a 
BOONE, «on 8 <i 0.035 | 0-046 | 0-024 
ee se | anges) | 0-139 
lee | 0189 =| | 
ll) a | a8 | 0°305 
| UN ie toad e8 | 0:56 0-514 
OOO). .......:. | 0646 0-77 0°734 
Peta00 2.4: | Bs | at 0-961 


| es 


We have thus, for each of the two nitrogen scales, ihe 
independent results of three writers using ditferent methods 
of computation. The various values of the thermodynamic 
corrections show an agreement, for each scale, which is closer 
than has been attained in independent determinations of high 
temperatures with the gas thermometer, and it seems probable 
that the means of the values represented by curves A, B,, and 
C, or A’, By’ and C’ of Plate XVII. give very nearly the true 


values of the corrections in question. 


ae 2. 


938 Anomalous Magnetic Rotatory Dispersion of Rare Earths. 


If the pressure in the thermometer, namely 7 or po, is 
less than 1000 mm., as it usually is in practice, the corrections 
are to be reduced in the sameratio. This process of reduction 
is not rigorous, but it is correct to a much higher degree of 
approximation than that to which the corrections for 1000 mm. 
are known. 


National Bureau of Standards, 
Washington, D.C., November 9, 1907. 


XLIX. On Anomalous Magnetic Kotatory Dispersion of Rare 
Earths. Remarks on Prof. R. W. Wood's recent paper. 
By G. J. Huras™. 


ge R. W. Woop having recently published an inves- 

tigation on anomalous dispersion of magnetic rotation 
in neodymiumyt, I should like to direct attention to expe- 
riments which I made in 1906f. 

For measuring the rotation I used a Lippich polarimeter, 
illuminated by light having passed through a spectrometer, 
and monochromatic to within 4 wy about. The rotation 
varying very rapidly within the absorption-band and near its 
edges, a high degree of homogeneity of the light is necessary; 
otherwise it may occur that the most important points of the 
curve are overlooked, as I actually found. Of course through 
the light not being absolutely homogeneous small errors 
will come in; nevertheless it seems to me that this direct 
method should give much better results than that used by 
Prof. Wood, which I rejected on account of its inaccuracy 
and the errors occurring necessarily in the neighbourhood 
of absorption-bands. Moreover, I consider it impossible to 


make measurements within the absorption-band in that way ;_ 


but this is most desirable in order to detect many peculiarities 
of the band’s structure and to find the sense of rotation on 
both sides of the maximum. 

In the case of the absorption-band of a solution of chloride 
of erbium, at about }X=520 pp, to which I confined my first 
measurements, I found considerable anomalies, and the 
sense of rotation in the immediate neighbourhood of the 
maximum positive, farther away negative, as is shown by 


* Communicated by Prof. H. du Bois. 

+ R. W. Wood, Phil. Mag. [6] xv. p. 270 (1908) ; also R. W. Wood, 
Physik. Zeitschr. vi. p. 416 (1905) ; ix. p. 148 (1908). 

{ G. J. Elias, Physik. Zeitschr. ix. p, 981 (1906). See also H. du Bois 
and G. J. Elias, Proc. Roy. Acad. Amsterdam, Feb. 29, 1908. 


A x | 


Irregularities in Radiation from Radioactive Bodies. 539 


the annexed curves; for further details my paper may be 
referred to. From this fact I will not draw any further 


a 


A 
500 505 510 515 520 525 530 WEE 


I=Double macnetic rotation of the plane of polarization in a solution 
of chloride of erbium, 3 to } normal, in glass vessel. Field 
7500 gauss, 
Ii=The same for water, in the same vessel. 


conclusions in this short notice. Of course the peculiar be- 
haviour of the rotation will be closely connected with the very 
complicated structure of the band, not to be seen in the liquid 
solution, but very clearly in solid solutions, especially at 
liquid air temperature *. 

I am busy continuing these investigations, as well as 
those on absorption, refraction, and related phenomena, with 
different salts of the rare earths, solid and dissolved, and am 
using for the purpose a monochromatic autocollimating 
illuminator, made by Zeiss in Jena, a description of which 


will shortly appear. 


Berlin, Bosscha Laboratory, 
Feb. 15, 1908. 


L. The Irregularities in the Radiation from Radioactive 
Bodies. By Hays Geicer, Ph.D., John Harling Fellow, 
University of Manchester t+. 


N all experiments in which the ionization currents due to 
two radioactive substances are balanced against each 
other by means of an electrometer, it is not found possible to 
obtain an exact balance. The needle of the electrometer 
always moves quite irregularly over a certain number of 
divisions on the scale. ‘This effect cannot be eliminated, no 
matter how much care is taken in the adjustment. Bronsont, 
who was troubled by this effect in the use of his steady 
* H. du Beis & G. J. Elias, loc. cit. 


+ Communicated by Professor E. Rutherford, F.R.S. 
t Bronson, Phil. Mag. Jan. 1906, p. 143. 


a 


deflexion electrometer, suggests that the effect may be due to 
an exceedingly small and rapid change in the ionization 
itself. EE. v. Schweidler* has in a special paper drawn 
attention to the fact, that according to the disintegration 
theory certain irregularities in the radiation from radioactive 
substances are to be expected. He calculates from the laws 
of probability that these irregularities should under certain 
conditions be within the limits of measurement. K. W. F. 
Kohlrausch t made some experiments to test the accuracy of 
the theory of v. Schweidler. A discussion of his results 
will be given later. 

In the course of some experiments, my attention wa 
attracted by the impossibility of obtaining a steady balanee 
of two opposite ionization currents due to the a rays. A few 
experiments will now be described which I have made to test 
the cause and the magnitude of these irregularities. 

It was of importance first to prove whether the effect was 
due to a real variation of the intensity of the radiation or to 
some secondary effect which might be eliminated. For this 
purpose, the following experiment was made :—T wo ionization 
vessels, A and B, were arranged as shown in fig. 1. Between 


540 Dr. Hans Geiger on the Irregularities in 


Piss 1. 
g A 


them, and insulated from them by the ebonite plugs e, ¢c and 
the guard-rings g and h, was fixed a piece of aluminium foil D, 
which was connected with an electrometer of the Dolezalek 


* E. v. Schweidler, Congrés international pour l'étude de la radiologie 
et de ionisation, Liége 1905. 


+ K. W. F. Kohlrausch, Wien, Ber. 1906, p. 673. 


the Radiation from Radioactive Bodies. 541 


type. The vessels A and B were closed at E and F by alumi- 
nium foil and connected with the opposite poles of a battery of 
400 volts, the middle point of which was earthed. A narrow 
pencil of 2 rays was sent through both vessels from the point 
‘ R, and as in practice the vessel B was somewhat larger than 
A it was easy to balance the two opposite ionization currents. 
Now in this case any effect of the irregularities of the 
radiation itself was eliminated, since each « particle con- 
tributed equally to the ionization currents both in A and B. 
The experiment, however, did not show an entire absence of 
the oscillations of the electrometer needle, but the effect was 
undoubtedly smaller than when the ionization currents of the 
same intensity were produced from two different sources at 
R and U, R producing ions only in A and Uonlyin B. The 


two curves given in fig. 2 correspond to the movement 


Fig. 2. 


gs 


(o) 


Electrometer Readin 


Time in Minutes. 


of the needle of the electrometer over a time of 4°5 minutes. 
Curve I. shows the effect due to one source of rays, Curve II. 
the effect of two separate sources. The intensity of the 
radiation was the same for both curves. The small oscil- 
lating effect shown in Curve I. was probably due to the 
irregular thickness of the aluminium foil D. Some of the 
particles were stopped in the leat and could not produce 
ionization in B, while other particles passed through the 
small holes, thus producing a stronger ionization in B than 
in A. 

The general arrangement used for the further experiments 
isshown in fig. 3. H is the electrometer of which one pair of 
quadrants was earthed and the other pair connected with two 
ionization vessels A and B. The outsides of the vessels were 
charged up to + 200 and —200 volts respectively. The radia- 
tion from suitable radioactive substances was allowed to pass 
into the vessels through two openings which were covered with 


Phil. Mag. 8. 6. Vol. 15. No. 88. April 1908. 20 


aluminium foil in order to avoid possible disturbances from 
air-currents. By means of a screw 8 the distance of the 
radioactive matter U from the vessel B could be varied and 
a balance obtained between the ionization currents in A 
and B. When the balance was obtained as closely as possible, 


D42 Dr. Hans Geiger on the Irregularities in 


Fig. 3. 


the needle of the electrometer showed small oscillations 
which were observed over a definite time, generally five 
minutes. 

With this arrangement, a comparison was made between 
the irregularities of the ionization currents produced by @ par- 
ticles with those produced by a current of the same intensity 
due to § particles. 

Two wires which had been exposed to the radium emanation 
were placed at R and U quite close up to the vessels A 
and B (fig. 3). The ionization in this case is almost entirely 
due to « particles. The oscillations of the needle over a 
space of five minutes are shown in the Curves I. and TT. 
of fig. 4. The intensity of the radiation was for Curve I. 
340 and for Curve IJ. 1100 divisions per minute on the 
electrometer-scale, where one division corresponds to the 
calculated ionization produced by about 22 @ particles. The 
Curves III. and IV. of the same figure were obtained 
under the same conditions except that the ionization in 
the cylinders A and B was produced by £ particles. For this 
purpose a little glass tube containing about 5 mg. RaBr,. 
was brought near to the vessels and its distance adjusted 
till the ionization in both cylinders was equal and of the same - 


Electrometer Readings, 


the Radiation from Radioactive Bodies. 543 


intensity as in the case of the a particles. The Curves III. 
and IV. correspond to I. and II. respectively as regards the 
intensity of the ionization currents. The Curve III. shows 


Fig. 4. 


a 
cS 


e 
Oo wm 


/ Zz 3 + Fi 
Time in Minutes. 


no oscillations at all, while IV. shows oscillations but com- 
paratively to a very slight extent. The fact that hardly any 
observable oscillations occurred in the case of 8 radiation 
shows, independently of any theory, that the effect observed 
with « radiation is really due to the irregular nature of 
the & radiation, and not to a secondary effect. 

The difference in the shape of the curves is to be expected 
from simple theoretical considerations. It is known that 
the absolute average error in a large number of observations 
for two events P and Q is given by 


e=+VNpq, 


where N is the number of observations and p.and gq 

are the probabilities for the events P and Q respectively. 

If we apply this formula to radioactive changes, taking N 

as the number of atoms present, the number of atoms 

breaking up during a given rae T (7 being small compared 
202 


* wala i, ry ree 
<7 we “a 
Vy 
a 
As 
4 “Pp 
ty 


with the period of the substance) is given by ArN, the 
number of atoms still unchanged after the time r is given by 
(1—A7)N. From this it follows that the probability of a 
single atom breaking up during the time rt is Xz, while the 
probability that the same atom will exist after that time is 
1—Ar. Hence the absolute average error is 


e=+./ NAT —=Azr), 


or, neglecting the square of 47 compared with Az itself, the 
error 1S 


544 Dr. Hans Geiger on the Irregularities in 


ext V/Nar or +VZ, 


where Z is the number of atoms disintegrating during the 
time r. This result was first deduced by H. v. Schweidler 
(loc. cit.) in a similar manner. 

According to the simple radioactive theory, the average 
number of atoms breaking up during the time T is given by 


Z = Nar. 


The actual number observed may show a deviation from 
this, or an average error equal to the square root of the. 
number of atoms breaking up during the time 7. The 
absolute average error increases therefore with the number 
of atoms breaking up, 2. e. with the intensity of the radiation ; 


while the relative error 7 decreases. The movement of 


the needle of the electrometer registers the absolute error. 
The correctness of this theoretical conclusion may be 
tested as follows :—Rutherford has shown that one « particle 
from radium itself produces 80,000 ions in its path of 
3°5 ems. in air at atmospheric pressure ; while Durack has 
found that each of the swifter @ particles from radium 
expelled at a speed approaching that of light makes a new 
pair of ions in every 6 cms. of air at 1 mm. pressure. Conse- 
quently, in the cylinders A and B which were about 12 cms. 
in length one 8 particle will produce at atmospheric pressure 
4x 760 or about 3000 ions. Therefore, in order to produce 
the same ionization current with 8 particles, about 25 times 
as many # particles as a particles are necessary. Hence the 
average error in the number of @ particles shot out is 


e= 4/252. 
The average error measured by the electrometer is 
e= tev 252, 


as one 8 particle produces only an effect 3; of that produced 


the Radiation from Radioactive Bodies. N45 


by one « particle. Therefore the error measured in the case 
of 8 particles, on the same electrometer and under the above- 
mentioned conditions, should be about one-fifth of that 
observed for a particles giving the same intensity of ioniza- 
tion. The smallness of the irregularities in Curves III. and 
IV. (fig. 4) compared with those shown in Curves I. and II. 
is thus to be expected. | 

A special series of measurements has been made to show 
how the average error depends upon the intensity of the 
radiation. At the points R and U (fig. 3) two wires were - 
placed which were made intensely active by exposure to the 
radium emanation. While the activity was decaying, the 
oscillations of the needle were observed at intervals, and 
curves were drawn showing the oscillations as accurately as 
possible over intervals of 54 minutes. From these curves 
the average error was determined by counting the divisions 
passed over by the electrometer needle during the attempted 
balance, and dividing that number by the number of swings 
observed during the same time. The average error was also: 
calculated theoretically. An example may be given. 


Haperimental Determination of the Error. 


The data are taken from Curve II. (fig. 4). The number 
of divisions passed over in 53 minutes, counting the maximum. 
divergence both positive and negative, was 380 and the 
number of oscillations during that time was 64. Hence the 
average magnitude of one oscillation was ae = 5'9 divisions, 
and the average deviation or the error +2°95 divisions. 


Theoretical Determination of the Error. 


The error is given by /Z where Z is the number of atoms 
breaking up during the time 7. This number gives also the 
number of 2 particles shot into the vessel during the time r. 
The time 7 is in this case the average time of swing, being 
5°8 sec. as an average taken from all the curves. The intensity 
of the radiation of one wire was 1100 divisions per minute, 
and therefore, as one division corresponds to the ionization 
produced by about 22 « particles, the number of particles. 
shot into the vessel is 2°4 x 10* per minute, and therefore the 
number shot into both vessels in 5°8 sec. is 

. 4 OLS : 
2x2'4x 10 x ay SH ETX IO. 


The square root of this is 68. The average error taken over 


546 Dr. Hans Geiger on the Irregularities in. 


5°8 sec. is thus +68 a particles or transformed into divisions 


ae: = = +3°1 divisions. 


The figures in the following table are all calculated in the 


same way as indicated in the above example. The difference 
between the theoretical and experimental error is about 


15 per cent. 


Absolute error determined. | 
Intensity of 
the radiation. 


Theoretically. | Experimevtally. 


diy. | 
500 6-4div. | 54 div. | 
min. 
2700 5 4:9 2° | 4°4 ” | 
1100 ,, | Saou 7 | 
DOD oc rat hee | as | 
100 > 09 9 | 1:3 9 | 
| | 


The agreement is better than one would expect considering 
the conditions of the experiment and the uncertainty of the 
data from which the number of @ particles is deduced. A 
slight correction ought also to be made since the electrometer 
needle was not quite dead-beat. The agreement between 
theory and experiment is quite as close if the error is deter- 
mined by measuring the magnitude of the oscillations of the 
electrometer-needle for any convenient time, for example, 
each half minute, instead of the time of swing of the 
electrometer, viz. 5°8 seconds. 

Kohlrausch (loc. cit.) did not find a numerical agreement 
between the theory and his experiments, but this seems to be 
due to an incorrect use of the formula, since on calculating 
the error as above from his data, quite a close agreement 
(10 per cent.) is obtained for saturation currents. If the 
current is not saturated, as was the case in some of Kohl- 
rausch’s experiments, the above formula cannot be applied. 
For if the current is only half saturated, half of the ions 
produced from each « particle are lost by recombination ; 
consequently each « particle produces only one half of its 
effect under ordinary conditions of saturation. Taking this 
fact into consideration, a close agreement between theory and 
experiment was also found by calculating the data given by 
Kohlrausch for non-saturated currents. 


a a! oa ©. 
. w 
a ae 
liens 


the Radiation from Radioactive Bodies. 547 


The agreement between theory and experiment in Kohl- 
rausch’s paper seems to me to be of special interest, for the 
method used by him differs from the method employed in 
this paper, while the intensity of the radiation in his experi- 
ments was nearly twenty times greater than the strongest 
used in mine. 

I have to thank Professor Rutherford for the kind interest 
he has taken during the progress of this research. 


Physical Laboratory, 
University of Manchester. 


Note added March 12, 1908.—Since the above was com- 
municated a paper has been published by E. Meyer and E. 
Regener in the Verhandlungen der deutschen physikalischen 
Gesellschajt, No.1,1908. The authors also find, using a dif- 
ferent method to the writer, that the error increases with the 
square root of the intensity of the radiation. Further, they 
state that by measuring the error e and the saturation current 
2 the charge of an ion may be determined. But the calcula- 
tion involves the number of ions produced by an a particle, 
and this number was determined by Rutherford under the 
assumption that the charge of an & particle is identical with 
the charge of an ion. This, however, is still an unsettied 
question. 

I may add here that the number of @ particles emitted per 
sec. from a given substance can be determined directly by 
simply measuring the error e and the saturation current 2. 
For the error ¢ in E.8.U. is given by 


e=+Ne aie. 


where N is the number of ions produced by one «e particle, e 
the charge of an ion, and Z the number of particles emitted 
per sec., while the saturation current is given in E.8.U. by 


1= NeZ. 


By division we get Z as function of eand 7 only. The agree- 
ment between the errors determined by theory and by 
experiment indicates that the calculated number of @ particles 
emitted per sec. from a radioactive body of known activity is 
of the right order of magnitude. 


X [ 548 ] 


\ 


LI. Further Measurements of Wave-lengths, and Miscel- 
laneous Notes on Fabry and Perot’s Apparatus. By 
Lord Rayuzien, O.M., P.RS.* 


tal a former paper t I described a modified form of apparatus 

and gave the results of some measurements of wave- 
lengths, partly in confirmation of numbers already put 
forward by Fabry and Perot and partly novel, relating to 
helium. I propose now to record briefiy some further 
measures by the same method, together with certain obser- 
vations and calculations relating thereto of general optical 
interest. 

The apparatus was arranged as before, the only change 
being in the interference-gauge itself. The distance-pieces,. 
by which the glasses are kept apart, were now of invar, with 
the object of diminishing the dependence upon temperature. 
The use of invar for this purpose was suggested by Fabry 
and Perot, but I do not know whether it has actually been 
employed before. The alloy was in the form of nearly 
spherical balls, 5 mm. in diameter, provided with projecting 
tongues by which they were firmly fitted to the iron frame. 
The springs, holding the glasses up to the distance-pieces,. 
were of the usual pattern. The whole mounting was con- 
structed by Mr. Hnock, and it answered its purpose 
satisfactorily. There is no doubt, I think, as to the advantage 
accruing from the use of invar. 

The measurements were conducted as explained in the 
earlier paper. The first set related to zine which was com- 
pared with cadmium. Both metals were used in vacuum- 
tubes, of the pattern already described, with electrodes merely 
cemented in. It was rather to my surprise that I found 
ordinary soft glass available in the case of zine, but no diff- 
culty was experienced. The former observations with the 
“trembler” suggested a wave-length for zinc red about 
one-millionth part greater than that (6362°345) given by 
Fabry and Perot. This correction has been confirmed, and I 
would propose 6362°350, as referred to Michelson’s value of 
the cadmium red, viz. 6438°4722. No difficulty was expe- 
rienced in identifying the order of the rings by the method 
formerly described and dependent upon observations with 
the gauge alone. 

The results of the measurements upon helium were not in 


* Communicated by the Author. 
t Phil, Mag. [6] vol. xi. p. 685 (May 1906). 


further Measurements of Wave-Lengths. 549) 


quite such close accord with the earlier ones as had been 
expected. Both sets are given below for comparison. 


Wave-lengths of Helium. 


EE: ie 
7065°192 7065-200 
6678-147 6678150 
5875618 5875:625 
5015-682 5015680 
A921-927 | 4921-930 

(4713-178) 4713-144 | 

(4471-480) | 4471-482 | 


The two last entries under II., enclosed in parentheses, 
were obtained with the 1 mm. apparatus, and could not be 
expected to be very accurate. Preference may be given to 
II. throughout. 

These measurements of wave-lengths were not further 
pursued, partly because it was understood that other ob- 
servers were in the field and partly because my own vision, 
though not bad, is less good than it was. In particular at 
the blue end of the spectrum I found difficulty. It is evident. 
that work of this sort should be undertaken only under the 
best conditions. 

One of the less agreeable features of the method is the 
complication which arises from the optical distance between 
the surfaces being slightly variable with the colour. In the 
earlier observations with a 5 mm. apparatus I was surprised 
to find the change amounting to 24 parts per million between. 
cudmium red and cadmium green. In the light of subsequent 
experience I am disposed to think that the silver surfaces 
must have been slightly tarnished. At any rate in the later 
measurements I found the difference much less, indeed scarcely 
measurable. It will be understood that no final uncertainty in 
the ratio of wave-lengths arises from this cause. Whatever 
the change may prove to be, it can be allowed for. 


Thirty Millimetre Apparatus. 


In this instrument the object was to construct a gauge with 
a much greater distance than usual between the plates, but 
otherwise on the same general plan as that of Fabry and 
Perot. The distance-piece A A, fig. 1, consisted of a 30 milli- 
metre length of glass tubing, each end being provided with 
three protuberances, equally spaced round the circumference, 


i 


550 Further Measurements of Wave-Lengths. 


at which the actual contacts took place. The removal of the 
intervening material and the shaping of the protuberances 
were effected with a file moistened with turpentine. 

Against this distance-piece the glass plates B B are held 
by the arrangement shown in fig. 1. The lower plate B 
rests upon a brass ring C to which the brass castings D are 
rigidly attached. The upper ring E is connected with the 
castings only through the steel springs F. Both rings are 
provided with protuberances in line with those on the glass 
cylinder, and the pressure is regulated by the screws G. The 
whole was constructed by Mr. Enock. Some little care is 
required in putting the parts together to avoid scratching the 
half-silvered faces ; but when once the apparatus is set up its 
manipulation is as easy as that of the ordinary type. 

In all interference-gauges it is desirable that the distance- 
pieces be adjusted as accurately as possible. For although 
a considerable deficiency in this respect may be compensated 
by regulating the pressures (see below), the adjustment thus 
arrived at is less durable, at least in my experience. Even 
when the distance-pieces are themselves well adjusted, it is 
advisable to employ only moderate pressures. 

Observations with the 30 mm. gauge have been made upon 
helium, thallium, cadmium, and mercury. In the first case 
the (yellow) rings are faint, the retardation being not far 
from the limit. Indeed when at first it was attempted to 
adjust the plates with helium, the rings could not be found. 
With thallium also the rings were rather faint, but with 
mercury and cadmium there was no difficulty. 


Magnifying Power. 


At a distance of 30 mm. the rings are rather small, and 
one is tempted to increase the magnifying power of the 
observing telescope. As to this there should be no difficulty 
if the aperture could be correspondingly increased. But 
although the plates themselves may be large enough, an 
excessive strain may thus be thrown upon the accuracy of the 
figuring and upon the adjustment to parallelism. If, on the 
other hand, the aperture be not increased, the illumination of 
the image falls and the extra magnifying may do more harm 
than good. 

A means of escape from this dilemma is to effect the 
additional magnification in one direction only, which in the 
present case answers all purposes. When straight inter- 
ference-bands, or spectrum lines, are under observation, 
there is no objection to astigmatism, and we may merely 


Ot Th 
4 a 
= | ‘A: cilia 
i) iat 
D ! | 


———— 
— $=, 
ee 
tf 
=" 
Ser SS | 


i ian i Cc 


D902 Lord Rayleigh: Further 


replace the ordinary eyepiece of the telescope by a cylin- 
drical lens or by a combination of spherical and cylindrical 
lenses. This arrangement can be employed in the present 
instance, but the result is not satisfactory. A complete 
focussing, leading to a pvint-to-point correspondence between 
image and object, may however be attained by suitably 
sloping the object-lens of the telescope. In this way excellent 
observations upon interference-rings are possible under a 
magnifying power which otherwise would be inadmissible, as 
entailing too great a loss of light. The subject will be more 
fully treated in a special paper. 


Adjustment for Parallelism. 


If the surfaces are flat, and well-adjusted, Haidinger’s. 
rings depend entirely upon obliquity. A slight departure 
from parallelism shows itself by an expansion or contraction 
of the rings as the eye is moved about so as to bring different 
parts of the surfaces into play. In making this observation 
the eye must be adjusted to infinity, if necessary with the aid 
of spectacle-glasses, and it may be held close to the plates ; 
but a telescope is not needed or even desirable. If the 
departure from parallelism be considerable, no rings at all 


are visible ; but there is an intermediate state of things where 


circular arcs may be seen by an eye drawn back somewhat 
and focussed upon the plates. 

The character of these bands is intermediate between those 
of Newton’s and Haidinger’s rings, the retardation depending 
both upon the varying direction in which the light passes the 
plates and reaches the eye and also upon the varying local 
thickness. If we take as origin of rectangular coordinates 
in the plane of the plates, the place corresponding to normal 
passage of the light, the retardation due to obliquity is as 
—(«?+y*). The retardation due to local thickness is repre- 
sented by a linear function of x and y, so that the variable 
part of it may be considered to be proportional to 2. Hence 
the equation of the bands is | 


au — x? —y*?=constant, 


a being positive if x is considered positive in the direction of 
increasing thickness. Accordingly the bands are in the form 
of concentric circles and the coordinates of the centre are 


L= fa, y=0. 


When curved arcs are seen by an eye looking at the plates 
perpendicularly, the greatest thickness lies upon the concave 


Measurements of Wave-Lengths. 553 


side of the arcs. The perpendicular direction of vision may 
be tested by observing the reflexion of the eye itself in the 
silvered surface. 


Behaviour of Vacuum- Tubes. 


The form of vacuum-tube described in the first paper, and 

depending on sealing-wax for air-tightness, continues to give 
satisfaction. As already mentioned, though made of soft 
glass, they are available ror zinc, and the cadmium tubes have 
lasted well with occasional re-exhaustion. It is advisable to 
submit them to this operation when the red light begins to 
fall off. After one or two re-exhaustions the condition seems 
to be more durable. 
_ With thallium my experience has been rather remarkable. 
The green light is very brilliant and offers a further advantage 
as being comparatively free from admixture with other 
colours*. But Fabry and Perot found thallium tubes to be 
very short-lived, sometimes lasting only a few minutes. [I have 
used but one thallium tube, of the same construction as the 
others, and charged witha little thallium chloride. This tube 
has been used without special care on many occasions—I cannot 
say how many, but probably sevén or eight times,—and it does 
not appear to have deteriorated at all. It looks as though 
the chloride had decomposed and metal had deposited upon 
the aluminium electrodes. But what the circumstances can 
be that render my experience so much more favourable I am 
at a loss to conjecture. 

The same form of tube answers well for mercury, but 
with this metal there is usually no difficulty. 


Control of the figure of the glasses by bending. 


Very good plates can now be procured from the best 
makers, but on careful testing they usually show some defi- 
ciency, mostly of the nature of a slight general curvature. 
Thus when in Fabry and Perot’s apparatus the adjustment 
for parallelism is made as perfect as possible, the rings may 
be observed to dilate a little asthe eye moves outwards in any 
direction from the centre towards the circumference of the 
plates. This indicates a general convexity. 

It occurred to me that an error of this kind might be 
approximately corrected by the application of bending forces 
to one of the plates—it does not matter which. ‘The easiest 
way tocarry out the idea is to modify the apparatus in such a 
way that the points of application of external pressure are 


* The green line is known to be itself double. 


504 . Lord Rayleigh: Further 


not exactly opposite the contacts with the distance-pieces, 
but are displaced somewhat inwards or outwards in the 
radial direction (fig. 2). If the plates are too convex, the 
points of pressure must be displaced outwards. Fic'2 

In this form I have tried the experiment with a er 
certain degree of success, but the displacements 

that I could command (1 mm. only) were too 

small in relation to the thickness of the plate. If 

it were intended to give this plan a proper trial, 

which I think it would be worth in order to 

render a larger aperture than usual available, 

the plates, or at least one of them, should be 

prepared of extra diameter, so that the bending 

forces could act with a longer leverage and at a 

greater distance from the parts to be employed optically. 
Such a construction need not involve a much enhanced cost, 
inasmuch as the outer parts would not need to be optically 
accurate. 

It may be worth while to consider the question here raised 
more generally. The problem is so to deform one surface, by 
forces and couples applied at the boundary, as to compensate 
the joint errors of the two surfaces and render the distance 
between them constant. If we take rectangular coordinates 
x, y in the plane of the surface with origin at the centre, the 
deformation obtainable in this way is expressed by terms in 
the value of ¢ (the other coordinate of points on the surface) - 
proportional to 2, y, a, ay, y’, 2°, xy, ay’, y®. For such 
terms are arbitrary in the solution of the general equation of 
equilibrium of a plate, viz. 7 


d? d? 2 
(72+ a) = 


Of these terms those in w and y correspond of course merely 
to the adjustment for parallelism, and those of the second 
degree to curvature at the centre. The conclusion is that 
we may always, by suitable forces applied to the edge, render 
the distance between the plates constant, so far as terms of 
third order inclusive. 

Another inference from the same argument is that, in any 
optical apparatus, approximately plane waves of light may be 
freed from curvature and from unsymmetrical aberration 
(expressed by terms of the third order) by means of reflexion 
at a plate to the boundary of which suitable forces are applied. 
And the surface of the plate need not itself be more than 
approximately flat. 


Measurements of Wave-Lengths. a 
Figuring by Hydrofluoric Acid. 


It would be poor economy to employ any but the best 
surfaces in measuring work needing high accuracy; but there 
are occasions when all that is needed can be attained by 
more ordinary means. Common plate-glass is rarely good 
enough*; but I have found that it can be re-figured with 
hydrofluoric acid so as to serve fairly well, and the process. 
is one of some interest. From what has been said already 
it will be understood that it is not necessary to make both 
surfaces plane, but merely to fit them together, which can 
be effected by operations conducted upon one only. 

Pieces of selected plate-glass, about 4} inch thick and of a 
size suited to the interference-gauges, were roughly shaped 
by chipping. The best surfaces were superposed and the 
character of the jit examined by soda-light. One glass being 
rotated upon the other, the most favourable relative azimuth 
was chosen; and by means of suitable marks upon the 
edges the plates were always brought back to the chosen 
position. 

The principles upon which the testing is conducted have 
been fully explained in a former papert. In the present 
ease the surfaces are so close to one another that no special 
precautions are required. With a little management the 
contact is so arranged that a moderate number of bands are 
visible. If the fit were perfect, or rather if the surfaces 
were capable of being brought into contact throughout, these 
bands would be straight, parallel and equi-distant. Any 
departure from this condition is an error which it is pro- 
posed to correct. The sign of the error can be determined 
without moving the glasses by observing the effect of 
diminishing the obliquity of reflexion, which increases the 
retardation. Thus if a band is curved, and the change in 
question causes the band to move with convexity forwards, 
it is a sign that material needs to be removed from the parts 
of the glass occupied by the ends of the band. Such an 
operation will tend to straighten the band. If, however, 
the movement take place with concavity forward, then 
material needs to be removed from the middle parts. In 
every case the rule is that by removal of glass the bands, or 
any parts of them, can be caused to move in the same 


Wt 
Or 


* If the surfaces are so shaped that the interference-bands presented 
on superposition are hyperbolic, much may be gained by limiting the 
aperture to a narrow slit corresponding to one of the asymptotes, espe- 
cially if the magnification used is in one direction only. 

+ “Interference Bands and their Applications,” ‘Nature,’ xlviii. 
p- 212 (1898) ; Scientific Papers, iv. p. 54. 


396 Lord Rayleigh: Further 


direction as that in which they move when the obliquity of 
reflexion is diminished. 

In carrying out the correction, the plate on which it is 
intended to operate is placed below, and it is convenient if 
it be held in some form of steady mounting so that the 
upper plate can be removed and replaced in the required 
position without trouble. The acid, two or three times 
diluted, is applied with a camel’s hair brush and after being 
worked about for a few seconds is removed suddenly with a 
soft cloth. Hndeavour should be made to keep the margin 
of the wetted region moving in order to obviate the forma- 
tion of hard lines. Success depends of course upon judge- 
ment and practice, and the only general advice that can be 
given is to make a great many bites at the cherry, and to 
Keep a record of what is done each time by marking suitably 
on one of a system of circles drawn upon paper and repre- 
senting the surface operated on. After each application of 
acid the plates are re-examined by soda-light and the effect 
estimated. ‘The difficulty is that in most cases the bands are 
not reproduced in the same form. In one presentation the 
error may reveal itself as a curvature of the bands and in 
another as an inequality in the spacing of bands fairly 
straight. Often by a little humouring the original form 
may be approximately recovered, and in any case the general 
rule indicates what needs to be done. 

By this method I have prepared two pairs of plates which 
perform very fairly well, but of course only when placed in 
the proper relative position. The operations, though pro- 
longed, are not tedious, and I doubt not that with perse- 
verance better results than mine might be achieved. The 
surface of the glass under treatment suffers a little from the 
development of previously invisible scratches in the manner 
formerly explained, but the defect hardly shows itself in 
actual use. I have not ventured to apply the method to 
surfaces already very good such as those supplied by the 
best makers for use in Fabry and Perot’s apparatus; but I 
should be tempted to do so if I came across a pair suffering 
from slight general concavity. The application of acid would 
then be at the outer parts. In the best glasses that I possess 
the error is one of convexity. 


Lifect of Pressure in Fabry and Perot’s Apparatus. 


The observation that the rings were more sensitive than 
had been expected to the pressure by which the plates are 


Measurements of Wave-Lengths. 557 


kept up to the distance-pieces, led to a calculation on Hertz’s 
theory of the relation between the change of interval and 
the pressure. If two spheres of radii 7, and rz and of 
material for which the elastic constants in Lamé’s notation 
are Dy, (4, Ae, My, are pressed together with a force P, the 
relation between P and the distance (a) through which the 
centres approach one another, as the result of the deforma- 
tion in the neighbourhood of the contact, is 


P a2) =-( YT 79 y az * 
37° by ae ap 6,+ 6, oi 


where 
Ay + 2 py No + 2 pbs 


5 a a a — 
‘ Amr pny(Ay + 1)’ °° Atrpin( Ae + fe) 


In the case of materials which satisfy Poisson’s condition, 
A=p, and we may take as sufficiently approximate 


5) 5) 
ain Sarpy” as Sm py | 


so that 
P= =( yy \® fy bg 2 
JD eit %7 bart Me 


In the application that we have to make, one of the spheres 
is of steel (invar) and of radius r,=°25 cm., while the other 
is of glass and of radius7,=«. Further, for the steel we 
may take p,=82x10", and for glass w,.=2'4x 104, and 
thus 


P= 3300 LOM, 
a being incm. and P in dynes. It will be convenient for 
our purpose to reckon « in wave-lengths (equal say to 


6 x 10-° cm.) and P in kilograms, taking the dyne as equal 
to a milligram weight. On this understanding 


P sd at; 


signifying that to cause an approach of one wave-length the 
force required is ‘15 kilogram. If P and « undergo small 


* See Love’s Math. Theory of Elasticity, § 139. 
Phil. Mag. 8. 6. Vol. 15. No. 88. April 1908. 2P 


598 Further Measurements of Wave-lenjths. 
variations, 
dP = $ (-15) a8 da = 3 (-15)3 P# da, 


dP/da being somewhat dependent upon the total pressure P. 

For the purposes of experiment a spring-balance was 
mounted upon the frame of the apparatus (carrying the 
distance-pieces) so as to diminish the pressure exerted over 
one of the distance-pieces, that is to diminish the pressure 
by which one of the plates was held up to one of the distance- 
pieces. tarting from perfect parallelism of the plates and 
keeping the eye carefully fixed so as to receive the light from 
the centre of the plates, it was observed that to cause a shift 
of one band (helium vellow) the spring-balance needed to 
exert a pull of ‘78 kilo. At this stage the plates were of 
course no longer parallel and a moderate shift of the eye 
would cause a displacement independently of any change in 
the spring-balance. At the same time the rings lost their 
sharpness. On this account it is hardly practicable to use a 
shift of more than one band, and indeed a smaller shift of 
half a band was usually preferred. The total force required 
to compensate the spring of the apparatus, and so to relieve 
the compression of the distance-piece on this side, was 2°4 
kilos. This is what is represented by P in the above formula, 
while dP=—’78. 

In order now to compare theory and observation we must 
remember that the one band (corresponding to half a wave- 
length) observed at the centre implies three times as great a 
shift at the particular distance-piece where alone the force 
was varied. Thus the observed dP corresponds to de=—3. 
For this da, the calculated dP is 


dP =—2 (-15)3 (2-4)} =—-85 kilo. 


The agreement with the observed —°78 is certainly as good 
as could have been expected. 

In considering what differences of distance are to be 
expected when the plates are adjusted to parallelism under 
different pressures, we must remember that the above calcu- 
lation and observation relates to the compression which may 
occur at the contact of a single distance-piece with a single 
plate. There are in all six contacts of this kind, and we 
may conclude that when no special pains are taken to regulate 
the absolute pressures employed, a shift of 6 bands or more 
on remounting need not cause surprise. 


ups ae 


[ 559 4 


| LI. The Positive Column in Oxygen. 
By the Rey. P. J. Kirxsy, Fellow of New College, Oxford”. 


as researches in which I have recently been engaged 

led me to investigate the electric force in the “ positive 
column” when a steady electric discharge passes through 
oxygen at low pressures between two plane parallel electrodes 
placed in a straight glass tube. 

The “positive column” in such a cylindrical discharge 
may, in view of common experience, be defined as a region 
of the discharge terminating at or close to the anode or 
positive electrode where the electric force—i. e. the potential 
difference per centimetre—is constant or nearly constant. 
Close to the cathode there is an abrupt fall of potential 
called the ‘“ cathode fall.” Here there is a bright glow 
lining the cathode which is unmistakable. Between this 
glow and the positive column there is nothing to catch the 
eye at pressures of a few millimetres of mercury. The 
positive column is easily recognized as a long faintly luminous 
column of a colour which varies with the gas through which 
the discharge passes, being in oxygen either pale violet or a 
curious green. | 

At pressures of thé order 1 mm. it begins at a distance of 
about 6 cms. from the cathode and extends right up to the 
anode. The nearest point of the positive column to the 
cathode may be called the foot of the positive column. The 
electric force is remarkably constant throughout this region: 
it depends chiefly on the pressure, but is said to vary to 
some extent with the diameter of the discharge-tube. 

The results of my experiments show that in oxygen the 


variation of the electric force in the positive column with 


the pressure presents exceptional features which do not 
appear to have been noticed. In the first place, the force is 
very much smailer than in the other common gases; secondly, 
instead of diminishing continually with the pressure it reaches 
a minimum at about 2:0 mms. pressure; and thirdly, there 
is a sharp discontinuity in its value at a pressure of ‘8 mm. 
as measured by the McLeod gauge. 

The methods hitherto employed to investigate the electric 
force in the positive column have generally involved the use 
of exploring wires fused into the sides of the tubes: their 
difference of potential during a discharge divided by their 
distance apart was assumed to give the electric force in the 
undisturbed gas. This method is open to the criticism that 


* Communicated by the Author. 
+ J. J. Thomson, ‘ Conduction of Electricity through Gases,’ ch. xv. 


560 Rev. P. J. Kirkby on the 


the introduction of the wires in the sides must disturb the 
uniformity of the surface of the discharge-tube, and pre- 
sumably affect the discharge itself. 

The method which I employed did not involve the use of 
an exploring wire. It is the same which I used to determine 
the force in the positive column in a discharge through 
electrolytic gas*. 

It depends on the assumption, suggested by experience 
and, I believe, fully justified by these experiments, that the 
positive column can be lengthened without making any 
other sensible change in the steady discharge. Throughout 
this paper ; 

D represents the distance in cms. between the electrodes, 
always exceeding the distance between the foot of 
the positive column and the cathode; 

C the steady current passing through the gas and always 
‘0025 ampere; 

X the voltage-difference of the electrodes during the 
discharge; 

Y the electric force in volts per cm. in the positive 
column; and 

‘p the pressure in mms. of mercury. 


Then if D is increased by d, the potential-difference of 
the electrodes required to maintain the same current C as 
before under the same conditions will be simply X+ Yd. , 

In fact if everything is kept constant in the discharge but 
X and D, X will obviously be a continuous function of D 
whatever D’s value. If D is taken as abscissa and X as 
ordinate a curve can be plotted, and it is easy to verify 
under favourable conditions that when D exceeds a certain 
value the curve becomes straight within the limits of ex- 
perimental error, and that the value in question is about the 
distance of the foot of the positive column from the cathode. 
The slope of this straight part of the D—X curve is obviously 
the electric force Y in the positive column. The advantage 
of this method is first that the exploring wire is dispensed 
with, and secondly that the electric force can be determined 
and verified by several observations with different values 
of D. The disadvantage is that the observations are not 
simultaneous, so that if a small change occurs in the region 
of the discharge near the cathode, where the electric force is 
very great, relatively large errors may occur in the observa- 
tions. This difficulty should only be felt when Y is small; 
but in oxygen Y is remarkably small, ranging for pressures 


* Phil. Mag. March 1907. 


Positive Column in Oxygen. 561 


less than 5 mm. between 4°5 and 20 volts per cm. in these 
experiments. Now the cathode fall is about 400 volts, so 
that if a small change of a few per cent. in the field near 
the cathode occurs between two observations, it can easily 
introduce an error of 1 em. in D. Moreover, it is necessary 
to keep the current accurately the same, since a very small 
error in reading it corresponds to a much larger one in D. 

The electrical arrangement is represented diagrammatically 
in fig. 1. ABPQ represents the discharge-tube, supported 
in such a position that its base dipped under the mercury 
filling a large jar J. The side tube T connected the dis- 
charge-tube through a drying-tube to a mercury pump, and 
to an electrolysis apparatus where oxygen was generated 
and dried. It may be stated here that the oxygen was pre- 
pared by the electrolysis of pure barium hydrate, which 
insures its purity *. 

Fig. 1, 


SSF 


BES SSS SSNS 


The figure represents the tube partly filled by the mercury 
—the shaded part—when the oxygen was reduced to a low 
- pressure. A is the fixed cathode connected to a platinum 


* H. B. Baker, Trans. Chem. Soc. 1902, vol. Ixxxi. 


562 Rey. P. J. Kirkby on the 


wire fused into the top of the tube. The anode B is attached 
to an iron rod, which, sliding in the fixed steel guides P, Q, 
and bent round the bottom, terminates at E. Thus by 
raising or lowering E, the movable anode could be ad- 
justed to any distance up to about 30 cms. The positive 
end of the battery Z was in metallic connexion with the 
mercury, and so with the anode; and the negative end was 
connected through a galvanometer G and the variable re- 
sistance R to the cathode A. Z was usually of about 1500 
volts, G was a high resistance and dead-beat voltmeter, and 
R several 100,000 ohms. 

With such an apparatus R could be varied and D deter- 
mined by adjusting the anode till the galvanometer recorded 
C. And then we may, as stated above, regard X as a linear 
function of D, C and p being constant. 

Now X and R are connected by the ohmic equation 


X = Z—C(R+G), 


if Z denotes the E.M.F. of the battery and G the resistance 
of the galvanometer, together with any other invariable 
resistance in the circuit including that of the battery. 
Therefore R is a linear function of D, and the points whose 
coordinates are D, R will determine a straight line, the slope 
of which multiplied by C is the electric force in the positive 
column, since 


The slope of the D—R line was determined throughout 
the earlier experiments by finding the values of D corre- 
sponding to several different values of R. Thus a number 
of points were plotted, which, if the discharge had been 
steady, lay upon or close to a straight line. If they did not, 
I often found that some lay very near one line and others 
fell close to a line parallel and clcse to the former. That 
showed in rather an interesting way that the discharge had 
shifted from one to the other of two slightly different states 
or positions without, however, affecting the magnitude of the 
force in the positive column. Sometimes three such parallel 
lines could be recognized, but of course, if there were more 
than two such shifts, the corresponding parallel lines cannot 
easily be detected, and the force Y can only be estimated, 
and with considerable error, by drawing the mean locus of 
the paints. Owing to this experience and in order to obviate 
such shifts, I ultimately determined Y, when possible, in a 
single discharge by diminishing R twice by the same amount 


Positive Column in Oxygen. 563 


r—i.e. by 2r altogether—and determining the corresponding 
increases d;, dj+d, to D. If d, was nearly equal to d,, it 
was assumed that no alteration had occurred in the discharge 
outside the positive column to invalidate the experiment ; 
the force Y was taken to be the mean of 2 and s. , and 
1 2 
the difference between d, and d, compared with either as an 
index of the experimental error. This method is useful when 
the discharge is unsteady. 

When the pressure is less than its value at which Y is 
discontinuous, d; is always nearly equal to d;, for the points 
whose coordinates are D, R fall accurately upon a straight 
line. But for pressures above ‘8 mm. it is seldom that the 
discharge remains constant enough to permit the determi- 
nation of Y with the same precision. The chief cause of 
variation is the flickering of the discharge which at the 
higher pressures does not cover the whole of the cathode nor 
remain on the same part of it. And if the electrodes become 
oxidized the trouble increases. 

After making experiments with zine and nearly pure silver 
electrodes and being much inconvenienced by their oxidation, 
I tried silver-gilt electrodes. These were much the most 
satisfactory, and as long as the gold lasted no sign of dis- 
colouration appeared on the cathode or discharge-tube. The 
anode deepened in colour, but only tarnished where the 
gold was obviously thinnest. The cathode closely fitted the 
discharge-tube whose diameter was nearly 2-4 cms. The 
diameter of the anode was 2 cms., so that it could move up 
and down the tube without touching it. 


TABLE I. 
i] II 

Pp ¥, | Dp. x / p NE 
2°52 DS Oe 19-2 3°94 121 
1-85 4-5 3-28 9°3 96 102 | 
1-48 £6 Sirieras 57 802 20:0 | 

oa. | Ite, Le 625 || , | 

02 | 192 56 15-4 | 1-21 73 | 

79 18:7 | : 


The results of experiments with these electrodes are given 
in Table I., each compartment of which applies to a fresh 
quantity of oxygen. The errors in the determination of Y 


564 Rey. P. J. Kirkby on the 


during a given discharge are estimated to be 4 or 5 per cent. 
except for pressures below *8 mm., when the results are 
much more accurate. The pressures were all measured upon 
a McLeod gauge. 

Table I. is illustrated by the curve in fig. 2, in which all 
the points lie satisfactorily near the curve except two, the 
determinations of which were more uncertain than that of 
the others. 


The curve is oe eee nS pee curves 
of the other common gases. When pis greater than 15 mm. 
the values of Y are only between one-tenth and one-fifth of 
the electric force in a similar positive column in the similarly 
dense gases nitrogen or carbon monoxide. Again, at the 
pressure 1°75 mm., Y reaches a minimum with the curiously 
small value 4°5 volts per cm. 

The third peculiar and most interesting feature of the 
curve is the violent discontinuity which Y undergoes at the 
pressure ‘8 mm. That the discontinuity is real and not 
apparent is shown by the fact that Y then passes suddenly 
from about 11 to 20 without seeming to assume any inter- 
mediate value, and even more by the fact that a complete 
change seems to come over the discharge, which now settles 
down into the greatest steadiness. Thus the curve is broken 
and not merely very steep. The change occurs during a 


: 
. 
4 
‘ 
; 
q 


Positive Column in Oxygen. 565 


discharge, if the pressure of discontinuity is approached very 
gradually. I have twice noticed immediately after the sudden 
rise in Y the appearance of peculiar strize so close together 
that there were 10 within 18cm. That was with the zinc 
electrodes. It is very seldom indeed that they appear with 
oxygen. 

To investigate the discontinuity in Y more thoroughly, I 
connected to the apparatus a large glass tube which could 
be filled partly or wholly with mercury like the barrel of a 
mercury pump. With this arrangement the pressure could 
be varied, and, without introducing new oxygen, could be 
made to pass backwards and forwards across the point of 
discontinuity. It was then observed that Y passed from its 
low value to its high and back again from high to low, 
keeping pace with the pressure. This is illustrated by the 
following simultaneous values of p and Y, which represent a 
series of experiments upon the same mass of oxygen :— 


‘95 mm.,10; -84,11-4; °765,19:2; +802, 20-0; 
-88,11°6; °81, 20. : 


It is evident, therefore, that the discontinuity is not brought 
about because the gas has been vitiated by the discharge. It 
must be due to some inherent instability in the oxygen when 
the pressure is slightly greater than ‘8 mm. 

But when a quantity of oxygen had been subjected toa 
long series of discharges, the pressure can be raised con- 
siderably beyond -8 mm. where the discontinuity first 
occurred. Thus on one occasion, after the pressure was 
reduced below the discontinuity point, it was gradually 
raised from *79 mm. to 1°07 without Y’s abruptly changing, 
and Y reached the abnormally high value 23. The same 
phenomenon was observed with the silver electrodes. 

The electric discharge therefore tends to prevent the 
oxygen from recovering the condition roughly defined by 
p='8, Y=11. It is natural to attribute this tendency to 
ozone ; especially since the fall of pressure during each of 
these two series of observations showed the conversion of a 
considerable amount—probably more than 15 per cent.—of 
oxygen into ozone. 

We may therefore conclude that the presence of ozone 
checks the repassage of the oxygen through the discontinuity. 

When the pressure falls below °8, the slope of the curve 

Y is very steep; but there is no discontinuity whatever 
here. On the contrary, Y is nowhere determined so easily 
as when p lies between *5 and‘8 mm. The discharges for 
this range of pressures are very steady: the readings of D 


566 Rev. P. J. Kirkby on the 


are repeated with little error, and the decrease in R, and 
therefore the increase in X, is strictly proportional to the 
increase in D—facts which prove that very little change 
occurs in the region of high force near the cathode during a 
series of discharges with the same current. This steadiness 
is shown by Table II., which gives the results of a series 
of experiments with gradually diminishing pressure upon 
freshly introduced oxygen during my earlier experiments 
with zine electrodes (diameter 2-1 ems.). The total dura- 
tion of the discharges used for these observations was about 


TasueE II. 

p. | ¥. 25°6 p+'9. 
Be js ee Je es 

804 | 21-4 21°5 
| 75 | 198 20°1 
| 72 | 19:2 19:3 | 
| 69 | 18:35 185 
| “66 17°85 177 

643 17-25 Cte | 
| 617 168 16-7 | 
| 58 158 157 
| 534 14-42 14:55 | 


3 minutes ; and it may be taken that ultimately 10 per cent. 
of the gas was ozone. The numbers show an extremely 
accurate agreement, and prove that the curve is very straight 
to the left of the discontinuity; for every value of Y butone 
differs from 25°6 p+°9 by less than 1 per cent. | 

Table II. exhibits great consistency in the observations, 
which were made rapidly, and all but the first and last with 
two observations only. In fact it is experiments like these, 
made with a steady discharge, that establish the validity of 
the method and prove that the positive column can be 
lengthened without other wise sensibly affecting the discharge. 
We can also infer from the table that the presence of a good 
deal of ozone does not greatly modify the force Y when p is 
less than °8. | 

Table III. contains the results of experiments, other than 
those in Table II., made with the zine electrodes. The per- 
centage errors in the values of Y, most of which were 
determined by several observations, is not more than 4 and in 


Positive Column in Oxygen. 567 


most cases less than 2 or3. Hach compartment of Table III. 
applies to experiments upon the same specimen of oxygen. 


TABLE III. 


p XM p | x p Y 
1:02 TD 6:26 15 1:02 79 
“495 131 4-2 66 Wo 20°4 
a 32 oo | ee 
2°02 4°6 2°42 4:04 |! “785 20°4 
os=SS See aes —|— = wis) 19-2 
40 Sieze> Tha Zee aes) f 17°6 
2:36 44 686 (eye || aaa eoelea ome 22 bs 2 
1-49 6°15 49 ior) hee 117 


The numbers in Table III. are with two exceptions suffi- 
ciently concordant, although the oxygen had in many cases 
been subjected to much discharge. They determine a curve 
which is nearly identical with that of fig. 2 for pressures 
below 2 mms., but falls much below it when p exceeds 2 mms. 
Thus, when p exceeds 4, Y is less than two-thirds of its cor- 
responding value in fig. 2. This large difference cannot be 
attributed to the small difference in the discharge-tubes ; 
especially since the latter difference should make—if we can 
be guided by the behaviour of nitrogen *—the values of Y in 
Table III. greater than those in Table I. 

Hence the conclusion is forced that the divergence is due 
to the use of different electrodes. This conclusion is con- 
firmed by my experiments with silver electrodes, which gave 
a general agreement with the foregoing results for pressures 
less than 2°5 mms., and at first gave the usual pressure of 
discontinuity. But partly perhaps owing to the cathode’s 
not fitting the tube, and chiefly, I believe, to the oxidation 
of the electrodes, these experiments left few numbers to rely 
on or worth recording. It may be remarked, however, that 
when the electrodes became badly attacked, the pressure of 
discontinuity was thrown back to the point ‘62 mms., and 
the values of .Y on both sides of it were then reduced to 8 
and 18. 


* Herz found that the electric force in nitrogen was 99'7 and 89°3 for 
tubes 10 and 20 mms. in diameter respectively.—J. J. Thomson, J. ¢. 
ch. Xv. 


568 The Positive Column in Oxygen. 


- The observation of the discontinuity in Y at the pressure 
*8 mm. immediately recalled the discontinuity of pu observed 
by C. Bohr * in oxygen at the pressure ‘7 mm., the history 
otf which is very interesting. Bohr’s observations were con- 
firmed by Baly and Ramsay + ; but in 1901 Lord Rayleigh ¢ 
was unable to detect any signs of such an effect with his 
extremely accurate manometer, and, at a loss to account for 
the difference between Bohr’s experience and his own, could 
“‘only suppose that it must be connected somehow with the 
quality of the gas complicated perhaps by interaction with 
the glass or the mercury.” 


Now, according to Bohr, Boyle’s law must be replaced, in 


the case of oxygen, for temperatures between 11 and 14, by 


(p+°109)v = k when p is greater than *7 mm., 
and by 
(p+°07)v = k when p is less than ‘7 mm. 


All the pressures given above were measured on a McLeod 
gauge: and the temperature of the laboratory ranged from 10° 
to 15°. Hence, if Bohr’s results apply to them, they must be 
diminished by -109 when p is greater than *7 mm., and by °07 
when p is less than:7 mm. Moreover, the pressure *7 would 
be given by the McLeod gauge either as *809 or ‘77. Thus 
the pressure of Bohr’s discontinuity agrees in the closest way 
with the pressure at which Y is discontinuous. Nevertheless, 
in view of Lord Rayleigh’s conclusions, I have not ventured 
to make any corrections to the estimated values of my 
pressures. 

The observed discontinuity in Y points to a molecularly 
unstable condition reached by oxygen at the pressure in 
question, such that under the discharge the gas—or possibly 
only that part of it which is the seat of the discharge— 
passes from one to the other of two states of molecular equi- 
librium. In this connexion the suggestion of Sutherland § 
in explanation of Bohr’s and Sir W. Crookes’s anomalies is 
interesting, that oxygen tends spontaneously to pass into 
ozone at about ‘7 mm. pressure. This view receives some 
support from the conclusion above that the presence of ozone 
checks the return of the oxygen to the state defined by smaller 
values of Y. I have twice observed at the point of Y’s dis- 
continuity curious falls of pressure which would be well 
explained by Bohr’s results, and cannot but conclude that his 

* Wied. Ann. vol. xxvii. pp. 459-479 (1886). 
+ Phil. Mag. vol. xxxviil. p. 301 (1894). 

Tt Phil. Trans. 1901, p. 205. 

§ Phil. Mag. vol. xlin. 1897, p. 201. 


EE eS 


Notices respecting New Books. 569 


discontinuity and the discontinuity in Y are intimately con- 
nected. Is it not a possible explanation of Lord Rayleigh’s 
and Prof. Bohr’s different experiences, that the carefully 
dried tubes of the latter may have developed a high state of 
electrification by the motion of the mercury, and that thus 
the gas became subject to electrical disturbances sufficient to 
effect the transition from one molecular state to the other ? 
These experiments were made in the laboratory of Prof. 


J. 8S. Townsend, Wykeham Professor of Physics, Oxford. 


LILI. Notices respecting New Books. 


Condensation of Vapor as induced by Nuclear and Ions. By 
Cart Barts, Hazard Professor of Physics, Brown University. 
Washington: Published by the Carnegie Institution. 1907. 
Pp. ix+164. 


[* this monograph the author, whose work on nuclei is well 

- known, describes a number of investigations carried out with 
his fog-chamber apparatus. The apparatus having been sufficiently 
improved, it was used for various experiments, including the 
growth of persistent nuclei, the production of water nuclei by 
evaporation, the results obtained when X-rays are allowed to 
strike the foo-chamber from different distances, the effect due to 
radium, &c. Other problems dealt with in the book are the dis- 
tribution of colloidal nuclei and of ions in media other than air- 
water, the simultaneous variation of the nucleation and the 
ionization of the atmosphere of Providence, and the variations of 
the colloidal nucleation of dust-free air in course of time. 


