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THE 


LONDON, EDINBURGH, anp DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


CONDUCTED BY 


SIR ROBERT KANE, LL.D. F.R.S. M.R.LA. F.CS. 


SIR WILLIAM THOMSON, Kw. LL.D. F.R.S. &e. 
AND 


WILLIAM FRANCIS, Pa.D. F.LS. F.RAS. F.GS. 


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


VOL. XXIIT—FIFTH SERIES. 


JANUARY SJ UNE 1887. 
Fa OKT 5 = NG, na™ 
fv el 
/ y \ 
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i a, 

f “> re) 
LONIAN DE pes 
LONDON: 


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Quid pariat nubes, veniant cur fulmina ceelo, 
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Tam vario motu.” 
J. B. Pinelli ad Mazonwm. 


CONTENTS OF VOL. XXIII. 


(FIFTH SERIES). 


NUMBER CXL.—JANUARY 1887. 


Mr. J. J. Coleman on Liquid Diffusion. (Plate I.) ........ 
Mr. O. Heaviside on the Self-induction of Wires.—Part V... 
Hon. R. Abercromby on the peculiar Sunrise-Shadows of 
feumerresk me coylon ! fe) bee ei el CL ads 
Prof. A. W. Riicker on thé Critical Mean Curvature of Liquid 
Peemeneeren, evo oo! Ve aan oats. 
Mr. T. Gray on Silk v. Wire Suspensions in Galvanometers, 
and on the Rigidity of Silk Fibre ..................0. 
Sir W. Thomson on Stationary Waves in Flowing Water.— 
Part IV. Stationary Waves on the Surface produced by 
Equidistant Ridges on the Bottom .................... 
Rev. T. K. Abbott on the Order of Lever to which the Oar 
EE tee aot nt ae ter ee due CW BSG aa 
Drs. W. Ramsay and 8S. Young on the Influence of Change of 
Condition from the Liquid to the Solid State on Vapour- 
PME e st pe a UL ARE POs. 
Proceedings of the Geological Society :— 

Prof. T. M*Kenny Hughes on the Drifts of the Vale of 
Clwyd, and their Relation to the Caves and Cave- 
RES se ee, Re A hal, 

Mr. I. Rutley on the Metamorphic Rocks of the Malvern 
LITE ee ees ogo Ps EE ae A era a A Pe ne 

On a nearly perfect Simple Pendulum, by J. T. Bottomley .. 


NUMBER CXLI.—FEBRUARY. 


Prof. H. E. Armstrong on the Determination of the Constitu- 
tion of Carbon Compounds from Thermochemical Data... . 

Prof. 8. U. Pickering on the foregoing Communication 

Sir W. Thomson on the Front and Rear of a Free Procession 
Seemed) tr Dee Water i... nce ie ee ete ees te aie ne as 

Prof. G. Carey Foster ona Method of Determining Coefficients 
Reema MEP TRCHIOR = 6. pints) i603 sos bs ne See OR 


70 
72 


1V CONTENTS OF VOL. XXIII.—-FIFTH SERIES. 


Page 

Drs. W. Ramsay and S. Young on the Nature of Liquids, as F 
shown by a Study of the Thermal Properties of Stable and 

Desociable Bodies. 0.0.0. Pete. ee 129 
Mr. W. N. Shaw on the Atomic Weights of Silver and 

COPP oe ede vee Se ae wields ce ety ete ee 138 
Prof. Tait on the Foundations of the Kinetic Theory of Gases. 

Part dD es oe oe senha e ed gene? oe 5 rn 141 
Rey. O. Fisher on the Amount of the Elevations attributable 
to Compression through the Contraction during Cooling of 

BV OOLIG Marth ie Abi. ie ee se ik wile aie Oe © rrr 145 

Mr. ih. H. M. Bosanquet on Silky. Wire <>....-g2 pee eee 149 
Mr. J. Walker on Cauchy’s Theory of Reflection and Refrac- 

thonof Light): 60293. etal 3 Pe Pe ee 151 

Mr. O. Heaviside on the Self-induction of Wires.—Part VI. . 173 

Notices respecting New Books :— 

Mr. T. Mellard Reade’s Origin of Mountain-Ranges, 
considered Experimentally, Dynamically, and in Rela- 

tion to their Geological History .................. 213 
Descriptive Catalogue of a Collection of the Economic 

Minerals of Canada... .«..!) 22 eee 216 
Journal and Proceedings of the Royal Society of New 

South! Wales for. 1885. 2. 1.20) Gays eee 216 

Capt. W. Noble’s Hours with a Three-Inch Telescope .. 218 
Prof. G. Chrystal’s Algebra: an Elementary Textbook 
for the higher classes of Secondary Schools and for 


Colleges is one es a lod cay ba OE ee 219 
Dr. B. O. Peirce’s Elements of the Theory of the New- 
tonian Potential Function ©......).3. 9) eee 220 


Proceedings of the Geological Society :— 
Mr. W. Whitaker on the Results of some deep Borings 
mm Kei aii.) cies wit alee at oer rh 2 
“To what Order of Lever does the Oar belong?” by Francis 
A. Tarleton, Fellow of Trinity College, Dublin.......... 222 
On the Specific Heats of the Vapours of Acetic Acid and 
Nitrogen Tetroxide, by Prof. Richard Threlfall.......... 223 


NUMBER CXULI.—MARCH. 


Lord Rayleigh’s Notes on Electricity and Magnetism.—IIL. 
On the Behaviour of Iron and Steel under the Operation of 
Heeble Magnetic Forces.” (Plate 11)... /) 5). 3 oases 225 
Mr. H. Tomlinson on the Permanent and Temporary Effects 
on some of the Physical Properties of Iron, produced by 
raise the Temperature to 100°C +...) a ee 245 


CONTENTS OF VOL. XXIII.—FIFTH SERIES. v 


Sir W. Thomson on the Waves produced by a Single Impulse 
in Water of any Depth, or in a Dispersive Medium ...... 252 

Sir W. Thomson on the Formation of Coreless Vortices by 

- the Motion of a Solid through an Inviscid Incompressible 


De eee al re OE SS bk eee te 255 
Prof. H. A. Rowland on the Relative Wave-lengths of the Lines 
MOY VCCI hy ga oie wiv cas Snelson eae os 257 
Mr. L. Bell on the Absolute Wave-length of Light ........ 265 
Prof. W. C. Unwin on Measuring-Instruments used in 
MR NN Se ies Peach cw om 6 ain ny walle. gic: comin 282 
Sir W. Thomson on the Equilibrium of a Gas under its own 
NUMMER, 23 a oo hic 8 dir we 1.08 enn dd Odd Oa, 287 


Mr. W. Brown’s Preliminary Experiments on the Effects of 
Percussion in Changing the Magnetic Moments of Steel 
TTS ete tO ek Chard oe Valle Ueda LA Dee 293 

Notices respecting New Books :— 

Annual Companion.to the ‘ Observatory,’ a Monthly 


Smee Cir A BUCONOIRY V5 aes ac sw ole Wee Woda 299 
Mr.G.8. Carr’s Synopsis of Elementary Results in Pure 
Pee cP MMEARIE. 22a Cease S. De awe t 300 


On the Action of the Discharge of Electricity of High Poten- 
tial on Solid Particles suspended in the Air, by A. von 


Sreeemaver and M. von‘Pichler) ..........0N 000i 301 
On a Simple and Convenient form of Water-Battery, by 
SEAT IRHOR Ui, Unis. diy oe ete? eA OL els 303 


On the Galvanic Polarization of Aluminium, by Dr. F. Streinz. 304 


NUMBER CXLIJ.—APRIL. 
Prof. Ludwig Boltzmann on the Assumptions necessary for 


the Theoretical Proof of Avogadro’s Law .............. 305 
Prof. 8. P. Thompson on an Arc-Lamp suitable to be used with 

mee uboscq Lantern. (Plate LIT.) 2... 2.0 ee ecien dace 333 
Mr. R. H. M. Bosanquet on Electromagnets.—VII. The Law 

of the Electromagnet and the Law of the Dynamo ...... 338 
Messrs. E. Gibson and RK. A. Gregory on the Tenacity of 

NEC Seo Roce sh ae ecidh oh anh s:crttueitg tat dol 
Mr. T. Gray on an Improved Form of Seismograph. (Plate 

Ee ee aa ene ee, ieee ane eee 353 
Mr. F. Y. Edgeworth on Discordant Observations ........ 364 
Dr. E. J. Mills on the Action of Heat on Potassic Chlorate 

EE cha bine siti Aid hs wo yt 9 ps ala! Ody Vad a 375 


Prof. J. J. Thomson’s Reply to Prof. Wilhelm Ostwald’s criti- 
cism on his paper ‘“‘ On the Chemical Combination of Gases” 379 
On certain Modifications of a Form of Spherical Integrator, 
by V. Ventosa 


v1 CONTENTS OF VOL. XXIII.—FIFTH SERIES. 
Page 
On the Strength of the Terrestrial AEG Field in Build- 
mes, by M. Aime! Witz) vi. cn cs aon ke ee ee 381 
On Metallic Layers which result from the Volatilization of a 
Kathode, by Bernhard Dessau -.,....4,..7. 044.82 oe eee 384 
On the Passage of the Hlectric Current through Air under 
ordinary circumstances, by J. Borgmann .............. 384 


NUMBER CXLIV.—MAY. 


Dr. W. W. J. Nicol on the Expansion of Salt-Solutions. 

CPlates V..65 VI.) oo. oe eee selene idles eines ee 385 
Prof. 8. U. Pickering on Delicate Thermometers .......... 401 
Prof. S. U. Pickering on the Effect of Pressure on Thermo- 

meter-bulbs and on some Sources of Error in Thermometers 406 
Mr. R. H. M. Bosanquet on the Determination of Coefficients 

of Mutual Induction by means of the Ballistic Galvanometer 

and Marth-Inductor...... 2... 3). 2 Je ee) eee 412 
Mr. W. Brown on the Effects of Percussion and Annealing on 

the Magnetic Moments of Steel Magnets.............. 420 
Prof. Tait on the Assumptions required for the Proof of 

Avogadro's Law ..ie. 00s. is sau iio. eee 433 
Drs. W. Ramsay and 8. Young on Evaporation and Dissocia- 

tion.—Part VI. On the Continuous Transition from the 

Liquid to the Gaseous State of Matter at all Temperatures. 


(Plates VIL. VIL. IX.,.& X.) 2.52.0. a eee 435 
Sir W. Thomson on the Stability of Steady and of Periodic 
Miaid Motion ...¢. 6... ees ees 459 


Notices respecting New Books :-— 

Profs. Oliver, Wait, and Jones’s Treatise on Algebra.... 465 
Proceedings of the Geological Society :— 

Mr. T. Roberts on the Correlation of the Upper Jurassic 


Rocks of the Jura with those of England.......... 466 — 
Rev. A. Irving on the Physical History of the Bagshot 
Beds ‘of the London Basin :... 2...) 75a 467 


On the Inert Space in Chemical Reactions, by Oscar Liebreich 465 
Apparatus for the Condensation of Smoke by Statical Elec- 


roeity, by HL Avnaury ../..0..0.......... 471 
On the Heating of the Glass of Condensers by Intermittent 
Hlectrification, by J. Borgmann ..........:.|., See 472 


On the Chemical Combination of Gases, by Prof. Ostwald .. 472 


NUMBER CXLV.—JUNE. 


Mr. R. F. Muirhead on the Laws of Motion.............. 473 
Mr. C. V. Boys on the Production, Properties, and some sug- 
gested Uses of the Finest Threads .............-...... 489 


CONTENTS OF VOL. XXIII.—FIFTH SERIES. vil 
Page 
Mr. 8. Bidwell on the Electrical Resistance of Vertically- 
TREE OD net Ia ener alate aie Saco a FA kw oh wiiale wiecn § © 499 
Prof. L. Meyer on the Evolution of the Doctrine of Affinity . 504 
Prof. R. Meldola’s Contributions to the Theory of the Con- 
stitution of the Diazoamido-Compounds................ 513 
Sir W. Thomson on the Stability of Steady and of Periodic 
Fluid Motion.— Maximum and Minimum Energy in Vortex 


NE ek Scans Sy rade roieaih © « waiinrehaiate nlela.cie eR 529 
Dr. G. Fae on the Variations in the Electrical Resistance of 
Antimony and Cobalt in a Magnetic Field.............. 540 
Capt. P. A. MacMahon on the Differential Equation of the 
most general Substitution of one Variable-.............. 542 
- Jurious consequences of a well-known Dynamical Theorem, by 
Dr. G. Johnstone Stoney...... NCAT ATM ile 544 
On the Gaseous and Liquid States of Matter, by Drs. W. 
UMP UIST, Pee QUI oes diac x hse oN blew eye se se 547 
Lecture Experiments on the Conductivity of Sound, by M. 
PiOeAUMS 6.6... eee Sees as i ae ae ainte waa ike eons aka © Vato iar 548 


PLATES. 
I, Tlustrative of Mr. J. J. Coleman’s Paper on Liquid Diffusion. 


II. Illustrative of Lord Rayleigh’s Paper on the Behaviour of Iron and 
Steel under the Operation of Feeble Magnetic Forces. 


IIL. Ilustrative of Prof. S$. P. Thompson’s Paper on an Arc-lamp. 

IV. Ilustrative of Mr. T. Gray’s Paper on an Improved Form of Seismo- 
eraph. 

V. & VI. Illustrative of Dr. W. W. J. Nicol’s Paper on the Expansion of 
Salt-Solutions. 


VIL, VIL, 1X., X. Illustrative of Drs. Ramsay and Young’s Paper on 
Hvaporation and Dissociation. 


THE 
LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[FIFTH SERIES.] 
JANUARY 1887. 


I. On Liquid Diffusion. 
By J. J. Coteman, f.1.C., F.C.S., F.RS.E* 


[Plate I.] 
()UR knowledge upon this subject is chiefly derived from 


Graham’s classical researches. His first paper was 
communicated to the Royal Society in 1849, and further 
papers in 1850 and 1851. 

About the year 1855, Fickf, commenting upon these in- 
vestigations, remarked that it was a matter of regret that in 
such an exceedingly valuable and extensive investigation the 
development of a fundamental law for diffusion in a single 
element of space was neglected, which (he added) it was quite 
natural to suppose would be identical with the law according 
to which diffusion of heat takes place in a conducting body, 
and upon which Fourier founded his theory of heat, and Ohm 
his theory of diffusion of electricity in conductors. Fick 
endeavoured to supply this omission so far as common salt is 
concerned, and Voit calculated the coefficient of diffusion of 
sugar. Professor Mach, of Prague, has also worked with 
these substances. Other experimenters have calculated the 
coefficients of diffusion of salts, or, rather, of a limited number 
of them, with not very concordant results, as may be seen by 
consulting the tables of Schumeister attached to the article 
“ Heat,” by Sir W. Thomson, in the ninth edition of the 
Encyclopedia Britannica, and comparing them with the results 
of Beilstein, who employed Jolly’s method, described in 

* Communicated by the Author. 
+ Phil. Mag. [4] x. 1855. 
Phil. Mag. 8. 5. Vol. 23. No. 140. Jan. 1887. B 


2 Mr, J. J. Coleman on Liquid Diffusion. 


Watts’s ‘ Dictionary of Chemistry,’ vol. iii. p. 710. Graham, 
however, in his latest paper upon the subject, communicated 
to the Royal Society in 1861, described a method of experi- 
ment which, to use his own words, affords a means of obtaining 
the absolute rate or velocity of diffusion. This he called “ Jar 
diffusion,” the saline solution being delivered by a pipette to 
the bottom of a column of pure water 127 millim. high, stand- 
ing in a cylindrical jar 87 millim. diameter, and the amount 
diffused being ascertained by drawing off the liquid from the 
top in equal fractions by a very fine syphon, the orifice of the 
short leg of which was kept close under the surface of the 
liquid being drawn off. 

Graham applied this method to a variety of organic com- 
pounds, but only to hydrochloric acid, sulphate of magnesia, 
and sulphates and chlorides of potassium and sodium amongst 
inorganic substances. He did not attempt the calculation of 
the coefficients of diffusibility in absolute measurements, but 
remarked that the method is extremely simple, and gives results 
of more précision than could have been possibly anticipated. 

I have recently made a considerable number of experiments 
with this method; but in place of using jars of 87 millim. in 
diameter I have employed glass cylinders of 36 millim. dia- 
meter, such as are used for “ Nesslerizing,’ and which are 
very uniform in bore. The annexed Table shows the actual 
amount of salt found in each section of the liquid after dif- 
fusion, expressed in millimetres ; the quantity of salt in the 
upper section being also calculated in percentages of that 
particular section which at the commencement of the dif- 
fusion was the point of junction of the saline solution and the 
pure waiter. | 

The concentrated saline solution was introduced below the 
pure water by a method slightly different to that of Graham ; 
namely, a fine-bore syphon with contracted orifices was first 
filled with water, the finger being placed so as to cover the 
short end ; the long end was thrust to the bottom of the water 
in the diffusion-jar, and the short end was uncovered in a 
vessel containing the saline solution at a higher level, and the 
syphoning continued until the height of the liquid admitted 
under the water in the diffusion-jar amounted to 50 millim. 

Many of the results agree with those of Graham, whilst 
others are additions thereto, such as the diffusions of mercurous 
nitrate, mercuric chloride, lithium sulphate, cadmium sul- 
phate, silver sulphate, manganese sulphate, nickel sulphate, 
and lead nitrate. 

Moreover they are twenty-five diffusions conducted for 
equal lengths of time, and under similar circumstances as to 
temperature and methods of experimenting. 


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Reetite MESO ss ZIS0%. CSOs CaCl. 
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1 5 Bere nin. | Sie Re | eee ee 
: Se eee | eee ee 
7 | ion 7 7 
1 | 92 14 198) 
3s | 739 92 || 280 
56 | 1310 57 | 504 
100 | 2309 100 813 Ee 


temp. 10°C, || temp. 10°C. 


PL(NO,),. 


Height, AgNO, | MgSO,. 


in millims. 


1693 100 


es the eight in milligrams of salt in each section syphoned off from the top. 


me : 
the approximate percentage of weight, assuming the bottom layer to contain 100 of salt. c 


1 


Mr. J. J. Coleman on Liquid Diffusion. 3 


Although from many points of view, and especially that of 
the chemist, most valuable and important deductions can be 
made from this class of experiments, some of which will be 
referred to in a Jater part of this paper, it must be admitted 
that, from the physicist’s point of view, all these experiments 
are vitiated from the fact that the diffusions should commence 
with the solid salt or the anhydrous acid or alkali, and end in 
an atmosphere of pure water. With Graham’s earlier expe- 
riments a saline or acid solution, the strength of which was 
gradually diminishing, was diffused into a weaker saline or 
acid solution which was constantly increasing ; and even with 
his latest method of “ Jar diffusion ”’ it is practically impossible 
to work with syphons in long columns of water such as are 
necessary to get a pure water atmosphere above, the salt being 
diffused. I have therefore turned my attention to improved 
methods of working, and have devised apparatus, illustrated by 
Plate I. fig. 1*. This apparatus allows of a very concentrated 
solution of the substance under examination being admitted 
at any desired rate of speed underneath a column of pure 
water of any desired length, and, further, of the liquids being 
drawn off in regulated quantities at the end of the time of 
diffusion. No doubt, with solid crystalline salts, theory in- 
dicates that diffusion should commence from the salt; but 
practice determines the fact that, with solid salts, air-bubbles 
are a difficulty; and, moreover, there are a very large number 
of salts that are not crystalline at all, and some that do not 
exist in the solid state, such as MgCl. 

Returning to the description of fig. 1, B B represents a 
Mohr’s burette 500 millim. long and 15 millim. diameter 
provided with a glass stopcock. 

The stopcock being closed, this burette is nearly filled with 
pure water, and is then connected by india-rubber tubing 
with an apparatus by means of which the air above the water 
ean be slightly rarefied, such as an air-pump or an aspirator, 
or, by what I find most convenient, an open glass tube C C 
standing in a jar of water. On opening the stopcock of the 
burette, about half its water escapes, say, to the level (a) 
(rarefying the air above), upon which a column of water 
rushes up the tube CC, say to z. The stopcock being now 
closed, the tube C C is raised and clipped in such a position 
that the column of water in the tube C C is much longer than 
the water-column in the burette, by which means the air above 
the water in the burette is sufficiently rarefied to admit of 
the concentrated saline or other solution being sucked up 


* A description of this apparatus and the principal results detailed in 


this paper were communicated to the Philosophical Society of Glasgow, 
16th April, 1886. 
B2 


4 Mr. J. J. Coleman on Liquid Diffusion. 


underneath the water by cautiously opening the stopcock. A 
column, say, 100 millim. deep can be drawn into the burette 
under a depth of water of 200 millim., which latter can be 
further added to by filling up the burette cautiously by a long- 
legged pipette discharged on a cork float. This, however, is 
seldom necessary except with very diffusive substances, such 
as hydrochloric acid, which barely reach a height of 200 
millim. in 21 days. To prevent leakages, an indiarubber 
cap is slipped over the point p of the burette. At the end of 
the diffusion-time the liquid is very slowly run off until the 
level where the water originally joined the saline solution is 
reached, after which equally-measured sections are carefully 
removed and reserved for further examination, each section. 
being, say, 25 millim. deep. The results are conveniently 
calculated in percentages upon the salt or other substance 
contained in the bottommost layer, where diffusion commenced. 
It was thought at first that some errors might arise from 
adherence of the saline solutions to the inside of the burette, 
down which the upper layers had to pass before estimation. 
This proved, however, not to be the case to any serious or 
appreciable extent. Most diffusions are carried on until at 
any rate 1 per cent. of the salt rises 50 millim. ; and it was 
found that sulphuric acid of 1:2 specific gravity, admitted in 
the burette under water coloured blue with litmus and then 
withdrawn carefully, had so little effect on the water 25 mil- 
lim. above that it came out of the stopcock barely reddened, 
and the water lying 50 millim. above the acid came out quite 
blue. Corroborative experiments were made with saturated 
cupric sulphate and ammonia as an indicator. 

Hydrochloric acid being the most diffusive substance ex- 
amined by Graham, comparative experiments were made with 
this compound at temperatures of 12°5 Centigrade, the 
results of which are shown in the curves, fig. 2. 

Fig. 3 shows in the form of curves the results of some 
comparative experiments made for a period of twenty days at 
a temperature of 12°°5 C., with the following substances :— 


1. Sulphuric acid containing 20 per cent. anhydride. 
20 


2. Hydrochloric acid __,, 43 “ 
3. Nitric acid gt eel) é, 5 
4. Potassic hydrate 5s ieee +9 = 
5. Sodic hydrate sh eeO es : 
6. Ammonia solution of *880 specific gravity. 

7. Magnesic sulphate . saturated. 

S$. Sodic ‘chloride jas +5;(s1,. «ditto: 


The ammonia solution was floated on the water, the rest 
admitted under the water, and, with the exception of the 


Mr. J. J. Coleman on Liquid Diffusion. 5 


magnesic sulphate, the sections of liquid run off were esti- 
mated by ordinary volumetric alkalimeter processes. 
A summary of the results may be conveniently added :— 


Comparative diffusion during twenty days; temperature 


12°5 Cent. 
One per cent. of the hydrochloric acid rose 250 millim. 
9 FS nitric acid reas 73, 
” m potassic hydrate ,, 225 _,, 
a 33 ammmonia a AAO) Fes 
¥5 n sulphuric acid Sa cia ee 


a i. sodic hydrate go 2hdt 
a is sodic chloride 5g 2 
bi s magnesic sulphate ,, 87 

These results are put forward merely as preliminary expe- 
riments with a method which is rapid in execution and sus- 
ceptible of great accuracy, and which it is hoped will afford 
or lead to the means of calculating the correct coefficient of 
diffusion of a large number of substances, Sir W. Thomson 
haying kindly promised his assistance in any mathematical 
calculations necessary. 

From the chemist’s point of view, several interesting con- 
siderations arise from a review of experiments already made, 
particularly when taken in connection with Newland’s and 
Mendelejeff’s periodic law, which was not put forward until 
some time after the date of Graham’s last paper. — 

The periodic law classifies the elements into vertical groups 
of the type R,O, RO, &c., and into series usually known as 
horizontal series. Most of our recent textbooks quote the 
ard series as typical ; thus :-— 

Ware (Ris, Al. Si. 1% = Cl. 
Atomic weight .23 24 273 28 31 32 = 35°5 
Atomic volume. 24 14 10 et Peo. Eo) ae 


It will be observed here that whilst there is a comparatively 
small difference between the atomic weights commencing and 
ending the horizontal series, the atomic volume commences 
very high at the extreme left, gradually diminishes to less 
than one half in the middle, and ends very high again at the 
extreme right of the series. Diffusibility of the compounds 
of these elements varies in the same way ; thus :— 


Na. Mg. AL Si. ike Ss. Cl. 
WaisO,, MesO0; ALSsO, SLO; P.0,. -8,0,.. Cl,0,,. 
Molec. vol. ...55 44. ? Alp 5 82 ? 
Diffusibility...15 4 2°8 ? 9-79 18°48 25 
a fF 
Coleman. Graham. Coleman. 


6 Mr. J. J. Coleman on Liquid Diffusion. 


The diffusion of chloric acid against sulphuric acid I had to 
make specially* ; the other figures are from the tables and 
data before given, and the atomic volumes are quoted from 
Ira Remsen’s recent book, the ‘ Principles of Theoretical 
Chemistry.’ 

Here then is evidence that, given a number of elements the 
atomic weights of which do not differ widely, the diffusibility _ 
will vary in some sort of proportion to the atomic or mole- 
cular volume, which latter, indeed, may be to some extent 
regulated by the energy of the molecules. 

Mendelejeff’s fourth horizontal series commences with K 
and Ca, the molecular volumes and diffusibilities of the chlo- 
rides being thus :— 


K,Cl, CaCl, 
Molecular volume . . . 74 44 
Pitrastbiliiy 6 sk ta ehees tyme 27 


Here again diffusibility corresponds with the larger molecular 
volume. 

The fifth series commences with copper and zinc, the mole- 
cular volume of the sulphates being exactly equal and the 
diffusibility nearly equal. 

The sixth series commences with rubidium and strontium, 
the metal rubidium possessing, with the exception of cesium, 
the largest atomic volume of any metal yet discovered. 3 

Graham has already shown that chloride of strontium is 
much less diffusible than chloride of potassium. If, there- 
fore, chloride of rubidium be as diffusible as chloride of potas- 
sium, then the diffusibility of chloride of rubidium will be 
greater than that of chloride of strontium. This has been 
proved to be the case by the following diffusions I have 
recently made. Chloride of rubidium, chloride of potassium, 
and chloride of sodium were diffused for 10 days at 12° C. in 
the apparatus figured on Plate I. 


* By new method of diffusion :— 


58 per cent. sulphuric acid rose 25 millim. in 7 days. 
29 0 


2) ”? ? 49 tp) 


12 ”) ? P) 15 9 ” 


70 per cent. chloric acid rose 25 is 
”) 99 ”) 50 7 ry) 


18 99 ” 9 75 ” 


temperature 12° Cent. 


Mr. J. J. Coleman on Liquid Diffusion. 7 


Height diffused, “hIovide, shlotida, silatide 
in millimetres. per cent. per cent. per cent. 
O2uty sso 1168 64 60 
afer: el aad 39 31 
Ge att bine Lal? 19 10 
een Sse shite TE fl 3 
Oot! A Fe BD 3°39 


It will be seen that rubidium chloride is quite as diffusive 
as potassic chloride, a salt which has hitherto been the most 
diffusive salt examined. This I anticipated from the large 
atomic volume of the metal, which is very much greater than 
even that of potassium ; but, on the other hand, its atomic 
weight is greater. 

If a large molecular volume indicates a tendency to rapidity 
- of diffusion, it may be suspected that a large atomic or mole- 
cular weight has a tendency to retard it; this, however, is 
not so easy to prove, from the extreme difficulty of getting 
groups of soluble compounds, the molecular volumes of which 
are identical, but the molecular weights of which differ. 
Moreover, it appears necessary to look for such groups in the 
family groups of Mendelejeft’s vertical series, or the isomor- 
phous groups of the older chemists. The sulphates of zine 
and magnesia are strictly isomorphous, and possess an identical 
molecular volume, viz. 44 when anhydrous, their molecular 
weights being as 120 to 161. 

Graham not only in this case, but in several other cases of 
isomorphous bodies, strove hard to prove that the rate of dif- 
fusion was identical, returning again and again to the subject, 
on the last occasion making seventy-two experiments with 
magnesic and zincic sulphates ; which he sums up by stating 
that the approach to equality becomes close in the 4-per-cent. 
and larger portions of salt, but differed as much as 8°75 per 
cent. in favour of the sulphate of magnesia in the 1-per-cent. 
solutions. 

I have recently diffused these substances for the long period 
of 50 days, and at a temperature of 15° C., in the apparatus 
described in an earlier part of this paper as an improvement 
upon Graham’s. ‘The results are that— 


9 per cent. of the magnesic sulphate rose 100 millim. 
7 Py ts zZincic + Si iidansed AY. Coad 


It therefore appears that these substances are not equi- 
diffusive, and that the one possessing the least molecular 
weight is the most diffusive. These experiments are very 


8 Mr. J. J. Coleman on Liquid Diffusion. 


decisive also on the matter of solubility not influencing dif- 
fusibility, since the sulphate of zinc is well known to be much 
more soluble in water than the sulphate of magnesia. 

Two substances from Mendelejett’s 6th group, viz. chromic 
acid and tungstic acid, were selected for diffusion; the 
molecular volumes of which are identical, and the molecular 
weights of which are as 100°5 to 232. They were diffused as 
normal salts of soda for thirty days, at a temperature of 
12°5 C., with this result (in which it will be again observed 
the one possessing the least atomic weight is most diffusive) :-— 


25 per cent. of chromate of soda rose 75 millim. 
1% 53 tungstate 4s 13, ae 


Molybdate of soda was diffused also at the same time, but 
as the atomic volume of molybdic acid is not identical but 
larger than that of either of the other two, there was a 
greater diffusibility, viz. :— 

28 per cent. molybdate of soda rose 75 millim. 

In the case of Mendelejeff’s 7th group we have chlorides, 
bromides, and iodides of the following molecular weights and 
volumes :— 


Mol. wt. Mol. vol. 
odie chloride. +8 2? «11585 26:0 
57 bromide... “aa. 10a30 33'0 
3) MOdIdS Wis. er Or La Oe) AS 
Potassic chloride . . . T74°6 37°0 
yy onomide@s vec. 20 ll Oo 44:0 
Wy)  WOGIO. «ia ge pie col ord 54:0 


In all these cases it will be noticed that there is an increase 
of molecular volume simultaneous with an increase of 
molecular weight, so that it is possible retardation of diffusi- 
bility, owing to an increased molecular weight, may be 
counterbalanced by an increased energy, indicated by the 
larger molecular volume ; and may account for the singular 
fact discovered by Graham that these substances are practi- 
cally equally diffusive, though further experiment may show 
some little variations. Similar remarks may be made in 
regard to the chlorides and nitrates of calcium, strontium, and 
barium, whicn Graham also showed to be practically equally 


diffusive. Mol. wt. Mol. vol. 
Caleicichloride «..1 5°) ata 44 
Barie. chloride, :is.0% “x. 220s 54. 
Qalciaigxide, steps 6 ae aoe 18 
Strontic oxide. 4.06 sy sn albsb 22 


partic oxide = § 6 «0s % \Roaw 28. 


Mr. J. J. Coleman on Liquid Diffusion. “8 


In regard to Mendelejeff’s 8th group, I have diffused ferrous 
sulphate, cobaltous sulphate, nickelous sulphate, and cupric 
sulphate, the molecular weights and volumes of which are as 
follows :— 


Mol. wt. Mol. vol. 
Ferrous sulphate. . . . 152 48 
Cobaltous sulphate . . . 155 44 
Nickelous sulphate . . . 155 ? 
Cupric sulphate. . . . 159°4 44 


The diffusions were carried on fifty days at 10° C. 


11 per cent. of the iron rose 100 millim. 
10 FA 5 cobalt SASS 
10 43 i nickel bpd ty, 
10 ” ” EOPPEE ” ” 


The diffusion rate of these substances, which closely approxi- 
_tnates those of Mendelejeff’s 2nd group, the Dyads, also 
corroborates the other experience as to the influence of the 
molecular weights and volumes. 

The difference in diffusibility, however, between the first, 
or Monad group of Mendelejeff, and his second, or Dyad 
group, is very striking, and can be approximately represented 
by the following figures :— 


Group I, Formula R,O [ Monads]. 
Sulphates. Nitrates. Chlorides. 


ing... ke 12 ? ? 
AM ge ge 15 28 34 
fetassium. . . « 24 ao 41 
Silver Nn Sati ie 17 35 insol. 
mverago:. s° 7.1% 32 38 


Group II. Formula RO [Dyads]. 
Sulphates. Nitrates, Chlorides, 
23 


Magnesium .. . 4. 21 
Cadmium :.o)%.3.:". 5 ? 14 
Zine Oe TOG Sts, a5 DS) 23 
Meretny 20.79) 6y 4 6 13 10 
Cateiamr ies 9.209240. insol. 23 D7. 
Strontium. 02 Ss Ol oditto 23 mAs 
ari = 3 Heals yk ALCO 23 27 
Average 22) 0 55 21 21 


Mendelejeff classes cuprous salts with Monads; and it is 
singular that very early in his experiments Graham pointed 


10 Mr. O. Heaviside on the 


out that cuprous chloride has only half the diffusibility of 
cupric chloride. 

Tam still prosecuting the investigation, and other interesting 
relationships may be detected, but the data as to the specific 
gravity and molecular volumes of compounds of many of the 
rarer elements are entirely wanting, and even in the case of 
those of well-known elements are incomplete. 


II. On the Self-induction of Wires.—Part V. 
By OLIveR HEAVISIDE*. 


HE mathematical difficulties in the way of the discovery 
of exact solutions of problems concerning the propaga- 
tion of electromagnetic disturbances into wires of other than 
circular section—or, even if of circular section, when the 
return current is not equidistantly distributed as regards the 
wire, or is not so distant that its influence on the distribution 
of the wire current throughout its section may be disregarded 
—are very considerable. As soon as we depart from the 
simple type of magnetic field which occurs in the case of a 
straight wire of circular section, we require at least two geo- 
metrical variables in place of the one, distance from the axis 
of the wire, which served before ; and we may have to supple- 
ment the magnetic force “of the current,” as usually under- 
stood, by a polar force, or a force which is the space-variation 
of a single-valued scalar, the magnetic potential, in order to 
make up the real magnetic force. 

There are, however, some simplified cases which can be 
fully solved, viz. when the external magnetic field, that in the 
dielectric, is abolished, by enclosing the wire in a sheath of 
infinite conductivity. It is true that we must practically 
separate the wire from the sheath by some thickness of 
dielectric, in order to be able to set up current in the circuit 
by means of impressed force, so that we cannot entirely abolish 
the external magnetic field; but we may approximate in 
a great measure to the state of things we want for pur- 
poses of investigation. The wire, of course, need not be a 
wire in the ordinary sense, but a large bar or prism. The 
electrostatic induction will be ignored, requiring the wire 
to be not of great length; thus making the problem an electro- 
magnetic one. 

Consider, then, a straight wire or rod or prism of any sym- 
metrical form of section, so that when a uniformly distributed 
current passes through it its axis is the axis of the magnetic 


* Communicated by the Author. 


Self-induction of Wires. 11 


field, where the intensity of force is zero. Leta steady cur- 
rent exist in the wire, longitudinal of course, and let the 
return conductor be a close-fitting infinitely conducting sheath. 
This stops the magnetic field at the boundary of the wire. 
The sudden discontinuity of the boundary magnetic force is 
then the measure and representative of the return current. 

The magnetic energy per unit length is }LO’, where C is 
the current in the wire and L the inductance per unit length. 
As regards the diminution of the L of a circuit in general, 
by spreading out the current, as in a strip, instead of concen- 
trating it in a wire, that is a matter of elementary reasoning 
founded on the general structure of L. If we draw apart 
currents, keeping the currents constant, thus doing work 
against their mutual attraction, we diminish their energy at 
the same time by the amount of work done against the 
attraction. Thus the quantity LC? of a circuit is the amount 
of work that must be done to take a current to pieces, so to 
speak ; that is, supposing it divided into infinitely fine fila- 
mentary closed currents, to separate them against their attrac- 
tions to an infinite distance from one another. We do not 
need, therefore, any examination of special formule to see 
that the inductance of a flat strip is far less than that of a 
round wire of the same sectional area; their difference being 
proportional to the difference of the amounts of the magnetic 
energy per unit current in the two cases. ‘The inductance of a 
circuit can, similarly, be indefinitely increased by fining the 
wire; that of a mere line being infinitely great. But we can 
no more have a finite current in an infinitely thin wire than 
we can have a finite charge of electricity at a point, in which 
case the electrostatic energy would be also infinitely great, for 
a similar reason ; although by a useful and almost necessary 
convention we may regard fine-wire circuits as linear, whilst 
their inductances are finite. 

Now, as regards our enclosed rod with no external magnetic 
field, we can in several cases estimate L exactly, as the work 
is already done, in a different field of Physics. The nature of 
the problem is most simply stated in terms of vectors. Thus, 
let h be the vector magnetic force when the boundary of the 
section perpendicular to the length is circular, and H what it 
becomes with another form of boundary; then 


H=h+F, and F=—yoO. .... |... (le) 


That is, the field of magnetic force differs from the simple 
circular type by a polar force F whose potential is QO. This 
must be so because the curl of H and of h are identical, re- 
quiring the curl of Fto be zero. To find F we have the datum 


12 Me Oi Fieavasidenes ie 


that the magnetic force must be tangential to the boundary, 
and therefore have no normal component; or, if N be the unit 
vector normal drawn outward, 


—FN=BN © 0)! °o a 


is the boundary condition. This gives F, when it is remem- 
bered that F must have no convergence within the wire. 

In another form, since we have h circular about the axis, 
and of intensity 2mrT,, at distance r from it, the current- 
density being I',; or 

h=2rl\Vkr, <,...4, ee 


if r is the vector distance from the axis in a plane perpendi- 
cular to it, and k a unit vector parallel to the current; we 
have 

hN=(27T',)(NVkr)=(27T,)(2VNk) 


Sa, wl 


if s be length measured along the bounding curve, in the 
direction of the magnetic force. The boundary condition (2a) 
therefore becomes, in terms of the magnetic potential, 
dQ de") 
Bi ea Hee 6.) Sat eee Mommas (5a) 
which, with V?Q=0, finds the magnetic potential. Here 


p, 1s length measured along the normal to the boundary 
outward. | 


Or we may use the vector-potential A. It is parallel to 
the current, and consists of two parts; thus, 


A= A! —(prlyr’)k, 2 2 oS eee 


where the second part on the right side is, except as regards 
a constant, what it would be if the boundary were circular, its 
curl being wh. ‘To find A’, let its tensor be A’; then 


V7A’=0, and A’=prl 77,3) eee 
the latter being the boundary condition, expressing that A is 


zero at the boundary. Comparing with (5a), we see that 
(7a) is the simpler. 


The magnetic energy per unit length of rod, say T, is 
T= SpH?/8r=zeh+F) er, . eee ee 


the summation extending over the section. But >FH=0, 
because F is polar and H is closed ; so that 


T= >ph?/8a —>pF?/8r 
= Zh? /8r+2phFBp. . . . . (Qa) 


Self-induction of Wires. 13 


Or, in Cartesian coordinates, let H, and H, be the w and y 
components of the magnetic force H, z being parallel to the 
current ; then 


H,=—2myT,— 5, Hy=2ne0,— 7 or (10a) 


express (la), and (8a) is represented by 


n= £ 3(/ +0, 
(11a) 

ET pay (g2 4 2) — HE (eS — =) 

= Daa? +y’) Z are oie 
the latter form expressing (9a). 

It will be observed that the mathematical conditions are 
identical with those existing in St. Venant’s torsion problems. 
Thus, if a and 6 are the y and « tangential strain components 
in the plane «, y in a twisted prism, andy the longitudinal 
displacement along z, parallel to the length of the prism, we 
have 


uA Se et ee 
b= Dea 2 0 bee a (12a) 


where 7 is the twist (Thomson and Tait, Part II. § 706, 
equation (9)). The corresponding forces aren times as great, 
if n is the rigidity (loc. cit. equation (10)); so that the energy 
per unit length is 


4nd(a’+ 6?) over section... . . . (18a) 
Also, to find y we have 


(14a) 


(loc. cit. equations (12) and (18)). Comparing (14a) with 
(5a), (12a) with (10a), and (18a) with the first of (11a), we 
see that there is a perfect correspondence, except, of course, 
as regards the constants concerned. The lines of tangential 
stress in the torsion problem and the lines of magnetic force 
in our problem are identical, and the energy is similarly 
reckoned. We may therefore make use of all St. Venani’s 
results. 

It will be sufficient here to point out that the ratio of the 
inductance of wires of different sections is the same as the 
ratio of their torsional rigidities. Thus,as L=4, in the case 
of a round wire, that of a wire of elliptical section, semiaxes 
a and b, is L=pab/(a’?+’) ; when the section is a square, it 
is 44174; when it is an equilateral triangle, -3627y, Ke. 


14 3 Mr. O. Heaviside on the 


That of a rectangle will be given later in the course of the 
following subsidence solution. | 

Consider the subsidence from the initial state of steady flow 
to zero, when the impressed force that supported the current 
is removed, in a prism of rectangular section. Let 2a and 
2b be its sides, parallel to x and y respectively, the origin 
being taken at the centre. Let H, and H, be the w and y 
components of the magnetic force at the time é Let H be 
the intensity of the magnetic-force vector E, which is parallel 
to z; then the two equations of induction ((6), (7). Part L.), 
or 

curl H=47F, —curl E=yH, 

are reduced to 


di ce POA 
dH, dH 
Te ~ gy =A... (16a) 


if [is the current density, & the conductivity, w the induc- 
tivity. [I speak of the intensity of a “force” and of the 
“density ’’ of a flux, believing a distinction desirable.| The 
equation of I‘ is therefore 


a eae 
(5 + Fp) P= Aeukl, 2oi iscsi 
of which an elementary solution is 
DI'=cos mz cos ny &, . |...) pee eee 
if 
4npkp=— (ne +177)... 2 


At the boundary we have, during the subsidence, H=0, or 
I'=0; therefore 


cos ma cos ny=0 at the boundary, 
or | 
cosma=0, cosnb=0, . . . «. . | (20a) 
or ma=$7, 37, 37, &e., nb=ditto. The general solution is 
therefore the double summation over m and n, 
T'= SA cos mz cos ny €, 
if we find A to make the right member represent the initial 
state. This has to be T=I, a constant. 
Now 
1=2(2/ma) sin macosma, from «=—a to +a, 
1=2(2/nb) sin nb cosny, from y=—b to +b. 


Self-induction of Wires. 15 


Hence the required solution is 


n2t 


sin nb --5, 
cos nye 4ruk, 


sin ma 
m 


m2t 
COS MH € 4Ampk, > 


4 
T= Ts 
or 


pes ri anne gn? cosmacosny &. . . (21a) 
ab mn 


From this derive the magnetic force by (15a). Thus 
167 


: t 
sin ma : eP 
2 a aay OS sin nb cos mz sin ny—— 
zt Fie m J m +n? 
16 sin nb ie 
li, 2 T>> sin ma sin mx cos ny ———>- 


(22a) 


The total current, say C, in the prism is given by 


b "a 

4a7rC= 2f Hody @=a) 7 2° Hyde ys) 
640r 

= Doh 

by line integration round the boundary. Or 


4. ept 


if C>=4abT,, the initial current in the prism. 

Since the current is longitudinal, and there is no potential 
difference, the vector potential is given by E=—A; or, A 
being the tensor of A, A is got by dividing the general term 
in the I solution (21a) by —pk ; giving 


ePt 


mn?’ 


(23a) 


men” 


167 sin ma sin nb 
A= a magna Ti cos mx cosny eP', . (24a) 


Since the magnetic energy is to be got by summing up the 
product AI over the section, we find, by integrating the 
square of I’, that the amount per unit length is 


e2pt 


is a®b® 22 mn? (m? oe) 

By the square of the force method the same result is 
reached, of course. We may also verify that Q+T=0, 
during the subsidence, Q being the dissipativity per unit 
length of prism. | 


The steady-flow resistance per unit length is the L in 


16 Mr. O. Heaviside on the 
T=4LC,?, which (25a) becomes when ¢=0; this gives 


Late s = 
(ma)%(nb)*4 %(nb)2+" (ma)? 


The lines of magnetic current are also the lines of equal 
electric-current density. That is, a line drawn in the plane a, y 
through the points where I’ has the same value is a line of 
magnetic current. Tor, if s be any line in the plane a, y, 


(26a) 


di : 

aa component of wH perpendicular to s, 

so that H is parallel to s, when dH/ds=0. The transfer of 
energy is, as usual, perpendicular to the lines of magnetic 
force and electric force. 

The above expression (26a) for L may be summed up 
either with respect to ma or to nb, but not to both, by any 
way I know. ‘Thus, writing it 

1 1 i 
Tides 2s they col en 
De b 
(mat ONS nd)? +2 (may? 
we may effect the second summation, with respect to nb, re- 
garding ma as constant in every term. Use the identity 


ae Oem a2 cos(tmx/20) 
72 1(e—™) ~ T ~ (ime? Gr[20 2 + PY? 
where i has the values 1, 3,5, &c. Take #=0, w/2l=nb, 
h=(b/a)(ma), =1, and apply to (27a), giving 


ye Ay 


b 6 
1 el al @®—e an) 
L=4912 Gap 3 (ma! Fi Io 


where the quantity in the : t is the value of the second > in 
(27a). The first part of (28a) is again easily summed up, 
and the result is 

1 fa Goo) 


met. La 
Lae 3) eae 


in which summation, we may repeat, ma has the values 
| dar 0 
a7 39 
that is, a/b changed to b/a, without altering the value of L. 
This follows by effecting the ma summation in (26a) instead 
of the nb, as was done. 


» eee) 


a, &c. The quantities a and b may be exchanged ; 


Self-induction of Wires. 17 


When the rod is made a flat sheet, or a/b is very small, we 
have L=47p/(a/b). 

Compare (29a) with Thomson and Tait’s equation (46) 
§ 707, Part II. Turn the nab’ outside the [ | to nab’, and 
multiply the } by 2. These corrections have been pointed 
out by Ayrton and Perry. When made, the result is in 
agreement with the above (29a), allowing, of course, for 
changed multiplier. [I also observe that the —7 in their 
equation (44) should be +7, and the +7 in (45), (the 
second t) should be —7.]| Such little errors will find their 
way into mathematical treatises ; there is nothing astonishing 
in that; but a certain collateral circumstance renders the 
errors in their equation (46) worthy of being long remem- 
bered. For the distinguished authors pointedly called atten- 
tion to the astonishing theorems in pure mathematics to be 
got by the exchange of a and 0, such as rarely fall to the lot 
of pure mathematicians. They were miraculous. 


I now pass to a different problem, viz. the solution in the 
case of a periodic impressed force situated at one end of a 
homogeneous line, when subjected to any terminal conditions 
of the kind arising from the attachment of apparatus. The 
conditions that obtain in practice are very various, but 
valuable information may be arrived at from the study of the 
comparatively simple problem of a periodic impressed force, 
of which the full solution may always be found. In Part II. 
I gave the fully developed solution when the line has the three 
electrical constants R, L, and S (resistance, inductance, and 
electrostatic capacity), of which the first two may be functions 
of the frequency, but without any allowance for the effect of 
terminal apparatus. It we take L=O we get the submarine- 
cable formula of Sir W. Thomson’s theory ; but although the 
effect of L on the amplitude of the current at the distant 
end becomes insignificant when the line is an Atlantic cable, 
its omission would in general give quite misleading results. 

There are some @ priori reasons against formulating the 
effect of the terminal apparatus. They complicate the for- 
mulz considerably in the first place ; next, they are various 
in arrangement, so that it might seem impracticable to for- 
mulate generally ; and, again, in the case of a very long sub- 
marine cable, we may divide the expression of the current- 
amplitude into factors, one for the line and two more for the 
terminal apparatus, of which the first, for the line, is always 
the same, whilst the apparatus-factors vary, and are less im- 
portant than the line-factor. But in other cases the terminal 
apparatus may be of far greater importance than the line, in 


Phil. Mag. 8. 5. Vol. 23. No. 140. Jan. 1887. C 


18 | Mr. O. Heaviside on the 


their influence on the current-amplitude, whilst the resolution 
into independent factors is no longer possible. 

The only serious attempt to formulate the effect of the 
terminal apparatus with which I am acquainted is that of the 
late M:. C. Hockin (Journal 8. T. E. and H., vol. v. p. 482). 
His apparatus arrangement resembled that usually occurring 
then in connection with long submarine cables, including, of 
course, many derived simpler arrangements; ard from his 
results much interesting information is obtainab'e. But the 
results ere only applicable to long submarine cables, on ac- 
count of the omission of the influence of the self-induction of 
the line. The work must, therefore, be done again in a more 
general manuer. It is, besides, independently of this, not 
easy to adape his formule, in so far as they show the in- 
fluence of terminal apparatus, to cases that cannot be derived 
from his. For instance, the effect of electromagnetic in- 
duction in the terminal arrangements was omitted. I have 
therefore thought it worth while to take a far more general 
case as regards the line, and at the same time have endeavoured 
to put it in such a form that it can be readily reduced to 
simpler cases, whilst at the same time the results apply to any 
terminal arrangements we choose to use. 

The general statement of the problem is this. A homo- 
geneous line, of length /, whose steady-flow resistance is R, 
inductance L, electrostatic capacity 8, and conductance of 
insulator K, all per unit length of line, is acted upon by an 
impressed force Vysin né at one end, or in the wire attached 
to it; whilst any terminal arrangements exist. ind the 
effect produced; in particular, the amplitude of the current 
at the end remote from the impressed force. If the line con- 
sist of two parallel wires, R must be the sum of their resist- 
ances per unit length. 

Let C be the current in the line and V the potential dif- 
ference at distance z from the end where the impressed force 
is situated. Then 

aV 


dC d 
~7, = (K+83,V, ——-=R'C, . . (1d) 


are our fundamental line equations. Here R’=R+L (d/dt) 
to a first approximation, and = R’+ L/(d/dt) in the periodic 
case, where R/ and L/ are what R and L become at the given 
frequency. Let the terminal conditions be 
V=Z,C at z=/ end, 2} 

—V,sinnt+V=Z,C atz=0end, vt (BB) 
so that V=Z,C would be the z=0 terminal condition if there 
were no impressed force. 


Self-induction of Wires. 19 


The solution is a special case of the second of (1620), 
Part IV., which we may quote. In it take 
Boop, te i EE pe 2 dy (30) 
p meaning d/di so far. Also put 2.=0, ¢=V_ sin nt, and 
—m?=F?=(K+S8p)(R’+L/p),. . . (40) 
and put the equation referred to in the exponential form. 
Thus, 
(F/S"+Z, )eFC-2 4 (F/G! — 7, went 2) ; 
ES + ZFS") "US"—Z)R+Z,) von CO) 
This is the differential equation of C in the line. Now in F, 
S”, Z,, and Z,, let d?/dt??=—n?. It is then ae to 


Pp! =e QS, (A/P!+ B'Q'n?) = (A'Q!—B'P) Z per 
ar sin nt = APE B 9 Sin nt, (60) 
ALB 


giving the amplitude and phase-difference anywhere; and 
the amplitude is 


Cy=V_(A2 + Bln?) 4(P? + Ql2n2)?; 2... (7B) 


A! and B’ are functions of z, whilst P! and Q! are constants. 


Put 
=P. +O: ; er gie4sd 
aes ns ae SG 1a (32) 
—Zo =Ro+ Lyni, l ; 
The values of P and Q are 
iP (2)? 4{(R2 4+ L/2n?)?(K2 + S2n2)? + (KR! — L'Sn?) be (9 b) 
Be OR + Lt Rt So + RI Sf 
possessing the following properties, to be used later, 
an Q?=(K?+ ox e). (BB+ L2n2)*, 
—Q?=KR'— . (100) 
* EPOe RISA KL. 
The expressions of R,!, Ry’, L,’, L,! can only be stated when 
the terminal conditions are fully given. ‘Their structure will 
be considered later. P and Q depend only upon the line. 
Let 
A=R!—§Sn?(R, L,'+ RL’) + K (BR, Ry'—L, li'n?)s 
Be Un Sn Re Ry! lid Lala) + Kn (Re! Ly! ++ Ry’ Ly!) 3 iin 
a= P(R)' + Ry’)—Qn( Li + Ly); | 
. 


b=Q(R,y' +R,') + Pn(Ly' + Ly). 
G2 


20 Mr. O. Heaviside on the 


The effect of making the substitutions (85) in (55) is to ex- 
press C in terms of the P,Q of (90) and the A,B, a,b of | 
(116); thus:— | 


C=[} (P—L/Sn?+KR,')cosQ—z)—(Q4+R/Sn+KL/n)sinQ(—z)} eP-*) 
Sg se eee +O.L, 1 e-PU-a) 
+i4 (..— .. + .. )smQU—z)+(..+ .. + .. JeosQ(Z—z)} eFC) | 


beg (eet es +) tee +0.=005 — ee te PU-a) 
x V, sin nt : 
+ [{(A+a)cPcosQl—(A—a)e~PleosQi—(B + b)eP/sinQl—(B—b)e*’sinQ/} 
+2{(B+b) .... —(B-8) .... +(Ata) .... +(e) 0) te 
| 


The dots indicate repetition of what is immediately above | 
them. Here we see the expressions for the four quantities | 
A', BI, P’, Q' of (66), which we require. (120) therefore 
fully serves to find the phase-difference, if required. I shall 
only develop the amplitude expression (7b). It becomes, 
by (128), 


[ e2Pt—-2) {(P24+-Q2)+(K2+82n?)(R?2+1,2n*) + 2Qn(R,'S+KL,!)+2P(KR,/—L,'Snl 
A as ok Ges gah caste =| +. ee | 
+2008 2Q(1—z) {(P?+.Q?)—(K2+ 8? n2(R,24+L,2n?)} 
—4 sin 2Q(0—z) {Pn(R,!+KL,')+ Q(L,'Sn?—KR,)t }? 


a [eP'S(A+a)?4+(B+b)?t +e-2P’ {(A—a)?+(B—b)?} 
—2 cos 2Q1. (A? + B?—a?— 6?) +4 sin 2Q7. (Ab—aB)} F, 
in terms of A, B, a, 6 of (1108). 


This referring to any point between z=0 and J, a very im- 
portant simplification occurs when we take z=/. It reduces 
the numerator to 2(P?+Q”)?. It only remains to simplify the 
denominator as far as possible, to show as explicitly as we can 
the effect of the terminal apparatus, which is at present buried 
away in the functions of A, B, a,b occurring in (130). 

First of all, we may show that the product of the coefficients 
of «7?! and e—?¥! equals half the square of the amplitude of the 
circular part in the denominator. This is an identity, m- 
dependent of what A, B, a, 6 are. (130) therefore takes 
the form 


C,=2V, (P+ Q?)} + [Ge?!+ He '—2(GH)? cos 2(QU+6)}. «(I 


Self-induction of Wires. 21 


The foliowing are the expansions of the quantities occurring 
in the denominator of (136) :— 
Let 


P=R?+LPn2, 12=Ri?+ Lo?n?, L2=Ry2+ Lyn. . (158) 
Then 
A24 B12 4 (K2 +. 82n2)121,24 2(R/R/—L,'L,'n2)\(KR! + L/Sn2) | 
4+2(R,L,! +B,'L,n(KL'—R'S), | 


a? +b? =(P?+ Q?) 4(R te Ry a (i! = L,!)2n? ; ; 
| \a+ BO=(R,/+R,')(B/P4L'nQ)4+(L,' + L)n(L/nP — P/Q) ; (16d) 


| 


A\b—aB=(R, +B,)(B/Q—L'nP) 4+ (L,! + L,n(B'P + L'nQ) 


a (R/T? a R,'T,?)(KP is SnQ) u (Le ty L,'T,?)n(KQ 6 SnP), 


+ (RJ1,2+R/12)(KQ—SnP)— (L,'12+L,'1,2)n(KP +8nQ). ] 


These may be used direct in the denominator of (145), which 

is the same as that of (13). But G and H may be each 

resolved into the product of two factors, each containing the 

apparatus-constants of one end only. Noting therefore that 

the @ in (140) is given by 
2(Ab—aB) 

tan 20> Se Baa 0" 


whose numerator and denominator are given in (160), it will 
clearly be of advantage to develop these factors. First 
observe that the expansion of H is to be got from that of G, 
using (160), by merely turning P to —P and Qto—Q. We 
have therefore merely to split up one of them, say G. If we 
put R,/=0, L,’=0 in G it becomes 


124 (P?+ Q2)1,2+2P(R/R! + Ly'Lin?) +2Q(UnR,!—R'nL;’). (188) 


If, on the other hand, we put R,’=0, L,'=0 in G it becomes 
the same function of R,! L,' as (180) is of R,, L,!. It is then 
suggested that G is really the product of (180) into the 
similar function of Rj’, Lj’; when the result is divided by I’. 
This may be verified by carrying out the operation described. 
But I should mention that it is not immediately evident, and 
requires some laborious transformations to establish it, making 
use of the three equations (10L). When done, the final 
result is that (14) becomes 


K? +S? 98 
C= 20 Rep re 


+ [G,G,e?’+ H Hye?’ —2(G,G,H)H,)? cos 2(Q/+ 6), (198) 


(178) 


23 Mr. O. Heaviside on the 


wherein G, and H, contain only constants belonging to the 
apparatus at z=0, and G, and H, those belonging to z=l, 
besides the line-constants. Only one of the four need be 
written ; thus 


iy 2 (P? + Q2)J,2+ 2P(RIRY + L! Lyn?) + 2Qn(Ry'L' RIL}. (208) 


From this get H, by changing the signs of P and Q. 
Then, to obtain G, and H,, the corresponding functions for the 
z=l end, change R,) to R,’ and L,’ to L,!.. These functions 
have the value unity when the line is short-circuited at the 
ends, (Zj=0, Z,=0). They may therefore be referred to as 
the terminal functions. Their form is invariable. We only 
require to find the R’ and L’, or the effective resistance and 
inductance of the terminal arrangements, and insert in (20)) 
and its companions. 

Thus, let the two conductors at the z=/ end be joined 
through a coil. Then R,’ is its resistance, L,' its inductance, 
the steady-flow values, and the accents may be dropped, ex- 
cept under very unusual circumstances, and J, is its impe- 
dance at the given frequency, when on short circuit. But if 
the coil contain a core, especially if it be of iron, neither R, 
nor L, can have the steady-flow values, on account of the 
induction of currents in the core. Their approximate values 
at a given frequency may be experimentally determined by 
means of the Wheatstone bridge. Of course R, and L, are 
really somewhat changed in a similar manner by allowing any 
induction between the coil and external conductors, the brass 
parts of a galvanometer, for instance; L going down and R 
going up, though this does not materially affect I. 

If, instead of a coil, it be a condenser of capacity 8, that 
is inserted at z=/; then, since 


C=8,V = SpV, 


Z, = (Sip)—! = —p/(S 7’). 
Therefore take Hi Cae 
Ry =0, and, Li! =— Cine. 


The condenser behaves, so far as the current is concerned, as 
a coil of no resistance and negative inductance, the latter 
decreasing as the frequency is raised, and as the capacity is 
increased ; tending to become equivalent to a short circuit, 
though this would require a great speed in general, as the 
guasi-negative inductance is large. [Thus n=100, S=10-% 
(one microfarad), makes Lj/=—10". To get the inductance 
of a coil to be 10% it must contain a very large number 
of turns of fine wire.] Thus, whilst the condenser stops 


we have 


Self-induction of Wires. 23 


slowly periodic or steady currents, it tends to readily pass 
rapidly periodic currents, a property which is very useful in 
telephony, as in Van Rysselberghe’s system. 

On the other hand, the coil passes the slowly periodic, and 
tends to stop the rapidly periodic, a property wuich is also 
very useful in telephony. A very extensive application of 
this principle occurs in the system of telepuonic interconmu- 
nication invented and carried out by Mr. A. W. Heeviside, 
known as the Bridge System, from the telephones at the vari- 
ous offices being connected up as bridges across from ore to 
the other of the two conductors which form the line. Whilst 
all stations are in direct communication with one arother, 
one important desideratum, there is no overhearing, which is 
another. For all stations except the two which are in corre- 
spondence at a certain time have electromagnets of high 
inductance inserted in their bridges, which electromagnets will 
not pass the rapid telephonic currents in appreciable strength, 
so that it is nearly as if the non-working bridges were non- 
existent, and, in consequence, a far greater length of bur’ed 
wire can be worked through than on the Sequence system, 
wherein the various stations have their apparatus in sequence 
with the line, whilst at the same time a balance is preserved 
against inductive interferences. When the two stations have 
finished correspondence, they insert their own electromagnets 
in their bridges. As these electromagnets are used as call 
instruments, responding to slowly periodic currents, we have 
the direct intercommunication. Of course there are various 
other details, but the above sufficiently describes the principle. 

As regards the property of the self-induction of a coil in 
stopping or greatly decreasing the amplitude of rapidly 
periodic. currents, or acting as an insulation at the first 
moment of starting a current, its influence was entirely over- 
looked by most writers on telegraphic technics before 1878, 
when I wrote on the subject (Journ. 8. T. H. & E. vol. vii.). 
A knowledge of the important quantity (R’+L?n’), which 
is now the common property of all electrical schoolboys 
(especially by reason of the great impetus given to the spread 
of a scientific knowledge of electromagnetism by the com- 
mercial importance of the dynamo), was, before then, confined 
to a few theorists. 

If the coil R, L, and the condenser §, be in parallel, we have 


C= (Sr + petp)Y) 


V_ R+{L—S,(R?+L?x’)} 
C7 ALS yn)? + (RSin)? ” 


or 


24 Mr. O. Heaviside on the 


which show the expressions of R,’ and Ly’, the second being 
the coefficient of , the first the rest. 

Similarly in other simple cases. And, in general, from the 
detailed nature of the combination inserted at the end of the 
line, write out the connections between the current and poten- 
tial difference in each branch, and eliminate the intermediates 
so as to arrive at V=Z,C, the differential equation of the 
combination, wherein Z is a function of p or d/di. Put 

"—=—n’, and it takes the form Z,=R,'+ ine "ene R,' 
and L,' are functions of the electrical constants and of n”, and 
are the required effective R,! and L,! of the combination, to 
be used in (208), or rather in its oy equivalent Gy. 

As regards the z=0 end, it is to be remarked that, owing 
to the current being reckoned positive the same way at both 
ends, when we write V=Z,C as the terminal equation, it is 
—Z, that corresponds to Z;. Thus —Z,=R,'+L,'p, where, 
in the simplest case, R,! and L,! are the resistance and induct- 
ance of a coil. 

So far sufficiently describing how to develope the effective 
resistance and inductance expressions to be used in the ter- 
minal functions G and H, we may now notice some other 
peculiarities in connection with the solution (19). First 
short-circuit the line at both ends, making the terminal func- 
tions unity and @=0. The solution then differs from that 
given in Part II., equation (82), in the presence of the quan- 
tity K, the former Sn now becoming (K? + $’n”)?, whilst P and 
A) differ from the former P and Q of (78), Part II., by reason 
of K, which, when it is made zero, makes them identical. If 
we compare ‘the old with the new P and Q, we find that 


L! becomes L'—KR//Sn’, 
R! becomes RY+ KLIS, } 

in passing from the old to the new. Then the function 

R24 L2n? Teeter (R’+ KL'/S)? + (L'—KR/Sn?)?n? _ RP + L?n? 

"Eee, 5 K24 S?n? “2 ae 

or is unaltered by the leakage. It follows that the equation 


(85) Part II. is still true, with leakage, if in it we make the 
changes (21) just mentioned, or put 


Balu BP) in OEE 


aes IIe n 


(210) 


(220) 


instead of using the v! and / expressions of Part IT. 
At the particular speed given by n?=KR’/L'S, we shall 
have 


P=Q= (})?(R?4 L2n2)#(K? + S’n?)? = (1)?(R/S + KL’) n, (230) 


Self-induction of Wires. 25 


making 


- 


a: ZENE) eC ts 


If we should regard the leakage as merely affecting the 
amplitude of the current at the distant end of a line, we should 
be overlooking an important thing, viz. its remarkable effect 
in accelerating changes in the current, and thereby lessening 
the distortion that a group of signals suffers in its transmission 
along the line. If there is only a sufficient strength of 
current received for signalling purposes, the signals can be far 
more distinct and rapid than with perfect insulation, as I have 
pointed out and illustrated in previous papers. ‘Thus the 
theoretical desideratum for an Atlantic cable is not high, but 
low insulation, the lowest possible consistent with having 
enough current to work with. Any practical difficulties in 
the way form a separate question. 

Regarding this quickening effect, or partial abolition of elec- 
trostatic retardation, I have (‘ Electrician,’ Dec. 18, 1885, and 
Jan. 1, 1886) pushed it to its extreme in the electromagnetic 
scheme of Maxwell. In a medium whose conductivity varies 
in any manner from point to point, possessed of dielectric 
capacity which varies in the same manner, so that their ratio, 
or the electrostatic time-ccnstant, is everywhere the same, but 
destitute of magnetic inertia (w=0, no magnetic energy), I 
have shown that electrostatic retardation is entirely done away 
with, except as regards imaginable preexisting electrification, 
which subsides everywhere according to the common time- 
constant, without true electric current, by the discharge of 
every elementary condenser through its own resistance. This 
being over, if any impressed force act, varying in any manner 
in distribution and with the time, the corresponding current 
will everywhere be the steady-flow distribution appropriate to 
the impressed force at any moment, in spite of the electric 
displacement and energy; and, on removal of the impressed 
force, there will be instantaneous disappearance of the current 
and the displacement. This seems impossible, but the same 
theory applies to combinations of shunted condensers, arranged 
in a suitable manner, as described in the paper referred to. 

Of course this extreme state of things is quite imaginary, 
as we cannot really overlook the electromagnetic induction in 
such a case. If we regard it as the limiting form of a real 
problem, in which inertia occurs, to be afterwards made zero, 
we find that the instantaneous subsidence of the electrostatic 
problem becomes an oscillatory subsidence of infinite frequency 
but finite time-constant, about the mean value zero ; which 


Co 


26 Mr. O. Heaviside on the 


is mathematically equivalent to instantaneous non-oscillatory 
subsidence. 

The following will serve to show the relative importance of 
R, S, K, and L in determining the amplitude of periodic 
currents at the distant end of a long submarine cable, of fairly 
high insulation : resistance :-— 


4 ohms per kilom. makes R=40*, 


: i 
A Tnieror.. 7, A S= Zoo 
100 megohms ,, yt KaOree 


Here, it should be remembered, K is the conductance of 
the insulator per centim. ‘The least possible value of L would 
be such that LS=v~’, where v=30"; this would make 
L=4/9 only. But it is really much oreater, requiring to be 
multiplied by the dielectric constant of the insulator in the 
first place, making L=2 say. It is still further increased by 
the wire, and considerably by the sheath and by the extension 
of the magnetic field beyond tue sheath, to an extent which 
is very difficult to estimate, especially as ‘it is a variable quan- 
tity; but it would seem never to become a very large number, 
as of course an iron wire for the conductor is out of the 
question. But leaving it unstated, we have, by (9b), taking 
Bah, 1/— L, 


n? Alt a2 
P= alae +L)! shat ahaa) +(gou—gem) b 


Lin? 
= } £ (1608+ Lene) ee +(400— Sei 


Now n/27 is the frequency, necessarily very low on an 
Atlantic cable. We see then that the first L?n? is quite 
negligible in its effect upon P, even when we allow L to 
increase greatly from the above L=2. The high insulation 
also makes the (RK —LSn?) part negligible, making approxi- 


mately 
P=O=(ln) 10? 
P being a little greater than Q, at least when L is small. Now 
this is equivalent to taking L=0, K=0, when 
P=Q=(4R8na)2, 2 a See 
.. (190) to 


Vo(Sn/R)?-+{GyGye?’ + H, Hye?’ — 2 (GG, H,H,)? cos 2PI}. 


(260) 


Self-induction of Wires. 27 


which is, except as regards the terminal functions I introduce, 
quite an old formula. It is what we get by regarding the 
line as having only resistance and electrostatic capacity. But, 
still regarding the line asan Atlantic or similar cable, worked 
nearly up to its limit of speed, PJ is large, say 10 at most, so 
that we may take this approximation to (260), 

C= 2Vi(seiye x Gh? GT) bs (278) 
where the first of the three factors is the line-factor, the 
second that due to the apparatus at the z=0 end, and the 


third to that at the <=/ end of the line ; thus, by (206) and 
(256), with L'=0 and R/=R in the former, 


Gy=1+ Ao{2PR(Ry—Ljn) +2P2(R,? + Ly?n2)} | 


L. (285) 
F G,= if + pot2Pk (R,'—L,'n) + 7p ais a ete + L,'2n?) } ‘ | 


This reduction to (27 b) is of course not possible when the 
line is very far from being worked up to its possible limit ; in 
fact, all three terms in the { } of (26 0), or, more generally, of 
(196), require to be used in general. for this reason a full 
examination of the effect of terminal apparatus is very labo- 
rious. Most interesting results may be got out of (196), 
especially as regards the relative importance of the line and 
terminal apparatus at different speeds, complete reversals 
taking place as the speed is varied whilst the line and appa- 
ratus are kept the same. The general effect is that, as the 
speed is raised, the influence of the apparatus increases much 
faster than that of the line. For instance, to work a land- 
line of, say, 400 miles up to its limit, we must reduce the 
inertia of the instruments greatly to make it even possible. 
In fact electromagnets seem unsuitable for the purpose, unless 
quite small, and chemical recording has probably a great 
future before it. But it would be too lengthy a digression to 
go into the necessarily troublesome details. 

The following relates to some properties of the terminal 
function G, which have application when (27 5) is valid. Con- 
sider the G, of (280). Let it be simply a coil that is in ques- 
tion. ‘Then R, is its resistance and Ly its inductance, dropping 
the accent. Keep the resistance constant, whilst varying the 
inductance so as to make G,a minimum, and therefore the 
current amplitude a maximum. ‘The required value of L, is 


I Pe a Es G95) 


depending only upon the line-constants and the speed, inde- 


28 On the Self-induction of Wires. 


pendently of the resistance of the coil. Taking P/=10, this 
makes L,=RJ/20n, where Ri is the resistance of the ‘line. 
The relation (29 6) makes 

DAR a2 ae 


Gia ger RP 1 


If the coil had no inductance, but the same resistance, G, would 
have the same expression, but with 1 instead of 4 in (30D). 
The effect of the inductance has therefore increased the ampli- 
tude of the current, and it is conceivable that G, could be 
made less than unity, though not practicable. 

Now the G,/R, of (806) is a minimum, with R, variable, 
when R=2PR,, and this will make G,=2, or the terminal 
factor to be Gy-?='7. Now if we vary the number of turns 
of wire in the coil, keeping it of the same size and shape, the 
magnetic force will vary as (R,/G)?, so it at first sight appears 
that R,=R/2P and RPE make the magnetic force a 
_ maximum for a fixed size and shape of coil. There is, how- 
ever, a fallacy here, because varying the size of the wire as 
stated varies L, nearly in the same ratio as R,, whilst (30d) 
assumes L, to be a constant, given by (290). It is perhaps 
conceivable to keep L, constant during the variation of R,, 
by means of iron, and so get (R,/G)? to be a maximum; but 
then, on account of the iron, this quantity will not represent 
the magnetic force. 

If, on the other hand, we vary R, in the original G, of 
(28 b) keeping L,/Ry constant (size and shape of coil fixed, 


size of wire variable) , G,/R, is made a minimum by 
Ry? + Lien? R3/2P? |) er 


giving a definite resistance to the coil of stated size and shape 
to make the magnetic force a maximum. Now Gy, becomes 


Gilg R ECR ays oF a a ee (326) 


where L,/R, has been constant. If this constant have the 
value n-1, we have G,=2 again, and Ry, L, have the same 
values as before. ‘There is thus some magic about G,=2. 

Again, if the terminal arrangement consist of a coil R,, 
L, and a condenser of capacity 8; and conductance K, joined 
in sequence, we shall have 


V/C=(Rit lyp)+ (i + Sip), 


= (4 oe Kes) + (ls ee bee 


=R,'+ Lp, say, 


Sunrise-Shadows of Adam’s Peak in Ceylon. 29 


if R,', L,! are the effective resistance and inductance, to be 
used in G,, making 


G,=1+ a {R-dnt te \ 


Ky? + Sn? 
oY? 2 22 f RK, == 8, Lyn? 


Variation of L, alone makes G, a minimum when 


Sin R q 
K?+8,72' 2P Pas tee Ren 
and if we take K,=0 (condenser non-leaky, and not shunted), 
we have the value of G, given by (806) again, independent of 
the condenser. Similarly we can come round to the same 
G,=2 again. These relations are singular enough, but it is 
difficult to give them more than a very limited practical appli- 
cation to the question of making the magnetic force of the 
coil a maximum, although the (305) relation is not subject 
to any indefiniteness. 


[In Part 1II. Hquation (103), ¢ represents or reduces to a 
negative resistance. In Part IV., for greater convenience, ¢ 
is always a positive resistance. 

Errata, p. 350. Equation (135), put the — sign before the 
>. Equation (137), for E read M.} 


III. The peculiar Sunrise-Shadows of Adam’s Peak in Ceylon. 
By the Hon. Ratpy Asercromsy, L.A. Met. Soc.* 


HERE are certain peculiarities about the shadows of 
Adam’s Peak which have long attracted the attention of 
travellers: a good deal has been written about them, and several 
theories have been proposed to explain the observed pheno- 
mena. In the course of a meteorological tour round the 
world, the author stopped in Ceylon for the express purpose 
of visiting the Peak, and was fortunate enough to see the 
shadow under circumstances which could leave no doubt as to 
the true explanation, and which also entirely disproved certain 
theories which have been propounded on the subject. 
The following account is taken from a paper by the Rev. 
R. Abbay, many years resident in the island, entitled “ Re- 
markable Atmospheric Phenomena in Ceylon,” which was 


* Communicated by the Physical Society: read November 13, 1886. 


30 Hon. Ralph Abercromby on the peculiar 


read before the Physical Society of London, May 27, 1876, 
and published in the Philosophical Magazine for July 1876. 
Writing from descriptions, for he himself had never witnessed 
the appearance, Mr. Abbay says:—-At sunrise apparently an 
enormous elongated shadow of the mountain is projected to the 
westward, not ony over the land but over the sea, to a dis- 
tance of 70 or 80 miles. As the sun rises higher, the shadow 
rapidly approaches the mountain, and appears at the same 
time to rise before the spectator in the form of a gigantic 
pyramid. Distant objects—a hill or a river (or even Colombo 
itself, at a distance of 45 miles)—may be distinctly seen 
through it; so that the shadow is not really a shadow on the 
land, but a veil of darkness suspended vetween the observer 
and the low country. All this time it is rapidly rising and 
approaching, and each instant becoming more distinct, until 
suddenly it seems to fall back on the spectator, like a ladder 
that has been reared beyond the vertical; and the next instant 
the appearance is gone. For this the following explanation 
is proposed :—The average temperature at night in the low 
country during the dry season is between 70° and 80° F., whilst 
that on the: summit of the Peak is from 30° to 40°. Conse- 
quently the lower strata of air are much less dense than the 
upper ; and an almost horizontal ray of light passing over 
the summit must of necessity be refracted upwards and suffer 
total internal reflection as in the case of an ordinary mirage. 

It will be remarked that Mr. Abbay does not allow for the 
difference of elevation, and the sequel will show that this 
theory cannot be maintained. 

Adam’s Peak is a mountain that rises in an abrupt cone, 
more than 1000 feet above the irregular chain to which it 
belongs; the summit reaches to 7352 feet above the sea. 
On the south side the mountain falls suddenly down to 
Ratnapura, very little above the sea-level; while on the 
north it slopes irregularly to the high valley of the Maskeliya 
district. The peak also lies near an elbow in the main chain 
of mountains, as shown in the diagram of the topography of 
the Peak (fig. 1), while a gorge runs up from the north-east 
just to the west of the mountain. When, then, the north- 
east monsoon blows morning mist up the valley, light wreaths 
of condensed vapour will pass to the west of the Peak and 
catch the shadow at sunrise only, if other things are suitable. 
The importance of this will appear later on. 

The only difficulty in getting to Adam’s Peak is the want 
of a rest-house within reasonable distance of the summit. 
Fortunately the kindness and hospitality of IT. N. Christie, 
Esq., of St. Andrew’s Plantation, Maskeliya, enabled the 


Sunrise-Shadows of Adam’s Peak in Ceylon. 31 


author, in company with Mr. G. Christie and Professor Bower, 
of the University of Glasgow, to make the ascent with great 


Fig. 1.—Diagram of the Topography of Adam’s Peak. 


Pe i a 


comfort and with a few necessary instruments. Our party 
reached the summit on the night of the 21st February, 1886, 
amid rain, mist, and wind. Towards morning the latter 
subsided, but at 5.30 a.m. the sky was covered with a con- 
fused mass of nearly every variety of cloud. Below and 
around us cumulus and mist ; at a higher level, pure stratus ; 
above that, wild cirro-stratus and fleecy cirro-cumulus. 

Soon the foreglow began to brighten the under surface of 
the stratus-cloud with orange ; lightning flickered to the right 
of the rising sun over a dense mass of cloud; upposite, a 
light pink-purple illumined an irregular layer of condensed 
vapour ; while above a pale moon with a large ill-defined corona 
round her, struggled to break through a softish mass of fleecy 
cloud. Below lay the island of Ceylon, the hills and valleys 
presenting the appearance of a raised relief-map ; patches of 
white mist filled the hollows ; true cloud drove at intervals 
across the country, and sometimes masses of mist coming up 
from the valley enveloped us with condensed vapour. 

At 6 A.M. the thermometer marked 52° F.; we had been 
told that the phenomenon of the shadow depended on the 
temperature at the summit falling to 30° or 40° F.; and 
when, shortly after, the sun rose behind a cloud we had 
almost lost all hope of seeing anything ; but suddenly at 
6.30 a.m. the sun peeped through a chink in the clouds, and 


32 Hon. Ralph Abercromby on the peculiar 


we saw the pointed shadow of the Peak lying on the misty 
land. Driving condensed vapour was floating about, and a 
fragment of rainbow-tinted mist appeared near the top of the 
shadow. Soon this fragment grew into a complete prismatic 
circle of about 8° diameter by estimation, with the red 
outside, formed round the summit of the Peak as a centre. 
The author instantly saw that with this bow there ought to 
be spectral figures, so he waved his arms about and immedi- 
ately found shadowy arms moving in the centre of the rain- 
bow. ‘Two dark rays shot upwards and outwards on either side 


Fig. 2.—Diagram of rainbow round the shadow. 


Shadow - 


Li: 


ma 


of the centre, as shown in the diagram fig. 2, and appeared 
to be nearly in a prolongation of the lines of the slope of the 
Peak below. The centre of the bow appeared to be just 
below the point of the shadow, not on it; because we were 
standing on a platform below a pointed shrine, and the sub- 
jective bow centred from our own eyes. If we did not stand 
fairly out in the sun, only a portion of the bow could be 
seen. Three times, within a quarter ofan hour, this appearance 
was repeated as mist drove up in proper quantities, and fitful 
glimpses of the sun gave sufficient light to throw a shadow 


Sunrise- Shadows of Adam’s Peak in Ceylon. 33 


and forma bow. In every case the shadow and bow were 
seen in front of land and never against the sky. The last 
time, when the sun was pretty high, we saw the characteristic 
peculiarity of the shadow. Asa thin wreath of condensed 
vapour came up from the valley at a proper height, a bow 
formed round the shadow, while both seemed to stand up in 
front of us, and then the shadow fell down on to the land, 
and the bow vanished as the mist passed on. 

Here, then, was an unequivocal explanation of the whole 
phenomenon. The apparent upstanding of the shadow was 
simply the effect of passing mist which caught the darkness 
of the Peak at a higher level than the earth, for when the 
condensed vapour moved on, the characteristic bow disappeared 
and the shadow fell to its natural plane on the ground. 
When the mist was low, as on the two first occasions, the 
shadow fell on the top as it were, and there was no appearance 
of lifting, only the formation of a bow. 

The well-known theory of the bow is that light diffracted 
in its passage between small water-globules forms a series of 
bows according to the size of the globules, their closeness, 
and the intensity of the illumination. Had the mist been so 
fine and thin as merely to catch and raise the shadow, but 
not to form a bow, there might have been some doubt as to 
the origin of the appearance. Our fortune was in the un- 
settled weather, which made the mist so coarse and close that 
the unequivocal bow left no doubt as the true nature of the 
cause. 

About an hour later the sun again shone out, but much 
higher and stronger than before, and then we saw a brighter 
and sharper shadow of the Peak, this time encircled by a 
double bow. Our own spectral arms were again visible, but 
the shadow was now so much nearer the base of the Peak, 
and we had to look so much down on it, that there was no 
illusion of standing up, and there were no dark diverging 
rays. ‘The inner bow was the one we had seen before; the 
outer and fainter one was due to stronger light. 

The bows were all so feeble and the time so short, that the 
author did not succeed in obtaining any sextant measure- 
ments of the diameters of the bows; but his thermometric 
observations conclusively disprove any idea of mirage. At 
6 A.M. the thermometer on the Peak marked 52° F., while at 
Colombo the temperature stood at 74°°85. The difference of 
22°°85 is just about the normal difference in temperature due 
to a height of 7352 feet. 

The Colombo figures were procured through the courtesy 
of the Surveyor-General for Ceylon. They are got as fol- 

Phil. Mag. 8. 5. Vol. 23. No. 140. Jan, 1887. D 


34 Sunrise-Shadows of Adam’s Peak in Ceylon. 


lows :—Colombo observations only give the minimum that 
morning as 73°°6 F., and the 7 a.M. reading as 75°°5. The 
mean curve of diurnal temperature for the month of February, 
as determined by the Office, gives a difference of 0°65 
between the 6 a.m. and 7 A.M. observations; and by subtract- © 
ing that correction from 75°°5 we get 74°85 as the 6 A.M. 
reading. 

The questions have been frequently asked—Why this lifted 
shadow should be peculiar to Adam’s Peak? ; whya similar 
appearance is not observed from any other mountain-top? ; and 
why the shadow is rarely seen at sunset? There are not many 
mountains which are habitually visited that are either over 
7000 feet, or that rise in an isolated, well-defined pyramid. 
Still fewer can there be where a steady wind, for months 
together, blows up a valley so as to project the rising morning 
mist at a suitable height and distance on the western side to 
catch the shadow of the peak at sunrise. The shadow is not 
seen during the south-west monsoon, for then the mountain is 
covered with cloud and deserted. Nowhere either do we find 
at sunset those light mists lying near the ground which are so 
characteristic of sunrise, and whose presence is necessary to 
lift the shadow. 

The combination of a high isolated pyramid, a prevailing 
wind, and a valley to direct suitable mist at a proper height 
on the western side of the mountain, is probably only rarely 
met with ; and at present nothing yet has been described that 
exactly resembles this sunrise shadow of Adam’s Peak in the 
green island of Ceylon. 

But there is another totally different shadow which is someé- 
times seen from Adam’s Peak, just before and at the moment 
of sunrise, that has been mixed up in some accounts with the 
shadow we have just described. The shadow of the base of 
the Peak stretches along the land to the horizon, and then the 
shadow of the summit appears to rise up and stand against 
the distant sky. The first part seems to be the natural 
shadow lying on the ground ; and the sky part to be simply 
the ordinary earth shadow of twilight projected so clearly 
against the sky as to show mountainous irregularities of the 
earth’s surface. As the sun rises, the shadow of the summit 
against the sky gradually sinks to the horizon, and then the 
ordinary shadow grows steadily shorter as the sun gets higher 
in the usual manner. This can only be seen at sunrise from 
Adam’s Peak, because the ground to the east is too high and 
mountainous to allow the shadow of the summit to fall on the 
sky before the sun is too far down. 

The author found a similar effect, only at sunset, on Pike’s 


Critical Mean Curvature of Liquid Surfaces of Revolution. 35 


Peak, Colorado, 14,147 feet above the sea, and nearly double 
the height of Adam’s Peak. There, towards sunset, the 
shadow of the mountain creeps along the level prairie to 
the horizon, and there begins to rise up in the sky till the sun 
has just gone down, and the anticrepuscular shadow rises too 
high to catch the outline of the Peak. The author only 
witnessed a portion of this sequence, for just about the time 
that the shadow stretched to the horizon, clouds obscured the 
sun, and the rise of the shadow could not be observed ; but 
from all the descriptions he heard, there can be no doubt 
that the character of the shadow is identical with that of 
Adam’s Peak, only that, as the order of sequence is reversed, 
it is more easy to follow the origin of the shadows. 

Since the above was written, the author’s attention has 
been called to the sketch of the shadow exhibited by the well- 
known traveller Miss C. F. Gordon Cumming, in the Colonial 
Hxhibition. This picture represents the shadow lying down, 
but not raised, on an irregular surface of white mist and 
mountain tops. The most interesting thing is a prismatic 
fringe of colour along the straight outside edges of the shadow 3; 
but there is no trace of a bow round its point. 

When we consider how much the appearance of the shadow 
depends on the height, size, and aggregation of the mist, we 
need not be surprised at the numerous phases of reflection and 
refraction that have been described by travellers; but the 
general principles which have been laid down in this paper 
appear to govern all. 


IV. On the Critical Mean Curvature of Liquid Surfaces of 
Revolution, By A. W. Rocker, M.A., F.RS.* 


| Gas a weightless mass of liquid, or a liquid film, be attached 
- to two equal circular rings, the planes of which are per- 
pendicular to the line joining their centres. It will forma 
surface of revolution ; and if it is in stable equilibrium, the 
longest or the shortest diameter will be half way between the 
rings. It is convenient to call this the principal diameter. 
At all points on the surface the sum of the reciprocals of the 
two principal radii of curvature is constant. Half this quan- 
tity may be called the mean curvature. Maxwell has, in his 
article on Capillary Action (ine. Brit., Ith edition), given a 
simple proof of the fact that if the film is a cylinder, a slight 
bulge will cause an increase or decrease in the mean curvature 
according as the distance between the rings is less or greater 


* Communicated by the Physical Society: read November 27, 1886. 
D 2 , 


36 3=6Mr. A. W. Riicker on the Critical Mean Curvature 


than half the circumference of either. If the distance between 
the rings is exactly half the circumference, an infinitely small 
change in the volume will modify the form of the surface, but 
will not alter the mean curvature. Thus the mean curvature 
of a cylinder, the length of which has this particular ratio 
(7/2\ to its diameter, is evidently a maximum or minimum 
with respect to that of other surfaces of constant mean curva- 
ture, which pass through the same rings at the same distance 
apart, and which differ but little from the cylindrical form. 
Hence the cylinder may be said to have a critical mean cur- 
vature when the distance between the rings is half their 
circumference. If the distance between the rings is altered, a 
similar property will be possessed by some other surface. It 
is proposed in *he present paper to determine the general 
relation between the magnitude and distance of the rings and 
the form of the surfaces of critical curvature. 

The expression for the change in the mean curvature of a 
film or liquid mass, under the conditions above laid down, has 
been investigated in a paper “On the Relation between the 
Thickness and the Surface-tension of Liquid Films,” lately 
communicated by Prof. Reinold and myself to the Royal 
Society. It was, however, applied only to the cases which 
were practically realized in the experiments therein described. 
It will be convenient, before discussing it more fully, to 
indicate the manner in which the equation is obtained. 

Beer has shown that if the axis of # be the axis of revolu- 
tion, the equation to a liquid surface of revolution is given by 
the expressions 


e=eH +8, y’=2' cos’? p+" sin? Oe eee 
where F and E are elliptic integrals of the first and second 
kinds respectively, of which the amplitude is ¢, and the 
modulus «=/a?—6?/a. 


As usual, 


A=/1l— sin?¢d, .-. 2 re 


whence y=aA ; and if c= sin0?, B=acos 0. 

Since a> 6,aand Pare the maximum and minimum values 
of y respectively : and the above equations implicitly assume 
that the origin lies on a maximum ordinate; for when ¢=0, 
x=0 and y=a. If we wish to transform to a minimum 
ordinate, ¢ is > 7/2, and 


e=a(H—E,)+@(E—F), » = ae 


where Hi, and F, are the complete integrals. 
It may be well, for the sake of clearness, to state that the 


of Liquid Surfaces of Revolution. 37 


surface is an unduloid or nodoid according as £ is positive or 
negative, 7. e. according as 7/2 >0> —/2 or 37/2>0>7/2. 

If 0 be supposed to vary continuously, and if one at least 
of the quantities « and @ is finite, the form of the surface may 
be made to pass through a continuous cycle of changes. 

Thus, between 0=0 and @=7/2 the surface isan unduloid, 
the limits being the cylinder when 0=0, and the sphere when 
9=7/2. As @ passes through the next "quadrant the surface 
is a nodoid, the limits being the sphere, and a circle the plane 
of which is perpendicular “to the axis of revolution, which 
is, as Plateau points out, a purely mathematical limit. In the 
third quadrant the surface is again a nodoid, the limits of 
which are the circle and catenoid. Finally, when 6 lies be- 
tween 37/2 and 2a the surface is an unduloid, the limits of 
which are the catenoid and the cylinder. 

If now 2X and 2Y are the distance between and diameter 
of the rings respectively, and if ¢, is the value of ¢ when y=Y, 
we have 


X=aH+F, Y?=a’ cos’ d,+ 6’ sin’ dy. 


Hence if a, 8, and ¢, vary, but so that X and Y are unaltered, 
we have, by differentiation, 


{ - ofa aan Pag \ be 
+ {o(°as fap+ BE” Oks f ap ag 


+ {adit = bag.no, . ees seh: do Silat ye 


and 
Qa cos” boa +28 sin? $,88—(«?—P”) sin 26\6¢,=0. . (4) 
But 


(et a-7 ae Z{\¢- Ft, 


Substituting these values in (3) and (4) and eliminating d¢, 
between them we get 


(oH —B’F + a’ A,cotd, de +2°(F —H +A, tang, )d8= 
or Ade + BéB=0. 


Now the mean curvature of a surface of revolution of mini- 
mum area has been proved by Lindeldf to be the same as 


38 Mr. A. W. Riicker on the Critical Mean Curvature 


that of a circle of radius (a +8). Hence 


sa(hen)=— Ee 


JASB fe, A= 


Bo +8)! (>) A ee 
Hence the mean curvature has in general a critical value 
when A—B=0. 

First confining ourselves to the case in which the prin- 
cipal ordinate is a maximum, and ¢, and @ are less than 
m/2, it is evident that, since F is always >H, B is always 
positive 


Also, by (5), 


Lak .. sin? pe 1 if 2 2 a 
fom) et ee 


whence, since dF /d« is positive and sin ¢, cos qd, is positive, 
ai — BF is positive, and therefore A is positive also. Further, 
B can only vanish if ¢;=0; and none of the terms in A or B 
become infinite unless ¢,=0 or 7/2, cases which it will be 
seen hereafter it is unnecessary to consider. 

Thus, 

(A—B)/e?=2H —F(1+4+ cos? @)+2A; cot 2¢,=0 . (7) 
is a relation which must be satisfied by ¢, and @ when the 
mean curvature has a critical value for changes in the form 
of the surface which take place, subject to the conditions that 
the radii of the rings and the distance between them are 
constant. 

Corresponding values of ¢,; and @ must be found by trial ; 
but it will now be shown that if such a pair of values is known, 
when 7/2 >6>0, the values of ¢, which are proper to 7—8O, 
a+, and 27—@ can be readily deduced without further trials. 

In the first place it is evident that, since the squares of the 
sine and cosine of @ alone enter into (7), the curve obtained 
by plotting the values of @ as abscissee and those of ¢, as 
ordinates is symmetrical with respect to the ordinate 0=7/2, 
and that the same value of ¢, corresponds both to @ and r—8. 

If, then, we conceive a film attached to two rings, the 
volume and length of which vary continuously in such a way 
that (7) is always satisfied, as the cylinder changes to the 
sphere and thus to a nodoid forms which correspond to the 
same value of sin’ @ will have the same value of ¢, also, and 
the lengths will be given by the expression 


A=«(h+ cos OP) 3 0.) 2 ee 


of Liquid Surfaces of Revolution. 39 


where @ is <7/2 for the unduloid and >7/2, but <a for 
the nodoid. 

All these forms will have a maximum diameter half way 
between the rings. If we now proceed to discuss cases 
where the principal ordinate is a minimum, we must all 
through the previous investigation consider the lower limit of 
the integrals to be 77/2 instead of 0, and ¢, to be >7/2. With 
this convention no change is produced in any of the equations ; 
as in equations (5) the quantities which are brought outside 
the sign of integration vanish, both when ¢,;=0 and when 
d,=7/2. 

Thus, writing as usual E, and F’, for the complete integrals, 
and taking ¢’ instead of ¢, as the upper limit of H and F, 
where q’ is >7/2, we have 


2(E—E,) —(F—F,) (1+ cos? 0) +2A(¢’) cot 2¢’=0. (9) 
Let y be an angle such that 
F(¢’) — Fi =F); 
then, by the addition theorem, 
H(¢')—H,=E (yr) — sin? @ sin d’ sin yp. 


Also 
tan d’ tan y= — sec 6, 
sin p= cos p/A (yy), 
: A(¢’)A(r) = cos 8, 
an 


1— cos’ 6 tan? 
f SS 
a 2 cos O tan 


Hence, substituting in (9), 
2E (yr) — Fb) (1 + cos? @) —2 sin” 6 See 
A (r) 2 cos @ tan 
or 


2E(v)—F(r)(1 + cos? @)+2A (yr) cot 2~=0, . (10) 


which is the same as (7). 

We thus conclude that, for every angle % or ¢, which 
satisfies (7), there is a corresponding angle ¢’ which satisfies 
(9) for the same value of x”, and that these angles are con- 
nected by the relation 


freee wit Ge HEC Ue ee ses. (11) 


If, then, we determine from (7) the values of 6, which cor- 
respond to certain values of 0 between 0 and 7/2, we can by 


+2 — 5 


40 Mr. A. W. Riicker on the Critical Mean Curvature 


(11) find the values of ¢’ corresponding to values between 
d7/2 and 2a; whence, since (7) and (9) depend on the 
squares of the sine and cosine of @, the values of ¢ and q’ 
between 7/2 and 37/2 are also known. 
Before making any numerical calculations it is convenient 
to discuss (7) more fully. 
Differentiating, we get 
1 sin” p 1+ cos? 6 (? sin? 
sinz0{ —| Re ger i} ar dp+k 
__ sin? d; cot 24, | f _ 1+ cos’@ 
ae Car 60+ 4 2A, oe 
__ sin? @ cos 2¢, 
Ay 
which, if we use equations (5) and simplify, becomes 
tan 0{ H—F cos? @—A, tan ¢,}80 + 4A, cosec?26,66,=0. (12) 
Hence 6¢,=0 if 0=nz, and if ¢d;=n7/2. Also, considering 
the case in which 
H—F cos’? @—A,tang,;=0, . . . . (18) 
we notice that, if we subtract (13) from (7), we get 
H—F +A; cot¢,;=0;  ... . 2) 
and these equations are satisfied if @=7/2 and ¢,=56° 28’. 
For, if 0=7/2, (18) is true identically, and (14) reduces to 


—4A, cosec? 2¢, \ 6p 0) 


vs 
log, tan (7 ae $)= cosec dy, 
which holds good when ¢,=56° 28’. 
Hence, when @=7/2, 6¢,/60 is of the form « x0, which 
is readily shown to be equal to zero. 
To find the corresponding value of 6¢//60; we have from (11), 


/ 
sec’ d, tan a + sec? f/ tan dy “= — sin @ sec? 9; 


and by substituting from (11) for tand/ and sec? ¢’, this 
becomes 


an dd, ,d¢’  _—_ sin sin 2¢, 
00s Og +t Sig > 
Putting 6 =7/2, | 


Tt must be remembered that 6 corresponds to ¢, and that if 
& corresponds to $’, 0’ =27—9, so that dd’/d0’= —dd'/dd. 


of Liquid Surfaces of Revolution. Al 


The question as to whether the critical value is a maximum 
or minimum has not yet been discussed. Since A—B=0O, 


ee) 


this depends on the sign of a aps eee if we write a= A/a”, 
b= B/a”, upon that of (a—b), where a and b are explicitly 


functions of @ and ¢, only. Now, putting a—b=y, 
Le = (% d0 (2x) aes 
es ca a ah ae | 
where (*) and (<<) are the coefficients of 64, and d¢, 
dé dy , 


in (12), with the signs changed. 
But since A=B, d8/de=—1, from (6). Hence, since 
cos 0=8/a ! 
sin 0 ap fey 
da ot” 
In like manner, from (4), 


df, _ 2(« cos? $,—B8 sin’ $y) 


so that da —— (@” — 8”) sin 24, 
By dy) 248 4 dx a cos’ $;—f sin’ d, 
da =(5 a” sin 0 db,) (a#?—’) sin2¢, ° 


Now as we pass from one surface which satisfies the condition 
A—B=0 to another, the value of X changes ; and it can easily 
be shown that if dX/dé@ be calculated subject to this condition, 
it is of the same sign as dy/da«. Hence if X increases with 8, 
dy/de« is positive and the critical value is a minimum; if X 
diminishes as @ increases, itis a maximum. If X is a maxi- 
mum or minimum the curvature has a stationary value, but 
it is not itself a maximum or minimum. 

I have calculated by trial the values of ¢, which satisfy (7) 
for a few angles between 0° and 90°. They are given, together 
with the corresponding values of H, F’, and A,, in Table I. 


TABLE I. 


42 Mr. A. W. Riicker on the Critical Mean Curvature 


The values in the last four columns are repeated in the 
reverse order as @ increases from 90° to 180°. 

In the next Table are given the values of 6’=a7—d’, and 
of E,—E($”) and F,—F(¢”). In representing the results 
graphically it is best to take ¢’—7/2 or d, as corresponding 
to ,, and therefore these values are also given. 


TaBueE II, 
| 
0. 9". ¢. | H—E(9"). | R—-F(¢"). 
360 45-00 45-00 0-785 0-785 
350 45-20 44-80 0-772 0-792 
330 47-20 42-80 0-664 0-841 
315 50-60 39-40 0519 0-913 
300 BTS 32-85 0326 1-018 
280 75°65 14:35 0-055 1-169 
270 90-00 0-00 0-000 1-200 


The values of A(¢’) are omitted because they are readily 
obtained by the formula A(¢’) A(¢,)= cos @. 

The curve obtained by means of these Tables, which shows 
the relation between ¢, or ¢, and @, is given in fig. I. 

Rectangular coordinates are perhaps the most convenient ; 
but it @ and @ be regarded as angle and radius vector, the 
curve assumes the symmetrical form shown in fig. IL. 

This result completes the solution of the problem ; but the 
nature of the conclusions at which we have arrived is more 
evident if we proceed to deduce the ratios of the lengths and 
principal diameters of the films to the radii of the rings. 

This is done by means of the following relations, where 
symbols with unity subscript refer to bulging films, and those 
with 2 subscript to films the principal ordinate of which is a 
minimum. 


/Y=1/A,, 
X,/e,=H+F cos 0; 
whence X,/Y is found. 
B2/Y =a, cos 6/Y = cos 0/A,=A,, 
X,/8.= (H’ —H,) sec 0+ (F’—F,) ;sx 


whence X,/Y is obtained. 
It is evident from these equations that «,8,= Y?; 2. e. the 


of Liquid Surfaces of Revolution. 43 


radius of the rings is a mean proportional between the prin- 
cipal ordinates of two surfaces in which the modulus of the 
elliptic integrals is the same, and the principal ordinates of 
which are a maximum and a minimum respectively. 


TaBLeE III. 


0. ee. Meo ae. g. Bai. hs Xe. | XX. 


—|,§ — | | | — | ————. 


0 1-000 1571 1571 180 1:000 0-000 | 0-000 
10 1-008 1-567 1579 190 0992 0-008 | 0:008 
30 1-074 1-527 1-640 210 0-931 0°073 | 0-068 
45 1-184 1-458 1-726 225 0:844 0-179 | 0-151 
60 1°372 1-333 1829 240 0-729 0366 | 0-267 
80 1-725 1-036 1787 260 0580 0°852 | 0494 
90 1-810 0834 1-509 270 0°552 1:200 | 0:663 

100 1-725 0630 1-086 280 0-580 1-486 | 0-862 

120 1-372 0316 0:433 300 0°729 LO7L. eh 218 

135 1-184 0°166 0-196 315 0°844 1-647 | 1:391 

150 1-074 0-071 0-076 330 0-931 1607 | 1:495 

170 1-008 0-008 0:008 350 0-992 1-576 | 1:563 

180 1-000 0-000 ()000 360 1-000 L57k.-} 1-571 


The ‘‘march”’ of the functions is shown by means of the 
curves in figs. III., 1V.,and V. Thus, if p be the length of the 
principal ordinate (whether it be a maximum or a minimum), 
fig. III. shows the relation between p/Y and @, fig. IV. that 
between X/p and @, and fig. V. that between X/Y and 0. 


180 210 240 370 300 330 0 30 60 90 120 150 180 
Fig. I. (=8, y=$1)° 


By plotting the values of X/Y we find that the maximum 
occurs when @=70°. ‘The corresponding value of dy is 54°15, 


44 Mr. A. W. Riicker on the Critical Mean Curvature 
and this gives 

afY=1:545, X/a=—1:2044, X/Y=1°860. 
Tf, then, we suppose the rings to approach to or recede from 
each other, and the volume and diameter of the film to be at 
the same time altered so that it always satisfies the conditions 
of critical mean curvature, it will undergo the following 
changes of form. 

Starting with the rings in contact, and supposing that as 
they separate the film has a slight bulge, it will first be a 
nodoid, and the length and principal diameter will increase 
together. When the length is a little more than one and a 
half (1:509) diameters of the rings the film is spherical, and 
the principal diameter is then a maximum (a/Y=1°810). As 
the diameter begins to decrease the film becomes an unduloid, 
but the length increases until it is 1-860 x diameter of rings. 
Thereafter length and diameter decrease together until, when 
the latter isa third proportional to the diameters of the sphere 
and of the rings, it reaches its minimum value (@/Y =0°552). 
The film is then a catenoid. As the length diminishes it 
becomes a nodoid, exerting a negative or outward pressure, 
and this continues until the cycle is completed by the rings 
meeting again. 

The whole of the above investigation has taken place subject 
to the condition that ¢,;<7/2, and without reference to the 
stability of the films, which is, however, secured by the 
condition as to d, except in the neighbourhood of 0=180°. 

The curves, when drawn on a larger scale, lend themselves 
to the solution of a number of problems with an accuracy 
quite sufficient for practical purposes. 

Thus, if we wish to determine the conditions of the film 
which has a critical curvature when the principal diameter or 
the length is a given multiple of the diameter of the rings, we 
have only to draw a circle with the origin in figs. III. or V. as 
the centre, and with the radius equal to the given ratio. The 

oints of intersection give the value of 0; ¢, is found from 
fig. II.; and thus the other quantities can be determined either 
by calculation or by means of the other figures. 

It is evident, since the maximum radius of the curve in 
fig. V. is such that X/Y=1°860, that the curvature cannot 
have a critical value for films such that the ratio of the length 
to the diameter of the rings exceeds this number, while for all 
less ratios there must be two critical points, a maximum and 
a minimum respectively. 

If, then, we suppose a film attached to two rings to be 
initially a nodoid with a diameter exceeding that of the sphere, 


of Liquid Surfaces of Revolution. 45 


and to contract gradually, its behaviour, as regards change of 
curvature, within the limits of the problem, would be as 
follows. 

If the length were >1°860 x diameter of rings, the film, 
after becoming a sphere, would always be an unduloid until it 
reached the limit at which the conditions no longer apply. 
The mean curvature would increase as the principal diameter 
diminished. 

If 1:360>X/Y>0°663, the film remains an unduloid 
_throughout all stages after it has become a sphere; but the 
mean curvature first increases, then diminishes, and finally 
increases again. The cylinder is the form of minimum mean 
curvature if X/Y=1:571. The sphere is the form of maxi- 
mum curvature if X/Y=1°509. 

If X/Y=0-'663 the last series of statements holds good, 
with the addition that the minimum mean curvature is zero. 
Hence the surface passes through the form of the limiting 
eatenoid, which is such that no catenoid can be formed be- 
tween the rings if the distance between them is increased. If 
the distance between the rings is diminished, two catenoids 
pass through them. 

If X/Y <0:663, the maximum mean curvature which is 
attained while the film is still a nodoid diminishes as the 
figure passes through the forms of the sphere, cylinder, and 
catenoid, and then becomes negative, 7. e. the pressure exerted 
by the film is directed outwards. The minimum is reached 
when the form of the film lies between the two catenoids 
which can be drawn through the rings. 

The calculations enable us also to solve another problem. 
If the interiors of two similar films be connected which are 
formed between equal and equidistant rings, and which are 
stable when separated from each other, the system will only 
be in stable equilibrium if a contraction in the principal 
ordinate, producing a decrease in volume, is attended by a 
decrease in the curvature. 

Hence no pair of similar films so arranged can be in stable 
equilibrium if the length is >1:860 x diameter of rings. 

Two cylinders cannot be in stable equilibrium if the length 
T 
2 
diameter of rings, 7. e. >0°834 x diameter of sphere. 


is > = x diameter, nor two spheres if the length is > 1°509 x 


fim 


V. On Silk v. Wire Suspensions in Galvanometers, and on the 
Rigidity of Suk Fibre. By THomas Gray, B.Sc., F.R.S.E.* 


i the last Number of the Philosophical Magazine there is 

a short article by R. H. M. Bosanquet drawing attention 
to some eccentricities of a galvanometer used by him. A 
determination of the rigidity of the suspending “ fibre” of the 
galvanometer-needle would have been interesting, as it would 
have thrown considerable light on the probability or improba- 
bility of the explanation offered. It must have caused no little 
surprise to many of the readers of the Philosophical Magazine 
to find that Mr. Bosanguvet based his condemnation of silk- 
fibre suspensions on the trouble he experienced with an instru- 
ment the suspending fibre in which was ‘left just stout 
enough to carry the weight,” aud which was of such a nature 
that it could possibly twist or untwist with stretching or 
with hygrometric changes in the atmosphere. Surely Mr. 
Bosanquet is scarcely in earnest when he writes about sus- 
pending the needles of a sensitive galvanometer with a twisted 
silk thread, or when he proposes to go back something like 
half century in the history of this subjectf, and adopt galva- 
nometers with needles seven inches long made of stout knit- 
ting needles and suspended by a wire five feet long. 

A galvanometer-needle should never be so heavy that it 
cannot be suspended by a single fibre of silk (that is, half an 
ordinary cocoon fibre), because such a fibre will bear easily, 
leaving a good margin of safety, two grammes; and it is an 
easy matter to so arrange such a mass that the period of vibra- 
tion will be not only so much as thirty seconds but even several 
minutes. With an astatic arrangement, especially if it be 
only “nearly astatic,”’ there will be changes of zero cer- 
tainly, but I can hardly see any thing comparable to a 
“ ghost ” in what could occur. 

About a year ago I made, in the Physical Laboratory of 
Glasgow University, a number of experiments on silk fibres, 
which included among other things some determinations of 
their rigidity. Mr. Bosanquet’s paper has suggested to me 
that possibly a few of the results may be worth publication. 
Some of the results of these experiments are in type in 
vol. iii. of the Reprint of Sir W. Thomson’s Mathematical 
and Physical Papers now in the press. 

Two methods were used for the determination of the rigidity. 

* Communicated by the Author. 

+ Some interesting experiments “ On the Suspension of the Magnetic 
Needle by Spiders’ Fibre” are described by the Rev. A. Bennet, F.R.S., 
in the R.S. Trans. vol. Ixxxi. 1792. 


On Silk v. Wire Suspensions in Galvanometers. 47 


The first method was almost identical with that introduced in 
this laboratory thirty-five years ago by Sir W. Thomson, and 
now commonly adopted for the determination of the rigidity 
of metallic wires. It consisted in suspending from a fixed 
support, by means of a measured length of the fibre, a thin 
circular rim of non-magnetic material and of easily calculated 
moment of inertia, and observing the period of the torsional 
vibrations. From this the torsional rigidity of the fibre can 
be readily calculated by a well-known formula. The second > 
method consisted in suspending a small mirror, to which was 
rigidly fixed a small magnetic needle of known magnetic 
moment by means of a measured length of the fibre, and 
observing the deflection of the mirror produced by twisting 
the top of the fibre through a measured angle. This gives a 
ready means of calculating the rigidity of the fibre in terms 
of the magnetic moment of the suspended needle, and the 
strength of the magnetic field in which it is suspended. 

The fibres were of Japanese floss-silk, which had been 
thoroughly washed in hot water to remove the gum which is 
always found in considerable quantity on cocoon fibres. The 
fibres were in all cases single fibres ; and it will be seen, both 
from the direct measurements by the microscope and from the 
rigidity, that they vary considerably in thickness. Hven a 
rough estimate of the rigidity per square centimetre section 
of the substance is impossible, as the fibre is not even approxi- 
mately circular in section, and its diameter not nearly regular 
along its length. The results of the experiments are given in 
the following Table, the headings of the different columns 
being sufficiently explanatory of the numbers. 


Vibrator Method. 


ac 

=i = = we 8 
a S wT aia, Dens, = PO . 
8 Sg aga a ean 248.2 
~n sm o = 2 Catt TOYS 
eS oO = ,~_f Ooh Sk os mM a 
Bey | eS ee = ay | sats 
Be | ee 8. | eo ics Se ieee iene 
a= Ss Ss o..8 oy ee as Zw aS 
Sa mas a ce os oo sea 
° nes ~ & Ysa al rs == 
aq ee st See 28 AOBs 
ae 3) og ad a a7 
Ss Ce) a) St ot oO oad 4 

ae = aw Bea oe 
0:0274 0-20 8°60 0-0008 29 0:00096 


00114 0-29 8:60 | 0-0010 16 0:00132 
| 


48 Mr. T. Gray on Silk v. Wire Suspensions in 
Magnetic Method. 


B | ga (5828 |ece [Bes | 8 

as Og Bislee ss 4 Cf LSE we gas ¢ 
SA | ee |S act i |e eas | 12 2 eee oe 
mel Se | 8 elo las ease Se 
so | 22 | o's ba i So Cl om See 
ap 2 sos CoSHe |@sa a Sa Bus, 19) 
Bo Seb. | et ee oe Se eel mee 

4 A Até*s A le Pm 

9:05 | 0:0010 8:0 117:0 00143 75°4 
9-20 | 00009 8:0 117-0 00090 61:3 
8°45 | 000145 21:0 117-0 00216 65°6 
9:55 | 0-0015 21°5 Ia 00250 73°5 


The following curve illustrates an experiment, and shows 
how nearly proportional the first deflection is to the torsion even 
after the elastic limit of the fibre has been far exceeded. In 


a 


MBL PADSI*IMN7IHN 


~— 


20 


2 st ae 


the first part of the curve the ordinates are the scale-readings, 
the abscissee the angle turned through by the torsion-head, 


* Thisis the ratio of the product of the pull applied to the fibre, and the 
length of the fibre, to the elongation produced by the pull, or, if E be the 
weight applied x length of fibre 


elongation. 


modulus, E= 


Galvanometers, and on the Rigidity of Silk Fibre. 49 


which we may, without appreciable error, assume to be 
the torsion of the fibre, as the angle turned through by the 
mirror is so small as to be negligible; in the last part the 
ordinates have the same meaning, but the abscissee indicate 
time. This second part of the curve shows the rate at which 
the fibre takes a set under the torsional stress ; the part of this 
curve below the zero-line shows the working out of the set 
after the fibre was untwisted. The length of the fibre in this 
experiment was 8°5 centim. and the average thickness about 
0:0015 centim. 

When a galvanometer is made sufficiently sensitive for the 
fibre to play an important part in directing the needle, the set 
of the fibre due to continued deflection always produces an 
apparent change of zero which, in exact measurements, it is 
somewhat difficult to properly allow for. LHxcept, however, 
in very special cases, as, for instance, in taking deflections 
with a Thomson’s “dead-beat” galvanometer in a weak 
magnetic field, the error is small, and it is not in any way 
capricious. It is important to bear in mind, however, that 
for very sensitive galvanometers to be used as deflectional 
instruments the suspension should be of considerable length, 
such, for example, as is provided in the Thomson’s astatic 
galvanometer. 

From the data given above we may very easily form an 
estimate as to when the rigidity of a silk fibre comes to be an 
important factor, affecting the sensibility of a galvanometer. 
If C be the current flowing through the galvanometer, K a 
constant depending on the coils, I and I’ the field at the upper 
and lower needles respectively, m and m’ the magnetic mo- 
ments of these needles, 7 the torsional rigidity of the fibre, 
and @ the deflection, we have 


Im—I'm’ 70 
O=K{ Rae Hr (m+m’) cos 05° 
When the needle system is perfectly astatic, m=’, and this 


reduces to saghF 
o=K {+ : tan 6+ ens 


2 Im cos S ° 
and for small deflections this may, without great error, be 


written =e 
C=Ke i= + x}. 


From this equation we see that the fibre becomes important 


when = is not small compared with I—I’, Now ina very 


sensitive instrument it is not unusual for I~I’ to be reduced 
Phil. May. 8.5. Vol. 23. No. 140. Jan. 1887. E 


50 Mr. T. Gray on Silk v. Wire Suspensions in 


to about ‘001, and m in such an instrument as we are consi- 
dering will not differ much from unity. Hence 7 must be 
much less than ‘001; and we find, from the tables given above, 
that, for a fibre of about the usual length, say 5 centim., 7 will 
be about ‘0003; or about one fourth of the total force is, in this 
case, due to the fibre. This, then, may be taken as about the 
limit of sensibility beyond which we cannot easily pass with an 
ordinary Thomson’s astatic galvanometer with small needles ; 
to get beyond it, attention must be directed to an increase of 
m*, The limit here indicated is, however, far beyond anything 
that can be reached with wire suspension, the smallest current 
which can be measured being about 10-9 ampere for a galva- 
nometer of 1 ohm resistance, and about 0°2 x 10—!° ampere for 
one of 10,000 ohms resistance. When I—I’ is as much as 
0:01, or between a tenth and a twentieth of the earth’s hori- 
zontal force in this country, the effect of the set of the sus- 
pending fibre is extremely small. With sucha value of I—I’, 
however, a properly constructed galvanometer, the resistance 
of which is as low as 1 ohm, will measure a current of 10-8 
ampere. When very high sensibility is absolutely necessary, 
it may be to some extent obtained by increasing the length of 
the fibre ; but if this prove insufficient, some alteration or 
other arrangement of the parts becomes necessary. Such an 
arrangement is described in the paper referred to in the foot- 
note ; but it may be remarked that, in so far as this arrange- 
ment is intended to increase m, it is only important when 
I—I’ is made practically zero. So long as I—I’ is consider- 
ably greater than 7, a high value of m is of no importance ; 
and the Thomson form is, because of the small inertia of its 
needle system, decidedly the best. 

Norre.—Since the above was written Mr. J. T. Bottomley 
has suggested to me that some interesting results might be 
obtained if the vibrational method, above referred to, were 
_ ¢earried out with the fibre and vibrator in a good Sprengel 

vacuum ; and in conjunction with him I have made some 
preliminary experiments, the results of which seem worth 
quoting. 

The vibrator used was the lighter of the two referred to in 
connection with the former experiments, and consisted of a 
small ring of brass 0°295 centimetre radius and 0°012 gramme 
in weight. It was suspended, as shown in the diagram, in- 
side a small spherical bottle provided with a long neck and a 
ground stopper, to the lower end of which the fibre was 
attached. A tube passed from the side of the bottle to one 


* On this subject see a paper “On a New Reflecting Galvanometer of 
Great Sensibility,’ by T. and A. Gray, Proc. Roy. Soc. No. 230 (1884). 


Galvanometers, and on the Rigidity of Silk Fibre. 51 


end of a U-tube, containing phosphoric acid and beads of 
glass, the other end of which was sealed to a tube leading to 
the Sprengel pump. The vibrator was attached to the fibre 
by means of three short single fibres, in the manner shown in 


TO PUMP 
»— 


PHOSPHORIC 
ANHYDRIDE 


/| 


VIBRATOR 


the sketch. The results are given in the following table, the 
meaning of the numbers in the different columns of which 
will be readily understood from the headings. In the-column 
headed “‘ numbers of vibrations observed ”’ the figures repre- 
sent roughly the number of periods which could be observed 
at the different degrees of exhaustion, shown in the preceding 
column, beginning in each case from an amplitude of about 
60°, and observing directly the transits of a black spot on the 
ring over a fixed mark until the amplitude fell to about 10°. 

The results are sufficient to show that the effect of the 
viscosity of the fibre in damping the vibrations is very small 
in comparison with the effect of the air friction; and it seems 
probable that a moderately heavy vibrator (say about 2 grammes 
in weight) with a small magnetic needle attached, and sus- 
pended by a single silk fibre, may prove a good arrangement 
for experiments such as have been carried out by Maxwell, 
Kundt and Warburg, Crookes and others on the friction and 
viscosity of gases. It certainly would have the advantage 
that the period would depend mainly on the strength of the 
magnetic field, and could be varied at pleasure. Should 
opportunity offer, Mr. Bottomley and the writer hope to con- 
tinue these experiments. 

K 2 


52 Sir William Thomson on Stationary 


s s . 2] 8 us cs Cees 
eevee | oe be cee |. Oo @ | eae 
geige| ge | 22 | ok | Se.) gee 
eS lge| ds | ee) g3 | 28 | eee 

eo a) 2 Ay om wo a =| 
1, |39 | -00095 | 105 | 1-00 7 
Plt ¢ 96 | 146x107] 14 00134 
9, | 37 |-00120 | 778 | 1-00 7% 
prices fh 764 | 0-066 Yq 
4s (Bee ‘ 7-42 | 1:46y107 | (?) 
as 4 7-42 | 040x107 | 60 00261 
3, | 3:65 | -00105| 125 | 1-00 5 
es i 11:92 | 7-45x107°| 7 
Pali cs 11:83 | 835x107°| 15 
sae Al aes i 11:63 | 053x10°| 40 
a eee 2 11:57 | 013x10-°| 50 
eee oe i 11:61 | 013x107} 50 00110 


VI. On Stationary Waves in Flowing Water.—Part IV. 
Stationary Waves on the Surface produced by Equidistant 
Fiidges onthe Bottom. By Sir Witu1AM THomson, F.R.S.* 


HE most obvious way of solving this problem is by the 
use of periodic functions, which we have been so well 
taught by Fourier in his ‘ Mathematical Theory of Heat ;’ and 
in this way it was solved in Part III. (formulas 1 to 15); the 
solution being (15) Part III. with 


K=1, m=2ne «4. 2 eee 


where a denotes the distance from ridge to ridge. Thus, 
reproducing (15) Part III. with the notation modified to shorten 
it in form and to suit it for numerical computation, we have 


re = 4 A/a. cos ip Me 


ly 4 p-i_ = (é —e-i) 


* Communicated by the Author. 


Waves in Flowing Water. 53 


where f denotes height above mean level of the water 
at distance # from the point over one of the 
ridges ; 
A denotes profile-sectional area of one of the 
ridges ; 
a denotes 27rx/a ; (3) 
é denotes 27/4; 
M denotes the g/mU? of Part ITT. (6) to (18) | 
or a/2mb ; 
b denotes U?/7; 
and D denotes the depth. J 


Thus, in (2) we have an expression for the surface-effect of 
an endless succession of equidistant ridges on the bottom. 
We shall see presently that if the succession of ridges is finite, 

the result expressed by (2) will not be approximated to by 
increasing the number of ridges. The difference in the effect 
of a million equidistant ridges from that of a million and one 
equidistant ridges, in respect to the corrugations on the surface 
of the fluid over any part of the series, may be as great as the 
difference between the effects of a thousand and of a thousand 
and one, or between the effects of ten and of eleven: and the 
absolute effect of four, or six, or eight, may be sensibly the 
same as, or may be greater than, or may be less than, the 
effect of a million, in respect to the condition of the surface 
over the space between the two middle ridges. The awkward- 
ness of the consideration of infinity for our present case is 
beautifully done away with, after the manner of Fourier, by 
substituting for an “infinite canal” an “ endless* canal,’ or 
a canal forming a complete circuitt: a circular canal as we 
may imagine it to be, although it might be curved, of any 
form, provided only that, whether it be circular or not circular, 
the radius of curvature at any point is very great compared 
with the breadth of the canal. This condition is all that is 


* It is curious that the word “ endless” should in common usage, and 
especially in technology, have so different a meaning from “ infinite.” 
Thus every one understands what is meant by an “endless cord.” An 
“infinite cord’? means, in common language, an infinitely long cord—a 
cord which has no limit to the greatness of its length. 

+ A curious piece of illogical usage in mathematical language, according 
to which an enclosing curve is called a “ closed curve,’ must henceforth 
be absolutely avoided. It has already led to endless trouble in electrical 
nomenclature, according to which, in common language, an electric cir- 
cuit is said to be closed when a current can pass through it, and to be open 
when a current cannot pass throughit. I believe all, or almost all, English 
writers on electrical subjects have been guilty of this absurdity. I doubt 
whether any one of them would say a road round a park is open when a 
gate on it 1s closed, and is closed when every gate on it is open. 


54 Sir William Thomson on Stationary 


necessary to allow the motion of the water in every part of 
the canal to be so nearly two-dimensional, that our formulas 
for two-dimensional motion in a straight canal shall be prac- 
tically applicable to the water in the curved canal. 

Now let there be any integral number n of equidistant 
ridges in the circuit, and let a be the distance from ridge to 
ridge. Superposition by simple addition of solutions of the 
formula (2) gives, for the surface effect, 


=0 


b= 


7=1 


‘—n—1 . 

4NJa. 3 cosi(p+ 

a NN eee 
eel ie es 

e+e*— 7 é—e *) 


The consideration of cases of different values of n, even or 
odd, leads to interesting illustrations both of mathematical 
principles and of practical results in dynamics; but for the 
present I confine myself to the case of n=1, for which (4) 
becomes identical with (2). 