Fourier’s Series and Integrals. By H. 8S. Carstaw. Macmillan 
and Co. London and New York. 1906. 


THE complete title is ‘“‘ Introduction to the Theory of Fourier’s 
Series and Integrals and the Mathematical Theory of the Conduction 
of Heat.” The book naturally falls into two parts, Part I. being 
concerned with the purely mathematical developments, and Part II. 
with the applications to the various practical and ideal problems 
of heat conduction. Basing on the modern theory of rational and 
irrational numbers, Professor Carslaw leads up through a logical 
discussion of the convergence of infinite series to the important 
forms associated with the name of Fourier, finishing in Chapter VIII. 
with the Integrals. This part occupies 187 pages in a book of 
430 pages. The remainder is devoted to the problems of thermal 
conduction. Professor Carslaw is to be congratulated on sup- 
plying a book which cannot fail to be of great service to several 
classes of readers. The student of pure mathematics will find 
his mind directed along the best modern lines of investigation of 


570 - Jotices respecting New Books. 


an important set of functions; and he will find opportunity of 
testing his powers by consideration of the examples which are 
appended to most of the Chapters in Part I. The physicist, again, 
who is face to face with a practical problem will find, if not 
exactly what he needs, something very near to it in one or other 
of the many solutions presented in Part II. The author does not 
claim to give all the methods which have been used for measuring 
conductivity ; but we think a reference might have been made to 
Yamagawa’s experiments, in which a sphere of stone was subjected 
to a periodic surface variation by alternate immersions in boiling 
water and melting ice. The results are certainly of a higher 
order of accuracy than those of Ayrton and Perry with the 
cooling sphere. 


Sechs Vortrage uber das Thermodynamische Potential. By J. J. 
van Laar. Friedrich Vieweg und Sohn. Braunschweig, 1906. 


THEsE lectures exemplify the use of the thermodynamic potential 
in the treatment of the outstanding problems of solution and 
saturation. They are prefaced by two polemical lectures in which 
the author criticises in a lively manner the worshippers of the 
osmotic pressure. 


The Scientific Papers of J. W1utaRp Gripes. In Two Volumes. 
Longmans, Green & Co. 1906. 


THESE volumes contain in attractive form the important contri- 
butions made to scientific journals and transactions by the late 
Professor Willard Gibbs, of Yale University. Undoubtedly his 
greatest work was in connexion with thermodynamics. His first 
paper on Graphical Methods appeared in 1873; and in 1876 and 
1878 appeared in two memoirs the remarkable and truly original 
discussion of the equilibrium of heterogeneous substances. It is 
a great advantage to have these papers together and easily acces- 
sible. The thermodynamic papers fill the first and larger volume. 
In the second volume the contents are of varied character; but 
probably the most interesting sections are those which concern 
vector analysis and multiple algebras. Here the mathematician 
will find some food for thought. The first volume is prefaced 
with a biographical sketch by H. A. Bumstead, and a complete 
bibliography. The portrait which forms the frontispiece shows a 
clear cut intellectual face of the best New England type. 


A Compendium of Spherical Astronomy. By Simon NEwcoms. 
The Macmillan Company, New York, and Macmillan & Co., 
London. 1906. 


Tue precise nature of the present volume is indicated by the 
continuation of the title,—‘‘ with its application to the determi- 
nation and reduction of positions of the fixed stars.” It is, as 
we learn from the preface, the first of a series, intended not only 


Sa a ee 


Notices respecting New Books. 571. 


for the student but for the working astronomer. It is from the 
point of view of the student that we consider the book here. The 
three opening chapters form Part I., and are devoted to preli- 
minary questions such as the meaning and use of small quantities, 
interpolation, and determination of probable errors. Chapters IV. 
to IX., which constitute Part II., deal with the fundamental prin- 
ciples of spherical astronomy, and discuss in order spherical co- 
ordinates, measurement of time, parallax and figure of the earth, 
aberration, astronomical refraction, and precession and nutation. 
Having thus laid the foundations strong and sure, Prof. Newcomb. 
proceeds to consider in the four remaining chapters which make 
up Part III. the reduction and determination of positions of the 
fixed stars. The value of this part is enhanced by the fact that 
real examples are worked out in full as illustrations of the methods 
described. At the end of each chapter is appended a set of 
historic notes which give with great conciseness the important 
lines of development. Where all is so admirable it is difficult to 
pick out particular points for special reference. The discussion 
of astronomical refraction seems, however, to merit special notice. 
The treatment is fresh, and should be read by all mathematical 
and physical students, even although they should have no intention 
to pursue astronomical studies further. 


Meteorologische Optik. Von J. M. Perntur. III. Abschnitt. 
Wien und Leipzig: W. Braumiiller. 1906. Pp. 346. 


In this volume the author continues his exhaustive account of 


_ meteorological optics. The first section of this monumental work 


appeared in 1902, and was soon followed by Section II., which 
appeared later in the same year. Both were noticed briefly in 
these columns. Section III., now under review, deals with 
phenomena produced by particles not always present in the 
atmosphere—ice crystals and water-drops of cloudsand rain. This 
section of the work is divided into three chapters, the division 
being based on the particular effects to which the phenomena in 
question are due. Thus, Chapter I., which fills the greater part 
of the volume, deals with phenomena caused by refraction and 
reflexion— halos and parhelia. Numerous illustrations and 
accounts of some very remarkable instances of such appearances 
are given, and a very thorough classification of snow crystals, with 
numerous plates illustrating their different forms, is incorporated 
in this chapter. Chapter II. treats of phenomena due solely to 
diffraction—corone and similar effects, while Chapter III. is devoted 
to rainbows. The same thorough method of treatment is followed 
by the author throughout. We first have a descriptive account of 
the appearances dealt with, then a detailed theoretical explanation, 
and lastly an account of experimental arrangements which enable 
us to reproduce the effects dealt with, and to supply a test of the 
soundness of the theoretical explanations. To the student of 
meteorological optics this work will be full of interest. 


572 Geological Society :— 


Two New Worlds. I. The Infra-World. Il. The Supra- World. 
By E. E. Fournier v’Axse, B.Sc. London: Longmans, Green, 
& Co. 1907. Pp. vii + 159. 


Tuts intensely interesting and strikingly bold essay on the relativity 
of time and space, and on man’s position in the universe, deserves 
to be widely read. ‘“ The main thesis of this work ”—to quote from 
the author’s Preface—‘‘ is that a universe constructed on a pattern 
not widely different from ours is encountered on a definite and 
measurable scale of smallness, and another on a correspondingly 
larger scale. To these universes I give the names Infra-W orld and 
Supra- World respectively.” Our “atoms” are the suns and stars 
of the “* Infra~W orld,” while our “ electrons ” represent its planets. 

An electron may, for aught we knew to the contrary, be of as 
complex a structure as our planet, and may be teeming with life. 
Lengths and times in this ‘“ Infra-World” are reduced in the 
same ratio of about 10° to 1, and absolute velocities are of the 
same order as thosein our world. Similarly, our planets form the 
electrons of the ‘‘ Supra-World,” our solar systems its atoms; the 
ratio of times and magnitudes being again 10” to 1, and absolute 
velocities being as before of the same order as ours. Our faculties 
enable us to perceive only three links of the chain of universes. 
The remaining links are concealed from us, and are (to us) at 
present non-material. 

The author writes in an interesting way, and the concluding 
chapters of the book contain many stirring passages. Even to those 
who may not be prepared to agree with “the author's conclusions, 
the book will be an intellectual treat. 


LIV. Proceedings of Learned Societies. 
GEOLOGICAL SOCIETY. 
[Continued from p. 204. ] 
December 4th, 1907.—Sir Archibald Geikie, K.C.B., D.C.L., Se.D., 
Sec. R.S., President, in the Chair, 
pee following communications were read :— 


‘The Faunal Succession in the Carboniferous Lieeeaae 
(iigtof Avonian) of the Midland Area (North Derbyshire and North 
Staffordshire).’ By Thomas Franklin Sibly, B.Sc., F.G.S. 
2. ‘Brachiopod Homceomorphy: “Spirifer glaber”.’ By S. §S. 
Buckman, F.G.S. 


December 18th.—Sir Archibald Geikie, K.C.B., D.C.L., Se.D., 
Sec.R.S8., President, in the Chair. 
The following communications were read :— 


1. ‘Some Recent Discoveries of Paleolithic Implements.’ By 
Sir John Evans, K.C.B., D.C.L., LL.D., F.R.S., For.Sec.G.8, 


*¢ ——— == 


Glacial Epoch in North America. 573 


2. ‘On a Deep Channel of Drift at Hitchin (Hertfordshire).’ 
By William Hill, F.G.S. 


Evidence is given, from nine borings running along a line slightly 
west of north trom Langley through Hitchin, of the existence of a 
channel of considerable depth, now filled with Drift, occupying the 
centre of an old valley in the Chalk-escarpment, which may be 
called the Hitchin Valley. For the first 3 miles it appears to 
be contained within narrow limits, persistent ridges of Chalk 
occurring on each side, and it might almost be compared to a 
Chalk-combe. At Hitchin, after passing between two Chalk- 
knolls, its confines become less clear, and there seems to be some 
evidence of broadening as it emerges on to the Lower Chalk-plain 
and leaves the higher ground of the main Chalk-escarpment. - The 
greatest depth to which the channel has been proved is at a boring 
in Hitchin, where the Gault was reached beneath Drift at a depth 
of 68 feet below sea-level. That the channel flowed northwards 
and belonged to a ‘ subsequent’ stream seems to be proved by the 
fact that at Braybury End, the only place where a southerly stream 
could pass, the space between bare Chalk-exposures is but 450 yards, 
and in about the middle of the space Chalk has been reached within 
50 feet of the surface (that is, about 200 feet above sea-level) in a 
well dug a few years back. The channel must be older than the 
Chalky Boulder-Clay, which still partly fills it as far south as Langley, 
and may have blocked it to the southward and given rise to the 
features now presented in the drainage on the northern slope of 
the escarpment. But the author is inclined to suggest that either 
glacier-ice or bay-ice must have played no unimportant part in 
damming up the old valley. The author suggests the existence 
of another channel, in this case draining southwards, buried under 
the broad area of Boulder-Clay and gravel which lies immediately 
south of Stevenage and to the north as fur as Letchworth and 
Wilbury Hill. Buta narrow space of bare Chalk, at an elevation 
of 240 feet O.D. connecting large areas east and west of it, pre- 
eludes the occurrence of a channel farther north than Letchworth. 


January 8th, 1908.—Sir Archibald Geikie, K.C.B., D.C.L., Sc.D., 
Sec.R.S., President, in the Chair. 


The following communications were read :— 


1. ‘Chronology of the Glacial Epoch in North America.’ By 
Prof. George Frederick Wright, F.G.S.A. 


In the case of Plum Creek, Lorain County (Ohio), the study of the 
activity of the stream and of the amount of work which it has done 
since a certain stage of the Glacial Epoch, has yielded important 
results. This stream began the erosion of its trough when the 
temporary lake, held up in front of the ice, was maintained for a 
considerable period at the level of its Fort-Wayne outlet ; it has 


Phil. Mag. 8. 6. Vol. 15. No. 88. April 1908. 2Q 


BYE! Geological Society :— 


never had anything more resistant than Till to act upon. From 
a given section, 5000 feet long, it has excavated 34 million cubic 
feet of Boulder-Clay, removing it from exposed banks 1600 feet long. 
Twelve years’ erosion of a 500-foot length of a part of the trough 
of the stream under observation, and from banks 1000 feet long, 
gives a rate of 8450 cubic feet per annum. Therefore, the removal 
of 34 million cubic feet from the 5000-foot section would give a 
period of 2505 years. Considerations tending to lengthen the 
estimate are the former forestation of the area and the increased 
gradient in the artificial cut-off. Those tending to shorten the 
estimate are the present wider flood-plain, the time taken for forests 
to grow, and the probably greater former water-flow. 

The erosion of the Niagara Gorge began considerably later than 
that of Plum Creek, and probably dates from midway between the 
disappearance of the ice from Northern Ohio and from Quebec. If 
conditions have been uniform, the age of the Gorge would be 7000 
years. As the Niagara Limestone is thinner at the mouth of the 
Gorge, and the Clinton Limestone has dipped out of sight at the 
Whirlpool, there is nothing in the stratigraphy to indicate a slower 
recession in the past than in the present. Moreover, nearly one- 
third of the erosion has been accomplished by two pre-Glacial 
streams, one from the south and a smaller one from the north. 
Therefore, the author concludes with considerable confidence that 
the Gorge is less than 10,000 years old, and that the ice of the 
Glacial Epoch continued down to that time, to such an extent over 
the lower St. Lawrence Valley and Central New York that it 
obstructed the entire eastern drainage of the Great Lakes. 

There is nothing which would lead to a longer estimate of the 
time which has elapsed since the Kansan stage of the Glacial Epoch 
than that approved by Prof. Calvin of Iowa, and agreed to by Prof. 
Winchell. These assume 8000 years as the limit for post-Glacial 
time, and that a multiple of this by 20, amounting to 160,000, 
would carry us back to Kansan time. This, however, would still 
leave as long a period still earlier, for the advance of the ice. The 
author’s impression is that the whole epoch may well have been 
compassed within 200,000 years. 


2. ‘On the Application of Quantitative Methods to the Study of 
the Structure and History of Rocks.’ By Henry Clifton Sorby, 
LL.D, FERS FE LSS fe 


The knowledge of the final velocities of material subsiding in 
water is of fundamental importance ; but the relation between size 
of particles and velocity is complex, and perhaps may be partly 
explained by a thin, adherent film of water. The angle of rest 
in the case of sand-grains of varying size and quality enables us to 
ascertain approximately the velocity of current necessary to keep 
such sand drifting, and that needed to move it when at rest. The 
comparison of this angle with that observed in sedimentary rocks 


Quantitative Study of the Structure of Rocks. 575 


made of similar materials may be used to determine the amount of 


vertical contraction of rocks since deposition, the average in cases 
studied in Tertiary and Secondary rocks being from 100 to 57. In 
studying the drifting of sand along the bottom by currents (on 
which the author experimented in a small stream many years ago), 
the results are found to vary, according to whether the water is 
depositing sand as well as drifting it, and according to whether ripples 
are or are not being formed on the bottom. The velocity of a current 
can be determined approximately in feet per second for different kinds 


of sand. The connexion between the structure of ‘ripple-drift’ 


and time is discussed ; and an equation is given, from which the 
rate of deposit in inches per minute can be deduced. The con- 
nexion between the structure of a deposit and depth of water is 
found to be difficult to study quantitatively. From the occurrence 
of ‘drift-bedding’ the depth of water may probably be determined to 
within a few feet, and on this being applied to particular rocks some 
interesting results come out, including the separation of sandstones 
into several different groups. The deposition of fine deposits, like 
clay, is a most complex subject, varying according to the amount 
of mud present in the water, and according to whether the grains 
subside separately or cohere together. When no pressure is applied, 
even when no further contraction takes place on standing for a 
year, the amount of water included in the deposited clay may be 
80 per cent., and, when dry, the minute empty spaces may still 
amount to 32 per cent. This leads to the conclusion that many of 
the older rocks must now be only 20 per cent. of their original 
thickness. In many cases there is produced by a gentle current a 
minute laminar structure from which probably the rate of depo- 
sition may be learned approximately, a common rate in the older 
rocks being from 9 to 18 inches per hour. But compiex and difficult 
experiments are very desirable on this question. The rocks classed 
as clays differ very much in structure, anal must have been formed 
under different conditions. 

Applying these conclusions to various Inds the author shows 
that in the green slates of Langdale there is good evidence that 
the volcanic eruptions sometimes occurred within a few weeks of 
one another, and at other times at more distant intervals. Now 
and then there were bottom-currents, probably due to volcanic 
disturbances, gradually rising to a rate of about 1 foot per second 
and’ gradually subsiding, the entire period being a a minutes, 
and deposition taking place in different cases at from ++, to 2 inches 
per minute. There is also good evidence that, when deposited, 
part of the rock was analogous to fine, loose sand, and part to 
semi-liquid mud. In the Coal-Measure sandstones deposition at 
the rate of 1 inch per minute was common, with intervals of little 
or no deposit. 

The volume of invisible cavities in rocks varies from 49 per cent. 
in some recent rocks, to nearly 0 in the ancient slates. The 
packing of grains is discussed mathematically and experimentally, 


576 i Geological Soctety :— 


the latter with round and flattened shot; and experiments with 
sand of various qualities, rapidly deposited and also when well 
shaken, show a good agreement with calculation. The methods of 
determining the volume of minute eavities in rocks are given, 
followed by a number of examples from recent and older deposits. 
It is found that in some limestones the cavities have been reduced 
by pressure to close on the mathematical minimum, whereas in 
others, even of Silurian age, the cavities were filled with carbonate 
of lime, introduced from without, not long after deposition. Some 
oolites have had their cavities filled in a similar manner; in others 
most of the material of the original grain has been removed, and 
the present solidity is due to the filling-up of the original cavities 
mainly by internal segregation. Among fine-grained rocks the 
Chalk probably was originally a sort of semi-liquid with fully 
70 per cent. of its volume water, and in its present state is about 
45 per cent. of its original thickness; the thickness of some clays 
must have diminished still more; while the amount of minute 
cavities in rocks with slaty cleavage is so small, that sometimes they 
are nearly sohd. 

By the measurement of green spots in slates it ean be deduced 
that the rock before cleavage was somewhat more consolidated 
than rocks of the Coal-Measures now are, and was then greatly 
compressed and the minute cavities almost completely squeezed up. 
The development of ‘ slip-surfaees* in cleaved rocks is very great, 
and furnishes an additional proof that the cleavage is of mechanical 
origin. ‘ Pressure-solution’ is also dealt with. 

In conclusion, the author discusses the volume of minute cavities 
in clay-rocks and their analogues of various ages, and shows that 
there is a distinet relation between it and the probable pressure to 
which the rocks have been exposed. Tables are given of the 
pressures so calculated for rocks of various geological ages, the 
volume of empty spaces decreasing in older rocks from the 32 per 
cent. existing in recent clays. In the Moffat rocks, with very little 
or no slaty cleavage, the pressure is calculated at about 7 tons to 
the square inch, while the Welsh slates, with very perfect cleavage, 
indieate a pressure of about 120 tons to the square ineh, 


January 22nd.—Sir Archibald Geikie, K.C.B., D.C.L., Se.D., 
Sec.R.S., President, in the Chair. 


The following communications were read :— 

1. ‘ The Origin of the Pillow-Lava near Port Isaac in Cornwall.’ 
By Clement Reid, F.R.S., F.LAS., F.G.S., and Henry Dewey, 
F.G.8. 

‘The Upper Devonian strata around Port Isaac consist of marine 
slates, in which oceurs a sheet of. pillow-lava over 200 feet in 


————eEE 


ie oe 


Origin of the Pillow-Lava near Port Isaac. Bye 


thickness. The pillows measure usually from 2 to 5 feet in 
diameter, but range up to 8 feet ; masses under 1 foot are rare. The 
individual pillows are quite disconnected, although moulded on one 
another and adherent where they touch. Where three pillows 
approached there was an angular vacant space, subsequently filled 
with calcite, which is often altered into chert. ‘Their mutual 
relations seem to prove that they were soft when deposited, but 
not sufficiently soft to squeeze into corners. 

ach pillow shows internally a central vacant space or very open 
sponge, often as much as 2 feet in length. This is succeeded by a 
thick shell of exceptionally vesicular Jaya, which is followed by an 
outer shell of banded, more or less vesicular rock. The whole mass 
is so vesicular that it must have been very light. 

If this lava were subaérial, the lightness would not help us to 
explain the origin of the isolated pillows; but the intimate asso- 
ciation with fine-grained marine strata shows that it was probably 
submarine. On calculating the proportion of cavity to rock in two 
of the pillows, the authors find that the specific gravity of the whole 
mass must have been very low, not greatly exceeding that of sea- 
water. The lava seems to have been blown out into thick-walled 
bubbles, kept from touching each other by the escaping steam. The 
whole mass was for a short time in the spheroidal state, and, 
although composed of a multitude of large plastic spheres, the sheet 
could flow likealiquid. This eruption seems to have been analogous 
to that of Mont Pelé, described by Dr. Tempest Anderson & Dr. Flett, 
except that it was submarine instead of subaérial. 


2. ‘On the Subdivision of the Chalk at Trimmingham (Norfolk),’ 
By Reginald Marr Brydone, F.G.8. 


The object of this communication is to lay before the Society a 
sketch-map showing the geographical distribution of the sub- 
divisions, with a brief account of their distinguishing features. 
Practically the whole of the Chalk exposed on the foreshore comes 
under two main divisions—one composed of (a) Sponge-beds, very 
largely yellow, 12 feet, resting on 8 feet of White Chalk; (6) White 
Chalk without Ostrea lunata, about 9 feet thick; (¢) White Chalk 
containing O. lunata, 20 feet ; and the other composed of (a) Grey 
Chalk, about 12 feet thick; succeeded by (0) White Chalk with 
Ostrea lunata, about 20 feet; (c) White Chalk without O. lunata, 
about 8 feet; (d) White Chalk with O. lunata, about.10 feet; and 
(e) Grey Chalk, about 25 feet. There is no evidence as to the 
relative positions of these two main divisions. The author seeks 
to justify his adoption of Yerebratulina gracilis and T. Gisei as the 
zone-fossils of the Trimmingham Chalk, in opposition to the proposal 
to adopt Ostrea lunata as the zone-fossil. Other important species 
are Pentacrinus Agassiz, P. Bronni, and Echinoconus Orbignyanus. 
The author still adheres to his view that these masses of Chalk 


978 Geological Society :-— 


ean only be a situ and must have once formed part of a large 
continuous mass, and that the bulk of this mass must have lain to 
seawards of the present coast-line. 


» February 5th.—Sir Archibald Geikie, K.C.B., D.C.L., Sc.D., 
Sec.R.S., President, in the Chair. 


The following communications were read :— 


1. ‘On Antigorite and the Val Antigorio, with Notes on other 
Serpentines containing that Mineral.’ By Prof. T. G. Bonney, 
pe,U., LIED. POR, 


It is by no means certain, as the author ascertained after his 
joint paper with Miss Raisin, published in 1905, that the first- 
described specimen of antigorite was really found in the Val 
Antigorio. So last summer he visited that valley, in company 
with the Rey. E. Hill, and after an examination, of which he gives 
an account, came to the conclusion that the rock most probably 
does not occur there im situ, though it is found in pebbles, ete. 
from tributaries. 

He next describes other specimens of antigorite-serpentine, 
examined since 1905: some from New Zealand, kindly sent to him 
by Dr. J. M. Bell, and others obtained in the Saasthal, especially 
from the Langefluh ; giving further particulars about specimens in 
the National Collection at South Kensington and in the University 
Collection at Cambridge. 

He then discusses the origin of the mineral. Pressure is 
apparently essential ; certainly it can be formed from augite, and, 
though he has not discovered residual olivine in his own rather 
numerous specimens, or typical antigorite in Alpine bastite-serpen- 
tines, he finds indirect evidence of its coming from this mineral, 
proofs of which are given by F. Becke, M. Preiswerk, and J. M. 
Bell. If, however, we suppose the former existence of two types 
of peridotite in the Alps, as at the Lizard and in the Vosges, and 
pressure sometimes to have preceded, sometimes to have followed 
serpentinization, we can account for the apparent conflict of 
evidence. 


2. ‘The St. David’s-Head “‘ Rock-Series ” (Pembrokeshire).’ By 
James Vincent Elsden, B.Sc., F.G.S. 


The St. David’s-Head and Carn-Llidi intrusions are of complex 
composition, ranging from a basic biotite-norite to an acid quartz- 
enstatite-diorite, and finally to soda-aplite. Throughout all the 
types, except the last, there is a high magnesia-percentage. The 
extreme types sometimes pass abruptly one into the other, and at 
other times they are mixed in various proportions. They do not 
represent a composite intrusion, but simultaneous intrusions of an 


————eo se < 


iit eal 
vi 


The St. David’s-Head Rock-Series. 579 


imperfectly-mixed magma. There is no evidence of differentiation 
in situ, but the facts suggest a common origin from a differentiated 
magma-basin. The aplite-veins may represent the most acid phase 
of the differentiated magma. 

Petrographically the rocks are of considerable interest, as 
exhibiting types not very commonly occurring in the British 
Isiands. They also afford unusual facilities for the study of both 
rhombic and monoclinic pyroxenes, and appear to throw light upon 
the origin of the sahlite-striation of the latter. Rhombic pyroxene 
generally crystallized earlier than the monoclinic pyroxene, but 
sometimes these relations are reversed, and often the two forms are 
erystallographically intergrown, sometimes as twins. There are 
two distinct varieties of augite, distinguished by the presence or 
absence of a basal striation. The relation of these two types lends 
support to the perthitic theory, that there is an ultra-microscopic 
erystallographic intergrowth of rhombic and monoclinic pyroxene. 
The probable age of the intrusions is not greater than that of the 
earth-movements which folded the Arenig Shales in this district. 
The observations recorded in the paper seem to point to the 
conclusion that acid streaks and cores in basic igneous rocks may 
not always be due to differentiation in situ. 


February 19th.—Sir Archibald Geikie, K.C.B., D.C.L., Se.D., 
Sec.R.S., President, in the Chair. 


The following communications were read :— 


1. ‘The Two Earth-Movements of Colonsay.’ By William Bourke 
Wright, B.A., F.G.S. 


The supposed Torridonian rocks of Colonsay exhibit in their 
folding and cleavage the effects of two movements analogous in 
their results to those proved by Mr. Clough in the Cowal district 
of Argyll. Not only the planes of the first or slaty cleavage, but 
also the quartz-veins formed along them have been folded by the 
second movement, and may be observed to be crossed at considerable 
angles by the cleavage produced during this second movement. An 
extensive series of lamprophyre-dykes, obviously later than the 
first cleavage, are found to be folded and cleaved by the second 
movement. Moreover, some of these dykes traverse and are chilled 
against a mass of syenite, which can also be proved to be later than 
the first cleavage. The distinctness of these two movements is, 
therefore, considered to be completely established. The second 
cleavage being of the nature of strain-slip, its development along 
the axial planes of the folds is of interest and is brietly discussed. 


2. ‘Notes on the River Wey.’ By Henry Bury, M.A, F.L.S., 
¥.G.S. 


The part of the River Wey within the Wealden area is divided 


a, rc ware — ares Sl «ee aw) 
i ; ? Lech Ba ; : = pee 8 Ses i ol 


580 we Geological Society. 


into six sections :—(1) The consequent river cutting the Chalk « : 


Guildford ; (2) the subsequent stream coming in from the east at 
Shalford ; (3) the western subsequent stream parallel to the Hog’s 
Back ; (4) the continuation of the last westward (the Tilford River), 
rising at Selborne and receiving many tributaries, including the 
Headley River, from between Blackdown and Hindhead ; (5) the 
short obsequent section from Farnham to Tilford (the Waverley 
River); and (6) the portion above Farnham coming from Alton and 
beyond (the Farnham River), Part I deals with the relation of 
sections 6, 5, & 4 to the Blackwater. It is assumed that there 
was a consequent river coming down from Hindhead, flowing north- 
wards along the ‘ Waverley River,’ and joined by the Farnham, 
Tilford, and Seale rivers. This seems to have been the original 
head of the Blackwater. But subsequently capture took place by 
section 3 of the Wey, with the result that the Tilford River passed 
into the Wey basin, and section 5 was thus beheaded. The de- 
velopment of an obsequent stream near the course of the last 
eventually tapped the Farnham River, but not the Seale. 

Part IT deals with the Paleolithic Gravels of Farnham. Their 
height and distribution is discussed, with a view of determining the 
river which originated the gravels. The ridges constituted by the 
gravel drop to a lower platform along the Waverley River: this is 
regarded as the left bank of the consequent valley before that was 
beheaded. If this were the case, the gravel would have been formed 
by the Farnham River while still tributary to the Blackwater. At 
this time, too, probably the Headley tributary drained into the 
Farnham, ‘and not the Tilford River, giving rise to the south-western 
portion of Alice Holt. 

Part IIT deals with the Farnham branch of the W ey and the 
Alton district, which is remarkable in that there is a complicated 
series of Chalk-valleys, which spread over some 50 square miles of 
country and discharge their waters into the Wealden area. One 
possible explanation is that this portion originally drained into the 
Whitewater over the present col of Golden Pot. In discussing this 
expiauation, it appears that the Tisted tributary has the characters 
of a consequent stream ; but there is no very good evidence, except 
alignment, of the former connexion of the two basins. On the 
other hand, the Farnham River rather appears to have originated 
in a Chalk-surface than in Wealden beds; and thus it and its 
tributaries may have been developed on the Chalk portion of the 
peneplain of the Weald. Thus the Farnham stream appears to 
present a case of the conversion of a Chalk-valley into a Wealden 
one in its lower part, while in the Caker stream the reverse is the 
case, and it is the upper part of the stream that has entered 
Wealden beds. 


Bis 


Phil. Mag. Ser. 6, Vol. 15, Pi. XV. 


» 
Pe 
_ 


a | 
B a 
a a 
| a 
‘a a 
| | T | 
H H 
| r 
= | a 
BoD a = 
Bag an 
ee : Ba 
EEEEo ae 
BEE EEE EEE Ba 
EEE EEE EH BB 
aaa 
as 
TT 
pris S 
pape — 
ga 
ae 


Sos tisstiitaaat Sees eer eaEGE 


g 


PRR Cee Eee ee epee eer ey 
DIDI SAD ORR RAR eRe a aso 
~ 2a epee Pps pp =f 


HEEL 
ELE 


Eee eSeEEEEEEESEGEGEGE 


1 | |_| ss 
JOD RS Ree e aR 
a BEER 
JESeae a tt 


er 


5a wal acest 
— Saena 


aoaeee ane Ba//SURBE pateial 
JD RSE eee PEEEEELEET Ler EEEEEEEH PEE Ric Ee 
JASE SRR RS EAE RRR $e fn Fee fp psf pe ft oe Pea 
Dea Sseae8 peep apa 


DeaDea Ss eRe SaaS [| 

DUAR SRR aR eee ARSE | | 

5 5 ae [| 
= Be saa 

He 


Sarpri PEER EEE EEEEEEEEEEE EE TTA PEE EEE EEE 
Lp) 
fae’ | 
| x 
| Pio Pe 
Bet’ an PEE 
= } ich eet ey to 
os | PEPER rare 
ae} 
a =f 
oe % 
E i 
P, - e~< 
Me "aaa Ra eeDE az 
x [ERRORS Sansa ee 
me bt + 
"Ts 2 
9d eee = = 
, , 7 : 


BuckineHn. a 


Rise 
im 


uate eeatatai iEscar tessa ttosastecat iced EEE EEE 
: ; EEPEEEEE TEECEEEEE 
aaat See ace 
4 a H H 
ee 2 
EEE EEEEE BE -EEEEEE HH o | BEEEEEEEEH 
cise He eit Beeseasavavavazaravoeeeesvanscs 
5 pee EEEEEEEH A 
EEE EECEEEEEt BEE EEE EEE EEE EEEEEEEEEE EERE a { EEE 
) SEE EEE BeaEEE EH FEE EEEEEEE EEE EEE 
fo : 
2 E HH aaa arotece 
a I Ht lett 
ey oh A =H 
250 ‘ t Ba 
. a3 aoe s 
| e aiseiees EEA 
| oa : FEEFEEEH 
aS cf EEE : 
oozes Siesedsteascesetensnattanststaae 
ex oqgap HFA 
CH 
eure oe 
Ht i zi 
Heo ESS 
Ht : 
fase He Eee 
A cet 


40 
70 
60 
50 
40 
30 


\ 
\ 
: 


sbatetetstetetariy 


+ 160 


Ease HHH Cy 
He REE Pee eee err er et Baise 
Sees SGRRSR!/ GEES 

SRESEESRSSS) GaQas 
BRaReDNMSaSeR 
GOESee ieaaes 

BREREEE IReaas 
BEBE SRESRS 
Bimmer Eee Eee iesie} a 
aie htsh false) ae a eee ee 

JIS SRE eS Ree Bee eee REESE AS 

ate 
Dawa ka Sa eae eases eaaee 
eee 

EeRReaRaa 


i . 
. 


bt] 
= 
4 


ai 


Siceesceeaed eaperdeeee Bao 


tcceaageeaae 


2? 


_— 
RS 


eee 


] 


6 


° 


= 
v 


“NYHONINIOG 


yb 
t 
+ 
J 
T { 
oor H 
a ia 
+ = +++ Sta [shal aah 
Py =| EEEEEH EEE HL | 
300 BEE PERE E EEE EES t 
At 4 | : EEE ela tla wi SESBESe! 
H CORSET Uh (a6) Kester 
Te fil +t into nko al ban na 
Pec Bias ia ®,CO, , Natanson 
: ie 
Ht sia OSgr HE x oC 0, , Joule- Kelvin 
E SOE iE Neos 66) + 
EEE haa 
ama tH A, Att : : iz 
ot at i. EEE ay NS 
a) See Ht ++ 1 2oeeeae Veale : >, Olszews Kt 
| 
| 
= Biber 
+ 
- 190 
<0 
60 
at fe 
aT) 
a) if : 


0 


T 10a ‘9 “0g “Bey “Trg 


® 


“pe i 
ei 


= os 
5 F 
a ee 


* 


Phil. Mag. Ser. 6, Vol. 15, Pl. XVII. 


eo 
I ae oe TV fale) 
CSS POM OF aes eae sa 
et a a i sa 


(200 


HE eee SSeS Sees Ba 
DOSS ab yaricapaea mapas 
CS BSIIcI IIs Scie anlaae a os 
cere 
Beas Ee He = 
Sreie PECEEE a sererae 
Niaee 4 PI SEE MMIII IE ‘e 
sei HI PSHE RIE a Par 
aaa ACEH 
poet coo = 
Peeece See ceeeeeee 
Seeeoeee eal 


G 


/| 


700 


Hae 


: BEECH 
Suee0 aid! quuneeeueeeuaeear/.anmnar “PPC ae anaes a 
SSDEEERSUGSSSEeHaCeeSed/ i suneeeereteseecen” Acces /seccesesssaseeeeseessn= 


\ ” taht. 
1? ot 
Bucking. on 
Phil. Mag. Ser. 6, Vol. 15 
: a | 
IL = HE 05 = : : J ae : PERCE £ 
- sala 
Too é : E 
| 
E EEE PPR EEEETE : 

ISsnoaesEos 

+ L a PEGOEr 

| DESAI IOGS 50 alee 


1.0 


[2 


= 
= 


x» 
> 


S 
~ 


»~ 
> 


Che vimddy wawmic Curyectiun, 


+ 


100 400 300 400 500 600 100 yo 700 1000 100 


able 


i 


oe 


A —. be 
a , E 
ntl 
a" ‘ 


Oe. 


THE 
LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


f. & - [SIXTH SERIES.] 
AY 

: 4 MAY 1908. 

\ ery fe > Aelia $59 ee 


LV. The Resonance Spectra of Sodium Vapour. By R. W. 
Woon, Professor of Experimental Physics, Johns Hopkins 
University *. 

[Plates XVII. & XIX.] 


FYXHE vapour of sodium, obtained by heating the metal in 

a highly exhausted steel tube to a temperature of about 
400 degrees, yields an absorption spectrum of great com- 
plexity. In addition to the D lines and the other lines (ultra- 
violet) of the principal series, which come out reversed, we 
find the entire visible spectrum, with the exception of a very 
narrow region in the yellow, filled with fine and sharp 
absorption-lines. This we shall call the channelled absorption 
spectrum, and we find it divided into two distinct regions, one 
extending from wave-length 4500 to 5700, and the other 
extending from about 5800 to the extreme limit of thered. It 
is probable that it extends out as far into the infra-red as 10 
or 12 y, for recent investigations by the writer, in collaboration 
with Professor A. Trowbridge of Princeton, have shown that 
the vapour exhibits heavy absorption in this region, though 
the bolometer strip was not narrow enough to resolve the 
lines. This remarkable absorption spectrum was investigated 
by the writer in collaboration with J. H. Moore a number of 
years ago ; but the precision which is now being obtained in 
the work upon the fluorescence of the vapour has made a 


* Communicated by the Author. 
Phil. Mag. 8. 6. Vol. 15. No. 89. May 1908. 2R 


582 Prof. R. W. Wood on the 


still more accurate stndy of it advisable, and this work has 
been carried on by Mr. Clinkscales, one of my students, 
during the past year. Photographs taken with the 21-foot 
grating in the second-order spectrum, show that it is even 
more complicated than was originally believed. We find, 
on the average, from 60 to 70 absorption-lines within a space 


only 12 Ang. units in width ; in other words, as many as 
30 lines in a region no wider than the distance between the 
D lines. This means that in the blue-green channelled 
absorption spectrum, which is about 1200 A.H. in width, 
there are roughly speaking about 6000 absorption - lines. 
Taken collectively these lines form themselves into a number 
of groups, which resemble the groups seen in the absorption 
spectra of iodine and bromine, and in certain banded emission 
spectra. 

A small portion of the absorption spectrum of sodium 
vapour in vacuo, taken with the 21-foot grating in the 
second-order spectrum by Mr. Clinkscales, is reproduced on 
Pl. XIX. figs.4 &5. The originals have been enlarged about 
six-fold, and the portions reproduced are thus on a somewhat 
larger scale than Rowland’s large map of the solar spectrum. 


Each of the strips reproduced is a little less than 20 Angstrém 
units in width, or considerably less than the distance between 
the series lines in the resonance spectra, the spacings of 
which vary from 386 to 38 units. The strong iron lines in 
this region appear on the plates for comparison. The 
absorption-lines are seen to be separated by distances which - 
in some cases are less than 0°15 unit in width. When the 
resonance spectra have been photographed with the 12-foot 
grating, an immense amount of information can be obtained 
by comparing the spectrograms with this magnificent map 
of the absorption spectrum. This map will enable us to 
determine whether a given exciting line strikes an absorption- 
line exactly, or falls midway between two. It must be 
distinctly understood that the spectrum reproduced is a 
positive and not a negative of the absorption spectrum. It 
resembles a bright-line emission spectrum so closely, that I 
deem it of importance to draw attention to this fact. It is 
most remarkable that we can absorb portions of a continuous 
spectrum and leave regions not much over 0°05 or 0-1 of an 
Angstrém unit in width, and it would be interesting to 
examine this residual light, after transmission through the 
vapour with a Fabry and Perotinterferometer. A discussion 
of this remarkable transmitted spectrum from the point of 
view of the pulse theory of white light should prove 


Resonance Spectra of Sodium Vapour. 583 


interesting, for the light transmitted appears to be as 
homogeneous as the light in the spectrum of the iron 
are. 

To attempt to unravel this spectrum, or find any regular- 
_ ities in it by the usual means, is quite out of the question, 
for the lines are so numerous and so close together that we 
could pick out series that would conform to any law that 
we might choose to invent. 

As I have shown in a previous paper, however, we possess 
a very beautiful experimental method of analysing the 
spectrum, and of determining just which lines belong 
together ; a method, moreover, which may in time yield 
results which will enable the theoretical physicists to tell us 
the exact nature of the piece of machinery which we call the 
sodium molecule. 

If we illuminate the vapour with a powerful beam of 
white light, it becomes strongly fluorescent, emitting a 
spectrum which I now believe to be the exact counterpart 
of the absorption spectrum. During the earlier part of the 
work, before the methods and apparatus had been perfected 
to the high degree which they have now reached, it was 
believed that comparatively few of the absorption-lines gave 
rise to fluorescence, as the fluorescence spectrum could not 
be photographed with the high dispersion which is now 
employed. ‘This fluorescent spectrum is obviously of little 
help to us, for it is quite as complicated as the absorption 
spectrum. If, however, we throw monochromatic light into 
the vapour, instead of white light, we observe a very remark- 
able phenomenon. We now have series of bright lines 
spaced at very nearly equal intervals along a normal spec- 
trum, and separated by a distance equal to about 37 Angstrém 
units. Various series of lines with varying distribution of 
intensity can be brought out by changing the wave-length 
of the exciting light. In every case, light of the same wave- 
length as that of the exciting light is emitted by the vapour, 
and in addition a large number of other frequencies, which 
bear a definite relation to each other. In the absence of the 
exciting light the vapour is non-luminous, that is the 
electrons, which we may perhaps regard as revolving in 
concentric rings, do not radiate any energy. The passage of 
an intense beam of monochromatic radiation through the 
medium disturbs this non-radiating system in some way, and 
causes it to emit a series of bright spectrum-lines. The 
phenomenon can of course be classed under fluorescence if 
we choose, but as we appear to be dealing with a much more 
tangible phenomenon than is usually the case, and as these 


2R2 


584 Prof. R. W. Wood on the . 


spectra are radically different from spectra excited by any 
other means, I propose that they be named Resonance spectra, 
since they are without doubt excited by the resonance of one 
or more of the electrons of the system when monochromatic 
radiation plays upon it. 

During the past year much additional information has 
been gathered regarding these remarkable spectra, and 
though the work is by no means completed at the present 
time, it seems advisable to place on record the results ob- 
tained thus far, in view of the fact that theoretical papers 
(notably those by G. A. Schott) are appearing from time to 
time on the nature of the radiation emitted by electron 
systems. 

I have already shown that the D lines are present in the 
emission spectrum when the vapour is excited by light of 
the same wave-length. This fact is of interest in connexion 
with statements recently made by some writers that the 
D lines are never present unless oxidation is going on, and 
in the air-tight steel tubes, highly exhausted and repeatedly 
washed out by the hydrogen evolved by the sodium, it is 
difficult to see how any oxygen can be present. 

The very remarkable fact has now been established that 
the D lines can be caused to appear in the emission spectrum 
by stimulating the vapour with a very intense beam of blue- 
green light, from which all yellow light has been completely 
eliminated. This proves that the mechanism which produces 
the principal series is a part of, or connected in some way 
with, that very complicated piece of machinery which gives 
rise to the channelled absorption spectrum with its thousands 
of lines. 

The second point of interest is the discovery of series of 
equidistant lines in the red fluorescent spectrum, excited 
by monochromatic red light, the photographic recording 
of which has been made possible by the panchromatic 
plates recently placed on the market by Wratten and 
Wainwright. 

The spectrum emitted when the vapour is stimulated by 
white light has been photographed under more favourable 
conditions with a large grating, and has been found to 
possess much more structure than was at first supposed, 
especially in the yellow-green region, where very little fine 
detail could be made out in the earlier photographs. 

The magnetic-rotation spectrum has been studied with a 
view of determining whether the rotatory power is positive 
or negative for the different electrons. The results obtained 


ice 


Resonance Spectra of Sodium Vapour. 585 


in the red region of the spectrum have been already 
published *. 
The resonance spectra obtained by stimulating the vapour 


with the radiations from the cadmium and zine quartz arc- 


lamps have been photographed with a large grating, and the 
wave-lengths of the bright lines which form the series have 
been determined to within a tenth of an Angstrém unit, a 
tremendous step in advance over the work recorded in the 
earlier paper, in which we could not be sure of the wave- 
lengths to within less than about 2 Ang. units. 

Many new sources of monochromatic light have been used, 
and the prism spectrograph improved by the addition of 
a new long-focus photographic objective by Cooke. The 
resonance spectra described in the previous paper have been 
photographed over again with this improved apparatus, and 
we now know the wave-lengths of the lines to within a little 
less than one unit. During the present year they are being 
photographed over again with a 12-foot concave grating 


with a collimating lens of six feet focal length. This work, 


which will take an enormous amount of patience, is being 
carried on in colijaboration with Mr. Felix EH. Hackett, 
Fellow of the Royal University of Dublin. 

Familiarity with the preceding paper above referred to 
will be presupposed in presenting the more recent results, as 
the details and methods have been already fully described. 
A few improvements have been made in the apparatus. 


Owing to the long exposures it has been found necessary to 


put a heating-coil and thermostat in the large three-prism 
spectrograph. This keeps the temperature constant to 


within a tenth of a degree, and improves the definition in no 


small degree. The telescope-lens has been removed and 
replaced by a Cooke photographic objective of 34 inches 
aperture and 40 inches focus, which was made to order. 
This lens was corrected so as to be as nearly as possible 
achromatic for the spectrum range 4500-5700, within which 
the resonance spectra, which are being studied, fall. It also 
gives fair definition over the range of the entire visible 
spectrum. The wet cotton jackets on the sodium tube have 


been replaced by tightly wound coils of lead pipe (5 mms. 


diameter) through which a current of cold water circulates. 
In working with open air arcs much difficulty was experi- 
enced from the wandering of the are over the surface of the 
electrodes. The image of the are on this account moved 


* “On the existence of Positive Electrons in the Sodium Atom,’ 
Phil, Mag. February 1908. 


586 Prof. R. W. Wood on the 


about over the aperture in the drum-retort of the sodium 
tube. The image of the carbons gave some trouble as well, 
exciting the complete fluorescence spectrum. Both troubles 
were obviated by focussing the arc on a small round hole in 
a screen by means of a large double-convex lens, and then 
throwing an image of the aperture into the sodium retort by 
means of a second lens. In this way it was possible to keep 
the white light away from the sodium vapour, and correct 
for the shifting of the arc by moving the lamp from side to 
side. As exposures of seven or eight hours are often 
necessary, it will be readily seen that the care of the arc is 
no light labour. A magnetic field was tried to keep the are 
in a fixed position, but nothing appeared to be gained by this 
expedient. 

The spectra excited by the radiations from the cadmium 
and zine arcs in exhausted quartz tubes were photographed 
with a large plane grating in combination with the Cooke 
lens, and as the wave-lengths of the resonance spectra excited 
in these two cases have been determined with consider- 
able accuracy, we will discuss them first. In each case 
the iron-spectrum was photographed in contact with the 
resonance spectrum. 


Cadmium Excitation. 


The wave-lengths of the lines in the resonance spectra 
excited by the monochromatic radiations from a cadmium- 
vapour lamp have been determined to within one or two 
tenths of an Angstrém unit. In the previous paper I showed 
that stimulation of the vapour with the radiation of wave- 
length 4800 caused it to emit a series of bright lines spaced 
at very nearly equal intervals along a normal spectrum. 

Whether the lines are in reality spaced at equal distances 
is a matter of considerable importance, since theoretical 
treatments of the radiation emitted by electron systems 
disturbed in various ways can only be verified by the test of 
experiment. Prof. Larmor has pointed out to me in a letter 
that a non-radiating system of electrons in steady orbital 
motion, when disturbed by the absorption of a radiation 
corresponding in frequency to the frequency of one of the 
electrons, should then emit radiations giving us a series of 
lines equally spaced along a normal spectrum. We must 
bear one fact in mind, however. The light emitted by the 
vapour is obliged to pass through a certain thickness of 
cooler vapour before reaching the spectroscope, and since 
the absorption-lines are very closely packed together, it is 


Resonance Spectra of Sodium Vapour. 587 


quite possible that the apparent position of an emission-line 
of finite breadth may be to a slight extent modified by 
absorption. If, for example, there is a strong absorption- 
line on the violet side of and close to the emission-line, the 
centre of the emission-line will appear a little on the red 
side of its true position. Very slight departures from the 
law of equal spacing need not be regarded as fatal. A portion 
of one of the spectrograms is reproduced on P]. XIX. 
fig. 2. Exciting line 5086 shows and 8 or 10 of the resonance 
lines—Comparison spectrum of iron. 

The wave-lengths of the lines in the series excited by 
stimulation at wave-length 4800 are given in the following 
table, together with the wave-length differences. 


Xi: d differences. 
AT GeO: Es ier Meas Mie oe. 34-5 
S000). TD om ents 558 
4835°8 a 
St Rar oie 7 = 36-2 
4 wD, . 
ate, i Qu awa ar 
OSU ly y Manat ures AAP ak 37°5 
5019-7 ve 
ci nnageb as _Suaide ope ihmer ahs ave 
USES LIMES MERLE SOE GRAY Biles 37-2 
as Whe er 3500 
SUG Or. Mee SVU SNM SY Sa 37-96 
SOUT AG, STO Ae itl ys 1 da 37-74 
PA) a mh 37-56 
Tes a ee cee nee 38°63 
USA GSS 4. ee eee ct 38:03 
5359°42 


Two only of the lines in the series are missing. In the 
previous investigation there were three that escaped detection. 
With longer exposure I feel confident that all would appear. 
It is clear that the lines are closer together in the region of 
shorter wave-lengths. ‘The slight variations in the differences 
above the line 5019°7 are, in my opinion, due to the cause 
mentioned above. If we take the differences between five 
lines in the series instead of between two, we find the spacing 
averages remarkably constant. Starting at the top (longest 
line) of the series, the differences between each line and the 
fifth below it run as follows :— 151°96, 151°89, 151:26, 150-93, 
151-46, 149°8, 149°3, and 149°17. 

The spectra are now being photographed with the 12-foot 
grating, and it is quite possible that still greater regularity 
will result. There seems to be no escape from the fact that 
the spacing widens slightly as we ascend the spectrum. 

We will now consider the resonance spectrum excited by 


we Prof. R. W.. Wosd.endle a es 


the green cadmium-line 5086, which, as I showed in the 
previous paper, consists of a ‘number of regularly nae a 
doublets. 

The blue lines of the cadmium lamp were cut off by a 
screen of nitroso-dimethylaniline. In more recent work a 
screen composed of a concentrated solution of a neodymium 
salt has been found to work better. 

The wave-lengths of the lines excited by the 5086 stimu- 
lation are given in the following table :— 


Lali 
4 
-_ 7 


4969-0 a 
( 4975-0 5 
5007-9 a 
5013-6 4 
50468 2 


5052-7 

5086-0 a DirFERENCES. 

pee 36 a. b. c. d. e. 
ia. 339 386 409 — 371 

51590 c 390 39-1 380 

a 392 386 0-381 

5l644a 39-0 39-0 
1 5169°5 6 < 

51952 d 

51999 c sa0 

5202-44 

5237°9 ¢ 

52685 e 

5276-0 ¢ = 

0305°6 é , ‘ 


In this table we see that in addition to the doublets which 
I have spoken of. and which are bracketed, there are a 
number of other lines. The lines which belong to the same 
series of equidistant lines are indicated by letters. We find 
in this case that as we approach the upper limit of the series, 
other lines appear on the short wave-length side of the line 
or lines at the upper end, a new series starting where the 
old one leaves off. This same thing occurs in many other 
instances, as in the resonance spectrum excited by the barium 
line 4934. E 

It is a little difficult to express in words exactly how these 
lines come in, and a much better idea can be gained by 
applying a pair of compasses to the chart and spacing off 
the series, and noting how the lines of the new series appear 
on the short wave-length side of the lines which are being 
spaced off. 

I have given the differences between the wave-lengths of 
the lines of the different series, a, b,c, and e in a separate 
table. The line d appears to be isolated. 


Resonance Spectra of Sodium Vapour. 589 


Zine Excitation. 


The wave-lengths in the resonance spectrum excited by 
the three blue zinc lines have been determined to within less 
than 0:2 A.E. Two negatives were made with the large 
grating and Cooke long-focus lens, one with an exceedingly 
narrow slit, which was used for the measurements, and one 
with a wider slit, which is reproduced on Pl. XVIII. fig. 2: 
the white-light fluorescence above, iron comparison between. 
On this Plate the lower end of the spectrum did not appear 
owing to the tact that a corner of a black cloth inside the 
spectrograph slipped down in front of the end of the plate. 
The reproduction gives a good idea of the sharpness of the 
lines in the resonance spectrum, although the exciting lines 
do not appear on the plate. it must be remembered too 
that this plate was made with what I have termed a wide 
slit. . 

The resonance spectrum contains a large number of series 
of equidistant lines, though the series are not as extended as 
in the case of the cadmium excitation and many others 
which will be considered presently. 

I have not, at the present time, photographed the spectra 
excited by the three lines separately, but by comparing the 
plate made with the grating with the one made in the earlier 
work, I have made a provisional assignment of the lines 
which I believe belong to the spectrum excited by the longest 
of the three zinc lines. ‘These lines I have marked with a 
star in the Table of wave-lengths, and have designated with 
letters a, b, c, &e., the lines which appear to belong to the 
same series. 

As many lines appear double in the present photographs 
which appeared as single lines in the earlier spectrograms, 
it will be understood that too much importance must not be 
attached to the present assignment of the lines. Many of 
the series appear to have missing lines, and an attempt is 
now being made to find them with longer exposures. 
Exposures of 8 or 10 hours were necessary with the quartz 
zinc lamp fed by a current of 5 amperes, and the lamps are 
of very little use after a run of 25 or 30 hours, owing toa 
black deposit (reduced silica probably) which forms on the 
inner surface of the tube. : 

The wave-lengths follow, together with the differences of 
wave-length of the lines which appear to belong to the same 
series. ‘he exciting lines are underscored, and the relative 
intensities are roughly indicated by numerals, 10 meaning 
very strong, and 1 very faint :— 


590 


4647-2 2 
46489 2 


Prof. R. W. Wood .on the 


754 


2 


766 


2 


5168°8 
5181°1 
5184°2 
5186°8 
5196°U 
5215°3 
5220°6 
5225°8 
5232°0 
5244:0 
5248°7 
5252°8 
5253°7 
5264-5 
5268°0 
52730 
5277°6 
5290°2 
5300-0 
5303°1 
5305°6 
5307°5 
5313°7 
53158 
5321°1 
5327-0 
5328°5 
63381 
5350°5 
5351°6 
5356°5 
5366°0 
5370°0 
5373°5 
5375°9 
5379°2 
5386°1 
5387-4 
5390°2 
5391-9 
5397°3 
5399'1 
9403°3 
54043 
5406°3 
5408°5 


* kK * 


x * 
Qs 


iv) 


CORN PNR WNWONRF OOP RR RNOOWWRNRrNNRrOHrF 
% 
Q& 


Qi 


DIFFERENCES. 


b 


=31°7 


38:2 


=38'3 


37°9 
39°0 
33°7 
58°6 


e 
375 
O74 
38'3 
38°5 


5410°0 
54140 


5418°4. 


54200 
5421°8 
9423°3 
5427°6 
54322 
5438-0 
5440°3 
0442-7 
5443°9 
5448:2 
5451°5 
5454-4 
5457°3 
5458°8 
5465°1 
5471-7 
54758 
5478-0 
5482°6 
5484-1 
5486'9 
5490-0 
54941 
54962 
5502-0 
5505°6 
5512°5 
55129 
5516°1 
5519°3 
5521-7 
9526°7 
5532°6 
5535-2 
5538-0 
5547°5 
55576 
5558°5 
5563°7 
55831 
5584°4 
5588°9 


36°0 
36°0 
35°1 
30°1 
30°4 
35°0 
34°2 
30°3 
34:5 


LO CO bY HO eR DO DD YE DD OD HO DO DO DD CO 9 ST OU G9 DOB LO Ht G9 DOLD OL HD DOS GH GDF So 


aaa .” 


Resonance Spectra of Sodiwm Vapour. d91 


The series are not as pronounced in the case of the zine 
excitation as in many of the other cases. Apparently the 
series which I have marked “a” and which is excited by the 
longest of the three zinc lines, appears to have its upper 
members excited by one of the other zinc lines, and not b 
the line which starts the series. In other words, the line 
4883°6 seems to be the last line-of the “a” series excited by 
Zn 4810°7, but we find a number of lines in the spectrum 
excited by the total radiation of the zinc lamp, which form a 
continuation of this series. Neither of the other zinc lines, 
however, falls at points determined by extending the series 
down to their region. We must remember, however, that 
when we stimulate the vapour in the blue region, the series 
which starts at the point of stimulation gradually dies out 
as we go up the spectrum, and other series make their 
appearance, a circumstance which explains the apparent 
difficulty just mentioned. There seems to be some evidence 
that as we ascend the spectrum the spacing between the lines 
of a series begins to grow less again, that is the spacing 
has a maximum value somewhere in the middle of the spec- 
trum. Compare the series “a” and “‘d” for example. Too 
much importance must not be attached to these tables of 
differences, however, as the lines which belong together have 
been determined only from the older spectrograms, which 
are very unsatisfactory, and by spacing off the new spectrum 
chart with a pair of compasses. The resonance spectrum has 
now been photographed with the 12-foot concave grating, 
and as soon as the new zinc lamps arrive the spectra excited 
by the isolated zine lines will be recorded with the same 
instrument. ? 

Lithium Excitation. 


The light of the lithium are stimulates a most remarkable 
group of resonance spectra. It is the only means that I have 
yet found of exciting definite series of lines in the red region 
of the fluorescence spectrum. There are four exciting lines 
in all, but they are so widely separated that the resonance 
spectra excited by them only overlap to a very slight degree. 
A photograph taken on a panchromatic plate of the complete 
spectrum excited by the lithium arc is reproduced on 
Pl. XVIII. fig. 1. The points of excitation are indicated by 
arrows. 

The blue line (A=4603) excites a resonance spectrum 
consisting of a nearly equidistant group of lines in the violet, 
and another group in the yellow-green. These two groups 
are indicated by the brackets “* A.” 

The blue-green line (A\=4972) excites a remarkable series 


592 Prof. R. W. Wood on the 


of lines midway between the groups “A.” This series is of 
especial interest, for, as will appear presently, the barium 
line 4934 takes hold of the same series, at the first line below 
the exciting line of the lithium series. The red and orange 
lines of lithium also excite two well-marked series in the red 
region of the spectrum, the individual lines of which appear 
to be very nearly equidistant. As yet I have not determined 
the wave-lengths of these lines, though they can be determined 
at any time from the plate which has an iron comparison 
spectrum running parallel to the resonance spectra. Taken 
as a whole, this plate is the most beautiful illustration of 
resonance spectra excited by monochromatic stimulations, 
that I have obtained up to the present time. The photograph 
was made with a small two-prism spectrograph, and has not 
been used for wave-length determinations owing to the small 
dispersion and the great width of the slit. Spectrograms 
best suited to the accurate measurement of the lines do not 
reproduce very well, consequently I have used only those 
taken with a wide slit for purposes of illustration, and the 
possible accuracy of the wave-length determinations must 
not be judged from the photographs which accompany 
this paper. 

We will now consider the series excited by the green-blue 
line 4972. The wave-lengths of the lines in this series are 
recorded in the following table. The wave-lengths I consider 
accurate to about three tenths of an Angstrém unit, as the 
plate was an especially good one, taken with a very fine slit. 


Differences. 
4862:°0 x ; 
4896°2 * ed ae 
4934°0 gp) ) PSS nessa 39°] 
4972-3 sk eee eee eee ee 
5007°4 r 
4009°6 rr 39 2 
5011‘7 x ; 
5049-9 we crete 38°2 
de ee ee 39°4 
5089-1 x 
5117°8 22. 
51Q37 00 eet 38°3 
5128°2 x 
5152 9 ~ 
51624000 cts 378 
2167°2 * 
5191°8 
5199-4 00 trteeeeeteeees 34°2 
5207-1 « ? 
5229°6 x * = 
mney Aetgy UO Roy.) ew 37°35 
5266°9 x * 37-4 


5304 Bax co 


o 
y 


Resonance Spectra of Sodium Vapour. 593 


In this table I have marked the lines which form the equi- 
distant set with a starx. The last three belong to a different 
series not continuous with the first. It will be seen that as 
we go up the spectrum other lines come in on the short wave- 
length side of the principal lines, as in the case of the series 
of doublets excited by the green cadmium line. 

A discussion of the spectrum excited by the blue-violet 
lithium line will be postponed for the present, as I prefer to 
wait until photographs have been obtained with the large 
concave grating. 


Barium Excitation. 


The excitation by the strong barium line 4934 is of great 
interest as the same series is brought out as in the case of the 
lithium excitation. The point of excitation is at the first 
line below to the exciting line of lithium, and the resonance 
spectra are identical except that the companion lines which 
appear on the short wave-length side of the principal lines 
are not identical. The wave-lengths are given in the 
following table, the principal lines being marked with stars. 


Differences. 


The three strong lines of a different series, which appeared 
with the lithium excitation (double starred in the table), do 
not appear with the barium excitation. 


594 Prof. R. W. Wood on the 


Thallium Excitation. 


_ The great line of the thallium arc excites a resonance series 
of lines some of which are accompanied by companion lines 
on the short wave-length side. A photograph of the spectrum, 
taken with a wide slit, is reproduced on P]. XIX. fig. 1. The 


wave-lengths are as follows :-— 


Differences. 
5206 x 
HORDE... Cea eee 36 
5242 x 
ROTB x | treet 36 
Ips si ek ELS VALS 2 486 
5314 x 
tt ee 
5387 * eeeteee eee eeeeetianne 
DATO Ya Lae 36°5 
5423°5 x 
AAU how ene) Sa ee See 36°5 
5460 x 


The thallium are burnt very steadily. The lower carbon, 
which was bored with a hole 2 mms. in diameter, was kept 
full o£ metallic thallium, and used as the anode. 

The lines of the series are a little closer together in this 
case, as appears from the table of differences. 


Magnesium Excitation. 


A great deal of time was wasted in trying to obtain a 
satisfactory magnesium are between metallic electrodes in 
hydrogen at low pressure. It was finally found that the 
simple expedient of introducing magnesium powder into a 
hollow carbon answered every purpose, the are burning 
steadily. Fresh powder was put in every four or five’ 
minutes. 

The green triplet (6 group) excite brilliant resonance - 
spectra, which are reproduced on Pl. XVIII. fig. 4. In this 
spectrum we find the triplet reproduced more or less perfectly 
at regular intervals. If the spacing of the individual lines 
forming the series was the same for each series, it is clear 
that we should have the triplet reproduced over and over 
again along the spectrum. ‘That this is so only to an 
imperfect degree is due to the difference in the spacing of 
the lines of the different series. No means has as yet been 
found of exciting the spectra separately, owing to the 
proximity of the three exciting lines. 


Resonance Spectra of Sodium Vapour. 595 


The wave-lengths are recorded below. 

It is easy to pick out the three series of equidistant lines 
excited by the three magnesium lines, but it is impossible to 
tell anything about the extra lines or to which series any one 
of them belongs. Lines belonging to the same series are 
indicated by the same letter, while the exciting lines are 
underscored. The differences for the three series a, b, and ¢ 
are given in a separate table. The series “‘a”’ is the best one 
obtained, so far as constancy of wave-length difference goes. 
The slight discrepancy at the middle of the table compensates 
itself, and is doubtless due to an error of half an Angstrém 
unit in the determination of one of the lines. 


5342 5b 5145°5 6 DIFFERENCES. 
5327 a 5142-7 

5305 5140 a. 6. é. 
5302°5 b 5136 a 39 39-5 38°8 
5297-2 5134-5 38 39 38°7 
5296 5133 38 39°5 39 
5288 a 5129 ¢ 38'5 39°5 385 
5286 5124 37-5 39 39-2 
5284 ¢ 5106°5 5 38 39 40 
-5263°5 5101 38 38:5 

5258 5098 a 38 38 

5250 a 5095-2 38 39 

5245-2c¢ ° 5089°8 ¢ 38 

5224 5 5075 

5222 5068 5 

5212 a 5060 a 

5206°5 c 5056 

5201-5 5049-5 ¢ 

5190 5030 3 

5184-55 5022 a 

E 4991 3 

peor 4984 a 

oe 4978°5 

eee 4957 

5167°5 c 4946 a 


Lead Excitation. 


The lead are causes the sodium vapour to emit a well- 
marked series of lines, with a wave-length difference of 
39 Angstrom units. These lines are accompanied by com- 
panion lines sometimes on one side and sometimes on both 
sides, as will be seen from the chart. The Jaw governing 
the distribution of the companion lines and the possible cause 
of their appearance cannot be determined until a larger 
collection of photographs of the resonance spectra has been 
made and the results compared and tabulated. 


+ Me 
: 


596 Prof. R. W. Wood on the 


The lines which form the series are marked with a star. 
5336°5 
5309°5 
5299 
5272 
5263 
5254 
5238 
5233 Differences. 
5201 x 
5195 
BIS 0 ete 39 
5162 x 


The line in the lead spectrum which stimulates this 
resonance spectrum has a wave-length of 5006: there is in 
addition a weaker line at 5002, and I am not sure whether 
the light of this line is responsible for any of the spectrum. 
It comes within one Angstrém unit of fitting into the series, 
but it is much fainter than the 5006 line. 


Bismuth Excitation. 


The bismuth arc is by far the best exciter of the vapour 
which we have, and it furnishes the most typical resonance 
spectrum. It has but a single line which is operative, and 
this line is located in the remote blue. The resonance 
spectrum consists of a series in the blue, which with long 
exposures can be traced well up into the green, and a host 
of lines at the upper end of the spectrum in the yellow-green 
region. 

The exciting line in the resonance spectrum is accompanied 
by two faint companions, one on each side, the first line above 
it is a close doublet, the second a wider one, and the first 
line below is a suspected doublet, judging by its width. In 
all probability the second and third lines below the exciting 


Resonance Spectra of Sodium Vapour. 597 


line are doublets, though they are so taint as to be only just 
discernible on the photographs obtained thus far. 

The wave-lengths in the blue and green region are given 
in the following table. ‘Two lines in the series appear to be 
missing. 

5007 - Differences. 

5004 

5 sala he ec 36 

4990 

4962 

4939 x 

Ge ema Pe ee gene 35°5 

4867 

4864 x ny 

AEE. Slagseeycmeneeendes cons 35°5 

4828°5 + 

4823-7 vai eR UD 2 Ae 30°59 

4795 x 2g9 

ATQOD — seceeecdecteteereeees 30°) 

AT57T-O * 

47565 

4797 eee Le ee 345 

4723 x 

JITSU) WWMM Wetec Or sce gee coca 33 

4690 x a 
3 


ee 


The lines in the yello w-oreen region are given below. 
The intensities are designated roughly by numerals, 1 meaning 
very faint. 


55344 1 54471 2 5391°9 
5530°3 1 54459 2 53911 2 
50091 5442°1 2 53883 1 
5904. 3 5440°9 5 53871 1 
5500°9 6 54383 1 538791 4 
5499°5 1 54367 1 538177 4 
54916 2 54311 53762 . 1 
5490°1 2 5429°5 93740 5 
54887 9 5426°8 SSO E 
54745 5 5425°4. 93872°5 1 
54735 6 5423'9 5358 = 2 
5470°5 2 54132 8 5357°3 2 
5469°0 2 54119 8 5309°7 1 
5463°8 1 5410-4 2 5336°8 6 
54623 1 54083 5 533596 2 
54594 7 54075 5 53341 1 
5458°0 3 53948 2 5299°8 2 
5456°5 5 5392'8 


Silver Excitation. 
In the earlier work it was believed that the silver line 
5209 excited the same series as the cadmium line 4800. 
This is now known to be an error due to the insufficient 


Phil. Mag. Ser. 6. Vol. 15. No. 89. May 1908. 28 


595 Prof. R. W. Wood on the - ~ 


power of the spectroscope employed. The silver line exe: es 7 
a series, the individual lines of which are about 1°5 Angstrim be 
units longer than the corresponding lines excited by the ~ 


cadmium line. The silver arc employed in exciting the 


resonance spectrum was formed between carbons, the lower 
one of which was furnished with a core of metallic silver. 
No prismatic separation was required, as the line 5209 was 
found to be the only one capable of exciting the vapour. 
The wave-lengths of the lines in the resonance spectrum are 
given in the following table. The first six lines in the table 
were very faint, and their wave-lengths are not so accurate 
as is the case with the remainder. 


Differences 
4947-5 
og eS ERS ICT “ 8 
nas I Se ae 
Se.) ee aT 2 
RT Or ae i‘o 
SO es veins 36 
5133-0 = 
TU a [eas Semin 2 oF 
51708 = 
AOR Wig eo ae 33.9 
5209-0 = 
BAYONNE: iiscuichcusomenee 38 
5247-0 + 
HIB T= are 33 
53 JA-5 PS ae Ee Te 38 vo 


The strong lines which have been measured with consider- 
able accuracy, and which form a series, I have indicated with 
stars. The first five limes in the table also belong to the 
same series I think, though there seems to be a slight 
discrepancy in the table of differences. Four of the strong, 
equidistant lines are accompanied by faint lines on the short 
wave-length side. These four companion lines are also equi- 
distant. They are marked “a” in the table, and their 
wave-length differences are exactly 35 A.E. 


Copper Excitation. 


The resonance spectra excited by the light of the are 
between electrodes of metallic copper are reproduced on 
Pl. XVIII. fig. 3. The three green lines are alone operative, 
and though the line 5293 appears on the plate I am of the 
opinion that no fluorescence is excited by it. Spectrograms 
have been obtained of the spectra excited by the three lines 
separately, but they are not very satisfactory, as the slits of 
the moncchromatic illuminator had to be made very narrow 
and there was but little light available. To separate close 


ry 


ny 


Resonance Spectra of Sodium Vapour. 599 


lines and still have enough light to work with is now the 
chief difficulty. Ihave recently found a chlorate of potash 
film which in transmitted light shows a very black interference 


band not much over 15 Angstrém units in width. The 
position of this band varies with the angle of incidence of 
the light, and it can be set so as to cut off either one of the 
copper lines at will. It cannot, however, do this for a 
divergent beam of light, owing to the variation of the 
incidence angle in this case; and as the crystal is less than a 
square centimetre in area, little or nothing is gained by its 
use over the monochromatic illuminator. If we could prepare 
crystal flakes similar to this, measuring several cms. in width, 
they could be used with hight made parallel by a lens, and 
the problem of separating spectrum lines would be solved. 
In the resonance spectrum excited by the three green 
copper lines I have already measured over one hundred and 
twenty-five lines, but until more satisfactory spectrograms 
have been obtained with the separated radiations I prefer 
not to publish this table. The spectrum is very complicated, 
and I do not feel at all sure of the lines which belong 
together, although there are a number of very well marked 
series. My surmise of a year ago that the shortest of the 
three lines excited a series of doublets has been verified. 


The Magnetic Rotation Spectrum. 


The present work has shown that there are not many coinci- 
dences between the bright lines of the magnetic rotation spec- 
trum and the lines of the resonance spectra. ‘The belief that 
these coincidences existed, which was expressed in the earlier 
paper, was due to the low dispersion used. There appear to be 
six or more series in the magnetic spectrum, but they do not 
coincide with any of the series excited by monochromatic 
radiations. The lines which show the magnetic rotation are 
the lines which come out with especial prominence in the 
spectrum of the fluorescence excited by white light. At the 
present time I am engaged in a study of the magnetic spec- 
trum by the method described in my recent paper on the 
possible existence of positive electrons in the sodium atom. 

I have photographed the spectrum over again under more 
favourable conditions, and have obtained a much larger 
number of lines, nearly 200 in all. The table of their wave- 
lengths will not be published until the final paper is ready 
for publication, as I have not yet determined whether all of 
the lines rotate the plane of polarization in the same direction. 


258 2 


600 Resonance Spectra of Sodium Vapour. 


White Light and Cathode-ray Excitation. 


At the close of the present year an attempt will be made 
to photograph the white-light fluorescence spectrum with 
the 21-foot grating in the second-order spectrum. This will 
give us a record on the same scale as the absorption-spectrum 
map. Ihave already found that the structure is much more 
complicated than was at first supposed, the darker regions 
between the strong lines being filled with a multitude of fine 
lines. The structure of the spectrum which I attempted to 
show with a drawing in one of the earlier papers is’ quite 
incorrect, the peculiar appearance being due to a peculiar 
distribution of the light in the spectrum, combined with a 
rather wide slit. The white-light fluorescence spectrum, 
which begins at about 4600, ends quite abruptly at the point 
where the yellow-green doublet of the first subordinate series 
appears in the are spectrum of the metal. 

This may be an accident of course, but it may also be due 
to some relation between the parts of the vibrating mechanism, 
with which we are at present unfamiliar. The general 
appearance of the upper half of this spectrum is shown on 
Pl. XVIII. fig. 2 (upper spectrum), with an iron comparison 
spectrum. The upper end of this spectrum is also shown on 
Pl. XIX. fig. 3 (upper spectrum), together with the D lines 
which come out bright, and a portion of the red and orange 
fluorescence spectrum. ‘The red-orange fluorescence extends 
a little below the D lines, but not quite down to the terminus 
of the yellow-green fluorescence, the region between the two 
spectra being destitute of light. The bands which border 
the edge of this black region give it the appearance of a deep 
chasm. I have already obtained some excellent photographs 
of this region, and they will be published with the final paper. 
Exciting the vapour with cathode rays causes it to emit a 
very remarkable spectrum. ‘The lines of the principal and 
subordinate series come out with great intensity, as well as 
the lines which appear in the fluorescence excited by white 
light. In addition there is an entirely new spectrum, which 
in a way appears to be symmetrical about the D lines, at 
least in their immediate vicinity. This spectrum is shown on 
P]. XIX. fig. 3 (lower spectrum). It is on the same scale as, 
and placed in coincidence with, the white-light fluorescence 
spectrum. The lines of the subordinate series I have marked 
with black dots. It isa very remarkable fact that the bright 
bands to the right and left of the D lines coincide with broad 
dark bands in the fluorescence spectrum. This scarcely 
shows on the print which I am reproducing, and it will in all 
probability disappear completely in the process of repro- 


Distribution of Excited Activity of Radium §c. 601 


duction. I have some plates made a year ago which show 
the bands most beautifully, but I have not yet enlarged them, 
and they are on too small a scale to reproduce well. 

The present paper must not be regarded as a complete 
presentation of the subject,and I am publishing these results 
now in order that they may serve as a guide to those who 
are at work at the theory of the radiation of electron systems, 
from the theoretical side. 


LVI. The Distribution in Electric Fields of the Active 
Deposits of Radium, Thorium, and Actinium. By 
SIDNEY Russ, Demonstrator of Physies, Victoria Uni- 
versity, Manchester * 


I ntroduction. 


HE distribution in an electric field of the excited activity 
produced by thorium and radium emanations has been 
studied by several observers. Working with thorium ema- 
nation, Rutherford 7 has shown that the amount of activity 
imparted toarod charged negatively decreases as the pressure 
in the containing vessel is reduced after a certain pressure is 
reached ; while experiments made by Makower { show that 
similar effects are obtained with the excited activity produced 
from radium emanation. Some further experiments by 
Rutherford § with the emanation from radium indicate that 
while at atmospheric pressure the greater part of the excited 
activity is directed to a cathode by moderate electric fields, 
yet a small fraction (about 5 per cent.) goes to an anode. 
Reasons are then (loc. cit.) given for supposing that some of 
the active deposit particles carry a negative charge, thus 
accounting for their transmission to the anode. 

It seemed of interest then to find out whether the quantity 
of active deposit that goes to an anode changes when the 
pressure is varied. 

A comparison has therefore been made over a range of 
pressure extending from -001 cm. to 10 cms. between the 
amount of activity - imparted to a rod charged positively and 
then negatively when exposed for the same interval of time 
to equal quantities of the radiam emanation. 

Tt was found that whereas the activity of the cathode 
decreased as the pressure was diminished, the anode showed 
a corresponding increase in activity. 

* Communicated by the Physical Society: read March 13, 1908. 
+ Rutherford, Phil. Mag. Feb. 1900. 


¢ Makower, Phil. Mag. Nov. 1905. 
§ Rutherford, Phil. Mag. Jan. 1905. 


602 Mr. 8. Russ on Distribution in Electric Fields of 


The rod which had served as cathode or anode was then 
made a neutral pole by connecting it to the containing vessel, 
and what may be called diffusion experiments, 7. e. with no 
electric field existing in the gas, were then made. Observatious 
showed that there was a marked decrease in the amount of 
active deposit that diffused on to the rod as the pressure was 
diminished. 

After a slight modification of the apparatus, a comparison 
of the activity obtained on a cathode and anode overa range 
of pressure extending from ‘1 mm. to 1 mm. was made 
with three gases having widely different densities, namely, 
hydrogen, air, and sulphur dioxide. 

The results in the case of air show that the activity of the 
cathode is considerably greater than that of the anode, but that 
they approach equality as the lower pressure is reached. 

This effect is more marked in sulphur dioxide, while for 
hydrogen no such difference in activity is observed, just as 
much active deposit being transmitted to the anode as to the 
cathode over this range of pressure. : 


Methods of Experiment. 


A comparison was first made between the amounts of 
excited activity obtained on a wire maintained at 200 volts 
acting as cathode and then as anode over a range of pressure 
extending from ‘01 mm. to 1:2 mm. 


Fig. 1. 
: ay 
¢ : 
ny FE | ; 
B W D 
rN U U 
Ain = | Pimp 
=1— GAUGE 
—=S= V 
Feso1un Sou UTION 


The apparatus used is represented in fig. 1. A brass wire 
‘W 10 cms. long, ‘55 mm. diameter, on which the active 


Active Deposits of Radium, Thorium, and Actinium. 603 


deposit was obtained, was suspended so as to hang symme- 
trically within the glass vessel V which was silvered on the 
inside, connexion to a battery of cells being made by means of 
a piece of platinum fused in the side of the vessel. 

The usual method of experimenting was as follows :— 

The whoie system, with the exception of the vessel con- 
taining the radium emanation evolved from a solution of 
radium bromide, was exhausted firstly by means of a Fleuss 
pump and then by a charcoal tube immersed in liquid air (not 
shown in diagram). When the desired pressure had been 
obtained the tap D was closed, and by turning the tap E the 
small capillary tube C (‘2 c.c. volume) was filled with the 
emanation at atmospheric pressure. 

The tap B was then closed and the glass spiral S surrounded 
by liquid air, E was then turned and the emanation contained 
in the capillary C was allowed to pass over a tube containing 
P,Q; for drying purposes, after which it condensed in the 
spiral 8. 

A few trials showed that nearly all of the emanation had 
condensed in 15 minutes. The tap A was then closed, 
B opened, the liquid air removed from the spiral, and the 
emanation allowed to diffuse into the vessel V and a McLeod 
gauge which was in connexion withit. The resulting pressure 
was then read. 

As the volume of the P,O; tube was several times that of 
the spiral, quite low pressures were obtainable in the spiral 
although the capillary C had been filled. at atmospheric 
pressures. The wire W was made either a cathode or anode 
as desired and exposed to the emanation for 1 hour. It was 
then removed from the vessel V, and its activity tested in the 
usual way by making it the central electrode of a cylinder 
connected with a Dolezalek electrometer and measuring the 
ionization produced by the @ rays emitted from it. The 
electrometer leak obtained in a measured interval of time 
14 minutes after the removal of the wire from the emanation 
was taken as a measure of the activity obtained on the wire. © 

The variation in activity with change of pressure is seen 
from Table I., also from fig. 2, where the abscisse represent 
pressures and the ordinates the corresponding activities of 
the wire. It will be seen from the diagram that there is a 
very large decrease in the activity of the cathode as the 
pressure is diminished, while the anode shows if anything a 
small increase; the amount obtained by diffusion alone, 
2. e. to aneutral rod, being practically constant over this 
range of pressure, 

It will be observed that whereas at a pressure of about 


b0£ Mr. 8. Russ on Distribution in Electric Fields of 


| 


TABLE 1, . 
Positive ELECTRODE. NurGative ELecrrope. 
|Pressure in mms. : Activity. | Pressure in mms. Activity. 
| Oll | 69 | 006 | 142 | 
| 013 | m9 | 008 189 | 
| 137 a “102 173 
125 157 
| "202 81 ‘476 425 
250 | 62 . 729 736 
. | 1-165 1118 
608 63 fill ‘ . 
| 799 | 67 i NEvuTrRAL ELECTRODE. 
Pressure in mms, Activity. 
| 865 | 56 pees Can of De) 
a / 4 "152 72 
| = | = 435 64 
1:102 57 i "722 57 
| 1:140 62 


1 mm. the activity of the cathode is about 20 times that of 
the anode, at ‘01 mm. it is only about twice as much. 

At this stage of the work it was decided to compare the 
activities of the cathode and anode at much higher pressures. 

Several trials with the apparatus just described were made 
at pressures higher than 1 cm., but discordant results were 
obtained. This was probably due to the emanation being 
blown back into the spiral on allowing air to enter the vessel 
V, which was initially well exhausted as in the previous 
experiments. 

For this reason observations were made with a modified 
system consisting simply of a brass cylinder down the centre 
of which passed a brass rod (provided with end-pieces to _ 
ensure uniformity of field) which was connected to cells giving 
the same voltage as before. The brass cylinder was connected 
on one side with a small vessel which was filled with the 
emanation at atmospheric pressure, and on the other with a 
pump and pressure-gauge. 

The small vessel containing the emanation was provided 
with metal electrodes, and ‘the precaution was taken of 
applying a strong electric field to the emanation before letting 
it into the brass vessel ; this removed any active deposit or 
dust particles. 


— 
a 


Active Deposits of Radium, Thorium, and Actinium. 605 


Fig. 2. 


| ae | 
1000 y 


ACTIVITY 


PF 23453 7 8 9TH te 
PRESSURE LN MMS 


The system was initially exhausted to about 1 mm. pressure, — 
the emanation then let in and the small vessel isolated from 
the cylinder, the pressure in which was then adjusted to 
whatever value was required. The activity obtained on the 
brass rod after one hour’s exposure was measured in exactly 
the same way as before. Observations were made for a range 
of pressure extending from 2 mms.to1l0cms. The numerical 
data are contained in Table II., and the results shown 


606 Mr.S. Russ te Distribution in Electric Fields of 


graphically in fig. 3, where abscissz denote pressures and 
ordinates the corresponding activities. 4 


TaBLeE II. 
| ; 
Positive ELscrrope. ) Nercative ELecrrope. 

Pressure in cms. | Activity. | Prsvencan cms, Activity, 
ie aid 152 | 13 780 | 
27 | 140 So bats ; 
i | fe 2°70 1400 | 
oY | 7-22 1540 
63 | 148 7°80 1470 

1:10 | 120 ) NevTrat Evectrope. 
3°78 94 | Pressure in cms. ) Activity. 
2ST) 79 so | = 
7:30 ee | fae Lcome: : 
| 3°14 ) 287 | 
10°96 | 75 | 8°44 447 
| | 8-82 437 
| : 


It will be seen that whereas the activity of the cathode 
decreases with diminishing pressure, there is a very marked 
imerease in the activity of the anode, while the amount of 
active deposit that diffuses on to the neutral rod decreases 
very considerably as the pressure is reduced. 

Dittusion experiments similar to those just cited are at 
present being made, in which, however, other metals than 
brass are used as containing-cylinder and rod. 

It will be seen from fig. 3 that at a pressure of about 
2 mms. the quantity of active deposit transmitted to the 
anode is approximately equal to that diffusing on to a neutral 
pole. 

Now the active deposit that diffuses on toa neutral pole 
we have seen to be in the main positively charged (the bulk 
of it going to a cathode in an electric field). Very little of 
this positively charged matter can make its way to the anode. 
In order to account for the quantity obtained on the anode 
we are led to the conclusion that some at least of the active 
deposit particles that make their way to the anode are 
negatively charged. 


Active Deposits of Radium, Thorium, and Actinium. 607 


The above results then go to confirm the view held by 
Rutherford already referred to, that the active deposit 
particles which are directed to the anode have a negative 
charge. 


Fig. 3. 


AcTIVITY 


O ao /O 


PRESSURE /N CMS. 


The activity of the cathode being much larger than the anode, the 
scale in fig. 3 for the cathode is one third that for the anode and neutral 
pole. 


It has been suggested by Rutherford*, in explanation of 
his results with thorium emanation, that the decrease in 
activity of the cathode as the pressure is reduced is due to a 
decreasing number of collisions between the active deposit 
particles and the gaseous molecules with which they are 
mixed. 

At the moment of expulsion of an a@ particle from the 
emanation atom, the residue, 7. e. the active deposit particle, 
acquires a velocity in the opposite direction, which, though 


* € Radioactivity,’ p. 319. 


608 Mr.S8. Russ on Distribution in Electric Fields of 


small compared with that of the a particle, is high compared 
with the velocity of an ordinary gaseous molecule. 

On the view above cited a certain number of collisions 
with the gas molecules is necessary to sufficiently reduce the 
velocity of the active deposit particles in order that moderate 
electric fields may direct them to cathode or anode. 

Hence we might expect the effectiveness of a molecular 
encounter to depend on the nature of the gaseous molecules 
with which the active matter is associated. Experiments 
to test this point were therefore made with three gases 
differing considerably in molecular constitution, namely, air, 
hydrogen, and sulphur dioxide, and a comparison between 
the activities of cathode and anode over a range of pressure 
extending from ‘1mm. to 1 mm. was made with the low 
pressure system already described. The silvered vessel was, 


Fig. 4. 


a 


MQW GQQG{O 


f 


however, replaced by another glass vessel containing two 
concentric cylinders as seen in fig. 4. 


| 
. 
| 
2 
: 


Active Deposits of Radium, Thorium, and Actinium. 609 


The inner one R, 7 cms. long and *36 cm. diameter, acted 
as the rod upon which the active deposit was obtained, C the 
outside cylinder, 8°95 cms. long and 2°7 cms. diameter, being 
suspended by two platinum wires PP which were sealed into 
the glass vessel and connected to either pole of the battery 
as required. The ebonite ring E kept the outer cylinder 
steady, and the rod R was supplied with a guard-ring at 
each end, one of which fitted into a glass tube G, thus serving 
to keep the rod central within the cylinder C. 

These modifications were adopted so as to ensure a uniform 
electric field between the rod and cylinder, and also to render 
the problem more accessible to mathematical treatment, the 
areas of the rod and cylinder being not very different. 
With this arrangement experiments with the gases already 
mentioned were made, and the results obtained are collected 


in Table III. and reproduced graphically in figs. 5 and 6. 


Taste III. 
AIR. | Sutpnur Drox1pe. 
Sign of Pressure > el). Sie Oe Pressure we 
Hlectrode. | in mms. | HSimulie | Electrode. | in mms, INE 
| Positive. | “115 |. 108 || Positive. | 302 11 
iA eo) | 122 i P20 Cee ee 
| ' ‘Gol | 95 9 a See | 
Negative. | ‘115 | 146 | Negative. |  -297 148 
| 33 “260 toe 53 | "462 184 | 
| : 902 | 211 Pahl elise 70 298 
. | HypDroGeEn. 
| 
| | Sign of Pressure in ae | 
| Electrode. mms. Activity. | 
Positive. | 084 184 
» “481 200 
A 828 194 
Negative. ‘116 212 
Pr ‘131 198 
" ‘462 192 
” "925 202 
i ‘970 239 
1-030 188 


610 Mr. 8. Russ on Distribution in Electric Fields of 


It will be seen that for SO, there is a much greater dif- 
ference in the activities of cathode and anode than is the 
case for air, while for hydrogen practically no difference in 
activity over the same range of pressure was obtained. 


Fig. 5. 


Activity 


O “ay | 
PressuRe in Mums. 


We should expect that the curves for cathode and anode 
in the case of hydrogen would show a divergence at higher 
pressures, for if an encounter between the active deposit 
particle and a hydrogen molecule is not so effective in 
reducing the velocity of the former as a sulphur dioxide 
molecule, then we shall require a higher pressure with 
hydrogen to observe the same effects as with sulphur 
dioxide. 


Active Deposits of Radium, Thorium, and Actinium. 611 


Fig. 6. 
~ HYDROGEN 
1@) 

< 

EE O 

a8 POO SO 
> - i) 
L. 

= 

= 

100 


Pressure IN Mos 


Thorium and Actiniun. 


A few experiments made with the emanations from thorium 
and actinium show a rather striking difference in the dis- 
tribution of their active deposits in an electric field. 

The experimental arrangement seen in fig. 7 was used. 


Fig. 7. 


Two thin brass wires.7°5 cms. long and ‘7 mm. diameter. 
were made to lie along the axis of a brass tube C 30 cms. 
long and 4:2 cms. diameter. This tube was fitted with a 
small capsule D, into which was placed a small quantity of 


612 Mr.S8. Russ on Distribution in Electric Fields of 


thorium oxide, or of a preparation of actinium kindly lent 
me by Professor Rutherford. 

The wire A was connected to the positive, B to the negative 
pole of a battery giving 320 volts, the brass tube being 
connected to the middle point. 

The exit-tube E was connected to a Fleuss pump and a set 
of tubes containing calcium chloride and cotton-wool, thus 
ensuring a supply of dry and dust-free air. 

When experimenting witb thorium the wires were exposed 
to the emanation for 24 hours, after which they were removed 
and their activities tested with an electrometer in the manner 
already described. 

The ratio of the activities of the cathode and anode at 
atmospheric pressure was found to be about 200: 1; on 
reducing the pressure to 2 mms. the ratio was diminished to 
25:1, this reduction being mainly due to the decrease in 
activity of the cathode already observed by Rutherford 
(loc. cit.). No certain increase in the very small activity of 
the anode was observed. 

In the experiments with actinium the wires were exposed 
usually for about 2 hours. 

At atmospheric pressures the ratio of the activities of the 
cathode and anode was about 2 to 1, while at 2 mm. pressure 
it was 22 to 1, the increase in the ratio being mostly due 
to an increase in the activity of the cathode. The active 
deposit on the two wires decayed at the same rate, namely, 
that of actinium A. 

These results show that while the greater part of the 
active deposit from actinium emanation has a positive 
charge, yet at amospheric pressure a very considerable 
fraction has a negative charge, and is therefore directed to 
the anode. 

For purposes of comparison the activities obtained on a 
wire when exposed to the emanations of Ra, Th, and Ac at 
atmospheric pressure are tabulated. PP ke 


Wire charged Wire charged 


negatively. positively. 
Thorium \ caus 200 1 
Radium.) incre 200 10 
Aectinigm— 4.2.42 200 100 
Conclusion. 


In view of the different distributions in an electric field of 
the active deposits of thorium, radium, and actinium, it is 
difficult to think of the whole series of events between the 


-_ 
—_ 


. 
EEE OO OO CCC EE 


Active Deposits of Radium,Thorium, and Actinium. 613 


formation of the active deposit and its transference to either 
electrode, as taking place in an exactly similar fashion. 

It has been shown by Miss Slater* that slowly moving 
8 particles accompany the « particles which are ejected 
from thorium and radium emanations. If there are two such 
B particles to every 2 particle, the remainder of the emana- 
tion atom, 7. e. the active deposit, would be left with a 
positive charge. 

This supposition would explain the almost complete trans- 
ference of the active deposit of thorium to the cathode: but 
the small, though quite definite, quantity of the active 
deposit of radium that is directed to the anode still presents 
a difficulty. It may be that a few of the active deposit 
particles gain negative ions from the gas in which they are 
moving, which must be present in very large numbers. 

In the case of actinium, although at a pressure of a few 
millims., far more activity is observed on the cathode than 
on the anode, as has already been found by Debierneft, yet 
we have seen that at higher pressures this inequality is very 
much reduced, there being almost as many negative carriers 
as positive. 

It is hoped that a more extended series of observations 
may throw some light on the apparently anomalous behaviour 
of actinium. 

In conclusion, I desire to thank Professor Rutherford for 
several suggestions during the latter part of the work. The 
research was begun with Mr. Makower, who, however, could 
not continue the collaboration owing to pressure of other 
work. I wish to express my thanks for the very considerable 
help he has given me in these experiments. 


Note added March 17th. 


Some recent experiments made with actinium tend to 
explain the anomalous behaviour referred to above. A pre- 
paration of actinium was placed at the bottom of a cylinder, 
and two parallel plates were suspended so as to hang verti- 
cally above it. These plates were connected to the terminals 
of a battery of 300 volts, and after an exposure at atmospheric 
pressure usually lasting several hours the activities of the 
two plates were compared by means of an electroscope. The 
distance of the plates.from the actinium preparation could 
be altered as required. 


* Phil. Mag. Oct. 1905. 
+ Comptes Rendus, vol. exxxvi. p. 671. 


Phil. Mag. 8. 6. Vol. 15. No. 89. May 1908. ya 


614 Prof. R. A. Lehfeldt on the Electrochemical 


As already observed in the experiments described, the 
activity of the cathode was always greater than that of the 
anode. 

It was found that when the plates were fixed at a distance 
of 4 cms. from the actinium, the ratio of the activity of the 
cathode to that of the anode was about 5 to 1; on bringing 
the plates nearer to the actinium this ratio rapidly increased, 
and when the distance was reduced to 2 mms. the activity 
of the cathode was more than a hundred times that of the 
anode. 

These observations and others of a similar nature at 
different pressures (details of which it is hoped will appear 
in a future number), indicate that the sign of the electrical 
charge exhibited by the active deposit particles is some 
function of the distance which they have travelled through 
the containing gas, before reaching the electrodes. 


LVII. The Electrochemical Equivalents of Oxygen and Hy- 
drogen. By R. A. Lenrept, D.Sc., Professor of Physies 
at the Transvaal University College *. 


ore present investigation is an attempt (a) to design a 
gas-coulometer (voltameter) suitable for exact mea- 
surement; (6) to find how far the electrolysis of various 
solutions gives the calculated yield of gas; (c) to use the 
instrument to determine the electrochemical equivalents of 
oxygen and hydrogen. 

In its general design the coulometer follows the accepted 
methods of gas-analysis. 


DESCRIPTION OF APPARATUS. 


The coulometer’ itself is shown in fig. 1. It is made 
entirely of glass. The tube A, in which the electrolysis 
takes place, is about 22 mm. diam. and 70 long. It is 
closed below by a rubber stopper B carrying the electrodes 
C and D and the inverted thermometer E. The electrodes 
are of platinum, usually about 3 sq. cms. each, welded to 
platinum wires which are sealed through the glass tubes. 
The tubes are about half a millimetre in internal diameter, 
with an enlargement at the top where the platinum wire is 
sealed through ; they can consequently hold mercury even 
when inverted, and current can be led in and out by means 
of copper wires (No. 30 gauge) pushed up from below. 

The thermometer E is required for the temperature of the 


* Communicated by the Author. 


| Equivalents of Oxygen and Hydrogen. 615 
waste space F' at the top of the wide tube, as the electrolyte 


rises in temperature considerably during passage of the 
current. 


Fig. 1. 
J 


The connecting tubes GG are about 1 mm. in bore, but 


‘there is an enlargement H to catch drops of electrolyte 


projected upwards by the current of gas. K is a side tube 
for introducing the electrolyte, and is closed by a rubber 
tube and glass stopper. 

I is a gauge to show when the gas in the measuring-tube 


-is at the pressure of the atmosphere. Mercury, water, and 


olive oil were tried in this gauge, and the last found to be 
much the best. The gauge-tube is about 6 mm. in bore, and 


-no error due to capillary action could ever be detected when 


oil was used. The gauge should be rather long, as when the 


.gas to be measured is left for some time in the apparatus 
there is a risk of oil being sucked into the coulometer, in 


AT 2 


616 Prof. R. A. Lehfeldt on the Electrochemical 


consequence of changes of temperature. The gauge-tube 
should, for convenience, have either a rubber tube and 
stopper, or a tap to close it at the top when necessary. 

The tap L allows of communication with the atmosphere. 
The descending tube M is connected by a rubber joint with 
the gas-burette cr measure-tube. __ 

The coulometer was, of course, thoroughly tested for 
leakage. — 

The measure-tube was about 35 mm. in diameter, and 
contained 200 c.c. It was drawn out at the top to connect — 
with M, and at the bottom to lead to a three-way tap. One 
side of the tap leads to a mercury reservoir, from which the 
mercury in the measure-tube is supplied before an experi- 
ment ; the other side of the tap allows the mercury, as it is 
displaced by gas, to flow out into a bottle. 

The measure-tube is surrounded by a glass water-jacket 
like a Liebig condenser, through which tap-water flows con- 
tinuously ; and the temperature is taken by a thermometer 
inserted at the top of the jacket. 

At first a graduated tube was used, but this was given up 
in favour of weighing the displaced mercury each time. 

The coulometer and measure-tube are supported by a 
retort-stand. uy 

The electrical connexions are shown in fig. 2. Current is 

Fig. 2. 


o 


Vv 


taken from a battery of ten accumulators, each of about 
60 ampere-hours capacity. It is led through an ammeter A, 
to the three-part plug-key K by means of which the coulo- 
meter V may be cut out of circuit if desired. During the 
actual experiments the current flows through the coulometer 
to one terminal of S, the standard resistance, and from the 
other terminal to the rheostats R, R’. Of these, R is a dial 


Hquivalents of Oxygen and Hydrogen. 617 


resistance-box giving tens, units, and tenths, and capable of 
standing the current used (0°6 ampere at the most). Risa 
fine adjustment, which though very simple is, so far as my 
experience goes, new, and proved extremely satisfactory. It 
consists of a glass tube (1 mm. bore, 25 cms. long) closed at 
the bottom ; a platinum wire is sealed through the bottom, 
the tube filled with mercury, and a steel wire inserted at 
the top. By pushing the wire down or pulling it up, the 
resistance of the circuit can be adjusted with ease and 


certainty to a thousandth of an ohm. 


By adjusting R and R’, the potential difference between 


p and g is kept censtantly equal to one, two, or three 


cadmium cells C by means of the galvanometer G. The 
cells, which were made in the laboratory, are kept in a 
water-bath. 

G is a direct-reading galvanometer, by R. Paul of London, 
indicating nearly one scale-division per microampere. 

K’ is an ordinary tapping key, kept down when desired by 


a weight. 


CoNDUCT OF AN HXPERIMENT. 


The electrolyte haying been filled in, current is started, 
with the tap L open. The current is then adjusted so that 
the cadmium cells are balanced, and is left flowing for a few 
minutes. The coulometer is usually covered with a wet rag 
to keep its temperature down. When the current is steady, 
all the temperatures are noted. Current is broken for fifteen 
seconds, during which time the tap L is closed, and a re- 
adjustment of the rheostat is made: this is necessary because 
the gas-bubbles in the coulometer increase its resistance ; 
hence, after the break of fifteen seconds the current would 
be stronger than before and out of balance. With practice, 
the rheostat can be adjusted so that the current is almost 
exactly balanced immediately the circuit is remade. The 
plug of key K is then inserted at a precise moment by the 
chronometer, and attention devoted instantly to adjusting 
the rheostats, while an assistant turns the tap of the measure- 
tube to allow the mercury to flow out at such a rate as to 
keep the gauge I level. After the first minute one observer 
can attend to both matters, and it is easy to keep the mercury 
flowing so that the pressure of the gas in the measure-tube is 
never more than a centimetre or two of oil above or below the 
atmospheric pressure. Under favourable circumstances the 
electric current can be kept steady as exactly as the galvano- 


aii 1 
meter can be read, which is about 30,000 Part ; and of course, 


even when there are sensible fluctuations, they are sometimes 


618 Prof. R. A. Lehfeldt on the Electrochemical 


on the side of too much current, sometimes too little. The 
galvanometer-key is kept down by a weight throughout. 

Shortly before the close of the necessary period the 
galvanometer-key is raised, tap of the measure-tube closed, 
so as to leave a slight excess pressure in the apparatus; the 
current is then stopped, and all temperatures taken again. 
The apparatus is left for a few minutes till the coulometer 
has regained its initial temperature, to within a degree or so; 
mercury is then run out till the oil-gauge is level, the 
temperature of the water-jacket taken, the barometer read, 
and the mercury weighed. | 


DIscUssION OF ACCURACY ATTAINABLE. 


1. Electromotive Force. 


The cadmium cells used as a standard were made in the 
laboratory ; the set of four on which most dependence was 
placed being of date August 1907. Of these, three were 
made with mercurous sulphate prepared electrolytically by 
Carhart’s method; the other with sulphate bought from 
Merck: there was no appreciable difference between the 
two makes, however. The cells were all made with cathodes 
of mercury deposited electrolytically on platinum, and in the 
usual H form. They were kept side by side in a metal water- 
bath, with a thermometer. Comparison of the cells, made 
from time to time, showed hardly any systematic differences. 
Such differences as were noted, averaging three or four 
hundredths of a millivolt, appeared to be due to temperature 
variations—despite the water-bath : this would hardly be the 
case in most laboratories, but the extreme temperature 
fluctuations in a temporary corrugated-iron building in such 
a climate as Johannesburg, constituted the greatest difficulty 
in the way of accurate measurement; the temperature would. 
sometimes vary from 8° to 22°C. in the course of the 
morning. The discrepancies between the cells do not really 
matter, however, as no combination of them would differ by 


so much as a from another. As to the absolute values, 


though the cells have not been compared with any in 
Europe, experience elsewhere seems to show that the 
cadmium standard is quite reproducible to one part in ten 
thousand. 

The value used is that given in a private communication 
from Mr. F. E. Smith, of the National Physical Laboratory, 


and based on experiments with the ampere balance :— 


E, = 1:01830 —0:000034,(t 17) —0°00000066 (¢—17)”. 


Equivalents of Oxygen and Hydrogen. 619 


2. Resistance. 


The working standards were a pair of ten-ohm coils by 
R. Paul, used either singly or in parallel. These are made 
of “eureka” (constantan), and tested between 0° and 25° 
seemed to have no temperature-coefficient exceeding one or 
two millionths. The coils are enclosed in brass cases, with 
terminals at the top. This is not the best pattern of standard: 
coil ; but the errors involved in the way of terminals and 
connecting straps would not exceed one or two ten-thousandths 
of an ohm at the most, and are therefore quite negligible. 

The largest current used through either coil was 0°3 ampere, 
the power spent being 0°9 watt. The heating due to this would 
not cause any thermoelectric error, as the coils are quite sym- 
metrical ; but it seemed possible that, with a combination of 
such large thermoelectric power as eureka and brass, the 
Peltier effect might introduceanerror. This is so in theory, 
for the current entering at p would heat that junction and 
cool g. Hence a thermoelectromotive force will be set up, 
acting against the current, and the potential difference 
between p and gq will be increased. 

A special experiment was made to test this, by means of , 
potentiometer; but nothing measurable was found. As ,, 
additional precaution, however, the brass case was filled w 
paraffin oil. 

The absolute values of the resistances were obtained in the 
following way :—The laboratory possesses two platinum- 
silver standards, of 1% and 100” respectively, made by 
Nalder Bros., and tested at the National Physical Laboratory. 
The values are stated to be 


octet 27 O00. 2170. Ph ki, 
So dat, ba 00. 217. NoPE. 


There were also two eureka coils of 1% and 100° similar 
to the working standards. The eureka 1” and 100° were 
compared with the platinum-silver standards by a Lehfeldt 
comparator. In this comparison the same thermometer was 
used as for the gas-measurement (vide infra), and the tempe- 
rature was within a degree of 17°. (It should be mentioned 
that this measurement was made in a well-built private house, 
where the temperature fluctuations were much smaller than 
in the laboratory.) The temperature coefficient of the 
platinum-silver was assumed to be 0:000247. 

The two ten-ohm working standards were compared with 
each other similarly, and then combined with the 1° and 
100% to make up a Wheatstone bridge, the 1° being shunted 


620 Prof. R. A. Lehfeldt on the Electrochemical 


till balance was obtained. In this way two equations were 
arrived at giving (10 4) x (108) and (10 4)+(10 B). 
The final results were 


(10 a) = 9°9986 
(108) = 9°9981, 


in terms of the ohm as determined at the National Physical 
Laboratory, and it is hardly possible that the error can 


exceed 1) te 
20,000° 


3. Time. 


This was determined by a chronometer (Carroll, London, 
No. 505) beating half-seconds. Its rate was not studied 
with care, but was only two or three seconds a day—a 
negligible amount. The duration of the experiments varied 
from 1350 sec. to 4050 sec. There seemed in this the 
possibility of a serious error in timing the make and break 
of circuit by a plug-key, so special experiments were made 
to test it—with somewhat surprising results. The current 
being adjusted beforehand as usual, electrolysis was carried 
on for a period of five or ten seconds, and the mercury 
weighed in the ordinary manner. This gives a value for 
the quantity of electricity flowing that would be affected by 
the error of starting and stopping, and also by the error in 
the initial adjustment of current already spoken of. The 
following short table shows that the two errors combined 
amount to much less than a tenth of a coulomb, 2. e., much 


less than in the main experiments. 


ies Sink 
10.000 


_ Quantity of Electricity. 


Pe yes | Be | Calculated. | Observed. 
seconds. | amperes. | coulombs. | coulombs. 

5 | 0-509 | 2°55 | 2°54 

7 | 0509 896 | 8:88 | 

5 /  0:509 | 2-55 | 2-54 | 

| 10 | 0-509 Fa: a 


I 


A further possibility of error lies in the adjustment of 
the current; but, as remarked above, the galvanometer 


allowed of detecting a want of balance amounting to 30 0 


and as during a long experiment the current was sometimes 


Equivalents of Oxygen and Hydrogen. 621 


too large and sometimes too small, this possibility may be 
disregarded. Altogether, reviewing the electrical measure- 
ments—E.M.F., resistance, time, and current balance, one 
seems justified in considering that the quantity of electricity 
flowing is known to within one ten-thousandth part. 


4, Volume. 


The mercury displaced by the gas was weighed on a large 
Oertling balance, capable of carrying five kilos and sensitive 
to a milligram. The weighing was only carried out to a 
decigram, however, as this is ample for the purpose, some 
2500 grams of mercury being used in an experiment. The 
arms of the balance were found to be sensibly equal : the 
weights consisted of a 2000 g. and 500 g. in brass from a 
rough set, and 200 g. downwards from an ordinary box of 
analytical weights ; the relative errors—amounting to 2 or 
3 decigrams, in the case of the largest—were allowed for, the 
analytical weights being taken as standard. ; The relative 
50,000 
but as to the absolute values I am unable to speak, there 
being no standard in this country. It is, however, hardly 
likely that the errors of the 200 g. and 100 g. ina box of 
weights of first quality would be large enough to matter. 

Tine volume of the mercury was calculated from the table 
in Landolt & Bornstein, p. 42: temperature taken being 
the mean between the initial and final temperatures of the 
water-jacket. In favourable cases this only varied one or 
two tenths of a degree during the experiment, so that the 


error in the density of mercury would be of the order of 
L 


60,000° 
flowing out could be adjusted was less than this, a single 
drop making a difference of one, ten, or twenty thousandth ; 
but it was sufficient. 


therefore : 


accuracy of the weighing may be put at 


The accuracy with which the amount of mercury 


5. Pressure. 


The barometer used was one of special design made for 
the author by Messrs. Baird & Tatlock, and filled in the 
laboratory. ‘The tubes, of 25 mm. bore, are large enough to 
eliminate capillary effects; the upper tube carries a glass 
pointer to which the mercury was adjusted * by means of an 
inlet and outlet tap, and a screw-clip compressing a short 
piece of rubber tubing. The mercury used was taken direct 
from the bottle in which it was supplied, and cleaned with 


* The barometer varies very little all the year round at Johannesburg. 


622 Prof. R. A. Lehfeldt on the Electrochemical 


nitric acid. The filling was done from the top by a Tépler 
pump, and the vacuum tested by the discharge between a 
pair of electrodes in a side tube. 

The height is measured by a pair of glass scales by Zeiss, 
each 20 cm. long, mounted on a strip of plate-glass which 
forms the back of the instrument. This arrangement was 
sent by the manufacturers to the National Physical Labora- 
tory to have the distance between the short scales measured. 
To read the barometer the level at the top was adjusted to. 
touch the glass pointer, and that at the bottom read by a 
telescope with micrometer eyepiece divided in hundredths 
of a millimetre. The mercury surface was illuminated from 
behind, through a little tissue-paper window into an adjoining 
room, where an incandescent lamp was placed. ‘The level of 
the pointer was found to be 87°84 mm. on the upper scale ; 
the distance between the zeros of the two scales 600°09 mm. 
at 15°:2, according to the certificate of the National Physical 
Laboratory. Hence the uncorrected reading at 15° is the 
reading of the mercury against the lower scale (s)-+600°09 
— 87°84. 

From this must be subtracted the expansion of the 
mercury between 0° and 15° (on an average height of 
622 mm.), 

15 x 622 x 0:0001819 = 1°70 mm., 
and for gravity, according to the provisional value obtained 
(Lehfeldt, Phil. Mag., Nov. 1906), 
978°7 ~ 980°62 
980°62 
The constant of the barometer is therefore 
600°09 — 87°84 — 1°70 —1°22 = 509°33 at 15°. © 
The temperature coefficient is 
(0°0001819 —0-0000087)622 = 0°108 mm./degree. 


Hence the pressure of the atmosphere is 
s+ 509°33 —0°108(¢—15), 


where ¢ is the temperature of the barometer. 

The pressure of the gas-burette could usually be adjusted 
to differ from that of the atmosphere by not more than about 
1 mm. of oil (=1/60 mm. mercury). 

The gas-burette was kept constantly moist by a few drops 
of water over the mercury, and the vapour-pressure of water 
at the temperature of the final adjustment was deducted from 
the observed pressure. 


S 622° = 22 am. 


Equivalents of Oxygen and Hydrogen. 623 


6. Lemperature. 


The most important temperature to record was that of the 
water-jacket of the gas-burette. This was taken by a 
thermometer (graduated in } degree) near the exit of the 
water. This thermometer had been compared with a 
standard that had been tested at the Reichsanstalt, and 
showed, over the atmospheric range of temperature, only 
irregular variations amounting to about 0°06 as a maximum. 
The standard itself was tested at the freezing and boiling 
points and found to be exact, as stated in its certificate. 

The water-jacket sometimes rose and sometimes fell in 
temperature during an experiment: in unfavourable cases 
as much as 1°5 degrees ; sometimes only one or two tenths 
of a degree. The final reading, taken immediately after the 
outflow of mercury had been adjusted, might sometimes be 
uncertain to the extent of 0°04 or 0°05: this is probably 
the most serious error in the experiments; it affects the 
reduction of the gas both directly and through the vapour- 
pressure of water, so that an error of 0°05 means about 


so00 1 — on the result. 

The other temperatures noted were :— 

(a) Barometer : owing to the rapid fluctuations in room- 
temperature, the mercury of the barometer may at times 


have been a degree or more different from the attached 
thermometer. This would affect the pressure by = 


(6) Cadmium cells: the influence of this is very small, 
and has been allowed for above. 

fc), Air. 

(d) Coulometer : a correction has to be made if the tem- 
perature of the coulometer is not the same at the beginning 
and end of the experiment. The coulometer itself was 
calibrated for volume, so that the amount of gas in it could 
' be determined. If this was say 2 c.c., and the burette held 
190 c.c., one degree makes a difference of 

I Znek tel 
290 * 190 ~ 28,000 
in the volume of gas. 

The temperature of the coulometer was allowed to fall 
after the experiment to nearly the same level as before, and 
the mercury then adjusted: the above correction was con- 
sequently always small. 

The uncertainties introduced by fluctuations of tem- 
peratures are certainly the most important, and show that. 
the results given below are about on the limit of the accuracy 
obtainable by this apparatus. 


624 Prof. R. A. Lehfeldt on the Electrochemical 


RESULTS. - 
(a) Coulometer jor exact measurement. 


The electrolysing cell and gas-burette as described are 
suitable for measuring quantities of electricity up to 800 or 
1000 coulombs : all the measurements can be made within 
ten minutes or so of the stoppage of the current, and the 
quantity determined to within one-tenth per cent. Further, 
the apparatus can be got ready for use again by the manipu- 
lation of a few taps, instead of the tedious chemical processes 
required by the silver voltameter. 

If the gas be measured against that in a compensating 
bulb, as described by Hempel (‘ Gas-analysis’), the measure- 
ment is still more easy and rapid, though probably with a 
slight loss of accuracy. 


(b) Electrolysis of various Solutions. 


The quantity of gas given off was measured in the cases 
of :— 


Strength. Per cent. of 

Per cent. theoretic yield. 
SOs ee. 10 99 
a ee oo 10 to 25 98°5 to 99°8 
Na,SO, .... 10 to 30 Exact. 
Nas O: ye. 10 53 (deposited sulphur). 
1d aig AS 10 97°5 to 99°7 
Natu. citne 10 99°5 
Oe eee 20 62 (turned brown, and 
KGr 0, 257 | Sao sg Exact. frothed). 
Na,C,O, .... Saturated 91 (frothed). 
Na,HAsO 10 97 
NaNO:) oVet 10 50 
NaClO 10 33 


It appears, therefore, that sodium sulphate and potassium 
bichromate are the best electrolytes to tise in the coulometer ; 
no systematic differences could be detected between them, or 
between solutions of different strengths, or at different tem- 
peratures. There is, then, every reason to suppose that the 
electrolysis of these solutions is quantitatively correct. 

Sodium hydroxide, contrary to what is usually stated, does 
not seem to yield better results than sulphuric acid. 


_ (ce) Electrochemical equivalent of Oxygen and Hydrogen : 
value of the “ Faraday.” 
One “‘faraday” of electricity (the amount associated with 
one gram-equivalent of matter) decomposes 9°0075 grams 
of water, yielding 8 grams of oxygen and 1°0075 of hydrogen. 


Equivalents of Oxygen and Hydrogen. 625 
If these be collected mixed under N.T.P. they occupy a 


volume 
8 10075 
0°0014290 ~— 0:0U008986 


using the best published values for the density of oxygen and 
hydrogen. 

But as the water has a volume of 9 c.c., the increase in 
volume is 16800°9 c.c. 

The experiments with sodium. sulphate and potassium 
bichromate detailed in the appendix—all the experiments to 
which no exception could be taken on account of the electro- 
lyte, or for other reasons, give the following values for the 
gas evolved per coulomb :— 


0°17396 0°17390 0°17398 0-17401 


= 16809°9 c.c., 


398 391 ag4 = B98 
389 387 401 388 
ag1 388 396 404 


The mean is 0°17394, with a probable error of -+0:00001. 
Hence the value of the faraday 


16800°9—0°17394 = 96590 coulombs. : 


The corresponding value for the electrochemical equivalent 
of silver is 
107°93 
96590 
This is lower than the value legally adopted (0001118) on 
the basis of Kohlrausch and Rayleigh’s experiments, but in 
close agreement with the latest work of Richards. 


= 0°0011174 gram per coulomb. 


P.S.—An attempt was made to allow for the deviations 
from the laws of gases. But the deviations for oxygen 
and hydrogen being of opposite sign, the total effect is too 
small to be worth taking into account. The uncertainty in 
the’ density of the gases is larger. 