Remark, now, that if M(é —e-‘)/(4 +e7*) is an integer, 
the denominator of (2) vanishes for the case of 7 equal to 
this integer. This is the case in which the length of the 
circuit of the canal is an integral number of times the wave- 
length of free waves in water of depth D. The interpre- 
tation is obvious, and is interesting both in itself and in 
its relation to corresponding problems in many branches of 
physical science. 

Meantime remark only that, when the value of 
M(e —e-*)/(e +e-*) approaches very nearly to any integer J, 
the chief term of (2) is that for which 7=j, and all the other 
terms are relatively very small. Thus the chief effect is 
forced stationary waves of wave-length a/j. Thus, if we con- 
sider different velocities of flow approaching more and more 
nearly to the velocity which makes M(e —e-*)/(e'+e7*) an 
integer, the magnitude of the forced stationary waves is 
greater and greater for the same magnitude of ridge, but the 
motion is still perfectly determinate. Suppose, now, we 
make the ridge smaller and smaller, so that the wave-height 
of the stationary wave may have any moderate value ; as the 
velocity approaches more and more nearly to that which makes 
M (e' —e-*)/(e +e-') an integer, the magnitude of the ridge 
must be smaller and smaller, and in the limit must be zero. 
Thus, with no ridge at all, we may have stationary waves 
of any given moderate value, in the limiting case,—that in 
which the velocity of the flow equals the velocity of a wave of 
wave-length a/j. 


Waves in Flowing Water. 55 


But now let us consider the case of M(é —e-*)/(é +-e7*) ag 
far as possible from being an integer ; that is to say, 

Me ayer eee se eS (5), 
where j is an integer. For all values of 2 less than j7+1 the 
denominator of (2) is clearly negative, with increasing abso- 
lute values up to 1=7; and for all values of 2 greater than 7 it 
is positive, with decreasing values from i=j7+1 to i=o., 
Thus the absolute magnitudes of the coefficients of cost in 
the successive terms of the series from the beginning arenega- 
tive, with increasing absolute values up to i=); and after that 
positive,with decreasing values converging ultimately according 
to the ratio e~!. Remembering that e=e*"?/*, we see that the 
convergence is sluggish when a, the distance from ridge to 
ridge (or the length of the circuit in the case of an endless 
canal with one ridge only,) is very large in comparison with 
the depth ; but that when a is less than the depth, or not 
more than five or ten times the depth (an exceedingly inte- 
resting class of cases), the convergence is very rapid. 

We shall find presently, however, another solution still 
more convergent, much more convergent indeed for the 
greater part of the configuration, whatever be the ratio of D 
toa; a solution which is highly convergent in every case 
except for values of # considerably smaller than the depth. 
The calculation for these small values of # is necessary to 
give the shape of the water-surface at distances on each side 
of the vertical through the ridge small in comparison with 
the depth : for this purpose, and for this purpose only, is the 
solution (2) indispensable. For investigating all other parts 
of the configuration the new solution is much more convenient, 
and involves, on the whole, very much less of arithmetical 
labour. It is found by summation from the solution of the 

single-ridge problem given in Part III. (40), (41), as follows. 
_ Let the whole number of ridges be 7+ ’+1, and let it be 
required to find the shape of the surface between the verticals 
through ridges numbers 7 and 7+1. Take the origin of the 
coordinate « in the vertical through number 7 ridge, and let 
number j+1 be on the positive side of it. The solution will 
be found by adding to the solution (40) Part III., 7 solutions 
differing from (40) only in having respectively «+a, x +2a, 
..., e+ a substituted for «; and 7’ solutions each the same 
as (40) Part LII., but having —#+a, —#+2a,.., —x+J/a, 
substituted for z Thus, denoting by Sthe sum of the effects 
of the 7+ 7’+1 single ridges, we find 
—- _ pithy pra i 4! pl—a/a 
S= 3 C, a Nae Aiited 3s 
i 


56 Sir William Thomson on Stationary 
where nae 
—1)'T* cos a; 
C, denotes $A/D. Sa ee ; } 
(é—3)9—«,Ja 6;a 
jf; denotese iD ore D; 

a; denotes (i—4)7—6;; or the numeric between 
zero and 7/2 which satisfies the equation 
[ @—4)7—a; | tane;—D/b=0; 

D denotes the depth ; 

b denotes U?/g ; 

U denotes the velocity of the flow; 

a denotes the distance from ridge to ridge ; 

A denotes the profile-sectional area of one of 
the ridges ; 

S denotes, for the horizontal coordinate x, the 
height of the water above the mean level 
of places infinitely distant either upstream 
or downstream from the ridges. 


ia 
vV“—~—-™ 

“I 
— 


Take first the case of b>D. In this case, as we have 
already remarked in Part III., a1, a,..., a; areallreal; and 
therefore 71, fo,.-+,/; are each real and less than unity. 
Hence in this case the 7 series and the 7’ series, of which the 
sums appear in (6), are each convergent, and if we take 7=00 
and j’=0 , (6) becomes 


soles iA + fie 

=> be 156 Bane 8 e 
x = me 1—/, (8) 
We have now the same expression for S whichever of the 
ridges be chosen for the origin of z; and the value for z=a 
is equal to the value for e=0. The water-disturbance is 
therefore equal and similar in all the spaces from ridge to 
ridge, and the solution (8), from e=0 to v=a, expresses 
within the period the height of the water above a certain 
level ; not now, as in (2), the mean level throughout the 


period, but a level at a height if S .dz/a above the mean level. 
0 
Now, by integration of (8), we find 


sey (Be i= 2C; 
i See=3 ap PME ek ey i (Oo): 


To evaluate the series forming the second member of this 


Waves in Flowing Water. 57 


expression, remark that by (7) above and (34) Part III., we 
have 


— (=1) cosas A/D-N (ay 
log (1/7;) CDs er LS. 7 ): 


Now by putting c=0 in Part III. (29) and (24), we find 
SNe Dee a My Oe PEy 
Hence, and by (10), (9) becomes 


t a 
1(% dz=A/D e . a e e (12). 
Denoting now, as before, by § the height above mean level 
from ridge to ridge, we find from (8), 


t=0 ee. be it 
b= 2G. AD... (13), 


=A/D 


The comparison between this and (2) above, two different 
expressions for the same quantity, (with, for simplicity, D=1), 
leads to the following remarkable theorem of pure analysis, 


27x 
Ala.cost ; 


1—2 


“aie: Leica tars: 
=! ef tet ~.—_ (@ —e7! 


i Qmb 
eet (—1)¥*1 cos a; €- 9 + e6ila-2) | 
Beare cee Te 9 


where 

a denotes any real positive numeric; 

b denotes any numeric > 1; 

e denotes 27/4; 

a; denotes the numeric between zero and 7/2, 

which satisfies the equation (15). 

[@—})r—a; | tan e,—b=0; 

@; denotes (i—43)7—a,; 

x denotes any real positive numeric <a; y 


The theorem (14) is easily verified by takin { ‘di - cos J as 
0 


of both members. The first member of the result is obviously 


2/[8 +eF— : : a (ei —e-J)]|. The second member, modified 


58 Rey. T. K. Abbott: Yo what Order 
by (34), (29), and (24) of Part III., is found to have the 


same value. For the particular case of 7=0 (that is to say, 
the mere integral 7 dz of each member), the equality is 


proved by (12). 

For the most interesting cases of our physical problem, the 
solution (13) converges with great rapidity, except for small 
values of x; and for these the form of the surface is more 
easily calculated by (2). Numerical illustrations and the 
working out of the solution corresponding to (13) for the case 
of b<a are reserved for Part V., which, I am sorry to say, 
must be set aside for some time. I hope it will appear in the 
April or May number, and that it, or Part VL., will contain 
practical illustrations, such as the stationary waves produced 
by a deeper place, or a less deep place, extending over a con- 
siderable length of the stream, which is very easily worked 
out from our solution (40) (48) Part IIL., for the effect of a 
single infinitesimal ridge. I hope to pass next to the effect of 
surface disturbance, with interesting applications to the ques- 
tion of the towage of a boat in a canal, and the beautiful 
practical discoveries of Mr. Houston and Mr. Scott Russell 
referred to at the commencement of Part III. If I succeed 
in carrying out my intention, this series of Articles on 
Stationary Waves will end with the investigation of the wave- 
group produced by a ship moving through the water with 
uniform velocity, promised at the commencement of Part I.; 
and suggestions for extension in the direction towards the 
theory of the effect of the wind in generating waves at sea. 


VII. To what Order of Lever does the Oar belong ? 
By Rev. T. K. Asport, Fellow of Trinity College, Dublin*. 


ao the above question every one who has learned even the 
elements of Mechanics will reply without hesitation ‘ To 
the second, the fulcrum being in the water and the resistance 
acting at the rowlock.”’ I propose to show that the answer is 
erroneous, and that the vulgar conception of the oar as a lever 
of the first order is correct. 

In fact, if the supposed answer were correct, it would fol- 
low—first, that the power would have the advantage over the 
resistance in the proportion of about 4 to 3 (this being usually 
about the proportion of the whole length of the oar to the part 
outside the rowlock) ; secondly, that if we moved the hands 
nearer to the rowlock we should gradually but slightly dimi- 


* Communicated by the Author. 


of Lever does the Oar belong? 59 


nish the advantage and increase the effort, until, when the 
hands were applied at the rowlock, power and resistance 
would be exactly equal, and the boat would still be moved, 
only with one third more effort. If we continued to move the 
hands till they were applied to the oar at one fourth of its 
length outside the rowlock, we should still propel the boat 
forwards, though with twice the original effort. Finally, if 
we contrived by a rope to pull at the blade of the oar (that is, 
at the fulcrum) we should produce no motion whatever. It 
is needless to point out how unlike these the actual facts are. 
If the power is applied to the oar at the rowlock no effect is 
produced ; if it is applied outside, the motion of the boat is 
reversed ; and the maximum effect in this direction is pro- 
duced if the power is applied to the blade. This is proof that 
the true fulcrum (relative to the rower) is at the rowlock, 
On one hypothesis only is the oar a lever of the second order; 
viz. if the rower stands on terra firma. In that case all the 
consequences above mentioned as resulting from a change in 
the point of application of the power will really result. 

Let us put another hypothetical case. Suppose the rower 
to be in the boat as usual, but the boat to be fixed, while the 
oar presses against another boat which is free to move. In 
this case the oar is manifestly a lever of the first order ; but 
the relative position and action of the oar, the power, and the 
resistance are in every respect precisely the same as in ordinary 
rowing. 

The fact is that writers on Mechanics have strangely enough 
neglected to consider that the rower is in the boat, and that 
therefore the reaction of his effort is against the boatitself. To 
him the boat is terra firma. It will perhaps be said that we 
have only to take this reaction into account, and that then the 
account given of the oar as a lever of the second order will 
appear to be correct. True; and we might similarly treat 
almost any lever of the first order as belonging to the second ; 
but when writers tell us that the oar is a lever of such and 
such a kind, they are supposed to mean that it is so relatively 
to the rower, not relatively to the universe. 

I find it stated in some books on Mechanics that the amount 
of work done in rowing has not been estimated ; and this is 
no wonder, if the nature of the effort has been so misconceived. 
It cannot be said that the difference is unimportant practically, 
if this means that it does not matter which theory we act on. 
If we suppose, as above, that the distance from the rowlock to 
the handle is successively one eighth, one fourth, one third, 
and one half the length of the oar, then the proportion of 
power to resistance would be successively :— 


60 To what Order of Lever does the Oar belong ? 


On the common theory ...... f ., 2, = 


On the true theory ......... 7, 3, 2, 1. 


Thus the effect of outrigging, so as to place the rowlock at 
one fourth the length of the oar instead of one eighth, would, 
on the common view, only be to increase the power by one 
sixth ; on the other view, it more than doubles it. 

Again, if we suppose the oar to remain with the rowlock at 
one fourth of its length, while the hands are successively at 
the handle, one eighth from the handle, at the rowlock, and | 
one eighth and one fourth outside, the successive proportions 
of power to resistance are :— 


6 6 Of eb 
On the common theory ...... 3 7 iY ar ee 
On the true theory ............ 3, 6, 0, —6, —3 


(the minus sign representing the reversal of the effect). It 
cannot be said that this difference is practically unimportant, 
except in the sense that practical men will be led by expe- 
rience, and not by a theory so manifestly erroneous as the 
former. It would be enough for such men to be told that, 
according to this theory, it is easier to move a boat by rowing 
than by dragging. 

Now let us look at the case of a canoe. Here we may sup- 
pose that the middle of the paddle remains at the same distance 
from the canoer’s body. Practically it is as if the paddle were 
attached at its middle point to a rigid rod resting against his 
body. If we are to apply to this case thesame sort of reason- 
ing which writers on Mechanics use in the former case, we 
must treat this middle point as the point of application of the 
resistance. Then we have the result that, as one hand moves 
forward and the other backward, they counteract one another. 
This is, I think, a reductio ad absurdum. There is no alter- 
native but to treat the middle point as the virtual fulcrum. 
The actual fulcrum is of course in the body of the canoer, but 
this does not affect the question. 

There are two analogous cases which seem worth consider- 
ing. Tirst, that of a paddle-steamer. Here there is an 
instantaneous lever of which the point that is actually at rest 
is the lowest float in the water ; and following the received 
theory of the oar, or rather on the same principle, we should 
regard this as the fulcrum, the resistance acting at the shaft 
or axis of the wheel. It would follow that when the 


Influence of Condition on Vapour-pressure. 61 


crank is in the lower part of its course, the power would be 
acting against the motion of the ship. Itis perfectly obvious 
that the true fulcrum (from a practical point of view) is at 
the shaft, and the resistance in the water. If we choose to 
treat the point of instantaneous rest as the fulcrum, we must 
introduce complications which will only result in bringing us 
back to the simple practical view. 

Another analogous case is that of a bicycle rider. I need 
not dwell on this, as, mutatis mutandis, the same considera- 
tions apply as in the case of the steamer. 

The true theory of the oar has been discussed recently in 
an appendix to a pamphlet entitled Ausa Dynamica, published 
under the pseudonym of ‘John O’Toole”*. The author 
remarks that ‘‘In certain emergencies it might be highly 
desirable for the captain of a vessel to know that the ‘power’ 
of an ordinary oar-lever is at a mechanical disadvantage. If 
he is unaware of this, or, still more, if he believes the opposite, 
he may send out an insufficient boat-crew to tow his vessel 
out of danger. It is highly probable,” he adds, ‘ that vessels 
have been actually lost in this way.” 


VIII. Influence of Change of Condition from the Liquid to the 
Solid State on Vapour-pressure. By W. Ramsay, Ph.D., 
and SypNEY Youne, D.Sce.t 


qe Wiedemann’s Annalen, vol. xxviii. p. 400, W. Fischer 
has published a paper on the above subject. After stating, 
in the course of a historical sketch, that no experimental 
work with a view to decide whether the vapour-pressure of a 
solid is identical with that of its liquid at the same tempera- 
ture below the melting-point of the solid has been carried out 
since Regnault’s time, he corrects himself in a footnote in 
which he refers to a paper published by us on this subject in 
the Philosophical Transactions in 1884. He there states, 
however, that he gained a prize through some work on the 
vapour-pressures of water and of benzene in 1883. Now it is 
generally understood that priority is determined, not by the 
date at which work is done, but by the date of publication ; 
and as Fischer’s work was not made public until July 1886, 
there can be no question of priority between us. But in the 
short sentence on our work he commits a grave error in stating 


* Dublin: Hodges and Figgis. 
+t Communicated by the Physical Society: read December 11, 1886. 


62 Drs. Ramsay and Young on the Influence 


that our work was merely qualitative ; on the contrary, it was 
rigorously quantitative ; and we then showed that the num- 
bers calculated for the vapour-pressure of ice, using as data 
an extension of the vapour-pressures of water below 0°, ex- 
trapolated from Regnault’s measurements above 0°, his deter- 
minations of the heat of vaporization of water and fusion 
of ice, and of the specific heat of ice, agreed closely with 
those found by us. As regards benzene, however, our work 
had no pretence to be quantitative. 

Fischer’s experiments were made by a process identical in 
principle with that employed by Regnault. The water, or 
benzene, on which he experimented, was introduced, by a pro- 
cess well devised for excluding air, into a vacuum connected 
with a gauge, and on alterating the temperature of the liquid, 
alteration of pressure was noted and registered. His experi- 
mental results are very regular, and, so far as water and ice 
are concerned, confirm ours, and agree well with theory. 

But although Fischer’s experimental results with benzene 
are equally regular, yet they present certain anomalies which 
are difficult to explain. From his results he calculated con- 
stants—one series to represent the relations of the pressure of 
vapour in contact with liquid, and the other to represent 
similar relations for vapour in contact with solid, employing 
formule of the general form p=a+lt+c’?. From the num- 
bers calculated by means of these constants, he concluded that 
the vapour-pressure of liquid benzene is not identical with that 
of solid benzene at the melting-point of the solid. This conclu- 
sion is evidently opposed to the second law of thermodynamics ; 
and, if it had not been apparently supported by Fischer’s really 
excellent experimental measurements, might have been dis- 
missed at once as absurd. But on revising Fischer’s results, 
we find that the constants employed by him, if used to calcu- 
late the vapour-pressures of the solid at low temperatures, 
give results which are by no means in accordance with his 
measurements. Indeed at —8° the calculated pressure is 
13°51 millim., whereas Fischer found 14°2 millim. ; and it is 
evident, from a graphic representation of his results, that the 
divergence would increase at lower temperatures. Now it is 
known that the relations of pressure to temperature are better 
expressed by means of a formula of the type suggested by 
Biot, p=a+ba'+cB*; or, for a small range of pressure, by 
the simpler form p=a+ba'. On calculating constants from 
Fischer’s results by means of this formula, we found that while 
a curve was obtained agreeing better with his experimental 
results, the anomaly which he supposed (viz. want of coinci- 


of Change of Condition on Vapour-pressure. 63 


dence at the melting-point) no longer existed. As in the case 
of water and ice, solid and liquid acetic acid, solid and liquid 
bromine, and solid and liquid iodine, which have formed the 
subject of our experiments (Phil. Trans. 1884, p. 461, and 
Trans. Chem. Soc. 1886, p. 453), solid and liquid benzene 
exert the same vapour-pressure at the melting-point. 

For recalculating Fischer’s results, his pressures at —7°, 
—2°, and +3° were taken. At —7°, the vapour-pressure of 
the solid determined by him was 15:273 millim.; at —2°, 
21:°679 millim.; and at +3°, 30°324 millim. The constants 
are a=4°81664; log b=0°5602315; log a=1:99628446; b is 
negative; and ¢ = temperature centigrade +7. Jor the 
vapour-pressures of liquid benzene, Fischer’s results agree well 
with his formula. As data are in existence whereby the 
vapour-pressures of solid benzene can be calculated for a short 
interval of temperature below the melting-point, provided 
those of the liquid are accurately known, it was deemed ad- 
visable to check Fischer’s results with liquid benzene by the 
dynamical method already described by us (Trans. Chem. Soc. 
1885, p. 42). 

This was accordingly done. A large quantity of commer- 
cial benzene was distilled, and the first half, boiling within 
five degrees, was frozen twice, the liquid portion being poured 
off each time. The solid portion was then shaken repeatedly 
with sulphuric acid until the acid was no longer coloured, 
thiophene being thus removed. The remainder was shaken 
with water and dried, and then fractionated until a product of 
constant boiling-point was obtained. The actual boiling-point, 
at a pressure of 753°4 millim., was 79°°9. 

The following determinations with the liquid were made. 
At the lower temperature, thermometers graduated in yy divi- 
sions were employed. ‘The zero-points of these thermometers, 
and the apparent lowering of temperature consequent on 
reduction of pressure, were determined, and corresponding 
corrections introduced. For higher temperatures (above 50°), 
a thermometer previously used for determination of vapour- 
pressures, and of which the corrections had been thoroughly 
investigated, was used. ‘The three +), thermometers we shall 
name A, B, and C; the one used at higher temperatures, D. 


64 Drs. Ramsay and Young on the Influence 


Liquid Benzene. 


Series I. Series IT. Series ITT. 
Temp. Pressure. || Temp. Pressure. || Temp. Pressure. 
2 millim. ts millim. 4 millim. 
A. —1°84 24°0 A. 5°30 35°05 || D. 31:4 125°85 
—0:97 25°1 6°25 37-2 33°6 137:9 
— 0:02 26°38 7:03 38°45 30° 150°8 
+0:98 27°9 CAL 39°85 44-7 2210 
1-23 28°15 7:98 40:9 50°1 267°0 
2:03 29°15 8°37 41°38 51°85 287°4 
3°08 30°95 9:21 43:0 - 54°25 313°9 
313 31:15 10°44 47:25. 56°4 339°5 
4°84 34-7 12°51 51:65 58°38 371°4 
5-06 35°2 14-66 57°8 61°2 402-7 
5°31 30°4 16°85 64'5 63°6 440:0 
5°48 35°9 18°72 70°75 67-2 497-6 
5°74 36°15 20°54 77°25 69°25 534°6 
B. 5°63 86°15 22°97 85°5 71°85 582°0 
A. 8°98 42'8 25°37 96°4 741 630°3 
B. 877 42°8 28°15 109°35 76°65 684°6 
30°87 123-45 79°6 743:1 
38°78 173°0 
39°35 174:95 
41°41 190°8 
43°71 209°6 
45°97 230°0 
47-94. 249-1 
49°36 263°3 


A curve was drawn to ‘represent these relations ; and from 
it three points were chosen, viz. 0°, 40°, and 80°; the corre- 
sponding pressures are: 0°, 25°54 millim. ; 40°, 180-2 millim.; 
and 80°, 755°0 millim. The constants for the formula 
p=at+bat are 

a=4:72452; log b=0°5185950 ; log «=1:996847125 ; 
b is negative. 


The following Table shows the calculated vapour-pressure 
for each 10°. 


Temperature. | Pressure. |/Temperature.| Pressure. 


ee | a ee 


0 millim. Ps millim. 
—10 14:97 40 180-20 
0 26°54 50 268°30 
10 45°19 _ 60 388°51 
20 74:13 70 548:16 


30 117-45 80 7550 


of Change of Condition on Vapour-pressure. 65 


These numbers agree fairly well with Regnault’s results at 
and above 10°, Below that temperature his constants are 
calculated from the vapour-pressures of the solid as well as of 
the liquid, and of course are therefore incorrect. 

We give a Table of comparison of our calculated results, 
with those calculated by Fischer, between 0° and 6°. 


Pressure. Pressure. 
ee emp 
de R. and Y. FE. R. and Y. 
= millim. millim. J millim. millim. 
0 26°40 26°54 4 32°84 82°99 
1 27:87 28-04 5 34°68 34°80 
2 29°43 29°61 6 36°60 36°69 
3 31:10 31:26 


lt will be seen that the agreement is a very close one ; and 
as our determinations were made by the dynamical method, 
while Fischer’s were obtained statically, there is a strong pre- 
sumption that the substance in both cases was pure. 

In order to calculate the vapour-pressures of solid from 
those of liquid benzene, the following formula was employed:— 


Vapour-pressure of solid at (¢(—1)=P,—(P’,—P’ )(3**5)), 
t—4 
P =vapour-pressure of solid ; 
P’=vapour-pressure of liquid ; 
V =heat of vaporization of liquid ; 
F =heat of fusion of solid ; 
¢ =temperature of solid and liquid. 


It is therefore necessary to know the heat of vaporization 
of liquid benzene at different temperatures, and the heat of 
fusion of solid benzene ; and in order to calculate these, the 
specific heats of liquid and of solid benzene. The following 
determinations are available :— 

1. Heat of Vaporization of Liquid Benzene.—Regnault 
(Mémoires de ’ Institut, xxvi. p. 881) has determined the total 
heats of vaporization of benzene at different temperatures, 
while Schiff (Annalen, ccxxxiv. p. 344) has made a single 
determination at the boiling-point. Regnault’s formula is 


H=a+bt+ ct’, 


where a=109, 6=0°24429, and c=—0°0001315. Schiff’s 
Phil. Mag. 8. 5. Vol. 23. No. 140. Jan, 1887. F 


66 ‘Drs. Ramsay and Young on the Influence 


single determination at the boiling-point, 80°35, at a pressure 
of 765°1 millim., is 93°4 to 93°5 calories. 

2. Heats of Fusion of Solid Benzene.—Peterson and Wid- 
mann (J. prakt. Chem. xxiv. p. 129) give the number 29°09 
calories; and Fischer (loc. cit.) found 30°085 calories. 

3. Specific Heat of Liquid Benzene.—This has been often 
observed. But isolated observations are for our purpose 
comparatively valueless, for the specific heat varies with the 
temperature. Formule are given only by Schiiller (Pogg. 
Ann., Erginzungs-Band, p. 5), and by Schiff (oe. cit.). We 
have used Schiff’s formula. It is, specific heat=a-+ bt; where 
a= (3834 and 6=0:001048. Between narrow limits of tem- 
perature this may be accepted as sufficiently correct. 

4. Specific Heat of Solid Benzene.—Fischer (loc. cit.) gives 

0°319. 
Calculating the heat of vaporization at the boiling-point 
under normal pressure from Regnault’s total heats of volatili- 
zation and Schiff’s specific heat of liquid benzene, the number 
93°67 is obtained; while Schiff found by direct measurement 
93:4 to 93°35. This is a strong presumption in favour of the 
correctness of the data. 

The mean of the two determinations of the heats of fusion 
of benzene was taken. 

The following Table summarizes the data for calculating 
the vapour-pressures of the solid. But this calculation in- 
volves the assumption that the heat evolved on solidification 
at any low temperature is equal to that evolved at the ordinary 
melting-point, minus the product of the specific heat of the 
solid into the difference of temperature; and that the specific 
volume of the vapour in contact with solid is equal to that of 
vapour in contact with liquid. It is certain that neither of 
these assumptions is true; hence it is not legitimate to calcu- 
late the vapour-pressures of the solid from those of the liquid. 
Still, for some degrees below the melting-point, the error 
involved in these assumptions is probably not very great. 


Ve—3t Fe—-3 


Temp.| P’. |P’s—P’t-1.| Ve—s. | Fe—-2. We cee P,P Ay ©. 
‘4 mm, mm. calor. | calor. se 
5°58 | 8589) 1.96 =| 108:3| 294/ 1-271 2866. [Poe 
458 | 8403) 17g | 108-4| 291 | 1-268 pope si) ee 
3:58 | 8225) 169 | 1085| 288 | 1-265 2139) | oe 
2°58 | 30°56) 1.62 | 1086] 28:5] 1-262 9045 | 2218 
158 | 28°94) 1-54 | 1088] 282/ 1-260 1oay* } 2808 


| 0-58 27-40 one 


67 


__ The vapour-pressures of solid benzene, determined by our 
method, are given in the following Table. As Fischer’s method 
was statical, while ours is dynamical, a comparison of the 


of Change of Condition on Vapour-pressure. 


results of both is therefore given. 


Temp. 


Pressure. | Temp. Pressure. | Temp. Pressure. 
Srrizs I. Serizs IT. 
Sertrs I, (cont.). (cont.). 

millim by millim, é moillim. A 
35°9 +5°43 21:8 —0-95 14:0 — 7-02 
35°4 5:21 21-4 —112 13°25 — (37 
35'2 5:07 21:0 —1:45 13°10 — 753 
34-7 4:89 19°95 —2:00 12:80 — 842 
31°15 3°62 19-0 — 2°54 116 — 877 
30°95 3°32 19:0 —2°97 11-2 — 9:90 
29°15 2°70 18-05 —2:98 10-2 — 972 
29°15 2°75 16°3 —4°63 10-1 —10°54 
28°7 2°41 16°25 — 5:45 9:95 — 11:03 
28°15 2°21 13°8 — 6°43 7°35 —1412 
27:9 1:99 12°35 — 8-07 
26°8 1:29 Serizes IIT, 
26°7 £33.43) £ 14-25 — 701 
26°65 SPAN er 140 | — 770 
25°9 OS ng. 219 — 1:26 10-4 —11:2 
25°1 0°64 211 —1-60 10-2 -—11-0 
24°45 0:23 20:05 —2°61 10-2 — 10°63 
24:0 0:0 18°75 —3:08 9:95 — 11:62 
23°9 —0-03 17:15 —4:38 9:3 —11-92 
22°75 —0°78 15-2 —5°78 9°3 —11:3 
22°2 —0°91 15:0 —6°02 87 —12:12 


For the first two series thermometers A and B were used ; 
for Series III. another thermometer, D ; and it will be seen 
that its readings confirm those of the other two. The indi- 
vidual results are not so concordant as Fischer’s. The 
reason is that the volatilization of solid benzene is so quick 
as to make it difficult to obtain an accurate reading before 
the solid has volatilized and partially exposed the thermo- 
meter-bulb. These numbers, near the melting-point, show 
close concordance with those of Fischer, but at lower tem- 
peratures they show signs of divergence. For example at 
— 8:42, if Biot’s formula be applied, the difference between 
Fischer’s results and ours amounts to 1°39 millim. ; and it is 
evident, from the graphical representation, that the difference 
would be an increasing one. It is to be noted that Biot’s 
formula agrees with Fischer’s own results much better than 
the formula employed by him. 

Our results were plotted on curve paper, and the constants 
for a formula calculated from points on a curve drawn to 
pass well through them. The constants are, for the formula 

F 2 ' 


68 Influence of Condition on Vapour-pressure. 


p=a+tba’, a=4-82602 ; log=0°5784772 ; log a=1-9959086; 
t=t° Centig.+10; 6 is negative. 

The following table comprises (A) the values calculated 
from the vapour-pressures of the liquid ; (B) those calculated 
from the formula and constants given by Fischer; (C) those 
calculated from Fischer’s results by Biot’s formula ; and (D) 
those calculated from our results by Biot’s formula. 


Temperature. A. B. C. D; 

55 millim. | millim. | millim. | millim. 
5°58 35°89 | 35°62 | 35°85 | 35°86 
4°58 33°52 | 33:36 | 33°62 | 33:39 
3°58 31:27 | 31:42 | 31:50 | 31:07 
2°58 29:13 | 29:54 | 29:50 | 28°00 
1-58 27°08 | 27-73 | 27-61 | 26°85 
0:58 25:14 | 25°97 | 25°82 | 24-94 

es 2 A | ar <tee 24:28 | 2414 | 23:14 
GSD ys, tlle sate oe 22°65 | 22:55 | 21:46 
AZT Dalai ieee 21:09 | 21:06 | 1989 
DADs) Al tedon teat 19:59 | 19°66 | 18-41 
2 ee Ie || one 1815 | 1833 | 17-04 
DA a 16-77 | 17-09%)" tae 
CADE: Se then 1545 | 15:92 | 14:56 
Wad ANS eee: 14:20 | 1482 | 13°44 
ro) 924MM | Re ae 13:01 | 13°79 | 12-40 
oo RMS a hg OA 11-89) 12°82) 7 hieas 

VODs «SOPRA rh ie eee a| veces 10°53 
LEAD... Me TR ae eee ieee 9°69 
LA: DM RAS NO ear BOR 506 8°91 


It is possible to calculate the heat of fusion of solid benzene, 
Dt the melting-point, from the equation 


dt 
ap pe / Cp ee san): Veen 
7g Solid) it guid) — ae 


where V is the heat of vaporization at the melting-point, and 
F the heat of fusion. 


With Fischer’s formula and constants . F= 6:29 calories 
With Fischer’s results and Biot’s formula, F=21:1 
With our results and Biot’s formula. . F=35-4 


from the value of 


The number found by Fischer. . . . F=30:085 _ ,, 
The number found by Peterson and 
Widman HG ge , L200 


3 

From these numbers it is evident that, although our con- 
stants are not perfectly correct, yet they agree better with 
experimental evidence than those of Fischer. 


[ 69 J 


IX. Proceedings of Learned Societies. 


GEOLOGICAL SOCIETY. 
[Continued from vol. xxii. p. 226. ] 


November 3, 1886.—Prof. J. W. Judd, F.R.S., President, 
in the Chair. 
3 following communications were read :— 
1. “On the Skull and Dentition of a Triassic Saurian, Galesaurus 
planiceps, Ow.” By Sir Richard Owen, K.C.B., F.R.S., F.G.S., &e. 


2. “The Cetacea of the Suffolk Crag.” By R. Lydekker, Esq., 
BlA., ¥.G.8., &e. 


3. “On a Jaw of Hyotherium from the Pliocene of India.” By 
R. Lydekker, Esq., B.A., F.G.S., &e. 


November 17.—Prof. J. W. Judd, F.R.S., President, in the Chair. 

The following communication was read :—- 

“On the Drifts of the Vale of Clwyd, and their relation to the 
Caves and Cave-deposits.” By Prof. T. M*Kenny Hughes, M.A., 
F.G.S. 

The Author divided his subject as follows :—I. Introductory Re- 
marks; II. The Drifts, viz. (i.) The Arenig Drift, (i1.) The St.-Asaph 
Drift, (iii.) The Surface-Drifts; III. The Caves, viz. (i.) The Caves 
themselves, (ii.) The Cave-Deposits ; IV. Conclusion. 

He exhibited a table showing the tentative classification he pro- 
posed. II. (i.) The Arenig Drift, he said, might be called the 
Western Drift, as all the material of which it was composed came 
from the mountains of Wales; or the Great Ice-Drift, as it was the 
only drift in the Vale which contained evidence of direct ice-action. 
He traced its course from the Arenig and Snowdon ranges by striz 
on the solid rock and by the included fragments, a large proportion 
of which were glaciated. There are no shells in this drift. 

IJ. i.) The St.-Asaph Drift might, he said, be called the Northern 
Drift, as it was the deposit in which fragments of north-country 
rocks first appeared ; or the Marine Drift, as it was, excepting the 
recent deposits at the mouth of the estuary, the only drift in the 
Vale which showed by its character and contents that it was a sea- 
deposit. It contained north-country granites, flints, and sea-shells, 
of which he gave lists. Most of them are common on the adjoin- 
ing coast at the present day, a few are more northern forms. 
None of the rocks are striated, except those derived from the 
Arenig Drift (i.). . 

II. (ii.) The Surface-Drifts included the older and newer alluvia 
of the rivers, the Morfa Rhuddlau Beds or estuarine silt, the recent 
shore-deposits or Rhyl Beds, and all the various kinds of deposits 
known as talus, trail, rain-wash, head, run-of-the-hill, &c., of which, 
in so Jong a time, very thick masses have accumulated in many 
places. He explained some methods of distinguishing gravels accord- 
ing to their origin. 

Turning to the subject of Caves, he thought they should be careful 


70 Geological Society a 


not to confound (III. i.) the question of the age and origin of the 
caves themselves with (III. i.) that of the deposits in the caves. 
He then described some of the more important caves of the district, 
explaining the evidence upon which he founded the opinion that 
the deposits in Pontnewydd Cave were Postglacial Paleolithic. He 
arrived at the same conclusion with regard to the deposits in the 
Ffynnon Beuno Caves. Combating the objections to this view which 
had recently been urged, he pointed out that the drifts associated 
with the deposits in those caves cannot have been formed before the 
submergence described under II. (ii.), because they contained north- 
country fragments and flints, and that, even if they were of the age 
of the submergence, they would not be preglacial; that they cannot 
have been formed during the submergence, as the sea would have 
washed away the bones &c. from the mouth of the cave, and its 
contents must have shown some evidence of having been sorted by 
the sea. He considered that the greater part of the material that 
blocked the upper entrance of the upper cave belonged to the 
surface-drifts described under II. (iii.), and were, as they stood, 
almost all subaerial. 

He further pointed out that, so far as paleontologists had been 
able to lay before them any chronological divisions founded on the 
Mammalia, the fauna of the Ffynnon Beuno Caves agreed with the 
later rather than with the earlier Pleistocene groups. 


December 1.—Prof. J. W. Judd, F.R.S., President, in the Chair. 

The President announced that he had received from Prof. Ulrich, 
of Dunedin, N. Z., the announcement of a very interesting discovery 
which he had recently made. In the interior of the South Island 
of New Zealand there exists a range of mountains, composed of 
olivine-enstatite rocks, in places converted into serpentine. The 
sand of the rivers flowing from these rocks contains metallic particles 
which, on analysis, prove to be an alloy of nickel and iron in the 
proportion of two atoms of the former metal to one of the latter. 
Similar particles have also been detected in the serpentines. This 
alloy, though new as a native terrestrial product, is identical with 
‘ the substance of the Octibeha meteorite, which has been called 
octibehite. Prof. Ulrich has announced his intention of communi- 
cating to the Society a paper dealing with the details of this inter- 
esting discovery—which is certainly one of the most interesting 
that has been made since the recognition of the terrestrial origin 
of the Ovifak irons. 


The following communications were read :— 

1. ‘On a new Genus of Madreporaria—Glyphastrea, with Re- 
marks on the Glyphastrea Forbesi, Kdw. & H., sp., from the Ter- 
tiaries of Maryland, U.S.” By Prof. P. Martin Duncan, M.B., 
PeBS.5 HiG.S., &e. 

2. “On the Metamorphic Rocks of the Malvern Hills.” Part I. 
By Frank Rutley, Esq., F.G.8., Lecturer on Mineralogy in the 
Royal School of Mines. 

Part I. is the result of conclusions arrived at in the field; Part II. 
will be devoted to a microscopic description of the rocks. 


Metamorphic Rocks of the Malvern Hills. 71 


The author referred especially to the paper by the late Dr. Holl, 
whose work he, in the main, confirmed. Dr. Holl’s object was to 
demonstrate that the rocks which had hitherto been treated as 
syenite, and supposed to form the axis of the hills, were in reality 
of metamorphic origin, and belonged to the Pre-Cambrian. Mr. 
Rutley restricted his observations to the old ridge of gneissic syenite, 
granite, &c., which coustitutes the main portion of the range, and, 
reversing the order of his predecessor, commenced at the north end 
of the chain, 

He considers that the beds of crystalline rock, mostly of a gneissic 
character, in the old ridge have been disposed in a synclinal flexure, 
which stretched from the north end of the chain to the middle of 
Swinyard’s Hill, where they receive an anticlinal flexure, and are 
faulted out of sight. The length of this synclinal fold would be 
over 53 miles. The lithological evidence is in favour of the rocks 
forming the north part of Swinyard’s Hill being a repetition of 
those on the Worcestershire Beacon. We might expect to find the 
older beds having the coarsest granulation, and being even devoid 
of foliation, and this is what occurs on the Malverns, where the 
northern hills are made up of the coarsest rocks, with finer schistose 
beds towards the south ; the exception is at Swinyard’s Hill; hence 
there are two groups of coarsely crystalline rocks at either ex- 
tremity of the presumed synclinal. The contrast between these and 
the fine-grained rocks of the other portions of the range has already 
attracted attention. The most northern of the coarse-grained 
masses is cut off towards the south by a fault near the Wych, while 
the other lies between a fault on the north side of the Herefordshire 
Beacon and the before-mentioned fault on Swinyard’s Hill. 

The metamorphic rocks of the Malverns seem, therefore, to be 
divisible into three series extending from the North Hill to Key’s 
End. A Lower, of coarsely crystalline gneissic rocks, granite, 
syenite, &c.,a Middle, of gneissic, granitic and syenitic rocks of 
medium and fine texture, and an Upper, of mica-schist, finely crys- 
talline gneiss, &c. A diagrammatic section shows the distribution 
of these ; the northern block, extending as far as the Wych, consists 
of the Lower and the lower part of the Middle; the central block, 
from the Wych to the fault in Swinyard’s Hill, consists chiefly of 
the Lower and upper Middle, but with a portion of the Lower at 
the southend. The southern block, south of the fault on Swinyard’s 
Hill, consists wholly of the Upper series. 

How far the foliation of these rocks and their main divisional 
planes represent original stratification must, the author thought, 
remain an open question. It has been held that the strike of foliation 
les parallel to the axis of elevation; but this is far from being the 
case in the Malverns. Still a once uniform strike may have been 
dislocated by repeated faulting. 

The author further discussed the Peierel question of how far 
foliation may or may not coincide with planes of sedimentation. 
He admitted that the absolute conversion of one rock into another 
by a process of shearing has been shown to occur, but doubted 
its application in this case. Although he is inclined to believe 


72 Intelligence and Miscellaneous Articles. 


that the divisional planes, with which the foliation appears to be 
parallel, may be planes of original stratification, yet, as a matter of 
fact, they are nothing more than structural planes of some sort, 
between which the rocks exhibit divers lithological characters. 

3. “On Fossil Chilostomatous Bryozoa from New Zealand.” By 
Arthur Wm. Waters, Esq., F.G.S. 


December 15.—Prof.J. W. Judd, F.R.S., President, in the Chair. 


The following communications were read :— 
1. “Notes on Nummulites elegans, Sow., and other English Num- 
mulites.” By Prof. T. Rupert Jones, F.R.S., F.a.8. 


2. “On the Dentition and Affinities of the Selachian genus 
Piychodus, Agassiz.” By A. Smith Woodward, Esq., F.G.S. 

3. “ On a Molar of a Phocene type of Hquus from Nubia.” By 
R. Lydekker, Esq., B.A., F.G.8. 


X. Intelligence and Miscellaneous Articles. 


ON A NEARLY PERFECT SIMPLE PENDULUM. 
To the Editors of the Philosophical Magazine and Journal. 


GENTLEMEN, 
N an article by Mr. Thomas Gray, communicated a few days ago 
to the Philosophical Magazine *, he has described experiments, 
carried out in conjunction with myself, on the oscillations, in a 
Sprengel vacuum, of a torsion vibrator hung by a single silk fibre. 
The performance of the tiny vibrator were so remarkable, that we 
are proposing to carry the experiments farther ; but in the mean- 
time they have led me in a somewhat different direction. With 
the assistance of Mr. Gray, I have suspended a small shot a little 
more than =, inch in diameter by a single silk fibre (half a cocoon- 
fibre) two feet long. This I have placed ina glass tube about three 
quarters of an inch in internal diameter, and have exhausted with 
the Sprengel pump to about 0-1 [¥J (one tenth of a millionth of an 
atmosphere). A most perfect ‘simple pendulum ” is thus obtained ; 
and I find that starting it with a vibrational range of 4 inch on 
each side of the middle position, the vibrations are very easily 
countable at the end of 14 hours. 

The weight of the lead shot used is only 3 gramme ; and a single 
silk fibre will bear nearly three grammes. Iam proceeding to make 
a ‘‘seconds ” pendulum, 39-1 inches long, with a heavier weight 
than that used at present, and it will be enclosed in a much better 
vacuum than 0:1 f¥J. With such a pendulum, I hope that I may 
obtain a still slower subsidence; and I propose to find, if possible, 
whether the subsidence observed is due to residual air, or to 
viscosity of the fibre. Your obedient servant, 

The University, Glasgow, J.T, BorroMnLeEy, 


December 14, 1886. 
* See p. 46. 


THE 


LONDON, EDINBURGH, anp DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[FIFTH SERIES.] 


FEBRUARY 1887. 


XI. The Determination of the Constitution of Carbon Com- 
pounds from Thermochemical Data. By Henry KH. 
Armstrone, F.R.S., Professor of Chemistry, City and 
Guilds of London Institute Central Institution *. 


| Mies past year has witnessed the publication of the 
fourth and concluding volume of Julius Thomsen’s 
Thermochemische Untersuchungen, an event of high import- 
ance on account of the magnitude of the work and of the 
reputation for accuracy which its author enjoys. Indeed the 
unwearying perseverance and great manipulative skill which 
have enabled him to accumulate the mass of data now pub- 
lished in a collected form must excite the admiration and 
wonder of all who are able to judge of the difficulties to 
be overcome in the execution of thermochemical determi- 
nations. 

In the present article I desire to direct attention to the 
fourth volume, which is wholly devoted to the description of 
determinations of the heat generated on combustion of no less 
than one hundred and twenty distinct carbon compounds— 
members of about twenty different groups, and also to a 
theoretical discussion of the results and their bearing on 
received views of the constitution of carbon compounds. As 
many of the conclusions are directly opposed to, and irrecon- 
cilable with, popular views, it is desirable that they should be 
brought more directly under the notice of British chemists, 
and that the interpretations which Thomsen puts upon his 
results should be carefully considered and criticised. I may 


* Communicated by the Author. 
Phil. Mag. §. 5. Vol. 23. No. 141. Feb. 1887. G 


74 Prof. H. E, Armstrong on the Determination of 


also add that I have been led to take a special interest in the 
work, as many of the results appear to me to lend considerable 
support to my own views of the nature of chemical affinity, 
serving to confirm the hypothesis that the affinity relations 
of the negative elements are altogether peculiar, emphasized 
by me in my Address to the Chemical Section of the British 
Association at Aberdeen in 1885, and more fully developed 
in a recent paper on ‘ Electrolytic Conduction in relation to 
Molecular Composition, Valency and the nature of Chemical 
Change: being an attempt to apply a theory of ‘ Residual 
Affinity ’” (Proc. Roy. Soc. 1886, vol. xl. pp. 268-291). 

(1) Manner in which Results are stated.—All the heats of 
combustion of the carbon compounds that will be referred to 
are calculated on the assumption that the substance burnt is 
in the state of gas at 18° C.,and that the products are gaseous 
carbon dioxide and liquid water at this temperature ; they are 
stated in gram-°C. units. The symbol / stands for ‘heat of 
combustion,” and when prefixed to a symbol or formula it 
denotes the heat of combustion of the weight in grams corre- 
sponding to the symbol or formula used. 

The heat of formation of a compound is the difference 
between its heat of combustion and the heat of formation of 
the products of combustion ; the heat of formation of carbon 
dioxide being taken as 96960 units and that of liquid water 
at 18° as 38360 units. The value thus calculated is the heat 
of formation at constant pressure ; that at constant volume is 
deduced by subtracting from it (n—2)x 290, n being the 
number of atoms of gaseous constituents which go to form 
the molecule of the compound. For exampie, 7. CH,=211930 
units. The heat developed in the combustion of the same 
quantity of the contained elements would be 96960 + 2. 68360 
= 233680 units, the difference between which and /. CH, is 
233680 —211930= 21750 units, being the heat of formation 
of methane at constant pressure ; as four atoms of its gaseous 
constituent go to form a molecule of methane, 21750—580= 
21170 units is the heat of formation of methane at constant 
volume. 

In previous publications Thomsen has disregarded the 
molecular composition of the elements, and has used equa- 
tions such as 


C,O.= 96960 units, H,,O=38360 units ; 
meaning thereby that in the formation of the compound CO, 
from the elements carbon and oxygen, or of the compound 


H,O from the elements hydrogen and oxygen, the specified 
amounts of heat are developed, the comma being used by him 


the Constitution of Carbon Compounds. 79 


always to denote combination of the elements between which 
it is placed in the proportions indicated by the symbols. This 
disregard of molecular composition has undoubtedly served to 
mislead. But the discussion of the amounts of energy to be 
expended in the separation of atoms is an important part of 
the present work ; and for this purpose Thomsen introduces 
the practice of representing atomic proportions of elements by 
means of small letters. Thus C is used by himas representing 
carbon as we know it, and c to indicate a carbon atom; so that 
if f.C denote the heat of combustion of ordinary (amorphous) 
carbon, 7. ¢ is to be taken as denoting the heat of combustion 
of carbon in the atomic condition. ‘This appears to me to be 
a most undesirable and unnecessary practice, and of set pur- 
pose I have not adopted it, but have preferred instead to write 
the equations so as to indicate the molecular composition of 
the gases dealt with. It is far better to use C, as the symbol 
of carbon of unknown molecular composition and thus remind 
the reader of our ignorance; and Q, is the proper symbol of a 
monatomic carbon molecule *. 

(2) Difference in the Heats of Combustion of Homologues.— 
The first noteworthy conclusion arrived at by Thomsen is that 
the heat of combustion increases from term to term in a series 
of homologous compounds by an almost constant amount: on 
an average by 157897 units. Of the forty-four differences 
cited (Table I.), the minimum is 155120 and the maximum 
159563, only five departing by more than one per cent. from 
the mean +. 


* The failure to realize the importance of molecular composition in 
relation to chemical change, which unfortunately is so clearly traceable 
in our chemical literature, probably is in a great measure due to our want 
of system in writing the formule of elements. I have more than once 
urged that an index should be used after the symbol of an element only 
to denote the number of its atoms in the molecule. If this practice be 
consistently followed and the index be used also in the case of monatomic 
elements, the absence of an index after the symbol of an element would 
at once serve to indicate that the molecular composition of the particular 
element (as gas) is unknown. 

+ It will be noted that all the compounds included in the Table are 
members of the paraffinoid class; also that the comparison is made be- 
tween closely related bodies, a large majority being true homologues, i. e. 
bodies of the same type formed from one another by the mere introduc- 
tion of CH, in place of H. If the heats of combustion of the benzenes 
examined by Thomsen be compared, it will be observed that whereas the 
difference between benzene and toluene 


C,H,—C,H,=955680 —799350 = 156330, 
is very near the average value (157897), the difference between toluene 
and pseudocumene or mesitylene, 


1281510—955680 =2 . 162915, [continued, p. 78. 
G 2 


76 Prof. H HE. Armstrong on the Determination of 


TABLE I, 


Paraffins. 


1 Uo] 1: Me ea ee es 1 CEL) onisecceeeecdeceer 211930 

TB LORS Yeo mee ere SI Oe Mire Oe CH, .CH, 370440 1.158510 

PROPANE) ca) uaa ae enone CHE(CE.). 529210 2.158640 

Trimethylmethane ......... CO TICOEIE ) ssesmecnanee 687190 3.158420 

Tetramethylmethane ...... CGH) hss ce bok 847110 4.158795 

DisOpLOpy!: spike ape acces C,H,(CH,;), 999200 5.157454 
Olefines. 

itlvlene Bs 2.5. Caccuapee varies Cpt CH, cocan settee 333300 

PEP PYIENS jsp coer ep chemedec ss CH,.CH.CH, ...| 492740 1.159390 

Hsobutyleme ie 0cc... sec os CH, . C(CH,), 650620 2.158635 

Trimethylethylene ......... CH(CH,). C(CH;).| 807630 3.158098 
Acetylenes, 

icetylenie tui. te kceaccsiercseves CH “CH On es 310050 

Ali Vlene CoP wach wss maces CH.C.CH 467550 1.157500 


Haloid Compounds, 


Methyl chloride. ............ CHCl par ive bide 164630 

Ethyl ra BETA 3t CH, . CH,Cl 321930 1.157800 

IE IRO) 0 ecard aaa 5 Se CH,.CH,.CH.Cl.| 480200 2.157785 

Usobuiyl (50 Ff es.\ioedeket (CH,),CH .CH,Cl.| 637910 3.157760 

Vinyl sig: aphtes se caeeeeee CH, . CHCl 286160 

Monochloropropylene ...... CH,.CCl.CH, ...| 441190 1.155685 

euliy chloride. 3...bo.ceeeee CH,.CH.CH,Cl..| 442500 \ ; 

Ethylene chloride............ CH,Cl. CH,Cl 272000 

Ethylidene chloride ......... CH, . CHCl, 272050 

Chlloracetol” irr 2 caeecee CCl,(CH,), 429520 1.157495 

Methyl bromide ............ CHAISE Peo mernccter 184710 

Ethyl pp ant SOA eee eee CH, .CH,Br 341820 1.157110 

PrOpyl itis Uh: eee eee GH. CH,:CH,Br.| 499290 2.157290 

Methyl iodide: i302 Gale. CHI APR ca aM 201510 eee 

Ethyl bf), ereeebe Ree CH, .CH,I 359160 1.157650 
Alcohols. 

Methyl alcohol . 2.0; 7acere CH OHS Best uet 182230 

By tr | ns oe C,H,.OH 340530 | 1.158300 

MAGOP VA si, shes cee enero @3H.. OH: 498630 2.158200 

HBObUEyE yee tas reac C,H, . OM ix. ines te 658490 3.158753 

oammayl 208 Oe ae CA Oe 820070 4.159460 
Ethers. 

Dimethyl ether............... (OH. );O Ratton 349360 

Methylethyl ether............ OBZ OSC 505870 156510 

Diethyletwerss 2. .scs ccc sn. (Clo) OA ce esee 659600 155120 


the Constitution of Carbon Compounds. — 77 


Table I. (continued). 


te Diffs. 
Aldehydes and Ketones. 
mectaldehyde. ........55...--5- OH. COM |. cccix.:. 281900 
Propionic aldehyde ......... CA COUP acad.ss 440720 1.158820 
Nsobutyrie?’! fpr) wikis. CHL (COB MAL 599940 2°159020 
GEIS OTE Sea Rs ee COUCH): dosccs.2 437250 
Methylpropy!l Ketone ...... CH, .CO;. CH. ~..|, 754190 2.158470 
Acids. 
MORNING ACIC Ste is 2s.cc acess: Te. COOH eeascc: 69390 
AGEING” AT SSeS ener CH, COOH -222 28 225350 1.155960 
Propionic: acid’ <........02.%>- CH. COOH is: 386510 2.158560 
Ethereal Salts. 
Methyl formate............... EE, COOCE er cet2z2 241210 
Fe PeSeebAte: fod o.e scl oet.. CH, . COOCH: 23232: 399240 1.158030 
fr eproplonate-”.<102..: C,H,.COOCH, ...| 553950 2.156370 
aren isobuty trate ..<..cc. C,H,.COOCH, ...| 716940 3.158576 
Hthyl formate .........:...-- EE. COO@GL He + 2es5h 400060 1.158850 
EOP Ye GN aa. cca vabtnss He. COOG, Hi tt22: 558800 2.158795 
ISS UL WES Se es Se FE, COOOAG# 2:22 719900 3.159563 
Dimethyl carbonate ......... CO(OCHe \erecccez3e: 357570 
Diethyl Ny kesseeones CO(OOsHe) i ses2535a: 674100 2.158265 
Sulphides. 
Methyl sulphydride ......... OTs SI Serna tise sce: 298810 
Ethyl air ghae Le (On) 5 Pec 5 Bae eee nar 455650 1.156840 
Dimethyl sulphide ......... S| (8) 3 1) same eee 457350 
Diethyl “inde pice reees SUCEE ys serssaincacs 772170 2.157410 
Nitriles, Amines, and Nitro- 
compounds, 
ACELODU PMS y oad. p Vitemicave eee CH SON G avenied. 312140 
ELOpIONIGTILSy F255 3.icsgenete CoH 2 GN jasaciweve: 471450 1.159310 
Meiny lamines.28680 os <ch eins « OL NE argc ahci 258320 
Hthylamine \...0.<.02-..c000- Cs ls eae 415670 1.157350 
IPEOD Yl mie 25 tai. g: sccecslens es © INT hanes 575740 2.158710 
SALAMA hace «acer aiee sacs (OA1 sess ree 890580 4.158065 
Dimethylamine............... (OH) a NE occas ace 420460 
Wrebhylamiine:' =). ..-05.020. 5. (CEHE)E = WES soo 734500 2.157020 
Trimethylamine§ ............ (CHE Nee eee. 582630 
Mriethylanawner’. 2352 ...c045.: (CRE ages see ace nt 1052380 3.156583 
Nitromethane ............... CHEPNOEs... 4850 180900 


Nitroethane ......0..0cc0000- Cle Nooo 337940 | 1.157040 


78 Prof. H. HE. Armstrong on the Determination of 


Important confirmation of the value found by Thomsen for 
z in the equation 


is afforded by Longuinine’s determinations of the heat of 
combustion of propyl, isobutyl, amyl and capryl alcohols in 
the liquid state, which gave as values of # 156400, 156900, 
and 156200 units ; and especially by Stohmann’s observations 
on the homologous phenols *. These are given in the fol- 
lowing Table, the numbers in the third column being each 


TaBie II. 
7. x. 
Phenol Nore sacs eh caso cece BREN. 723659 
5 cae BULGER ss asin Rate eS 726002 
Orthocresol juquid. i. 70 -ece- 0-8 883008 157006 
‘, SOlIG: fsa eeeaee she 879758 156099 
HeMetderasol, liquid fi. vss. 2ne.: <8. < 880956 154954. 
Raracresol liquid. i be)oc nese sea 882900 156898 
“ HONEA Gives sono edkeee 1 880441 156782 
Orthoxylenol, solid ............35-: 1035434 155887 . 2 
Metaxylenol, liquid.................. 1037499 155748 . 2 
Paraxylenol,solid ....5.-:i:ss42 9e4- 1035638 155989 . 2 
Pseudocumenol, solid ............... 1191451 155931 . 3 
Carvacrol, liquid.......... aaa See ne 1354819 | £157204.4 
Why), Liquighs.s.. 020 s0-0. ag obese 1353790 em. 156937 . 4 
Gt ONG ses i oust alee pe teen 1349982 156581 . 4 
Resorcinol, solid ............eeeeeeee 670780 
Orncimolmsoltd) ike hawt ce eevee 824724 1538944 
Means dccciv je at oie aon ence Gee eae ante 156152 


is considerably above the highest of the forty-four differences in Table I. 
Thomsen himself, however, calls attention to the probable inaccuracy of 
the number for pseudocumene and mesitylene, the determination being 
very difficult in the case of bodies so rich in carbon and of such high 
boiling-point. 

It should here be mentioned that Stohmann, Rodatz and Herzberg 
(Journ. fiir praktische Chemie, 1886, xxxili. p. 241) have called in ques- 
tion the number 799350. given by Thomsen. These chemists find 787488 
units as the heat of combustion of benzene-vapour at 17°, that of liquid 
benzene at 17° being 779262 units—a value which differs from that given 
by Berthelot (776000 units) only in the ratio of 100-4 to 100, and is 
almost identical with the value found—but rejected—by Thomsen in the 
second of his four sets of determinations of the heat of combustion of 
benzene. It is perhaps noteworthy that the difference between Stoh- 
mann’s value for benzene and Thomsen’s value for toluene, viz. 168192= 
955680 — 787488, is very nearly the same as half the difference found by 
Thomsen to obtain between the heats of combustion of toluene and 
trimethylbenzene. 

* Journal fiir praktische Chemie, 1886, xxxiv. p. 3826. 


the Constitution of Carbon Compounds. 79» 


the difference between the heat of combustion of phenol and 
that of the homologue when both are in the same state of 
aggregation. 

(3) Equality of the Four Affinities of Carbon.—On reference 
to Table I., it will be noticed that on passing from methane 
to ethane or methylmethane, from ethane to propane or 
dimethylmethane, and thence to tri- and tetramethylmethane, 
the heat of combustion increases at each step to the same 
extent: the displacement of hydrogen by methyl, in fact, 
appears always to involve the same development of heat; and 
hence it may be inferred that the four affinities of the carbon 
atom are of equal value. Thomsen also finds that ethylene 
and ethylidene chlorides have identical heats of combustion 
(272000 units); and that there is practically no difference 
between those of allyl chloride, CH,.CH.CH,Cl (442500 
units), and the isomeric monochloropropylene, OH,. CCl. CH; 
(441190 units). These facts, and also the practical identity 
of the heats of combustion of isomeric phenols established 
by Stohmann’s determinations (Table II.), may be adduced 
as confirmatory of the above deduction. 

(4) Heat of Combustion of Gaseous “ Atomic” Carbon and 
the Amount of Heat required to separate “Doubly-linked”’ Car- 
bon Atoms.—On comparing the heats of combustion of bodies 
differing in composition by one or more atoms of carbon 
(Table III.), it is seen that the heat of combustion of carbon 
in its compounds must be greater than that of the ordinary 
amorphous variety of carbon (96960 units). The average 
value deduced from fourteen comparisons is 121085 (in round 
numbers 121090) units, the extremes being 115610 units (the 
difference between ethyl- and allylamine), and 125750 units 
(the difference between diethyl and diallyl ether) —a somewhat 
wide range be.it remarked. 

In the instances given in the Table the comparison is in- 
stituted between a saturated and an unsaturated compound, 
the latter being formed from the former by the addition of a 
carbon atom which, according to the popular view, becomes 
doubly linked with another carbon atom ; the amount of heat 
developed by this “ double-linking”’ of two carbon atoms (vy) 
is the difference between the heat of combustion of gaseous 
“atomic 7 carbon (7. Q,) and the factor 121090, 2... 


4 Ae C,=121090 units + Vp. 


Now if carbon dioxide were capable of combining with an 
atom of carbon, it is to be supposed that it would form an 
unsaturated compound, CO: CO, bearing the same relation 
to it that ethylene bears to methane, and that the heat of 


80 Prof. H. EH. Armstrong on the Determination of 


TaBeE IIT. 
fF: 
Methane © ailj.egss.doscdes. ot CH, 211930 
121420 
Hfliyleme Mi. .30. 04 ee ok C,H, 333350 
MEGANE Le aes: «asectnes saree C,H, 370440 
122300 
Propylene|.:....4s2$..0once CT 492740 
Propane VL) AA C,H, 529210 
121410 
Isobutylene ...scc..sevsh.. 0%: C,H, 650620 
Trimethylmeth C,H 687190 
rimethylmethane pes } 10t46 
Tsoamylene ...............06: C;H,, 807630 
Trimethylmethane C,H 687190 } 
2.12281 
MD EaMlivAl, Mele ceveNeivcideconee scars C,H, 932820 2 
Methyl chloride ............ CH,Cl 164770 
} 121390 
Monochlorethylene C,H,Cl 286160 
Ethyl chloride............... C,H,;Cl 321930 
} 119260 
Chloropropylene ............ C,H;Cl 441190 
Tetrachloromethane CCl, 75930 
} 119140 
Tetrachlorethylene C,Cl, 195070 
Hthyl chloride .....:..:...... C,H,Cl 3219380 
120570 
Allyl chloride ............-.. C,H;Cl 442500 - 
Ethyl bromide............... C,H;Br 341820 
12030 
Allyl bromide’ 2.202:5.i82). C,H,Br 462120 : 
Methylethyl1 ether CH, ..0. @,H, 505870 - 
121 
Methylallyl ether ......... CH, .O. C,H; 627200 j is 
Diethyl ether -........:....0. (C,H;),0 659600 
2.125750 
Digtly Lether,s...---.ee02.c3 (C,H;),0 911100 } 
Hthyl alcohol. ...........2... Cie OF: 340530 
} 124230 
Allyl alcohol........ vanementee C,H; .OH 464760 
HiplyAaM NO) ...5.0.3+07-4+0 2002 C,H, . NH, 415670 
115610 
Milydamrime (15. f2. 14.3325 snot C,H; . NH, 531280 


the Constitution of Carbon Compounds. 81 


combustion of this compound would exceed that of carbon 
dioxide by 121090 units ; whence it follows, the heat of com- 
bustion of carbon dioxide being nil, that the heat of com- 
bustion of the product in question should be 121090 units. 
In point of fact, however, two molecules of carbon monoxide 
are produced by assimilating an atom of carbon with a mole- 
cule of carbon dioxide, the double linkage becoming annulled, 
while at the same time the volume is doubled. In the act of 
expansion 580 units are absorbed (§ 1): hence the heat of 
combustion of the product of the union of an atom of carbon 
with a molecule of carbon dioxide exceeds 121090 units by 
the amount absorbed in the separation of “ doubly-linked ” 
carbon atoms plus 580 units. The heat of combustion of 
2. CO is 135920 units ; therefore 


f . Cy =121090 + v9 = 135920 —580=135340 units, 
It being thus determined that 


f.C,=135340 units, 


it follows that 
= 14250 units. 


(5) Heat absorbed in the production of Gaseous “ Atomic” 
Carbon.—The heat of combustion of gaseous “ atomic” carbon 
being 135340 units, while that of amorphous carbon is 96960 
units, the amount of heat expended in passing from the 
solid amorphous state to the gaseous atomic state will be 
38380 units for each gram-atomic-proportion (12 grams). 
The heat developed in the formation of a compound at con- 
stant pressure from gaseous atomic carbon may hence be cal- 
culated by adding for each atom of carbon in the molecule 
38380 units to the uncorrected heat of formation (§ 1) cal- 
culated for carbon in the amorphous state. All the heats 
of formation subsequently to be given are deduced in this 
manner. 

(6) Heat developed in the Combination of Hydrogen with 
Carbon.—The heat of formation of methane, CH,, at constant 
volume, calculated on the assumption that it results from the 
combination of ordinary hydrogen with gaseous “ atomic ” 
carbon—of two hydrogen molecules with one carbon atom—is 
59550 units ; that of ethylene, C,H,, is 73470 units. Both 
contain the same number of hydrogen atoms, but in the latter 
there are two “doubly-linked”’ carbon atoms—assuming 
ethylene to have the constitution popularly assigned to it: 
the heat of formation of the two hydrocarbons should, there- 


82 Prof. H. E. Armstrong on the Determination of 


fore, differ by vp, 7.e. 14250 units (§ 4). The difference 
73470 —14250=59220 units is the heat developed in the 
combination of two atoms of carbon with two molecules of 
hydrogen ; while the heat of formation of methane—59550 
uvits—is that developed in the union of the same quantity of 
hydrogen with a single atom of carbon. ‘The two values may 
be regarded as identical ; and hence, argues Thomsen, it may 
be concluded that the four atoms of hydrogen are equally 
firmly held, whether they are associated with a single or with 
two carbon atoms *. 

Halving the numbers 59220 and 59550, we have 29610 
and 29775 units as the heat developed in the combination 
of a gram-molecular proportion of hydrogen with gaseous 
atomic carbon. The value 29775 is denoted by Thomsen by 
the symbol 2r. 

(7) Heat developed in the Combination of Carbon Atoms by 
single, double and treble Affinities—The amounts of heat 
developed in the combination of (gram-atomic proportions of) 
carbon atoms by one, two, and three “ affinities’ of each, i. e. 
in the manner in which they are assumed to be associated in 
the paraffins, in ethylene, and in acetylene respectively, may 
be designated by the symbols 2%, vs, v3. 

It has been previously shown that v.-=14250 units. 

As regards the value of v,, the heat developed in the 
formation of ethane, C,H,, results from the combination 
of three hydrogen molecules with two carbon atoms, and 
of these carbon atoms with each other by single affinities ; 


hence : 
(2C, 3H,) =3 . 2r+v,=104160 units. 


*, vj =104160—89325= 14835 units. 


Or, comparing ethane with benzene, and assuming that there 
is one single linkage between carbon atoms in the former and 
nine such in the latter (§ 8), 


(6C, 3H,) =216740=38. 2r+4 9u, 
(2C, 3H.) =104160=38.2r+ 7), 


~. 8v,;=216740—104160=112580 units. 
*, v7= 140738 and 2r=30029. 


* Tt should be noted, however, that the difference between the heats 
of formation of methane and ethylene is the value v,; and that the 
heat of formation being dependent on f.C,, as f.C, and v, are both given 
by the same equation, the calculated and found differences must be 
identical ; Thomsen’s conclusion as regards the relation of the hydrogen 
atoms in ethylene to the two carbon atoms must, therefore, he held to be 
“not proven.” 


the Constitution of Carbon Compounds. 83 


Discussing a series of cases in this way, Thomsen arrives 
at the mean values 27= 380130 units ; 7;=14056 units. 

It will, however, be observed that the value of v, is prac- 
tically identical with that previously found for v2; whence it 
follows that the same amount of heat is developed in the 
combination of carbon atoms, whether they become singly or 
doubly linked ; or, in other words, that there is no difference 
between the two modes of union. 

The heat developed when carbon atoms become trebly 
linked may be deduced from the heats of formation of ace- 
tylene, CH: CH, allylene, CH;.C:CH, and dipropargyl, 
CH: C.CH,.CH,.C : CH, viz. 28990, 74610, and 133080 
units. Heat may be assumed to be developed in their for- 
mation in the manner indicated in the following equations:— 


(20 Hs) = -28990=  2r+ tt, 
(3C, 2H.) = 74610=2 .2r+ 2s, 
(6C, 5H) — 133080= 3 . Qr+ 2v3+32,. 


Substituting for 27 and v, the mean values above given, three 
values of v3 are found, viz. —1140, 294, and 261 units. The 
mean of these is —81 units, a value so small that it may be 
neglected ; and it would therefore appear that the so-called 
treble linking of carbon atoms is unattended with the deve- 
lopment of heat. 

The data recorded by Thomsen afford, in his opinion, 
abundant proof that,a single carbon atom may retain two, 
three, or four others with the same degree of firmness as one. 
The following Table contains several examples, the values of 
v (=v, or v2) being deduced by subtracting the heat due to 
the combination of the hydrogen from the total heat of for- 
mation, 30000 units being taken as the mean value of 27 :— 


TABLE IV. 


v. 
Methylmethane, CH,(CH,)...... 104160 — 90000 = 14160= wv _ 14160 
Dimethylmethane, CH,(CH,),.. 148510 — 120000 = 28510 = 2u 14255 
Trimethylmethane, CH(CH,),.... 193690 — 150000 = 43690 = 8 14563 
Tetramethylmethane, C(CH,),.. 286850 — 180000 = 56850 = 4v 14212 
Bropylene, (C,H. c.csac3 ws.da 117200 — 90000 = 27200 = 2v 13600 
Bpenmene,, Osby .b4i songs de esk 4 216740 — 90000 = 126740 = 9v 14052 


But the one hundred and twenty compounds burnt do not 
afford a single instance from which it can be inferred that 
two carbon atoms are ever held together by a force exceeding 
that equivalent to about 14200 heat- units. 

Hence Thomsen concludes that one carbon atom cannot be 
united to another by several affinities ; and that compounds in 
which the carbon atoms are popularly supposed to be doubly 


84 Prof. H. E. Armstrong on the Determination of 


linked, must, in fact, be regarded as unsaturated; and that this 
is more especially the case with compounds with so-called triple 
bonds. For the present it must be left undecided how the 
carbon atoms are held together in compounds of this last- 
mentioned class; their union is certainly unattended with 
development of heat. This conclusion, Thomsen considers, is 
in agreement with experience : bodies with “ trebly-linked ”’ 
carbon atoms are in a condition of unstable equilibrium, which 
is easily disturbed by external influences, and such compounds 
are easily decomposable ; this could not be the case if the 
carbon atoms were firmly held together (comp. § 24). 

As the force with which carbon atoms are held together in 
gaseous compounds—expressed in heat-units—never exceeds 
about 14200 units, the separation of gaseous diatomic carbon 
molecules into atoms would involve the absorption of only 
this amount of heat per gram-atomic proportion. The much 
greater absorption of heat in converting amorphous into 
gaseous ‘‘ atomic” carbon may be explained by assuming that 
each atom of carbon is combined with several others, that is 
to say, that the molecule of carbon is complex ; if the mole- 
cule contain five or more atoms, each atom may be in direct 
connection with four other atoms: in such a case the number 
of linkages will be twice the number of atoms, and the heat 
developed on combination will be 2.14200 units per gram- 
atomic proportion. The difference between this number and 
38380 units—the amount of heat absorbed in converting 
ordinary amorphous into gaseous “atomic”? carbon—viz. 
9980 units, will be the amount required to gasify the carbon 
as distinct from that absorbed in producing molecular dis- 
ruption. 

(8) Heat of Combustion of Isomeric Hydrocarbons. Con- 
stitution of Benzene.—From the foregoing explanations it 
will be evident that the heat developed in the formation of a 
gaseous hydrocarbon, C,H», from gaseous atomic carbon and 
ordinary hydrogen at constant volume, may be expressed by 
the formula 


(Oz Hy») = 26 r+ dv 
=b. 380000+n. 14200. 


If ordinary amorphous carbon be taken, the formula becomes 


(Ca Hes) =b . 30000 +n. 14200—a. 38380 ; 


while the heat of formation under constant pressure may be 
calculated from the equation 


(C., Hy») =b . 30000-+n . 14200 —a. 38380 + (b—1) 580. 


the Constitution of Carbon Compounds, 85 


The heat of combustion of a hydrocarbon C,H», is expressed 
by the equation 


H 
AAAS & ei Ee 


CEOs: 24t,O; 
v Zz, 


Inyerting the values for and , viz. 96960 


and 68360 units, and simplifying as far as possible, we obtain 
the equation 


f.CHa=a. 135340 +6.37780—n. 142004 580, 


from which the heats of combustion of hydrocarbons generally, 
in the gaseous state, may be calculated. 

By comparing experimental values with those calculated 
from the above equations, it is possible to distinguish between 
alternative formule in the case of hydrocarbons in which there 
may be a difference in the number of “single (or double) 
bonds.”” For example, Thomsen argues that, if benzene has 
the constitution represented by Kekulé’s formula, its heat 
of formation would be 3.380000+6.14200=175200 units ; 
whereas, if each carbon atom be regarded as associated with 
three others (as in the prism formula), its heat of formation 
should be 3.30000+9.14200=217800 units. There is 
thus a difference of 42600 between the two values. Actually 
Thomsen finds 216610*, which agrees well with the higher 
value; and he therefore concludes that the six atoms of 
carbon in benzene are linked together by nine bonds, A 
comparison of the observed and calculated heats of forma- 
tion of chlorobenzene, aniline and methylphenyl ether serves 
entirely to confirm this conclusion. In the case of dipro- 
pargyl, the observed value, 1383080 units, agrees well with 
that calculated on the assumption that it has the formula 

CE; G.. Cy Cie C L@r, 
viz. 
132600=3 .30000+ 3.14200 units. 


Thomsen cites 18 cases of hydrocarbons, in 15 of which the 
heat of formation thus reckoned is within 3 per mille of the 
observed heat of combustion, a degree of accuracy which 
closely approaches that obtainable experimentallyf. 

_ * Compare footnote, § 2, p. 78. 

+ In three cases the difference is somewhat greater— 

Cale. Observed. Diff. 


Diusopropyl) ....'. 2: 281000 287880 6880 
Mesitylene ........ 350400 343010 — 7390 
Pseudocumene...... 350400 348830 — 6570 


But is very difficult to obtain diisopropyl pure, unless a large quantity be 


86 Prof. H. H. Armstrong on the Determination of 


(9) Heat of Combustion of Haloid Compounds.—The heat 
produced by the combination of the halogens may be deduced 
by subtracting from the observed heat of formation of the 
haloid compound from gaseous atomic carbon at constant 
volume (P) that due to the combination of the carbon atoms 
and of the hydrogen with the carbon atoms; the values 
to be assigned to v and A in the case of the haloid com- 
pounds are, however, slightly different from those assigned in 


the case of the hydrocarbons, viz. = 15720, v= fouun. 


As regards the values for chlorine thus arrived at, it will 
be seen, on reference to the last column of Table V., that six 
of the eight compounds containing a single atom of chlorine 
give values of 13180 to 13750 units ; in the case of allyl 
chloride and the isomeric monochloropropylene the numbers 
are somewhat higher, viz. 14560 and 15870 units: hence the 
mean value is 13827 units*. The mean values for compounds 
containing 2, 3, and 4 atoms of chlorine are 33090, 47320, 
and 59050 units respectively. It would appear, therefore, 
that the fixation of the four atoms of chlorine with which a 
carbon atom may combine, involves the development of unequal 
amounts of heat. Thomsen supposes that in the case of two 
of the atoms the heat developed amounts to about 16500 units, 
but to only about 13500 units in the case of the other two. 
Thus : 

ke ie 


= 13500 —13500. Found 13830. 
2 ue = 2. 16500=33000. 33390: 
3 2 = 13500 + 2. 16500=46500. 47420. 
422.1350 +2 . 16500= 60000. 59950, 


As regards the bromine compounds, it is only necessary to 
point out that of the heat developed in the formation of 
methyl, ethyl, and propyl bromide, the portion attributable to 


at disposal; and it is probable that the numbers for the two benzenes are 
inaccurate (compare footnote, § 2); therefore no great weight can be 
attached to these exceptions. 

* This, of course, is not the amount of heat developed in the combina- 
tion of chlorine atoms with the hydrocarbon radicals, as heat is absorbed 
to an unknown extent in separating the atoms composing the diatomic 
chlorine (or bromine or iodine) molecules. 


87 


the Constitution of Carbon Compounds. 


OLOL9 
O&88G 
ONP8P 
O6T9V 
OVGEE 
06668 
OFOEE 
099ET 
O9cFI 
OL8ST 
OsTéT 
OIGET 
O0GET 
OGLET 
O61ET 


"oUILO[YO OT} JO 
CREP RS ie 
pedojaaop yeoyy 


Z 


‘sytun Q96eT = “a= "a Ssqun OZ/GT =< =u 


G 


096gT = a 
OZ119 = Ta-+ug 
OzL6T = ub 
OFeZGT = = 'ae+49 
OF89L = Ta-bulp 
OFs9L, = Tatup 
orzhos = = 'aG+ug 
Oce90T = “a+ 'a+ug 
Ozeg90T = “a+la+ug 
03119 = “aug 
ogeest = ‘aet+ug 
O9GLET = ‘2G 4-4, 
09cc6 = Taba 
OOLLr = Wg 


‘[Rorpea uoga«voorpsq 
oY} JO TOLJLUILOF Of} UT 
podoyossp 4vozyT 


4G 
oL0cL seretereerereeesrereeteees Tati 
029g eo rea) 
ogceol rreseeeeseeeesesere ees BL EETE a 
OI6T9 srretseeenersesrersesee Bre 


OSFSCT Gee ag gritiic "10 ‘"100 °*HO 
0egGot eeeae tsar hen rer oe Cry 
OggGor Dregs eran pe Germ chy 
006L1% GGT Tae eaanee ava Tey Sere Ly 
OSO0IZT an COC fo \rava ye fayeia 1a 
OGEZZT EC a aE favo Ra = 9) 
OOeFL savicinanng e+e behets kas ney Ore Gey 
014961 satan eee Reece Every 
O9ITIST eR ede negt  -¥s| Terre 


Seer eer ene ones evenee euefAoLOTyOV.IYy, 


HreseeeeererereeeeeesouretaUOLOTYORLIO IT, 


See uS UC | ofeucay nc sue, Aq4e.10Tq9 
SOC CO rere meee eres eeseseneeees TUALOJOLOTYL 


Ba indie ao CUTE Terni (09 (9) 
ssipededskse nabs" oOTEO THO eueprTAqIoL 
dein prataisreceaetiasios eprixoryo eu, Aq 
CPE SED Stace CTT 3) oH CVI) [Aueyg 


See nn 2) 0) C0) 89) LITy 
wee nsec reece sereesseeens euop{doado.z0p yO 


pee c ec eeresseeeeesseeeesee eueTAT 040 YO 


ceuee oenese AES e+9 aie SO TLIOTUO pAynqosy 
seem reece sees eres ceeesee epr4qoyyo jAdorg 


OLE90T sees eee ee ec eeeerscryeces 10° HO vine aaa Cee See es wes HTT EO qua 
0Gg09 Soe err rrr) 10°HO were cee er ees esse erenene eprisoryo [AqIOTW 


"A WIEVY, 


88 Prof. H. EH. Armstrong on the Determination of 


the combination of the bromine is 5430, 6040, and 6290 units ; 
whereas in the case of allyl bromide, as in that of allyl chloride, 
there is a somewhat larger amount of heat developed, viz. 
7120 units. : 

A comparison of the heats of combustion would appear to 
show that isomerism such as obtains between allyl chloride and 
chloropropylene and between ethylene and ethylidene chlorides 
has no influence : 


Allyl ‘ehloride:s: sy-n. 442500. 
C3H;Cl Chloropropylene ... 441190. 
C,H,Cl, Hthylene chloride... 272000. 


Hthylidene chloride. 272050. 


(10) Heat of Combustion of Alcohols—The thermal be- 
haviour of isomeric alcohols of the C,Hon+,;.OH series is 
very different from that of isomeric paraffins and their haloid 
derivatives, as their heats of combustion are by no means the 
same—the primary having higher heats of combustion than 
the secondary, and the secondary higher than the tertiary. 
The heat developed in the formation of the group C. OH may 
be calculated by deducting from the observed heat of formation 
at constant volume of the alcohol that of the hydrocarbon 

2 

radical, assigning to r= = as before, the value 15000 units, 
and to v, (=v,) the value 14200 units. The numbers thus 
calculated are given in the last column of Table VI. It will 
be noticed that nearly the same values are found for the six 
primary alcohols and (which is remarkable) phenol, and that 
considerably higher values result in the remaining cases, 
especially in that of trimethylcarbinol. 

(11) Heat of Combustion of Ethers—An ether being re- 
presented by the formula C,H,.O.C,H,g, its heat of formation 
may be expressed by the equation 


P=(64+B)r+nv+n, 


where 7 represents the number of single and double bonds, 
and 7 the heat developed in the fixation of an atom of oxygen 
by two carbon atoms. 

Deducting the heat of formation of methylethyl ether from 
that of methylallyl ether, the difference will be the value 
of v2, 


185570 — 171560 =14010=v,. 


In like manner, deducting the heat of formation of methyl- 
ethyl ether from that of methylphenyl ether, 


282490 —171560=110930=8v 5 .*. v=13870. 


of Carbon Compounds. 


won O 


the Constitut 


oT9TO Gigl in = serene ozeeg cea HO: “CHONG: ieee emedees weeoes jourqaes pAqjeunzy 
CeB0G XG 00GFL = ‘a+tup OUGSH ieee un (HO)"HO *(HO)*HO [rte strerr eres Joodys ouop Ay GT 
Garo 00z69 = %e-+te-tue Guaiic oes hee “HO "HOO Hg [tereeeeeeneensnses joyoors 4Brudorg 
OOFTS 00ST = = 'aF+wTT O0geLg HO" CH‘O)CHO)O |r: Jourqavo Aqo-Aqyouurg 
aio’ oonteT =" tor tu), GEESE re tetris oe IO” “CHO NED masessuerers he an joyoore 7Adoadosy 
OLIFP 008z0z = Lag+uc oleope. eee Seer eeeeeees FIO FO) est tenite aise en ders crtsenn cena joueyg 
O6F IP OOFSOT = °e+ "auc O68 — ec" HO *“HO' HO ‘°HO |e" OETA ITI Joyoore TAITy 
OOSTF OO8IZS = ‘ap+uIT 00989 HO CHO) “CHO)HO ecieee Seco as [oyooye [Aureosy 
OFF OO9LLT = ‘ne+ug Ondgsg HO * “HO *“°(°HO)HO [ovvttttrrtreeeseeess foyooye [Aqnqosy 
OOFGE OOFSSI = ag-tu) Gisele men [ee errno 18K@) ois re) ero eekeoc: poeerrepLoarc cence: joyoore [Adoag 
OSS 00c68 = ‘a-+tug OBLEGTH ee HO" “HO HQ) 28 ite tenant sangria: jouoore Ayggy 
096&F O00ce = ue 09688 Poem eerercereerassnns HO’ "FLO vvlaistatyisio.apis alstelaiels eieia Serslefetnie JOyooyR | AMOI 

‘TeoIpel FO'O ‘Teorpea toqavoorpcy eae Oe aaa 

JO UOT}eULLOY Ur ey} JO UoIyeUIIO; ‘d 

pedojoaep yvoxy oy} ul pedojoaop yvozyT 


Se ee oe, a a ee ne ee Ween SN Ieee | 


‘TA TEV, 


—$—$—— eee 


H 


Phil. Mag. 8. 5. Vol. 23. No. 141. Feb. 1887. 


90 Prof. H. E. Armstrong on the Determination of 


It therefore appears that v, and v, have practically the 
same values in ethers as in hydrocarbons and alcohols. If, 
now, from the heat of formation, P, of an ether, the value nv 
be deducted, the difference will be the heat developed in the 
combination of b+ 8 hydrogen atoms and of the two oxygen 
atoms to the carbon atom. On reference to Table VII. it 
will be seen that the quotient of P—nv by 6+ 8+2 is prac- 
tically a constant, the mean value being 15757 units, which is 


Y 
very nearly the value of eek hence Thomsen concludes 


that the amount of heat developed in combination of an atom 
of oxygen with two carbon atoms is about the same as is 
developed by combination of a hydrogen molecule with a 
carbon atom. 


Tasxe VII. 

C5, O10, P—n. b-:6:.|, ee 

e 6+B+2 
Dimethyl ether............ CH,.0O.CH, ....%=0 124950 6 15619 
Methylethyl ether ...... CH230) 40, 0e... 1 157690 8 15769 
Miethylether \/ 5.26.25: CAE ORC Ee... 2 193210 10 16101 
Methylpropargyl ether...| CH,.O.C,H, ... ty 127290 6 1591] 
Methylallyl ether......... CHI OFC HESS. 2 157790 8 15779 
Diallyl ether .......c.c... C,H,.0.0,H,..| 4 184650 | 10 | 15388 
Methylphenyl ether ......| CH, .0.C,H, . 9 157660 | 8 | 15766 
Ethylene oxide ......... { pa On ae 1 -goli0 | 4 | 18352 


In this section occurs one of the most noteworthy of 
Thomsen’s conclusions, viz. that ethylene oxide is in reality a 
dimethylene oxide of the formula CH,.O.CH,. The heat of 


formation of a compound of this formula is 
(442) 15757=94540 units, 


which agrees well with the observed value 93980 units ; 
whereas, assuming the formula to be that ordinarily accepted, 


the heat of formation should be greater by v, or about 
14,000 units. 


* It is not quite evident why the value of 7 is calculated in this 
manner ; if deduced from the equation 


n=(b—nv)—(at+B)r, 
giving 27 the usual value (80000 units), its value is considerably higher, 


viz. 2.17475 units in the case of dimethy] ether, for example, which is 
much above the value of 27 found in previous cases. 


the Constitution of Carbon Compounds. 91 


Thomsen also points out that, whereas the difference 
between the heats of formation of methane and methoxy- 
methane or dimethyl ether is 65400 units, that between 
methoxymethane and methylal—which is usually regarded as 
dimethoxymethane—is 76110 units, and that between methy- 
lal and methylic orthoformate—which is commonly represented 
as trimethoxymethane CH(OMe);—is 79730 units. Compar- 
ing the heat of formation of methylal, C;H,O,, with that of 
methylethyl ether, C;H,O, there is a difference of 29500 
units; between methylic orthoformate, C,H,)03, and diethyl 
ether, C,H,,O, there is a difference of 2.29920 units; so 
that the heat of formation is increased by the addition of an 
atom of oxygen by 29500 to 29920 units. This value very 
nearly corresponds to the difference between an alcohol and 
the hydrocarbon from which it is formed, thus 


Methyl] aleohol—methane= 29410 units. 

Hthyl » ethane =29620 _ ,, 

Propyl ,, —propane =30290 _,, 
Thomsen therefore suggests that possibly the function of the 
oxygen in bodies such as methylal and methylic orthoformate 
is the same as that of the oxygen in alcohols. 

(12) Heat of Combustion of Aldehydes, Ketones, Acids, 
Anhydrides and Ethereal Salts—The observed heats of for- 
mation of a number of these compounds are given in Table 
VIITI., and also the values deduced by subtracting the amounts 
of heat which it may be assumed are developed in the formation 
of the hydrocarbon radicals. 

The mean values which thus result, representing the heats 
of formation of the characteristic groups in the aldehydes, 
ketones and acids, are as follows :— 

Aldehydes . COH . . 65400 units 
Rerones i SCO ae.  oAZOD. 
OW gen OC . LESIGU.. 5. 

It will be observed that 65400 +54250=119650: in other 
words, the amount of heat developed in the formation of the 
carboxyl group of acids is the sum of that developed in the 
formation of the group COH and CO. Hence Thomsen con- 
cludes that the aldehydes are unsaturated compounds of the 
formula R..C(OH), inasmuch as it is to be supposed that the 
ketones and acids are respectively constituted as represented 
by the formule 

R,C:O and R.C: O(O8). 

Deducting from the heat of formation of acetic anhydride, 
254340 units, the value 6r+ 2v, the difference (165940 units) 
will represent the heat developed in the formation of the 

H2 


92 Prof. H. E. Armstrong on the Determination of 
Tasxe VIII. 


P. | mr+nv.| P—(mr+nv). 


——_—_— 


Acetaldehyde............ CH) COR eon 124630 | 8r+v |COH = 65430 
Propionic aldehyde ...| O,H COM)... ..- 168930 | 5r+2v 65530 
Isobutyric aldehyde .... C,H,.COH ...... 212830 | Tr+3v 65230 
Dimethyl ketone ...... CH, .CO.CH, .| 172400 | 6r+2v | CO = 54000 
Methylpropyl ketone .| CO { ae \ al 261300 | 10r+-40 54500 
ga27 | gue 
Bormic acid. ©... sen HP COOER..ctes: 133730 | + CO. OH=118730 
Weetie ACI, -2.-25.sccr se CHE COOR YT . 180890 | 3r+v 121690 
Propionic acid .-.-. a. C,H,.COOH ...; 222850 | 5r+2u 119450 
Acetic anhydride ...... O(CO.CH,), ...| 284340 | 6r+2v |(CO),0 =165940 
Dimethylic carbonate .| CO,(CH,),......... | 251500 | 6r 161500 
Diethylic carbonate ...| CO,(C,H,), ...... 341200 | 10r-+2u 162810 
Methylic formate ...... Fi ACOOCH 5.0. 165030 | 4r 105030 
Methylic acetate ...... CH, .COOOH,...| 210120 | 6r+v 105920 
Ethylic formate......... H .COOC,H, ...| 209800 | 67+ 105100 
Propylic formate ...... H .COOC,H, ...| 253680 | 8r+2u 105280 
Methylic isobutyrate...| C,H, .COOCH, .| 298660 | 107+3u 106060 
Ethylic acetate ......... CH, .COOO,H, .| 265910 | 8r+2v 117510 
Methylic propionate...) C,H,. COOCH, .| 258530 | 87r-+-2v 110130 
Isobutylic formate ....H.COOC,H, ...| 295700 | 107+3u 103100 
Allylic formate......... H.COOC,H, ...| 216800 | 6r+2v 98400 


group O:C.0.C:0. This value, however, is very nearly 
thrice (165940=3.55310) that found for the group CO in 
ketones and acids, viz. 54250 units; so that it may be assumed 
that equal amounts of heat are developed in the formation of 
the groups C:O and C.O.0, both of which contain an 
oxygen atom united by two bonds to carbon. The values 
found for methylic and ethylic carbonate afford confirmation 
of this conclusion, inasmuch as 162155 units—the mean of 
the two values 161500 and 162810 units—which is the amount 
of heat developed in the fixation of the three oxygen atoms 
in the group C.0.CO.0.0, is thrice 54052. 

It will be observed on reference to the table that the heats 
of combustion of ethylic acetate and of methylic, isobutylic, 
and allylic formates give values for P—(mr+ nv) very different 
from those found for the five preceding ethereal salts in the 
list. Thomsen states that in the first instance the specimens 
of ethylic acetate examined gave very variable results, and 
that he had great difficulty in obtaining a pure substance by 
fractional distillation*. Ultimately two samples were prepared 


* R. Schiff has also recently called attention (Liebig’s Annalen, 1886, 
vol. 234. p. 808) to the great difficulty of obtaining pure ethereal salts. 


the Constitution of Carbon Compounds. 93 


—one from absolute alcohol and crystallized acetic acid, the 
other from ethyl iodide and argentic acetate—boiling at 
77°-4—77°5, the heats of combustion of which were 548080 
and 549250 units, respectively. ‘The value thus arrived at, 
however, is about 12000 units lower than that found for the 
allied ethereal salts, such as the metameric propylic formate, 
for example. Thomsen is therefore inclined to think that the 
formula CH;.COOC,H; does not correctly represent the con- 
stitution of ethylic acetate, as the heat of formation of such 
a compound would be 253880 units, or 12030 units less than 
the value found experimentally—a difference of over two per 
cent. of the heat of combustion. Assuming the formula to be 
CH,;.CH(OH).CO.CHs, and calculating the heat of forma- 
tion with the aid of the constants previously deduced for acids, 
aldehydes and ketones, the value found, viz., 


Tr + 8v, + 54250 + 65400 = 267250, 


differs by only 1340 units, or a quarter per cent. of the heat 
of combustion, from the theoretical number. 

Thomsen leaves unexplained for the present the cause of 
the discrepancy in the case of methylic and _ isobutylic 
formates ; he does not regard it as remarkable that the value 
for the allylic salt should be somewhat abnormal, as in other 
cases values are found for allyl compounds which differ 
somewhat from those for corresponding compounds of 
C,Hon+1 radicals. 

(13) Heat of Formation of Nitrogen Molecules—Thomsen 
bases his determination of the heat developed in the formation 
of nitrogen molecules from nitrogen atoms on the argument 
that probably, as in the case of carbon, the two atoms are 
held together by single affinities. Assuming that in nitrogen 
peroxide, N,O,, the nitrogen atoms are directly associated as 
represented by the formula 


os Bey 
ae i XO 
or by the formula 
O—N—O 
Niel Fee 
O—N—O 


the amount of heat absorbed in converting the peroxide into 
nitrogen dioxide (NO,) molecules is the quantity sought to be 
determined. Now, according to Berthelot and Ogier, the heat 
of dissociation of nitrogen peroxide between 27° and 198° is 
10608 units ; as about 20 per cent. is already decomposed at 


94 Prof. H. H. Armstrong on the Determination of 


the former temperature, the entire amount may be set down 
as 13250 units. Boltzmann in a recent paper (Wiedemann’s 
Annalen, vol. xxii. p. 71) gives the value 13920 units: the 
mean value, in round numbers 13600 units, may be regarded 
as not far from the truth. But in the conversion of N,O, 
into 2NO,, the volume becomes doubled, so that the heat of 
combination of nitrogen atoms at constant volume is 


N .N*=13600—580=13020 units. 
(14) Heat of Formation of the Oxides of Nitrogen.—F rom 


his previous determinations, and taking into account the heat 
of dissociation of nitrogen peroxide, Thomsen calculates that 
the heat of formation of pure nitrogen peroxide gas at 18° is 


N,, 20, =N,0,= —3810 units at constant volume, 
while that of nitrogen dioxide is 
N,, 20,=2NO,=2(—8415) units at constant volume. 
On the assumption that nitrogen dioxide has the constitution 


Ye 
ING: 


and that the amount of heat developed by the combination of 
two similar atoms does not exceed that developed by their 
union by single affinities, it is possible to determine the 
heat of formation of oxygen molecules by comparison of the 
heats of formation of nitrogen monoxide and dioxide at con- 
stant volume. The manner in which the heat-disturbance 
which attends the formation of these two oxides is effected 
may be expressed by the following equations :— 


Ny, Op=2NO =2(—21575) =2N:O—N .N—O. 0, 
Ny, 20,=2NO,=2(—8415) =4N.O—N.N. 


Or,in words—the heat-disturbance of 2(—21575) units, which 
attends the formation of 2 gram-mol. props. of nitric oxide 
from 1 gram-mol. prop. of nitrogen and 1 gram-mol. prop. of 
oxygen, is the difference between the heat absorbed in the 
separation of the nitrogen atoms of the nitrogen molecules, 
and of the oxygen atoms of the oxygen molecules—the 
atoms in both cases being united by single affinities—and that 
given out in the combination of the nitrogen and oxygen 
atoms by two affinities of each. In the case of the dioxide, 
as the oxygen atoms remain united by single affinities as they 


* The dot is used merely to indicate the number of affinities of each 
atom engaged in holding the atoms together. 


the Constitution of Carbon Compounds. 95 


were in the original oxygen molecules, heat is absorbed only 
in the separation of the nitrogen atoms. On the other hand, 
the heat developed arises from the association of the nitrogen 
atoms each with two oxygen atoms, as exhibited in the 
formula above, the expression 4N.O signifying merely four 
single nitrogen-oxygen affinities. On the assumption that 
N:O=2N.0, 2.¢. that the amount of heat developed when a 
single atom of nitrogen combines by ¢wo of its affinities with 
a single atom of oxygen is double that to which the associa- 
tion of an atom of nitrogen by but one of its affinities to an 
atom of oxygen gives rise, the difference between the heats 
of formation of nitrogen monoxide and dioxide will be the 
amount of heat developed in the formation of the oxygen 
molecule : 
(No, 20,=2NO,)—(N., O,=2NO) =2.13160. 
“. O. O=26320 units. 


Provided that the heats of formation of nitrogen and oxygen 
molecules thus deduced be accepted, the heat developed in the 
formation of nitric oxide from nitrogen and oxygen atoms 
may also be ascertained, thus :— 
N,, 205 = 2NO,=2(—8415)=4N ° O—N ° N 
~ —16830—13020= —3810=4N. 0. 
“ N.O=—952 units. 


The value thus deduced is very small, and probably may be 
regarded as zero, it being unlikely—remarks Thomsen—that 
there are atoms which exercise a negative affinity. 

(15) Heat of Formation of Cyanides—Comparing homo- 
logous cyanides, the difference between the heats of formation 
of acetonitrile (60500) and propionitrile (104810) is 48810 - 
units, or very nearly the value which in other cases corre- 
sponds to a difference of CH,; but the difference between 
hydrogen cyanide (10900) and acetonitrile is considerably 
greater, viz. 49600 units, pointing to a difference in consti- 
tution between the nitriles and hydrogen cyanide. 

Assuming the formula of acetonitrile to be CH;.C:N, 
the heat of formation of the radical C: N from atomic carbon 
and ordinary nitrogen will be the difference between the heat 
of formation of acetonitrile, 60500 units, and 3r+v, =59200 
units, 7. e. 1300 units. Adding to this half the value N . N 


C : N=1300 a 6510=7810 units. 


A similar calculation for propionitrile gives 910+6510= 


7420 units. 
In like manner, adding 6510 to the heat of formation of 


96 Prof. H. E. Armstrong on the Determination of 
hydrogen cyanide (10900 units), the heat developed in its 


formation from ordinary hydrogen, gaseous “‘ atomic’’ carbon 
and atomic nitrogen willresult ; deducting from this 15000 
units on account of the hydrogen, the remainder (17410- 
18000= 2410) will be the amount of heat developed in the 
formation of the radical CN. It is much lower than that found 
for either of the nitriles. 

The heat of formation of cyanogen from gaseous atomic 
carbon and atomic nitrogen is 24080 units. Assuming that 
it has the constitution N:C.C: N, 


24080=2N:C+C.C=2N:C+y, 
“. 2N ? C=24080—14200= 9880. 


The value for N: C (4940 units) thus deduced is much lower 
than was calculated from the heat of formation of the nitriles. 
If it be assumed that cyanogen has the formula N:C:C:N 
the value 4940 would be the heat developed in the combina- 
tion of carbon and nitrogen atoms by double affinities. But 
more probably, says Thomsen, its constitution is to be repre- 
sented by the formula C: N.N: QC, in which case 


24080 = 2N:C4+N.N; 


whence 


NEC 5530: 


It would appear, therefore, that the affinity of carbon for 
nitrogen varies according to the mode of combination, the 
values obtained being 


From acetonitrile i: ge ie { 7810 


»» propionitrile 7420 
3 cyanogen. seer yIN GC 5530 
» hydrogen cyanide (?)N.C 2410 


Very probably the three values are related as 8: 2:1, and 
the heat developed on combination of carbon and nitrogen 
atoms is proportional to the number of affinities satisfied, 
being on an average 2600 units per affinity. 

(16) Heat of Formation of the Amines.—The difference be- 
tween the heats of formation of homologous amines is some- 
what greater than is usually found between homologues, 
especially in the case of secondary and tertiary amines, but 
some uncertainty attaches to the values for these latter on 
account of their high heats of combustion. The heats of 
formation of primary amines are greater than those of the 
isomeric secondary and tertiary amines. Thus 


the Constitution of Carbon Compounds. 


Methylamine. 
Hthylamine . 
Propylamine. 
Amylamine . 


46760 ! 
92530 


45770 


" 135560 9 44400 
" 996990 $ 4. 44882 


97 


Dimethylamine . 87740) 5 
Diethylamine amaelene oe 
Trimethylamine. . 128690 
Triethylamine : 368300 $ as 


Primary amines may be regarded as formed by the with- 
drawal of a molecule of hydrogen from a molecule of ammonia 
plus a molecule of the corresponding hydrocarbon, and 
secondary as formed in like manner from primary, and ter- 
tiary from secondary amines :— 

C,H; == NH; = C.H3-1 ° NH, a5 H, 5 &e. 
The heat-disturbance in such reactions may be deduced from 


the heats of formation of the three compounds, that of am- 
monia at constant volume being 11310 units (Table IX.). 


TaBLe IX. 
| CaHo. P. | CoH _NH,. P| P'—(P+11310). 
Methane...... 21170 =| Methylamine .. 8380 — 24100 
| Ethane ...... 27400 | Ethylamine...... 15770 — 22940 
Propane...... 33370 | Propylamine ... 20420 | — 24260 
| Pentane ...... 44950 ||Amylamine...... 35080 || — 21180 
Benzene ...... —13670  |/Aniline ......... —19190 —16830 
Propylene ... 2060 ~=—‘||Allylamine ...... — 2880 — 16250 
EET pes... 29100 | Piperidine*...... 24090 —16320 
Isobutane...) 40130 |Isobutylamine .| 35560 —15880 
| 


On inspection of the last column of the table, it will be 
seen that the primary amines form two groups; the difference 
is too great and too constant within each group to be the 
result of accident. Hence Thomsen is of opinion that the 
amines of the two groups differ materially in constitution. 
The numbers for secondary and tertiary amines (Table X.) 
closely resemble those afforded by the members of the second 
group of primary amines, which may be regarded as indicative 
of similarity in constitution. 

It is commonly held that the amines are formed by the 
displacement of the three hydrogen atoms in ammonia one by 


* Thomsen regards piperidine as a primary amine. 


98 Prof. H. EH. Armstrong on the Determination of 


TABLE X. 

Components. Product. pr, || pre (P+P’). 
Methylamine ...... P’= 8380 | Dimethylamine ..., 10980]/ = —18570 
Methane............ P =21170 ; 
inoitivlamins PAN OEEN } Grimethylamine ...| 18550]/ — —18600 
Ethylamine ...... aie \ Diethylamine ... .. 26420 — 16750 
HGhame - 3.2. cy.etee P =27400 
Diethylamine...... P’—26420] } Teiethylamine.....,| 88020], —15800 


one by hydrocarbon radicals, on which view it is to be ex- 
pected that equal amounts of heat would be developed 
at each stage. This, however, is not the case (Table X.): 

the displacement of the first hydrogen atom by methyl 
occasioning a much smaller heat-disturbance than that of 
either the second or third atom, the amounts being very 
nearly the same in the case of these latter however. The 
difference may arise either because the hydrogen atoms in 
ammonia are of different value, or because the change is 
not of the nature supposed. In the former case one of the 
hydrogen atoms must be regarded as more firmly held than 
the other two, and in the formation of methylamine that which 
is most firmly held would be first displaced ; but it is not 
easy to explain why this should be the case. Moreover, in 
the formation of the group C. NH,, on the one hand in the 
primary amines containing methyl, ethyl, propyl and amyl, on 
the other in aniline and allylamine, there is a difference in the 
amounts of heat evolved of about 7000 units. It is also im- 
probable, if the constitution of the amines be that commonly ~ 
supposed, that the introduction of the first methyl or ethyl 
into ammonia should involve an increase in the heat of neu- 
tralization, and that of the second and third a considerable 
diminution*. Thomsen therefore seeks to deduce the heat of 
formation of the primary amines of the first group on the 
assumption that they differ in constitution from those of the 
second group. Assuming the nitrogen to be pentadic, the 
constitution of methylamine may be represented by the for- 
mula H,C : NH3, i.e. it may be regarded as formed by direct 
combination of methylene with ammonia. The heat of for- 


<2 it MC TEGOXOT OWE Bee Spm 12270 . 
: Units of heat developed on 
Methylamine...... 13115 neutralizing with chlor- 
Dimethylamine .... 11810 hydric acid 


Trimethylamine... 8740 


the Constitution of Carbon Compounds. 92 


mation of methylamine from atomic carbon being 46760 units, 
that of ammonia 11310 units, and that of methylene 2r or 
30000 units. 


46760 = (GC, H,) S- 4(N,, 3H.) + (CH, NH;) 3 
“. (CH, NH3) =5450 units. 


An almost identical value (5530 units) was previously de- 
duced from the heat of formation of cyanogen for C:N. 
Hence Thomsen concludes that the interpretation thus put upon 
the reaction involved in the formation of methylamine is 
justified. 

If methylamine be represented by the formula HC: N Hs, 
the formule of di- and trimethylamine will be 


HC . 
The heat of formation of the group associated with the hydro- 


carbon radical in each of these amines may be calculated 
from the following equations :— 


C: NH;=2C.N+38N.H—4N.N = 16510 units. 


¢'}NH,=30.N+2N.H-1N.N aed rea 
C 


O. ENE =40.N+ No HEN 98380. -:,, 
C. 


The value of C.N has previously been determined; that of 
N . H is given by the equation 
6N.H = (N,, 8H3)+N.N+3H.H; 
but we have no means of determining H.H. Hence 
N.H = 5940+4H.H and 4(2N.H,)=5940. 
Adding the values for the hydrocarbon radicals, the heats of 
formation of the three methylamines will be as follows :— 
Found. 
Methylamine = 2r+16510= 46510 46760 
Dimethylamine == 5r+13170= 88170 87740 
Trimethylamine= 8r+ 9830=129830 128690 


If aniline were constituted like methylamine, it would be 
represented by the formula O,H,: NH;; but in a body of 
this formula the nitrogen atom would be united to two carbon 
atoms, as each carbon atom in benzene carries but a single 
hydrogen atom, so that if it be assumed that in the amines 
the nitrogen is always combined with only a single carbon 


i H.C : 
= ‘UNH, HC Love 


100 Prof. H. H. Armstrong on the Determination of 


atom of the radical such a formula is impossible, and the 
ordinary formula must be assigned to aniline. The heat of 
formation would then be 


5r+9v+N.C+2N.H—iN.N=210770 units. 


The value found experimentally is 211090 units, thus con- 
firming the formula NH, .C,H;. 

(17) Heat of Formation of Pyridine——Pyridine is usually 
represented by the formula 


HC—CH—CH 


| | 
HC= N —CH 


The heat of formation of such a compound would be 
by si 204 ais 25 + C= N—C= 131800 + C=N—C. 


The difference between the heat of formation calculated from 
the heat of combustion, P=171370 units, and 131800 units, 
viz. 39570 units, should represent the amount of heat deve- 
loped in the fixation of the nitrogen atom ; but it so exceeds 
the values found in the case of all other nitrogen compounds 
that the assumption that doubly-linked carbon atoms are 
present in pyridine must be abandoned. If a formula similar 
to the prism formula for benzene be adopted, the heat of for- 
mation may be expressed as follows :-— 


br + 60; oa N — C= 160200 + N == Cx 
The value of N=C; thus found, 11170 units, is also unusually 


high ; pyridine therefore must differ from benzene in consti- 
tution ; Thomsen suggests the formula 


/CH-——CH 
Pi ORT 
NCH ae 
Wee 
\CH-—CH 


the calculated heat of formation of such a compound being 
very nearly that found for pyridine (1713870 units), thus: 


5r+Tv+C.N=170490 units. 


(18) Nitro-paragins.—Nitromethane and nitroethane are 
commonly regarded as compounds of the form R.NO,. De- 
ducting the heats of formation of the hydrocarbon radicals 
from the observed heats of formation of the compounds, the 
difference will be the heat developed in the formation of the 
radical C . NQ,. 


the Constitution of Carbon Compounds. 101 
i 5 R. P—R. 
Nitromethane. 55820 3r = 45000 10820 
Nitroethane . 101900 5r+2v = 89200 12700 


The mean value of P—R is thus 11760 units: deducting from 
this the heat developed on combination of a carbon and a 
nitrogen atom by a single affinity, 2600 units, the remainder, 
9160 units, is the heat developed in the formation of nitrogen 
dioxide ; but this has previously been stated to be —8415 
units (§ 14). So that the value deduced from the two nitro- 
compounds under consideration differs to the extent of 17575 
units from that determined experimentally by means of nitric 
peroxide. ‘This result is again suggestive of a constitution 
different from that commonly attributed to the nitroparafiins. 
Thomsen points out that by formulating them as nitroso- 
alcohols, heats of formation may be deduced which agree with 
those found ; thus :— 


Nitromethane= CH, (NO).OH =2r+C.0H+C.N+N.0 
= 55820 units. 
Nitroethane =CH(CH;)(NO).OH =4r+v7+C.0H+C.N+NO 
=101900 units. 


Putting r=15000, v=14200, C. OH=44520 (§ 10), and 
C.N=2600 (§ 15), the heat of formation of NO will be— 


If calculated from Nitromethane, — 21300 units, 
be » Nitroethane, —19420 ,, 


These values differ but slightly from that found, viz. —21575 
units. : 

(19) Heat of Formation of Sulphur Compounds.—lt will not 
be necessary to discuss Thomsen’s conclusions regarding these 
compounds ; it will suffice to callattention to the one instance 
in which he arrives at a result at variance with received 
opinion. This is in the case of thiophen, which is usually 
represented by the formula 


/CH=—CH 
S< | 
\CH=CH 


On grounds similar to those advanced in the case of benzene 
and pyridine, Thomsen concludes, however, that the carbon 
atoms cannot be doubly linked, but that they are united by 
five single bonds, and that the sulphur atom is united to a 
single carbon atom by two affinities, as shown by the formula 


/CHN 
S= CC ~ POH: 


102 Prof. H. E. Armstrong on the Determination of 


(20) Thus far I have only endeavoured to give an abstract 
of Thomsen’s arguments and conclusions, quoting as far as 
possible his own words. Many of his deductions are in the 
highest degree remarkable. Recalling the more striking, he 
not only finds that the same amount of heat is developed in 
the combination of carbon atoms as they occur in ethylene 
and as in the paraffins—in other words, that even in the ole- 
fines the carbon atoms are united only by single bonds—that, 
in fact, there are no such things as “double bonds”; but he 
also arrives at the startling conclusion that in the formation 
of acetylene the carbon atoms unite without any evolution of 
heat : so that we are forced to assume that in an acetylene 
not only is there no treble or double bond, but not even a 
single bond! Then ethylene oxide is pronounced to be methy- 
lene oxide ; the aldehydes are hydrovy-compounds ; and pro- 
bably methylal and methylic orthoformate are also alcoholic 
bodies. Lastly, the amines are to be regarded as derivatives of 
pentad nitrogen; and pyridine is not analogous in constitution 
to benzene. 

(21) Now, admitting even that a more careful consideration 
of the chemical evidence might result in our acknowledging 
the correctness of Thomsen’s conclusions in some few cases, 
it is impossible to do this in the majority of instances: we 
cannot admit that the carbon atoms in ethylene oxide are dis- 
united, and that this compound is in reality a methylene oxide; 
and every chemist must regard Thomsen’s formule for the 
amines as altogether lacking probability. Moreover, the 
method by which the constitution of bodies like ethylene oxide 
and the amines is ordinarily arrived at, is the method by 
which the constitution of compounds generally is determined. 
Hence, if we accept Thomsen’s conclusions in their entirety, 
results arrived at by the strict application of the same method 
throughout are to be accepted in some instances but rejected 
in others. To admit this would be to acknowledge that our 
entire system of constitutional formule is based upon a false 
conception, to which there is no possible key. That current 
views of structure require modification in some not unessential 
particulars, I have long been of opinion; but that they will 
have to be moditied to the extent which Thomsen’s arguments 
indicate appears to me altogether improbable. 

(22) It remains, therefore, to seek for some explanation of 
his anomalous results, and one of the first questions to be 
answered is: Are we justified in regarding the value 135340 
units as the true heat of combustion of gaseous atomic carbon? 
Thomsen’s determination of this value involves the assump- 
tion: that when a molecule of oxygen combines with two 
molecules of gaseous carbon monoxide, the same amount 


the Constitution of Carbon Compounds. 103 


of heat is developed as if combination were to take place 
between an oxygen molecule and a single carbon atom, due 
allowance being made for the change in volume. But it is 
by no means certain that this is the case—that the addition 
of the first and second atom of oxygen to a carbon atom 
involves the development of the same amount of heat. The 
argument that it does, which has been based on the results 
obtained for certain solid oxides in cases where both higher 
and lower oxide are ultimately obtained in the same state of 
ageregation as the element oxidized, cannot, in my opinion, 
be accepted in evidence, for the very reason that in the cases 
in question solids are dealt with throughout. If we consider 
what are the properties of carbon monoxide, they are such as on 
the whole favour the contrary view, viz. that, of the total heat 
developed in the formation of carbon dioxide, the larger pro- 
portion is given out in the combination of the carbon atom 
with a single oxygen atom. It is especially noteworthy, in 
fact, that carbon monoxide does not appear to be so markedly 
unsaturated, combining directly with but a limited number of 
other bodies and, as a rule, only under special conditions. If 
this view be accepted, 135340 units is too low a value for 
f.Q, (§ 4); in other words, /. C,=135340+ 2; and I think 
facts justify the conclusion that w has a high value. 

(23) As the value 7.C, is made use of in deducing the 
amount of heat developed in the formation of carbon com- 
pounds generally, the calculation of the amount of heat 
developed in the combination of hydrogen with carbon from 
the heat of formation of methane and other hydrocarbons 
must also be affected by the error which possibly has been 
made in determining /. C,. 

(24) Thomsen’s conclusion that v3;=0 (§ 7) is also, I 
imagine, evidence of a flaw somewhere in the argument by 
which so improbable a conclusion is arrived at. Now, it is 
assumed by him that the heat developed in the formation of 
methane (59550 units) is given out in the combination of two 
molecules of hydrogen with a single atom of carbon ; as pre- 
viously explained (§ 6), it amounts in round numbers to 60000 
units=2(27). The oe value for the heat of formation of 
methane will therefore be 60000+ 2 units, and the corrected 
value of 2r will consequently be 30000+4w units. But if 
this be granted, it follows that, in order to determine V3, it is 
necessary to add to the heat of formation of acetylene calcu- 
lated by Thomsen (28990 a) twice the value of x, and to 
deduct only 27-+4x=30000+ 4a. This will give —1010-+ 1d, 
instead of zero, as the value of v3; so that the heat of forma- 
tion of acetylene may well be a fairly high positive value. 


104 Prof. H. H. Armstrong on the Determination of _ 


Although acetylene is especially prone to undergo change, I 
think it is incorrect to speak of it as an “‘ easily decomposable ”’ 
hydrocarbon (p. 84); its apparent instability is probably the 
direct outcome of its ‘‘unsaturatedness’”’—of the readiness 
with which it consequently enters into combination. 

(25) I certainly believe, with Lossen and Thomsen, that 
compounds of the olefine and acetylene type are truly un- 
saturated bodies ; and that the affinities of the carbon atoms 
cannot be supposed to have satisfied each other in the manner 
indicated by the formule usually employed*. But, on the 
other hand, I hold it to be both possible and probable that in 
the formation of the olefines, for example, there is a partial 
neutralization—a partial outgoing of energy—beyond that 
which occurs in the formation of paraffins. What appears to 
be direct confirmation of this view is afforded by Thomsen’s 
observations that the heat of combustion of trimethylene 
exceeds that of its isomer propylene by 6690 units. Tri- 
methylene is regarded by many as a “closed chain ”’ hydro- 
carbon, consisting of three CH, groups, while propylene is 
methylethylene, the formule being 


CH, CH. CH; 
oN. I 
H,C—CH, CH, 
Trimethylene. Propylene. 


Trimethylene combines much less readily with bromine than 
does ordinary propylene; but, in my opinion, there is no 
chemical evidence to justify the assumption that the former 
is a “closed-chain”’ compound. The properties of trimethy- 
lene may be explained by regarding it as an open-chain 
hydrocarbon of the formula CH,.CH,. CH, which exhibits 
two of the carbon atoms as possessing each a “ free affinity;”’ 
and the sluggish behaviour with bromine may be attributed 
to the fact that the “ free affinities’’ are not associated with 
contiguous carbon atoms. Thomsen’s observation, assuming 
it to be correct, not only affords evidence in support of this 
conclusion, but may also, I think, be held to prove that in 
propylene the affinities of the carbon atoms partially satisfy 
each other beyond the point which would be typified by the 
formula H,C.CH.CH;, as the difference of 6690 units 
between the heats of combustion of the isomers is considerably 
below the probable value of v,. If this view be correct, the 
value of v, is not determined by deducting the difference 
between the heats of combustion of a paraffin and the corre- 
sponding olefine (§ 4) from the heat of combustion of gaseous 
* See also Bruhl, Liebig’s Annalen, 1882, ccxi. p. 162. 


the Constitution of Carbon Compounds. 105 


atomic carbon : the value thus calculated is less than v,, but 
greater than 1. 

(26) As regards Thomsen’s conclusion that v,=v,—in 
other words, that there is no such thing as a double bond, it 
is to be remarked also that no great confidence can be placed 
in the determination of the value of v2, involving as it does 
the use of the heat of combustion of carbon monoxide. More- 
over it cannot be assumed, as a matter of course, that if a 
compound C,0, did exist (§ 4), it would be strictly speaking 
an analogue of C,H,; 7. e. that the energy of combination of 
two carbon atoms would be the same in the two cases, whether 
they were associated with oxygen or hydrogen—this, in fact, 
is the point to be proved. The non-existence of a compound 
C,O, may even be regarded as disproving any such conclusion. 
The agreement between theory and practice, i.e. between the 
calculated and observed heats of formation of olefines, might 
be claimed on behalf of Thomsen ; but the amount of evidence 
of this kind is too small at present, added 1o which marked 
discrepancies actually do occur among olefine derivatives— 
as in the case of the allyl compounds, to which attention has 
already been more than once directed. 

(27) Finally, another argument against the correctness of 
Thomsen’s conclusions may be based upon the values which 
he has put forward (it is right to say, with reservation) as 
representing the affinities of certain elementary atoms, viz.:— 


C.C= 14200 units 
N.N= _ 18020 ne 
OS OS] 213970 
eee ooh Oe nee 


Except in the case of carbon, these values are supposed to 
be the amounts of heat developed in the formation of the 
molecules from the atoms. In the case of carbon, as pointed 
out in § 7, Thomsen considers it probable that the atoms in 
the molecule are united in such a manner that at most two 
affinities are exerted per atom: so that of 38380 units—being 
the difference between the heat of combustion of ordinary 
amorphous carbon and the hypothetical heat of combustion 
of gaseous atomic carbon—2.14200 units are required to sepa- 
rate the carbon atoms from each other; the remaining 9980 
units are supposed to be absorbed in gasifying the carbon. 
But if we take into account the properties of the four elements 
in question, it is inconceivable that the numbers given by 
Thomsen can represent the amounts of energy to be expended 
in effecting their molecular simplification. Iodine we know, 
from VY. Meyer’s experiments, and from those of Meier and 


Phil. Mag. 8. 5. Vol. 23. No. 141. Feb. 1887. I 


106 Prof. H. E. Armstrong on the Determination of 


Crafts, is resolved into monatomic molecules with compara- 
tive ease, although far less readily than nitrogen peroxide is 
resolved into nitrogen dioxide ; but the entire behaviour of 
oxygen, nitrogen and carbon would seem to show that the 
stability of their molecules is enormously greater than that of 
diatomic iodine molecules. Thomsen’s conclusion also appears 
the less likely to command acceptance when the mode of de- 
ducing the values for nitrogen and oxygen is considered, and 
it is remembered that the value for nitrogen is nothing more 
than the heat of dissociation of nitrogen peroxide. All the 
facts, so far as I can interpret them, appear to me to show 
that nitrogen peroxide and nitrogen are to be placed at the 
very opposite extremes in the scale of molecular stability. 
Another instance of the unsatisfactory nature of the argument 
upon which Thomsen’s determinations of the affinity values 
for nitrogen compounds is based is afforded by nitric oxide, 
the formation of which, from its atoms, would, it is supposed, 
take place without evolution of heat. Meyer and Ziiblin have 
shown, however, that this gas is stable at 1200°, although it 
is entirely decomposed at about 1700°. 

(28) It remains to consider those results which are inde- 
pendent of any correction in the value of f.C,, &e. Amon 
the most interesting are those relating to the chlorides (§ 9), 
and to isomeric alcchols ($10). As regards the former it 
will be recollected that whereas the addition of a single atom 
of chlorine is attended by a heat-evolution of only 13500 
units, that of two involves the production of no less than 
2.16590 units, that of three of 18500 + 2.16500 units, and that 
of four of 2.18500+2.16500 units. It appears to me that 
the greater amount of heat developed in the formation of the 
symmetrical di-derivative (symmetrical as regards the relation 
of the chlorine atoms to the carbon atom or atoms) is pro- 
bably due to the partial neutralization of the (residual) affinity 
of the one chlorine atom by that of the other. It is unfortu- 
nate that no chlorine compound, in which the two chlorine 
atoms are associated with non-contiguous carbon atoms, has 
been examined. According to the view here put forward it is 
scarcely to be expected that in such a case there will be the 
extra development of heat which Thomsen has noticed, - 
although it is conceivable, however, that the association of an 
even number of chlorine atoms either with a single carbon 
atom, or with contiguous or symmetrically placed carbon 
atoms, may involve an extra outgoing of affinity. 

(29) Passing to the alcohols, we have to note Thomsen’s 
conclusion that the primary have the highest, and the isomeric 
tertiary the lowest heats of combustion, the secondary occu- 


the Constitution of Carbon Compounds. 107 


pying an intermediate position. To judge from the compara- 
tively greater stability of the primary alcohols, an opposite 
result might fairly have been looked for.—Here, again, I con- 
tend that it is to the local accumulation of negative elements 
that we must attribute the increased development of heat ob- 
served in the formation of tertiary alcohols, &c. 

The peculiar behaviour of the amines may, I think, be ex- 
plained on similar lines. 

Thomsen’s observations on ethylene oxide and aldehyde are 
also of great interest from this same point of view; they 
clearly prove that these compounds are not only unsaturated 
in the ordinary sense, but that they are actually to a con- 
siderable extent unexhausted bodies. Retaining the formula 
ordinarily assigned to ethylene oxide, I am inclined to think 
that it is a direct consequence of the structure of the com- 
pound—especially of the relation in which the single oxygen 
atom stands to the éwo carbon atoms, that the affinities of the 
oxygen atom—and probably of the carbon atoms likewise— 
are far from being satisfied. And in the case of aldehyde, 
whether the oxygen atom is or is not associated by two affinities 
with the carbon atom, as ordinarily supposed, there can 
be little doubt that any formula in which the oxygen atom is 
represented as ‘‘ fully combined” with the carbon atom is 
more or less misleading ; meanwhile, therefore, it is advisable 
to formulate the aldehydes as containing the radical C(O)H. 

The very low values found for the heats of formation of the 
simpler cyanogen compounds would appear to furnish addi- 
tional evidence in support of the view that the polyad nega- 
tive elements are endowed with peculiarities which limit the 
extent to which on entering into combination they can 
mutually satisfy or neutralize their affinities. 