The experiments were carried out in the temporary 
physical laboratory of the Transvaal University College ; 
and some trials have been made there with a double 
apparatus, which, it is hoped, will avoid some of the 
sources of error. As, further, a well-constructed permanent 
laboratory is in course of erection, it may be possible to get 
still more concordant results. The author wishes to express 
his indebtedness to Miss Winifred Judson, B.Sc., for much 
assistance in the work. 


December 1907, 


Prof. R. A. Lehfeldt on the Electrochemical 


626 


88EL1-0 L8EL1-0 T6EL1-0 O6ELT-0 T6ELT-0 68ELT-0 86EL1-0 96ELT-0 SS eS ee 


LL-FE8 68-FZ8 18:FG8 60-468 21-428 60-428 0-928 1-928 sseees Ajortqooyo Jo AqrGQueng 
IP-8FT GF-EFT GP-EFT SF-SFI O¢-SF1 OF-PL 64-SFI. EC-SFT mrreererees” QUINTOA Poon por 
0-6F9% P4193 6 8696 B-919G GEIS €.8Gz 8-899% F-96496 rereeeeee KIMOLOUL JO IULIO AA 
26-61 28-81 69-61 aaa 03-21 29-FT 08-41 0Z-FI ae eS eee 
GLB GG 6.9% 83 CG 6 &3 st Bog “re7** doqouto[nod ‘7, 
6.95 8:36 9 GG OCT C.9T 8-91 9.41 easy 7 Avereesee  gegoutoreg ‘y, 
GO-LOT 88-01T £9-80T 82-911 PP OT 1¢-PIT LE-S1T 00-FIT seremmangas ease “4s Te19 MOA G 
OG0F 0046 00L6 GZ0G GOB | = G20 O0LE O0LZ eS Se S08) Otay, 
6-61 2.6L | 9-81 $1 G81 | 2-61 99 Z-8T | 4-21 079-21 | E31 OF 8-11) 9-FT 1091] 9-¢1 01 3-1 GET 8 @ qeypoul-royem 
9B 0} GLE O€ 9} GZ 209 O.1% | G16 °F 8G | 4-66 01 3G | 9-98 97 62 8G 94 &Z G Se “aajouoqnoo =“ 
12% G81 P 6r 8-21 8-11 GT 8-41 G11 Be ae Seep. 
LEV LG | G-FG 99.2 OZ) GBS 91 F-1G | S-GT 91 9-91! G-9T 91 9-81 |8-9T 9} 9-91 9-L1 ia! east "euteaye ‘drop 
‘tunurgeyd ‘ eto “mos be oc ae qystaq ‘utd "bs T Eee Snes Sok SOO LION, 
‘op 00 G ‘0 op | 00 Z Aree Cc eonceleia oveevcece eectceccasenncncece oords O98 AA 
VOI+ X90 ‘VOI 5 (6) (s po | ‘(qetyeced ut # QT 9 VOT)+ +(e) (po ‘VOT ‘s0r+(e) 9 OO Beep ie ee ee ONG 
“/) 08 "OS"8N */o OL “O° M “toy “9° OOT Ul O° HOT’ 'OS*UN “9 08 Regn Oe sh eB OM ORTEL 
‘UST Boni ‘USL “L0'S-Z1) uel *L0°8'OT] ‘aS ‘aPT “LOLS | uSl LOLT| “uot STOlee LD heme ice Serer ana 
dal) gaat Msi es ae ‘eM a ia ‘Tv a Tey aL ye align ga i Sie ane 5S ATOR) 
‘B/, LL ‘BL ‘OL ‘69 ‘GG "2G IG rreeesees  ATIOUITLIAXO JO ‘ON 


‘T wild Li XIGNidd'V 


Equivalents of Oxygen and Hydrogen. 


FOPLT-O 
68-768 
G¢-§I 
61696 
OV-61 

0-F6 
G-06 
GL-LIT 
OOF 
F-61 9} ¢-0G 
GG °F GGG 
8-06 94 9-02 
6G 9} GG 


‘Op 


vor+@) (po 


‘uGT “LOOTG 
BV" aL 


S8ELT-0 
LI-Sh8 
19-9F1 
8:-G186 
06-61 

¢-€§ 
F-1G 
F8-E11 
O8éI 
6-61. °F G06 
SE OF 9¢ 
L-16 
96 °F €G 


‘op 


86ELT-0 
SI-EF8 
OL-9F1 
0.9896 
09-06 
GEE 
9-16 
68-§11 
O8sI 
6-06 94 €-06 
8E 07 ¢-E¢ 
1:06 
GG 9} GG 


‘unurzyezd 


| ‘0g 


10FLI-0 96EL1-0 LOPLT-O F6EL1-0 
66-768 {6-768 16-468 OT-9¢8 
g¢-€F1 1¢-€FI G¢.F1 GG-SP1 
0:899¢ 1-0996 L-LE&S G-GES6 
09-ST F8-CT 69-GT 80-F1 
8-GP €-86 9G = 92 
L61 G61 6-61 &-GT 
69-911 F9-G1T €L-¢CTT 6-911 
OSST COG GGOG GZ0G 
9-GT 94 8-CT | 8-GT 99 0-91 | L-GT 99 2-41] 0-FT 9} G-FI 
CY 91 G-GP | GE 9 G.8G GE 9} 9G 9€ 99 9G 
0-ST GLI L-ST 6-61 
0-6T 9} G-6T 'G-GI 940-61 0-61 °F G81! 91 °F OL 
A ote Wolke Oise 
‘op gO een liy PRONG: Sera Oran 


“O19 V OT-+ (ot) (8) (@) PO 


20°10 0 °/o OL 


“u9Tt “LO OTT 
‘L'oM 
‘06 


‘UST “LOO 
“LM 
‘68 


‘ull “L0°6'FS 


‘LM 
88 


“LM 
‘L8 


“LoM 
‘98 


‘TV 


86E41-0 
86:F68 
69-SF1 
61896 
GL LT 
96 
P81 
L9-F11 
{606 
FLT 99 6-91 
GE 94 9G 
6-91 
G-61 9} 6-61 


‘0'0 @ 


‘(joqexed ur -# gt vot) + (ot) (8) PO 
Tos’en °/o OT 
"u9L “LO'6'FZ|'uST °L0°6' FS, uGL 10°68 


1) 0S "OS°EN 
"ul ‘L0'°6 LT 
Tey 


"(panurjuoo) "| Wavy, 


Ce eo ee i 


q[nsoy, 


oor rereseee 


9tnjoaA peon pet 


AUNDIOUT JO JUSTO AA 


tee estes eres eeseees DG ale 


:, nereeeees TaqouO[NOd “7, 
a Teeeeeceeees TOJOUIOTLG “7. 
sousberteesO¥006es TaqOUIULEG) 
SODOOUDOAIOU Ove > (‘098) OuuLT, 
¢ ‘goyoul-tojum = 

ae ““Joyomopno0o =“ 

EE eeesae S][90 po 66 

5 Kaen came Ie ‘dwuay, 
Heerreeeeeeeereess san ogoaT AT 
teeesecressoorsess) ORCS OIST AA 
eereesseeseesereoeens ag gang 
sence neces eeneence 94A470.1400]5] 
tiereseeseertereeseseenen gam 
eRigtiey cate MeN 194.089 


quowittedxo Jo ‘oN 


LVIII. Inverse Interpolation by Means of a Reversed Series. 
By C. H. Van OnstRAnpD *, 


HE formulas developed by Newton, Bessel, and Stirling 
for the direct interpolation of a value of a function from 
values tabulated at equal intervals of the argument, are con- 
sidered to be among the most important of their contributions 
to the science of applied mathematics. The converse problem, 
that of finding the argument when a value of the function is 
given, although of about equal importance, seems not to have 
received the thorough treatment which mathematicians have 
given to the subject of direct interpolation. Apart from 
methods applicable only to special cases, there are two general 
methods+ now in use which are to a certain extent satis- 
factory. Hach of the methods is in reality based upon the 
same principle—that of diminishing the interval of interpo- 
lation. In the one case this is accomplished by computing 
values of the function at equal intervals of the argument for 
values preceding and following the required value. If the 
interval of the new series of tabular values is sufficiently 
small, the correct argument can be found by taking second 
differences into account. In the second case, the function 
and its first and second derivatives are computed for a value 
of the argument (7m) true to the nearest tenth of a unit. It 
is then generally sufficient to write the interpolation formula 
as a Taylor's series involving only the first and second 
derivatives. The correction to the approximate value of the 
argument is then found by reverting this series of two terms. 
Each of the methods has the serious disadvantage of being a 
tentative process, and neither of them provides a satisfactory 
check on the computation, without the aid of additional 
quantities. | 


As a means of avoiding the difficulties noted above, it is 
desirable to call attention to the use which may be made of a 
reverted series as a formula for inverse interpolation. To 
derive this formula, let the tabular values of the function 
(F_3, F_s,...) and its successive differences (the a’s, b’s.. .) 
be represented by the following schedule :— 


* Communicated by the Author. 
+ Rice, ‘Theory and Practice of Interpolation,’ pp. 192-5. 


—T- 
i ¢ 
1 


j 
J 


Inverse Interpolation by Means of a Reversed Series. 629 


| T F(t) ny Att a Alv AV 
t — 3w | F_3 
| | bare 
t— Qu Fo b_. 
a_» G_» 
_ t-—w Boy b_y dai 
@ 4 C_y e_} 
t F, by d, 
| & sy bat 
é +w F, b, d, 
|} ay Cy 
t+ 2w F, | b, 
as 
t + 3w F, 
Put also 
1 P Cc e 
a= =(a,+a-_ a’ =a——+— 
Aaa gel 6 * 30 
1 rf bo dy 
¢ = 5 (4 +e.) b 0 ee 
if »1E e 
C=. = ( C Gx € SSS. = SS 
g (1+ e-) 6 24 
d 
a if 22°50 
AF =F. —F, a= 5 
pegs: 
120. 


Then, using Stirling’s expression for the successive derivatives, 
because of their simplicity and rapid convergence, and writing 
the interpolation formula as a Taylor’s series, 

AF=a'n+b'n?+c'n?+d'n*t+en+..... .-. (1) 
The solution of the problem consists in finding the value of n 
when AF, a’, 0’, c',... are given. To accomplish this, we 
revert equation (1) by means of the usual method of equating 
coefficients, or by means of the expression obtained by 
Professor McMahon for the general term of a reverted series*, 
and obtain 


_ AF bead (ey cf! = 
morta? a! 7) +([2(9 —S\ (= 
eae Paes 
+> s CRcaen eee, 
! ! , 5 
+ [3(5) +6y9—9-™ (6) ot 4G) IG 


* Bull. Am. Math. Soc. iii. p. 170 (1893-1894). 
Phil. Mag. 8. 6. Vol. 15. No. 89. May 1908. 2U 


630 Mr. C. E. Van Orstrand: Jnverse 


Equation (2) may be put into a more convenient form 
for computation by rearranging the terms and putting 
ny=AK—a’ and r=n,—a’. Then 

n=n+[—rb!+2(rb')?—5(rb')3 +14 (rb!)4] 1, 
+ [rd (—14+5 rb'—21(10')? |n/? 
+ |[rd'(—1+ 67b'/) 4+ 3 (rd)? n,3 
+[—ré|ny+.... - «+ er 

Now substitute 

fr=—rb' +2 (rb!) —5 (rb!)? +14 (rb')4 
fo= rel —1+57rb!—21 (rb')?] 

fp= ral —1+6rb'j +3 (rc)? 
Jsa=—re 


in (3), and the equation for inverse interpolation, inclusive 
of fifth differences, is 


n=ntfimtfonetfanitfint . . - (4) 


For backward interpolation m, is negative. A _ slightly 
different form of (4) convenient for logarithmic computa- 
tion is 
n=mytfin tre fon? +rd/fs'nye+ 3(re'yPne+fi... (5) 
in which 
Jo =—14+5 rb’—21 (rb')? 
Ts =—1+6 rb’, 


The expressions for the /’s, as obtained by developing the 
reverted series to the same order of powers as the given series, 
are represented by the following groups, according as each 
series terminates with the second, third, or fourth power : 


fAH—rh'=— rl +2 (rb! P=—rb' +2 (rb!) —5 (rb/)? 
(— =—rc' = ré(—1+5rb’) 
Vase =—rd', 


Analytically, there are an infinite number of the /’s and each 
contains an infinite number of terms. The number of terms 
required in any particular reversion is easily ascertained 
from the data. As a complete formula for inverse interpola- 
tion, applicable in all cases, the reversed series is unfortunately 
not always sufficiently convergent when a limited number of 
terms is used. However, the most important applications of 
reversion obtain when the numerical magnitudes of the 
successive orders of differences gradually diminish. It will 


be found for such functions that rb'<0'1, and since n <3, the 


Interpolation by Means of a Reversed Series. 631 


doubly infinite series will always converge*. The cases in 
which the series fails to converge with sufficient rapidity are 
therefore exceptional and are easily discovered. In fact it is 
only necessary to see if the terms at the end of each hori- 
zontal series, such as 14 (rb’)'m, 21 re! (rb’)’n?, ..., are small 
im comparison with 7. 

The quantities (a/ble ‘d'e') are easily computed when the 
first derivatives are tabulated. Representing the functional 
values, the first derivatives, and the successive differences of 
the first derivatives by the scheme : 


T F(T) wk" A! As gent AIv 
t -- 3w Ue a—3 
eerie, | a. | °° 
ze B : Y-2 
t—w F C27 wis mn C_9 
rh B ¥ t | 3 ei 
t F, a, St y 6-1 
| 3 g €o 
t+w 1s a B, é. 
B N €) 
t — 2w 1 a, : é, 
| ak: ie : 
| pou) F, | a ; 
| 


these auxiliary quantities are represented by the simple 
forms :— | 


ay 

(=F — 75 

a= “ B=3(8,+6-.)=hi- > 
d= 755 $= 36,464) =0-2 


when Stirling’s formulas for the successive derivatives are 
used. Evidently, 
2h LI wo “il o' iv Fava Vv 

| a ee OR irk yt spon 0 

Sea ae bem a en 8 30" 
‘The computation of a!, b!, c', d’, e'... can be checked in various 
ways. In general it 1 is quite sufficient to duplicate the work. 
An independent check is of course obtained by computing 


* Harkness & Morley, ‘ Treatise on Theory of Functions,’ p. 116, 
aU, 


6 
= 


632 Mr. C. E. Van Orstrand : Jnverse 


the derivatives from the formulas of either Newton or Bessel, 

preferably the latter for arguments other than at the beginning 

or end of a series of tabular values. 
As an application of formula (4) let it be required to find 

the interpolation interval (7) from the following tabular 

values of the function and its differences, when the logarithm 

of Mercury’s distance from the Earth equals F,,=9°7968280. 
Date Log. Dist. of | 
1898. from @. 
May 8 9°7560706 


+91669 
10 9°7652375 +24839 
116508 —4748 
12 9°7768883 20091 +382 
136599 4366 + 75 
14 9°7905482 15725 457 
152324 3909 +135. 
16 9°8057806 11816 +592 
164140 — 3317 
18 9°8221946 + 8499 
+ 172639 


20 9°8394585 
We take from the table, 
F, = 9°7905482, 
a =4(a,ta_1) = + 1444615 


b, =+4 15725 
e =4s(qte-1) =— 413875 
e =4(@q+e.1) =+ #10350 
and then compute, 7 a 
es 3 —+ 78435 3894510 _—rb! = +.0°233773 
dat 2 = 69819 284130» —re"== —0°0020681. 
ne a oe 4 D0 9 Mido TER rd! = +0°0000566 
e! = 755 =+ 09  9:954_15 re! = +-0°0000027 
AF=F,—-F,  =+ 627980  4:7979458 f=—0-0223441 
a'=a ~£ 46 = 41451546 51618308 fa= +.0-0018500 
iy al ae 9-6361150-1 _fs= +0°0000359 


r=ny—a' = 4-4.742842_1  f,= —0°0000027. 


Interpolation by Means of a Reversed Series. 633 


Substituting in equation (4) we compute n ; then substituting 
n in equation (1) we compute AF asa check. Thus, 


n, = +0°4326284 a'n = + 614447 
fm = —0°0096667 b'r2=+ 1405-4 
fyny? = + 0°0003463 cv=— 52:6 
fn; = —0-0000029 dnt=+ 0°6 
én,4= —0-0000001 en = 0-0 

n=>= +0°423305 AF=>=+ 62798'1. 


By means of the second method of inverse interpolation 
previously explained, Rice* finds n=0°423303. ‘This dis- 
crepancy is due to neglecting units in the seventh decimal 
place. The value of AF computed from this value of m and 
Stirling’s derivatives is AF =62797°7. 

There are a great number of tabulated functions in which 
second differences are approximately constant and in which 
the term 2n,(rb’)? is negligible. In such cases the following 
rule applies: Divide the increment of the function (AF) by 
the mean of the first differences (a, +409), and likewise divide 
this quotient (n,) by the same quantity; then the product 
of these two quotients (n;r) and one-half the second differ- 
ence (40,) is the correction to be applied algebraically to the 
first quotient (7) in order to obtain the required interpolation 
interval (n). Thealgebraic signs of the quantities need not 
be taken into account; for it is plain that if the numerical 
values of the first differences are increasing, the quotient (n,) 
is too large and the correction is therefore negative, while 
if the numerical values of the first differences are decreasing, 
the value of n, is too small and the correction is positive. 
This method is very convenient when the numbers are so 
large as to require the use of a logarithm table in making the 
interpolation. For example, let it be required to find 6 from 
Vega’s ten-place table when log sin 0=8°910 7867 247. 

The tabular values are as follows :— 

0. log sin @. A’. At, 

A° 40" O" 8°910 4038 653 

+2578 594 
10 -910 6617 247 — 1537 
2577 057 
20 "910 9194 304 — 1538 
4.2575 519 
30 ‘911 1769 823 


The quantity to be computed is 
n=n,—rb'ny. 


* Loe, cit. p19, 


634 Mr. C. E. Van Orstrand: Inverse 


We disregard the algebraic sign of the last term, for a mere 
inspection of the first differences suffices to show that the 
correction to n, is positive. Following is the computation :— 


Fn, = 8910 7867 247 
1 a 8:910 6617 247 
An +1250 000 6:096 9100 
@'=-0,4-6' = +2577 825 6:411 2534 
2,=AF+a’= 0°484 9049 9°685 6566—10 
(Mg 3'2744—10 
b= : = ~768 28854 Check. 
2,70! = +0:000 0701 58455 a’n =1250 181 
— 0:484 9750 9°685 7198 b'n?= —181 
a= 4° 40' 14'-84975 AF=1250 000 


The value of the term 2n,(rb’)? is 2x 10-°, a negligible 
quantity. The formula here suggested will frequently be 
applicable when the differences are small numbers. The 
computation may be made by actual multiplication and 
division ; but in general, logarithmic computation requires 
less labour. In either case, the method just explained is 
about as easy of application as any other, except that in mis- 
cellaneous computations, some care must be exercised to see 
that the term 2n,(rb')? may be omitted. On the other hand, 
if the higher differences need be taken into account, the com- 
plete formula has the important advantage of putting into 
evidence the terms necessary to obtain a required degree of 
accuracy. Thus, although third differences occur in that 
portion of Vega’s ten-place table of the logarithmic sines for 
arguments = or > 2°, the terms in rc’ are negligible and 
the maximum value of 2n,(rb')’, which occurs in the imme- 
diate vicinity of 2°, is five units in the seventh decimal place. 
Consequently, the correction to the simple formula 


—n=n—n,rb! 


is so small that it may be omitted except when extreme 
accuracy is required for arguments a little greater than 2°. 
A problem of frequent occurrence in many branches of 
applied science is the reversion of an empirical formula 
expressed in the form of a power series. Since the relation 
is given in the form (1), it is only necessary to substitute in 
(3) to obtain the value of the argument or independent 
variable. As an illustration, find the temperature of quick- 
silver when its volume is 1°01825409. The empirical formula 
given by Chappuis* expressing the increase of volume of 


z- Landolt=Bornstein: Tabellen, p. 209. 


Interpolation by Means of a Reversed Series. 635 


quicksilver as a function of the temperature, for tempe- 
ratures ranging from 0° to 100° ©., may be written 
Av= [6°259332_1 ]é— [1°470008 19 |t? + [0°059041_19 | 2. 
Arranging the computation as before, 
b’=1:470008_1 To (212704 256 
e’ =0°059041_19 f= 1 oki oto 
AE = Av —8"261360_ 16 wo (ellie, 
a’ =6°259332_10 Ia * DOULA 
ti —t, = 2°002028 t, = 100-4681 
ie oote Aiti= +0°1645 
fot? = —0°6394 
¢t=>=100°:007 C 
Instances may arise where it will be more convenient on 
account of the relative magnitude of the quantities to com- 
pute the coefficients of equation (2) instead of the auxiliary 
quantities 7;, /, f3,.... Written in this form, the inverse 
of Chappuis’ equation is ~ 
t=[3°740668 | Av -+ [ 2°692012 | (Av)? —[5:021349 | (Av). (a) 
Sometimes, however, the inverted series converges so slowly 
as to be useless. Probably the best manner of handling such 
tabulated data or empirical relations is to select or compute 


nm values of the function corresponding to n values of the 
independent variable and substitute in an equation of the form, 


t=A,(Av) + A,(Av)?+ A;(Av)?+....A,(Av)”, 
and then evaluate the n unknowns (A,, A:,...An) by solving 


the n linear equations. Proceeding in this manner and using 
the values 


B=, O. Av= [7°736488_10] 
t=60° Cx Av= [8°038045_10 | 
i — 90° C Av=([8°215155_10| 


there results for the last equation, 
t= [3°740678] Av + [ 2°655719] (Av)? — [5°009834](Av)*. (6) 

The coefficients of (a) differ slightly from those of (6). 
Substituting Av=0:0182541, the values found for ¢ are 

i= 100-007 Ge 

and ig SURO We 

The value ¢=100°-007 C. is the same as found from 
equation (3), the modified form of the general formula for . 


inverse interpolation. Hither of these three methods is suffi- 
ciently accurate for this problem. The last two methods are 


636 Mr. C. HE. Van Orstrand : Jnverse 


to be used in preference to the first when a great number of 
values is to be interpolated. Inthe many instances in which 
only one or two values are to be obtained, the first method is 
the most convenient. Functions determined by any of the 
preceding methods may not of course be very reliable for 
extrapolation. The last formula, for example, is applicable 
between the limits 30° and 90°, but is uncertain for values of 
t<30° and >90°. 


It is desirable to ascertain if some other arrangement of 
the terms of the reverted series would facilitate computation. 
The numerous possible combinations of the quantities are 
best exhibited by again making use of Professor McMahon’s 


expression for the general term of a reverted series. Putting 
b/ c da’ e' 
MS: eS a, bs= — 5 bbz i.e ey 
a a a | 


the (m—1)th term of the reverted series may be written 


eo 
m(m+1)(m+2)...(ptq+..(m—2)) bee 


in which the exponents and subscripts are subject to the 
condition pitgit...=m—2. 

Giving m successive values, say from 3 to 8, we easily find 
the terms containing all possible combinations of 0403... bg. 


They are 
M=—=o Di 

Ab, bs 

D WOaer Osos bs 

6 by by? by by bg by by by 

7D Dio DPbS 3D? Be Davos 140, 04 bs 

8 0° by, 6°03 “byby O26, bby) 070505" Bs? Oe 

Hach termis of weight m—2. There are two combinations 

of the terms of nearly complete symmetry in which functions of 
the same quantities appear in a form suitable for computation. 


Thus, assuming that the given series terminates with the fifth 
power, one arrangement of the terms is 


+6,+67 +b) +5)'+62+.... 1 
“+ bal 1 by be ae | 
+63[1 +0,+0,7+6+....] a 
+b[1+0,+b7+.... | | 
+bs[l+b;+.... ]t..--Og[botb3+...-] J 


a 


and another is 


feet o.-t 0, +..<. 
+l, [b; +bo+b3+ etal ye .| 
+b [b; +b, +634. aad iy) 
ger 10, 0,4 O34... 4... Oo[Oo thst. ..-s| 
The elements have been selected firstly with respect to 
the verticals, and secondly with respect to the diagonals. 
Of other approximately homogeneous and symmetrical com- 


binations there are none. Group (1) has been used in 
equation (3). Group (2) gives the formula 


b! ¢! d' e' 
te ni |) 1 — — yn 2 — n= ] 
1 qa’ 1 | b! 1 b! } b/ 1 
/ 


b! 2 2 , / '\Q : 
+(5)m | +2455 m +65 mP+... +3(F) nt... | 


ENTS tl 
+(G)e[-wengnn 


a re 
== (7) ny [+14+4+. sie sil 


Interpolation by Means of a Reversed Szries. 637 


(2) 


or 
b! TRAE ee b/\4 
NSN, +); 4 ns? +o (5) nN, +73 G) Ny +f ia hie s (6) 


Asin equation (3), symmetry fails in the term 6,?n,°.. The 
functions within the brackets are not dependent upon a single 
quantity as in the other formula, but their evaluation involves 
only a summation of simple functions of the same variables. 
The distribution of algebraic signs is unique. 

Computing again the interpolation interval (n) from the 
data of the first problem, there results : 


! 
ae (3) n?= 800490910 ny = +0°4326284 


ft : b! 
log (7) ny? =8'58290,-10 fh (7) n?= —0-0097313 


a 


I. 2 : b! 2 
log (§;) n2=T16580-1 fe (3 nye = +.0°0004293 


I 
eql PEENS 
les (5) BESGOIEG Yai OK 3 :) m= —0-6000232 
e! b! 4 ; 
log (G)mP=4967-10 ( ) nj? = +0°0000018 


n = > = +0°423305, 
the value obtained by the first method. 


638 Mr. R. D. Kleeman on Different Kinds of y Rays 


The reversed series necessarily involves the quotients, 
powers, and products of the differences, and for this reason is 
difficult of computation. This difficulty is largely overcome 
in the above formulas by taking advantage of the homo- 
geneity of the reverted series, and expressing certain factors, 
the 7/’s, as functions of the same quantities. If the values of 
the j’s, or preferably their logarithms, were tabulated, the 
labour involved in inverse interpolation would be not much 
greater than that of direct interpolation ; for each of the 
terms in (4), (5), or (6) would then be given in the form 
of simple product expressions suitable for logarithmic com- 
putation. This method has the further advantage of being 
straightforward ; and in so far as the reversion of the series 
is concerned, a check is provided without the introduction of 
any new quantities by the substitution of n in equation (1). 

The possibility of using the reversed series as a formula 
for inverse interpolation was suggested to me by Dr. George 
F. Becker of the U.S. Geological Survey, with whom 
I have cooperated in preparing a volume of mathematical 
tables * entitled ‘Tables of Hyperbolic Functions,’ and now 
in course of publication by the Smithsonian Institution of 
Washington, D.C. The formulas here suggested have been 
used toa certain extent in the preparation of these tables. 

Washington, D.C., 

February 1908. 


LIX. On the Different Kinds of y Rays of Radium, and the 
Secondary y Rays which they produce. By R. D. 
Kuireman, B.A., B.Sc., 1851 Exhibition Research Scholar 
of the University of Adelaide, and Research Student of 
Emmanuel College; Emmanuel College, Cambridge ft. 


[* a paper published in the Philosophical Magazinef the 
writer showed that part of the y radiation of radium 
could be approximately divided into three groups of rays. 
Hach of these groups of rays is selectively absorbed by one 


* In this volume are tabulated the natural and logarithmic hyperbolic 
sines, cosines, tangents, and cotangents to five places; the natural and 
logarithmic circular sines and cosines to five places; the ascending and 
descending exponential to seven significant figures with log,, e to seven 
places ; the natural logarithms of the integral numbers from 1 to 1000 to 
five places; the gudermannian to seven places and the corresponding 
anoular equivalents; the anti-gudermannian to hundredths of a second ; 
and other tables of minor importance. The arguments for the most part 
advance by ten-thousandths from 0 to0:1, by thousandths from 0:1 to 3:0, 
and by hundredths from 3:0 to 6:0. 

+ Communicated by Prof. J. J. Thomson, F.R.S. 

J Phil. Mag. Noy. 1907, p. 618. 


of Radium, and Secondary Rays which they produce. 639 — 


of three groups of substances. Thus, if the three groups of 
rays be denoted by A, B, and ©, then the group A is better 
absorbed by one of the three groups of substances than either 
of the groups B and C, and the group B better absorbed by 
one of the other groups of substances than either of the 
groups A and C, and lastly the group of rays C is better 
absorbed by the remaining group of substances than either 
of the groups of rays A and B. To give an illustration, the 
ratio of the secondary cathode radiation from lead to that 
from zine is decreased if the y rays are passed through a 
thick screen of lead before falling on the radiators, showing 
that the rays producing the larger part of the radiation from 
zinc are decreased in a less proportion by the lead screen 
than the rays producing the larger part of the radiation from 
lead. 

It became of interest, therefore, to investigate the pro- 
perties of the secondary y rays from different substances 
exposed to the y rays of radium, and to compare their pro- 
perties with those of the primary rays. Moreover, it was 
thought that such an investigation might lead to an indirect 
confirmation of the above result. The experiments described 
in this paper were accordingly undertaken. 

It appeared from the experiments described in the paper 
mentioned, that the amount of secondary y radiation, as 
measured by its ionizing power, is small in comparison with 
that of the incident primary radiation. It was necessary, 
therefore, to design a sensitive apparatus for measuring the 
secondary radiation. 


A vertical section of the apparatus used is shown in fig. 1. 
A is a quantity of radium (about 30 mgrs.) in a glass tube 
surrounded by lead sheeting 2 mm. thick to cut off the 


640 Mr. R. D. Kleeman on Digerent Kinds of y Rays 


B rays. The y rays fell in part on the radiator B, which in 
consequence emitted secondary 8 and secondary y rays. 
Some of the secondary rays penetrated into the ionization 
chamber D, ionizing the air that it contained. The selective 
absorption of the secondary y rays was investigated by 
placing successively screens of different substances at C, and 
measuring in each case the ionization in the chamber D, the 
ionization in the chamber being produced by the secondary 
rays not absorbed by the screen. By using radiators and 
screens of different substances the properties of the secondary 
radiation from different substances could thus be investigated. 
The screen served also to screen the ionization-chamber from 
the secondary @ rays emitted by the radiator B. This was 
necessary since the ionization-chamber was made of thin 
sheet iron which would have been penetrated to some extent 
by the 8 rays. The object of having the ionization-chamber 
of thin material was to introduce as little absorption of the 
secondary y rays as possible other than that due to the screen. 
The screen C and the ionization-chamber were screened from 
the action of the primary rays by the lead block E 16°5 cm. 
long, 9°5 em. broad, and 5°5 em. thick. 

But since the y rays possess great penetrating power, an 
object can be screened approximately only from the action 
of y rays, and the leak in the ionization-chamber was there- 
fore caused in part by the primary y rays which penetrated 
the lead block. It was therefore found convenient to partially 
compensate the leak in the chamber by a leak in the opposite 
direction in another chamber F. The air in this chamber 
was ionized by a layer of uranium oxide, and the chamber 
placed at a distance of one metre from the radiator B and 
screened from its secondary y rays by a lead plate 1°5 cm. 
thick. The leak in the chamber was therefcre not affected 
when the radiator was replaced by another, and was there- 
fore constant. The chamber F was connected to a negative 
potential of 200 volts, while the chamber D was connected. 
to a positive potential of the same magnitude. 

On the right-hand side of fig. 2 a horizontal section of 
part of the apparatus is shown. It will be seen that the 
radiator was placed so that it made an angle of about 45° with 
the screen. This was done because the amount of radiation 
received from the radium was greatest in this position. 

On the left-hand side of the figure the arrangement is 
shown by means of which the radiator or screen was kept in 
a fixed position. It consisted’ in each case of three fixed 
blocks of wood provided with slits whose width was slightly 
greater than the thickness of the radiator or screen. Both 
radiator and screen were kept in a fixed position by being 


of Radium, and Secondary Rays which they produce. 641 


held to one of the sides of each slit by means of a wedge. 
Therefore, when one of them was removed and again replaced, 
it always occupied its previous position. 


Fig. 2. 


In order to obtain as large a leak as possible, the radiator 
B and the ionization-chamber D were made as large as was 
possible without causing any inconvenience. The leak in D, 
it will be observed, is approximately proportional to the 
product of the capacity of the ionization-chamber and the 
area of one side of the radiator, if the thickness of the 
radiator is kept constant. The length and breadth of each of 
the plates actually used as radiators were 28 and 24 cm. 
respectively. The ionization-chamber was of a cylindrical 
shape, its diameter was 18 cm., and its height approximately 
20 cm. 

The leak in tbe chamber D, as has already been pointed 
out, is partly due to the direct action of the y rays pene- 
trating the lead block E. If we denote by N the leak per 
second due to these primary rays, and by M the leak due to 
the secondary rays, then the leak when the radiator is in 
position is given by (N+M). By observing the leak with 
no radiator we obtain the value of N. The value of M is 
then obtained by subtracting N from (N+ M). 

In practice, the leak N or (N+M) was obtained by ob- 
serving the time taken by the electrometer-needle to reach 
a given defiexion. The deflexion for a given time could 
then be easily calculated. In this manner, the error due to 
the change of capacity of the electrometer with deflexion of 
needle was obviated. It will be observed, however, that if 
N is much greater than M (in these experiments this was 
the case), the times required to obtain a given deflexion 
when observing the leaks N and (N+M) would differ from 
one another by much less than the time of either leak. The 
possible error that is attached to the value of this difference 
is therefore large; and the error that is attached to the value 
of M, which depends on this difference, is therefore also 
large. This defect in the method was obviated by the use 
of the uranium chamber. The leak due to the primary 


642 Mr. R. D. Kleeman on Different Kinds of y Rays 


rays could be wholly or partially compensated by a leak in 
the opposite direction in the uranium chamber, and the 
difference in the times of leak thereby rendered as large as 
convenient. 

The compensation was, however, rendered only partial, 
this affording a method of eliminating the loss of leak due 
to leakage over the insulation, and leakage in the screening 
tubing due to imperfect shielding from the y rays. “Thus, if 
the times required to obtain a given deflexion with the leaks 
N and (N+M) be measured, the leaks for a given time (not 
corrected for loss of leak) can be calculated. Let these 
calculated leaks be denoted by N; and (N,+M,). Then the 
true value of M unaffected by leakage is given by sub- 
tracting N, from (N,+M,), as will now be shown. When 
the system is charged, the loss of charge at any instant is 
approximately proportional to the deflexion of the electro- 
meter-needle. Therefore the loss of leak for a time dé is 
DA .dé, where D is the deflexion of the electrometer-needle 


and Ais a constant. If n denote the leak per second if - 


there were no loss of leak, then, since the loss of leak is 
small in comparison with the leak, D=tn approximately. 
If the leak be taken for a time 4,, the loss of leak is approxi- 


mately equal to is inA .dt, that is to-¢’?An/2. If ¢, denote 


the time required to obtain some given deflexion D,, t;=D,/n 
approximately. The loss of leak for the time ¢, is therefore 
DA/2n. Therefore, when the leak is calculated for a given 
time t., the value obtained is less than the correct value, the 
Di ANG 3 * 
difference being equal to fil ae that is AD,t,/2. Since 
a 1 
D, and ¢, are taken the same in all the measurements, and A 
is a constant, the difference between the correct leak and the 
calculated leak is the same in each case. Consequently, 


when the calculated leak obtained when no radiator is used 


is subtracted from the calculated leak obtained when a 
radiator is used, the correct value of M unaffected by leakage 
is obtained. This method of eliminating the loss of leak was 
especially advantageous in these experiments since the loss 
of leak could not have been easily obtained directly, and it 
could not be neglected since its magnitude was not negligible 
‘in comparison with that of the leak. 

A set of measurements will now be described. <A screen 
of some substance was placed in position, and the secondary 
radiation from a set of substances in terms of that from a 
standard substance determined. . The screen was then re- 
placed by one of different material, and the whole process 


of Radium, and Secondary Rays which they produce. 643 


repeated, and so on. . The measurements for determining 
the radiation from a substance in terms of that from a 


standard substance were carried out in the following 


manner :—TI'wo readings were taken without any radiator. 
Two readings were then taken with the radiator under in- 
vestigation placed in position. Lastly, two readings were 
taken with the standard radiator. This process was repeated 
at least seven or eight times. The mean of the first set of 
readings was then substracted from each mean of the two 
other sets of readings, the result giving the relative secondary 
radiation from the substance under investigation and the 
standard radiator. | 

The following numbers will give an idea of the relative 
magnitude of the leaks measured. A deflexion of 1800 
scale-divisions in 4 minutes was obtained when the screen 
was of iron and ‘6 cm. thick, and no radiator was used. 
Deflexions of the same amount were obtained with a lead 
radiator 1°8 mm. thick in 3 minutes, and with a zinc radiator 
3mm. thick in 2 minutes 8 seconds. An electrometer of 
the Dolezalek type, fitted up with a telescope and scale, was 
used in these experiments. Since the leaks were all very 
small the experiments were very tedious, the determination 
of the amount of secondary radiation from a substance in 
terms of that from a standard substance requiring about 
three hours. 

Table I. contains the results of some measurements carried 
out in the manner described. The first vertical column gives 
the nature of the radiators and their thickness. The first 
horizontal column gives the nature and thickness of the 
screens used. The numbers in each vertical column give 
the relative amounts of secondary radiation from the sub- 
stances given in the first column, when the screen given at 
the top of the column was used. The radiation from lead, 
the standard substance, has been put equal to 100 for each 
screen. The relative radiations obtained with the same 
screen depend, it will be observed, on the selective partial 
absorption of the radiations by the screen, and the nature 
of the secondary radiation emitted by each substance, and 
on the relative masses of the substances used. Since some of 
the radiation generated in a substance is absorbed by the 
substance, the nature of the radiation emitted by a given 
mass of substance must depend somewhat on its shape and 
size. ‘The shape of each radiator was therefore made the 
same as that of the standard substance with which it was 
compared. The radiation from mercury was compared with 
that from lead by placing a quantity of mercury in a thin 


644 Mr. R. D. Kleeman on Different Kinds of y Rays 


celluloid dish 14 cm. long and 11 em. broad, and comparing 
the radiation from the mercury with that from a lead plate 
having the same length and breadth as the dish. The depth 
of the mercury in the dish was 10 mm. and the thickness of 
the lead platewas 6mm. In these measurements the screens 
were placed in a horizontal position underneath the ioniza- 
tion-chamber, and the mercury dish or lead plate placed in a 
horizontal position underneath the screen. The radiation 


TABLE I. 
ae) | | 
Me ee ai Fe. fy ready C. bias ! 
eh idleunce 1:8 mm. | 6°5 mm.|} 6°5 mm.!| 9 mm.) 7°5 mm.| 20 mm. pein, 
un... see 100 100 100 | 100 | 100 100 100 
2 eee 641 | 641 | 676 | 685 | 670 | 676 | G41 
=e aes a ee ees = a | St 
Yn 3 mm....| 156 268 953 | 225 | 214 186 186 
Gar 6 MMe 330 303. | 278 | 257 233 923 
Hoes 4 ty ee Si 304 | 264 | 244 214 230 
Sa SO 260 24g) fo. 223 186 187 
Ni ee 267 273 | 239 ag 209 208 
: —- > —— ~—-—_ <= — | — 
C 20 mm....| 167 343 377 | 347_| 334 274 284 
TRL 
Rafiatant Bhs Zi, Fe. S. vr | C. Aye; 
ce 641 37:3 395 | 44-4 467 53:7 53:8 
He ...... 41-1 939 | 267 30:4 31:3 36°3 345 
gn...) 100 | 190 | 100° 1° too |} 100-"| ‘too 9 | ane 
Cn) 7 123 | 120 22") | $20 125 120 
eee 110 116 | 120 117 114 115 127 
tbo Gimme as The 97 | 984 me 104. 998 | 101 
Vie: ie 96°8 996 108 106 “i 112 112 
Cn ee 107-2 | 18 140 | 154 | 1562 | 147 | 158 


| | 


from the lead plate was taken equal to 100. The radiation 
from each of the other radiators was compared with that 
from a lead plate 1°8 mm. thick, and of the same length and 
breadth as these radiators, each of which was 28 cm. long 
and 24 cm. broad. The radiation from lead was taken, as 
before, equal to 100. | 

Table II. gives the same observations as Table I. except 


of Radium, and Secondary Rays which they produce. 645 


that the value of the radiation from zinc has been put equal 
to 100 for each screen, and the values of the radiations from 
the other substances changed correspondingly. 

It will be seen from Table I. that when Pb is put equal to 
100, the values of the radiation from Pb with the different 


screens (given in the horizontal column containing Hg) 


become approximately equal to one another. From Table II. 
it will be seen that when zinc is put equal to 100, the values 
of the radiations (with the different screens) from each of 
the substances Al, S, Fe, Cu, Zn, become approximately 
equal to one another. But the values of the radiations from 
C do not become equal to one another in either case. Thus 
the substances appear to fall roughly into three groups. 
The interpretation of this is that each member of a group 
radiates approximately the same kind of secondary rays, but 
the radiation from a group is as a whole different from that 
from some other group. Thus, if each of the substances of 
a group radiates the same kind of rays,a screen will diminish 
the radiation in the same proportion in each case whatever 


the selective absorbing nature of the screen. The relative 


radiations will therefore not be altered by a screen, and 
therefore when the radiations with the different screens from 
one of the substances of the group is put equal to 100, the 
values of the radiations from each of the other substances of 
the group will become approximately equal to one another. 
These considerations also show that the substances which do 
not belong to a group radiate rays which differ as a whole: 


- from the rays radiated by the group. 


For convenience of reference we wiil denote by Group I. 
the substances Pb and Hg, including the substances which 
radiate the same kind of rays as Pb or Hg, if any exist. 
Similarly we will denote by Group II. the substances which 
radiate the same kind of rays as Zn, Cu, Fe, 8, and Al; and 
by Group ILI. the substances which radiate the same kind 
of rays as OC. 

The problem that presented itself next was to determine 
if there is any relation between the rays from the various 
groups of substances. This was done by the following method. 
‘The radiations from two substances A and B belonging to 
two different groups were compared—when the screen used 
was thin, and when it was much thicker. Now, if one of 
the substances, say A, radiates more rays which are well 
absorbed by the screen, than the other substance B, the ratio 
of radiation from A to that from B will decrease when the 


thickness of the screen is increased. In that case, we can 


Phil. Mag. 8. 6. Vol. 15. No. 89. May 1908. 2X 


646 Mr. R. D. Kleeman on Different Kinds of y Rays 


say that the substance A at least radiates rays which are well 
absorbed by the substance of the screen. In this manner, 
substances belonging to different groups were compared with 
one another, using successively screens belonging to different 
groups. It was found that when there is an indication that 
a radiator radiates more rays which are well absorbed by the 
screen, than some other radiator, the effect may be made 
more pronounced by increasing the thickness of either 
radiator or of both. The reason for this will appear when 
the results of this investigation are discussed. It is obvious 
that the effect can also be increased by increasing the ratio 
of the thick screen to the thin one. For our present purpose: 
it is only necessary to make sure that there is a change in 
the ratios with increase of thickness of screen. 

Table III. gives the results of some experiments carried 
out in the manner described. Let us first consider the 
resuits obtained when the substance of the screen used 
belonged to Group I. The first and third vertical columns. 
of this part of the-table give the radiators and their thick- 
nesses. The column on the right of each of these columns 
gives the values of the secondary radiations corresponding to: 
two different thicknesses of the screen, the nature of the 
screen being given at the top of the column. The radiation 
from lead has in each case been put equal to 100. It will 
be seen that there is a decrease in the ratio of the radiation 
from zinc to that from lead with an increase of thickness of 
the lead screen. Thus zinc radiates a greater proportion of 
rays which are well absorbed by lead than lead itself. The 
reason for this, as will afterwards be seen, is that the rays 
generated in a radiating plate, of the kind best absorbed by 
the plate, are almost entirely absorbed by the plate. Further, 
since the substances which we have denoted by Group II., 
of which zinc is one, radiate approximately the same kind of 
rays, and the substances of Group I., of which lead is one, 
also radiate the same kind of rays, it follows that the sub- 
stances of Group II. radiate a greater proportion of rays 
well absorbed by lead than the substances of Group I. If 
the substances of a group resemble one another in their 
absorption properties, this would be true for all the substances 
of Group I. besides lead. It will presently be shown in this 
connexion that the substances of Group II. possess the same 
absorption properties, and we may therefore take it for 
granted that this is true for all groups. Moreover, it will 
be seen, when the primary and secondary rays are compared, 
that the substances of each group possess the same absorption 
properties for the primary rays. 


} of Radium, and Secondary Rays which they produce. 647 
| TaBLe III. 


| Screen belonging to Group I. 
Radiator ) Pb screen. Radiator | Pb screen. 
and its and its | 
thickness. thickness. | 
lmm., 4mm. 1mm. |} 4mm. 
eo f-S8.mwmi. ...... 100 1? Eb VS mm. ....... | 100 100 


| 
Ress fied pails. Sate Ge | 240 | 178 
| 


Screens belonging to Group II. 


| Radiator Fe screen. Radiator | Fe screen. 
and its and its 
thickness. thickness. | 
13mm. 4:3 mm. |1°3 mm. | 43 mm. | 
Pees | 100 | 100 | 65imm........ 100 | 100 
one oid... basal) head pene’) ss 355 | 329 
Al screen. | | Cu screen. 
| Scie Pl | 
38 mm. 75 mm. /16 mm./|4°6 mm. 
Pb 1:8 mm. ...... 100 | 100 | Pb18mm. ...... 100 100 
| | | | 
Bee ceee Gil tins. 190 722% | B65, », 5570 see. | 329 35d 


Screen belonging to Group III. 


Radiator | C screen. Radiator C screen. 
) aud its and its 

thickness. thickness. 
| | 4mm. | 24 mm. 4mm. | 24 mm. | 
a ee —|—_—| —_——_ 
Phyo) min, ...-..... | SO. ; 100 |} 4m 65 mm. ...... — 100 100 
) / 
BAD) 1) je cebne.- | 230 237 C 20 a eee 85 92 

C screen. 


4mm. | 24 mm. 


7 ge ee 100 100 


648 Mr. R. D. Kleeman on Different Kinds of y Rays 


It will be further seen in the table that the ratio of the 
radiation from lead to that from carbon is also increased 
with an increase of the thickness of the lead screen. Thus 


carbon radiates a greater proportion of rays which are well’ 


absorbed by lead than lead itself. It follows, therefore, 
reasoning in the same manner as before, that the substances 
of Group III. also radiate a greater proportion of rays which 
are well absorbed by a substance of Group I., than the 
substances of Group I. themselves. 

Let us next consider the results obtained when the substance 
of the screen used belonged to Group IJ. (The values for 
lead and zine are put equal to 100 in each case.) It will be 
seen that the ratio of the radiation from lead to that from 
zinc is decreased with an increase in the thickness of the 
iron screen. Thus lead, a substance of Group L., radiates a 
greater proportion of rays well absorbed by iron, than zine, 
a substance of Group Il. It follows, therefore, that the 
substances of Group I. radiate a greater proportion of rays 
well absorbed by iron, a substance of Group II., than the 
substances of Group Il. This was proved by experiment to 
be true for some other substances of Group II. besides iron. 
Thus, it was found (see table) that there is a decrease in the 
ratio of the radiation from lead to that from zinc, substances 
of Group L. and II., with an increase in the thickness of a 
screen of copper or aluminium, substances of Group II. 
From this it tollows that the foregoing result is also true for 
aluminium and copper, and it may therefore be taken to be 
true for all the substances of Group II. 

Further, the ratio of the radiation from carbon to that 
from zine is decreased, as will be seen in the table, with an 
increase in the thickness of an iron screen. It follows, 
therefore, that the substances of Group III. also radiate a 
greater proportion of rays well absorbed by a substance of 
Group II., than the substances of Group II. themselves. 
~-Jt remains to consider the results obtained when the sub- 
stance of the screen used belonged to Group III. (The 
values for lead and zine are put equal to 100 in each ease.) 
It will be seen that the ratio of the radiation from lead (a 
substance of Group I.) to the radiation from carbon (a sub- 
stance of Group III.) is decreased with an increase in the 
thickness of the carbon screen. The decrease, it will be 
observed, is greater for the thicker of the two lead radiators 
used. Further, the ratio of the radiation from zine (a sub- 
stance of Group II.) to the radiation from carbon, is also 
decreased with an increase of the thickness of the carbon 
screen. It follows, therefore, in the same manner as before, 


NE ——eE——eeeeeEEO——eE——EE—EE _ 


of Radium, and Secondary Rays which they produce. 649 


that the substances of both Group I. and Group II. radiate a 
greater proportion of rays which are well absorbed by a 
‘substance of Group III., than the substances of Group III. 
themselves. 

Thus we have the general result that the substances of any 
two groups radiate a greater proportion of rays which are 
well absorbed by the substances of the remaining group, than 
ihe substances of this group radiate themselves. 1t will now 
be shown that it follows from this result that each group of 
substances radiates two groups of rays, each of which is best 
absorbed by one of the remaining groups of substances. 
Let us begin with considering the radiation from the sub- 
stances of Group I. We are given that Group I. and 
Group II. radiate a greater proportion of rays which are 
well absorbed by Group IIT., than Group III. itself; and also 
that Group I. and Group III. radiate a greater proportion of 
rays which are well absorbed by Group II., than Group II. 
itself. Hence Group I. radiates rays which are well absorbed 
by Group III., and rays which are well absorbed by Group II. 
These two groups of raysare different, since the former group 
is radiated in a greater proportion by Group II. than by 
Group II1., and the latter group is radiated in a greater 
proportion by Group III. than by Group II. In a similar 
manner, the above statement may be proved for the other 
groups of substances. 

Thus the substances of Group I. radiate two groups of 
rays, one of which is best absorbed by the substances of 
Group II. and the other best absorbed by the substances 
of Group III., and the substances of Group II. radiate two 
groups of rays, one of which is best absorbed by the sub- 
stances of Group III. and the other best absorbed by the 
substances of Group I. It follows from this that the sub- 
stances of Groups I. and II. radiate one group of rays in 
common, the group being best absorbed by the substances of 
Group III. In the same manner it can be shown that any 
other pair of groups of substances radiate a group of rays in 
common. Since there are three groups of substances, each 
of which radiates two groups of rays, there are at least three 
different groups of secondary y rays. But since the sub- 
stances in Tables I. and IJ. fall approximately only into 
three groups, it follows that the rays of a group consist of 
different kinds of rays, but which ditfer less from one another 
than from the rays radiated by some other group. We will 
return to this point about the grouping of the rays later on. 
It will be convenient to distinguish these groups of rays 
from one another by names. Thus the secondary rays which 


650 Mr. R. D. Kleeman on Different Kinds of y Rays 


are best absorbed by the substances of Group I. will be 
called Group I. rays, and a similar name given to each of 
the other groups of rays corresponding to the group of sub- 
stances by which the rays are best absorbed. According to 
this notation the substances of Group I. radiate secondary 
rays of Group II. and Group III., and so on. 

We have seen (Table IIJ.) that the increase in the ratio of 
the radiations from two substances with an increase of thick- 
ness of screen, becomes more marked when the thickness of 
one of the radiators (or of beth) is increased. This is due 
to secondary rays of all three groups being generated in a 
substance, but one of the groups being more easily absorbed 
by the substance than either of the other groups. The 
radiation from a substance would then consist principally of 
two groups of rays, and a small proportion of rays of the 
remaining group; and the difference in the nature of the 
radiations from two substances would therefore become more 
marked the greater their thickness. 

The changes observed in the ratios of the radiations with 
the thickness of screen are small, but they can be accurately 
measured. The reason for their smallness will now be ap- 
parent from the deductions made from the results. Since 
two substances of two different groups radiate (at least) one 
group of rays in common, this group of rays will mask to 
some extent the relative change in the radiations produced 
by an increase of the thickness of the screen. Moreover, the 
fact that all the groups of rays are absorbed more or less by 
a substance will also have the effect of making the change 
small. 

It will be profitable now to compare the results so far 
obtained with the result quoted in the beginning of tbis 
paper from a previous paper, namely, that the primary 
y rays of radium consist in part of three groups of rays. In 
the paper mentioned it was shown that one of the groups of 
rays is best absorbed by lead, mercury, and bismuth, and one 
group best absorbed by tin, zinc, copper, nickel, iron, sulphur, 
and aluminium, and the remaining group best absorbed by 
carbon. It will be seen that each of these three groups of 
substances includes one of the three groups of substances 
defined in this paper. Itappears, therefore, that the primary 
y vays of radium consist of at least three groups which 
possess the same property of selective absorption as the three 
groups of secondary rays discussed in this paper. 

It became of interest next to investigate if the secondary 
y rays possess the same penetrating power as the primary 
rays. The measurements for determining the coefficients of 


of Radium, and Secondary Rays which they produce. 651 


absorption of the secondary rays from different substances 


were carried out in the following manner :—Readings were 
taken, using no radiator, with a thin screen of the substance 
whose coefficient of absorption was to be found placed at C 
(see fig. 1), and then with a thicker screen of the same sub- 


stance in place of the thin one. The radiator, which was 


28 cm. long and 12 em. broad in each case, was then placed 
at B, and “the readings repeated. Hach of the readings 


feken without the radiator was then subtracted from ihe 


corresponding reading taken with the radiator, the result 


giving the relative radiations corresponding to the two 


different thicknesses of the screen. The radiating plate was 
made as small as was possible without reducing the leak to 


an inconveniently small quantity. This was done in order 


that the maximum angle which the secondary rays made 
with the normal to the screen might be as small as possible, 
in the actual case the angle was about 45°. If X denote the 


coefficient of absor ption, we have then I,=I, e-™, and 


I,= 1) e~*, where I, is the leak in the chamber when the 
thickness of the screen is x,, and I, the leak when the 
thickness of the screen is &, and I, the leak that would be 
obtained without a screen. The elimination of I) gives 


an equation from which A is obtained. The value of the 


coefficient of absorption is somewhat influenced by the fact 
that the secondary rays traversed the screen at various angles. 
But it was found on calculation that this did not introduce 


any very appreciable error. Since the nature of the radiation 


from a substance depends to some extent on its thickness, 
the coefficient of absorption depends to some extent on the 
thickness of the radiator. Thus, in the case of a very ihin 
radiator, the radiation would consist of the three groups of 
rays in almost equal amount; but as the thickness of the 
radiator is increased, the proportion of one of the groups is 
much decreased on account of being more easily absorbed 
by the plate than the other groups. It was on account of 


these considerations that the coefficient of absorption was not 


determined with greater accuracy than that given by two 
different thicknesses of the screen. 

Table 1V. gives the results obiained with the secondary 
rays from the radiators lead, zine, and carbon, each of which 
belongs to one of the three groups of substances defined in 
this paper. The first column contains the substances whose 
coefficients of absorption were found. The coefficients of 
absorption of a substance corresponding to the different 
secondary rays radiated by the substances given in the first 
horizontal row, are given in the same horizontal row which 


652 Mr. R. D. Kleeman on Different Kinds of y Rays 


TABLE LY. 
Abeorbiig | eee sa eee EON aa 
(of Group IT.).| (of Group I.). | (of Group III.)| ™9* 
| | 
Pb (Group 1) .. A\=484 A=1°85 N= X= ae 
‘Fe | d= 365 r=628 | . sae | A=-28 
Cu , Group IL.).| A=-794 XE -BG4 WY OTR a | A=B1 | 
2m) ee, eee ee mee = 28 | 
Ne (Group Tene ae ie CEO ee. | 


contains the substance. It will be seen that on the whole 
the coefficients of absorption of the secondary rays are much 
greater than those of the primary rays. The secondary rays: 
we are dealing with, it will be observed, consist in each case 
of at least two different groups. Thus it follows that the 
three groups of primary rays of radium with which we are 
dealing in this paper are transformed by a substance into 
three groups of secondary rays, which possess the same 
property of selective absorption as the primary rays, but 
which are, on the whole, softer or more easily absorbed than 
the primaryrays. This result shows that we must distinguish 
between selective absorption and softness of y rays. The 
distinction is best illustrated as follows :—Suppose we were 
able to isolate a group of primary y rays, and then measure 
its absorption by three different substances, each of which 
belongs to one of the groups defined in this paper. We 
would then find that each of the substances absorbs the 
group of rays to some extent, but one substance to a greater 
extent than either of the remaining two substances. In the 
case of the corresponding group of secondary rays, we would 
find that each of the substances absorbs the group of rays to 
a greater extent than the primary group, and that the same: 
substance as before absorbs to a greater extent the group 
than either of the remaining twa substances. 

It may be pointed out that the results contained in 
Table IV. bear out some of the conclusions drawn from the 
results in Table III. Thus, the secondary radiation from 
lead, a substance of Group I., is better absorbed by iron and 
copper, substances of Group II., than the secondary radiation 
from zinc, a substance of the latter group. Also, the 
secondary radiation from carbon, a substance of Group IIL., 


a 
‘ 7 * 
aan 
3 
ee 


is better absorbed by iron,a substance of Group II., than the 
radiation from zinc, a substance of the latter group, and so 
on. This shows, as before, that the substances of a group | 
radiate those rays in least amount which they best absorb. 
It should be observed that we can only compare the coefti- 
cients of absorption for the same substance. 

We have seen that the three groups of primary rays possess 
the same properties of selective absorption as the three 
groups of secondary rays. But it will be observed that 
it does not follow from this that a primary group of rays 
gives rise to the corresponding group of secondary rays, 
although this is very probable. Some further experiments 
were therefore made to demonstrate this. The experiments 
were carried out in the following manner. The coefficient 
of absorption of the secondary rays from a substance of one 
of the groups—say Group II].—was determined for a 
substance belonging to a different group—say Group II. 
A thick screen of a substance belonging to Group Il. was 
then placed over the radium and the coefficient of absorption 
again determined ; the secondary rays being now produced 
by the primary rays which penetrated the thick screen 
placed over the radium. Now, since the radiator belongs to 
Group III., it radiates rays of Group II. and Group L.; 
and since the substance whose coefficient of absorption is 
measured belongs to Group II., the part of the secondary 
radiation belonging to Group II. is absorbed to a greater 
extent by the substance than the remaining part. Therefore, 
if the secondary rays of Group II. are produced by the 
primary rays of Group II., the coefficient of absorption should 
decrease when the primary rays are passed through a screen 
of asubstance of Group Il. In the same manner, obviously, it 
may be investigated whether there is a relation between the 
primary and secondary group for other groups of rays. The 
magnitude of the decrease of the coefficient of absorption will 
be influenced by a large number of things. Thus it depends 
on the relative strength of the group of primary rays under 
investigation to that of the remaining groups. It also 
depends on the extent the screen over the radium absorbs 
the other groups besides the one which is best absorbed, and 
so on. 

In Table V. some experimental results are given. The 
substances whose coefficient of absorption was determined 
under the different conditions, are given in the first vertical 
column, and their coefficients of absorption in adjacent hori- 
zontal columns. The horizontal column above the column 
containing the coefficients of absorption for a substance gives 


of Radium, and Secondary Rays which they produce. 653 


654 Mr. R. D. Kleeman on Different Kinds of y Rays 


TABLE V. 
Absorbing Secondary rays from a carbon plaie 


substance. 2 em. thick. (Rays of Groups I. & IT.) 


| 
No screen over Screen of lead 1:2 em, 


| radium. thick over radium. 
Pb i ag 
(of Group I). A=75 A=5'17 
No sereen over Screen of zine 2°] em. 
radium. | thick over radium. 
Fe | ee 
(of Group IT.). A=2'72 A="80 . 
| Secondary rays from a zine plate 
65 em. thick. (Rays of Groups I. & TII.) | 
No sereen over (|Screen of lead 1:2 em. 
radium. thick over radium, 
Pb 
(of Group I.). A=4°82 A=45 


the conditions under which each coefficient was determined. 
It will be seen that the coefficient of absorption of each 
substance is decreased with a screen of the same substance 
placed over the radium. The results, therefore, show that 
the primary rays of Group I. and Group II. are transformed 
into corresponding groups of secondary rays. We may take 
it for granted, therefore, that this is the case with each 
primary group of rays. | 

Since the secondary y rays consist of at least three groups, 
and, as we have just seen, a group of secondary rays is 
caused by the transformation of a primary group possessing 
similar properties, it follows that the primary rays of radium. 
consist of at least three groups, each of which 1s selectively 
absorbed by one of three groups of substances. We have 
thus established in an entirely different manner a result 
obtained in a previous paper. 

Since the substances in Tables I. and II. fall approximately 
only into groups, the grouping of part of the primary and 
secondary radiation into three groups is therefore only a 
rough one. It appears very probable, therefore, that there 
are other primary and secondary rays which are not included 
in any one of these groups. Some of these rays or groups 
of rays would become apparent by investigating the secondary 
radiation from other substances than those dealt with in this 


of Radium, and Secondary Rays which they produce. 655 


paper. It may be pointed out in this connexion that the 
selective absorption properties of the elements probably do 
not change gradually with the atomic weight. Thus, 
aluminium and zinc possess approximately the same property 
of selective absorption, while that of carbon is different, 
though the ratio of the atomic weight of zine to that of 
aluminium is about the same as the ratio of the atomic 
weight of aluminium to that of carbon. It will be observed 
that the existence of other rays or groups of rays than those 
dealt with in this paper, does not invalidate any of the 
deductions made. The existence of other rays or groups of 
rays when two groups of rays are compared, has the effect of 
making the properties of the latter appear less marked. 
Thus, it is evident that the larger the number of different 
kinds of rays in a beam of rays, the smaller will be the 
decrease in the strength of the beam when a screen which 
absorbs best one kind of rays is placed in the path of the 
beam. This indicates some measurements from which a 
rough indication of the distribution of the y rays amongst 
the different kinds of rays may be obtained, and which will 
now be described.. The amount of secondary radiation from 
a zinc plate which penetrated a plate of lead 1:8 mm. thick 
was found to be equal to 795, in arbitrary units ; and the 
amount which penetrated a zinc plate 3 mm. thick (of about 
the same mass as the lead plate) was found to be equal to 
2258. Let the total radiation from the zine plate be denoted 
by (A+B), where A denotes the radiation of Group I. and 
B denotes the remaining radiation. Suppose that the lead 
plate (a substance of Group I.) allows the fraction wu of the 
radiation A to penetrate the plate. Since the lead and zine 
plates are approximately of equal mass, we may suppose as a 
first approximation that the same fraction w of the radiation 
B penetrates each plate, where w, it will be observed, must 
be larger than vu. The amount of radiation measured when 
the zine and lead plates were successively used as screen was 
therefore (wA+wB) and (uA +wB) respectively, and there- 
fore in this experiment : 

uA+wB=795 and wA+uB = 2258. 


From these two equations we have 


wA—uA = 1468, 
and therefore 
wA—uA _ 1463 


wB+uA 795° 
Now, if w=0, that is, if all the radiation of Group I. is 


absorbed by the lead plate, the right-hand side of the last 
equation gives the ratio of the radiation of Group I. to the 


656 Mr. R. D. Kleeman on Different Kinds of y Rays 


remaining radiation. If only part of the radiation of Group I. 
is absorbed, as is actually the case, the ratio A/B is greater 
than this value. For it will be easily seen that if wis increased 
and A kept constant, B must decrease in order that the value 
of the right-hand side of the equation may remain the same. 
The value of B on the above assumption is therefore smaller 
than that of A. There is therefore little room for the radia- 
tion B to consist of a number of groups of rays of the same 
intensity as A, or the secondary rays from zine to consist of 
a number of groups (of intensity A) much greater than two. 
It seems therefore that a large part of the primary and 
secondary radiation is represented in each case by the three 
groups of rays defined in this paper. 

The relative total amount of y radiation produced per 
unit mass in a substance, a quantity which will now be 
discussed, cannot be obtained, since each radiating substance 
absorbs to a large extent one of the groups of rays generated 
in the substance. Moreover, the proportion of y rays 
absorbed by the screen which is used to cut off the 6 rays, is 
not the same with each radiating substance but depends on 
the nature of the substance. But since the substances 
belonging to any one of the groups defined in this paper 
radiate the same two groups of rays, the values of the 
radiations obtained for a group of substances give the 
relative amounts of radiation of these two groups of rays 
produced per c.c.,if the proportion of this radiation absorbed 
by the radiator is the same with each. It will be seen that 
in this case the screen absorbs the same proportion of the 
radiation from each substance, and therefore does not alter 
the values of the radiations relatively. Table VI. gives the 


Tape vt: 
| M F | Radiation Radiation 
Radiator, | ae oe | from per 
«he A | radiator. | unit mass. 
. { 
Bb keine | 1652 ems. >| 53°'8 ) a 
“AGM ais SAME RA oa: 1622 gms. | 100 100 
Cute eee er ON 121 | 1087 
| | : Hl 
Fo [2 | Aes: 1LY, 104°3 
Si... eee eee 1505 tee 99°3 23:3 
| 
::\ ree yo! 1455 _,, 105°7 1176 
MRED aa tk 2 2076 gms. 153 119-4 


=i 
* 
Ao 
’ . 


relative amounts of radiation of a number of substances. 
The first two columns in the table give the nature and mass 
of the radiator, and the third column gives the values of the 
secondary radiation. The fourth column gives the radiation 
per unit mass calculated on the assumption that the amount 
of radiation absorbed by each radiator is proportional to its 
mass. The values for carbon and lead are those obtained 
with the cardboard screen, while the values for the substances 
of Group II. are the mean of the values given in Table II. ; 
they differ little from those obtained with the cardboard 
screen. The values of the radiation per c.c. of tne substances 
of Group II. only in the table can be compared with one 
another, as just explained. They give approximately the 
relative amounts of radiation of Groups II. and ILI. produced 
per c.c. in each substance. It will be seen that the values do 
not differ much from one another. This is a result that 
would be expected if the mass of an atom is proportional to 
the number of electrons it contains. The values obtained 
would probably be more nearly equal to one another if the 
masses of the radiators used had been equal to one another. 
The difference in the calculated amount of radiation per unit 
mass for lead and a substance of Group II. cannot be 
altogether due to the selective absorption of the cardboard 
screen, or to a difference in the absorption of the radiated 
rays. It seems to indicate, therefore, that the radiation of 
Groups II. and III. produced in a unit mass of lead is less 
than the radiation of Groups I. and III. produced in unit 
mass of a substance of Group II. This is probably due to 
the amount of radiation of Group I. being greater than that 
of either Group II. or Group III. We obtained some evi- 
dence of this, it will be remembered, when discussing the 
number of different primary and secondary groups of rays. 


of Radium, and Secondary Rays which they produce. 657 


We will now discuss at some length the transformation of 
primary rays into secondary. When a beam of primary 
y rays passes through a plate, it generates secondary rays in 
the plate corresponding to the different groups of primary 
rays ; the amount of radiation in a secondary group ot rays 
depending on the absorption of the corresponding primary 
group. Since one of the primary groups of rays will be 
better absorbed by the plate than either of the remaining 
groups, the amount of secondary radiation generated corre- 
sponding to this group must be larger than that corresponding 
to either of the remaining groups. The amount and nature 
of the secondary radiation which is not absorbed by the plate, 
and which is therefore radiated away from the plate, depends 
on the selective absorption of the different groups of rays by 


658 Mr. R. D. Kleeman on Different Kinds of y Rays 


the plate. And since the primary and secondary rays possess 
the same properties of selective absorption, that group of 
secondary rays is best absorbed which corresponds to the 
best absorbed group of primary rays. Thus the group of 
secondary rays which is produced in a larger amount in a 
plate than the other groups, is best absorbed by the plate. 
Now, the experiments described in this paper show that the 
amount of secondary radiation from a thick plate corre- 
sponding to the group of primary y rays which is best 
absorbed, is small in comparison with the secondary rays 
corresponding to either of the other groups of primary rays.. 
If the amount of secondary rays produced is proportional to. 
the absorption of the primary rays, and the proportion 
absorbed of each group of secondary rays by the plate is the 
same multiple of the proportion absorbed by the plate of the 
corresponding group of primary rays, this wouid obviously 
not be the case. Let us therefore investigate this more 
closely. We will make the following assumptions and compare: 
the conclusions that can be drawn from them with the results 
of experiment. Suppose that the coefficient of absorption of 
a group of secondary rays generated in a. substance is for a 
distance dx in that substance n times that of the primary 
rays, and the coefficient of absorption of the tertiary rays. 
n times that of the secondary, and soon. Suppose also that. 
the fraction m of the energy absorbed of the primary rays in 
a distance dx is converted into secondary radiation, and the 
same fraction of the energy absorbed of the secondary radia- 
tion converted into tertiary radiation, and so on. Let us 
further suppose that m and m are the same for all substances. 
Tt will now be shown that on these assumptions the amount 
of secondary radiation from a plate corresponding to the 
group of primary rays which is best absorbed, would be 
greater than that corresponding to any other primary group. 
Let us consider the secondary radiation produced by two 
primary groups of rays which we will denote by A and B, ~ 
where the group A is better absorbed than the group B.~ 
The amount of radiation trom a given plate will not be 

altered if it be compressed to a less thickness, for the amount 

of radiation produced depends on the number of absorbing 

and radiating molecules only, and not on their distance of 
separation. Suppose then that a plate is compressed to such 

a thickness that the coefficient of absorption of the rays of 

group B (for a distance dw) is the same as that of the rays 

of group A in an uncompressed plate. Then it follows that 

the coefficients of absorption of the secondary rays, and rays 

of higher order, corresponding to group B in the compressed 


of Radium, and Secondary Rays which they produce. 659 


plate, are the same as the coefficients of absorption of the 
secondary and higher rays corresponding to group A in the 
uncompressed plate. The value of m, it will be observed, 
will not be altered by the compression. Therefore the 
amount of secondary radiation corresponding te group B 
radiated by the compressed plate is equal to the amount of 
secondary radiation corresponding to group A radiated by 
an uncompressed plate of the same thickness and area. The 
amount of secondary radiation of group B from an uncom- 
pressed plate must therefore be smaller than the amount of 
secondary radiation of group A from the same plate ; that is, 
the amount of secondary radiation corresponding to the more 
absorbable primary group of rays is greater than that corre- 
sponding to the other less absorbable group. But this result 
is exactly opposite to that obtained by experiment. Let us. 
see what this means. According to the foregoing, the amount 
of secondary radiation generated by a group of primary rays 
in an element of volume of a substance is proportional to ma, 
where 2X is the coefficient of absorption of the rays. The 
amount of secondary radiation which is not absorbed by the 
element of volume, and which is therefore radiated away 
from the element, is approximately proportional to 


(mA—(mA)nr), or mdr? (« —n). 


Since ? increases more rapidly with an increase of X than 
1 1 — : ; | 
G. —n) decreases, md? (5 —n) increases with an increase of 


A, and we obtain as before that the amount of secondary 
radiation increases with the absorption of the primary rays. 
Tt follows, therefore, from the experiments that m and n 
cannot be the same for each group of primary rays. And 


iL: E : 
since ™? = —n) must decrease with X, either m must decrease 


or n increase with X, or both vary in this way with». It 
seems improbable though that the decrease of the amount of 
secondary radiation with increase of absorption of the 
primary group is due to a decrease of m; that is, to 
a decrease of the fraction of the absorbed energy which 
is converted into secondary rays. It is more probably 
due to an increase of n, that is, an increase in the 
relative absorption of a secondary group to a primary group 
with increase of absorption of the primary group. This may 
also be expressed by saying that the selective absorption is 
more marked for the secondary rays than for the primary. 
If the transformation of primary rays into secondary is a 


660 Mr. R. D. Kleeman on Different Kinds of y Rays 


resonance effect, we would expect that the absorption of the 
secondary rays would be, on the whole, greater than that of 
the primary, and that the selective absorption of the secondary 
rays would be more marked than that of the primary. It 
becomes of interest, therefore, to investigate what the results 
of these experiments indicate on the eether-pulse theory of the 
y rays. 

Boia to this theory, one ether pulse may differ from 
another in three different ways, namely—thickness of pulse ; 
magnitude of the electric force at, say, midway between the 
edges of the pulse ; and the function which expresses the 
relation between the ratio of the force at any point in the 
pulse to that midway between the edges of the pulse and 
the distance of the point From one of the edges of the pulse. 
This function will be called throughout the paper the toree- 
relation of the pulse. It follows also from the theory that 
the force-relation of a pulse and its breadth remain unchanged 
‘as the distance of the pulse from the source increases, but the 
force in the pulse is everywhere decreased. The selective 
absorption of some members of a number of different zether 
pulses must now be explained by differences between the 
pulses in the magnitude of one of the properties of an ether 
pulse. It was observed in these experiments that the selective 
absorption of the y rays is independent of the distance of the 
rays from their source. It follows, therefore, that selective 
absorption cannot be due to the different groups of rays 
differing from one another in the magnitude of the force in 
the pulse, for, if selective absorption were due to this, a group 
of y rays at a given distance would possess the same. properties 
as some other group at some other distance. It is also im- 
probable that it is due to the groups of rays differing from 
one another in the breadth of the ether pulse. For there would 
be nothing in this case to distinguish a broad pulse from a 
narrow one, as far as the absorbing electrons are concerned, 
till both had passed completely over the electrons. One 
would expect then that an electron which absorbs a pulse of 
certain breadth well, would absorb to a greater or less extent 
other pulses also. It is difficult to see, therefore, if selective 
absorption is entirely due to differences in breadth of pulse, 
why a narrow pulse is better absorbed than a broad one by 
one substance, while the opposite is the case with some other 
substance. We conclude, therefore, that selective absorption 
is probably due to the existence of groups of rays which 
differ from one another in the force-relations of their pulses. 
Since the secondary rays possess the saine properties of 
selective absorption as the primary, the form of the force- 


“ae 
eee 


ae ho) a 2 


PL XVI 


ade, F~ 


? 


15 


x 


ddo7 


R\yat 


Sexr6. Vol. 


> aad tof 


¢ san, Sac nae at et semen lg are He 
. ng 


= 
= 
< 


‘y bi4 zest 


5 
= =z 


oy @) S &y y) 


‘ 
\ . 
° ; " t . 
. - ; 
’ ’ i 7 ) : 
- w ’ 
q y ia : | 
| ) : . 
: ~ ; My ’ ; ; t 
; 3 ; 
7 | ’ | ‘ | | ; 
>. ; ; ‘ 
: ; | ; 
cf ; 


9168b 74 6 060% eo eleb 7S ber 


Phil. Mag. Ser.6.Vol.15.Pi.XIX. 


—— 


; 
| 
| 


“UsaAr) 


. SAU/7 GI 
2] ONY IEOS ‘9! ‘YIOXT wniyseYy/ 


of Radium, and Secondary Rays which they produce. 661 


relation is not altered when a group of primary rays is 
transformed into a group of secondary rays. 

The difference in the absorption of a primary and secondary 
pulse (which, as we have seen, must be distinguished from 
selective absorption) is probably due in part or on the whole 
to the form of the force-relation of the secondary pulse satis- 
fying the condition for absorption better than the primary 
pulse. But there is probably some other cause operating 
besides this one. Now, since the coefficient of absorption of 
a y pulse is independent of the distance from its source, the 
difference in the absorption cannot be due to a difference in 
the force of the pulse. The difference in the absorption is 
therefore probably due in part to a difference in the breadth 
of the pulse, the pulses in a group of secondary rays being 
probably broader than those in the corresponding group of 
primary rays. 

The production of secondary rays and their properties may 
then be explained on general linesas follows. Each electron 
in an atom is in equilibrium under the action of the forces in 
the atom, and when displaced from its position of equilibrium 
by the force in a pulse passing over it, the atomic forces tend 
to bring it back to its previous position. The maximum 
absorption of energy from the pulse by the electron will take’ 
place if the force acting on the electron due to the electric 
force in the pulse is always proportional to the force of restitu- 
tion of the electron, whether both forces act in the same or in 
opposite directions. This is the condition, it will be observed, 
that acceleration of motion of the electron must take place all 
the time the pulse is acting upon it. That this condition 
may be satisfied depends on the form of the force-relation of 
the pulse as well as its breadth. The form of the force- 
relation of a pulse and its breadth, it may be pointed out in 
passing, may easily be such that at the instant the electron 
ceases to be under the influence of the pulse after having 
been deflected, it is at rest in its position of equilibrium. If 
a pulse passes over an electron which does not exactly satisfy 
the conditions for best absorption (we may suppose the pulse 
narrower for one thing than required by these conditions), 
the pulse radiated by the electron will satisfy these conditions 
much better, and will therefore be more easily absorbed by a 
similar electron than the primary pulse. If the pulse satisfies 
the conditions for best absorption to a small extent only, it 
will be only slightly absorbed, and the secondary pulse will 
not be so soft in comparison with the primary as in the fore- 
going case. The selective absorption of a group of primary 
y rays by a substance may then be explained by the force- 

Phil. Mag. 8. 6. Vol. 15. No. 89. May 1908. Pe 


662 Different Kinds of y Rays of Radium. 


relation and breadth of pulse of the pulses of the group 
satisfying very approximately the conditions for best absorp- 
tion for a number of electrons, while the other groups of rays 
do not satisfy these conditions and are therefore not absorbed 
by these electrons. But since the other groups of rays are 
also absorbed to some extent by the substance, the pulses 
satisfy to some extent the conditions for best absorption for 
some other electrons in the atom. The secondary rays pro- 
duced in the substance under these conditions will then be on 
the whole softer than the primary rays, and possess the same 
properties of selective absorption as the primary rays, but in 
a more marked degree. 


Conclusions. 


It has been shown in this paper that the substances Pb, 
Hg, Zn, Cu, Fe, 8, Al, and C, whose secondary radiation 
was measured, could be roughly divided into three groups, 
each group of substances radiating rays which differ asa 
whole from those radiated by the other groups. From this 
result and the results of other experiments described in the 
paper, it was deduced that a part of the primary y radiation 
from radium and the secondary radiation which it produces 
could be roughly divided into three groups. The rays of 
each of these groups are not homogeneous, but consist of 
different kinds of rays which may be said to differ less from 
one another than from the rays of some other group. It was 
shown that the three secondary groups of rays possess the 
same properties of selective absorption as the corresponding 
primary groups, but are on the whole more easily absorbed. 
Thus, one of the groups of primary rays is better absorbed 
than the other primary groups by the substances Pb and Hg, 
and one of the other primary groups is better absorbed than 
the other groups by C, and the remaining group of primary 
rays is better absorbed than the other groups by Zn, Cu, Fe, 
S, and Al. The three corresponding secondary groups 
possess the same property, except that each secondary group 
is softer or more easily absorbed than its corresponding 
primary group. ‘There are probably other rays or groups of 
rays in a primary or secondary beam of rays besides those 
investigated in these experiments. 

It was also found that a substance radiates those secondary 
rays in least amount which it best absorbs. This is due to 
the substance absorbing to a large extent that part of the 
radiation generated in the substance which is easily absorbed 
by the substance. The absorbed radiation—that is, the 


Investigation of the Nature of the y Rays. 663 


radiation which is not able to get out of the plate—is 
probably transformed into secondary cathode radiation and 
other forms of energy. 

It is very probable that similar relations exist between 
primary and secondary X rays. The writer hopes to make 
some experiments in the near future to test this point. 

In conclusion I wish to thank Prof. Thomson for his keen 
interest and ever ready advice during these experiments. 


Cavendish Laboratory, February 18, 1908. 


LX, An Experimental Investigation of the Nature of the 
y Rays. By W. H. Brace, W.A., F.RS., Elder Pro- 
fessor of Mathematics and Physics in the University of 
Adelaide, and J. P. V. Manseyn, D.Sc.,. Lecturer on 
Electrical Engineering *. 


ie papers recently published in the Proceedings of the 

Royal Society of South Australia (May and June 1907) 
and in the Philosophical Magazine (October 1907) an attempt 
was made to show that the ether-pulse theory of y and X 
rays might prove to be incorrect after all, and that most of 
the known properties of these rays could be explained more 
simply and directly on the supposition that they were 
material and consisted of neutral pairs. The arguments 
were based on a comparison of known phenomena with 
deductions from each of the two opposing hypotheses. At 
that time there did not seem to be any opportunity of appeal 
to a decisive experiment. 

The object of this paper is to give a preliminary account 
of an investigation which appears to us to give the final 
answer as regards the y rays, and to show that they are 
material in nature. 

The argument is as follows :— 

Secondary radiation which is excited in an atom by a 
passing wave or pulse must be distributed symmetrically 
with regard to a plane passing through the atom perpen- 
dicular to the direction of motion of the pulse. If we speak 
of the primary pulse as going forwards, the secondary radiation 
is just as likely to go backwards as forwards. This isa 
well-recognized principle. For example, J. J. Thomson 
divides the secondary radiation due to y rays into two equal 
parts which he supposes to move away symmetrically in 
opposite directions, and, for convenience of calculation, 
parallel to the direction of the primary rays (‘ Conduction of 


* Communicated by the Physical Society: read April 10, 1908. 
2¥ 2 


= 


664 Prof. Bragg and Dr. Madsen: An Experimental 


Electricity through Gases,’ p. 406). Suppose, therefore, a 
pencil of y rays to pass normally through a plate so thin 
that its absorption may be neglected, the secondary radiation 
should be exactly the same on the two sides of the plate in 
amount, in quality, and in distribution ; and it ought not to 
be possible to discover, by any comparison of the secondary 
radiations on the two sides, which is the face of entry and 
which of emergence. 


Fig. 1. 


A 
8 


Consider now the ionization-chamber represented in fig. 1. 
The two ends are closed by plates, of which A and A’ are 
alike ; so also are Band B’. The material of A and A’ is 
different to that of Band B’. The nature of the side walls 
is of no consequence. A pencil of y rays passes along the 
axis of the chamber, which is represented by a dotted line. 
The ionization current within the chamber is measured as 
usual by inserting a high-potential electrode connected to an 
electroscope. 

When the plates A and B are inverted there is a change 
in the amount of the current: so also when A’ and B’ are 
inverted. By an extension of the principle already stated. 
it ought not to be possible, on the eether-pulse theory, to 
discover which way the rays are going (up or down in the 
figure) by comparing the consequence of inverting A and B 
with that of inverting A’ and B’. 

As a matter of fact the direction can be discovered with 
ease ; the more easily the greater the difference between the 
atomic weights of A and B. 

For example, in one experiment of ours the chamber was 
of cylindrical form, 7°5 ems. high and 25 cms. diameter, 


Investigation of the Nature of the y Rays. 665 


The plates used were aluminium and lead ; the thickness of 
each plate was a little less than 2mm. Inversion of the top 
plates A and B made a difference in favour of Al of less 


Fig. 2. 
Ra. 
i P) 
‘ 4 
5 
6 
‘ 

Ae Pl A 
—=————- - _. B 
‘SS 
a ee 0 eee: 

— : 

; ; 

H a ‘ 

Berea 

@eeeeurv ©wmeweoevwuse aaa ® =u == 2 @@oe w c 

“-+-B 


than 1 per cent.; 2. e., the current was slightly larger when 
the Al was next the chamber. On the other hand, inversion 
of the bottom plates made a difference of 44 per cent. in 
favour of Pb; 2.e.,the current was 44 per cent. larger when 
the Pb was on top. The details are shown in the figure. 
Allowance was made for all radiation other than that which 
proceeded down the conical opening in the lead block. 

It may be well to point out that this effect cannot be 
ascribed to any complication due to secondary or tertiary 
rays. No doubt the radiation in the chamber is very com- 
plex; but the fact isimmaterial. Provided that the chamber 
is symmetrical in the first place, then the secondaries must 
be symmetrical also if the ether-pulse theory is correct, and 
therefore the tertiaries and so on. Nor is it necessary to 
consider whether the secondary radiations are @ rays or 
scattered y rays. 7 

Also it must be remembered that the secondary radiations 
which enter the chamber have their origin almost entirely in 
a very few millimetres of material bordering on the chamber. 
Therefore the y rays are in almost exactly the same condition, 
both as to quality and as to quantity, when they excite 
secondary radiations from the top plate as they enter the 
chamber, and secondary radiations from the bottom plate as 
they leave. 

The details of the experiment may be varied greatly ; but. 


666 Prof. Bragg and Dr. Madsen: An Experimental 


in all the cases we have tried the want of symmetry is obvious. 
In fig. 3 are shown the details of one other case, in which 


--- Pb lnm (A) 
r---C lem. B) 
a 
C 
P | Q 
(1) Current with plates arranged as above.... 59°8 


carbon and lead were the materials used, and the form of 
the chamber was different. It seems unnecessary to give 
more, because in the first place the experiments are easy to 
repeat : and in the second place, the complete quantitative 
analysis of the figures depends on several factors, the in- 
fluence of which is imperfectly understood, such as the 
previous screening of the rays, the form of the chamber, and 
the respective parts played by the original y rays, cathode 
rays, and secondary y rays if any such exist. The experi- 
ments as they stand show how far away is that symmetry 
which the sether-pulse theory demands. It seems to us that 


there is no escape from the conclusion that the y rays are 


not ether pulses. 

Let us therefore proceed to consider the hypothesis that 
the y rays are material. In the paper already mentioned it 
was argued that they might well consist of neutral pairs, 
liable to be broken up on encountering atoms or parts of 
atoms ; and that the secondary cathode radiations might be 
the negative particles. thus set free. Let us suppose, pro- 
visionally, that the particles when set free move at first in 
the direction of the y stream, but are subsequently scattered 
in the usual manner of @ rays. [It is here that the absence 


- ~~ — 


Investigation of the Nature of the y Rays. 667 


of symmetry arises. On the pulse theory the particles should 
go equally backwards and forwards; indeed, if they were 
ejected by atomic explosions, the result of energy accumu- 
lated by passing pulses as suggested by J. J. Thomson in 
the case of X rays, they would move equally in all 
directions. ] 

Wigger gives a table (Jahrbuch der Radioaktivitdt, Bd. ii. 
p- £31) showing that the y rays are absorbed according to a 
density law pretty strictly, except for the smaller thicknesses 
in the case of the substances of larger atomic weight. 
Assume this law to hold good: and also assume for the 
present that the absorption of 8 rays follows the density 
law. The latter is only roughly true, of course ; but we 
may deal with quantities in a broad fashion first, and make 
the proper amendments afterwards. 

We can now compare the quantities of cathode radiation 


Fig. 4. 
A so 
Pr 
a ad & Ga aaa A 
A. Oe 2 


Pp’ 


.S 
~--4-~---}]--4----"3-7 


which should emerge from the far sides of two plates of 
different densities, p and p’. Let these be represented by 
AD and A’D’ in the figure; and let BC and B/C’ be cor- 
responding strata of equal weight, in fact let AB/A’B’= 
BC/B’/C’=CD/C’D’/=p’'/p. Let the plates be crossed by 
equal pencils of y rays, as shown in the figure. A certain 
quantity of y radiation is absorbed in crossing BC: in the 
language of our present hypothesis we should say that a 
certain number of y particles are stripped of their positives, 
and the negative remainders go on. An equal number of 


668 Prof. Bragg and Dr. Madsen: An Experimental 


negatives are set free in B’C’ because the two strata are of 
equal weight. Of those set free in BC only a certain 
number emerge from the face D because of the absorption 
of the plate CD. Since CD and (’D’ are of equal weight a 
similar absorption occurs in the case of the particles set free 
in B/C’. Thus the same number emerges from each plate. 
Integrating for all effective strata, the whole cathode radia- 
tions emerging from the two plates are equal. 

We thus find that if the absorptions of 8 and vy rays both 
followed the density law, the secondary cathode radiation on 
the far side of a plate—we may call it the ‘emergence ” 
radiation—would be the same for all materials. There should 
be no such relation between the amount of the radiation and 
the atomic weight of the plate as various observers have 
shown to be true for the secondary cathode radiation of 
“incidence,” a relation which is closely parallel to that 
found in the case of 8 rays. 

Experiment is in agreement with this theory, for it shows 
that no such relation exists in respect to the emergence 
radiations : in marked contrast to what happens in the case 
of the radiation from the front sides of the plates of various 
materials—the incidence radiations. 

It is true that the emergence radiations are not all equal, 
but this is to be expected, because (1) the amount of secondary 
cathode radiation depends, as Kleeman has shown, on the 
previous screening of the y rays, (2) the @ rays are not 
absorbed strictly according to a density law, (3) the y rays 
also depart from this law. We have made no serious attempt 
as yet to disentangle the effects of these various disturbing 
factors. In fact the task promises to be long and intricate, 
for it will be necessary to find out how much of the ioniza-~ 
tion in the chamber is due to each class of rays: to discover 
the law of distribution of the radiations in space so that the 
form of the chamber may be allowed for, if necessary: to 
find out the nature of the departures from the density law of 
those @ and y rays which are in question, and so on. Never- 


theless the results are satisfactory, so far as we have gone, - 


The amount of emergence radiation is found to depend on 
the previous screening of the rays. In one case the in- 
version of a ©, Pb pair jof plates” from ——-—~ Canes 
——= Pb, C altered the current in the ratio 1: 1:11 when 
the rays had been previously screened by Pb; but in the 
ratio 1:°96 when the screen was changed to C. Again, 
when the rays had previously passed through an iron screen, 
the inversion ——s Pb Fe to ——s» Fe Pb changed the 
current in the proportion 1: 1:12, but when a lead screen was 


—— 
Py q 
1 ‘ 
‘ 
° fu 


Investigation of the Nature of the y Rays. 669 


substituted for an iron one the change was 1: 1:04. In 
illustration of the effect of the second disturbing factor 
mentioned above, we have found that, other things being 
equal, the substances of small atomic weight give the most 
secondary radiation, in a general way; and it may be no 
coincidence that in some cases we have found Sn and Fe to 
give surprisingly small amounts. This is in agreement with 
what is to be expected ; for it is clear, on consideration of 
the argument already given, that the greater the §-ray 
absorption of a substance in proportion to its density, the less 
‘‘ emergence” radiation should issue fromit. Some observers 
have found Sn and Fe to possess exceptional absorbing powers. 
We do not wish, however, to lay any stress upon these last 
observations, some of which we may not have interpreted 
correctly ; but we mention them in order to show that the 
inequalities that are found to exist between the emergence 
radiation of various substances promise to be reducible to 
order as soon as the difficulties of interpretation have been 
surmounted. 

Let us now consider the cathode radiations on the front 
sides of the plates. Of the cathode particles set free in BC 
and moving at first in the direction of the y rays, a certain 
proportion, say p, is returned by what is beyond. These move 
towards the face A, and a certain number of them succeed 
in reaching it and emerging therefrom. In the case of the 
other plate the proportion returned is p’; the absorption in 
B’A’ is the same as in BA because the weights are the same. 
Comparing the two plates stratum by stratum, we find that 
the ‘‘incidence ” radiation of one plate is to the incidence 
radiation of the other plate as pto p’. Now pand p’ are 
the well-known constants of the 8 rays. 

When a stream of y rays is allowed to fall upon a plate, 
the cathode radiation which issues from the place of inci- 
dence must be divisible into two parts. One consists of 
scattered 8 particles derived from the stream of such particles 
which was travelling with the y rays before incidence, and 
which was formed during the previous transit of the screens 
employed—solid, liquid, or gaseous. This part is scattered to 
an extent which depends on the atomic weight of the plate, 
according to the usual (McClelland’s) law of 8 particles. 
The other part is originated in the plate itself in the manner 
just described, and the amount of it is also regulated ac- 
cording to the B-ray law. When, therefore, observers have 
measured the secondary radiation due to y rays, and have 
found a law corresponding to that for @ rays, the reason of 
the correspondence has been that they really were measuring 


670 Prof. Bragg and Dr. Madsen: An Experimental 


the secondary radiation due to 8 rays. Properly speaking, 
the secondary radiation produced by y rays, or rather from 
y rays, is proportional to the density of the substance traversed 
(cf. Wigger’s table) ; and this is only another form of the 
law of absorption of y rays. 

The relative importance of the two parts of the incidence 
radiation just mentioned must depend on the circumstances 
of the experiment*. The researches of Kleeman (Phil. Mag. 
Noy. 1907) show very well how the second part, which is 
influenced by previous screening, modifies the effect of the 
first part, which is not so influenced, but which follows the 
law of @ rays strictly. 

It is easy to show, by comparing corresponding strata at 
the front and back of one plate, that the incidence radiation 
should be somewhat less than p times the transmitted radia- 
tion,—somewhat less, because the cathode radiation which 
is turned back is scattered and softened in the process. 

To sum up :— 

On the ether-pulse theory we ought to find perfect 
symmetry in the secondary radiations from the two sides of 
a plate; but experiment shows nothing of the kind. 

On the material, or neutral pair theory, the “ incidence ” 
radiations should follow the 8-ray law. This is known to 
be the case. The “ emergence” radiations should not follow 
the 8-ray law; and experiment shows that they do not. If 
the density law held for both @ and y rays, and if the y rays 
were homogeneous, the emergence radiations should all be 
equal. As already explained, experiment shows that the 
observed inequalities give promise of ready explanation on 
the ground that no one of these suppositions is quite true. 


It is perhaps better not to extend the preliminary account 
of these experiments by any lengthy discussion of the issues 
arising from them. Many points that invite consideration 
have been discussed already in the papers first referred to. 


Moreover, our own further experiments are incomplete ; and 


their full interpretation isnot yet certain. We will therefore 
confine ourselves to one or two questions which seem to be 
of special interest. 

The X rays resemble the y rays so closely that it is 
practically inconceivable that the two radiations should be 
essentially different. The secondary cathode radiations which 
are set free when X rays impinge on any material must 


* In a recent letter addressed by one of us to ‘ Nature,’ too much 


stress was laid on the part played by the first part under all 
circumstances. . 


| 


Investigation of the Nature of the y Rays. 671 


therefore have been part of the X-ray stream, and must start 
their independent existence by moving on in the line of the 
X-ray motion. Their velocity is much smaller than that of 
the secondary cathode rays due to y rays, and they are much 
more readily scattered. It may still remain an open question 
whether or no the X-ray stream contains «ther pulses. 
Perhaps their existence must be supposed in order to explain 
the velocity experiment of Marx, and the diffraction experi- 
ment of Haga and Windt. Possibly they are also required 
in order to explain Barkla’s polarization experiments; but 
we do not think that the experiment described by Barkla in 
‘Nature’ (Oct. 31, 1907) is in any way decisive. 

It seems proper to consider a possibility that the negative 
particle, when it moves on in the original line of motion of 
the pair from which it came, retains also its original velocity. 
It is a striking fact that the cathode particle due to the 
y rays has the same speed, very nearly. as the 8 particle 
issuing from the original radioactive material. And it looks 
quite unlike a coincidence that similar comparisons can be 
made in the case of the X rays. The secondary cathode 
radiations due to these rays have velocities which, at the 
least, are of the same order as the velocities of the cathode 
particles in the X-ray bulb. If we examine the table given 
by Innes (Proc. Roy. Soc. Aug. 2, 1907, p. 461), and if we 
may be allowed to adopt an interpretation differing somewhat 
from the author’s, but more natural, it seems to us, in view 
of the conclusions of this paper, we find that the velocities 
of the electrons emitted by all the metals are practically the 
same, zinc being an exception because it is unable to break 
up the hardest rays. We find that the velocities range from 
about 6 x 10° to 7°5 x 10° for soft rays, and 6 x 10° to 8x 10° 
for hard rays. Remembering that bundles of X rays are 
very heterogeneous, the natural conclusion seems to be that 
the softest rays give the slowest speeds, and that the velocity 
of the secondary rays increases with the hardness of the 
X rays from which they are derived. Now the hardness of 
the rays grows with the speed of the cathcde particles in the 
bulb. Is it then possible that the cathode particle is first 
set in motion by the electromotive force in the bulb, strikes 
the anticathode and picks up a positive there, becomes neutral 
and is now called an X ray, is subsequently stripped of the 
positive and becomes a secondary cathode particle, the identity 
of the negative remaining the same throughout and its speed 
invariable or nearly so? The difficulty comes in when we 
try to consider the part played by the mass of the positive. 

And, again, may not the 8 and ¥ forms be interchangeable 
at times? <A y particle which had been stripped of it 


672 Prof. Bragg and Dr. Madsen: An Experimental 


positive and beccme a secondary cathode or 8 ray would be 
lost to measurement as a y ray ; and we should thus have an 
explanation of how the y rays are “ absorbed,” and why the 
absorption follows an exponential law. And in the same 
way, if a 8 particle picked up a positive it would disappear 
from view as a § particle ; it would be “absorbed.” 

Although we have made a few experiments with magnetic 
fields, we have not yet come to any conclusion as to whether 
or no there are y pairs which have become loosened in the 
attachment of positive to negative, forming a softer and 
more ionizing radiation. Their existence might be suspected, 
since there is an analogous effect in the case of X rays ; and 
probably they would be found more at the back of the 
penetrated plate than in front of it. 

A few further experimental illustrations are shown dia- 
grammatically in figs. 5 and 6 with the explanations 
attached. 


Fig. 5.—The upper figure shows the general arrangement. 
The lower figures are diagrammatic, and show the currents 
for different arrangements of the Pb and Al at the bottom of 
the chamber, and at the top with the exception of the plate 
through which the v rays enter. Inverting the top plates 
makes little difference when the upper of the two plates at 
the bottom is Al; but an appreciable difference when it is 
Pb, because in the latter case a good deal of secondary 
radiation is thrown up by the Pb, and there is a tertiary 
from the top plate. 

The same, when the conical opening is completely filled 
by a Pb stopper :— 


12°6 12°6 Ney 18°3 


The differences show the effects of those rays only which 
are stopped by the Pb stopper :— 


2H ik 28° ong 44-7 


These show the effect of inverting that portion only of the 
top plate where the y rays enter. Three Pb plates=-55 cm.; 
Al plate=°16 cm. 

The same, with Pb stopper inserted :-— 


14°] 14:7. 


—_~— wee a oe a a ee eee eee el 
J 


r>> Se --- > 
pias Wale eres los. | pair 


O.€9 ! | 


SS 
! \ 
1 
{ 


4 el 


674 Prof. Bragg and Dr. Madsen: An Experimental 


Se an ae ates oe ante a aa | geese | | hia i | oo iaaliataatan | 


' \ ! } \ ' ' ' 
ses EFS| |F27| | 3-8 rarer stance! E82! 12 
_ eee ee ee eee ee eee 


89 
secre 


4229 O/ 


It 
' 

' 

! 

' 

! 

' 
Ni 


‘9 SIT 


Investigation of the Nature of the y Rays. 675 


Fig. 6.—The upper figure shows the general arrangement. 
The wall of the cylindrical vessel was of brass: a Pb or an Al 
lining could be inserted as shown. The lower figures are 
diagrammatic, and show the currents for different arrange- 
ments of Pb and Al at top, bottom, and sides. Inversion of 
the plates through which the y rays pass into the chamber 
makes little difference ; but there is a great change if the 
material is changed on which the y¥ rays fall, or the “ emer- 
gence’ radiations from the top plate. The base is of less 
importance than in fig. 5; but the sides of more importance. 
This should clearly be so, for geometrical reasons. When 
the conical opening was filled by a Pb stopper the currents 
were all reduced considerably, but retained the same propor- 
tions pretty nearly. 

When a small pencil of @ rays was admitted through a 
hole in the centre of the top plate a change of the material 
otf the bottom became more effective, and of the sides less 
effective than before; but this difference became smaller 
when thin Al sheets were so placed as to scatter the 8 rays 
on their entry into the chamber. 


In conclusion we should like to add that Wigger was the 
first, so far as we know, to show clearly that the secondary 
radiation of Al, on the far side of the plate, was greater than 
that of Pb. A comparison of the emergence radiations of 
different metals was made by Dawes (Phys. Rev. xx. p. 182), 
who showed that they did not follow the law of the incidence 
radiations. The same effect was indicated in the experiments 
of Hve (Phil. Mag. Dec. 1904). We have little doubt that 
the interesting experiments of Mackenzie (Phil. Mag. July 
1907) are to be explained on the lines indicated in this 
paper. In fact it is clear that this is the case in a broad 
sense ; but it is difficult to give a complete explanation until 
the laws are so completely worked out that they can be 
applied to the interpretation of experiments which are really 
very complicated, although at first sight they may seem to 
be simple. 


| 676 J 


LXI. Notices respecting New Books. 


The Axioms of omen Geometry and the Axioms of Descriptive 
Geometry. By A. N. WurrenuaD, Sc.D., F.R.S. Cambridge 
University Press, 1907. 


HESE tracts, which form Nos. 4and 5 of the Cambr ie Tracts 
in Wat homatice and Mathematical Physics, deal with a branch 
of mathematics which has lately received much attention” from 
mathematicians, and which, although in its present form of com- 
paratively recent origin, has alr eady attained a high state of 
development. So important has this subject—the Foundations 
of Mathematics—become that it is imperative on every mathe- 
matician to have some acquaintance with its developments in 
order to be in sympathy with much of the mathematical work of 
the present day. Dr. Whitehead’s two tracts deal with the geo- 
metric side and form an excellent introduction to a detailed study 
of the whole field. They contain a clear exposition of the founda- 
tions of geometry, comprising as much of the theory as is of 
interest to the general mathematical reader, to whom presumably 
they are especially intended to appeal. They are, however, equally 
valuable to the specialist, since they give a fresh and connected 
view of a subject which is particularly confusing from the variety 
of ways in which it may be approached. 


Theory of Sets of Points. By W. H. Youne, M.A., Se.D., and 
Grace CuisHo~m Youna, Phil. Doc. (Gott.). Cambridge Uni- 
versity Press, 1906. 


THIS is In many respects a unique book. Not only is it the 
first of its kind which has ever appeared in English, but in no 
other language apparently has an attempt been made to givea 
systematic exposition of the subject. The subject itself does not 
demand on the part of the reader a large stock of mathematical 
knowledge. The book may therefore be taken up at almost any 
point of his mathematical training, probably the sooner the better. 
After a brief account of rational and irrational numbers and of 
the manner in which numbers are represented.as to order by points 
on a straight line, the authors enter upon the theory of linear sets 
and sequences leading up to the conceptions of potency, content, 
and order. Plane sets are introduced in Chapter VIII.; and in 
Chapter IX. a succession of theorems deals with the conceptions 
of region, domain, boundary, and rim, leading up to the definition 
ofacurve. It is here and in the later discussion of plane content 
and area that the student familiar with the ordinary geometrical 
assumptions regarding curves and areas will probably for the first 
time be fully impressed with the value of the theory of sets of 
points. There can be but one opinion as to the great service 
rendered by Dr. and Mrs. Young in placing in our hands this 
systematic treatise on the foundations of mathematics. An 
appendix running to fifteen pages gives the bibliography of the 
subject. 


BAN dee) dads A ARB 


THE 


LONDON, EDINBURGH, anno DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE, 


[SIXTH SERIES.] 
JUNE 1908. a 


LXII. Hamilton's Principle and the Five Aberrations of 
von Seidel. By Lord Rayurten, 0.1L, Pres.RS.* 


' ARGELY owing to the fact that the work of Hamilton, 
and it may be added of Coddington, remained unknown 
in Germany and that of v. Seidel in ‘England, it has scarcely 
been recognized until recently how easily v. Seidel’s general 
theorems relating to optical systems of revolutions may be 
deduced from Hamilton’s principle. The omission has been 
supplied in an able discussion by Schwarzschild, who expresses 
Hamilton’s function in terms of the variables employed by 
Seidel, thus arriving at a form to which he gives the name 
of Seidel’s Kikonal tf. It is not probable that Schwarz- 
schild’s investigation can be improved upon when the object 
is to calculate complete formule applicable to specified com- 
binations of lenses; but I have thought that it might be 
worth while to show how the number and nature of the five 
constants of aberration can be deduced almost instantaneously 
from Hamilton’s principle, at any rate if employed in a 
somewhat modified form. 

When we speak, as I think we may conveniently do, of 
five constants of aberration, there are two things which we 
should remember. The first is that the five constants do not 
stand upon the same level. By this I mean, not merely that 
some of them are more important in one instrument and some 


* Communicated by the Author. 
+ The word Eikonal was introduced by Bruns. 


Phil. Mag. 8. 6. Vol. 15. No. 90. June 1908. 224 


N . = = 42°) a 


678 Lord Rayleigh on Hamilton’s Principle and 


in another, but rather that the nature of the errors is different. 
In earlier writings the term aberration was, I think, limited 
to imperfect focussing of rays which, issuing from one point, 
converge upon another. Three of the five aberrations are of 
this character ; but the remaining two relate, not to imper- 
fections of focussing, but to the position of the focus. It is, 
in truth, something of an accident that, e. g. in photography, 
we desire to focus distant objects upon a plane. The second 
thing to which I wish to refer is that, although Seidel did 
much, four out of the five aberrations were pretty fully dis- 
cussed by Airy and Coddington before his time. To these 
authors is due the rule relating to the curvature of images, 
generally named after Petzval, so far, at any rae as it 
refers to combinations of thin lenses. 

Some remarks are appended having reference Pe systems 
of less highly developed symmetry. 


According to Hamilton’s original definition of the charac- 
teristic function V, it represents the time taken by light to 
pass from an initial point (2’, y’, 2’) to a final point (2, y, 2), 
and it may be taken to be \ pds, where yw is the refractive 
index and the integration is along the course of the ray which 
connects the two points. If the path be varied, the integral 


is a minimum for the actual ray ; and from this it readily 
follows that 


Ll = dV /dz, m =dV/dy, n = dV /de, ee 
= dV/da', —m' =dV/dy, —n'=dV/de’,. (2) 


where J, m,n, l',m',n' are the direction-cosines of the ray 
at the end and beginning of its course, the terminal points 
being situated in a part of the system where the refractive 
index is unity. 

In his communication to the British Association (B. A. 
Report, Cambridge 1833, p. 360) Hamilton transforms these 
equations. As his work is so little known, it may be of 
interest to quote in full the principal paragraph, with a 
slight difference of notation :—‘‘ When we wish to study the 
properties of any object-glass, or eye-glass, or other instru- 
ment in vacuo, symmetric in all respects, about one axis of 
revolution, we may take this for the axis of z, and we shall 
have the equations (1), (2), the characteristic funcivon V being 
now a function of the five quantities, 2*+y", 2a’ +yy/, 

x'?+4y'?, z, 2', involving also, in general, the colour, and 
having its form determined by the properties of the instru- 
ment of revolution. Reciprocally, these properties of the 


the Fire Aberrations of von Seidel. 679 


instrument are included in the form of the characteristic 
function V, or in the form of this other connected function, 


T=le+my+nz—Ve' —m'y'—n'2'—-V, . «. (8) 


which may be considered as depending on only three inde- 
pendent variables besides the colour ; namely, on the incli- 
nations of the final and initial portions of a luminous path to 
each other and to the axis of the instrument. Algebraically, 
T is in general a function of the colour and of the three 
quantities, [7+ m?, Il/+mm’', /?+m; and it may usually 
(though not in every case) be developed according to 
ascending powers, positive and integer, of these three latter 
quantities, which in most applications are small, of the order 
of the squares of the inclinations. We may therefore in most 
cases confine ourselves to an approximate expression of the 
form 


Jig ATLO 7 ane 


in which T® is independent of the inclinations ; T® is small 
of the second order, if those inclinations be small, and is of 
the form 


T@=P(? +m?) +P, (ll! +mm')+P(l? +m"); . (5) 
and T“ is small of the fourth order, and is of the form 
T= Q(? +m’)? + Q,(? +m?) (Ul +mm') 

+ Q'(P +m?) (U2 +m?) + Qy(ll’ + mm’)? 

+ Q,'(Ul' +mm’) (U? +m") + Q"(1?2 +m'?)?;. (6) 
the nine coefficients, P P, P’ Q Q, Q’ Q1, Q,’ Q”, being either 
constant, or at least only functions of the colour. The 
optical properties of the instrument, to a great degree of 
approximation, depend usually on these nine coefficients and 
on their chromatic variations, because the function T may in 
most cases be very approximately expressed by them, and 


because the fundamental equations (1), (2) may rigorously 
be thus transformed ; 


qtaahy nak ash rae ' 
merit ater Os Ohne ee 

aS 

mbes dT ai, te aT 2 Mi) 


ie mene Ie Goon oe ee a) gf 3 
n dl. n dm p 


The first three coefficients, P P, P’, which enter by (5) 
into the expression of the term T®), are those on which the 
focal lengths, the magnifying powers, and the chromatic 


aberrations depend: the spherical aberrations, whether for 
ye 


680 Lord Rayleigh on Hamilton's Principle and 


direct or inclined rays, from a near or distant object, at 
either side of the instrument (but not too far from the axis), 
depend on the six other coefficients, Q Q, Q’ Q, Qi’ Q”, 
in the expression of the term T“. Here, then, we have 
already a new and remarkable property of object-glasses, 
and eye-glasses, and other optical instruments of revolution ; 
namely, that all the circumstances of their spherical aber- 
rations, however varied by distance and inclination, depend 
(usually) on the values of SIX RADICAL CONSTANTS OF ABER- 
RATION, and may be deduced from these six numbers by 
uniform and general processes. And as, by employing 
general symbols to denote the constant coefficients or elements 
of an elliptic orbit, it is possible 1o deduce results extending 
to all such orbits, which can afterwards be particularized for 
each ; so, by employing general symbols for the six constants 
of aberration, suggested by the foregoing theory, it is possible 
to deduce general results respecting the aberrational properties 
of optical instruments of revolution, und to combine these 
results afterwards with the peculiarities of each particular 
instrument by substituting the numerical values of its own 
particular constants.” 

Equations (7) are easily deduced. So far as it depends 
upon the unaccented letters, the total variation of T is 


dT=ldxe+mdy+ndz+aedl+ydm+zdn 
dV dV dV 
= alee wee Phe 
or regard being paid to (1), 
dT =adl+ydm+zdn, 
in which 
ldl+mdm+ndn=0, 


so that 
AT len, aoa: mz 


Ay A wn dem ate 
and in like manner by varying the accented letters the second 
pair of equations (7) follows. 
If we agree to neglect the cubes of the inclinations, we 
may identify n, n' with unity, and (7) becomes 
we =(z2+2P)l4+ Pil’, y =(2+2P)m+ Pym’, 
a’ = —Pyl+(2—2P', y'=— Pim t (2 — 2 Pm, 


determining z, z' in terms of 2, z’, /, l’ supposed known, or 
conyersely J, /' in terms of z, 2’, x, w' supposed known. The 


the Five Aberrations of von Seidel. 681 


case of special interest is that in which w, y, z and 2’, y', 2’ 
are conjugate points, 7. e. ima ges of one another in the optical 
system, ‘The ratio x: #2 must then be independent of the 
special values ascribed to J, iF In order that this may be 


possible, z.e. in order that z, 2’ may be conjugate planes, the 
condition is 


Geert ieee ) Ae 05) ws sy Sard & a(S) 
and then 
ee) = Meee Py 
ee FOP ee a 
giving the Res 
Hquations (8), (9) express the theory of a symmetrical 


instrument to a és i approximation. In order to proceed 
further we should have not only to include the terms in (7) 
arising from T, but also to introduce a closer approximation 
for n. Thus even though T“)=0, we should have additional 
terms in the expressions EON com a! equal respectively to 
tlze(P +m) and 4$l’e’ (l?+m"). 

If the object is merely to express the aberrations for a single 
pair of conjugate planes, we may attain it more simply by a 
modification of Hamilton’s process. 


Supposing that the conjugate planes are <=0, 2'=0, we 
have V a function of the coordinates of the initial point 2’ y ; 
and of the final point v, y. And if as before J, m, n, I!, m', n' 
are the direction-cosines of the terminal portions of the ray, 
we still have 


T= aN ida psi |g Ce, Miata dana al a 14) 

l’= —dV/dx’, Mi NAY, oy op M1 LN) 

But now instead of transforming to a function of J, m, 
l', m’, from which 2’, y', vz, y are eliminated, we retain 2’, y’ 


a independent le eliminating only «, y, the coor lines 
of the final or image point *. Jor this purpose we assume 


Wh Ta mp Ne od) 
The total variation of U is given by 
dU =axdl+ldx+ydm+mdy 
PS gs ll a SpA: 
or with regard to (10), (11) 
dU =xdl+ydm4+l'dz'+m'dy’, . . . (13) 


dV 
Coe an dy i UY’ 


* Compare Routh’s ‘Elementary Rigid Dynamics,’ § 418. 


682 Lord Rayleigh on Hamuiton’s Principle and 


from which it appears that U is in reality a function of a’ ets 
l,m. As equivalent to (13), we have 
e =aU/dl y= dU/dm,” .. 7 
=A /de, m = dU /dy'. 2 4.) pee 
So far U appears as a function of the four variables 2’, y’; 
l,m; but from its nature, as dependent upon /x-+ my and V, 
and from the axial symmetry, it must be in fact a function of 
the three variables 
at y’?, ?+m?, and le'+my’, 


the latter determining the angle between the directions of 


z’, y’ and 1, m. When these quantities are small, we 


may take 7 
U= U® +08 4094...) Ss” 
where U® is constant and 
U® = SL(? 4 m?) + M(v'lty'm) +43N(@?4+y”), . AT) 
L, M, N being constants. If we stop at U™, equations (14) 
give 
c= LIi+M2’, y = Lim+ My’, eee 
determining #, y as functions of a’, y', l,m. We have next 
to introduce the supposition that x,y is conjugate to a’, 7/. 
Hence L = 0, for to this approximation z, y must be deter- 
mined by w’, y’ independently of 7, m. Accordingly, 
oo Mes y= My’? 

We are now prepared to proceed to the next approximation. 
In order to correspond, as far as may be, with the notation of 
Seidel * we will write 
U® = 4A (? +m’)? + BP? +m’*)(le’ + my’) 

+43(C—D) (le’+ my’)? +$D(C? 4m?) (vx? +”) 

+ E(le’ + my’')(@? +y?) +F(v?+y?y, . . . (20) 
which is the most general admissible function of the 
fourth degree. 

From (20) we obtain by use of (14) the additional terms 
in 2 and y dependent on U®. No generality is lost if at 
this stage we suppose, for the sake of brevity, y'=0. 
Accordingly, 

v= Al(?+m?)+ Ba’(8? +m’) + Cx?l+ Eal?, 2 (21) 

y = Am(? + m?)+2Ballin4+ De?m. 1)... (eee 

* Finsterwalder, Miinchen. Sitz. Ber. xxvii. p. 408 (1897). 


the Five Aberrations of von Seidel. 683 


In order to complete the value of « we must add the 
expressions in (19) and (21). 