(30) Thomsen has contended, time after time, that his 
results afford a solution of the problem which has of late 
years very properly excited so much attention, viz. that of 
the constitution of benzene. As pointed out in § 8, he 
maintains that the six atoms of carbon in this hydrocarbon 
are linked together by nine “single bonds,” and not, in the 
manner indicated by Kekulé’s well-known formula, by three 
single and three double bonds. Although of opinion that we 
cannot at present accept so absolute an interpretation of the 
thermochemical data, I yet think that Thomsen’s results, 
taken together with all that is known of benzene, must be 
held to prove that benzene is in no sense a compound of the 
same order as an olefine ; and that Kekulé’s formula, if used 
at all, must be literally interpreted as indicating that the 
carbon atoms are held together by nine affinities, there being 


108 Prof.H. EH. Armstrong on the Determination of 


abundant evidence to show that in the olefines the carbon 
atoms are not held together by double bonds. In other 
words, if we employ Kekulé’s benzene formula, we are bound 
to abandon the use of the conventional formula for olefines, 
From this point of view I see little difference between 
Kekulé’s symbol and the prism formula or the modification 
of the latter quite recently advocated by Thomsen (Deut. 
chem. Ges. Ber. xix. p. 2944). Objections have, however, 
been urged against the prism formula which appear to be 
justified ; the symbol advocated by Thomsen can scarcely be 
regarded as marking any particular advance; and Kekulé’s 
symbol is open to the oft-raised objection that it indicates 
the existence of four distinct di-derivatives. I venture to 
think that a symbol free from all objections may be based on 
the assumption that of the twenty-four affinities of the six 
carbon atoms twelve are engaged in the formation of the six- 
carbon ring and six in retaining the six hydrogen atoms, in 
the manner ordinarily supposed ; while the remaining six 
react upon each other,—acting towards a centre as it were : 
so that the ‘‘ affinity ”’ may be said to be uniformly and sym- 
metrically distributed. I would, in fact, make use of the 
following symbol :— 


The only difference between this symbol and those employed 
hitherto arises from the fact that I do not consider that, 
apart from its connexion with the other carbon atoms owing 
to their association in a ring, any one carbon atom is directly 
connected with any other atom not contiguous to it in the 
ring; my opinion being that each individual carbon atom 
exercises an influence upon each and every other carbon 
atom. ‘The idea here expressed is, I believe, essentially diffe- 
rent from that embodied in the somewhat similar symbol 
used by Lothar Meyer (Modernen Theorien der Chemie, ed. 4, 
1883, p. 262 ; comp. also Briihl, loc. cit. p. 12): as, according 
to my view, there is an excess of (negative) affinity beyond 
what is required to maintain the C,H, ring ; and as I do not 
consider that each carbon atom can be supposed to have 
an affinity free. 


the Constitution of Carbon Compounds. 109 


(31) While of opinion therefore that the arguments ad- 
vanced by Thomsen, as a rule, do not suffice to substantiate 
his views, I would yet express my conviction that even in 
their present form his results are of the very greatest value, 
and that it is already possible to base important theoretical 
conclusions upon them. Jt may be doubted whether it will 
ever be possible to determine the value of /. C, in the manner 
he suggests ; but fortunately many of the problems of che- 
mical affinity which press for solution may be solved without 
the knowledge of this value by comparing isomeric compounds, 
as in the case of the alcohols, for example §§ 10, 29: and itis 
in this direction that further investigation appears to be most 
likely to lead to valuable results. 

It is before all things important, however, that independent 
investigators should occupy themselves with thermochemical 
studies, in order that the fundamental values may be placed 
beyond doubt and that accurate thermochemical data may 
be accumulated for discussion ; moreover, it is essential to 
extend the field of study so as to include members of every 
class of compound. 


XII. Note on the foregoing Communication. By SPENCER 
UmFreEVILLE PicKERING, M.A., Professor of Chemistry at 
Bedford College*. 


N the foregoing communication Professor Armstrong 
makes a suggestion which must recommend itself to 
most chemists—that the heat evolved on the fixation of one 
atom of oxygen by one atom of carbon is greater than that 
evolved on the subsequent addition of a second oxygen atom 
to the compound thus formed. The fundamental value, 7. C,, 
on which all Thomsen’s calculations are based, will thus be 
erroneous; and the only meaning which can then be attached 
to his values for v1, v2, and v3 will be the differences between 
their actual values. | 
Thus, vz is given by the equation 


vy=f.C,—[ (CH, 30.) —(CHy, 20.) ] 
=f. C,— (833350—211920);. . . . . (1) 
and v, is deduced from the heat of formation of ethane, 
(2C, 8H,)=6r+r, 
3(H2, $02) + 27. C,—(C2He, 502) 
3 x 68360 + 2f. C,—370440=6r+y,; .. (IL) 
* Communicated by the Author. 


110 Prof. 8. U. Pickering on the Determination 
in which the value of 7, as deduced from the equation 
(C, 2H,)=47r=2(H, 0,) +7. ©, —(CH,, 203) 
=2~x 68360 +/.C,—211930, 
may be substituted, thus giving 
3 x 68360 + 2f. C,—370440=3 x 68360 4- 


whence 


3f.C, 38x 2119380 +y,; 

2 2 
3 x 211930 | 

2 3 
and, by combining this equation with (I.), we get the differ- 
ence between the values of v, and v,, independent of the 
value of f.C,, or of any quantities other than those deter- 
mined experimentally, 
20, —v,=13920. 

Similar results are obtainable with 13. 

Now, adopting Professor Armstrong’s view that Thomsen’s 
value for f. C, should be increased by some unknown quantity, 


jf dls vi ~ 3704404 (III.) 


x, it will follow that the value for r will become 7+ a while 
those for V1, Vo, and v3 will be converted into v,+ 5 Vat 2, 


and v3+ a respectively ; or 


2 10 
v,=14056 + 5" 
V2 >= 14250* + Uv, 

OL 

U3= —8l1 + Oo. 


The importance of these values can scarcely be overrated. 
If, as seems very probable, w represents some number con- 
siderably larger than 14000 cal., the heat evolved in the union 
of two carbon atoms will be very nearly, though not quite, 
proportional to the number of bonds by which they are united; 
a view which, I think, must be admitted to be highly probable. 
We at once obviate the necessity for denying that carbon atoms 
can be joined by anything but single bonds, and that com- 
pounds containing trebly-bound carbon atoms are not really 
compounds at all; while the slight loss of energy entailed in 
substituting a double bond for two single ones, and a treble 
bond for a double and single one, gives an explanation of 
the relative instability of unsaturated hydrocarbons. Thom- 
sen’s conclusions tend to entirely destroy the bond theory ; 


* 14056 X 2—14250 gives 13862, instead of 13920 as above, owing to its 
being deduced from mean, instead of special results. 


of the Constitution of Carbon Compounds. 114 


whereas his results, when studied from the present stand- 
point, not only confirm it, but endue it for the first time with 
a clear kinetic meaning. 

Passing on to other cases affected by the introduction of «: 
it will be seen that the union of the carbon atoms in the 
(hypothetical) gaseous molecular carbon by single bonds only, 
and the non-existence of the fourfold bond, is no longer 
tenable; and that the whole basis of the argument on which 
the surprisingly small values for the formation of molecular 
nitrogen and molecular oxygen rest is thus destroyed, and the 
conclusions which are based on these quantities, in some cases 
so opposed to all accepted views, will be destroyed also. 

x, however, will not affect the values calculated for the heat 
of formation of benzene, except in so far as the difference in 
these values will be comparatively a much more insignificant 
quantity than formerly. But the difficulty experienced in 
adopting Kekulé’s formula will vanish if we accept in its en- 
tirety the kinetic conception of bonds here developed. All that 
Thomsen has proved is that in certain classes of compounds— 
so-called open-chain bodies—v, is somewhat less than twice 73 
but it by no means follows that, in compounds constituted on 
such a different principle, and possessing such perfect sym- 
metry as benzene, this should be the case: in benzene the 
second bond has probably the full value of the first bond, 
V,=2v,, and hence the stability of the substance. The con- 
ception that the value of a bond, between even the same 
atoms, has a somewhat variable value dependent on the nature 
of the other atoms present in the compound, has already been 
developed in another direction by the author (‘ Atomic 
Valency,”’ Chem. Soc. Proc. 1885, p. 122), and has been in- 
dependently brought forward by Frofessor Armstrong, and 
used by him to explain many of the apparently anomalous 
results arrived at by Thomsen in other parts of the work 
under discussion. 

As with the value of 7, so also with the heat evolved on the 
combination of a chlorine atom with carbon and of hydroxyl 


with carbon, Thomsen’s numbers will have to be increased by i 
for each Cl or (OH). With the ethers, the value assigned to 
C—O—C will be 5 greater ; in the aldehydes, H—C=—O will 
become = greater ; in the ketones HC=O will be 5 greater ; 


and in the acids O—C—OH will become 4 


* In all these cases the numbers represent the heat of formation from 
gaseous atomic carbon and the other elements in the molecular condition. 


greater™. 


112 On the Constitution of Carbon Compounds. 


It will be remembered that, on the strength of the fact that 
the heat of formation of H—C=O in the aldehydes together 
with that of C—O in the ketones was found equal to that of 
O—C—(OH) in the acids, Thomsen arrived at the startling 
conclusion that the aldehydic radical consists of hydroxyl ; 
so that 

Aldehyde. Ketone. Acid. 
C—(OH) + O=C may be equal to O—C—(OH). 
Now the introduction of x destroys this equality; for the sum 
of the heat of formation of the aldehydic and ketonic radicals 


will exceed that of the acid radical by = and this excess brings 


the results into full accordance with the generally accepted 

views concerning the constitution of these bodies. The actions 

concerned in the formation of these groupings will be 
Aldehyde. Ketone. Acid. 
O—C—H O=C O—C—(OH) 


(O=C) + (C—H) (O—C) (O=C) + (C—( OH)). 


But (C—H) has been shown equal to (C—(OH) ); and hence 
the aldehyde + ketone will exceed the acid by (C=Q), which 


quantity, as shown above, will be within 15000 cal. of > 


It is important to remark that the value obtained by 
Thomsen for (C=O) in the ketones (54250) is by no means 
identical with that obtained for (C—O) in carbonic oxide 
(77670), showing that the heat developed is certainly depend- 
ent on the presence of other atoms in the compound, and not 
on the number of bonds concerned only. ‘This difference is 
still further increased if x be introduced into the calculation ; 


for (C=O) becomes 54250 + = whereas (C=O) becomes 
T7670 +2. 

A fuller study of Thomsen’s results can scarcely fail to 
bring to light a large number of important conclusions which 
have been omitted here ; and there can be few chemists who 
will not appreciate the services which Professor Armstrong 
has rendered to science in criticising a work of such extreme 
importance, and indicating the direction in which some of the 
apparently anomalous results to which it leads may be brought 
into accordance with views which we cannot afford to reject. 


Py ao 4 


XIII. . On the Front and Rear of a Free Procession of Waves 
in Deep Water. By Sir Witu1AmM Tuomson, /.R.S.* 


PRELIMINARY. 


General Problem of Deep-Sea Wave-Motion, in two dimensions. 
(Infinitesimal Motion.) 


et « horizontal, and y vertically downwards ; let 

(7+&, y+) be, at time ¢, the position of the particle 
whose position at time 0 is (z, y) ; let ® denote the velocity- 
potential at (z, y, t) ; and let P denote its time-integral, 


‘, dt®. We have 
t d® dP 


ANGD 
E= Coe Win? and n=( a? = 


dP 
af @.); 


Let p be the pressure at (2+&, y+). (The motion being 
infinitesimal,) we have 


db 
p=Ctgolytn—a ° > ° a ° ° (2 
or, in virtue of (1), 


ee aee 
PHOdsiteg 12 (3) 
The kinematical conditions are, the equation of continuity, 
a lo 
ae a dy i} ° ° ° e ° ° ° (4) ; 


and the boundary equa‘ion, in two parts—one relating to the 
upper surface, the other to the bottom. The latter, for our 
present case of infinitely deep water, is simply 


Oy Wile i ae te We ss Ne 


To find the former, or upper-surface kinematical equation, at 
time ¢, let it be y=0 at time 0, and let § be the height at 
time ¢ above the level y=0, of the upper-surface particle 
whose coordinates at time 0 are (z, 0). Remembering that 
y positive was taken as downwards, we have, by (1), 


=-(F) “yp he Ree eee Cp 


* Communicated by the Author; having been read before the Royal 
Society of Edinburgh, Friday, January 7th, 1887. 


114 Sir W. Thomson on the Front and Rear of 


The most general upper-surface dynamical condition which 
can be imposed is 
Py nA) oo ok A 


where f denotes an arbitrary function of the two independent 
variables. 

Suppose now the water to be at rest at time 0. It is clear 
from dynamical considerations that the solution of (4), subject 
to the conditions (5), (7), (3), is fully determinate : and when 
it is found, (1) gives the position at time t of the fluid-particle 
which at time 0 was in any position (#, y) ; and so completes 
the solution of the problem. 

The particular solution which we are now going to work 
out to represent a uniform procession of waves commencing 
at time 0, and produced and maintained by the application of 
changing pressure to the surface in the neighbourhood of the 
zero of #, must, as its appropriate form of (7), fuifil the 
condition 

Py=n= (2) snot+F (x) cosot . . (8); 


where (x) and F(x) denote functions which vanish for all 
large positive or negative values of 2. 

If we wish to make only a single procession, in the direc- 
tion of w positive for example, we may take 


S(2) =F @—to7/o*) >. eee 


A perfectly general formula is easily (by the Fourier- 
Poisson-Cauchy method) written down to express the value 
of P; and so, by (1) and (6), the complete solution of the 
problem: for §§ and F any given arbitrary functions. 

It is obvious that, so far as f is concerned, the general 
solution for w any considerable multiple of +/, and exceeding 
+/ by not less than two or three times the wave-length, 
27q/w’, must, for values of ¢ great enough to have let the 
front of the procession pass the place wz, be 


r 


for x positive, 


(10); 


and 
2 2 
b= sin | et te i +/)| — A cos [ot (—atf)] | 
for x negative, 


where I and f denote quantities calculable from the form of §;. 
and A and f similarly from F. Further, it is obvious that 
the front of each procession will, for any value of ¢ not less 


a Free Procession of Waves in Deep Water. 115 


than several times the period and not less than several times 
the time one of the wave-crests takes to travel through a 
space equal to /, be independent of the particular forms of § 
and F. From the theory of Stokes, Osborne Reynolds, and 
Rayleigh, we know that it advances at half the speed of a 
wave-crest ; but their theory, so far as hitherto developed, 
does not teach us the law according to which the front, as 
it advances, becomes longer and longer in proportion to /f, 
nor even the fact that it does become longer and longer. All 
the details of this interesting question are explicitly given 
in what follows: having been found with great ease for the 
particular case, 
oP +)F+b UR 
F@)=0, and g@)= {ES . an, 


where b denotes a length of any magnitude, which we shall 
take to be very small in comparison with 27g/w*, the wave- 


length. We shall in fact find that 


2 pes 1 bys. 
py-n=0t {© aes pin ft A), 


in the particular processional case of the general equations 
(1)...(6), which we now go on to work out. 

Remembering Cauchy and Poisson’s discovery that every 
surface of particles which are in a horizontal plane when un- 
disturbed, fulfils the condition of a free upper-surface (so that 
if all the water above it were annulled the motion of the water 
remaining below it would be undisturbed,) in the case of free 
waves of infinitely deep water; we see that when p,,_,)= const., 
we have also, in our notation, p = const., for every constant 
value of y. Hence, looking to (3) above, we must find, in 
the case of free waves, 

giP ad’ PR . ; 

dyad ae 3 Nash ar 6 25) f 

for every value of y, and not only at the upper surface, y=0. 
Thanking Cauchy and Poisson for this as a suggestion, but 
not assuming it without the proof of it which we immediately 
find; and borrowing now from Fourier* his celebrated 


“‘ instantaneous plane-source”’ { solution of his equation 
dv dr . ee 
a ca for thermal conduction, assume, as an imaginary 


* Théorre Analytique de la Chaleur. 
t+ W. Thomson’s Collected Papers, vol. ii. p. 46. 


116 Sir W. Thomson on the Front and Rear of 
type-solution of (4) and (13) for free waves, 


i] ae 


(b+y+ue)) Ds ole oot 


where s denotes»/—1. Whence, asa real solution by adding 
the values of (14) for ¢ and —s, and dividing by v2, 


—g@(y+6) 
p(t)= : { (7 acy + b)? cos ~ Ae: ens (r— y—b)! sine cae 


where r=[(y+))?+22} 


Curves representing calculated results of this solution for 
free waves were shown at the meeting of the British Associa- 
tion (Section A) at Birmingham in September, and at the 
last meeting (December 20). of the Royal Society of Edin- 
burgh. To build up of it a solution for a uniformly maintained 
procession of waves (a double procession it shall be, of equal 
and similar waves travelling in the two directions from «=0) 
take 


B= | deg) 0 a a 


and 
ee -f dt sin wt! P(t—t')= — {a sin w(¢—?’) Pi’)... (17). 


Since #(é), as we have seen, satisfies (13), P(¢) must satisfy it 
also. Hence 


dP(t) _ d°P(t) 
a ae ME Le 


for all values of z, y, and t. Now by differentiation of (17) 
we find, because P(0)=0, and by (16), 


ey t pone pa —t’) =—{: / / — 
aa (a sin wl! 7 P(t t dt’sin at’ p(t aes (19); 


and differentiating this, we find, because $(0)= (r+y+ by r-}, 


a ee ae |) i 
mare ee fa sin wt 7 p(t—t’) 


: a t 
ate ea Rae sin vt av sin w(¢— vy? —— - (20), 


(15 


a Free Procession of Waves in Deep Water. 117 
From this and the second form of (17), we find 
qe @FP (r+y +b)? 


Jy (ie wt 
Haba dP(t’)  d?P(t’) 
—i dis enue iS NEI pai atlee oN S, 
{ isin o(t—2) |g — Fe | - @1)s 
whence, by (18), 


dP d?P_ (r+y+b)’ 
Tay PY Gt NN eae pee A 2 2) 


and therefore finally, by (3) above, we have, for the surface 


pressure, 
bie (22402)? +5)2 . 
Fe Sint . ° (23), 


as promised in (12) above. 


To work out our solution, remember that dP/dt is the 
velocity-potential of the motion ; and calling this ®, we find, 


by (19), 
t 
o=—{ dé sintalé=t') b(@) 20d) ss 24)s 
0 
and by (22), (3), and (2) we find 
_1fd@ , (r+y +b)? 
hg dt , 


What we chiefly want to know is the surface-value of 7, 
which we have denoted by —; and we shall work this out 
for the case b=0. But it is to be remarked that the assump- 
tion of b>=0 does not diminish the generality of our problem, 
because the motion at any depth, c, below the upper surface 
with b=0, is the same as the motion at the surface, with b=c. 

Put now 6=0 and y=0 in a we find 


o (i) =27 (cost +sin 4 oa )=/2 sin( + i) @ 


Using this in (24), and ae a’ =gt/4a, we find 


= =2/- ce sin (wt—204 / £2) sin(o?+7) : PROT), 
aril bac onlle—e ys) ogee 
— cos [(c+e \/ 2) oS at “| } - (28). 


sin ot} Lp eek (2a). 


SL /E-/eon(oa/2)] (Sener 


The interpretation of this is eased by putting it into the form 


118 Sir W. Thomson on the Front and Rear of. 
Using now the following notation, | 
say 9—( ‘a0 sin 6 ; cay a=('a6 cos.6*'' _, SC2aye 
for two situ which have been tabula by Airy* through © 
the range from 0 to 554 / 7 we reduce (28) to 


" Ever -oy/*) )+say(o o/*)| [sin (F»—at—§) 
— | eay(¢ (/ L+o/f5 )—cay ( oy/*)| 008 (2+ at—4) 
-[oa 9/ Se +0y/s)-moy/2)] iat) 


p= ves Q cos (e—or—e)—R Cos (“2+0t~/) } a (31), 


where 


O= {Low (/Z-or/5) tenor) 
+ [ev(en/se-or/ too) Fe 
wal £--\/)@C/) Sa 
oa'4/ £-04/ 2) +en(04/ 2)” 
BaF [ow (4. /Et0a/3)- eo /OY 
+ [sou (n/a ren/5)— we (on /D)] FO 
De an Bie 3) ae 
oy (6 /Z t0a/5) ov (0/5) 


* “Tracts ” (Undulatory Theory of Optics, last page). 


e= tan 


an 


i Gam 


‘ 


a Free Procession of Waves in Deep Water. AS) 


Now, remembering that cay (©) = say («% =F g» We see 


that if oa/* is large, and if ¢ Paes \/" is large 
g ° 4a g ° 


positive, we have 
Oe te Ce ha he GOO) 


and therefore 


: 27 wx ) 
P— 7 08 (= an ONL Gs? CEs 


whence, and by (25) with b=0, 


b— 7 on/ = [sin in (2 at) — 2 " a ot | (38) ; 


or, since ad (z/g) is very large, 


pga 24/77 sin(=2 ot) 4 Fe ey 


This represents a uniform procession of free waves, of which 
the wave-length, X, and the wave-velocity, U, are as follows: — 


Ra eg a tee ee EOD 


To explain the meaning of “ very large” 


now used it, let 


; x aoag 1 
x=nX, which makes o/* = V2rn, and */ yo= lhe (41). 


as we have just 


Hence the term of (38) omitted in (39) is 1/4aV/n of that 
retained. And the value of the R, omitted by (86) in (37), 


is of the order 1/2V 2n of the Q een is retained, because 


cay (©) —cay (v 2arn)==— a 
and say (%©)—say (V 2an)== ed ae - (42), 
" 2V 2a0n 


when n is very large. 
In (36) and its consequence (31), we supposed ¢ so large 


that i, f-o/* =e large positive: let us next suppose 


¢ so small that it is large negative ; that is to say, let 


t= 2000 y—my [= Spee re (48); 


120 Free Procession of Waves in Deep Water. 


where m is a large positive numeric. Thus, remarking that 
cay (—@)=—cay (6), and say (—0) = —say (6), we have, by 
(43) and (41) in (82), . 


Qa a /= {eay (m) —cay (v Bmn)]?+ [say (m) ~say(v Bara) >} (44) 


and therefore, when m and n are each very large, Q=0. 
Because n is large we still, as in (86), have R=-0; and 
therefore the motion is approximately zero, at any consi- 
siderable number, n, of wave-lengths from the origin, so long 
as m in (43) remains large. As time advances, m decreases 
to 0, and on to —o«: and, watching at the place e=n\X, we 
see wave-motion gradually increasing from nothing, till it 
becomes the regular procession of waves represented by (39); 
and continues so unchanged for ever after. When m=0, 
that is to say, at the time 


t=2oz/g 9 . %\ . eee 


Q has attained half its final value. The point «x where this 
condition is fulfilled at time ¢ may be called the mid-front of 
the procession. It travels at the velocity 4q/o, or half the 
wave-velocity ; which agrees with the result of Stokes. 

We may arbitrarily define “ the front”’ as the succession of 
augmenting waves which pass between the times corresponding 
to m= +10 and m=-—10 (or any other considerable number 
instead of 10). Thus the time taken by the front, in passing 
the place z=nd, is 40m-1V 2an. The space travelled by the 
mid-front in this time is 20g@-!W 2an, which may, arbitrarily, 
be defined as the length of the front. It increases in pro- 
portion to ./n; and therefore in proportion to /t, as said 
above. The effect upon phase of the changigg waves in the 
front ; due to the fluctuations of e, and to the law of augmen- 
tation of Q from zero to its final value ; is to be illustrated by 
calculations and graphic representations, which I hope will be 
given on a future occasion. 

The rear of a wholly free procession of waves may be quite 
readily studied after the constitution of the front has been 
fully investigated, by superimposing an annulling surface- 
pressure upon the originating pressure represented by (12) 
above, after the originating pressure has been continued so 
long as to produce a procession of any desired number of 
wayes. 


Ror) 


XIV. Note ona Method of Determining Coefficients of Mutual 
Induction. By G. Carry Foster, £.L.S.* 


HE determination in absolute measure of the coefficient 

of mutual induction of two electric circuits by the ordi- 

nary method founded on the throw of a ballistic galvanometer 
is a somewhat complicated matter, necessarily occupying a 
good dealof time. But the process may be greatly simplified 
if we have available a condenser of which the capacity is 
accurately known. For instance, if P and § are two coils 
whose coefficient of mutual induction is required, let them 
first be joined up, as indicated in fig. 1—p in a primary 


Fig. 1. 


circuit containing a battery, B, and make-and-break key, K ; 
and § in a secondary circuit of total resistance r including a 
ballistic galvanometer G. Then, on closing or opening the 
key K, a momentary defiection of the galvanometer will 
occur in consequence of its being traversed by a quantity 


of electricity Q given by 
Q=M = 


where M is the coefficient of mutual induction between P and 

s, and y the strength of the primary current. Next, leaving 

the primary circuit unaltered, let connections be made as in- 

dicated by fig. 2, where c isa condenser of known capacity 
igs 2; 


C, and A and D are two points in the primary circuit sepa- 
rated by aresistance R. Then, on making or breaking contact 
* Communicated by the Physical Society: read November 27, 1886. 

Phil. Mag. 8. 5. Vol. 23. No. 141. Feb. 1887. K 


122. Prof. G. C. Foster on a Method of Determining 


at K, the galvanometer is traversed by a quantity Q’, such that 
QO’ = 70. 
If the points A and D are found by trial so that the deflection 
of the galvanometer is the same in both cases, we have 
‘M = CRr. 

This mode of working, however, has the obvious defect that 
the result, as stated, implies that the current in the battery- 
circuit is of exactly the same strength during each part of the 
experiment. As this cannot be looked for, it would be needful 
to include a measuring galvanometer in the battery-circuit, so 
as to take account of the variation of the current. The re- 
quired coefficient is then obtained in the form 


M = “ORr. 
V1 


But, instead of making two separate experiments, as above, 
it is simpler to adopt an arrangement of apparatus which is 
very nearly a combination of the two arrangements just 
described. A single experiment then takes the place of two, 
and, instead of having to reproduce a particular deflection of 
the galvanometer, we have to adjust a resistance so as to 
prevent deflection. Tbe connections will be understood by 
reference to fig. 8, where, so far as the reference-letters used 


Fig. 3. 


in the previous figures recur, they have the meanings already 
given to them. The observation consists in adjusting a set 
of resistance-coils at F, between the galvanometer and the 
coil s, until there is no throw of the galvanometer on making 
or breaking contact at K. 

Let the resistances ASFE, AGE, AD be represented by 
P, 4, 7 respectively, and the corresponding currents by 4, y, z. 
Further, let the current in the battery and primary coil be 


Coefficients of Mutual Induction. 123 


denoted by y, the coefficient of self-induction of the coil s by 
L, and the potentials at the points A and E by A and H 
respectively. 

When the battery-current has attained its steady value, it 
is evident that the currents w and y will both be nothing, and 
therefore that A=W, and that the charge of the condenser 
will be Cyr. But if there has been no throw of the galvano- 
meter-needle, the average value of the current y during the 
whole time of establishment of the battery-current has been 
=(0. Consequently, the total quantity conveyed by the cur- 
rent 2 has been equal to the charge of the condenser, or 


{ zdt = Cyr. 
0 


But, if the average strength of the current y=0, the average 
difference of potentials A—H=0, and the effective electro- 
motive force in the conductor p is that due to the mutual 
induction of the coils P and s (for the integral value of the 
electromotive force of self-induction must vanish). Hence 


a M = Cpr. 

In order that the galvanometer-current y may be zero at 
every instant, as well as on the average, during the establish- 
ment of the primary current, it is essential that the coefficient 
of self-induction, L, of the coil s should be equal to the 
coefficient of mutual induction M. ‘This may be proved 
as follows:—Since, in the case supposed, we have always 
A—H=0, we may write 

dx dy 
pet at M ae ==(() 
But wz=y—z (since y=0, always), consequently the instan- 
taneous value of the current wz is 


pak dry dz 
ae [ (ML) 2 + be 


and the simultaneous charge of the condenser is 
é 1 
C re ( aadt =F [ -L) y+Le |, 


or 


(M—L)y = (Cpr—L)z. 


But, since it has been already proved that M=Cpr when the 
K 2 


124 = Prof. G. C. Foster on a Method of Determining 


average current through the galvanometer vanishes, the last 


equation becomes 

(M—L)y = (M—L)z, 
which requires either (1) that y=z (that is, that the primary 
current has become steady), or (2) that M—L=0. 

When the last-mentioned condition is satisfied, we ought to 
be able to replace the galvanometer between the points A and 
E by a telephone, and to employ a rapid make-and-break *. 

By using a pair of coils of known and invariable coefficient 
of mutual induction, the experimental process described above 
may afford a ready way of determining the capacity of a 
condenser; but for this purpose the method does not appear 
to possess any advantage over the well-known methods of 
De Sauty and Sir William Thomson for determining the 
capacity of condensers by comparison with a known standard, 
more especially as a known condenser is more frequently at 
hand than a pair of coils of known coefficient f. 

The limit of accuracy attainable in measurements by the 
method here described depends essentially upon the sensi- 
tiveness of the galvanometer employed; but with a given 
galvanometer, the method is susceptible of various degrees of 
accuracy according to the relative values given to the two 
variable resistances p and r. 

These should be so adjusted that, for a given value of the 
difference Cpr—M, that is, for a given error in the adjustment 
of the product of the resistances p and r, the quantity of elec- 
tricity traversing the galvanometer may beas great as possible. 
This requires that the resistances should fulfil the condition 


—— — ——_ 


rx? 
where 7 is the resistance between A and p through the 
primary coil and battery. Let X and Y stand for the 


* Experiments in this direction, made since the reading of this paper 
before the Physical Society by Mr. F. Womack, have not yet led to a 
fully satisfactory result. 

+ Since this paper was read, Professor Roiti, of Florence, has kindly 
sent me a copy of a paper communicated by him to the Royal Academy 
of Sciences of Turin (Memoirs of the Academy, series ii., vol. xxxviii.), in 
which he describes a method, very similar to that indicated in the text, 
for the absolute measurement of the capacity of condensers, using for the 
purpose a pair of coils whose coefficient of mutual induction is accu- 
rately known from their dimensions and relative positions. The chief 
difference between Professor Roiti’s arrangement and that given in this 
paper is that, instead of inserting the galyanometer between the points 
marked a and # in fig. 3, he includes it in the branch, aF E, containing 
the secondary coil. Between A and £ he places a contact-key H, which, 
as well as the key K, is opened and closed mechanically, the two keys 
being moved by cranks attached to the same axle at right angles to 
each other. 


Coefficients of Mutual Induction. 5) 


integrals , wat and Syat respectively, that is, for the total 


currents through the secondary coil and through the galvano- 
meter. Then, equating the integral values of the electromotive 
force between A and E derived by considering the paths 
AGE and AFE, respectively, we get 
iH 
pxX q¥=M > 

Hi being here the electromotive force of the battery B. 
Again, the final charge of the condenser is 


X+Y=CE—.. 
[oa We 
Hliminating X from these two equations, 
1D) 
Cpr we LF jpg): 
and this is to be as great as possible. 
The denominator on the right may be written 


/! J 
pits £4); 
ep la els 
the last term inside the bracket is the product of the second 
and third, and may be taken as constant, since q, the resistance 
of the galvanometer, and 7’, the resistance of the part of the 
circuit containing the primary coil, are practically determined 
by the apparatus employed, while pr has the constant value 
M/C. Hence, Y/(Cpr—M) is greatest when (p+q)(7 +7") 
is least, or when p/g=7/r', which is identical with the con- 
dition of maximum sensibility given above”. 


In conclusion, I may give a few numerical results as 
examples of the applicability of the method ; they are derived 
from experiments made in the laboratory of University College, 
London, by Mr. F. Womack, B.Sc. 

A. Small Induction- Coil (without iron core). Approximate 
dimensions :—Primary: length 11°5 cm.; mean radius 2 cm.; 
wire, No. 20 B.W.G.; resistance 1°65 ohm. Secondary : 
length 10:4 cm.; radius, inside 2°55 cm., outside 3°83 cm. ; 
wire, No. 30 B.W.G.; resistance 194 ohms. Battery, 2 
Groves. Condenser, 4°926 microfarads (by direct measure- 
ment with ballistic galvanometer). The secondary coil could 
slide endways while remaining coaxal with the primary. The 
first measurements were made with the centres of the primary 
and secondary as nearly coincident as possible, so as to give a 
maximum coefficient of induction. The following are the 
results obtained :— 


* For the mathematical theory of the method, so far as it is giver 
above, I am greatly indebted to my friend Dr. A. H. Fison. 


126. Prof. G. C.-Foster on a Method of Determining 


Ohms 
ee Product jp 
r. _p (=secondary coil+resistance-box). (absolute units). 
15 194+ 217 6165 x 10'* 
14 +247 6174 
13 + 282 6188 
12 +322 6192 
11 +367 6171 
10 +493 6170 
) +490 6156 
8 +576 6160 
7 +688 6174 
6 +835 6174 
Mean value of M/C = _ 6172-4x 10 


Hence 
M=4-926 x 10-® x 6172 x 10°=3-0403 x 107. 
In these experiments the value of g/r’ was about 
135/(1°65 + °6) = 60. 
Hence the greatest sensibility would be with p=60r. This 
condition was nearly fulfilled with 
e—10 and p= 1944+ 423=617. 

In the same way the values of M were obtained for the 
same pair of coils after displacing the secondary coil endways 
through various distances. The following are the results 
obtained :— 

Distance between 


centres of coils. Value of M. 

centim. 

0°55 304°0 x 10° 

155 292°4 

rai: 240-5 

3°DD 246°4 

4°55 Palins 

5°dD 187°8 

6°55 158-4 

T'd5 122 

8°55 97:3 

9°55 71:1 
10°55 49°7 
11°55 33°0 
12°55 2a3 
13°55 16°5 
14°55 Irae 
15°55 9°48 


These values are represented graphically in the curve (fig. 4), 


Coefficients of Mutual Induction. 1a 
Fie. 4, 


Ordinates x 10°=Coefficients of Mutual Induction. 
ry 
ler} 


1 2 3 £ 5) 6 7 8 SE LOE a eet 2 3) 14 15 NG 


Abscissee Xx 2= Distances between centres of Coils. 


where ordinates denote values of M and abscisse distances 
between the centres of the coils. 

B. Induction-Coil, by Apps, capable of giving a 7-inch 
spark in air (presented to the University College Laboratory 
by Mr. J. Rose-Innes, B.Sc.).  Juength of secondary coil 
21 cm., diameter (measured outside velvet covering) 11°3 cm. 
Resistance of primary wire=0°278 ohm (at 16°5); resistance 
of secondary wire 7394 ohms (at 16°5). Battery, 1 Grove. 
q==135°6 ; 77=0°58 (about). Condenser, 4:°926 microfarads. 


Ohms. | 


Pehle WURONAR TS TA 2) "hs pr 
ie pp. (absolute units). 
27 7394 + 1550 2-415 x10? 
28 +1250 2°420 

29 + 940 2°417 

30 + 650 2°413 

31 + 390 2°413 

32 +- 150 2°414 


Mean value of M/C = 2-415 x 10 


128 Method of Determining Coefficients of Mutual Induction. 


Hence 
M=4:926 x 10-® x 2-415 x 102 
= 1°1896 x 10°. 

As a further test of the accuracy of the method, the 
secondary wires of Apps’s coil and of the small coil (A) were 
connected in circuit with each other and with a galvanometer, 
and the two primaries were connected so that the battery- 
current was divided between them, as shown in fig. 5. The 


Fig. 5. 


connections being arranged so that the induced electromotive 
forces opposed each other, it was possible, by the proper 
adjustment of the resistances of the branches containing the 
two primary coils, to prevent the galvanometer being deflected 
when the battery-circuit was made or broken. When this adjust- 
ment is made, it is evident that the ratio of the resistances of 
the two primary circuits is the same as the ratio of the two co- 
efficients of mutual induction. In this way, 39°45 was obtained 
as the ratio of the coefficients ; whereas the condenser method 
118-96 
30403 
about 1 per cent. As, however, the current in the primary 
of the smaller coil has to be about forty times as strong as 
that in the large coil in the comparison experiments, in order 
that the induced electromotive forces may balance—and as, 
moreover, the resistance of the copper wire of the former was 
a comparatively large fraction, at least one sixth or one seventh, 
of the whole resistance in circuit with it, while the copper 


described above gives 


= 39°13, giving a difference of 


On the Nature of Liquids. 129 


resistance of the larger primary coil was not much more than 
one fifteen-hundredth of the whole resistance in circuit with it 
—the ratio of the resistances was no doubt somewhat disturbed 
by the unequal heating of the two primary circuits, and was 
in reality rather less than what was inferred from the marked 
values of the coils used. ‘That this was the case was shown 
by the fact that the apparent ratio decreased progressively 
from 40°3 to 39°45, as the strengh of the testing current was 
diminished from its first value to rather less than one sixth. 
A better arrangement of the apparatus would have been to 
put the two primary wires in series with the battery, and to 
have connected the two secondaries in parallel circuit; but 
the matter was not thought important enough to require a 
repetition of the measurements. 

The method of measuring coefficients of mutual induction 
described in this paper may perhaps be of use in the experi- 
mental study of dynamo-electric machines, whose whole action 
depends upon the variation of the coefficient of mutual induc- 
tion between the field-magnet coils and the armature coils, as 
the latter take various positions during the course of a 
revolution. 


XV. On the Nature of Liquids, as shown by a Study of the 
Thermal Properties of Stable and Dissociable Bodies. By 
Wiuram Ramsay, Ph.D., and SypNEY Youne, D.Sc.* 


Peas fundamental concept of Chemistry, as well as of 
Physics, is the molecular and atomic constitution of 
matter. This concept serves to represent to the chemist the 
_ definite composition of compounds, and, to some degree, the 
nature of isomerism, while all attempts to realize and explain 
the progress of chemical change depend on its adoption. 
This concept also furnishes to the physicist the means of 
conceiving the relations of heat, light, magnetism and elec- 
tricity to matter ; and where the action of one of these agents 
involves not merely a change in the form, but also in the 
nature of the matter, the problem becomes of deep interest 
to both chemist and physicist. The action of heat on matter, 
from the physical side, involves an increased molecular 
motion, tending to separate individual molecules from each 
other, on the one hand ; or, on the other, if this separation be 
opposed by confining walls, to increase the momentum and 
number of impacts on those walls, and therefore to raise the 
pressure. But this increased molecular motion is accom- 


* Communicated by the Physical Society : read December 11, 1886. 


130 Drs. Ramsay and Young on 


panied by greater internal vibration, which eventually leads, 
in almost all cases, to a simplification or rearrangement of the 
molecules, involving chemical change. When increased 
molecular motion is imparted to gases at temperatures much 
above their points of condensation, and at moderate pressures, 
the problem is a comparatively simple one ; and has been 
solved with great success by Clausius, Maxwell, Thomson, 
and others, from the physical side, and from the chemical side 
by Pfaundler, Naumann, and Willard Gibbs. But near their 
condensing points, and also at high pressures, Boyle’s and 
Gay-Lussac’s laws no longer hold, owing partly no doubt to 
the mutual attraction of the molecules, and also to the fact 
that the absolute size of the molecules is no longer insigni- 
ficant relatively to the space which they occupy. Both these 
causes of deviation may be relegated to the class “ physical,” 
inasmuch as the mutual attraction alluded to is not confined 
to any small number of molecules, but is exercised by each 
molecule on all its neighbours, and limited in absolute amount 
only by the relative masses of the attracting molecules 
and by their distances from each other. But it is also con- 
ceivable that this attraction may be wholly or in part of a 
chemical nature, tending towards the formation of complex 
molecules, resulting from combination of two or more simple 
molecules. Now as this deviation from the simple gaseous 
laws occurs both with what are commonly termed “ stable” 
and with ‘ dissociable’’ substances, it is of importance to 
enquire whether the abnormality of the vapour-density of 
stable substances is at all due to chemical association of mole- 
cules ; and how much of the abnormality of dissociable sub- 
stances is to be ascribed to purely physical attraction of the 
molecules for each other, due to mere propinquity. 

At any temperature below the critical one, when the volume 
of gas is decreased, pressure rises until a certain maximum is 
attained, when it becomes constant, and change of state 
occurs. It is conceivable, on the one hand, that the liquid 
condition is a purely physical one, and that a liquid consists 
of molecules similar in all respects to those of its gas, but, 
owing to their closer proximity, exhibiting that form of at- 
traction which is known as cohesion. And on the other hand, 
it has been advanced by Naumann and others that the gaseous 
molecules, in changing to liquid, form molecular groups of 
definite complexity, exercising cohesive attraction on each 
other ; and, according to this view, the problem is both a 
physical and a chemical one. According to the first view, if 
heat be imparted to a liquid, work is done in expansion 
against pressure, and in overcoming cohesion ; and, according 


the Nature of Liquids. 131 


to the second view, additional work is done in dissociating the 
complex molecules into their simpler constituents, and in 
imparting increased velocity and internal motion to those 
constituent molecules (see ‘ Evaporation and Dissociation,” 
part i., Trans. Roy. Soc. 1886, Part I.). 

When a substance, such as chloral hydrate or ammonium 
chloride, passes from the solid or liquid into the gaseous state, 
the physical change is obviously accompanied by a chemical 
one, for dissociation into simpler molecules occurs. There is 
an obvious analogy between evaporation and such cases of 
dissociation ; and we have recently undertaken experimental 
work to test whether this analogy is a real one. 

In part i. of this series of papers the phenomena attending 
the volatilization of such solids as dissociate wholly or partially 
on their passage from the solid to the gaseous state have been 
studied. There are two ways of measuring the vapour- 
pressure of a stable substance, which have been termed by 
Regnault the statical and the dynamical respectively. The 
first consists in measuring the pressure exercised by the 
vapour of the substance kept at a uniform temperature ; and 
the second in measuring the highest temperature attainable 
by the substance at given pressures, when evaporation freely 
takes place. It has been shown by Regnault, and by nume- 
rous other observers, that these methods give identical results 
with liquids, and by ourselves with solids (Trans. Roy. Soc. 
Part I. 1884, p. 87). But in the case of the majority of the 
dissociable bodies examined, the results of the two methods 
were not identical; indeed, in many cases in which dissociation 
is complete, or nearly so, the temperature of volatilization is 
independent of pressure. With nitrogen peroxide, acetic 
acid, and ammonium chloride, however, the two methods 
gaye identical results. This method, therefore, cannot be re- 
garded as a means of deciding the question of the analogy 
between evaporation and dissociation, unless, indeed, two 
kinds of chemical combination be conceived, one of which 
may be termed ‘‘molecular combination” as distinguished 
from “ atomic combination.” 

In parts ii. and i. the thermal behaviour of stable liquids 
has been investigated, as exemplified by alcohol and ether. 
For a complete account of these researches reference must be 
made to the original papers (Trans. Roy. Soc. 1886, Part 1.*). 
We are here concerned chiefly with the densities of the 
saturated vapours, and with the heats of vaporization. We 
found, with alcohol, that the density of the saturated vapour 
was normal at temperatures below 40° or 50°, and remained 


* The constants for ether will be published shortly. 


——— 


= == 


132 Drs. Ramsay and Young on 


normal down to a temperature of 13°, the lowest temperature 
at which observations could be made. With ether the vapour- 
density was approaching normality at 13°, and from the form 
of the curve would have doubtless become normal at a lower 
temperature. In both cases, with increase of temperature 
and corresponding increase of pressure, the density of the 
saturated vapour increased towards the critical point with 
great rapidity, until at the critical point the weight of unit 
volume of the saturated vapour was equal to that of the 
liquid. 

At the critical point the heat of vaporization of a stable 
liquid is theoretically zero ; below that temperature we found 
it to increase with alcohol and with ether as the temperature 
fell; with ether the increase was found to be continuous to the 
lowest observed temperature 13°; whereas, with alcohol, it 
becomes practically constant below about 20°. Our calculated 
numbers correspond well with direct measurements by various 
observers at the boiling-points under atmospheric pressure. 


Fig. 1. 


Vapour-density (H=1 at ¢° and p millim.). 
Alcohol. 


With acetic acid the results were very different. With 
rise of temperature above 150° the density of the saturated 
vapour increased, as with other liquids ; but below that tem- 
perature (at which the vapour-density was 50:06, the calcu- 
lated density being 30) the vapour-density, instead of con- 
tinuing to fall, rose more and more rapidly with fall of 
temperature, until at 20° the vapour-density was approxi- 
mately 59, and apparently, from the form of the curve, was 
continuing to rise more and more rapidly, with fall of tem- 


the Nature of Liquids. 133 


perature (see figs. 2 & 3). It may be mentioned that direct 
observations by Bineau at 20° give nearly the same value. 
Fig. 2. 


°o 
$00— 
ate ne 30000 
IS mms 
40090 
7 
200|— { 


100] ns | 
100 


IS Z20mms 
| 
30 40 50 60 
Vapour-density (H=1 at ¢° and p millim.). 
Acetic Acid. 


The curve representing heats of vaporization of acetic acid 
at various temperatures also differs entirely in form from those 
of alcohol and ether, for it exhibits a maximum at 110°, and 
decreases both with rise and with fall of temperature. It is 
difficult to draw any conclusion from a comparison of our 
measurements of this quantity at the boiling-point under at- 
mospheric pressure with those of other observers ; but it may 
be stated that our result differs far less from the observation 

Fig. 5. 


0° = 
300calories 200 100 0 


of Favre and Silbermann than theirs does from that of 
Berthelot (see fig. 3). 


134 Drs. Ramsay and Young on 


It appears to us that these results negative the “‘ chemical ” 
explanation of the constitution of liquids, or, to confine our- 
selves to known cases, of the liquids alcohol and ether. The 
molecules of these liquids cannot, we think, be regarded as 
complex, consisting of gaseous molecules in chemical combi- 
nation with each other, as, for example, n(C.H,O), where 7 is 
any definite number. We believe, rather, that the physical 
explanation of the nature of liquids is the correct one, and 
that the difference between liquids and gases les merely in 
the relative proximity of their molecules. 

The chief argument for this view is that it is difficult to 
conceive that the rise of vapour-density of acetic acid, both 
at high and at low temperatures, can be due to the same 
cause, under conditions so radically different; for at high 
temperatures we have conditions unfavourable to chemical 
combination, but owing to the necessarily high pressure, the 
molecules are in close proximity ; whereas, at low tempera- 
tures, the conditions are favourable to chemical combination, 
while the molecules, owing to the corresponding low pres- 
sures, are very far apart. Now we have shown that, with 
alcohol and with ether, a rise of density does not accompany 
fall of temperature ; indeed, the saturated vapour of alcohol, 
at low temperatures, obeys the laws of Boyle and Gay-Lussac; 
while the rise of vapour-density at high temperatures is com- 
mon to all bodies. But with acetic acid, the lower the tem- 
perature the higher the density of its saturated vapour—a fact 
which indicates the formation of complex molecules ; at high 
temperatures, however, it forms no exception in behaviour to 
ordinary liquids. 

We have shown that with stable substances there is proof 
of the absence of complex molecules in their vapours; but it 
might be asserted that in the passage from the gaseous to the 
liquid state, combination might occur. ‘That this cannot be 
the case, is evident from a consideration of the behaviour of 
liquids near their critical point. For the specific volumes of 
liquid and gas just below the critical point are nearly equal; 
and were the liquid to consist of congeries of gaseous mole- 
cules, there would necessarily be fewer molecules in unit 
volume of the liquid than in unit volume of the gas—an im- 
probable conception. 

It is impossible to decide from our experiments whether 
the higher limit of vapour-density of acetic acid is 60; and 
the difficulty of measuring small pressures with sufficient 
accuracy renders an answer to this question apparently im- 
possible ; but it is a remarkable circumstance that our observa- 
tions, as well as those of Bineau, should so closely approximate 


the Nature of Liquids. 135 


to this limit. Although the curves representing the density 
of the saturated vapour in figs. 1 and 2 apparently point to a 
vapour-density greater than 60, yet a trend in the curve is 
not impossible ; and it is conceivable that at lower tempera- 
tures than those represented, the density might remain normal 
for C,H,Q,. 

If there is a definite limit to the vapour-density of acetic 
acid, then the following considerations will hold. It has been 
pointed out in our paper on acetic acid, that condensation 
took place before pressure ceased to rise ; and the same phe- 
nomenon was observed with chloral ethyl-alcoholate, where 
dissociation is known to occur. Now with alcohol and with 
ether absolutely no sign of this behaviour was observable ; 
condensation occurred the moment the vapour-pressure was 
reached, but not till then. This behaviour corresponds to that 
of a mixture. If an indifferent gas, to take an extreme 
instance, is compressed along with the vapour of a conden- 
sable liquid, pressure continues to rise after condensation has 
commenced, until the gas, if possible, has been dissolved, or 
has itself condensed. On the other hand, if a small quantity 
of liquid of high boiling-point be present along with a large 
quantity of liquid of low boiling-point, the liquid of higher 
boiling-point separates out first, on reduction of volume, 
while pressure continues to rise. This was indeed noticed 
with an impure sample of ether; and the absence of this 
behaviour affords proof of the homogeneity of a liquid. 

Supposing the vapour of acetic acid to consist of molecules 
of two different degrees of complexity, it is probable that the 
more complex would be first condensed, and that pressure 
would rise until the less complex molecules had also con- 
densed. This was in fact observed. But below a certain 
temperature the substance would consist almost wholly of 
more complex molecules, and the phenomenon would then be 
less visible. This is indeed the case with the isothermals at 
50° and at 78°-4. At higher temperatures the phenomenon 
becomes evident. That this behaviour is not the effect of im- 
purity has been proved by the fact that the vapour-pressures 
at low temperatures, measured by the statical and by the 
dynamical methods, were identical. 

Formule representing the dependence of dissociation on 
pressure and temperature have been proposed from thermo- 
dynamical considerations by Prof. Willard Gibbs*. The 


formula is for acetic acid 


Joo 2073(D—2:073) __ 3520 
8 (£146—D)? t+ 278 


* American Journal of Science and Arts, 1879, p. 277. 


+ log p—11°349. 


136 Drs. Ramsay and Young on 


The numbers 2°073 and 4°146 are the densities referred to 
air of the molecules C,H,O, and C,H,O, respectively ; D is 
the observed density; and 3520 and 11°349 are constants 
deduced from the determinations of Cahours and Bineau. 
This formula, of which the constants in its author’s opinion 
can claim only approximate correctness, is quite inadequate 
to represent actual facts at high temperatures and high 
pressures where cohesion becomes marked. For example, it 
gives at a temperature of 280° for the density of the saturated 
vapour 35°13 instead of the observed number 62°62. 

If our opinion be correct, and if the abnormal density of 
saturated vapours and of vapours near their saturation-points 
and also above their critical points, at high pressures, of stable 
substances, be due to mere molecular proximity, and not to 
any form of molecular combination ; then a dissociating sub- 
stance must exhibit a vapour-density which may be partly 
due to this cause. With such a substance as ammonium 
chloride, which, we have shown, is almost compietely disso- 
ciated at 280°, the products of dissociation (hydrogen chloride 
and ammonia) are under such conditions of temperature and 
pressure that they would probably behave as perfect gases ; 
the relatively few molecules of ammonium chloride which 
remain undecomposed in the gaseous state are under such low 
pressure, that their density is probably normal for the formula 
NH,Cl; and in this case it is probable that the chemical 
factor alone determines the vapour-density. But with acetic 
acid the increase of density above 150° is evidently wholly 
due to the physical cause ; while the abnormality is partly 
due to a physical, partly to a chemical, cause. It is, how- 
ever, impossible in this case to ascertain at what temperature 
the physical cause begins to operate. It is evidently to be 
wished that, from a study of the behaviour of stable substances, 
some general law could be discovered which would embrace 
all instances of physical abnormality ; and many attempts 
have been made in this direction, but as yet with only partial 
success. Willard Gibbs, on the other hand, has attacked the 
problem from the chemical side ; and we have shown that his 
formula ceases to apply when the physical change becomes 
predominant. 

Messrs. E. and L. Natanson* have recently published a 
research on the vapour-densities of nitric peroxide (N,O, or 
NO,), which, taken in conjunction with experiments of ours 
on the vapour-pressures of that body (Phil. Trans. 1886, 
Part I.), affords a striking confirmation of the correctness of 
our views. They give an isolated observation at —12°6; 


* Wiedemann’s Annalen, 1886, p. 606, 


the Nature of Liquids. 137 


and isothermals at 0°, 21°, 49°-7, 73°°7, 99°°8, 129°-9, and 
151°°4. The limit of pressure was 800 millim. Now the 
boiling-point of nitric peroxide is, from our measurements, 
21°°8; hence the densities of the saturated vapour are de- 
ducible from only the first three of the Messrs. Natansons’ 
isothermals. We have plotted their results on curve-paper ; 
this has shown us the regularity and trustworthiness of their 
observations ; and by continuing the curves in the direction 
in which they run until they intersect the straight lines 
denoting vapour-pressures at the temperatures at which their 
measurements were thade (using for this purpose the vapour- 
pressures determined by us), the density of the saturated 
vapour is determined with but small error. 
The Natansons’ numbers are as follows :— 


Temp. Pressure. | Density. Temp. Pressure. | Density. 
a millim. millim. ° millim. millim. 
—126 115-4 52°54 21 491-60 38°74 
0:0 37°96 35°84 516°96 39°01 
86°57 38°59 55350 39°15 
172-48 40-71 639°17 39°64 


250°66 41:90 


At —12°:6 the vapour-pressure of nitric peroxide is 125 mm. 
The density of the saturated vapour must therefore be a little | 
above 52:54. Now the theoretical density of N.O,is 46. It 
may be that the higher density is due to experimental error; 
but from graphic representation of the Natansons’ results 
this appears improbable. If the measurement is correct, it 
would imply that the chemical combination of molecules of 
NO, is not complete when the molecular complexity is repre- 
sented by the formula (NO,)2, but may extend to (NO,)3, or 
even further. At 0° the vapour-pressure is 255 millim.; again 
the density found by the Natansons must be nearly that of 
saturation. At 21° the vapour-pressure is about 700 millim. ; 
and a prolongation of the curve constructed from the above 
numbers would cut the line representing the large alteration 
of yolume with no rise of vapour-pressure at a vapour-density 
of about 40. It is evident, then, that with nitric peroxide, as 
with acetic acid, the density of the saturated vapour rises with 


fall of pressure and temperature. Now it is known that nitric 


peroxide dissociates, for the physical properties (colour, &c.) 
change, on change of (NO,)n into n(NO,) ; and the similarity 
of behaviour between nitric peroxide and acetic acid renders 


Phil. Mag. 8. 5. Vol. 23. No. 141. Feb. 1887. L 


138 Mr. W. N. Shaw on the Atomic 


the dissociation of acetic acid from (C,H,O,), into n(C,H,O2) 
no longer conjectural. 

If it be granted that our thesis is proved, that the molecules 
of stable liquids are not more complex than those of their 
gases, it follows that the difference between liquids and gases 
is one of degree, not of kind; is quantitative, and not quali- 
tative. 

University College, Bristol, 

November 18, 1886. 


ERRATA 27 article in January number :—Page 62, lines 5 and 4 from bot- 
tom, for p=a+bat+c@t read log p=a+bai+cpet; for p=a+bat read 
log p=a+bet. Same correction on p. 64, fifth lme below table; and 
p- 68, line 1. P. 64, line 3 below table, for 25°54 millim. read 26°54 
millim. P. 68, line 1, for log =0°5784772 read log b=0°5784772. 


XVI. On the Atomic Weights of Silver and Copper. 
By W. N. Suaw, I.A.* 


I lia the table of atomic weights given by Landolt and Born- 

stein in their Physikalisch-Chemische Tabellen, p. 1, that 
of copper is quoted as 63°18 from L. Meyer and K. Seubert, 
and as 63°17 from Clarke. These numbers give the chemical 
equivalent of dyad copper as 31°59 and 31°585, respectively. 
If we take the value of the atomic weight of silver as 107-66, 
we get for the ratio of the chemical equivalents of silver and 
copper 3°4080 or 3:4086. 

The atomic weight of silver is given as one of those whose 
accuracy is of the first order, with a possible error less than 
‘05 ; that of copper is, on the other hand, placed in Class II., 
for which the possible error may reach 0'5. The possible 
limits assigned by this to the ratio of the chemical equivalents 
are 3°381 and 3°435. The principal determinations of the 
atomic weight of copper are by Berzelius, Hrdmann and Mar- 
chand and Hampe ; and the methods used may be called strictly 
chemical. Hampe used an electric current to extract all the 
copper from a solution of the metal, and further tried to de- 
termine the ratio of the equivalents by comparing the electro- 
lytic deposits of silver and copper in the same circuit ; but he 
abandoned the method as unsatisfactory. I have communi- 


-eated to the Hiectrolysis Committee of the British Association 


the details of a number of experiments carried out at the 
Cavendish Laboratory under my direction, designed to deter- 
mine the amount of copper deposited in cells with different 
current-densities from a nearly saturated solution of copper 


* Communicated by the Author. 


Weights of Silver and Copper. 139 


sulphate, and, if possible, to establish some formula of correc- 
tion by which the deposit of copper at any current-density 
could be reduced to that at a standard density. The results 
show that the amount of copper deposited tends to a maximum 
value when the current-density reaches the highest limit with 
which it will give coherent deposits, this limit being about 
‘13 ampere per square centimetre. The amount of deposit 
does not, however, vary appreciably from this until the 
current-density is below ‘025 ampere per square centimetre. 
For current-densities less than ‘02 and down to ‘0014 the 
amount of deposit can, with fair concordance between the 
results of different experiments, be reduced to that at the 
00002 

d 7 
where d is the current-density ; although it is possible that a 
slightly different value may have to be assigned to the nume- . 
rator of the fraction when all the circumstances of the state 
of the solution, temperature, and other variables are taken 
into account. 

The fact that the amount of deposit tends to a limiting 
_ value for high current-densities, but not for low ones, has led 
me to examine the results of adopting this as a method of 
determining the atomic weight of copper from that of silver. 
And the first result that appears is that the ratio of the che- 
mical equivalents of silver and copper is given, as the mean 
of fifteen final experiments with current-densities above °025, 
as 3°39983 ; which differs so little from 3°4000 that this value 
may be adopted as the true ratio within the limits of error of 
the experiments. The amounts of copper deposited were 
generally speaking, about 1 gramme, and were weighed to 
0-1 milligramme. The greatest error from the mean of the 
fifteen experiments is (0058, or nearly *2 per cent.; while the 
mean error is ‘00175, or less than ‘06 per cent. 

The ratio has been determined experimentally by Lord 
Rayleigh in three experiments with platinum bowls, quoted 
in the paper by himself and Mrs. Sidgwick in the Philoso- 
phical Transactions, Part II. 1884, p. 458. His results are 


inet with current-density about ‘012 ampere per square 


higher current-densities by multiplying by a factor 1 + 


centimetre, and 3°404 with a current-density about -026. 
And the subject was brought to the notice of readers of the 
Philosophical Magazine by the paper by Mr. T. Gray, last 
November. Mr. Gray’s number for the ratio is 38-4013, when 
the current-density is ‘02 ampere per square centimetre. The 
number that I have here given is derived from experiments 
with platinum-wire cathodes, generally with higher current- 


140 Atomic Weights of Silver and Copper. 


densities, and is correspondingly lower. At any rate the 
results point to a limiting value for the ratio differing very 
little from 3°4000. 

Now if the atomic weight of copper be taken to be 63°18, 
as determined by purely chemical methods, the ratio would 
be, as stated above, 3°4080. This is the ratio obtained, as the 
result deduced from the experiments I am referring to, when - 
the current-density is ‘(0085 ampere per square centimetre. 
The variation with current-density is, however, quite con- 
tinuous through that value ; and there seems to be no good 
reason for supposing that there are a number of secondary 
actions which exactly balance each other at that value. More- 


~ over the number 3°400 lies well within the possible limits 


3°381 and 3°435, assigned in Landolt and Bornstein’s table. 

Accepting, then, the value of the ratio of the chemical 
equivalents of silver and copper as 3°4000, it follows that the 
ratio of the atomic weight of silver to that of copper is 17: 10; 
and if the atomic weight of silver be 107°66, the atomic weight 
of copper is 63°333. These two numbers are evidently thirds 
of whole numbers, and we get the following numerical rela- 
tions, which are somewhat remarkable :— 


323 17x19 


Cu = 63:53— ee 
Ag 17 
Cu #10 


There are many lines of speculation which start from the 
grouping of whole numbers suggested by these results. For 
instance, if we refer the atomic weights to a unit which is one 
third of the weight of the hydrogen atom, the numbers for 
silver and copper are whole numbers resolvable into factors. 
The resolution of the numbers into factors may correspond to 
the different valencies of the elements: thus the only possible 
valencies for silver would be 1, 17, and 19; for copper, 1, 2, 
5,and 19. The several groups might then be regarded as 
practically separate atoms, each associated with the same 
charge of electricity in electrolysis as is always associated with 
a monad atom. The valencies 1 for silver and 1 and 2 for 
copper do actually occur. If the atomic weight of oxygen 
referred to this unit be regarded as 48, the number of possible 
valencies would be very great ; and the same may be said of 
carbon, with the atomic weight 36 referred to the same unit. 


Foundations of the Kinetic Theory of Gases. 141 


If a compound of silver with a valency 17 or 19, or one of 
copper with a valency of 19, were known, this suggestion 
might become a somewhat plausible hypothesis; in their 
absence it must, I fear, be regarded merely as an idle specu- 
lation. The only reaction of silver that has occurred to me 
_as likely to give any evidence of suck a compound is its 
behaviour with oxygen when melted. I have not, however, 
been able to find the composition of the compound or mixture 
given more accurately than by the statement that silver gives 
out twenty-two times its volume of oxygen when it is nearly at 
the point of solidification. Taking this rough statement of 
the composition and assuming values for the density of the 
silver and oxygen, the formula for the compound works out 
to be Agi,O, which is sufficiently near to AgO_., to make one 
wish that the composition were more accurately known. 

It is perhaps merely a remarkable coincidence that, taking 
the atomic weight of potassium as 39-1, and that of sodium as 
23, the ratio of the two gives 


| rae | re 

NG = 10 Cxactly 5 

or the same as the ratio of the atomic weights of silver and 
copper ; and in the absence of more complete certainty in the 
determinations of atomic weights it may be unwise to speculate 
about the matter. 


Cavendish Laboratory, 
December 1886. 


XVII. On the Foundations of the Kinetic Theory of Gases. 
Part II. By Professor Tarr*. 


se a former paper (of which a brief abstract appeared in 

the Philosophical Magazine for April 1886, p. 348, and 
which has since been printed in full in Trans. Roy. Soc. 
Kdin.) I showed that the recovery of the “ special ”’ state by 
a gas supposed to consist of equal hard spheres takes place, at 
ordinary pressures and temperatures, in a period of the order 
of 10~° seconds, at highest. 

This forms the indispensable preliminary to the present 
investigation. For it warrants us in assuming that, except 
in extreme cases in which the causes tending to disturb the 
“special ”’ state are at least nearly as rapid and persistent in 


* Abstract of Papers read to the Royal Society of Edinburgh, Decem- 
ber 6, 1886, and January 7, 1877, Communicated at the instance of Sir 
W. Thomson. 


142 Prof. Tait on the Foundations of 


their action as is the tendency to recovery, a local “ special ” 
state is maintained in every region of the space occupied by a 
gas or gaseous inixture. This may be, and in the cases now 
to be treated is, accompanied by a common translatory motion 
of the particles (or, of each separate class of particles) in the 
region—a motion which at each instant may vary conti- 
nuously from region to region, and may in any region vary 
continuously with time. 

A troublesome part of the investigation is the dealing with 
a number of complicated integrals which occur in it, and 
which (so far as I know) can be treated only by quadratures. 
All are of the form 

yur 


é 


0 

where v is that fraction of the whole number of particles of 
one kind per cubic unit whose speeds (relatively to those of 
the same kind, in the same region, as a whole) lie between v 
and v+dv; and 1/e is the mean free path of a particle whose 
speed is v. Throughout the paper regard has been had,to the 
fact that e must be treated as a function of v. So longas the 
particles are of the same kind, or at least of equal mass if 
of different diameters, such integrals are easy to evaluate ; 
but it is very different when the masses differ in two mixed 
gases. In what follows, the merely numerical factor of the 
expression above will be denoted by C,, so that the value of 
the expression is, when the masses and diameters are equal, 
C,./nms?h"?, and the introduction of different diameters merely 
introduces another factor. Here 3/2h is the mean square speed, 
n the number of particles per cubic unit, and s their common 
diameter. 

When the masses are unequal there will, in general, be 
different mean free paths for particles of two different kinds, 
and the integrals cannot be simplified in the above way. In 
this case the integrals will be expressed as , ,, of, 

(1) In the first part of the paper I showed that the Virial 
equation is, for equal hard spheres exerting no molecular 
action other than the impacts, 

nPv?/2=3 p(V —2n7s*/3), 
where ” is the number of particles, P the mass of one, s its 
diameter, v? the mean-square speed, p the pressure, and V the 
volume. The quantity subtracted from the volume is four 
times the sum of the volumes of the spheres ; and I pointed 
out that this expression exactly agrees in form with Amagat’s 
experimental results for hydrogen, which were conducted 


the Kinetic Theory of Gases. 143 


through wide ranges of pressure, and between 18° C. and 
100° C. 

In a mixture of equal numbers of two kinds of particles, of 
diameters s,, so, I find that for s* in the above formula we 
must put 


+(5,° + 25° + s,°), 


where s=(s,+58,)/2. Thus the “ultimate volume”’ is in- 
creased if the sizes of the particles differ, though the mean 
diameter is unaltered. 

(2) For the coefficient of viscosity in a single gas the value 


found is 
en ph 
Barns? / h a WE 


where p is the density, and X the mean free path. The pro- 
duct pA is the same at all temperatures, so that the viscosity 
is as the square root of the absolute temperature. 

(3) The steady linear motion of heat in a gas is next 
considered, temperature being supposed to be higher as we 
ascend, so as to prevent complication by convection. It is 
assumed, as the basis of the inquiry, that :— 

Hach horizontal layer of the gas is in the “special”’ state, 
compounded with a vertical translation which is the same for 
all particles in the layer. 

The following are the chief results:— 

(a) Since the pressure is constant throughout, we have 


Pn 
Oh. 
so that 7/h is constant. 
(b) Since the motion is steady, no matter passes (on the 
whole) across any horizontal plane. This gives for the speed 
of translation of the layer at x, 


“=| oF [n+ =P v Jo/Be 


(c) Equal amounts of energy are (on the whole) transferred 
across unit area of each horizontal plane, per unit of time. 


Be PC” el (BJne ese) 


By the above value of p, and its consequence as to the ratio 


144 Foundations of the Kinetic Theory of Gases. 


n/h, these expressions become 


a= Ls nae ¢ cs )=5 gh aca P* 0-06, 


da Opis 
Lighe 25 eae \ dha ae 
iD = ie h (gae e C, —5C; + C;)= Fe h pr0 45. 


Since Hi is constant, by the conditions, we see that « also 
must be constant. Hence, as hr (where 7 is absolute tempe- 


LOT 
rature ) 1s constant, we have 7? ae constant, or 


rT? =A+Ba, 


which, when the terminal conditions are assigned, gives the 
steady distribution of temperature. The motion of the gas is 
analogous to that of liquid mud when a scavenger tries to 
sweep it into a heap. The broom produces a general transla- 
tion which is counteracted by the gravitation due to the 
slope, just as the translation of the gas is balanced by the 
greater number of particles escaping from the colder and 
denser layers than from the warmer and less dense. 

In thermal foot-minute-centigrade measure, the conductivity 
of air, at one atmosphere and ordinary temperatures, appears 
from the above expressions to be about 


a — 
ean ° 


or about 1/28,000 of that of iron. No account, of course, is 
taken of rotation or vibration of individual particles. 

(4) In the case of diffusion, in a long tube of unit section, 
suppose that we have, at section « of the tube, m, Pjs and 
n, P,s per cubic unit, with translational speeds a, and ap, 
respectively. If G, be the whole mass of the first gas on the 
negative side of ae section, it is shown that the rate of flow 
of that gas is 


dG 

ie =—P, (ma Ge 1/8), &e. 
Obviously 

ns =P &e. 


The motion of the layer of Pys at « is (if approximately steady) 
given the equation 


Pin, es 8 D) t(h, + hy) Poke ee 
dx a (Te hy )= 7g ee / Ahy Pyt+P, (“ee 


Contraction during Cooling of a Solid Earth. 145 


where the right-hand side depends on the collisions between 

the two kinds of gas in the layer, s being the semi-sum of the 

diameters. From these we obtain 

aG. / 3 P,+P, De lel Gy 

di. it (= s? MV trhyhy (hy + hy) p oF 3n (ng 1G, =F Ny 2G,)) da2 1 
In the special case, when the masses and diameters are 

equal in the two gases, the diffusion-coefficient (the multiplier 
d2 

of 


da? 


! above) has the value 


3 Te x r 
(G4/5 ii a= Th 185, 


It is therefore inversely as the density, and directly as the 
square root of the absolute temperature. And in the case of 
two infinite vessels, connected by a tube of length / and sec- 
tion 8, and containing two gases whose particles have equal 


masses and diameters, the rate of flow of either is a 1°785 


in mass per unit of time. 

Other cases are treated ; and among these it is shown that 
with equal masses, and constant semi-sum of diameters, differ- 
ence of diameters favours diffusion. 


XVIII. On the Amount of the Elevations attributable to Com- 
pression through the Contraction during Cooling of a Solid 
javth. By Rey. O. Fisusr, M.A., F.G.S.* 


T is now thirteen years since I first published in the 
‘Transactions of the Cambridge Philosophical Society’f, 
and, at a later date, in my ‘ Physics of the Harth’s Crust’ tf, 
an attempt to estimate the mean height of the elevations 
which compression, resulting from the contraction due to 
cooling, might give rise to upon the earth considered as a solid 
lobe. 
: A remark from my friend Mr. Davison, who is working on 
this subject, has suggested to me that the investigation I have 
given is not quite satisfactory ; and I now offer the following 
as an Improvement. 

If we are to attribute the corrugations which we meet with 
in the earth’s crust to compression arising from the secular 
cooling of a solid globe, we must assume, as I have tacitly 

* Communicated by the Author. 


+ Vol. xii. pt. 2. Read Dee. 1, 1878. 
{ Macmillan’s, 1881, 


146 Rev. O. Fisher on Elevations atiributabie to 


done in my former work, that the matter in each layer retains 
its horizontal extension during the settlement into its present 
position. On this supposition the corrugations will clearly be 


- influenced by the sphericity of the surface. But if we make 


use of Sir W. Thomson’s expression™ for the temperature at 
any depth, we must recollect that he neglects the sphericity. 
Still it seems probable that his law of cooling for an infinite 
plain will be sufficiently applicable to the globe to make the fol- 
lowing of some value. For itis evident that the temperature- 
curve for the sphere will be of a similar character, though not 
exactly of asimilar form ; the more rapid escape of heat to- 
wards the convex surface causing the ordinates to decrease 
somewhat more rapidly as the free surface of the sphere is 
approached. 

Let a layer of the globe at a distance z’ descend, by cooling 
of the matter beneath it, to the distance z from the centre C. 
Then our assumption, that this layer retains its horizontal 
extension, necessitates that we suppose the voluminal con- 
traction to take place wholly in the vertical dimension. 

Let E be the coefficient of voluminal contraction. If, then, 
the layer in question has fallen through @° since it solidified, 
we must have 


dz =(1—H@)dz; 
or dz' = (1+ H@)dz, approximately. 


The volume of this layer on first solidifying was 
Acar 2? dz!. 


And, after cooling, the thickness of this layer has contracted 
to dz, but has retained its horizontal extension. Its volume 


therefore becomes 
Aq 2'2 dz. 


Also, the proper volume of the spherical layer of the same 
thickness at this depth is 
Amr z2* dz. 


And the difference between these volumes will be the contri- 
bution to the surface-corrugations from this particular layer. 
Call the volume of the whole corrugations 4ar?h; we then 
shall have 

Anrry? ble dz = 4mz'2dz—4Ame7dz. 


dz 
But since every layer beneath the one in question has cooled 


* Trans. Roy. Soc. Edin. vol. xxii, pt.1, p. 157. Also Nat. Phil. 
Appendix D; and Phil. Mag. 4th series, vol. xxv. p. 1 (1863). 


Compression during Cooling of a Solid Earth. 147 


through 6°, and contracted in the vertical dimension only, 
es a (1+ EH@)dz ; 


=2+H( dz, 
©/0 


ne =228( “aa: neglecting H?. 


e/ 


Diagram of temperature-curve, adapted from Sir W. Thomson’s paper. 


x 
7000° # 


ON the depth below the surface =z. NP the excess of temperature 
above the temperature of the surface =v. OQ the excess of the melt- 
ing temperature above that of the surface = V. Pn the temperature 
through which matter at the depth x has cooled = 6. Oa= a. 


Let « be the depth of the layer under consideration from 
the surface at O. ‘Then, using Sir W. Thomson’s notation, 
we shall have 


d=V-—v, 
or b Zz a2 
d=V— ( e dz; 
a 


«0 


V being the temperature of solidification, and pein Wa ; and 
Dy Ww TT 
a a depth, which may be determined at once (without know- 
ing the conductivity or the time of cooling) by means of the 
oe 
formula eS ee from the temperature-rate near the 


die anr/a 


148 Contraction during Cooling of a Solid Earth. 


surface, which, taken at 1° Fahr. per 51 feet, gives 


is 38 
Sl ana 
If we follow Thomson in assuming 7000° Fahr. for the tem- 
perature of solidification (a very high value) this gives | 
a=420832 feet, 
and at twice this depth, 0=V x0-00468; below which the 


cooling will be small. Let us then separate the integral into 
two portions at the point A, at the depth 2a ; and 


Zz CA 
od =2:h(( +{ Jods 
dz JCA 0 


the second integral being the contribution to the surface- 
corrugations from matter below A. | 
Setting this aside, since z=r—a, .. dz=—dzx ; and, sup- 
posing @ now expressed in terms of wx as above, reversing the 
order of the limits, 
dh 


2a 
Tae =— 2(r— a) ( Oda ; 


a 2a 
He (1 ao ) ({ Ode) de-+ cane 
Lr 1h af 


The integral begins at A, where e=2a and h=0; 


)) (°2a P 2a 
i ae (1-)(| Ode de, 
L Pe F 


0 


wherefore 


gives the mean height of the corrugations formed out of the 
compression of the matter down to the depth 2a, which, with 
the assumed constants, will be about 160 miles. | 

If we substitute for @ the value given above, expand the 
exponential and integrate between the limits (see ‘ Physics ’ 
&e. p. 63 et seg.), putting H=0-0000215*, r=20902500 
feet, the above gives 

h=938 feet. 


The effect of the contraction below A need not be considered, 
being at most not two feet. 

The value obtained for / implies that, if all the elevations 
which would have been produced by compression, through 
the contraction of the earth cooling as a solid, were levelled 


* This is the coefficient of contraction obtained from Mallet’s experi-- 
ments on slag. See ‘ Physics’ &c. p. 68. 


Silk v. Wire in Galvanometers. 149 


down, they would form a coating of about 900 feet in thickness 
above the datum level, which would be the surface, had the 
matter of the crust been perfectly compressible so that com- 
pression would not have corrugated it. 

The value obtained for this quantity in my former work 
was 866 feet. 

Practically, these two numbers do not materially differ ; and 
they show that, if we take into consideration the land and the 
ocean-basins, the existing inequalities of the surface are greater 
than can be accounted for by the theory of compression 
through contraction by cooling of a solid globe, even upon the 
too highly favourable suppositions made in the present paper. 
The strictly geological arguments against this theory stand 
upon their own merits. 

The result of the above emendation of the demonstration in 
my ‘Physics of the Harth’s Crust’ is therefore simply to 
confirm the arguments I have built upon the less satisfactory 
calculation given in chapter vi. of that book. 


XIX. Sik v. Wire. By R. H. M. Bosanquer. 


To the Editors of the Philosophical Magazine and Journal. 


GENTLEMEN, 
ac a note in the December number of the Philosophical 

Magazine for 1886, entitled “Silk v. Wire, or the ‘Ghost’ 
in the Galvanometer,” I mentioned reasons for distrusting 
silk, and alluded amongst other things to the way in which it 
untwists when stretched. Condensation of expression has its 
inconveniences, and in the January number for 1887 Mr. 
Gray infers that I used a twisted silk thread, by which, I 
presume, he means an artificially-twisted silk thread ; but 
that isnot so. The thread used was prepared from suspension- 
silk supplied by Hlliott Brothers. This consists of a small 
number of fibres more or less aggregated together, and pre- 
sents no appearance of twist. ‘This is picked to pieces until 
the substance desired is left. It is then fine enough to be 
hardly visible. 

I abandoned the use of cocoon-fibres and very small needles 
years ago in consequence of the impossibility of accurately 
determining the error introduced by the fibre. 

I made at one time a great many observations on silk fibres 
of various descriptions. The phenomenon, to which I alluded 
in speaking of the untwisting when stretched, may be de- 
scribed as structural twist: it has shown itself as follows, in 
all silk fibres I have ever examined. 


150 Silk v. Wire in Galvanometers. 


Ifa small weight with a pointer is suspended from a silk 
fibre, and it is brought to rest and left covered under glass, 
the position will continue slowly to change for some days. 
This is what I regard asa consequence of structural twist. It 
and the taking of set combined lead to the phenomenon of the 
change of zero in galvanometers, leaving the ‘ ghost” out of 
consideration for the present. 

With me, sensitiveness is far from being the first require- 
ment in galvanometers. The elimination of elements of an 
inconstant character stands first. Now, when we so greatly 
diminish the moment of the needles as to admit of the pro- 
perties of the silk suspension causing changes of the zero, which 
is admittedly the case, we have a demonstration that changes 
depending on the silk are able to influence the results. The 
increase in the length of the silk, according to my experience, 
increases some at least of the irregular effects instead of dimi- 
nishing them. By the elimination of the silk we can obtain 
instruments quite sufficiently sensitive for practical purposes, 
and entirely free from errors which I regard as introduced by 
an incorrect identification of refinement with accuracy. 

Since my former note was written I have wound a pair of 
coils for the galvanometer in question, consisting of about 
500 turns of 16 B.W.G. By employing a cement of shellac 
varnish thickened with red lead I was able to wind these solid 
without any frame, and so to get all the thick wire into the 
same space as before. I thus obtain nearly the same delicacy 
as with the old silk suspension. The clearness of the indica- 
tions, as compared with those of the old galvanometer, is 
evidenced by the facility with which the thermoelectric cur- 
rents, arising from the binding-screws of the circuit, are 
identified, a considerable deflection being produced by laying 
the finger on the brass binding-screw. With the silk such 
effects were not so easily isolated, movements often continuing 
even though the circuit was not joined up. 

Mr. Gray deprecates going back, ‘‘ something like half a 
century,” to galvanometers with large needles. I doubt 
whether the older experimenters realized the advantages of 
wire suspension. But I have always wondered that so little 
weight is now attached to the deliberate opinion of Gauss, 
that accuracy in such measurements is to be best attained by 
enlarging the dimensions of the apparatus. In this matter I 
am occasionally tempted to think that the old is better. 


XX. An Account of Cauchy’s Theory of Reflection and Refrac- 
tion of Light. By James WALKER, M.A., Demonstrator at 
the Clarendon Laboratory, Oxford*. 


. hom theory of reflection and refraction of light holds such 
an important place among the problems of Optics which 
await their solution that it is advantageous to have a clear 
idea of the work which has been previously done in the subject. 

The theory advanced by Green has been so thoroughly 
discussed by Lord Rayleigh and Sir W. Thomson that all 
questions connected with it may be considered as completely 
settled. But this is by no means the case with Cauchy’s 
work on the subject; and some account of it may be of in- 
terest, even though the theory cannot be said to contribute 
much towards a solution of the problem. 

Several “reproductions” of Cauchy’s work have indeed 
appeared in French and German, but in most of them the 
elegance, and therewith the clearness, of Cauchy’s method 
have been given up; while they leave in more or less ob- 
scurity the reasoning which led him to enunciate his “principle 
of continuity,’ and make no mention of a point of considerable 
interest, viz. the mistake which originally led to his adoption 
of a theory involving the strange assumption of a negative 
value for the coefficient of compressibility of the ether. 


x: 


Cauchy, at different periods, gave three distinct theories of 
reflection: the first two, however, require only a passing 
notice, as they were afterwards rejected by him as in no 
respect affording a complete solution of the problem. 

The first theory was published in the Bulletin de Férussuc 
of 1830. It rested on the true dynamical basis of the equality 
of pressures} at the interface of the media; but was vitiated 
by the neglect of the pressural waves, which must take part 
in the act of reflection and refraction. The method led, on 
the assumption of the equality of the density of the ether in 
the two media, to the formule given by Fresnel §. 

The second theory was based on a method of obtaining the 

* Communicated by the Physical Society: read December 11, 1886. 

tT A. y. Ettingshausen, Pogg. Ann. 1. p. 409; Sitzb. der Wren. Akad. 
xvill. p. 369. Beer, Poge. Ann. xci. pp. 268, 467, 561; xcii. p. 402. 
Eisenlohr, Pogg. Ann. civ. p. 346. Briot, Liowv. Journ. (2nd) x1. p, 305; 
xii. p. 185. Lundquist, Pogg. Ann. clii. pp. 177, 398, 565. 

{ Cauchy’s reasons for rejecting the principle of the equality of pres- 
sures at the interface are given in Comptes Rendus, xxviii. p. 60. 

§ Cauchy, Mémoire sur la Dispersion, § 10. 


152 Mr. J. Walker on Cauchy’s Theory of 


equations of condition at the interface, which was given in a 
lithographed memoir published in 1836. This method assumes 
a change in the equations of motion near the interface to a 
distance comparable with the radius of the sphere of activity 
of a molecule, and leads to the following theorem :— 

“ Htant donnés deux milieux ou deux systemes de molécules 
separés l’un de l’autre par le plan de yz, supposons que des 
équations d’équilibre ou de mouvement généralisées de maniére 
a subsister pour tous les points de l’un et de l’autre systeme 
et méme pour les points situés sur la surface de séparation, 
Pon puisse déduire une équation de la forme 


d?s 
de? 
g, © désignant deux quantités finies, mais variables avec les 
coordonnées wy z. On aura, pour z=0, 
dz __ ds’ 
dx dx’ 
en admettant que l’on prenne pour premier et pour second 
membre de chacune des formules les resultats que fournit la 
réduction de # a zéro, dans les deux valeurs de la fonction 
= ou & relatives aux points intérieurs du premier et du 
second systéme.”’ 

The equations of condition resulting from the application of 
this theorem were published in Cauchy’s memoir on Dispersion 
in the same year*™. They express that the linear dilatation of 
the zether normal to the interface is the same for both the 
media, and that the rotations in the three coordinate planes 
of a particle at the interface is the same, whether the particle 
is considered as belonging to the first or second medium. 

The method of deducing these conditions was given in a 
memoir presented to the French Academy on October 29, 
1838}. This memoir has never been published; and all we 
know is that the method involved the assumption that the 
velocity of propagation of the pressural waves is very great 
compared with that of the distortional wavest. In 1842 Cauchy 
showed that these conditions lead to Fresnel’s formulee§. 

The final theory was published in detail{| in the years 1838 
and 1839, and is contained in the 8th and 9th volumes of 
the Comptes Rendus, and in the Exercises d’ Analyse et de 

* Mém., sur la Dispersion, § 10. +t Comptes Rendus, vii. p. 751. 

{ Ibid. x. p. 905. § Ibid. xv. p. 418. 

|| The idea seems to be prevalent that we are indebted to the German 
reproductions for our knowledge of the details of Cauchy’s method. 


=@, 


/ 
s=s, 


Reflection and Refraction of Light. 153 


Physique. Later volumes of the Comptes Rendus contain 
re-statements of it; and in 1850 an extension of the method 
was made to rotatory isotropic media* and to anisotropic 
mediat ; but this later work was never completed. 


IT. 


Cauchy’s final methodt of determining the conditions at the 
interface of the media depended on finding the relations 
which must exist between the known values of the displace- 
ments in the interior of the medium, and the values, consistent 
with the conditions of the problem, which these displacements 
take when the change in the form of the equations of motion 
near the interface is taken into account. 

Treating the ether as an isotropic elastic solid, for which 
the density is p, and the coefficients of compressibility and 
rigidity are k, n, the cee of motion are 


d2 
p os mane 4 nV%t, 
Q 
Pp = - + ny 77, a . e ° e (1) 
d? i 
pag =™ ma a nVv"6 J 


where 

dé dyn , at 

a det dy dy ' dz’ 

Sir W. Thomson§ has shown that all possible solutions of these 
equations are included in 

db 

35 dx 


where ¢, wv, v,w are some functions of 2, y, z, ¢ and u, v, w such 


that a + ms + Ga =(; further that, making these substitu- 
te ay ide 


tions, equations (1) may be replaced by 
d? du d?v 
= =(m+n)V*>, pas =nV*Uu, p Ga=nVr, 


d?w 
Paa PV. 


and m=k+n. 


dp 


+U, ee et or? ge 


“Ew, 


So that there are two modes of waves possible : a condensa- 


* C. R. xxxi. pp. 160, 225. + Ibid. xxxi. pp. 257, 297. 
t Ibid. vill. pp. 374, 432, 459. § Baltimore Lectures, p. 32. 


Phil. Mag. 8. 5. Vol. 23. No. 141. Feb. 1887. M 


154 Mr. J. Walker on Cauchy’s Theory of 


ae “, and for which 


tional wave, propagated with velocity we 
the velocity-potential is ¢; and a distortional wave, propa- 


gated with the velocity a7 “, and for which the components 
p 


of the displacement are wu, v, w. 

Let the interface of the media be the plane of yz, and sup- 
pose the first medium on the side of positive a. 

Considering only plane waves, which have the same period 
of vibration 27/m and the same trace by +cz=0 on the inter- 
face of the media, the values of ¢, u, v, w satisfying the 
equations may be taken as 


ee ee 


— ll (aa by + ez —wt) V7 He M_ o(—a,2-+by+ez—wt) ¥—=1*, 


eTipay Ey 


—wi)V-1 ae qh cal 
u= NAC wt)V—1 a A, a ax+ by +ez—wt) a 
as =, 33 —wt)V 1 
i B artbytez wt)V—=1 as B, a ax-+by+cz—wt) : 
=f Vo] = RP hyn 
w= Cc lant byez wt) 1 uit C, a ax+ byt+cz—wt)* i, 
where 


2 HAL 
Chee ce AY Ss ee, a= Ne —b?—¢?, 


and A, B, C; A,, B,,C, are connected by the relations 
Aa+Bd+Cc=0, —Aa+B04+C0c=0. 
The corresponding values of £, 9, € are accordingly 
E—A lar bytez—wt)V—1 as A, e(— ant by tez—wt) aoe B, ay pleut toy tez—ot)V 1 


ie C, aye ; 


(—a,x+by+cz—ot)V=1 
n= Beat t by +z - wt) voi as Bee cz—wt) V=1 au B By (a,,2-+by+ez—wt)V—1 
r 
| 


(2) 


+ O,be at by-+ez—wt)V sale 
¢ oe Cela t by + cz—at) v=} i Ce ee v=) ae Bee ee V1 


a Be eee 


Now within the medium the displacements are those due to 
the distortional waves alone, and hence the values of &, 9, ¢ 
are 


* The V —1 is inserted for convenience. 


Reflection and Refraction of Light. 155 
Bah ert ees = ce A a eee 
7— pe iver a)) vai as Bie ene ‘eae P 5 (3) 


8 — Cet tiytee—at) -1 ue Cb eke si, 


whence, comparing these values with equations (2), 
By =O, =0". 


Near the interface, and for values of x less than the small 
quantity ¢, the differential equations change form by the 
addition of terms whose coefficients are functions of x, which 
vanish when x exceeds the small quantity e. These additional 
terms may be reduced to linear functions of &, n, & and their 
differential coefficients with respect to zT, since the equations 


will still be satisfied by taking the Daplacenonts proportional 


—wt)V —1 
to the same exponential Oy tez—wt) 


We require now to determine the values of E, n, € which 
satisfy these altered equations. 

Cauchy’s method of doing this depends, as v. Httings- 
hausen{ has pointed out, on the method of the variation of 
parameters: by this method the constants At Aas Bee 
treated as functions of 2; and a first condition imposed : upon 


them is that a a a must remain unaltered in form, so that 
CL Ok 
oe at Art teed V4 ee N pt aaa) ee ale | 


‘ b —wt eae ie ts NE ——— | 
+ Ba, 2 ue y +ez—wt) "+ C,,a,2¢ a,2+by+ez—ot) yy 


dn = {Bae (ax+ by+cz—wt)V 1 —Bae (—ar+by--ez—wt)V =1 
dz 


r (4) 


LEW ay —wt)V —} 
+ Bay permit byez wt) '—C, a, be a, 2+ by+cz—wt) A =r 


dx 
+ Bya,ce 


dt eat Ngee ete a 1st ae at hy tee atv =I 
— i 


(—a,x+by+ez—wt)V —1 eo . 


(a,c+bytez—wt)V—1__ a, ce 
iT] 


Consider now any one of the parameters, say B,,; its value 
deduced from equations (2) and (4) is of the form 


AON BEES ae 
By=(AE+untott eS +p ol +o%*)e apathy tee—at)y— 


1x 


Differentiating this equation with respect to x, and substitu- 


* C. R. viii. p. 440. + Tom. cit. p. 461. 
t Pogg. Ann. 1. p. 409 


156 Mr. J. Walker on Cauchy’s Theory of 
are bt mS 
da de? da 
all the terms will cancel out except those which depend on 
the change of form; and we shall have 


Pu 


ting for >, from the changed differential equations, 


—(a,,x2 cz—-wt)V —1 
= (Le+My+NE+PS +RD +8 Ne tii 


where L, M,... vanish for finite values of x. 
Now the values of &, 7, § will differ but slightly from those 
given by (3), so that this last expression may be written 


dB 


whence the variable part of B, is 
leyjex {cat +BM +... )ee-ude”ldy 

+ ten L+BM+...)e~@tade’=I1da,, 
Similar values are obtained for the parts of A, A... which 
depend on «. NowlL, M,... vanish for finite values of aa 


so that if (tae, (* Mdzaz,...are very small relatively to 
0 e 0 


A, #,¥,...", the variable part of B, may be neglected if 
—(a+a,)/—1, (a—a,)/ —1 have no real positive part ; so 
that those among the coefficients A, A, ... will remain un- 


altered, when the change in the medium near the interface is 


taken into account, which have the coefficient of « in their 
exponential factor with a real part not less than that of a\/ —1. 

In the present case this will be so for all the parameters 
except B,; and hence, calling & 7, € the corrected values 
of &, n, €, we have 


t= (ALA BM PS ey A ee 


= +6 —wt)V=1 —ar+b Sob) V a1 b Ee eae | 
Eg Ne PFC!) ae A é ax+ by+cz—wt) + Bien y +cz—wt) : 
n= Belert by tee—at) 7 =1 1B pia an bby te2— wt) V --1 Bb pleut by +ez—at) V =1 
ee / ta 9 
y _ (¥, (artbyt ce-wt)V—=1 , (NX (—axtby+cz—wt)V =1 (a,,2+by+ez—wt)V —1 
gaa Go +Cre + Bee ; 


* This necessitates, first, that the coefficients of the added terms in 
the altered differential equations are all finite, and their product by e 
very small; secondly, that the thickness of the modified layer is small 
compared with the wave- -length (Comptes Rendus, vill. p. 439; ix. p. 5). 


Reflection and Refraction of Light. 157 
as 


= EN agli NED ta are = 
(a,,a-+by+cz—wt)V¥ —1 ete ii 
ia ere a t/—1, 
dn 


—_ = { Bae” +by+cz—wt) 4 =) -_ B gel att by + c2—at)V 1 
“4 / 


1 —wt Mi 
de — aa tee ss wt C ae at thy beet) V1 
1 


dx (a,,x-+by+ | ges 
a, cz—wt)V — 
1. B ace Le j J abs 


or, if 


ot: S 
ep In © fae the components of the displacements in the 


iw Dip Si 


incident ‘eS 
reflected - wave, £, 7, € are such that 


pressural 
E=E+E +E, n=nt+n+ C=C4+646,; 
dé _ de dé, , dey Eee On dg _& de, My 


dz dx dz ' dx’ dx dx dx de dx dx dx dx 
In the same way, for the second medium the corrected 
values of the displacements are such that 


f= Ely gl =" 4 n!', v= C4 gl. 
aa dé dé"! dn] _ dy dr! dt J dé! dc! 


dz dz dz da d«z' da’ dx dx dz’ 
where 
al | i] 


a Mi oi } are the components of the displacements in the 1 oe } wave. 


Finally, assuming*, that for «=0, 
‘sn A ee a a a eS lé = am + zi 
) F=f, n=7/, 0=%, oa a=s. — (5) 
we have as the interfacial conditions, that for <=0, 
Er ait Sus & +e; gtaty= ttn", | 
C4545, = O40", 
dé dé, dey _ di? dé" adm day dny _ dy! dr!’ 


reir icudkes bidet hein tda: dz iene? (6) 


a dG, dy a, ae 
dz d« dae dx dx- 


* OC R. ix. p. 94. 


158 Mr. J. Walker on Cauchy’s Theory of 


These equations express Cauchy’s principle of the continuity 
of the motion of the ether, according to which the incident 
waye passes into the reflected and refracted waves “sans 
transition brusque.”’ 

Judging from the historical sequence of Cauchy’s papers, 
there can be little doubt that he enunciated this principle as 
the physical interpretation of the result arrived at by reason- 
ing analogous to the above; it is, however, impossible to 
agree with v. Ettingshausen that “ Cauchy hat diese Gleich- 
ungen (6) anfinglich aus Griinden gerechtfertigt, die sich 
auf das Verfahren der Variationen der Constanten zuriick- 
fiihren lassen ;”?* as the principle is already involved in the 
assumption (5)f. 

All that the above analysis really leads to, and all that 
Cauchy} claimed to have established by it, is the necessity for 
including the pressural waves in the problem of reflection and 
refraction. 

Since the true dynamical equations of condition, given by 
the equality of displacements and pressures, are that for <=0, 


F=7, g=7, €&8, | 
is 1é I INR TG dé 
(m—n)o+ antt — (m'—n')d' + 2n a (7) 


dx dz dx dz 


Z 


(Ea y) an(Ey o (Ss = =n(% 4 | 


dy dz aye) dy 
it is clear, as has been often pointed out, that Cauchy’s 
assumption involves that of the identity of the statical pro- 
perties of the zther in the two media. Lundquist§, however, 
considers that ‘‘ Cauchy has established his principle of con- 
tinuity by the aid of analysis, the exactitude of which it is 
not easy to contest ;” and hence that this result, combined 
with the dynamically exact conditions (7), proves “‘ the legiti- 
macy of Green’s assumption of the equality of the compres- 
sibility and the rigidity of the sether in the two media.” 
Cauchy himself did not see that this was involved in his 


* Sttzb. der Wien. Akad. xviii. p. 371. 

+ I do not think Cauchy contemplated a continuous rapid transition 
of one medium into the other (cf. C. R. x. p. 847); neither does 
vy. Ettingshausen in his paper. Supposing the assumption justified on 
these grounds, yet, as von der Muhl has pointed out, the former assump- 
tion respecting the coefficients of the additional terms in the modified 
equations precludes the assumption of a finite change in the statical pro- 
perties of the media (Jlatt. Ann. v. p. 477). 

CLR. x. p. 847. § Poge. Ann. clii. p. 185. 


Reflection and Refraction of Light. 159 


conditions ; and so in what follows the compressibilities and 
rigidities of the two media will be considered as unequal. 


IES 


Taking, as before, the interface of the media as the plane 
of yz, and the first medium on the side of positive 2, let the 
axis of z be parallei to the plane of the waves, so that the 
plane of xy is the plane of incidence ; then, if &  € and &' a ¢! 
denote the components of the displacements in the first and 
ezond medium respectively, § n ¢, & 7! ¢' will be independent 
of z. 

(1) Let the incident vibrations be perpendicular to the 
plane of incidence. 

The general equations of motion are in this case 

Com (Cont, OS apes ae 
tae aet ay) ae =" (ae a) 
and the principle of continuity gives for the interfacial con- 
ditions that for z=0, 
dG dv’ 
ee A 
s=o, da dx’ 
Assuming 
t=O glartby—at)v =o a Cel art tye) V 1 


¢' = e(v'atby— wt)/—1 


we get at once 


ata a—a! 
C— C= 
za? yea? 
aad __ sin(t—r) 
ata sin(t+r)’ 


since 
b ee D 
=——tan?, -j;—ball 7 
a a 
where 2, 7 are the angles of incidence and refraction. 


(2) Let the incident vibrations be in the plane of incidence. 
The equations of soe a in the first medium are 


(GE dak ay, (ECE 
Pap" da\de dy tn(iat dy)? 

} @n _ a(dé dn) Pn &n cay 

L? ae dé Ca da’ dy cee niet dy’ 


* C. R. viii. p. 985; ix. pp. 1, 59, 91, 676, 726, 727 ; x. p. 847. Ex. 
d’ An, et de Phys. i. pp. 2153, 212, 


160 Mr. J. Walker on Cauchy’s Theory of 


Using Green’s* method of separating the distortional and 
condensational parts of the solution, and assuming 


the equations of motion become 


Ph (Pb Pb & Pap Bap 
eat at) aenVaet a) 


where 
g=(m+nj/p, Y=n/p. 
Similar equations apply to the second medium. 
The principle of continuity gives for the interfacial con 
ditions that for e=0, | 


dp, dy_ dg’ a 


———— 


da. dy dee dy (8) 
dp_dyp_ dd ay | ae 


TE EER TA Cie a 
a2 d? a ! DEN Yi 
$, Py _@P$', dy | 


da dady dz dady 


ao ay _ Pp dy 
dzdy aa T dicdy, (jae an 


Since these equations are true for all values of y, we may 
differentiate with respect to it, and hence, by means of the 
equations of motion, replace (9) by ; 

Lo Mra Nae a 


fa Pa fd. f 4 
It may here be noted, that if we take the general equations 
of condition (7) and assume the equality of the rigidities of the 
ether in the two media with no assumption respecting the 
compressibilities, we get, instead of (9a), 
a oO) dq? op’ d? Va day! 
Pp “TP aE? Pe =p’ dpe Ss i (90) 


(9) 


Assume 
ths (a, jw+tby—wt)V¥ —1 
co) S15 2 ’ 
a — A trtby—otV=1 , ny part ly—wt) V1 
/ ’ 
1 — pir (alz+by—wt)V¥ =1 
Gf = Biderete-oo/ =, 


apy! = giaetby—ot)V = 1 


* Collected Works, p. 261. 


Reflection and Refraction of Light. 161 


The equations of motion give 
ow? =2(a2 +0) =9? (a, 2 +12) =y'2(al?2 + 82) = 92a! +02); 


whence 


sy Wie —/ —L= bujp/ =I, say; 


P= 5,/\- = aie Y —1=—bu)/—1, say, 


the negative sign being taken, as the second medium corre- 
sponds to negative . 


From equations (8) and (9a) we get 


u, B,” —1+ (A+A) =—u" BY VY —141, ) 
dB, —a(A—A,) =bB’—a’, 


B= 7B", r + (10) 
1 if 
of (A+A))= 15 


The last two of these cc give 
A+A,= = 2 = ps? ’ 


where yp is the refractive index, and 


mB = OE Wm, ay 
Substituting in the first two of equations (10), we get 
(u,B,,+u" BY) V —1=1—p%, 


whence 
1) pet aT 
oe, My ty, ul! e 
and 
/ 
asa 428,39 
na 0 2 
=2 42 yt 1M Vv =I, 
where 


162 Mr. J. Walker on Cauchy’s Theory of 


Hence 
Goud — 
2A=p?+ “+ (we —1)MV¥—1 


12 2 I [22 eee 
_ aw?+bh | ene e MVDI 


ate iat a @+b 


/ Q ee naer as 
fe (aa’ +6?) +6(a a)MV L(g +a) =2REY=1, say ; 


a(a? +b?) 
: aa’ —b*) —b(a' +a)MV —1 5 


Then R, R, denote the amplitudes of the incident and 
reflected vibrations, and 6, 6, the difference of phase between 
the incident and refracted, the reflected and refracted waves 
respectively. 

Hence, if « is the azimuth with respect to the plane of inci- 
dence of the incident vibration, the reflected vibration will in 
general be elliptical with a difference of phase 6,—6 between 
the components in and perpendicular to the plane of incidence ; 
and if this difference of phase is destroyed, the azimuth 6 of 
the resulting rectilinear vibration will be given by 


cot B=R/C,. 
Hence 
cot 8 saw (ago) = ble tee ay 
cot a (aa! +b?) +b(a! —a) MV —1 
and 


cot? 8 (aa!—b?)? +67(a' +.a)?M? 
cot?a (ua! + b?)? +. 0?(a! —a)? M? 
_ cos* (i+7) + M? sin? +7), 
cos? (t—7r) + M? sin? (i— 7)’ 
also 
_ Mitan (i+r)+tan (i—r)} 
tan (0-9) = 7 _ Mian Gtr) tan Gr) 


Total Reflection’. 


If » is less than unity, we may write w=sin I, and we 
get 


2 Agr” \ ute 
ee ae cos? r= =r sin (I—7) sin (I+2). 


* C. R. ix. p. 764; xxx. p. 465. 


Reflection and Refraction of Light. 163 


Hence, if i>I1, the value of a’ becomes imaginary, and the 
refracted ray will die out as it leaves the refracting surface. 
Writing 
Ussin? (i—I) sin (i+ J), 


we must substitute in the formule obtained above 
d=— ce Ue Vai 


the negative sign being taken, as the second medium is on the 
side of negative wz. 

Substituting this value, we find that the reflection is total 
both for the vibration in the plane of incidence, and for the 
vibration perpendicular to the plane of incidence, and for the 
difference of phase between the components of the reflected 
ray we get from (11) 


cot B s—-oV=1_ Sint (sini + UM)+cosi(Msini+U)V —1 
cot a sin i (sini + UM)—cosi(Msini+U)V —1’ 
whence 


ao cost {M sin i+sin® (i—I) sin? ((+])} 


ta eile ray ay 
2 sinz {sint+ M sin? (i—I) sin?(¢+])} 
1, te an aye Mite: 
M sanz? iB (¢—1I) sin? (¢+ 1) 


hawt MMe 8 ee - COS 2, 
if the square and higher powers of the small quantity M are 
neglected. 

Cauchy has sin?z instead of sin? I in the numerator of the 


last expression; the correct formula was first given by 
Beer*. 


TVET 


Before proceeding further, it will be as well to discuss the 
value of the expression denoted above by M. 

Cauchy, not seeing that his equations of condition involved 
the assumption of the identity of the statical properties of the 
ether in the two media, adopted tha following relations, 


mtn=—en, m+n =—e' al, 


! 


where e, e are very small numerics. 


* Poge. Ann. xci. p. 274. % C. &. ix. pp. 691, 727. 


164 Mr. J. Walker on Cauchy’s Theory of 


These relations give 


7 2 
5, eee a P = 
° p m! +n! ae 
n ih 1 
U)/= Ji bet ee ee Ee 
ll vs = Jad ae ae 
(m+n) sin? 2 \ e*sin?7 esin? 
y= uf 
pea ont 7 = es 
e sin’ 
Hence 
pont 
€ 
2 
ey 2p 1 e ° fees ° e 
M=__ ee ein ie sin 7 in, 
o€ it 1 
e? esinz eésinr 
7 
: € 
if H=e— —*., 


No attempt has been made, so far as | am aware, to indi- 
cate the reasons which led to Cauchy’s adoption of the above 
remarkable relations between the coefficients of compressibility 
and rigidity of the zther in a medium. 

In order to find a relation between the coefficients, Cauchy 
considered the condition which must be {fulfilled if the 
incident light is completely polarized by reflection. 

This condition is that M=0, giving since 
N" 


eerie! Way ae 
eo Uy + U l—uj,u 


Mie 2 Whit 


where wu, uw are both positive, that 
mtn m+n 


In his first memoir on the subject, Cauchy}, forgetting to 
take into account the fact of the media being on different sides 
of the plane of yz, wrote 


al = bull / —1, 


! 


where w!! is positive. 
Hence he obtained 
M= ae 7 = Bip 
fugrtt,—ul Lu, aul”? 
where u,, vw! are both positive, giving as the condition for 
complete polarization 
Uj, =ul=o0, or m+n=0=m 4. 


* C. R. xxvii. p. 64. Originally Cauchy took e=0. 
t+ Ibid. ix. p.94, Ex. d’An, et de Phys. i. p. 167. 


Reflection and Refraction of Light. 165 


He then argued that incomplete polarization must be due 
to the fact that these expressions differ slightly from zero, and 
that their value must be negative, in order that the pressural 
waves should be insensible at a distance from the interface for 
all angles of incidence. 

In a memoir published in 1840 and in the Fzercises 
d@ Analyse et de Physique*, this mistake was corrected, and the 
true condition p/(m-+n)=p!'/(m' +n’) was given; but, appa- 
rently led astray by his original mistake and by a desiret 
(afterwards given up, Compé. Rend. xxviii. p. 125) to make 
complete polarization depend on the properties of the refracting 
medium alone, and not on any relation between the two 
media, he still adopted the solution 


mt+n=0=m'+7'; 


though he mentionedf also the true solution, viz. that the coeffi- 
cient of compressibility of the zether is infinite, and the wave- 
lengths of the pressural waves in the two media are equal. 

Assuming that the ether is incompressible, the polarization 
of the reflected ray will be elliptical when the wave-lengths of 
the pressural waves are Aone and we get 


= 1) | te 1), 


where X,, A” are the wave-lengths of the pressural waves in 
the two media. This is Hisenlohr’s suggestion; but the 
form in which he made it does not show that it involves the 
absolute incompressibility of the ether. If ),/A"=A/d', we 
get Green’s formula. Hisenlohr$ says that this assumption 
is absolutely untenable: it is, however, as Green shows, a 
direct consequence of the assumption made by him, and in- 
volved in Cauchy’s conditions, viz. the identity of the statical 
properties of the ether in the two media. 

Further than that, if we assume only the equality of the 
rigidities, the equations of condition become (8), (9b) ; whence 


and if the ether is incompressible, 


_ wal 
Tee 
C. R. x. p. 857. Ex. d’ An, et de Phys. 1. p. 233. 


C. R. ix. p. 727. t Lbid. x. p. 388. 
Pogg. Ann. civ. p. 388. 


+ 
§ 


—— 


SSS SSS 


SSS SSS 


166 Mr. J. Walker on Cauchy’s Theory of 


Haughton’s* suggestion that the coefficient of compressi- 
bility is very great, but not infinite, does not help matters ; 
so that it would appear that the only way to escape the diffi- 
culty is by one of Lord Rayleigh’st suggestions :— 

(1) That “although the transition between the two media 
is so sudden that the principal waves of transverse vibrations 
are affected nearly in the same way asif it were instantaneous, 
yet we may readily imagine that the case is different for the 
surface-waves, whose existence is almost confined to the layer 
of variable density.” 

(2) That ‘the densities concerned in the propagation of the 
so-called longitudinal waves are unknown, and may possibly not 
be the same as those on which transverse vibrations depend.” 

Hisenlohr ¢{ gives another (it appears entirely empirical) 
value for M: it involves, as Cauchy’s, a negative value for 
the coefficient compressibility of the sether, and leads to 
formulee closely agreeing with experiment ; as, however, they © 
contain a third disposable constant, this close agreement is 
hardly to be wondered at. 


V. 


Cauchy’s formule for metallic reflection were originally 
published on April 15, 1839§, and thus were obtained from his 
second set of equations of condition, in which the pressural 
waves were neglected. The formule were republished on 
January 17, 1848]|, and apparently no attempt was made to 
obtain equations in which the influence of the pressural waves 
was included. 

Canchy considers the peculiarities of metallic reflection to 
be due to a complex value of the refractive index. 

Writing iy 

= be” =, 
we get 
2 2 
g?= a (@2e2” —1 — sin? i)= 8 é Ue say; 
whence 
U2 sin 2u=6’ sin 2e, cot 2u—e=cot € cos (2 tan—** (12) 


Substituting 


ei 20r ‘ Dar Cea 
d= 7 Wear, C= rs ee 1, b = 5 d, 
* Phil. Mag. [4] vi. p. 81. + Ibid. xlii. pp. 96, 97. 
t Poge. Ann. civ. p. 356. § C. R. viii. p. 553. 


|| Zoid, xxvi. p. 86. 


Reflection and Refraction of Light. 167 
in the values of ©,/C, A,/A, and making M=0, we get at once 


Cauchy’s well-known formule. 
Making these same substitutions in (11), we get 


cotB y= _ sin?itcos? Ue | 
eae 2S ip A gan en ae & 
cota sin? 7—cos7 Ue"! 
whence 
2U sinucosisin?i . U cos7 
a oll u tan ( 2 tan-! — ae | 
sin* i1— U* cos? 2 sin? 7 


cot? 8  sin*7+cos?z U?—2 sin?z cosi U cos u 
SS eo, 
cot?a@ ~~ sin*z+cos?2 U?+4+ 2sin?icosi U cosu 


= cot(w—45°), 


where 
: Ucosi 
— —1 
cotyr=cos u sin (2 tan aes ), 
er uw a—A5°, 
: _,Ucosi 
cot 28=cos u| sin 2 tan~! —,— }. 
sin? 2 


At the polarizing angle I, for which A=7r/2, we have 
U=tanism i, w=28, 


where # is the azimuth of the reflected vibrations, when the 
incident vibrations are in an azimuth 45° with respect to the 
plane of incidence. 

These values substituted in equations (12) give the values 
of the constants 6, e, and then these same equations serve for 
the determination of u, U for any other angle of incidence. 

While the above equations can at the best be only considered 
incomplete, objections have also been made to the complex 
value of the refractive index involved in them. 

Lord Rayleigh’s criticism* that the real part of ~? should 
be positive, while the results of experiment substituted in 
Cauchy’s equations give a value of y* with its real part nega- 
tive, seems not so much an argument against Cauchy’s idea, 
as an “argument against the attempt to account for the 
effects on a purely elastic solid theory ”’ f. 

The valne of pv” resulting from Sir W. Thomson’s theory of 
light is a real negative quantity ; this value substituted in 


* Phil. Mag. [4] xliii. p. 325. 
+ Eisenlohr, Wied. Ann. i. p. 204; Glazebrook, Brit. Assoc. Report, 
1885, p. 197. | 


168 Mr. J. Walker on Cauchy’s Theory of 


Green’s equations gives the reflection total at all angles of 
incidence. For this result there is no experimental evidence 
at present, except in the case of silver. The same will result 
from Lord Rayleigh’s extension of Green’s theory, unless, as 
seems scarcely probable, the refractive index of the pressural 
waves is a complex quantity. 


VL 


In August 1850* Cauchy published the outlines of the 
result of applying his method to the case of reflection at 
the surface of an isotropic medium which possesses rotatory 
power. 

The displacements in the upper medium are taken as 


a= Abert ty—ot)V 1 A, be(—aetby—wt)V =I B eer t ey =I, 


n= —Aacertey—wt)Y—1 4 A geet ty—wtV—1 4 B belauetty—olv 


2ar 


¢=57 Ce(artty—wt)V—1 4 A" yu e- ax+by— —wt)V =] 1, 


and those in the lower medium, since there will be two 
refracted waves circularly polarized in opposite directions, 


= Ay!bela'etby—wt)¥ 1 ae A lbeleizt byt) ¥ =1 + Bilal! la’a+ by—wt)V 1 


of = — Aylaylela'e+2y-wt)¥—1 A lg ealetty—of) "1 4 Bil elatatby—wt)v =1, 
= — y a Je At Rae ane —1 =e / | Al 20 7 x+by—wt) v=1, 
Ay 


Substituting these values in the equations of condition re- 
sulting from the principle of continuity, we get 


b(A+A,—A,!—A,’) Ip "_B al, 
—(A—A,)a+ Aja! + A,'a,!=b(B"—B,), 
b { (AS A,) a— Ay'a'—A,/ay} = (BiG? ith Ba); 
—(A+A,)a? + Ajay? + Agay? = i "_Bia'), 


C+C =(-> Lae Ad) eer 


| 
la 

| | 
(C—C,)u=(->. gf Cl aot a 5031!) V—l. | | 


ANC. ft. Xx. pp. 160, 225. 


Reflection and Refraction of Light. 169 


The last two of these equations give 


=e  » fe ! Xx ! Y gare \ 
Ba = | (arbor) 55 Al— (ater) rAd $I, L. (14) 
2aU,= } (a—as X As—(a-a!) SAY} VHT | 
/ 2 re! 2 ay r d! ‘ ) 
From the first and fourth we get 
A 2 . 
A+A,= BA t ye Ae ee ete), 


and from the second and third 
qli2 
=p 
fam a2 +0? + §2 B'= = [5° De 3 


whence, writing as oe 


where M is the coefficient of ellipticity, and eliminating B, Bl’ 
between the first two of equations (13), 
(a—Mb / —1)A—(a+ Mb V—1)A,= (ay! —Mb / —1)A;! 
+ (a! —Mb / —1)Ay, 

and from (15) 

9aA= U,A,’ + U,A,!, 2aA, = V,A,! + V,AJ, Py (16) 
where 
ay’ +a 
U,={ (aa, +0?) + Mb(a,/—a) V¥—1} 24 


_ 2a 
a {cos(i—r,) + Msin (t—7,) ¥—1} ae 
ay’ —a 
a -MB(a/ +a) Y=1} t=" 
ain (ery, 
woos (i+7,)—Msin (i+7,) ¥—1} =e 
and U, V, are similar expressions with (,) written instead 


of (;). 


First, consider the case in which the incident vibrations 
are perpendicular to the plane of incidence. 
Then A=O, and equation (15) and the first of equations 


(16) give 
see ~ > 12 bom +x, 12 A, y) U, A! + DAS == 
Phil, Mag. 8. 2 Vol. 23. No. 141. Feb. 1887. N 


170 Mr. J. Walker on Cauchy’s Theory of 


whence from (14) 


Ay a 2a »/—10 
Ne ? TEN r 
2 Uz x2 Us mE (a+ay))Uy com (at+ay')U, 
it 2a/ ~—10, 
m (a—a ‘Ope (a—ay') U2 
Ae! : Ay! 


and writing for U,, U2, a, ', a.', b their values in terms of 
the angles of incidence and refraction, 

___ sin 2isin 5. [cos 2R—M sin2R ¥ 1] 
A, =— V —1-— J 


CG; 
D | sin” (i+ R)—sin? aed 


D sin (i—R) sin (+ R)+ D’ sin? al 
C+ arscay peo 
D | sin" +R)— in S| 
2 

where bt 

D = cos (iQ— R) + M sin ((— R) V — 1, 

D’= cos ((+R)—Msin ((+ R) /—1, 

R= ae, the mean angle of refraction. 
Omitting squares and products of the small quantities 


BO a iene 
M, sin 5 2, the formule become 


a = . cos 2R 

a ae [cos ((—R)+M sin (i—R) W—1] sin? G+ R) ’ 
sin Gh) 
Ce siniGa RR). 

Hence the reflected ray will be in general elliptically pola- 
rized, except for an angle of incidence such that the angle of 
mean refraction is 7/4, in which case the refiected ray will be 
plane-polarized with vibrations perpendicular to the plane of 
incidence. In all cases the component perpendicular to the 
plane of incidence is practically the same as if the medium 
had no rotating power, the other component being very small. 


sin 22 sin 


Reflection and Refraction of Light. Or 


Next consider the case in which the incident vibrations are 
in the plane of incidence. 
Then C=0, and from equation (14) 


! I 
yes ok Ore I 
r_! a+a,! p) 


and hence 
rn2 
C, Adah: 


7 ee ee Xv 
(a a) x! (a+a') Vy + wl (a+a;') Ve 


at ree 


r? a 
nie Vv—1 


x x 
Xe! (a+a,!)U,+ al (a+ay')U, 


W hence 
Piet 


alana wih Gi alt onal ae 
D [sin? (+B) — sin? a) 


sin 22 sin 


i 


D’sin (i+ RB) sin (i—R) + D sin? 
a ey eer et ene ee A ; 


D [ sim (+R) — sin? 15 
or, to the same degree of approximation as in the former case, 


Bae 
2 
[cos(i— R) +Msin(i—R) /—1] sin’?(i+ R) 


sin 


C= V¥—1sin 2 


A — 008 (+R) —Msin (+B) /—1 sin (i—R) & 


‘~ cos ((— R) + Msin (i—R) /—1 sin (i+ R) 


Hence the reflected ray will be in general elliptically 
polarized, the component of the vibration in the plane of 
incidence being practically the same as if the refracting 
medium had no rotating power, the component of the vibration 
perpendicular to the plane of incidence being extremely small. 
At the polarizing angle for which R+i=7/2, the reflected 
vibration is plane-polarized, and the vibrations will be at an 


N2 


3 


172 Reflection and Refraction of Light. 


azimuth with respect to the plane of incidence given by 


77 
i 


tan B=tan 21. a ae 


VII. 


In the same year (1850) Cauchy extended his method to 
the problem of crystalline reflection: the complete solution 
was given in a memoir presented to the French Academy on 
September 16, 1850*. 

This memoir was never published, though it was announcedT 
to appear in the 23rd volume of the Mémoires de ? Académie ; 
and we have only slight indications of Cauchy’s manner of 
dealing with the problem. 

In accordance with the results of his theory of double 
refraction, Cauchy does not suppose the vibrations to be 
necessarily strictly transversal and longitudinalf. In order 
to eliminate the amplitudes of the latter vibrations, he assumes 
as an approximation the strict transversality of the former, 
and thus obtains§$ four equations between the quasi-transversal 
amplitudes, which contain three coefiicients, whose values are 
known when coordinate axes are taken depending on the re- 
fracting surface and the plane of incidence. 

A second memoir|| is devoted to the determination of the 
value of these coefficients, when fixed directions in the crystal 
are taken as the axes. The value of this determination is 
lessened by the fact, that at the very commencement an 
approximation is made depending on the peculiar relation 
between the coefficients of elasticity, which we have considered 
above. 

This is all that has been published, except some notes indi- 
cating a few of the results of his analysis; it is, however, 
probablef that Cauchy first obtained a solution on the assump- 
tion of the strict transversality of the luminous vibrations, 
and then proceeded to apply corrections to the values thus 
obtained, and it is possible** that he adopted in the solution 
MacCullagh’s idea of uniradial directions. 

There is no need to enter further into this part of Cauchy’s 
work, as Briottt has employed both these methods in his excel- 
lent adaptation of Cauchy’s theory to the problem of Crystalline 
Reflection. 


~ 0, R. xxxi, p. 422, + Tom. cit. p. 509. 
t Tom. cit. pp. 258, 299. § Tom. cit. p. 257. 
| Tom. eit. p. 297. q Tom. cit. p. 160. 


*& Tom. cit. p. 532. tt Low. Journ. [2] xii. p. 185. 


Ror 


XXI. On the Self-induction of Wires.—Part VI. 
By OLIVER HEAVISIDE*. 


: ea most important as well as most frequent application 

of Mr. 8. H. Christie’s differential arrangement, known 
at various times under the names of Wheatstone’s parallelo- 
gram, lozenge, balance, bridge, quadrangle, and quadrilateral, 
is to balance the resistances of four conductors, when sup- 
porting steady currents due to an impressed force in a fifth, 
and is done by observing the absence of steady current in a 
sixth. But its use in other ways and for other purposes has 
not been neglected. Thus, Maxwell described three ways of 
using the Bridge to obtain exact balances with transient cur- 
rents (these will be mentioned later in connection with other 
methods); Sir W. Thomson has used it for balancing the 
capacities of condenserst; and it has been used for other 
purposes. But the most extensive additional use has been 
probably in connection with duplex telegraphy ; and here, 
along with the Bridge, we may include the analogous differ- 
ential-coil system of balancing, which is in many respects a 
simplified form of the Bridge. 

On the revival of duplex telegraphy some fifteen years ago, 
it was soon recognized that “the line” required to be balanced 
by a similar line, or artificial line, not merely as regards its 
resistance, but also as regards its electrostatic capacity—ap- 
proximately by a single condenser ; better by a series of smaller 
condensers separated by resistances ; and, best of all, by a more 
continuous distribution of electrostatic capacity along the 
artificial line. The effect of the unbalanced self-induction 
was also observed. ‘This general principle also became clearly 
recognized, at least by some,—that no matter how complex a 
line may be, considered as an electrostatic and electromag- 
netic arrangement, it could be perfectly balanced by means 
of a precisely similar independent arrangement ; that, in fact, 
the complex condition of a perfect balance is identity of the 
two lines throughout. The great comprehensiveness of this 
principle, together with its extreme simplicity, furnish a strong 
reason why it does not require formal demonstration. It is 
sufficient to merely state the nature of the case to see, from 
the absence of all reason to the contrary, that the principle is 
correct. 

Thus, if AB,C and AB,C be two identically similar inde- 
pendent lines (which of course includes similarity of environ- 


* Communicated by the Author. 
+ Journal S. T, E. and E, vol. i. p. 394. 


174 Mr. O. Heaviside on the 


ment in the electrical sense in similar parts), joined in parallel, 
having the A ends connected, and also the C ends, and we join 
A to C by an external independent conductor in which is an 
impressed force e, the two lines must, from their similarity, 
be equally influenced by it, so that similar parts, as B; in 
one line and B, in the other, must be in the same state at the 
same moment. In particular, their potentials must always be 
equal, so that, if the points B, and B, be joined by another 
conductor, there will be no current in it at any moment, so far 
as the above-mentioned impressed force is concerned, however 
it vary. The same applies when it is not mere variation of 
the impressed force e, but of the resistance of the branch in 
which it is placed. And, more generally, B, and B, will be 
always at the same potential as regards disturbances origina- 
ting in the independent electrical arrangement joining A to C 
externally, however complex it may be. 

There is, however, this point to be attended to, that might 
be overlooked at first. Connecting the bridge-conductor from 
B; to B, must not produce current in it from other causes 
than difference of potential ; for instance, there should be, at 
least in general, no induction between the bridge-wire and the 
lines, or some special relation will be required to keep a balance. 
This case might perhaps be virtually included under similarity 
of environment. 

If we had sufficiently sensitive metheds of observation, the 
statement that one line must be an exact copy of the other 
would sometimes have to be taken literally. But the word 
copy may practically be often used to mean copy only as 
regards certain properties, either owing to the balance being 
independent of other properties, or owing to our inability to 
recognize the effects of differences in other properties. Thus, 
in the steady resistance-balance, we only require AB, and AB, 
to have equal total resistances, and likewise B,C and B,C; 
resistances in sequence being additive. But evidently, if the 
balance is to be kept whilst B, and B, are shifted together 
from end to end of the two lines, the resistance must be 
similarly distributed along them. 

If, now, condensers be attached to the lines, imitating a sub- 
marine cable, though of discontinuous capacity, we require 
that the resistance of corresponding sections shall be equal, as 
well as the capacities of corresponding condensers, in order 
that we shall have balance in the variable period as well as in 
the steady state; and the two properties, resistance and ca- 
pacity, are the elements involved in making one line a copy 
of the other. 

In case of electromagnetic induction, again, if AB,C and 


Self-induction of Wires. 175 


AB,C each consist of a number of coils in sequence, they will 
balance if the coils are alike, each for each, in the two lines, 
and are similarly placed with respect to one another. But 
the lines will easily balance under simpler conditions, coefti- 
cients of self-induction being additive, like resistances ; and 
it is only necessary that the total self-inductions of AB, and 
AB, (including mutual induction of their parts) be equal, and 
likewise of B,C and B,C. Again, if a coil a, in the branch 
AB, have another coil 0; in its neighbourhood (not in either 
line, but independent), and az be a copy of qj, in the branch 
AB,, we can complete the balance by placing a coil 2 which 
is a copy of 0, in the neighbourhood of the coil a,, so that the 
action between a, and 0, is the same as that between dg and bg. 
But it is not necessary for J, and b, to be copies of one another 
except in the two particulars of resistance and self-induction ; 
whilst as regards their positions with respect to a, and ag, we 
only require the mutual induction of a, and 0, tu equal that of 
dg and by. 

On the other hand, if 0, be a piece of metal, not a coil of 
fine wire, that is placed near the coil a, many more specifica- 
tions are required to make a copy of it. ‘The piece of metal 
is nota linear conductor ; and, although no doubt only a small 
number (instead of an infinite number) of degrees of freedom 
allowed for would be sufficient to make a practical balance, 
yet, as we have not the means of simply analyzing pieces of 
metal (like coils) into a few distinct elements, we must generally 
make a copy of 6, by means of a similar piece of the same 
metal, b,, and place it with respect to az as 6, 1s to a, to secure 
a good balance. But very near balances may be sometimes 
obtained by using quite dissimilar pieces of metal, dissimilarly 

laced. 
; So far, copy signifies equality in certain properties. But 
one line need be merely a reduced copy of the other. It is 
only when we inquire into what makes one line a reduced copy 
of another, that we require to examine fully the mathematical 
‘conditions of the case in question. In the state of steady flow 
the matter is simple enough. If AB, has times the resist- 
ance of AB,, then must B,C have n times the resistance of 
B,C to keep the potentials of B, and B, equal. If condensers 
be connected to the lines, as before mentioned, we require, 
first, the resistance-balance of the last sentence applied to 
every section between a pair of condensers ; and next, that 
the capacity of a condenser in the line AB,O shall be, not 
n times (as patented by Mr. Muirhead, I believe), but 1/n of 
the capacity of the corresponding condenser in the line AB,C*. 


* “On Duplex Telegraphy,” Phil, Mag. January 1876. 


176 Mr. O. Heaviside on the 


If the lines are representable by resistance, self-induction, 
electrostatic capacity, and leakage conductance (R, L, 8, K 
of Parts IV. and V., per unit lengths), one line will be a 
reduced copy of the other if, when R and L in the first line 
are n times those in the second, 8 and K in the second are 
n times those in the first, in similar parts. 

After these general remarks, and preliminary to the con- 
sideration of the quadrilateral, let us briefly consider the 
general theory of the conjugacy of a pair of conductors in a 
connected system, when an impressed force in either can cause 
no current in the other, either transient or permanent. The 
direct way is to seek the full differential equation of the cur- 
rent in either, when under the influence of impressed force in 
the other alone. Let V=ZC be the differential equation of 
any one branch, C being the current in it, V the fall of 
potential in the direction of C, and Z the differential operator 
concerned, according to the notation of Parts III., IV., and 
Y. If there be impressed force e in the branch, it becomes 
e+V=ZOC. We have > V=0 in any circuit, by the potential 
property; therefore 2e=>ZOC in any circuit. Also the cur- 
rents are connected by conditions of continuity at the junctions. 
These, together with the former circuit equations, lead us to a 
set of equations :— 

FC, = Sues + fiz@2+- So 

BC, = frei tfortet--- (1c) 
C,, O,,..., being the currents, and ¢&, ¢,... the impressed 
forces in branches 1, 2, &c.; F being common to all, and it 
and the f’s being differential operators. We arrive at similar 
equations when the differential equation of a branch is not 
merely between the V and © of that branch, but between 
those of many branches ; for instance when 


Vi = ZO; =F Z42Co = .'s ial) Serene (2c) 


is the form of the differential equation of branch 1. 

Now let there be impressed force e in one branch only, and 
C be the current in a second, dropping the numbers as no 
longer necessary. We then have 


FC = fe). . . . ee 


Conjugacy is therefore secured by /(e) =0, making C inde- 
pendent of e. Therefore /(e)=0 is the complex condition of 
conjugacy. If, for example, 

Fe) = qetayetaet+... 5 2s a eg 
where the a’s are constants, functions of the electrical con- 


Self-induction of Wires. 177 


stants concerned, then, to ensure conjugacy, we require 
Ay = 0, Qn= 0, ag = 0, Xe. ieee ve (5c) 
separately ; and if these a’s cannot all vanish together we 
cannot have conjugacy. 
What C may be then depends only upon the initial state of 
the system in subsiding, or upon other impressed forces that 


we have nothing to do with. As depending upon the initial 
state, the solution is 
DS Go ag aah ae ees 575) 


the summation being with respect to the p’s which are the 
roots of F(p)=0,p being put for d/dét in F; and the A 
belonging to a certain p is to be obtained by the conjugate 
property of the equality of the mutual electric to the mutual 
magnetic energy of the normal systems of any pair of p’s. 
As depending upon e, the impressed force in the conductor 
which is to be conjugate to the one in which the current is C, 
let e be zero before time ¢=0, and constant after. Then, 


by (3c), 
_f(d/dije_ » f(p)e ot 
he F(d/dt) hy aes At ee ) 
=C)— > ee :eidirivnionen ite) 


if Cp is the final steady current, and F’=dF/dp, the summa- 
tion being with respect to the p’s. 
If there is a resistance-balance, aa=0, Cp =0, and 


O = BOLT WP dence Bee 3) 
Now, subject to (4c), calculate the integral transient cur- 
rent :— * 
\ Cdt = > Ae 
= value of /( p)e/pF (p) when p=0, 
roe UPR Hraeae Cot geen eh Coe eo ars wee Lp a ene 7) 


if Fy is the p=0 value of F. If then a,=0 also, we prove 
that the integral transient current is zero. 
Supposing both a=0, a;=0, then 
2 
gu App +... nae 
therefore P pee es 


t 
{ Cdr 3 SP ane ea ae 
: = 


178 Mr. O. Heaviside on the 


and therefore 


0 t 
dt {, dt = 3 OF -F 
0 


Thus, if ag=0 also, we have 


ity ts 
{ ar Cat = 0. na 
0 0 


Similarly, if a3=0 also, then 


00 *y 
[ae (ae | Ode = 0, 
0 0 0 


and soon. The physical interpretation of a=0 and a,=0 is 
obvious, but after that it is less easy. 

If F contain inverse powers of p, the steady current may 
be zero. But in spite of that, it will be found that to secure 
perfect conjugacy for transient currents, we must have a true 
resistance-balance, or that relation amongst the resistances 
which would make the steady current zero, if we were to 
allow the possibility of a steady current by changing the 
value of other electrical quantities concerned. I will give an 
example of this later. 

I have elsewhere* pointed out these properties of the func- 
tion F, in the case where there is no mutual induction, or 
V=ZC is the form of the differential equation of a branch. 
Let n points be united by $n (n—1) conductors, whose con- 
ductances are Ky, Ky,3, &c., it being the points that are 
numbered 1, 2, &c. Then the determinant 


Ku, Ky, O90) Kin 

ig GING on wictegt ny 
is zero, and its first minors are numerically equal, if any K 
with equal double suffixes be the negative of the sum of the 
real K’s in the same row or columny. Remove the last row 
and column, and call the determinant that is left F. It is the 
F required, and is the characteristic function of the combina- 
tion, expressed in terms of the conductances. If every branch 
have self-induction, so that R+L(d/dt) takes the place of 
K-}, then F=0 is the differential equation of the combination, 
without impressed forces, and =O is always the differential 
equation subject to the condition of no mutual induction. In 


* ¢ Hlectrician, Dec. 20, 1884, p. 106. 
+ Asin Maxwell, vol. i. art. 280. 


Self-induction of Wires. 179 


the paper referred to cores are placed in the coils, giving a 
special form to K. 

When K is conductance merely, the characteristic function 
contains within itself expressions for the resistance between 
every two points in the combination, which can therefore be 
written down quite mechanically. For it is the sum of pro- 
ducts each containing first powers of the K’s, and therefore 
may be written 

B= Ky. Xyo + Yig=Ky3Xo3+ Yos=..-5- » (14e) 
where X23, Y23do not contain Ky3, and Xj, Yy, do not contain 
Ky. (lt is to be understood that the diagonal Ky,, Kos, ..., 
is got rid of.) 

Then 

R/q2= X42/Yj.=resistance between points 1 and 2, 15 

Rvo3= Xg3/Yo3= ” ” » 2 and 3, ag) 
&c., it being understood that these resistances are not Rj, 
R,3, &c., but the resistances complementary to them, the com- 
bined resistance of the rest of the combination ; thus, if e, be 
the impressed force in the conductor 1, 2, the current (steady) 


in it is m ae Be s 
be alas Recancnic nat, 
The proof by determinants is rather troublesome, using the 
K’s, but, in terms of their reciprocals, and extending the 
problem, it becomes simple enough. Thus if we turn K to 
R-! in F, and then clear of fractions, we may write F=0 as 
Ri2X12 sia Y= 0, Ro3X'o8 oie Y'o3 =()) &C., Phe (17) 
where X!j., Y'j2, do not contain R,,; &. From this we see 
that the differential equation of the current Oy, in 1,2, sub- 
ject to ey. only, is 
(Ryg + Bo) Cre aa C1oy oe ee whe (18c) 
if Rly=Y'j2/X'12. For this make the dimensions correct, 
and that is the only additional thing required, when we 
observe that it makes the fixed steady current 
Oyo= ey2/( Rag + B’51), mihny mh lak Cae tA (19¢) 
so that R’., is the resistance complementary to Ryp. 

Although it is generally best to work in terms of resist- 
ances, yet there are times when conductances are preferable, 
and, to say nothing of conductors in parallel arc, the above 
is a case in point, as will be seen by the way the characteristic 
function is made up out of the K’s. There is also less work 
in another way. Thus, $n(n—1) conductors uniting n points 
give 4(n—1)(n—2) degrees of freedom to the currents. It 
is the least number of branches in which, when the currents 
in them are given, those in all the rest follow. Thus, if 10 


180 Mr. O. Heaviside on the 


conductors unite 5 points, the currents in at least 6 conductors 
must be given, and no four of them should meet at one point. 
The remaining conductors are n—1 in number, or one less 
than the number of points, and n—1 is the degree of the 
characteristic function in terms of the conductances. Now 
put F=0 in terms of the resistances, by multiplying by the 
product of all the resistances. It is then made of degree 
4 (n—1)(n—2) in terms of the resistances, which is the num- 
ber of current freedoms. If n=4, the degree is the same, 
viz. three, whether in terms of conductances or resistances; 
- but if n=5, it is of the sixth degree in terms of resistances 
and only of the fourth in terms of the conductances; and if 
n=6, it is of the tenth degree in terms of the resistances, but 
only of the fifth in terms of the conductances, and so on; so 
that F becomes enormously more complex in terms of resist- 
ances than conductances. 

When every branch has self-induction, Z=R-+ Lp, and the 
degree of p in F=0 is the number of freedoms, so that there 
are n—1 fewer roots than the number of branches. It is the 
same when there is mutual induction. The missing roots 
belong to terms in the solutions for subsidence from an arbi- 
trary initial state which instantaneously vanish, producing a 
jump from the initial state to another, which subsides in time. 

On the other hand, if every branch (without self-induction) 
is shunted by a condenser of capacity 8), S., &e., K becomes 
K-+8p, so that the degree of p in F=0 is the same as that 
of K, or 4(n—1)(n—2) fewer than the number of con- 
densers *. 

Coming next to the Wheatstone quadrilateral self-induction 
balance, let there be six conductors, 1, 2, &c., uniting the four 
points A, By, B,, C in the figure. AB,C and AB,C are the 
lines referred to in the beginning. Let R be the resistance 
and L the inductance of a 
branch in which the current is 
C, reckoned positive in the 
direction of the arrow, and the 
fall of potential V in the same 
direction ; thus R,, L,, Vy, C, Af 
for the first branch. The six 
branches may be conjugate in 
pairs, thus: 1 and 4, or 2 and 
3, or 5 and 6. In the follow- 
ing 5 and 6 are selected always, 
the battery or other source 
being in 6, and the telephone 


* «Klectrician,’ Jan, 1, 1886, p. 147. 


e 


(o— 


Self-induction of Wires. 181 


or other indicator in 5. Mutual inductances will be denoted 
by M; thus, M,,. C, is the electromotive impulse in 2 due to 
the stoppage of the current C, in 1; similarly Mj, Cy is the 
impulse in 1 due to stopping Cp. 

Deferring mutual induction for the present, though not 
confining self-induction to be of the electromagnetic kind 
only, but to include electrostatic if required, the condition of 
conjugacy is that the potentials at B, and B, be always 
equal. Therefore 


RV amd ou 5 V Vn oe Ya C20) 

om. V = ZC, 
TiC i= 7, 05,; andy, 7,03 = 7,00 2) oo. (ile) 
But, by continuity, C;=C3, and C,=C, at every moment 


(including equality of all their differential coefficients); so 
that (21c) becomes 


Fi C=O 5, ai Zar Lo ae kal ih. (226) 
consequently 
Z,Z,—Z,43;=0=f aoe ies hehe. ek te cine (23¢) 
is the complex condition of conjugacy. This function is the 
f of the previous investigation. 


When the self-induction is of the electromagnetic kind, 
Z=R-+ Lp; so that, arranging fin powers of p, 


(R,R,— RRs) + (RL, + R,L;—R,L3;—R3Le)p + Ly Ly- LeLs)p’. 


Therefore, if c= L/R, the time-constant of a branch, we have 
three conditions to satisfy, namely, 


RR = RRs, ° ° ° > - (25¢) 
Ly +Xy~=ly+ Xs, (26c) 
LS Le Ls. (27¢) 


“Tf the first condition is fulfilled, there will be no final 
current in 5 when a steady impressed force is put in6. This 
is the condition for a true resistance balance. 

‘‘ Tf, in addition to this, the second condition is also satis- 
fied, the integral extra current in 5 on making or breaking 6 
is zero, besides the steady current being zero, (25c) and (26c) 
together therefore give an approximate induction balance 
with a true resistance balance. 

“Tf, in addition to (25c) and (26c), the third condition is 
satisfied, the extra current is zero at every moment during 
the transient state, and the balance is exact however the im- 
pressed force in 6 vary. 


(24¢ 


182 Mr. O. Heaviside on the 


“ Practically, take | 
R=, and VL, =L, f°. ee 


that is, let branches 1 and 2 be of equal resistance and induct- 
ance. Then the second and third conditions become identical; 
and, to get perfect balance, we need only make 

R,=Ry, and L;=b,: 02 20 See) 

“ This is the method I have generally used, reducing the 
three conditions to two, whilst preserving exactness. It is 
also the simplest method. The mutual induction, if any, of 
1 and 2, or of 3 and 4, does not influence the balance when 
this ratio of equality thats is employed (whether L,=L, or 
not) *. So branches 1 and 2 may consist of two similar wires 
wound together on the same bobbin, to keep their tempera- 
tures equal.’’ + 

Of the eight quantities, four R’s and four L’s, only five 
can be stated arbitrarily, of which not more than three may 
be R’s, and not more than three may be L’s. We may state 
the matter thus :—There must first be a resistance-balance. 
Then, if we give definite values to two of the L’s, the cor- 
responding time-constants become fixed, and it is required 
that the other two time-constants shall be equal to them ; 
thus 

either Si—aee and! (25 — 
or else Cy) PAM es 

Thus the remaining two L’s become usually fixed. In fact, 
eliminating R, and L, from (26c) by (25c) and (27c), the 
second condition may be written 

(a —&_) (a — 23) =0. 

Suppose R,, R,, Rz; given, then R, is fixed by (25c). 
Two of the inductances may then be given, fixing the 
corresponding time-constants. If these inductances be L, 
and L,, then we must have (unless #,;=,) 

X= U3, Vg. 
But if L, and L; be given, then we require (unless 2,=.3) 
LU, =X) U3 V4. 

These two cases present a remarkable difference in one 

respect. The absence of current in 5 allowing us to remove 9 


* The words in the () should be cancelled. The independence of M,, 
and M,,, which is exact when L,=L., L;=L, and sensibly true when’ 
the inequalities are small, becomes sensibly untrue when the inequalities 
L,—L, and L,—Ly, are oreat, 

"tf Electrician, April 30, 1886, p. 489. 


Self-induction of Wires. 183 
altogether, we see by (18c) that the differential equation of CO, is 


Z, + Zs) ery. 
— 7, + (41+ Zs) (Ze + Ly) 1 3) 2 4 
: { 8 Ly + Ze+ Zs + Ly Cs, 


manipulating the Z’s like resistances. The absence of 
branch 5 thus reduces the number of free-subsidence systems 
to two. Now, if we choose 7j= 22, we shall make 


(1, + Ls)/(R; + Rs) = (Le + Ly)/(Re+ Ry), 
or the time-constants of the two branches 1+3 and 2+4 
equal. Then one of the p’s is 


sini Ba aes 

Le Lee Ls 3 
and this is only concerned in the free subsidence of current 
in the circuit AB,CB,A. Consequently the second p, which is 


p= (R, + Rs)R, + Re(R, + Re) 
(L, + L;) Ry + L,(R, + Re)’ 


is alone concerned in the setting-up of current by the im- 
pressed force in 6; and the current divides between AB,C 
and AB,C in the ratio of their conductances, in the variable 
period as well as finally. In fact, the fraction in the above 
equation of C, will be found to contain Z,+ Zs; as a factor in 
its numerator and denominator, thus excluding the p, root, 
so far as e is concerned. On the other hand, if we choose 
%,=23, we do not have equality of time- constants of AB,C 
and AB,C, so that there are two p’s concerned, which are not 
those given ; and the current C, does not, in the variable 
period, divide between AB,C and AB,C in the ratio of their 
conductances, but only finally. 

In the above statement it was assumed that when L, and 
Ly, were chosen, it was not so as to make 2;=2,. When this 
happens, however, it is only the ratio of L; to L, that becomes 
fixed, for we have #,;=#, = anything. 

Similarly, when L, and L3 are so chosen that 7,=23, we 
shall have v,=«, = anything, so that only the ratio of L, to 
Ly, is fixed. 

And if L3, L, be so chosen that #3;=.,, then #,=x, = any- 
thing, only fixing the ratio of L, to Ly. But should 2; not 
=,, then we require 7,=43 and #,=4,, thus fixing L, and Lp. 

And if L:, Li, be so chosen that 4,=a,, then 2; =2;= any- 
thing, only fixing the ratio of R, to R3. But if so that x2 not 
=, then #;=a and #,=., fix L, and Ls. 

There are yet two other pairs that may be initially chosen, 
and with somewhat different results. Letit be L, and L, that 
are chosen ; if not so as to make 4;=.,, there are two ways 


184 Mr. O. Heaviside on the 


of fixing L, and Ls, viz. either by & = 3 and #,=4a,, or by 
fiw, and 22, ; (but i so that #, =, in the first place, then 
they must also =#7,=23. 

Similarly the choice of L, and L; so as not to make #,=4 
gives two ways of fixing L, and Ly, by vertical or by hori- 
zontal equality of time-constants, as before ; whilst 2.=.; 
produces equality all round. 

The special case of all four sides equal in resistance may be 
also noticed. Balance is given in two ways, either by hori- 
zontal or by vertical equality in the L’s. 


Leaving the mathematical treatment for a little while, I 
proceed to give a short general account of my experience of 
induction-balances. I did not originally arrive at the method 
of equal ratio just described through the general theory, (20c) 
to (27c), but simply by means of the general principle of 
balancing by making one line a copy of the other, of which I 
obtained knowledge through duplex telegraphy, and inves- 
tigated the conditions (25¢) to (27c) more from curiosity than 
anything else, though the investigation came in useful at 
last. In 1881 I wished to know what practical values to 
give to the inductances of various electromagnets used for 
telegraphic purposes, and to get this knowledge went to the 
quadrilateral. Not having coils of known inductance to start 
with, I employed Maxwell’s condenser method *, with an 
automatic intermitter and telephone. let 1, 2, and 3 be 
inductionless resistances, and 4 a coil having self-induction. 
Put the telephone in 5, the battery and intermitter in 6. We 
require first the ordinary resistance-balance, R,R,=R,R3. 
But the self-induction of the coil will cause current in 5 when 
6 is made or broken. This will be completely annulled by 
shunting 1 by a condenser of capacity 8,, such that 

RS; = L,/R, 
signifying that the time-constant of the coil on short-circuit 
and that of the condenser on short-circuit with the resistance 
R, are equal. 

The method is, in itself, a good one. But the double 

adjustment is sometimes very troublesome, especially if the 

capacity of the condenser be not adjustable. For when we 
vary Ry, to approximate to the correct value of R,S,, we 
upset the resistance-balance, and have therefore to make 
simultaneous variations in some of the other resistances to 
restore it. But the method has the remarkable recommenda- 
tion of giving us the value of the inductance of a coil at once 
in electromagnetic units. 


* Maxwell vol. 1. art. 778. 


Self-induction of Wires. 185 


In the course of these experiments I observed the upsetting 
of the resistance- and induction-balance by the presence of 
metal in the neighbourhood of the coils, which is manifested 
in an exaggerated form in electromagnets with solid cores. 
So, having got the information I wanted in the first place, I 
discarded the condenser method with its troublesome adjust- 
ments, and, to study these effects with greater ease, went to 
the equal-ratio method with the assistance that I had obtained 
by the condenser method, the values of the inductances of 
various coils, to be used as standards. 

“To use the Bridge to speedily and accurately measure the 
‘inductance of a coil, we should have a set of proper standard 
coils, of known inductance and resistance, together with a 
coil of variable inductance, 7. e. two coils in sequence, one of 
which can be turned round, so as to vary the inductance from 
a minimum toa maximum™®. The scale of this coil could be 
calibrated by (12a), first taking care that the resistance- 
balance did not require to be upset. This set of coils, in or 
out of circuit according to plugs, to form say branch 3, the 
coil to be measured to be in branch 4. Ratio of equality. 
Branckes 1 and 2 equal. Of course inductionless, or prac- 
tically inductionless, resistances are also required to get and 
keep the resistance-balance. The only step to this I have 
made (this was some years ago) .... was to have a number 
of little equal coils, and two or three multiples; and get 
exact balance by allowing induction between two little ones, 
with no exact measurement of the fraction of a unit.” f 

Although rather out of order, it will be convenient to 
mention here that although I have not had a regular induction- 
box made (the coils, if close together, would have to be closed 
solenoids), yet shortly after making these remarks, I returned 
to my earlier experiments by calibrating the scale of the coil 
of variable inductance. As it then becomes an instrument of 
precision, it deserves a name ; and as it is for the measure- 
ment of induction it may, I think, be appropriately termed 
an Inductometer. Of course, for many purposes no calibra- 
tion is needed. 

I found that the calibration could be effected with ease and 
rapidity by the condenser method more conveniently than by 
comparisons with coils. Thus, first ascertain the minimum . 
and the maximum inductance, and that of the coils separately. 
Suppose the range is from 20 to 50 units (hundreds, thou- 


* Prof. Hughes’s oddly named Sonometer will do just as well, if of 
suitable size and properly connected up. It is the manner of connection 
and use that give individuality to my inductometer. 

+ ‘Electrician,’ April 30, 1886, p. 490. 

Phil. Mag. 8. 5. Vol. 23. No. 141. Feb. 1887. O 


186 Mr. O. Heaviside on the 


sands, millions, &c. of centimetres, according to the quite 
arbitrary size of the instrument). It will then be sufficient 
to find the places on the scale corresponding to 20, 21, 22, &., 
49,50. Starting at 21, set the resistance-balance so that Ly, 
should be 21 units; turn the movable coil till silence is 
reached, and mark the place 21. Then set the balance to suit 
22, turn again till silence comes, and mark again; repeat 
throughout the whole range. Why this can be done rapidly 
is because the resistance-balance is at every step altered in the 
same manner. We have thus an instrument of constant 
resistance and variable known inductance, ranging from 


l + be _ 2m to L, + b + 2M, 


if 1, and J, are the separate inductances and m, the maximum 
mutual inductance. The calibration is thoroughly practical, 
as no table has to be referred to to find the value of a certain 
deflection. 

I formerly chose 10° centim. as a practical unit of in- 
ductance, and called it a tom; the attraction this had for me 
arose from L toms—R ohms equalling L/R seconds of time. 
But it was too big a unit, and millitoms and microtoms were 
wanted. Another good name is mac. 10% centim. might be 
called a mac. Since Maxwell made the subject of self- 
induction his own, and described methods of correctly mea- 
suring it, there is some appropriateness in the name, which, 
as a mere name, is short and distinctive. 

The two coils of the inductometer need not be equal; but 
it is very convenient to make them so, before calibration, by 
the equal-ratio method, which, of course, merely requires us 
to get a balance, not to measure the values. Let 1 and 2 be © 
any equal coils; put one coil of the inductometer in 3, the 
other in 4, and balance. It happened by mere accident that 
my inductometer had nearly equal coils; so I made them 
quite equal, to secure two advantages. Tirst, there is facility 
in calculations ; next, the inductometer may be used with its 
coils in parallel or in sequence, as desired. When in parallel, 
the effective resistance and inductance are each one fourth of 
the sequence values. Thus, let’ V=ZC be the differential 
equation of the coils in parallel, C being the total current, 
and V the common potential fall ; it is easily shown that 


_(mthp)(ret ls Do) — mp 
Z— MATo+ (, +l,—2m) p ae 5 * (30¢) 


when the coils are unequal ; 7, and r, being their resistances, 
1, and /, their inductances, and m their mutual inductance in 


Self-induction of Wires. 187 
any position. Now make 7;=7., and l,=/,; this reduces 
eo PPA lp se, on ses Gla) 
whilst, when in sequence, we have 


Lex one 2 em yp toate eh i.90 JOCSBe) 


thus proving the property stated. We may therefore make 
one inductometer serve as two distinct ones, of low or high 
resistance. 

There does not seem to be any other way of making the 
two coils in parallel behave as a single coil as regards external 
electromotive force. Any number of coils whose time- 
constants are equal will, when joined up in parallel, behave 
as a single coil of the same time-constant ; but there must be 
no mutual induction. (An example of the property * that any 
linear combination whose parts have the same time-constant 
has only that one time-constant.) This seriously impairs the 
utility of the property. This reservation does not apply in 
the case of the equal-coil inductometer. 

Having got the inductometer calibrated, we may find the 
inductance of a given coil, or of a combination of coils in 
sequence, with or without mutual induction, nearly as rapidly 
as the resistance. Thus, 1 and 2 being equal, put the coil to 
be measured in 3, and the inductometer in 4.. We have to 
make R;=R, and L;=Ly, or to get a resistance-balance, and 
then turn the inductometer till silence is reached, when the 
scale-reading tells us the inductance. This assumes that L; 
lies within the range of the inductometer. If not, we may 
vary the limits as we please by putting a coil of known 
inductance in sequence with branch 3 or 4 as required, putting 
at the same time equal resistance in the other branch. 

Or, the inductometer being in 4, and 1, 2 being induc- 
tionless resistances, put the coil to be measured in 3. If ithas 
a larger time-constant than the inductometer’s greatest, insert 
resistance along with it to bring the time-constants to equality. 
The conditions of silence are Rj R,=R,R; and L;/R3= L,/Ry4. 
Here a ratio of equality is not required. The method is 
essentially the same as one of Maxwell’st, and is a good one 
for certain purposes. 

Or, 1 and 2 being any equal coils, put one coil of the 


* This property supplies us with induction-balances of a peculiar kind. 
Let there be any network of conductors, every branch having the same 
time-constant. Set up current in the combination, and then remove the 
impressed force. During the subsidence all the junctions will be at the 
same potential, and any pair of them may ae aes be joined by an 
external conductor without producing current 1 in it. 

+ Maxwell, vol. ii. art. 757. 

O 2 


188 Mr. O. Heaviside on the 


inductometer in 6 and the other in 4, the coil to be measured 
being in 8. Then 

. 7 L3=L,—2Mig . . . « « (Bae) 
gives the induction-balance, L, being here the inductance of 
the coil of the inductometer in 4, and M,, the mutual induc- 
tance of the two coils, in the position giving silence. This is 
known in all positions, because the scale-reading gives the 
value of /,+l,4+2m (or else 2(/+m) if the coils are equal), 
and j,+/,is known. If the range is not suitable, we may, 
as before, insert other coils of known inductance. 

There are other ways ; but these are the simplest, and the 
equal-ratio method is preferable for general purposes. I 
have spoken of coils always, where inductances are large and 
small errors unimportant. When, however, it is a question of 
small inductances, or of experiments of a philosophical nature, 
needing very careful balancing, then the equal-ratio method 
acquires so many advantages as to become the method. 

“So long as we keep to coils we can swamp all the irregu- 
larities due to leading wires &c., or easily neutralize them, and 
can therefore easily obtain considerable accuracy. With short 
wires, however, it is a different matter. The inductance of a 
circuit is a definite quantity : so is the mutual inductance of 
two circuits. Also, when coils are connected together, each 
forms so nearly a closed circuit that it can be taken as such ; 
so that we can add and subtract inductances, and localize 
them definitely as belonging to this or that part of a circuit. 
But this simplicity is, to a great extent, lost when we deal 
with short wires, unless they are bent round so as to make 
nearly closed circuits, We cannot fix the inductance of a 
straight wire, taken by itself. It has no meaning, strictly 
speaking. The return current has to be considered. Balances 
can always be got, but as regards the interpretation, that 
will depend upon the configuration of the apparatus. 

“‘ Speaking with diffidence, having little experience with 
short wires, | should recommend 1 and 2 to be two equal 
wires, of any convenient length, twisted together, joined at 
one end, of course slightly separated at the other, where they 
join the telephone wires, also twisted. ‘The exact arrangement 
of 3 and 4 will depend on circumstances. But always use a 
long wire rather than a short one (experimental wire). If 
this is in branch 4, let branch 3 consist of the standard coils 
(of appropriate size), and adjust them, inserting, if necessary, 
coils in series with 4 also. Of course I regard the matter 
from the point of view of getting easily interpretable 
results.’’ * 

* Electrician,’ April 6, 1886, p. 490. 


Self-induction of Wires. 189 


Consider the equations (24c) to (27c). Three conditions 
have to be satisfied, in general, the resistance-balance (25c) 
and the balance of integral extra-current (26c) not being 
sufficient. To illustrate this in a simple manner, let 2 and 3 
be equal coils, by previous adjustment, and 1 and 4 coils 
having the same resistance as the others, but of lower induc- 
tance, or else two coils whose total resistance in sequence is 
that of each of the others, but of lower inductance when 
separated. The resistance-balance is satisfied, of course. 
Now, if the next condition were sufficient to make an 
induction-balance, all we should have to do would be to make 
L,+L,=2L;. For instance, if L, is first adjusted to equal 
L, and Ls, then, by increasing either L, or Ly, to the right 
amount, silence would result. It does result when it is Ly 
that is increased, but not when it is L,. If the sound to be 
quenched is slight, the residual sound in the L, case is feeble 
and might be overlooked ; but if it be loud, then the residual 
sound in the L, case is loud and is comparable with that to 
be destroyed, whilst in the L, case there is perfect silence. 

The reason of this is that in the L, case we satisfy only the 
second condition, whilst in the L, case we satisfy the third as 
well. 

Another way to make the experiment is to make 1, 2, and 
3 equal, and 4 of the same resistance but of lower inductance, 
much lower. Then the insertion of a non-conducting iron 
core in 1 will lead to a loud minimum, but if put in 4 will 
bring us to silence, except as regards something to be men- 
tioned later. 

Supposing, however, we should endeavour to get silence by 
operating upon L,, although we cannot do it exactly, yet by 
destroying the resistance-balance we may approximate to it. 
Thus we have a false resistance- and a false induction-balance, 
and the question would present itself, If we were to wilfully 
go to work in this way in the presence of exact methods, 
how should we interpret the results? As neither (25c) nor 
(26c) is true, it is suggested that we make use of the formula 
based upon the assumption that the currents are sinusoidal 
or pendulous, or 8.H. functions of the time. Take p?=—n? 
in (24c), the frequency being n/27, and we find 


R,Ru(#, + 2%.) =RRs(2.-+23), . . «+ (d4e) 

(R,R,—R,R;) =n?(L,L,—L,L;3) . . (85e) 

are the two conditions to be satisfied; and we can undoubtedly, 

if we take enough trouble, correctly interpret the results, if 
the assumption that has been made is justifiable. 

I should have been fully inclined to admit (and have no 


il 


190 Mr. O. Heaviside on the 


doubt it is sometimes true) that, with an intermitter making 
regular vibrations, we might regard the residual sound as 
due to the upper partials, and that n/2a could be taken as 
the frequency of the intermitter, and (34 c), (35 c) employed 
safely, though not with any pretensions to minute accuracy, 
if circumstances compelled us to ignore the exact methods of 
true balances, were it not for the fact that this hypothesis 
sometimes leads to utterly absurd results when experimentally 
tested. Of this I will give an illustration, and, as we have 
only to test that intermittences may be regarded as 8.H. 
reversals, simplify by taking R,=R,, L,=L,, which makes 
an exact equal-ratio balance, R;=R,, L3= 

Since a steady or slowly varying current does not produce 
sound in the telephone, if a battery could be treated as an 
ordinary conductor, we could put it in one of the sides of the 
quadrilateral and balance it, just like a coil, in spite of its 
electromotive force. So, let 1 and 2 be equal coils, 3 the 
battery to be tested, and 4 the balancing coils. I find that a 
good battery can be very well balanced, though not perfectly, 
with intermittences, as regards resistance, which is, however, 
far less with rapid intermittences than with a steady current*. 
Thus: steady, 24 ohms; intermittent (about 500), 14 ohm. 
Another battery : steady, 166 ohms ; intermittent, 126 ohms. 
The steady resistances are got by cutting out the intermitter, 
using a make-and-break instead ; the deflection of a galvano- 
meter in 5 must be the same whether 6 is in or out. If we 
leave out the battery in 6, it becomes Mance’s method. ‘The 
sensitiveness is, however, far greater when the battery is not 
left out, although other effects are then produced. 

So far regarding the resistance. As regards the inductance, 
or apparent inductance, of batteries, that is, I find, usually 
negative. That is to say, after bringing the sound to a 
minimum by means of resistance-adjustment, the residual 
sound (sometimes considerable) may be quenched by inserting 
equal coils in branches 3 and 4, and then increasing the 
inductance of the one containing the battery under test. I 
selected the battery which showed the greatest negative induc- 
tance, about 4 mac, or 590,000 centim., got the best possible 
silence by adjustment of resistance and inductance, and then 
found the residual sound could be nearly quenched by allowing 
induction between the coil in 3 and a silver coin, provided, at 
the same time, R, were a little increased. 

It was naturally suggested by the negative inductance and 

* Tam aware that Kohlrausch employs the telephone with intermit- 


tences to find the resistance of electrolytes, but have no knowledge of how 
he gets at the true resistance. 


Self-induction of Wires. 191 


lower resistance that the battery behaved as a shunted con- 
denser, or as a shunted condenser with resistance in sequence, 
or something similar ; and I examined the influence of the 
frequency on the values of the effective resistance and induc- 
tance. ‘The change in the latter was uncertain, owing to the 
complex balancing, but the apparent resistance was notabl 
increased by increasing the frequency, viz. from 125 to 130 
ohms, when the frequency was raised from about 500 to about 
800, whilst there was a small reduction in the amount of the 
negative inductance. The effect was distinct, under various 
changes of frequency, but was the opposite (as regards 
resistance) of what I expected on the 8.H. assumption. To 
see whereabouts the minimum apparent resistance was (being 
165 steady), I lowered the speed by steps. The resistance 
went down to 113 with a slow rattle, and so there was no 
minimum at all. The 8.H. assumption had not the least 
application to the apparent resistance, as regards the values 
165 steady, 113 slow intermittences, although it no doubt is 
concerned in the rise from 113 to 130 at frequency 800. 
The balance (approximate) was some complex compromise, 
but was principally due to a vanishing of the integral extra- 
current. Of course in such a case as this we should employ 
a strictly S.H. impressed force ; a remark that applies more 
or less in all cases where the combination tested does not 
behave as a mere coil of constant R and L. 

The other effects, due to using a battery in branch 6 as well, 
are complex. It made little difference when the current in the 
cell was in its natural direction ; but on reversal (by reversing 
the battery in 6) there was a rapid fall in the resistance—for 
instance, from 46 ohms to 18 ohms in half a minute in the 
case of a rather used-up battery, but a comparatively small 
fall when the battery was good. 

Besides the advantage of independence of the manner of 
variation of the impressed force (in all cases where the re- 
sistance and inductance do not vary with the frequency), and 
the great ease of interpretation, the equal-ratio method gives 
us independence of the mutual induction of 1 and 2 and of 
3 and 4; and this, again, leads to another advantage of an 
important kind. If the arrangement is at all sensitive, the 
balance will continually vary, on account of temperature 
inequalities occurring in experimeating, caused by the breath, 
heat of hands, lamps, &. Now, if the four sides of the 
quadrilateral consist of four coils, equal in pairs, it is a 
difficult matter to follow the temperature changes. To restore 
a resistance-balance is easy enough; but more than that is 
needed, viz. the preservation of the ratio of equality. But, by 


192 Mr. O. Heaviside on the 


reason of the independence of the self-induction balance of 
M,,, we may, as before mentioned, wind them together, and 
thus ensure their equality at every moment. There is then 
only left the mequality between branches 3 and 4, which 
must, of course, be separated for experimental purposes, and 
that is very easily followed and set right. When a sound 
comes on, holding a coin over the coil of lower resistance will 
quench it, if it be slight and due to resistance inequality, and 
tell us which way the inequality lies. If it be louder, the 
cancelling will be still further assisted by an iron wire over or 
in the same coil, or by a thicker iron wire alone, for reasons 
to be presently mentioned. 

On the other hand, a small inequality in the inductance may 
be at once detected by a fine iron wire, quenching the sound 
when over or in the coil of lower inductance ; and when the 
resistance- and induction-balances are both slightly wrong, a 
combination of these two ways will show us the directions of 
departure. These facts are usefully borne in mind and made 
use of when adjusting a pair of coils to equality, during 
which process it is also desirable to handle them as little as 
possible, otherwise the heating will upset our conclusions and 
cause waste of time. But a pair of coils once adjusted to 
equality, and not distorted in shape afterwards, will practically 
keep equal in inductance; for the effect of temperature- 
variation on the inductance is small, compared with the 
resistance change. 

Regarding the intermitter, I find that it is extremely de- 
sirable to have one that will givea pure tone, free from harsh 
irregularities, for two reasons: first, it is extremely irritating to 
the ear, especially when experiments are prolonged, to have 
to listen to irregular noises or grating and fribbling sounds ; 
next, there is a considerable gain in sensitiveness when the 
tone is pure”. 

Coming now to the effects of metal in the magnetic field of 
a coil, the matter is more easily understood from the theoretical 
point of view in the first instance than by the more laborieus 
course of noting facts and evolving a theory out of them—a 
quite unnecessary procedure, seeing that we have a good 
theory already, and, guided by it, have merely to see whether 
it is obeyed and what the departures are, if any, that may 
require us to modify it. 

Virst, there is the effect of inductive magnetization in 
increasing the inductance of a coil. Diamagnetic decrease is 


* I. e. pure in the common acceptation, not in the scientific sense of 
having a definite single frequency, which is only needed in a special class 
of cases, when no true balance could be got without it. 


Self-induction of Wires. 193 


quite insensible, or masked by another effect, so that we are 
confined to iron and the other strongly magnetic bodies. 
The foundation of the theory is Poisson’s assumption (no 
matter what his hypothesis underlying it was) that the 
induced magnetization varies as the magnetic force; and 
when this is put into a more modern form, we see that 
impressed magnetic force is related to a flux, the magnetic 
induction, through a specific quality, the inductivity, in the 
same manner as impressed electric force is related to electric 
conduction-current through that other specific quality, the 
conductivity of a body. Increasing the inductivity in any 
part of the magnetic field of a coil, therefore, always increases 
the inductance L, or the amount of induction through the 
coil per unit current in it, and the magnetic energy, $LC’. 
The effect of iron therefore is, in the steady state, merely 
to increase the inductance of a coil, without influence on its 
resistance. J have, indeed, speculated* upon the existence of 
a magnetic conduction-current, which is required to complete 
the analogy between the electric and magnetic sides of 
electromagnetism ; but whilst there does not appear to be any 
more reason for its existence than its suggestion by analogy, 
its existence would lead to phenomena which are not ob- 
served. 

But this increase of L by a determinable amount—deter- 
minable, that is, when the distribution of inductivity is 
known, on the assumption that the only electric current is 
that in the coil—breaks down when there are other currents, 
connected with that in the coil, such as occur when the latter 
is varying, the induced currents in whatever conducting 
matter may be in the field. LL then ceases to have any 
definite value. But in one case, that of 8.H. variation, the 
mean value of the magnetic energy becomes definite, viz. 
11/C,’, where L’ is the effective L, and Cy the amplitude of 
the coil-current, the change from 4 to + being by reason of 
the mean of the square of a sine or cosine being 4. This 
definiteness must be, because the variation of the coil-current 
is S.H., as well as that of the whole field. That L’ is less 
than L, the steady-flow value, may be concluded in a general 
though vague manner from the opposite direction of an in- 
duced current to that of an increasing primary, and its 
magnetic field in the region of the primary; or, more dis- 
tinctly, from the power of conducting-matter to temporarily 
exclude magnetic induction. 

In a similar manner, the resistance of a coil, if regarded as 
the R in RC’, the Joulean generation of heat per second, 


* ‘Electrician,’ January 4, 1885, p, 219 et seg. 


194 Mr. O. Heaviside on the 


ceases to have a definite value when the current is varying, if 
C be taken to be the coil-current, on account of the external 
generation of heat. But in the 8.H. case, as before, the mean 
value is necessarily a definite quantity (at a given frequency), 
making $R/C,’ the heat per second, where R’ is the effective 
resistance. That R’ is always greater than R is certain, and 
obvious without mathematics ; for the coil-heat is 4RC)’, and 
there is the external heat as well. It is suggested that, in a 
similar manner, a non-mathematical and equally clear demon- 
stration of the reduction of L is possible. The magnetic 
energy of the coil-current alone is +LC,’, and we have to 
show non-mathematically, but quite as clearly as in the 
argument relating to the heat, that the existence of induced 
external current reduces the energy without any reference to 
a particular kind of coil or kind of distribution of the external 
conductivity. Perhaps Lord Rayleigh’s dynamical generali- 
zation* might be made to furnish what is required. 

When the matter is treated in an inverse manner, not 
regarding electric current as causing magnetic force, but as 
caused by or being an affection of the magnetic force, there is 
some advantage gained, inasmuch as we come closer to the facts 
as a whole, apart from the details relating to the reaction on the 
coil-current. Magnetic force, and with it electric current, a 
certain function of the former, are propagated with such 
immense rapidity through air that we may, for present pur- 
poses, regard it as an instantaneous action. On the other 
hand, they are diffused through conductors in quite another 
manner, quite slowly in comparison, according to the same 
laws as the diffusion of heat, allowing for their being vector 
magnitudes, and that the current must be closed, thus pro- 
ducing lateral propagation. The greater the conductivity and 
the inductivity, the slower the diffusion. Hence a conductor 
brought with sufficient rapidity into a magnetic field is, at the 
first moment, only superficially penetrated by the magnetic dis- 
turbance to an appreciable extent ; and a certain time—which 
is considerable in the case of a large mass of metal, especially 
copper, by reason of high conductivity, and more especially 
iron, by reason of high inductivity more than counteracting 
the effect of its lower conductivity—is required before the 
steady state is reached, in which the magnetic field is calcul- 
able from the coil-current and the distribution of inductivity. 
And hence a sufficiently rapidly oscillatory impressed force 
in the coil-circuit induces only superficial currents in a piece 
of metal in the field of the coil, the interior being com- 
paratively free from the magnetic induction. 


* Phil. Mag. May 1886. 


Self-induction of Wires. 195 


The same applies to the conductor forming the coil-circuit 
itself; it also may be regarded as having the magnetic dis- 
turbance diffused into its interior from the boundary, and we 
have only to make the coil-wire thick enough to make the 
effect of the approximation to surface-conduction experi- 
mentally sensible. But in common fine-wire coils it may be 
wholly ignored, and the wires regarded as linear circuits. 
There is no distinction between the theory for magnetic and 
for non-magnetic conductors ; we pass from one to the other 
by changing the-values of the two constants, conductivity and 
inductivity. Nor is there any difference in the phenomena 
produced, if the steady state be taken in each case as the basis 
of comparison. But, owing to copper having practically the 
same inductivity as air, there seems to be a difference in the 
theory which does not really exist. 

A fine copper wire placed in one (say in branch 3) of a 
pair of balanced coils in the quadrilateral, under the influence 
of intermittent currents, produces no effect on the balance. 
Its inductivity is that of the air it replaces, so that the steady 
magnetic field is the same ; and it is too small for the diffu- 
sion effect to sensibly influence the balance. On the other 
hand, a fine iron wire, by reason of high inductivity, requires 
the inductance of the balancing-coil (say in 4) to be increased. 
The other effect is small in comparison, but quite sensible, and 
requires a small increase of the resistance of branch 4 to balance 
it. A thick copper wire shows the diffusion effect ; and if 
we raise the speed and increase the sensitiveness of the 
balance, its thickness may be decreased as much as we please, 
if other things do not interfere, and still show the diffusion 
effect. If thick, so that the disturbance is considerable, the 
approximate balancing of it by change of resistance is insuffi- 
cient, and the inductance of coil 4 requires a slight decrease 
or that of 3 a slight increase. A thick iron wire shows both 
effects strongly : the inductance and the resistance of branch 3 
must be increased. These effects are greatly multiplied when 
big cores are used ; then the balancing, with intermittences, 
at the best leaves a considerable residual sound. The in- 
fluence of pole-pieces and of armatures outside coils in 
increasing the inductance, which is so great in the steady 
state, becomes relatively feeble with rapid intermittences. 
This will be understood when the diffusion effect is borne in 
mind. : 

If the metal is divided so that the main induced conduction 
currents cannot flow, but only residual minor currents, we de- 
stroy the diffusion effect more or less, according to the fineness 
of the division, and leave only the inductivity effect. In my 


196 Mr. O. Heaviside on the 


early experiments I was sufficiently satisfied by finding 
that the substitution of a bundle of iron wires for a solid 
iron core, with a continuous reduction in the diameter 
of the wires, reduced the diffusion effect to something 
quite insignificant in comparison with the effect when the 
core was solid, to conclude that we had only to stop the 
flow of currents to make iron, under weak magnetizing 
forces, behave merely as an inductor. More recently, on 
account of some remarks of Prof. Hwing on the nature 
of the curve of induction under weak forces, | immensely 
improved the test by making and using nonconducting cores, 
containing as much iron as a bundle of round wires of the 
same diameter as the cores. I take the finest iron filings (sift- 
ings) and mix them with a black wax in the proportion of 1 of 
wax to 5 or 6 of iron filings by bulk. After careful mixture 
I roll the resulting compound, when in a slightly yielding 
state, under considerable pressure, into the form of solid round 
cylinders, somewhat resembling pieces of black poker in 
appearance. (4 inch diameter, 4 to 6 inches long.) That 
the diffusion effect was quite gone was my first conclusion. 
Next, that there was a slight effect, though of doubtful 
amount and character. The resistance-balance had to be very 
carefully attended to. But, more recently, by using coils 
containing a much greater number of windings, and thereby 
increasing the sensitiveness considerably, as well as the 
magnetizing force, I find there is a distinct effect of the 
kind required. Though small, it is much greater than could 
be detected ; but whether it should be ascribed to the cause 
mentioned or to other causes, as dissipation of energy due to 
variations in the intrinsic magnetization, or to slight curvature 
in the line of induction, so far as the quasi-elastic induction is 
concerned, is quite debateable. ‘To show it, let 1 and 2 be 
equal coils wound together (L=3 macs, R=47 ohms), 3 and 
A equal in resistance (R;= R,=93 ohms), but of very unequal 
inductances, that of coil 3 (L;=24 macs) being so much 
greater then that of coil 4 that the iron core must be fnlly 
inserted in the latter tomake Ly=Ls. (Coils3 and 4; 1 inch 
external, 4 inch internal diameter, and 3 inch in depth, Fre- 
quency 500.) The balancing of induction is completed by 
means of an external core. Resistance of branch 6 a few 
ohms, H.M.F’. 6 volts. There is, of course, immense sound 
when the core is out of coil 3, but when it is in there is merely 
a faint sored sound which is near ly destroyed by increasing 
R; by about g}5 part, a relatively considerable change. On 
the other hand, pure self-induction of copper wires gives 
perfect silence, and so does Mes, a method I haye shown to be 


Self-induction of Wires. 197 


exact*, [I may, however, here mention that in experiments 
with mere fine copper-wire coils there are sometimes to be 
found traces of variations of resistance-balance with the fre- 
quency of intermittence, of very small amount and difficult to 
elucidate owing to temperature-variations.] Balancing partly 
by M,, and partly by the iron cores, the residual sound in- 
creases from zero with M,, only, to the maximum with the 
cores only. Halving the strength of current upsets the 
induction-balance in this way. ‘The auxiliary core must be 
set a little closer when the current is reduced. This would 
indicate a slightly lower inductivity with the smaller magne- 
tizing force, and proves slight curvature in the line of induc- 
tion. But, graphically represented, it would be invisible 
except in a large diagram. 

It is confidently to be expected, from our knowledge of 
the variation of mw, that when the range of the magnetizing 
force is made much greater, the ability of nonconducting iron 
to act merely as an increaser of inductance will become con- 
siderably modified, and that the dissipation of energy by 
variations in the intrinsic magnetization will cease to be 
insensible. But, so far as weak magnetizing oscillatory 
forces are concerned, we need not trouble ourselves in the 
least about minute effects due to these causes. Under the 
influence of regular intermittences, the iron gets into a 
stationary condition, in which the variations in the intrinsic 
magnetization are insensible. It seems probable that pw 
must have a distinctly lower value under rapid oscillations 
than when they are slow. The values of w calculated from 
my experiments on cores have been usually from 50 to 200, 
seldom higher. I should state that I define yu to be the ratio 
B/H, if B is the induction and H the magnetic force, which is 
to include h, the impressed force of intrinsic magnetization. 
(See the general equations in Part I.+) It is with this u, not 
with the ratio of the induction to the magnetizing force as 
ordinarily understood, that we are concerned with in experi- 
ments of the present kind. 

Knowing, then, that iron when made a nonconductor acts 
merely as an inductor, when we remove the insulation and 
make the iron a solid mass, it requires to be treated as both a 
conductor and an inductor, just like a copper mass, in fact, of 
changed conductivity and inductivity. When the coil is a 
solenoid whose length is a large multiple of its diameter, and 
the core is placed axially, the phenomena in the core become 
amenable to rigorous mathematical treatment in a compara- 

* ¢ Electrician,’ April 30, 1886. 
* Phil. Mag. August 1886. 


198 Mr. O. Heaviside on the 


tively simple manner. [In passing, I may mention that on 
comparing the measured with the calculated value of the 
inductance of a long solenoid according to Maxwell’s formula 
(vol. ii. art. 678, equations (21) and (23)) i in the first edition 
of his treatise, I found a far greater difference than could be 
accounted for by any reasonable error in the ohm (reputed) 
or in the capacity of the condenser, and therefore recalculated 
the formula, The result was to correct it, and reduce the 
difference to a reasonable one. On reference to the second 
edition (not published at the time referred to) I find that the 
formula has been corrected. I will therefore only give my 
extension of it. Let M be the mutual inductance of two long 
coaxial solenoids of length /, outer diameter cy, inner ¢, 
having n, and n, turns per unit length. Then 


M=4a'nyngty"(l—2 1), « . = 2) 8 (20m) 


where, if p=c,/¢2, 


ga=1—8 (148 (1478 (1438 (1+ Ge (1+3et - Meee Cues 


8 56 
When 


(=e, %=1—'149=-851. 


As regards Maxwell’s previous formula (22), art. 678, how- 
ever, there is disagreement still. | 

References to authors who have written on the subject of 
induction of currents in cores other than, and unknown to, 
and less comprehensively than, myself, are contained in Lord 
Rayleigh’s recent paper”. So far as the effect on an induction- 
balance is concerned, when oscillatory currents are employed, 
it is to be found, as he remarks, by calculating the reaction of 
the core on the coil-current. This I have fully done in my 
article on the subject. Another method is to calculate the 
heat in the core, to obtain the increased resistance. This I 
have also done. When the diffusion effect is small, its in- 
fluence on the amplitude and phase of the coil-current is the 
same as if the resistance of the coil-circuit were increased 
from the steady value R to + 


R=R+4lhyrpkn’'e? 
= R+4 2lmk(mNe?un)?=R+ R, say } Ws 
Many phenomena which may be experimentally observed when 


rods are inserted in coils may be usefully explained in this 
manner. Here yw and & are the inductivity and conductivity 


* Phil. Mag. December 1886. 
t ‘ Electrician,’ May 31, 1884, p. 55. 


(48c) 


Self-induction of Wires. 199 


of the core, of length /, the same as that of the coil, n/2a the 
frequency, ¢ the core’s radius, and N the number of turns of 
wire in the coil per unit length ; whilst 


hi, = (27Nc)?pl 


is that part of the steady inductance of the coil circuit which 
is contributed by the core. 

The full expression for the increased resistance due to the 
dissipation of energy in the core is to be got by multiplying 
the above R, by Y, which is given by * 


ou 
iy y y ( y ( 
Bt eld 2.6.8" (1 xT Fae ee 


where y= (47pkne’)”. The value of R’ is therefore R+ R,Y. 
The series being convergent, the formula is generally appli- 
cable. The law of the coefficients is obvious. I have slightly 
changed the arrangement of the figures in the original to show 
it. We may easily make the core-heat a large multiple of the 
coil-heat, especially in the case of iron, in which the induced 
currents are so strong. When y is small enough, we may 
use the series obtained by division of the numerator by the 
denominator in (49c), which is 


eee ey 11 . 437 
lee fe alien 05 lea ae 


Corresponding to this, I find from my investigation + of 
the phase-difference, that the decrease of the effective induc- 
tance from the steady value is expressed by 


y (4 19y 2299? ) 
Lx fa(1 ai tig tee he + Cle) 


(50c) 


When the same core is used as a wire with current longi- 
tudinal, and again as core in a solenoid with induction longi- 
tudinal, the effects are thus connected. Let L, be the above 
steady inductance of the coil so far asis due to the core, and 
L/, its value at frequency n/27, when it also adds resistance 
R’, to the coil. .Also let R, be the steady resistance of the 
same when used as a wire, and R’, and L’, its resistance and 
inductance at frequency n/27, the latter being what du then 


* ‘Electrician,’ May 10, 1884 p. 606, 
+ Ibid. May 14, 1884, p. 108. 


t+ oP (1 T 940.127 (1 T 374.182 (1 T 778.20 (14. = 


,(49e) 


200 Mr. O. Heaviside on the 
becomes. Then 
Amp N72? /k = RL, — bys + RU, } 
RRL =U, Lin’. 
I did not give any separate development of the L’, of the core, 
corresponding to (48c) and (49c) above for RY, but mer ged 
it in the expression for the tangent of the difference in phase 
between the impressed force and the current in the coil-circuit. 


The full development of L’; is 


(52c) 


y y ‘ee 
ee tea ae aaa (1+ rt £13.16 He 


the denominator being the same as in (49¢). 


The high-speed formule for R’; and L’; are 
Ln 
(22)? 
if y=16z’._ When z is as large as 10, this gives 
RY = L'n=-2234 Iyn, 
whereas the correct values by the complete formule are 
no aS Lyn, iG 225 L,. 


It is therefore clear that we may advantageously use the 
high-speed formule when ¢ is over 10, which is easily reached 
with iron cores at moderate speeds. 

The corresponding fully developed formule for R’, and L’s, 


when the current is longitudinal, are 


Y ie I PRIS 
BR’, _ 14+ Ao (14+ ay, (1+ 35 ual 


1+ 


1a = hin = 


JE eeR ens 
x6i6(1+ soriore(!+ gania. crtaag(lt- 


showing the laws of formation of the terms, and 


Oe Fay Sie 
ies 1+ ore rae(! pa ae be 3414.16 (1+... 
er e e e e e e e e ) 


the denominator ae as in the Jide pe formula. At 
z=10, or y=1600, these give 


ie: 507 R,, Lin, =1px'442; 


Self-induction of Wires. 201 
whereas Lord Rayleigh’s high-speed formulz, which are 
R’,=L/n=R, ($2)! 
make 
R,=2°2384 R,, L’/,=4p x 447. 


This particular speed makes the amplitude of the magnetic 
force in the core case, and of the electric current in the other 
case, fourteen times as great at the boundary as at the axis of 
the wire or core (see Part I.). As, however, we do not ordi- 
narily have very thick wires for use with the current longi- 
tudinal, the high-speed formule are not so generally applicable 
as in the case of cores, which may be as thick as we please, 
whilst by also increasing the number of windings the core heat- 
ing per unit coil-current amplitude may be greatly increased. 
If the core is hollow, of inner radius Cy, else the same, the 
equation of the coil-current is, if e be the impressed force 
and C the current in the coil-circuit whose complete steady 
resistance and inductance are R and L, whilst Ly, is the part 
of L due to the core and contained hollow (dielectric current 
in it ignored), 
2  J,(sc) —gK, (sc) 
sc J,(sc) —g Ko (sv) 


when g depends upon the inner radius, being given by 


e—RC+ (L—L,jC+ EG: 4 /(58e) 


__ ¥8C oJ 0(8¢o) —J1(seo) ‘ 
1 Fy se,Ko(8¢y) — Ku (8¢0) Borah ol Vine ae. 


(whose value is zero when the core is solid), and 
s°= —4Ampk(d/dt). 


There may be a tubular space between the core and coil, and 
R, L include the whole circuit. In reference to this (53c) 
equation, however, it is to be remarked that there is consider- 
able labour involved in working it out to obtain what may be 
termed practical formule, admitting of immediate numerical 
calculations. The same applies to a considerable number of 
unpublished investigations concerning coils and cores that I 
made, including the effects of dielectric displacement ; the 
analysis is all very well, and is interesting enough for educa- 
tional purposes, but the interpretations are so difficult in 
general that it is questionable whether it is worth while either 
publishing the investigations or even making them. 
Professor Hughes* has also devoted some attention to 
induction in cores, and has arrived at the remarkable conclu- 


* Proc. Roy. Soc. 1886. 
Phil. Mag. 8. 5. Vol. 23. No. 141. Feb. 1887. P 


202 Mr. O. Heaviside on the 


sion that he has obtained experimental evidence of the exist- 
ence of induced currents therein. Now, although when it 
is considered that although induced currents in wires were 
known to exist, yet the possibility of their existing in metal 
not in the form of wires was only a matter of the wildest 
speculation, Professor Hughes’s conclusion must be admitted 
to be very comforting and encouraging. 


Leaving now the question of cores and the balance of 
purely electromagnetic self-induction, and returning to the 
general condition of a self-induction balance Z,Z,=Z,Zs, 
equation (23c), let the four sides of the quadrilateral consist 
of coils shunted by condensers. Then R, L, and 8 denoting 
the resistance, inductance, and capacity of a branch, we have 


Z= {8p+(R+Lp) 3. . . . (dde) 
so that the conjugacy of branches 5 and 6 requires that 


{Sip + (Ry + Lyp)-1} {8p + (Ry + Lip)-"} 
= {S.p+(R, + Lep)-1} {8:9 + (Rs+-Lyp)-1}, — (56e) 


wherein the coefficient of every power of p must vanish, 
giving seven conditions, of which two are identical by having 
a common factor. It is unrecessary to write them out, as 
such a complex balance would be useless ; but some simpler 
cases may be derived. Thus, if all the L’s vanish, leaving 


condensers shunted by mere resistances, we have the three 
conditions 


R,R,=R,Rs, 
8,/R, ao S,/R, = 5) bes oh S3/Ra, of (8 Fil fe (5%c) 
8,8,= S283, 


which may be compared with the three self-induction condi- 
tions (25c) to (27c). 

If we put RS=y, the time-constant, the second of (57) 
may be written 


Wnt Y4 = Yot+Y35 «ee lel aetna (58c) 


which corresponds to (26c). If S,=0=8,, the single con- 
dition in addition to the resistance-balance is y,=y;. If 
5 0= ho, it 1s Y3=Y4- 

Next, let each side consist of a condenser and coil. in 
sequence. ‘Then the expression for Z is 


Z=R4+bp+(Sp)7,.. . 2 2 as 


Self-induction of Wires. 203 
which gives rise to five conditions, 
SiSu= S2Ss, i) 
Yr Ys=Yot Ya; 
5,5,(R, Ry — RoR3) = L,8; + L38;—L,8,— LS, 


(60c) 
fs oe Biond 
Vy Ne i Bo, 
i igeaihs bs. J 


Here it looks as if the resistance-balance were uuneces- 
sary; and, as there can be no steady current, this seems a 
sufficient reason for its not being required. But, in fact, the 
third condition, by union with the others, eliminating 
Ss, L;, 8, and Ly by means of the other four conditions, 
becomes 


= (RR, —B,Ry) Si82(RSi—R,S,} (La Re—R,Ls) — (Lo8)—L,8))" 


-. . (6le 
(RS, — B,S,) (2 —B,L,) oF 


So the obvious way of satisfying it is by the true resistance- 


balance. 
If there are condensers only, without resistance-shunts, 
we have 
Ae lOP ho reer Bute tnmok Queee) 
so that : 
rows Cede in Gal) adenine? (de) 


as the sole condition of balance. 
If two sides are resistances, R, and R,, and two are con- 
densers, S3 and 8,4, we obtain 


Boe Sal Setvcus Pucnveiie bo aloe) 


as the sole condition. The multiplication of special kinds 
of balance is a quite mechanical operation, presenting no 
difficulties. 


Passing now to balances in which induction between diffe- 
rent branches is employed, suppose we have, in the first 
place, a true resistance-balance, R,R,=R,R;, but not an 
induction-balance, so that there is sound produced. ‘Then, 
by means of small test coils placed in the different branches, 
we find that we may reduce the sound to a minimum in a 
great many ways by allowing induction between different 
branches. If the sound to be destroyed is feeble, we may 
think that we have got a true induction-balance; but if it is 

Ei2 


204. Mr. O. Heaviside on the 


loud, then the minimum sound is also loud, and may be com- 
parable to the original in intensity. We may also, by upset- 
ting the resistance-balance by trial, still further approximate 
to silence, and it may be a very good silence, with a false 
resistance-balance. The question arises, Can these balances, 
or any of them, be made of service and be as exact as the 
previously described exact balances? and are the balances 
easily interpretable, so that we may know what we are doing 
when we employ them ? 

There are fifteen M’s concerned, and therefore fifteen ways 
of balancing by mutual induction when only two branches at 
a time are allowed to influence one another, and in every case 
three conditions are involved, because there are three degrees 
of current-freedom in the six conductors involved. Owing 
to this, and the fact that in allowing induction between a pair 
of branches we use only one condition (i.e. giving a certain 
value to the M concerned), whilst the resistance-balance makes 
a second condition, I was of opinion, in writing on this sub- 
ject before *, that all the balances by mutual induction, using 
a true resistance-balance, were imperfect, although some of 
them were far better than others. Thus, I observed experi- 
mentally that when a ratio of equality (Ry=R,, L,=L,) was 
taken, the balances by means of M,3; or Me, were very good, 
whilst that by M,s was usually very bad, the minimum sound 
being sometimes comparable in intensity to that which was 
to be destroyed. 

I investigated the matter by direct calculation of the in- 
tegral extra-current in branch 5 arising on breaking or 
making branch 6, due to the momenta of the currents in the 
various branches, making use of a principle I had previously 
deduced from Maxwell’s equations t, that when a coil is 
discharged, through various paths, the integral current divides 
as in steady flow, in spite of the electromotive forces of in- 
duction set up during the discharge. This method gives us 
the second condition of a true balance. 

But more careful observation, under various conditions, 
showing a persistent departure from the true resistance- 
balance in the M,; method (due to Professor Hughes), and 
that the M,; and M,, methods were persistently good and 
were not to be distinguished from true balances, led me to 
suspect that the second and third conditions united to form 
one condition when a ratio of equality was used (just as in 
(28c), (29¢) above) in the M,,; and M,, methods, but not in 
the M,; method. So I did what I should have done at the 

* ¢ Electrician,’ April 30, 1886. 
+ Journal 8. T. E. 1878, vol. vii. p. 303. 


Self-induction of Wires. 205 


beginning : investigated the differential equations concerned, 
verified my suspicions, and gave the results in a Postscript. 
I have since further found that, when using the only practi- 
cable method of equal ratio, there are no other ways than those 
described in the paper referred to of getting a true balance of 
induction by variation of a single L or M, after the resistance- 
balance has been secured. This will appear in the following 
investigation, which, though it may look complex, is quite 
mechanical in its simplicity. 

Write down the equations of electromotive force in the 
three circuits 6+1+8, 1+5—2, and 3—4—5, when there 
is impressed force in branch 6 only. They are (p standing 
for d/dt), 

eg= (Ry + Lg p)Cg+ (R, + Ly p)C,+ (R; + L; p)O3 | 
+p (MgC, + MgeO, + Me3C3 + Mg,C,+ Mg5C;) 
+ p( My2C. + My303 + MyC,+ M505 + My6Cg) 
+ p(Ma 0; + Map. + M3,0, + M350; + MgC). 
0=(R, + Lyp)C, + (CR; + Lsp)C;—(R, + Lyp) Cy | 
+ p(My2C, + My303 + My,C, + MysC; + MygCg) : (650) 
+ p(MsC; + MsoC, + M5303 + M540, + MseCg) | 
— p(Mz;C, + My303 + Ma,C,+ M,;C;+ Mo¢Cg). 
0=(R3+ Lp) C3— (Ry + Lyp)C,—(R; + L;p) Cs | 
+ p( MgO; + Mg. + M,C, + Ms5C; + MgeCo) | 
—p( May, + MyC.+ Mys03+ MysC;-+ MigCe) | 
—p(MaC,+ MaaCs+ MisCs-+ MssC,+ MoCo). J 


Now, eliminate C,, C,, Cs by the continuity conditions 


Q,=C03;+0;, C,=C,—C;, Cy=C3+Cy . . (66c) 
giving us 
Cg = X03 + Xy20, + Xy3C;, 
O= X_,03 + X90, + Xo30s, Sh a Oh PEGE) 


ee X3,C3 3= X30, oe X33C;, 


where the X’s are functions of p and constants. Solve for 
C;. Then we see that 


DE Ge Keo Na oS es eweirer tre (68¢) 


is the complex condition of conjugacy of branches 5 and 6. 
This could be more simply deduced by assuming O;=0 at 


206 Mr. O. Heaviside on the 


the beginning, but it may be as well to give the values of all 
the X’s, although we want but four of them. Thus 
Xy= R,+R34+ Re + (Ly + Ly + Lg + 2M + 2Moy +2Ma:)p, ) 
Xig=  Re+ (Lig + Moo + Moa + My + Myst Mig + Myo + Mg, + Mae)p, | 
Xyg= Ry + (Ly + Mey — Meg + Mos — Myo + Mas + Mz, — Moo + M5) p, 
X= Ry+ CE, + Mis + Mys + Mig + Mss + Msg — Mo; —Mo3— My.)p, | 
Kop = — Rot (—Ly + My + Mus + Mug + Mop + Mya + Mss—Mes—Moe)p, 7 (68 
X= R,+R,4+R4+ (,4+1,4+ L;+2M,;—2M,;—2Myo)p, 
Xs= Rs + (s+ Mg, + Mgg— May — Mas — Mag — M5, — M3 — Mo), | 
Xgo= — Ry t+ (— Lg + Moe + Moy + Mig — Mag — Mag — Msp — Mss— Mose )p, | 4 
X33= —R;+ (—L; + Ms —Myy + Ma;s—Ma + Mig—Mus + Msp—My)p. J 
Now, using the required four of these in (68¢),and arranging _ 
in powers of p, it becomes 
0O=A,+A,p+A.p% . °c 
So A,=0 gives the resistance-balance ; A,=0, in addition, 
makes the integral transient current vanish ; and A,=0, in 


addition, wipes out all trace of current. 
There is also the periodic balance, 


A,=90, Ay= Asn", . . 7. 
if the frequency is 7/27. 
The values of Ay and A, are 
A,=R,R;—R, hy, : ° . . 5 (72c) 
Ay = R,Ls + R;L, <— R,L,- R,L, 
+ Re( Moy + Mag — My, — Mags — Mgg — Ms — M3 — Mog) 
+ R3(Mo4+ Msg—My,—M,,—Mi¢— Msp—M,,— Msg) 
+ Ry (Moy + Mog + Mogg — M2 — Mg — Ms — M54 — Mig) 
+ Ry(Moi + Mos + Mog—My3—Migs—My;—Ms3—Msg). . (78e) 
In this last, let the coefficients of R,, Rs, R,, Ry, in the 
brackets be qs, 93, 91, ga Then the value of A, is 
Ay= LeL3— LL, + Lege + Lyg3t+ Lyq)+ Lugs t+ qo93—-Mge (74e) 


It is with the object of substituting one investigation for a 
large number of simpler ones that the above full expressions 
for A, and A, are written out. 

If we take all the M’s as zero, we fall back upon the self- 
induction balance (25¢) to (27c). Next, by taking all the 
M’s as zero except one, we arrive at the fifteen sets of three 
conditions. Of these we may write out three sets, or, rather, 


Self-induction of Wires. 207 


the two conditions in each case besides the resistance-balance 
condition, which is always the same. 

All M’s=0, except Mg. 

R,Ry( 2, + 24— %— 23) = (R, + Re) Mg, 
L,i,—L,L;= (L, a L,) Mg. 

All M’s=0, except Myg. 

R, Ry (x, + v4—%—23) ie Malt | : (76c) 
L,L,—L,L3= — (L, + Lz) Myg. 

As these only differ in the sign of the M, we may unite 
these two cases, allowing induction between 6 and 3, and 6 
and 4. The two conditions will be got by writing M3,— My. 
for Mz, in (75c). 

All M’s=0, except M;, (Prof. Hughes’s method). 

0O=K,R,(2, + L4— Xy— X3) Se Mz,(R, + Re + R; + a (770) 

i= L,U,—L,L; “+ Ms6(L, -~ Le bt Ls + Ti). 

Now choose a ratio of equality, R,=R,, L,=L,, which is 
the really practical way of using induction-balances in 
general. In the M;, case the two conditions (75 ¢) unite to 
form the single condition 

Ly— L3=2M3,, Se es! 1) preg iiss: Ee (78c) 
and in the M,, case (76c) unite to form the single condition 

L,—L;= —2 Mic. e e e e ° (79c) 
We know already that the same occurs in the simple Bridge 
(29¢), making 


(75c) 


1 a ane as al RNa (10/6) 
so that we have three ways of uniting the second and third 
conditions. Now examine all the other M’s, one at a time, 
on the same assumption, R,=R,, L,=L, With M,. we 
obtain 

(L,—Ls)(L, —Mj,) =9, and =Ly=L;. 
But L,—My, cannot vanish ; so that 

iat We crits ge Peas COLE) 
is the single condition. Similarly, in case of My, 

foi e e e ° ° ° ° (82c) 


again. All these, (77 ¢) to (82¢), were given in the paper 
referred to ; the last two mean that M,. and M;, have abso- 
lutely no influence on the balance of self-induction. 

All the rest are double conditions. Thus, in A, and A, 


+ 2(Myg—M35) + 1 — 
O=L,(L,— Ls) + Ls(M,,-+ Mag + Mig + Moo + Moy + a 
+ L(y; + My5+ My,+ M;;+ Ms5— 

+ Ly (Mg + Ma + M5; + Ms. + Ms3 + Mz,— 
+ (Ma3+ Mis + Mig + Ms; + Ms_— 
X (May + Mag + Moa + Moo + Mas 

+ (May + Mas + Mag + Ms; + M53 + fe 


ieee 


Mr. O. Heaviside on the 
put Rj=R,, R;=R,, and L,=L,; then the two conditions 


eee as 

a tie ot 

=i ed ine ) 

—(0=L,—L;+ Mul + R,/R)) 
10=L,—L, + My,(1 + L,/l,)— M2,/L, 

—M,;(1 + Ry R)) 

— My3 (1 + Ly/L,) + M3,/Ly 


O=L,—L;+ (1+ R,/R,) (My,—Mo3-+ My; + Moy + Ms3+ Ms, + 2Mse) 
—M,,) +2(R/R,) (Mas— 


— Ms.) ; 


— My) 
— Mz.) 
— Ms» + 2M, + 2M;.— 2M) 


— Me) 


—Mse) s 


which are convenient for deriving the conditions when several 
M’s are operative at the same time. 
excepting the few already examined :— 


O=L,—L;+M;,(1+ R,/R,) | 
O=L,—L,+ M;,(14 L,/L,) 
0O=L,—L;+ M,.(1+ R,/R,) 
O=L,—L,;+ M;.(1 + L;/L,) 
=L,—L;+ M;3(1+ RR) 
"(0=L,—L, + M;3(1 + L,/L,) 
=L,—L;+ M;,(1+ R,/R,) 
=L,—L;+ M;,(1+ L,/L,) 
= L,—L;+2Ms.(1 + Ry/R,) 

O=Ly—Ls + Msg {2+ (Ls + L3)/Ly} 
O0=L,—L;+2M,,R,/R, 

0=L,—L;+M,,.(L;4+ L,)/L, 
0O=L,—L,;—2M,,R,/R, 

=l,— Fe aes )/L, 


Thus, one at a time, 


(85¢) 
(860) 
(87¢) 
(88¢) 
(89) 
(900) 
(910) 
(92e) 
(8c) 


(94c) 


\ (95¢) 


Self-induction of Wires. 209 


If we compare the two general conditions (83c), (84¢), we 
shall see that whenever 


1194— 9293=9, 
we may obtain the reduced forms of the conditions by adding 
together the values of L,—L, given by every one of the M’s 
concerned. We may therefore bracket together certain sets 
of the M’s. To illustrate this, suppose that Mjg and M,, are 
- existent together, and all the other M’s are zero. Then (92c) 
and (93c) give, by addition, 


R 
L,—L4= (Mx~M,)(1— j), 


L L 
L,—L,=M,—M,3+ Miz, Mat 


which are the conditions required. 

Similarly M,, and M,, may be bracketed. Also Mg, Mga, 
Megs, Mg,, and M,.. Also Ms, Mso, Mzg, Mss, and Meg. But 

y, and M,, will noé bracket. 

As already observed, the self-induction balance (28c) (29c) 
is independent of M,, and M;3,, when these are the sole mutual 
inductances concerned ; that is, when R, = Ry, L, = L,, R3= Ry, 
L3;=L, By (92c) and (93c) we see that independence of 
M,3 and M,, is secured by making all four branches 1, 2, 3, 4 
equal in resistance and inductance. 

But it is unsafe to draw conclusions relating to inde- 
pendence when several coils mutually influence, from the 
conditions securing balance when only one pair of coils at a 
time influence one another. Let us examine what (83c) and 
(84c) reduce to when there is induction between all the four 
branches 1, 2, 3, 4, but none between 5 and the rest or 
between 6 and the rest. Put all M’s=0 which have either 
5 or g in their double suffixes, and put Ly=L;. Then we may 
write the conditions thus :— : 


0=(1+R,/R,)(My,— My3) +(1— R,/R,)(Ma—Mis), » - - ee (96 
O=(L, + Ly) Mis—Mys) + (L, — Ly) (Ma,— Mis) + M3,—Mi, 
+ (Mo, —Mys) (Mgs— Miz) + (Maz— Mos) (Mog + Mis—Miz—M3,), (97 
The simplest way of satisfying these is by making 
Nei “Mand © Mb Ma ee (982) 


If these equalities be satisfied, we have independence of My, 
and M,,. : 
Now, if we make the four branches 1, 2, 3, 4 equal in 


210 Mr. O. Heaviside on the 


resistance and inductance, so that in (96c) and (97c) we have 
R,=R, and L,=Ly,, the first reduces to 


0= Mu M3, PEO gor Sot, (9 9c) 


so that it is first of all absolutely necessary that M,,= Mg, if 
the balance is to be preserved; whilst, subject to this, the 
second condition reduces to 

O= (Ma—Myj3)(Msi—Mye), . . « (1000) 
so that either M.,=M,3, or else Mz,=My. Thus there are 
two ways of preserving the balance when all four branches 
are equal, viz. M,,=M,3 and M.,=M,3, independent of the 
values of My. and Ms,; and My,=M>,3 and M3,= My, inde- 
pendent of the values of M,, and M,3. 

The verification of these properties, (98c) and later, makes 
some very pretty experiments, especially when the four 
branches consist, not merely of one coil each, but of two or 
more. The meanings of some of the simpler balances are 
easily reasoned out without mathematical examination of the 
theory; but this is not the case when there is simultaneous 
induction between many coils, and their resultant action on 
the telephone-branch is required. 

Returning to (96c) and (97c), the nearest approach we can 
possibly make to independence of the self-induction balance 
of the values of all the M’s therein concerned, consistent with 
keeping wires 3 and 4 away from one another for experi- 
mental purposes, is by winding the equal wires 1 and 2 
together. Then, whether they be joined up straight, which 
makes M,;= M.3 and M,,= Mg, identically, or reversed, making 
M,3= —M,3 and My,= —M,,, we shall find that 

My=M,; 
is the necessary and sufficient condition of preservation of 
balance. 

At first sight it looks as if M3; and M3. must cancel one 
another when wires 1 and 2 are reversed. But although 1 
and 2 cancel on 3, yet 3 does not cancel on 1 and 2 as regards 
the telephone in 5. ‘The effects are added. On the other 
hand, when wires 1 and 2 are straight, 3 cancels on them as 
regards the telephone, but 1 and 2 add their effects on 3. 
Similar remarks apply to the action between 4 and the equal 
wires 1 and 2 when straight or reversed ; hence the necessity 
of the condition represented by the last equation. 

On the other hand, M,, and M,, cancel when 1 and 2 are 
straight, and add their effects when they are reversed ; whilst 
M;, and M;, cancel when 1 and 2 are reversed, and add their 
effects when they are straight, results which are immediately 


Te 
el ni 


Self-induction of Wires. 211 


evident. But wires 1 and 2 must be thoroughly well twisted, 
before being wound into a coil, if it is desired to get rid of 
the influence of, say, Mg, and Mg, when it is a coil that 
operates in 6, and this coil is brought near tol and 2. [This 
leads me to remark that a simple way of proving that the 
mutual induction between iron and copper (fine wires) is the 
same as between copper and copper, which is immensely more 
sensitive than the comparison of separate measurements of the 
induction in the two cases, is to take two fine wires of equal 
length, one of iron, the other of copper, twist them together 
carefully, wind into a coil, and connect up with a telephone 
differentially. On exposure of the double coil to the action 
of an external coil in which strong intermittent currents or 
reversals are passing, there will be hardly the slightest sound 
in the telephone, if the twisting be well done, with several 
twists in every turn. But ifit be not well done, there will 
be a residual sound, which can be cancelled by allowing in- 
duction between the external or primary coil and a turn of 
wire in the telephone-circuit. A rather curious effect takes 
place when we exaggerate the differential action by winding 
the wires into a coil without twists, in a certain short part of 
its length. The now comparatively loud sound in the telephone 
may be cancelled by inserting a nonconducting iron core in 
the secondary coil, provided it be not pushed in too far, or go 
too near or into the primary coil. This paradoxical result 
appears to arise from the secondary coil being equivalent to 
two coils close together, so that insertion of the iron core does 
not increase the mutual inductance of the primary and secon- 
dary in the first place, but first decreases it to a minimum, 
which may be zero, and later increases it, when the core is 
further inserted. Reversing the secondary coil with respect 
to the primary makes no difference. Of course insertion of 
the core into the primary always increases the mutual induc- 
tance and multiplies the sound. The fact that one of the 
wires in the secondary happens to be iron has nothing to do 
with the effect. | 

Another way of getting unions of the two conditions of the 
induction-balance is by having branches 1 and 3 equal, instead 
of land 2. Thus, if we take R,=R;, L,=L;, R-=R, in A, 
and A, (78c) and (74¢), we obtain fifteen sets of double con- 
ditions similar to those already given, out of which just four 
(as before) unite the two conditions. Thus, using M,; only, 
we have 


WHY sell, Lhd. 4oa0 Ae ORS) 


and the same if we use M,, only, and the same when both Mj; 


212 On the Self-induction of Wires. 


and M,, are operative. That is, the self-induction balance is 
independent of Mj; and M,,. This corresponds to (81c) and 


(82). 
The other two are M,; and M,;. With M,,; we have 
O0=L,—L,—2M,;, Et Bets (102c) 
and with M,,, : 
0=L,—L,—2M,;. e e ° e (103c) 


The remaining eleven double conditions corresponding to 
(85c) to (95b) need not be written down. 

Several special balances of a comparatively simple kind 
can be obtained from the preceding by means of induc- 
tionless resistances, double-wound coils whose self-induction 
is negligible under certain circumstances, allowing us to put 
the L’s of one, two, or three of the four branches 1, 2, 3, 4 
equal to zero. We may then usefully remove the ratio of 
equality restriction if required. This vanishing of the L ofa 
branch of course also makes the induction between it and any 
other branch vanish. 

For instance, let L,=L,=L,=0 ; then 

O=R,L3;+ Mz (Rj+R.) . . . . (104c) 
gives the induction-balance when M3, is used, subject to 
R,R,—R,R3. And 

O=R,L;—M;,(Ro.+R,) . . . . (105e) 


is the corresponding condition when M;; is used. But Mg, 
will not give balance, except in the special case of 8.H. cur- 
rents, with a false resistance-balance. The method (104c) 
is one of Maxwell’s. His other two have been already 
described. 

In the general theory of reciprocity, it is a force at one 
place that produces the same flux at a second as the same 
force at the second place does at the first. That the reciprocity 
is between the force and the flux, it is sometimes useful to 
remember in induction-balances. Thus the above-mentioned 
second way of having a ratio of equality is merely equivalent 
to exchanging the places of the force and the (vanishing) 
flux. We must not, in making the exchange, transfer a coil 
that is operative. For example, in the M,, method (79c), 
there is induction between branches 6 and 4; M,; (equation 
- (88c)), on the other hand, fails to give balance. But if we 
exchange the branches 5 and 6, it is the battery and telephone 
that have to be exchanged; so that we now use M;,, which 
gives silence, whilst M,, will not. 

I have also employed the differential telephone sometimes, 
having had one made some five years ago. But it is not so 


Notices respecting New Books. 213 


adaptable as the quadrilateral to various circumstances. I 
need say nothing as to its theory, that having been, I under- 
stand, treated by Prof. Chrystal. Using a pair of equal coils, 
it is very similar to that of the equal-ratio quadrilateral. 


December 29th, 1886. 


XXIT. Notices respecting New Books. 


The Origin of Mountain-Ranges, considered Experimentally, Struc- 
turally, Dynamically, and in Relation to their Geological History. 
By T. Metuarp Reavz, CLL, GS, fAIBA. London: 
Taylor and Francis, 1886. 


[ is now twenty years since Mr. George L. Vose published his 

‘Orographic Geology,’ containg an admirable review of all 
that had, up to that time, been done in the way of explaining the 
structure and origin of mountain-chains. Strange to say, the 
author of the work now before us does not appear to be acquainted 
with the labours of his predecessor in the same field; but the large 
amount of original research bearing upon the subject in question, 
which has been carried on in the interval, fully justifies the pre- 
paration of this new book by one so competent to undertake it as 
Mr. Mellard Reade has shown himself to be. 

The author aims at nothing less than framing a complete and 
consistent theory of the origin of mountain-ranges; and whatever 
divergences of opinion may arise as to the soundness of particular 
portions of that theory, or of the force or value of certain of the 
arguments by which they are supported, there can be no hesitation 
among candid readers in admitting the great value of the mass of 
facts relating to the question which have been obtained by the author 
by ingenious experiment and patient observation, or the interest 
attaching to the conclusions which he has founded: upon those 
facts. 

If the theory, as a whole, can lay no claim to absolute novelty, 
there are certain new and striking features introduced into it by 
the author, and the principles on which it is based are certainly 
exemplified and enforced by him with much freshness, ingenuity, 
and vigour. 

Mr. Mellard Reade insists on the principle so well recognized by 
Hall, Rogers, Dana, Le Conte, and most recent authors who have 
treated on the subject, that the first stage in the origination of a 
mountain-chain consists in excessive sedimentation. After giving 
an outline of the main facts made known by recent researches 
concerning the Appalachians, the Rocky Mountains, the Andes, the 
Himalayas, the Alps, and the mountains of our own islands, he sum- 
marizes his conclusions as follows :—‘ No great range of mountains 
was ever ridged up excepting in areas of great previous sedimenta- 
tion. Out of these sediments the mountains are mostly built and 
carved, but along with the newer and originally horizontal sedimen- 
tary beds, the older gneissic and Archean rocks are usually thrust 


914 Notices respecting New Books. 


up, and often enclose in their folds strata of a newer age, which 
have become thereby considerably metamorphosed. 

“Tt is only in the great or in the old mountain-ranges that these 
old gneisses and schists are seen, because it is by denudation alone 
that they become exposed ” (p. 84). 

While agreeing with his predecessors as to the proofs of great 
sedimentation prior to the formation of a mountain-chain, the 
author to some extent differs from many of them in questioning 
the necessity for that progressive subsidence which the majority 
of geologists believe must have gone on side by side with the depo- 
sition. In support of this view, Mr. Mellard Reade cites the case 
of the accumulation of strata probably of great thickness in deep 
water off the mouth of the Amazon ; but he would probably himself 
admit that such an explanation is only capable of being applied to 
the occurrence of great thicknesses of clays, and not to alternating 
strata of coarse- and fine-grained sediments, such as so constantly 
constitute the materials out of which mountains are made. 

The second stage of mountain-making is explained by the 
author as arising from the upward displacement of the isogeo- 
therms, consequent upon the sedimentation in a particular area. 
In this he follows the line of reasoning previously suggested with 
ereater or less precision by Scrope, Babbage, Herschel, and other 
authors. 

It is in applying this well-recognized principle to the explanation 
of the contortion and crumpling of the thick masses of sediments 
that the author shows much novelty in his treatment of the 
question. Mallet and many other authors have insisted that the 
tangential thrusts, by which the folding of the strata was evidently 
produced, must have resulted from the contraction following from 
the secular cooling of the globe, whereby the outer crust is con- 
tinually tending to accommodate itself tothe central nucleus. Our 
author not only combats this view with many arguments that de- 
serve to be very carefully weighed, but offers an alternative hypo- 
thesis, which does not appear to be open to the objections to which 
the older theory is lable. We cannot do better than allow the 
author to explain this hypothesis in his own words. After insist- 
ing that the rise of the isogeotherms is the necessary consequence 
of excessive sedimentation, he goes on to say :— 

‘‘The rise of temperature exerts a tendency to expand the new 
sedimentaries, in every direction, in proportion to their extent and 
mass. The tendency to expand horizontally is checked by the 
mass of the Harth’s crust bounding the locally heated area. The 
expanding mass is therefore forced to expend its energies within 
itself ; hence arise those foldings of lengthening strata, repacking 
of beds, reversed faults, ridging up, and elevatory movements which 
occur in varied forms, according to the conditions present in each 
case. 

“The upper layers of the Harth’s crust being less and less 
affected by these variations in temperature as the surface is neared, 
are by the ridging-up thrown into a state of tension, while the 
lower beds of the sedimentaries are in a state of energetic com- 


Notices respecting New Books. 215 


pression. The mean rise of temperature of the whole sedimentary 
mass is half the total rise of the lowest beds.” (P. 326.) 

The author then goes on to show that at a certain depth the 
“‘ cubical expansion” of the mass must cause the heated though 
still solid materials to actually flow, and in so doing they will pene- 
trate along the lines of least resistance, giving rise to the gneissic 
axial cores so constantly exhibited in all great mountain-chains. 
The actual transfer of this flowing material adds corsiderably to the 
solidity and the consequent permanency of mountain-ranges. 

That this rise of temperature in the lower por-ions of a sedi- 
mentary mass is competent to produce the results he ascribes to 
it, the author illustrates by many genious experiments, and en- 
forces by very cogent reasoning. He shows that flat masses of 
lead, stone, and other material ridge up during heating in their 
centre if their edges be not free to move outwards; and he further 
insists upon a consideration which has not hitherto received the 
attention which ic deserves, namely, that the effects of repeated 
heating and cooling are to a great extent cumulative. A local in- 
crease of temperature causes expansion ; but in the subsequent fall 
of temperature, the contraction, or drawing back of the particles is 
very partial, and thus the changes all work towards the same end. 
This principle is iJustrated by a number of ingenious experiments, 
and it is argued that in the corrugation of strata we have illustrated 
the results of accumulations 0: small effects from simple causes. 

Wherever it is possible, the author endeavours to test his own 
results and those of others by calculations based on data obtained 
by actual experiment. He has determined the coefficients of ex- 
pansion of a number of sandstones, marbles, slates, and granites, 
and the results of these experiments show a very satisfactory agree- 
ment with those previously published by Adie and Totten. The 
mean of the whole of his results is a linear expansion of 2°77 feet 
per mile for every 100° Fahr. This the author points out is equal 
to about 8°25 feet of cubic expansion ; that is to say, the surface of 
a cubic mile of rock, if the base and sides were not free to move, 
would be raised, not 2°75 feet, but 8°25 by a rise of temperature of 
100° F. In amass of rock 500 iniles square and 20 miles thick— 
one which would equal only the spagth part of the bulk of the 
globe —an increase of 1000° F., or, what would ainount to nearly 
the same thing, a series of alternations in temperature amounting 
to 1000°, would, it is calculated, cause an expansion of no less than 
52,135 cubic miles ! 

The latter part of the work before us is occupied by descriptions 
of varieties of mountain-structure, and an explanation of the 
manner in which these may be accounted for on the author’s 
theory, as outlinedabove. Incidentally, many important geological 
problems are discussed, such as the origin of cleavage, foliation 
and jointing, the causes of ordinary and reversed faulting, the 
significance of the “ fan-structure,” the connection between vol- 
canic activity and mountain-building, the time required for the 
formation of mountain-chains, the cause of earthquakes and earth- 


216 Notices respecting New Books. 


tremors, &c. On all of these questions Mr. Mellard, Reade advances 
views which are well worthy of the consideration of geologists. 

The work is very amply illustrated by no less than forty-two 
plates containing many figures. Some of these are reproductions 
of the sections and maps published by the United-States Geolo- 
gical Survey and the Second Geological Survey of Pennsylvania, 
while a few are taken from the sections published by our own Geo- 
logical Survey. But the majority of the illustrations are repro- 
ductions by photo-lithography of the author’s own drawings and 
sepia-sketches. 

We very heartily recommend this valuable work to the attention 
of geologists, as an important contribution to terrestrial dynamics. 


Descriptive Catalogue of a Collection of the Economic Minerals 
of Canada. 8vo. London, 1886. 


Tuts Catalogue of one hundred and seventy pages is one of the 
many useful works published in connection with the Colonial and 
Indian Exhibition. It is compiled by the Geological Corps of 
Canada, acting under the direction of Dr. A. R. C. Selwyn. 
Although essentially a list of the specimens exhibited in the Cana- 
dian Collection, the minerals and rocks are fully described, with 
their properties, localities, and the geological horizon from which 
they were obtained, thus rendering the work of permanent value as 
a reference book to the mineralogist, geologist, and prospector. 
The districts represented are—British Columbia, North-west 
Territories, Manitoba, Ontario, Quebec, North-east Territory, New 
Brunswick, Prince-Edward Island, and Nova Scotia; and from 
these upwards of 700 specimens are described. The more impor- 
tant minerals, as Coal and Apatite, receive especial attention ; and 
the notes on the latter are rendered more valuable by the addition 
of plans and sections illustrative of the occurrence of the mineral 
and of the two finest examples that were exhibited. The work is 
divided into eleven sections, the most noteworthy of which are the 
following :—I. Metals and their Ores; II. Materials used in the 
Production of Heat and Light; III. Minerals applicable to certain 
Chemical Manufactures and their Products; IV. Mineral Mantire ; 
VII. to X. Materials applicable to various Constructions, Fine Arts, 
Jewellery, &c. 


Journal and Proceedings of the Royal Society of New South Wales 
for 1885. Vol. XIX. 8vo. Sydney, 1886: 240 pp. 

Tu1s volume contains eleven papers, besides the usual valuable 
Meteorological appendix, and a Rainfall Map for 1884. In the 
Anniversary Address the President, Mr. H. C. Russell, gives some 
important notes on the movement of the “ Hast Coast, if not all 
Australia,” quoting the late Rev. W. B. Clarke, Mr. John Kent, 
Mr. Ellery, and others, and giving the results of his own observa- 
tions for twelve years past. ‘The evidences,” Mr. Russell says, 
“ for elevation and subsidence of the land are about equal ;” and, 
as accurate observations have as yet been made only at Sydney, where 
in the twelve years no appreciable change has been noticed, it is 


Notices respecting New Books. 217 


difficult to say whether the movement has an upward or a down- 
ward direction. Some account of Lake George, and interesting 
notes on the gradual rise and fall of its waters, are given, together 
with a description of a self-recording gauge, which gives a con- 
tinuous record of changes of level by evaporation and otherwise. A 
remarkable “impulse” was recorded on the 14th April, 1884, when 
the water had been unusually still for the three previous days. At 
11 a.m. the observer at the gauge saw a thunderstorm coming 
from the North, and, watching the instrument, saw that the 
lake rose at that point four inches in thirty minutes. As soon 
as the storm passed, the water began to fall, reaching its —previous 
level in fifteen minutes, sank two inches more, and began to 
rise again. ‘The whole series of pulsations lasted five days. The 
Lake is referred to as ‘‘a body of water eighteen miles long, five 
wide, and 15 or 20 feet deep,” and in its wet period “ at least forty 
miles long and ten or twelve wide.” The ‘“ seiches”’ of the Lake 
of Geneva are noticed in connection with this subject. 

Mr. Russell has also a paper on “ Local Variations and Vibra- 
tions of the Earth’s Surface,” in which he especially deals with the 
effect of lunar attraction upon the solid portion of the globe. 
After quoting from the British-Association Reports for 1831 and 
1882, he proceeds to give the result of his own observations, more 
particularly those taken with the Lake-George tide-gauge, and he 
notes the “‘ Level-errors of the Sydney Transit-Instrument” in 
relation to the sandstone hill upon which the Observatory is built. 
Important photolithographic copies of the sheets from the recording 
instrument at Lake George, together with Level-, Temperature-, 
Azimuth-, and Barometer-curves at Sydney Observatory are given. 
This paper and Mr. Russell’s Address form very important con- 
tributions to our knowledge of the level-changes of the Harth’s 
surface. 

The Rev. P. MacPherson deals with “ Some causes of the Decay 
of the Australian Forests.” After discussing the various theories 
advanced to account for the decay, the author dismisses ‘“* Wet 
eround,” “Drought,” “ Bush Fires,” ‘“ Differences of Soils,” “Sheep 
manures,” ‘ Caterpillars,” and “ White Ants,” as inadequate to 
effect the mischief observed; and he refers the majority of the 
damage done to the Opossums and a “ Copper-coloured Beetle,” 
the name of which is unfortunately omitted. <A plate is given in 
illustration of the remarks. Another paper by the same author 
deals with the ‘‘ Stone Implements of the Aborigines of Australia 
and some other countries” in Australasia. After describing the 
specimens exhibited, the author discusses their antiquity, and 
comes to the conclusion that “up to date, direct evidence for a 
geologic antiquity on behalf of the Australian Aborigines seems to 
be very scanty.” Three illustrations accompany the paper, one 
of which represents the incisions made by the natives into trees, 
to get food and for other purposes, and shows the permanent 
effect on the tree so treated. 

In a “ History of Floods in the Hawkesbury River,” Mr. J. P. 
Josephson gives a table of “heights of floods from years 1795 to 


Pinks Mog. S. 5. Vol. 235. No. 141. feb. 1337. Q 


218 Notices respecting New Books. 


1881,” and two maps showing the extent of the floods around the 
town of Windsor in 1867, with contours of the district. 

Arundinaria falcata, Nees, and A. spathiflora, Trinius, of the 
Himalayas, as suitable for cultivation in New South Wales, are 
treated of by Dr. Brandis; and a note on the Adelong Gold-Reef, 
with a sketch-plan, is given by Mr. 8. H. Cox. 

Microscopy is represented by Mr. William Morris, who has 
devoted twelve pages to the various methods employed in mounting 
the rare diatom Amphipleura pellucida. 

‘“ A Contribution to the Study of Heredity ” by an inquiry into 
the family- and life-history of the idiotic and imbecile, by Dr. F. 
Norton Manning,—“ A system of Accurate Measurement by means 
of long Steel Ribands,” instead of the one-chain tapes, in rugged and 
undulating districts, by Mr. G. H. Knibbs,—and some “‘ Notes on 
Flying-Machines,” by Mr. L. Hargrave, complete the volume. 


Hours with a Three-Inch Telescope. By Capt. WM. Nosuz, /.R.A.S., 
FRM S., Honorary Associate of the Liverpool Astronomical Society. 
London: Longmans, Green, and Co., 1886. 

“ Tats little book,” the author tells us at the outset, ‘is written 

to furnish the very heginner in observational Astronomy with such 

directions as shall enable him to employ, to the greatest possible 
advantage, the kind of instrument with which he will, in all 
probability, at first provide himself.” He therefore only pre- 
supposes the possession, on the part of the reader, of a small 
telescope mounted on the ordinary pillar-and-claw stand, of some 
such work as the ‘ Stars in their Seasons,’ and of an ardent desire 
to become familiar with the beauties and glories of the celestial 
vaults. To such beginners—for whose guidance there has been 
hitherto no perfectly suitable book—Capt. Noble will be a true 
friend. He thoroughly understands the needs of those for whom 
he writes, and he gives them the information they require in 
a simple and straightforward manner, free from perplexing techni- 
calities. No pains, too, have been spared to make the book 
accurate ; the descriptions and drawings were all made at the eye- 
end of a telescope of three inches’ aperture, and are not taken at 
secondhand from any other work, however trustworthy. There 
are twelve chapters : the first treating of the instrument (the three- 
inch telescope) itself, and here Lord Crawford’s device for giving 
an approximately equatoreal motion to an altazimuthly mounted 
instrument is described. ‘The sun and moon form the subjects of the 
two next chapters; the latter is one of the most interesting and 
important in the book; the principal lunar formations being 
passed in review during seven nights’ work. A short chapter on 
the observation of occultations follows. Then come the planets, 
Jupiter naturally receiving most attention; whilst the eleventh 
chapter gives some useful hints on drawing the planets. The last 
chapter is devoted to the fixed stars and nebulae, and the work of 
nine nights is described in detail. Here, we fear, the student 
will find Capt. Noble’s help least effective, for there are no star- 
maps given. It is true he supposes his readers to possess some 


Notices respecting New Books. 219 


elementary star-atlas, but that is hardly what is required so much as 
a series of little simple diagrams giving all the brightest stars in 
the various districts under examination with the relative positions 
of the objects to which attention is called. Such diagrams as 
those given of several of the constellations in Sir R. S. Ball’s 
‘Story of the Heavens’ are what are required, and could be easily 
and cheaply supplied. We trust Capt. Noble may see his way to 
introduce something of the kind in a second edition; he has 
already provided for the study of the Moon by an excellent 
reproduction of Webb’s well-known map. 

The book is to a great extent a reprint of papers which 
originally appeared in the columns of ‘ Knowledge,’ and there are 
occasionally references to the positions of the planets which were 
appropriate enough at the time of writing, but which of course no 
longer apply, and read rather strangely when published in this more 
permanent form. There are one or two other incidental points 
which might be criticised : it is to be hoped, for example, that the 
student will not imagine that the rather rough and ready method 
on p. 15 for computing the longitude of the Moon’s terminator gives 
it correctly to 0":1, though certainly it would be excusable for him 
to draw thatinference. But these little flaws are not serious enough 
to detract from the value of the work as a whole; it is useful, 
clear, and practical, and will be of the most essential service to many 
a young beginner, and, without doubt, will, as its author hopes, 
prove in many an instance a suitable introduction to works of a 
more advanced character, such as Webb’s ‘ Celestial Objects.’ 


Algebra: an Elementary Teatbook for the higher classes of Secondary 
Schools and for Colleges. By G. Curystan, M.A. Part I. 
Edinburgh: A. & ©. Black, 1886; pp. xx+542. 

Tuts first part is composed of twenty-two chapters, which treat 

of the subject of Algebra (and of allied subjects which do not 

usually come into an Elementary Treatise) under the following 
heads. 1. Fundamental Laws and processes of Algebra: a valuable 
introduction which discusses the laws of Association, Commutation, 
and Distribution, and the properties of 0 and 1, and closes with an 
interesting historical note. 2 treats of Monomials, the laws of 
Indices and Degree. 3, with the heading “Theory of Quotients, 
first principles of Theory of Numbers,” lays open to view the 
fundamental properties of Fractions, treats of prime and composite 
numbers, and gives several theorems connected with factors and 
primes. 4 is a most important chapter on distribution of Pro- 
ducts, elements of the Theory of Rational Integral functions, 
in which are explained the 3 and z Notations, the principles of 
Substitution, Homogeneity, Symmetry, and the principle of Indeter- 
minate Coefficients. This chapter was written ‘‘as a suggestion to 
the teacher how to connect the general laws of Algebra with the 
former experience of the pupil. In writing this chapter I had to 
remember that I was engaged in writing, not a book on the philo- 
sophical nature of the first principles of Algebra, but the first 
chapter of a book on their Consequences.” 5 discusses at some 


220 Notices respecting New Books. 


length the transformation of the Quotient of two integral functions, 
treating under this head of the Remainder theorem and its appli- 
cation to Factorisation, of a new basis for the principle of Indeter- 
minate Coefficients and of Continued Division. 6 gives much 
useful matter under the heads of Greatest Common Measure and 
Least Common Multiple. 7, on the Factorisation of Integral 
Functions, introduces the consideration of surd and imaginary 
Quantity and of Complex Quantity. In 8 are discussed Rational 
Fractions ; in 9 we have a continuation of Theory of Numbers, com- 
prising Scales of Notation and Lambert’s Theorem ; in 10 we have 
a general discussion of irrational functions .. interpretations of #?/2, 
x, «2, and a general theory of rationalization; and in 11 is an 
account of the Arithmetical theory of Surds. One of the best 
chapters to our thinking is 12, which gives an excellent account of 
complex numbers and herein of Argand’s diagrams and of De- 
moivre’s theorem. 131s good on Ratio and Proportion. 14 at some 
length discusses conditional equations in general; 15 at equal 
length treats of the Variation of a function (a good introduction to 
the theory of Maxima and Minima for a student of the Calculus) ; 
and 16, 17 are concerned with equations of the first and second 
degree respectively. In 18 is an account of a general theory of 
Integral Functions, in which figure Symmetric functions of the 
roots of an Equation, Newton’s theorem regarding sums of powers 
of roots, special properties of Quadratic Functions (including La- 
grange’s Interpolation formula), and Variation of a Quadratic 
Function for real values of its Variable (analytical and graphical 
discussion of three fundamental cases, Maxima and Minima). 19 
is devoted to the Solution of Problems by means of Equations. 20 
discusses the Arithmetic, Geometric, and allied series ; 21 is oceu- 
pied with Logarithms (interpolation by first differences); and 22 
closes the work with an account of the Theory of Interest and 
Annuities. Numerous historical notes impart considerable interest 
to the perusal of the text. 

It is evident that there are many subjects handled which do not 
come within the range of an elementary student’s reading; but 
these are all handled in such a way as to be most valuable to more 
advanced students and to teachers. The author warns us on the 
very threshold, that his work is not intended for the use of 
absolute beginners. His great object in laying down the three 
fundamental laws is to imtroduce the idea of Algebraic Form, 
“ which is the foundation of all the modern developments of Algebra 
and the secret of analytical Geometry, the most beautiful of all its 
applications.” We advise higher-form boys and others, who have a 
desire to go more deeply into the subject than they can do with the 
aid of ordinary textbooks, to get or borrow Prof. Chrystal’s splendid 
treatise and make a careful study of its well-arranged contents. 
It is the pure Mathematical Elementary textbook of the year 1886. 


Elements of the Theory of the Newtonian Potential Function. 
By Dr. B. O. Perrce. [Boston: Ginn & Co., 1886.] 


Tue Compiler of these Lecture Notes adopts the term employed 


Geological Society. 221 


by Neumann (‘‘ Untersuchungen iiber das Logarithmische und 
Newton’sche Potential”’)*. 

There is no attempt made at original investigation, for the notes 
are professedly elementary in character, and to a great extent 
reproduce work given in Todhunter’s ‘ Analytical Statics’ and 
Minchin’s ‘Statics.’ Dr. Peirce has, however, carefully consulted 
numerous other authorities, the most important works both by 
English and Continental writers, and derives much of his matter 
from original Memoirs. There are five chapters inall. In the first 
the author treats of the Attraction of Gravitation ; in the second of 
the Newtonian Potential Function in the case of Gravitation; in the 
third of the same function in the case of Repulsion; in the fourth 
of the properties of Surface Distributions and of Green’s Theorem 
(also of Thomson’s Theorem); and in the last Chapter he applies 
his preceding results to Hlectrostatics. It will be thus seen that 
the writer has endeavoured to meet the difficulty experienced by 
his class of getting “‘ from any single book in English a treatment of 
the subject at once elementary enough to be within their easy 
comprehension, and at the same time suited to the purposes of such 
of them as intended eventually to pursue the subject farther, or 
wished, without necessarily making a speciality of mathematical 
Physics, to prepare themselves to study Experimental Physics 
thoroughly and understandingly.”’ To the list of 20 works recom- 
mended to students, we would add such parts of Todhunter’s 
‘ History’ as bear upon the subject. We have detected a few simple 
misprints, but the “get up” maintains the reputation already 
acquired by the printers and publishers. 


XXIII. Proceedings of Learned Societies. 


GEOLOGICAL SOCIETY. 
[Continued from p. 72.] 


January 12, 1887.—Prof. J. W. Judd, F.R.S., President, 
in the Chair. 


eas following communications were read :— 
1. “The Ardtun Leaf-beds.” By J. Starkie Gardner, Esq., 
F.G.S., F.L.S.; with Notes by Grenville A. J. Cole, Esq., F.G.S. 


2. «On the Echinoidea of the Cretaceous Strata of the Lower 
Narbada Region.” By Prof. P. Martin Duncan, M.B., F.R.S., F.G.S. 


3. **On some Dinosaurian Vertebre from the Cretaceous of India 
and the Isle of Wight.” By R. Lydekker, Esq., B.A., F.G.S. 


* In the Index to his ‘ History of the Theories of Attraction...’ Mr. 
Todhunter gives numerous references to articles in his book which treat 
of the Potential. §789 assigns the introduction of the function, on the 
authority of Legendre, to Laplace, but it is in §790 (to which he omits to 
refer in the Index) that we learn that the term was first used by Green 
(1828, Papers, p. 22), and apparently independently by Gauss (1840) : ef. 
also Thomson and Tait, Nat. Phil. vol. i. §482. 


222 Intelligence and Miscellaneous Articles. 


4, “Further Notes on the Results of some deep Borings in Kent.” 
By W. Whitaker, Esq., B.A., F.G.S. ; 

This paper contained some details on the borings at Chattenden 
Barracks and at the Dover Convict Prison, in addition to those 
already published in the Quarterly Journal of the Society for 1886. 
Sections of the new borings, one at Strood the other at Lydd, were 
also given. 

The Chattenden boring had been successful in reaching the Lower 
Greensand, and a supply of water had been obtained. This result 
showed that on the section accompanying the previous paper the 
beds of the Lower Greensand should have been carried rather fur- 
ther to the northward. 

The Dover boring was abandoned at 931 feet from the surface. 
The examination of the specimens showed that the thickness 
formerly assigned to the Lower Greensand should be reduced to 
31 feet, the upper 5 feet referred to that stage belonging to the 
base of the Gault, whilst the bottom, 13 feet, together with an addi- 
tional 69 feet, mostly of clay, subsequently cut through, were, for 
reasons given, assigned to the Wealden series and probably to the 
Hastings beds. 

The results of these additional details went to show (1) that, 
though the Lower Greensand itself was rather thicker at Chatham 
than at Dover, comprising two divisions, the Folkestone and the 
Sandgate beds at the former place, and only the Sandgate at the 
latter, the Lower Cretaceous beds, as a whole, were ‘much thinner at 
Chatham, owing to the disappearance of the Wealden series; and (2) 
that in passing to the eastward the Weald clay thinned out before 
the Hastings beds, instead of the reverse, which was previously 
suggested. 

The Strood and Lydd sections were merely of importance as fur- 
nishing details. The paper concluded with some remarks on the 
best site for additional borings at Dover, in order to prove the deeper- 
lying rocks. 


XXIV. Intelligence and Miscellaneous Articles. 


‘moO WHAT ORDER OF LEVER DOES THE OAR BELONG?” BY 
FRANCIS A. TARLETON, FELLOW OF TRINITY COLLEGE, DUBLIN. 


he a paper published in the Philosophical Magazine for January, 

1887, Mr. Abbott has discussed the question, “ To what order 
of Lever does the Oar belong ? ” 

The conclusion at which Mr. Abbott has arrived seems to me 
substantially correct ; but he has, I think, stated it in language 
which, to say the least, is rather paradoxical, and has supported 
it by arguments which leave the mind of the reader in a somewhat 
unsatisfied state. 

To simplify the discussion of the problem, I shall suppose the 
boat to be moved by two oars on which equal pulls are exerted, and 
to be perfectly symmetrical on both sides. 

If weregard the rower, boat, and oars as one system, the only forces 


ee 


Intelligence and Miscellaneous Articles. 223 


external to the system, which need be considered, are the pressures 
of the water on the blades of the two oars and the resistance of the 
water to the motion of the boat. If the boat be on the point of 
moving, these forces must equilibrate each other, and as they 
are parallel, the semi-resistance of the water to the motion must be 
equal and opposite to the pressure on the blade of one oar. Again, 
for the equilibrium of this oar, the moments round the rowlock of 
the pull exerted by the rower and of the pressure against the blade 
must be equal and opposite. Hence the moments round the row- 
lock of the pull of the rower, and of a force equal to the semi- 
resistance of the water to the motion, supposed to act at the blade 
of the oar, must be equal. 

This is the proposition which Mr. Abbott seeks to establish, and 
which he has, I think, paradoxically, if not inaccurately, expressed 
by stating that the oar is a lever of the first order. 

The word lever has a kinematical reference, and implies a rigid 
body having a fixed point. The division of levers into different 
orders is an obscure way of stating the relative position of the 
fixed point, and of two forces which are supposed to act upon the 
body. 

Now in the case of the oar, the fixed point is the blade, and the 
resistance of the water to the motion of the boat does not act on 
the oaratall. To say, then, that the oar isa lever of the first order 
acted on by the pull of the rower and the semi-resistance of the 
water to the motion of the boat applied at the blade of the oar, the 
rowlock being the fulcrum, cannot be regarded as accurate, except 
we look upon the rower, not as moving on the boat, but as moving 
back the world. 

The investigation of the question, regarding the oar as a lever 
whose fixed point is the blade, can be easily accomplished. 

Let P be the pull of the rower, R the reaction between the oar 
and the boat at the rowlock, S the semi-resistance of the water to 
the motion of the boat, and a and 6 the distances of the rowlock 
from the hand of the rower and the blade of the oar. 

We have then, from the equilibrium of the oar, 

ee pO en ee oe beet 

Again, as the boat is Just about to move, the forces acting on it 
equilibrate each other, and these are 2S, 2R, and a force equal and 
opposite to 2P, exerted by the feet and body of the rower against 
the boat. As these forces are parallel, we have S+P=R. Sub- 
stituting for Rin (1) and reducing, we get 

Pa=Sb6 ; 
the same result as before. 


ON THE SPECIFIC HEATS OF THE VAPOURS OF ACETIC ACID AND 
NITROGEN TETROXIDE. 


To the Editors of the Philosophical Magazine and Journal. 


GENTLEMEN, 
Amongst others, it has lately been my business to read a very 


224 Intelligence and Miscellaneous Articles. 


interesting paper by Berthelot and Ogier in the Annales de Chimie 
et de Physique for 1883, in which the authors set forth the result of 
their careful experiments on the Specific Heats of the Vapours of 
Acetic Acid and Nitrogen Tetroxide. The quantities of heat absorbed 
at different temperatures are compared with the percentage disso- 
ciation, as deduced from the vapour-density experiments, of Nau- 
mann, Soret, and other experimenters. ‘The experiment*is very 
close (as may easily be seen by placing the several series of num- 
bers on curves), and well within the limits of experimental error. 
In spite of the valued opinion of M. Berthelot, I therefore venture 
to consider that the experiments referred to do afford additional 
evidence in favour of the hypothesis of the dissociation of these 
gases ; the alternative theory involving, so far as I can see, the 
relinquishing of Avogadro’s law as far as these gases are con- 
cerned. 

Assuming therefore, for the moment, that dissociation does take 
place, and that the thermal changes méasured by Berthelot and 
Ogier are the natural expression of this effect, some information, 
it seems to me not altogether devoid of interest, may be gained by 
a comparison of these results with those obtained by Regnault for 
the specific heats of those gases which undergo a condensation in 
the course of their formation from their elements. As is well 
known, the specific heats of nitrous oxide and carbonic acid have, 
according to Regnault, a small temperature-coefficient, showing 
that, as the temperature rises, the specific heats increase, at all 
events up to 200° Centigrade. This, however, is precisely what 
has been found to hold true in a more complicated form for 
nitrogen tetroxide and acetic-acid vapour, where the temperature- 
coefficient is much greater and reaches a maximum. If, however, 
we decide to attribute this to a molecular change tending from 
greater to less molecular complexity in the case of acetic-acid vapour 
and nitrogen tetroxide, why should we not apply the same argu- 
ment to the gases examined by Regnault, and regard the tempera- 
ture-coefficient of the specific heats of nitrous oxide and carbonic 
acid &c. as evidencing a similar molecular change? Might we not 
argue, in fact, that at low temperatures the molecular composition, 
as expressed by the ordinary formule for carbonic acid and nitrous 
oxide, only refers to the vast majority of the molecules ; and that 
the number of molecules of a higher degree of complexity, as well 
as of a lower (7. e. dissociated), is notnegligibly small. On raising 
the temperature, those of higher than normal complexity might 
decompose and account for the extra absorption by heat, evidenced 
by Regnault’s temperature-coefficient. Such a view seems to me 
to be in harmony with the views set forth by Clausius and Wil- 
liamson, which are daily gaining ground among physical chemists. 

I am, Gentlemen, 
Your obedient servant, 


RicHAaRD THRELFALL. 
University of Sydney, 
November 8, 1886, 


THE 


LONDON, EDINBURGH, ann DUBLIN 
PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[FIFTH SERIES.] 
MARCH 1887. 


XXV. Notes on Electricity and Magnetism.—IIl. On the 
Behaviour of Iron and Steel under the Operation of Feeble 
Magnetic Forces. By Lord Rayweicn, Sec. £.S.* 


[Plate II.] 


eee question whether or not iron responds proportionally 
to feeble magnetic forces is of interest not only from 
a theoretical point of view, but from its bearing upon the 
actual working of telephonic instruments. Considerable dif- 
ference of opinion has been expressed concerning it, several of 
the best authorities inclining to the view that a finite force is 
required to start the magnetization. Prof. Ewing remarkst:— 
“As regards the hysteresis which occurs when the magnetism 
of soft iron is changed, my experiments confirm the idea 
already suggested by other observers, that when the molecular 
magnets of Weber are rotated they suffer, not first an elastic 
and then a partially non-elastic deflection as Maxwell has 
assumed, but a kind of frictional retardation (resembling the 
friction of solids), which must be overcome by the magneti- 
zing force before deflection begins at all.’’ In a subsequent 
passaget Prof. Ewing treats the question as still open, re- 
marking that though his curves suggest that the initial value 
of & (the susceptibility) may be finite, they afford no positive 
proof that it is not initially zero, or even negative. 
My attention was first called to the matter about a year 
and a half ago in connection with the operation of iron cores 


* Communicated by the Author. 
+ Phil. Trans. 1886, p. 526, § 5. 
t ZL. c. § 61. 


Phil. Mag. 8. 5. Vol. 23. No. 142. March 1887. R 


226 Lord Rayleigh on the Behaviour of Iron and Steel 


in the coils of an induction-balance. Hxperiment showed 
that iron responded powerfully to somewhat feeble forces ; 
and I endeavoured to improve the apparatus in the hope of 
being able thus to examine the subject more thoroughly. 
Two similar long helices were prepared by winding fine insu- 
lated wire upon slender glass tubes. These were connected in 
series with a battery, a resistance-box, and a microphone- 
clock, so as to constitute a primary circuit. The secondary 
consisted of a large quantity of copper wire, mounted upon a 
bobbin, through the opening in which both primary coils were 
inserted. The circuit of the secondary was completed by a 
telephone. When neither primary coil contained a core, 
silence at the telephone could readily be obtained. The iron 
cores used were those described in Part II.*; and it was found 
that all of them (including the bundle of seventeen very fine 
wires) disturbed the silence until the resistance was so far 
increased that the magnetizing force was less than about 34 of 
the earth’s horizontal force (H). Moreover, there was no 
indication that the absence of audible effect under still smaller 
magnetizing forces was due to any other cause than the want 
of sensitiveness of the apparatus. 

I did not pursue the experiments further upon these lines, 
because calculation showed that the feeble magnetization of a 
piece of iron could more easily be rendered evident directly 
upon a suspended needle (the magnetometric method), than 
indirectly by the induction of currents in an encompassing coil 
connected with a galvanometer. Nearly all the results to be 
given in this paper were obtained by a form of the magneto- 
metric method, specially adapted to the inquiry whether or not 
the magnetization of iron continues proportional to the mag- 
netizing force when the latter is reduced to the uttermost. 

The magnetizing-spiral first used was one of those already 
referred to. It consists of a single layer of fine silk-covered 
copper wire wound on a glass tube and secured with shellac 
varnish (A, Plate I]. fig. 1). The total length of the spiral 
is 17 centim., its diameter is about 6 centim., and the wind- 
ings are at the rate of 32 per centim. The resistance is about 
54 ohms. 

The magnetometer was simply a small mirror backed by 
steel magnets (B), and suspended from a silk fibre, as supplied 
by White for galvanometers. It was mounted between glass 
plates at about 2 centim. distance from the magnetizing-spiral. 
The earth’s force was compensated by steel magnets, which 
also served to bring the mirror perpendicular to the helix in 
spite of the influence of residual magnetism in the iron core. 


* Phil. Mag. December 1886, p. 490. 


under the Operation of Feeble Magnetic Forces. 227 


The deflections were read in the manner usual with Thomson’s 
galvanometers, by the motion of a spot of light thrown upon 
a scale after reflection by the mirror. ‘The division is in 
millimetres, and with the aid of a lens a displacement of -J5 of 
a division can usually be detected with certainty. 

The direct effect of the magnetizing-spiral upon the sus- 
pended needle was compensated by a few turns of wire OC, 
7 centim. in diameter, supported upon an adjustable stand D. 
This adjunct might have been dispensed with; but what is 
essential is the larger coil, EH, by which the effect of the iron 
core is compensated. This coil consisted of 74 convolutions, 
of mean diameter 18 centim., tied closely with string, and 
mounted upon an independent stand, F. By sliding this 
stand, and ultimately by use of the screw, G, the action of 
this coil upon the suspended needle can be adjusted with 
precision. All the coils are connected in series; and pro- 
vided that the magnetic condition of the iron under given 
force is definite, matters may be so arranged that the imposi- 
tion of the force produces no movement of the suspended 
needie, or, more generally, the compensation may be adjusted 
so as to suit the transition from any one magnetic force to any 
other. Ifthe susceptibility (4) and permeability w (=47k+1) 
were constant, as has often been supposed in mathematical 
writings, the compensation suitable for any one transition 
would serve also for every other, and the magnetometer-needle 
would remain undisturbed, whatever changes were permitted 
in the strength of the magnetizing current*. The question 
now presenting itself is, How far does this correspond to fact ? 
or, rather, How far is it true for magnetizing forces which are 
always very small? for we know already that, under the ope- 
ration of moderate forces exceeding (say) 1 or 2 C.G.S., not 
only is ~ not constant, but there is no definite relation at all 
between magnetic induction and magnetizing force, whereby 
the one can be inferred from the other without a knowledge 
of the previous history of the iron. 

The magnetizing force of the spiral is of course easily cal- 
culated. The difference of potential in passing through n 
convolutions of current C is 4anC. If the n convolutions 
occupy a length /, the magnetizing force is 


AarC : 
or, in the present case, 198,7(). 


* The idea of compensating the iron is not new. The method was 
employed by Koosen (Pogg. Ann. Bd. Ixxxv. S. 159, 1852) to exhibit the 
phenomena of “ saturation.” 

R 2 


228 Lord Rayleigh on the Behaviour of Iron and Steel 


C is here expressed in 0.G.S. measure, on which scale the 
ampere is ‘1. 