Since F disappears from the values of 2 and y, we see that 
there are jive etfective constants of aberration of this order, 
as specified by Seidel. The evanescence of A is the Eulerian 
condition for the absence of spherical aberration in the 
narrower sense, 7. ¢. as affecting the definition of points lying 
upon the axis(v’=0). If the Hulerian condition be satisfied, 
B=0 is identical with what Seidel calls the Fraunhofer 
condition *. The theoretical investigation of this kind of 
aberration was one of Seidel’s most important contributions 
to the subject, inasmuch as neither Airy nor Coddington 
appears to have contemplated it. The conditions A=0, B=0 
are those which it is most important to satisfy in the case of 
the astronomical telescope. 

To this order of approximation B=0 is identical with the 
more general sine condition of Abbe, which prescribes that, 
in order to the good definition of points just off the axis, a 
certain relation must be satisfied between the terminal 
inclinations of the rays forming the image of a point situated 
on the axis. The connexion follows very simply from the 
equations already found. By (15), (16), (17), (20), with 
mn = 0, 

l’= Ml+ Bi?+ terms vanishing with wz’, 7’; 
so that for the conjugate points situated upon the axis 
ee BOB ya. 5 on 5 yy, COD 


The condition B=0 is thus equivalent to a constant value 
of the ratio l'/1, that is the ratio of the sines of the terminal 
inclinations of a ray with the axis. And this is altogether 
independent of the value of A. 

On the supposition that the two first conditions A=O, 
B=0 are satisfied, we have next to consider the significance 


of the terms multiplied by C and D. Since 
dz/di— Cz", dy/dm = Dz”, 


we see that C and D represent departures of the primary and 
secondary foci from the proper plane. In fact if 1/p,, 1/p. 


* If A be not equal to zero, it can be shown that the best focussing of 

points just off the axis requires that 

Al,+Bza'=0, 
where 7, is the value of 7 for the principal ray. For example, if the 
optical system reduces to a combination of thin lenses close together, 
1,=2/f, where f is the distance of the lenses from the image plane. 
Since by (19), x=Mz2’, the condition may be written 

AM+Bf=0. 


684 Lord Rayleigh on Hamilton’s Principle and 


be the curvatures of the images, as formed by rays in the 
two planes, 


1/p, = 2C, l/p, = 2D.  . 2 ae 
The condition of astigmatism is then 
C=D; »j1 + uteri eae 


but unless both constants vanish the image is curved. 
Finally the term containing E represents distortion. 
1f we impose no restriction upon the values of the constants 
of aberration, we have in general from (21), (22) 


da/dl = ABP +m?) + 6Ball + Cx”, 
dy[dm = A(?+3m?)+2B e'l+ Da”. 


These equations may be applied to find the curvatures of the 
image as formed by rays infinitely close to given rays, as for 
example when the aperture is limited by a narrow stop placed 
centrally on the axis, but otherwise arbitrarily. The principal 
ray is then characterized by the condition m=0, and 
we have 


da/dl = 3A’?+6Be'l+ Ce? =3H+K,. . . (26) 

dy/dm= AP+2Ba'l+ Dz? =H+K, .... 2 

equations which determine the curvatures of the images as 

formed by rays in the neighbourhood of the given one, 

and deviating from it in the primary and secondary planes 

respectively. 

According to (26), (27), 

2H = 2A’?+4Bel+(C—D)2”, . . . (28) 

2K = (BDO) e.g wee os) de 

The requirement of flatness in both images is thus satisfied 

if H=0, K=0. The former is the condition of astigmatism, 

and it involves the ratio of #’:/, which is dependent upon the 

position of the stop ; but the latter does not depend on this 

ratio. It corresponds to the condition formulated by 


Coddington and later by Petzval. From (28), (29) we 


may of course fall back upon the conditions already laid 
down for the case where A=0, B=0. 

The further pursuit of this subject requires a more parti- 
cular examination of what occurs when light is refracted at 
spherical surfaces. Reference may be made to Schwarz- 
schild *, who uses Hamilton’s methods as applied to a special 
form of the characteristic function designated as Seidel’s 
Hikonal. A concise derivation of the Coddington-Petzval 


* Gottingen ADA. iv. 1905. 


the Five Aberrations of von Seidel. 685 


condition by elementary methods will be found in Whittaker’s 
tract *. 

Before leaving systems symmetrical about an axis to which 
all the rays are inclined at small angles, we may remark 
that, as U™ contains 6 constants, in like manner U® 
contains 10 constants f and U® 15 constants, of which in 
each case one is ineffective. 

The angle embraced by some modern photographic lenses 
is so extensive that a theory which treats the inclinations as 
small can be but a rough guide. It remains true, of course, 
that an absolutely flat field requires the fulfilment of the 
Coddington-Petzval condition ; but in practice some com- 
promise has to be allowed, and this involves a sacrifice of 
complete flatness at the centre of the image. It will be best 
to fulfil the conditions dv/dl=0, dy/dm=0, or, what are 
equivalent, 


PU/de? = 0, dU/dm? = 0, 


not when / is very small but when it attains some finite 
specified value. If we suppose y’=0, U is a function of 
zw”, P+m?, and lx’, or say of u, v, w. Hence 


on Ud. at, aU 
eee gee tg? 
ZU a?U dU 


Se 2 ea Ue y A, 
dm? ae dv? ae dv’ 
After the differentiations are performed, we are to make 
m=O; so that the two conditions of astigmatism and focus 
upon the plane, analogous to (28), (29), are 
BW 0 Ua Wika, a7 U 
— = 0, A] Eee Bae asia 


dv O 
in which v is to be made equal to ?. But it is doubtful 
whether such equations could be of service. 


Let us now suppose that the system is indeed symmetrical 
with respect to the two perpendicular planes of x and y, but 
not necessarily so round the axis of z. In the expression 
for U no terms can occur which would be altered by a 
simultaneous reversal of x’ and I, or of y’ and m. For U®) 
we have 

U® = a? + Bm? + ya'l + dy'm 
+ terms independent of / and m. 


* “Theory of Optical Instruments,’ Cambridge, 1907. The optical 
invariants, introduced by Abbe, are there employed. 
Tt Schwarzschild, loc. cit. 


686 Hamuilton’s Principle and the Five Aberrations of v. Seidel. 
Hence, by (14), | | 
w= 2al+ye', y= 2Bm+6y’. 
If x, y is conjugate to 2’, y', we must have 


a=0, B=0; 
so that 
2sye', y =6y’. . = ee 


These are the equations of the first approximation, and 
they indicate that the magnification need not be the same in 
the two directions. 

There are no terms in U. As regards U“, we have 


U® = Alt + Bl?m? 4+ Cm! 
+ Dea’? +-Ea'lm? + Fy'm’ + Gy/Pm 
+ He? ? + 1e?m? + Se'y'lm+ Ky? + Ly'?m? 
+ Mwv'*l+ Ne'?y'm+ Oa'y"7l + Py”’m 
+ terms independent of /andm. . . . . (81) 


In (31) there are 16 effective constants as compared 
with 5 in the case where the symmetry round the axis 
is complete; so that such symmetry implies 11 relations 
among the constants of (31). For example, in the terms 
of the first line representing Hulerian aberration, axial 
symmetry requires that 


O=1B=A.......... sai 


We will next suppose that the only symmetry to be 
imposed is that with respect to the primary plane y=0:; 
so that U is unchanged if the signs of y/ and m are both 
reversed. U®) is of the same form as in the case of double 
symmetry, and 


& = alta’, y=2Bm+6r/. 


If z,y is the image of 2’, y', formed by rays in both 
planes, e=0, B=0, as before. But it may happen, e. g. in 
the spectroscope, that there is astigmatism even in the first 
approximation. If the points are images of one another as 
constituted by rays in the primary plane, «=0, but £ is left 
arbitrary. | 

The next term in U may be denoted by U®. If no 
conditions of symmetry were imposed, U® would include 
16 effective terms, i. e. terms contributing to 2, y; but the 


The Problem of a Spherical Gaseous Nebula. 687 
symmetry with respect to y=0 excludes 8 of these. We 
may write 

U® = al + bln? 
+ cv’? +dx'm’ + elm 
+ fell+ga'y'm+hyl, . . . . (88) 
and, by (14), 


wv = 8al? + bm? + Qea/l +ey'’m+fu +hy?. . (84) 


If the rays all proceed from the point wv =0, y'/=0, the 
conditions for a well-formed primary focal line are 


eet Oi gan Bec Oy biG 1 8 LR ER) 


of which the first expresses that there is no aberration of 
this order for rays in the primary plane, 7. e., that the focal 
line is thin, while the second is the condition that the focal 
line is straight *. 

But if, while «'=0, y' be left arbitrary, so that the source 
of light is linear, the evanescence of (34) requires, in addition 


TOveod),, wnat 
ROE =O Oe its sid wt ete er(B6) 


Terling Place, Witham, 
April 20, 1908. 


LXIIT. The Problem of a Spherical Gaseous Nebula. 
By the late Lord Kevin f. 


THis paper was begun about the close of 1906, in order to fulfil 
a promise given at the end of the paper “On the Convective 
Equilibrium of a Gas under its own Gravitation only,” published in 
the Philosophical Magazine, 1887; and part of it was commu- 
nicated by Lord Kelvin to the Royal Society of Edinburgh at its 
meeting on the 21st January 1907. Since then, however, important 
additions have been made to it, and the subject has been dealt 
with more fully than was originally intended. Unfortunately the 
manuscript was left incomplete at Lord Kelvin’s death. It ended 
with § 35. 

However, from information which I received from Lord Kelvin 
while carrying out the earlier work connected with the paper, I have 
been able to write the sections from § 36 totheend. These complete 
all that Lord Kelvin desired to include in this communication ; 


* Compare Phil. Mag. vol. vii. p. 481, 1879; Scientific Papers, i. 
p. 440. 
+ Communicated by Dr. J. T. Bottomley, F.R.S. 


688 Lord Kelvin: The Problem of 


and they express, I believe, the views he held while writing the 
earlier sections. 

The statement of mathematical solutions and numerical results 
separately, as an Appendix to the paper, under my own name, is 
in accordance with Lord Kelvin’s wishes. 

GEORGE GREEN, 
Secretary. 


Slee ie a fluid globe were given with any arbitrary dis- 

tribution of temperature, subject only to the 
condition that it is uniform throughout every spherical 
surface concentric with the boundary, the cooling, by radia- 
tion into space, and consequent augmentation of density of 
the fluid at its boundary, would immediately give rise to an 
instability according to which some parts of the outermost 
portions of the globe would sink, and upward currents would 
consequently be developed in other portions. In any real 
fluid, whether gaseous or liquid, this kind of automatic 
stirring would tend to go on until a condition of approximate 
equilibrium is reached, in which any portion of the fluid 
descending or ascending would, by the thermodynamic action 
involved in change of pressure, always take the temperature 
corresponding to its level, that is to say, its distance from 
the centre of the globe. The condition thus reached, when 
heat is continually being radiated away from the spherical 
boundary, is not perfect equilibrium. It is only an approxi- 
mation to equilibrium, in which the temperature and density 
are each approximately uniform at any one distance from 
the centre, and vary slowly with time, the variable irregular 
convective currents being insufficient to cause any cunsider- 
able deviation of the surfaces of equal density and temperature 
from sphericity. 

§ 2. The problem of the convective equilibrium of tempe- 
rature, pressure, and density, in a wholly gaseous, spherical 
fluid mass, kept together by mutual gravitation of its parts, 
was first dealt with by the late Mr. Homer Lane, who, as we 


are told by Mr. T. J. J. See, was for many years connected 


with the U.S. Coast and Geodetic Survey at Washington. 
His work was published in the American Journal of Science, 
July 1870, under the title “ On the Theoretical Temperature 
of the, Sun, 242 


* §1 is extracted from “On Homer Lane’s Problem of a Spherical 
Gaseous Nebula,” Nature, Feb. 14, 1907. 

+ The real subject of this paper is that stated in the text above. The 
application of the theory of gaseous convective equilibrium to sun heat 
and light is very largely vitiated by the greatness of the sun’s mean 
density (1'4 times the standard density of water). Common air, oxygen, 
and carbonic acid gas show resistance to compression considerably in 


a Spherical Gaseous Nebula. 689 


In a letter to Joule, which was read before the Literary 
und Philosophical Society of Manchester, January 21, 1862, 
and published in the Memoirs of the Society under the title, 
“On the Convective Equilibrium of Temperature in the 
Atmosphere” *, it was shown that natural up and down 
stirring of the earth’s atmosphere, due to upward currents 
of somewhat warmer air, and return downward flow of 
somewhat cooler air, in different localities, causes the average 
temperature of the air to diminish from the earth’s surface 
upwards to a definite limiting height, beyond which there 
is no air. It was also shown that, were it not for radiation 
of heat across the air, outwards from the earth’s surface, 
and inwards from the sun, the temperature of the highly 
rarefied air close to the bounding surface would be just over 
absolute zero ; that is to say, temperature and density would 
come to zero at the same height as we ideally rise through 
the air to the boundary of the atmosphere. Homer Lane’s 
problem gives us a corresponding law of zero density and 
zero temperature, at an absolutely defined spherical bounding 
surface (see § 27 below). In fact it is clear thatif in Lane’s 
problem we first deal only with a region adjoining the 
spherical boundary, and having all its dimensions very small 
in comparison with the radius, we have the same problem of 
convective equilibrium as that which was dealt with in my 
letter to Joule. ° 

§ 3. According to the definition of “ convective equi- 
librium ” given in that letter, any fluid under the influence 
of gravity is said to be in convective equilibrium if density 
and temperature are so distributed throughout the whole. 


= 
fluid mass that the surfaces of, equal temperature, and of 


excess of the amount calculated according to Boyle’s Law, when com- 
pressed to densities exceeding four, or five, or six, tenths of the standard 
density of water. There seems strong reason to believe that every fluid 
whose density exceeds a quarter of the standard density of water resists 
compression much more than according to Boyle’s Law, whatever be 
the temperature of the fluid, however high, or however low. We may 
consider it indeed as quite certain that a large proportion of the sun’s 
interior, if not indeed the whole of the sun’s mass within the visible 
boundary, resists compression much more than according to Boyle’s 
Law. It seems indeed most probable that the boundary, which we see 
when looking at the sun through an ordinary telescope, is in reality a 
surface of separation between a liquid and its vapour; and that all the 
fluid within this boundary resists compression so much more than 
according to Boyle’s Law that it does not even approximately satisfy 
the conditions of Homer Lane’s problem; and that in reality its density 
increases inwards to the centre vastly less than according to Homer Lane’s 
solution (see § 56 below). ° 

* Republished in Sir William Thomson’s Math. and Phys. Papers,. 
yol. ili, p. 250, 


690 Lord Kelvin: The Problem of 


equal pressure, remain unchanged when currents are produced 
in it by any disturbing influence so gentle that changes of 
pressure due to inertia of the motions are negligible. The 
essence of convective equilibrium is that if a small spherical 
or cubic portion of the fluid in any position P is ideally 
enclosed in a sheath impermeable to heat, and expanded or 
contracted to the density of the fluid at any other place P’, 
its temperature will be altered, by the expansion or con- 
traction, from the temperature which it had at P, to the 
actual temperature of the fluid at P’. The formulas to 
express this condition were first given by Poisson. They 
are now generally known as the equations of adiabatic 
expansion or contraction, so named by Rankine. They may 
be written as follows, for the ideal case of a perfect gas :— 


. a (2) | Jon divke Se 
on eh 
B=(4)F pie) Aha 


where (t, p, p), (t’, p’, p’) denote the temperatures, densities, : 
and pressures, at any two places in the fluid (temperatures 
being reckoned from absolute zero) ; and & denotes the ratio 
of the thermal capacity of the gas when kept at constant 
pressure to its thermal capacity at constant volume, which, 
according to a common usage, is for brevity called “the 
ratio of specific heats.” For dry air, at any temperature, 
and at any density within the range of its approximate 
fulfilment of the gaseous laws, we have 


~ 


k—1 k 
Alig ‘ ALES) : mee ) 
k=1°41 ; : 291 ; Fac 3°44...) 2 
For monatomic gases we have 
wo Shs eo a i 
an : ky Be ea oe (5) 


For real gases, we learn from the Kinetic Theory of Gases, 
and by observation, that & may have any value between 
1 and 12, but that it cannot have any value greater than 13, 
or less than 1. 

§ 4. To specify fully the quality of any gas, so far as 
concerns our present purpose, we.need, besides &, the ratio 
of its specific heats, just one other numerical datum, the 
volume of a unit mass of it at unit. temperature and unit 


a Spherical Gaseous Nebula. 691 


pressure. This, which we shall denote by 8, is commonly 
called the specific volume; and its reciprocal, 1/S, we shall call 
the specific density (D) of the gas. Interms of this notation, 
the Boyle and Charles gaseous laws are expressed by either 
of. the equations 


ee eta. OF == pOb se he 


where p, v, p, denote respectively the pressure, the volume 
of unit mass, and the density of the gas at temperature ¢, 
reckoned from absolute zero. Our unit of temperature 
throughout the present paper will be 273°C. ‘Thus the 
Centigrade temperature corresponding to ¢ in our notation 
is 273(¢—1). | 

$5. In virtue of §4, what is expressed by (1), (2), (3), 
equivalent as they are to two equations, may now, for working 
purposes, be expressed much more conveniently by the single 
formula (6), together with the following equation— 


where A denotes what we may call the Adiabatic Constant, 
which is what the pressure would be, in adiabatic convective 
equilibrium, at unit density, if the fluid could be gaseous at 
so great a density as that. 

§ 6. Looking to (6), remark that p being pressure per 
unit of area, the dimensions of pv are L-? x L® or L, if we 
express force in terms of an arbitrary unit, as in § 10 below; 
therefore S, though we call it specific volume, is a length. 
It is in fact, as we see by (9) below, equal to the height of 
the homogeneous atmosphere at unit temperature, in a place 
for which the heaviness of a unit mass is the force which we 
eall unity in the reckoning of p. 

§7. In the definition of what is commonly called the 
“height of the homogeneous atmosphere,” and denoted by 
H, an idea very convenient for our present purpose is intro- 
duced. Let p be the pressure and p the density, at any 
point P within a fluid, liquid or gaseous, homogeneous or 
heterogeneous, in equilibrium under the influence of mutual 
gravitation between its parts ; and let g be the gravitational 
attraction on a unit of mass at the position P. Let 


gpH=p ee ne (8). 

This means that H is the height to which homogeneous 

liquid, of uniform density p, ideally under the influence of 

uniform gravity equal to g, must stand in a vertical tube to 
give pressure at its foot equal to p. 

§ 8. The idea expressed by (8) is useful in connection with 


692 Lord Kelvin: The Problem of 


questions connected with internal pressure throughout a 

spherical liquid mass, such as the sun. It is also useful 

when we are considering pressure and temperature in 

gaseous fluids, such as our terrestrial atmosphere, or the 

outermost parts of the sun; which may be regarded as 

practically gaseous where the density is anything less than ‘1. 
§ 9. For a perfect gas, (8) divided by p, becomes 


gi=s |! 


By this we see, what is interesting to remark, that for the 
same temperature and same gaseous material, the “height 
of the homogeneous atmosphere” is the same for the air at 
the earth’s surface and for the air at any height above the 
surface ; and is the same for different barometric pressures. 
For different temperatures, it varies as the absolute tempe- 
rature. For different gases at the same temperature, it is 
proportional to their specific volumes. For different forces 
of gravity, it is inversely proportional to them. 

§10. Even for cosmical reckonings in respect to our 
present subject, and in many and varied terrestrial reckon- 
ings, it is convenient to take as unit of force the heaviness 
in mid-latitudes of the unit of mass. The unit of mass, for 
all nations and peoples of the earth, must for general con- ~ 
venience be founded on the existing French Metrical System. 
The unit may, according to the particular magnitude or 
character of substance of which the mass or quantity is to 
be specified, be conveniently taken as a milligram, or a gram, 
or a kilogram, or a metric ton (one thousand kilograms), 
or 10° tons. 

§ 11. The choice of unit force as mean terrestrial heaviness 
of unit mass is very convenient for ordinary earthly pur- 
poses, but language in which it is adopted is, unless properly 
guarded and tacitly understood, always liable to ambiguity 
as to whether force or quantity of matter is meant. Thus 
if (using for a moment the moribund British Engineering 
reckonings in pounds, inches, etc.) we speak of 73 pounds 
of lead, there is no doubt that we mean quantity of a 
particular kind of matter ; but if we speak of 73 pounds 
per square inch (which might be 73 pounds of lead, or of 
iron, or of stone) we mean a force. If we call the pressure 
on the boiler of a ship 73 pounds per square inch, we mean 
a somewhat greater pressure when the ship is in middle or 
northern latitudes than when she is on the equator; though 
the difference is, for pressures on safety-valves, practically 
negligible, being for example three-tenths per cent. between 
the equator and the latitude of Glasgow or Edinburgh. 


a Spherical Gaseous Nebula. 693 


§ 12. In the present paper we shall take as unit of mass 
the mass of a cubic kilometre of water at standard density 
(which is 10° metric tons); and we shall take its heaviness 
in mid-latitudes as unit of force. This means taking for g 
in (8) and (9), and in all future formulas, the ratio of gravity 
at the place under consideration to terrestrial gravity in 
mid-latitudes. Hence (remembering that in § 4 we have 
chosen for our unit temperature reckoned from absolute zero 
the temperature of melting ice, being equal to 273° Centi- 
grade above absolute zero) we see by (8) that § is simply 
the height in kilometres of the Homogeneous Atmosphere 
in mid-latitudes, at the freezing temperature. Thus, from 
known measurements of densities, we have the following 
table * of values of S for several different gases :— 


Gas. S. 

Air Mee ra aldi. | ace, KMOMICLE ES. 
mmm ©. . : .. ... La 414 be 
Pe ns OL =A ribs 
Garnom dioxide ~. ...  3°232 5 
Carbon monoxide. . .  8:370 . 
Sumer ek... | OUT ‘2 
Helium Beal fe Ld eS 0 25 if 
beuromen, 2... CC LIA TG vs 
Biteesen. . .  . . ... 8206 ie 
Oxygen “Aes Ula Gi RE rok Ys ie 


Ewlpiur dioxide . . . , 3°109 a8 


§ 13. Consider now convective equilibrium in any part 
of a wholly gaseous globe, or in any part of a fluid globe 
so near the boundary as to have density small enough to let 
it fulfil the gaseous laws. Let z be depth measured inwards 
from any convenient point of reference. The differential 
equation of fluid equilibrium is 


aoe ee eg! ut 2 (EO). 
Now, if the equilibrium is convective, we have by (3) 
apc Bay Ma ve 
Pampa lates) ~~) (12): 


Using this, and (2), in (10), and dividing both members 
1 


by (GP. we find 
t dt _k-1 gp't' 12 
Hailey at ° ° . . ° e e ( 5) 


* Tf instead of taking 10° tons as our unit of mass we take a oram, 
the numbers in this table must each b2 multiplied by 10°, and they will 
then be the values of S in centimetres instead of in kilometres. 


Phil. Mag. 8. 6. Vol. 15. No. 90. June 1908. oA 


AE? ee re i 


694 Lord Kelvin: The Problem of 
Whence, by (6), we find 
dt_k—lg 
aot ae ok” ce ae oe (13) A 


and, ( (2) repeated ) 


P=) a 


§ 14 These are exceedingly important and interesting 
results. By (13) we see that in any part of a wholly 
gaseous spherical nebula, or in a gaseous atmosphere around 
a solid or liquid nucleus, in convective equilibrium, suffi- 
ciently stirred to have the same chemical constitution 
throughout, the temperature-gradient of increase inwards is 
in simple proportion to the force of gravity at different 
distances from the centre. We also see that in gaseous. 
spherical nebulas of different chemical constitutions, or in 
gaseous atmospheres of different chemical constitutions, 
around solid or liquid nucleuses, the temperature-gradients — 
at places of the same gravity are simply proportional to the 
values of (k—1)/(KS) for the different gases or gaseous 
mixtures. 

§ 15. For the terrestrial atmosphere we have by (4) 
we = 3°44, and by the table in § 12, S=7:988 kilometres. 


The temperature-gradient according to (13) is therefore, at 
the rate of our unit of temperature, or 273 degrees Centi-. 
grade, per 27'5 kilometres ; or 1° C. in 100°6 metres. This 
is much greater than the temperature-gradient found by 
Welsh, in balloon ascents of about fifty years ago, which 
was only 1°C. in 161 metres*. Joule, with whom I had 
been in discussion on the subject in 1862, suggested to me. 
that the discrepance might be accounted for by the conden- 
sation of vapour in upward currents of air. In endeavouring 
to test this suggestion, I made some calculations of which 
results are shown in the following table, extracted from 


a table given in my paper of 1862, referred to in § 2: 
above. | 


* Mr. Shaw informs me that much investigation in later times gives. 
a general average mean gradient of 1° C. per 164 metres. This is very 
nearly the same as it would be with no disturbance from radiation in. 
air saturated with moisture, at 4°C. 


a Spherical Gaseous Nebula. 695 
: Elevation from Earth’s 
Temperature centigrade surface required a cool 
| erie 20d i moist air by 1°C. 
| 
| dx 
q Sat 
0 Metres 152 
5 168 
10 186 
15 207 
20 229 
| ao 22 
. 30 274 
3d 284 


_ §16. From this we see that an ascending current of moist 
air at 3° C. would sink in temperature at about the rate 
of 1°C. in 161 metres of ascent. This is exactly Welsh’s 
gradient ; ‘“‘and we may conclude that at the times and 
places of his observations the lowering of temperature 
upwards was nearly the same as that which air saturated 
with moisture [at 3° C.] would experience in ascending” *. 
But it is not to be supposed, indeed it cannot have been the 
case, that his observations were made in a single ascent 
through cloud. “It is to be remarked that except when 
the air is saturated, and when, therefore, an ascending 
current will always keep forming cloud, the effect of vapour 
of water, however near saturation, will be scarcely sensible 
on the cooling effect of expansion ” f. 

17. But, considering our terrestrial atmosphere as a 
whole, and the complicated circumstances of winds, and rain, 
and snow, and its heatings by radiations from the sun, and 
its coolings by radiation into starlit space, and its heatings 
and coolings by radiations to land and sea in different 
latitudes, we may feel sure that Joule’s suggestion shows a 
cause contributing importantly to the general average tempe- 
rature-gradient being less than it would be in dry air in 
convective equilibrium. 

§ 18. For the solar atmosphere, we have approximately, 
g=28 (28 times middle latitude gravity at the earth’s surface). 


* Quoted from the Manchester paper above referred to, Math. and 
Phys. Papers, vol. 111. p. 260. 
t Lbid. 
3A 2 


Mes me hs _ 


696 Lord Kelvin: The Problem of 


By way of example, we may take S and & the same as 
for the terrestrial atmosphere, as we have not sufficient 
knowledge from spectrum analysis to allow us to guess other 
probable values of S and & for the mixture of gases consti- 
tuting the upper parts of the sun’s atmosphere, than those 
we know for the mixture of Oxygen, Nitrogen, Argon, and 
Carbonic Acid, which in the main constitutes our terrestrial 
atmosphere. Thus in the upper atmosphere of the sun, if in 
purely convective equilibrium, and undisturbed by radiations 
and other complications, the temperature would increase 
at the rate of 280 degrees Centigrade per kilometre down- 
wards, and, looking forward to § 27 below, we see that the 
increase of temperature would start from absolute zero at 
the boundary, where density, pressure, and temperature, are 
all zero. It would require very robust faith in the suggestion 
of convective equilibrium for the gaseous atmosphere of the 
sun to believe in +7° C. being the actual temperature of 
the sun’s atmosphere at one kilometre below the boundary. 
Iam afraid I cannot quite profess that faith. It seems to 
me that the enormous radiation from below would, if the 
upward and downward currents were moderately tranquil, 
overheat the air in the uppermost kilometre of the sun’s 
atmosphere to far above the temperatures ranging from 
— 273° Centigrade to +7° Centigrade, calculated as above 
from the adiabatic convective theory. 

§ 19. Keeping, however, for the present by way of example, 
to the calculated results of this theory, with the data for § 
and k& chosen in § 15, we find at ten and at fifty kilometres 
below the boundary, the temperatures, reckoned in Centi- 

‘rade degrees above absolute zero, would be respectively 
2800 and 14000. Calling these temperatures ¢’ and ¢, and 
the densities at the same places p’ and p, we find by (14) 


pi lef AA 000 \2 whee 
B=( a) = 350. 
Suppose for example p’ to be ‘001 (1/1000 of the density of 
water), we shall have p=056. This last is nearly but not 
quite too great a density for approximate fulfilment of the 
gaseous laws for the same gaseous mixture as our air. Thus, 
if not too much disturbed by radiation of heat from below, 
the uppermost fifty kilometres of the sun’s atmosphere might 
be quite approximately in gaseous convective equilibrium ; 
with density and temperature augmenting from zero at the 
boundary, to density ‘056, and temperature 14000 Centi- 
grade degrees above absolute zero, at the fifty kilometres 


a Spherical Gaseous Nebula. 697 


depth. But, going down fifty kilometres deeper, we find 
that the temperature at one hundred kilometres depth would 
be 28000°, and the density would be °316. This density is 
much too great to allow even an approximate fulfilment of 
the gaseous laws, by any substance known to us, even if its 
temperature were as high as 28000°. This single example 
is almost enough to demonstrate that the approximately 
gaseous outer shell of the sun cannot be as much as 100 kilo- 
metres thick—a conclusion which may possibly be tested, 
demonstrated, or contradicted, by sufficiently searching 
spectroscopic analysis. The character of the test would be 
to find the thickness of the outermost layer from which the 
bright spectrum lines proceed. If it were *1” as seen from 
the earth, it would be 73 kilometres thick. 

§ 20. Considering the great force of gravity at the sun’s 
surface (about 28 times terrestrial gravity), it is scarcely 
possible to conceive that any fluid, composed of the chemical 
elements known to us, could be gaseous in the sun’s atmo- 
sphere at depths exceeding one hundred kilometres. I am 
forced to conclude that the uppermost luminous bright-line- 
emitting layer of our own sun’s atmosphere, and of the 
atmosphere of any other sun of equal mass, and of not greater 
radius, cannot probably be as much as one hundred kilo- 
metres thick. 

§ 21. There must have been a time, now very old, in the 
history of the sun when the gravity at his boundary was 
much less than 28, and the thickness of his bright-line- 
emitting outermost layer very much greater than one hundred 
kilometres. Going far enough back through a sufficient 
number of million years, in all probability we find a time 
when the sun was wholly a gaseous spherical nebula from 
boundary to centre, and a splendid realization of Homer Lane’s 
problem. The mathematical solution of Homer Lane’s problem 
will, for a spherical gaseous nebula of given mass, tell exactly 
what, under the condition of convective equilibrium, the 
density and temperature were at any point within the whole 
gaseous mass, when the central density was of any stated 
amount less than ‘1; on the assumption that we know the 
specific volume (S) and the ratio of specific heats (£) for 
the actual mixture of gases constituting the nebula. It will 
also allow us to find, at the particular time when any stated 
quantity of heat has been radiated from the gaseous nebula 
into space, exactly what its radius was, what its central 
temperature and density were, and what were the temperature 
and density at any distance from the centre. Thus, on the 
assumption of S and & known, we have a complete history 


698 Lord Kelvin: The Problem of 


of the sun (or any other spherical star) for all the time before 
the central density had come to be as large as ‘1. 

§ 22. To pass from the case of convective equilibrium in 
a gaseous atmosphere so thin that the force of gravity is 
practically constant throughout its thickness, to the problem 
of convective equilibrium through any depth, considerable in 
comparison with the radius, or through the whole depth 
down to the centre, provided the fluid is gaseous so far, we 
have only to use (13) and (14), with the proper value of g, 
varying according to distance from the centre. Remembering 
that we are taking g in terms of terrestrial gravity, and that 
the mean density of the earth is 5°6, in terms of the standard 
density of water, which we are taking as our unit density, 
we have the following expression for g, in any spherical 
mass, m, having throughout equal densities, p, at equal 
distances, r, from the centre :— 


a \drr? 
m/r? 3 i P 

a i) 
J Hye TS6.2" 29 (18 


where E denotes the earth’s mass, and e the earth’s radius. 
This expression we find by taking g as the force of gravity 
due to matter within the sphere of radius 7, according to 
Newton’s gravitational theorem, which tells us that a 
spherical shell of matter having equal density throughout 
each concentric spherical surface exerts no attraction on a 
point within it. Using this in (13) of § 13, with dz=—dr ; 
multiplying both members by 7”, and introducing m to denote 
the mass of matter within the spherical surface of radius 7, 
we find 


it 3 k—1(" 3 k-1lm 
5°6.e k T 


— Pe = ty =e 
7 ie eee Cees get - 


Differentiating (17) with reference to 7, we find 


z at ee oe 2 
al in| = 562 WS Ae. e 
§ 23. By (6), and (7), of §§ 4, 5, we find 
Sa e 
pale). + ison ee 
where 
a iy 


= Ot 
e k—1 20) 


a Spherical Gaseous Nebula. 699 
Eliminating p from (18) by (19), we find 


d gilt rt” : 
pars ier ce PN C2) 
where 
sa tore.atve + TL yA* 
og? = vate Ate An Sea Ail 7-1) 
§ 24. By putting 
Oo 
== 2 
ey eke hit Leads Lae (23), 
we reduce (21) to the very simple form, 
Pt e 
Se 2 re 24); 
dx? ae ( 4); 


the equation of the first and third members of (17), modified 
by (20) and (23),%gives 

m (<+1)Se dt (25> 

Wee ors, yk 

§ 25. Let t= (x) be any particular solution of this equa- 


tion ; we find as a general solution, with one disposable 
constant C, | 


f—Creiatar pe iii (ol bls 2B), 


which we may immediately verify by substitution in (24). 
Here §(x) may denote a solution for a gaseous atmosphere 
around a solid or liquid nucleus, or it may be the solution for 
a wholly gaseous globe, in which case §(x) will be finite, and 
& (x) will be zero, when e=co. Hach solution §(a) must 
belong to one or other of two classes :-— 

Class A: that in which the density increases continuously 
from the spherical boundary to a finite maximum at the 
centre. In this class we have dp/dr=0 (dt/dr=0), when 
7=0; or, which amounts to the same, dp/dx=0 (dt/dz=0), 
when r=. 

Class B: that in which, in progress from the boundary 
inwards, we come to a place at which the density begins to 
diminish, or is infinite ; or that in which the density increases 
continuously to an infinite value at the centre. 

With units chosen to make §(« )=1, we shall denote the 
function & of class A by Ox, and call it Homer Lane’s 
Function ; because he first used it, and expressed in terms of 
it all the features of a wholly gaseous spherical nebula in 
convective equilibrium, and calculated it for the cases, e=1°5 
and e=2°5 (k=12 and k=1-4). He did not give tables of 


700 Lord Kelvin: The Problem of 


numbers, but he represented his solutions by curves*. He 
did give some of his numbers for three points of each curve, 
and Mr. Green, by very different methods of calculation, has 
found numbers for the case « = 2°5, agreeing with them to 
within 5th per cent. 

§ 26. By improvements which Mr. Green has made on 
previous methods of calculation of Homer Lane’s Function, 
and which he describes in an Appendix to the present paper, 
he has calculated values of the function ©x(x), and of its 
differential coefficient ©’«(z), which are shown in five tables 
corresponding to the following five values of «, 1°5, 2°5, 3, 
4,«. For the four finite values of « the practical range of 
each table is from w=q to r=~, g denoting the value of x 
which makes t=0. 


§ 27. There is such a value of x which is real in every case 
in which « is positive and less than 5. This we see exem- 
plified in the four diminishing values of g found by Mr. Green 
(2737, °1867, -1450, -0667) + for the four finite values of «, 
1°5, 2°5, 3, 4, and in the zero value of g for x = 5, the 
case described in § 29 below. In this case equation (24) has 
a solution in finite terms, which gives t = 3.x for infinitely 
small values of #, and therefore makes g=0 for =0. 

§ 28. Two interesting cases, c=1 and «=5, for each of 
which the differential equation (24) is soluble in finite terms, 
have been noticed, the former by Ritter t, the latter by 
Schuster §. Ritter’s case yields in reality Laplace’s cele- 
brated law || of density for the earth’s interior (sin nrfr), 
which Laplace suggested as a consequence of supposing the 
earth to be a liquid globe, having pressure increasing from 
the surface inwards in proportion to the augmentation of the 
square of the density. With Ritter, however, the value of n 
is taken equal to 7/R, so as to make the density zero at the 
bounding surface (r=R). With Laplace, n is taken equal 
to 27/R to fit terrestrial conditions, including a ratio of 
surface density to mean density which is approximately 
1/2°5. The ratio of surface density to mean density given 
by Laplace’s law, with n=377/R, is in fact 1/2°4225, which is 
as near to 1/2°5 as our imperfect knowledge of the surface 
density of the earth requires. 


_* American Journal of Science, July 1870, p. 69. 
+ See Appendix to the present paper, Tables I. .1V. 
t Wiedemann’s Annalen, Bd. xi. 1880, p. 388. 
§ Brit. Assoc. Report, 1883, p. 428. 
|| Mécanique Celeste, vol. v. livre xi. p. 49. 


a Spherical Gaseous Nebula. 701 


§ 29. For the case «=5, Schuster found a solution in finite 
terms, which with our present notation may be written as 
follows :— 

A a “its 1 DY Qn 

t=@0;(2) ~ \/(Ba® +1) Slee elite (27). 

This makes t=1 at the centre (¢/r=a#=a). At very great 
distances from the centre (w = 0) it makes 


’ 3 Bye (S./3\ S,/3\o° 
savin, md o=(SfH(S2)e (ELIE cn 


Using (27) in (25), we find 


m__ (k+1)So 3 99). 
E a ni eee (3a? + 1)3? e ° ) ° (2 ie 
and if in this we put «=0, we find 
M_ («+1)8a,/3 
ee (30), 


where M denotes the whole mass of the fluid. Thus we see 
that while the temperature and density both diminish to zero 
at infinite distance from the centre, the whole mass of the 
fluid is finite. 


§ 30. It is both mathematically and physically very inter- 
esting to pursue our solutions beyond «=5, to larger and 
larger values of x up to x=: though we shall see in § 43 
below, that, for all values of « greater than 3 (or £<14), 
insufficiency of gravitational energy causes us to lose the 
practical possibility of a natural realization of the convective 
equilibrium on which we have been founding. But notwith- 
standing this large failure of the convective approximate 
equilibrium, we have a dynamical problem of true fluid 
equilibrium, continuous through the whole range of « from 
—l1 to —#,and from +2 to 0; that is to say, for all 
values of & from 0 to «. In fact, looking back to the 
hydrostatic equation (10), and the physical equations (1), or 
(7), and (16), we have the whole foundations of equations 
(17) to (26), in which we may regard ¢ merely as a con- 
venient mathematical symbol defined by (6') in § 4. Any 
positive value of k is clearly admissible in (1), if we concern 
ourselves merely with a conceivable fluid having any law of 
relation between pressure and density which we please to 
give it, subject only to the condition that pressure is increased 


702 Lord Kelvin: Zhe Problem of 


by increase of density. It is interesting to us now to remark, 
what is mathematically proved in § 44 below, that unless 
k>1, the repulsive quality in the fluid represented by & in 
equation (1) is not vigorous enough to give stable equilibrium 
to a very large globe of the fluid, in balancing the con- 
glomerating effect of gravity. 


§ 31. As to the range of cases in which « has finite visti 
greater than 5, we leave it for the present and pass on to 
x=0,o0rk=1. In this case equation (1) becomes 


eee 


which is simply Boyle’s law of the “ Spring of air,” as he 
called it. It was on this law that Newton founded his caleul- 
ation of the velocity of sound, and got a result that surprised 
him by being much too small. It was not till more than a 
hundred years Jater that the now well-known cause of the 
discrepance was discovered by Laplace, and a perfect agree- 
ment obtained between observation and dynamical theory. 
But at present we are only concerned with an ideal fluid 
which, irrespectively of temperature, exerts pressure in 
simple proportion to its density. This ideal fluid we shall 
call for brevity a Boylean gas. 

§ 32. For this extreme case of « = «©, our differential 
equation (24) fails ; but we deal with the failure by express- 
ing tin terms of p by (19), and then modifying the result 
by putting «=co. We thus find 


d? logo 5° Ween teat inh ink 
ie oe where =~ . | ae 


o denoting a linear constant given by (37) below. Equation 
(32) is the equation of equilibrium of any quantity of 
Boylean gas, when contained within a fixed spherical shell, 
under the influence of its own gravity, but uninfluenced by 
the gravitational attraction of any matter external to it. The 
value of o might, but not without considerable difficulty, be 
found from (22) by putting c=. But it is easier and more 
clear to work out afresh, as in § 33 below, the equation of 
equilibrium of a Boylean gas, unencumbered by the exuvie 
of the adiabatic principle from which our present problem 
emerges. 


s | 
§ 33. Let pS Boies. oer aie 


where B denotes what we may call the Boylean constant for 


a Spherical Gaseous Nebula. 703 
the particular gas considered; being its pressure at unit 
density. According to our units, as explained in §§ 10, 11, 
12, B is a linear quantity. The analytical expression of the 
hydrostatic equilibrium is 

apd, oh ee tae 
where | (16) repeated | 


die 
ee ll (35) 
pr Bee Be, ; 


Eliminating p from (34) by (83), and multiplying both 
members by 7”, we find 


llog 3 ; e=m 
gqhi0OrPp _ ae a 
oe ae soap} aes BE (S87 


Differentiating this with reference to 7, and then trans- 
forming from 7 to 2 as in equations (21)... . (24) above, 
we find (32), with the following expression for o :— 


PUAN Ske fl oo Oe SONG 


The equation of the first and third members of (36) gives 


m_ Bodlogp 
Brie ae 


(38). 


§ 34. Let now p=F (2) be any particular solution of (32); 
we find as a general solution with one disposable constant C, 


p=CE (J) Sa Hag BA: bee 


which we may immediately verify by substitution in (32) 
(compare §25 above). The particular solution F must 
belong to one or other of the two classes, class A and class B, 
defined in § 25 above. _ . = 

§ 35. We shall denote by V(wv) what F(a) of § 34 becomes, 
when the particular solution of (32), denoted by F, is of 
class A, with units so adjusted as to make V(o)=1; that 
is to say, central density unity. Mr. Green in his Appendix 


704. Lord Kelvin: The Problem of 
to the present paper has calculated V(«) and W'(z)/V(z), 


through the range from z=x to e='l. His results are 
shown in Table V. of the Appendix. Thus we may consider 
WV(az) and its differential coefficient W(x) as known for all 
values of « through that range. 


§ 36. Using this solution, V(), instead of F in (39) above, 
we find that the solution of class A, which makes the central 
density C, is 


p= C¥(,,) _.. 


and when we insert this expression for p in equation (38) we 
obtain 


= eV) . a 


§ 37. From equations (40) and (41), with values of 


¥( vu) and ibe, ¥(%) obtained from the curves of 
WV(a) and V’(z)/V¥(z) in the range from z= to e="1, and 


with the relation — where o is given by (37) above, we 


can tell exactly the density at any point of a spherical mass 
of an ideal Boylean gas, and the mass of gas within each 
spherical surface of radius 7, when the gas is in equilibrium 
under its own gravitation only,.and has a density at its 
centre of any stated amount C. It is interesting to examine 
by means of these solutions the changes in p and m at any 
given distance from the centre when the central density C_ 
increases by any small amount dC; and to find also the 
changes in the radius of the spherical cell enclosing a given 
mass m, required to allow the mass to continue in equilibrium 
when the central density is increasing or diminishing con- 
tinuously. The following table shows the values of p or 


ow(a), and em/EBo or ae vCv( so): for 


several of the larger values of 7, corresponding to the central 
densities 1 and 1°21 respectively. 


: S Boal 
r i Ebe - p EBo 
@ fi 0 1:21 G | 
‘279 -2AQT 6°697 2511 TA9 
*250 -2076 7905 2069 8°39 
"225 "1673 9°38 "1647 9°86 
°200 °1295 P20 *1260 11°64 
"195 *1223 11°61 "1189 12°03 
"190 SHEDS 12°04 1118 12°46 
"185 "1085 12°50 "1048 12°89 
"180 1017 12°97 "0982 13°35 
al "0952 13°47 "0918 13°83 
“170 “0889 13-99 0855 14°34 
"165 "0828 14:53 "0795 14°86 
"160 ‘0769 15°10 ‘0738 15:40 
“155 0712 15°71 “0681. 16°10 
"150 ‘0657 16°34 “0528 16°59 
"145 “0605 17-01 0577 17°22 
"140 "0554 iver al! "0529 18°04 
"13D "0506 18°45 °0483 18-61 
"130 "0461 19:25 "0439 19°35 
=“§25 0418 20-06 °0398 20°14 
°120 "0377 20:95 "0359 20°98 
*115 "0339 21°89 0322 21°88 
“10 *0303 22°88 ‘0288 22°82 
"105 “0269 23°95 *0257 23°83 
"100 0238 25°10 "0227 24-94 


§ 38. From this table we see that it is possible to have 
the same mass of an ideal Boylean gas (e*m/EHBo=21°'9) 
distributed in two different equilibrium conditions within a 
given sphere (o/r=°115). We see also that in all smaller 
spheres the mass has increased, and in greater spheres it has 
decreased, through the alteration of density at the centre 
from 1 to 1:21. Indeed, when we trace the changes in the 
condition of any stated mass of a Boylean gas as its central 
density ideally increases from very small to very great values, 
we find that its radius diminishes till a certain central 
density has been reached, after which it increases till it 
becomes infinite. 

§ 39. By taking any two values of C in equation (26) 
above, and comparing the two solutions thus obtained as in 
§ 37, it may be verified that results similar to those found 
in the case of a finite mass of an ideal Boylean gas, are 
found also in the case of a finite mass of any gas for which 
x>3, or k<1} ; while for any finite mass of a gas for 
which «<3, an increase in the density at the centre is 
always accompanied by a decrease in the radius of the shell 
enclosing the mass in equilibrium. ‘These differences in the 


706 Lord Kelvin: The Problem of 


behaviour of the Buylean gas from that of gases for which 
«<3, and the resemblances of the Boylean gas and of gases 
for which «> (of which it may be regarded as the limiting 
case, k=), become of interest when we come to the 
question of the possibility of equilibrium of a mass of gas 
which is gradually losing energy by radiation into space. 
The result found above that there are two equilibrium con- 
ditions of a mass of any gas for which «>3, and one — 
equilibrium condition of a mass of any gas for which «<3, 
within a given sphere, makes it desirable to investigate the 
nature of the equilibrium in each case, and leads. us to 
the consideration of the energy required to maintain a mass 
of gas in equilibrium, within a sphere of radius R, in 
balancing the condensing influence of gravity. 

§ 40. Let K, denote the thermal capacity at constant 
volume of the particular gas considered. The energy within 
unit volume of the gas at temperature ¢ is K,pt; and the 
total energy I, within a sphere of radius R, is given by 


R wy 
t= ink, { dre pt =i, ( dmt .. 3 (42) 
0 AUU ae 


By using equation (6), and then integrating by parts, we 
obtain 


Ame (Pi, ) ea I(; 3 a 1%, 34p : 
I= 5 { arr p= g 3”) —3 dn | (43) ; 


0 0 


and since p=0 at the outer boundary of the sphere and r=0 
at the centre, we have 

PAIRS dp 

[=— 38), dry ae 


(44). 


Substituting now the expression given for — in the 


_ equation of hydrostatic equilibrium (84), we obtain finally 
ATG ac, 
1= 35" |, dr gp 1) OS ae 


§ 41. The work which is done by the gravitational attraction 
of the matter within any layer of gas 47rr’pdr in bringing 
that layer from an infinite distance to its final position in the 
sphere is given by 

dw=Anr'pdr sgn. he Le CE 


and the work done by gravity in collecting the whole sphere 


a Spherical Gaseous Nebula. 707 


of radius R is therefore 
R A (ia 2 
W=ir| dnr'gp= 5 | Bins Ss aphid 
Se : Dla A ? 


§ 42. From equations (45) and (47) we obtain, as the 
ratio of the intrinsic energy within the sphere of gas to 
the work done by gravity in collecting the whole mass from 
an infinite distance, 

a ae 
Ws 3s 
If K, be the specific heat of the gas at constant pressure, we 


have S=K,—K,, and equation (48) may now be written in 
the form 


(48). 


ts. KK: ih, t Weak (49) 
SG Cal a9 ne ae A 


§ 43. According to this theorem, it is convenient to divide 
gases into two species: species P, gases for which the 
ratio (k) of thermal capacity pressure constant to thermal 
capacity volume constant is greater than 13; species Q, 
gases for which & is less than 14. And the theorem ex- 
pressed mathematically in equations (48) and (49) may be 
stated thus :—‘ A spherical globe of gas, given in equi- 
librium, with any arbitrary distribution of temperature having 
isothermal surfaces spherical, has less heat if the gas is of 
species P, and more heat if of species Q, than the thermal 
equivalent of the work which would be done by the mutual 
gravitational attraction between all its parts, in ideal shrink- 
age from an infinitely rare distribution of the whole mass to 
the given condition of density” *. 

§ 44. It is easy to show from the theorem of $$ 42, 43 
that the equilibrium of a globe of Q gas is essentially 
unstable. Let us first suppose for a moment that by a slight 
disturbance of the equilibrium condition the ratio I/W for 
the globe of Q gas becomes greater than that required 
for equilibrium by equation (49). Unless the excess of 
internal energy were quickly radiated away, the repulsive 
force which the globe of gas possesses by virtue of its 
internal energy would more than balance the condensing 
influence of gravity, and the globe would tend to expand. 
Since the internal energy lost in expansion is exactly equi- 
valent to the work done against gravity, we see that the 


* Quoted from ‘‘On Homer Lane’s Froblem of a Spherical Gaseous 
Nebula,” ‘ Nature,’ Feb. 14, 1907. 


708 Lord Kelvin: The Problem of 


ratio I/W would continue to increase and the globe would 
become farther from an equilibrium condition than before. 
The expansion of the globe would therefore go on at an ever 
increasing speed till the density of the gas becomes infinitely 
small throughout. 

If, on the other hand, through a slight disturbance of the 
equilibrium condition, the ratio I/W becomes less than that 
required for equilibrium, the globe of gas would in this case 
tend to contract. The increase in the internal energy due 
io any slight condensation would be exactly equal to the 
thermal equivalent of the work done by gravitation; and 
the ratio I/W would therefore go on diminishing instead of 
increasing, as it would require to do if the gas is to be 
restored to a condition of equilibrium. 

§ 45. “ From this we see that if a globe of gas Q is given 
in a state of equilibrium, with the requisite heat given to it 
no matter how, and left to itself in waveless quiescent ether, 
it would, through gradual ioss of heat, immediately cease to 
be in equilibrium, and would begin to fall inwards towards 
its centre, until in the central regions it becomes so dense 
that it ceases to obey Boyle’s Law; that is to say, ceases to 
be a gas. Then, notwithstanding the above theorem, it can 
come to approximate convective equilibrium as a cooling 
liquid globe surrounded by an atmosphere of its own 
vapour” *. | 

§ 46. But if, after being given in convective equilibrium 
as in § 45, heat be properly and sufficiently supplied to the 
globe of Q gas at its centre, the whole gaseous mass can be 
kept in the condition of convective equilibrium. 

§ 47. The theorem of §§ 42, 43 is given by Professor 
Perry on page 252 of ‘Nature’ for July 13, 1899 ; and in 
the short article ‘‘On Homer Lane’s Problem of a Spherical 
Gaseous Nebula,” published in ‘ Nature,’ February 14, 1907, 
I have referred to it as Perry’s theorem. Since this was 
written, however, I have found the same theorem given by 
A, Ritter on pp. 160-162 of Wredemann’s Annalen, Bd. 8, - 
1879, with the same conclusion from it as that stated in 
§ 44 above, namely, that when £<1} the equilibrium of the 
spherical gaseous mass is unstable. 

§ 48. In the theorem of Ritter and of Perry, given in 
§ 42, convective equilibrium is not assumed. For the pur- 
poses of our problem, indicated in § 21, it is desirable to 
obtain expressions for the energy and the gravitational work 
of a mass M in equilibrium with a stated density at its 


* Quoted from “On Homer Lane’s Problem of a Spherical Gaseous 
Nebula,” ‘ Nature,’ Feb. 14, 1907. 


a Spherical Gaseous Nebula. 709 


centre, in terms of the notation of §§ 23...25 above. 
Thus, taking as our solution with central temperature C 
(equation 26), 


eee (OD (2) MND RRM aaa arial 639d 
where 
e=al 2); pao) /z ; 


and where ¢o is given in terms of the Adiabatic Constant, A, 
by (22) ; we have from equations (25) and (50) 


m (K+ 1) is eee 


e 


CP Cay Te a re (EE), 
and by differentiating this we obtain— 


—1(«—3) 
a ee 


§ 49. With these values of t and dm substituted in the 
third member of equation (42), the expression for the internal 
energy, i, of the gas within a sphere of radius r becomes 


r —3(K—5) 
ee Sethe | 4:0"(0(2) (58) 
0 


x 
By putting @"(z) )=—-[O¢ z) |*/z 2* in this, and then integrating 
by parts as in § 40, equation (43), we may write iin the 
form— 


a # pee pel a lee oe sal: @'( 2) | (54). 


Similarly, from the third member of (47), with the values of 
m and dm given in (51) and (52) above, we obtain the 
following expression for the gravitational work, w, done in 
collecting the gas within a sphere of radius 7 from infinite 
space— 


: ge) ez) . (55). 


& 


_E(«t+ = Aa al | = [0] 
3 


It is easy to verify from these equations for 2 and w that 
with S=K,—K,, as in § 42, 


_K,E(«+1)8eC7=*" [@(z) ]**? , 
See +50 ‘ (56) F 


Phil. Mag. 8. 6. Vol. 15. No. 90. June 1908. a 


710 The Problem of a Spherical Gaseous Nebula. 


_ §50. For the complete mass of gas, M, which can be 
in convective equilibrium under the influence of its own 
gravitation only, with central temperature C, we have the 
following results :— 


M_(«+1)8e07?*- 
B a ) 


@.(¢) . . . eee 


é 
OE ad ‘ 
SOOT it oh are 
(58) 
; —1(K-5) Pw K 
p-BBlet D Sat i iOEl aie) EH: 
e 4 e 
202, (13 (K—5) 2 ~)|* | 
wa Bet Sef er. 
? 


with 
nye 5°6.e(k +1)A* 


a Ge [(22) repeated]. 


The two equations (59) and (60) give as before | 
il 
wes )%o): 


§ 51. The equations of §§ 48...50, with equation (19), give 
the solution of Homer Lane’s problem for all values of « for 
which the function @,(z) and its derivative ©’.(z) have been 
completely determined, namely for «=1 and «=5, referred 
to in §§ 28, 29 above, and for the values 1°5, 2°5, 3, 4, for 
which the Homer Lane functions and their derivatives are 
given in the Appendix to the present paper (Tables I....IV.). 
It is important to remark that these equations indicate clearly 
the critical case x =3, and that they also reveal some inter- 
esting peculiarities of the case «<=5 ; which we have found 
to be the smallest value of « for which a finite mass of gas 
is unable to arrange itself in equilibrium within a finite 
boundary (see §§ 27, 29). 

Equation (57) shows that in spherical nebulas for whose 
gaseous stuff «<=3 the total mass of any gas which can exist 
in the equilibrium condition corresponding to a definite 
central temperature, when so distributed throughout its 
whole volume that the temperature and density at every 
point are related to each other in accordance with a chosen 
value of the adiabatic constant A, can also be brought into 
the equilibrium condition corresponding to any smaller 
central temperature, through gradual loss of energy, without 


Laterally loaded Struts and Tie-rods. 711 


disturbing the relation of temperature and density at any 
point of the mass. 

Equations (59) and (60) show that in spherical gaseous 
nebulas for whese gaseous stuff «=5 the total internal 
energy, and the gravitational work, corresponding to each 
equilibrium distribution of gas, has the same value, whatever 
be the central temperature or total mass, provided tempera- 
ture and density at each point within the mass are related to 
each other in accordance with the same value of the adiabatic 
constant in each case. 

(To be continued. | 


LXIV. Laterally loaded Struts and Tie-rods. By ARTHUR 
Mortezy, W.Sc., Professor of Mechanical Engineering at 
University College, Nottingham”. 

4 kee frequent occurrence of laterally loaded struts and 

ties in structures and machines makes the subject one 
of someimportance. very horizontal strut or tie-rod carries — 

a lateral load in its own weight, every vertical stanchion 

which carries a horizontal wind load, every beam which is 

not horizontal, and the coupling-rods of locomotives loaded 
transversely by their own centrifugal force, are common 
examples. The increase in the maximum intensity of stress 

due to a small transverse load on a strut which has a 

considerable axial thrust is very marked in long struts. 

Prof. Perry has given a method of finding the stresses 
in such cases, his solution embracing conditions of the most 
general kind. 