It may be objected that the magnetic force of the spiral is 
not the only external force operative upon the iron. It is 
true that the compensating-coils must have an influence, and 
in the opposite direction. But calculation shows that the 
influence must be small. The radius of the large coil is 
9 centim., and (to take an example) the distance of its mean 
plane from the suspended needle in one set of experiments on 
hard iron was 13°6 centim. Under these circumstances the 
magnetic force in the spiral, even at the nearer end, is influ- 
enced less than 2 per cent. by the large compensating-coil. 
The effect of the smaller coil is about the same. For the 
present purpose it is hardly worth while to take these correc- 
tions into account. 

As has been remarked, the coils of the apparatus were 
always connected in series; but a reversing-key (serving also 
to-make and break) was introduced so as to allow of the re- 
versal of the compensating-coil in relation to the others. In 
one position of the key (—) the action of the coil and of the 
magnetized iron are opposed; in the other (+) the actions 
conspire. When the currents to be used were not exceedingly 
small, the whole apparatus was in simple circuit with a Daniell 
cell and such resistance-coils as were necessary. Hxclusive 
of the cell and of the added resistances, the whole resistance 
was 74 ohms. 

As an example, I will now give the details of some obser- 
vations on December 6 made to test the behaviour of unan- 
nealed Swedish iron wire. The diameter of the wire is 
1°6 millim. ; it is from the same hank as a piece used in the 
experiments of Part II.* The compensating-coil was adjusted 
until it made no difference whether the key was open or 
closed (—), the additional resistance being 1000 ohms. In 
stating the result it will for the present be sufficient to give 
the German-silver resistances, that of the apparatus and of the 
battery being relatively of noimportance. The corresponding 
current is about 10-*C.G.S., and the strength of the magnetic 
field in the spiral is given by 

128%7C=:04 C.G.S. 
We shall have a better idea of this if we recall that, on the 
same system of measurement, 
Heaws 3 
so that the force in action is about 3 of that which the earth 


exercises horizontally. 
* L. e. p. 488. 


under the Operation of Feeble Magnetic Forces. 229 


When the resistance was altered to 11,000 ohms, the com- 
pensating-coil of course remaining undisturbed, contact (—) 
produced no visible motion, showing that the same compen- 
sation is suitable for the much smaller force. But at this 
point we require to be assured that the absence of disturbance 
is not due merely to want of sensitiveness. The necessary 
information is afforded at once by making reversed contact 
(+), which (with 11,000 ohms) gave a swing of 57 divisions. 

To diminish the magnetizing force still further, a shunting 
arrangement was adopted. ‘The current from the Daniell was 
led through 10,000 ohms and then through a box capable 
of providing resistances from 1 to 1000. The circuit of the 
apparatus included another coil of 10,000 ohms, and its ter- 
minals were connected to those of the box. The battery- 
current was thus about ‘0001 ampere, or 10-5 C.G.8. Ifa 
be the (unplugged) resistance in the box, the H.M.F. at the 
terminals of the apparatus-circuit is ax 1074 volts; and the 
current C through the magnetizing helix and compensating- 
coil isa x 10-9 C.G.S. 

When a=1000 ohms, (—) gave no visible deflection, while 
(+) caused a swing of 5 divisions. 

At this stage recourse was had to the “ method of multipli- 
cation ” in order to increase the sensitiveness*. A pendulum 
was adjusted until its swings were synchronous with those of 
the suspended needle. It was then easy to make and break 
contact in such a way as to augment the swing due to any 
outstanding force. Thus, when a=1000, the swing was in- 
creased by the use of the timed contacts and ruptures (+) 


* The advantage of the method of multiplication seems to be hardly 
sufficiently appreciated. It is not merely that the effect is presented to 
the eye ina magnified form. That object can be attained by optical 
appliances, and by diminishing the directive force upon the suspended 
parts, whether by using a nearly astatic system uf needles, or by com- 
pensatiug the field. For the most part these devices augment the un- 
avoidable disturbances (which exhibit themselves by a shifting zero) in 
the same proportion as the effect to be measured, or at any rate rendered 
apparent. The real ultimate impediment to accuracy of measurement is 
almost always the difficulty of distinguishing the effect under examination 
from accidental disturbances, and it is to overcome this that our efforts 
should be directed. The method of multiplication is here of great service. 
The desired etfects are largely magnified, while the disturbances, which 
are not isoperiodic with the vibrations of the needle, remain unmagnitied, 
and therefore fall into the background. 

It is obvious that, in order to secure this advantage, the vibrations 
must not be strongly damped. No doubta highly damped galvanometer- 
needle is often convenient, and sometimes indispensable. But it seems to 
be a mistake to use it where a null method is applicable, and when the 
utmost delicacy is required. In such a case the inertia of the needle, and 
the forces both of restitution and of damping, should all be made small. 


230 Lord Rayleigh on the Behaviour of Iron and Steel 


until it measured 26 divisions instead of 5 only. Buta 
similar series of operations with reversed currents (—) caused 
no swing amounting to j!, division ; so that we may consider 
the compensation proved to be still perfect to about 1 per cent. 

In applying the method to still smaller forces we cannot 
avoid a loss of sensitiveness. With a=100, (+) gave 3 di- 
visions, while the effect of (—) remained insensible. The 
correctness of the compensation is thus verified to about 6 per 
cent. of the separate effects. Had the iron, even at this stage, 
refused to accept magnetization, the fact would have mani- 
fested itself by the equality of the swings obtainable in the 
two ways, (+) and (—), of making the connections. 

In the last case mentioned the current was 10-7 C.G.8., 
and the magnetic force was 4x 10-5 C.G.8. We may there- 
fore regard the proportionality of magnetic induction to mag- 
netic force over the range from }H to s,55H as an experi- 
mental fact. In view of this, neither theory nor observation 
give us any reason for thinking that the proportionality 
would fail for still smaller forces. 

Quite similar results have been obtained with steel. On 
December 13 a piece of drill steel (unannealed) was examined, 
the delicacy of the apparatus, as evidenced by the ( + ) effect, 
being about the same as in the above experiments on hard 
Swedishiron. No failure of proportionality could be detected 
with forces ranging from about 4 H to zodo9 H. 

Annealed iron is a much less satisfactory subject. With 
unannealed iron and steel the compensation for small forces 
may be made absolute, so that neither at the moment of closing 
the circuit nor afterwards is there any perceptible disturbance. 
This means that (so far as the magnetometer-needle can 
decide) the metal assumes instantaneously a definite magnetic 
condition which does not afterwards change. But soft iron 
shows much more complicated effects. The following obser- 
vations were made upon a piece of Swedish iron (from the 
same hank as the former) annealed in the flame of a spirit- 
lamp. When an attempt was made to compensate for the 
imposition of a force equal to 4 H, no complete balance could 
be obtained. When the coil was so placed as to reduce as 
much as possible the instantaneous effect, there ensued a drift 
of the magnetometer-needle represented by about 170 divi- 
sions of the scale, and in such a direction as to indicate a 
continued increase of magnetization. Precisely opposite 
effects followed the withdrawal of the magnetizing force. 
The settling down of the iron into a new magnetic state is 
thus shown to be far from instantaneous. On account of the 
complication entailed by the free swings of the needle, good 


under the Operation of Feeble Magnetic Forces. 231 


observations on the drift could not be obtained with this ap- 
paratus ; but it was evident that, whilst most of the anomalous 
action was over in 3 or 4 seconds, the final magnetic state 
was not attained until after about 15 or 20 seconds*. 
_ The operation of feebler forces was next examined, rather 
with the expectation of finding the drift reduced in relative 

importance. But the imposition of 4; H was followed by a 
drift of 13 or 14 divisions, no very small fraction of the whole 
action ; as was seen from the observation that the (+) effect 
was now 300 divisions, of which 150 are due to the iron. 
With 20,000 ohms in circuit, giving a force equal to ;4, H, 
the drift was 6 or 7 divisions. By still further diminishing 
the force the drift could be reduced to insignificance ; but it 
appeared to maintain its proportion to the instantaneous effect. 
Apart from the complication due to the drift, the magneti- 
zation was proportional to magnetizing force from 1, H to 
5000 H or lessT. 

The question now presents itself, What is the actual value 
of the permeability which has been proved to be a definite 
constant for small forces? In consequence, however, of the 
nearness of the operative pole to the suspended needle in the 
preceding experiments, no moderately accurate value of pu 
can be deduced. But the observations described in Part II. 
are sufficient to show that the constant permeability for hard 
iron has some such value as 90 or 100, the forces then opera- 
tive being within the prescribed limits. The fact that the 
initial value of w is so large is obviously of great theoretical 
and practical importance. Further evidence will be brought 
forward presently in connection with observations made with 
an arrangement better suited to an absolute determination. 

Too definite a character must not be ascribed to the above- 
mentioned limit of 3H. Below this point the deviations from 
the law of proportionality, though mathematically existent, are 
barely sensible. In order to understand this, it is well to con- 
sider what happens when the limit is plainly exceeded. If a 
force of the order H be imposed, the compensating-coil (ad- 
justed for small forces) appears to be overpowered, and a 


* Prof. Ewing (loe. cit. § 52) describes “a time lag in magnetization,” 
especially noticeable in the softest iron and at points near the beginning 
of the steep part of the magnetization-curve. It should have been statea 
that my apparatus was very firmly supported, and, being situated under- 
ground, was well protected from vibration. The dzift or creeping did 
not appear to be due to this cause. 

+ The results here set forth were announced in a discussion following 
Prof. Hughes’s address to the Society of Telegraph Engineers on Fe- 
bruary 11, 1886 (Journ. Tel. Eng. xv. p. 39), on the strength of prelimi- 
nary experiments tried towards the close of 1835. 


232 Lord Rayleigh on the Behaviour of Iron and Steel 


large deflection occurs. If the force be now removed, the 
recovery is incomplete, indicating that the iron retains residual 
magnetism. Subsequent applications and removals of the 
force produce a nearly regular effect, and always of such a 
character as to prove that the magnetic changes in the iron 
exceed those demanded by the law of proportionality. As 
might be expected, the excess varies as the square of the 
force ; and thus, when the force is small enough, it becomes 
insignificant, and the law of proportionality expresses the 
facts of the case with sufficient accuracy. But the precise 
limit to be fixed to the operation of the law depends neces- 
sarily upon the degree of accuracy demanded. 

The readings with and without the force being tolerably 
definite, it would of course be possible, by pushing in the 
compensating-coil, to bring about an adjustment in which the 
application or removal of the force causes no deflection. But 
this state of things must be carefully distinguished from the 
compensation obtainable with very small forces, in that it is 


_ limited to one particular step in the magnitude of force. If 


we try a force of half the magnitude, we find the compensation 
fail. Not only so, but the reading will be different under the 
same force according as we come to it from the one side or 
from the other. The curve representing the relation between 
force and magnetization is a loop of finite area. 

Except for the purpose of examining whether the whole 
magnetization is assumed instantaneously (absence of drift), 
there is little advantage in the compensation being adjusted 
for the extreme range under trial. It is usually better to 
retain the adjustment proper to very small forces. Hven 
though it fails to give a complete compensation, the coil offers 
an important advantage, which will presently appear; and its 
use diminishes the displacement to be read upon the scale. 

We have seen that when the forces are very small there is 
a definite relation between force and magnetization, of such a 
character that one is proportional to the other: the ratio k 
(the susceptibility) is a definite constant. When, however, 
certain limits are exceeded there is no fixed relation between 
the quantities ; and if & is still to be retained, it requires a 
fresh definition. It is not merely that 4, as at first defined, 
ceases to be constant, but rather that it ceases to exist. Upon 
this point the verdict of experiment is perfectly clear. There 
is no curved line by which the relation between force and 
magnetization can be unambiguously expressed, and which 
can be traversed in both directions. As soon as the line 
ceases to be straight, it ceases also to be single. I have 
thought it desirable to emphasize this point, because the term 


under the Operation of Feeble Magnetic Forces. 233 


“ magnetization-function,” introduced by Dr. Stoletow, rather 
suggests a different conclusion. 

The curves given by Stoletow and by Rowland in their 
celebrated researches are not exactly magnetization-curves in 
the more natural sense; that is to say, they do not exhibit 
fully the behaviour of a piece of iron when subjected to 
a given sequence of magnetic forces. Buta number of such 
curves have been drawn by Ewing which afford all necessary 
general information. Among these we may especially dis- 
tinguish the course followed by the iron in passing from 
strong positive to strong negative magnetization and vice 
versd, and that by which iron starting from a neutral con- 
dition first acquires magnetization under the action of a force 
constantly increasing. 

Attention is called by Ewing to the loops which are formed 
when the forces are carried round a (not very small) cycle of 
any kind. ‘ Hvery loop in the diagram shows that when we 
reverse the change of magnetizing force from increment to 
decrement, or vice versd, the magnetism begins to change 
very gradually relatively to the change of § (the force), no 
matter how fast it may have been changing in the opposite 
direction before. So much is this the case that the curves, 
when drawn to a scale such as that of the figure, appear in all 
cases to start off tangent to the line parallel to the axis on 
which §§ is measured whenever the change of § is reversed in 
sign.” 

“The question here raised as to the direction of the curve, 
after the force has passed a maximum or minimum, is one of 
great importance. If it were strictly true that this direction 
were parallel to the axis, it would follow generally that iron in 
any condition of magnetization would be uninfluenced by 
small periodic variations of magnetic force ; for example, that 
in many telephone experiments iron would show no magnetic 
properties. The experiments already detailed prove that when 
the whole force and magnetization are small (they were not 
actually evanescent) very sensible proportional changes of 
magnetization accompany small changes of force, the ratio 
being such as to give a permeability not much inferior to 
100. Nothing is easier than to show that this conclusion is 
not limited to very small mean forces and magnetizations. 
As regards the latter, we may apply and remove a force 
(say) of 5H. By this operation the iron is left in a different 
magnetic condition, and the zero-reading of the magnetometer 
is altered, probably to the extent of driving the spot of light 
off the scale. But if we bring the needle back with the aid of 
external magnets, we can examine, as before, the effect of 


234 Lord Rayleigh on the Behaviour of Iron and Steel 


imposing a small force (under }H). If this be in the opposite 
direction to the previous large force, it will produce, in spite 
of the compensating-coil, a very sensible effect; for in this 
case the movement from 0 to —}H is in continuation of the 
previous movement from 5H to 0. But subsequent appli- 
cations and removals of +H produce no visible effect upon the 
needle, as would have happened from the first had the small 
force operated in the positive direction. We may conclude, 
then, that the compensation for small forces suitable when the 
iron is nearly free from magnetization is not disturbed by the 
presence of considerable residual magnetism. 

To examine the action of a small increment or decrement, 
when the total force is relatively large, we must either intro- 
duce a second magnetizing helix or effect the variation of 
current otherwise than by breaking the circuit. I found it 
most convenient simply to vary the resistance taken from the 
box, so arranging matters that the small alteration of current 
required could be effected by the insertion or removal of a 
single plug. The corresponding change of current is obtained 
by inspection of a table of reciprocals; and it was readily 
proved that within the admissible range of the apparatus the 
compensation was just as effective whether a step (not ex- 
ceeding +H) was made from zero or froma force (say) of 5H, 
20 or 30 times as great as the increment or decrement itself. 
It need scarcely be repeated that there is an exception as 
regards the first step, in the case where it is in the same 
direction as the large movement preceding it. 

At this stage the original magnetizing-coil, having been 
arranged for the investigation of the smallest forces, was 
replaced by another at a greater distance from the suspended 
needle. When the magnetization of the iron in its various 
parts fails to vary in strict proportion to the force, the effective 
pole is liable to shift its position ; and this is an objection to the 
horizontal arrangement adopted in the earlier experiments. 
The helix was therefore placed vertically, the lower end of the 
iron core being a trifle below the level of the magnetometer-~ 
needle. The upper pole was at such a distance as to give 
but little relative effect. The length of the new helix, wound 
like the other upon a glass tube, is about 30 centim. The 
windings are in four layers, at the rate altogether of 65 per 
centim.; so that (under the same current) the magnetizing 
force is about twice as great as before. The resistance is 
4-75 ohms. 

A large number of observations have been made upon a 
core of rather hard Swedish iron, 3°30 millim in diameter. 
The same compensating-coil as before was found suitable, and 


under the Operation of Feeble Magnetic Forces. 235 


the arrangements were unaltered, except that an additional 
reversing-key was introduced, by which the poles of the 
Daniell cell could be interchanged. The total resistance of 
the circuit, independently of the box, was 7 ohms. The 
length of the core—or, rather, of the part exposed to the 
magnetizing force*—being about 100 diameters, is scarcely 
sufficient for an accurate determination ; but from the observed 
position necessary for the compensating-coil we can get at 
least a rough estimate of the susceptibility for small forces. 
Thus, on December 28th, there was compensation for small 
forces when the distances of the needle from the mean plane of 
the compensating-coil and from the operative pole of the iron 
core were respectively 17°2 centim. and 9:3 centim. The 
magnetic force at the needle, due to unit current in the com- 
pensating-coil, is 

2a x 74. x 9? 

{9°+17-2°}2 
The magnetizing force in the interior of the helix for unit 
current is 


ie 


4a x 65 = 817. 
If k be the susceptibility, the strength of the pole is 
dm x 330? x 817 xk; 


and since the distance of this from the needle is 9°3 centim., 
we have, to determine f, 
Makerere XD" 
mam x3007 x 817 


== 6°30; 


so that 
p=1+4rk=81. 


This is probably an underestimate. 

In order to obtain results comparable with those of Stoletow 
and Rowland, the iron was submitted to a series of cycles of 
positive and negative force. According to Ewing, the 
behaviour is simplest when the iron is first treated to a 
process of “‘ demagnetization by reversals.”’ This was effected 
in situ as a preliminary to the experiments of January 4th, 
the resistance in the bex being increased by small steps from 
a few ohms to a thousand ohms; while at each stage the 
battery was reversed several times. It must be remarked, 
however, that the iron was all the while under the influence 
of the earth’s vertical force ; so that the resulting condition 
was certainly not one of demagnetization. But even as thus 


* At the upper end the iron projected beyond the coil. 


236 Lord Rayleigh on the Behaviour of Iron and Steel 


carried out, the operation was probably advantageous as obli- 
terating the influence of the previous history of the iron 
core. 

The compensation was in the first place adjusted so that no 
displacement could be detected, whether the resistance was 
infinity or 2007 ohms*. ‘This, of course, was in the position 
of the reversing-key denoted (—). When the iron and the 
compensating-coil acted in the same direction (+), the dis- 

lacement was 8 divisions. 

In Table I. the first column gives the total resistance of 
the circuit in ohms, and the second gives the reciprocals of 
the first, numbers proportional to the current or magnetizing 
force. Repetitions of a cycle are shown on the same hori- 
zontal line, for greater convenience of comparison. Thus the 
first application of current +197 gave the reading 242; a 
second application, after the cycle +197, 0, —197, 0, gave 
2414. After two of these cycles had been completed, the 
current +3826 gave the reading 245. To the readings as 
entered a small correction to infinitely small arcs has been 
applied. The letters R, L in the first column indicate the 
alternative positions of the battery reversing-key. It will 
be seen that very nearly the same numbers are obtained 
on repetition of a cycle, and that even the first application 
of an increased force gives a normal result. 

The first question which suggests itself is the law connecting 
the magnitude of a current with the alteration of magneti- 
zation caused by its reversal. The quantities under conside- 
ration are exhibited in Table II., where the first column gives 
the current (a) and the second column the displacement (y) 
due to reversal. The relation between w and y is well ex- 
pressed by the formula 


y= —'005384+ 107222, oo a ee (1) 


of which the whole of the second member is shown in column 3, 
and the two parts separately in columns 3 and 4.. Column 6 
gives the differences between the observed displacements and 
those calculated from the formula; they do not much exceed 
the errors of observation. 

It will of course be borne in mind that the magnetization 
exhibited here is additional to the part rendered latent by the 
compensating-coil, and that the existence of the smail linear 
term may be attributed to a defective adjustment of that coil. 
The calculated value of y for the step from infinite resistance 


* For greater delicacy, recourse was had to the “method of multipli- 
cation,’ assisted by a pendulum, as already described. 


under the Operation of Feeble Magnetic Forces. 237 
TABLE I.—Jan. 4, 1887. 


Resistance. Current. Corrected Readings. 
Coe ee ee 0 240 
HOOT Bio .c. tas + 099 241 
50) hd Oe ae 0 241 
10/074 Dipeaae — 099 240 
ico” er 0) 240 
15 (7a 2 a aaa + 197 242 2412 
3S 4 a O 2413 241 
BOP iy cc ol ea LOE 2384 2381 
C3) SE Se 0 239 239 
——— 
SLE Scteiertine + 326 245 245 
Co) eee ets eee 243 
307 Teens — 326 235 235 
SO LE Se aa 0 aa | 237 
——- SS 
UE IB. d vsw ston + 483 2502 25021 
be at Scan wacks 0 246 246 
OA 7M ae eee — 483 228 228 
63). Ree eee 0) 23822 2322 
LOR eS + 934 2831 2833 284 
Cone eee 0 2643 2643 265 
10a ee ae — 934 1952 1952. 1952 
CD a Se eae 0 214 214 2133 
Srgckan Was exis be +1149 306} 3072 307 
Ce a oe ile 0 2765 2762 277 
SM cela cae —1149 1723 ype 1713 
Cincy. Late ess 0) 2012 201 201 
1 27)7 i Sete to Boot: alee: 2381 
7 (5 ha +1298 3253 3253 
1 Lr Gal i eee es 0) 0 ey bd re 3152 
FIOM no Se tieat dis 0 2863 286 
liv eae =O De Mr Poe es 2373 
7p OO ewes tr — 1298 1513 1503 
UF (al Visas ar DOM ea ce ve 160 
0 hee ae 0 1903 1883 
GG Puke ccos can AOE ake dle Ve ccenel Weta ias 2323 2323 
Git. wdeeae +1493 ooes 353 3Dax 3533 
HG Bocsicasess fe OU) Pm dine hans ee ae ns 3372 3082 
the Ae Sh 0 299 2992 300 3014 
GT basse ede. SOOT cs. ENN enceseee Teer pees 2413 
oy Al De ea —1493 1213 1212 1212 
Cyd ae eS AS Cre a ee ee a eee 136 
cot eee ee 0) 1748 1743 1733 


to 2007 ohms, which is one quarter of the first step in the 
table (from 1007R to 1007L), is 


y= —'138+°06=—:07 division. 
This is the step for which the coil was adjusted ; and the dif- 
ference between the calculated and observed (zero) value of y 


238 Lord Rayleigh on the Behaviour of Iron and Steel 


is perhaps as small as could have been expected. It is fair to 
conclude that, if the compensating-coil could have been per- 
fectly adjusted for a very small step (the actual step was 
scarcely small enough), the uncompensated effects visible with 
larger currents would have been expressible by a quadratic 
term simply. 

The currents (#) given in the tables are reduced to C.G.S. 
measure when divided by 10°. On the same system the mag- 
netizing force is 


82x10 "xe; 


so that the force due to the strongest current referred to in 
the table is 1:2 C.G.8., or about 7H. When the current is 
reversed, the change of magnetic force is of course the double 
of this quantity. 

In extending the definition of susceptibility to cases in 
which the force is not very small, we might proceed in more 
than one way. If we take the ratio of the change of mag- 
netization to change of force when the force is reversed, we 
are following good authorities; and we get a definition which 
is at any rate consistent with the definition necessary when 
small forces are concerned. The values of k for different 
forces are not given by a direct comparison of the numbers in 
Table LI., since the magnetometer-scale is arbitrary ; but we 
may find for what force the susceptibility is (for example) the 
double of that applicable to infinitely small forces. 


TABLE II. 

i oe nk 00532. | 1-072x?. rep Diff. 
99 1 0-52 1-05 0-5 405 
197 33 1-0 4-9 3-2 0 
326 10 17 11-4 9-7 40:3 
483 293 2:6 250 | 204 0-2 
934 gsi 4-9 93-7 88:8 _03 

1149 136 6:1 141:5 135-4 406 
1298 174 69 180°6 173°7 403 
1493 231 79 238-9 231-0 0 


For this purpose we must note that the conjoint effect of 
the magnetization due to current 50, simply applied or re- 
moved, and of the compensating-coil, was 8 divisions, of 
which half is due to each cause. ‘The effect of the coil for a 


under the Operation of Feeble Magnetic Forces. 239 


reversal of current 50 is thus 8 divisions, and being propor- 
tional to the current can be deduced for any other case. At 
the bottom of the table, where the current is 1493, the dis- 
placement rendered latent by the coil is thus about 240 divi- 
sions ; and since at this point the uncompensated displacement 
is nearly of the same amount, we see that the value of & (as 
above defined) is here doubled. Thus, if denote the mag- 
netizing force in C.G.S. measure, we have 
k=6°4 (14°85). 

The form of the relations of k to § for small forces is pretty 
accurately demonstrated by the observations. On the other 
hand, the reduction to absolute measure is rather rough*—a 
point of less consequence, inasmuch as the constants may be 
expected to vary according to the sample and condition of the 
iron. 

The observations in Table I. givea good deal more than the 
extreme range of magnetization due to the reversal of a force. 
In all cases the two residual magnetizations (when the force 
is zero) are recorded ; while in the two latter, where the range 
is greatest, further intermediate points are included. The 
results are plotted in Plate II. fig. 2, where it will be seen 
that the curves start backwards in a horizontal direction after 
a maximum or minimum of force. Special observations (not 
recorded in the table) were directed to this point. Neither at 
the maxima nor at the zeros of force was there any evidence 
of failure of compensation when a small backward movement 
was made. 

The curves do not differ much from parabolas ; and in 
other cases, where the applied magnetic forces were all of 
one sign, I have found that after a large movement in one 
direction, the curve representing a backward movement 
coincides somewhat closely with a parabola whose magnitude 
is nearly the same under different circumstances, and which 
is placed so that its axis is vertical and vertex coincident with 
the point where the backward movement commences. The 
reader will not forget that to obtain the real curves fully ex- 
pressing the relation between magnetization and force, we must 
add the effect, proportional to the force, rendered latent by the 
compensating-coil. 

On the basis of this parabolic law we may calculate the 
influence of hysteresis in the magnetization of iron upon the 
apparent self-induction and resistance of the magnetizing- 
coil, when periodic currents of moderate power are allowed 


aie all probability the number 6°4, applicable when $=0, is too 
small 


240 Lord Rayleigh on the Behaviour of Iron and Steel 


to pass. If we reckon from the mean condition, we may 
express the relation between the extreme changes of magneti~ 
zation and force by the formula 


JS =a +85", 2. 
where a and £ are constants, corresponding with the 6-4 and 
64x °8 of the example given above. But no such single 
formula can express the relation for the rest of the cycle. 
When § is diminishing from H=’ to H= — SH’, 
=af) + BO"{1—-30— $/91)"h5 

but when § is increasing from H=—H! to H=H/, 
=aQ+PH"{—1+3 0+ $/91)t. 

These expressions coincide at the limits H=+5, but differ 


at intermediate points. Since the force is supposed to be 
periodic, we may conveniently write 


H = H'cosd; 
whence, putting also for brevity a’ in place of af’, 8’ in place 
of BH", we get 
=a! cos 0+ B'{cos 8+4 sin? 7} 
from 0=0 to 0=7, 
$=a’ cos 0+ 6! {cos 0—3 sin? 6} 
from 0=7 to 0=2r. 

We have now to express ¥ for the complete cycle in 
Fourier’s series proceeding by the sines and cosines of @ and 
its multiples. The part 

a' cos 6 -+- B' cos 8, 
being the same in the two expressions, is already of the 
required form. Tor the other part we get 

+4 sin? 6=B,sin@+ B;sin 36+ B;sin50+...., . (8) 
where only odd terms appear, and B, is given by 
—4 

Ler ry . ° ° e ° e (4) 

Thus 


= (a! + £') cos 048" 4 5 sin 8— = sin 86— — sin 50—.. }. 


If the range of magnetization be very small, Q’ vanishes, 
and the influence of the iron upon the enveloping coil is 
merely to increase its self-induction ; but if @! be finite, the 
matter is lesssimple. The terms in sin 30, sin 50, &e. indicate 
that the response of the iron to a harmonic force is not even 


(5) 


under the Operation of Feeble Magnetic Forces. 241 


purely harmonic, but requires higher components for its ex- 
pression. If we put these terms out of account as relatively 
small, we must still regard the phase of 3 as different from 
that of §. The term in sin @ will show itself as an apparent 
increase in the resistance of the coil, due to hysteresis, and 
independent of that which may be observed even with very 
small forces as a consequence of induced currents in the 
interior of the iron. The augmentation of resistance now 
under consideration may be expected to be insensible when 
the extreme range of magnetizing force does not exceed one 
tenth of the earth’s horizontal force. 

In the absolute determination (p. 235) of the susceptibility 
to very small forces of the hard Swedish iron wire (3°30 
millim. diameter), the length (about 100 diameters) was 
scarcely sufficient for an accurate estimate. Similar experi- 
ments on a thinner wire (1°57 millim. diameter) of the same 
quality of iron gave £=6°85, corresponding to w=87. This 
is in the hard-drawn condition. After annealing the same 
piece of wire gave a higher result, but in this case the obser- 
vation is complicated by the assumption of the magnetic state 
occupying a sensible time. The susceptibility applicable to 
the final condition is as high as 22-0, more than three times 
as great as before annealing. But a lower number would 
better represent the facts, when the small magnetic force is 
rapidly periodic ; and it may even be that under forces of 
frequencies such as occur in telephonic experiments, most of 
the difference due to annealing would disappear. Such a con- 
clusion is suggested by the slight influence of annealing in 
the experiment described in Part II.,* where is determined 
the increment of resistance of an iron wire due to the concen- 
tration of a variable current in the outer layers. But the 
matter is one requiring further examination under better ex- 
perimental conditions. 


The sensitiveness of the magnetometer-needle in the ex- 
periments directed to prove the constancy of susceptibility to 
small forces, suggests the inquiry whether iron should be 
used when the object is purely galvanometric. An attempt 
to produce a sensitive galvanometer by hanging a mirror and 
needle between the pointed pole-pieces of a large electro- 
magnet, arranged as in diamagnetic experiments, was not 
very successful. A better result was obtained with an astatic 
needle system, and an electromagnet on a much smaller scale. 
This was of horseshoe form, the core being of hard Swedish 


* Phil. Mag. Dec. 1886, p. 488. 
Phil. Mag. 8. 5. Vol. 23. No. 142. March 1887. S 


242 Lord Rayleigh on the Behaviour of Iron and Steel 


iron wire 3°35 millim. diameter. The insulated copper wire was 
in three layers, of resistance °34 ohm, and the total weight of the 
electromagnet was 283 grams. It was held so as to embrace 
the upper needle system. When the time of swing from rest 


to rest was 4 seconds, the movement due to a current of about 
1 


55000 ampere was 100 divisions. The zero was steady enough 
to allow a displacement of half a division to be detected with 
tolerable certainty in each trial; so that, as actually used, the 
arrangement was sensitive to a current of +x 10~° ampere. 
The addition of a similar electromagnet embracing the lower 
needle system, and connected in series, would double the 
sensitiveness, and raise the resistance to °68 ohm. A galva- 
nometer thus constructed, and of resistance equal to 1 ohm, 
would show a current of 10-7 ampere. Using finer wire, we 
might expect an instrument of 100 ohms to show a current of 
10-® ampere, and so on. 

For comparison with the above I tried, in as nearly as 
possible the same way, the sensitiveness of a good Thomson 
astatic galvanometer of resistance 1:3 ohm. With an equal 


time of vibration, a current of wave ampere produced a move- 
ment of 300 divisions. The zero was perhaps a little steadier 
than before ; but it will be seen that the sensitiveness was of 
the same order of magnitude. In both cases, by taking pre- 
cautions and by using repetition, the delicacy might have been 
increased, probably tenfold. 

The experiments show that there is no difficulty in con- 
structing a galvanometer of high sensitiveness upon these 
lines. According to theory, with ideal iron of permeability 
100, it should be possible to attain a much higher degree of 
sensitiveness than without iron. But the tendency to retain 
residual magnetism would certainly be troublesome, and pro- 
bably neutralize in practice most of the advantage arising 
from the higher permeability, which allows of windings more 
distant from the needles being turned to good account. Another 
inconvenience may be mentioned. If the iron poles are 
brought at all close to the needles, there is a strong tendency 
to instability at moderate angles of displacement. 


Tixperiments already described proved conclusively that the 
response of iron and steel to small periodic magnetic forces is 
not affected by the presence of a constant force, or of a re- 
sidual magnetization, of moderate intensity. At the same time 
it appeared in the highest degree probable that the indepen- 
dence was not absolute, and that the response to a given small 
change of force would fall off as the condition of “ saturation” 
is approached, even though we admit, in accordance with 


under the Operation of Feeble Magnetic Forces. 243 


recent evidence, that saturation is attainable only in a very 
rough sense. The question was too important to be left un- 
decided, but it was difficult to deal with by the magneto- 
metric method. If the arrangement is sensitive enough to 
allow the effect of the small force to be measured with reason- 
able accuracy, it is violently disturbed by the occurrence of 
high degrees of magnetization. Moreover it is undesirable 
to depend so much, as in this method, upon what may happen 
near the free extremities of the iron rod, where the magnetic 
forces must vary rapidly. The “ballistic method,’ in which 
the changes of magnetization are indicated by the throw of 
a galvanometer-needle in connection with a secondary coil 
embracing the central parts of the rod, has the great advan- 
tage for this purpose, that the reading is independent of the 
stationary condition of the iron. In the first experiments by 
this method the magnetizing helix was similar to one already 
described (p. 234); and the small, as well as the large, altera- 
tions of force were effected by varying the resistance of the 
circuit. By suitably choosing the resistances from a box, the 
small alterations of current could be obtained with sufficient 
suddenness by the simple introduction or removal of a plug, 
and weretaken of the same order of magnitude at different parts 
of the scale. A comparison of effects (with the aid of a table of 
reciprocals) proved that a pretty strong total force* or mag- 
netization did not interfere much with the response of the 
iron to a given force of small magnitude. 

This arrangement did not well allow of the investigation 
being pushed further so as to deal with stronger magnetizing 
forces. If, with the view of increasing the current, we cut 
down the german-silver resistance too closely, the estimate of 
total resistance depends too much upon the battery, and the 
current becomes uncertain. This difficulty is evaded by the use 
of a double wire—one conveying the strong current, of which 
the measurement does not require to be very exact; the other 
conveying the weak current, of which the effect at different 
parts of the scale is to be examined. 

In order to obtain a satisfactory ratio of length to diameter, 
without the loss of sensitiveness that would accompany a 
diminution in the section of the iron, a helix was prepared 
of length 59:6 centim. It was wound upon a glass tube 
with a double wire in three layers, the whole number of turns 
of each wire being 1376. The magnetizing force due to unit 
current in one wire is therefore 


Aor X 1376/59°6= 290'1. 


* Up toabout 6C.G.S. The iron was unannealed Swedish, 3°3 millim. 
in diameter. 
82 


244 Lord Rayleigh on the Behaviour of Iron and Steel. 


The resistance of each wire is 3°2 ohms; and thus when 
two Grove cells are used in connection with one of the wires, 
a current of about an ampere (‘1 C.G.S8.) can be commanded. 
Smaller currents were obtained by the insertion of resistances 
from a box. 

Although the secondary coil, connected with a delicate 
galvanometer, contained a large number of convolutions, the 
sensitiveness was insufficient to allow of the small magnetizing 
force being taken as low as would otherwise have been de- 
sirable. It was obtained by means of the second wire of the 
helix, which was included in the circuit of a Daniell cell and 
200 ohms from a resistance-box. When the circuit was com- 


pleted (or broken) at a key, the force brought into operation, 
or removed, was 


290-1 
5040 = 14 C.G.8. 

In making a series of observations it was usual, after each 
alteration of the strong magnetizing force, to apply and re- 
move the small magnetizing force several times before 
attempting to take readings. 

The results obtained by this method were of a pretty de- 
finite character. The small force produced a constant effect 
upon a wire of unannealed Swedish iron, 3°3 millim. in 
diameter, until the large force was increased from 0 to about 
5 C.G.8. At about 10 C.G.S. the effect of the small force 
fell off 5 per cent. The highest force used, about 29 C.G.S., 
reduced the effect to about 60 per cent. of its original amount. 
On complete removal of the force due to the Grove cells, there 
was but a partial recovery of effect, doubtless in consequence 
of residual magnetization. After the wire had been removed 
from the helix and well shaken, the small force was found to 
have recovered its full efficiency. 

The wire was then annealed and submitted anew to a similar 
series of operations. The magnetization due to the alternate 
application and removal of the small force was found to be at 
first, 7. e. in the absence of a constant force, twice as great as 
before *. 

The increase, however, is not long maintained, a steady force 
of 2 C.G.8. being already sufficient to cause a marked falling 
off (of about 20 per cent.). Under the operation of 29 C.G.S., 
the effect of the small force fell to about 4 of its original 


* It should here be remembered that any part of the change of mag- 
netization which lags behind for more than a second or two, fails to 
manifest itself fully in the indications of the galvanometer. 


Mr. H. Tomlinson on the Physical Properties of Iron. 245 


amount. Removal from the helix and shaking in a zero 
field sufficed to restore the wire to its initial condition. 

Similar experiments upon an annealed wire of “ best spring 
steel?” showed no sensible change of effect when the steady 
force was varied from 0 to about 16 C.G.S. In this case the 
ratio of length to diameter was about 300. 

We may now regard it as established :— 

That in any condition of force and magnetization, the sus- 
ceptibility to small periodic changes of force is a definite, and 
not very small, quantity, independent of the magnitude of the 
small change. 

That the value of the susceptibility to small changes of force 
is approximately independent of the initial condition as regards 
force and magnetization, until the region of saturation is ap- 
proached. 


Terling Place, Witham, Essex, 
Jan. 24, 1887. 


XXVI. The Permanent and Temporary Effects on some of the 
Physical Properties of Iron, produced by raising the Tempe- 
rature to 100° C. By Hersert Tomurnson, B.A. * 


Introduction. 


Ko many years I have been carrying on researches re- 

specting the effects of stress on the physical properties of 
matter, and during this period I have become acquainted with 
certain phenomena, which, though pertaining more or less to 
most metals, are so conspicuous in iron as to render it worthy 
of special attention. As these phenomena bear importantly on 
what Sir William Thomson has designated the thermodynamic 
qualities of metals, the investigation of which seems to be 
attracting daily more and more attention, I propose to lay 
before the Physical Society, from time to time, such informa- 
tion concerning them as a patient study has enabled me to 
acquire. 

On the present occasion I would invite attention to certain 
remarkable effects produced on some of the physical proper- 
ties of iron, by merely raising the temperature to a degree 
not exceeding 100° C. 


The Internal Friction of Iron. 


If an iron wire be suspended vertically with its upper extre- 
mity clamped to a rigid support, and its lower one clamped or 
soldered to the centre of a horizontal bar of metal, attached 


* Communicated by the Physical Society: read January 22, 1887, 


246 Mr. H. Tomlinson on some 


to which, at equal distances from the wire, are two cylinders 
of equal mass and dimensions, and the whole system be set 
in torsional oscillation, the amplitude of the vibrations will be 
found to diminish more or less gradually until finally rest 
ensues. This diminution of amplitude is almost entirely due 
to two causes, namely, the friction of the air, and the internal 
friction of the metal. The internal friction of the wire may 
be measured by the logarithmic decrement of are for a single 


log A—log B 
n 


vibration, or by , where A is the initial arc, and 


B is the arc after n vibrations. By the aid of Prof. G. G. 
Stokes’s mathematical formule *, and an experimental deter- 
mination by myself + of the coefficient of viscosity of air, I 
have been able to eliminate the resistance of the air, and to 
compute the damping effect due to the internal friction of the 
metal. Where the deformations produced are sufficiently 
small, I have proved the following laws respecting the loga- 
rithmic decrement of are t :— 

1. It is independent of the amplitude ; 

2. It is independent of the vibration-period. 

These laws only hold good when the wire has been allowed 
to rest for a considerable time after any change has been 
made in the arrangements, and when there has been a large 
number of oscillations executed previously to the actual 
testing. 

What is the nature of this so-called internal friction of the 
metal? It cannot resemble fluid friction ; because for such 
velocities as we have here the friction of fluids is proportional 
to the velocity. Neither can it resemble altogether the ex- 
ternal friction of solids; because the latter is not nearly so 
independent of the velocity as is the internal friction, nor 
would the logarithmic decrement be independent of the am- 
plitude. Some experiments by Prof. G. Wiedemann§ throw 
light on the subject. Let A, By represent the original posi- 
tion of equilibrium of the axis of the bar to which the wire is 
attached, and let a torsional couple be applied so as to bring 
the bar to A, B,. On reducing the torsional stress gradually 
to zero the bar will not come back to Ay By, but remain in a 
new position A, B,, however small may have been the angle 
of torsion A,0 A>. Again, if the bar be twisted by an equal 
torsional stress in the opposite direction to A; B; and the stress 
be then reduced to zero, the bar will remain permanently 

* Camb. Phil. Soc. Trans. vol. ix. no. x. (1850). 

+ Phil. Trans. 1886. 

t “ The Internal Friction of Metals,” Phil. Trans. 1886. 


§ Wiedemann’s Annalen, 1879, No. 4, vol. vi.; Phil. Mag. January 
and February 1880. 


of the Physical Properties of Iron. 247 


twisted in the position A, By. Now if we keep on applying 
and removing the torsional couple in this way, first in one 


direction and then in the other, the region A,0 A, will gra- 
dually diminish until a minimum is reached. According to 
Wiedemann, this is exactly what takes place when we allow 
the wire to vibrate freely ; the permanent position of equili- 
brium is constantly shifted to and fro. Within the regions 
A,0 A, and A;0 A, the elasticity is perfect, and there is on 
the whole no gain or loss of energy. The loss of energy ex- 
perienced in a torsionally vibrating wire arises from the work 
expended in the region A,0 A, in shifting the permanent 
position of equilibrium from A, to A, and back again ; and, 
provided the amplitudes of the oscillations do not exceed a 
certain limit, the extent of the region A, 0 A, is proportional 
to the amplitude. Wiedemann goes further than this; for he 
says what is true with respect to the wire as a whole is true 
with respect to each molecule of the iron, and that the internal 
friction is really due to the rotation to and fro of the perma- 
nent positions of equilibrium of the molecules. We need not 
stop to discuss here this last point ; but what does seem pro- 
bable is, that the main part of the loss of energy is experienced 
as the bar swings from A, to Ay. When, as in my own ex- 
periments, the deformations produced by the oscillations are 
very small, it would seem that the positions A., A, are really 
subpermanent rather than permanent ; and if time were given 
and the molecules agitated, the bar would of itself return to 
the position Ay, when the torsional couple was reduced to zero. 

I have said, that as the wire oscillates, the region A, 0 A, 
becomes narrower and narrower, and Wiedemann speaks of 
the period during which the diminution takes place, as “ the 
accommodation period.” My own experiments have verified 
the results of those of Wiedemann and Sir William Thomson *, 


* Proc, Roy. Soc. May 18, 1805. 


248 Mr. H. Tomlinson on some 


in showing that repeated oscillation will reduce the internal 
friction ; but they also show a very large influence to be 
exerted by long rest, either with or without oscillation, and have 
further proved that considerable diminution, both temporary 
and permanent, can be produced by merely raising the 
temperature of the wire to 100°C. Thus a well annealed 
iron wire, when tested about ten minutes after suspension, 
was found to have a logarithmic decrement due to the 
internal friction of the metal of :003011, after one hour of 
001195, and after one day of 001078. After the last period 
the friction became sensibly constant, and after four days was 
found to be still the same ; the wire had apparently ‘‘ accom- 
modated ”’ itself as far as possible. 

Great, however, as was the reduction of the internal friction 
produced by oscillation and rest, the minimum had by no means 
been reached ; for on repeatedly heating the wire to 100° C., 
and then allowing it to cool, the logarithmic decrement rapidly 
diminished, until after six days, on each of which the wire was 
heated to 100°C., and then allowed to cool slowly, it became only 
000412, when further repetition of the above process ceased to 
sensibly affect the friction. ‘The greater part of the diminution 
occurred after the first heating and cooling, but several repeti- 
tions were necessary to produce the minimum mentioned above. 

Still more marvellous is the temporary effect of a rise of 
temperature not exceeding 100° C. on the internal friction of 
annealed iron. <A careful examination of the above specimen 
at temperatures ranging between 0° C. and 100°C., revealed 
the astonishing fact that, at a temperature of 98°C., the 
logarithmic decrement was only ‘000112, and was considerably 
less than one fourth of its amount at 0O°C.* At 98°C. the 
friction was a minimum, further rise of temperature resulting 
in increase of the logarithmic decrement. It may perhaps 
assist us in forming some notion of the very small amount of 
internal friction in the above specimen of iron at 98° C., if we 
estimate the number of vibrations which would be required 
before the amplitude would be reduced to one half of its initial 
value by molecular friction only : this number is nearly 3000. 
It follows that if we could make the wire and its appendages 
vibrate im vacuo, and maintain the temperature constantly at 
98° C., with a vibration-period of ten seconds, more than eight 
hours would elapse before an initial amplitude of 100 would 
be reduced to 50. 

The internal friction of the wire when reduced to its 
minimum by all the above-mentioned processes was only one 
thirtieth of its original amount. 


* The temporary diminution of the internal friction of annealed iron 
was shown at the meeting of the Society. 


of the Physical Properties of Iron. 249 


When the wire has fully “ accommodated”’ itself, a very 
small cause will disturb the accommodation: a mechanical 
shock, a change of load, a slight rise or fail of tempera- 
ture, or even rotation of the molecules by. magnetic stress, will 
necessitate fresh oscillations before the friction again reaches 
its minimum. Consequently, if a wire be raised to 100°C., 
and be then cooled again rather quickly, it does not imme- 
diately regain the accommodation which it had before heating. 
The time taken by the wire to reaccommodate itself when 
the accommodation has been disturbed by change of tem- 
perature, depends considerably upon the direction of the 
change. Thus, when the wire is raised from the temperature 
of the room to 100°C., the reaccommodation is effected in a 
very much shorter time than when the accommodation has 
been disturbed by lowering the temperature from 100°C. to 
the temperature of the room. 

I am inclined to regard both the temporary and permanent 
alterations of the internal friction of iron, which are produced 
by rise of temperature not exceeding 100° C., to be partly due 
to mere agitation of the molecules, but the permanent effects 
do not seem to be entirely due to molecular agitation ; for the 
maintaining of the temperature at 100° C. for some time does 
not bring down the friction anything like so much as repeated 
heating and cooling. It would seem that the slow shifting 
backwards and forwards of the molecules induced by the last 
process is ina great measure to becredited with the permanent 
diminution of friction, in the same manner that the shifting 
backwards and forwards of the molecules caused by torsional 
oscillation has been shown to produce permanent diminution. 


The Longitudinal and Torsional Elasticity of Iron. 


It might readily be imagined that since the internal friction 
of iron is so considerably altered by change of temperature, 
the elasticity would be correspondingly affected. This, how- 
ever is not so; both the torsional and longitudinal elasticity 
of iron are affected by raising the temperature to 100° C., but 
not nearly to the same extent as the internal friction. Thus 
an annealed iron wire, when suspended ready for torsional 
vibration, was heated slightly by passing the flame of a 
Bunsen’s burner rather quickly up and down it several times. 
The time of vibration before heating was 1:154 second ; and in 
5 minutes, 3) minutes, and 245 minutes after cooling was 
1:147, 1:142, and 1°136 second respectively. Here we have 
a small, though distinct, permanent increase of elasticity ; and 
it will be noticed that time is an important element in the 
amount of increase. Again, by a very carefully conducted 
set of observations, I have shown that the torsional elasticity 


250 -  _ Mr. H. Tomlinson on some 


of annealed iron is temporarily decreased to the extent of 2°693 
per cent., when the temperature is raised from 0° to 100° C.* 

Again, an annealed pianoforte-steel wire, when tested with 
a certain load at the temperature of 12° C., was temporarily 
elongated to the extent of 1:502 half-millimetres; when heated 
to 100° C, in an air-chamber the elongation was 1-487 half-milli- 
metres; and when cooled again and tested 24 hours afterwards, 
was elongated by 1:450 half-millimetres +. Thus, as with the 
torsional elasticity, there was a permanent increase of elasticity, 
and a temporary decrease of 2°58 per cent. Time in this case 
also is an important element, for the elasticity immediately 
after cooling was very appreciably less than when a long rest 
had been given. 

Similarly, I have shown both for longitudinal and torsional 
elasticity that an iron wire, after having been permanently ex- 
tended by traction, has its elasticity very perceptibly increased 
by long rest. Itis also well known that the portative power of a 
magnet can be considerably increased by putting on the load by 
small quantities ata time, with long intervals of rest between. 

These and other considerations prove beyond a doubt, that if 
the molecular arrangement of iron be disturbed by any kind of 
stress whatever, exceeding a certain small limit, the molecules 
will not assume at ordinary temperatures those positions which 
will secure a maximum of elasticity, until a rest of many hours 
has been given. 


The Velocity of Sound in Iron. 


According to Wertheim, the velocity of sound in iron and 
steel is increased by a rise of temperature not extending beyond 
100° C.t Now in no sense whatever is this statement correct. 
It is true that the longitudinal elasticity of iron, as determined 
by the method of statical extension, may be found greater at 
100° C. than at 0° C., provided we begin with the lower tempe- 
rature first and the wire has not been previously tested at 
100° C. But, as we have seen, the apparent temporary 
increase of elasticity is really a permanent one ; and if the wire 
be repeatedly heated to 100° C. and then cooled, subsequent 
tests will always show a less elasticity at the higher tempe- 
rature than at the lower, if sufficient rest after cooling be 
allowed. When, however, we come to such small molecular 
displacements as are involved in the passage of sound through 
a wire, even the apparent increase of elasticity, mentioned 
above as taking place at the first heating, vanishes. I have 
been able to prove that, when an iron or steel wire is thrown 

* Proc. R. 8. No. 244, 1886. 


+ Phil. Trans. Part I., 1883, p. 180. 
} Ann. de Chimie et de Phys. 3m° série, 1844, p. 421. 


of the Physical Properties of Iron. 251 


into longitudinal vibrations, so as to produce a musical note, 
the pitch of this note becomes lower as we raise the temperature 
even when the wire is heated for the first time.* 

It seems rather strange that the error should have been so 
long allowed to remain uncorrected ; for it has been known 
for many years that the pitch of a steel tuning-fork is lowered 
by small rises of temperature to a greater extent than would 
follow from mere change of dimensions. Calling the fre- 
quency of the fork n we havet 


b 
n= am, . gees ee yo 


where m is an abstract number, 
b is the velocity of sound in steel, 
lis the length, 
« the radius of gyration of the section about an axis 
perpendicular to the plane of bending. 
If D be the thickness of the fork, 
D 
k= ——=. 
. 12 
We may therefore obtain 3 = 
4rV/3P 
b= be AL Se Wea Oe CET MR (2) 

From (2) and the value of the coefficient of thermal expan- 
sion of steel, it follows that if the pitch of the note is lowered 
by rise of temperature to the extent to which it is known to 
be, the velocity of sound must be lowered also. Indeed from 
the coefficient of thermal expansion of steel and from my own 
determination of the effect of change of temperature on the 
longitudinal elasticity of steel, I have calculated what would be 
the lowering of pitch of a certain fork, and find it in sufficient 
accord with the-lowering of pitch as determined by direct 
experiment. 

Wertheim inferred the increase of velocity of sound in iron 
and steel from the apparent increase of longitudinal elasticity 
produced by rise of temperature. From his experiments on 
the longitudinal elasticity of these metals I have collected the 
following :— | 


Increase per cent. of elasticity 


Metal. between 15° to 20° and 100° C, 
Annealed iron Ui SG RR OR Bias 
mamedied iron wire . 2 Nop yank. er eo eed 
mmniealed. cast steel = fs ' 1. 2 aN OL e AO 
Amnested steélhwire . 4.5.0 7 3. as 6 28°90 


Bicel tempered blue. 9 4. 7. 2: | 8 
* The lowering of pitch produced by vise of temperature was shown 
before the Society. 
T Lord Rayleigh’s ‘Theory of Sound,’ vol. i. p. 219. 


252 Sir W. Thomson on the Waves produced by 


whereas my own experiments show that both the torsional 
and longitudinal elasticities of iron and steel are decreased by 
about 24 per cent. when the temperature is raised from 
OaG: to 100° GC: 

I have dwelt longer than I should otherwise have done on 
this part of my paper because I find that even our best text- 
books relating to elasticity and sound still retain what I am 
convinced is an error. 


XXVIT. Onthe Waves produced by a Single Impulse in Water 


of any Depth, orina Dispersive Medium. By Sir WILLIAM 
THomson, LL.D., F.RS.* 


i yee brevity and simplicity consider only the case of two- 
dimensional motion. 

All that it is necessary to know of the medium is the rela- 
tion between the wave-velocity and the wave-length of an 
endless procession of periodic waves. The result of our work 
will show us that the velocity of progress of a zero, or maxi- 
mum, or minimum, in any part of a varying group of waves 
is equal to the velocity of progress of periodic waves of wave- 
length equal to a certain length, which may be defined as the 
wave-length in the neighbourhood of the particular point 
looked to in the group (a length which will generally be inter- 
mediate between the distances from the point considered to its 
next-neighbour corresponding points on the preceding and 
following waves). 

Let /(m) denote the velocity of propagation corresponding 
to wave-length 27/A. The Fourier-Cauchy-Poisson synthesis 
gives 


v= | ameosmir—t (in)] 


for the effect at place and time (, ¢) of an infinitely intense 
disturbance at place and time (0,0). The principle of inter- 
ference, as set forth by Prof. Stokes and Lord Rayleigh in 
their theory of group-velocity and wave-velocity, suggests the 
following treatment for this integral :— 

When a—t/f(m) is very large, the parts of the integral (1) 
which lie on the two sides of a small range, w—a@ to p+ae, 
vanish by annulling interference ; « being a value, or the 
value, of m which makes 


djdmim|_2—if(m)|}=0 7 ee 


* Communicated by the Author, having been read at the Meeting of 
the Royal Society, 3rd February, 1887. 


a Single Impulse in Water of any Depth. 253 


so that we have 


e=t{f(w)+uf’(u)}=Vt. . . . (8), 
Nesp iy iy eee ie on 4); 


and we have by Taylor’s theorem for m—yp very small, 

m| e—tf(m) |=ple—tf(e) ]—t[ef’ (H) + 2f'(#) 13 (m—)’ (5)5 
or, modifying by (8), 

m| e—tf(m) |=2t{u2f'(e) +[— ef" (H) —2f"(H) (mm —#)"} (8). 


Put now 


where 


~ oe Shp (7) 
[ —wf’(u)—2/"(m) | 

and using the result in (1), we find 
/ 25° do cos [tu?f'(u) +07] (8); 

w= BL af" (eH) —27"(e) 

the limits of the integral being here —«# to «, because the 
denominator of (7) is so infinitely great that, though +a, the 
arbitrary limits of m—vy, are infinitely small, a multiplied by 


it is infinitely great. 
Now we have 


‘ do cos = (" do sin 0? = ft j 


Hence (8) becomes 
_ cos[ tu?’ (w) | —sin[tu?/” (w) ae: / 2 cos[ tu?’ (uw) +477] 
Lae) are) ela ( 7b )} 
To prove the law of wave-length and waye-velocity for any 
point of the group, remark that, by (3), 


tw’ f'(w)=h[o—tf()], 


and therefore the numerator of (10) is equal to /2cos@, 


where 

@=y[e—t/lu)]+3a . . . . (104, 
and by (2) and (3), 

d/du {pl e—tf(u) ]} =05 
by which we see that 


dO/de=p, and d0/dt=—pf(u) . . (10”), 
which proves the proposition. 


m— pb 


(10). 


* This is the group-v elocity according to Lord Rayleigh’s generaliza- 
tion of Prof. Stokes’s original result. 


254 Waves by a Single Impulse in Water of any Depth. 


Example (1).—As a first example take deep-sea waves ; we 
have 


fim=y/2 . . 2. 2 


170 


which reduces (4), (3), and (10) to 
Va3/2 . one 


and waiy/ Se. Meee 
be 


Le a t 
g/2 “a lcos * in) = gut mies i. a a (i) 
which is Cauchy and Poisson’s result for places where 2 is 
very great in comparison with the wave-length 27/u ; that is 
to say, for place and time such that gt?/4@ is very large. 
Example (2).—Waves in water of depth D, 


fon)=a/ £5} ae 


Example (3).—Light in a dispersive medium. 
Example (4).—Capillary gravitational waves, 


jm=a/ (L410) | ee 


Eaample (5).—Capillary waves, 
fim=/ (im)... . 


Example (6).—Waves of flexure running along a uniform 


elastic rod 
fm) =ma/= MO 


where B denotes the flexural rigidity and w the mass per unit 
of length. 

These last three examples have been taken by Lord Rayleigh 
as applications of his generalization of the theory of group- 
velocity ; and he has pointed out, in his “ Standing Waves in 
Running Water’ (London Mathematical Society, December 13, 
1883), the important peculiarity of example (4) in respect to 
the critical wave-length which gives minimum wave-velocity, 
and therefore group-velocity equal to wave-velocity. The 
working out of our present problem for this case, or any case 
in which there are either minimums or maximums, or both 
maximums and minimums, of wave-velocity, is particularly 
interesting, but time does not permit its being included in the 
present communication. 

For examples (5) and (6) the denominator of (10) is ima- 


Formation of Coreless Vortices. 255 


ginary ; and the proper modification, from (7) forwards, gives 
for these and such cases, instead of (14), the following:— 


fy aaiele [ew? if’ (u) | + sin [tu 7” (w) | (19) 
[wf (m) + 27" (mu) |2 sore 


The result is easily written down for each of the two last 
eases [| Hxamples (5) and (6)]. 


XXVIII. On the Formation of Coreless Vortices by the Motion 
of a Solid through an Inviscid Incompressible Fluid. By 
Sir W. Tuomson, LL.D., P.RS* 


NAKHE the simplest case: let the moving solid be a globe, 

and let the fluid be of infinite extent in all directions. 

Let its pressure be of any given value, P, at infinite distances 

from the globe, and let the globe be kept moving witha given 
constant velocity, V. 

If the fluid keeps everywhere in contact with the globe, its 
velocity relatively to the globe at the equator (which is the 
place of greatest relative velocity) is 3V. Hence, unless’ 
P>2V’, the fluid will not remain in contact with the globe. 

Suppose, in the first place, P to have been >3V’, and to 
be suddenly reduced to some constant value <2 V’. The fluid 
will be thrown off the globe at a belt of a certain breadth, and a . 
violently disturbed motion will ensue. To describe it, it will 
be convenient to speak of velocities and motions relative to the 
globe. The fiuid must, as indicated by the arrow-heads in fig. 1, 
flow partly backwards and partly forwards, at the place, I, 
where it impinges on the globe, after having shot off at a tan- 
gent at A. The back-flow along the belt that had been bared 
must bring to EH some fluid in contact with the globe; and 
the free surface of this fluid must collide with the surface of 
the fluid leaving the globe at A. It might be thought that 
the result of this collision is a “‘ vortex-sheet,”’ which, in virtue 
of its instability, gets drawn out and mixed up indefinitely, 
and is carried away by the fluid further and further from the 
globe. A definite amnout of kinetic energy would thus be 
practically annulled in a manner which I hope to explain in 
an early communication to the Royal Society of Edinburgh. 

But it is impossible, either in our ideal inviscid incom- 
pressible fluid, or in a real fluid such as water or air, to 


* Communicated by the Author, having been read at the Meeting of 
the Royal Society, 3rd February, 1887. 
+ The density of the fluid is taken as unity. 


256 Formation of Coreless Vortices. 


form a vortex-sheet, that is to say, an interface of finite slip, 
by any natural action. What happens in the case at present 
ie. 1 


under consideration, and in every real and imaginable case of 
two portions of liquid meeting one another (as, for instance, 
a drop of rain falling directly or obliquely on a horizontal 
surface of still water), is that continuity and the law of con- 
tinuous fluid motion become established at the instant of first 
contact between two points, or between two lines in a class of 
cases of ideal symmetry to which our present subject belongs. 

An inevitable result of the separation of the liquid from the 
solid, whether our supposed globe or any other figure per- 
fectly symmetrical round an axis, and moving exactly in the 
line of the axis, is that two circles of the freed liquid surface 
come into contact and initiate in an instant the enclosure of 
two rings of vacuum (G and H in fig. 2, which, however, 
may be enormously far from like the true configuration). 

The ‘‘ circulation ’’ (line-integral of tangential component 
velocity round any endless curve encircling the ring, as a 
ring on a ring, or one of two rings linked together) is deter- 
minate for each of these vacuum-rings, and remains constant 
for ever after: unless it divides itself into two or more, or 
the two first formed unite into one, against which accidents 
there is no security. 


Wave-lengths of the Lines of the Solar Spectrum. 257 


It is conceivably possible* that a coreless ring-vortex, with 
irrotational circulation round its hollow, shall be left oscillating 
‘in the neighbourhood of the equator of the globe ; provided 
(3V°—P))P be not too great. If the material of the globe 


Fig. 2. 


be viscously elastic, the vortex settles to a steady position round 
the equator, in a shape perfectly symmetrical on the two sides 
of the equatorial plane; and the whole motion goes on steadily 
henceforth for ever. 

If @V’—P)/P exceed a certain limit, I suppose coreless 
vortices will be successively formed and shed off behind the 
globe in its motion through the fluid. 


XXIX. On the Relative Wave-lengths of the Lines of the Solar 
Spectrum. By Prof. Henry A. Row.anpf. 


OR several years past I have been engaged in making a 
photographic map of the solar spectrum to replace the 
ordinary engraved maps, and I have now finished the map 
from the extreme ultra-violet, wave-length 3200, down to 
* If this conceivable possibility be impossible for a globe, it is certainly 
possible for some cases of prolate figures of revolution. 
-.+-Communicated by the Author, 


Phil. Mag. 8. 5. Vol. 23. No. 142. March 1887. E 


258 Prof. H. A. Rowland on the Relative Wave-lengths 


wave-length 5790. In order to place the scale correctly on 
this map, I have found it necessary to measure the relative 
wayve-lengths of the spectrum and to reduce them to absolute 
wave-lengths by some more modern determination. I have 
not yet entirely finished the work; but as my map of the 
spectrum is pow being published,,and as all observers so far 
seem to accept the measures of Anystrém, I have decided 
that a table of my results would be of value. For as they 
stand now, they have at least ten times the accuracy of any 
other determination. This great accuracy arises from the use 
of the concave grating, which reduces the problem of relative 
wave-lengths to the measure of the coincidences of the lines 
in the different spectra by a micrometer. 

The instrument which I have employed has concave gratings 
5 or 6 inches in diameter, having either 7200 or 14,400 lines 
to the inch and a radius of 21 tt. 6in. By my method of 
mounting, the spectrum is normal where measured, and thus 
it is possible to use a micrometer with a range of 5 inches. 
The spectrum keeps in focus everywhere, and the constant of 
the micrometer remains unchanged except for slight variations 
due to imperfections in the workmanship. The micrometer 


has no errors of run or period exceeding 3,5 inch. The 


probable error of a single setting on a good clear line is 


about sain of the wave-length. 1/ of arc is about ‘0012 inch. 


The D Jine in the second spectrum is ‘17 inch or 4°4 millim. 
wide. Determinations of relative wave-length of good lines 
seldom differ 1 in 500,000 from each other, and never exceed ~ 
1 in 100,000, even with different gratings. This is, of course, 
for the principal standard lines, and the chance of error is 
greater at the extremities of the spectrum. The interpolation 
of lines was made by running the micrometer over the whole 
spectrum, 5 inches ata time, and adding the readings together 
so as to include any distance, even the whole spectrum. The 
wave-length is calculated fora fixed micrometer constant, and 
then corrected so as to coincide everywhere very nearly with- 
the standards. I suppose the probable error of the relative de- 
terminations withthe weight 1 in my table to be not far from 
1 in 500,000. Angstrom thinks his standard lines have an 
accuracy of about 1 in 50,000, and ordinary lines much less. 

> As to the absolute measure, it is now well determined that 
Angstrém’s figures are too small by about 1 part in 6000. 
This rests, first, on the determinations of Peirce made for the 
U.S. Coast Survey with Rutherford’s gratings, and not yet 
completely published ; secondly, on an error,made by Tresea in 
the length of the standard meter used by Angstrém*, which 


* Sur le Spectre du Fer, Thalén. 


of the Lines of the Solar Spectrum. 259 


increases his value by about 1 in 7700 ; thirdly, on a result 
obtained in my laboratory with two of my gratings by Mr. 
Bell, which is published with this paper. Mr. C. 8S. Peirce 
has kindly placed his grating at our disposal; and we have 
detected an error of ruling which affects his result and makes 
it nearly coincide with our own. The wave-length of the 
mean of the two E lines is— 


Angstrém ChblS eT sale iiiects Ssh 2o200 12 "OS 
r (Corrected by Thalén) . . 5269-80* 


emcee ain sar 04s eee Nae ae a on bs PB 2DROLG 
» (Corrected by Rowland and Bell) 5270-00 
et taney choy Talis? aA Ce as Sp O2TOOk 


These results are for air at ordinary pressures and tempera- 
tures. The last isreduced to 20°C. and 760 millim. pressure. 
To reduce to a vacuum, multiply by the following :— 
Fraunhofer line...... A. C. E. G. 13 
Correction-factor .. 1:000291 1:000292 1:000294 1:000297 1-:000298 
- the relation between my wave-lengths and those of 
Angstrom are given by the following, Angstrém’s values 
being from p. 31 of his memoir :— 


a A (edge). B (edge). 0. 

Auestrom ...:..... 75975 6867°10 6717°16 6562°10 6264-31 
Oi ae 759397  6867°38 6717°83 656296  6265°27 
Difference ......... —3'5 28 67 86 96 
s D,. Dr. Peirce’s line. 
Angstrom ......... 5895:18 5889-712 570845 5623°36 5454-84 
Rowland ......... 589608 5890°12 5709°56 562470 5455°68 
Difference ......... ‘95 1°00 - TE 1:34 84 
" EB. E. b. FE. 
Angstrom ......... 5269'°59 526867 5183:10 5138-78 4860°74 
Rowland \s...2s <0: 527043 526965 518373 5138947 4861°43 
Difference ......... 84 98 63 ‘69 69 
5 G. 

Angstrom ......... 4702°44 4307-25 

Rowland ......... 4703-11 4807°96 

Difference ..,...... ‘67 cra t 


The greatest variation in,these differences is evidently due 
to the poor definition of Angstrém’s grating, by which the 
numbers refer to groups of lines rather than ,to single 
ones. Selecting the best figures, we find that Angstrém’s 
wave-lengths must be multiplied ,by 1°00016 to agree with 
Bell’s, while the correction for Angstrém’s error of scale 
would be 1:000110. 

* Sur le aa du Fer, Thalén. 
2 


————————— 


a eee 


aT 


260 Prof. H. A. Rowland on the Relative Wave-lengths 


It is impossible for me to give at present all the data on 

which my determinations rest ; but I have given in Table I. 
many of the coincidences as observed with several gratings, 
the number of single readings being given in the parentheses 
over each set. 
- Table LI. gives the wave-lengths as interpolated by the 
wicrometer. It is scarcely possible that any error will be 
found (except accidental errors) of more than ‘02; and, from — 
the agreement of the observations, I scarcely expect to make 
any changes in the final table of more than ‘01, except in the 
extremities of the spectrum, where it may amount to -03 in 
the region of the Aand H lines. The wave-lengths of weight 
greater than 1 will probably be found more exact than this. 
The lines can be identified on my new photograph of the 
spectrum down to 5790. Below this there is little trouble in 
finding the right ones. All maps of the spectrum, especially 
above IF’, are so imperfect that it is almost impossible to 
identify my lines upon them. ‘The lines can only be properly 
identified by a power sufficient to clearly divide b3 and by. 
Some of them are double, and most of these have been 
marked; but as the table has been made for my own use, I 
have not been very careful to examine each line. This will, 
however, be finally done. Micrometric measures have now 
been made of nearly all the lines below 6, with a view of 
making a map of this region. 

Table I. gives the coincidences of the different orders of 
the spectra as observed with several concave gratings on both 
sides of the normal, the numbers in the brackets indicating 
the number of observations. The observations have been 
reduced as nearly as possible to what I consider the true 
wave-length, the small difference from the numbers given in 
Table I. being the variation of the observations from the 
mean value. The true way of reducing these observations 
would be to form a linear equation for each series and reduce 
by the method of least squares. A simpler way was, how- . 
ever, used, and the relative wave-lengths of the standard lines 
(marked 8 in Table II.) were obtained ; however, some other 
observations were also included. 

Table II. gives the wave-lengths reduced to Bell’s value for 
the absolute wave-length of the D line. These were obtained 
by micrometric measurement from the standards, as described 
before. The weights are given in the first column, and some 
of the lines, which were measured double, have also been 
marked. But the series has not yet been carefully examined 
for doubles. The method is so much more accurate than by 
means of angular measurement, that the latter has little or no 
weight in comparison. 


(2) 

7039 :969 
7027°658 
7023°676 
5269°656 
5270°448 


(6) 
7039°963 
4922-336 
7027-627 
5270-429 
5269-647 
4215°627 
7023°632 


(8) 
4691°516 
4690°260 
5624-696 
5624:184 


(6) 
4508-402 
4501-387 
5624-696 
5624181 
4496-990 
4494-677 


(18) 
6430-993 
6439-222 
4293°181 


(9) 
6439-222 
4293-181 


(2) 
4824-240 
4823-640 
7184-701 


(4) 
7247569 
4824-249 
4823°630 


(4) 
7240°868 
4824°243 
4823636 
7234-854 
7233063 


6) 
4501:377 
6750°332 


of the Lines of the Solar Spectrum. 


(4) 
4691°517 
7035:056 
7027-665 
7015'641 
7015:256 


(4) 
4691-517 
7027°675 


(10) 
4501377 
6750-308 


(8) 
4508381 
4504-921 
4502°791 
6752-830 
4501°377 
6750°306 

(4) 
6013'682 
4508407 
6003-173 
4501-377 
4496-982 
4494°652 


(10) 
4215-618 
6322'820 


(4) 
6562960 
6564-341 
4376-052 

(4) 
4222-309 
4215613 
6322'817 
6318-165 
6278-255 
6252-698 
6246-451 


(6) 


TABLE [.—Coincidences. 


6562°960 


4376°041 


(3) 
4691-517 
7023-706 

(2) 
4691:517 
7027-655 
(1) Phot. 
5624-691 
3754-63 
3747-09 


(1) Phot. 
5914°32 
0942°72 


(1) Phot. 
5914-32 
3942°70 


(1) Phot. 
5890-12 
5896-10 
3926-12 


(4) 
6883'994 
4590-051 
4588°306 


(2) 
4823-638 
4824-261 
7233'103 
7240-902 


(2) 
5288623 
6609-215 
6593°992 
6593°038 
5270°419 
5269-651 
6567-645 
6564313 
4376-050 
5250°751 
5250-329 
6562°970 
6569°353 
6546393 
6495119 
6493°931 
6462 760 
5162 492 
5159-171 
6439°215 
6430°984 
6421°513 
6420-090 
5183-805 
6411-776 
5110-506 
5109-754 
5068°878 
5060°191 
F049-932 
6322°833 
6318°150 
6265°256 
6261-22] 


(1) Phot. 
6024-2 
6016-8 
5948°7 
3953-9 
3950-4 
5916-4 
5914-3 
3916-8 
5862°5 
5859°8 
3897-5 
5791-1 
5788:1 


(1) Phot. 
4789°7 
4789-4 
4788:8 
47541 
3064'6 
35499 
4727°5 
3545'2 
3540°2 
4691°5 
4690°3 


(14) 
5162-394 
6883:995 


(6) 
6594-016 
6593072 
5270°427 
5269 647 
6569-348 
6562 970 
5250 325 
5250°752 
4376 052 


(4) 
6569°370 
4924-889 
4924-045 
6562-965 
6546-409 
4903°419 
4859866 
4861-428 
6462-744 
4824-255 
4823631 


(6) 
5914-323 
5896-084 
5890°125 
4691:517 
4690°266 
5862°522 
5859-75] 
4683°691 


(6) 
4376-039 
5269-632 
5270°420 

e) 
4222-301 
4215613 
5270-485 
5269-654 

(12) 

5914314 

222-325 
4924-889 
4924-052 
4215-613 
5896-083 
5890°125 
4903-411 


(4) 
4508°397 
4501380 
5624-696 
5624°180 
4496-979 
4494-647 
(10) 
4508:396 
5405-901 
4501-377 
6750299 
4496-977 
4494-660 
(15) 
4215°613 
6322'848 
(6) 
5896-091 
5890: 124 
4703112 
4691-517 
4690-272 
4683-694 
6) 
5791-124 
4824-255 
4823°632 
5788-064 


261 


(4) 
5914314 
5896-086 
5890°121 
5862-514 
5859°753 
5857-613 

(6) 
6564'330 
5270°430 
5269 647 


(8) 
5068°880 
6335-486 
4222-330 
5064779 
4215-613 
5060-194 
6322:842 
6318-168 


(4) 
7027-671 
5624-696 
7035°107 
7039-989 


(6) 
4891523 
5624-696 
5624-184 
4686°345 
4678-971 

(4) 
5896-080 
4691-517 
5862'506 


(Sy: 
7039 975 
4691-532 
5624-696 
5624198 
7027657 
7023686 

ee 
0270-431 
6322°825 
4508°397 
4501°393 


(14) 
6322-825 
5270-431 

(16) 
4222328 
5624-696 
5624-194 
4215-612 


' 1 
a ET 


262 Prof. H. A. Rowland on the Relative Wave-lengths 


TaBLE I1.—Wave-Lengths of Standard Lines. 


Weight. | Wave-length. 


Lime 3169-4 
Ait, 32616 
ieee 3347 9 
Ae 3329°50 
ieee 3406°50 
ee 354024 
Be eeee 3545°31 
ieee 3549-97 
Thelen 3564-64 
Tie 3747-09 
Wises 3754-63 
ees 3897-54 
arses 3916'82 
oe 3924°70 
1p sais 3925°38 
ee 3925°81 
ees 3926-15 
Bhai 394254 
Geen 3950-45 
ee 3953-93 
hee 3984-08 
Pee 398554 
Tee 3987-000 
inate 4005-261 
eye 4035°764 
et 4048-821 
2 ana, 4055-626 
aed 4073°83 
Bau. 4083°748 
Baa. 4107-578 
SHES 4114-580 
OP s 4157-893 
he a 4184-992 
Dire ts 4199-190 
Sain 4215613 
See 4929-393 
oat ee 4254-452 
geibetsr 4267-974 
Bacaes 4293-201 
7 ae 4307-961 * 
Oeit: 4318-782 
Bs 4325-924 
Bid oes 4337-148 
Bes 4343-304 
Bai! 4352: 865 
2(?)...| 4859-715 
eer 4369-887 
Se 4376-089 
Denn: 4391-089 
Be 4407-797 
Delt: 4447-848 
Sits: 4494-667 


* Fraunhofer’s G. 
§ Fraunhofer’s 6,. 
** Fraunhofer’s E. 


Weight. | Wave-length. 


Sd(?).| 4496°984 
Tice 4499-022 
1 eae 4499267 
Sea 4501384 
Sanne ee 4508°400 
ieee 4504-154 
PE ASR 4571-214 
Pa he ‘| 4572-092 
Pays eR 4578663 
ra tree 4588°320 
AR 4590-055 
Ot gabe. 4602°107 
Dee «ica 4611-376 
Pah SBA 4629-445 
yest 4630-218 
DAceine 4643:580 
2 .....| 4668°230 
Spann! 4678:970 
Di hsadaie 4683 688 
Sees 4686°344 
Sipe 4690°262 
Wibeense 4691°520 
2 tees 4703°110 
Digeentine 4703:910 
Dr trccteat 4727 565 
reine 4754159 
Bien sane 4805°186 
Siesenes 4823'630 
S) ange 4824°256 
Stocker 4859°864 
Suiwae 4861-428 t+ 
2 .....| 4890°885 
dL Siseiewies 4900-039 
Vera 4900°237 
Seas 4903°409 
bes ete 4907-869 
2 ereker 4919111 
Bote ser 4920°632 
SD etaae 4924-050 
Seta. 4924-887 
PA pis 4934-18] 
a Saad 4973294 
cana 4 4978°712 
iD otlaate 4980292 
Deiat 4981°836 
os a 4994-251 
Degen Ors 4999-626 
FASB eR 5005°838 
PRP 5006-239 
D se cietiee 5007-370 
eee: 5014-350 
DB tenpase 5020:139 


+ Fraunhofer’s I’, 
|| Fraunhofer’s 6,. 


Weight. | Wave-length. 
2 Seiseae 5036 029 
3 suave 5049:944 
Sve 50607188 
Sia 5064773 
Sc eenee 5068°879 
PRET 5083'460 
Pes 5090-897 
2. sae 5097-071 
Dee 5105663 
BN ar. 5109-760 
eds 5110502 
Di clean 5115-495 
2 seen 5121-730 
cea 5126°309 
aes 5127-468 
4S ee 5133°812 
Ole wrod 5139-472 
2 ental 5141-845 
PAL 5142°986 
3) secu 5146-612 
D  waehe 5150°957 
2 caer 5154157 
ae: 5155°864 
ote 5159-171 
Gi asaae 5162-486 
2h heaps 5165°518 
2G. sates 5167°499 
20 ae 5169-094 § 
eee, 5171-714 
Pe 5172-795 
3 i ee 5173°838 
Ohne 5183-735 J 
Qibiactisee 5188-892 
2 aaa 5193-071 
2 nade 5198819 
Dyke 5202-422 
Desa 5204646 
qteae 5210-492 
Danae 5215-277 
Disa ee 5217-488 
ce at 5225°617 
Py ss 5229-950 
Oe eee 5233-047 
2 isk 5241-599 
Si nse 5250°334 
S cdot 5250-759 
aera: 5253-558 
Disbhsteae 5261°815 
Svc les 5269°649 ** 
Sd ...| 5270-429 ** 
26 ine 5273°379 
2 disaee 5276:138 


{ Fraunhofer’s 6,. 
4] Fraunhofer’s db. 


a) 


of the Lines of the Solar Spectrum. 
Table II. (continued). 


263 


i 
Weight. | Wane-length. Weight. | wave-length | Weight. | Wave-length. 


——<—- 


5281-908 ye eae 


ea 5641595 || 2 ...... 5956-853 
eed 5283°747 cain 5645-751 || 2 ...... 5975°508 
Bs. 5288-64 rime hie 5655645 || 3... 5976-934 
Spe 5296-798 E ipiales 5658-019 De veo 5984-977 
ee 5300:843 roles, 5662679 as 5987-214 
Reece 5307-478 ae 5675593 grins 6003173 
3d...... 5316:803* || 2 ...... 5679184 || 2 ...... 6008-700 
Be) 5324311 Fated 5682-894 eae 6013-682 
Pee. 5333-038 Ea Sot 688370 || 4 ...... 6016-776 
ee 5353°530 pe 5701-708 || 2d...) 6020-278 
oa. 5361-752 to 5709565 idee 6021-948 
re 5362970 pi penta 5709-700 Axa Se 6024-207 
eee 5367600 || 3 ...... 5715-244 || 4... 6042-241 
a. 5370-093 PMs 5731-909 Ean 6056:153 
aren 5371-622 hit peal 5741-994 Gale 6065-635 
ae 5379-704 Bde 5752188 || 6 ...... 6078-635 
eee 5383-497 E ele 5753278 finan 6079:146 
Sic ta 5389-611 Patol 5754-819 Eee 6102864 
ated 5393-298 Evo se 5763-153 re 6103-346 
a 5397-268 ene 5772-299 Seen 6108-262 
Sema 5405-914 E none 5175235 tat 6111-206 
ees 5415°341 ate ee 5782-285 alte 6116°345 
AS. 5424-203 || 2 ...... 5784015 Be) 6122-357 | 
Chee 5434-656 eaeh 5788-075 BaD 6136-760 | 
alee 5447-046 goon 5791-137 Faas. 6141-882 | 
2 ge 5455682 Gli 5798:330 Aids 6162°319 
ee 5462-666 pinbbae 5809:357 Rae 6169-699 
Be 5463-090 yee 5816°504 7 seebes 6173-477 
2 Sie 5463-408 = ee 5853°838 || 2 ...... 6176-943 
ales 5466-521 || 4 ...... 5857°606 | 2 ...... 6180-337 
ela 5477-040 || 6 ...... 5859°741 || 3 ...... 6191-324 
2 ER 5497-660 || 7 .....- 5862511 || 4 ou... 6191-695 
er. 5501-609 ae 5883-971 || 5 ...... 6200°455 | 
i, Ales 5506-920 || 3 ...... 5889°804 || 3 ...... 6213-569 
ee 5513-122 || §...... 58907125 D,|| 4 ...... 6219-420 
2 eee 5528560 || 3 ...... 5893-026 || 4 ...... 6230876 
ae 5534-990 || § ...... 5896-:080D,| 2 .....- 6237-452 
ae 5543-339 || 2 ...... 5898-327 || 3 o..... 6246-460 
a 5044-073. || 2 oo... 5901630 || 3...... 6252-706 
eek? 5555085 || 3 ...... 5905820 || 2.0... 6256:500 
4 iN 5569°772 || Sd...... 5g14S1S |). Bee: 6261-234 
| ane 5576222 || 3 ...... | 5916-409 i es 6265-271 
Baik 5582120 || 2 ...... |-5919°795 || 6 22.2): 6270-370 
ee 3 5588-910 || 3 ...... 5930339 || 5 ...... 6278-225 
2. 5603-019 || 2 ...... 5934-809 || 4...... 6281°315 
Dee 5615-451 |} 2 ...... 5946-130 roe 6289:542 
Pees. 5615809 || 4 ...... 5948-685 y. Rae 6293-077 
Ses 5624-181 || 2 ......| 5951-710 || 2....... 6296-066 
ae 5624-696 t || 2 ...... 5955106 | 2 ...... 6314-798 


| 


* Kirchhoff’s 1474. 

+ Peirce’s standard, given by him 5624825 (Amer. Journ. of Science), 
later corrected by him to 5624:86, and finally corrected by Rowland and 
Bell for error of ruling of grating and of standard to 5624-66. The latter 
can be considered as very near to what the final corrected value of Peirce 
will be, though it may be even so high as 5624-76. 


iN 
Wi 
Mi 


264 


6318-160 
6322-830 
6335-479 
6336°968 
6344297 
6355°184 
6358'°834 
6380-889 
6393°751 
6400-453 
64087163 
6411-793 
6420°103 
6421:498 
6430°993 
6439-224 
6449-951 
6462°762 
6471°805 
€480:198 
6482031 
6493:921 
6495°127 
6499°896 
6516-226 
6518 514 
6532°496 
6534 090 
6546-400 
6552°758 
6562°965* 
6564-338 
6569°360 
6572-245 
6575-090 
6592725 
6593'068 
6594-016 
6609:253 
6633°898 
6643°787 
6663-601 
6678141 
6703°719 
6705:262 
6717:°833 
6722-005 
6726°835 


* Fraunhofer’s C. 
1 First line in what may be ealled the head of Fraunhofer’s B. 

{ Single line between what may be called the head and tail of B. 
§ Edge of what may be called the head of A. 
|| Single line between the head and tail of A. 


.| Wave-length. 


Weight. | Wave-length. 
S ates 6750°325 
A rots ox 6752°876 
Diveieaes 6767°945 
Digests 6772479 
Dosti 6 oe 6787-051 
Dvn scdh 6807-007 
2 6810°432 
A feoatia 6828:770 
Di ieeideies 6841-518 
D iswienis 6855°348 
Oeicersott 6867°382t 
De wahese 6867-717 
Diora 68707123 
Be shine 6875°742 
Digi sa 6876°879 
Di. ao: 6877-797 
Giyac 2: 6879-212 
Os cores 6880:102 

LO rece. 6883-992 
Opes ea 6885925 
Groen 6886°898 
es ares 6896°211 
Qe 6897-103 
7 eat 6901:032 
ieee 6909°597 
cSt 6919-160 
Assets 6923°488 
Aaa ae 6924°340 
AGN a 6929-687 
Bea neues 6947°685 
A ik es 6956-609 
eae 6959-634 
Dee eons 6961-437 
ABR ee 6978-586 
Open 6986:755 
PIs 6989-172 
Sas dees 6999°104 
Pee, 7006-069 
PA 7011481 
Sita 7015:253 
Buc crse 7015639 
Acar 7016-616 
Sie 7023°675 

|) Sige saeee 7027659 
Rise cielo ds 7035-083 
|) ae eae 7088°398 
See 7039-968 
AEM 7090612 


Wave-lengths of the Lines of the Solar Spectrum. 
Table II. (continued). 


Weight. | Wave-length. 


Lee 7122-431 
Lae 7147-893 
Tee. 7148-276 
ioe 7168134 
oe 7176-279 
tee 7184-401 
4 7184°705 
a 7186-470 
| 7194-805 
ie 7199:689 
ie 7200673 
Se 7216693 
bien. 7219-282 
ae 7223'830 
ea. 7227-686 
io 7232419 
heli 7233092 
Sees. 7234868 
Se 7240°879 
ona 7243-800 
eae. 7247590 
Os. 7264770 
io 7265°750 
Pies 7273138 
Lae 7287590 
Thee. 7289844 
Pie 7290°621 
he 7299-993 
Ds. 7304-382 
2 3 7318678 
1 ee 7331-101 
Tine: 7335°532 
1s 7442-574 
pe 7445-941 
oie: 7495-248 
oe 7511-188 
tee 7545817 
Ste 7593-9758 
Sp 7621-183] 
2 ne 7623-425 
Dents: 7624-737 
fae 7627-259 
Pee 7628°605 
Taal 7659:550 
eee 7660°679 
ie 7665°683 
pie. 7671-412 


On the Absolute Wave-length of Light. 265 


This table is to be used in connection with my photographic 
map of the normal spectrum, to determine the error of the 
latter at any point. The map was made by placing the pho- 
tograph in contact with the scale, which was the same for 
each order of spectrum, and enlarging the two together. In 
this way the map has no local irregularities, although the 
scale may be displaced slightly from its true position and may 
be a little too long or short, although, so far as I have tested 
it, it seems to have very little error of the latter sort. The 
scale was meant in all cases, except the ultra-violet, to apply 
to Peirce’s absolute value, and so the correction is generally 
negative, as follows :— 


Approximate Correction to the Photographic Map of the Normal 
Spectrum to reduce to latest absolute value. 


Correction. 
Strip 3200 to DOW \gesaes —'05 
Je rO2t D tO DO BOY cabiaks —'05 
» 475 to 8730 ...... — 02 
spb ODM AOLOOOUE sis sles —'10 
OOCd CO ALO 5% vce —'16 
» 4075 to 4330 ...... — ‘04 
5, 4275 to 4530 ...... —'08 
59 1 4480 104739» caver —°‘10 
5 AGSD4O/A9AO rs Saces —'18 
ip O Os kG: DLO i memes —'14 
» 9075 to 5330 ...... —'15 


» 9215 to 5595 ... about —°05 
» 0415 to 5790 ... about —:04 
ok LO ora LO, «shir —°20 
gy -8810\t0. 4000  osasissc —14 


It is to be noted that the third spectrum of the map runs 
into the second, so that it must not be used beyond wave- 
length 3200, as it is mixed with the second in that region. 


XXX. On the Absolute Wave-length of Light. By Louis 
Be, Fellow in Physics in Johns Hopkins University.* 


U P to the present time, Angstrém’s map of the solar 
spectrum, and with it his determination of absolute 
wave-length, has remained the final standard of reference in 


all spectroscopic matters. But since Angstroém’s work was 

published, optical science, particularly that part of it which 

deals with the manufacture and use of diffraction-gratings, 
* Communicated by the Author. 


266 Mr. L. Bell on the Absolute 


has made enormous progress. It is now possible with the 
concave grating to measure relative wave-lengths with an 
accuracy far greater than can be claimed for any one of the 
absolute determinations. The numbers given by Angstrom 
are now known to be too small by as much as one part in 
seven or eight thousand, as has been shown by ,Thalén, in 
his monograph Sur le Spectre du Fer; and since Angstrém’s 
work but one careful determination has been made. This is 
by Mr. C. 8. Peirce, and was undertaken some eight years 
since for the U.S. Coast and Geodetic Survey. No full 
report of this work has as yet been published, though it is 
evidently very careful, and has already consumed several 
years. Certain results were communicated to Prof. Rowland 
of this University, to serve as a standard of reference for his 
great map of the solar spectrum now nearly completed ; and 
it was to serve as a check on these results and to furnish a 
value of the absolute wave-length as nearly as possible com- 
mensurate in accuracy with the micrometrical observations, 
that the experiments detailed in the present paper were under- 
taken. Only the work with glass gratings has been as yet 
completed ; but since the relative wave-lengths, which are 
intrinsically of far greater importance, are now ready for 
publication, and have been reduced by the value herein given, 
the result is published, leaving for further work with speculum 
metal gratings its final confirmation or correction. _ 

This portion of the determination is delayed awaiting better 
facilities for carrying it out, but the writer intends under- 
taking it at the earliest possible moment, and hence leaves for 
a future paper the complete discussion of the problem. 

The writer desires here to express his deep obligations to 
Prof. Rowland, under whose guidance the work has been 
carried on, and to whom a very important correction is due ; 
and to Profs. W. A. Rogers and C. 8. Peirce for information 
given and courtesies extended. 


Haperimental. 


The determination of absolute wave-length involves two 
quite distinct problems—first, the exact measurement of the 
angle of deviation of the ray investigated, and second, the 
measurement of the absolute length of the gratings used. 
Hach portion of the work involves its own set of corrections, 
frequently quite complicated and difficult, but it is the latter 

art that is peculiarly liable to errors, which will be treated 
in detail further on. As to the former part, several import- 
ant questions arise at the very outset. Tirst is the choice 
between transmission- and reflection-gratings. The principal 


Wave-length of Light. 267 


work heretofore has been done with the former ; but metallic 
gratings possess certain advantages, notably from the ease 
with which their temperature can be accurately measured, and 
the fact that they can easily be made of a size much larger 
than glass gratings, and consequently a small inaccuracy in 
measuring them involves much less error in the result. 

On the other hand, the coefficient of expansion of speculum 
metal is more than twice as great as that of glass, and being a 
good conductor it is far more sensitive to small changes of 
temperature. And this property increases the liability to 
irregularities in the ruling, particularly in large gratings 
which require several days for completion. In ruling on 
glass change of temperature is less serious, but this advantage 
is more than offset by the faults caused by the wearing away 
of the diamond point, which breaks down so rapidly that it 
is enormously difficult to produce a glass grating free from 
flaws and at all comparable in optical excellence with those 
upon speculum metal. The determination of absolute wave- 
length should rest on measurements made with both classes ; 
and with sufficiently exact instruments and very careful ex- 
perimentation, the better results can probably be obtained 
from the metallic gratings. For the reasons previously stated, 
this paper is confined to the results from glass ones. 

Now there are two quite distinct ways of using transmission 
gratings—first, perpendicular, or nearly so, to the collimating 
or the observing telescope; and second, in the position of 
minimum deviation. The method in the rat case is familiar; 
the properties of the second are as follows :— 

The general relation between the incident and the diffracted 
ray is 

aay anys mar 
sin 7+sin Sans eerie © 


When i=0°, this gives the ordinary formula for normal in- 
cidence. Putting it in the form 

2(a mY 

n= 2) sin : cos (-5), 
the deviation represented by the angular term will evidently 
be a minimum when i= : ; and the wave-length will then be 
given by the formula 

sin 9° 


It is not easy to say which method of procedure is prefer- 


Se 


268 Mr. L. Bell on the Absolute 


able; but on the whole the ordinary plan of normal incidence 
offers fewer experimental difficulties, and therefore was 
adopted particularly as the spectrometer used was specially 
well suited to that method. It is quite certain that either 
method will, with proper care, give the angular deviation with 
a degree of exactness far surpassing that attainable in the 
measurement of the gratings. 3 


The Spectrometer. 


This was a large and solid instrument by Meyerstein, with 
a circle on silver 32 centim. in diameter divided to tenths of 
a degree. ‘This is read by two micrometer-microscopes 180° 
apart. The pitch of the micrometer-screws is such that one 
turn equals about 2’; and as the head is divided into sixty 
parts, each of these represents 2”. The micrometer can, 
however, be set with certainty to less than half this amount. 
The collimating and observing telescopes are of 4 centim. 


clear aperture and 35 centim. focal length, and the lenses are 


well corrected. ‘The collimator is fixed to the massive arms 
which carry the reading microscopes ; while the observing 
telescope is attached to a collar on the axis of the main circle, 
and moves freely upon it or can be firmly clamped so as to 
move with the circle. The grating is carried on an adjustable 
platform with a circle 12°5 centim. in diameter, divided to 
30’, by verniers to 1’, and moving either upon or with the 
large circle. 

This arrangement of parts does not admit of fixing the 
grating rigidly normal to the collimator ; so in all the experi- 
ments it was placed normal to the observing telescope, a 
position which was particularly advantageous in the matter 
of adjustment. The instrument was set up ina southern 
room in the physical laboratory, and throughout the experi- 
ments the collimator pointed about south-south-east. With 
the eye-piece used, the observing telescope had a power of 
very nearly sixteen diameters. 


Gratings. 


Very few glass gratings have ever been ruled on Prof. 
Rowland’s engine, since for most purposes they are much 
inferior to the metallic ones, and are very much more diffi- 
cult to rule, as they run great risk of being spoiled by the 
breaking down of the diamond-point. A very few, however, 
were ruled in 1884 with special reference to wave-length 
determination ; and of these the two best were available for 


Wave-length of Light. 269 


these experiments. ‘They are both ruled upon plane sextant 
mirrors, and are of very nearly the same size—thirty milli- 
metres long, with lines of about nineteen millimetres. Hach 
hundredth line is longer, and each fiftieth line shorter, than 
the rest ; so that the gratings are very easy to examine in 
detail. The ruling of bothis smooth and firm, without breaks 
or accidental irregularities, and almost without flaws. They 
were ruled at different temperatures and on different parts of 
the screw; and while one was ruled with the ordinary arrange- 
ment of the engine, the other was ruled to a very different 
space by means of a tangent-screw. This great diversity of 
conditions in the two gratings is far from favouring a close 
agreement in the results; but tends to eliminate constant 
errors due to the dividing-engine, and hence to increase the 
value of the average result. It must be remembered that 
two gratings ruled on the same part of the screw are in most 
respects little better than one. ‘The grating designated I. in 
this paper contains 12,100 spaces, at the rate of very nearly 
400 to the millimetre, and was ruled (by tangent-screw) at 
a temperature of 6°°7 C. inJanuary 1884. It gives excellent 
definition with almost exactly the same focus for the spectra 
on either side, and is quite free from ghosts or other similar 
defects. 

The grating designated IJ. has 8600 spaces, at the rate of 
about 7200 to the inch, and was ruled in November 1884, 
at 11°°6 C. Its definition and focusing are very nearly as 
good as in I., and, like it, it shows no trace of ghosts or false 
lines. They are both exquisite specimens of the work which 
Prot. Rowland’s engine is capable of doing, though, as the 
event showed, I. is decidedly the better grating, in the 
matter of regularity of ruling. 


Angular Measurements. 


At the beginning of the work a serious question of adjust- 
ment arose. ‘There are two ways of using a grating perpen- 
dicular to one of the telescopes. In the first place it may be 
placed and kept accurately in that position ; and, secondly, it 
may be placed nearly in the position for normal incidence, 


and the error measured and corrected for. Angstrém used. 
the latter method, which involved a measurement on the 
direct image of the slit as well as on the lines observed. 
Using Angstrém’s notation—let « and a’ be the readings on 
the spectra, and M that on the slit. Let also 
ata’ 
2 


—M=A and =¢. 


DO) Mr. L. Bell on the Absolute 


Then if y is the angle made by the incident ray with the 
normal to the grating, and N the order of spectrum, 


a =cos (y+A) sin ¢ ; 


also 
sin y=sin (y+A) cos ¢, 
one tan ee Teesoe 
VR eas ee. 
But from the second of the above equations, 
é sin y 
sin (y+A)= cand: 


Now it was found that with the collimating-eyepiece belong- 
ing to the spectrometer, y would never exceed and seldom 
reach 10’, while the angles of deviation observed were about 
45°. Substituting these values in the last equation, it at 
once appeared that the cosine of (y+A) was a quantity dif- 
fering from unity by considerably less than one part in a 
million, and hence entirely negligible. Further, it was found 
that the grating once set could be trusted to remain perpen- 
dicular through a series of measurements ; and though at the 
end of each series the grating was adjusted to a new part 
of the circle, and a close watch kept for its slipping out of 
adjustment, it was never found necessary to reject a series 
from that cause. | 

The grating was centred and adjusted with reference to 
the circles and their axes by the ordinary methods. Through- 
out the experiments the light was concentrated on the slit by | 
an achromatic lens of about half a metre focus, which was 
placed behind a sheet of deep yellow glass, which served to 
cut off the overlapping blue rays, which might otherwise have 
proved troublesome. A heliostat enabled the sun’s image to 
be kept centrally upon the slit. 

The method of observation was as follows :—When instru- 
ment and grating were in exact adjustment, readings were 
taken on D, in the spectra on either side of the slit, and the 
angle measured from three to six times in rapid succession, 
the last reading being of course on the same side as the first. 

Then the grating was rotated about ten degrees, readjusted, 
and the process repeated. 

The angles observed in one series were combined to elimi- 
nate errors of setting, while the use of all portions of the 
circle served to correct errors of subdivision, since the num- 
ber of independent series of observations was quite large. 


s 


Wave-length of Light. 271 


To eliminate any errors which might be due to imperfec- 
tions of figure in the gratings, they were used in all the four 
possible positions. Nosuch error, however, became apparent 
either from critical examination of the gratings themselves, or 
from the results obtained in the different positions. 

Observations with grating I. were begun late in October, 
1885, and occupied the clear days fora month. Forty-eight 
series of measurements were made, and the agreement be- 
tween them was very satisfactory. After correcting for tem- 
perature, thirty-six of the number fell within a range of three 
seconds, and the rest were clustered closely about them. 
Observations on the various days were as follows:— 


Date. Re a Angle, 
Oct. 19 1 AH? WO ATED 
20 1 Aa AAS. 4 
22 2 Ad t, AS. 92 
23 1 Ad, LAD: °S 
26 4 452 + 4993 
Dib 3 45 48:2 
ak 1 4h $50.71. 
Nov. 3 1 45 1 48 °6 
4 3 Ay et Ag A 
5 2 Ad LACS 
10 4 Ly ed OR 
11 6 ADL AG ef 
16 8 45 1-48) 2 
Le 5 45) 1 Ah 
20 6 Ad. 1p Ae oD 


All the above were in the third spectrum, to which mea- 
surements were in the main confined, as in it the definition 
was particularly good ; and it being the highest order which 
could be conveniently observed, an error in the angle would 
produce the minimum effect. The spectra on both sides of the 
slit were about equal in brilliancy and definition. 

The observations were weighted as nearly as possible accord- 
ing to the favourable or unfavourable conditions under which 
they were made ; and when finally combined, gave as the value 
of the angle of deviation for grating I.:— 


p= 45° 1! 482440411, 


The above probable error is equivalent to a little less than 
one part in a million, and can introduce no sensible error into 
the resulting wave-length. 

Other work intervened, and the measurements with grating 


272 Mr. L. Bell on the Absolute 


IT. were not taken up until early in the succeeding March. 
Precisely the same method of observation was employed, and 
the results were nearly as consistent and satisfactory. 

The observations on the various days were as follows :— 


Date, 1886. ae Angle. 

March 6 2 AD by! eile 
10 il 42 4 58 °6 
11 7 42 Oe 
15 1 42s 5- AsO 
16 6 AQ 4°08 
17 6 42 4 58°5 
18 7 42, 4 ge 
23 6 42 4 58:3 


When collected thus by days, the observations do not 
appear to agree nearly as well as those made with grating L., 
particularly since a solitary wild reading, that of March 15, 
is retained. The distribution of the various readings, how- 
ever, is such that, after weighing and combining, the final 
result is by no means deficient in accuracy. It is 


b=42° 4! 59!-28 + 0"-2. 


The above probable error amounts to about one part in six 
hundred thousand. The observations with grating LI. were 
uniformly in the fourth order of spectrum. 

Throughout the measurements with both gratings, the tem- 
perature was kept uniform within a few degrees. 20°C. had 
been selected as the standard temperature, and the variation 
was rarely more than two or three degrees on either side of 
that figure. The question of temperature determination is a 
serious one in case of glass gratings; for it is very hard to 
tell what heating effect the incident beam has on the grating, 
and equally hard to measure that effect. It is hardly safe, 
without extraordinary precautions, to assume that the grating 
has the same temperature as the air near, and it is sucha 
bad conductor that it would not easily assume the tempera- 
ture of the apparatus. In these experiments a sort of com- 
promise was effected. A small thermometer was attached to 
the thin metallic slip that held the edge of the grating, and 
shielded by cotton from air-currents, which of course would 
affect it much more than they would the grating. The 
thermometer was a small Fahrenheit graduated to quarter 
degrees, and quite sensitive. It was carefully compared, 
throughout the range of temperatures employed, with thermo- 
meter Baudin 7312, which served as a standard in all the 


Wave-length of Light. 273 


measurements regular and linear, and during part of the 
time was placed directly over the grating to give a check on 
the attached thermometer. This expedient was finally aban- 
doned as unlikely to be of much use. 

The corrections for temperature were deduced from the 
assumed coefficient of expansion of glass, which was taken as 
0:0000085. This was reduced to angular correction for the 
approximate value of @, and applied directly to the observed 
angles. Since the temperatures at which observations were 
made varied little from 20° C., and were quite equally distri- 
buted on both sides of that figure, any error in the assumed 
coefficient would hardly affect the average result, but would 
appear, if at all, as a slight increase in the probable error. 

760 millim. (reduced) was taken as standard pressure, and 
the values for the days of observation were taken from the 
U.S. Signal-Service observations for the hours of 11 a.m. 
and 3 P.M.on those days. The average for the measurements 
made with grating I. was 761 millim., and for those with 
grating II. 760 millim.; so that corrections for pressure were 
uncalled for. 

The effect of the velocity of the apparatus through space is 
a subject concerning which there has been much discussion. 


Angstrém deduced a correction; but Van der Willigen, in 
quite a lengthy discussion of the whole matter, came to the 
conclusion that there was no error due to the above cause. 
Since that period the question has been raised from time to 
time, but no decisive investigations on the subject have yet 
been published. At present, however, it seems to be tolerably 
well settled that no correction is needed, as the error, if there 
be any, is of an order of magnitude entirely negligible, and 
in the present paper none has been applied. 

The angular measurements, after all corrections were ap- 
plied, may thus be regarded as determined with a high degree 
of accuracy—most probably to less than one part in half a 
million. 

Measurement of the Gratings. 


The exact determination of the grating-space is by far the 
most difficult portion of a research on absolute wave-length, 
and has been uniformly the most fruitful source of errors. 
Besides the experimental difficulties of the task, it is far from 
an easy matter to secure proper standards of length. The 
standards used in various former investigations have proved 
to be in error, sometimes by a very considerable amount; and 
indeed very few of the older standards are above suspicion. 
As Peirce has very justly remarked in connection with this 


Phil. Mag. 8. 5. Vol. 23. No. 142. March 1887. U 


274 Mr. L. Bell on the Absolute 


subject—“ All exact measures of length made now must wait 
for their final correction until the establishment of the new 
metric prototype.’’ Short standards of length are in some 
respects peculiarly liable to error, since they must be com- 
pared with the subdivisions (often not sufficiently well deter- 
mined) of secondary standards; and small sources of uncer- 
tainty, such as poor defining-lines, slight changes in the 
apparatus and the like, of course are much more serious as 
the length is less. 

Fortunately, there were available for the measurement of 
the gratings two standard double decimetres, which have been 
determined with almost unprecedented care by Professor W. 
A. Rogers. They are upon speculum metal; were graduated 
and determined by Professor Rogers early in 1885, and were 
purchased by the University late in the same year. They are 


_ designated respectively S¢ and 8é%, and are discussed at length 


in the ‘ Proceedings’ of the American Society of Muicrosco- 
pists for 1885. 

The bar 8, is 23 centim. in length. Near one edge is the 
double decimetre 8% divided to centimetres, the 5-centim. 
lines being triple. SS, is 27 centim. in length, and graduated 
in the same way. ‘The defining-lines in both are fine and 
sharp, and the surfaces are accurately plane. They are stan- 
dard at 16°°67 C.; and from an elaborate series of compari- 
sons with four different standards, the coefficient of expansion 
was found to be 


17:946 mw per metre per degree C. 


S?% and S$ depend for their accuracy on a long series of inde- 
pendent comparisons with Professor Rogers’ bronze yard and 
metre R,, and steel standards whose relation to R, was very 
exactly known. R, has been determined by elaborate com- 
parisons with various standard metres and yards, and is 
described and discussed at length in the ‘ Proceedings’ of 
the American Academy, vol. xviii. The length of the metre 
was determined, both directly and through the yard, by com- 
parison with the following standards :— 

I. The metre designated T, copper with platinum plugs, 
traced and standardized by Tresca in 1880 from the Conser- 
vatoire line-metre No. 19, which bears a very exactly known 
relation to the Métredes Archives. 

II. The yard and metre designated C.S., brass with silver 
plugs, belonging to the Stevens Institute. The yard was 
compared with the Imperial Yard in 1880, so that it is directly 
and exactly known. It was afterwards sent to Breteuil; and 


Wawe-length of Light. 275 


the metre was determined with great exactness by elaborate 
comparisons with Type I. of the International Bureau of 
Weights and Measures. 

Ill. “Bronze 11,” a primary copy of the Imperial Yard, 
presented to the United States in 1856. It was taken to 
England in 1878; and finally determined by direct compari- 
son with the Imperial Yard, Bronze Yard No. 6, and Cast 
Iron Yards B No. 62 and C No. 63. 

The subdivisions of R, have been determined with very 
great care; and thus Sj and $3, whose lengths relative to R, 
are accurately known, may finally be referred to the ultimate 
standard Type I. of the International Bureau. 

Only the 5-centim. spaces of Sj and S83 were investigated 
by Prof. Rogers; but these were determined by various methods 
under widely different conditions, and their relations to the 
standards with which they were compared may be regarded 
as definitely known. From a combination of all results the 
subdivisions of Sj have the following lengths at the standard 
temperature :-— 

millim, 

Standard Si=199-99918 
Gitte, r= 99 9995 
GM Sp==, Goo 2S 
5-em., Si= 50:00010 
5-cm., Sj= 49°99985 
5-cm.3 Si= 49°99901 
5-cm., Si= 50°00022 


Similarly the following values were derived for 82 :— 
millim. 

Standard 83=199-99968 

dm.,; 83=100-00001 

dm., S3= 99:99967 

5-em., 83= 50:00020 

5-cm.,g S3= 49°99981 

5-om.3 So= 49°99931 

5-em., S2= 50:00042 
As to the degree of accuracy attained in determining Si 
and $3, Prof. Rogers says that, including all sources of un- 
certainty, either standard may have an error of +0:3y; but 
the mean of the two, since the determinations were inde- 


pendent, ought to be even more reliable. Taking all things 
into consideration, it seems very improbable that the mean 
U2 


276 Mr. L. Bell on the Absolute 


value of Sj and 83 can be in error by as much as one part in 
half a million. 

So much for the standards of length. The comparator used 
in the measurements was a very efficient instrument, particu- 
larly suited for the purpose. It consisted essentially ‘ofa long 
carriage (4 metre) running on V-shaped ways and carrying 
the microscope. This carriage slides against adjustable stops, 
and is pressed against them with perfect uniformity by means 
of weights. An adjustable platform below carries the stan- 
dards and objects to be measured. The ways of both carriage 
and platform had been ground till they were perfectly uniform 
and true, and the working of the instrument left little to be 
desired in the way of accuracy. Throughout a long series of 
measurements the stops would not be displaced by so much 
as O'lw if proper care were used in moving the carriage. 
The microscope was attached so firmly as to avoid all shaking, 
and was armed with a half-inch objective and an excellent 
eyepiece-micrometer. The objective was made specially for 
micrometric work, and was fitted with a Tolles’ opaque illu- 
minator. Measurements were made as follows :—The standard 
bar, and the grating mounted on a polished block of speculum 
metal, were placed side by side—or sometimes end to end— 
on the platform and very accurately levelled. The stops were 
set very nearly three centimetres apart, one end of the grating 
brought under the microscope resting against one of the steps, 
and the micrometer set on the terminal line. Then the 
carriage was brought against the other stop, and the micro- 
meter again set. The same process was then gone through 
on three centimetres of the standard, and then going back 
to the grating it was compared in the same manner with 
succeeding triple centimetres till the fifteen-centimetre line 
was reached, thus eliminating the errors of the single centi- 
metres and making the determination rest only on the fifteen- 
centimetre line. The temperature was given by a thermo- 
meter placed against the standard bar or the block that carried 
the grating. In this manner each grating was repeatedly 
compared with the first. fifteen centimetres of each bar, at or 
near 20°, the temperature at which the gratings had been 
used. The micrometer constant was determined by measuring 
tenths of millimetres ruled on Prof. Rowland’s engine; but in 
practice the stops were so adjusted that it was almost elimi- 
nated. Hach division of the micrometer-head equalled 
0°28, and the probable error of setting was less than half 
that amount. 

All measurements were reduced to 20° C., as in the case of 
the angular determinations. The line along which the linear 
measures were made was that formed by the terminations of 


Wave-length of Light. 277 


the rulings. It was therefore necessary to know very exactly 
the angle between this line and the direction of the individual 
rulings ; in other words, the angle between the line of motion 
of the grating and the direction of the diamond stroke in the 
dividing-engine. This was ascertained by means of two test 
plates each some twelve centim. long ruled in centims., and 
then superimposed line for line. By measuring the minute 
distances between each end of a pair of superimposed lines, 
the length of the lines and the amount by which their ends 
overlapped at each end of the test plate, the required angle 
could be deduced with great exactness. It differed so little 
from 90°, however, that the correction produced, barely one 
part in a million, was entirely negligible. 

After all reductions and corrections, the following series 


of values were obtained for the grating-spaces of gratings I. 
and II. :-— 


Series, Crating I. Standard. 
millim. 
f. 0:00250023 83 
2. 0:00250016 i 
a 000250013 i 
4. 0:00250015 ik 
5. 0:00250018 ‘ 
6. 0°00250021 Si 
7. 0-00250023 : 
8. 0-00250023 ‘ 
9, 0:00250023 if 


Mean value adopted after weighting and combining the 
above observations was 


0:002500194 millim. + 10. 


The probable error thus appears to be not far from one part 
in two hundred and fifty thousand. The difference in the 
results obtained from the two standards seems to be purely 
accidental, as appears from the measurements on grating II. 
Series. Grating II. Standard. 
millim. 
0:00351888 Si 
0:003851883 
0:00851885 
000351886 
0:00351883 sf 
0:00351893 2 
0:00351888 
0:00351888 
: 0:00351888 é, 
Mean adopted, 0:003518870+10, 


rer eee 


278 Mr. L. Bell on the Absolute 


The probable error appears to be rather less than in the 
measurements of grating I. As, however, the angular deter- 
minations made with I. are the better, so far as probable 
errors of observation are concerned, the results from the two 
gratings are about equal in value. 

Computing now the wave-lengths corresponding to the given 
values of dé and A for each grating, we have finally for the 
wave-length of D, at 20° C. and 760 millim. pressure :— 


From grating I. uncorrected, 5896-11 tenth metres. 
From grating II. “ 0895°95 - 


The difference in the above results is by no means large 
compared with the results obtained from different gratings by 
other investigators, but it certainly is enormously great com- 
pared with the experimental errors alone. 

As nearly as can be judged, these ought not in either 
grating to exceed one part in two hundred thousand, while 
the above discrepancy is about one part in thirty-five thousand. 

Its cause must be sought in the individual peculiarities of 
the gratings, rather than in the method of using them. 

All gratings are subject to irregularities of ruling, and the 
effects of these are various, according to the nature and 
magnitude of the defects. Linear or periodic errors in 
ruling, unless very small, will make themselves apparent by 
changing the focus of the spectra or producing ghosts, re- 
spectively ; and if such errors are large, render the grating 
totally unfit for exact measurement. Accidental errors, such 
as a flaw or break in the ruling, are also serious, but are 
easily detected and may be approximately corrected, as was 


done by Angstrém in the case of one of his gratings. Any 
marked and extensive irregularities of spacing will produce 
bad definition or false lines, and in most cases both. If, then, 
a grating on microscopical examination is free from flaws 
and on the spectrometer gives sharply defined spectra, alike 
in focus and free from ghosts, it is safe to conclude that it is 
tolerably free from the errors above mentioned ; but, unfor- 
tunately, there is one fault that does not at once become 
visible, while it introduces a very serious error in the measure- 
ments: this isa rather sudden change in the grating-space 
through a portion of the grating, usually at one end. Such 
an error is usually due to abnormal running of the screw 
when the dividing-engine is first started, and may in this 
case be avoided by letting the engine run for some time 
before beginning to rule. Thus grating L., ruled with this 
precaution, is nearly free from this error. Sometimes, how- 
ever, it is the terminal or an intermediate portion of the 


Wawe-length of Light. 279 


grating that is thus affected, in which case the error may be 
due to a change of temperature or to a fault in the screw. If 
an error of this kind is extensive, it will produce the effect of 
two contiguous gratings of different grating-space, injuring 
the definition and widening or reduplicating the lines. When, 
however, the abnormal spacing is confined to a few hundred 
lines, it produces no visible effect when the whole grating is 
used, but simply diffuses a small portion of the light, and 
increases or decreases the average grating-space. For it is 
evident that such a portion of the grating must possess little 
brilliancy and less resolving power ; and the more its spacing 
differs from that of the rest of the grating, the less chance of 
visible effect and the greater error introduced. Such a fault 
is compatible with the sharpest definition, but can be detected 
by cutting down the aperture of the grating till the spectrum 
from the abnormal portion is relatively bright and distinct 
enough to be seen. ‘The effective grating-space, producing 
the spectra on which measurements are made, is, of course, 
that of the normal portion only. Both the gratings used in 
these experiments were affected by the above error, No. I. 
very slightly, No. II. somewhat more seriously. Not only 
the discrepancies between different gratings, but those between 
different orders of spectra in the same grating are due to this 
cause. For while in one order, where the effect due to the 
abnormal portion is imperceptible, the spectrum as measured 
is produced by the effective grating-space alone, in another 
order there may be produced a slight shading-off of the lines, 
so that their apparent centres may correspond approximately 
to the average grating-space. In any case, it is quite clear 
that a combination of the results from different orders of 
spectra will not eliminate the error. 

The remedy lies either in stopping out the imperfect por- 
tion of the grating, or measuring it and introducing a cor- 
rection. As the work of angular measurements was nearly 
finished before the study of the gratings was begun, owing 
to a delay in getting apparatus, the latter course was adopted 
in these experiments. Hach grating was examined in detail, 
and the relation of the grating-spaces in the various portions 
of it carefully determined. From these data a simple gra- 
phical method gave the correction to be applied to the wave- 
length. In each grating the fault was confined to a small 
portion ; and as the order of the spectrum employed in each 
was selected on account of its good definition and freedom 
from anything like haziness or shading-off of the lines, it seems 
safe to assume that the abnormal portion produced no visible 
effect, and that, consequently, the correction above mentioned 


280 Mr. L. Bell on the Absolute 


counteracts the error quite effectually. In grating I. the 
correction was one part in 800,000, and in grating II. one 
part in 60,000. Applying these to the wave-lengths we have 
er grating I. :— 


Wave-length 2)... ww BON 
Correction a ae eee ee, ie —'02 


Corrected w.-l. . . . . 5896-09 
And for grating II., 


Wave-lenoth,  ...45 606 .  «) soSobeD 
Cormectionatiyuscse ei keh sake +°10 


Corrected ayzels Ge iielic -5896:05 


Combining these, and giving to seatine I. the greater 
weight on account of its very small error of ruling, we have 
finally for the wave-length of D, at 20° C. and 760 millim. 
pressure, 5896:08 
or in vacuo, 9897-71. 

It is no easy matter to give any well-founded estimate of the 
probable error of the above result. So far as experimental 
errors are concerned, the result with either grating should be 
correct to one part in two hundred and fifty thousand; but 
the error in the gratings introduces a complication by no 
means easy to estimate. As nearly as the writer can judge, 
however, it seems probable that the error of the final result 
does not exceed one part in two hundred thousand. For 
comparison, the values deduced from the work of Peirce and 


of Angstrém are subjoined :— 
Micrometer measure by Rowland, from Peirce’s 
preliminary result; 054) 720 ae ee 5896°22 
Thalén’s correction of AngsirOm. . . | = a ueeenaee 


Both being in air at ordinary temperature and 760 millim. 

As neither result was corrected for errors in the gratings, 
the cause of the discrepancy is obvious. 

Two determinations of absolute wave-length have been 
published since this work was undertaken by the writer. 
One is a very elaborate one by Miiller and Kempf, who 
employed four gratings by Wanschaff, and used the method 
of minimum deviation. Their results were as follows :— 


Grating. /.,.1).°) .. 7 (2k (5001) (8001) (80014) 
Wave-length . 589646 5896714 5895°97 5896-33 

By a correction founded on the unwarrantable assumption 
that the mean value was correct, the above results are br ought 
into apparent agreement. Nothing, however, short of a study 


/ 


Wave-length of Light. 281 


in detail of each grating can furnish data for obtaining any- 
thing like an accurate result from the above figures. It 
would seem that (5001), which had the smallest probable 
error, should show but a trifling error of ruling, while one 
would expect to find a portion or portions of (2151), in which 
the grating-space is abnormally large. Corresponding errors 
of ruling should appear in (8001) and (80014). A similar 
study of the gratings used by Angstrém would be of no little 
interest. 

The other determination alluded to is one by M. de 
Lépinay, using a quartz plate and Talbot’s bands. Without 
discussing the method, it is sufficient to say that the result 
obtained depends on the relation of the litre to the decimetre, 
a ratio not at present exactly determined. 

The results detailed in this paper are in a certain sense 
preliminary. The writer hopes that in the near future, ex- 
periments with metallic gratings will enable him to lessen the 
probable error very materially, and therefore defers, for the 
present, further discussion of the problem. 

Through the courtesy of Mr. Peirce, the writer has been 
enabled to test the legitimacy of the above correction and, at 
the same time, check his own results. Mr. Peirce kindly 
forwarded his gratings and standard of length for examination 
and comparison, and the results were decidedly instructive. 

Grating “ H,” with which a large part of the work was 
done, showed, as was suspected, a local error, equivalent to a 
correction of one part in 55,000 in the resulting wave-length. 
Tested in the spectrometer, the portion including the error 
showed a grating-space distinctly greater than that of the 
erating taken as a whole, showing thus both the necessity for 
and the algebraic sign of the correction. The other gratings 
showed similar errors varying in amount, but the same in 
sign, the correction requiring in every case a reduction in 
the wave-length. The abnormal portion was invariably at 
one end or the other of the grating concerned, never in the 
middle. 

The standard of length used by Mr. Peirce (No. 3” a 
glass decimetre) was compared with 8% and 8%; and the pre- 
liminary results show that the length assigned to it was too 
great by very nearly 2u, 1 part in 50,000. Now the wave- 
length of D,, as deduced from grating H, was 


589626 
Mess:error of rulme ...  —10 
Messemor of’ No.3”: °° "12 
Corrected value. . . . 5896-04 in air at 30 in. pressure 


and 70° F.; 


=> et SS a a eS eS a 


982 Prof. W. C. Unwin on Measuring-Instruments 


which shows a tolerably close correspondence with the results 
obtained by the writer. A more complete discussion of 
Peirce’s results is reserved until the relation between “ No. 3”’ 
and S? and 8% shall be more exactly known. The latter 
standards would appear to be the more trustworthy, since 
they are based on various independent determinations ; while 
“No. 3” is based on an indirect comparison with metre 
“No. 49,” a standard concerning the exact length of which 
there seems to be some little doubt. 


XXXII. Measuring-lnstruments used in Mechanical Testing. 
By Prof. W. C. Unwin, #.4.S.* 


HE determination of the exact distance between two fine 
marks on a standard of length is an operation of some 
difficulty, as is well known to physicists. But that operation 
is free from many of the difficulties which attend the measure- 
ments which have to be made in the engineering laboratory. 
Among these the determination of the modulus of elasticity 
(Young’s modulus) of a bar by measuring its change of length 
by stress is one of the most important. Now the bars sub- 
jected to test are usuaily, in the part which can be measured, 
not more than 10 inches in length ; and the whole elastic ex- 
tension of such a bar is generally only about 0-007 inch. It 
is obvious, therefore, that measurements must be made with 
considerable accuracy and refinement to be of any value. 
But the bar cannot be placed in a position convenient for 
measurement; and the attachments to the testing-machine are 
more or less in the way of the measuring-apparatus to be 
applied. The bar itself is a somewhat rough bar, the form of 
which must not be interfered with to facilitate the measure- 
ments. Then also bars of very different forms have to be 
tested, flat and round, of various widths and diameters; and 
the measuring-apparatus must be applicable to all these with 
equal readiness. Last, but not least, the work of an engi- 
neering laboratory is pressing, and measurements must be 
carried out with rapidity. 

In some cases, two diamond scratches have been made on 
the bar, and the distance between these measured by two 
micrometer-microscopes. Apparatus of this kind is awkward 
to apply on the testing-machine, and tedious to adjust and 
read. 

A cathetometer has been used. But then two adjustments 


* Communicated by the Physical Society: read January 22, 1887. 


used in Mechanical Testing. 283 


have to be made, and two readings taken for each elongation. 
Also the limit of accuracy of the cathetometer is hardly suf- 
ficient for the purpose. 

Very often mechanical magnification by a lever is adopted. 
But there are some difficulties in satisfactorily attaching a 
lever-apparatus to the bar : ifa leverage of 100 to 1 is adopted, 
the fulcrum distance becomes very short, and the range of the 
apparatus is limited. There is also some difficulty in the 
calibration of the instrument to determine the value of the 
readings. 