Three methods are in common use for the determination of 
the maximum intensity of stress in such cases. In each 
of these the resultant intensity of stress is found by taking 
the algebraic sum of that due to the axial thrust or pull 
and the intensity of bending stress. The methods differ in 
the calculation of the bending stress only ; the usual as- 
sumptions are made and the simple Bernoulli-Euler theory 
employed; but the calculation of the maximum bending 
moment differs in the three methods as follows :— 

(a) The maximum bending moment is taken as that due 
to the transverse loads only. This will evidently be nearly 
correct in a very short stiff beam having a small thrust 


or pull. 
(6) To the bending moment which the lateral loads would 


* Communicated by the Author. 
tf Phil. Mag. March 1892. 


3B2 


113 Prof. A. Morley on 


alone produce is added that which would be caused by the 
axial load if the deflexions of the beam were due to the trans- 
verse loads only and were unaffected by the axial loads. 
This is the method usually given in the text-books for other 
than very short beams. 

(c) The bending moment resulting from the axial loads 
is estimated by means of the deflexions calculated from the 
axial and lateral loads jointly, and is added to that resulting 
from the lateral loads. This is the method adopted in the 
paper mentioned above ; but in the common case chosen for 
illustration an approximation is made in estimating the 
deflexion. 


Object of this Paper. 


The present paper is mainly concerned with the simple but 
most important cases of uniformly distributed and single 
concentrated loads and simple conditions of end support ; its 
main object is to record the more exact solutions and to 
examine under what circumstances the simpler methods of 
calculation are approximately correct and to indicate the 
degree of approximation. 


Notation. ~ 


The axis of zis taken through the centres of area of the 
two ends of the bar and the origin is midway between these 
points. The length of the strut or tie-rod in all cases is taken 
asl. The axial force is +P a thrust in the case of a strut, 
and —P a pullin the case of a tie-rod. The radius of gyration 
of the area of cross section A about a central axis perpen- 
dicular to the plane of bending is k, and the moment of 
inertia, k2A, of the area A about the same axis is I. Only 
sections symmetrical about this axis are considered and the 
depth of section is taken as d. The average intensity of 
tg 
t 
Since the curvature is always small within the limits of 
d*y 
tat 


reckoned negative when it tends to bend the bar concave 


stress over the section is pp= + 
and the bending moment, 


elasticity, it is taken as 


2 
towards its undeflected position, is equal to KI where E 


is the modulus of direct elasticity. 


Laterally leaded Struts and Tie-rods. 18 
Case I. 


Uniform straight strut with uniformly distributed load w 
per unit length, and ends freely hinged. 


Cys 4 wl 
es Cae ae (1) 
we w (P 
ta’ s lz —*) 2) 


+4. ee l di 
The conditions being y=0 for «== and ““=0 for w=0 
the solution is aan 2 da ; 


wi, wl? whl 


i Patel Gaile aif. of 
Y= gp gp ~ “pr (1-805 prey, a’) - @) 
and at the origin 


we wEI l ny 
Y= 8p = pe (1—see5 ‘ mi) 20 ee (4) 


and the bending moment with sign reversed is 


' EI l G24 
My=Pyptgul'="h (1-se054/ a2) Oe sc 


or _ wEI mer ) 
Mo= “p- (see P. I Pe ter ce get eee (6) 


where P, is Euler’s limiting value of P for the ideal strut, in 


: i) re aie 
this case p which value makes M and y infinite. 


The actual working value of 2 will rarely reach as much 
as + when there is no lateral load, and with a lateral load 
1 may be taken as an extreme value. As the resistance to 
bending under end thrust alone in a plane perpendicular to 
that of the lateral loads will often be less than that to bending 


under end thrust in the plane of the lateral loads, a will 


often be much less than 4, as for example in a rectangular 
section the depth of which is considerably greater than the 
breadth. } 


714 Prof. A. Morley on 
Applying the expansion 
Oy OO, GLO’ pigeon: 


sec O@— ee Gh.? eae 


2 y Slo (PY |. 2ife ee 
e {ht ae 48 p cos 958048 p) + 1.60. } me 


the coefficients of the various powers of s differing but little 


+...&c. to (6), 


from unity. This or (5) may also be written 


4 ue > “alt 617? 4 21a"! Py? : 
Moe aaa oe 600 P. + 56580 (B:) +..de b. (8) 


The form (8) shows in a striking manner the relation of 
the approximate method (b) to the corréct method of @aleu- 
lating the bending stress and the order of the error involved 
in the former; the first two terms of (8) represent the 
maximum bending moment as estimated by method (0), and 


the coefficients of the powers of = are nearly unity; also the 


first term of (7) or (8) represents the maximum bending 


moment as estimated in the method (a). 
The equal-and opposite intensities of maximum bending 


stress In a sy mmetrical section are 


M, wkd a & -1): 
Z = op AV? ee 


" 
| 
| 


| Nam 


; 
where Z= = is the Stee of section. 


The maximum intensity of the compressive stress is 


whd l Do 
=o po= Fok (seogpa/ 2 -1) tT Pos 


and the maximum intensity of the tensile stress is 


wkd l Do 
S.=Po—Po= ooh (8° Aye -1) —Pos 


which may be positive or negative. 


1 
wen Te the error involved in calculating the bendint 


stress intensities by method (a) is 20 per cent., and by method 
(b) 4 per cent.; practically the same proportions it may be 
noticed in either case that the first neglected term in the 


M and y, being evidently infinite for P= 


Laterally loaded Struts and Tie-rods. 715 


series bears to the first term. The errors involved in calcu- 
lating f, the maximum intensity of stress due to the axial and 
lateral forces combined will be less than these amounts, the 
error depending on the amount of the lateral compared with 
the axial load, and the shape of cross section. Similarly the 
error in calculating fe will be greater than the above 


- amounts. 


Prof. Perry’s ae mentioned above consists in 
substituting - ws cos | a for (5 eat *) in (1), which is equi- 


valent to Piericiine a smaller but not a uniformly dis- 


tributed load Tel for the actual lateral load wi. This gives 


wl? : Be 
p= UEP) and M)>=4wi?. PP’ 


the deflexion being rather below the true value for all values 
of P. The most serious error arising from this approxi- 
mation is that in calculating the bending stress for high 


ratios oe assuming the limits of elasticity not to be exceeded. 


P. 


For = =0°9 the error is 34 per cent. on the deflexion and 


Pa 
somewhat less on the bending stress; that on the maximum 
tensile stress /; is proportionally more than that on the bending 
stress. 


Case Il. 


Uniform straight strut with central lateral load W and ends 
freely hinged. , 
The equation in this case is 


d*y af 

Tet BEY = smi (3 — v). Sy aS) 
and the conditions being as before, 

var | am NY a 

Noe ei een) ap? 7, OP 
and at the centre 


Sirs Dig. Ay age 
== / Fan 34 /F - ar OVS GaE verry oan i) 


7’ HL 
[? 


eile 


7 oe a 


716 Prof. A. Morley on 


Using the expansion 


== 4, 3 es 17 
tan 0=0+ 50 Kiki 50+. 


ENE Ae | 
so a * ole) 20160 p) + Keb (2x 


| 

} 

: 

| 

4 lint | 

TST ein PqltT ae Y 1680 p) +. bo. b, - (13) © 


which very clearly shows the relation of the calculation by 
the method (6) to the true bending moment. For bab! 
the error is under 4. per cent. Pe 3 


Case III. 


Uniform straight strut with uniformly distributed load w 
per unit length and ends fixed in the direction of the axis 


of x. 
The equation is 
2 
lc ak ee (14) 


da? ' Hl’? ont A ns ine 
where M, is the bending moment at the ek we con- 
ditions being eu =) tor 00% —-O0lior 2=5 and § a = =0 for 


“= : the solution 


w 3 HI l Py eee l 2) 
y= Ye ays ee Gs OS BI? °%9 am) 


gives 


d? Me, -2 il 
E> aE p cose es 5¢ cosa / es (15) 


The points of inflexion are ata distance from the centre 


given by 
= / Boost {4 / et 2 ye 
b= P pp sing ay 


l 2 
which yaries from ——~= 2/8 woen P=0 to — “when P=P- 4n° EI 


(? 
HKuler’s limiting value for the ideal ‘ae fixed at both ends. 


Laterally loaded Struts and Tie-rods. pith 
At the origin 


wHI _ wi ay ou ee 
ep cosee 5A Er (16) 


and the bending moment of greatest magnitude is at the fixed 
ends where 
3 Oa 151 Re: diss 
aa on 2 P cot 2 BI’ ers (17) 
If the forms (16) and (17) are expanded they show a 
similar relation to those in cases I. and II. between the 
exact method of calculating the bending moment and method 
(6) if deflexions above or below the points of inflexion are 
used. These points as mentioned above vary somewhat 
in position according to the value of P and method (6) 
becomes less simple than the exact method, 


Case LV. 


Uniform straight strut with central load W and ends firmly 
fixed in the direction of the axis of w. 


The equation is ! 
yg ae W (| - M,, 
met my=—sm(—*) +e + 08) 


where M, is the bending moment at the fixed ends. The 
solution gives the bending moment with sign reversed, 


W El l 
M= eu f tan bn reo / ar —sin / a | (19) 


which vanishes for all values of P when w= s 


cy ny oa hae 
i De p tang 7 ns (20) 
By 
which becomes infinite for P= ane 


And expanding (20), 


-* Wi An? lia 
My=—M.=5-+ sgrpy Pf 14 an )+ 7580 (P ye be. b (21) 


where Pea Kuler’s limiting value of P for the ideal 
strut. 
This again shows the relation of the method (b) to the more 


exact method, the deflexions being reckoned above or below 


a 5 ‘a "To 


718 Prof. A. Morley on 


the points of inflexion which are at ¢= + L The coefficients 


of powers of, are nearly unity, and with any assigned ratio 


Jee 
Pp the error is easily estimated and with usual working ratios 
will be very small. 

Case V. 


ee straight tie-rod with uniformly distributed load w 
per unit length and ends freely hinged. 

This is the ; same as case I. with the sign of P reversed. The 
solution of the equation is 


er toy ye BE agel AJ ea r) . 
y= — poet oe or (1 —sech 5 F cosh , (22) 


and at the origin- 


wil? |. wHI l P ; . 
————— = e e e e s i] . e g 
Yo= gp a pr (sech5; EI =1) (23) 


and at the origin the bending moment with sign reversed, 


2 whl P 
My= ~~ —Py= “> (1- —sech er ap, (1—sech™4/F), (24) 


T- 
where P = Br 


And expanding (24), 
‘wE 5a? P| 6lat;P\? 
= OS ye" 4 Oe) de 
My= “541-45 -p. + 3700 (B) de. (25) 
per SS hale a ty fl ca eee athe FS ee ke. b. 
ie 3 Soa re i 600 (=) 36880 P) : (26) 


The errors invelved in the use of the method (0) are here 


Pry ae 
about the same as in case I. when is small, a condition 


which would not be fulfilled in a tie-bar carrying a reasonable 
pull unless / is small or the lateral loadis great. For tie-bars 
of considerable length (say.l greater than 20d for circular 


sections) : is generally greater than unity, and method (6) 


is no longer of any use when the lateral load is only the 
weight of the bar; but, on the other hand, the bending 
stresses are then small compared with the tension resulting 
from the axial pull P. 


Laterally ioaded Struts and Tie-rods. 719 
Prof. Perry’s approximation gives — 


wl? wi? fag 
Fa (PeePy eo TOR PP 


For P=0 this underestimates the deflexion by 24 per cent. 
but gives the correct bending moment ; for other values of P 
it overestimates the value of the bending moment by a pro- 
portion which increases as P is increased; ¢. g., if P=P, the 

sal 
error is 2°6 per cent., and if P, =9, a very possible value in a 
tie-rod1 of circular section, the error is over 12 per cent. If 
] is increased indefinitely M, according to this approximation 


approaches the limit ae instead of aa as in (24), a 


limit of error of 23 per cent. on the bending moment. But 
in cases where / is great the stress arising from the bending 
moment is usually unimportant compared with the axial 
tension, w being small. 

The intensity of bending stress 


M Kd l Do 
P= be = + FG (1—sech gy Tf), 


The maximum intensity of tensile stress is 
wkd / i Po 
=" = ——_ — Se == 3 Z 
cy P,+Po Beye sech 57 om) + Po ; ( 8) 


and the maximum intensity of compressive stress is 


wkd l Do 
Yom) Po PBs (1—sech 34/7) —Po + (29) 


which when negative gives the minimum intensity of tensile 
stress. 

The pull pp per unit area of cross section which with given 
load and dimensions makes f/, a minimum may be found from™ 
(28), and the value of py) which makes /, zero may be found 
from (29). 


Case VI. 3 


Uniform straight tie-rol with central load W and ends 
freely hinged. 
The equation is 


oe ~ ig —*)- tT ey 


720 Prof. A. 8. Eve on Changes in Velocity in an 


The conditions rae as in cases I., II., and V., 


Bl a eye 
-> eR fe taahs4/ ap - - CD 


and the central bending moment 


W EI l bE | ual 
—-M)= > ?P tanh 5 , EI : Pt tat (32) 


and expanding this 


_W!l a (P wt (P\2 170° (P ‘i 
ts 4 43- ole) . fie) sig F) i }. ee 


wv HI 
a! 


where Ey — 


or 


ee ot ~ gem? 41- - Ole ‘+ Tae) ~ oh. G2 


Here as in case V. for calculation of bending stresses 


method (0) is evidently a good approximation only when ee 1S 


small, 7. : when the bar is short or the lateral load great : 
for ~~ Te the error is 3 per cent. 

Other cases with the same simple loadings as the above 
naturally suggest themselves, such for instance as those in 
which clamps fixing the ends at any assigned inclination to 
the axis of w increase or decrease the flexure due to the lateral 
and axial loads, and those in which the longitudinal loads not 
passing through the centres of area of the ends of the strut 
or tie-rod may increase or decrease by their eccentricity the 
deflexion and bending moment resulting from the lateral 
loads. 


LXV. The Changes in Velocity, in an Electric Field, of the 
a, B and Secondary Rays from Radioactive Substances. By 
A. 8. Evz, M.A., Assistant Professor of Mathematics, and 
Lecturer in Radioactivity, McGill University, Montreal*. 


i Lge a paper on Secondary Radiation communicated to this 

Magazine in December 1904, it was stated by me that 
the secondary rays from substances, due to the 6 and y rays 
of radium, were homogeneous in character, and on that 


* Communicated by the Author, 


Electric Field of the «, B, and Secondary Rays. 721 


supposition values of the coefficients of absorption by alu- 
minium were given for various radiators. It has since been 
shown by 8. J. Allen*, and others, that the secondary elec- 
trons really move with velocities as widely different as those 
of the 8 rays themselves. The mistake in my paper arose 
partly from working with too thick an aluminium face 
to the electroscope, and from using too light a substance, 
namely paper, as radiator. Allen also proved that the 
secondary radiations from lead or zinc had penetrating powers 
and, therefore, velocities almost as great as those of the 
particles from radium ©; but that is not the case for light 
substances such as wood, paper, aluminium or paraffin. 

It seems desirable in the first place to give corrected values Tf 
for the coefficients of absorption by aluminium in the case of 
some different secondary radiating substances, because these 
results have considerable bearing on the subsequent and main 
work described in this communication. 

The values of » for secondary rays have been determined 
in the following manner :— 

About fourteen milligrams of pure radium bromide were 
placed in two sealed glass test-tubes. The primary @ and 
y rays struck the radiating plates, which were several centi- 
metres thick, and had an area of 22 x 22 square centimetres. 
The secondary incident rays were tested in a gold-leaf 
electroscope of small natural leak, measuring 10 x 10 x 16 c.c. 
One of the larger faces was removed, a few fine wires were 
stretched across, and the open side was then covered with thin 
aluminium leaf 0°00031 cm. thick. Rutherford has found that 
passage through such a sheet of aluminium reduces the length 
of the range in air of the a particle by 0°5 cm. oniy. It is 
certain, therefore, that such a layer of aluminium will 
scarcely affect the passage of 8 or of secondary rays. In 
front of the electroscope aluminium screens could be placed ; 
both these and the electroscope were separated from the radium 
by 15 cms. thickness of lead. The coefficients of absorption 
were calculated in the ordinary way between the various thick- 
nesses stated in the first column of Table I. In the first 
and second rows of figures the atomic weights and secondary 
radiation values are stated. In the second column are the 
values of X for primary rays, and in the other columns the 
values for secondary rays from the various substances named. 


* Phys. Review, August 1908. 

+ Since this paper was forwarded for publication I have read acom- 
munication by H. Starke in Le Radium (Feb. 1908). He had obtained 
results with which those given in the first part of this paper are in general 
agreement, 


eR is: 


722 Prof. A. 8. Eve on Changes in Velocity in an 
Tase I. 


Values of for aluminium as absorber. @ and y rays. 


| 


} 


| PRIMARY. | SEconDARY, | 
Substance ...... eae Lead. Tron. Brick. | Carbon. 
Atomic sand see 207 56 eee 12 
Secondary 
ee } ee 106 72 48 36 
Thickness | 
in cms. Al. 
0 —-009 48 48 71 70 90 
Ogee 0! 24. 29 40 47 50 
-021—-042 19 20 31 23 28 
042—-063 14 16 20 18 16 
063— 084 12 14 15 10 
| 084—-105 19) i Pala 14 


These figures show that the secondary rays from lead are 
nearly similar to the primary rays from radium, as Allen 
proved. The rays from the lighter substances are more easily 
absorbed, especially at first. The lighter the substance, the 
less is the radiation, the less is the grcup velocity, and the 
quicker the absorption by the screens. 

It will be proved later by an electrical method that this 
is a case not of selective absorption, but of difference in 
velocity. The table shows also that the more penetrating rays 
from the various substances approximate in character to one 
another and to the primary rays. As the lighter substances 
emit electrons which, regarded as a group, have lower velocities 
than those from the heavier substances, we must conclude that 
the secondary rays are to a large extent intrinsic and projected 
from the radiating substance ; they are not merely dittusely 
scattered primary rays. It may be remarked, however, that 
Allen has shown that the tertiary rays, from lead-lead-radium, 
are more easily absorbed than the secondary rays from lead. 
It is not difficult to arrange lead plates in such a way as to 
observe tertiary and even quaternary radiations. The relative 
amounts observed by me in one arrangement of apparatus 
were— 

Secondary ... .. 70°5 divisions a minute, 
Tertiary Jo ese Ox rere “4 
Quaternary: .<:fa, Ue ae * 


Electric Field of the a, B, and Secondary Rays. 723 


The values of A are more difficult to determine, but using an 
aluminium screen 0:021 cm. thick, the values were about 


Prtanby eee ates suis: 24 
S CUO OF gece teas 29 
PERL ye eae A 3 60? 


It is strange that whilst secondary rays from lead are so like 
the primary rays which cause them, yet the tertiary should be 
different in absorbability from both. Judging from the curves 
published by Allen, the tertiary rays from lead are absorbed 
at the same rate as the secondary rays from paper, and my 
results seem to agree; for the tertiary from lead and the 
secondary from carbon have values for X not far apart. 


Penetrating Secondary Rays. 


If lead and brick are respectively examined as radiators, 
and their radiations are in turn cut off from the electroscope 
by a thick book, or by a sufficient thickness of aluminium, it 
will then be found that brick becomes a more efficient radiator 
than lead. If curves are plotted with the thickness of screens 
as abscissee and the ionizations in the electroscope as ordinates, 
then the curves cross one another. The ionization values 
obtained in one experiment were :—. 


~ Tapue II. 


Thickness of Aluminium 


Gosden, Lead. Brick. 

cm. 
0 106 48 

04 37 15 

08 20 10 
"16 83 Cid. 
nae 57 ta 
‘50 4°3 6-9 


The lead and brick radiators were both a little more than 
5 cms. thick, and of nearly the same area. Again, using 
two stout books as screens (Drude’s ‘ Optics’? and Wood’s 
‘ Optics’), protected in all cases from the radium by 15 cms. 
of lead, the figures obtained were— 


No iradiator” / i420. 4-1 divisions a minute, 
Lead radiator ...... A°8 
Brick radiator ...... 6°8 


93 99 


724 Prof. A. 8S. Eve on Changes in Velocity in an 


Using another book as screen the values for the radiators 
named, for a different arrangement from the above, were— 


Radiator. Ionization. 
Nios 6. dv4e0, Gate ee ee 3°00 
head oo og ia ec eee 3°44 
Books. c., Serre Slab Leste A*15 
Carbon: fone se eee AS 50) 
Trg 2," 0 See ant eee 4°60 
Ceniant.: 255... eee 4°80 
Briel) he en ee 5°00 


These results agree with the statement made in my first 
paper, that thick blocks of brick, slate, cement, granite, give 
rise to very penetrating rays. It can be proved that these 
rays come from considerable depths. Witha book screen and 
carbon radiators the effects in divisions a minute were— 


Carbon. 
No. of Radiating Plates. Secondary Radiation. 
Oy tik hie Sian Lae 3°00 
Hey basa cts eae dc Aetg 3°48 
Dash le sha terae ena: aD 4-00 
D Wie eRe Ble 4°52 


3+ a brick behind... 4°68 


Now these carbon plates were each 1°5 cm. thick. 

Again, with wood or slate radiators, 0°75 and 0°5 cm. 
thick respectively, the measured movements of the gold-leaf 
were, when the screen was 4 sheets of aluminium each 0:04 
cm. thick, as follows :— 


RADIATORS. 


No. of Sheets of :— | Wood (each 0°75 cm.).| Slate (each 0°5 em.). 


Lee aeekede ss eee 6-42 6:96 
Dy caetconanet Aen. 6°48 7°68 
BD see atte etna 6°50 3:37 
A Se ake sn dearest eo 8:94 
Dhiba tegekeeaaee ss 6°70 

O Bvwikepeeocue bee 9°42 
MO bs ceenplcosteen 6°85 

Pit Ris ses oeaeeee 2°80 


DO aaetveae eee: 6°97 — 


Electric Field of the «, 8, and Secondary Rays. 725 


Thus these penetrating rays came from a depth of about 
6 ems. of wood or 4 cms. of slate. Itis difficult to decide 
whether these rays are high velocity scattered primary § rays, 
or secondary negative rays, or secondary y rays due to the 
primary y rays, or secondary y rays due to the expulsion of 
secondary 8 rays, or secondary y rays due to the stopping of 
primary 8 rays. There is some evidence that they are 
secondary y rays, but it is not conclusive. The paper by 
A. 8. Mackenzie, in this Magazine, July 1907, shows that 
there are remarkable conditions, depending on direction, when 
y rays traverse matter. 

In the experiments made by Cooke on the penetrating 
radiation due to the earth, he found that brick gave rise to a 
very penetrating radiation. As clay generally contains radium 
it was natural to attribute the effect observed by Cooke to the 
presence of radium in the brick. It is probable, however, 
that the penetrating rays were due to the secondary radiation 
set up in the bricks by the radium and thorium in the ground, 
and not chiefly by the minute amount in the bricks employed. 
[have not succeeded in obtaining these penetrating rays from 
lead ; they appear to arise in the lighter substances. 


Electrical Method. 


By an important experiment it was proved by Lenard* 
that the cathode rays could be accelerated or retarded in an 
electric field, when the particles moved along the lines of 
force. He found that the value of e/m remained nearly 
constant, whilst the change in the velocity amounted to more 
than 20 per cent. for a difference of potential of 29,000 volts. 

It seemed possible to apply a similar method to the « and 
8 rays from the radioactive substances, and to the secondary 


rays produced by the @ and y rays. 
a Rays. 


A metal plate, 10 x 8 cms., was charged negatively and 
exposed to the emanation from thorium hydroxide until an 
active deposit of maximum. activity was obtained. ‘The plate 
wasthen placed parallel and opposite to the thin aluminium face 
of the gold-leaf electroscope, and was insulated on a block of 
paraffin so that it could be charged to about 30,000 volts, 
either positive or negative, by a Wimshurst influence-machine. 
One pole of the machine was earthed, a Leyden-jar was used 
to steady the potential of the other, and the potential, which 


* Lenard, Wired. Ann. xlv, (1898). 
Phil. Mag. 8. 6. Vol. 15. No. 90. June 1908. aC 


726 Prof. A. 8. Eve on Changes in Velocity in an 
could be modified by a spark-gap, was measured by a Kelvin 


‘ 


electrostatic voltmeter. 


| RANGE IN EMS (-0'5 ) 


_ IONIZATION — DIVS.PER MIN. | 


The range of the @ particle from ThC was found to be 
| 8:5 cms. by Rutherford and Hahn; and the bend at 
| 8 ems., shown in (fig. 1) is due to the entry of the « rays into 
| the electroscope at that range, for it is known that the 
| aluminium face of the electroscope is equivalent to 0°5 em. of 
! air. The curve (fig. 1) is drawn with the ionization effects 
as abscisse and the ranges as ordinates, following the well- 
known method of Bragg. 
| At 9 cm. no @ rays enter the electroscope, and a negative 
| charge given to the active plate accelerates the @ rays, and 
' a positive charge retards them. Oblique rays are bent towards 
| or from the electroscope according to the sign of the charge. 
The charge actually observed was 12 per cent. of the mean 
value, and the effect for negative was greater than the effect 
| for the positive charge. 
| At 7 cms. the a rays were accelerated by a positive charge 
: and retarded by a negative charge. ‘The difference observed 
was 3 to 3°5 per cent. for potential-differences of +30,000 
volts, and the effect for a positive charge was then greater. 
At 8 cms. no change could be observed because the accelera- 
tion of the « rays balanced the retardation of the f rays, 
and conversely. Only the very ends of the ranges of the « 
rays were inside the electroscope at this distance. 

If the field were uniform the « particles, projected in all 
directions, would describe parabolas, bent towards the electro- 
scope for a positive charge, and away from the electroscope 
for a negative charge. Yor the £ particles the reverse state- 
ment is true. The effect may be compared to the motion of 


Electric Field of the «, B, and Secondary Rays. 727 


water issuing from a rose at the end of a hose directed to or 
from the earth. : 

It is easy to estimate the change of velocity for an « particle 
moving by any path from the plate at 30,000 volts to the 
electroscope at zero potential, for the change of kinetic energy 
is equal to the work done, so that 


a 
dmv? —4tmuv?= +eV, 


e 
or vu? +2— V. 
Sy 7 


Here wu is the velocity of projection, v of arrival, V is the 
potential-difference, e the ionic charge, and m the mass, 
expressed in H.M. units,and grams. The sign selected 
depends upon that of the charge. It has been proved 
by Rutherford and Hann that for Th C, 

u=1-98 x 19°, <= 5:6 x 103, and V=3x 10". 
Hence 


= (aa +a30)10 and v=19-91 x 10°, or 19°74x 108 ; © 


so that the total calculated difference per cent. for a change 
of sign of the potential is 0°8. 

The effect of absorption by the air has not been considered. 
The change in ionization observed is not necessarily pro- 
portional to the change of velocity produced. The large 
change observed at 7 cms. (3 per cent.) was due in part to 
the bending of the rays, and to the consequent increased 
length of path, of the oblique rays, which is in the electro- 
scope. Moreover, oblique rays which barely reach the 
electroscope when the plate is uncharged would be bent inside 
when the charge is given. The fact that no effect was observed 
at 8 ems. for a reversal of potential shows that the a ray 
change then balanced the 6 ray change, and that the electric 
field altered the velocity of the normal « rays, and that the 
actual range of the @ particles was increased or decreased 
according to the direction of the lines of force. I have 
endeavoured, hitherto without success, to observe this change 
of range with a Crookes spinthariscope, using an ebonite 
cylinder ; the eyepiece was earthed, and a wire, covered with 
radium ©, was placed inside at the bottom on a plate raised to 
+30,000 volts. It is easy to show that the expected change 
of range must be small and difficult to detect, about 1:4 mm. 
for a reversal of potential *. 


* Rutherford, Phil. Mag. Oct. 1906. 
3 C2 


728 Prof. A. 8. Eve on Changes in Velocity in an 


A Kelvin dynamo-static machine, kindly lent by Professor 
Owens, enabled me to work ata higher potential than with 
the Wimshurst machine. But sparks passed from the Th C 
plate and perforated the thin face of the electroscope, and 
ions then rushed inside. The holes could be repaired with 
very thin aluminium leaf, but observations were difficult to 
make. : 

It may be here stated that it is important to have the mica 
windows of the electroscope as small as possible, otherwise 
the readings of the gold-leaf are not the same for positive 
and negative charges of the leaf, owing to induction effect, 
electric wind, &. It is well to have a large well-earthed 
sheet of wide-meshed wire-netting just in front of the electro- 
scope. This screens off the induction effect, and it does not 
interfere with the success of the experiment. ‘The motion of 
the gold-leaf was remarkably steady and unaffected by the 
high potential in its neighbourhood. 

This experiment with the « rays shows in a simple manner 
the fact, proved by Rutherford, and later by Becquerel, that 
the a@ particle carries a positive charge, and it indicates 
that the length of the range of these particles can be increased 
or decreased to a slight extent by an electric field with lines 
of force in the direction of motion. 


B Rays. 

Active matter containing radium, spread over a brass plate 
10 cms. in diameter, was covered by just sufficient aluminium 
to absorb the « rays, and to prevent the escape of emanation. 
This, made by Dr. Riimelin, serves as a convenient @-ray 
standard. The plate was placed in front of the electroscope, 
about 20 cms. from it, and it was charged to +30,000 volts. 
The total change in the ionization amounted to about 12 per 
cent. of the value when the plate was uncharged. A con- 
siderable part of this change is due to the bending of oblique 
rays, in or outof the electroscope according to the sign of the 
charge ; secondary rays from brass must be present. Using 
the dynamo-static machine and obtaining a higher voltage, 
about +50,000, a difference of more than 15 per cent. was 
produced by a reversal of the electric field. As the potential 
is gradually increased from zero, the ionizations for positive 
and negative charges on the brass plate are not equidistant 
from the mean value (fig. 2). It appeared easier to retard 
and decrease the rays entering the electroscope than to 
accelerate and increase them. It is unfortunate that there is 
so little evidence as to the relationship between the magnitude 
of the velocity of electrons and the ionization produced by 


. 
] 
- 
a) 
i ° 


Electric Field of the «a, B, and Secondary Rays. 729 
them. According to Townsend the ionization decreases as 


the velocity increases. Again, in the case of high velocity 


HO > 


O05 — 


JON/IZAT/ON. 


700. 


D5 pe 


90. 
POTENTIAL /N VOLTS 


electrons it is difficult to increase their speeds because, as 
Kaufmann and others have shown, there is an increase of 
mass as the velocity approaches that of light. There are then 
two reasons why the outer curves, shown in fig. 2, should 
not be symmetrical about the axis of «. 

It is easy to calculate the velocity of the @ particle on its 
arrival at the electroscope. As before, we write 


v=ut+2—V. 
ane 
But here we may take 


ae eid 6 x 10°. 
m 


Sees =6T x 10 and 2% V=6 x10", 
so that v—=(60-.6)10" 

=O her! Fa:; 31, 
and y= 7-A46,.0m,2740,;%, 10“. 


There is then a calculated difference equal to 9°2 per cent. 
of the mean value. But the difference produced in the 
velocity need not cause a proportional difference of ionization 
in the electroscope. The result calculated above is inde- 
pendent of the path of the electron. Butif the number of 
electrons entering the electroscope varies for positive and 


| 


Te 


730 Prof, A. 8S. Eve on Changes in Velocity in an 


negative charges on the active plate, then the observed effect 
will not correspond with that calculated. 

In order to test this point experimentally, a capsule con- 
taining 1°5 mgs. of radium was placed so that the @ rays 
passed through a small hole in a block of wood, and entered 
normally the centre of the face of the electroscope. The 
diameter of the hole was 1°5 cm., and the length of it was 
10 cms. The block of wood was covered with thin aluminium 
foil, was insulated on a block of paraffin, and charged as 
previously described. The observed differeace for positive 
and negative potentials was 6°3 per cent., or about two-thirds 
of the calculated value (inner curves, fig. 2). It must 
be borne in mind that the 8 rays of radium always include 
secondary rays, and that a group of 6 rays has a very wide 
range of velocity, from almost that of light to that of the 


slowest electrons produced by secondary, tertiary, or higher 


order radiation, The value for the velocity of projection of 
the 8 particles assumed in the above calculation was selected 
from Allen’s paper*, as being that derived under conditions 
most similar to those obtaining in my work. 

In this case, as with the & particle, no allowance has been 
made for the absorption and scattering by air between the 
plate and the electroscope. Under more rigorous experimental 
conditions, it is possible that this method might throw light 
on the ionizations produced for various velocities of the 
electrons. 


Secondary Rays. 


The method described in this paper was next applied to 
the investigation of the nature ot the secondary rays from 
various substances due to the 8 and y rays from radium. 
About 14 mgs. of pure radium bromide were sealed in two 
thin glass test-tubes. The radiators were 5 cms. thick and 
measured 22 x 22 ems. (fig. 3). They were insulated on 
blocks of paraffin and charged by the Wimshurst machine. 
A line from the radium to the centre of the radiating plate 
made an angle of 60° with the normal. When the radiator 
is charged to a high potential the primary rays will be repelled 
from, or attracted to, the radiator to an extent depending on 
the sign and the magnitude of the potential. This variation 
of the primary rays might have been avoided by using y rays 
only ; but y rays are always accompanied by secondary rays 
from the screen which cuts off the 8 rays. Moreover, if all 
the radiators are of the same size and shape, and if they are 
placed in turn in the very same position, then, for any given 


* Phys. Review, Aug. 1906. 


Electric Field of the «, 8, and Secondary Rays. 731 


potential of whichever sign, the amount of primary @ and v 
rays falling on the plates will also be the same for all 


Fie. 3, 


To EARTH 


[MIRROR | 


|ELECTROSCOPE | 


[| PADIATOR {3h 


ON i Fiebal fishes 
PARAFFIN | |70 WIMSHURSTT 


substances used as radiators. Thus we have a thoroughly 
satisfactory comparative test of the secondary radiations from 
the various materials employed. 

The results obtained are shown in fig. 4 (p. 732), and it will 
be seen that the curves for each substance are unexpectedly 
symmetrical *. The abscissee denote potentials and the ordi- 
nates the percentage ionization effects observed. The upper 
curves were obtained when the radiators were negatively 
charged, and the lower curves for the reverse charge. 

It will be noted that the lighter substances emit rays 
which, regarded as a group, have a relatively low velocity. 
These slow electrons are readily retarded and made to 
diverge when the radiator has a positive charge, and easily 
accelerated and concentrated by a repulsion towards the 
electroscope. The numerical values are given in Table III., 


p. 732. 


* The details of these curves (fig. 4) would repay further investigation. 
They diverge near the origin somewhat more rapidly than is shown in 
the figure, and there are some indications of points of inflexion on the 
upper curves. The details will necessarily depend on the angles and 
distances between the principal parts of the apparatus. 


732 Prof. A. 8. Eve on Changes in Velocity in an 
Fig. 4. 
SECONDARY . RAYS 


: po 
7s Pe 
ia Tee ge 


LLOe 


5 
i 


eye) 


PERCENTAGE JONIZATION. 
ito) 
ig, 


90 


RADIATOR > 


8s 
80 
75 
TABLE ITT. 
Difference per cent. for | 
Substance. | Density. ee r. 
adiation.| +34000 | -+50,000 
volts. volts. 
League eee 11:4 113 16:5 24 28'3 
ron: acc eee 78 95 246 36 42°0 
} Aluminium ... 2°6 80 35'8 ioe | nae 
Bmnekws Nags D2 79 340 er 50:0 
i Paraffin ..... 09 65 41-0 48 | 


Electric Field of the a, B, and Secondary Rays. 738 


The third column gives the secondary radiation actually 
observed from the blocks used as radiators. The fourth 
column shows the differences per cent. on the mean value 
when the radiators were charged to 30,000 volts, first positive 
and then negative. In the fifth column are the differences 
per cent. for a higher potential, obtained with the dynamo- 
static machine. In the last column are the values of the 
coefficients of absorption by aluminium, 0°021 em. thick, 
requoted from the first part of this paper. In this experiment 
the bricks were slightly moistened to make them conductors, 
and ihe paraffin was covered with aluminium foil, ‘00031 cm. 
thick. This thin layer of aluminium did not affect the 
radiation to a measurable extent, and it made the surface a 
conductor. 

By a comparison of the fourth and fifth columns it appears 
that it becomes increasingly difficult, as the potential rises, 
to magnify the percentage differences for opposite charges 
of the radiator. The values of the coefficients of absorption 
in Table I. are also in accord with the view that together 
with the slower electrons, which form the bulk of the radia- 
tion from the lighter substances, there are also swifter 
electrons, which are with greater difficulty absorbed by 
matter or affected by an electric field. 

These experiments as a whole indicate that the secondary 
rays from various substances have a distinctive group velocity 
depending on the density or atomic weight of the substance 
of the radiator. The values of the secondary radiation follow 
the order of the atomic weight, as shown by McClelland : 
it is now seen that the values of ) are also in inverse order 
of the secondary radiations, and that the percentage changes 
of ionization, due to the reversal of the electric field, are in 
that same inverse order. 

Hence, secondary rays are in the main intrinsic, released 
from the atoms of the radiating substance, with distinctive 
group velocities depending on the density of the radiator. 
It appears that the secondary rays are for the most part not 
due to dispersed primary rays, which have entered the - 
radiator, and by changes of path re-emerged ; but it is not 
improbable that the secondary rays may include an unknown 
fraction of such primary rays. Thus 8 and y rays appear 
merely to release the electrons, which issue from the radiators 
with velocities possibly depending on those which they had 
in the radiator*, ~ 


* Since writing the above I have received a paper by Brage and 
Madsen (Trans of R. Soc. of S. Australia, 2 Jan. 1908; Phil. Mag. May 
1908) advocating a different view. 


734 Prof, A. S. Eve on Changes in Velocity in an 


If we take, for any radiating substance, the square root of 
the double ordinate, or percentage difference, at 20,000 or 
at 30,000 volts, and multiply by the actual secondary radia- 
tion from the plate, we get, with no great accuracy, a 
constant, 


For example, with +34,000 volts :— 


| 


Substance, | Bereontage | Square | Secondary | Product 
Carton s65.25 35°6 5°96 69 410 
Paraffin ......... | 41-0 6-40 65 415 
IBTICK.ic.58eeeeee / 34:0 5°84 79 470 
Aluminium ... 30'8 5°98 80 476 
Tren tue ee. 24-6 4:96 95 470 
Whead > acteaceae 16°5 4:06 113 460 


This approximate rule emphasises the relationship between 
the velocity of the secondary rays and the amount of secondary 
radiation. The rule requires further investigation before it 
ean be affirmed. 

At some distance from the origin the curves shown on 
fig. 3 crudely resemble the parabola «?=ky, where k 
is a constant for each radiator. In the case of the less 
dense substances, it is found that when either the potential, or 
the distance between radiator and electroscope, is increased, 
there is not symmetry about the # axis, but the positive 
ordinates are numerically less than the negative ordinates for 
the same abscissee. With a sufficiently high positive voltage 
the ionization could be reduced to zero, and a very large 
negative potential would cause each curve to approach an 
asymptote parallel to the # axis. 

Some preliminary experiments were made to compare the - 
quantity and quality of the secondary rays from hot iron 
plates and from hot bricks. The results were but slightly 
different, if at all, from those obtained from cold bricks and 
cold iron. Observations were made on radiators from a red- 
heat to the temperature of the room. 

It is usual to suppose that the secondary rays, which start 
from some depth within the radiator, must lose velocity on 
their passage through matter before they make their escape. 
The remarkable experiment of H. W. Schmidt * throws doubt 


* Phys, Zeit. June 1, 1907. 


Electric Field of the a, 8, and Secondary Rays. 735 


on this view. The result obtained by him was not for a wide 
range of thickness, amounting only to 0°5 mm. of aluminium, 
but within that limit he proved that the @ rays from radium 
E continued to move with undiminished speed until they 
were actually absorbed. There is no doubt that after passage 
through matter the swifter electrons emerge, because the 
slower are first of all absorbed. The question remains an 
open one, whether the @ particle loses velocity during its 
passage through a considerable thickness of matter. The 
question is always complicated by the attendant train of 
secondary rays which everywhere arise in the path of the 
particle through matter from the moment of its emission to 
the time of its complete absorption. In the case of the « 
particle, Rutherford has clearly proved that velocity is lost 
during passage through matter. 


In conclusion, one more experiment may be described. 
Fourteen milligrams of radium bromide were enclosed in a 
cylindrical block so that the y rays passed through 2°5 ems. of 
lead before emerging to the air. The lead block was insulated, 
and placed before the electroscope. The potential was 
changed from + to —30,000 volts. The difference of the 
ionization observed was 3 per cent. of the mean value. The 
inner curves (fig. 2), otherwise obtained, almost represent 
the results of this experiment. The total ionization observed 
was* mainly due to y rays, and the change of ionization, 
on reversal of the sign of potential, was due to the electrons, 
which are projected from the surface of matter traversed by 
the y rays*. 

This paper must be considered as preliminary, and there 
are some criticisms which may be applied to the method and 
to the arrangement of apparatus. Thus it is difficult to 
estimate the effect of the scattering and absorption by air 
of the rays employed, but it is not relatively large. Again, 
it might be expected that some of the ions generated outside 
the electroscope near the active plate would be accelerated 
in the electric field sufficiently to enter the electroscope. 
This either did not take place, or the effect was relatively 
too small to be noticed. 

It is hoped that these experiments may be extended to 
other substances and that higher potentials will be employed. 
The electric method may throw some light on the ratio of the 
amounts of secondary X rays and of negative rays from 


* Phil. Mag. Dec. 1904. 


736 = Changes in Velocity of a, B, and Secondary Rays. 


radiators exposed to primary Réntgen rays*. ‘The investiga- 
tion of transmitted rays, and particularly of the rays from 
hot and cold bodies, should prove interesting when this 
method is applied. 

Some of the work done overlaps that by Allen, with 
magnetic and electric deflexion methods. I trust that full 
justice has been done in making references to his valuable 
paper. 


Summary. 


1. The coefficients of absorption by aluminium of the incident 
secondary radiations, due to the 6 and vy rays of radium, 
produced from various substances have been determined. The 
lighter substances emit negative rays not only less in 
quantity than the dense, but these rays are initially more 
readily absorbed. (Allen.) 

2. Some substances, such as brick, slate, wood, paper, and 
carbon, give rise to penetrating incident secondary rays 
originating from several centimetres depth, which are either 
secondary y rays or high velocity negative rays. After 
screening, the secondary radiation from brick or paper may 
exceed that from a heavy substance, such as lead. 

3. In an electric field, with the lines of force parallel to 
those of direction of motion, the velocity of the « rays can 
be increased or diminished, and there is apparently a slight 
variation of range. 

4, The same method may be applied to primary 8 rays or 
to secondary rays due to 8 and y¥ rays. 

5. Ihe amount of secondary radiation is known to follow 
an order depending on the atomic weights of the radiators. 
It is shown by the method herein described that the velocities, 
recarded as a group, as well as the initial absorptions by 
aluminium, follow the same order, so that the main secondary 
negative rays are projected with velocities, which are a 
function of the density, or the atomic weight, of the radiating - 
substance, the velocities being greater for the more dense. 
The main group of electrons therefore emerge with less 
velocity from the lighter substances. 


* Recent experiments have been made, using the electric method 
described in this paper, with the secondary rays due to (1) X rays and (2) 
y rays. In the case of X rays the cathode secondary rays are so rapidly 
absorbed by the air that they do not reach the electroscope. Hence a 
change from + to —50,000 volts in the potential of the secondary 
radiator made no difference in the ionization current. Perhaps there was 
a slight effect in the case of lead. The secondary rays, due to y rays, 
gave percentage changes about equal to those observed in the secondary 
rays due to both B and y rays of radium. 


Electrical Charge of the Active Deposit of Actinium. 737 


6. The square root of the percentage differences of ioniza- 
tion, due to a reversal of potential of the radiator, is inversely 
proportional to the secondary radiation from the radiator. 
This is a rough approximation for radiators of equal areas 
made of different substances. 


McGill University, Montreal, 
February 1908. 


Note added 25th April, 1908. 


In this paper I have upheld the old view that the 
secondary rays due to radium are not merely scattered 
primary. The experiments of H. W. Schmidt (Phys. Zeit. 
June 1, 1907) have to some extent weakened this theory. 
But the same observer has also shown that the @ particle 
does not lose speed in passing through matter; and, if that is 
the case, the secondary electrons should have the same velocity 
as the primary; and they have not. Again, y rays (we will 
suppose them to be ether pulses) give rise to secondary rays 
with velocities nearly equal to those of the secondary rays due 
_ to @ particles. There seems no reason to suppose that the 
8 rays could not cause the same projection of electrons from 
the atoms of the secondary radiator which we know that the 
y rays cause. It is true that the view, advocated by Bragg, 
of the material nature of the y rays is not out of harmony 
with observed facts; and, if that theory is correct, we may 
regard all secondary radiations, consisting of electrons with 
high velocities, as scattered primary rays. 


LXVI. On the Electrical Charge of the Active Deposit of 


Actimum. By Sipney Russ, Demonstrator in Physics, 
Manchester University*. 


ee by several observers have shown that 
the active deposit of Thorium is aimost entirely directed 
to a negative electrode. 

If into a vessel containing thorium emanation two wires 
be inserted, one charged negatively, the other positively, the 
amount of active deposit obtained on the negative wire is 
more than one hundred times that on the positive; this 
number, however, shows a considerable reduction if the 
pressure in the vessel is diminished to a few millimetres. 


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


738 Mr. 8. Russ on the Electrical Charge 


While working along similar lines with Actinium an 
exactly opposite effect was observed. Under the conditions 
of these experiments the activity of the negative was only 
twice that of the positive pole at atmospheric pressure, this 
ratio being increased to twenty-two on reducing the pressure 
to a few millimetres. (Phil. Mag. May 1908.) 

It was found, however, that by varying the experimental 
arrangements, the activity of the negative electrode could be 
made to vary from twice to more than a hundred times that 
of the positive electrode, without altering the pressure. It 
appears from the observations to be described, that the 
variable which determines the electrical charge of the 
carriers of the active matter, is the distance which they 
travel before reaching the electrode. If this distance ‘be 
very small (a few millimetres) the particles are almost com- 
pletely directed to the negative pole, the ratio mentioned 
above therefore being very large; as the distance of travel is 
increased this ratio shows a marked diminution. 

The experiments indicate that the collisions of the active 
deposit. particles with the gas molecules or ions with which 
they are mixed, determine the sign of the electrical charge 
which they exhibit. 


Methods of Experiment. 


The first experiments, which have already been partly 
described in the paper referred to above, were made in the 
following manner. 

Fig. 1. 


E 


JO Ge 


ic haig i 


Two thin brass wires (A and B, fig. 1) 7°5 cms. long and 
‘7 mm. diameter and 4 cms. apart, were made to lie along 
the axis of a brass tube C 30 cms. long and 4°2 cms. diameter. 
This tube was fitted with a small capsule D into which was 
placed a small quantity of a preparation of actinium. 

The wire A was connected to the positive and B to the 


Ww 
NE 
<i 
ei > 
» 
Lie 


JPATIO 


of the Active Deposit of Actinium. 739 


negative pole of a battery of 320 volts, the brass tube being 
connected to the mid-point. 

The exit-tube E could be connected to a Fleuss pump and 

a set of tubes containing calcium chloride and cotton-wool, 
ie ensuring a supply of dry and dust-free air. 

The wires were exposed usually for about two hours, when 
they were removed and their activities compared by measuring 
the saturation currents, which they separately produced in a 
cylinder connected to an electrometer. 

Observations were made for pressures between 2 mms. and 
76 cms. of mercury. The numerical results are collected in 
Table I. and shown graphically in fig. 2. 


Fig. 2. 


fo) 


o ne 20 30 40 50 60 70 


PRESSURE 1N C45 


TasueE I. 
| ee | 

| Activity a eee | Ratio of | 
ee cca : Activities. 

| _ in cms, | Sek. 

Cathode. Anode. +yve 
ee i ee 15 75 2 | 

5 | SRP ear AEnE ae 35 87 + 

i) pee ee ete 62 52 12 

| 5, pean 137 | 57 24 

Gy 22 Diss) 200 55 36 
BR oa hel Sys ees 270 4°5 60 
fr Mea 88 4:0 22 | 


See eee oe eee Ee POE ee 
It will be observed that the ratio of the activities of the 
two wires increases steadily as the pressure is diminished, 


740 Mr. 8. Russ on the Electrical Charge 


attaining a maximum value at about 5 cms. pressure, then 
falling abruptly at a pressure of about *2 cm. 

Owing to the very short life of the emanation from actinium 
(3°9 seconds) the distribution of the emanation in the cylinder 
C must be very different at the different pressures employed. 
At atmospheric pressure the emanation, owing to its rapid 
change into actinium A, will be confined to a comparatively 
small region round D, and the active deposit particles will have 
a greater distance to go before reaching the wires A and B 
than at the reduced pressures. Itis probable that the decrease 
in the ratio of the activity of the cathode to that of the 
anode between pressures of 5 cms. and 2 mms. corresponds 
to the similar effect with thorium which has been already 
mentioned. 

It is worth noting that the activity observed on the anode 
is almost entirely due to some of the active deposit particles 
being negatively charged. This was shown as follows :— 
An experiment was made in which no field existed between 
the wires and the containing cylinder, the activity observed 
on either of the wires was only about one-seventh part of 
that observed on the positive electrode. Hence the above 
conclusion. 

At this stage it was resolved to vary the conditions of the 
experiment. : 

A preparation of actinium was placed at the bottom of a 
cylindrical vessel A (fig. 3) 9 ems. long and 4°5 cms. diameter, 


Fig. 3. 


=> Pump 


Actimiu/t. 


whichiwas:closed by a rubber stopper through which passed 
two brass rods carrying two vertical brass plates B and C ; 


of the Active Deposit of Actinium. TAL 


the brass plates, 1°5 cms. apart, were fixed parallel to one 
another and connected to the terminals of a battery as before; 
and their distance from the actinium could be varied as 
required. 

Two series of observations were made, at 76 cm. and 2 mms. 
pressure, for different distances of the plates from the 
actinium. After an exposure of about two hours the plates 
were removed and their activities compared by means of an 
a ray electroscope. 

The numerical results are collected in Table II., and the 
variation with distance of the ratio of the activity of the 
negative to that of the positive plate is reproduced graphically 
in fig. 4 (p. 742). 


TABLE II. 
Activity in Arbitrary Units. | Batio. 
Paice) Swami |e... Fee 
Actinium. | Negative plate.| Positive plate. +ve 
ae al Pressure. 
pews v2.2. | 300 BE Oy) (vv AO 
BERR seacctaex 250 3°0 | 83 
eee 220 ot 72 
ia [Rak 0g 4-2 48 
a teers a, 54 22 
aan 100 77 13 
ee co. sig ae ce en 
ees | 10 4:3 2:3 
| 2 mms. Pressure. 
SM esae sce: | co 16 4-9 
Ls TA E5 4:8 
yan ss en 1 9-6 
12, i ree 22:0 | 1:0 22-0 


An inspection of the second and third columns of Table II. 
shows that at 76 cms. pressure, whereas the active matter 
deposited on the negative plate decreases as the distance of 
the plate from the actinium is increased, the amount obtained 
on the positive plate increases till a distance 2°7 ems. from the 


Phil. Mag. 8. 6. Vol..15. No. 90. June 1908. 3 D 


742 Mr. 8. Russ on the Electrical Charge 


actinium is reached, after which a diminution occurs, pro- 
bably owing to the fact that the total quantity of active matter 
collected on the two plates is then very small. 

At 2 mms. pressure the active matter obtained on the 
negative plate shows an increase as the plate is removed from 
the actinium, while the quantity on the anode shows a steady 
decrease, 


Fig. 4, 


NEGATIVE 
POSITIVE 


RATIO 


Distance in Cms. 


Crosses refer to 76 cms. pressure. 
Circles refer to 2 mms. pressure. 


It will be seen from fig. 4 that at atmospheric pressure 
the negative plate is more than one hundred times as active 
as the positive for small distances from the actinium, a very 
large diminution in the ratio resulting as the distance is 
increased. The opposite effect holds at a pressure of 2 mms. 

It has been shown by Debierne* that the amount of 
emanation obtained from a uniform layer of actinium falls to 
half value on going ‘55 cm. from the layer. We should, 
therefore, expect a diminution in the activity of the plates 
as their distance from the actinium is increased; but the 


Le Radium,’ 1907, p. 218. 


of the Active Deposit of Actinium. 743 


RATIO of the activities on the two plates should remain the 


same. This is, however, far from being the case, and more- 
over the positive plate shows signs of increased activity as 
its distance from the layer of actinium is increased. It 
appears then that in their passage through the containing gas, 
the positively charged active deposit particles may become 
neutralized, and in fact may even become reversed in sign. 
To account for the increase in the ratio observed at the low 
pressure, it must be remembered that at the moment of 
formation the active deposit particles have a very high 
velocity. Rutherford* has suggested that a certain number 
of collisions may therefore be necessary to reduce their speed 
so that the electric field may be capable of directing them to 
the electrodes; hence as the distance of the plates from the 
actinium is increased at the low pressure, their total activity 
should increase, as is found to be the case. 

It might be expected that if the distance of the plates from 
the actinium were made sufficiently large, even at the lower 
pressure, similar effects to those obtained at the high pressure 
would be observed. With the small cylinder described this was 
not possible ; a larger one 40 cms. long and 7:5 cms. dia- 
meter was used, and experiments at an intermediate pressure 
(1:2 cms.) were conducted in an exactly similar way to that 
already described. 

The numerical data are collected in Table III. andirepro- 
duced graphically in fig. 5. 


TaBue ITT. 
Ratio of Activities. 

| Distance from Negative 
Actinium. | Positive 

| Beni 7. .....,,| 26 

Ree os bt apie | 44 

LET Oh) a ee ee | 62 

SO cus ht iipeeeeced 76 

ON) (ote: | 82 

REO Ty! y Rey 28 | 63 

Bel AN 5 BRS! 37 

och Ho see weess 25 


* * Radioactivity,’ p. 319. 
DEP 2 


744 Mr. 8. Russ on the Electrical Charge 


The amount of active deposit obtained on the plates decreased 
as their distance from the actinium was increased, but the rate 
of decrease was different for the two plates. 

Inspection of the figure shows that at first there is an 
increase in the ratio of negative to positive activity as at the 
lower pressure, which subsequently shows a decrease similar 
to that observed at the high pressure. 


Fig. 5.—1'2 ems. pressure. 
0 * 
° 


70 


‘3 


NEGATIVE 
POSITIVE 


AATIO 


ra) Ss 10 1° 20 25 3O 


Distance 1w Cs. 


It must be noted that in the region between 11 and 18 ems. 
from the actinium it was impossible to obtain concordant 
results, sometimes a high and sometimes a low value for the 
ratio resulting. 

It has been observed by Debierne in the paper already 
referred to, that there is a position of maximum activity in a 
vessel containing a preparation of actinium, but the writer 
does not know whether the experiments here described have 
any bearing on that point. 

The question arises as to whether the apparent loss of the 
positive charge of the particles can be explained by their 
diffusion to the sides of the vessel, but considering the re- 
latively small distance travelled necessary for the observed 
effects, it is difficult to see how loss by ditfusion can account 
for them; moreover, it would not explain the marked increase 
in the activity of the positive plate in the experiments at 
atmospheric pressure. 


of the Active Deposit of Actinium. 745 


Conclusion. 


It may be useful here to draw attention to some of the 
differences that have been observed in the transmission in 
electric fields of the active deposits of the different radioactive 
substances. 

In dry and dust-free air at atmospheric pressure the active 
deposit of Thorium is almost completely confined to the 
negative pole ; on reduction of the gaseous pressure to a few 
mms. a marked diminution in the amount transmitted to the 
negative pole is observed, some of the active deposit then 
being found on the positive boundary of the system. 

In the case of Radium the transmission of its active deposit 
to the negative pole is not so complete as with thorium ; at 
atmospheric pressure about 95 per cent. of the active deposit 
is directed to the negative pole, the remaining 5 per cent. 
going to the positive pole. With reduction of the gaseous 
pressure similar effects are observed to those mentioned for 
thorium, the activity of the negative boundary decreasing, 
that of the positive increasing. 

With Actinium the active deposit may at atmospheric 
pressure be almost completely directed to the negative pole, 
but by suitable variation of the experimental conditions, the 
distribution may be rendered much less complete, the amount 
obtained on the negative pole being reduced to as little as 
twice that on the positive. 

At low pressures, there is the same possibility of the 
phenomena observed depending on experimental conditions. 

From the results of several independent observers it seems 
clear that the electrical charge exhibited by the active deposit 
particles of the radioactive substances is mainly determined 
by the numerous collisions between these particles and the 
ions or molecules of the gas with which they are mixed. 

Whether the active deposits are electrically charged at the 
moment of formation from the emanations, still remains an 
open question. Although experiments with the active deposit 
from Radium show that at very low pressures (about ‘01 mm.) 
almost as much is directed to a positive as to a negative pole, 
this does not preclude the possibility of the active deposit 
particles being really positively charged, for at this pressure 
the velocity of the particles is so high that no ordinary electric 
field would be capable of directing them to the negative 
boundary of the system. 


| 746 J 


LXVII. On Molecular Aygregations produced in Gases by 
Sudden Cooling. By Gwitym Owen, J/.A., WSe., 
Assistant Lecturer and Demonstrator in Physics, and 
A. Lu. Hueuss, B.Se., Oliver Lodge Fellow, University 
of Liverpool*. 


N the Phil. Mag. for Oct. 1907 we described some experi- 
ments which showed that certain gases after passing 
through a process of severe cooling contained large numbers 
of nuclei, the presence of which was shown by their ability 
to act as centres for the condensation of supersaturated water 
vapour. The following paper is an account of the continua- 
tion of those experiments. 


I. THE CONDITIONS GOVERNING THE PRODUCTION 
oF THE NUCLEI. 


In the paper referred to, it was shown that the gas had to 
be cooled below a certain temperature (which we called the 
“critical temperature”) before the nuclei were produced. 
Since the term “critical temperature ” has another meaning 
in Physics, it may be advisable to avoid using this term in 
future ; we shall therefore throughout this paper employ the 
term ‘‘ nucleating temperature” instead. 

As stated in the previous paper, the number of nuclei 
produced depends upon the pressure of the gas and the 
temperature to which it is cooled. We have since found thai 
the effect depends also upon the rate at which the gas 1s cooled. 
In fact the predominating factor in the formation of the 
nuclei appears to be suddenness of cocling. 

It may be well here to describe the modification of the 
original apparatus which enabled the cooling of the gas to 
take place very rapidly. Figure 1 represents the original 
type of apparatus used. In performing an experiment, the 
bend C (which we call the “ tester”) was surrounded by a 
cold liquid and as the temperature of the gas fell, the pressure 
was kept constant at any desired value by running the 
mercury up in the reservoir B. Fig. 2 represents the new 
form of apparatus designed to allow the gas to be very 
suddenly cooled. As is seen from the diagram the tester X 
is made in the form of a spiral. A glass tap M is introduced 
between the tester and the reservoir B, while the tap D is 
replaced by a three-way tap D'. This new form of apparatus 
was used as follows:—With M closed, the tester is first 


* Communicated by the Authors. 


Nuclei produced in Gases by Sudden Cooling. TAT 


exhausted through D! to a low pressure by means of a water- 


pump and is then surrounded by the cold liquid. When the 


Fig. J. Fig. 2. 


tester has attained the temperature of the liquid, the tap M 
is quickly turned and the gas rushes from B into it while the 
mercury is run up to bring the pressure to the desired value. 
After some thirty seconds, during which the temperature 
has risen about one degree, the cooling liquid is removed. 
The gas is then given a minute and a half to regain its 
normal temperature; at the end of which it is driven into 
the cloud-chamber @ where it is tested for the presence of 
nuclei. This method of cooling we shall denote as the 
“ sudden’? method, while the original method may be called 
relatively the “ slow” method. The “sudden” method of 
cooling possesses one great advantage over the “slow ” 
method, viz., that it enables the temperature to which the 
gas is cooled to be determined much more accurately ; for 
when the tester is surrounded by the cold liquid which is 
kept vigorously stirred by a small rotating screw, the 
temperature of the liquid rapidly rises six or seven degrees, 
and then remains nearly constant, rising only a degree or so 
per minute. With the “slow” method of cooling the mean 
of the initial and final temperatures (which differed by 
several degrees) has to be taken as the temperature to which 
the gas is cooled, an assumption which is perhaps hardly 
justifiable. But with the “sudden” method the gas is not 
admitted into the cooled evacuated tester until the temperature 
of the latter has become practically constant. Thus, by this 


748 Messrs. Owen and Hughes on Molecular 


new method, any given experiment can be repeated time 
after time with the certainty of always dealing with the same 
temperature to within half a degree. Consequently the 
results obtained by this method are much more consistent 
than those obtained by the original ‘“ slow ” method. 

The temperature of the cooled petroleum ether was given 
by a pentane thermometer reading down to the temperature 
of liquid air. The thermometer is correct at 0° and —190° 


but is incorrect at intermediate temperatures, having a 


maximum error of 12° at —100°. The temperatures there- 
fore given in the previous paper, especially those for COs, 
require correction. In the present paper corrected tem- 
peratures are given. 


Comparison of the Effects obtained in Air by the “ Sudden” 
and “ Slow” Methods of Cooling. 


The air was drawn from outside through potassium per- 
manganate, soda lime, P,O; and a spiral six feet long 
immersed in liquid air and then through a plug of cotton 
wool. : 

The following table shows the effects obtained at different 
temperatures by the two methods :— 


Papi I: 
(TestER X. Vol. 3 c.c.) 


Temperature. “Sudden” Method. ‘“‘ Slow” Method. 
126° @. 5. eee | 0 0 
ISLES ets Few drops. 0 
PBS A eee Fair shower. 0 
=IA02 6.5 ..te ae _ Good shower. 0 
sos AO Pee ase at na | Heavy shower. Fair shower. 
—190° ..............., Tinted rain-cloud. Rain cloud. 


The above table illustrates two facts :—The more sudden 
the cooling the higher is the “nucleating temperature” and 
the larger is the number of nuclec produced at any given 
temperature. 

In the course of the experiments it was noticed that the 
values of the “nucleating temperatures” obtained depended 
somewhat upon the shape and size of the tester used. In 
order to investigate this effect systematically, the double 


Aggregations produced in Gases by Sudden Cooling. 749 


tester shown in fig. 3 was made. Y had a volume of 6 c.c. 
and Z of 36 c.c. The following table gives the results 
Fig. 3. 


obtained with the apparatus in the case of air purified as 
described above. 


TABLE II. 
Temperature. Tester Y. 6 ce. Tester Z. 36 c.c. 

—— | | 
Beek re eine ten dee | 0. 0. | 
=] alee eee PLP EL Se 0. Several drops. | 
POO | dass secu sese sy 0. Fair shower. | 
SEI acne icy cute tas | 0. Heavy shower. ' 
| eee 2 _ Several drops. Very heavy shower. | 
= 1 een Good shower. 


SIEM dusdeaestairaayd | Heavy shower. 


750 Messrs. Owen and Hughes on Molecular 


Comparing this table with Table I. it will be noticed that 
the “ nucleating temperature ”’ for the spiral tester X of 3 c.c. 
and for the testers Y and Z are respectively —131°, —97°, 
— 80°. It will be noticed that there is a considerable difference 
between the nucleating temperatures obtained with testers X 
and Y while there is a much smaller difference between the 
values obtained for testers Y and Z, although in the latter case 
the ratio of volumes is 6: 1 whereas in the former only 2:1. 
Now the internal diameter of the tube forming tester Y was 
7 mm. while that of the tube forming the spiral tester X was 
only 3 mm. The rush of the gas into X is therefore not 
nearly so sudden as into the wider tube Y. On the other 
hand since Y and Z are both wide the difference in the rate 
at which the gas rushes into them will not be so marked. 
The above variations are therefore due, we think, not so 
much to the differences in the volumes of gas cooled as to 
variations in suddenness of cooling due to differences in 
the size and shape of the testers. | 

The different effects obtained by the “slow” and “ sudden ”” 
methods of cooling, as illustrated by Table L, naturally 
suggested trying a very slow rate of cooling. We used the 
following method to cool the gas very slowly right down to 
the temperature of liquid air. 2 

The tester X with tap M open (see fig. 2) was enclosed 
in a stout brass box, but not actually in contact with it. 
Through the lid of the box passed a thermometer, and a 
propeller for stirring up the air inside. By surrounding 
the box with liquid air the temperature fell very slowly 
until it became almost stationary at —152°. The box 
was then quickly removed and replaced by a vessel of liquid 
air. The whole process took about twenty minutes. The 
result obtained was a few drops, practically a no effect, whereas 
by the ordinary “slow” method a rain cloud was obtained 
(see Table I.). In all probability the few drops actually 
obtained were due to the more rapid fall of temperature from 
— 152° to —190°. 


Il. Tae Errect 1n Pure GASES. 


Carbon Dioxide. 


In the first paper we mentioned that the effect is much 
more marked in carbon dioxide when some of the gas is 
actually solidified during the cooling process, and suggested 
that the nuclei are produced, not while the gas is approaching 
the solid state, but while subliming from the solid back 


Aggregations produced in Gases by Sudden Cooling. 751 


again into the gaseous condition. In order to settle this 
point, further experiments have since been carried out 
on CQO,. 
The gas was prepared in two ways. In the first method 
it was obtained by heating pure sodium bicarbonate. It was 
then passed through calcium chloride, P,O;, a tight plug of 
cotton-w ool, and finally solidified in a glass tube surrounded 
by a tall Dewar vessel containing liquid air. When a 
quantity of CO, was required the liquid-air vessel was lowered, 
and after the desired amount had sublimed the flow of gas 
was checked by replacing the Dewar vessel. The tube 
containing the solid CO, formed therefore a very convenient 
gas-holder of large capacity. On its way from this gas- 
holder to the apparatus the gas passed through a plug of 
cotton-wool. 

In the second method, the gas was obtained direct from 
the steel tubes in which it is supplied commercially and 
purified and condensed as before. The CQO, obtained by 
these two methods gave the same results. 

It was found that the gas could be cooled to any tempera- 
ture not lower than its condensing point, both by the 
“sudden” and “slow” methods, without a single nucleus 
being produced. (By the “sudden” method the gas could 
be cooled with certainty to within a degree of its condensing 
point.) As the expansions used for the detection of the 
nuclei (where not otherwise stated) were about 1°10 some 
expansions nearly large enough to catch the ions in CO, 
were tried, but with the same result. This proves that for 
CQ, (in contradistinction to air) no nuclei at all are pro- 
duced when the gas is suddenly cooled right down to its 
condensing point. 

It was found, however, that when condensation actually took 
place and the condensed gas was allowed to sublime, nuclei 
were present and their number increased with the amount 
of CO, which had been condensed. The nuclei obtained when 
CO, sublimes are considerably bigger than those obtained by 
cooling air or oxygen. ‘The slightest supersaturation in the 
expansion apparatus caused by the adjusting of the pressure- 
drop preparatory to the expansion, brings down large numbers, 
but if care be taken to obviate the slightest fall in pressure, 
no drops are seen until an expansion is actually made. This 
means that the nuclei produced on the sublimation of the 
solid gas are bigger than those produced in air, but the fact 
that a certain degree of supersaturation (though small) is 
needed to initiate condensation on them proves them to be 
entirely different in character from that class of ‘ chemical 


752 Messrs. Owen and Hughes on Molecular 


nuclei” (such as those produced by intense ultra-violet 
light) in which clouds are formed without any supersaturation 
at all. 

Ethylene. 


Ethylene was prepared by the action of pure alcohol upon 
syrupy phosphoric acid and was passed through a condenser 
in ice, a strong solution of caustic potash, concentrated sul- 
phuric acid, and P,O;. 1t was then condensed in the tube as 
described in the section on CO. After this liquefied gas had 
been allowed to boil under reduced pressure for a short time 
to remove any air or carbon monoxide, a quantity was dis- 
tilled over into a second evacuated tube surrounded by liquid 
air until two-thirds of the liquid ethylene in the first tube had 
evaporated, the remaining third being rejected. Thus the 
second tube (our gas-holder for these experiments) contained 
ethylene of a high degree of purity. 

The following summarizes the effects obtained :— 

(1) The gas which comes off liquid ethylene is quite 

nuclei-free. | 

(2) No nuclei are produced in ethylene when cooled either 

by the “sudden” or the “slow” methods. The 
temperatures tried were —95°, —110°, —145°, and 
—190°, condensation taking place at the last three 
temperatures. 

Thus there is no “nucleating temperature ” for ethylene. 


Methane. 

Methane was prepared by heating sodium acetate with 
soda lime. The gas was passed through caustic potash, strong 
sulphuric acid, condensed and distilled as in the case of 
ethylene. 

The following results were obtained :— 

(1) The gas which comes off liquid methane is nuclei-free. 

(2) Table IIT. gives the results obtained in methane by the 

“* sudden ” method of cooling. $ 


TABLE III. 
Temperature. Effect. Tusrer Y. 
gp Is, BR tea 0. 
= Nees ce aeens tae eee eee Few drops. 
= OFC. a eee ee Thin shower. 
eel Ue pA Ane NO Is) Te Good shower. 
=a 11 es a Tinted cloud. 


Aggregations produced in Gases by Sudden Cooling. 753 


Comparing the above with Table II. it will be seen that. 
the effects obtained in methane are very similar to those 
obtained in air. 


TIT. Tae Errect 1x MIxtTUREs. 


In the hope of getting further light on the origin of the 
nuclei we performed a series of experiments on mixtures of 
eases of varying proportions. Three mixtures were tried, viz., 
air and carbon dioxide, air and ethylene, and air and water- 
vapour. The apparatus shown in fig. 4 was devised to mix 
the gases in known proportions. 


Fig. 4. 


RY 38 
Shs 
S y 
er 8 


> Aur A 
es = J 
—> COz Te Tap $ fight 
(ee eer ee Ta 


The apparatus consists of two large glass globes of 13 litres 
capacity connected together by a glass tube. The tube F 
leads to tap S in fig. 2. Enough concentrated sulphuric acid 
is introduced to fill one globe. By means of a three-way 
tap A the globe B can be connected to a water-pump or to 
the atmosphere as desired. The mixing of the gases is 
effected thus:—The globe B is connected to the pump and 
the sulphuric acid is drawn into it to within a centimetre of the 
tap, which is then closed. Globe C is now exhausted through 
the tap D’ (see fig. 2) to a pressure of about 4 cm. and air 
let in through P until the pressure falls a definite amount. 
Carbon dioxide is then admitted and, finally, more air until 
the pressure is atmospheric. The proportions of the con- 
stituents are determined from their respective partial pressures. 
Some thirty minutes are taken over this process to allow the 
gases to mix thoroughly. In order to diminish any 


754 Messrs. Owen and Hughes on Molecular 


inaccuracy in the estimation of the composition of the mixture 
due to the 4 cms. of gas originally in the globe, it is once 
more exhausted and the gases mixed as before. The results 
obtained show that this precaution was unnecessary as the 
effects change but slowly with the composition of the mixture. 
The latter could be drawn into the apparatus (see fig. 2) 
at approximately atmospheric pressure by connecting the 
globe B to the atmosphere through the tap A. About twenty 
tests could be performed with the mixture before the supply 
in the globe C became exhausted. By means of a side path 
(not shown) pure air could be drawn into the apparatus for 
purposes of comparison. 


Air and Carbon Dioxide. 


The following table summarizes the results obtained in 
mixtures of air and carbon dioxide. 


TasuE LV, 
“Sudden”? Method of Cooling. 
| l / 
Temperature. Pure Air. | 5 per cent. CO,. 50 per cent. CO,,. 
eS ae ee 0 | 0 | 0 
Soh Ce eee 0 0 Fair shower. 
ae MO) a aoe, 0 0 Good shower. 
me OEE te ate ae Few drops. Few drops. 
= 1289 eee Thin shower. Thin shower. 
— 190m... Meee | Tinted cloud. Tinted cloud. 
| 


It will be seen from the above table that the effects in pure 
air and air mixed with 5 per cent. of CQ, are indistinguishable 
both in respect to the value of the “ nucleating temperature ” 
and to the magnitude of the effects obtained. oh. 

In the above experiments the mixture was maintained at 
a pressure of 80 cms. ‘Thus the partial pressure of the CO, 
in the 50 per cent. mixture at atmospheric temperature was 
40 cms. At this pressure the condensing temperature is 
approximately —85°; on cooling down, however, the partial 
pressure of the CO, when near the condensing point would 
be less than 40 ems., and therefore its condensing temperature 
would be somewhat below —85°. Now from the above table 
it is seen that the effect in the 50 per cent. mixture starts 
somewhere between —86° and —92°. On making allowance 
for the reduced partial pressure, it is seen that CO, in the 


Aggregations produced in Gases by Sudden Cooling. 755 


mixture behaves just the same as if the air were absent, the 
nuclei being produced only when condensation occurs. 

The same result is borne out by experiments performed 
with a mixture containing 90 per cent. of CQy. 

That the effects in pure air and air mixed with 5 per cent. of 
CO, are practically indistinguishable is sufficiently accounted 
for by the fact that only very few nuclei would in any case 
be produced by the complete condensation and subsequent 


sublimation of the small quantity of CO, present in this 
mixture. 


Air and Ethylene. 


As pure ethylene itself gives no effect at all when cooled 
to every temperature we have tried, the effect in air mixed 
with ethylene was investigated. 

The following table shows the results obtained in three 
different mixtures of air and ethylene. 


TaBieE V. 
| . 5 per cent. 50 per cent. 90 per cent. 
Sea | aes Ethylene. Ethylene. Ethylene. 

—125° C 0 0 
—127 ie 0 Few drops. ~ 
— 128 ...| Very thin shower.| Thin shower. | 0 0 
—131 _...| Thin shower. Goodishower. »") 2 ‘ished. 0 
—133 ...| Good shower. Good shower. Fair shower. | Very thin shower. 
—143 PERCY SHOWER Ff ncccncsateds <P) awowwasceens Heavy shower. 
—190 ...| Tinted rete oun Beanie Tinted rain-cloud, | Tinted rain-cloud. 


Here again it will be noticed that the effects in pure air 
and in air mixed with 5 per cent. of ethylene are practically 
identical. The “nucleating temperature ” in the mixture, it 
is true, is afew degrees higher than for pure air, but the 
difference is not sufficiently marked for any great importance 
to be attached to it. On the other hand, the results ina 
50 per cent. mixture are readily understood, the effects 
obtained being simply those which would be obtained in 
air at a pressure equal to its partial pressure in the mixture. 

The nucleating temperature in the 90 per cent. mixture 
bears out the same view. It will be seen, however, from the 
table that when the 90 per cent. mixture is cooled by liquid 
air just as many nuclei were produced as in the case of pure 


756 Messrs. Owen and Hughes on Molecular 


air. Now considering the fact that ethylene itself is in- 
effective in the production of nuclei and the total quantity of 
air in this mixture is small, this is somewhat surprising. This 
may possibly be accounted for by the fact, that when the tap 
between the reservoir and the evacuated tester in liquid air 
is opened, the instantaneous condensation of the ethylene that 
takes place carries the air so rapidly into the tester that the 
latter is subjected to an extremely sudden fall of temperature. 


Water- Vapour in Air and CO,. 


In the experiments described in the previous paper we had 
taken great care to obtain the gases quite dry. Since then 
we have examined the effect in wet gases. 

An apparatus was devised in which the gas after having 
been thoroughly dried could be drawn through either of two 
paths before entering the reservoir and tester. In one path 
was an arrangement by which the gas bubbled through 
water, thus saturating it with water-vapour. With dry air 
the effects started at about —129°. With wet air we 
obtained a very small effect (never more than a “few 
drops’) at temperatures between —122° and —129°. At tem- 
peratures below the “ nucleating temperatures” the effects 
in dry and wet air were identical. ~ 

With wet CO, we obtained a small etfect (a ‘very thin 
shower ”’) at some five or seven degrees above the condensing- 
point, whereas with dry CO, no effects were obtained unless 
condensation occurred, as has been described above. But 
below the condensing temperature the effects in wet and dry 
CO, were indistinguishable. 


LV. Tuer PERSISTENCY OF THE NUCLEI AT DIFFERENT 
TEMPERATURES. 


Air Nuclei. 


It had often been noticed that when air containing nuclei 
produced by sudden cooling was left over night in the tester, 
no effect was obtained on passing it into the cloud-chamber 
on the following morning, showing that the nuclei had in 
some way or other disappeared. On making a similar expe- 
riment with dusty air from the room there was only a small 
diminution in the effect in the same interval of time. This 
led to a more thorough investigation of the persistency of 
the nuclei. Several forms of apparatus were tried, but the 


Aggregations produced in Gases by Sudden Cooling. 757 


form found to be most suitable for these experiments is shown 
in fig. 5. 


Fig. 5 
§ i S 
S | Cotton Woo/ = 
t “A ji 
E | 
QQ aN ans} ras 
| 

om es 
one, 
D 


The tester used was the spiral tester of volume 3 c.c¢. 
The nuclei were produced in this by the ‘sudden ” method 
of cooling, the temperature being that of liquid air. They 
were then driven into the evacuated spiral “oven” D of 
capacity 35 c.c., where they were kept for different lengths 
of time and at different temperatures. From this “oven” 
they were finally driven into the cloud-chamber by means of 
filtered air from the auxiliary reservoir E. 

The following table illustrates the rate at which the air 
nuclei disappear at atmospheric temperature and pressure. 


Tasie VI. 
Length of Time in the “ Oven.” Effect obtained. 
1 minute. Tinted rain-cload, 
3 minutes. Heavy shower. 
11) ae Thin shower. 
7h) De ye Several drops. 
14 hours. 0 


We next compared the persistency of the nuclei at different 
temperatures. The “oven” could be maintained at any 
desired temperature up to 200° C. by hot glycerine. In 
order to have initially in the “oven” always the same 
number of nuclei the nucleated gas was driven into it at 


Phil. Mag. 8. 6. Vol. 15. No. 90. June 1908. i) 


758 . Messrs. Owen and Hughes on Molecular 


atmospheric temperature, and immediately the tap K was 
closed, the “ oven” was surrounded by hot glycerine for a 
definite length of time. A similar experiment was then made 
in which the “ oven”? was kept at atmospheric temperature 
for the same time. Before admitting into the cloud-chamber, 
the gas in the “ oven ’’ was allowed one minute in which to 
regain atmospheric temperature. 

Table VII. compares the persistency of air nuclei at 
different temperatures. 


TABLE VII. 


Time and Tempe- Effect, Time and Tempe- Effect. 


rature in ‘“ Oven.” rature in ‘‘ Oven.” 
| 
i - | in. em : 
2 min. at 15°...... Heavy shower. | : ae ah . BS } Fair shower. 
‘ 2 _ || 3 min. at 100°.1 | Very thin 
4 min ab LD? ce Good shower. | 1 min aha? Huw 
. | 8 min. at 100°. 
9 min. ab 1pe Lt: Few drops. | 1 ie, ak je: 0 


Time and Tempe- 
rature in “ Oven.” Effect. 


1 min. at 185° 
1 min. at) 15° | 


2 min. at ibe. f 0 


} Fair shower. 


2min. at 15°, 


The above table shows clearly that keeping the nuclei at 
a high temperature aids their disappearance. This suggested 
that the nuclei would be still more persistent if kept at a low 
temperature. This was verified, for it was found that after 
thirty minutes at —75° a fair shower was obtained which ‘is 
rreater than the effect obtained after a much shorter period 
at 15° (see Table VI.). 

Thus raised the interesting question as to whether the 
nuclei would persist indefinitely if kept at a temperature at 
which they were produced. To investigate this point an 
experiment was carried out with tester Z (see fig. 3). When 
the nuclei were preduced by the “sudden” method at the 
temperature of liquid air, the effect obtained a minute and a 
half later was a thick white fog. On producing the nuclei 
in the same way, but keeping them for two hours in the tester 


‘ 
\. 


Aggregations produced in Gases by Sudden Cooling. 759 


surrounded by liquid air, the effect now obtained was only a 
‘fair shower”; this shows that, provided a sufficiently long 
time be given, the nuclei disappear even at the temperature of 
production. 


Carbon Dioxide Nuclei. 


Table VIII. shows the rate of disappearance of CO; nuclei 
at 15° and 80 cms. pressure, the nuclei having been produced 


by the sublimation of CO, solidified by liquid air. 
Taste VIII. 


Time in “ Oven” at 15°. Effect. 


DO SEG 1:35 heeds Tinted rain-cloud. 
SAIBISIS. 5.0 cv edcddede ees, Rain cloud. 

NO re, advecdgananacqaces Very heavy shower. 

=) eget RRS py Sa Good shower. 


The following Table IX. shows the effect of high tempera- 
ture upon the life of these nuclei. In this case, the nucleated 
CO, was driven into the “oven” at a pressure of 55 cms. 
The showers are therefore not so dense as those given in 


Table VIIT. 


TABLE IX. 
Time and Time and 
Temperature in “ Oven.” pleat Temperature in “ Oven.” Hees 
. = (2 ey eel aa iT 
; gs “ ne \ Pies Heavy shower. 2 min at bao icc Dense shower. 
50 
3 ” a foc Fair shower. 4. ,, Beg Sie Heavy shower. 
ix hi 4 
ae a \ bis hes tee ee eG REE ot Good shower. 


fp? - 


{ 


Evidently the disappearance of the CO, nuclei is also 
aided by high temperature. Comparing Table VIII. with 
Table VI. we see that the CO, nuclei are somewhat more 
persistent than air nuclei. 


V. ELectricaL ConDITION oF THE NUQLEI. 


Experiments were made to see if the nuclei were electrically 
charged. Nucleated air produced by cooling to the tempera- 
ture of liquid air was passed into a chamber in which it could 


rn 


760 Messrs. Owen and Hughes on Molecular 


be subjected to an intense electric field. This chamber was 
a glass tube about 10 cms. long and 3 cms. in diameter. Inside 
was a closely fitting brass tube of nearly the same length, 
which in turn surrounded a thick brass rod coaxial with it. 
There was thus left an annular space about 5 mm. wide between 
the brass tube and the central rod. Most of the gas driven 
into the tube was located in this space. After remaining in 
this space for one minute under the action of different electric 
fields, the nucleated air was driven into the cloud-chamber. 
The effect was found to be the same whether the electrodes 
were both earthed or at potential-differences of 230, 1000, or 
4000 volts. 
This proves the nuclei to be uncharged. 


VI. Discussion oF RESULTS. 


The additional investigations published in this paper have 
not furnished evidence necessitating a change in the views 
expressed in the previous paper. Still much remains obscure 
and difficult to account for completely. We restate our 
explanation of the effects as follows :— } 

When the temperature of the gas falls sufficiently, and not 
too slowly, molecular aggregations are formed, most probably 
of those slowly moving molecules whose kinetic energy is less 
than their mutual potential energy. On the kinetic theory, 
this means of course that the aggregations approximate more 


to the liquid phase than to the gaseous. Possibly the effects — 


may be regarded as pointing to incipient liquefaction taking 
place in the gas at a temperature well above the real liquefying 
temperature. 

The number of molecular aggregations is increased to a 
remarkable degree by increasing the suddenness of cooling. 
We find this difficult to explain, possibly it may be connected 
with the zrreversibility of the phenomenon as evidenced by 
their persistency. , 

It has already been stated that the nuclei are of considerable 
size. This does not necessarily mean that smaller nuclei are 
not produced under the same circumstances, for such aggre- 
gations might be produced but, owing to smaller stability, 
have disappeared before the expansion can be made. 

The more rapid disappearance of the nuclei at higher 
temperatures is readily explained by the more vigorous 
bombardment of the aggregations by the molecules of the 
heated gas. 

The results in CO, are interesting as they suggest a 
fundamental difference between evaporation from the liquid 


Sa? 
:Y 


Aggregations produced in Gases by Sudden Cooling. 761 


phase and sublimation from the solid. We have found that 
the gas evolved from liquid air, oxygen, ethylene, methane is 
nuclei-free, confirming the ordinary view that the evaporation 
of a liquid consists in the escape of separate molecules. But 
the fact that a mass of solid CO, (previously condensed in a 
perfectly dust-free state) continues to give off gas containing 
enormous numbers of nuclei until the whole mass has dis- 
appeared, suggests that the sublimation of solid CO, (and 
possibly of other substances which can pass from the solid 
direct into the gaseous phase) consists in the escape of separate 
molecules together with numerous molecular aggregations. 

We will in conclusion meet some possible objections to 
the above views. As the effects obtained are larger when 
the gas is allowed to rush into the cooled evacuated chamber 
it might be urged that the nuclei are simply dust particles 
dislodged from the walls of the vessel by the sudden rush of 
gas. This view, however, is quite untenable from the following 
considerations :—(1) The effect shows no signs at all of 
diminution however often the experiments are made with any 
particular tester. (2) No effect is obtained when the gas 
rushing into the evacuated tester is pure ethylene, whereas 
a considerable effect is obtained when the ethylene contains 
a small percentage of air. (3) The nuclei are rapidly 
destroyed at a temperature of about 200°C. 

Nor can it be argued that the nuclei are due to insufficient 
drying of the gases, as there is very little difference between 
the behaviour of wet gases and of those dried with the utmost 
precautions. 

Then again it might be urged that the nuclei are due to 
the presence of some impurities in the gases. This point was 
discussed in the previous paper. If the impurity be, let us 
imagine, CQ, it is not likely that the effects in purified air 
and in air mixed with five per cent. of CO, would be 
indistinguishable as was found to be the case (see Table IV.). 

Supposing on the other hand the impurity to be some gas 
only slightly more easily liquefied than air, we meet with the 
same difficulty in accounting for the production of the nuclei 
in this impurity as we do in the case of air. If the effects 
are due to some impurity, then the only gas we obtained pure 
must have been ethylene. The precautions which we have 
taken, however, to obtain pure’ gases, especially in the case 
of oxygen and air, render this “impurity” explanation 
highly improbable, certainly not more probable than the view 
we have expressed above. 


George Hoit Physics Laboratory, 
The University of Liverpool. 


62 


LXVIII. On the Effect of the Position of the Grating (or Prism) 
upon the Resolving Power of a Spectroscope. By ALFRED 
W. Porter, B.Sc., Fellow of and Assistant- Professor of 
Physics in Ui niversity of London, University College™. 


f ha paper is an extension of an interesting paper by 
Mr. H. Morris-Airey in this Journal for March 1906 
(p. 414). In that paper it was shown that even wide slits 
may be employed if the light fall at nearly grazing incidence 
upon a grating or prism. But the closeness together of two 
spectral lines of different wave-length also varies with the 
position of the grating ; and it is necessary to examine the 
variation of this proximity before it can be definitely decided 
that a gain in purity is obtained by inclining the grating or 
rism. 
We will consider in detail the case of a grating 
Let ¢ be the angle of incidence of the light cre ee. colli- 
mator upon the grating and @ the angle of “diffraction ; both 
measured positive when on the same side of the normal. 


Rigs 1, 


Then, if e is the grating-interval, and nm the order of 
spectrum referred to, 
e(sin@ + sing) = m2. 
The effect of the finiteness of the width of the slit is allowed 
for by considering a small variation in the angle of. incidence 


* Communicated by the Author. 


On the Resolving Power of a Spectroscope. 763 


@. The angular width of the image resulting therefrom is 
given by A@ where 
A@ _cosd 
~ Ad cos 0° 
It must be observed that whether this decreases or not as 
decreases can only be decided when it is known which side 


of the normal the image is on. 
Two spectral lines for which the wave-lengths differ by 
6x will be at an angular distance apart 6@ where 


Ca n 
oN ecosd- 


In these formule, A@ is a measure of the broadness of each 
image and 6@ is the separation between the centres of neigh- 
bouring images (or dispersion). 

Now, in estimating the purity of the spectrum, it is clearly 
the ratio of the second to the first of these two quantities 
which is important. Greater broadening of the image would 
be consistent with greater purity if the dispersion was 
increased in proportionately greater amount. The ratio of 


pa. | AG 
SX Ad” 


_aratio which may be spoken of as the tilt-advantage, is 


— ne ° 

~ @Cosd’ 
whence it is seen that the advantage of the arrangement 
increases as ¢ increases, whether the spectrum examined is on 
the same side of the normal as the incident beam or not. The 
relative advantage compared with the case of normal incidence 
is 1/cos ¢. 

In illustration of the meaning of this result, it may be 
pointed out that the case in which greatest separation between 
the centres of the images is obtained is not the case in which 
greatest advantage is gained. Greatest separation between 
the centres occurs when cos 0 = 0; that is when 


— e(1 + sing) =n. 


This case can be obtained by starting from normal incidence, 
selecting any spectrum and turning the grating so as to 
increase @ numerically. The relative advantage in this case 
can be only moderate. On the contrary, in order to increase 
the angle of incidence to 90°, the grating must be turned the 
other way ; in this case greater purity is obtained although 


764 On the Resolving Power of a Spectroscope. 


the dispersion is not nearly so great. This latter is the case 
in which very wide slits may advantageously be used. 
The corresponding equations in the case of a prism are :— 
Aé@ _cosr'cosh _ Bo 
Ad” cosrcos@ B' 
OF Sing ae 
Sus cosOcosr—«&B! 


sina  dw_b' dp 
cos’ cos¢ dX B dar 


where w is the refractive index and the other data are 
explained by the accompanying figure. 


Fig. 2. 


Occasionally the question of purity may not be the most 
important one. It is notorious that the focal length of the 
telescopes usually furnished with spectroscopes is almost 
always much less than it advantageously might be. In a 
particular single-prism spectroscope which | have in mind, 
the sodium lines (D,D,) are only just separable when the 
prism is in the position of minimum deviation. This is simply 
not owing to a lack of purity, but because the ultimate images 
are so close together that it would be impossible to detect 
intermediate images (with the telescope provided), if such 
existed, because the separation is below the minimum visible. 
By turning the prism so as to increase the angle of emergence, 
the dispersion becomes so great that in spite of the simul- 
taneous increase in he breadth of each line ene would easily 
detect several intermediate lines. 


‘? 


a 


[ AOo ey \ 
LXIX. Note on certain Dynamical Analogues of Temperature 
Equilibrium. By G. H. Bryan (Bangor)*. 
JN the Archives Néerlandaises for 1900 (Livre jubilaire 
dédié 4 M. H. A. Lorentz) I described under the title 
ot “ Energy Accelerations ” a method of studying problems 
dealing with the partition of energy in systems of particles in 
which some kind of statistical equilibrium exists. This 
method consists essentially in writing down expressions for 
the second differential coefficients with respect to the time of 
the squares and products of velocities or momenta of the 
system considered. 

The method in question appears to have been previously 
employed by Mr. Burbury, who, however, did not employ the 
term ‘‘ energy-accelerations” in connexion with the second 
differential coefficients in question. 

I was, and still am, of opinion that a further study of 
energy-accelerations must have an important bearing on all 
problems in the Kinetic Theory dealing with questions of 
temperature-cquilibrium, and the fact that no attempt seems 
to have been made to follow the problem up, may be taken as 
evidence of the large congestion of unsolved problems which 
presses heavily on the shoulders of the mathematician of 
to-day. The present note deals with two simple applications 
which happened to remain unnoticed when the paper was 
written. 

Consider in the first place an ideal medium formed of 
material particles uniformly distributed in space, both as 
regards position and as regards direction, and attracting or 
repelling one another according to any law of force as a 
function of the distance between them. 

If we confine our attention to one particular particle, the 

effect of the other particles will be to produce varying fields 
of force acting on the particle in question. Take now the 
equation of energy-accelerations which may be written in 
the more general form :— 
d? ——- meee z av oY a’V a! 
aden a m =) Rage de de dy UW dn dz? 
where w, v, w stand for velocity components, V the potential 
of the field due to the other particles. 

The assumption that the field produced by the other 
particles is independent of the motion of the particle under 
consideration, and that this field has on an average no 
directional properties, shows that the mean values of the last 
two terms vanish, and the assumption that a stationary state 

* Communicated by the Physical Society : read March 13, 1908. 


766 Prof. G. H. Bryan on certain Dynamical 


exists requires that the mean value of the term ud*V/dt de 
should also vanish, reducing the energy-acceleration equation 
in this case to the ordinary standard form, brackets denoting 
mean values :— 


d? 1 fedvVy* a?V 
i (dm) | — on i <-) | — [ u? | laa 

For energy equilibrium the left-hand side must vanish. 
This is only possible when the mean value of d*V/dz? is 
positive. Now the assumed absence of directional properties, 
or in other words the assumed isotropic character of the 
medium, requires that the average values of d?V/dzx?, d?V/dy’?, 
and d?V/dz* shall be equal ; therefore each is equal to one- 
third the average value of AV. We conclude that statistical 
energy equilibrium cannot exist in a system of particles 
possessing the assumed properties unless AV is positive. 

For the Newtonian law of foree AV=0. In this case 
the mean value [d?V/dz?] therefore also vanishes, and the 
energy-acceleration equation takes the form :— 


[ie m= Lae) I 


This shows that the only kind of statistical equilibrium is 
the statical state of unstable equilibrium defined by dV/dz=0, 
dV/dy =0, and dV/dz =0 on each particle, and in the 
absence of this state an acceleration of kinetic energy will 
take place. 


It is thus impossible with the Newtonian law to build up | 


a medium of material particles, either attracting or repelling 


one another, and possessing properties of energy equilibrium. 


analogous to those of a system of gas molecules. 

It need not be pointed out that this investigation does not 
preclude the possibility under the Newtonian Law, of 
stationary motions such as those occurring in the solar 
system. 

Exactly the same conclusions hold good if the law of force 
is such as to make AV negative. The equations of energy 
equilibrium here require the further condition that [u?] [v?] 
[w?}| should be zero, which can only happen if the system 
always remains at rest. 

The necessary condition AV >0 becomes, for the particular 
case in which the field is due to an attracting particle placed 


at the origin, 
1 *( ae 
n—-)>0, 


showing that r?dV/dr must increase with r. In other words, 
the force if repulsive must vary according to a higher power 


Analogues of Temperature Equilibrium. 767 


of the inverse distance than the square (the inverse fifth 
power would represent a possible law of force if repulsive). 
If the force is attractive it must vary according to a lower 
power of the inverse distance than the square. 

The second point to be noticed is that unless the expression 
for the kinetic energy of a dynamical system is a quadratic 
fuuction of the velocities with constant coefficients, the 
equations of energy equilibrium no longer assume the form 
of linear relations connecting the mean squares and products 
of velocities, but they also involve terms of the fourth degree 
in these velocities. An illustration of this fact is afforded by 
the motion of a particle in a plane when referred to polar 
coordinates. If we write down the expressions for the energy- 
accelerations 


i at deja ots 
pla”) and qp ar”) 


in terms of the velocities and coordinates, using the equations 
of motion 


_ a er aN LE RE Te dV 

r—re?=— ae and eras 0) ae? 
we shall obtain expressions involving the velocity components 
up to the fourth degree, and the energy components there- 
fore up to the second degree. The conditions of energy 
equilibrium between the transversal and radial components 
will therefore no longer take the form of linear relations con- 
necting the mean values of the energy components in question. 

On the other hand, when we refer the motion to # and y 
coordinates, the equations of energy equilibrium are linear in 
the energy components. 

It appears, therefore, that if dynamical systems are to 
represent the phenomena of temperature-equilibrium consis- 
tently with the commonly accepted hypothesis that tempe- 
rature is a quantity of the nature and dimensions of molecular 
kinetic energy, the expression for the kinetic energy must 
in general have constant coefficients or must at least satisfy 
certain conditions which are fulfilled in the case of constant 
coefficients. While the analogy of kinetic energy with tem- 
perature may hold good in the case of a system of particles, 
there must exist dynamical systems for which this analogy does 
not hold good. Asa purely negative conclusion this result 
is not inconsistent with Stefan’s law in which we have the 
energy of radiation proportional to the fourth power of the 
energy of molecular motion. But whether it is possible to con- 
struct a dynamical model whose energy-partition is analogous 
to Stefan’s Law must be a question for future investigation. 


[968 


LXX. Notices respecting New Books. | 

Bulletin of the Bureau of Standards. Vol. IV. No. 3, Jan. 1908. 
Washington: Government Printing Office, 1908. 
THE contents of this issue include an important paper by W. A. 
Noyes and H.C. P. Weber on the Atomic Weight of Chlorine. 
The method employed consists in weighing the hydrogen absorbed 
in palladium and the chiorine in the form of potassium chloro- 
platinate. The hydrogen is passed over the heated chloroplatinate, 
hydrochloric acid being formed which is condensed and weighed. 
The mean value found is 35:184 (H=1). 

The part contains also a critical paper by W. W. Coblentz on 
instruments and methods used in radiometry, in which the relative 
merits are discussed of a radiomicrometer, linear thermopile, 
radiometer and bolometer. The thermopile is recommended on 
account of its greater steadiness for measuring very weak sources 
of radiation, e. g. the extreme ultra-violet aud infra-red region of 
the spectrum. The other articles are the preparation of chloro- 
platinic acid by electrolysis of platinum black, by H. C. P. Weber ; 
the self-inductance of a coil of any length and any number of 
layers of wire, by E. B. Rosa; the self-inductance of a solenoid 
of any number of Jayers, by Louis Cohen; a quartz compensating 
polariscope with adjustable sensibility, by F. Bates. 


LXXI. Intelligence and Miscellaneous Articles. 
ON THE MIXING OF GASES. 
To the Editors of the Philosophical Magazine. 
GENTLEMEN,— 
if REGRET that my absence in Egypt during February and 
March prevented my seeing Professor Orr’s paper in your 
February number till quite recently. JI have since contributed to 
‘Science Progress’ an article in which my views on the questions 
which he raises are explained more at length. I do, however, 
doubt whether there is any proof that the mixture of two gases 
by diffusion is an irreversible process. It is of course true that if 
we have two mixtures of (say) oxygen and nitregen at the same 
pressure and temperature, in one of which oxygen, and in the 
other nitrogen, preponderates, then if we remove the partition 
which separates them, they at once begin to mix more uniformly. 
But as that would equally be the case if the process of diffusion 
were periodic, the observed fact is no proof that the process is not 
periodic. Theoretically I maintain that no irreversible motion is 
possible among the molecules of a gas, if the kinetic theory be true, 
and if the gas be completely isolated, that is completely protected 
from the influence of external bodies. Professor Orr, as I gather, 
will not accept the kinetic theory. Inthat he may be right. But if 
we do not accept it, is there any other theory that can be applied ? 
To prove experimentally that the diffusion process is irreversible 
would require that the experiment be continued for a time at least 
half as long as the period of the motion if it were periodic. How 
long a time would Professor Orr consider sufficient? Also to 
make the experiment at all conclusive, the diffusing gases must at 
every instant during the experiment be completely protected from 
all external influences. Has such an experiment ever been made? 
S. H. Burgury. 


Peer] 


INDEX to VOL. XV. 


—————— 


Actinium, on the distribution in 
electric fields of the active deposits 
of, 601 ; on the electrical charge of 
the active deposit of, 737. 

Adams (Dr. F. D.) onthe Laurentian 
system of Canada, 201. 

Adsorption, on, 499. 

Alpha-rays, on changes of velocity 
of the, in an electric field, 720. 

Atmosphere, on the thermally ex- 
cited vibrations of an, 147. 

Atom, on the existence of positive 
electrons in the sodium, 274. 

Atomic origins, on the formation of 
concrete matter from, 397. 

—— weight of radium, 280. 

Bakker (Dr. G.) on the theory of 
surface-forces, 415. 

Barkla (Dr. C. G.) on X-rays, 288. 

Bars, on the lateral vibration of, 497. 

Barton (Prof. E. H.) on spherical 
radiation and vibrations in conical 
pipes, 69. 

Basedow (H.) on beds of Cambrian 
age in §. Australia, 203. 

Bessel integrals, on certain, 332. 

Beta rays, on changes of velocity of 
the, in an electric field, 720. 

Books, new :—Chéneveau’s Recher- 
ches sur les Propriétés optiques des 
Solutions et des Corps Dissous, 
200; Rivista di Scienza, 200; 
Seidell’s Solubilities of Inorganic 
and Organic Substances, 201; 
Lehrbuch der  Gerichtlichen 
Chemie, 393 ; Clark’s The Polarity 
of Matter, 394; Fischer’s Guide 
de Préparations Organiques 4 
VYUsage des Etudiants, 394; von 
Rohr’s Die Binokularen Instru- 
mente, 594; Annuaire du Bureau 
des Longitudes, 395 ; The Science 
Year-Book and Diary, 395; Bril- 
louin’s Lecons sur la Viscosité des 
Liquides et des Gaz, 395; Bulletin 


of the Bureau of Standards, 396 ; 
Barus’s Condensation of Vapor as 
induced by Nuclei and Ions, 559 ; 
Carslaw’s Fourier’s Series and 
Integrals, 569; van Laar’s Sechs 
Vortrage uber das Thermodyna- 
mische Potential, 570; The Scien- 
tific Papers of J. Willard Gibbs, 
570; Newcomb’s Compendium of 
Spherical Astronomy, 570; Pern- 
ter’s Meteorologische Optik, 571 ; 
Fournier d’Albe’s Two New 
Worlds, 572; Whitehead’s The 
Axioms of Projective Geometry 
and the Axioms of Descriptive 
Geometry, 676; W. H. & G. C. 
Young's Theory of Sets of Points, 
676; Bulletin of the Bureau of 
Standards, 768. 

Bonney (Prof. T. G.) on antigorite 
and the Val Antigorio, 578. 

Bowlker (T. J.) on the direction 
of sound, 318. 

Bragg (Prof. W. H.) on the nature 
of the y rays, 663. 

Bryan (Prof. G. H.) on certain 
dynamical analogues of tempera- 
ture equilibrium, 765. 

Brydone (R. M.) on the chalk at 
Trimmingham, 577. 

Bucherer (Prof. A. H.) on the 
principle of relativity and on the 
electromagnetic mass of the 
electron, 316, 

Buckingham (E.) on the thermo- 
dynamic corrections of the nitrogen 
scale, 526. 

Bumstead (Prof. H. A.) on heating 
effects produced by Réntgen rays 
in lead and zinc, 482. 

Burbury (S. H.) on the mixing of 
gases, 768. 

Burton (Dr. C. V.) on the thermally 
excited vibrations of an atmo- 
sphere, 147. 


7 
H 
i 
: 
Bi 
sd 
: 
: 


770 INDEX. 


Bury (H.) on the River Wey, 579. 

Campbell (A.) on the use of variable 
mutual inductances, 155. 

Canal-ray group, on the, 372. 

Coils, on the coefficients of mutual 
induction of coaxial, 352. 

Comstock (Dr. D. F.) on the relation 
of mass to energy, l. 

Dewey (H.) on pillow-lava near 
Port Isaac, 576. 

Dickson (J. D. H.) on the Joule- 
Kelvin inversion temperature, 126. 

Dispersion, on the anomalous mag- 
netic rotatory, of neodymium, 270; 
of rare earths, 538. 

Dyeing, on the theory of, 499. 
Elastic properties of steel, effect 
of combined stresses on the, 214. 
Electric charge of the active deposit 

of actinium, on the, 737. 

fields, on the distribution in, 
of the active deposits of radium, 
thorium, and actinium, 601. 

Electron, on the electromagnetic 
mass of the, 316. 

Electron theory of matter, on the, 
172. 

Electrons, on the existence of positive, 
in the sodium atom, 274; fre- 
quencies of the free vibrations of 
quasi-permanent systems of, 438. 

Elements, on the evolution and 
devolution of the, 21, 396. 

Elias (G. J.) on the anomalous 
magnetic rotatory dispersion of 
rare earths, 538. 

Elsden (J. V.) on the St. David’s- 
Head rock-series, 578. 

Energy, the relation of mass to, 1. 

Equipotentials, on a freehand graphic 
way of determining, 237. 

Erbium, on the anomalous magnetic 
rotatory dispersion of, 538. 

Eve (Prof. A. 8.) on changes in 
velocity in an electric field of the 
a, 8, and secondary rays, 720. 

Fabry and Perot’s apparatus, notes 
on, 548. 

Focometry of a concaye lens, on the, 
198, 300. 

Gamma rays, on the, of radium, 638 ; 
on the nature of the, 668. 

Gaseous nebula, on the problem of a 
spherical, 687. 

Gases, on the mixing of, 297, 7€8 ; 
on molecular aggregations pro- 
duced in, by sudden cooling, 746. 


Geiger (Dr. H.) on the irregularities 
in the radiation from radioactive 
bodies, 539. 

Geological Society, proceedings of 
the, 201, 572. 

Glass, on figuring, by hydrofluoric 
acid, 555. 

Goldstein (Prof. E.) on the canal- 
ray group, 372. 

Grating, on the effect of the position 
of the, on the resolving power of 
a spectroscope, 762. 

Hamilton’s principle, on, 677. 

Hancock (Prof. Ek. L.) on the effect 
of combined stresses on the elastic 
properties of steel, 214. 

Havelock (Dr. T. H.) on certain 
Bessel integrals and the coefficients . 
of mutual induction of coaxial 
coils, 352. 

Helium, wave-lengths of, 549. 

Helix, on the curvature and torsion 
of a, on any cylinder, 55. 

Hill (W.) on a deep channel of drift 
at Hitchin, 573. 

Honda (K.) on the secondary 
undulations of oceanic tides, 88. 
Houstoun (Dr. R. A.) on a new 

spectrophotometer, 282. 

Howchin (Rev. W.) on beds of 
Cambrian age in S. Australia, 202. 

Hughes (A. LI.) on molecular agere- 
gations produced in gases by 
sudden cooling, 746. 

Hydrogen, on the electrochemical 
equivalent of, 614. 

lliffe (J. D.) on beds of Cambrian 
age in S. Australia, 203. 

Inductances, on the use of variable 
mutual, 155. 

Induction, on the mutual, of coaxial 
coils, 332; on mutual, 564. 

Ingersoll (L. R.) on phenomena 
exhibited by small particles on a 
Nernst glower, 205. 

Interpolation, on inverse, by means 
of a reversed series, 628. 

Inversion temperature, on the Joule- 
Kelvin, 126. 

Isitani (D.) on the secondary undula- 
tions of oceanic tides, 88. 

Isomorphisms, on the groups of, 
of the groups whose degree is less 
than eight, 225. 

Jessup (A. C. and A. E.) on the 
evolution and devolution of the 
elements, 21. 


INDEX. v7 


Joly (Prof. J.) on the radioactivity 
of sea-water, 385. 

Kelvin (Lord) on the formation of 
concrete. matter from atomic 
origins, 897 ; on the problem of a 
spherical gaseous nebula, 687. 

Kirkby (Rey. P. J.) on the positive 
column in oxygen, 559. 

Kleeman (R. D.) on different kinds 
of y rays of radium, 638. 

Lead, on the heating effects pro- 
duced by Réntgen rays in, 482. 
Lehfeldt (Prof. R. A.) on the electro- 
chemical equivalents of oxygen 

and hydrogen, 614. 

Lens, on the focometry of a concave, 
198, 300. 

Lewis (W. C. M.) on Gibbs’s theory 
of surface-concentration, 499. 

Lowell (Prof. P.) on the tores of 
Saturn, 468. ; 

Madsen (Dr. J. P. V.) on the nature 
of the y rays, 663. 

Magnetic induction in an elongated 
spheroid, on, 65. 

properties, on the explanation 

of, 172. 

rotatory dispersion, onthe anom- 
alous, of neodymium, 270; of rare 
earths, 538. 

Mallik (Prof. D. V.) on a potential 
problem, 63; on the experimental 
determination of magnetic induc- 

- tion in an elongated spheroid, 65 ; 
on mutual induction, 364. 

Mass, relation of, to energy, l. 

Materials, on the rupture of, under 

. combined stress, 81. 

Matter, on the electron theory of, 
172; on the formation of concrete, 
from atomic origins, 397. 

Meadoweroft (L. V.) on the curva- 
ture and torsion of a helix on any 
cylinder, 55. 

Mendenhall (C. E.) on phenomena 

_ exhibited by small particles on a 
Nernst glower, 208. 

Miller (G. A.) on the groups of 
isomorphisms of the groups whose 
degree is less than eight, 223. 

Morley (Prof. A.) on laterally loaded 
struts and tie-rods, 711. 

Morrow (Dr. J.) on the lateral vibra- 
tion of bars, 497. 

Moss (H.) on the contact potential- 
differences determined by means 
of null solutions, 478. 


Nebula, on the problem of a spherical 
gaseous, 687. 

Neodymium, on the anomalous mag- 
netic rotatory dispersion of, 270. 
Nernst glower, on phenomena ex- 

hibited by particles on a, 205. 

Newton’s rings, on the ettect of a 
prism on, 345. 

Nitrogen scale, on the thermo- 
dynamic corrections of the, 526. 
Optical systems of revolutions, on 
von Seidel’s theorems relating to, 

677. 

Orr (Prof. W. McF.) on the mixing 
of gases, 297. 

Orstrand (C. E. van) on inverse 
interpolation by means of a re- 
versed series, 628. 

Owen (G.) on molecular aggregations 
produced by sudden cooling, 
746. 

Oxygen, on the positive column in, 
559; on the electrochemical equi- 
valent of, 614. 

Phosphorescence, mechanical, 352. 

Pipes, on spherical radiation and 
vibrations in conical, 69. 

Porter (Prof. A. W.) on the effect 
of the position of the grating on 
the resolving power of a spectro- 
scope, 762. 

ee column in oxygen, on the, . 

59. 

Potential differences, on the contact, 
determined by means of null solu- 
tions, 478. 

problem, on a, 63. 

Prism, on the effect of a, on Newton’s: 
rings, 345. 

Radioactive substances, on the 
irregularities in the radiation 
from, 589; on changes of velocity 
of rays from, 720. 

Radioactivity of sea-water, 385. 

Radium, on the atomic weight of, 
280; on the distribution in electric 
fields of the active deposits of, 601 ;. 
on the y rays of, 638. 

Ramage (H.) on the evolution and 
devolution of the elements, 396. 
Rayleigh (Lord) on the effect of a 

prism on Newton’s rings, 345; 
measurements of wave-leneths and 
notes on Fabry and Perot’s appa- 
ratus, 648; on Hamilton’s prin- 
ciple and the five aberrations of 

von Seidel, 677. 


772 


Reid (C.) on the pillow-lava near 
Port Isaac, 576. 

Relativity, on the principle of, 316. 

Resonance spectra of sodium vapour, 
on the, 581. 

Richardson (L. F.) on a freehand 
graphic way of determining stream- 
lines and equipotentials, 237. 

Richardson (Prof. O. W.) on the 
displacement of spectral lines by 
pressure, 204. 

Rontgen rays, on the heating effects 
produced by, 432. 

Rose-Innes (J.) on the thermo- 
dynamic scale of temperature, 301. 

Rupture of materials under combined 
stress, on the, 81. 

Russ (8.) on the distribution in 
electric fields of the active deposits 
of radium, thorium, and actinium, 
601; on the charge of the 
active deposit of actinium, 737. 

Saturn, on the tores of, 468. 

Schott (G. A.) on the electron theory 
of matter and the explanation of 
magnetic properties, 172; on the 
frequencies of the free vibrations 
of quasi-permanent systems of 
electrons, 438. 

Sea-water, radioactivity of, 385. 

Smith (S. W. J.) on the contact 
potential-differences determined by 
means of null solutions, 478. 

Sodium atom, on the existence of 
positive electrons in the, 274. 

vapour, on the resonance 
spectra of, 581. 

Sorby (Dr. H. C.) on the structure 
and history of rocks, 574. 

Sound, on waves of, in conical pipes, 
69; on the factors serving to de- 
termine the direction of, 318. 

Spectra of sodium vapour, on the 
resonance, d81. 

Spectrophotometer, on a new, 282. 

Spectroscope, on the effect of the 
position of the grating on the 
resolving power of a, 762. 

Spectrum lines, on the explanation 
of, 438. 

Stability, on induced, 233. 


END OF THE FIFTHTEENTH VOLUME. 


Printed by Taytor and Francis, Red Lion Court, Fleet Street. 


INDEX. 


Steel, on the effect of comb .- 
stresses on the elastic properties 
of, 214. * 

Stephenson (A.) on induced sta- 
bility, 233; on mechanical phos- 
phorescence, 352. 

Stream-lines, on-a freehand graphic 
way of determining, 237. 

Struts, on laterally loaded, 711. 

Surface-concentration, on (Gibbs’s 
theory of, 499. . 

forces, on the theory of, 413. 

Temperature, on the thermodynamic 
scale of, 301. 

Temperature equilibrium, on certain 
dynamical analogues of, 765. 

Terada (T.) on the secondary undu- 
lations of oceanic tides, 88. 

Thermodynamic corrections of the 
nitrogen scale, on the, 526, 

scale of temperature, on the 
practical attainment of the, 301. 

Thorium, distribution in electric 
fields of the active deposit of, 601. 

Tides, on the secondary undulations 
of oceanic, 88. 

Tie-rods, on laterally loaded, 711. 

Tomkins (J. A.) on the focometry 
of a concave lens, 198, 300. 

Vibrations, on, in conical pipes, 69 ; 
on the thermally excited, of an 
atmosphere, 147. 

Wave-lengths, measurements of, 548. 

Wilde (Dr. H.) on the atomic 
weight of radium, 280. 

Williams (W. E.), rupture of ma- 
terials under combined stress, 81. 

Wood (Prof. R. W.) on the anom- 
alous magnetic rotatory dispersion 
of neodymium, 270; on the ex- 
istence of positive electrons in the 
sodium atom, 274; on the reson- 
ance spectra of sodium, 581. 

Wright (Prof. G. F.) on the glacial 
epoch in N. America, 573. — 

Wright (W. B.) on the two earth- 
movements of Colonsay, 579. 

X-rays and scattered X-rays, on, 
288. 

Zinc, on the heating effects produced 
by Rontgen rays in, 432. 


Bprrweb on 


ee ae 
we) Set ary 


ee 
- _ 
a ee el 


SMIT| HSONIAN INSTITU 


UENO 


re 


3 9088 01299 4816