A micrometer-screw is sometimes used as a means of me- 
chanical magnification. With this there is, again, the diffi- 
culty of suitable attachment to the bar; and, as generally 
used, it is difficult to ascertain when exact contact of the 
screw is obtained without excessive pressure. 

There is a special difficulty in measuring the elongation of 
ordinary test-bars which has been overlooked in the construc- 
tion of most of the apparatus of this kind. It is difficult to 
get test-bars which are rigidly straight. Hven if the test-bar 
is strictly straight, it is difficult to hold it in the testing- 
machine, so that the resultant of the stress on any cross section 
passes strictly through the centre of figure of the section. 
Now if this condition is not satisfied, the bar becomes curved 
during the test. The straightening ofan initially curved bar, 
or the curving of an initially straight one, introduce errors 
in the measurements of very considerable amount. 

If the measurements could be made at the axis of the bar, 
the errors of this kind with any amount of curvature likely 
to occur would not be very serious; but this is of course 
impossible. The best that can be done is to measure at the 
surface of the test-bar. But, in straightening, the surface of 
the bar on one side lengthens and on the other shortens, and 
thus introduces a not inconsiderable error of measurement. 
If, as in many forms of elongation measuring-apparatus, the 
measuring-points are two inches or more from the axis of the 
bar, the errors become very large relatively to the elongations 
to be measured. 

Let fig. 1 represent a bar bent in the plane of the paper, 
the centre of curvature being O. ‘Then, if measurements 
could be made on the axis of the bar, between the points ad, 
the straightening of the bar would introduce an error equal 
to the difference of the length of the chord a6 and are acb. 
With any amount of curvature likely to occur in a test-bar, 
this error would not be very serious. (Generally, however, 
the best that can be done is to measure the distance between 
points a,b, on the surface of the bar. Then, since by 


ne reer a 
SSS = a 


=a eS 


Se eee See any pe ene SSE 


SSS See 


284 Prof. W. C. Unwin on Measuring-Instruments 


straightening the lines aQ, 6 O become parallel, the error 
introduced is the difference between a, b, and the arc acb; 


Fig, 1. 


745 


¢ 


A LO I I EY OO I, AT eH = 
M 


ey- 
iw) \ 


and this is much more serious. Most commonly, however, 
measurements are made between points on clips fastened to 
the bar at 1 or 2 inches distance from its surface, such as 
dy b,. Then the error introduced by straightening is the 
difference between a, b, and the arc acb; and this may bea 
serious error, even with a very small amount of initial cur- 
vature. 

If simultaneous measurements are taken of a,b, and az bz, 
the mean of these will have no greater error than the mea- 
surement of ab. That is, the mean of measurements on two 
sides of the bar reduces the error due to initial or induced 
curvature to the same amount as a measurement actually 
made at the axis of the bar. 

Prof. Bauschinger, of Munich, appears to have been the 
first to recognize the importance of this double measure ment 
symmetrically on the two sides of the bar. He has always 
used an apparatus in which a finger, or touch-piece, attached 
to one end of the bar, presses on a roller attached to the other 
end. As the bar extends, the roller rotates by friction against 
the finger. A mirror is attached to the roller ; and the amount 


used in Mechanical Testing. 285 


of rotation is observed by noting the image of a scale in the 
mirror through a reading-telescope. In this way measure- 
ments to >5,—th of an inch can be taken. To eliminate 
errors due to curvature, two rollers are placed, one on each 
side of the bar, and two sets of readings are taken. This 
involves the adjustment of two instruments and the taking of 
two sets of readings. But the principle is perfect; and no 
more accurate measurements than Bauschinger’s have pro- 
bably been made. 

Touch-Micrometer Katensometer.—The first instrument used 
by the author was a kind of callipers. Two bars, one sliding 
in the other, could be set by touch to the distance between 
two fixed clips on the test-bar. A scale was engraved on 
silver on one bar ; and the distance of the nearest division from 
a fixed zero-mark on the other was taken by a microscope- 


micrometer. Readings could be taken to : — aa th of an inch. 


The instrument is easy and rapid to use. Headings can be 
taken on both sides of the test-bar; and the readings are direct 
on toa carefully graduated scale, so that no calibration of the 
instrument is necessary. 

Screw-Micrometer Extensometer.—This aims at obtaining 
the extension along the axis of the bar by a single reading. 
Two clips are fixed on the bar, each by a pair of steel points, 
one on each side, gripping the bar in a plane through its axis. 
If, then, these clips can be made to preserve the same relative 
position to the bar, the middle points of the clips will move in 
the same way as points on the axis of the bar. Fig. 2 isa 
diagrammatic sketch of the apparatus. aa and 0b are the 
clips on the test-bar, fixed to it by points in its middle plane. 
c¢é are projections on the clips, to which are fixed delicate 
spirit-levels ; dis.a small screw which just touches the test-bar; 
e 1s a micrometer-screw with graduated head, which supports 
the upper clip on the lower clip. In use the lower clip is 
first levelled by the screw d; then the upper clip is levelled 
by the micrometer-screw, and a reading taken. The clips being 
always accurately levelled, in a plane perpendicular to that in 
which the four points attaching the clips to the test-bar lie, 
the micrometer-readings are the distances between the middle 
points of the two parallel clips ; and their differences are the 
mean of the elongations on the two sides of the test-bar, or 
virtually are readings at the axis of the test-bar. Readings 
tO 7poo0th of an inch can be taken. 

Roller-and-Mirror Eatensometer.—-Fig 3 is a diagrammatic 
sketch of another instrument on the same principle. a and b 
are two clips similar to those in fig. 2; the lower clip is sup- 


286 Measuring-Instruments used in Mechanical Testing. 


ported on the test-bar by a screw d; the upper clip is sup- 
ported on the lower by a stay-bar with knife-edges, e. Atr 


Fig. 2, Fig. 3, 


and mare the roller and mirror, the axis of these being at the 
same distance from the knife-edge of the stay-bar as the set 
screw of the clip. A touch-piece or finger, 7, attached to the 
lower clip presses on the roller. If the bar extends, the roller 
approaches the lower clip by an equal amount; it turns 
against the finger 7; and the amount of rotation is read by a 
telescope and scale. This instrument will easily read to 


sooo of an inch. The roller being at the centre of the 


clip, its movement is the mean of the elongations on the two 
sides of the test-bar. 

The author showed a third instrument on the same principle, 
for obtaining the compression of small blocks of stone. 


fos? 


XXXII. On the Equilibrium of a Gas under its own Gravita- 
tion only. By Sir W. THomson*. 


as problem, for the case of uniform temperature, was 
first, I believe, proposed by Tait in the following 
highly interesting question, set in the Ferguson Scholarship 
Examination (Glasgow, October 2nd, 1885) :—“ Assuming 
Boyle’s Law for all pressures, form the equation for the 
equilibrium-density at any distance from the centre of a 
spherical attracting mass, placed in an infinite space filled 
originally with air, Find the special integral which depends 
on a power of the distance from the centre of the sphere 
alone.” 

The answer (in examinational stvle!) is:—Choose units 
properly ; we have 


dlp p). pr? dr 
a, = Drees er - ° - . ° : as, 


where p is the density at distance r from the centre. Assume 
: (Dy eee s BRA Coie eM ota ene ean 0 
We find A=2, e=—2; and therefore 


Bs ens enol ee 


satisfies the equation in the required form. 

Tait informs me that this question occurred to him while 
writing for ‘Nature’ a review of Stokes’s Lecture t on 
Inferences from the Spectrum Analysis of the Lights of 
Sun, Stars, Nebule, and Comets; and in the ‘ Proceedings 
of the Edinburgh Mathematical Society’ he has given some 
Transformations of the equation of Hquilibrium. The same 
statical problem has recently been forced on myself by con- 


* Communicated by the Author, having been read before the Royal 
Society of Edinburgh on the 7th and 2ist February, 1887. 

Note of February 22, 1887.—Having yesterday sent a finally revised 
proof of this paper for press, I have today received a letter from Prof, 
Newcomb, calling my attention to a most important paper by Mr. J. 
Homer Lane, “On the Theoretical Temperature of the Sun,” published 
in the American Journal of Science for July 1870, p. 57, in which pre- 
cisely the same problem as that of my article is very powerfully dealt 
with, mathematically and practically. It is impossible now, before going 
to press, for me to do more than refer to Mr. Lane’s paper; but I hope to 
profit by it very much in the continuation of my present work which I 
intended, and still intend, to make.—W. T. 

7 Lecture III. of Second Course of “Burnet Lectures,’ Aberdeen, 
Dec. 1884; published, London, 1885 (Macmillan). 


eS ee ee 
—— 


288 Sir William Thomson on the Equilibrium of 


siderations which I could not avoid in connection with a 
lecture which I recently gave in the Royal Institution of 
London, on “ The Probable Origin, the Total Amount, and 
the Possible Duration of the Sun’s Heat.” “4 

Helmholtz’s explanation, attributing the Sun’s heat to 
condensation under mutual gravitation of all parts of the 
Sun’s mass, becomes not a hypothesis but a statement of 
fact, when it is admitted that no considerable part of the heat 
emitted from the Sun is produced by present in-fall of meteoric 
matter from without. The present communication is an 
instalment towards the gaseous dynamics of the Sun, Stars, 
and Nebule. 

To facilitate calculation of practical results, let a kilometre 
be the unit of length; and the terrestrial-surface heaviness 
of a cubic kilometre of water at unit density, taken as the 
maximum density under ordinary pressure, be the unit of force 
(or, approximately, a thousand million tons heaviness at the 
earth’s surface). If p be the pressure, p the density, and ¢ the 
temperature from absolute zero, we have, by Boyle and 


Charles’s laws, 
p=...” . ) eee 


where ¢ denotes absolute (thermodynamic” ) temperature, with 
O° Cent. taken as unit; and H denotes what is commonly, in 


- technical language, called “the height of the homogeneous 


almosphere” at 0° ©. For dry common air, according to 
Regnault’s determination of density, 


H = 7-985 kilometres +.) ieee o: 


Let @ be the gravitational coefficient proper to the units 
chosen ; so that Bmm'/D? is the force between m, m’ at 
distance D. The earth’s mean density being 5-6, and radius 
6370 kilometres, we have 


= 6870 5-681; and therefore 47,8 = 1/1 ie ean 
Let now the p, p, ¢ of (4) be the pressure, density, and tem- 
perature at distance » from the centre of a spherical shell 
containing gas in gross-dynamic+ equilibrium. We have, by 


* The notation of the text is related to temperature Centigrade on the 
thermodynamic principle (which is approximately temperature Centigrade 
by the air-thermometer), as follows :— 


= = (temperature Centigrade +273) ; 


see my Collected Mathematical and Physical Papers, vol. i. Arts, xxxix., 
and xlviii. part vi. §§ 99, 100; and article “ Heat,” §§ 35-88 & 47-67, 
Encyc. Brit., and vol. iii. (soon to be published) of Collected Papers. 

+ Not in molecular equilibrium of course; and not in gross-thermal 
equilibrium, except in the case of ¢ uniform throughout the gas, 


a Gas under ils own Gravitation only. 289 


elementary hydrostatics, 


- Zs —p(u + ( arany’p) Big Yona one, CO), 
whence d é dp 


where M denotes the whole quantity of matter within radius a 
from the centre; which may be a nucleus and gas, or may be 
all gas. 

If the gas is enclosed in arigid spherical shell, impermeable 
to heat, and left to itself for a sufficiently long time, it settles 
into the condition of gross-thermal equilibrium, by ‘‘ conduc- 
tion of heat,” till the temperature becomes uniform through- 
out. But if it were stirred artificially all through its volume, 
currents not considerably disturbing the static distribution of 
pressure and density will bring it approximately to what I 
have called convective equilibrium* of temperature—that is to 
say, the condition in which the temperature in any part P is 
the same as that which any other part of the gas would acquire 
if enclosed in an impermeable cylinder with piston, and dilated 
or expanded to the same density as P. The natural stirring 
produced in a great free fluid mass like the Sun’s, by the 
cooling at the surface, must, I believe, maintain a somewhat 
close approximation to convective equilibrium throughout the 
whole mass. The known relations between temperature, 
pressure, and density for the ideal “‘ perfect gas,’’ when con- 
densed or allowed to expand in a cylinder and piston of 
material impermeable to heat, aret 


1 Gores alllys Pumsgaeg ae toma ate ey Ree NG!) 
pre vad Ue Te oom Cays 


where & denotes the ratio of the thermal capacity of the gas, 
pressure constant, to its thermal capacity, volume constant, 
which is approximately equal to 1°41 or 1-40 (we shall take 
it 1-4) for all gases, and all temperatures, densities, and pres- 
sures ; and T denotes the temperature corresponding to unit 
density in the particular gaseous mass under consideration. 


Using (8) to eliminate p from (7) we find 


dp ,d(et)) __ 4mB(k—1) 
al “p |= ETE’? 5 oe te (10); 


* See “ On the Convective Equilibrium of Temperature in the Atmo- 
sphere,” Manchester Phil. Soe. vol. ii. 3rd series, 1862; and vol. iii. of 
Collected Papers. 

+ See my Collected Mathematical and Physical Papers, vol. i. 
Art. xlvii. note 3. 


Phil. Mag. 8. 5. Vol. 23. No. 142. March 1887. X 


290 Sir William Thomson on the Equilibrium of 


which, if we put prt sw 
V(b) =e 6. 0. 
and a bold bye ae 
. aR = 2... ys So 
takes the remarkably simple form 
du uk 
ae TB sae 


Let f(a) be a particular solution of this equation ; so that 
f(a) =— [flay e4 
(15). 


Sma) = = [ fna) | m-427* 
From this we derive a general solution with one disposable 
constant, by assuming 


and therefore 


u=Cf (mz) ys (16); 
which, substituted in (14), yields, in virtue of (15), 
1 Cnr 
so that we have, as a general solution, 
w= Of [a@0r 7] is Be ee 


Now the class of solutions of (14) which will interest us 
most is that for which the density and temperature are finite 
and continuous from the centre outwards, to a certain 
distance, finite as we shall see presently, at which both vanish. 
In this class of cases w increases from 0 to some finite 
value, as # increases from some finite value too. Hence if 
u=f(x) belongs to this class, u=Cf(mz) also belongs to it; 
and (18) is the general solution for the class. We have 
therefore, immediately, the following conclusions :— 

(1) The diameters of different globular* gaseous stars of 
the same kind of gas are inversely as the $(«—1)th powers 
(or ? powers) of their central temperatures, at the times when, 
in the process of gradual cooling, their temperatures at places 
of the same densities are equal (or “'T'”’ the same for the dif- 
ferent masses). Thus, for example, one sixteenth central 
temperature corresponds to eight-fold diameter: one eighty- 
first central temperature corresponds to twenty-seven fold 
diameter. 

* This adjective excludes stars or nebulee rotating steadily with so 
sreat angular velocities as to be much flattened, or to be annular; also 
nebulee revolving circularly with different angular velocities at different 
distances from the centre, as may be approximately the case with spiral 
nebule. It would approximately enough include the sun, with his small 
angular velocity of once round in 25 days, were the fluid not too dense 
through a large part of the interior to approximately obey gaseous law. 


It no doubt applies very accurately to earlier times of the sun’s history, 
when he was much less dense than he is now. 


———  ———=—— 


a Gas under its own Gravitation only. 291 


(2) Under the same conditions as (1) (that is, H and T the 
same for the different masses), the central densities are as the 
«th powers (or 3 powers) of the central temperatures ; and 


therefore inversely as the vei OF des or = , powers of the 


rs ? 

diameters. IR 
(3) Under still the same conditions as (1) and (2), the 
quantities of matter in the two masses are inversely as the 


( —3)th powers, (inversely as the cube roots) of their 


diameters. 

(4) The diameters of different globular gaseous stars, of the 
same kind of gas, and of the same central densities, are as the 
square roots of their central temperatures. 

(5) The diameters of different globular gaseous stars of 
different kinds of gas, but of the same central densities and 
temperatures, are inversely as the square roots of the specific 
densities of the gases. 

(6) A single curve [y=/(r—')] with scale of ordinate (r) 
and scale of abscissa (y) properly assigned according to (18), 
(17), and (11) shows for a globe of any kind of gas in mole- 
cular equilibrium, of given mass and given diameter, the abso- 
lute temperature at any distance from the centre. Another 
curve, {y=| /(7—') |“}, with sca'es correspondingly assigned, 
shows the distribution of density from surface to centre. 

It is easy to find, with any desired degree of accuracy, the 
particular solution of (13), for which 


w—A, and = = WHEE ar ss, ae Le 


a denoting any chosen value of 2, and A and A’ any two 
arbitrary numerics, by successive applications of the formula 


win=A—(de(a/—['ae) . . . eo); 


the quadratures being performed with labour moderately pro- 
portional to the accuracy required, by tracing curves on 
“section’’-paper (paper ruled with small squares) and counting 
the squares and parts of squares in their areas. To begin, up 
may be taken arbitrarily; but it may conveniently be taken 
from a hasty graphic construction by drawing, step by step, 
successive arces* of a curve with radii of curvature calculated 
from (13) with the value of du/dx found from the step-by- 
step process. If this preliminary construction is done with 

* This method of graphically integrating a differential equation of the 
second order, which first occurred to me many years ago as suitable for 
finding the shapes of particular cases of the capillary surface of revolution, 
was successfully carried out for me by Prof. John Perry, when a student in 


X2 


292 Equilibrium of a Gas under tts own Gravitation only. 


care, by aid of good drawing-instruments, uw, calculated from 
wu by quadratures will be found to agree so closely with wo, 
that uo itself will be seen to be a good solution. If any dif- 
ference is found between the two, wu; is the better: u, is a 
closer approximation than u,; and so on, with no limit to the 
accuracy attainable. 

Mr. Magnus Maclean, my official assistant in the University 
of Glasgow, has made a successful beginning of working-out 
this process for the case w=16 where z= ; andhas already 
obtained a somewhat approximate solution, of which the pro- 


duce useful for our problem is expressed in the following 
table. 


: , d7u oy ee 
Numerical Solution of ag he ee 


dx 
Mass within dis- 
Dishes Eronk Reciprocal of tance 7 from 
distance from| Temperature} Density the centre 
co : centre =U. —y2'5, |=dujdax 
ay G ={ OL U mee 
ate - 
0 oO 16:00 1024 ‘00 
“100 10 14-46 795'2 ‘28 
ay -9 14-14 7516 38 
125 8 13-71 6958 "52 
143 7 13°10 621-2 Toul 
"167 6 12:20 520°0 1-056 
‘200 5 10°92 394°1 1°566 
250 4 9:00 243°0 2°336 
"333 3 6:15 93°81 3°436 
‘500 2 2°25 7595 4°366 
"667 15 0 0 4-49 


The deduction from these numbers, of results expressing in 
terms of convenient units the temperature and density at any 
point of a given mass of a known kind of gas, occupying a 
sphere of given radius, must be reserved for a subsequent 
communication. 

One interesting result which I can give at present, derived 
from the first and last numbers of the several columns of the 
preceding table, is, that the central density of a globular 
gaseous star is 224 times its average density. 


my laboratory in 1874, in a series of skilfully executed drawings repre- 
senting a large variety of cases of the capillary surface of revolution, 
which have been regularly shown in my Lectures to the Natural Philo- 
sophy Class of the University of Glasgow. These curves were recently 
published in the Proc. Roy. Instit. (Lecture of Jan. 29, 1886), and 
‘Nature, July 22 and 29, and Aug. 19, 1886; also to appear in a volume 
of Lectures now in the press, to be published in the ‘Nature’ series. 


pees. | 


XXXII. Preliminary Experiments on the Effects of Percus- 
sion in Changing the Magnetic Moments of Steel Magnets. 
By Wiutram Brown, Thomson Experimental Scholar, 
Physical Laboratory, Unwersity of Glasgow*. 


Part I. 


| ae experiments described in this paper were made in 

the Physical Laboratory of Glasgow University. They 
were first suggested by some casual observations made while I 
was assisting Mr. T. Gray during his recent determination of 
the horizontal intensity of the Harth’s magnetic force, and I 
am greatly indebted to him for many hints and much valuable 
advice during the progress of the experiments. 

This paper is an account of some preliminary observations 
on a subject which is at present being investigated in this 
Laboratory, and I hope to give the results of further experi- 
ments in an early number of this magazine. 

The effects of percussion in changing the magnetic moments 
of steel magnets have not (so far as I know) been made the 
subject of special observation. The results hitherto published 
have, for the most part, formed less important sections of other 
investigations. 

A number of interesting experiments on magnets were made 
by Joule at intervals from 1864 to 1882. Among other results, 
he gives an interesting set on ‘the effect of mechanical 
violence on the intensity of magnetic bars,’’ an account of 
which is published in his Scientific Papers, vol. i. p. 596. 

In 1878, Mr. T. Grayt published a very accurate series of 
investigations on magnetic moments in absolute measure. 
One of the objects of his investigation was to obtain informa- 
tion as to the permanence or non-permanence of magnetism 
in steel bars when left undisturbed for a length of time. On 
page 328 of the same paper, however, he gives a short series 
of observations on the direct effects of percussion, which, so 
far as they go,agree with my own results. 

There has been published quite recently an excellent series 
of papers on the “ Electrical and Magnetic Properties of the 
Tron Carburets,’’ by OC. Barus and V. Strouhal, of the U.S.A. 
Geological Surveyt. 

- * Communicated by Sir W. Thomson, having been read before the 
Mathematical and Physical Section of the Philosophical Society of 
Glasgow, November 30, 1886. 

+ “On the Experimental Determination of Magnetic Moments in 


Absolute Measure,” Phil. Mag. November 1878, pp. 321-331, 
{ Bulletin No. 14, 1885, Department of the Interior. 


=== === 


=e ee 


ae 


294 Mr. W. Brown on the Effects of Percussion in 


In one of these papers the authors treat incidentally of 
magnetic retentiveness, but more with respect to the effects 
of annealing than of direct percussion. Annealing appears to 
play an important part in the ultimate retentive power of 
magnets, and on the constancy of their magnetic moments. 
My own experiments on annealed magnets are not yet com- 
plete enough to be put into presentable form, and with two 
exceptions the results given are for magnets tempered glass- 
hard. Regarding the effects of annealing generally, the 
results of Barus, Strouhal, and Gray appear to agree on the 
whole. 7 

On pages 326 & 327 of T. Gray’s paper it appears that an- 
nealing increased the magnetic moment. This is the result 
stated generally in the text; but a marginal note and curve 
in the copy in my possession show that the magnetic moment 
did not increase continuously as the annealing went on, but 
increased at first, then diminished, then again increased— 
thus passing a maximum when the annealing temperature was 
about 150° C., and a minimum when the temperature was 
about 230° C. 

Another set of results given in this paper is a series of 
magnetic moments for bars of the same steel tempered in oil, 
the temperature of which was varied so that the bars should 
be suddenly cooled only to the same temperature as that to 
which in the first set they were heated in the annealing pro- 
cess. The results in this second set show precisely the same 
characteristics as those of the first set, only that the maximum 
and minimum points are much more pronounced. 

The curves given by Mr. T. Gray indicate an interesting 
peculiarity which does not seem to have been noticed by 
any previous observer, inasmuch as they show that the 
minimum point may be preceded by a maximum, the effects 
of annealing depending greatly on the kind of steel used. 
The steel used by Gray was almost a pure charcoal steel, 
whereas that used by Barus and Strouhal was of the kind 
known as ‘‘ Kinglish silver steel;” that is to say, of a kind 
similar to that which I have been experimenting on. 

Mr. Gray in his paper refers very briefly to some experi- 
ments on other steels, chiefly, it would appear, for the purpose 
of showing how very different results may be expected from 
different specimens. In one set, which took an average mag- 
netic moment of about 50 per gramme, the magnetic moment 
was slightly diminished by hardening; whilst in another set 
the direct opposite was arrived at in a very marked manner, 
showing how very much the effect of annealing depends on 
metallic impurities in the steel. The diminution of the mag- 


Changing the Magnetic Moments of Steel Magnets. 295 


netic moment of one of his specimens by annealing, Mr. Gray 
tells me, he believes to be due to the known presence of 
manganese in it. 

The behaviour, however, of alloys in the annealing and 
magnetic retentiveness of steel magnets needs further eluci- 
dation before anything very definite can be said on the subject. 
The effects of small quantities of tungsten in increasing the 
magnetic retentiveness of steel are well known; whereas the 
recent experiments of Dr. J. Hopkinson, J. T. Bottomley, 
and others tend to show that a very moderate quantity of 
maganese in steel almost totally destroys, not only the mag- 
netic retentiveness, but even the magnetic susceptibility. 

With respect to the “silver steel,” of which my magnets 
are made, | am in doubt as to whether it really contains silver. 
Some well-known steelmakers say it is only a trade name. 

In preparing the magnets, great care was taken to have 
them made straight, and the ends made as accurately as pos- 
sible at right angles to their length. In tempering them they 
were put into an iron tube having one end closed, and the 
whole put into a brisk coal fire and left there till they attained 
a bright red heat. The tube, with the magnets inside, was 
then taken out, and, with the open end temporarily closed by 
a glass plate, was held vertically above a vessel of water at a 
temperature of 15° C. and about 20 inches deep. The glass 
cover was then quickly withdrawn, and the magnets were 
allowed to drop perpendicularly into the water, thus making 
them all glass-hard. 

A greater number of magnets than were actually required 
were treated in this manner, and only the straightest and most 
uniform in temper were chosen for the experiments. ‘This 
was the method employed in tempering all the glass-hard 
magnets, and was adopted mainly in order to obtain an indi- 
cation of what kind of results were to be expected. In sub- 
sequent experiments, however, a method somewhat similar to 
that used by T. Gray and Barus will be employed. The two 
exceptions to glass-hardness already referred to (those tem- 
pered blue and yellow) were first tempered glass-hard along 
with the others ; they were then laid on the top of a hot 
metallic plate, where they were allowed to remain till they 
exhibited the oxide tints characteristic of those tempers. 

The magnets were all magnetized to saturation by placing 
them between the poles of a large Ruhmkorff electromagnet, 
excited by a dynamo giving a potential of between 80 and 90 
volts. During the process of magnetizing, the magnets were 
reversed several times between the poles of the electromagnet 
and then finally magnetized. This was done in every case 


296 Mr. W. Brown on the Hfects of Percussion in | 


for the first set of six magnets, when they were remagnetized 
after a series of observations. 

In order to obtain the deflections for calculating the mag- 
netic moments, the apparatus used consisted of a lamp and 
scale, a magnetometer, and a cradle for holding the magnets. 
The magnetometer was of the ordinary Bottomley type, con- 
sisting of a small circular mirror, with two short magnetic 
needles attached to the back of it, and suspended by a single 
torsionless silk fibre, the whole being enclosed in a slot cut in 
a pyramidal block of wood, and the slot covered by a plate of 
thin glass. On the base of this pyramid were fixed three 
conical feet, which fitted accurately into the conical hole, 
groove, and plane arrangement of Sir Wilham Thomson. The 
hole and groove were cut out of a piece of thick plate glass, 
which was firmly fixed to the table in a position where the 
horizontal component of the Harth’s magnetic force was 
known. When the magnetometer was put in position, the 
mirror and the attached needles of course placed themselves 
in the magnetic meridian. 

Immediately to the west of the magnetometer, at a distance 
of 40 centim., was placed a cradle for holding the magnets 
during the deflection observations. The base of this cradle 
i was made on the same geometrical principle as that of the 
magnetometer, and was so arranged that the magnet could be 


reversed relatively to the magnetometer, without touching the 
: magnet by hand. This cradle, I may say, was made and used 
i by Mr. Gray in his recent determination of the Harth’s hori- 
iH) zontal magnetic force; the whole arrangement is fully 
iM described and illustrated by a drawing in his paper”. 
MH To the east of the magnetometer, at a distance of 129 centim., 
i was a glass scale divided to millimetres, and having a lamp 
i placed immediately behind it. The deflection of the spot of 
light from the lamp when reflected by the mirror of the mag- 
netometer upon the scale could be read to 4/5 of a millimetre, 
by means of the shadow cast by a fine wire stretched across 
the orifice in the side of the copper funnel of the lamp. 

The magnet and magnetometer being placed in position, 
the magnetic moment of the magnet, M, is given by the 
following equation :— 


M 


— Htand’—/)? | 

ai 2r : 

where 7= the distance of the centre of the magnet from the 
centre of the magnetometer-needle ; 


| 
* “Qn the Measurement of the Intensity of the Horizontal Component 
i of the Earth’s Magnetic Field,” Phil. Mag. December 1885, pp. 484-497, 


Changing the Magnetic Moments of Steel Magnets. 297 


1 = half the distance between the poles of the deflecting 
magnet (in these experiments, taken as half the 
actual length of the magnet) ; 

H = the horizontal component of the Harth’s magnetic 
force ='153 C.G.S. unit ; 

@ = the deflection, in degrees, of the magnetometer- 
needle. 


In order to test the constancy of the magnetic field during 
the experiments, the deflections given by a standard magnet 
were occasionally taken. 

The method employed to obtain the effects of percussion, 
with the least possible amount of handling of the magnets, 
was as follows :—A series of glass tubes, wide enough inside 
to allow the magnets to fall through them freely, were fixed 
on a long narrow board by means of brass clamps, which were 
just loose enough to allow the tubes to slip easily through them. 
This was for the purpose of raising the tube vertically in order 
to take the magnet out after falling through it. A thick shelf 
was firmly fixed at one end of this board and a thick plate of 
glass fastened to it. The magnets were held in the hand and 
allowed to fall vertically through the tube upon the glass plate 
at the bottom, and always with the true north end of the 
magnet downwards. 

A few trial experiments were made in letting the magnets 
fall through a height of a half metre and one metre respec- 
tively ; but to give uniformity in the results, the 1:5 metre 
height was adopted throughout. 

The plan of experimenting was as follows:—The magnets 
were magnetized and laid aside undisturbed for the periods of 
time specified in Table I. Gne of them was then taken, and 
the deflection for calculating its magnetic moment was ob- 
served. It was then allowed to fall once through the height 
of 1°5 metre, and the deflection againtaken. It was then let 
fall three times in succession through the same height, and the 
deflection again taken. Hach magnet in turn was put through 
the same series of operations. 

The percentage loss in the magnetic moment due to the 
one fall, and that due to the three falls, and, finally, that due 
to the whole four falls, were all calculated, and are shown for 
the fourteen glass-hard magnets in Table II. 

In Table I. the percentage loss due to the four falls alone 
is given. The magnets specified in Table II. had been lying 
aside for a period of six months after being magnetized, and 
before they were experimented on. 


298 


Magnetic Moments of Steel Magnets. 


TABLE I. 
43 3 Rend “ i | 
a 4 sa |2 Bs + |Percentage loss of Magnetism due to 
oo Sp A SS) ney aM a 5 od percussion after lying aside for dif- 
| =| sete) eal woes 5 =i ke) A ferent periods of time before being 
a os Zs Zz 2 2 ss = = | experimented upon. 
e e © SA “ma ¢ 
EB) 2 jaf] 53/3) 35 | 
E = Sp q | 8 Pate ap 3 1 44 20 1 
S ar 83 ea = si months. |month.| hours. | hours. | hour. 
aL: ae 10x-3) 33 | 5200] 41 | 08 | 104) 194] 20 | 198 
2. i 102} 5bO | 2084) 45 0:0 1:00 | 148). 3:2 | 2°96 
3. | Yellow. |10x°3! 33 | 5195| 44 6:2 54 | 48 61 | 6:03 
4, as 10x°3; 50 2:087| 46 4:0 26 | 376! 35 | 40 
5, Blue. 103) 33. 5'240| 54 75 1V8) | O97 AOS aes 
6. - 10x:2} 50 | 3095] 71 87 75) | Sls Srae ae 
TABLE II. 


The Magnets in this Table were all tempered Glass-hard. 


i eel 


o a ot 

te cata = 
=| pele 2 8 
Sal owe) 8 a oO 
2 Sgo osm 
Byileea || 

hy PES ha 

iv 15°25 60 
op 15x 25 60 
or 10x°3 33 
4, 6xX°3 20 
ian lice at) 10 
6. 10x :2 50 
7. xe? 30 
8. 10x«°15 70 
9. Lx<aly AT 
0. 10~x°:1 100 
ale 7X'1 70 


Weight of magnet, 
in grammes. 


SvSuor 
aa 
Sao 


SCOrerhy rw 
1c ©) CO CO G3 © 
Sa GO CO ~I bD Od CO 


0:39 


Magnetic moment, 
per gramme. 


tye : . 
CHNOOHNN 


Pee 


Percentage loss of Magnetism 
due to percussion after fall- 
ing the number of times 
specified. 


| 
3 times. Total loss. 
| 


By inspection of Table I., we see that in the case of the 
two glass-hard magnets the percentage loss diminishes the 
longer they are left undisturbed before they are subjected to 
percussion ; it would also appear that the smaller the dimen- 
sion ratio the greater is the loss. 
The other four magnets, so far as the experiments go, 
appear to show that the greater the amount of annealing the 
greater is the loss ; the observations, however, have not been 


Notices respecting New Books. 299 


sufficiently extended to allow any detailed deductions to be 
made; and the effects of annealing will form one of the chief 
parts of subsequent investigations. 

From Table II. it will be seen that the greater part of the 
magnetism has been shaken out by the first fall; this holds 
throughout, except in the case of magnet No. 10. I am in- 
clined to think that this is due to an error in placing the magnet 
in position ; in this case also the percentage loss was calcu- 
lated from a diminished deflection of one scale-division only. 


XXXIV. Notices respecting New Books. 


Annual Companion to the ‘Observatory, a Monthly Review of Astro- 
nomy. Edited by HK. W. Mavnoer, F.R.AS., A. M. W. Down- 
ine, MA., F.R.AS., and T. Lewis, f.R.A.S. London: Taylor 
and Francis. 1887. 


(eee contemporary ‘ The Observatory,’ which was started, some 

ten years ago, by the present Astronomer Royal, to supply a 
want then very greatly felt, used to supply its readers every month 
with an Astronomical Ephemeris. 

This Ephemeris was useless to its many subscribers in the 
Colonies and other distant parts of the world. Partly for this 
reason, and partly in order to extend the scope of the work, the 
Editors resolved to prepare all the information desirable more than 
a year in advance, and this they have carried out in the ‘ Companion 
to the Observatory.’ The idea suggested itself to them towards 
the end of 1885, and, acting upon it at once, the ‘Companion’ for 
1886 appeared in time. The present number, for 1887, is so com- 
plete, so well arranged, and so admirably suited to the wants of the 
practical Astronomer, that it is difficult to suggest any improve- 
ment to it. 

It begins with a short Introduction, stating the sources whence 
the Ephemerides have been devived, frequently such as are not 
generally available to the English reader ; such as the Annuaire du 
Bureau des Longitudes, the ‘American Nautical Almanac,’ the 
Astronomische Nachrichten, &c. 

The first part of the work is a Calendar, giving for every day the 
times of rising, culminating, and setting of the Sun and Moon ; 
the Equation of Time, in the form of Mean Time at Apparent 
Noon ; the Sidereal Time at Mean Noon ; the fraction of the year 
elapsed since January 1; the quarterings of the Moon; and the 
principal showers of Shooting-stars. 

Next follow the places, diameters, times of rising, culminating, 
and setting of the major Planets; particulars of Kclipses and of 
Occultations of fixed Stars by the Moon, visible at Greenwich, in 
which a better mode of reckoning the angles is introduced. The 
phenomena of Jupiter’s Satellites are given im catenso from the 


300 Notices respecting New Books. 


‘Nautical Almanac,’ than which no better exist. They are founded 
on Damoiseau’s Tables, with the modifications introduced by Adams 
and Woolhouse. 

The complete Ephemerides of the Satellites of Saturn, Uranus, 
and Neptune, together with the diagrams of their orbits, will be 
duly appreciated by the now sufficiently numerous possessors of 
large telescopes. For Physical observations of the Sun, we are 
given the positions of the Sun’s axis, and the latitude and longitude 
of the centre of disk for every fifth day of the year. For Jupiter, 


the time is given of every third passage of the famous Red Spot 


over the central meridian of the planet. 

Stars with remarkable spectra are treated at some length, and 
the information in regard to Variable Stars is more complete than 
exists anywhere else in the English language. The work closes 
with a table of the Selenographical Latitudes and Longitudes of 100 
lunar formations. 

In this work the public is supplied, at a merely nominal cost, 
with a vast amount of accurate, well chosen, and well arranged 
information, some of which is original. Hven the professional 
Astronomer will frequently turn to its pages from mere conye- 
nience, while to the amateur it is almost indispensable. We can 
only hope that the care and judgment displayed in its compilation 
will be appreciated by the numerous class for whom it is intended. 


A Synopsis of Elementary Results un Pure Mathematies ; containing 
Propositions, Formule, and Methods of Analysis, with abridged 
Demonstrations. By G. 8S. Carr, M.A. London: Francis 


Hodgson, 1886; pp. xxxvili+ 9386+ 285 diagrams. 


TxoueH the first sections of this book were issued in 1880, the 
compilation of it commenced about twenty years ago, many of 
the abbreviated methods and mnemonic rules having been drawn 
up for the use of the author’s pupils. 

The completed work admirably serves the object Mr. Carr set 
before himself, viz. that of presenting in a moderate compass the 
fundamental theorems, formule, and processes in the chief branches 
of Pure Mathematics. We hope he will be encouraged by the 
reception accorded to the present venture to complete his original 
plan, and supplement this work by a similar one on Applied Mathe- 
matics. We will let the author speak as to his intentions. “The 
work is intended, in the first place, to follow and supplement the 
use of the ordinary textbooks, and it is arranged with the view of 
assisting the student in the task of revision of book-work. To 
this end I have, in many cases, merely indicated the salient points 
of a demonstration, or merely referred to the theorems by which 
the proposition is proved. I am convinced that it is more bene- 
ficial to the student to recall demonstrations with such aids than 
to read and re-read them. Let them be read once but recalled 
often....In the second place, I venture to hope that the work 
may prove useful to advanced students as an atde-mémoire and 


Intelligence and Miscellaneous Articles. 301 


book of reference. The boundary of mathematical science forms, 
year by year, an everwidening circle, and the advantage of having 
at hand some condensed statement of results becomes more and 
more evident. ‘To the original investigator occupied with abstruse 
researches in some one of the many branches of Mathematics, a 
work which gathers together synoptically the leading propositions 
in all, may not therefore prove unacceptable.” In an undertaking 
of such magnitude it would not be difficult to detect faults, but 
these are in the main corrected by the lists of Errata; and some 
others, as the unfortunate wrong numbering of a limited number 
of sections in Section viii., which is indicated on p. 473, are allowed 
for in the Index. Every subject that can be classed under the 
head of Pure Mathematics, with the exception perhaps of Quater- 
nions, appears to us to have been carefully treated on the author’s 
lines. A little difficulty is at first experienced in working with 
such a vast Index; but it will be found with use that this part is 
as carefully done as the rest of the work. To many of our readers 
a most useful part will be found to be comprised in that portion 
which is represented by the sclosmg words of the titlepage, 
“« Supplemented by an Index to the Papers on Pure Mathematics 
which are to be found in the principal Journals and Transactions 
of learned Societies, both English and Foreign, of the present 
Century.” These are thirty-two in number, are tabulated with 
great care, and references to the British Museum Catalogue are 
appended. This is such a valuable feature of the book that we 
should like to see the list considerably extended, so as to include for 
instance references to our own columns, in which from time to time 
most important articles on Pure Mathematics have been furnished 
by Cayley, Sylvester, Boole, and others of our leading men. 

The typography, arrangement of text, colour of paper, and 
figures leave little, or we would rather say nothing, to be desired, 
for readers can consult the book with comfort under almost any 
hight. 


XXXV. Intelligence and Miscellaneous Articles. 


ON THE ACTION OF THE DISCHARGE OF ELECTRICITY OF HIGH 
POTENTIAL ON SOLID PARTICLES SUSPENDED IN THE AIR. BY 
A. VON OBERMAYER AND M. VON PICHLER. 

ae purification of air from dust by electrical discharges has been 

observed by Aitken (‘ Nature,’ vol. xxviii. p. 322), and Lodge 

(Phil. Mag. [5] xvii. p. 214). The authors have examined the dis- 

charges of a double-influence machine in turpentine-smoke; this 

was contained in a glass tube, 11 centim in diameter and 111 cen- 
tim. in length, provided with brass mounts. The discharge took 
place between rods provided with combs at the ends, and which 
were supported in insulated mounts nearly parallel to the axis of 
the tube. The smoke was deposited in large flakes near the combs 
in less than a second. The spark of an induction-coil and of a 


302 Intelligence and Miscellaneous Articles. 
magneto-electrical machine acted more slowly. In like manner; 
when the brush-discharge was produced from sixteen points in the 
chimney-pipe of a stove towards the sides, the smoke was almost 
completely deposited, especially if the points were connected with 
the negative, and the pipe with the positive electrode of the ma- 
chine ; and two double induction-machines connected together were 
used. 

Kundt’s dust-circles were also obtained, when the points and 
the plate were connected with the poles of an induction-machine, 
and the discharge continued for some time. They are equal for 
the two electricities. If the metal disk is not dusted with sulphur 
or lycopodium until the discharge takes place, dust-rings are formed 
instead of dust-circles. Here also the rings are equal, whether the’ 
point is positive or negative. If the point is very close, dust- 
circles appear surrounded by an annular surface almost entirely 
free from dust, and again surrounded by slight dust near the edge 
of the plate. 

With a positive plate 215 millim. in diameter, with a fine nega- 
tive point at a distance s, the magnitude of the rings is as follows 
(D, and D; being the outer and inner diameter of the dust-circle or 
dust-rings, D that of the dusted surface :— 


Be 5 9) 45” 99, ko. | 5 
i. ee 30°55 “We 90 110° —. 12a 
Dee O50 BB ao 385 ee 

if eae 70 100 130 160 Jo18 aa 


The explanation of the various figures is ascribed by the authors © 
to the electrification of the badly-conducting particles of dust 
which lie on the plate, and which in consequence adhere to the 
plate as in Kundt’s figurec. If to this is added the electrical aura, 
it blows the particles away from the middle and lodges them 
towards the edge, where they adhere in consequence of permanent 
electrification. 

If, instead of metal plates, vertical networks of 1 centim. width 
of mesh are used, a horizontal point being brought to within a dis- 
tance of 7 centim., while the net and the point are connected with 
the conductors of two influence-machines connected with each 
other, and if sulphur-powder is dusted on the net while the brush- 
discharge takes place from a negative point, a dust-ring is formed of 
38 centim. external and 15 centim. internal diameter. If the powder 
blown through the net is caught on a horizontal paper, a tongue 
about 15 centim. in width appears free from dust, to which fol- 
low zones on each side of tolerably dense powder. With a positive 
point the same result is obtained, except that the dust-free zone is 
not so distinct. Lycopodium flies further away. 

If a fine point, and a wire gauze of i centim. width of mesh, is 
placed ata distance s parallel in the electrodes of an induction- 
machine, and these latter are adjusted at a distance D so that 
sparks alternately pass between the electrodes, and brush-discharges 
take place from the point, the following values were obtained for 


Intelligence and Miscellaneous Articles. 303 


the potentials V corresponding to s, D, and for V/D the following 
numbers :— 


s Eee a oe 20 30 40 50 centim. 
D Oh MND 2, Ot A At Ooo OO. 
0-3 V ee 28 44 60 69 74. 


mee) 13d 2a. 185" 12-1. 9:2 8-8 

The equivalent striking distances D approach a limit as s increases ; 
they increase more rapidly than the corresponding potentials. 

In the discharge of electricity from points towards a wire gauze, 
the velocity of the electrical aura could be determined by means of 
an anemometer placed behind the wire gauze. 

When the point was at a distance s = 7 centim. between the 
point and the wire gauze, the following velocities, in metres and 
seconds, were obtained :— 

ae Ls Oe. <6 LOD 157 207 centim. 
1g Oe See 1: FE <) 1:69) A- 19 C617 1.0 

Hence, by means of the electrical aura, fine particles of dust can 
be carried to great distances.— Wiener Berichte, xciii. p. 408 (1886); 
Beiblatter der Physik, vol. x. p. 641. 


ON A SIMPLE AND CONVENIENT FORM OF WATER-BATTERY. 
BY HENRY A. ROWLAND. 

For some time I have had in use in my laboratory a most simple, 
convenient, and cheap form of water-battery, whose design has 
been in one of my note-books for at least fifteen years. It has 
proved so useful that I give below a description for the use of other 
physicists. 

Strips of zinc and copper, each two inches wide, are soldered 
together along their edges so as to make a combined strip of a little 
less than four inches wide, allowing for the overlapping. It is 
then cut by shears into pieces about one fourth of an inch wide, 
each composed of half zinc and half copper. 

A plate of glass, very thick and a foot or less square, is heated 
and coated with shellac about an eighth of an inch thick. The 
strips of copper and zinc are bent into the shape of the letter U, 
with the branches about one fourth of an inch apart, and are 
heated and stuck to the shellac in rows, the soldered portion being 
fixed in the shellac, and the two branches standing up in the air, so 
that the zine of one piece comes within one sixteenth of an inch of 
the copper of the next one. A row of ten inches long will thus 
contain about thirty elements. The rows can be about one eighth 
of an inch apart, and therefore in a space ten inches square nearly 
800 elements can be placed. The plate is then warmed carefully 
so as not to crack, and a mixture of beeswax and resin, which melts 
_ more easily than shellac, is then poured on the plate to a depth of 
half an inch to hold the elements in place. A frame of wood 
is made around the back of the plate with a ring screwed to the 
centre, so that the whole can be hung up with the zinc and copper 
elements below. 

When required for use, lower so as to dip the tips of the ele- 


304 Intelligence and Miscellaneous Articles. 


ments into a pan of water and hang up again. The space between 
the elements being ~, inch, will hold a drop of water which will 
not evaporate for possibly an hour. ‘Thus the battery is in opera- 
tion in a minute, and is perfectly insulated by the glass and cement. 

This is the form I have used; but the strips might better be 
soldered face to face along one edge, cut up and then opened.— 
Silliman’s American Journal, February 1887. 


ON THE GALVANIC POLARIZATION OF ALUMINIUM. 
BY DR. F. STREINZ. 


Plates of aluminium were investigated by a method described in 
Wiedemann’s Annalen, vol. xxix. p. 181, and modified by opening 
and closing the polarizing-current by a tuning-fork instead of by the 
polarization-current itself. Phenomena were thereby observed 
which were very surprising. 

The original difference of potential between amalgamated zinc in 
a concentrated solution of zinc sulphate, and the bright metallic ~ 
aluminium plates of the voltameter, amounted to 0°32 volt. 

If, now, the latter were polarized, and first one and then the 
other plate compared with the zine plate at the electrometer, it 
was found that the difference of potential of the plate charged with 
oxygen increased within very wide limits with the electromotive 
force of the polarizing-cell, while the plate charged with hydrogen 
at small electromotive forces showed none, or only a very small 
polarization ; but at considerable electromotive forces, a difference 
of potential opposite in direction to ordinary hydrogen polarization. 

A few of the numbers found are collated in’proof of this. For 
the two polarizations (Zn/ Al+0O and Zn /Al+H.,), fifteen minutes 
after alternate closing and opening of the primary current by the 
tuning-fork, there was found for the electromotive forces of 


Zn/Al-+0. Zn/Al-+H,, 
Hem armel cco, 6 Ree 1:12 volt. 0-31 volt. 
Pe wmamiells 6e°) os. 1eQAe ie. 0°32 ,, 
oaniells 4 00.20. 0. igen has) O29 ies 
evemiellisy ees oe Bd 1M,, O36 
qe Maniells ee FA Bo2 0:49 ,, 
iMDanielis ees) eo TSO ARs 0:76 


99 

If the primary current was permanently open, the high values 
for oxygen polarization at once fell considerably, while the values 
for hydrogen polarization usually changed but little. It may still be 
observed that the disengagement of gas at the aluminium elec- 
trodes, even when the primary current has great electromotive 
force, is very small compared with that observed in other metals. 
The cause of this phenomenon is obviously to be sought in the 
great opposing force of oxygen polarization. To confirm this, 
measurements of the intensity in the primary current were made. 
On this, and on the further results of the investigations, a report 
will be made.—Kaiserl. Akad. der Wiss. in Wien, December 16, 
1886. 


THE 
LONDON, EDINBURGH, ayn DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[FIFTH SERIES. ] 
AV ei he VSS i. 


XXXVI. On the Assumptions necessary for the Theoretical 
Proof of Avogadro’s Law. By Prof. Lupwia BourzmMann 
(of Graz).* 

er OR TAIT+, by making a number of special 

assumptions, has given a very exact proof of Avogadro’s 

Law—or, rather, of the proposition that in the case of the 
heat-equilibrium of two mixed gases the mean potential 
energy of the molecules must be equal. Prof. Tait does not 
appear to me, however, to have given the least proof that the 
special assumptions which he makes as the basis of this pro- 
position are necessary, or even that the more general pro- 
positions stated by myself { and Maxwell § are incorrect. 

I believe that in my calculations I have stated precisely 
the assumptions made, and have invariably drawn the logical 
conclusions from them; and therefore that I have not 
deserved the reproach which Prof. Tait makes, that they are 
“rather of the nature of playing with symbols than of reason- 
ing by consecutive steps,” which must also apply to the just- 
mentioned treatise of Maxwell, when he accepts and further 
develops the propositions of mine upon which Prof. Tait throws 
doubt. The contradiction of these propositions by certain expe- 
rimental facts seems to me to have resulted only from too great 


* Translated from an advance proof, communicated by the Author, 
from the Sitzber. d. Wien. Akad. d. Wissensch. vol. xciv., having been 
read at the Meeting on the 7th October, 1886. 

t Phil. Mag. [5] vol. xxi. p. 343 (1886). 

1 Sitzber. d. Wren. Akad. d. Wissensch. vol. lviii. (1868); vol. Ixiii. 
March 9 and April 18, 1871; vol. Ixvi. October 10, 1872. 

§ Camb. Phil. Trans. vol. xii. part 13, p. 547 (1879). 


Phil. Mag. 8. 5. Vol. 23. No. 143. April 1887. OG 


306 Prof. L. Boltzmann on the Assumptions necessary 


a generalization thereof. We must remember that analysis 
can deal only with systems more or less analogous to the 
molecules of nature, but not with the molecules themselves. 

The behaviour of hot bodies is certainly influenced by heat- 
radiation, probably also by movements of electricity &.— 
conditions which have not been taken into account by me, 
nor in any other mechanical theory of heat. Absolute agree- 
ment with facts cannot therefore be expected. It is therefore 
only possible to consider (1) whether the propositions stated 
really follow logically from the assumptions made, and 
(2) whether the analogy between the properties of the 
system considered and those of hot bodies undoubtedly exists. 

This analogy can only be made perfectly clear by reference 
to all my papers, and in particular to those on the second law 
of Thermodynamics. 

Although Mr. Burbury* has made well-founded objections 
to the propositions of Prof. Tait, it appears to me desirable to 
discuss still more rigorously the question, Which of Prof. 
Tait’s assumptions are really necessary to the proof? In doing 
this I will follow Prof. Tait’s method, and at first will treat 
only some special cases, in order not to become unintelligible 
by too great a generalization. Although he does not expressly 
say so, Prof. Tait yet implicitly assumes that two molecules 
upon impact behave like elastic spheres. For under any other 
law of mutual action Prof. Tait’s equations on p. 346 would 
only hold good in case the quantities which he denotes by u 
and v were the components in the direction of the apsides ; and 
the calculations on p. 847 would then not be applicable to 
these equations. I will make the same assumption, and use 
the notation of my Theory of Gaseous Diffusion, part i.T 

I have there treated the impact of two elastic balls as 
generally as possible with reference to the Theory of Gases. 
The figures and formule are certainly a little copious, but 
are generally applicable, and, as I believe, also clear, when 
once their meaning has been comprehended. 

Let two elastic spheres (molecules) of masses m and M 
impinge upon each other. Let v=Qv and V=OV be their 
velocities before impact (see fig. 1), and v/=Qv! and V'=OYV! 
their velocities after impact, 6 ‘the sum of radii of the spheres, 
rv=vV their relative velocity before impact, and suppose their 
relative velocity after impact to have the direction 7=v'V’, 
Let QC be the line of centre of the spheres at the moment of 
impact. 


* Phil. Mag. [5] vol. xxi. D. 481 (1886). 
+ Sitzber. d. Wren. Akad. d, Wissensch. vol. lxxxvi. p. 63 (June 1882). 


for the Theoretical Proof of Avogadro's Law. 307 


Let ¢vV=T, f{eor=G, ¢{rr'=28, <7, OC=90°4+S, 
ToN =. ter = Ge 


Fig. 1. 


Let O be the angle of the planes 7, v, and r, QO; and O’ 
the angle of the planes 7”, v', and 7’,QC. I denote the sines 
and cosines of the angles by the corresponding Roman and 
Greek letters. For the sake of clearness I include in fig. 1 
those lines of fig. 2 of my Theory of Gas Diffusion, just 
mentioned, which are here necessary. Then, according to 
formula (21) of p. 72 of that treatise, 


_ mv”? mv 2mMovr 


ee 


2mM? ro" 
pet pn ess eae y EES = = ee ee 
2 Dns ek mM be? (m+M)?° ~ (1) 


Sah 


808 Prof. L. Boltzmann on the Assumptions necessary 
The right-hand side agrees completely with the expression — 
which Prof. Tait finds on p. 346 for cea | since, 


according to equation (20) of my treatise, the quantity which 
Prof. Tait denotes by-u has the value v(—ga+vyso), where 
u—v is the projection of 7 on QO, which is equal tore. Now 
let there be very many molecules of mass m (molecules of the 
first kind), as also of those of mass M (molecules of the second 
kind), uniformly mixed in a space and uniformly distributive 
with reference to the directions of their velocities. Of all 
possible impacts which in general may occur, we will first 
select only those for which the variables v, V, T, 8, O lie 
between the infinitely close limits, 


vandv+dv, Vand V+dV, T and T+dT, (2) 
Sand S+ds8, Oand 0+d0. t 


In fig 1 a sphere of radius 8, supposed concentric with 
~ one of the molecules of mass M, is drawn. All straight lines 
are drawn dotted from their intersections with the surface of 
this sphere. The arcs are arcs of great circles of this sphere. 
The molecules of mass m fly against this sphere, so that their 
centres describe the straight line AC relative to the first 
molecule which is parallel to vV and OR. At the point C 
the centres are reflected in the direction CB, which is parallel 
to v'V and OR’. The relative motion before impact, with 
which alone we are concerned, remains unaltered if we imagine 
the molecules of mass m at rest and the sphere drawn in fig. 1 
moving with the velocity r=vV in direction opposite to OR. 
If its surface be divided by a plane passing through its centre 
at right angles to QR, then, with our last-mentioned con- 
ception of relative motion, the preceding haif-sphere would, 
in unit time, pass through a space bounded by two half-spheres 
and a cylindrical surface of volume 7é’r. Of this whole 
space a small portion is now to be cut out, as follows :—OQC 
is that radius of the sphere which is parallel to the direction 
of the line of centres. Whilst the point C is so moved on the 
surface of the sphere that 8S increases by the amount dS without 
change of direction of QR, C describes a linear element of 
length éd8._ If, on the other hand, we move C so that O 
increases by dO, C describes a linear element of length dsdO. 
These two determine an element of area on the surface of the 
sphere of area 6°sdSdO. This is inclined at an angle 8 to the 
direction of r. Since the half-sphere moves with the velocity 
r in the direction —QR, this element of surface moves through 
a prism of volume ré’sadSdQ. So soon as the centre of a 


for the Theoretical Proof of Avogadro’s Law. 309 


molecule of the first kind lies within this prism, it impinges in 
the manner described upon a molecule of the second kind. 
Of all molecules of the first kind lying within this prism, 
we have to consider only those whose velocities lie between 
the limits v and v+dv, and which are moving in directions 
making angles with the direction of V which lie between 
the limits T and T+dT. If there are in the unit volume 
Arv’/(v)dv molecules of the first kind which fulfil the first 
condition, then, in the above-described prism, there are 


2rv"f(v)ré*rsc . dvdS dT dO 


molecules of the first kind which also satisfy the second con- 
dition. 

Let us further suppose that in the unit volume there are 
AmwV?F(V)dV molecules of the second kind whose velocities 
lie between the limits V and V+dV. Then, by multiplying 
the above expression by this factor, the followi ring expression 
for the total number of impacts which occur in unit time in 
unit volume between a molecule of the first kind and a 
molecule of the second kind, so that the conditions (2) are 
fulfilled, 


AL=8r'v’V? fv) EF (V)re'tsodvdV dT dSdO,. . (8) 
is obtained. 

In order to obtain from this the number Z of all the 
impacts which may occur altogether in the unit volume in 
unit time between a molecule of the first kind and one of the 
second kind, we have to integrate with reference to O from zero 
to 2a, with reference to T from zero to 7, with reference to S 


T . 
from zero to 5? and with reference to v and V from zero 


to «© ; which we may express by Z=\dZ. 
J = (0 dZ is the energy which is brought by all these 


impacts to molecules of the first kind. If we integrate the 
expression D dZ first with reference to O and 8, we obtain 


erie 5 oe mM ( Maan, 
8rv"*f(v) . V7 ECV )rd?r de av alo mg +a) . (4) 


whilst dZ integrated with reference to the same variables 
gives 


Sav" f(v) V7 F(V) rd? du dv aT. 


The quotient of the two expressions may be considered to be 
the energy which, on the average, is transferred from mole- 
cules of the first kind to molecules of the second kind, when 


310 Prof. L. Boltzmann on the Assumptions necessary 


the magnitudes of the velocities v and V and their angle T 
are given. This quotient is 


mM. Mr? ] 
ea aan , 

This agrees exactly with the first expression for 1,/—h, 
which Stefan* finds in his treatise on the dynamical theory 
of the diffusion of gases towards the end of the second part, 
and which, after correction of some printer’s errors, 1s as 
follows :— 


2m. Be 


for, in Stefan’s formula, 


ie = — 


ve V2 ; 
L=m 9? i,=M g7 te + YYot+ yeo=vVi=vrg tr, 
and 
VW=v' +r? + org. 

But the first conclusions of Stefan’s would apply only 
where the molecules act upon each other with a force in- 
versely proportional to the fifth power of the distance. In 
this case the factor r disappears from the expression dZ, and 
on further integration in respect of dT the terms 

MX9 + Y1Y2 + £42 
disappear. 

With elastic spheres, on the other hand, the impacts are so 
much the more probable the greater the relative velocity r ; 
therefore, for each of the products x,t, yi, 422, positive 
values are more probable than negative values, and we cannot 
therefore suppose 
&yLy t+ YYot 242. =0 
on the average. In fact, if we wish also to integrate express- 
sion (4) with reference to T, we must observe that 


re=vr+V2—2v0Vti and rg=Vt—v. 
Integration gives, therefore, 
87°? mM V?—v?  M—mr?) , 


the upper limit is V+v, the lower V—v or v—V according 
as V>v or the reverse. In the first case the integral has 
the form | 


M—m 
— dy? —2y°V2 4+ 0Vi+ Meer, (Je? + 20°V?-b oN") ae 


* Satzber. d. Wien. Akad. d, Wissensch. vol. Ixy. April 1872. 


jor the Theoretical Proof of Avogadro’s Law. 311 


In the latter case V and v must be exchanged. If, there- 
fore, the functions f and F are chosen quite arbitrarily, 
\DdzZ will in general not disappear if the molecules of the 
first and second kinds have equal energy. In other words, 
even though the molecules of the first and second kinds have 
equal energy, yet, if f and F are chosen arbitrarily, energy 
may at first be transferred from molecules of the first kind 
to molecules of the second kind, or vice versd. It would be 
transferred to them continuously if by any external action 
the originally chosen values of the functions f and F were 
maintained constant. 

If, for example, with uniform distribution in space and 
equal probability of all directions of velocity, the molecules of 
the first kind had all the same velocity v, the second kind all 
the same velocity V>v, then there would be at first no 
transference of energy to the molecules of the first kind by 
the molecules of the second kind if expression (5) should 
disappear. If we put the positive quantities 


3Ma2=2m—M + / ; (5m2—2mM +2M2). 


If m is only a little larger than M, say m=M(1+e), then 
this gives nearly 

2 mad EY: 

MV?=mwv (145) ; 


for m=4M, we have 
Bi eg 77s 0 ear h ie tae 
If m is very much greater than M, we have nearly 
MV?=4mv?. 


The condition that, on the average, no energy shall be 
communicated in the first moment to the molecules of the first 
order, requires then that the molecules of smaller mass should 
have greater energy than the others. But it must be observed 
that this only holds good for the first moment ; the velocities 
even of the molecules of the first kind immediately become 
different amongst themselves as the result of impacts, and so 


312 Prof. L. Boltzmann on the Assumptions necessary 


also those of the molecules of the second order; so that 
the conditions at once become completely altered. 

If we assume Maxwell’s law of distribution of velocities, by 
putting f(v) =Ae”, F(V)=Be~’”’ we find, by precisely the 
same calculation as that of Prof. Tait, that there is no exchange 
of energy between the molecules of the first kind and those of 
‘the second, if the mean energy of both is the same—which, 
(following Prof. Tait) we will call Maxwell’s Theorem. 

So far all Prof. Tait’s conclusions are, without doubt, cor- 
rect. But it does not yet follow from this that the existence 
of Maxwell’s law of distribution must be assumed in order to 
prove Maxwell’s theorem. It may rather be shown that, 
whatever the masses and ratio of diameters may be, if only the 
molecules of the first order come generally into collision with 
those of the second order, then Maxwell’s distribution of velo- 
cities is spontaneously brought about both amongst the former 
and also amongst the latter molecules. 

In this it is not even necessary to assume that the molecules 
of the first order are generally in collision amongst themselves, 
nor that the molecules of the second order are in collision 
amongst themselves. The only assumptions are: that both 
the molecules of the first and also those of the second order 
_ are uniformly distributed over the whole space; that through- 
out they behave in the same way in all directions; and that the 
duration of the impact is short in comparison with the time 
between two impacts. I have given* the proof of this in my 
paper on the Thermal Hquilibrium of Gases acted on by Ex- 
ternal Forces, at the conclusion of § 1; but as I have there 
only briefly indicated the mode of calculation and have only 
given the result, Prof. Tait has probably quite overlooked the 
passage, and I will be more explicit on the present occasion. 

Let the molecules of the first and second kinds of gas be at 
the beginning of the time (¢=0) uniformly distributed in a 
space enclosed by rigid perfectly elastic walls; but let the 
distribution of energy among them be altogether arbitrary. 
Exactly as before, let 

Anv’f(v, 0) dv 
be the number of molecules of the first kind of gas in the unit 
volume whose velocities lie between the limits 

v and v+dv. 6) es has 
In exactly the same way, let 47 V’F(V, 0) dV be the number 
of molecules whose velocities lie between 


V and V4av."°. ) oe 


* Wren. Sitzd. vol. lxxii., October 1875. 


for the Theoretical Proof of Avogadro's Law. 313 


Since no direction in space, and also no element of volume, 
has any advantage over another, we may assume that the dis- 
tribution of energy also remains uniform for all succeeding 
time. But in general this is prevented by the collisions 
which occur. At the time ¢, let there be in the unit volume 
Amv" (v, t) dv molecules of the first kind of gas whose velocities 
lie between the limits (6), F(v,¢) having a similar signifi- 
cation. 

Hvidently the problem is conceived in its utmost generality 
if we imagine /(v, 0) and F(V,o0) having any given values, 
and determine the changes of these functions in course of 
time. Evidently we have first of all to determine the increase 


a t) oad gol (Y, t) 


ot 

which the functions f and F undergo, whilst the time increases 
from ¢ tot+6. During the element of time 0, let n molecules 
in the unit volume out of the 47rv?/(v, t) dv molecules of the 
first kind therein contained, whose energies lie between the 
limits (6), enter into collision with other molecules of the first 
kind, and N molecules with those of the second kind. Let 
us imagine @ so chosen that 7 and N, although large numbers, 
are yet small in comparison with 47rv’ f(v, t) dv. 

Since the number of those molecules for which the velocities 
after impact also lie between the limits (6), or for which the 
velocity of the second impinging molecule lies between the 
same limits, is of an order higher only by an infinitely small 
amount, we may assume that the velocities of all these  mo- 
lecules and also of the N molecules after lapse of the time 0, 
no longer lie between the limits (6). +N is therefore the 
number of those molecules whose velocities were at the be- 
ginning of the time @ between those limits, but at the end of 
that time were not between the same limits. 

But during the time @, other molecules whose velocities 
were not previously between the limits (6), in consequence of 
impacts acquire a velocity lying between these limits. Let, 
then, p molecules of the first kind acquire a velocity lying 
between these limits by impacts with other molecules of the 
first kind, and P molecules of the first kind by impacts with 
molecules of the second kind. Then 


4it9 VOD <p + P—n—N, ga et Saag 


We found before the expression (3) for the number of 
impacts which in unit time and unit volume so occur that the 
variables v, V, 'T, 8, and O lie between the limits (2). 


314 ~~‘ Prof. L. Boltzmann on the Assumptions necessary 


Integrating this expression for all the other differentials up 
to dv and multiplying by 0, we have 


0©¢0 

The kind and mode of impact is completely determined by 
the values of the variables v, V,T,S, and O. The magnitudes 
vw’ and V’ of the velocities of both molecules after impact, their 
angle T’ as well as the angle O/ which the plane ROR’ of the 
two relative velocities makes with the plane v'Q/V’ of the two 
absolute velocities vu’ and V’ after the impact, may therefore 
be expressed as functions of v, V, T,S,and O. For all impacts 
for which the latter variables lie between the limits (2), 
vw’, V’, T’, 8’, and O’ will also lie between certain infinitely 
close limits, which we may denote by 


v and v'+dv'’, V‘ and V’+dV’, 
T’ and T’+dT’, O/’ and O’+d0’, Sand 8+d58. . (10) 


The angle 8 has the same meaning before and after impact, 
and is therefore after impact also included within the limits 8 
and S$+d8. But it is now clear that each impact may also 
be taken in inverted order. If, therefore, inversely the values 
ot the variables before impact lay between the limits (10), 
then they would also after impact lie between the limits (2). 
Exactly as with expression (3), so also will — 


dL! = 8v"V"? fv! ,t)E(V’, t)0r 821! so du! dV’ dl’ dO'dS! (11) 


be the number of impacts which, in the unit volume during 
the time @, occur between a molecule of the first order and 
one of the second, so that after impact the variables shall lie 
between the limits (2). Since v’, V’, T’, and O’ are known 
as functions of v, V, T, O, and S, we may here again intro- 
duce the differentials of the latter variables, and obtain] 


AL! = 8770? Vf (v',t) F (V’, t) Ord’7’sco AdvdV dTdOd8, (12) 


where 


N= 80? /(v, t) dv 8°0 i ("| { "WF (V,t)rrscdV dT dSdO. (9) 
0./0 


av G0) 4 Ol Oe, 

A=2t 30 0V aT 30? 

the partial differentials being taken under the supposition 
that 8 is constant. If we integrate expression (12) for all 
variables except dv, we obtain all impacts which occur in unit 
volume during the time 9 between a molecule of the first and 
a molecule of the second kind, so that after impact the velo- 
cities of the molecules of the first kind lie between the limits (6). 


“{43) 


for the Theoretical Proof of Avogadro's Law. 315 
Hence 7 


P= 8r°63%do { : { j ("wv fv’, t)F(W/, t)rr'soA dVaTaSdO. (14) 
0 00 0.0 


Maxwell* has already given an equation which in our 
notation is 


DAN (UN Tel aie AES SUS weer ys CUO) 


He has, it is true, indicated a proof of this equation, but 
has not clearly shown its truth. 

I have proved this proposition more fully, and obtained a 
similar one for polyatomic molecules ; on which account I 
have also assumed it in my treatise on the thermal equilibrium 
of gases upon which external forces act. The most general 
proposition, in which all similar propositions are included, is 
shown in Maxwell’s paper “ On Boltzmann’s Theorem.” ft 
But since all the necessary formule are now at our disposal, 
I will here verify equation (15) by direct calculation of the 
functional determinants. 

If we first of all introduce, instead of V and T, the variables 
ry and G, we obtain V*7dVrdT=?r"drydG, as is geometrically 
evident. So also 


V?dV' aT’ =r'dry'dG’. 
Since 7 is not altered by impact, we introduce, instead of dv, 
dG, dO, the differentials of the three variables, 
AMvur, , AM?7?a? 
pee M (go —Yyso0) + (m+ My” 
y=(m+M)v?+ 2Mourg + Mr, 


Z=vYyO. 


fav yt 


The calculation of the functional determinants gives 
AMv*ry 
m+M 
Further, from the spherical triangle RK FR’ (fig. 1), we have 
® : = sin ROK: sin ROK ;W 

and from the triangles POQw and POv’, 

sin ROK : y =v: OF, 

Sim by QIK Fry’ ==o OP ; 


du dy dz= [ (m+ M)o(2gso—ys’0 + yoo) + 2Mrsa|dv dG dO. 


* Phil. Mag. [4] vol. xxxv. (March 1868). 
7 Cambridge Phil. Trans. vol. xii. part 3, p. 547 (1879). Wied. Bevbl. 
v. p. 405 (1881). 


316 Prof. L. Boltzmann on the Assumptions necessary 
therefore 

vyo =v'y'o' =z; 
whence 


|2_,/2 


INDO ID OS a Hes 2 
(v'9/0')? =v? —v"9" —v yo { vase ySO-+ YOO) + ———e 
Therefore 

da dy dz=4Mv'v'ryy'o'dv dG dO. 
Since, further, 7 

e=v", y= (m+ M)v? + 2Mo'rg + Mr’, z=0'qo’, 

it follows that 

da dy dz=4Mv ryo'dv'dG'dO’; 
whence the relationship to be proved between dvdGdO and 


dv! dG! dO!' follows. But, according to equation (15), it 
follows from (14) that 


. © Pr(°2 C2 | 
P= ameatado | { i) { V2 ful, t) F(V', t)rtsodV dT dS dO. (17) | 
0 oY 0e¢0 { 


n and p can be found from N and P by simple exchange of 
the function F and the mass M for fand m, and in place of 
6 we have A the diameter of a molecule of the first kind. | 

Hence 


© Cr (2 C20 
n= 877" /(v, t) dv ro ( { { V/V, t)rtsa dV dT dS dO, 
7~0v¢%0¢a~0 


0 


‘ (18) 
2 LEC2 C27 
p= 8n'vdv n20( i} ) ) V7f(u', O/V',Ortsc dVdT dS dO, 
0 Jowo,Jo 


and thence 


oe = 2nd? { : { : { i} "(ft —ff,)VrrsedV dT dS dO 
0 0 0 0 


2 pT t Qer 
+ 2nd { i} i} (f'F'| fF) V?rrso dV dT dS dO, 
0 0 020 


= 7 : 20 
ome = ann { . i} (E’E" —FF,)v?rrso dvdTdSdO 
0 0 0 0 


O (1 5 Qar 
+ 28 ( ( { (fF —fE,)v*rrso dv dT dS dO. 
©o0 «<0e0 


(19) 


(20) 


for the Theoretical Proof of Avogadro’s Taw. 317 


In this f=/v,t), A=AV,t), f=fo',d), fl=V',0, 
ian), We, 2), Fy (V,2), FI=F(V', 2)... A is 
the diameter of a molecule of the second kind. 

By means of these equations we can show that the quantity 


B= | “Ay—1y'a+ ( “RE.-DVv-dv . . (21) 


(whose intimate connection with the magnitude called by 
Clausius “ Entropy’? [have remarked elsewhere) can only de- 
crease or (in an extreme case) remain constant. J/ denotes 
the natural logarithm. We have 


By Lotavs (ye, vay, 
Oh Fs Wiest Cay fh, Dt 3 


therefore, with reference to the equations (20), 


1 3E 7} 
Qa Ot | 


T 


={ fe f° ((" NP. (fF | ff) V2rrso dv dV dT dS dO 

0 0 0 0./0 

+ ( ; { ’ ( : { { "AE, (FF —FE,)e2V2rrsodvdVdTdSdO ' (22) 
eo 0 e090 0 0 


i i ; (" { i { - “SI PE! —fE,eVerrse dv dV dT dS a0 | 

0 Jovovoro0 | 

¥ ( ; (" ( ‘ ( ‘ ( “SF, (fF —fF,)v°V2rrsadv dV dT dS do ! 
e 0 e920 


e0 ¢0 


From the circumstance that every impact may take place 
also in inverted order, it follows at once that the value of any 
definite integral which, according to the above, is to be 
extended over all possible values of the variables before 
impact, does not alter if we exchange the value of the variables 
before impact for their values after impact, and vice versd, 
and, finally, again integrate for all possible values. 

We shall have, therefore, for each function Y compounded 
in any way of the variables under the functional sign, 


318 Prof. L. Boltzmann on the Assumptions necessary 


or ae cic 4 2a 
( ( (* ('v%, V, T, 0, o', W, 8) dvdV dT dS dO 
O00 aL 


2 0 ‘ 
s Ds" far. 5 2 
=( ( ( ) V(v', V’, T’, O' », V, T, O, 8) Adu dV dV dO ds @| 
e a a 0 | 


0 Jo. 
where 
Ne veN er eV 27. 
Therefore the third line of the equation (22) is also equal to 


SY eer: = an 
( ( { (| bf’ (fF, —f' Fy) v? V2rrso dv dV dT dS dO ; 
e0e0 0 '0 v0 | 


also the fourth is equal to 


To © pr = 27 
( { (" OF, (fE,—/' Fy)v?V2rtsc dv dV dT ds dO. 
v0 /0 02/0 .J0 


For the sum of the third and fourth lines we find, by 
taking the arithmetic mean of the expressions found now and 
previously, 


2{ { ((? “3?(f By —f Fy) 1 1 v°V'rrsa dv dV dT d8 dO. 
0 0 0,/0 0 fF, 


The same transformations are to be applied to the first and 
second lines of equation (22). But in the latter expressions, 
moreover, the two impinging molecules play exactly the same 
part ; so that we may here exchange the quantities which 
refer to the first molecule for those which refer to the second, 
and vice versd. If both molecules belong to the same order, 
we have again, generally, 


oo Cay 5 20 
( i} { ty Vv, V, T, O, 2’, V’, T’, O', 8) de dV dl dS dO 
0 0 7020 oI 


eo car : Qn 
ce ( ( { ) : ( V(V, v, T, 0, Vv, T, O', 8) dvdV dl d8 dO. 
30 = Jo; e/0 <).0<)0 
Therefore the first line of equation (22) may also be written 
ee 0 Cr = 27 
{ \ { ("| MACS A’ ff) v*V2rrso dv dV dT dS dO. 
0 Yo JoJo Jo 


If we once more apply to this the transformation mentioned 
on the previous page, we obtain a fourth value for the first 


for the Theoretical Proof of Avogadro's Law. 319 


line of equation (22); and we have now to substitute the 
arithmetic mean of all four values. If we treat in the same 
way the second line of this equation, we obtain finally 


ie =| ("("¢ rc v?V2rrsa dv dV dT dS dO 
aw dt Avi Ue 


: EF’ F ! 
x MOAT) ot a A2 (EB — PE) =e WU FY 


fF 
$93%(f' Fy -fF) . ae 
For the condition of equilibrium of energy, fand F and 
also KE must therefore be independent of the time ; we must 


therefore have —(0. But we see at once that the last-found 


integral for a represents a sum of infinitely small members, 
which are all negative or at tie ee to zero; for if 


t' fi —ff, 3s positive the factor yd. Fi = is negative, and vice 


vers. 


Therefore = can only vanish when each of these members 


itself vanishes. 

If the molecules of the first kind were very small in 
comparison with those of the second kind, then would 
rA=0, A=6d. We may assume, still more generally, that 
the molecules of the first kind are perfectly permeable for 
each other, and so also those of the second kind for each 
other, and that each of the latter molecules onlv is surrounded 
by a sphere of radius 6, at which the centres of the molecules 
of the first kind are reflected like infinitely small elastic 
spheres ; in the latter case we should have A=A=0. As 
soon as 5 differs from zero, /’ Fy’ must always be equal to 
FF, for all values of the variables under the functional si on. 

ince v, V, and v! are quite independent, and only V’ is 
determined by the equation of energy, we find, without 
difficulty, 

f= Nes [Pie Rye aN 


But, in consequence of the impact of molecules of the first 
kind with those of the second, there will therefore be pro- 
duced Maxwell’s distribution of velocities and equality of mean 
energy amongst all the molecules. I take the present oppor- 
tunity of remarking that I do not understand how gravitation 


320 Prof. L. Boltzmann on the Assumptions necessary 


is to be explained if we ascribe to Lesage’s ultramundane 
particles the properties of gaseous molecules ; for, whilst the 
sun protects the earth from certain impacts, it reflects other ; 
particles towards the earth which would not otherwise have 
reached it. Only, if the particles are not reflected by the ~ 
earth and sun, but absorbed or perhaps reflected with a loss 
of energy, would attraction be produced between the sun and 
the earth. 

I have carried out calculations similar to the above in my 
treatise, ‘‘ Further Studies on the Thermal Hquilibrium of 
Gaseous Molecules,’ * for a single kind of molecules after a 
somewhat different method, which, I believe, is characterized H ' 
by great clearness; and I venture to recommend to the | 
reader who is so disposed a perusal of the first section of this 
treatise. 

We can easily apply the method there described to a mixture 
of two kinds of gas. For the first of these let »/x . 6(a, t)da 
be the number of molecules in the unit volume whose energy 
at the time ¢ lies between the limits x and #+dz. Let 
(x,t) have a similar meaning for the second kind of gas. I 
have then used 


J/ aX (a, t)ded(X, t)dXw(ex, X, 2')da’ 


- to express the number of impacts which occur in unit time 
and volume between two molecules of the first kind, so that 
before impact their energies shall lie between the limits 
xand «+dz, X and X+dX, whilst after impact that of the 
one molecule lies between «' and #' + da’. 

By variable distribution of conditions we are always to 
understand the number of impacts during a very short time 
divided by that time; has a similar meaning for the impacts 
of the molecules of the first kind with those of the second 
kind. 


Then we easily convince ourselves that 


ee ee 


Telly Oe sata an ya aes 


Vr et Bt) aii fi dXda'| $(a', t)d(a#+ X—a’, t) 


Val(a + X—2')(a’, e+ X—a', x2) — h(a, t)b(X%, t) VaxX(2, X, a) 
+ (a, t)P(a+X—a', t) x Val (a+X—a'!)y (2, 2+ X—2', 2) 
—(a, t)O(X, t) WzXy(e, X, z')] 


PD ae 
wherein we obtain / rg See ‘ 2, by interchange of @ and ®, 


eae epee na 


—w 


ag Rates wag 


— ee 


* Sitzber. d. Wien. Akad. d. Wissensch. vol. lxvi., October 1872. 


for the Theoretical Proof of Avogadro's Law. 321 
wand X,yand VW. Inplace of w' we have X’, the energy of 


the second molecule after impact. 
We have further, except for a constant factor, 


=|, Nala, t)[1p(2, t) —1]da | | 
i { ‘VX@(K, t)[16(K, t)—1]ds, | (24) 


= =") Vaiies0) a 2D dat (. VXI(X, i) SP Oa, 

0 

Before we substitute the above values in equation (24), 
we have still to establish two properties of the functions w, 
W,and y. In the functions ~ and V both impinging mole- 
cules play the same part. Itis then just as probable that 
before impact the energy of the first should lie between « 
and #+dz, that of the second beween X and X+dX, and, 
after impact that of the first between 2 and «'+dz’, as that 
inversely before impact the energy of the first molecule should 
lie between X and X + dX, that of the second molecule between 
ze and «+dz, and after impact that of the second between a! 
and w'+dz'. Therefore that of the first lies between «+ X—.! 
and #«+X—.z'+dz'. Or, in algebraic language, 

(tN eo) W(X, # c+ X—ey,. . .. (2d) 
W(X, a, X\)=WV(e, X,2+X—X'). . . (26) 

The second property may be obtained as follows. We found 
the value (3) for the number dZ of impacts which occur in 
unit time and volume between a molecule of the first and one 
ot the second kind in such a way that the variables v, V, T, 
S, O lie between the limits (2). We will first introduce ‘the 
variables 7, G instead of V and T. Then, instead of G, twice 
the energy of both molecules, 

=(m+ M)v? + Mr? + Mung = (m+ M)v? + Mr? + 2Mo'rg’; 
this gives 


ioe ii = ur’scdvdrdydSdOf(v) F(V)&. 


sp 


We will now for the constants v, 7, y, and 8 introduce the 
variable v’ instead of O. Since in this g is also constant, 


AMvur AM?r?0? 


A ee he 
= mim 9° ysoo) + bi (m+ My? 
d7= ee BREET OF al dard Sitar 


Phil. Mag. 8. 5. Vol. 03, No. 143. April 1887. Z 


322 Prof. L. Boltzmann on the Assumptions necessary — 


If, lastly, instead of v, vo! we introduce the energies « 
and a’, 


ia m+ M Q17r8? 


n?M? vyo 
Now 4arv®/(v)dv= “/x2$(2, t)dx ; therefore 
mV m MVM 

= = i), EX ) = = 

fo) = @% $a, 8), FW) = Fs 

then, since for constants wand 2! evidently dy=dX, 


it m+M $ 
~ 16 VmM 


dadaldydrd8f(v) F(V). 


P(X, t) 3 


2 
aZ (a, P(X, #) oe dnd dXard8. 


Hence the quantity formerly denoted by V wXy (a, X, 2’) 


is equal to 
_m+M a ( (es 
16 ¥mM vy@ 


where the integration is to be extended over all possible values 
with the given x, X, a’. If we interchange the values before 
and after impact, we have 


vy’ ® 


et 1+M 
V «(a+ X—2')y(a', e+ X—a', 2) = — #( rdrd 


Since according to equation (16) v'y'o'=vye, and also the 
limits of the two double integrals are the same, it follows at 
once that 


V aXe, X, 2')= V a (a—X—z2')y(2', e+ X—2', x). (27) 


Two analogous equations hold for y and V. 

The further calculations are now purely algebraical trans- 
formations of definite integrals, and are effected exactly as in 
the first section of the already mentioned “ Further Studies.” 
I will therefore only briefly indicate the method to be adopted. 
After substituting the values of ae and OP(X, 2) a 


fe) Ot 


equation (24), we have a term with 


Ip(z, t). Loe, t)o(a+X+2',t)—(a, t) P(X, t)]. 

This, with the aid of an equation for Wy analogous to equa- 
tion (27), is to be transformed into a similar term, with the 
factor —Id(2#’, t) before the square bracket. Both are, by 
means of equation (25), to be transformed into two terms, 


having the factors /f(X, ¢) and —/ (a+ X—z’', t) before the 


for the Theoretical Proof of Avogadro’s Law. 323 


square bracket. The arithmetic mean of the four expressions 
thus obtained is to be substituted in equation (24) for the 
transformed term, The term 

I@(X, t) . | P(X’, t)B(w+ X—X, t) —B(X, t) B(x, t) 
is to be treated in exactly the same way. 

The remaining terms have now only to be once transformed 
by means of equation (27), so that the factors —/d(z’, ¢) and 
—l®(“x+ X—z2',t) appear instead of the factors /d(x, t) and 
I@(X, ¢); and again the arithmetic mean of the original and 
transformed expressions is to be taken. In this way again ~ 
oe proves to bea sum of terms of which each vanishes sepa- 
rately if a vanishes. afr and VY may be equal to zero. As soon 
as x vanishes for no definite set of variables, that is only so 
soon as the molecules of the first kind impinge freely upon 
those of the second kind, these terms only vanish separately if 
Maxwell’s distribution of velocities holds amongst the mole- 
cules of the first kind as well as those of the second kind. 

The foregoing considerations serve also to completely 
establish the still more general proposition which Mr. Burbury 
has stated in the place already referred to. In fact, let us 
assume that the first kind of gas consists, as above, of very 
many molecules (molecules A) which do not impinge amongst 
themselves ; but that the second kind, on the other hand, 
consists but of a single molecule B which comes into collision 
with the molecules of the first sort. Let the time of its free 
motion be great in comparison with the time of a collision. 
Let the whole be enclosed in a vessel R with rigid elastic 
walls. Let R denote also the volume of the vessel. There 
must at length ensue a stationary condition in which the 
molecules of the first kind are, on the average, uniformly dis- 
tributed through the vessel, and in which any direction is as 
probable as another for their velocity. In this condition, let 
there be in unit volume 4zvf,(v)dv molecules whose velo- 
cities lie between the limits (6). The molecule B will of 
course continually change its velocity; but if we take a very 
long time after the commencement of the stationary condition, 
then during that time its velocity, on the average, will, with 
equal probability, have assumed all possible directions in 
space; and the probability that its velocity will lie between 
the limits (7) will be expressed as some function of V, which 
we will denote by 


AV?RE, (V)dV. 


Let us now imagine a very great number of similar vessels 
1,2 


324 Prof. L. Boltzmann on the Assumptions necessary 


R, in each of which let there be an equal number of molecules 
A constituted as before, and each with the same distribution 
of velocities. Let 4arv’f(v)dv be the number of molecules in 
the unit volume whose velocities lie between the limits (6). 
In each of the vessels R let there be a single molecule B 
which in each vessel shall be now here, now there, now moving 
in this direction, now in that with equal probability. Let the 
number of vessels R for which the velocity of the molecule 
B shall lie between the limits (7) be 47N V7RE(V )dV, where 
N isthe number of all the vessels, of which each has the volume 
R. Further, let there be no other impacts than those of the 
molecule B in each vessel with the molecules A. Then eyi- 


dently, at least for the first moment of time, oe is determined 


by the same equation by which on) was determined 
above, since it is quite indifferent whether all the molecules 
B are in the same vessel or whether each is in a separate one. 

But if we now imagine all the N vessels of volume R brought 
together into one, we obtain a vessel of volume NR in which 
in the unit volume there are 47V’?F(V)dV molecules of the 
second kind whose velocities lie between the limits (7). In 
order that this may not be a proper fraction, but a very large 
number, we may imagine the unit volume to be very small in 
comparison with R; that is, R as any large number, which 
must, however, be still very small in comparison with N. 

The change in fwill no doubt be different in the different 
vessels ; let us, however, denote by /the arithmetic mean of all 
the values of f for the different vessels; then again, at least in the 


first moment of time, or will be given by the same equation 
as before, oft) In the former expressions we have of 
course A =A=0, since neither the molecules A impinge upon 
each other nor the molecules B. We convince ourselves most 
easily of the truth of the above assertion by imagining all the 
vessels R united into one large vessel of volume NR, in which 
in the unit volume there are 47 V?F'(V)dV molecules B whose 
velocities lie between the limits (7), and 47/(v)v’dv molecules A 
whose velocities lie within the limits (6). We may now sus-. 
pect that in course of time / assumes different values in dif- 
ferent vessels, and not until the end becomes again equal in all. 
This suspicion is most easily removed by supposing that 
at the beginning of the time / and F' have the values denoted 
above by 7; and Fy. Everything then remains as it was; 


| for the Theoretical Proof of Avogadro’s Law. 325 
of oF 


—, and di will vanish. Nevertheless the former two 


Ot Of dt 
magnitudes must be given by the equations (19) and (20) 


(where X=A=0). Therefore also = must be given by the 


same equation (22) ; and, from the condition that it must 
vanish at the same time, it follows exactly as before that 


— —hmv? rat —hAM V2 


by which Mr. Burbury’s proposition is proved. In this it is 
simply assumed that the number of molecules A is very large. 
This produces the effect that, so soon as the condition has 
become stationary, the distribution of velocities in each sepa- 
rate vessel is scarcely perceptibly influenced by the condition 
possessed by the molecule B in that vessel. 

It is only necessary to further remark here in passing, that 
the proof may be obtained in exactly the same way if, instead 
of regarding the molecules as elastic spheres, we assume any 
other law of mutual action; if only, in the first place, the 
Lagrange-Hamilton equations of motion are applicable, and, 
in the second place, if the time of perceptible mutual action 
for each molecule is vanishingly small in comparison with the 
time of free motion. 


First Appendix. 


I have just received a treatise by H. Stankewitsch*, which 
has for object to prove an equation which essentially is iden- 
tical with equation (15) of this paper. I have long ago, in 
my treatise ‘‘Some General Propositions on the Equilibrium 
of Heat’’t, called attention to the connection of a still more 
general equation with Jacobi’s principle of the last multiplier. 
H. Stankewitsch arrives at the proof of his equation in an 
altogether different way, which, however, is in every respect 
similar to Jacobi’s proof of the principle of the last multiplier. 
However ingenious the method employed by Stankewitsch, 
I hope to show in the following lines that the equation in 
question may be proved much more simply in the way indi- 
cated by Maxwell. I will first show that the equation of H. 
Stankewitsch is only an altered form of our equation (15). 
If A be the angle between the velocity v and the axis of 
abscissee, B the angle made by the XZ plane with the plane 
which is parallel to the directions of v and OX, and, lastly, K 
the angle made by the last plane with the plane parallel to the 


* Wiedemann’s Annalen, Bd. xxix. p. 153 (1886). 
+ Wiener Sttzber. Bd. Iviii. May 1871. 


326 Prof. L. Boltzmann on the Assumptions necessary 


directions of v and V; if, further, &,7, € be the components 

of v; &,m, & those of V in the directions of the axes of 
coordinates: then 

dé dyd&=vadvudAdB, . . . . 426) 

dé, dn, dG=V'rdV dl dK: 5 yeaa 


If we denote the magaitudes with reference to the velocities 
v and V’ after impact by a dash, we have 
dé dy! da =v%2" du dA’ dB’). eae) 
dé) dy’ AC4= Vr! AV! aT! dK) a ea 
Let us in fig. 2 denote the 
points of intersection of all Fig. 2. 
these lines drawn from the 
centre © of a sphere of radius 1 
with the surface of the sphere 
in the same way as the lines 
themselves. The variables v, 
V, T, 8, O determine simply 
the magnitude and relative 
position of the lines determi- 
ning the impact; they deter- 
mine what I have called the 
form of the impact; v’, V’, 
T’, OY are therefore simple % 
functions of the first-named 
variables. We will leave these variables constant, so that the 
whole form of the impact remains unaltered. Only its position 
in space, and so the variables A, B, and K are to alter; and 
the product of the corresponding changes in the variables 
A’, B', K’, viz. 
dA’ dB’ dK'=dA dBdK .>+ 


dA' dB' dk’ 
dA dB dK” 
is to be determined. Itis geometrically evident that dAdBdK 
must be equal to dA'dB'dK'; for both sets of differentials 
may be supposed to be obtained by supposing that, for fixed 
position and magnitude of v, v', V, V', the axis of abscissze 
describes the whole interior of a cone of infinitely small aper- ° 
ture ; and the system of coordinates revolves about the axis 
of abscissee at a very small angle. This follows analytically 
in the following way. Wesee from fig. 2 that B’'=B+ {vXv'. 
{vXv' is simply a function of A, K, and the now constant 
angles. If, therefore, we now introduce A’, K’, B’ instead of the 
variables A, K, B, we have dB’=dB. Therefore 

5, DAY OB’ OK! _y, A! OK! 

~ ON AOR ORAS ih? Ore 


for the Theoretical Proof of Avogadro’s Law. 327 


In the latter functional determinant, besides the angles 
already put constant, B’ is to be regarded as constant, 
Further 
a’ =al+anrcosh, 
sinj: sinh=a : @’; 
whence 
asin h asin h 


V1i—a?—asinh ad—alcosh 
We see, further, from the figure that 
180°—K=h— gv'eV, 


when the latter angle depends simply on the form of the im- 
pact, and is therefore to be regarded at present as constant. 


So also 
j+180=K’+ ¢€V'v'r. 


The latter angle again is constant ; whence it follows, since 
nothing here depends upon the sign, that 


Bee ee 15 Oe. OF. 


tan7= 


me OK aA... oe 
Since in the equations for a’ and tan j also the angle L, 
which equally depends only on the form of the impact, plays 
the part of a constant, the determinant can be calculated 


without difficulty, and we obtain for it the value “ We 


might also have obtained this result without any calculation 
by imagining the points v, v', V, and V’ as fixed. Since A 
and h are spherical polar coordinates of the point X of the 
spherical surface, so also A’, 7; the element of area ad Adh 
expressed by the former polar coordinates must be equal to 
the element of area a'd A'dj expressed by the latter. We 
have then 
a dA'dB' dK'=adA dB dK. 


For a fixed position of the points v, v', V, and V’, A, K 
and then A’, K’ may be regarded as spherical coordinates of 
the point X, which would give at once 


adA' dK =e' dA' dK’. 
Since, further, from the definition of A (equation 13), 
dv' dV' dT’ dO'=Adv dV aT dO, 
it follows from equations (28), (29), (30), (81) that 
dé dy’ de’ d&y' dn, d&'dO' vw? V"r'A 


eal 


dé dnd€dé,dn,djdO ~~ = vV*r 


328 Prof. L. Boltzmann on the Assumptions necessary 
Hquation (15) is therefore proved by proving the equation 
dé’ dy! dt dk! dn! dt! dO'=d dn dEdE, dy, dt, dO, . (32) 


and vice versa. : 

O is here the angle between the planes ROR’ and ROvw of 
fig. 1. If on the right-hand side of the equation (32) we 
introduce, instead of O, the angle yy, which the former plane 
makes with the plane ROX (compare fig. 3), & 7, & &1, m1, &, 
and therefore also the angle between the planes ROX and 
RQv remain constant ; and since this is equal to the difference 
between ¢O and Ww, it follows that dyy=dO. If in the same 
way we introduce upon the left-hand side of the equation (32) 
vy’ instead of O', it follows that 


| d0'=dy’, 
and equation (32) becomes 
dé dn} dd dé! dy! dey’ dyy! = dé dy dE dé, dm doy dp, 
which is exactly the form which H. Stankewitsch gives to the 
equation. 
We will, however, further multiply each side by od8, by 
which at the same time we indicate that S is to be chosen as 


the eighth independent variable. The equation thus assumes 
_ the form 


dé dy! dt’ dé,’ dn, dt,’ dy’ odS= dé dn dg dé, dn, at, dip ods. (33) . 


We now again draw all the 
lines from the centre © of a 
sphere of unit radius, and de- 
note in fig. 3 the points of 
intersection of the two relative 
velocities before and after im- 
pact with the surface of the 
sphere by Rk and R’; the ends 
of the two relative velocities 
by R, and R,'. Let H be the 
middle point of the arc R R’ of 
a great circle, X the point in 
which the axis of abscissa in- 
tersects the surface of the 
sphere. We now for constants 
E,n, & &1, m, $ introduce the 
angles ¢(N=XH and H=ZXH instead of ¢S=RH and 
w=XRR'. Since, again, for a fixed position of the points 
X, Z, and R, both 8 and wW as well as N and B are spherical 


for the Theoretical Proof of Avogadro’s Law. 329 
polar coordinates of the point H of the sphere, we have 
_ vdN dh=cd8 dy. 
The left side of equation (33) is next transformed into 
dé dn d¢d&,dn,df,vdNdE. . . . . (34) 


If, now, we denote the projections of the relative velocity 
QR, before impact on the axes of coordinates by w, y, z, and 
also the projections of the relative velocity OR,’ after impact | 
on the axes of coordinates by 2’, y’, 2’, and with constant &, 7, ¢ 
introduce the variables 


a=&—§& y=m—n, 2=&-4 
expression (34) becomes 
d&dnd¢dxdydzvdNdH. . . . . (85) 


Then we leave wz, y, z, N, Ei constant, and instead of &, n, € 
introduce the variables &', y', ¢. If x,y, z be the projections 
of the line R, R,’ of the relative velocities drawn from © on 
the axes of iene we have 

Mz, 


or m+ MM 
. Since, now, all the lines drawn in fig. 3 remain altogether 


unaltered in magnitude and position, a, ¥,, and z, are also 
constant, and we have 


dé dy! dt! =dé dn dé. 
Hence expression (35) becomes 
dé dy dG dadydzvdN dW...) ~ (86) 


The next step consists in introducing for constant &', 7’, ¢, 
N, E the variables wz’, y', z' instead of xz, y, z; that is, the 
coordinates of the point Ry,’ instead of the sootdtnates of the 
point R,. It is at once seen from fig. 3 that the element of 
volume described by the point R, on change of its coordinates 
is exactly equal to that which the point R,' describes for the 
position of the point H remains unchanged. it follows, 
therefore, that 


Wes My, s (pees 
; U ie Came m+M’ v=c— 


LEO EET OME Sa EPS Mactan, aaa 
and expression (36) becomes 
dé dy dc da dy dz sv..dN.dB.... °. (38) 


Now, again, inversely 


E/=E Gh, m =n +y', Cit +2! 


330 Prof. L. Boltzmann on the Assumptions necessary 7 


are introduced instead of w',y’,z', so that the expression (38) 
becomes | 
dé dy dé’ dé,' dn, dt! vdN di. oF las a0 (39) | 
Lastly, we introduce, instead of the spherical polar coordi- | 
nates N, E of the point H, its spherical polar coordinates 8, W’; 
so that we obtain 
vadN .dH=od8S. dw’. 


Lastly, expression (380) becomes therefore 
dé’ dy’ de’ d&y' dn,' d&' odS.. df’, 
by which equation (33) is proved. 
If we prefer to prove equation (37) analytically, fig. 3 


would give 
L=rm, y=rpsnd, z=rpcosd, 
where 0. Ry=OR,'=r. 
d:d=pic, 6: p=p'ioa, p'd'=pd, 
s=mnt+prf, oA? =o'—p' : 
=1—(mn-+ pif)’ — pw’? = (mv —pnf)’, 
m=ns+vod=mMn+ bs f, 
where 
N= cos2N, v= sin 2N. 
From : 
s=m'n-+ plyf'=mn-+ py, 
it follows that 
pf! — MVo5 ne pf Ng. 
If in this equation and in the equation w/¢’=d we put 
f'=ecos+esin@’, g'=esin O'—ecos G, 
it follows that 
pL! cos 6! =mev,— pefn, — ped, 
wi! sin 6’ = mev,— pen, + wed. 
By multiplying by 7 and observing that 
rm =z', rp sinf'=y', rp'cosf' =z’, 
f=ecos@+esin@, h=ecos é—esin 8, 
rpfaeytez, rup=—ey +ez, 
we obtain 
x! = Nyx + Voey + V9€Z, 
y' =Voen — (e? + €’na) y + 2veez, 


2’ =veex + 2vecey —(e? + e’ng)z, 


for the Theoretical Proof of Avogadro's Law. 331 


and we can then convince ourselves directly that 


02 Oy 02 _ 
2+ Qa" Oy 8 
Although I have already deduced a great variety of rela- 
tions from fig. 1, yet it would probably furnish several 
other equations which might be of use in particular circum- 
stances, ¢. g. by denoting the magnitude and position of the 
straight lines v, V, v', V’ symmetrically by the magnitude 
and position of the straight lines ©, P and of the line joining 
the point P with the middle point of the straight line W’. 
Symmetrical relationships of this kind are particularly conve- 
nient when we wish to obtain equations in which the magni- 
tudes before and after impact play the same part. as the 
equation we have used. 


a 


Ver. x(a, X, 2')= V2(a+X—2’').y(2', c+ X—w', 2). 


Second Appendix. 


After correcting the foregoing for the press, I became ac- 
quainted, by the kindness of the author, with Prof. Tait’s 
paper “On the Foundations of the Kinetic Theory of 
Gases”’*. While reserving for a future occasion my remarks 
on Prof. Tait’s observations on the mean path, and on the case 
when external forces act, I will here mention only one point. 
Ifin a gas on which no external forces act,and whose molecules 
are elastic spheres, F(z, y, z) dx dy dz be the probability that 
components of the velocity of a molecule parallel to the axes 
of coordinates shall at the same time lie between the limits 2 
and #w+dz, y and y+dy, z and ¢+dz, then Maxwell bases 
the first proof which he givesf of his law of distribution of 
velocities on the assumption that F(a, y, z) is a product of 
these functions, of which the first contains only 2, the second 
only y, the third only z. Thisis the same as the assumption 
that, fora given component of velocity at right angles to the 
axis of abscissze, the quotient of two probabilities, viz. the pro- 
bability that the component of the velocity of a molecule in 
the direction of the axis of abscissz lies between x and x+dz, 
and the probability that the same quantity lies between certain 
other limits € and &+dé, is altogether independent of the 
given value of the component of the velocity of the same 
molecule at right angles to the axis of abscisse. In a 


* Trans. Roy. Soc. Edin. xxiii. p. 65 (1886). 
+ Phil. Mag. [4] vol. xix. p. 19 (1860). 


332 Theoretical Proof of Avogadro's Law. 


later paper* Maxwell himself speaks of this assumption as 
precarious; and therefore gives a proof resting on a quite 
different foundation. In fact, we should expect that greater 
velocities in the direction of the axis of abscissee in comparison 
with the smaller ones would be so much the more improbable 
the greater the component of velocity of the molecule at right 
angles to the axis of abscissee. If, for example, 


mr ty ye ACen 3 e—hlatt 202(y2+27)] , 
— i : 


I'(z, y, 2) =ce 
then the quotient just mentioned would be 


F(x, y,2z)\dx _ dx 
WUE, y, 2)d& dé 


The larger wf: y’? +2", the more would small values in com- 
parison with large ones gain in probability. Now, by means 
of the law of distribution of velocities, which is to be proved, 
we obtain the proof of the very remarkable theorem: that the 
relative probability of the different values of x is altogether 
independent of the value, supposed to be given,which 4/7? + 2? 
has for the same molecule; that therefore the quotient F(z, y,z) : 
Fé, y, 2) is independent of y and z; or, what is the same, 
since the three axes of coordinates must play the same part, 
that F(z, y, z) may be represented as a product of three 
functions of which the first contains only #, the second only 
y, the third only z. 

It is therefore an altogether inadmissible circulus viliosusto 
make use of this assumption to prove Maxwell’s law of distribu- 
tion of velocities. This therefore also holds good of the proof 
which Prof. Tait has given (pp. 68 & 69 of the paper quoted), 
and which is only a reproduction of Maxwell’s first proof, 
which he himself later rejected. or, from the circumstance 
that the distribution of velocities must be independent of the 
special system of coordinates chosen for its calculation, we can 
never show that F(w, y, z) must have the form f(x) d(y)W(), 
only when this has been already proved. One might make use 
of the circumstance to show the similarity of form of the three ~ 
functions /, @,and yr. I do not even need to enter upon known 
geometrical investigations if the value of a function of three 
rectangular coordinates x, y, z is independent of the choice of 
the system of coordinates. For Prof. Tait has already shown 
of the function denoted above by F, that it can only bea — 
function of Vz2’+y?+2; but the value of the expression 
V2 +y" +2 is already quite independent of the special posi- 


* Phil. Mag. [4] xxxv. p. 145 (1868). 


oh(Et—at) ph 2—a2)(y2-+ 22), 


Arc-Lamp suitable for use with the Duboscg Lantern. 333 


tion of the system of ened therefore evidently any 
fanction whatever of /2?+y?+2’ fulfils the same condition, 
and by this condition no other further property of the func- 
tion F' can be disclosed. As, for example, the value of the 
above-used function e—*@?++2"" is also entirely independent of 
the special choice of the system of the coordinates, although 
it does not permit of being reduced to the form /(z), $(y), 


v(2). 


XXXVIT. On an Arc-Lamp suitable to be used with the 
Duboscg Lantern. - By, Professor Sirvanus P. THompPson, 
D.Se* 

[Plate IIL.] 


HE lamp devised by Foucault and Duboscq, and supplied 

for so many years by the famous house of Duboscq, 

fails to fulfil the electrical requirements of the modern physi- 

cal laboratory, though it has rendered excellent service in the 

past. Yet the lantern and optical adjuncts of the standard 

pattern of Duboseq are so widely used that it seemed desirable 

to find some other arc-lamp which, while fulfilling the elec- 

trical requirements of the case, could be used with the 
Duboseq lantern. 

Before describing the lamp which I have for twelve months 
employed for this purpose, I propose to state the conditions 
to be fulfilled, and the reasons why the old Duboscq lamp 
fails to fulfil them. 

The modern physical laboratory is usually supplied with 
electric energy under one of two alternative conditions, namely 
either at constant potential or with constant current ; more 
usually under the former condition. If supplied from a 
dynamo the dynamo may be either series-wound, shunt- 
wound, or compound-wound. If supplied from accumulators 
the accumulators will work at constant potential, and will 
have a very small internal resistance. 

The arc-lamp for laboratory use must be capable of working 
under the given conditions. No doubt the Duboseq lamp 
worked fairly when supplied with current trom 50 Grove’s 
cells. But in a laboratory where there is another and better 
and less wasteful source of supply, 50 Grove’s cells are not 
desirable. Though 40 accumulators have an electromotive 
force almost exactly equal to that of 50 Grove’s cells, the 
Duboseq lamp does not work well with them unless a resist- 
ance of several ohms is intercalated in the circuit to represent 


* Communicated by the Physical Society. 


334 Prof. 8. P. Thompson on an Arc-Lamp 


the internal resistance of the Grove-cells ; and even then the 
Duboseq lamp does not, for certain reasons, work as satis- 
factorily as the lamp to be described, and its cost is about 
three times as great. 

In every arc-lamp for optical purposes there must be 
mechanism adapted to perform the four following actions:— 

1. To bring the carbons together into initial contact. 

2. To part the carbons suddenly, and with certainty, to a 
short distance—about 3 millimetres—apart. This action is 
technically called “ striking ” the arc. 

3. To supply carbon as fast as it is consumed, by moving 
one (or both) of the pencils forward into the are. This 
action is called “ feeding ” the arc. 

4. To so move the carbons, or their holders, that the lumi- 
nous points retain the same position in space at the proper 
focus of the optical system. This action is called “ focusing” 
the arc. 

It may be remarked, in passing, that the feeding mechanism 
of many lamps also performs the action, set down as No. 1 of 
the above, of bringing the carbons into initial contact pre- 
paratory to striking the are. 

In many arc-lamps the attempt is made to unite the striking 
and feeding mechanisms in one; but in many lamps, and in 
the one I have to describe, the striking and feeding mechanisms 
are distinct. The striking mechanism in all the arc-lamps of 
commerce consists of an electromagnet or solenoid arranged 
in the main circuit of the lamp, the armature or plunger of 
the same being mechanically connected with one or both of 
the carbons, so that when, by the turning on of the current 
through the touching carbons, there is a great rush of current, 
the attraction of the electromagnet or solenoid shall instantly 
part the carbons and strike the arc. In the majority of the 
commercial arc-lamps it is the upper carbon only that is 
raised to strike the arc ; in a few other lamps, and in the one 
I am using, the lower carbon is depressed. In one of the 
older patterns of the Duboscq lamp the lower carbon was also 
thus directly acted upon, its holder being attached to the 
armature of an electromagnet beneath it. The same is true 
of the Serrin lamp. But in the Duboscq-Foncault lamp the 
arc is struck in a different way. ‘The two carbon-holders are 
connected by racks to a clockwork gearing which either 
parts them or brings them together, the movement being 
driven by a double train of wheels, either of which can be 
released in turn. The weight of the upper carbon-holder 
drives the train that moves the carbons together; a coiled 
spring drives the train that parts the carbons. Whether 


suitable to be used with the Duboscq Lantern. 339 


either of the trains, or neither of them, shall be released is 
determined by the position of a double-toothed detent which, 
placed between the final spur-wheels of the two trains, locks 
both of them when in its mean position, but releases one or 
other when shifted to right or left. The position of this de- 
tent is determined by the current through the lamp, it being 
attached to one end of a three-arm lever, the two other ends 
of which are respectively attached to the armature of the con- 
trolling electromagnet and to an opposing spiral spring. 
When the moment of pull of the electromagnet upon its 
armature is greater than that of the opposing spring, the 
detent is pulled over one way, releasing the approximating 
train of wheels while retaining locked the parting train. 
When the moment of the pull of the opposing spring exceeds 
that of the electromagnet on its armature, the detent is pulled 
over the other way, locking the approximating train and 
releasing the parting train. When the pull of the electro- 
magnet exactly balances that of the opposing spring, both 
trains are locked. Now when the current is at first turned on, 
there isa sudden pull upon the armature of the electromagnet; 
but the carbons are not instantly parted, partly because of 
the inertia of the train of wheels, and partly because of the 
backlash of the mechanism. Two or three seconds may 
elapse before the arc is struck. This delay is serious, either 
when working with dynamo or with accumulators. If the 
dynamo is shunt-wound, the shortcircuiting even for this short 
period demagnetizes the field-magnets. If the dynamo is 
series-wound or compound-wound, or if accumulators are 
being used, there is overheating during the period of delay. 
Supposing, however, the arc to be struck, then the inertia of 
the train of wheels makes itself evident in another way ; for 
it parts the carbons too far, producing a long arc of consider- 
able resistance ; and as the current then drops below its normal 
value, the armature goes over the other way, and the other 
train of wheels is momentarily released. ‘This alternation 
between the two trains, which often lasts for some time, pro- 
duces a disagreeable instability. 

The feeding mechanism of arc-lamps next comes in for con- 
sideration. The object of the feeding mechanism is to supply 
carbon as fast as it is consumed, and so keep the light constant. 
But the light cannot be kept constant unless the consumption 
of electric energy in the arc is constant. The electric energy 
is the product of two factors—the current through the arc, and 
the difference of potential between the electrodes. Calling 
the current i and the potential difference e, it is the product 
ei which is to be kept constant. Now, as remarked at the 


336 Prof. 8. P. Thompson on an Arc-Lamp 


outset, the very conditions of modern electric supply are that 
either ¢ or 7 is maintained constant, the usual arrangement in 
commercial lighting being 7 constant for arc-lamps in series, 
and e constant for glow-lamps in parallel. One of the two 
factors being a constant by the conditions of the supply, the 
other factor must be kept constant by the feeding mechanism. 
Or, in other words, the variations of the other factor should 
be made to control the action of the feeding mechanism. The 
mechanical part of the feed may consist of a train of wheels 
driven by the weight of the carbon-holder or by a spring, or 
it may consist of a friction-clutch holding the carbon from 
sliding forward, or of a worm-gearing or any other; but it 
must be controlled by an electric mechanism of one of the two 
following kinds. For keeping i constant, the feeding mecha- 
nism must be controlled by an electromagnet (or solenoid) 
placed in the main circuit, working against an opposing spring 
or weight. For keeping e constant, the feeding mechanism 
must be controlled by an electromagnet (or solenoid) placed 
asashunt to the arc, and working against an opposing spring 
or weight. In the latter case,if for any reason the arc grows 
too long, the potential at the terminals will rise, more current 
will flow around the shunt, which will then overcome its op- 
posing spring (or weight), and will release the feeding 
machinery until balance is restored. The use of the shunt, 
introduced first by Lontin, enables arc-lamps to be connected — 
two or more in series in one circuit. A less perfect solution 
is the differential principle introduced by Von Hefner Alteneck, 
where the difference between the attractions of a series and a 
shunt-solenoid maintains constant, not the product e, but the 
difference e—i. 

The only perfect solution of the problem is a feeding 
mechanism which, by combining in itself a shunt-coil and a 
series-coil, shall keep the product e: a constant, however either 
factor may vary. All the commercial arc-lamps for lighting 
in series have shunt-circuits to control the feeding mechanism; 
though often the arrangement takes the form of a shunt-coil 
wound (differentially) outside the series-coil of the striking 
mechanism ; so that feeding is accomplished by the shunt-coil 
demagnetizing the striking electromagnet and momentarily 
un-striking the arc. 

Returning to the Duboscq lamp, it may be observed that, as 
it possesses no shunt-coil, it can only feed by a weakening of 
the current in the main circuit. Hence it is obvious that a 
Duboseg lamp cannot possibly work in a constant-current 
circuit. Also two Duboscq lamps will not work in series with 
one another, as their individual feeding is not independent of 


suitable to be used with the Duboscg Lantern. 337 


the other. Neither will two work in parallel with one another; 
for the weakening of current in one throws more current 
through the other, and the instability before alluded to—called 
“hunting” by electric engineers—becomes yet more pro- 
nounced. | 

The lamp that Ihave adapted to the Duboscq lantern is one 
known in commerce as the “ Belfast ” arc-lamp, its principles 
of construction being due to Mr. I’. M. Newton; but I have 
had the design altered to suit the special work. In this lamp, 
as previously mentioned, the striking and feeding mechanisms 
are separate. ‘The arc is struck by means of an electromagnet 
Hi of the tubular pattern, having as its armature an iron disk 
A, which, when no current is passing, is held up by a short 
spiral spring at about 3 millim. from the end of the electro- 
magnet. The lower carbon-holder is mounted upon this disk, 
so that the are is struck by the downward movement of 
the lower carbon. The feeding mechanism is both simple 
and effective. The upper carbon-holder is along straight 
tube of brass: it passes through a collar in the frame of the 
lamp, and also through a metal box Babove. This metal box 
contains a piece of curry-comb with the steel bristles of the 
comb set to point obliquely inwards and downwards. They 
grip the carbon-holder and allow it to be pushed downwards, 
but not upwards. ‘The box itself is mounted upon a strong 
brass lever, L, close to the point of the lever. One end of this 
lever is drawn downwards by an adjustable spiral spring 8, 
whilst the other carries an iron armature which stands imme- 
diately above the poles of an electromagnet, which is wound 
with fine wire and placed as a shunt to the lamp. Above the 
lever there is a contact-screw, platinum-tipped, making con- 
tact with the lever, exactly as in the ordinary trembling electric 
bell, and the lever and contact-screw are included in the shunt- 
circuit. The attraction of the shunt-magnet for its armature 
is opposed by the pull of the spiral spring. Whenever, by 
reason of the resistance of the arc, a sufficient current flows 
through the shunt-circuit, the opposing spring is overcome, 
and the lever is set into vibration like the lever of an electric 
bell, but more rapidly. The vibratory motion is thus com- 
municated to the box containing the steel wire comb, which 
at once, by an action well known in mechanism, wriggles 
the carbon-holder downwards by innumerable small successive 
impulses. So soon as the motion of the carbon has reduced 
the resistance of the arc, the shunt-current diminishes and 
the feeding action ceases, to reeommence when required. It 
is found best for lantern-purposes to send the current upwards 
through the lamp, the lower carbon being the positive one. 

Phil. Mag. 8. 5. Vol. 23. No. 143. April 1887. 2A 


338 Mr. R. H. M. Bosanquet on Electromagnets. 


A thick cored carbon of 13 to 15 millim. diameter is preferred, 
as it gives a good luminous crater and burns slowly. A 10- 
millim. copper-plated carbon is used for the upper electrode, 
and it is adjusted so that its centre falls slightly in front of 
the centre of the lower carbon, thereby causing the crater to 
send its light forward. 

The lamp as used in commerce has no focusing-arrange- 
ment. In adapting it to the Duboscq lantern, the frame was 
made narrow; so that when the inner chimney of the Duboseq 
lantern was removed, the lamp could be dropped entire down 
the outer chimney, a metal sleeve of the same diameter as 
the inner chimney being added to the lamp as a guide. At 
the bottom of the lamp a gun-metal tube was added, tapped 
inside with a screw-thread, into which works a steel screw 
having a small hand-wheel near its lower end and a pointed 


‘pivot at the extremity. The lamp slides down the chimney 


of the lantern until the pivot touches the base-board. When 
the are burns down the lower carbon, so that the luminous 
crater is no longer in the optical focus, a turn given by hand 
to the wheel suffices to raise it to the proper position; but 
the lamp will burn for ten minutes without requiring any 
readjustment on this account. The lamp shown to the Physical 
Society was constructed by Mr. E. Rousseau, Instructor in 
the Physical Workshop of the Finsbury Technical College, 
assisted by Mr. A. D. Raine, now Demonstrator in the City . 
and Guilds Central Institution. 


XXXVITI. Hlectromagnets—VII. The Law of the Elec- 
tromagnet and the Law of the Dynamo. By R. H. M. 
BosanquEtT, St. John’s College, Oxford. 


To the Editors of the Philosophical Magazine and ee 


GENTLEMEN, 

| ae present communication will consist of two parts. First, 

the application of the measures of the bars with pole- 
pieces, contained in No. V. of this series (Phil. Mag. [5] xxii. 
p. 298), to the establishment of the type of law which governs 
electromagnets of this description, and the comparison of 
this law with the various assumptions which have been made 
on the subject, and in particular with Frdlich’s law and the 
law of tangents. 

Secondly, a few propositions will be stated which offer a 
general method of discussing the action of dynamos, inde- 
pendent of the assumption of any particular law of magneti- 
zation, and based on a consideration of the dynamic action. 


Mr. R. H. M. Bosanquet on Electromagnets. 339 


These will be applied to the law first obtained. A discussion 
of the actual behaviour of my Gramme dynamo will follow in 
a future paper. | 

The average values of the magnetic resistance of a number > 
of bars with pole-pieces throughout the whole course of 
magnetization were given in the investigation above cited. 
The bars in question all had cores of the same shape, viz. 
length : diameter :: 20:1. Thenumbers for other shapes will 
no doubt be different, and the laws which deal with the shapes 
will be the subject of future investigation; but it is not likely 
that the general type of the law will differ materially from 
that here discussed. 

It has been a matter for some consideration what scheme 
should be adopted for the representation and comparison of 
the different laws. I have adopted a scheme in which the 
magnetic inductions are measured horizontally, and the 
permeabilities, or, as I prefer to call them, conductivities, 
vertically. 

The reason for adopting the magnetic induction as the 
chief variable, is the fact that the magnetic properties of the 
metal depend only on the induction. It is enough to glance 
at the figures which follow, or, better, at the reciprocal figures 
representing magnetic resistance at p. 303 of the paper above 
cited, to see that, whether the metal be in the form of rings, 
or of bars with or without pole-pieces, the resistance to 
magnetization (or its reciprocal the conductivity) changes 
always in much the same way at the same values of the 
induction. Any representation which overlooks this, overlooks 
the principal law so far known as to the variation of the 
magnetic properties of iron. And I dissent from the position 
of those who say that the magnetizing current, or magneti- 
zing force, has the chief claim to be regarded as the 
independent variable in such a representation. 

I have in this case chosen conductivity, rather than the 
magnetic resistance, to be combined with the magnetic 
induction, mainly because this combination represents F'ré- 
lich’s law, which is so generally accepted, as a straight line ; 
and this facilitates the comparison of Frolich’s law with other 
laws. 

A few words as to the precise meaning of the expressions 
permeability and conductivity. 

The word permeability was originally used in connection 
with the old theory, and was the ratio 


% magnetic induction 

= or = ; 

a) magnetizing force ” 
2 


340 Mr. R. H. M. Bosanquet on Electromagnets. 


but the magnetizing force existing within the metal of the 
bar, for instance, was supposed not to be the same as the 
external magnetizing force, but to be diminished by demag- 
netizing forces, which resided on the ends of the bar. In 
rings, however, where there were no ends, the theoretical 
magnetizing force was the same as the external magnetizing 
force, and the permeability of a magnetic metal could be 
determined by measures of rings. 

In dealing practically with bars, what we want to know is 
the connection between the magnetism developed and the 
external magnetizing force. The ratio of these two quantities, 
which I call conductivity, is a quantity analogous to the 
permeability of the theory, but not identical with it ; the 
conductivity does not take into account the supposed demag- 
netizing forces at the ends of the bar. The system that I 
have adopted prefers to attribute the diminished magnetism in 
the case of bars with ends to the increased resistance experi- 
enced by the magnetism in traversing space external to the 
magnetic metal; and I employ the word conductivity in 
connection with an entire magnetic circuit, with its air 
resistances ; leaving permeability with its original application 
to circuits or parts of circuits lying wholly within the 
magnetized substance. 

The permeability of rings and the conductivity of bars 
have, then, precisely the same meaning. In both cases the 
meaning is, 

magnetic induction 
external magnetizing force’ 


If we suppose the magnetizing force uniformly extended 
along the bar, as by a uniformly wound coil, we have 
Hxternal magnetizing force x length = external potential. 


Conductivity __ magnetic induction _ 1 
length =~ ~‘magnetic potential ~ p’ 
where p is magnetic resistance, according to our definition of 
magnetic resistance. 


Whence permeability of rings or conductivity of bars 
_ length 


p 
I invariably reserve * for the permeability, since it is thus 


* In a paper presented to the Royal Society, Dec. 20, 1880 (Proce. R. 
S. vol. xxxiv. p. 445), in which the system of broken magnetic circuits 
with air resistances, now so generally used by practical men, was first 
developed, I distinguished insufficiently between permeability and conduc- 
tivity, and used » to represent both ideas. 


Mr. R. H. M. Bosanquet on Electromagnets. 341 
used by Maxwell and the best writers. For conductivity I 


write either ; or cy. 


Having the magnetic resistances of the magnets with pole- 
pieces mentioned above, we can obtain their conductivities ; 


.) 

p 

The formula known as Frdélich’s may be obtained by 
assuming that the conductivity of the magnet is proportioned 
to its defect of saturation. (See my letter to ‘The Electrician,’ 
vol. xvi. p. 247, February 1886; and Prof. 8. P. Thompson, 
Phil. Mag. xxii. p. 290, Sept. 1886.) Ifwe measure the mag- 
netism as % (magnetic induction), we may write this, 


Conductivity = k (B, — B), 


which represents a straight line on the scheme of conduc- 
tivities and inductions ; see figs. 1 & 2. 

For the sake of clearness I have drawn on fig. 1 the 
permeabilities of Rowland’s table i., and of my ring H, and 
also the average conductivities of plain bars and bars with 
pole-pieces, deduced from the magnetic resistances given in my 
paper first above cited ; also both in figs. 1 & 2 the applica- 
tion of Frélich’s law to the bars with pole-pieces. 

It will be seen that, if a real state of things be represented 
by any curve, a tangent drawn to that curve at any point will 
represent a Frélich’s law, which will be true only so far as the 
curve and the tangent coincide. In the present case there 
appears to be a point of inflexion on the curve just before 
approaching the region of what I may call super-saturation 
(tendency of % to increase without actual limit ; see paper 
first cited). The tangent drawn through this point of inflexion 
coincides with the curve for a considerable distance in the 
neighbourhood of % = 15,000 ; and I shall show later that the 
excitation in the cores of a dynamo with such magnets may 
be confined in actual practice within very moderate limits on 
either side of this value. 

The use of Frolich’s law to deduce consequences where 
wide variations of the magnetic intensity take place, as, for 
instance, where the magnetism is supposed to be reduced to 
half its maximum value, appears to be fallacious in such cases 
as the present. : 

The curve in fig. 2 is the same as the curve marked “ Bars 
with PP ”’ in fig. 1, but drawn to a larger vertical scale. 


342 Mr. R. H. M. Bosanquet on Electromagnets. 


Conductivity-= 


2,500 


His, 1, 


Rowland’s Table No.1. 


2,000 


bt ae 


500 


me with. ieee. mae 


700 
1,000 


=<, 
= 
IS 


Magnetic Conductioity g& Bars 1-20 with pole pieces 


Conductivity 2 


200 


700 


1,000 


Plain- Bars ee 
Sie | 


5,000 10,000 


Fig. 2. 


o 


AUN 
EEEEPTP EEE 
FEECEEEELE EEE 


15,000 


5,000 10,000 


15,000 


[Ss 


-..| _ |3$ 
20,000 


20,000 


Mr. R. H. M. Bosanquet on Electromagnets. 343 


The following Table contains the conductivities of the bars 
with pole-pieces, as obtained from the experimental magnetic 
resistances, and their comparison with some of the different 
methods which may be or have been adopted for representing 
the law of magnetism. These methods are enumerated in the 
following statement, the table only containing such compa~ 
risons as seem necessary. 


Numerical Comparison with Experiments of some of the 
Calculations representing the Magnetic Conductivity of 
Bars 1 : 20 with Pole-pieces. 


| | 
| | 


Conduc- : 

a : s III. Rule with 

pny or | II. Fourier Series. approximate p. IV. Tangent law. 

B. = Oe seh Sth Uae eit eal Py ebue ne case 
E 1 i I 

eee: | Das. oe eT A Dilts, 

periment. p p p 
0,000 169 187 +18 171 + 2 
1,000; 271 245 —26 252 —19 614 
2,000; 301 290 --11 289 —12 610 
3,000; 319 316 — 3 310 — 9 603 
4,000} 332 326 — 6 323 9 592 
5,000| 337 332 — 5 330 — 7 578 
6,000| 340 337 — 3 334 — 6 561 
7,000; 340 340 0 338 — 2 541 
8,000} 340 340 0 309 — 1 518 
9,000; 339 339 0 338 — 1 490 
10,000} 336 337 +1 334 — 2 459 
11,000} 332 331 — 1 330 — 2 424 
12,000| . 324 327 +3 323 — 1 385 +61 
13,000} 310 312 + 2 310 0 341 +31 
14,000} 286 283 — 3 289 + 3 292 + 6 
15,000| 288 233 — 5 252 +14 238 0 
16,000 171 175 +4 171 0 17k 0 
17,000} 136 130 — 6 110 — 26 
18,000 91 104 +13 36 —55 


Statement of Methods for representing Conductivities by 
Calculation. 


I, By value of D where p is derived from my theory: see 


Phil. Mag. vol. xxii. p. 8308 (September 1886). (The formule 
at the head of the tables at pp. 307 & 308 have been unfor- 
tunately wrongly copied. I give the correct headings com- 
plete in the Hrrata at the end of this paper.) 

This representation is quite close, but the computations are 
rather too laborious for ordinary use. It is not necessary to 
exhibit the comparison. 

Il. Empirical representation of conductivities by Fourier’s 


344 Mr. R. H. M. Bosanquet on Electromagnets. 


series. This is fitted specially in the region from %% =7000 
upwards, which includes the dynamo range. 


= 210-+150 sin 6+26 sin 30+ 8sin 50+2 sin 70, 
where 6=-0124%3 —400}°. 
III. Representation by means of my rule that 
p = shape-constant + ° 


with rough approximate value of pw. The shape-constant 
for unit length 00252 in this case is derived from my in- 
vestigation cited above. General values for this will be the 
subject of further investigation. 


Dee ae 
Di isl 
00252 + — 
be 
where p= 300 + 2000 sin @, 
— 800 13 e 


This fits fairly up to % =16,000, but fails above. It is a 

useful formula for an approximation to the general outline. 
IV. Representation by means of the tangent law. 

G g° 


Magnetizing force = conluelney =k tan 505" 
Fitting to the 15,000 entry, 


k=1-28051 3 
then | 


The differences are only entered when there is some approxi- 
mation to the truth. 
V. Frolich’s law. 


Conductivity = = 


p 7 magnetizing force 


=k(B,—B). 


This is shown by the straight line on the figures. It is 
unnecessary to exhibit the calculation of the numbers. The 
correspondence with the truth is about the same as in the 
tangent law ; but the error increases more rapidly, and ulti- 
mately becomes much larger, as the magnetism diminishes. 


Mr. R. H. M. Bosanquet on Electromagnets. 345 


Thus the tangent and Frolich’s laws, upon one or other of 
which almost all treatment of the theory of dynamo machines 
has been based, are shown to be far from representing the 
true laws which govern electromagnets. 

In a paper by the Messrs. Hopkinson, reprinted in ‘ The 
Hlectrician,’ Nov. 19th, 1886, we have an example of another 
way in which it has been attempted to fit Frdlich’s law to 
represent the law of magnetization. The intersection of the 
Frélich law with the true law in diagram A there given is 
made to take place at about % = 5500. If the case be 
represented by a scheme of conductivity and induction, the 
straight line representing [rélich’s law crosses the curve of 
the true law at a considerable angle, and by the end of the 
representation in about % = 11,000 the two diverge widely. 

Now it seems unlikely that Frolich’s law, so used, can have 
any bearing upon the action of the dynamo machine. The 
advantage of the law is that, being easily manipulated, it can 
be made to coincide exactly with the true law in the part of 
the dynamo range in actual use. Such a case is represented 
by a tangent drawn to the curve in my scheme. It is very 
unlikely, however, that any dynamic action, such as to be of 
practical utility, could take place in the region of B = 5000. 


I shall now proceed to a few propositions, suitable for 
application to the true laws of electromagnets as embodied in 
series of numbers rather than in formule, founded chiefly on 
the dynamic action itself. 

The outlines of the theory have been explained to some 
extent in my paper on Self-regulating Dynamo machines, 
Phil. Mag. [5] xv. p. 275. But the application to laws 
expressed numerically, and the line of reasoning now adopted 
are new. 

For the present I confine myself to the series dynamo. 

According to the mode of statement now usually adopted, 
the H.M.F. developed in an armature at n revolutions is 

Bye AAAs ite ee 
where 9%, is the field-intensity within the coils of the armature. 
This differs only in arrangement of units from the formula 
adopted in my previous paper. 

The first thing is to express %, in terms of the % developed 
in the field-magnets. We have measures of the % across the 
equatorial sections of the field-magnets, and can connect it to 
some extent with the potential of the magnetizing current. 
In the present approximate purpose we assume 


et fe a 


346 Mr. R. H. M. Bosanquet on Electromagnets. 


where f may be called the coefficient of efficiency ; it will 
depend on the build of the machine, and may probably range 
from 4 to 3/5 or less. Nothing in the theory depends on it, so 
far as our present purpose is concerned. We neglect varia- 
tions of distribution, which would give rise to change of f. 

We then put H=CR, from Ohm’s electrical law in the 
circuit, and the equation stands 


Cl 4n ASB. ee 
The next step is to express 93 in terms of the magnetizing 
current. If we make here the usual assumption, 
%3 = conductivity x magnetizing force, 


4 
or G = cy x a . 8 Oe eee 


and substitute in (3), the current disappears from the equation, 
and we have 


[R = I6amn Af Xx cys: 4) a 


a relation between the coefficients for a given value of the 
conductivity, which is not without use, but does not help us 
in the general problem. 

The assumption (4) isnot, however, called for by the nature 
of things, for it is clear that % is not generally proportional 
to the magnetizing current. And our present treatment will 
be founded on the assumption that, so long as the conditions 
of the machine vary but little, there must be some power of 
the magnetizing current or of its magnetic potential to which 
$3 may be regarded as proportional. Assume, then, a general 
form of law which can be fitted to any part of the range of 


magnetization, 
B= KO ch eee (6) 


Substitute this in (3), and gather up the constants into the 
coefficient ; then, 


Cry 
1 
eae 62k at. 


which expresses the current as a power of the velocity of 
rotation. 


: : 1 
Here we may conveniently put #= i and assume Cy and 


nm, to be a pair of corresponding values differing little from C 
and n, so that és 
n HH 
Cami, ) (8) 


By means of this formula we can determine the value of « 


Mr. R. H. M. Bosanquet on Electromagnets. 347 


experimentally, for any condition of the machine. We vary 
the speed slightly, and measure the two speeds and the two 
currents. Then we have « from the equation 


2 (log n—log m)=log C—log GQ. . . . (9) 
A rough determination of the values of 2 for my Gramme 
machine, by this method, is given at Phil. Mag. [5] xv. 
p- 285, It is as follows * :— 
Current Amperes. 
BUS SRT, ge yak 
10 


f oe 
aC Zoe a 


bo oo & 


cH tolHoxito “SS 


x=1, y=0, correspond to the condition of saturation, accor- 
ding to the theoretical assumption of a saturation limit, which 
we know is not quite justified in practice. . 


I have now to show how the assumption (6) can be fitted 
to a law of magnetization when the law is given by a series 
of numbers representing the magnetic resistances or conduc- 
tivities of the magnets of the machine, for the different induc- 
tions used, 

Rearrange (6) as follows :— 


95 —K’ (magnetizing force)’; . . . . (7) 


(yk, SRST at ets 


magn. force 


then 


or 
%-Y (conductivity )¥== ys) |. sei ie (9) 


or, if %, cy; %o, cyo are pairs of values differing but little, 
BB ~1(cy)¥=Bo (eyo), ~ » - « + = (10) 


and 
ae es a 
Ee =xr—1; 
fy Aes. rep} 
° B, a * >] 
whence 
log B— log B . 


Smt hime log cy’ ween) 


* These numbers are so far justified by my later determinations that 
it is not worth while to amend them at present. The conditions and 
limitations to which they are subject will be touched ae in the discus- 
sion of my dynamo. 


848 Mr. R. H. M. Bosanquet on Electromagnets. © 


| where « is that power of the velocity with which the current 
: varies, in the given state of the machine. 
We immediately infer some propositions of interest with 
regard to 2. 
x—1 can only be finite and positive so long as the conduc-. 
tivity diminishes as 9% increases. | 
It is infinite when the conductivity is constant. It is_ 
negative when % and the conductivity increase together. . 
Following the course of the changes usual in electromag- 
nets, which we may illustrate by fig. 2, we then have 


z—l. Sg oe ae i : Conductivity. 
- - »« « Oup to about 5000 © increasing. 
ie ie, “9000 to about 7000 maximum. 
+0 
“= Scant above 7000 diminishing. 


Limit 0 at saturation 18,000 to 20,000. 


- Thus the range of possible dynamic equilibrium is from 
about %=7000 upwards. In the lower part of this z is great, 
| or the current changes violently for small changes in the 
velocity. In the saturation region # approximates to 1, and 
| the current is more nearly proportional to the velocity. 

I have calculated the values of x given by the successive 
pairs of numbers on which fig. 2 is based (bars with pole- 
pieces). » These. are:— , 


BB. ie 
8/500) uae eae 3) Qe t 
F500 me mee Eh) ese . 
10,500: edge Bo 3 whos 
11,500 eee ia dene 
12,500 Rae ea a 
13,500 eee eae teal 
14,500; tee oe ilar 
15,500.10 ee a Gale 
16,500. /.k7. 5 Weg cee ileal 


Plotting these on a scale, I took out the values of % corre- 
sponding to those of w obtained from the Gramme machine. 
And from the experimental conductivities (fig. 2) I calculated 
the magnetizing forces which would be required by magnets 
such as ours to produce this condition. These are :— reid 


Mr. R. H. M. Bosanquet on Electromagnets. 349 


Power of velocity Inductions in dynamo | Magnetizing forces 
to which current having electromagnets required by bars 
is proportional, such as in fig. 2. with pole-pieces. 

2. 8. 

3 12,500 39:4 
2 13,640 455 
1:25 15,320 729 


We cannot of course assume that the magnets of the Gramme 
machine follow the same law as that of our bars; in fact they 
do not do so at all approximately, probably in consequence of 
the large amount of cast iron in the machine. But these 
numbers are enough to illustrate the limited nature of the 
variations of the induction which may be possible during the 
working of a dynamo, while the current produced varies in a 
ratio of more than 2:1. By carrying the magnetization 
higher still, we get a further considerable range of current 
with a small change of the induction. Looking at fig. 2, we 
see that it would be possible to draw a secant through the 
point of inflection, representing a Frélich’s law, and deviating 
but little from the curve from about 12,500 up to18,000. In 
this way the law of magnetization would be approximately 
represented by a Frélich’s law over a very wide range. 

A word as to the physical meaning of the quantity y, which 


is connected with w by the relation ~ =u. This y may be 


said to be what determines the dynamic action. The dynamic 
action consists of the summation ofan infinite number of ele- 
ments, whether of magnetism, current, or H.M.F., which 
originate in one small change of velocity. It is only where 
these elements are successively less and less, and so form a 
convergent series, that their sum is finite, and gives rise to a 
definite behaviour of the machine, or to what we may call a 
state of dynamic equilibrium. 

Let y be then the ratio of diminution of the successive 
elements, due to the additional element of induced magnetism 
being less than the element of inducing current. It is easy 
to see that we may express the whole change of C, say, due 
to a small change of velocity dn, thus :— 


AS =(ltyte+...)@ 


nr 


Lek) On 
~ l-y 2 


390 Mr. R. H. M. Bosanquet on Electromagnets. 
(see Phil. Mag. [5] xv. pp. 285 & 286) ; whence 


log.n= (1—y) log C+ const.,. 
or 
N= comeby si! >, 
or 
n*= const. O; 


and we have deduced the form of equation before assumed, 
from the principles of the dynamic action of the machine. 
In practice these considerations and laws are much modified, 


chiefly by the enormous magnetic retentiveness of the field- 
magnets. 


I shall have to deal with this subject in discussing the 
performance of my Gramme machine. 


HRRATA in recent papers. 
Phil. Mag, vol. xxii. p. 307, for the heading of the Table substitute 
7 Bars with P.P. without Magnetizing-Force term. 
pcale, ="00252-4 * 1=1centim., log < = 235933, 
p='415 (18,366 — 33) cos 6, log f= ‘16500, 


vat __ Kk 60°—w 
Se aang tamer 


Phil, Mag. vol. xxii. p. 308, for the heading of the Table substitute 
Bars with P.P. with Magnetizing-Force term. 


pcale. =-00252+ /, 7=1centim, logn=1-08160, 
pe 
u='69 (= +16,421-33) cos 6, loge : =2°57346, 


78, 38 ra log f = ‘16878. 


Phil. Mag. vol. xxii. p, 536, line 3, for ‘526 centim. in diameter read ‘526 
inch in diameter. (This clerical error appears in the description of the 
instrument, but does not affect the calculations.) 


fBaE J 


XXXIX. Note on the Tenacity of Spun Glass. 
By H. Gisson and R. A. GREGORY *. 


1 is well known that the tenacity of metallic wires increases 
as the diameter diminishes, so that very fine wires will 
carry much larger loads than those obtained by calculation 
based upon the assumption that the breaking weight varies 
as the square of the diameter. As glass can be drawn into very 
fine fibres, we have made some observations on the tenacity 
of this material, comparing the strength of very thin threads 
with that of rods made from the same glass, but of much 
greater thickness. 
_ The experiments were carried out in the course of our work 
in the Physical Laboratory of the Normal School of Science 
and Royal School of Mines. 

In dealing with a substance so brittle as glass, it is evident 
that special care must be taken to ensure that the observation 
is not vitiated by rupture due to a shearing stress, at or near 
the points of support. Precautions were taken to prevent 
this in all cases, and no experiments are quoted in this paper 
in which rupture took place near the points of support, or of 
attachment of the weight. 

Three different thicknesses of glass were subjected to 
experiment: viz., fibres the diameters of which were about 
0-002 and 0-004 centim. respectively, and rods with dia- 
meters varying between 0:05 and 0:09 centim. 

The fibres were attached at theends of two strips of paper by 
means of shellac varnish ; this on setting was found sufficiently 
strong to carry more than the breaking weight, without 
allowing the fibre to slip. A small paper basket suspended 
from the lower strip carried the load, consisting of fine shot 
and silica, the latter being added when the fibre was near its 
breakin g-point. 

The diameter of the thread was measured at the place of 
rupture by means of a Compound Microscope with micrometer 
eye-piece. From data thus obtained the tenacity was calcu- 
lated with the following results :— 


Tenacity, in dynes 


Diameter, in centims. B. weight in grs. per sq. centim. 
0-00186 11°76 424 x 10° 
0:00159 8°70 425 x 10° 
0:00315 32°26 405 x 10° 
0-003840 43°23 466 x 10’ 


~ * Communicated by the Physical Society: read February 12, 1887, 


352 Note on the Tenacity of Spun Glass. 


Some observations were next made on rods about 1 millim. 
in diameter ; the method of support and the loading being 
changed. Two pieces of angle brass, each about 8 inches 
long, were substituted for the slips of paper. Through a hole 
drilled near the end of the angles, a piece of }” wire was 
passed, turned up and soldered to the back. The free 
extremities of the wires were plaited into rings, which served 
to support the load and suspend the whole from a hook above. 

The ends of the rod were laid in the angles, leaving the 
glass free for about 12 inches. Small pieces of red-ochre 
cement (a compound consisting of resin, red ochre, and bees- 
wax) were placed at intervals along the glass, and a Bunsen 
flame applied. The cement speedily melted, and imbedded 
the glass; on cooling, the whole was suspended vertically. 
A bottle was hung on the wire attached to the lower angle- 
piece, into which a fine stream of mercury flowed from a 
reservoir above. ‘The apparatus was so arranged that when 
the rod broke mercury would no longer fall into the bottle. 

The mode of measuring the diameters of the rods differed 
from that adopted in the case of the fibres. About half an 
inch of rod was broken away at the place of rupture, and 
mounted in wax on a piece of looking-glass, the broken section 
being upwards. Its diameterwas then measured by means of 
a microscope-cathetometer, and the tenacity found as in the 
ease of fibres. The following are the results of four 
experiments :— 

Tenacity, in dynes 


Diameter, in centims. Weight, in grs. per sq. centim. 
0-090 3908 60 x 107 
0°082 4443 83 x 10’ 
0-050 1948 OT x 10F 
0-042 1731 126 x 10° 


These observations show, in the first place, that the tenacity 
of fine fibres is very considerably greater than that of thick 
rods, and that the strength of rods increases as the diameter 
diminishes. It may be interesting to point out that the 
tenacity of glass fibres studied by us is nearly as great as that 
assigned by Wertheim to many ofthe metals ; e. g., the tenacity 
given by him for annealed steel wire 1 millim. in diameter i 1s 
499 x 10’ cent.-dynes, and even in the case of drawn steel the 
tenacity is not greater than twice that of a glass fibre, viz. 
998 x 10° cent. -dynes. 

With steel pianoforte-wire the tenacity is, however, con- 
siderably greater ; according to Sir William Thomson (Art. 


On an Improved Form of Seismograph. 353 


‘ Hlasticity,’ Encyc. Brit., new edition) the breaking-stress is 
Cent.-dynes. 


Best pianoforte steel-wire . . . . . 2318 x 10! 


The question as to what is the most probable cause of this 
increase in strength as the diameter diminishes, presents some 
difficulty. 

Quincke (Comptes Rend. de Vl Acad. de Berlin, 1868, 
p. 132) has suggested that the great increase observed in the 
case of metals is due to a surface tension, analogous to that 
observed in liquids. If this were the true explanation, the 
breaking-weight could be expressed by the sum of two terms 
which vary as the diameter and the square of the diameter 
respectively. This suggestion does not receive much support 
from our observations, as the results cannot be satisfactorily 
expressed by means of such a formula. It is, perhaps, more 
probable that the heating and rapid cooling undergone by the 
glass when it is drawn out into a fine fibre produces an 
increase in tenacity ; and it is at all events certain that no 
comparisons can be made between the strengths of different 
materials unless they have undergone similar treatment, and 
unless the sizes of the rods or wires submitted to experiment 
are the same. 


XL. On an Improved Form of Seismograph. 
By Tuomas Gray, B.Sc., F.RS.L* 


[Plate IV. ] 


| ee apparatus described in this paper is an improved 

form of a seismograph which was made for Prof. Milne 
in the beginning of 1883, to be used by him in his investi- 
gations for the Committee appointed by the British Associa- 
tion to “ Investigate the Harthquake Phenomena of Japan.” 
That apparatus was exhibited to the Geological Society of 
London, and a description of it by the present writer was 
published in the Quarterly Journal of that Society in May of 
the same year. It consisted of a combination of instruments 
which had been devised by Prof. Milne and the writer, and 
descriptions of which had appeared from time to time in the 
‘ Transactions of the Seismological Society of Japan,’ and in 
the ‘ Philosophical Magazine.? The object of the apparatus 
was to determine the time of occurrence, the amount, the 
period, and the direction of the different motions in an earth- 
quake shock. Arrangements were made for recording three 
components of the motion, one vertical and two horizontal, at 


* Communicated by the Author. 
Phil, Mag. 8. 5. Vol. 23. No. 143. April 1887. 2B 


Se o-- 


354 Mr, T. Gray on an Improved 


right angles to each other, on a band of smoked paper which 
covered the surface of a cylinder. The cylinder was intended 
to be kept continuously in motion round its axis by clock- 
work ; and the recording points were, on the supposition of 
no motion of the earth, expected to trace continuoasly the 
same line on the smoked paper in a sim‘lar manner to that 
introduced by Prof. J. A. Hwing, and used by him in his 
experiments in Japan*. Prof. Hwing used smoked glass for 
his record-receiving surfaces, and that is a very good arrange- 
ment when it can be conveniently adopted. It had been pre- 
viously used by Prof. Milne in apparatus in which the 
record-receiver was either stationary or automatically started 
into motion by the earthquake ; and it has since been much 
used by him and the writer in earthquake investigations. 
Smoked paper was adopted in the apparatus here referred to, 
and, when smoked surfaces are used, it is still recommended 
for the present form, because it is desirable to obtain straight 
records, written side by side and to the same scale, of all the 
three components. ‘This, combined with continuous motion, 
could only be got on a cylindrical surface ; and, considering 
the risk of breakage, cylinders of glass sufficiently true and 
inexpensive could not be readily obtained. 

The apparatus used for recording the motions was in prin- 
ciple the same as that described in this paper, but differed 
considerably in detail. A separate clock was provided for 
the purpose of recording the time of occurrence, the record 
being made on the dial of the clock, which was, at the time 
of an earthquake, automatically pushed forward into contact 
with ink-pads fixed to the ends of the hands, a mark being at 
the same time made on the record-receiver to show at what 
part of the earthquake the time was recorded. In subsequent 
instruments this method of recording time was abandoned 
because, with the improved form of record-receiving appa- 
ratus, it became unnecessaryt. This will be more particularly 
referred to when the method of recording time now adopted 
is being described. 

The instrument above referred to was set up in the Meteo- 
rological Observatory in Tokio, where it is still in use. Hx- 
perience with it, however,soon suggested many improvements, 


* See “A new Form of Pendulum Seismograph,” Trans. Seis. Soc. 
Japan, vol. i. part 1, p. 38; and “On a New Seismograph,” Proc. R. S. 
no. 210 (1881). 

+ This refers only to the instruments here described, which are made 
in this country by White, of Glasgow. In a less complete form of the 
apparatus made in Japan, and a considerable number of which are in use 
in different parts of that country, the clock with movable dial is still used. 


Form of Seismograph. 355 


which have been introduced into later instruments. It was 
found that when the “ conical pendulums”’ (see below, p. 362) 
used for actuating the recording indices were, as in that in- 
strument, made to turn with very little ee and were 
adjusted to have a long period of free oscillation, that is to 
say to have very little positive stability, the lines traced by 
the recording points gradually broadened to a very incon- 
venient extent. This rendered good records of small motions 
impossible after the record-receiver had been in motion for a 
short time, and introduced a risk that such records might be 
obliterated after they had been obtained. Such considerations 
as these led Prof. Milne to abandon the continuous motion 
element, and adapt the instrument to the comparatively old 
method of automatic starting at the time of the earthquake. 
There are, besides the difficulty experienced due to the 
broadening of the lines by the recording points, several other 
important objections to the use of a band of paper of such 
limited length as that provided by a single turn round a 
cylinder of moderate dimensions. The record may, for ex- 
ample, extend more than once round the cylinder ; that is, 
the earthquake may last longer than the time taken by the 
cylinder to make a complete turn. This produces great con- 
fusion in the record, rendering it difficult to interpret. 
Again, two earthquakes may occur before the record-sheet 
has been changed ; and in such a case both records are practi- 
cally lost. Considerations such as these have led us to adopt 
one or other of the forms of apparatus described in this paper. 

The new form of apparatus has for its object the determi- 
nation of the same elements as have been already enumerated 
with reference to the old instrument. Provision is, however, 
now made for the whole of the record being obtained on fresh 
surtace, and for any number of earthquakes which may occur 
within a limited period, say a week, being recorded on the 
same sheet. The record-receiver is kept continuously in 
motion at a very slow rate, and time is marked on it at 
regular intervals by means of a good clock; the object being 
to secure with perfect certainty that most important element 
in earthquake investigation—the time of occurrence of the dis- 
turbance. In the most complete form of the apparatus the 
record-receiving surface is a long ribbon of thin paper, which 
is gradually unwound from a supply drum on to another, 
which may be called the hauling-off drum, by means of a 
weight or spring and a train of wheelwork. The speed is 
rendered uniform by taking the paper in its passage from the 
one drum to the other round a third drum, which is kept 
continuously in uniform motion by a train of clockwork and 

2B2 


396 Mr. T. Gray on an Improved 


a suitable governor. A somewhat simpler arrangement is 
obtained by using a single drum covered with paper, or a 
smoked glass or metal cylinder, and giving to this cylinder a 
slow motion of translation in the direction of its axis, so that 
the record takes the form of a spiral line round it. As, how- 
ever, the rate of motion must be such as to give the time of 
occurrence with fair accuracy within a second of time, it is 
difficult to obtain a good record on a cylinder of moderate 
size, which will extend over more than twelve hours with 
this arrangement. It is of course easy to adapt the apparatus 
to be used either way, if that were desirable; but the con- 
tinuous ribbon of paper is so much the better form of re- 
ceiving-surface that the description given in this paper, in so 
far as it refers to earthquales, only includes that form. The 
spiral record has some advantages in apparatus adapted to 
record slow changes of level of the earth’s surface ; and it will 
be again referred to in that connection. For such purposes 
the rate of motion may be made excessively slow; end hence 
the records for a considerable length of time may be written 
on one sheet. 

At the time of occurrerce of an earthquake, the rate of 
motion of the paper is automatically greatly increased, and a 
chronographic reed is simultaneously set into vibration, and 
made to mark equal intervals of time on the ribbon, thus - 
showing accurately the rate of motion at any instant. The 
actual rate of motion of the paper on the slow speed may be 
varied from about a quarter of an inch to an inch per minute, 
and on the fast speed from about 25 to 50 inches per minute, 
with the present form of instrument. This change of speed 
is generally obtained by including in the driving clockwork 
two governors, one of which can be automatically thrown out 
of gear, either electromagnetically or mechanically. The 
latter method has been found the best and the simplest in 
practice. The arrangement commonly used is described 
below, page 361, and need not be more particularly referred 
to here than in a general statemert of the operations it is in- 
tended to perform. At the time of an earthquake three 
operations take place simultaneously. One is the introduction 
in train with the clockwork of an adjusting mechanism which 
is intended to readjust the starting apparatus, wuatever that 
may be, so thati t may be in readiness for another earthquake 
should that occur. Another is to throw out of gear the slow- 
speed governor, or, if that method is adopted, to work a change- 
wheel lever, so as to shorten the train between the driving 
power and the governor. A third is to close the circuit of 
the chronographic reed, so as to cause it to mark time on the 


Form of Seismograph. 357 


record sheet. It will thus be seen that the instrument is 
intended to be absolutely self-acting, so long as its supply of 
paper lasts and the driving mechanism continues to go. The 
supply-drum can take as much paper as Is required in a week 
on the slow speed. 

The record is made in ink by means of fine glass siphons, 
in very much the same manner as that which was introduced 
by Sir William Thomson in his siphon-recorder for submarine 
telegraph-cable work. This is extremely well adapted for the 
continuous ribbon method of working, and, besides, gives an 
excellent clear record which requires no further preparation 
before it is filed for reference ; and, what is of great im- 
portance, the record is obtained with exceedingly little dis- 
turbance from friction at the marking-point. 

The siphons which write the horizontal components of the 
motion are controlled by two pendulums, the suspending wires 
of which are held out of the vertical by horizontal struts ter- 
minating in knife-edges which rest against the bottoms of 
flat V-grooves fixed to a cast-iron pillar rigidly attached to 
the sole plate of the instrument. These pendulums, when set 
in vibration, describe cones, and hence they have been called 
“conical pendulums.”’ The degree of deflection from the 
vertical can be varied from about one and a half inches toa 
foot, by sliding the pendulum-bob along the strut. The strut 
is made in two pieces, so that a part of it can be removed 
when high sensibility is required, and in consequence the 
mass is used near the knife-edge. The bob of the pendulum 
is suspended by a fine platinum or steel wire from an arrange- 
ment which permits the suspending wire to be lengthened and 
shortened, and also allows the po‘nts of suspension to be put 
in such positions above the knife-edges as causes the struts to 
place themselves in positions at right angles to each other, 
and at the same time provides the means of adjusting their 
periods of free vibration to any desired length *. 

It is of great importance in apparatus of this kind that the 
mass which, through its inertia, enables the record of the 
motion of the earth to be written, should be as far as possible 
from the knife-edge or poins fixed to the earth; a long 
period of free vibration can thus be obtained combined with 
considerable stability of position, while the greatest motion to 
which the knife-edge is likely to be subjected does not turn 
the strut through a large angle. If this latter condition be 


* This pendulum isa modification of one designed by the Author in 
the beginning of 1880, in which the weight was supported by a thin wire 
in line with a rigid vertical axis fixed to the end of the strut and resting 
against bearings so as to keep the strut horizontal. 


358 Mr. T. Gray on an Improved 


not provided for, the interpretation of the record becomes 
exceedingly difficult ; and this difficulty is likely to be greatly 
increased by the mass acquiring oscillations in its own free 
period of such large angular amplitude that the direction of the 
component whichis being recorded becomes avariable quantity. 

The siphon which writes the vertical component of the 
motion is controlled by a compensated horizontal lever instru- 
ment, on the same principle as that introduced by the present 
writer and exhibited to the Seismological Society of Japan, 
and described in the Transactions of that Society, vol. i. 
part 1, p. 48, and vol. ii. p. 140, and also in the Philo- 
sophical Magazine for September 1881. This instrument 
consists of a horizontal lever carrying near one end a heavy 


mass, and provided at the other end with knife-edges in a 


line at right angles to the length of the lever. The lever 
is supported by two flat springs, acting, through a link, on 
a knife-edge attached to it at a point between the mass and 
the knife-edges before mentioned, which are by this means 
held up against the apex of inverted V-grooves rigidly 
fixed to the framework. In the form of this instrument 
previously described in the Philosophical Magazine, the 
supporting springs were of the ordinary spiral type; but in 
subsequent instruments two flat springs have been adopted , 
because for the same period of oscillation of the lever with- 
out compensation they give a more compact arrangement. 
These springs are now made of such variable breadth between 
the fixed and the free ends that, when they are supporting the 
lever, each part is equally bent. They may either be initially 
straight, and bent into a circular form when in use, or they 
may be initially set to a circular form and straight when in 
use. When the lever is supported in this way it has a fairly 


long period of free vibration ; and this may be increased to — 


any desired extent by means of a second pair of springs, 
which pull downwards on a light bar fixed vertically above 
the axis of motion of the lever. This second pair of springs, 
besides providing the necessary compensation for the positive 
stability of the lever and supporting-spring system, gives a 
ready means of obtaining a fine adjustment for bringing the 
lever to the horizontal position. This is accomplished either 
by giving to the points of attachment of the compensating 
springs a screw-adjustment so that they can be moved a short 
distance backward or forward, or by making the point of 
attachment of one spring a little in front of, and of the other 
a little behind, the vertical plane through the knife-edge. 
The lever can then be raised or lowered by increasing the 
pull on one spring and diminishing that on the other. Sir 


Form of Seismograph. 399 


William Thomson has recently suggested to the writer that a 
flat spring, which in its normal state is bent to such a curva- 
ture that it is brought straight by supporting a weight on its 
end, might be found a good arrangement for a vertical motion 
seismometer. This would certainly have considerable advan- 
tage in the way of simplicity, and with proper compensation 
applied, say to the index-lever, so as to lengthen the period, 
may be found very suitable. The only doubtful point seems 
to be whether the want of rigidity in the spring may not lead 
to false indications in the record due to the horizontal motions. 

The application of a rigid horizontal lever, pivoted on knife- 
edges and supported by springs as a vertical-motion seismo- 
meter, was first described in the earlier of the two papers to 
the Seismological Society of Japan, quoted above. The ad- 
vantage of this arrangement, as rendering it possible to obtain 
a long period of free vibration by placing the intermediate 
point of support below the line joining the other two, was also 
pointed out. The advantage obtained by the lever itself, 
without compensation, over an ordinary stretched string was 
more specifically pointed out in the other papers referred to ; 
and a method of obtaining very perfect compensation, either 
for a lever or an ordinary spring arrangement, by means of a 
liquid, was then given. The idea of increasing the period of 
a vibrating system by the addition, as it were, of negative 
stability, which was first brought forward in these papers, has 
been worked out in various ways ; but the method described 
in this paper is the most perfect yet adopted. Its application 
to the ordinary pendulum was also brought forward and dis- 
cussed at a subsequent meeting of the Seismological Society 
of Japan*. 

The apparatus above referred to for recording the horizontal 
components of the motion during an earthquake may, when 
properly adjusted, be used for registering minute tremors and 
slow changes of level of the earth’s surface. It is, however, 
absolutely necessary for such a purpose that friction of the 
different parts should be reduced to a minimum ; and hence 
the siphons, or the marking-points when a smoked surface is 
used, are only brought for a few seconds at a time into contact 
with the paper, thus recording a series of dots close enough 
together to form practically acontinuous line. Anothermethod, 
which gives excellent results and is simple, has been much 
used by Prof. Milne in Japan. It consists in passing from the 
point of the index, through the paper, to the drum a series of 
sparks from an electric induction-coil. The sparks can be 


* “On a Method of Compensating a Pendulum so as to make it 
Astatic,” by Thomas Gray, Trans, Seis. Soc. Japan, vol. iii. p. 145. 


360 Mr, T. Gray on an Improved 


made to pass at regular intervals by a clockwork circuit- 
closing arrangement; and, by the perforations they leave, a 
record both of their position and the corresponding time is 
obtained*. This method is absolutely frictionless so far as the 
recording-point is concerned, and has the advantage that the 
sheet can afterwards be used as a stencil-plate for printing 
copies of the record. An ordinary simple pendulum, furnished 
with a very light vertical index of thin aluminum tube giving 
a multiplication of 200, has been for some time in use. The 
record of the position of the end of the index is taken on two 
strips of paper which are being slowly pulled along, in direc- 
tions at right angles to each other, under it. The sparks per- 
forate both sheets simultaneously, thus automatically breaking 
up the motion into two rectangular components. The details of 
some forms of apparatus for this purpose will form the subject 
of a separate communication. 


Mechanical Details. 


The record-receiver consists of a long ribbon of thin paper, 
about five inches broad, which is slowly wound from the 
drum A, situated behind the drum C (Plate IV. fig. 1), on 
to the drum, B, by means of a train of clockwork driven 
by a spring or a weight of sufficient power to keep the 
ribbon taut. The rate at which the paper is fed forward 
is governed by a second train of clockwork, driven by a 


separate weight and governed by means of two Thomson 


spring-governors. In gear with this train of wheelwork 
there is a third drum, C, round which the paper is taken as 
it passes from the drum A to the drum B. This drum is 
kept moving at a uniform rate, and serves to regulate the 
motion of the paper. The object of the double set of clock- 
work mechanism is to render the rate at which the paper is 
fed forward independent of the size of the coil on the drums 
A and B. The surface of the drum C is covered with several 
thicknesses of blotting-paper for the purpose of giving a soft 
surface for the siphons to write upon, and of preventing the 
ribbon blotting or adhering to the drum in consequence of ink 
passing through the paper. This blotting-pad is of some 
importance, because a cheap kind of thin paper is found to 
answer perfectly for the siphons to write upon. They move 
with less friction on a moderately rough surface and on paper 
which rapidly absoros the ink. Under ordinary circumstances 
the paper is fed forward from a quarter of an inch to an inch 

* This method of recording the motions of an index was used by 


Sir William Thomson in his “Spark Recorder.” ‘Mathematical and 
Physical Papers,’ vol. ii. p. 168, 


Form of Seismograph. 361 


per minute, this being kept up continuously for the purpose of 
allowing the magnitude and the time of occurrence of any dis- 
turbance, which is of sufficient amplitude to leave a record, to be 
accurately obtained. This obviates the unavoidable uncertainty 
which exists as to the action of any automatic contrivance de- 
signed to come into action at the time of the disturbance. The 
time of occurrence is obtained by causing the siphon, D (figs. 1 
and 3), to mark equal intervals of time on the paper ribbon. 
The siphon is fixed to a light index-lever which is pivoted on 
the end of the lever, H, and the link, F. The lever E turns 
round an axis at G, and rests with its end in contact with the 
wheel, H, which is fixed to the end of the hour-spindle of the 
clock, K (fig. 1). As each tooth of the wheel H passes the 
end of the lever H a mark is made on the paper, and the end 
of the hour is distinguished by putting a larger or a double 
tooth at that part of the wheel. ‘The time at which an earth- 
quake has occurred can thus be found by measuring the dis- 
tance of the record of the disturbance from the last time-mark, 
then counting the number of intervals from the last hour- 
mark, and then the number of hours to a known point. It is 
convenient to mark the hour once or twice a day on the paper, 
so as to save trouble in the reckoning should an earthquake 
occur. 

The ordinary rate of motion is much too slow for the record 
to show the motions of the earth in detail; and, as has been 
already stated, this is obtained by automatically increasing 
the speed at the commencement of the shock. The arrange- 
ment for doing this is shown at O (fig. 1), and is also illustrated 
diagrammatically in fig. 2. Referring to the diagram, a and 
b represent two levers, which are pivoted at ¢ and d respec- 
tively. On the right-hand end of the lever 6 a ball ¢ is fixed, 
and the weight of this is counterpoised by another ball /, 
which rests on a rocking platform g, pivoted on the other end 
of the lever. Opposite the end of the rocking platform g, and 
fixed to the end of the lever a, there is another platform, h, 
which receives the ball / when it rolls off the platform g. The 
ball is prevented from rolling sideways by light springs, 7 i, 
fixed to the sides of the platforms. On the end of the lever a, 
or on another lever connected with it, the end of the spindle 
of the wheel) is supported. This wheel is in gear with the 
pinion &, which is on the shaft of the most distant of the two 
governors from the driving-power. The ball / is so adjusted 
over the pivot of the rocking platform g that an exceedingly 
slight disturbance causes it to roll forward on to A, tilting g 
over, and at the same time pushing down the end of a and 
raising the wheel 7 out of gear with the pinion 4, thus allowing 


362 Mr. T. Gray on an Improved 


the clockwork to run on without the governor which regulates 
the slow speed. The rate of motion then rapidly increases 
until the second governor acquires sufficient velocity to con- 
trol the speed, after which the paper moves forward at a rapid 
but uniform rate. In order to again reduce the speed after a 
sufficient interval has elapsed, the rolling forward of the ball 
f allows the unbalanced weight of e to bring a wheel /, on the 
spindle of which a “ snail,’ m, is fixed, into gear with the 
pinion, », which forms part of the clockwork mechanism. 
The spindle of / rests on a spring, 0, which is adjusted so as 
to push the lower part of the “snail” just into contact with a 
pin, p, fixed in the lever b. The weight of e acting through 
the pin p on the “ snail” deflects the spring o and brings the 
wheel / into gear with the pinion. The “snail” is then gra- 
dually moved round and raises the ball e and the end of the 
lever b, at the same time lowering the rocking platform g. 
After this has proceeded so far as to cause the platform g to 
come below the lever of A the ball rolls back to its original 
position; and, as the “snail”? moves round, the platforms 
are gradually raised to their original positions, the wheel 7 
again comes into gear with the pinion &, and the speed is re- 
duced. The wheel / remains in gear with the pinion n for a 
short time after the speed is reduced, so as to allow the final 
adjustment in position of the platform g and the ball f to be 
made gently. After this is accomplished a hollow in m allows 
the spring o to push the wheel / out of gear, and everything 
is left in readiness for the next disturbance. 

In order to obtain the rate at which the paper is moving at 
any instant during the transition period between the slow and 
the quick speed, the lever a is made to close an electric circuit 
at g, which causes an electromagnetic vibrator, indicated at J 
(fig. 3), to come into action and write equal short intervals of 
time on the record-sheet. ‘The short intervals are sometimes 
given by a vibrating reed, which is the most convenient 
arrangement if the intervals are to be fractions of a second ; 
but, for marking seconds, a break-circuit arrangement worked 
by the clock, &, is preferable. The way in which the siphon, 
D, is made to record both the long and the short time-intervals 
is sufficiently explained by the diagram, fig. 3. 

One of the “conical pendulums”’ used for actuating the 
siphons which record the two horizontal components of the 
motion is illustrated in plan in fig. 4, and in elevation in 
fig. 5. It consists of a thin brass cylinder 7, filled with lead, 
and held deflected by a light tubular strut, s, furnished with 
a knife-edge at ¢, which rests against the bottom of a vertical 
V-groove fixed to the support wu. The weight of the pendu- 


Form of Seismograph. 363 


lum-bob and strut is supported by a thin wire, v, attached at 
the lower end to a stirrup, w, pivoted at wa little below and 
in front of the centre of gravity of 7, and taken at the upper 
end over a small wheel, y, to a drum, z, round which the wire 
may be wound, so as to adjust the level of the strut, s. The 
position of the pivot, w, is so arranged that the knife-edge at 
t has little or no tendency to rise or fall, no matter at what 
part of the strut the cylinder 7 may be clamped. The wheel 
y is provided with adjusting screws, a, and 6,, by means of 
which the top of the wire can be placed vertically above the 
knife-edge, or as much in front of or behind that point as may 
be necessary to make the period of free vibration of the pen- 
dulum have any desired length. A light aluminium lever 
is hinged to the strut s at d,, and is provided at its outer end 
with a small hollow steel cone e,, which may be placed over 
one or other of a series of sharp points /;, fixed to the vertical 
arm of the cranked lever g,. The lever g, turns round a 
horizontal axis at h; in bearings fixed to the ink-well 7,, and 
the vertical arm is hinged at j,, so as to be free to turn in a 
direction at right angles to the plane of the crank. A siphon, 
ky, is fixed to the horizontal arm of the lever g,, and, drawing 
ink from the well 7, writes a continuous line on the paper 
ribbon. The horizontal arm of the lever g, is made very 
flexible in a horizontal direction, and besides can be turned 
round a vertical axis to such an extent as allows the pressure 
of the point of the siphon on the paper to be adjusted until it 
is only sufficient to give a record. 
The horizontal-lever pendulum used for actuating the 
siphon which writes the vertical motion is illustrated dia- 
grammatically in fig. 6. It consists of a horizontal lever, /,, 
carrying at one end a cylindrical weight m,, and free to turn 
round knife-edges m,, fixed to the other end of the lever. 
The lever is supported in a horizontal position by two flat 
springs, clearly shown in fig. 1, and indicated at 0, fig. 6. 
A light aluminium index, p,, pivoted at q,, and connected by 
a thin wire or thread to the end of the lever /,, carries a fine 
siphon, 7, which rests with one end in the ink-well, s,, and 
the other end touching the surface of the paper. The end of 
the index is weighted sufficiently to cause it to follow the 
motions of the lever. This arrangement gives a period of 
free vibration of about two seconds in the actual instrument ; 
and in order to increase this period a second set of springs, 
indicated at t, are made to act on knife-edges, w, fixed ver- 
tically above 7,,so as to add negative stability to the arrange- 
ment. When the lever is deflected downwards the pull on 
the supporting spring is increased, but at the same time the 


= Sa es 
SSS eee 


= = 2S = —— 
SS 


2 a ee 


364 Mr. F. Y. Edgeworth on 


knife-edge u, comés in front of the vertical plane through 1; 
and, since the lower point of attachment of the compensating 
spring ¢, is far below 7, a couple is introduced which com- 
pensates for the greater upward force. The same is the case 
in the reverse order, when the lever is deflected upwards. 
Hence if the pull exerted by ¢, and the other conditions 
mentioned below be properly adjusted, the horizontal lever 
may be made to have any desired period of free oscillation. 
In actual practice some positive stability must be given to 
the lever in order that its position of equilibrium may be 
definite ; but its period may be made so great that, even if 
oscillations of considerable amplitude in its own period are 
set up, they will be so slow compared with those of the earth- 
quake, that the undulating line so drawn will still be practi- 
cally straight, so far as the earthquake record is concerned. 
In order to insure good compensation, the condition must be 
fulfilled that the rate of variation of the compensating couple 
is always the same as that of the supporting couple. If 
this be not the case, the pendulum must either be left with 
excessive positive stability for small deflections, or it will be 
continually liable to become unstable by the compensating 
couple becoming too great when the deflection exceeds a cer- 
tain limit. In the present instance, let the modulus of the 
supporting spring be M, the arm at which it acts a; let the 
modulus of the compensating spring be M,, and the distance 
between 1, and wu, be a;. Then for a deflection of the lever 
equal to @ we have, on the supposition that the length of the 
supporting spring and link is great compared with a,, for the 
return couple Ma’ cos 0 sin 0@— Mya,’ cos 6 sin 0—M,8 sin 8, 
where 8 +a, is the total elongation of the spring for the hori- 
zontal position of the lever. Now our condition necessitates 
B being either zero or negative; and in order to keep within 
this condition the length of the unstretched spring and link 
are made to reach a little above m, and the height of w, is 
made adjustable, so that M,a,” can be adjusted to be as nea 

Ma? as may be desired. 


XLI. On Discordant Observations. By F. Y. Hpgnworta, 
M.A., Lecturer at King’s College, London*. 


ANT observations may be defined as those which 

present the appearance of differing in respect of their 
law of frequency from other observations with which they are 
combined. In the treatment of such observations there is 
great diversity between authorities ; but this discordance of 


* Communicated by the Author. 


Discordant Observations. 365 


methods may be reduced by the following reflection. Different 
methods are adapted to different hypotheses about the cause of 
a discordant observation ; and different hypotheses are true, or 
appropriate, according as the subject-matter, or the degree of 
accuracy required, is different. 

To fix the ideas, I shall specify three hypotheses : not pre- 
tending to be exhaustive, and leaving it to the practical reader 
to estimate the & priori probability of each hypothesis. 

(a) According to the first hypothesis there are only two 
species of erroneous observations—errors of observation proper, 
and mistakes. The frequency of the former is approximately 


represented by the curve y= a e—?; where the constant h 


J 7 

is the same for all the observations. But the mathematical 
law* only holds for a certain range of error. Beyond certain 
limits we may be certain that an error of the first category 
does not occur. On the other hand, errors of the second 
category do not occur within those limits. The smallest 
mistake is greater than the largest error of observation proper. 
The following example is a type of this hypothesis. Suppose 
we have a group of numbers, formed each by the addition of 
ten digits taken at random from Mathematical Tables. And 
suppose that the only possible mistake is the addition or sub- 
traction of 100 from any one of these sums. Here the errors 
proper approximately conform to a probability curve (whosef 
modulus is 4/165), and the mistakes{ are quite distinct from 
the errors proper. 

Here are seven such numbers: each of the first six was 
formed by the addition of ten random digits, and the seventh 
by prefixing a one to a number similarly formed— 


Wein 9B 431, 50. (49. 45, | 136) 


* This follows from the supposition that an error of observation is the 
joint result of a considerable, but finzte, number of small sources of error. 
The law of facility is in such a case what Mr. Galton calls a Binomial, or 
rather a Multinomial. (See his paper in Phil. Mag. Jan. 1875, and the 
remarks of the present writer in Camb. Phil. Trans. 1886, p. 145, and 
Phil. Mag. April 1886.) 

+ I may remind the reader that I follow Laplace in taking as the 
constant or parameter of probability-curves the reciprocal of the coefficient 


of x: that is 2 according to the notation used above. It is 2 times 


the “Mean Error” in the sense in which that term is used by the 
Germans, beginning with Gauss, and many recent English writers 
(e.g. Chauvenet) ; and it is 7 times the Mean Error in the (surely more 
natural) sense in which Airy, after Laplace, employs the term Mean Error 
(Chauvenet’s Mean of the Errors). 

{ In physical observations the limit of errors proper must, I suppose, 
be more empirical than in this artificial example. 


366 Mr. F. Y. Edgeworth on 


The hypothesis entitles us to assert that 23 is an error-proper 
—an accidental deviation from 45; though the odds against 
such an event before its occurrence are considerable, about 100 
to 1. On the other hand, we may know for certain that 136 
is a mistake. | 

(8) According to the second hypothesis, the type of error 
is still the probability-curve with unvarying constant. But 
the range of its applicability is not so accurately known before- 
hand. We cannot at sight distinguish errors proper from mis- 
takes. We only know that mistakes may be very large, and 
that the large mistakes are so infrequent as not to be likely to 
compensate each other in a not unusually numerous group of 
observations. This hypothesis may thus be exemplified :— 
As before, we have a series of numbers, each purporting to be 
the sum of ten random digits. But occasionally, by mistake, 
the sum (or difference) of two such numbers is recorded. The 
mistake might be large, but it would not always exceed the 
limits of accidental deviation (100 and 0); which need not be 
supposed known beforehand. Here is a sequence of seven 
such numbers, which was actually obtained by me (in the 
course of 280 decades) — 


d0, 54, 41, 78, 46, 38, 49. 
The hypothesis leaves it doubtful whether 73 may not be a 
mistake ; the odds against it being an ordinary accidental 
deviation being, before the event, about 250 to 1. 
(y) According to the third hypothesis all errors are of the 


type y = eine But the A is not the same for different 


observations. Mistakes may be regarded as emanating from 
a source of error whose / is very small. This hypothesis may 
be thus illustrated. Take at random any number n between 
certain limits, say 1 and 100. ‘Then take at random (from 
Mathematical Tables) digits, add them together and form 
their Mean (the sum + 7), and multiply this Mean by ten. 
The series of Means so formed may be regarded as measure- 
ments of varying precision ; the real value of the object mea- 
sured being 45. The weight, the h’, being proportionate to n, 
one weight is a priori as likely as another. In order that the 
different degrees of precision, the equicrescent values of h, 
should be & priori equiprobable, it would be proper, having 
formed our 7 as above, to take the mean of (and then mul- 
tiply by 10), not n, but n? digits. Here is a series formed 
in this latter fashion :— 


lOp oe So oseeanenon” Dalida 6 Lo Oe as 1 
PAu teeue sana 25 49 36 1 100 64 1 


n 
10x Mean ofm" (3) 45 43 100 48 47:5 100 
random digits ; 


Discordant Observations. 367 


In this table the first row is obtained by taking at random 
ten digits from a page of Statistics, 0 counting for ten. The 
second row consists of the squares of these numbers. The 
third row was thus formed from the second :—I took 25 
random digits, and divided their sum by 25; then multiplied 
this mean by 10. I similarly proceeded with 49 (fresh) digits, 
and so on. It will be noticed how the defective precision of 
the fourth and seventh observations makes itself felt. It was, 
however, a chance that they both erred as far as they could, 
and in the same direction. 

In the light of these distinctions I propose now to examine 
the different methods of treating discordant observations. For 
this purpose the methods may be arranged in the following 
groups :— 

I. The first sort of method is based upon the principle that 
the calculus of probabilities supplies no criterion for the cor- 
rection of discordance. All that we can do is to reject certain 
huge errors by common sense or simple induction as distin- 
guished from the calculation of a posteriori probability. 

II. Or, secondly, we may reject observations upon the 
ground that they are proved by the Calculus of Probability 
to belong to a much worse category than the observations 
retained. 

IIT. Or, thirdly, we may retain all the observations, affecting 
them respectively with weights which are determined by 
a postertort probability. 

IV. In a separate category may be placed a method which, 
as compared with* the simple Arithmetical Mean, reduces the 
effect (upon the Mean) of discordant observations—the method 
which consists in taking the Medianf or ‘“ Centralwerth’’ t of 
the observations. 

I propose now to test these methods by applying them in 
turn to all the hypotheses above specified. 

I. (a) The first method—which is none other than Airy’s, 
as I understand his contribution$ to this controversy—is 
adapted to the first hypothesis. Upon the second hypothesis 
(@) the first method is liable to error, which, as will be shown 
under the next heading, is avoidable. (y) Upon the third 
hypothesis the first method is not theoretically the most 
precise ; but it may be practically very good. 

II. Under the second class 1 am acquainted with three 


* This is pointed out by Mr. Wilson in the Monthly Notices of the 
sa enmicel Society, vol. xxxviii., and by Mr. Galton, Fechner, and 
others. i 

+ Cournot, Galton, &c. 

{ Fechner, in Abhandl. Sax. Ges. vol. [xvi.]. 

§ Gould’s Astronomical Journal, vol. iv. pp. 145-147. 


368 Mr. F. Y. Pilsen orth on 


species : the criteria of Prof. Stone*, Prof. Chanvoneli and 
Prof. Peircet. 

IT. (1) Prof. Stone’s method is to reject an observation 
when it is more likely to have been a mistake than an error 
of observation of the same type as the others. In deter- 
mining this probability he takes account of the a prior 


probability of a mistake. He puts for that probability = 


admitting that m cannot be determined precisely. The use of 
undetermined constants like this is, I think, quite legitimate§, 
and, indeed, indispensable in the calculation of probabilities. 

This being recognized, Prof. Stone’s method may be justified 
upon almost any hypothesis. Hypothesis (a) presents two 
cases: where the discordant observation exceeds that limit of 
errors proper which is known beforehand, and where that 
limit is not exceeded. For example, in the instance|| given 
above—where 45 is the Mean, and the Modulus is about 13— 
the discordant observation might be either above 100 (e.g. 110) 
or below it (e.g. 84). Now let us suppose that the a priori 
probability of a mistake is not infinitesimal, but say of the 
order yj55: Since the deviation of 110 from the Mean is 
about five times the Modulas, the probability of this deviation 
occurring under the typical law of error is nearly a millionth. 
This observation is therefore rejected by Method II. (1), which 
so far agrees with Method I. Again, the probability of 84 
being an accidental deviation is less than a forty-thousandth; 
84—45 beingabout three times the Modulus. Therefore 84 also 
is rejected by the criterion. And we thus lose an observation 
which is by hypothesis (a) a good one. But this loss occurs 
very rarely. And the observation thrown away is, to say the 
least, not** a particularly good one, though doubtless it may 
happen that it is particularly wanted—as in the case of Gen. 
Colby, adduced tf by Sir G. Airy. 

II. (1) (@) The second hypothesis is that to which Prof. 
Stone’s criterion is specially adapted. Upon this hypothesis, 
84 may be a mistake. In rejecting such discordant observa- 
tions, we may indeed lose some good observations, especially if 


* Month. Not. Astronom. Soc. Lond. vol. xxviii, pp. 165-168. 

+ ‘Astronomy,’ Appendix, Art. 60. t Ibid. Art. 57, 

§ See my paper on @ priort Probabilities, in Phil. Mag. Sept. 1884; also 
‘Philosophy of Chance,” Mind, 1884, and Camb. Phil, Trans. 1885, 
pp: 148 ef seg. | Page 365, 


q 165, exactly. As determined empirically by me from the mean- 


square- -of-error of 280 observations (ze. sums of 10 digits), the Modulus 
was 4/160. 

** See the remark made under II. (2) ({). 

+t Gould’s Astronom. Journ. vol. iv. p. 138. 


Discordant Observations. 369 


we have exaggerated the a priori probability of a mistake. But 
it may be worth while paying this price for the sake of getting 
rid of serious mistakes. specially is this position tenable 
according to the definition of the quesitum in the Theory 
_ of Errors*, which Laplace countenances. According to this 

view, the destderatum in a method of reduction is not so 
much that it should be most frequently right, as that it should 
be most advantageous ; account being taken, not only of the 
Frequency, but also of the seriousness, of the errors which it 
incurs. Prof. Stone’s method might diminish our chance of 
being right (in the sense of being within a certain very small 
distance from the true markt); and yet it might be better than 
Method I., if it considerably reduced the frequency of large 
and detrimental mistakes. 

II. (1) (¥) Prof. Stone’s method is less applicable to the 
third hypothesis. ‘Though even in this case, if the smaller 
weights are @ priori comparatively rare, it may be safe enough 
to regard (m—1) of the m observations as of one and the 
same type ; and to reject the mth if violently discordant with 
that supposed type. 

The only misgiving which I should venture to express 
about this method relates, not to its essence and philosophy, 
but to a technical detail. Prof. Stone says:—“ If we find that 


value which makes J-| eVdy = wy [ where p is the devia- 
Te) P. nr 


tion of a discordant observation, and a is the modulus of the 
probability-curve under which the other observations range, 


and = is the & priori probability of a mistake], all larger 
values of p are with greater probability to be attributed to 
mistakes.” But ought we not rather to equate to > not the 
left-hand member of the equation just written, which may be 
called (2), but 6” = where m is the number of observa- 


tions. Jam aware that the point is delicate, and that high 
authority could be cited on the other side. There is some- 
thing paradoxical in Cournot’s{ proposition that a certain 


* See my paper on the “ Method of Least Squares,” Phil. Mag. 1883, 
vol. xvi. p. 363; also that on “ Observations and Statistics,” Camb. Phil. 
Tr. 1885; and a little work called ‘ Metretike’ (London: Temple Co., 1887). 

+ The sense defined by Mr. Glaisher, ‘ Memoirs of the Astronomical 
Society,’ vol. xl. p. 101. 

_{ Exposition de la théorie des Chances, Arts. 102, 114, “Nous ne nous 
dissimulons pas ce qu'il y a de délicat dans toute cette discussion,” I 
may say with Cournot. 


Phil. Mag. 8. 5. Vol. 23. No. 143. April 1887, 2C 


370 Mr. F. Y. Edgeworth on 


_ deviation from the Mean in the case of Departmental returns 
of the proportion between male and female births is signifi- 
cant and indicative of a difference in kind, provided that we 
select at random a single French Department; but that the 
same deviation may be accidental if it is the maximum of the. 
respective returns for several Departments. There is some- 
thing plausible in De Morgan’s* implied assertion that the 
deficiency of seven in the first 608 digits of the constant 7 is 
theoretically not accidental; because the deviation from the 
Mean 61 amounts to twicet the Modulus of that probability- 
curve which represents the frequency of deviation for any 
assigned digit. I submit, however, that Cournot is right, and 
that De Morgan, if he is serious in the passage referred to, has 
committed a slight inadvertence. When we select out of the ten 
digits the one whose deviation from the Mean is greatest, we 
ought to estimate the improbability of this deviation occurring 
by accident, not with De Morgan as 1—@(1°63), corresponding 
to odds of about 45 to 1 against the observed event having 
occurred by accident ; but as 1—6"(1°63), corresponding to 
odds of about 5 to 1 against an accidental origination. 

II. (2) Prof. Chauvenet’s criterion differs from Prof. 
Stone’s in that he makes the & priori probability of a mistake 
—instead of being small and undetermined—definite and con- 
siderable. In effect he assumes that a mistake is as likely as not 
to occur in the course of m observations, where m is the number 
of the set which is under treatment. Itis not within the scope 
of this paper to consider whether this assumption is justified 
in the case of astronomical or of any other observations. It 
suffices here to remark that this assumption coupled with 
hypothesis («) commits us to the supposition that huge mis- 
takes occur on an average once in the course of 2m observa- 
tions. Upon this supposition no doubt Method II. (2), is a 
good one. Hypothesis (8) expressly t excludes this suppo- 
sition ; the mistakes which, according to II. (2), are as likely 
as not, must, according to this second hypothesis, be of 
moderate extent. Thus, in the case above put of sums of ten 
digits, suppose that the number of such sums under observa- 
tion is ten. According to Prof. Chauvenet’s criterion we 
must reject any sum which lies outside 45+, where 

k 2n—1 19 
3) Ton) 0 ee 
* Budget of Paradoxes,’ p. 291. 


t If we take many batches of random digits, each batch numberin 
608, the number of sevens per batch ought to oscillate about the Mean 61, 


according to a probability-curve whose Modulus is a a 608 = 10-4, 
t Above, p. 366. 10 


Discordant Observations. 371 


This gives for the required limit about 15. According, then, 
to II. (1) (@), any observation greater than 60, or less than 30, 
is more likely than not to be a mistake in the sense of not 
belonging to the same law of frequency as the observations 
within those limits. But why on that ground. should the 
discordant observation be rejected ? Suppose there were not 
merely a bare preponderance of probability, but an actual 
certainty, that the suspected observation belonged to a different 
category in respect of precision from its neighbours, the best 
course certainly would be if possible (as Mr. Glaisher in his 
paper ‘On the Rejection of Discordant Observations” sug- 
gests) to retain the observation affected with an inferior weight. 
But if we have only the alternative of rejecting or retaining 
whole, it is a very delicate question whether retention or re- 
jection would be in the long run better. There is not here 
the presumption against retention which arises when, as in 
IT. (1), the discordant observation is large and rare ; so that, 
if it is a mistake, it is likely to be a serious and an uncom- 
pensated one. However, Prof. Chauvenet’s method may 
quite possibly be better than the No-method of Sir G. Airy. 
Much would turn upon the purpose of the caleulator—whether 
he aimed at being most frequently right* or least seriously 
wrong. ‘The same may be said with reference to hypo- 
thesis (7). 

There is a further difficulty attaching particularly to this 
species of Method II. In its precise determination of a limit, 
it takes for granted that the probability-curve to which we 
refer the discordant observation is accurately determined. 
But, when the number of observations is small, this is far 
from being the case. Neither of the parameters of the curve, 
neither the Mean, nor the Modulus, can be safely regarded as 


C 
accurate. The “probable error” of the Mean is *477 a. 


e 
where ¢ is the Modulus. The probable error of the Modulus 
is conjectured to be not inconsiderable from the fact that, if 
we took m observations at random, squared each of them and 
formed the Mean-square-of-error, the “‘ probable error ” of that 

2 


Mean-square-of-error would be ‘477 — }. This, however, is 
vn , 


not the most accurate expression for the probable error of the 
Modulus-squared as inferred { from any given n observations. 


* See the remarks above, p. 369. 

+ Todhunter, art. 1006 (where there is no necessity to take the origin 
at one of the extremities of the curve). 

} LI allude here to delicate distinctions between genuine Inverse Pro- 
bability and other processes, which I have elsewhere endeavoured to 
draw, Camb. Phil. Trans, 1885, 

2C2 


372 Mr. F. Y. Edgeworth on 


To appreciate the order of error which may arise from these 
inaccuracies, we may proceed as in my paper of last Octo- 
ber*. First, let us confine our attention to the Mean, sup- 
posing for a moment the Modulus accurate. Let k have 
been determined according to Prof. Chauvenet’s method, so 


; EO. 


To determine more accurately the probability of an observa- 
tion not exceeding a we must put for a, a+z, where z is the 
error of the Mean subject to the law of frequency 


mz2 
i J m ia 
WV 16 
The proper course is therefore to evaluate the expression 


{ a(“7*) = vie me ae 


Expanding 0, and neglecting the shee powers of zt, we 
have for the correction of a(t ) the subtrahend See he 


where £ is put for = e Call this modification of 8, 00. To see 
how the primd facie limit 8 is affected by this modification, 


let us put 
[0+00](8+46) =}, 
whence eae tale i 
Beal eae Ret? 
Whence A vem a i: 


an extension of the limit which may be sensible when 7 is 
small. 

In the example given by Prof. Chauvenet the uncorrected 
limit as found by him is 1:22. This divided by the Modulus 
[which= V 2e=°8] is 1°5. This result, our 6, divided by 15 
the number of observations, gives ‘1 as the correction of 8 ; 
08 as the correction of the limit a. The limit must be ad- 
vanced to 1:30. This does not come up to the discordant 
observation 1:40. But we have still to take into account 
that we have been employing only the apparent Modulus (and 
Mean Error), not the real one. In virtue of this consideration 
I find—by an analysis analogous to that given in the paper 


* Phil. Mag. 1886, vol. xxii. p. 371. + See the paper referred to. 


Discordant Observations. - aes 


just referred to—that the limit must be pushed forward as 
much again; so that the suspected observation falls within 
the corrected limit. I have similarly treated the example 
given by Prof. Merriman in his Textbook on The Method of 
Least Squares (131). The limit found by him is 4°30, and 
he therefore rejects the observation 4°61. But I find that 
this observation is well within the corrected limit *. 

II. (3) Prof. Peirce’s criterion is open to the same objec- 
tions as that of Prof. Chauvenet. Indeed it presents additional 
difficulties. If by y the author designates that quantity which 


Prof. Stone calls 2 and which I have termed the “a priori”’ 


probability of a mistake, I am unable to follow the reasoning 
by which he obtains a definite value for this y. But I am 
aware how easy it is on such subjects to misunderstand an 
original writer. 

III. We come now to the third class of method, of which 
Tam acquainted with three species. (1) There is the procedure 
indicated by De Morgan and developed t by Mr. Glaisher ; 
which consists in approximating to the weights whic are to be 
assigned to the observations respectively, after the analogy of 
the Reversion of Series and similar processes. (2) Another 
method, due to Prof. Stone {, is to put 

es iieas ke are Pe eo OX dig Dhgea 
as the a posteriori probability of the given observations having 
resulted from a particular system of weights h,’ h,” &e., and a 
particular Mean 2 ; and to determine that system so that P 
should be a maximum. (3) Another variety is due to Prof. 
Newcomb §. 

Ill. (1) & (2) Neither of the first two Methods are well 
adapted to the first two hypotheses. Both indeed may success- 
fully treat mistakes by weighting them so lightly as virtually to 
reject them. But both, I venture to think, are liable to err in 
underweighting observations, which, upon the first two hypo- 
theses, have the same law of frequency as the others. Both, in 
fact, are avowedly adapted to the case where the observations 


* These corrections may be compensated by another correction to which 
the method is open. In determining whether the suspected observation 
belongs to the same type as the others, would it not be more correct to 
deduce the characters of that type from those others, exclusive of the 
suspected observation? The eftect both on the Mean and the Modulus 
would be such as to contract the limit. 

+ Memoirs of the Astronomical Society. 

t Monthly Notices of the Astronomical Society, 1874. This Method 
was proposed by the present writer in this Journal, 1883 (vol. xvi. 
p. 360), in ignorance of Prof. Stone’s priority. : 

§ American Journal of Mathematics, vol. viii. No. 4. 


374 On Discordant Observations. 


are not presumed beforehand to emanate from the same source 
of error. The particular supposition concerning the a priort 
distribution of sources which is contemplated by the De- 
Morgan-Glaisher Method, has not perhaps been stated by 
its distinguished advocates. The particular assumption made 
by the other Method is that one value of each h is as likely as 
another over a certain range of values—not necessarily between 
infinite limits. I have elsewhere* discussed the validity of 
this assumption. I have also attempted to reduce the in- 
tolerable labour involved by this method. Forming the equa- 
tion in x of (n—1) degrees, 


nx”-!— (n—1)Sa, a-? + (n—2)Sax, 4, x" —&e.=0, — 


I assume that the penultimate (or antepenultimate) limiting 
function or derived equation will give a better value than the 
last-derived equation |nv—jn—1S2,, which gives the simple 
Arithmetic Mean. Take the observations above instanced 
under hypothesis (+), | 


31, 45, 48, 100, 438, 47:5, 100. 
For convenience take as origin the Arithmetical Mean of 


these observations 58°5, say 58. Then we have the new 
series 


429, 18) 216, 492 15 


Here S2,2,= —2494. And the penultimate limiting equa- 
tion is 
7x6xX5xX4x8a74+5xX4x%38xX2x1x —2494=0, 


Whence. #?=119. And #=+11 nearly. To determine 
which of these corrections we ought to adopt, the rule is to 
take the one which makes P greatest; which ist the one 
which makes («—.)(@—2) (w—a3) . . . (@—2;) smallest ; 
each of the differences being taken positively. 

The positive value, +11, gives the differences 


38, 24,26," 21). 36; 22a 
For the negative value, —11, the differences are 
16,0 25) 4,..03,): 4 HOR ae 


(where 0 of course stands for a fraction). The continued 
product of the second series is the smaller. Hence —11 is 


* Camb. Phil. Trans. 1885, p. 151. 
Tt See Phil. Mag. 1888, vol. xvi. p. 371. 


Action of Heat on Potassic Chlorate and Perchlorate. 375 


the correction to be adopted. Deducting it from 58, or rather 
58°5, we have 47:5, which is a very respectable approximation 
‘to the real value, as it may be called, viz. 45. 

III. (8) Prof. Newcomb* soars high above the others, in 
that he alone ascends to the philosophical, the utilitarian, 
principles on which depends the whole art of reducing obser- 
vations. Here are whole pages devoted to estimating and 
minimizing the Hvil incident to malobservation. With Gauss, 
Prof. Newcomb assumest “that the evil of an error is pro- 
portional to the square of its magnitude.”” He would doubt- 
Jess admit, with Gauss, that there is something arbitrary in 
this assumption. Another somewhat hypothetical datum is 
what het describes as the “distribution of precisions.” In view 
of this looseness in the data, it becomes a nice question 
whether it is worth while expending much labour upon the 
calculation. The answer to this question depends upon an 
estimate of probability and utility, concerning which no one 
is competent to express an opinion who has not, on the one 
hand, a philosophical conception of the Theory of Errors, and, 
on the other hand, a practical acquaintance with the art of 
Astronomy. The double qualification is probably possessed 
by none in a higher degree than by the distinguished astro- 
nomer to whom we owe this method. 

IV. It remains to consider the fourth Method. But the 
length and importance of this discussion will require another 


paper. 


XLII. On the Action of Heat on Potassic Chlorate and Per- 
chlorate. By Kyuunp J. Miuts, D.Sc., AS. 


|? has been pointed out by Teed||, and subsequently by 

P. Frankland and Dingwall{, that potassic chlorate and 
perchlorate may be decomposed by heat in such a manner as 
to lead in each case to various relations among the products 
of decomposition. 

It has occurred to me that both of these chemical changes 
are instances of Cumulative Resolution**, from which point 
of view they admit of very simple, and at the same time 
perfectly adequate, representation. 


* American Journal of Mathematics, vol. viii. No. 4. 

Tt § 3, p. 348. t § 9, p. 359. 

§ Communicated by the Author. 

| Proc. Chem. Soc. xii. p. 105; xvi. p. 141; xxxiil. pp. 24 & 25. 

q Ibid. xvi. p.141; xxx. p. 14; and Trans, Chem. Soe. 1887, p, 274. 
** Phil. Mag. [5] ili. p. 492 (1877). 


376 Dr. E. J. Mills on the Action of Heat 


Action of Heat on Potassic Chlorate. 

The products of this action are potassic chloride, oxygen, 
and perchlorate. All known relations among these products 
may be expressed by the cumulative equation 

2n KC1O3— (n—2) O.= (n+ 1) KC1O,+ (n—1) KCI. 

In order to compare theory with experiment, I have selected 
the quotient of the percentage of chloride produced by that 
of the oxygen formed as the specific measure of the change ; 
the percentage being calculated on the weight of chlorate 
taken for trial. If this quantity be called 7, the equation 
alleges that 


OFT Der! 
Y RCAC) 20. 
or 
AGTH oe 
n—2 


It will be seen from the following Table that this is the case ; 
a rational value ‘of m always corresponding to the specitic 
measure r. No attention has been paid to instances in which 
perchlorate is known to have been decomposed. Whenn= @, 
the equation becomes 


2K'C1O;—O,=KCI10,+ KCl. 


Taste I. 
| 
No. of| Oxygen, |Chloride, ss i Authority. 
exp. | per cent.) per cent. 
166 | 526 | 31687 | 47910 Teed. 
3°49 | 10°86 J 1117 | 49949 Ps 


6:00 | 18:25 3°0417 | 52906 i 

2°66 |x 9°4916| 3°5684 | 3°8879 | Frankland and Dingwall. 
5:19 |x18°566 | 3°5774 | 38744 i 

647 |*21°609 | 3:3399 | 4:3164 55 

6°89 /*21°533 | 3:1253 | 4:9438 ‘S 

6°78 |x20°147 | 2°9715 | 5°6523 i 

36! | 11:58 3°2167 | 4°6392 Teed. 

10. 4:27 4-73 3°7244 | 36764 3 

Ie 161 6:00 3°7267 | 3°6736 0 


£9 OC ST Se St BO DO 


12. 1:60 6:14 3°8375 | 3°5502 . 
13. 1:47 4°84 3°2925 | 4:4307 ” 
14, 0°80 2°18 2°7250 | 7:9488 5 


It is remarkable that the value of n should, amongst so 
many experiments, prove to be so very restricted in its range. 
There seems to be some tendency for r to be preferably about 
equal tom. The exact fulfilment of this condition requires 


* Recalculations. 


on Potassie Chlorate and Perchlorate. BTL 


r=n=3'7028 or *6300,—values which indicate the reduction 
of the chemical change to a mere action of mass. __ 


Action of Heat on Potassie Perchlorate. 
The equation of Cumulative Resolution is 
(n+1)KCIO,—(2n—1)0,=2KCI1O; + (n—1) KCI; 


the products of the reaction being chlorate, chloride, and 
oxygen. Its starting-point is a point in the chlorate equa- 
tion, viz., (n+1)KCIiO,. In this case the percentage of 
chlorate cannot exceed a certain amount, viz., that indicated 
by the relation given by n=1, or 

. 2KC1O,—O0,=2KCI1Os;, 
= 88°46 per cent. 

A comparison of theory with experiment can be made on 
a basis similar to that previously taken, viz. :— 


Oz ~ nol 
eb, eee 

or n—1 
42867 r = apa i 


When n = «, the equation reduces to 


KC10,—20, = KCL. 


TaB_e ILI. 
Number of | Oxygen, | Chloride, : 
Experiment. | per cent.| per cent. e oo 
310 | 297 | -95806| 32992 Teed. 
447 4-41 ‘98658 | 3°7430 Ss 
7°30 7°82 10712 | 66275 “i 


35°21 40°33 1:1454 |28:278 Bs 
6°34 * 67148 | 10591 | 59348 | Frankland & Dingwall. 
780 | * 82600 | 1:0590 | 5-9300 ‘5 

24:05 | *27:145 | 11287 |15-970 


STS Ure Chore 


In this case r cannot be equal to n. As regards the pro- 
portion of chlorate formed, it has been stated by all three 
investigators that this diminishes as the reaction proceeds. 
Frankland and Dingwall have made actual determinations of 
itsamount. In order to compare this part of their work with 
theory, I have taken their experimental ratio p-of chloride 
to chlorate, and calculated it from the estimations, made in 


* Recalculations. 


378 Action of Heat on Potassie Chlorate and Perchlorate. 


Table II., of the corresponding values of n. The relation 
required for this purpose is 


Oe Pa Da 
or 4:2860 p = n. 
TaBLE III. 
= eae a (Chloride p cale. N. 
eR | =ehilonasbes) 
5 1:5630 1:3847 59348 
6. 15781 1:3836 59300 
7 68442 37261 15-970 ; 


There is a fair agreement in comparisons 5 and 6. The 
discrepancy in 7 arises in great part from the fact that the 
form of the function renders it difficult to deduce accurately 
such high values of n as 15:970 from experiments of not 
exceptional accuracy. If, for example, n=30, r=1:1275, 
which differs very little Hdesd from r=l1' 1287, when n= 
15:970r. It is probable also that the chlorate (never actually 
exceeding more than about 4 per cent. of the perchlorate) 
was decidedly underestimated. Additional experiments on 
this subject are much to be desired. 


Equal Weight Relations. 


It is usual in chemical change for a critical relation to be 
established when certain of the reagents are present in equal 
weights. Thus, in the chlorate reaction, if the ratio of 
chloride to oxygen be that of equality in weight, r=1; and 
the equation 
42867 r ==> 
n—2 
then gives n='24970. 

Similarly, in the case of the perchlorate, where 


asa 
2n—1’ 
if r=1, n=4:0048—i. e. the reciprocal of the previous value 
of appears then that, subject to the condition indi- 


cated, the reaction whereby perchlorate is decomposed is the 
exact inverse of the chlorate reaction. 


42867 r= 


Biiaes 


XLII. Reply to Prof. Wilhelm Ostwald’s criticism on my 
paper * On the Chemical Combination of Gases.” 


To the Editors of the Philosophical Magazine and Journal. 
(GENTLEMEN, 
ROFESSOR WILHELM OSTWALD, in a work en- 
titled Lehrbuch der Allgemeinen Chemie (Bd. i. 
p- 745), has criticised my paper on the Chemical Combina- 
tion of Gases published in the Philosophical Magazine, Octo- — 
ber 1884, in which I applied the Williamson-Clausins theory 
of dissociation to the solution of several problems in the theory 
of the combination of gases. I wish in this letter to answer 
this criticism, and, in order to make my meaning clear, I 
must recapitulate one part of the paper. According to the 
Williamson-Clausius hypothesis, the molecules of a gas are 
continually splitting up into atoms, so that the atoms are 
continually changing partners. I defined the “ paired”’ time 
of an atom to be the average time an atom remained in part- 
nership with another atom, and the ‘free time ” the average 
time which elapses between the termination of one partner- 
ship and the beginning of the next. Now the free time will 
evidently depend upon the number of free atoms in the unit 
volume, for before an atom can be paired again, it must come 
into collision with another atom ; and though it need not get 
paired at the first collision, yet it is evident that the time it 
remains “‘ free ” will be proportional to the time between two 
collisions, and, therefore, inversely proportional to the number 
of free atoms in unit volume. But after the atom has got 
paired with another, there is no reason why the time they 
remain together should depend upon the number of molecules, 
unless we assume that the atoms are knocked apart by colli- 
sion with other molecules. 

As one of my reasons for undertaking the investigation 
was, that an eminent spectroscopist had mentioned to me that 
there was spectroscopic evidence to show that the molecules 
got split up independently of the collisions, and as I wished 
to see if I could get any evidence of this from the phenomena 
of dissociation, it would have been absurd on my part to beg 
the question by assuming that the paired time was inversely 
proportional to the number of atoms. I therefore made no 
supposition as to the dependence of the paired time on the 
number of atoms, except when the dissociation was produced 
by an external agency, such as the electric discharge, but left it 
to be determined from the experiments. 

The above reasoning seems to me to be clear enough, but 
as it is substantially the same as that in my paper, and Prof. 
Ostwald says it is difficult to conceive how it is that I have 


380 On the Chemical Combination of Gases. 


not noticed that the paired time is inversely proportional to 
the number of atoms, I must endeavour to find some way of 
explaining myself which shall not entail the necessity of form- 
ing any abstract conceptions. Let us then illustrate the 
pairing of a molecule by the act of getting into a cab, and a 
gas by a number of men and cabs, the men riding about in the 
cabs, getting out,and after walking about for a time getting into 
acab again. To fix our ideas, let us suppose that after leaving 
a cab, each man gets into the sixth cab he meets. Then it is 
evident that the time he spends on foot (his “ free time’’) 
will depend upon the number of cabs, the more cabs the 
shorter the time ; and if the cabs are evenly distributed, his | 
“‘ free time ” will be inversely proportional to the number of 
cabs. But after getting into a cab, unless he is upset by a 
collision with another cab, there is no reason why the time 
he stays in his cab should depend upon the number of cabs. 
Prof. Ostwald’s remark, when applied to this case, is—it 
is difficult to conceive how it is that I have not noticed that 
the only way of getting out of a cab is to wait until one is 
shot out by the collision of one’s own cab with another. But 
difficult as the conception is, Prof. Ostwald is equal to it, for 
in a footnote he suggests that the reason is that I knew 
what the result ought to be, and so “ cooked’’ my equations 
accordingly. Now I should not have thought it worth while 
to reply to criticism of this order had it not been that the 
subject of the application of mathematics to chemistry is only 
dealt with in a few text-books, so that it is important to point 
out any misrepresentations and misstatements in those which 
profess to explain this subject. The amusing part of Prof. 
Ostwald’s criticism is that when, after his tirade, he attempts 
to obtain one of my equations, he implicity assumes that the 
molecules are not split up by the collisions, for he assumes 
that the number of molecules split up in a given time is pro- 
portional to the average number of molecules. Now, if we 
refer to the illustration of the cabs, it will be evident at once 
that this is equivalent to assuming that the collisions have 
nothing to do with the breaking-up of the molecules, for if 
the men were shot out of their cabs by collisions with cabs 
with men inside, the number leaving their cabs in any time 
would be increased fourfold if the number of men in cabs 
were doubled, for the number of men in cabs would be 
doubled, and the average time they spend in the cabs would 
be halved. 

It may illustrate the care with which the book has been 
written, and the reliance to be placed on its contents, if I 
mention that within about half a page Prof. Ostwald makes 
three misstatements. He says that an equation he obtains by 


Intelligence and Miscellaneous Articles. 381 


a process of his own is the same as one of mine, though it is 
not ; he says that I sometimes suppose the free time to be 
constant, and sometimes to depend on the number of atoms, 
when I do not; and, lastly, that I have not stated what 
meaning I attach to r, when on page 238, line 44, I have 
defined it to be the free time multiplied by the number of atoms. 
Iam, Gentlemen, 
Your obedient servant, 


Trinity College, Cambridge, J.J. THOMSON. 
Feb. 14, 1887. 


XLIV. Intelligence and Miscellaneous Articles. 


ON CERTAIN MODIFICATIONS OF A FORM OF SPHERICAL 
INTEGRATOR. 


To the Editors of the Philosophical Magazine and Journal. 


GENTLEMEN, 
if HAD occasion recently to read in the Philosophical Magazine 

(August 1886) the very interesting description of a ‘‘ Spherical 
Integrator,” designed by Mr. Frederick John Smith, and which 
appears to be a modification of that of Prof. Hele Shaw. But the 
first conception of these apparatus, and it is to this that I wish to 
call your attention, belongs without doubt to me, as in No. 630 of 
‘Nature’ (Nov. 24, 1881) I gave a description of an ‘‘ Anemometer 
Integrator” founded on the same principle, and which was after- 
wards mentioned in the Quarterly Journal ot the Royal Meteoro- 
logical Society, No. 48(1882), by Mr Laughton (‘‘ Historical Sketch 
of Anemometry and Anemometers ”). 

The modification designed by Mr. F. J. Smith tending to do 
away with or lessen, as much as possible, the moment of inertia of 
the sphere, appears to me excellent, especially if it is to transmit 
velocities of small magnitude. But when it is simply required to 
register that of the wind upon a moderate scale, I believe that the 
primitive form suffices; and after several trials which I have made, 
an ivory ball rolling on bronze cylinders is that which gives the best 
results. 

I beg, Gentlemen, that you will allow this claim of priority to 
appear in your valuable Journal, and also that you will accept my 
most sincere thanks and the assurance of my marked regard. 


Madrid Observatory, March 12, 1887. V. VENTOSA. 


ON THE STRENGTH OF THE TERRESTRIAL MAGNETIC FIELD 
IN BUILDINGS. BY M. AIME WITZ. 


In consequence of the removal of my laboratory to a new 
building in which the joists and framework are of iron, I have 
been led to determine exactly the values of the horizontal com- 
ponent in the various rooms used for Physics, with a view to 
certain researches which I have undertaken. I have observed 


382 Intelligence and Miscellaneous Articles. 


astonishing discrepancies ; and I think it useful to draw the attention 
of physicists to this subject, which has been but little studied. 

A simple method of measuring the horizontal intensity consists 
in passing a constant current through a circuit containing a weight 
voltameter, and a tangent-galvanometer. By determining the 
absolute strength of the current on the one hand by the results of 
electrolysis, and on the other by the deflection of a compass- 
needle, and equating these two values, we can get the value of T at 
the spot where the galvanometer was placed. This method was 
of sufficient exactitude for the work of comparison in which I was 
engaged. 

A Poggendorff’s battery may be used; this is a very constant 
source when the chromic liquid is strongly acid, and the external 
resistance is great. As an electrolyte I took a 10 per cent, 
solution of pure copper sulphate; the copper electrodes at a 
distance of about 30 mm. had 12 square centimetres immersed ; 
from this resulted a favourable density of current, and therefore a 
beautiful deposit of metal which was continuous and _ perfectly 
adherent. The loss of the soluble electrode was always equal to 
within 5 mer. to the gain of the negative electrode. The intensity 
of the current, which was about + of an ampere, was determined to 
within 5,55 of an ampere; it was assumed that 1190 mgr. was 
deposited per ampere-hour. Two good tangent-galvanometers 
were used simultaneously ; their constants are as follows :— 


Length Mean Number 
Galva- -——_—*-—oa~ radius. of : 
nometer. of needle of wire R. windings. 20 
mm. mm. cm. 
PA VGN 15 1258 16°68 12 0-221 
Lae ae 20 1114 16°12 11 0-233 


The needles are suspended to a cocoon-thread ; the long pointers 
of aluminum enable us to read ;4, of a degree. The relative 
dimensions of the needles and of the frames are in these two 
instruments in such a ratio, that we may dispense with the use of 
the term of correction, which I have considered proportional to the 
tangent of the deflection 6. 

The manipulation was very simple; the element having been 
shortcircuited for a few minutes, the current was passed for an 
hour through the voltameter and the galvanometer. Two double 
readings were made after five and twenty-five minutes; the current 
being reversed in the galvanometer after thirty-five and fifty-five 
minutes. The mean of these eight readings gives the value of the 
mean deflection of the needle in the course of the operation. It 
remained to weigh the electrodes, and to take the mean p of the 
loss and gain of the plates in miligrammes. The formula 


Pate ha 
10 1190 ~ 2xn 
leads to the value of T in C.G.S. units; and the same operations 
repeated in various places enable us to discover considerable 


T=tan 6 


Intelligence and Miscellaneous Articles. 383 


variations of the horizontal component in a building where much 
iron has been used in the construction. 

The following are the details of an experiment; they enable us 
to judge of the value of the method, and the agreement of the 
observations. This experiment was made on the 13th April 18386, 
at La Solitude in the suburbs of Lille, in the centre of an open 


space of several acres, and at a distance from any buildings and 
from water- or gas-pipes. 


Deflections of the Galvanometer. 


Right. Left. 
Time. a we FT! 
m ° ) fo) ro) 

BPE Osea a, 0 Wee ie ae 
PN ss as 3o°/0 34:15 

Bi .. 43 OO ad 33°50 is LE 
24 55 Weel i ay 33°85 33°50 
4 55 30°40 33°00 


General Mean 33°- Bae 33° 33, 


Observations of the Voltameter. 


mer. 
Loss of the Soluble Electrode .......... 360 
Gain of the Negative Electrode ........ 355 
Veer, Ate. ORAS ee ged 307'°O 
WEE Gs 
10 1190 = 0°233 T tan Soo 
T = 0:187. 


This value of T will serve as a point of comparison; it is less 
than the value observed at Paris on the 1st January 1886, as was 
to be expected from the position of Lille. We consider it exact to 
within 3 or 4 thousandths ; in fact an observation made after the 
first gave 0:185, and a great many experiments made in the 
laboratory aes that the result of an experiment never differs 
by more than zoo0 from the mean of a month of investigations. 


The table given below gives the value of T obtained in various 
parts of our septs sc 


Date. aps 
La patos mieaw: Dille: wi taeulailly. April 13 0°186 
esquinimearullle i G4 ay oo aed brit 0-191 
Outer court of the Faculty ...... us 1 0°183 
Inner court of the Faculty ...... May 21 0-190 
Protessor's Room 1.7. oh wue elo I 0:152 
nyisical:Cabimeti jetties os Soe a oes Mar. 23 0-134 
PARA VAY sae is SY HOS YU Wiel 3 29) 0-133 
Peles eR Sey A. cihaite Sand OR dhol a oh 330 0-114 
Vaulted Hall ....... Y FA, July 21 0-194 


It follows from these researches that T may be reduced by 40 or 
50 per cent. in a building made of iron ; hence the same current 
will give in the same galvanometer a ‘deflection of 33° to 45° 


+ 


384 Intelligence and Miscellaneous Articles. 


according to its position. It will thus be seen that the calibration 
of instruments of this kind must not be forgotten when they are 
moved from one place to another.—Journal de Physique, Jan. 1887. 


ON METALLIC LAYERS WHICH RESULT FROM THE VOLATILIZA- 
TION OF A KATHODE. BY BERNHARD DESSAU. 


The results of the present investigation may be summed up as 
follows :— 

By appropriate electrical discharge in highly rarefied spaces, the 
metal which acts as kathode is volatilized and settles on a glass plate 
as a reflecting layer or mirror. If the oxygen has not been most 
carefully removed, all metals seem to undergo oxidation under these 
circumstances. There is perhaps in all cases a combination with 
the traces of residual gas (hydrogen or nitrogen), yet the mirrors 
obtained in hydrogen are not materially different from those of pure 
metals. With suitable arrangement of the electrodes the layer of 
metal is obtained as a flat cone ; and when viewed in reflected light, 
under as acute an angle as possible, coloured interference-rings are 
obtained, which prove the presence of a dispersion in the metals. 1t 
may be concluded with some certainty that this dispersion is normal 
in platinum, iron, nickel, and silver, and abnormal with gold and 
copper. The layer directly produced by the discharge, whether it 
be metal or oxide, is always double refracting, probably in conse- 
quence of an electrical repulsion between the particles expelled, and 
the regular stratification thereby produced; in the metals the ray 
which vibrates tangentially is accelerated in respect of the others. 
In the metals the cross of double refraction was also observed in 
reflected light, and in reflection from the metal side the action 
was the reverse, and from the glass side the same as in transmitted 
light. Double refraction disappears on oxidation of the double- 
refracting metals, as well as by reduction of the layers of oxide, 
while heating without any chemical change has no effect.— Wiede- 
mann’s Annalen, No. 11, 1886. 


ON THE PASSAGE OF THE ELECTRIC CURRENT THROUGH AIR 
UNDER ORDINARY CIRCUMSTANCES. BY J. BORGMANN. 


One end of the coil of a Wiedemann’s galvanometer is connected 
with the earth, and the other with a platinum wire, which is placed 
in the flame of an insulated spirit-lamp. Ata distance of 14 metre 
from this lamp is an ordinary Bunsen burner, which is connected 
with a conductor of the Holtz machine ; the other conductor is put 
to earth. 

When the lamp is lighted the galvanometer indicates no current ; 
but when the disk is rotated a distinct current at once appears in 
the galvanometer, and the deflection of the needle does not alter so — 
long as the machine works at a uniform rate. If the Bunsen 
burner is connected with the other conductor of the machine, a 
current in the opposite direction is at once set up.—Bezblatter der 
Physik, January 1887. 


THE 


LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[FIFTH SERIES.] 
MAY 1887. 


XLV. On the Expansion of Salt-Solutions. By W.W. J. 
NicoL, M.A., D.Sc., F.R.S.E., Lecturer on Chemistry, 
Mason College, Birmingham*. 


[Plates V. & VI.] 


NHIS is a subject which has at no time attracted much 
attention. With the exception of the experiments of 
Bischoff, Muncke, Despretz+, and Rosetti t, which deal 
with special cases, such as the expansion of sea-water, we are 
indebted to Gerlach§ and Kremers|| for the whole of our 
knowledge of the subject ; and this may be summed up as 
follows :— 

1. The rate of expansion of a salt-solution is the more 
uniform the more concentrated the solution. Thus, while the 
line representing the volume of pure water at various tempe- 
ratures is very pronounced in curvature, the lines correspond- 
ing to the volumes of various solutions of a salt approximate 
more and more to a straight line the stronger the solution 
(Gerlach, loc. cit.). 

2. As a consequence of the above it follows that salt-solu- 
tions expand faster than water at low temperatures ; but that 
at high temperatures, on the other hand, the rate of expansion 
is less than that of water. 


* Communicated by the Author. 

Tt Pogg. Ann. xli. p. 58. 

t Ann. de Chim. et Phys. (4) xvii. p. 370 (1869). 
§ Spec. Gew. der Salzlosungen. Freiberg, 1859. 
|| Pogg. Ann. vols. c.—cxx. (1857-62). 


Phil. Mag. 8. 5. Vol. 23. No. 144. May 1887. 2D 


386 Dr. W. W. J. Nicol on the 


3. So markedly is this the case that at or below 100° C. 
the difference between the volumes of water and of a salt- 
solution of the same volume at 0° C. actually changes its sign 
in many cases. 

4. It is therefore possible in the case of every salt-solution 
to find a temperature at which its coefficient of expansion is 
the same as that of water at that temperature. According to 
de Heen* this temperature is dependent only on the nature 
of the salt, and is the same whatever be the strength of the 
solution. Kremerst, on the contrary, holds that this last has 
some slight influence. 

5. No connexion can be traced between the expansion of 
‘a salt in the solid state and that of its solution (Gerlach, loc. 
cit.). 

In the course of experiments on the nature of solution it 
appeared to me probable that there exists a connexion be- 
tween the increase of solubility with rise of temperature and 
the rate of expansion of solutions of the salts. With the 
object of ascertaining whether or not such a connexion exists 
I made a series of experiments on the subject; for I found 
that the results obtained by previous experimenters either 
were not numerous enough or required confirmation. While 
in all cases they were not suited for my purpose, owing to 
the solutions experimented with being of percentage com- 
position, and thus requiring recalculation into terms of mole- 
cules of salt per 100 H,O. Even when this was done the 
differences in the strengths of the various solutions were so 
irregular that any conclusions derivable from the results were 
extremely unsatisfactory. 

The present paper contains the results of experiments on 
solutions of the four salts NaCl, KCl, NaNO, and KNOs, at 
temperatures between 20° C. and 80° C. The solutions were 
as nearly as possible molecular, and differed from one another 
in the case of each salt by two molecules of salt; for the 
reasons given below, it will be seen that it is almost impossible 
to use solutions of the precise composition aimed at. Still 
the error thus introduced is practically eliminated by the 
method of calculation employed. 

In the determination of the expansion of a liquid two 
general methods are available, each presenting certain points 
of advantage over the other. I finally decided to employ the 
dilatometric in preference to the pycnometric method ; and I 
had the less hesitation in doing so, as I found it possible to 


* Physique Comparée. Bruxelles, 1883, p. 76. 
+ Pogg. Ann. evi. p. 882 (1858). 


Expansion of Salt-Solutions. 387 


construct a constant-temperature bath, which removed all the 
difficulties and inaccuracies attending the use of long tubes. 

As I wished, so far as possible, to experiment with solu- 
tions of definite molecular composition, and at the same time 
to avoid the multiplication of calibration and other corrections, 
it was not possible to employ dilatometers with tubes suf- 
ficiently large in internal diameter to permit of the introduc- 
tion of the solution through the graduated tube. I therefore 
modified the form of dilatometer devised by Kremers (Joc. cit.), 
so that it presented the appearance shown in Pl. V.fig.1. The 
bulb A is furnished with a tube at either end. One of these, 
B, is short and bent round parallel with the side of the 
bulb; it is about 3 millim. external and 1 millim. internal dia- 
meter, but at the free end is thickened and narrowed to about 
0:2 millim. A shoulder is formed about 20 millim. from the 
free end, by which the closing apparatus is attached. The 
measuring-tube, C, is about 700 millim. long, and is divided 
into millimetres from —10 millim. to 600 millim. The gra- 
duated tubes were obtained from Geissler, of Bonn, and after 
calibration were sealed on to bulbs of suitable capacity. 
The dilatometer is filled through the short tube, the end of 
which is flat, and is closed by an indiarubber pad screwed 
down by the clamp shown in fig. 2. 

The calibration of the tubes was performed as follows:— 

A short thread of mercury was passed through the tube 
and measured at every 20 millim. It was found that the 
bore was extremely uniform in all the tubes, no abrupt change 
being perceptible. As the tubes were so long and so uniform, it 
was considered unnecessary to do more than calibrate them 
for more than every 100 millim. Thus, in the case of one 
of the dilatometers Dv,a thread of mercury had the following 
lengths at various parts of the tube :-— 


At Omillim. length was 110 millim. 


HODES 0 ah, nhl WALA a, 
200 ” oD 111 ” 
POU iy imctinas volt beets 
A00 aK % Tray 
500 ” ” 110 ” 


Mean value 110°7. 


The mean value in grammes of mercury of each millimetre 
was obtained by weighing the mercury required to fill nearly 
the whole of the graduated part of the tube. In the case 
above 594 millim. contained 2°723 grm. mercury, or 1 millim. 
contained 0:00459 grm. 

2D 2 


388 Dr. W. W. J. Nicol on the 


From the data thus obtained the bulbs were proportioned 
to each tube, so that the value of each millim. in terms of the 
total capacity of the bulb should lie between 0-00004 and 
0:00006. When the bulbs had been sealed on, the dilato- 
meters were filled with mercury at 20° C. up to the zero 
mark on the stem, and the weight of the mercury was deter- 
mined. In the case above this was 79°93 grm. Thus the 
mean value of 1 millim. of the stem was 5:74, that of the bulb 
and stem up to the zero being 100,000. This was then cor- 
rected according to the calibration results for every 100 millim., 


giving :— 


0-100=5°778 300-400 =5°713 
100-200 =5°713 ~ 400-500 = 5-701 
200-300 =5°718 900-600=5°778. 


The coefficient of expansion of the glass was determined in 
each case by both mercury and water. With the above dila- 
tometer the apparent expansion of mercury between 20° C. 
and 78°°8 C. was found to be 100,914°6. Calculated from 
Landolt’s tables the true volume is 101,069, difference 154°4. 
The volume of water was 102,483°8, calculated 102,637°8, 
difference 154:0, giving coefficient for glass=0-00002°62. 

Of the various dilatometers thus made only three were used 
in the following experiments. In these the mean value of 
1 millim. of the stem was Dr=4°81, Div=5'51, Dv=5:74. 

The thermometers employed were two by Geissler and two 
by Negretti and Zambra. These last were verified at Kew. 
Those by Geissler were from 20°-60° C., and from 40°-100° C., 
and were divided into 10ths; one of the others was from 
—10°-40° C., also in =1,ths, and the fourth from —10°-110° C., 
divided into half degrees (a very open scale). These were 
carefully compared together and corrected at 20°, 45°-46°, 
50°-51°, 56°-57°, 61°-62°, 67°-68°, 72°-73°, and 78°-80°— 
the temperatures at which determinations were to be made. 
The comparisons were made in two constant-temperature 
baths, one at 20° C., the other being the one employed for 
heating the dilatometers. 

The constancy of temperature was in this last case obtained 
by means of the vapour of a liquid boiling under a constant - 
pressure variable at will. The liquid in this case was a mixture 
of alcohol and water boiling at about 82° at 760 millim. The 
apparatus consists of two parts, the dilatometer-bath with boiler 
and condenser, and the pressure-regulator. 

The bath is shown in fig. 3. It consists of two glass tubes, 
one within the other, diameters 65 and 45 millim., and re- 
spective lengths 900 ‘and 700 millim: The longer and wider 
of the tubes is drawn out at one end to about 15 millim. 


EKzpansion of Salt-Solutions. 389 


diameter ; the shorter tube is closed and rounded at the lower 
end. A brass cap, firmly cemented on to the wide end of the 
outer tube, carries the side tube bent at right angles, by which 
communication is made with the Y-shaped condenser. The 
inner tube is secured air-tight in the cap by an indiarubber 
cork, the lower end being kept in the centre of the outer tube 
by a ring with three projecting arms. ‘The free upper end 
of the Y-condenser communicates with the pressure-regulator, 
a Woulff’s bottle being interposed to retain any liquid boil- 
ing over. The lower end of the Y passes to the bottom of the 
boiler, which is a stout copper cylinder 150 millim. high and 
120 millim. in diameter, and stands on a solid flame-burner. 

When the boiler has been one third filled with alcohol, the 
whole apparatus is made as nearly air-tight as possible, and 
connected with the pressure-regulator. The inner tube is 
filled with water, the gas is lighted, and the pressure is re- 
duced the desired amount. The vapour of the boiling alcohol 
passes up between the two tubes, entirely surrounding the 
inner for its whole length. At first the condensed alcohol 
flows back into the boiler; but as the temperature of the water 
in the inner tube rises, the alcohol vapour passes into the 
condenser and thus back to the boiler, complete condensation 
being insured by the second limb of the Y-tube. 

The pressure-regulator is shown at fig. 4: it is based partly 
on that proposed by Meyer, and partly on the modification 
introduced by Stadel and Schummann”*. It consists of a 
firm wooden stand some 900 millim. high by 200 millim. 
wide. <A gauge, standing side by side with a barometer in a 
trough of mercury in front of a mirror-millimetre scale, is 
connected with a tube passing across the stand near the top, 
and hent down as shown on one side, ending in a wide closed 
cylinder. The tube is furnished with two stopcocks, the upper 
to establish communication with the air when desired, the 
other communicating with the bath. On the other side of the 
gauge there is a T’-piece, the vertical limb of which passes 
_ through the stand, while the horizontal limb leads to the 
‘cut-off.’ This arrangement is shown on a larger scale at 
the side A. It consists of a narrow tube some 300 millim. 
long, opening at the top into the wide tube furnished with a 
narrow side tube. Into the upper end of the wide tube is 
fastened air-tight another tube, only slightly narrower ; it is 
contracted at the lower end to a small orifice, and a small 
hole is made at the shoulder. (The delicacy of the regulation 
depends to a large extent upon the proper size and position 
of this opening.) To the upper end of this tube, which is 


* ZLeitschrift fiir Instrumentenkunde, 1882, p. 391. 


390 Dr. W. W. J. Nicol on the 


about 400 millim. long, is attached a narrow tube which 
passes down behind the stand and leads to the pump, a small 
Woulff’s bottle being interposed to catch any mercury which 
may splash over. A reservoir of mercury, capable of being 
raised and lowered, attached to the lower end, makes the “ cut- 
off’’ complete. 

On the back of the stand is pivoted a wooden bar about 
800 millim. long, and capable of being fixed at any angle by 
the thumb-screw passing through the stand. ‘This bar carries 
a tube about 10 millim. internal diameter, and closed at one 
end with a side tube about 20 millim. from the other. The 
tube is filled to a depth of about 700 millim. with mercury, 
and a capillary passing through the cork which closes the 
end and reaching to the bottom establishes communication 
with the air when the pressure is sufficient to overcome that 
of the column of mercury in the tube (the height of this 
column being determined by the inclination of the tube). 
Connexion with the remainder of the apparatus is made by 
an indiarubber tube attached to the side tube and to the end 
of the T-piece which passes through the stand. 

The pressure-regulator is used as follows :—The water- 
pump is set in action, and the mercury-reservoir being lowered 
both stopcocks are closed. When the pressure as indicated 
by the gauge is about that which is required, the reservoir is 
raised till the mercury just reaches the end of the “ cut-off,” 
and a screw-clip on the tube to the pump is closed till only a 
few bubbles of air pass the “ cut-off.”’ Then the tube at back 
of stand is slowly inclined till air begins to bubble through 
the mercury. The clip is slightly opened ; and if the pressure 
be too great, the reservoir is lowered, and the inclined tube 
placed more upright. With a little care, and with at least 
50 millim. of mercury above the side hole of the “ cut-off,” 
it is possible even with very variable water-pressure (provided 
the pump have a Bunsen valve) to maintain the pressure in 
the apparatus itself constant within 1 or 2 millim.; while if 
an air-reservoir be interposed between the regulator and a 
separate gauge, the pressure in this last can be kept absolutely 
constant. 

With this regulator I succeeded in maintaining a constant 
temperature in the dilatometer-bath for any required time ; 
while the temperature of the whole column of water, 700 
millim. long, was so uniform, that, without stirring, the differ- 
ence between the top and bottom did not exceed 0°°2 C. The 
dilatometers were used in pairs: two placed side by side in a 
small wire cage along with the thermometers being lowered 
into the bath to within 50 millim. of the bottom. 


Expansion of Salt-Solutions. 391 


I have entered thus minutely into the details of the appa- 
ratus employed, as it seems to me that it supplies the means 
of filling up the gap left by the paper of Ramsay and Young* 
on the attainment of constant temperatures by the employ- 
ment of liquids boiling under constant pressures; for while it is 
easy to make a small piece of apparatus air-tight, this becomes 
well nigh impossible when the apparatus is complicated, and 
then a simple and efficacious pressure-regulator is a deside- 
ratum. 

The solutions with which the experiments were carried out 
were prepared by weighing definite quantities of water, and 
adding the amount of salt necessary to make a solution con- 
taining n molecules salt per 100 molecules of water. The 
solution thus prepared was placed in the vessel shown in fig. 5. 
This was exhausted of air, placed in a water-bath, and boiled 
for ten minutes. ‘The vessel was then quickly cooled, and one 
portion of its contents was transferred to the dilatometer, 
another to a Sprengel tube, in which its density was deter- 
mined. The solutions were thus so far deprived of air that 
no bubbles made their appearance during the experiments. 
The composition of the solutions, as indicated by the density, 
was very nearly that indicated by nsalt 10OH,O. A slight 
error was unavoidable, owing to the necessity for expelling the 
alr. 

Before giving the results of the experiments it will be well 
to indicate the degree of accuracy obtained. A solution of 
6NaCl 100H,O, gave in two distinct experiments the results 
given in Table I. The millimetre divisions on the dilatometer- 


TABLE I. 

te Volume. i Volume. 
45-2 101-099 450 101-093 
50°1 101:°339 50°8 101°370 
56:0 101-626 56°3 101-640 
61°5 101°903 61:3 101898 
67:3 102-217 67-1 102-211 
723 102°488 72-6 102°502 
78:2 102°823 78-1 102°828 


stems were read roughly to tenths, 7. e. to 0°000005, and the 
temperature to 0°1. The degree of accuracy aimed at was 


* Journ. Chem. Society, 1885. 


392 Dr. W. W. J. Nicol on the 


+0:00005; and the results show that this was always attained, — 
and in most cases greatly exceeded. 

It was found that a simple interpolation-formula V = 100,000 
+t/a+t"@, where t!= (t°—20°), satisfactorily expressed the 
experimental results. The following Table gives the experi- 
mental and calculated results for the 18 solutions examined. 
These comprise the four salts NaCl, KCl, NaNO;, and KNO;, 
dissolved in water in molecular proportions. The initial tem- 
perature is 20° C., and seven other determinations were made 
in each case. The constants for the formule are given at the 
foot of each section of the Table. It will be seen that out of 
125 separate determinations between 45° and 80°, the calcu- 
lated volume differs from the found in five cases only by more 
than 9 in the 100,000 ; while the sum of the + and — differ- 
ences is (+212—230)=—18. There is, therefore, little doubt 
that the formule accurately represent the experimental re- 
sults, and that the latter are correct as.a rule to within +2 in 
100,000. 

The next point to be examined is the concentration of the 
solutions. These were, as stated, made of definite molecular 
composition, but the expulsion of the air caused a change 
in the strength which is necessarily most marked in the strong 
solutions. ‘Table III. contains the observed densities at | 
20° C., and the molecular volumes on the supposition that 
the solutions were of the strength given in the formula. 
While in the last column of the Table are given the true 
molecular volumes, where these have been determined. In 
other cases the volumes calculated from the formule given in 
my paper on saturation* are inserted, though these are 
necessarily only approximate. 


TaBLeE II. 
2 NaCl 100 H,0. 4 NaCl 100 H,0. 
é; Found. | Calculated.| A. & Found. | Calculated.) A. . 

20 100,000 100,000 0 100,000 100,000 
45°4 954 958 —4 || 45°7 1,059 1,054 +5 
50°4 1,188 1,188 0 |} 509 1,300 1,301 —1 
56:5 1,495 1,487 +8 || 56°5 1,582 1,578 +4 
61:7 Wats} 13752 +6 || 61:5 1,839 1,837 +2 
67:6 2,079 2,081 —2 || 67:6 2,168 2,167 +1 
72:5 2,364 2,366 —2 || 72°5 2,445 2,443 +2 
78:0 2,699 2,699 0 || 7&8 2,812 2,813 —Il 

a=30°86; B=0 2703. a=35'7 ; B=0°2061. 


* Phil. Mag. January 1886. 


Expansion of Salt-Solutions. 


Table II. (continued). 


6 NaCl 100 H,0. 


(ip Found. Caleculated.| A. 
20 | 100,000 | 100,000 | ~ 
45-1 | 1,096 1,095 | 41 
505 | 1.355 ene at 
56-2 | 1.633 1640 | —7 
61-41 1.901 1909. || 8 
672) 2914 9911 | +3 
72:5 | 2495 9510 | —15 
731) 2827 2.897 0 

a=380°8; B=0'1522. 
10 NaCl 100 H,O. 

20 | 100,000 | 100,000 
A56)| 1,178 1,178 0 
509 | 1,440 1,440 0 
Be3 0 L715 1712 | +3 
61-7} 1.989 1985 | +4 
671 | 2,279 9975 | +4 
72:3.| 27555 9554 | +1 
782| 2878 2877 | +1 

a—43°36; B=0-105. 
3 KCl 100 H,O. 
20 | 100,000 | 100,000 

45°6 958 ORO) a) eu 
5031 L171 173) 4\" 29 
565 | 1.470 1472 | —2 
61-7 | 1,737 1,733 | 44 
67-41 2051 9045 | +6 
723 | 2393,) 2304 | -1 
7821 2671 2674 | —8 

a=30°83 ; B=0-2598. 
7 KCl 100 H,0. 

20 | 100,000.| 100,000 
45:7 1,036 1,035 | +1 
50'6 1951 1957 | 6 
563 1.528 1.525 | +3 
61:6 1.782 e784 ||: 
67-3 2.074 9.072 | +2 
72:3 2.335 2332 | +3 
78-5 2.670 2.668 | +2 


a=3612; B—0-1624. 


8 NaCl 100 H,0. 


393 


b. Found. | Calculated.| A. 
90 | 100,000 | 100,000 
451 1,132 1,134 | —2 
50:5 | 1,398 1,398 0 
565 | 1,690 1,699 | --9 
ee | G58 POGE Gs 
675 | 2,971 9973 | —92 
720 | 92,518 2516 | +2 
730| 2,848 2.848 0 
a—42°22; B=0°1185. 
KCl 100 H,O. 
20 | 100,000 | 100,000 
45-4, 870 873 | —8 
5051 1,101 00°) a 
56-21 1.374 1.374 0 
61:3.| 1,689 1,637 | +2 
672 | 1,965 1.963 | +2 
72:41 92971 9268 | +3 
785 | 2641 9.649 | —8 
a=26-04; B=0-3288. 
5 KCl 100 HO. 
20 | 100,000 | 100,000 
45°6 992 994 | — 2 
503] 1,205 1217 | 232 
565 | 1.518 1,503 | +10 
61:6 | 1.760 Tyee tua atretag 
67-41 2,063 2064. | avy 
72:3) 2397 2332 | — 5 
7821 2.664 9669 | — 5B 
a—33-31; B—0-2157. 
Vs Na NO; 100 H,O. 
20 | 100,000 | 100,000 
45°6 1,088 1,086 | + 2 
50-4 1,325 11399 )| =2"4 
561 1,633 1,633 0 
616 1,936 1046 1 10 
671 2.977 2068 | + 9 
72-2 2.593 9586 | + 7 
18-5 2.999 2995 | + 4 


a=35'64; B=0-266. 


394 Dr. W. W. J. Nicol on the 
Table IT. (continued). 


4 NaNO, 100 H,0. 6 NaNO, 100 H,0. 
és Hound. |Calculated.| A. iE Found. |Caleulated.| A. 
30 | 100,000 | 100,000 20 | 100,000 | 100,000 
45-4 L175 1,180: | —5 |] 455 1333 1333 | 0 
50:5 1.457 1.459 | —2 || 501 1.597 1,598 | 21 
56-2 1,786 1,782 | +4 || 56-4 1.970 1,973 | —3 
61:3 2091 2086 | +5 || 61-2 2.976 2,267 | +9 
67-2 2.455 9.454 | +1 || 67-1 2.639 2640 | —1 
72-4 2.796 2795 | +1 || 72:3 2.979 9980 | —1 
785 31912} 3,208 | —172|| 78:5 3,395 3396 | —1 
a=39-99; B—0-2545, a=4787; B=0-174. 
8 NaNO; 100 H,0O. 10 NaNO, 100 H,O. 
20 | 100,000 | 100,000 20 | 100,000 | 100,000 
45:5 1,396 1.395 | 451) jl) 456 1.435 1437 | —2 
50-1 1,665 1,667 | —2 || 50-7 1,739 1.742 | —3 
56-4 2,047 2050 | —3 || 566 2,102 2103 | —1 
61-2 2.351 2,350 | +1 || 616 2,495 2416 | +9 
671 2,725 2726 | —1 || 67-9 2,893 2819 | +4 
72:3 3,068 3.067 | +1 || 72-7 3.140 3.132 | +8 
78:5 3,486 3485 | +1 || 78-0 3,480 3.494 | —4 
a—50-96; B=0-147. a=53:00; B—01219. 
12 NaNO, 100 HO. KNO, 100 H,O. 
30 | 100,000 | 100,000 90 | 100,000 | 100,000 
45:6 1,456 (55 ell 459 933 937 | —4 
50°7 L755 176l | 64, 504 1,182 1,180 | +2 
56-6 2.119 9194 | — 5 || 56-2 1.473 1,468 | +5 
61:6 2446 2.436 | +10 || 61-6 1.763 1754 | +9 
67-9 9,842 2338 | + 41] 67:8 2.107 2107 | 0 
73-7 3.156 3.149 | + 7 || 72:2 2368 | 2372 | —4 
78-0 3,499 3,499 0 || 781 2.736 2.745 | —9 
a=54-08; B=0:1075. a= 29-49 ; B=0°3057. 
3 KNO; 100 H,0. 5 KNO, 100 HO. 
30 | 100,000 | 100,000 30 | 100,000 | 100,000 
45-6 1,122 1120 | 42 || 45-4 1.201 1,200 | +1 
50:8 1,380 1388 | —8 || 50-2 1.458 1455 | =9 
56:7 1.694 1702 | —8 || 565 1801 1,803 | —2 
615 1.973 1.972 | +1 || 61:3 2.078 2078 | 0 
67:8 2.345 9,343 | +2 || 67-2 9,433 2498 | +5 
72:3 2,629 9.620 | +9 || 72:4 2,748 2748 | 0 
78:5 3,023 3,019 | +4 


a=3761; B=02889. a=42°38 ; B=0:1919. 


Expansion of Salt-Solutions. 395 
Taste III. 


M.V. found. M.V. 
1836°75 1836°42 
1876°93 1876°74 
1920-06 1919-44 
1963°83 1963°93 
2010-70 2009°66 
1827°70 1827-70 
1884:58 1886°8  cale. 
1949-04 1949-84 
2013°36 2012 cale. 
1858-22 1858:60 
1921-31 1922°6  cale. 
1987-77 19896 
2051°38 20d8'4 =, 
2125-07 Zoe O's \, 
2196°35 220204) 
1839°18 1839-07 
1920°83 1921-15 
2007°74 2006°74 


DISCUSSION OF THE RESULTS. 


Rate of Expansion.—In every case examined the expansion 
is a constantly increasing value. This follows from the form 
of the expression for the volume, 

V=100,000+¢a+t*B; 
for were the expansion uniform it would have the form 
V=100,000 +¢'e. 

It is, however, to be noted that the more concentrated the 
solution the more nearly does the curve of volume approach 
a straight line, as will be seen from Table IV., which contains 
the ratio of the two constants a and £#, and this increases 
with the concentration. In fig. 6 these ratios are plotted, and 
it will be seen that the lines are practically straight, showing 
the uniform effect of increase of concentration. The volume 
of water between 20° and 100° C. cannot be expressed by a 


too constant formula, but if 20°, 60° and 100° be taken, then 
the expression is 


V = 100,000 + 23:8 + ¢70°35, 
and the ratio of 2 is 68:0; that is, a value lower than any in 


Table TV. It thus follows that the volume-line of water is 
more curved than that of any of the salt-solutions examined, 
even the most dilute. In fig. 7 the volume-curves for water 
and for the strongest and weakest solutions are given. Again, 


396 Dr. W. W. J. Nicol on the 
TaBLeE IV. (see also fig. 6). 


a 


a. 2 ae 
B B 
2NaGhki its 30°86 0:2703 114-0 
BN ics BREE 35°70 0:2061 L73-0%- 
Ge ike Res 39-80 0°1522 261°5 
Sat she nie 49,92 0:1185 3856'0 
11 Meee Naar ern 43°36 0:1050 413:0 
KG) e388 96-04 03288 . 7992 
SRM oe 3 GE DE ne 30°83 0:2598 118-6 
Ses a ee Dor 0°2157 1545 
"(iis bra 36:12 01624 2226 
2, NaNO, ae 35°64 0:2660 134-0 
aa ot 89-99 0:2554 1bf-5 
Gis ae, 47°87 0-1740 275-0 
8 + abe 50-96 0:1470 339'6 
10 a «ke 53:00 071219 434-7 
Os welt 54:08 0:1075 503°2 
KONO, ks 29°49 0°3057 96°5 
eo tenes es 37°61 0°2389 157°5 
De. veh 42°38 0-1919 221-0 


as the constant a for water is less than that for any of the 
solutions, it is evident that at low temperatures the amount 
of expansion of salt-solutions is greater than that of water, 
and that at some temperature the volume-difference between 
a salt-solution and water reaches a maximum ; that is, at that 
point the rate of expansion of both is the same. In order to 
ascertain what temperature this is for the various salts, and 
how far it is dependent on the strength of the solutions, I 
calculated the-volumes of the solutions at every 5° between 
20° and 100° C., and compared them with those of water. 
How far I was justified in doing so is open to question ; still 
the close agreement of the experimental and calculated num- 
bers seem to warrant extrapolation to this extent without 
there being much risk of introducing serious errors. In any 
case this affects only NaNO; and the most dilute solution of 
KNO,;. The volumes for water are calculated from the 
figures given by Volkmann™* and Rosettit as the means of 
the results obtained by previous experimenters. Plate VI. 
contains the differences between the volumes of the various 
salt-solutions at various temperatures. The result of this 
comparison was that the maximum differences lay between 
55°-60° for all solutions of NaCl; at or about 50° for all KCl 
solutions; while 2NaNO;, 4NaNO;, and KNO; give no- 


* Wied. Ann. xiv. p. 260 (1881). 
} Pogs. Ann. Erg. Bd. v. p. 268 (1871). 


Expansion of Salt-Solutions. 397 


maxima, but the other solutions have maxima lying lower on 
the temperature-scale the stronger the solution ; thus:— 


6NaNO;, 90°-95°; 8NaNO; at 90°. 
10NaNOs, 80°-85° 3 12NaNO, at 80°. 
3KNOs;, 80°-85° 3 5KNOs,, 75°-80°. 


Again, with 8 and 10 NaCl and 3, 5, and 7KCl the volume 
of the solution at 100° C. is less than that of water ; but such 
is not the case even with the strongest solutions of the other 
two salts. Comparing the volumes of the various solutions of 
the same salt at 100° C., we find that with NaCl and KCl the 
stronger the solution the smaller the volume ; but with KNO; 
the reverse is the case; while NaNO; forms a connecting- 
link, the weaker solutions 2,4, and 6 behaving as those of 
KNO;; the stronger, 8, 10, and 12, as those of NaCl or KCL. 

This is all that can be gathered from the results in the 
above form, and it is clear that the conclusions arrived at by 
former experimenters are in the main correct. ‘The results of 
Kremers with solutions of the same salts give figures which, 
where the solutions are the same strength, completely agree 
with mine at low temperatures, though the discrepancy is 
marked at high temperatures ; the cause of this being probably 
in the use of too high a correction for the exposed dilatometer 
column (see Thorpe, Chem. Soc. Journ. 1880). 

The questions now remain to be considered: Is it permis- 
sible to compare together equal volumes of solutions of 
different strengths? Is it probable that such a comparison will 
lead to any conclusion that can be in any sense considered 
general? A little reflection will show that the answer must 
be no. For equal volumes of various solutions of a salt con- 
tain more salt and less water the more concentrated the 
solution ; and when solutions of different salts are brought 
into the comparison, the proportions of water and salt vary 
according to the molecular volumes of the various dissolved 
salts. The bearing of the expansion of salt-solutions on the 
question of solution must necessarily be obscure until mole- 
cular, not unit, volumes are compared ; then, and not till then, 
can we reasonably expect to gain trustworthy information. 

In order to convert results for equal volumes into those for 
molecular volumes, it is only necessary to multiply by the 
molecular volume of the salt-solution at 20° C., which is 


1800 +nM.W. | 


M.V.= ‘ 


398 Dr. W. W. J. Nicol on the 


where 
M.V. = molecular volume of salt-solution ; 
1800 = 100H,0; 
nM.W. = molecular weight of the salt multiplied by 


the number (7) of salt molecules present 
per 100 water molecules ; 

5 = density of solution at 20° referred to water 
at 20%. 


‘When this is done we obtain the volume occupied at ¢° C. by 
m molecules of salt and 100 molecules of water. 

In the following Table I have employed the values of M.V. 
at 20° for the various solutions, as given by calculation 
according to the method described in a previous paper*. 
These values are sufficiently correct for the purposes of this 
paper, and no error is introduced by employing them, or by 
considering the salt-solutions as possessing the precise mole-~ 
cular composition aimed at. In the table the molecular 
volumes of water at intervals of 10° are given in the first 
column, and in the others are the molecular volumes of the 
salt-solutions at the corresponding temperatures. On com- 
paring the volumes of the salt-solutions with those of water, 
it is found that the maxima are moved up the temperature- 


scale. Thus all the NaCl solutions and 3, 5, and 7 KCl have 
TABLE V. (see also fig. 8). 


v°. Water. | 2NaCl. | 4NaCl. | GNaCl. | 8NaCl. | 10NaCl. 


——S S§= —— | 


20 18000 | 18361 | 1876°6 | 1920-0 | 19645 | 2008-4 


30 46 42°3 83:7 27-9 73°0 17°3 
40 10°7 49-4 91°5 36°5 82:0 26°7 
50 18-4 576 | 1900-2 45°6 91:5 36-4 
(010) ie | ani a or 66°7 9°6 55°3 | 2001-4 46°6 
70 Sia 769 19°8 65°5 11:8 57-2 
80 48°7 83:0 30.7 76-4 22:7 68:2 
90 61-1 | 1900-1 42-5 87°8 34:0 79°6 
é 100 74:6 13:2 550 99°8 458 91°5 


©. | KCl. | 3KCL| 5KCL | 7KCl. |2NaNO,.|4NaNO,,. 


9 
20 1827°8 | 1886°8 | 19488 | 20118 | 1859:0 | 1922°5 
30 302 93:1 5o°7 19-4 66:1 30°6 
40 39°7 | 1900-4 63°5 27°6 74:2 39°8 
60 47.5 8:7 72:1 36°5 83°3 49-9 
60 56-4 17°9 81°5 46:0 93°4 61:0 
70 66°6 28°2 918 56:3 | 1904-5 731 
80 78:0 39°4 | 2002-9 67:1 16°6 86:2 
90 90°6 51:5 148 78:7 29°6 | 2000-2 
100 1904-4 64-7 27°6 90°8 43°7 15:2 


* Phil. Mag. January 1886. 


Expansion of Salt-Solutions. 399 


Table V. (continued). 


1°. 6NaNO,. 8NaNO,, 10NaNO,.|12NaNO,.| KNO,. |3KNO,. | 5KNO,. 
[e} 
20 | 1989°3 | 20588 | 21300 | 22021 | 1839-0 | 1921-7 | 2006-2 
30 992 | 696 41-5 143 | 450 | 294 | 15-1 
40 | 20097 | 81:0 | 536 26°9 521 | 380 | 248 
50 21:0 | 930] 662 40:0 603 | 47:5 | 35:2 
60 | 329 | 2105-7 | 793 53:5 697 | 580 | 46-4 
70 | 456 | 1901] 930 67-6 g02 | 693 | 583 
80 | 589 | 32:9 | 2207-1 82:1 918 | 816 | 712 
Se, 29 | 474 |. 7 971 | 19045 | 948 | 846 
100 | 876! 625 | 369 | 23125 18-4 | 20089 | 98-9 


maxima at 70° C.; while KCl and the solutions of NaNO; and 
K NO, now show no maxima at all. 
Examining now the amount of expansion per 10° C. for 


each solution, we meet with some interesting points. The 
data are given in Table VI. 
TABLE VI. 
t—z', H,0. 2QNaCl. 4NaCl. 6NaCl. 8NaCl. | 10NaCl. 

wear. eer sak i aie ere EGY 
20— 30 4-55 6:0 Feil, 79 85 89 
380- 40 6:18 fill) 79 8:5 9-0 9:3 
40- 50 7:62 8:2 86 9-1 9°5 9-8 
50- 60 8:89 9:2 9-4 9°7 9:9 10°2 
60-— 70 10°12 10°1 10:2 10°3 10-4 106 
70-— 80 10 eas} 111 11:0 10°9 10°9 11:0 
80- 90 13°42 leet LF 11°5 res j1-4 
90-100 13°48 desig 12°5 12:0 11:8 11:9 
t—t'. KCl. | 3KCl. 5KCI. 7KCl. 2NaNO,. 4NaNO,.|6NaNO,. 
20-30| 54] 63 6-9 7-6 7-1 1 99 
30- 40 6°6 ia 7 8:3 81 9-2 10°6 
40-— 50 78 8:3 86 89 9:1 10°1 Pies 
50- 60 89 9:3 9°5 9°5 10-1 1 cil 11:9 
60-— 70 10-2 10:2 10:3 10°3 len 121 126 
70- 80 11-4 Li-2 11:2 10°8 194 d Fee 13:3 
80- 90 12°8 122; 11:9 11°5 13:0 14-0 14-0 
90-100 13°8 S32 12'8 12:3 14:1 15:0 14:7 

¢—t'. | 8NaNO,,. |LONaNO,.|12NaNO,.| KNO,. | 3KNO,. | 5KNO,. 

20- 30 | 108 Ties 12-2 6-0 7-7 89 
30- 40 11-4 12°1 12°6 Heil 86 9-7 
40-— 50 12:0 12°6 13:1 8:2 95 10°4 
50- 60 12:7 13-1 135) 9-4 10°5 ele? 
60-— 70 sks Loy ail 10°5 11-4 12:0 
70- 80 13:9 14°1 14°5 11:6 1s: 12°8 
80- 90 14:5 146 15:0 12-7 13°2 13°5 
90-100 15:1 15:2 15°4 13-9 14°1 14:3 


400 On the Expansion of Salt-Solutions. 


At low temperatures, 20°-30° and 30°-40, the behaviour 
of all the salts is the same ; they all expand more than is due 
to the 100 H,O contained in the solutions, and as the strength 
increases the amount of expansion increases. Ascending the 
temperature-scale, we come to points at which the expansion 
per 10° is the same as that of water in the case of NaCl and 
KCl,as pointed out above. Then at high temperatures NaCl and 
KCI solutions expand less the more concentrated the solution. 
With NaNO; and KNO; this not the case ; though even here 
there is a close agreement observable in the expansion of the 
stronger solutions between 90°-100°, as compared with the 
marked difference between 20° and 30°. 

The different behaviour of these four salts, which separates 
them into two classes, NaCl and KCl, as compared with. 
NaNO; and KNOs, lies entirely in the effect of temperature 
on their solubility. According to Mulder”, the solubility at ~ 
different temperatures is as follows :— 


20° 60° 100° 
NAOT ec rer 36:0 3:3 39°8 
Molecules ...... Tet 11:5 12:3 

isk ite hy ae } Ste ne Ba : Fal 
NCCE tie. She 34:7 45°5 56:6 
Molecules ...... 8-4 11:0 13:7 
NaNO. ocetads eee 87:5 122-0 180:0 
Molecules ...... 18:5 25°9 388°1 
HEN oc. oer 31:2 111:0 247-0 
Molecules ...... 5:0 19:8 44:0 


Thus, while NaCl increases in solubility between 20° and 
100° in the ratio 1 : 1:11, KCl increases as 1: 1:16, NaNO; 
1: 2:06, KNO; 1: 7°92. It thus follows that a ine of 
NaCl or KCl is almost as nearly saturated at 100° as at 20° ; 
while NaNQ, solutions become on heating, so to speak, rapidly 
more dilute ; and this is even more markedly the case with 
KNO, solutions. 

Itis therefore not to be expected that these salts should all 
behave in precisely the same way ; but they do resemble one 
another in this, that the volume-line is straighter than that of 
water when merely equal volumes are considered. When mo- 
lecular volumes are compared this similarity disappears ; and it 
remains now to ascertain why in some cases the expansion of a 
solution is actually less than that of the water it contains ; 


* Bidragen tot de Geschiedenis van het schetkundig gebonden Water 
(Rotterdam, 1864). 


‘On Delicate Thermometers. AO1 


for in the case of 10NaCl, while the apparent volume of the salt 
is at 20° r=208°4, at 70° r=219°8, falling again to r=216°9 
at 100°. Now, as pointed out by Ostwald*, this decrease in 
the apparent volume of the salt does not necessarily imply a 
contraction of the salt, but only that between 70° and 100° 
the solution, as a whole, expands at a slower rate than that of 
pure water; but below 70° it expands faster, for 7 increases 
from 20° to 70°. This, however, is only apparent; for the 
form of the interpolation-formula shows that the expansion 
increases at a uniformly increasing rate ; the cause of the 
apparent irregularity lies in the expansion of the water, which 
is not uniform. 


XLVI. On Delicate Thermometers. By SPENCER UMFREVILLE 
PrickeRInG, M.A., Professor of Chemistry at Bedford 
Collegef. 


OME months ago I had the honour of bringing before the 
notice of this Society (Phil. Mag. 1886, vol. xxi. p. 330) 
the fact that with very delicate thermometers the temperature 
registered was never exactly the same when the column had 
risen to the point of rest as when it had fallen to it. ‘The in- 
vestigation was made by placing the instrument in a large 
calorimeter of water, removing the bulb at intervals, cooling 
or heating it slightly, and then replacing it and observing 
the reading. About eight separate observations were made 
in order to determine the difference between the falling and 
rising readings at any particular point on the stem; and 
throughout the observations the instrument was tapped con- 
tinuously on the upper end { to overcome the inertia of the 
mercury in the tube. ‘The difference which was noticed was 
explained at the time by the bulb not being of precisely the 
same shape while the mercury was being forced upwards 
through the fine tube as when it was being dragged downwards; 
nevertheless there were several points which rendered such an 
explanation not altogether satisfactory. One out of the two 
instruments examined (No. 62839) gave differences propor- 
tional to the height of the column in the tube; in the other 
(No. 63616) no such proportionality was noticed: moreover, 
the instrument which showed this defect to the greatest ex- 
tent (63616) was the one which had the smallest and strongest 
bulb. I was subsequently indebted to Lord Rayleigh for a 
suggestion that these differences should be attributed to the 
* Allgememe Chemie, vol. i. p. 392. 


+ Communicated by the Physical Society: read April 23, 1887, 
¢ I now employ a clockwork tapping apparatus for this purpose. 


Phil. Mag. 8. 5. Vol. 23. No. 144. May 1887. 245 


402 Prof. 8. U. Pickering on 


capillarity of the tubes acting on the expansibility of the 
bulbs, and not to the action of the bulbs only, and thus the 
differences observed between the behaviour of the two instru- 
ments might easily be accounted for by slight differences in 
the shape or size of the bore at various points. In order to 
test the validity of such an explanation, which at first sight 
appeared highly probable, other experiments were performed 
with these instruments, taking care to make all the obser- 
vations at exactly the same temperature, removing some of 
the mercury into the upper chamber, so as to make the : 
different points on the scale correspond to the same Cen- | 
tigrade temperature. The somewhat bulky details of the 
experiments may here be omitted, and the general results 
only given. ‘Table I. contains these results; those which were 
obtained originally, and have been given in the previous | 


TABLE I. | 
Thermometer No. ’39. | 
1 aaa: eee 465 457x 270% 162 142% mm. 
ifference in falling : ‘ i : : 
and rising ors 05 46% 27% 17 15*mm. 
Relative capacity of 
tube at the point } 461 461 45 45°6 45°6 
ULL 0 i Se 


Thermometer No. 616. 
Scale-Reading ..... wees 962 458° 458% 289 154 142 140x mm. 
TEA caine voodinge | O18 42 (6 | 78. | “6 aaa 
Relative capacity of 
tube at the pont | 40°3 40-40 393 < 401 401 408 
PAREN ch As bbs te. Suelo 
communication, are marked by an asterisk; and any small 
discrepancies observed between them and the fresh experi- 
ments may be attributed to differences in the actual tempera- 
ture of the observations, the recent experiments only being 
strictly comparable with each other, having all been performed 
at 10°°5 C. The readings and the differences are expressed here 
in mm. of the column, instead of cm. (or scale-degrees) as in 
the former communication. 

To whichever thermometer we confine our attention, it is 
perfectly obvious that the magnitude of the differences bears 
no relation to the size of the bore. Taking the original 
experiments with ’39, the difference as judged by the size of 
the bore should be least at 457, and greatest at 270, and of 
intermediate value at 142 mm.; whereas they are found to 
be greatest at 457 and least at 142 mm. Again, with 
No. ’616, the differencés should be greatest at 289, and least 
at 562 mm.; whereas they are practically equal at these two 
points, and greater (taking the later experiments only) than 


Delicate. Thermometers. 403 


at any other points. It may be objected, however, that the 
capacity of the tube at these points, as given by the length 
of a comparatively long (2 cm.) thread of mercury in that 
position, affords but a very rough, and perhaps quite 
erroneous, measure of the diameter of the tube at the parti- 
cular point in question. As a crucial test, therefore, it was 
determined to alter the bulb of one of the instruments. This 
was done with Thermometer 39. The capacity of the bulb 
was reduced to half its former value, so as to hold 18 instead 
of 36 grams of mercury; and a determination of the coefficients 
of expansion of the two bulbs under pressure (given in column 
6 of Table II.) shows that what may be termed the apparent 
expansion, or the effect of pressure on the height of the 
mercury in the tube, was thereby reduced to nearly half its 
former value, from ‘042 to ‘025 mm. per mm. of mercury- 
pressure, so that the differences in the readings with fallin 

and rising columns should have been diminished in that 
proportion, if it depended on the expansion of the bulb by 
pressure; but on examination it was found that this difference 
was even greater now than it had been previously—at 270 
mm. it amounted to ‘59 instead of ‘27 mm., and at 457 mm. 
it was ‘39 instead of ‘46 mm. (col. 7 of Table II.). Another 
thermometer (No. 783) was then examined in a similar 
manner. Originally it exhibited no difference whatever in 
the falling and rising readings; but when the bulb was altered, 
without increasing the coefficient of apparent expansion to 
any considerable extent (from ‘011 to °015), a small though un- 
mistakable ditference in the readings was observed (No. 783 B 
in the table). This second bulb was then removed, anda third 
and smaller one substituted for it (No. ’83 C), by which the 
coefficient of apparent expansion was reduced by one half its 
former value; but, instead of the differences being diminished, 
they were actually increased, although no accurate measure 
could be made of them, for the column of mercury in the 
tube kept breaking off when the instrument was tapped. It 
was evident, therefore, that the cause of these differences in 
the readings did not lie in the bulbs of the instruments, but 
in the stems, that each time the instrument was opened and 
air admitted into the stem, the defect was increased, till the 
tube eventually became entirely ruined. The moisture and 
gases present in the air, no doubt, affect the glass and adhere 
so strongly to it that the heating to which the stem is sub- 
jected is quite incapable of removing it, and the interior of 
the tube remains coated with an elastic covering which 
destroys the working capabilities of the instrument. The 
researches of Bottomley (Proc. Roy. Soc. xxxviii. p. 158) and 
others on the absorption of air, and especially carbon dioxide, 

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Delicate Thermometers. 405 


by glass, prove, in a striking manner, the extent to which such 
absorption occurs, and the persistence with which the absorbed 
gases are retained. 

Both the delicate instruments (’39 and ’616), in which the 
difference of the readings was first observed, had had small 
temporary bulbs attached for the purpose of calibration; and 
it therefore seemed probable that the second opening of the 
tube may have been the sole cause of the defects which they 
exhibited. To settle this question, and to ascertain whether it 
was possible to make instruments of such delicacy entirely 
free from this defect, two other thermometers were manu- 
factured, Nos 708 and ’61. The delicacy of these was some- 
what less than in the former instruments, owing to the 
impossibility of procuring sufficiently fine tubes; the size of 
the bulbs, however, was increased, that of ’61 containing as 
much as 46 grams of mercury. An estimation-figure, ‘05 
mm.,represented about 0:0005°C. On examining these instru- 
ments, in the same manner as previously, it was found that 
they worked perfectly, the mercury registering exactly the 
same temperature whether the column had risen or fallen to 
the point of rest, equally satisfactory results being obtained 
whatever portion of the stem was examined. Instruments of 
this excessive delicacy are therefore perfectly workable; it is, 
however, only by observing the utmost precaution in making 
them that success can be obtained. The tube must on no account 
be opened till the last minute, when the bulb is finished and 
ready to be attached without a moment’s delay; the bulb, as 
soon as it is attached, must be warmed so as to fill the tube with 
mercury and prevent the access of air through the upper end. 
If any failure occurs in the attachment of the bulb at the first 
trial, the stem must be rejected ; a second attempt would be 
attended with the same results as putting on a second bulb 
after the instrument had been made up. When once the 
stem is filled with mercury, the tube may apparently be 
opened several times at the top without damage being done, 
and the bulb itself may be made 24 hours before it is attached 
to the stem without being injured by exposure to air for that 
time. 

It is a common practice of thermometer-makers to examine 
the bore of a tube before it is made into a thermometer by 
passing a thread of mercury along it, and often, indeed, the 
stems are divided and fully calibrated before the bulb is 
attached and the tube closed. From what has been ascertained 
as to the effect of the air on the interior of the tube, it is obvious 
that a tube which has been treated in such a manner will be 
utterly useless for any really delicate instrument. 


406 Prof. 8. U. Pickering on the 


XLVI. On the Effect of Pressure on Thermometer-bulbs, and 
on some Sources of Error in Thermometers. By SPENCER 
Umrrevitte Picxerine, M.A., Professor of Chemistry 
at Bedford College*. 


ae great difficulty which exists in obtaining exact con- 

cordance between two thermometers throughout a con- 
siderable range of their scales must have been experienced by 
all who have had occasion to require such concordance. In 
the course of a series of experiments, in which the tempera- 
ture-disturbance in a calorimeter was measured simultaneously 
with two instruments, I was much struck by the appearance 
of a certain amount of regularity in the difference in the re- 
sults yielded by the two instruments, which, according to 
direct comparisons with each other through longer ranges of 
temperature, should have been absolutely concordant. The 
instruments being open in the scale and having large bulbs, 
I was led to seek for an explanation of these discrepancies in 
irregularities in the expansion of the bulbs under the pressure 
of the column of mercury in the tube. 

The effect of pressure on a thermometer-bulb has been in- 
vestigated by Egen (Pogg. Ann. xi. p. 283), and by Mills 
(Roy. Soc. Hdin. xxix. p. 285), with the general result of 
showing that the expansion experienced is directly propor- 
tional to the pressure. But although the pressures employed 
in Mills’s experiments were considerable (ranging up to 134 
atmospheres), the thermometers which he examined were not 
of the most delicate character, and the coefficient of expansion 
was but small in comparison with that possessed by most calo- 
rimetric instruments. In the present experiments the bulb 
of the thermometer was enclosed in a small thin brass cylin- 
der, which was connected with a pump, by means of which a 
vacuum, or pressure up to two atmospheres, might be pro- 
duced. ‘The case enclosing the bulb was previously filled 
with melting ice and placed in a large vessel of the same, the 
zero point. of the thermometer having been previously ad- 
justed so as to be slightly above the level of the ice. 

Three instruments were investigated in this manner, 
Nos. 62839, 68616, and 65108, of which the details of con- 
struction are given in the table il. of the preceding commu- 
nication (see also Phil. Mag. 1886, vol. xxi. pp. 331, 340). 
About twenty observations at different pressures were made 
in each case, and the results are given in Table I., where are 
entered the observed pressures and readings in millim., and also 


* Communicated by the Physical Society: read April 23, 1887. 


:' 


407 


Liffect of Pressure on Thermometer-bulbs. 


TABLE I.—Hffect of Pressure on the Bulbs of Thermometers. 


a a a ee ee ee ee ee 


Thermometer 62839. 


External 
pressure 
in mm. of 
mercury. 


1986 
1718 
1438 
1254 
1093 
928 
851 
723 
665 
596 
516 
416 
316 — 
216 
173 
118 
82 
47 
1 
—22°5 
—55 


Observed | Calculated 
reading. reading. 
mm. mm. 
97°80 99°40 
86:60 89:00 
75°35 alts, 
68°70 69°35 
62°65 62°50 
56:25 55°50 
52°85 52:25 
48:00 46°80 
45°55 44°35 
42°55 41°45 
38°15 38°05 
33°80 33°80 
29°45 29°55 
25°50 25°30 
23°55 23°50 
21:20 21°15 
19°55 19:60 
18:15 18°15 
16°55 16:60 
15:10 15°20 
14:00 13:80 


41:20 


Difference. 


mm. 

—1:60 
— 2°40 
—1:80 
—0°65 
+0°15 
+0°75 
+0:60 


41-20 
+110 
40:10 
0 
—0:10 
+0:20 
+005 
+0:05 
— 0-05 
0 
—0-05 
—0'10 
+0:20 


Average points, 92°5 mm. at 1800 P, and 
16:68 mm. at 12°5 P. 


External 
pressure 
in mm, of 
mercury. 


1960 
1738 
1433 
1180 
1009 
874 
724 
667 
596 
515 
414 
320 
216 
163 
Iles 
66 
16 
—25 
—56 


Thermometer 63616. 


Observed | Calculated 


reading. | reading, 
mm. mm. 
94:00 94°80 
87°60 88°10 
78°75 78:80 
11°45 71:05 
66:05 66:00 
61:65 61:85 
57:25 57°30 
55°85 55°60 
53°60 53°45 
51:25 51:00 
48:05 47°90 
44-95 45:05 
41:90 42:00 
40:25 40:30 
38°65 38°90 
37°40 37°35 
39°90 35°85 
34:60 34°60 
33°80 33°65 


Difference. 


Average points, 67:03 mm. at 1044 P. and 


36:07 mm. at 23°6 P. 


External 
pressure 
in mm. of 
mercury. 


1771 
17388 
1660 
1597 
1521°5 
1435 
1328 
1228 
1131°5 
1052 
932 
832'5 
7345 
634 


Average points, 


Thermometer 65108. 


Observed | Calculated Differ 


reading. | reading. 
mm. mm. mm, 
43:00 42-95 0-05 
42:45 49:50 —0:05 
A135 41°35 0 
40°40 40:45 —0:05 
39°35 39°35 0 
38°15 38:10 0:05 
36:55 36:45 +0:10 
35:15 35:10 +0:05 
33°65 33:70 — 0:05 
39:95 32:55 —0:30 
30°85 30°80 +-0:05 
29:35 29:35 0 
28-00 28:00 0 
26:40 26:50 —0:10 
24-80 24°85 Sue 
23:45 23°40 0:05 
92°15 22°00 +0:15 
20:60 20:50 +0:10 
19:15 19°15 0 
17°75 17:70 +0:05 
16:90 17:10 —0:20 
16°40 16°30 +0°10 
15°75 15°75 0 


17:20 mm. at —8°5 Pi 


See SS 


eeu 
| 


ence. 
j 


41°32 mm. at 1657:5 P. and 


SS 


t 
— 


TT TT 


408 Prof. 8. U. Pickering on the 


the readings, calculated on the assumption that the alteration 


in height of the column is directly proportional to the pres- 
sure; these calculated values being deduced from the average 
points given at the foot of the table. With the last-mentioned 
instrument (65108) the effect of pressure on the bulb would 
appear to cause a very regular expansion; there are only 
three observations which differ from the calculated values by 
more than 0'1 millim., while the average difference amounts 
to about 0:05 millim. only. With the other two instruments 
the observations at the higher pressures cannot be much relied 
on, since the pressures were ascertained by means of a Bour- 
don’s gauge, instead of a column of mercury as in the other 
cases; and this gauge was afterwards found to be untrust- 
worthy. Omitting these observations, we find that, in case of 
No. 63616, where the coefficient of apparent expansion of the 
bulb was twice as great as that of 65108, the error in the 
readings is considerably larger, amounting to as much as 
0:12 millim. on the average, while the grouping together of 
the positive and negative differences is well marked. It is 
only with No. 62839, however, which possessed a still greater 
coefficient of expansion, that the differences become so large 
as to render it quite impossible to attribute them to mere ex- 
perimental error. As the pressure is increased above 500 
millim., the bulb begins to contract far more than it should, 
causing a difference of as much as 1°2 millim. between the 
observed and calculated readings; it then contracts less, crosses 
the line representing the calculated values and for pressures 
from 1100 up to 1986 millim. (if these higher results may 
be trusted) does not contract as much as it should do*. The 
action of the bulb under pressure is evidently not regular. 
As the differences in the calorimetric results above men- 
tioned were observed with the thermometers 616 and ’08, 
which behaved normally, or nearly so, under pressure, as well 
as with other instruments with bulbs of considerably smaller 
expansibility, it is impossible to attribute these differences to 
the cause suggested; at the same time, however, the present 
investigation leads to results of considerable practical im- 
portance. It is evident that where the coefficient of expan- 
sion of the bulb is large, as with ’89, irregularities in expan- 
sion sufficient to introduce considerable errors may occur; 
the bulb, when subjected to pressure, would appear to behave 


* Care was of course taken that the thermometer should not be read till 
the column had attained a position of equilibrium. The top of the in- 
strument was tapped throughout the experiments by means of the tapping 
apparatus, and the column read at intervals of one minute till it was 
found to be perfectly stationary. 


Effect of Pressure on Thermometer-bulbs. 409 


in a manner analogous to that often noticed with thin tin 
plate vessels, where a small addition or removal of pressure 
will cause a sudden and considerable alteration in the form of 
the vessel. It would certainly be advisable, in the case of 
any thermometer required for very delicate work, to examine 
it under pressure to ascertain whether its action be uniform 
or not. In order to reduce the chances of irregular action, 
it is necessary to render the bulb as inexpansible as possible. 
From the fact that the thermometer 63616 showed a slight 
amount of irregularity in its action, we may place the limits 
of expansibility desirable in a very fine instrument at a number 
between that of this instrument and that of 65108 ; the co- 
efficient of expansion should not exceed 0:000,000,03, or the 
apparent expansion 0:02 millim. of the mercurial column per 
millim. of pressure. 

An examination of table i. of the preceding communica- 
tion will show at once the increase of rigidity obtained by 
having the bulb made out of a glass cylinder instead of being 
blown before the lamp ; the instruments ’83 and 739 B are the 
only ones mentioned in this table which had blown bulbs, 
and the coefficients of expansion in their cases are higher 
than in any other case, although their bulbs were only 4 and 
4 as big as those of most of the other instruments. Again, the 
thermometers ’83 and ’16 were identical in all respects except 
as regards their bulbs, and here the blown bulb (’83) possesses 
only half the strength of that made out of cylinder. 

Although it seems probable, prima facie, that a blown bulb, 
however well constructed, would not be so uniform as a 
cylinder bulb, these facts of course do not prove that such is 
necessarily the case, as the thickness of the walls of the bulbs 
was not known; but it does prove that, by ordering a thermo- 
meter with a cylinder bulb, we should in all probability get an 
instrument possessing nearly twice the strength of one with a 
blown bulb. A further very considerable addition of strength 
may be gained by having the bulb made double instead of 
single. The instruments Nos. ’616, ?08, and ’61 possessed 
double bulbs made out of glass cylinder ; and a comparison of 
them with ’83 B and °39, which had single bulbs made out of 
cylinder also, will show the advantages of the double bulb. 
For thermometers to be used in a liquid which is stirred in a 
thoroughly efficient manner, the thickness of the walls of the 
tube forming the bulb may be very considerable. The instru- 
ments Nos. ’08 and ’61, which contain between 40 and 
50 grams of mercury, will take the temperature of the liquid 
in the calorimeter in about 5 seconds, although their bulbs 
are 0°75 millim. thick in the walls. 


410 Prof. 8. U. Pickering on the 
Although the effect of different pressures on the bulb did 


not appear to produce irregularities such as would account 
for all the difficulties experienced in getting two instruments 
to correspond perfectly throughout their scales, it was thought 
possible that the expansibility of the bulbs under pressure 
might be influenced to a considerable extent by the tem- 
perature of the experiment, and that this might produce dis- 
cordance between two instruments which had been compared 
with the same standard but at different temperatures. To 
investigate this, the behaviour of No. ’08 under pressure was 
examined at 12° C. As it was impossible to keep the tem- 
perature of the bath absolutely constant throughout the series 
of experiments, the reading at any pressure P, was compared 
with that at P by taking the mean of two observations at P, 
made immediately before and immediately after the observa- 
tion at P,;. In the following Table the value of the constant 


Thermometer ’08. Under pressure at 12°. 


Rk, —R 

pap PoP 
millim 

— 500 0:01445 

— 400 0:01445 

— 300 0:01434 

— 200 0:013880 

+200 0:01470 

S5 0:01469 

+300 0:01438 

3 0:01462 

+400 0:01447 

+500 001455 

+600 0:01457 

Mia 263) 20 5.25235 =: 0:01453 

obtained from Pak (R, and R being the readings at P, 


and P respectively) is given in the second’ column, while 
P,—P is given in the first column. P was in all cases the 
atmospheric pressure. ‘The mean value of the constant (the 
coeflicient of apparent expansion) is 0°0145, a number abso- 
lutely identical with that found at 0° (table ii., preceding 
paper), which shows that small alterations of temperature, 
such as would occur in standardizing delicate calorimetric 
thermometers, produce no appreciable alteration in the rigidity 
of the bulbs. The causes producing non-concordance of ther- 


Effect of Pressure on Thermometer-bulbs. A11 


mometers under certain circumstances still remains to be 
discovered *. 

I may mention one source of error in thermometric work 
which attains considerable dimensions when dealing with tubes 
of very fine bore. These tubes, even when of the most perfect 
description obtainable, generally possess a few points at which 
the mercury column experiences a difficulty in passing: the 
mercury, when it has reached such a point, sticks there an 
appreciable time, and then passes it suddenly with a jerk ; 
sometimes, even, the mercury sticks so persistently that the 
column will separate sooner than pass it. These points do not 
indicate any contraction which is sufficient to affect the results 
of calibration, and are probably due to some difference in the 
nature of the glass, for they may be developed by heating 
the thermometer-tube externally with a very small flame up 
to about 400°. The error of taking a reading while the 
mercury is sticking at such a point may, I estimate, amount 
sometimes to as much as (5 millim. All delicate thermo- 
meters should be carefully examined in order to ascertain the | 
position of such points, and they should be avoided, if possible, 
in any work with the instrument. 

In any thermochemical work in which the effect of tempe- 
rature on a given reaction is being studied, many of the 
sources of error inherent in the use of thermometers may be 
avoided by using the same portion of the stem of the instru- 
ment, whatever the actual temperature may be. To effect 
this, the zero-point is altered in each experiment by removing 
some of the mercury into the upper chamber of the thermo- 
meter. Hormerly I removed the requisite amount of mercury 
by the application of a very small flame to a point just below 
this chamber; but I now adopt a method which is much 
safer, more expeditious, and equally exact. A fine tube, 
somewhat wider than the stem of the thermometer, is affixed 
to the upper end of the stem, and in this tube there is a small 
contraction or “ knife-edge,” sufficiently wide to permit of the 
mercury passing it either upwards or downwards, but yet so 
narrow that a slight swing of the instrument will cause the 
column of mercury to break off at it, that portion of the 
mercury which is above the knife-edge passing up into the 
upper chamber. By this means any point on the stem may 
be adjusted with ease to within 0°02 of any given tem- 
perature. 


* For an unexplained instance of non-accordance of results obtained 
with different thermometers, and also with different portions of the stem 
of the same thermometer, see Chem. Soc. Trans. 1887, pp. 304, 322. 


f 41gh J 


XLVI. On the Determination of Coefficients of Mutual 
Induction by means of the Ballistic Galvanometer and 
Earth-Inductor. By R. H. M. Bosanquet*, 


Y attention was drawn to this subject by the paper 

recently read before the Society by Prof. Fosterf. I 

observed at once that the appliances which I am in the habit 
of using afford a very simple solution of the problem. 

The ballistic galvanometer has a resistance less than 2 B.A. 
units, and about 500 turns. I have three earth induction- 
coils, all having a mean diameter of about half a metre, and 
the following constants. The resistances are approximate, as 
they vary so much with temperature. The wires are all 
cotton-covered copper. 


Number of Total fice Yagn creme APHORIN Wi 
log NA. ference. BA. B.W.G. 
42 ASOT te 2 167°325 6 16 
250 5:74200 166°583 10°8 20 
1000 6°36182 170:005 42 20 


The circular channels are turned in wooden rings framed 
together in many pieces. They were turned in the Royal 
Society’s lathe in my laboratory. They are mounted so as to 
turn on vertical axes through half a revolution, in doing which 
they are reversed with respect to the horizontal component 
of the earth’s magnetism, and so experience an electrical 
impulse equal to 2 NAH. 

If the resistance of the circuit be R, a transfer of electricity 
then takes place, such that 


2 NAH 
o— R : 


The various windings of the coils were measured with great 
care during the construction. The areas were calculated for 


each layer separately; and the above values of log NA are 


probably certain to the fourth place of decimals. 

There are two tangent-galvanometers, both on Helmholtz’s 
pattern, having mean circumferences of exactly 1 metre. The 
one has 2 coils, the other 18. The currents in them are 
measured by the formula 


C=GH tan 6; 


* Communicated by the Physical Society: read February 26, 1887, 
+ Phil. Mag. February 1887, p. 121. 


Determination of Coefficients of Mutual Induction. 413 


where log G has the values:— 


No. of coils. log G. 
7g RR Oo 4 hie ia calle Mg 72 Yo aL 


1S ie | het cen aaa p25 i, 


We can now proceed to the problem. 

Two coils, P, 8, are placed side by side, and a current passed 
through P ; then, on making the contact, a ballistic galvano- 
meter in the circuit of 8 is deflected. The earth induction- 
coil forms part of the circuit of 8. 

So far we have the equations 


CSG lbtand 2 srs Ae oie 
he om 


with reading = « minutes of arc. 
If we now give the earth induction-coil a half turn, we have 


INAH ‘ 
a a ee Ree 


with reading = 8 minutes of aoe 


MG tan 0 
Whence : = TONGA Tt 
a 
my: 
by the principle of the ballistic galvanometer. And 
a 2NA 


fall Kok A et otaabecinay etait ces Go 


The 2 disappears from the numerator if the observation is 
made by reversal. 

In order that H may be the same for the tangent-galvano- 
meter and the earth induction-coil, it is necessary to remove 
the tangent-galvanometer and put the earth induction-coil 
in its place before making the observation 8. 

There are two classes of cases which cannot be treated by 
this most simple form of the method :— 

(a) Where the quantity of electricity to be dealt with is too 
large for the galvanometer, even with the coil of highest 
resistance. And 

(6) where the resistance of one of the coils to be determined 
is so great that no earth induction-coil could throw a sufficient 
quantity of electricity through it. 

Tn both these cases we have to introduce a large resistance 


ee 


SE ee are meen. 


— 


ee eed es 


A I RO a oF NR 


eT 


Sr rg SN a Ee 
a 


ee 


So SE a 


414 Mr. R. H. M. Bosanquet on the Determination 


into the circuit 8, after calibrating the galvanometer without 
it. The value of the deflection is then altered in the ratio of 
the change of resistance. 


- Let R be the total increased resistance, 
R, the resistance without the addition. 


Then equations (2), (3), (4) become 


gee 
Q= AES . it aia 


2aRNA 
“BR Gtan 0° . e . ° e (7) 


The 2 disappears, as before, when reversal is employed. 

An example of case (a) is given later ; where, in the second 
determination of my Gramme machine, a resistance of 1000 
B.A. units was introduced into the secondary circuit, to mo- 
derate the deflection. The determination of the resistances, 
however, can hardly ever be accurate, as the copper earth 
induction-coil is liable to great alterations through change of 
temperature. 

An example of case (6) wouid be afforded by the determi- 
nation of such an induction-coil as the one by Apps, with high 
resistance S, mentioned in Prof. Foster’s paper. In this case 
the galvanometer would be calibrated by the earth induction- 
coil, the high resistance-coil then introduced, and the known 
values of R, R, employed in formula (7). 

The determinations | have made are mostly of the mutual 
coefficients of the coils of my earth-inductors, placed close 
together. I have also madea couple of determinations of the 
coefficient of field-magnets and armature in my A Gramme 
machine. 

The determination of February 21 was rather rough; the 
coils were placed together in their frames, but they could not 
be brought very near. The numbers are useful, as illustrating 
the increase of the discrepancy of Maxwell’s formula, with 
the increase of b, the distance of the central planes. In fact, 
unless 6 is small compared with the diameters, the formula is 
not properly applicable. Here it is about half either of the 
radii, and calculated and observed values are nearly as 4: 5. 

On February 22 the coils were dismounted and brought as 
close together as possible, 6-being now between a quarter and 
a third of either of the radii. The calculated and observed 
values are nearly as 5:6. In order to see what part of the 


v= 


of Coefficients of Mutual Induction. 415 


error was due to the dimensions of the section of the coils, 
the coefficient was then calculated by the more accurate for- 
mula referred to below; but the improvement is very slight. 
It appears that this formula does not in any way cure that 
defect, which arises from the distance of the central planes. 
So that in coils similar to mine we cannot.expect to get nearer 
by calculation than the number last referred to, which is to 
the observed number nearly as 6: 7. 

On February 23 experiments were arranged with a view 
to test the consistency of the method, the coils examined 
being made primary and secondary alternately. Though the 
mode of observation does not admit of great accuracy, it 
appears that there is a systematic difference between the 
results of the two arrangements, amounting to about 1 per 
cent. This I have not been able to explain. The coefficients 
of self-induction of the coils should have no influence ; but 
that of the 250 will be very much greater than that of the 42, 
and this is the only source of error that I can suggest. The 
number calculated by Maxwell’s original formula is to the 
mean of observation nearly as 6: 7. 

In all the experiments difficulty was experienced in con- 
sequence of the continual fall of the battery-current. With-. 
out care in charging, and freshly charged cells, the experi- 
ments could hardly be made, as the fall of the current affects 
the galvanometer in the intervals. 

Rowland’s method of control, with a magnet and coil, was 
employed. 

The determinations of the coefficient of the Gramme machine 
were very rough. In fact the fundamental equation (2) is 
not really applicable to the dynamo at all, except perhaps in 
the upper part of its range, where the current is, say, 10 
amperes or more. Tor small currents do not do more than 
shake the subpermanent magnetism, which is considerable 
compared with the magnetism due to small currents. And 
in motion the machine makes use of this subpermanent mag- 
netism, although it does not enter into the electrical coefficient 
of mutual induction. 

The observations of Feb. 24 were very irregular ; it ap- 
peared as if the current sometimes shook the subpermanent 
magnetism and sometimes not. Still a mean was fairly 
deducible. 

On Feb. 25 a stronger current was employed, and the ob- 
servations were fairly regular ; but this current, though it 
moves the subpermanent magnetism, is quite insufficient to 
reverse it as the reversal of a large current does; so that 
even here we do not get a representation of the whole effect. 


Ea SSy = <a. oP = = Sar - ~ ~ 
Se = SS SSS SS SSE 2 
SE Se ey ee SF 


TMP Bt 


SSS SS 


See = 


SS 


EAE eS 


Ss rN ae es ee 


SS Se Ee aS ee 


nS 
~ 


416 Mr. R. H. M. Bosanquet on the Determination 


In the higher part of the range of the machine, say for 
unit-current (10 amperes), we can assume, without serious 
error, that the fundamental equation (2) holds. That being 
so, we can deduce the value of M from the resistance and 
number of revolutions with which a stable current is pro- 
duced. 

It will be convenient to assume that all the coils of the 
armature are gathered up into two coils at right angles. 
Then, if one be parallel and the other perpendicular to the 
axis of the magnetic field, the determination of the coefficient 
of induction at rest affects only one of the coils, or half the 
armature. 

Then, as to one of these coils, or half the armature :—In a 
half revolution the same effect is produced as if the exciting 
current were reversed, and a quantity of electricity passes, 


_2MC 
= 
In a whole revolution ave passes. 
The same applies to the other half of the armature, so 
8M 
that altogether R 
If there be n revolutions per second, the quantity passing 
in n revolutions=the current numerically, and this divides 
out, so that 
R 


M=—. 
In an old paper of mine on Practical Electricity (Phil. 
Mag. xiv. p. 246) I find data from which, by interpolation, 
I obtain the corresponding values for this machine :— 


passes in each revolution. 


m' per minute. R C 


840 10°x 5°55 1=10 amperes, 
Whence M=10! x 4:95. 


The assumption made above is rough, but we cannot 
specialize further unless the distribution of the field is known. 
The two coils may then be supposed to be in any required 


proportion instead of being equal. 


Regarding the coefficient as the product of the number of 
windings of armature and magnets, and of the magnetic induc- 
tion common to both, due to unit-current, we can find the com- 
mon magnetic induction in this case. There are 1700 turns in 
the armature, half of which count, and 234 turns on each of the 
four magnets, two of which receive each part of the common 


of Coefficients of Mutual Induction, 417 


induction. Whence the common magnetic induction is 
107 x 4:95 
468 x 850 

This seems very small. But the dimensions of the armature 
are such that it can have hardly any core. 

But I must not now pursue the subject of the dynamo. 

The method of the earth induction-coil and ballistic galva- 
nometer is susceptible of numerous applications. I have in 
my mind the direct determination of the capacity of a con- 
denser, and a method following the lines of that used by 
Weber, for the determination of resistance. He employed 
earth induction-coils on this principle, though I do not know 
how they were arranged in detail. I cannot conceive how a 
tangent galvanometer could be employed for the ballistic 
work, as seems to be contemplated in Maxwell’s account. I 
should determine the constant of the ballistic galvanometer 
for the purpose by a shunt comparison with a tangent galva- 
nometer. I believe Mr. Glazebrook has done something of 
this kind in another case. In fact I have been through the 
work of such a method; but where there is a shunt to be 
verified, in this case 1 : 10,000, as well as other determina- 
tions of resistances, the errors of temperature are so trouble- 
some to deal with, that in the present state of the subject I 
doubt whether it is worth while to spend much labour on it. 

The numerical results of the experiments referred to are 
appended. 


Coefficients of Induction of 250 and 100 coils. 
Feb. 21. Distance between central planes, 6=13°97 centims. 
8 by half turn of 1000 coil. 


=a 


Arrange- Galvano- 

ae a. B. 0. meter. ' j M. 

250 ‘ Z Geen make an ee 
1000 § fa2-o $64") 15745 18 eae 10’ x 7°6903 


*Calculation by Maxwell’s original formula . . . 10’x6:164 


Feb. 22. Distance between central planes, b= 7:925 centim. 
: B by half turn of 1000 coil. 
| Arrange- Galvano- 


950 P a. B. 0. meter. k j M. 

* - 1@) make an 8 " 
1000 § 203-7 170! 12° 33! 18 Bicscte 108 x 1°2593 
| *Calculation by Maxwell’s original formula . . . 108x1:070 


| *Calculation by formula for dimensions of section of coils 10° x 1:104. 


* New edition of Maxwell’s ‘ Electricity and Magnetism, vol. ii. 
pp- 314, 320. 


Phil. Mag. 8. 5. Vol. 23. No. 144. May 1887. 2F 


- 2 ae 
ae 
bec 


418 Mr. R: H. M. Bosanquet on the Determination 


Data for this last formula :-— 


n= 1000 nm =| 250 
a=27:056 O=zZooue 
2h=2°18 2h'= +562 
2k= 5°45 2 =5°77 


Coefficients of Induction of 42 and 250 coils. 


Distance between central planes, b>=6°35 centim. 
& by half turn of 1000 coil, in 8, throughout. 


Galvano-|Make & break 


Arrange- | 
Date. | Set. ment. a Bb. 0. meter. or reversal. M. 

49P A 1 Olen oe a j 
Feb. 23.] 1] osgg |2022 | 142 | 3156 2 m. b. 10° x 5-9390 | || 
2) 2207 | e82| 173! |iaeoo |) IB rev. x61075 | | 
Hobe 24 sla) oo. 86:5 | 175 | 43.55°5 a ne x 60030 | | 
MTR fae 85:2 | 175 | 435 nie 3 x6 0893 | || 

3 | oayg |1902 | 1447] 2948 2 m. b. x 59862. 


Calculation by Maxwell’s original formula . 10° x 5-275 } | 


Comparison of arrangements :— 


49 'P 250 P 
2508 428 ; 
Feb. 23. 1 10°x5-9390 Feb. 23. 2 10°» 6:1075 i il 
Bade, he x 59862 sip), ee peal x 6:0030 
| ag LOE ” 99 2 x 6:0893 


Mean x5:9626 | ae 
Mean x 6:0666 — 
Mean of both arrangements 10° x 6:0146. 


Determinations of Coefficient of Induction of Dynamo Field- 
magnets and Armature. (A Gramme.) : 

Feb. 24. P field-magnets, with two bichromates and 10 
B.A. units. } 

S armature, ballistic galvanometer, and 1000 coil. 
By reversal. 


Galvano- ei! 

ae B. 0. meter. Amperes. Me} q 

Meamvalue li i. | L464 (0990 27° 11! 18 181 107 x 1:85 
Highest value . 176 Ee ecco 4 ie pais x 2°23) 


Lowest _,, Se) Levee as nie0's 23 aul x1 9 


of Coefficients of Mutual Induction. 419 


Feb. 25. P field-magnets, with two bichromates. 
S armature, ballistic galvanometer, and 1000 coil, 
with and without 1000 B.A. units, so that 


R=1045, Ro=40. 
By reversal. 


Galvano- 
meter. Amperes. M. 


a. B. 6. 
ies = 179 39° 24 2 9°54 10'x 3-685 
By formula M=~ 100. 1074-95 


ADDENDUM. 
March 5. Supplementary Notes. 


1. Calculation of coefficients by the elliptic integral table 
in the new edition of Maxwell, combined with the formula 
for approximation to the effect of the sections of the coils. 

I have now calculated the experiments of Feb. 22 and 
Feb. 23-24 by this method, which is the most complete that 
exists, short of calculating all the single combinations of 
circles in the two coils. The accordance with experiment is 
somewhat better, but still far from close. 


Observed. Calculated. 
Hen 22.1108 x 1-259 10° x 1:176 
yy 28-24. 10°x 6-015 10° x 5-585 


2. Simple formula for approximate calculation of the 
coefficient. 


Assume that the field in a due to A is everywhere the same 
as at the centre of a. Then the total lines of force for unit 
current are Dar A? 

A alan 
(A? == b?)# 
where b is the distance between the planes of the circles, or, 


if estan 6, this becomes 
27a” sin °0 
o% 7 
which is simpler to calculate than Maxwell’s approximate 
formula. 


__ The following is the comparison of this formula with 
observation. 


Observed. Calculated. 
Web.) 21. 40%x 7690 10’ x 8-991 
«22. 108 x 1-259 108 x 1:133 
», 28-24. 10°x 6015 10° x 5:036 


22 


420 Mr. W. Brown on the Effects of Percussion and 


3. Course of values of the coefficient of field-magnets and 
armature of a dynamo. 

The numbers stated in the paper may possibly be mislead- 
ing, as it is not sufficiently explained that the number deter- 


mined from the motion of the machine, ( =) is not of the 


same nature as the two results of determinations made at rest, | 
which precede. 3 

The number determined from (=) is necessarily infinite 
when the current is evanescent, if there is any retention of 
magnetism in the machine, and diminishes continually as the 
current increases. The following corresponding values of 
this coefficient and current are given by the data referred to 
in the paper. 


Coefficient. Current in amperes. 
10° x 7-054 3°8 
x 6°250 D°D 
x 5°600 76 
x 4°732 10°5 


XLIX. The Hfects of Percussion and Annealing on the 
Magnetic Moments of Steel Magnets. By Witu1AmM Brown, 
Thomson Haperimental Scholar, Physical Laboratory, Uni- 
versity of Glasgow”. , 


Parr II. 
A Part I. of this paper, which appeared in the March 


number, certain preliminary results were given, showing 
the effects of percussion on the magnetic moments of steel 
magnets. In the present communication these effects are 
considered in greater detail, with tables giving the results of 
an extended series of experiments, and the question of an- 
nealing is treated with respect to exact measurements of the 
annealing temperature. 

The steel experimented on in this case was furnished to Sir 
William Thomson for experimental purposes by two different 
steel-makers, 

The following Table gives approximately the relative per- 
centage proportions of all the substances found in the steel, 
the quantities in specimen I. being taken as unity. They are 
taken, not from analyses of the particular pieces experimented 
on, but from a general analysis of the sample in each case. 


* Communicated by Sir William Thomson. 


Annealing on the Magnetic Moments of Steel Magnets. 421 


The proportions are on this account probably only roughly 
approximate, and until special analyses are obtained it seems 
unnecessary to give the actual quantities. 


TABLE I, 


Comparative Composition of the Specimens. 


Number of specimen. 


Substance. 
iL, II. III. 
SILC T1T a aR ee 1:00 0:08 0-17 
Manganese ............ 1:00 1:28 3°25 
Phosphorus .j)....-0:5- 1:00 eral 1:55 
SUN OLA0 eens ame 1-00 0:00 0-00 
Gambon: cesses cen sg 1-00 0:25 0-25 


MOM ac deen iL” 1:00 0-994. 0:987 


All the specimens contain, as a matter of course, nearly the 
same amount of iron, but the other constituents differ con- 
siderably. The magnets were prepared in the same manner 
as those referred to in Part I. of this paper. They were all 


made glass-hard to begin with ; and this was done by bringing 


them to a bright red heat, and then dropping them, with their 
lengths vertical, into a vessel 60 centim. deep, which was filled 
with water at a temperature of 7° C. 

A greater number of magnets than were actually required 
were prepared, but only those which were found to be straight 
and of uniform glass-hardness throughout, chosen for the ex- 
periments. The hardness was tested by means of a file run 
longitudinally along and around the magnet ; in this way any 
marked divergence from uniformity in hardness was detected. 

Also, to make sure that all the pieces of the same sample 
should be as nearly as possible alike, they were one by one 
let fall on to a block of hard wood, and those which gave the 
same kind of metallic ring were taken for the experiments. 
They were then thoroughly cleaned and polished, and their 
lengths, diameters, and weights accurately determined ; these 
measurements being, for ease of reference, given below in 
Table IT. 

There were fifteen magnets in all, 7. e. five samples of each 
specimen, and each one was made exactly 10 centimetres in 
length. 


422 Mr. W. Brown on the Effects of Percussion and 
TasiE I1.—Dimensions of Magnets. 


Length of | Diameter of ; 
Number of | magnet, in | magnet, in | Dimension, Weight of 


specimen. | centimetres, | centimetres, | ratio d/d. be aa 
7 in grms. 
Tee ae 10 0:300 i Oo 5D 
TT: fos 10 0-265 | 38 4:3 
ALT dees 10 0-270 OF uk 4:5 


The five pieces of each sample were then magnetized by 
placing them between the poles of a powerful Ruhmkorif 
electromagnet, which was excited by a current from twenty- 
four Thomson tray-cells joined in series. The magnetizing 
current was approximately 5:3 amperes, producing a field of 
900 C.G.S. units intensity. The field was measured by rota- 
ting a coil of known dimensions between the poles of the 
magnet and observing the deflection produced on a ballistic 
galvanometer ; and this was reduced to absolute measure by 
comparing with the deflection (on the same galvanometer) 
obtained by rotating another coil of known dimensions in a 
field the strength of which was known. 

When the field due to the electromagnet was being measured, 
there was nothing between the poles except the meagsuring- 
coil. In the process of magnetizing, the magnets were reversed 
three times between the poles of the electromagnet and then 
finally magnetized. This was also done in every case when the 
magnets were remagnetized between two sets of experiments. 

After being magnetized the magnets were laid aside for a 
period of eighteen hours, and then the deflections were taken 
for the purpose of calculating their magnetic moments. They 
were put through the same series of operations as the magnets 
used in the former experiments, described in Part I. ; that is 
to say, the deflection produced by each magnet on a magneto- 
meter-needle was observed ; each was then allowed to fall 
once perpendicularly through a height of 150 centimetres, 
with the true north end downwards, on to a thick glass plate; 
and the deflection on the magnetometer again taken with each 
magnet in exactly the same position. Hach was then allowed 
to fall three times in succession through the same height, and 
the deflection again taken. 

The following Table gives the results obtained for the 
magnets when they were all glass-hard ; and also after they 
had been magnetized and left undisturbed for a period of 
eighteen hours. | 


Annealing on the Magnetic Moments of Steel Magnets. 423 
Taste III.—Glass-hard. 


Specimen I, 


Percentage loss due to 


Magnetic : 
ever ot moment, falling Total loss. 
ore per gram. : ; 
one time. three times. 
i es ae 0-79 0-40 1-19 
i.e 60°42 0:90 0:20 1:10 
2 ae 60°18 111 0°20 1°31 
Bae 3fh), 60°96 . 0:49 0:30 0°79 
es. oo oe 59:03 1:64 0°83 2°46 
Mean ...... 60°33 _ 0:99 0:39 1:37 
Specimen II. 
Rees 72°10 1-72 0:87 257 
hee 72:70 213 1:30 3°53 
= peas 71°50 1:30 0°88 2°16 
AI ais css 72°70 2°13 0:87 2:98 
2) one 71:80 2°15 0:88 3:02 
Mean ...... 72°16 1°88 0:96 2°85 
Specimen ITI. 
eh 68°89 4-72 2°25 6°89 
7, 69°78 1°27 1-72 2°96 
eee 68-10 4:78 1°83 6°52 
73 Ee: 70°80 2°49 2-13 4:56 
5 re 72:40 4:08 1:28 531 
Mean ...... teas OO Ne ty BA | 1:84 5:25 


The above table, as far as it goes, seems to show that the 
percentage loss in the magnetic moment varies in the order 
of the quantity of manganese which the specimen contains. 
Thus specimen III. has a mean total loss of 5:25 per cent., 
and it has about three times as much manganese as either of 
the other two ; and specimen II. has about 20 per cent. more 
manganese than I., and its loss is 3 per cent. nearly, whilst 
that of I. is approximately 1-4 per cent. 

Specimen I., however, differs very much from the other 
specimens in the quantity of silicon it contains, and it alone 
contains sulphur. 

These same fifteen magnets were now all fastened to a 


424 Mr. W. Brown on the Effects of Percussion and 


piece of wood by means of soft copper wire, and annealed for 
one hour in a bath of linseed oil at a temperature of 100° C. 
They were then taken out and allowed to lie at the ordinary 
temperature of the room (8° C.) for a period of 6 hours, 
after which they were magnetized with the same battery- 
power, and every precaution taken, as formerly. Then, after 
lying aside undisturbed for a period of 20 hours, they were 
put through a similar series of observations for the purpose 
of finding the effects of percussion in changing their magnetic 
moments. The results are given in the following Table:— 


Taste TV. (Annealed for one hour at 100° C.) 


Specimen I. 


Magnetic Percentage loss due to 


Number of moment, fallin Total loss. 
magnet. per gram. 
one time. three times. 
ilo 63-4 0-76 1-92 2:67 
AREA 62°4 1:94 0°39 2°32 
2 ee 617 274 | 0:60 3:33 
AINA So a 62°6 1°54 0°78 2°70 
Le eens 61:4 1:97 1:20 319 
Mean ...... 62°3 1-79 0:98 2:84. 
Specimen II. 
Eee teh W1-2, 1°74 1°33 2:67 
ae trscccs 72-1 2°57 1°32 3°86 
Be fiers 7271 3°43 0:89 4:99 
Aen, 72°4. 171 0°87 2:56 
Wap es 72:4 2:99 0°88 3°85 
Mean ...... 72:04 2°49 1:06 38°45 
Specimen III. 

LEAS aapelle 66°8 2°54 0°92 4:42 
Fr 69°8 2°54 1°74 4:24 
See 65:1 4:09 237 6°36 
(5 TAN AR 67:4 2°63 1:80 4:38 
SOAR ter 68°3 2°16 1:33 3°46 
Rica i 67:5 2:79 1-63 457 


From the above Table we see that annealing for one hour 
in an oil-bath at temperature 100° C. has slightly raised the 


Annealing on the Magnetic Moments of Steel Magnets. 425 


magnetic moment of specimen I. and lowered it in III, 
whilst that of II. remains unaltered. Also that the total 
percentage loss in I. and II. is increased, whilst in ILI. it is 
slightly diminished ; indeed, we find it is doubled in speci- 
men I., and in II. it is increased 17 per cent., whilst it is 
diminished 12 per cent. in specimen III. 

We must remember, however, that specimen I. alone con- 
tains sulphur, and has the least quantity of manganese, and by 
far the most silicon of the three, while II. contains the least 
amount of silicon. 

The same fifteen magnets were again annealed for a period 
of two hours in the same oil-bath at a temperature of 100° C. 
They were allowed to cool and lie for six hours, as formerly, 
at the ordinary temperature of the room. They were also 
magnetized and treated similarly in every way as in previous 
experiments. ‘Then, after lying aside undisturbed for a period 
of twenty hours, they were put through the same series 
of observations for determining the loss in their magnetic 
moments. ‘The results are given in the following Table:— 


Tasie. V. 
(Annealed for two hours at 100° C.) 


Specimen I. 
; Percentage loss due to 
Rneiher of meee falling Ro? 
ones moment, otal loss. 
per gram. 
One time, three times. 
: 62:09 156 1-19 2-73 
De 0 ae 60°72 1:99 1:02 2-99 
5. pease 60°72 1:99 061 2°59 
Le 62:40 1:94 0°39 252, 
5 aaa 61:20 2°37 Lot 2°56 
fe oe 61:42 1-99 ROBeHt ben) cet 
Specimen IT. 
1) eae ai ay 2-13 1-70 3°83 
rt 72°72 2:97 0°88 3°83 
2 SA ak 72-10 Dey) lire 3°86 
0) Nee 73°00 2°96 0:87 3°81 
i 72°46 3°42 0°88 4°27 


mee. 72:60 2:8] 113 3-92 


ee I 


ee. Ee EF 


426 Mr. W. Brown on the Effects of Percussion and 


Table V. (continued). 


Specimen IIT. 


Magnetic Percentage loss due to 


jie moment, falling. Total loss. 
peretan. one time. three times. 

RE See 63°86 1°85 0:94 2°78 

ee 71°55 5:00 1:09 5-99 

St aS 68°89 4:72 5:40 9°87 

ZA ae he 71:55 5:00 1°74 6°61 

Sys ek ae W25 3'O2 2°14 5°39 
Mean ...... 69°42 3°98 2°26 6°13 


From this Table we see that the second annealing for two 
hours has had no effect on the magnetic moment per gramme 
in the case of specimens I. and fI., and has only slightly in- 
creased that of specimen III. We also see that the total 
percentage loss is unaltered in I., and but slightly increased 
in II., but in specimen III. there is an increase of about 
33 per cent. 

All the magnets were now annealed for a period of thirty 
minutes in an oil-bath at a temperature of 236° C.; they were 
then taken out and allowed to cool, as usual, to the ordinary 
temperature of the room (8° C.). Then, after lying aside 
for six hours, they were magnetized in the same manner and 
with the same battery-power as in the previous operations. 

The temperature of the oil was at first determined approxi- 
mately by means of a mercury in glass thermometer ; it was, 
however, accurately determined by an air-thermometer con- 
structed on a method introduced by Mr. J. T. Bottomley, and 
communicated by him to the Birmingham Meeting of the 
British Association in 1886. ‘This method will be explained 
further on. 

After being magnetized, the magnets were laid aside for a 
period of twenty hours and then put through another series 
of observations, the results of which are given in the following 


Table :— 


Annealing on the Magnetic Moments of Steel Magnets. 427 


TABLE VI. 
(Annealed for half an hour at 236° C.) 


Specimen I. 


Magnetic Percentage loss due to 
Ae moment, coals Total loss. 
magueb per gram. 
one time. three times. 
BS saa 63°48 514 321 8°19 
7, ee 61-90 5°27 3°08 8:20 
Sharepee 61-60 7°45 2:96 10:20 
AN isibe 62°90 6°73 2°94 8:84 
Dara oeeee 61°66 5:49 4°56 9°80 
Mean ...... 62°32 6-01 3°35 9°04 
Specimen IT. 
eee 69:13 671 BVP lr LOSI 
2 ae 69°13 7°80 7:28 | 14°54 
Bie tiaras 68-05 6°36 7°76 13°63 
2 Scere 69°60 L377 2°83 16°22 
Be aaa 68°36 12°88 4-93 17:20 
Mean ...... 68°85 9:50 5:37 (14-42 
Specimen ITI. 
i Ae 65°8 17-97 4-11 21°34 
2) eee 67°9 10°43 5°34 15:22 
Ser | 64:7 17:80 5°55 20°10 
A Telateiwes 65:0 13°63 4:2] 17-27 
DS aeeern 64:0 13°85 617 19°16 
Mean ...... 65:5 473 | 507 | 1861 


Here we get a very interesting result: we find that, by 
annealing for half an hour at approximately three times the 
temperature, we get three times more percentage loss. It is 
also interesting to note that in every case the total percentage 
loss is almost exactly tripled ; but the three specimens still 
preserve the same relative behaviour throughout. 

The same magnets were again immersed for half an hour in 


428 Mr. W. Brown on the Hffects of Percussion and : 


an oil-bath at temperature 236° C. and then allowed to cool 
in the air as formerly. But this time they were allowed to 
lie for three weeks, and then magnetized in a manner every 
way similar to that formerly employed. After being mag- 
netized they lay undisturbed for a further period of twenty 
hours, and were then put through the same series of observa- 
tions as on the previous occasions. The results are contained 
in the following Table :— 


Taste VII. 
(Annealed for half an hour at 236° C.) 


Specimen I. 


Magnetic | Percentage loss due to 


Number of moment, falling Total loss. 
ME = 
; one time. three times. 
Eerie 57-0 76 105 14:6 
PAE ene 60°9 Ia 5:4 16°8 
SS ep sigan 61:1 10°8 4:4 14:8 
4 Oh aan 60:4 86 53 13:6 
Ey ane sete 60°9 15°4 54 19°8 
Mien, G00 | aoe 56 15-9 
Specimen II. 
1 Pees eae 60°3 22:0 70 29°5 
ey es 60°9 16°4 14-1 27:8 
Se aa 58:9 10°6 16:1 25:0 
2 et 60:4. 17:0 16°9 aH es | 
Ryn cecsan 58°5 25:0 10°8 32'°8 
Mean’... ... 59°8 18:2 12:9 29-2 
Specimen III. 
LS ae 5671 16°6 9°5 24-6 
CA aaa 56°9 10°6 14°6 22°71 
2 th ie ee 56'1 17°4 9:9 25°9 
Di ea 582 23°0 6:2 27°6 
Posh hate 58:2 20:0 14:2 31:0 
Meant ee 571 17-5 10°8 62 


We here find that the second annealing at a high tempera- 


Annealing on the Magnetic Moments of Steel Magnets. 429 


ture has diminished the magnetic moment per gramme by 
13 per cent. in specimen II., and by about 12 per cent. in 
III., whilst in specimen I. itis decreased by nearly 4 per cent. 
We also find that the total percentage loss due to falling four 
times through a height of 1:5 metre has increased above the 
results of the last experiment as much as 100 per cent. in 
specimen IJ., and 70 per cent. in I., and 40 per cent. in the 
case of specimen III. 

The magnets were not again magnetized, but were allowed 
to lie undisturbed in the varying temperature of the room for 
a period of nine months, that is from May 15, 1886, till 
February 12, 1887. This was done merely to see what would 
be the effect of time upon them in their annealed condition. 
They were put through a similar series of observations, with 
the exception that they were not remagnetized. The follow- 
ing Table contains the results :-— 


TasLe VIII. 


Magnets not remagnetized and left undisturbed for 9 months. 


Specimen [. 


: . Percentage loss due to 
Number of HEGEL 


moment, ee Total loss. 
magnet. 
per gram : . 
one time. three times. 
i so av7g | «1-02 1-03 2:04 
5) ae ~ 49-48 0:50 0:50 0:98 
aes 48:51 1:50 0:00 1:50 
AG! 4% 50:70 0-96 1:45 2-40 
ht a 48:01 1:01 0:00 1:01 
Mean ...... 48:89 1-00 0:59 1:58 
Specimen IT. 
egy a's 41:47 | 2-25 4-23 5-64 
Fh) ee 43:02 | 2:89 1:50 4-34 
SO veal 39-28 1:58 0:80 2:38 
Jay Sea 41:00 1:14 0:00 ue Pace! 
ee 38°66 0:80 2:52 3-22 


4380 Mr. W. Brown on the Effects of Percussion and 
Table VIII. (continued). 


Specimen III. 


. Percentage loss due to 
Magnetic Lee 
Number of moment, falling Total loss. 
magnet. a ; 
acess one time. three times. 
‘peyote 8 42°31 211 2°16 4:22 
De, Bee 43°45 1:37 1:38 2:82 
oe park a 4111 0:00 0:00 0:00 
2a eens 41-11 1:45 1:47 2°90 
Lap eats 39°36 IE 0-77 2:27 
Mean ...... 41-46 1-34 1-15 2:44 


From the above Table we find that the relative losses of 
magnetism in the different specimens due to lying undisturbed, 
as indicated by the diminished magnetic moments, is in the 
reverse order to what has taken place throughout the whole 
series when the magnets were subjected to percussion. 

The total percentage loss all through these experiments due 
to percussion has been in the order of the number of the spe- 
cimen. ‘Thus specimen I. has always decreased the least and 
specimen III. the most; but, in the case of lying undisturbed 
for nine months, the decrease in the magnetic moment of spe- 
cimen I. is 3°5 per cent., and of II. 3:7 per cent., while ILI. 
has diminished only 1:6 per cent. Specimen III., however, 
contains about three times as much manganese as either I. 
or II. 

In Joule’s Scientific Papers, vol. i. page 591, some results 
are given on the effects of time and temperature on hard mag- 
nets. ‘The magnets used by him were either one inch long or 
half an inch, and were made up of a number of thin bars 
placed side by side so as to form compound magnets of ya- 
riously shaped sections but with plane ends; the magnetic 
moments of these magnets diminished about 33 per cent. on 
lying aside for a period of eighteen years. 

The rate of diminution of magnetism in different kinds of 
steel, with annealing, time, and temperature, is at present 
under investigation in this laboratory. In connection with 
this investigation, further experiments are being made on the 
same kind of steel as is referred to in this paper, and it is 
hoped that further results will be ready for publication at an 
early date. I will now give a tabular view of the results 
obtained up to this point. 


431 


Annealing on the Magnetic Moments of Steel Magnets. 


TaBLE IX.—Showing the changes in the magnetic moment per gramme, and the total percentage loss due to the 
whole four falls through a height of 150 centims.; also showing the effects of annealing on the different specimens. 


Not remagne- 


Length Annealed Annealed other | Annealed half Annealed Ficeel acl et 
and Dinaiaien Weight Glass-hard. one hour two hours at an hour at another half adi eee ee 
Speci- | diameter |" "atin | OF at 100° C. 100° ©. 236° 0. _| hour at 236° ©, | UNdisturbed for 
peci- of ratio magnet 9 months. 
men. U/q 2 z 
tape a Per Per Per Per Per 12 
In brane. Mag. Mag. Mag. ; Mag. Mag Mag. iss 
cents mom. | | mom, | gE | mom SEM | mom. | $2" | mom [PE | mom | Se 
Te ep LOX O'S 33 5D 60°33 | 1:37 62°3 2:84 61:42 | 2°84 62°32 | 9:04} 60:00 | 15:9 | 48:9 16 
ncecsul sO >< O:26D 38 4'3 72:16 | 2°85 | 72°04 | 3-45 72°60 | 3:92 68°85 | 1442] 59:80 | 292 | 407 | 3°34 
IIT.......| 10 0:27 37 4:5 70:00 | 5:25 675 | 457 | 69°42 | 611 65°50 | 18°61) 57:10 | 26:2 | 41°5 | 2:44 
TABLE X.—Showing the effects of the separate falls. 
Gisacand Annealed one hour | Annealed two hours Annealed half an Annealed another After being left un- 
: at 100° C. at 100° C. < hour at 286° C. half hour at 286° C. |disturbed for 9 months. 
. - No. of - : No. of : : Nowot Sis : No. of f : No. of : ; No. of j 
Speci-| 8 falls. | 2 | & falls, |-2 | & falls. | 2 | 8 falls. g g falls. | @ | 9 falls. | 
men. q as S| Bo S| ee | = 5 = q a 
Pel se lk a. 8 See ee ee a) ee eal a eel eles 
Slee oie ee SLES erste ee eee Sep ake E 
T....| 60°33 | 99 | -39 /1:37 | 62:3 {1°79 | 98 |2:84 | 61:42 |1-99 |1-08 |2:84 | 62:32 | 6:01 |3:35| 9-04! 60:00 |10°9| 5°6|15°9| 48-9 |1:00 0-59 |1°6 
II....} 72°16 [1-88 | -96 |2:85 | 72-04 |2-49 /1-06 |3-45 | '72°6 /2°81 /1-13 |3-92| 68-85 | 9:50 [5-37 |14-42 | 59-80 |18-2 112-9 29-2] 40:7 1-73 |1-81 [3-34 
TIT....| 70-00 [3:47 |1-84 |5:25 | 67-5 |2°'79 |1-63 |4:57 | 69-42 [3:83 |2-26 |6-11 | 65:50 |14-78 |5-07 |18°61 | 57-10 {17-5 |10°8 |26:°9 | 41:5 [1:34 )1-15 [2°44 


432 Effects of Percussion and Annealing on Steel Magnets. 


Mr. Bottomley’s modification of the air-thermometer, re- 
ferred to above, which was used for measuring the high tem- 
peratures, is constructed and employed as follows :— 

Suppose a glass tube, 4 inch or ? inch internal diameter, 
is made to the shape shown in fig. 1, which Fie. 1 
is a quarter of the full size of the tubes used a ; 
in these experiments. | 

The parts AB and DC are drawn out to 
fine capillary tubes, very small in volume in : 
comparison with the bulb BD of the ther- 
mometer. When ready for use it is com- 
pletely filled with pure dry air and closed at 
C, but open at A. 

The parts CDB and the greater portion of 
AB are now inserted into the liquid, the 
temperature of which we wish to measure ; B 
and when it has been in long enough to be : 
at the same temperature as the liquid, it is 
sealed at A with a blowpipe flame, thus en- 
closing a sample of the air at the required 
temperature. The height of the barometer 
at the time of closing is also noted. 

It is then taken out and allowed to cool, and also thoroughly 
cleaned, with alcohol if the bath has been of oil, as it was in 
the case under consideration. 

It is now carefully weighed in a chemical balance ; then 
the end C is opened under water at a known temperature; the 
height of the barometer being again noted. 

By this operation the water is allowed to rush into the bulb 
BD and to compress the contained air to the volume consistent 
with the barometric height and temperature at the given instant. 

The thermometer with the contained air and water is again 
carefully weighed, at the same time taking care to add the 
small piece of tube which was broken off in the act of open- 
ing the end C. The remaining part of the tube AB is now 
filled with water by breaking off the end A, and the whole 
again carefully weighed. 

In the following calculation the weight of the air displaced 
during this last operation is assumed to be so very smail that 
for our present purpose we may neglect it. 

Let now : 

g= Weight of the glass, in grammes. 
g+w,= Weight of the glass and the contained air, in 
rammes. 
g+w.= Weight of the glass and water, in grammes. 
t= Temperature of the water employed. 


D 


Assumptions required for the Proof of Avogadro's Law. 433 


T=The absolute temperature of the oil-bath. 
H=Observed barometric height at the time of 
sealing. 
H’=The barometric pressure at the time of opening, 
corrected for pressure of vapour of water, at the 
. temperature of the water used in filling the tube. 
Then we have 
H! Wg—- Wy, _2id+t 
He pe OS AR Gh 
p— Uwe (273 +t) 
H! (w2—wy) © 
In these experiments the observed values were, after making 
all corrections, 


H=752:4 millim. 


i 78: 
W2== 9°310 grammes. 
W4= 2:278 ” 
R= de ©. 
oe 752-4 x 5°31 x 288 


See S02 Ce 


And the temperature of the oil was therefore 509—273= 
236° C, 


L. The Assumptions required for the Proof of Avogadro’s 
Law. By Professor Tart*. 


1 ee months ago (in consequence of a chance hint in 

‘Nature’) I managed to procure a copy of Prof. Boltz- 
mann’s paper (anté, p. 305), and inserted a reply to it in the 
(forthcoming) Part II. of my investigations ; but, as there 
may be some delay in the publication, I send a short abstract 
to the Philosophical Magazine. 

Prof. Boltzmann says that I do not expressly state that my 
work applies only to hard spheres. This is an absolutely 
unwarrantable charge, as I have taken most especial care 
throughout to make this very point clear. 

Prof. Boltzmann, while objecting to my remark about 
“playing with symbols,’ has unwittingly furnished a very 
striking illustration of its aptness. His paper bristles through- 
out with formule, not one of which has the slightest direct 
bearing on the special question he has raised ! 

He asserts that, in seeking a proof of Clerk-Maxwell’s 
Theorem, I have made more assumptions than are necessary. 
To establish this, he proceeds to show that the Theorem can 


* Communicated by the Author. 
Phil, Mag. 8. 5. Vol. 23. No. 144. May 1887. 2G 


OT ae a 


434 Assumptions required for the Proof of Avogadro’s Law. 


be proved by the help of a different and much more compre- 
hensive set of assumptions! “ “Hrép@ ye tud@, Avoyeves”! He 
allows that my proof is correct ; and I am willing (without 
reading it) to allow asmuch for his. The point at issue, then, 
is :— Which of us has made the fewer, or the less sweeping, 
assumptions? Another question may even be :— Whose as- 
sumptions are justifiable ? 

My assumptions are (formally) three, but the first two are 
expressly regarded as consequences of the third, which is thus 
my only one, viz. :— 

There is free access for collision between each pair of par- 
ticles, whether of the same or of different systems; and the 
number of particles of one kind is not overwhelmingly greater 
that that of the other. 

From this I conclude (by general reasoning as to the be- 
haviour of communities) that the particles will ultimately 
become thoroughly mixed, and that each system (in conse- 
quence of its internal collisions) will assume the ‘‘special state.” 

Prof. Boltzmann denies the necessity for internal collisions 
in either system, and assumes that (merely by coliisions of 
particles of different kinds) uniform mixing, and distribution 
of velocities symmetrically about every point, will follow! 
Surely this requires proof, if proof of it can be given. So 
sweeping is the assumption that it makes no proviso as to 
the relative numbers of the particles in the two systems! The 
character of this absolutely tremendous assumption is so totally 
different from that of mine that 1¢ is impossible to compare 
the two. My assumption has, to say the least, some justifica- 
tion ; but I fail to see even plausible grounds for admitting 
that of Prof. Boltzmann. There is noneed to inquire as to its 
truth, at present; for I am not now discussing his extension 
of Maxwell’s Theorem which, of course, is implied in it. The 
question is :—Is Prof. Boltzmann’s assumption, even if cor- 
rect, sufficiently elementary and obvious to be admitted as an 
axiom? It is so wide-reaching as, in effect, to beg the whole 
question ; and I venture to assert that, on grounds like these, 
it cannot possibly be shown that any of my assumptions are 
unnecessary. 

The objection raised in Prof Boltzmann’s “Second Ap- 
pendix ”’ (which is not in my German copy) was made long 
ago to me by Prof. Newcomb and by Messrs. Watson and 


Burbury.* I have replied to this also in my Part II., and 


I will not discuss it now. I need only say that Prof. Boltz- 
mann, while causelessly attributing to me a silly mathema- 
tical mistake, has evidently overlooked the special importance 
which I attach to the assumed steadiness of the “ average 
behaviour of the various groups of a community.” 


a BOBS 


LI. On Evaporation and Dissociation—Part V1.* On the 
Continuous Transition from the Liquid to the Gaseous State 
of Matter at all Temperatures. By WitttaAM Ramsay, 
Ph.L)., and SypNEY Youne, D.Sc.F 


[Plates VIL, VIIT., IX., & X.] 


ie was proved by Boyle, in 1662, that the volume of a gas, 
provided temperature be kept constant, varies inversely 
as the pressure to which it is subjected ; this relation may be 
expressed by the equation p= - , or pv = constant, where p 
and v respectively stand for pressure and volume. But sub- 
sequent experiments by Van Marum, Oersted, Despretz, and 
others showed that certain gases do not obey this law; and it 
is now well known that Boyle’s statement is only approximate; 
for it has been proved by experiment by Regnault, Natterer, 
and more recently by Amagat, that no gas, under high pres- 
sures, is diminished in volume in inverse ratio to the rise of 
pressure. Indeed Boyle’s law could hold only on the assump- 
tion that the actual molecules of matter possess no extension 
in space and exert no attraction on each other. A gas, such 
as hydrogen, at low pressures, and consequently at large 
volumes, fills a space very great when compared with the 
space occupied by the actual molecules ; and these molecules 
are comparatively so distant from one another, that the attrac- 
tion which they mutually exercise is inappreciable. But, on 
compression, the actual space occupied by the molecules bears 
an increased ratio to the space which they inhabit; and, by 
their approach, the attraction which they exert is also increased. 
The gas, then, deviates appreciably from Boyle’s law. 
Gay-Lussac, in 1808, enunciated the law that the volumes 
of all gases increase by a constant fraction of their volume at 
0° for each rise of 1° in temperature. It was subsequently 
ascertained by Magnus, and confirmed by Regnault, that cer- 
tain gases deviate from this law, expanding more rapidly than 
others. Such gases, as a rule, are at temperatures not far re- 
moved from those at which they condense to liquids ; that is, 
their volumes are comparatively small, and the actual size of 
the molecules and their mutual cohesion begin to manifest 
themselves within the range of experimental observation. 
Again, it is evident that no gas can perfectly follow Gay- 
Lussac’s law; but the larger the volume it occupies the 
smaller is the influence of the disturbing factors. The usual 
* Parts I. and IL., Philosophical Transactions, parti. 1886, pp. 71 and 
123; Part III., ibid. part ii., 1886, p. 1; Part IV., Trans. Chem. Soe. 


1886, p. 790; Part V., in the hands of the Royal Society. 
+t Communicated by the Physical Society: read February 26, 1887. 
2 G2 ; 


re 


a 


if 


| 


436 Drs. Ramsay and Young on 


expression for Gay-Lussac’s law is v= =c(1+at), or, if the 
absolute temperature-scale be employed, v=cT. 


As a deduction from these laws, it follows that if the volume 


‘of unit mass of a gas, supposed to follow them rigorously, be 


kept constant, the pressure varies dinecer as the absolute 
temperature ; or p=clT. 

Now, so long as the volume of unit mass of a gas is kept 
constant, the average distance of its molecules from one an- 
other will remain constant; and it is a fair assumption that 
the attraction of the molecules for each other will not vary. 
It may, of course, be the case that the effect of a rise of tem- 
perature on any individual molecule is to alter its actual 
volume ; but of this we know nothing; and, in default of 
knowledge, it has been assumed by us that no such alteration 
takes place. If these assumptions are correct, it follows 
that the temperature and pressure of gases—and indeed the 
same assumptions may be extended to liquids—should then 
bear a simple relation to each other. We have obtained ex- 
perimental proof of a convincing nature that this is the case ; 
and in a preliminary note to the Royal Society, read on 
January 6, we promised such a proof. ‘This proof is the sub- 
ject of the present paper; and we must ask for indulgence 
in quoting a large array of figures, some of which have 
already been published, on the ground that such an important 
generalization requires as much experimental evidence as can 
be brought to bear on it. 

The relation between the pressures and temperatures afie a 
liquid or-a gas at constant volume is expressed by the equation 
p=bT—a; 
where pis the pressure in millimetres, T the absolute tempera- 
ture, and b and a constants. The values of these constants 
depend on the nature of the substance and on the volume. It 
follows from this, that if a diagram be constructed to express 
the relations of pressure, temperature, and volume of liquids 
and gases, where pressure and temperature form the ordinates 

and abscissee, the lines of equal volume are straight”. 

We have proved this to be the case for ethyl oxide (ether) 
between the temperatures 100° and 280°, and for volumes 


varying from 1°85 cubic centim. per gram to 300 cubic centim. 


per gram. This proof we now proceed to give. 

The data for the calculations are at present in the press, and 
will shortly appear in the Philosophical Transactions for 1886, 
p. 10. A diagram (which will accompany that memoir) was 
constructed with the greatest care, showing isothermal lines, 
' * Amagat (Comptes Rendus, xciv. p. 847) has stated a similar relation 


for gases; his data are, however, imperfect, and he expressly states that 
the law does not apply to liquids. 


Evaporation and Dissociation. 437 


the ordinates and abscisse being respectively pressures and 
volumes. It was possible to read pressure accurately to 
within 20 millim.; and volume, up to a volume of 3:1, to 
within 0°001 cubic centim. per gram ; and, at volumes greater 
than 3:1 cubic centim. per gram, to 0:01 cubic centim. per 
gram. Pressures corresponding to each isothermal were then 
read off on the equal volume-lines, from curves constructed to 
fit the experimental points as accurately as could be drawn 
with the help of engineers’ curves. These pressures and 
temperatures were then mapped as ordinates and abscisse ; 
and it was found that points corresponding to each volume 
lay in a straight line. Again, two points were chosen on 
these equal volume-lines, as far apart as the scale of the dia- 
gram would permit, and the values of the change of pressure 


: dp : 
per unit change of temperature, a were ascertained for each 


separate volume chosen. To eliminate irregularities, these 
values were smoothed graphically ; but it was difficult to find 
any very satisfactory method. The method employed for 
ether, which we found to give the best results, was to map as 
ordinates the ratios between these values, and similar values 
calculated on the supposition that the gas or liquid followed 
the usual gaseous laws, against the reciprocals of the volumes 
as abscisse. A curve was then drawn, taking a mean course 


d 
among the actual points, and the values of - were calculated 
from readings at definite volumes. This expression, ~ is the 


6 of our formula. Having thus obtained the most probable 
value of b for each volume, the value of a at each volume was 
ascertained by calculation from each individual point read 
from the original curves, and at each volume the mean of all 
was chosen. 

Isothermals were then calculated by means of the equation 
p=bl—a, T being kept constant ; and those values of a and 
6 corresponding to the volumes required being selected. These 
calculated isothermals are shown on Plate VII. ; and the lines 
of equal volume, or isochors*, on Plate VIII. It is evident, 
from inspection of the former, that the calculated lines corre- 
spond as closely as possible with the actual observations. 

Tt is necessary now to give the data on which these deduc- 
tions are based. The following Table gives those points 
corresponding to lines of equal volume read from the diagram 
constructed from experimental observations. 

* From icos, equal, and yawpeiv, to contain. Another suitable word 
would be “:soplethe,” but we have Professor Jowett’s preference for the 


one selected. Hither of these terms seems preferable to that ( zsometrics ) 
already proposed. 


d Young on 


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Evaporation and Dissociation. 


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0006 
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8-o861 
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8-o86T 
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8-o861 
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8-861 
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Drs. Ramsay and Young on 


440 


GE-0086 
Orrel 


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GE: 0086 
OFITS 


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“(panunquos) J e[qeL 


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‘UUN]O A. 


Evaporation and Dissociation. 44]- 


Lines of equal volume, or isochors, were then mapped from 
these results, pressures and temperatures being ordinates and 


abscissee. The values of = or 6, were then read for each 


separate volume and smoothed, as before described. The fol- 
lowing Table shows the read values, and the values after 
smoothing. ‘The values of a are also given, calculated by 
the equation a=bi—vp, from the results shown on the previous 
table ; the means of all the individual results for each volume 


are stated. TABLE II. 
| | } 
b | 6dcaleu- | 
Volume. found. | lated | log b. ae 
eg oi 3. 2034 30826 826860 
1:90 1746 1861 326986 767670 
ch he oe ae 1716 323452 715860 
210 Sch ee 1597 320335 672820 
2-05 1561 1492 317372 633070 
21 1500 1405 314777 600110 
2-15 1342 1320 3-12061 566170 
2-2 1211 1243 309462 535100 
2-25 1155 1175 307022 507170 
- 23 1117 1115 304744 482160 
2-4 1020 1010 300423 437240 — 
25 920°8 919-7 269363 397970 
2-75 732-0 732-0 2-26451 313605 
3-0 623-5 621-7 2:79357 262917 
3:3 585-7 5328 | 2°72659 221630 
37 453°1 454°6 | 265762 185109 
4:0 413-7 413-7 | 261669 165996 
5:0 321-5 319-1 | 2-50892 121895 
6-0 257-15 254-2 2-40511 91906 
7 2136 208-7 2-31959 71464 
8 178-0 176-1 | 9-94584 57203 
9 150-95 1514 | 918092 | 46742 
10 130-8 132°7 2°12280 39079 
11 116-25 117°6 2-07027 33037 
12 104-45 1055 | 2.02336 28401 
13 94-29 95°54 | 1:98017 24659 
14 85:00 87:09 1:93998 21567 
15 78°82 80-06 1:90339 19125 
16 72-29 73-95 1:86893 17049 
17 67°58 68°76 1:83733 15313 
18 62:53 64-24 1-80780 13854 
19 59:23 60-11 1:77893 12533 
20 55°70 56-43 175151 11386 
25 43°75 43-26 163612 7529 
30 35-60 34-99 1:54393 5412 
40 26:23 2509 1:39945 3159 
AO 20:38 19-46 1:28925 2077 
75 12:82 12°50 1:09674 994 
100 9-32 9-119 095997 571 
150 5:87 5-923 077254 270 
200 4°38 4-396 064306 160°5 
250 3-46 3-483 0-54198 105 
300 2-93 2-858 0°45605 59 


OF9ZG | OLEOF | 0906E | OIFFE | OGIZE | 0166G | OS48z | OF9LT | 060LZ | OLGIG | 09EGS | OGOES | OSE81| OGLTT | OSTL OSSeT— jOLII9— | L€ 
OSIGP | OOTY | OELGE | OLOEE | OOFOE | 0088E | OF LZ | OOLLG | O€T9Z | OLOGS | OLFGS | OSOLT | 0806 O9LE O88ZG— |OLT9L— | BE 
G6PSP | GE9EP | GSPLE | STEVE | COSTE | GESES | G608T | SFELZ] | GZZIZ | G8GFE | GLSTT | G99GT | CEO GIT + | G960E— \cETg6— | 0-€ 
OFG6P | OGELP | OSOOP | OLEVE | OTLZE | OLGOE | 0206G | OL 18 | O489Z | O8E8G | OFLIG | OLFFI | 62FE 0068— | O6FOF— \OOLETT— |SL-6 

OSZOF | OLOTF | OLOLE | OSSFE | OFFSE | OGETE | 0962 | O82 | OFZES | OFOFT | 0SZ O<68— | O€6FS— \0069FT—| S-G 
O6EOF | OIELE | OFELE | OFTHE | OLESE | OBZOE | OGSGG | OGTST | 0 OOTOT— | 06909— JOELT9T—} FG 

060GF | O986E | NSG8E | OLSIE | O8GFE | OTL8G | OSELT | 0Z8 OVEOT— | OTT99— JOG9LLT—| Se 

OLGGF | OSETF | OSF6E | OSOLE | OTZTE | OGFGT | O88G OF66— | OTL89— |O9Z98T —|&G-G 

OSEGF | OGOEF | OLIOPF | OOFVE | O96IG | OLEE O€I6— | O0ETL— |0G9E6T—| 6% 

0900E | O89LF | OVOGF | OGF8E | OFGGG | OFF 0994— | OLLEL— |08LG06 — |S T-@ 

O&GEP | OLFG6G | 0668 OL9E— | 0&6¢L— |O9P9TE—| T-6 
OLGGE | OOGGT | O0GOE— | OLL9L— |008SEG — |G0-6 
OGLEF | O9L8T | O8Lz OLOLL— |O089EG—| 0-6 

O8TZG jOTTOL | 06LEL— \06EL¥G —\G6-T 
O968E |OPLEL | O€SEL— |O6F6EE—| 6-1 
osgee+ | OFSSE+ | O€E89— |OOLTLZ— [48-1 


“Wun “UU “TU “ULL “UU “THU “TOU “TUL “TUL “UU “TUL “TUL “UU “TUUL moogoet “TOUT “WUuUr “ULUL 


Drs. Ramsay and Young on 


——ooo—--eeeeeeee— —— a eeeee_=_ SO OO OO OO OOO | CC |_| | | 


0086 | 0096 | o€66G | o0GZ | oOIG | G0G | .00G | L6T | oG6T |8-o861 | oG61 | oO6T | of8T | o841 | O9T o0GT o00T 00 “A 


‘TIT S1S4VL 


ON 
bs 
H 


443 


Ewaporation and Dissociation. 


18861 
86091 
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66196 
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89 10€ 
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99666 
PS806 
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80196 
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1698 

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19891 
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6161 
2006 
98606 
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9896 

6687 

6002 

6998 

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TS9E 

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688¢T 
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9098 

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8$89 

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L8CE 

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TE86 
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989T1— (99089 — 


OOT 


444 Drs. Ramsay and Young on 


The lines of equal volume calculated from these values of 
a and 6 are reproduced graphically in Plates VIII. and IX. 
We propose to discuss the form of the diagram later. 

We have calculated the values of p for various isothermals, 


at the above volumes. These are shown in Plate VII.; the 


actual experimental observations, of which a detailed account 


is given in the ‘ Philosophical Transactions,’ 1886, p. 10 et 


seq., are represented by circles. It is evident that the curves 
through the calculated points represent the actual measure- 
ments very closely, indeed as nearly as unavoidable error of 
experiment allows. It is to be noticed that the greatest 


divergence is at the temperatures 250° and 280°, but the de- 


viations are in opposite directions, and must therefore be 
ascribed to experimental error. 

Table III. (pp. 442, 448) gives the data from which the cal- 
culated isothermals were constructed. 

As volumes above 30 cubic centim. per gram are not given 


in the diagram, we have thought it advisable to show the cor- 


respondence of calculated and observed results by a table ; 
the calculated numbers were read from curves specially con- 
structed from the formula p=6t—a, and the observed results 
are those actually furnished by our experiments. It will be 
seen that the correspondence is very close. 


Table III. (continued). 
Temperature 100°. 


Waolures Pressure Pres. cal- Wee Pressure | Pres. cal- 
: found. culated. ; found. culated. 

Cc. ¢. millim. millim, c. ¢. millim. wmillim. 
54:06 4817 4875 74:42 3668 3685 
55°50 4720 4760 77:34 3546 3550 
56°95 4627 4670 83°14 3334 3320 
59°84 4434 4470 86:02 3243 3215 
62°73 4265 4290 88 89 3150 3120 
65°63 4102 4100 94-63 2978 2970 
68°56 3946 3970 O77bL 2893 2900 
71:48 3818 3815 | 

Temperature 150°. 

31:09 9066 9110 68°64 4651 4675 
33°94 8497 8520 74-51 4304 4310 
39°66 7484 7505 80°35 4027 4010 
45°43 6674 6745 86:13 3783 3760 
51-23 6024 6100 91:87 3564 3530 
57-02 5480 5560 97:63 3375 3045 
62°81 50384 5090 


Evaporation and Dihociation. 445 
Table III. (continued). 


Temperature 175°. 


Pressure | Pres. cal- i Pressure | Pres. cal- 


7465 5232 5225 97°81 4044 4020 


Volume. found. culated. | Volume. found. culated. 
c. ¢. millim. millim. ib - €..G millim. millim. 
31-11 9906 9975 | 68°69 4987 5000 
33°96 9248 24) ee 74:56 4626 4630 
39°68 8065 8130 80-40 4307 4310 
45°46 7208 7205 86:18 4035 4030 
51°26 6485 6510 91:93 3803 3800 
57:06 5903 5940 97-69 3589 3600 
62°85 5396 5450 
Temp. 185°. Temp. 190°. 
31°12 10284 10300 || 381-12 10455 10460 
45°47 7434 7460 45°47 7549 7545 
59:97 5811 5865 | 59°97 5902 5940 
74:58 4755 4770 74:59 4828 4810 
86°21 4159 4140 86:21 4210 4190 
91-95 3915 3900 
97°72 38692 3690 
Temp. 192°. | Temp. 193°°8. 
Fe 10544 10520 | 44 al 10587 10590 
45°48 7590 7600s 45°48 7627 7630 
59°98 5930 5970 59°98 5951 6600 
74:60 4847 4850 74:60 4867 4875 
86°42 4230 4200 | 86°42 4252 4220 
Temp. 195°. | Temp. 197°. 
31°12 10631 10620 33°17 10108 | 10055 
45°48 7651 MOTO «| 38:07 8972 8965 
59:98 5969 6020 47°84 7312 7320 
74:60 4884 4895 | 57:59 6166 6130 
86°42 4260 4230 67°33 5356 5340 
91:97 4007 4000 77:05 4718 4735 
97°74 3797 38775 | 86°75 4219 4220 
96-44 3820 3820 
| Temperature 223°°25. 
| 31-15 11567 11550 || 8050 4870 4880 
45°51 8260 8240 86:29 4556 4550 
| 60:02 6412 6420 92°04 4289 4270 
i 


it 
q 
H 


446 Drs. Ramsay and Young on 


These complete the data for ether. It appeared necessary 
to examine the relations between volume, temperature, and 
pressure for carbon dioxide, because it is chemically entirely 
different from ether, and also because the data are furnished 
by Dr. Andrews, whose experimental skill was very great; and 
we shall prove that his results entirely corroborate our views. 

It was first necessary to correct the pressures given by him 
in atmospheres by means of Amagat’s results, so far as that is 
possible. As Amagat’s experiments on the compressibility of 
dry air do not extend beyond 65,000 millim., no correction was 
possible above that pressure ; and extrapolation is inadmissible, 
inasmuch as the minimum value of pv for air is at60,000 millim. 

Data for obtaining the actual volume of carbon dioxide used 
are given by Andrews. The weight was calculated in the 
following manner :—Andrews gives the volume of carbon di- 
oxide filling his tubes at 0°and076 millim.,and from Regnault’s 
data the weight was calculated. This refers to Andrews’s first 
paper (Phil. Trans. 1869, i. p. 575). In his second paper 
(Phil. Trans. 1876, p. 421) no direct data are given from 
which the weight can be determined ; but we succeeded, by 
combining the results given in his various tables, in arriving 
at the weight without any serious error. 

His results are reproduced in an available form in the 


following Tables :— 


TasLE IV. (The first quantity weighed 0:000612 gram.) 


emp. Vd": Temp. 21°°5. Temps" aly 
Vol. of Vol. of Vol. of 
= 1 gram i 1 gram. 1 gram. 
millim. c. ¢. millim. Cc. €. millim. Cc. ¢. 
68726 * 1-038 46600 1-232 63462 © 1:389 
56333 1EOS9 46383 1:241 61416 1:425 
40725 1:104 45490 1-484 59540 1:470 
37631 1:124 45155 2:270 57847 1-495 
37459 1°145 44962 3'124 56178 1812 
37074 1377 44787 4-760 55010 2234 
36942 1547 34907 8118 54588 2°338 
36816 1-972 53089 3814 
36719 2°758 51709 4-192 
36668 3°733 50390 4-534 
36610 5004 49118 4°855 
36528 6:554 47910 5147 
36483 6595 46725 5471 
35497 6:964 ; 45622 5°750 
44577 6009 
43611 6°265 
43182 6°515 
41761 6°765 
40895 7:003 


* The pressures to which an asterisk is affixed are not corrected. 


Evaporation and Dissociation. 447 
Table IV. (continued). 
Temp. 32°°5. Temp. 35°°5. | Temp. 48°°1. 
| 
_ Vol. of Vol. of || Vol. of 
| - hit gram. = 1 gram. | = 1 gram. 
millim | €..€. millim. c. ¢. | millim. c. ¢. 
63246 1461 || 81775* 1:330 83144 * 1-999 
59425 1612 || 75673* 1:392 72344 * 2:999 
58501 1-318 | ‘70406* 1-476 62837 4-061 
56813 2954 || 68035 * 1-532 56311 5-057 
55151 3543 | 64516 1-624 51012 5991 
54839 3°629 | 60552 2-511 46675 6°887 
53292 4034 | 57057 2°549 
42808 6°595 53977 4-233 
51166 4-831 | 
48625 5°396 
| 46340 5935 | 
| 44263 6-432 
| 42383 6917 


Table IV. (continued). 
(The weight of the second quantity was 0°0018075 gram.) 


Temp. 0°. Temp. 6°-6°°9. || Temp. 63°°6-64°. || Temp. 99°:5-100°°7. 
Vol. of Vol. of Vol. of Vol. of 
1 gram. e- 1 gram. o 1 gram. | = 1 gram 
millim. | ¢. ¢. millim. | ¢. ¢. millim. | ¢. ¢. | millim. €, e. 

25865 | 10:53 || 25870 | 11:25 ||169420*| 1-401 |} 169910*| 1-816 
23325 | 12°36 23291 | 13:09 |/110610*| 1:907 | 110530*} 3-161 
20800 | 14°53 20822 | 15°16 81229* | 3363 || 80323*| 5:057 
18674 | 16°77 18681 | 17-11 60422 5-477 || 59985 7-192 
15155 | 21°73 || 16777 | 19-90 || 48425 | 7-485 | 47910 | 9301 

12277 | 27-83 || 15161 | 22°54 40554 9-462 | 40169 11°52 

9081 | 39°09 || 12907 | 27-16 34806 | 11:45 | 34382 13°72 

| 11093 | 32:18 30351 | 13-48 29910 15°96 

| 9994 | 36°12 26212 -| 15:99 25930 18°74 

9087 | 40:06 23568 | 18-00 23325 21-03 

21106 | 20-41 || 20876 23°70 

18866 | 23°08 | 18711 26°65 

17000 | 25°86 || 16858 29°74 

15356 | 28°88 15214 33°09 

13288 | 33°74 13153 38°58 


As with ether, these numbers were plotted graphically up 
to a pressure of 75,000 millim. Above 65,000 millim. no true 
correction for deviation of Andrews’s air-gauge was possible; 


* The pressures to which an asterisk is affixed are not corrected. 


448 Drs. Ramsay and Young on 


but approximate corrections were introduced. It was possible 
to read pressures to within 30 millim.on the scale employed, and 
volume to within 0:02 cub. centim. per gram. Pressures corre- 
sponding to even volumes were read off, as with ether ; and on 
mapping the isochors with temperatures as abscissze and pres- 
sures as ordinates, the gens points lay in straight lines. The 


values of 0, i.e. of ——, were then read off, and smoothed, 


by mapping them against the reciprocals of the volumes. After 
smoothing, the values of a were calculated at each volume, 
making use of the pressures previously read from the curve 
representing isothermals. The diagram (1) on Plate X. was 
then constructed from these smoothed values. The crosses 
denote our readings of pressures at the temperatures chosen 
by Andrews for his isothermals. These values of a and 6 
were then made use of in recaiculating isothermals at the 
above temperatures, and the diagram (2) on Plate X. repre- 
sents the curves complete so far as Andrews’s data allow. The 
circles represent Andrews’s actual measurements ; and it is 
evident that no better concordance could be expected. The 
tables which follow give the data afforded by Andrews’s 


experiments. 
TaBLe V. 


Pressures read from Curves originally drawn from Andrews’s 
experimental data, and represented by circles in the 


diagram (2), Plate X. 


Temperature. 

Vol. |) 

0°. Go. WTSEr VATS be Bel, (8205. 35°°5. | 489°1.| 64°. | 100°. 
Cc. c. mm. mm. | mm, mm, mm. mm. mm, mm, mm. | mm. 
30 11430) 41790) 2.2 yen oe te ie ... | 14820} 16710] 
45) 13480 | 13830]... As Bis aS se .-- | 17500] 19860 
20 16200| 16570)... pa gig Je he .-. | 21450 | 243800 
15 20300 | 20970| ... a Me a pis 3M .-- | 27700} 31570 | : 
12 23730 | 24800]... ae ok Ne ra ..- | 000900 | 88730 | 
10 26715} 28000] ... tae a: as Ae ... | 38970 | 45330]: . - 
8 id. one ... |853840 |87560?.37800?/38400?| 41930 | 46200 | 54630 | 
ff hs ... | 85340 |38220?|40970 41370 |42060 | 46200 | 50850 | 61500 | 
6 sens Le, ... |41100?/44700 45060 |46000 | 51000 | 56700 baa | 
5 ne ne ... |44100?/48600 49200 |50370 | 56760 
4:5 see Bs ae ... {50550 |51400 [52830 | 60000 | 
4:0 Bee nan ie ... |02420 |538430 |55140 | 63500 . | 
oD Re is ae ... |54250 155260 {57240 | 67000 
30 55800 |56700 |59130 | 72400 


Evaporation and Dissociation. 


Tasxe VI. 
Read and Smoothed Values of 4, and Values of a. 


449 


Vol. b, read. 6, smoothed. log b. a. 
Cc. CG 
30 52:3 52°5 1-72016 2877 
25 63°3 64:0 1:80618 4024 
20 81-9 82:0 1913881 6256 | 
Peck 113°85 114°5 2:05881 10990 
pith 150-0 149-9 2°17580 17103 
188°5 2:27531 24718 
8 255°0 25271 240175 39120 
7 300°0 302°0 2°48001 50970 
6 368°1 3730 2°57171 68877 
5 472°3 475°5 2°67715 96008 
45 5d3°7 548°5 2°73918 116230 
4:0 6540 638°0 2°80482 141525 
3°5 7500 759°5 2°88053 176860 
| 30 933°6 936°5 2°97151 229420 
TaBLE VII. 
Calculated Pressures on Isothermal Curves, at definite 
volumes. 
Temperature. 
Vol. . react WOON DPN BW oy SES 
0° Grete lay 2h Ot heli 20D: | OD Oe aor dst, Gao.) LOOP. 
c. Cc mm. mm mm mm mm. mm. mm. mm. mm. mm. 
30 11456 | 11771 ... | 13088 13981 | 14816 | 16706 
25 13448 | 13832 ... | 15439 16526 | 17544 | 19848 
20 16130 | 16622 os 18680 20074 | 21378 | 24330 
15 20269 | 20956 .. |23830 ... | 25776 | 27597 | 31719 
12 23719 | 24719 | 25783 | 27043 | 28482 29140 | 31030 | 33413 | 38810 
10 26742 | 27873 | 29212 | 30796 | 32605| ... | 33433 | 35809 | 38806 | 45592 
8 29731 | 31244 | 33036 | 35155 | 37576 | 37928 | 38684 | 41862 | 45872 | 54952 
7 31476 | 33288 | 35433 | 37971 | 40870 | 41291 | 42197 | 46003 | 50800 | 61680 
6 32973 | 35190 | 37843 | 40973 | 44553 | 45073 | 46193 | 50893 | 56820 | 70250 
5 33802 | 36652 | 40032 | 44032 | 48592 | 49262 | 50682 | 56672 | 64232 
45 | 33510 | 36800 | 40700 | 45310 | 50570 | 51340 | 52980 59890 | 68620 
4:0 | 32645 | 36475 | 41005 | 46365 | 52495 | 53385 | 55295 | 63335 | 73485 
3°5 | 30480 | 35040 | 40440 | 46820 54110 | 55170 | 57440 | 67020 
3°0 | 26240 | 31860 | 38520 | 46380 | 55370 | 56680 | 59620 | 71290) . 


It will be seen that the highest calculated pressure is about 
73,500 millim, Andrews gives measurements at much higher 
pressures ; but these are few in number and uncertain, and 
the correction for the compressibility of air is moreover 
unknown. Hence it was impossible to make use of them in 
determining the values of 0. 

On reference to Andrews’s paper (Phil. Trans. 1876, p. 435) 


Phil. Mag. 8. 5. Vol. 23. No. 144. May 1887. 


2H 


RTE SS ee Se ee ee OAS SS ae ee 


Se re i Fe I ye ES TS ME a ee Bie oe Tn as. ee 


450 Drs. Ramsay and Young on 


it will be seen that he compared the relation of increase of 
pressure to temperature-ditference at constant volumes, and 
came to a conclusion opposed to ours. This is owing to his 
having made very few observations, and having accidentally 
chosen those which support his statement. If the coefficients of 
increase of pressure for unit rise of temperature be calculated 
by means of Table V., it may be noticed that, although irre- 
gular, there is no tendency towards a rise or fall of the 
coefficient. 

Regnault has measured the rise of pressure of gaseous car- 
bonic anhydride at constant volume. He gives the results of 
four experiments, none of which are available for our purpose, 
inasmuch as the volumes of a gram are too large. 

Reverting to the behaviour of ether, as shown on Plate VII., 
it will be seen that the curves have been drawn in the region 
where measurements are impossible. These curves have all 
the same general form. After rise of pressure and decrease of 
volume have proceeded for some distance, the curves bend 
downward, presenting the abnormal feature of decrease of 
volume with fall of pressure. The pressure continues thus to 
fall, and at 160° the isothermal touches zero-pressure. At 
lower temperatures, with small volumes, the pressure becomes 
negative, and may even represent an enormous tension. At 
0° the isothermal at vol. 1°85 cub. centim. per gram reaches 
the almost incredibly great tension of —271,700 millim. ; 
and it has at that volume (the smallest our results allow us 
to calculate) by no means reached its limit. At still smaller 
volumes the tension would doubtless still increase, until the 
curve turned, and further decrease of volume would be repre- 
sented, as it is at higher temperatures, by increase of pressure. 

The existence of these unrealizable portions of such iso- 
thermal curves was, we believe, first suggested by Prof. James 
Thomson, in a paper in the ‘ Proceedings of the Belfast 
Natural History and Philosophical Society,’ Nov. 29, 1871. 
Since that time attempts have been made to express relations 
between the pressure, temperature, and volume of gases and 
liquids by Van der Waals and by Clausius ; and the formulze 
which they propose, and which we hope to consider in a sub- 
sequent paper, give isothermals of similar form. Portions of 
these curves have, indeed, been experimentally verified. In 
Professor Thomson’s paper, above referred to, he points out 


that Donny, Dufour, and others have observed the phenomenon | 


commonly alluded to as ‘boiling with bumping.” This is 
usually the effect of a rise of temperature at constant pressure: 
But it may equally well be produced, as we have frequently 
had occasion to remark, at constant temperature by lowering 


Evaporation and Dissociation. 451 


pressure. If the diagram on plate ill. in our memoir on 
‘alcohol (Phil. Trans. 1886, part i. p. 156) be referred to, it 
will be seen that our actual measurement of such reduced 
pressure was made on the isothermal 181°°4. Mr. John Aitken, 
in an extended series of experiments on this subject (Trans. 
Royal Scott. Soc. of Arts, vol. ix.), has shown that such 
“ superheating”’ can take place only in absence of a free 
surface, 7. e. the existence of gaseous nuclei in the liquid, into 
which evaporation may take place. And Mr. Aitken has also 
shown that a gas may be compressed to a volume smaller 
than that at which liquefaction usually occurs, at any given 
temperature, without formation of liquid. The space, again, 
if no nuclei be present on which condensation may take place, 
remains “ supersaturated with vapour.’ It is evidently, there- 
fore, only the instability of such conditions which prevents 
their complete realization*. 

The formule: of Clausius and Van der Waals are based on 
the assumption that two causes are in operation—those 
referred to in the beginning of this paper—viz. the actual 
size of the molecules, and their mutual attraction. It is 
possible, by help of these assumptions, to realize the nature of 
the continuous change from the gaseous to the liquid state of 
matter. When a gas at a given temperature is reduced in 
volume its molecules necessarily approach each other, and 
their attraction for one another increases. This attraction 
aids the increase of pressure in reducing volume. When a 
certain volume is reached, the attraction has become so marked 
that further reduction of volume is accompanied by fall of 
pressure. If a certain volume be chosen on the descending 
portion of an isothermal, a state of balance may be imagined 
where pressure and cohesion unite in maintaining the volume 
constant against the kinetic energy of the molecules, tending 
to cause expansion. 

The conception of negative pressure, or tension, is that at 
low temperatures and small volumes the cohesion is such that, 
‘In order to overcome ‘it and increase volume, it would be 
necessary to apply tension to each molecule. But after the 
lowest pressure or greatest tension has been attained, the 
actual size of the molecules presents a bar to closer approach; 
and to cause further decrease of volume pressure must again 


* The reasoning of a recent paper by Wroblewski (Monatsheft der 
Chemie, Wien, July 1885, p. 383) rests on the assumption that such con- 
ditions are inconceivable. He supposes lines of equal density to be curves, 
and on their close approach to the vapour-pressure curve to run parallel 
with it. His conclusions are therefore not borne out by experimental 


facts. 
20a ge 


452 Drs. Ramsay and Young on 


be applied. It is not to be supposed that at any given volume 
only one of these factors is operative ; the actual size of the 
molecules exerts its influence even at large volumes, and the 
cohesion does not disappear, but no doubt immensely increases, 
as the volume is reduced, even when that reduction requires 
rise of pressure Still, a mental picture of the process may, 
we think, best be attained by directing attention to cohesion, 
when volume is being decreased with fall of pressure, and to 
the influence of the actual size of the molecules when volume 
is small. 

When a liquid is converted into gas, heat is absorbed, or 
work is done on the liquid. We have previously (loc. cit.) 
given tables showing the heats of vaporization of ether at 
various temperatures. Our experiments have confirmed the 
prediction that the heat of vaporization of stable liquids 
decreases with rise of temperature, and in all probability 
becomes zero at the critical temperature. Now the volume of 
a fluid may be changed, either keeping the pressure constant 
or allowing it to vary during the operation ; but if the initial 
pressure and final pressure are the same, the variation of 
pressure during the operation does not affect the total work 
done. A liquid may be changed into saturated vapour at any 
given temperature in the usual manner, when the intermediate 
states are represented by non-homogeneous mixtures of liquid 
and saturated vapour. ‘The area enclosed between the vapour- 
pressure line and lines drawn vertically from its terminal points, 
cut by the line of zero pressure (or pressure xchange of 
volume), represents graphically the external work performed 
in evaporating a liquid. If, however, the change of condition 
be not abrupt, but continuous, the area enclosed by the iso- 
thermal below the vapour-pressure line must be equal to that 
above the vapour-pressure line (see Plate VII.). If this were 
not the case, the amount of work required to effect the con- 
tinuous change would differ from that required for the abrupt 
change of state. 

Now it is evident that a slight alteration in the position of 
the vapour-pressure line would have great influence on the 
relative areas enclosed by the isothermal above and below the 
vapour-pressure line ; and it may also be seen that, when these 
areas are rendered equal by a horizontal line, the position of 
that horizontal line must represent the true vapour-pressure. 

We have determined the position of the horizontal line in 
the following manner :— 

Knowing approximately the position of the vapour-pressure 
line at a given temperature, three pressures were chosen—the 
highest above and the lowest below the experimentally deter- 


Evaporation and Dissociation. 453 


mined vapour-pressure ; and by means of a planimeter (by 
Stanley of Holborn) the areas enclosed between each hori- 
zontal line and the curves respectively below and above it, 
formed by the isothermal lines, were measured. To ascertain 
what position of the horizontal line would render these areas 
equal, the values of each set of three areas were mapped on 
sectional paper as abscissz, the pressures corresponding to the 
position of the horizontal lines being ordinates. The curves 
passing through the resulting points cut each other at a point 
which represents the true pressure and the true area. This 
method is rendered clearer by inspection of the following 
figure :— 


20,400 
Pf ime ST) ON Po ft 
pf Nf euuged 
HA NE ff 
LN Bees ene Des a eae 


TN ERE SERRE ARE RE SRSA 
at aS J eee ef 
See ERE 


It is evident that these vapour-pressures ey Lae on 
measurements represented in the diagram outside the area 
bounded by the curve representing orthobasic volumes of gas 
and liquid. It will be seen, on reference to the table on p. 444, 
that the agreement between these calculated vapour-pressures 
and those experimentally determined is a very close one, the 
greatest difference being about 1 per cent. This agreement 
between experimentally observed vapour-pressures and those 
depending on the formula p=bt—a is very remarkable, and 
it is difficult to believe that, if the isochoric lines were curves, 
such an agreement could exist. 

What are usually termed vapour-pressures, then, are those 
pressures at which horizontal lines drawn through them render 
the areas enclosed by the isothermal lines below the horizontal 
lines equal to those above them. But there are other two 
conditions of matter, each of which has its characteristic 
pressures. One of these is represented by the highest pressure 
attainable on any isothermal, or the summit of the curve above 
the vapour-pressure line ; and the other the apex of the curve 
below the vapour-pressure line. Hach temperature chosen has 


454 Drs. Ramsay and Young on 


its particular value for each of these conditions ; and it is 
evident that the relations between the temperatures and 
pressures corresponding to the inferior or reversed apices, as 
well as those corresponding to the superior apices, would each 
form a special curve. 

The following Table gives the final results of the calculation 
of vapour-pressures by the method of areas; and, for the 
sake of comparison, the actually found vapour-pressures are 
appended. ‘The pressures at the superior and inferior apices 
of the isothermal curves, and also the enclosed areas, are 
given”. 


TABLE VIII. 


Vapour- Vapour- P ep ese e ee Area 
Tempe-| pressures, pressures, motes. sa are, | above or below 
rature. deduced mean of superior | mrerior  |yapour-pressure 
from areas. observed. aie a line. 
- millim. millim. | millim. millim. sq. in. 
192 26350 26331 26490 26125 0-0425 
190 25554 25513 25870 24960 0°1245 
Old, 23623 r ; Bae 
185 | 23708 {| Noy, d3yeg }| 24510 | 21660 04550 
i a Old, 20189 ) ‘ 
ws | 20259 {| Nov Snort t| 22100 | 14060 1-6520 
160 15900 15778 19090 |— 20 4-710 
150 13405 13262 17380 | —10400 7551 


The three pressure-curves—which we shall name the “ ordi- 
nary’ vapour-pressure curve, the “ superior’’ vapour-pressure 
curve, formed by the superior apices of the isothermal lines, 
and the “ inferior’? vapour-pressure curve, produced by 
the lower apices of the isothermal lines—must, it is evident, 
meet at the critical point; and on mapping them, it was 
found that this was the case. Points were chosen on these 
curves at equal intervals of temperature, and the constants 
for formule of the type logpy=a+ba‘ were calculated for 
each. As the pressures on the inferior curve below a certain 
temperature were negative, it was found convenient to add 
30,000 millim. to each, which was subsequently subtracted 
from the result. The constants for the curves are— 


* The areas are in square inches; the scale was 2000 millim. and 
2 cub. centim. per gram to the inch. It would be easy, if necessary, to 
conyert these data into actual work. 


ee 


Evaporation and Dissociation. 455 
Superior curve : . 
a=3°59797 ; log b=1°8343195 ; log «a =0:00257762. 
Ordinary vapour-pressure : 
a=6°72909 ; log b=0°4027232 ; log «2 =1:99876897 


(b is here negative). 
Inferior curve : 


a=4:867404 ; log b=1:5913793 ; log a=1-98382413 
(b is again negative). 
In each case t=%° Cent. —160°. 


The results are given in the following Table :— 


TABLE IX. 
Ordinary Superior Inferior 
Tempe- vapour-pressures. curye-pressures. curve-pressures. 
rature. | Ss 
Read. Calculated.| Read. Calculated. | Read. Calculated. 
millim. millim. millim. millim. millim. millim. 
150 13405 13084 17380 17437 | —10400 —10185 
160 15900 15900 19090 19090 — 20 — 100 
Lis 20259 20259 22100 22100 +14060 +14060 
185 23703 23678 24510 24549 21660 21704 
190 25554 25556 25870 25935 24960 24900 
192 26350 26341 26490 26523 26125 26065 
ia! ZOO, WEN Mace 2OSZH) Se 1) 8a 26624 
13331 DAO Dikaale py d Se seSe PAROLE rata sks 27077 


With exception of the lowest temperature, the agree- 
ment between the read and the calculated pressures is close. 
The extrapolation amounts to only 3°°83. The agreement 
is close at 192°, and above that temperature the extrapolation 
is only 1°83. It will be seen that at that temperature 
(193°'83) the pressures coincide. The apparent critical point 
was 193°8, 


Isochoric Lines. 


Plate IX. represents the whole of the isochors which we 
have calculated between the volumes of 1°85 and ,300 cub. 
centim. per gram. 

If the gas followed Boyle’s and Gay-Lussac’s laws abso- 
lutely, under all conditions, the isochoric lines would all radiate 
from zero pressure, and would become more and more vertical 
as the volumes decreased ; and the tangents of the angles 
formed by these lines with the horizontal line of zero pressure 


456 Drs. Ramsay and Young on 


would be proportional to cin the equation p=ct, where c varies 
inversely as the volume. But our equation, p=bt—a, intro- 
duces another term, a, which is negative. These values of a 
are represented on the diagram by the extremities of the 
isochoric lines, where they cut the vertical line representing 
absolute zero of temperature. The tangents of the angles 
made by these lines with a horizontal line are proportional to 
the values of b in our equation. 

On referring to Plate IX. it will be noticed that, beginning 
at the largest volume, two adjacent isochors cut each other at 
a point, as regards pressure and temperature, not far above 
zero. With decreasing volumes the points of intersection of 
adjacent isochors occur at higher and rapidly increasing tem- 
peratures and slowly increasing pressures ; and this proceeds 
until the critical volume is reached. With still smaller volumes, 
however, the points of intersection of adjacent isochors oceur 
at lower and decreasing temperatures and pressures ; the ~ 
former decrease slowly, but the latter with great rapidity, and 
soon extend into the region of negative pressures. 

It is evident from the diagram that each isochor between 
the largest and the critical volume is the tangent of a curve, 
representing the relations of pressure to temperature ; while 
the isochors below the critical volume are tangents to another 
curve, also exhibiting the like relations. Neither of these 
curves is identical with the vapour-pressure curve, which falls 
in the area between them. | ais 

Tt will be noticed that, in the area included between the 
line of zero pressure and these two curves, each isochoric line 
is cut by two others at every point along its whole length; 
but outside this surface, and above the line of zero pressure, 
no two lines cut each other, and below the line of zero pressure 
each isochor is cut at each point by one other. The physical 
meaning of the fact that within the first-mentioned region 
three isochors intersect each other at one point is, that a gram 
of the substance may occupy three different volumes at the 
same temperature and pressure. Now, on referring to the 
diagram on Plate VII., representing the experimentally un- 
realizable portions of the isothermal curves, it is evident that 
on each isothermal line, at pressures limited by the superior 
or inferior apices of the isothermal, there are, corresponding 
to each pressure, three volumes. At any pressure above or 
below these pressures the isothermal line is cut only once, by a 
horizontal line of equal pressure ; so that, for each pressure, 
there is only one corresponding volume. At each apex a 
horizontal line of equal pressure cuts the isothermal line 


Evaporation and Dissociation. 457 


at one point, and is also a tangent to the apex. There are, 
therefore, two volumes corresponding to each of these pres- 
sures. Since no gas can be submitted to a negative pressure, 
those portions of an isothermal line representing the truly 
gaseous condition of matter never extend below the horizontal 
line of zero pressure ; only those portions of the isothermal 
which proceed towards the inferior apex fall below this line. 
An isothermal line below zero pressure is therefore cut only 
twice by a line of equal pressure, and there are therefore two 
volumes corresponding to each pressure. At each inferior 
apex, however, the horizontal line is a tangent to the curve, 
and there is therefore only one volume corresponding to a 
given pressure. 

On referring back to Plate IX., it will be seen that the 
pressures corresponding to the superior apices of each iso- 
thermal line, when mapped, produce the curve AC; and 
those corresponding to the inferior apices, the curve BC. 
The surface bounded by these curves and the line of zero 
pressure corresponds to portions of the isothermal lines, 
including pressures between the two apices, and each point in 
the surface is the locus of intersection of three isochoric lines. 
Below the line of zero pressure the isochoric lines cor- 
responding to the gaseous state are absent ; and hence each 
point is the locus of intersection of only two isochors. The 
isothermal lines above and below the limits of pressure given 
by the apices are cut only once by any line of equal pressure; 
hence the isochors outside the area ACD, and above the line 
of zero pressure, do not intersect. The apex C of the curvi- 
lateral triangle ACD is the point of highest temperature and 
pressure at which intersection can take place, and therefore 
represents the critical point ; it is also the common point of 
intersection of the three pressure-temperature curves. 

Referring now to Plate VIII., in which the isochoric lines 
in the neighbourhood of the critical point are shown on a 
larger scale, it will be seen that the isochoric lines above 
a volume not far removed from 4 cub. centim. per gram cut 
the ordinary vapour-pressure curve CH on one side, while 
those below the volume 3°75 evidently cut the vapour-pressure 
line on its other side. There must therefore be an isochoric 
line which does not cut the curve at all, but forms a tangent 
to its end-point. That isochor gives the critical volume. It 


may be determined by calculating the value of s at the 


critical temperature. This value of 2 is identical with 


458 On Evaporation and Dissociation. 


the value of } in our equation p=bt—a at the critical volume. 
Until a mathematical expression is discovered, representing b 
as a function of volume, the only means at our disposal for 
ascertaining the true volume corresponding to 0 is by inter- 
polation of the original curve by which the values of 6 were 
smoothed. The common point of intersection of the three 
pressure-temperature curves has been shown on p. 450 to lie 
at the temperature 193°°83. The value of a on the vapour- 
pressure curve at this temperature, calculated by the formula 
of which the constants have already been given, is 405 millim., 
which is also the value of } at that temperature. The volume 
corresponding to this value is 4°06 cub. centim. per gram ; 
and the specific gravity of ether at its critical point is there- 
fore 0:2463. | 

Unfortunately, Dr. Andrews’s measurements of the constants 
of carbon dioxide are not sufficiently numerous to warrant an 
attempt to obtain the critical temperature, pressure, and volume 
by this method. The critical volume of carbon dioxide is 
evidently less than 3 cub. centim. per gram ; but the values 
of b below that quantity are unascertainable. It may be 
noticed that the curves below volume 3 are inserted in broken 
lines, showing a probable course ; but no reading from them 
would be permissible. 

The two liquids, ether and carbon dioxide, have no chemical 
analogy with one another ; and we therefore feel justified in 
concluding that the law which is the subject of this paper is 
generally applicable to all stable substances. We have, how- 
ever, other less complete data for methyl and ethyl alcohols, 
which, so far as they go, are confirmatory of the results 
described. We have also data available for the examination 
of acetic acid—a substance which differs from those men- 
tioned, inasmuch as it undergoes dissociation when heated ; 
and we hope shortly to be able to communicate an account of 
its behaviour. | 

Professor Fitzgerald, to whom we gave a short account of 
this law, has recently communicated to the Royal Society a 
paper in which its thermodynamical bearings are considered. 


Bristol, 12th February, 1887. | 


[ 459 J 


LII. On the Stability of Steady and of Periodic* Fluid Motion. 
By Sir Witi1am THomson ft. 


1. FAXHE fluid will be taken as incompressible; but the 
results will generally be applicable to the motion of 

natural liquids and of air or other gases when the velocity is 
everywhere small in comparison with the velocity of sound in 
the particular fluid considered. I shall first suppose the fluid 
to be inviscid. The results obtained on this supposition will 
help in an investigation of effects of viscosity which will follow. 
2. I shall suppose the fluid completely enclosed in a con- 
taining vessel, which may be either rigid, or plastic so that 
we may at pleasure mould it to any shape, or of naturai solid 
material and therefore viscously elastic (that is to say, return- 
ing always to the same shape and size when time is allowed, 
but resisting all deformations with a force depending on the 
speed of the change, superimposed upon a force of quasi- 
perfect elasticity). The whole mass of containing-vessel and 


* By steady motion of a system (whether a set of material points, or a 
rigid body, or a fluid mass, or a set of solids, or portions of fluid, or a 
system composed of a set of solids or portions of fluid, or of portions of 
solid and fluid), I mean motion which at any and every time is precisely 
similar to what itis at one time. By periodic motion I mean motion 
which is perfectly similar, at all instants of time differing by a certain 
interval called the period. 

Example 1. Every possible adynamic motion of a free rigid body, 
having two of its principal moments of inertia equal, is steady. So also 
is that of a solid of revolution filled with irrotational inviscid incompres- 
sible fluid. 

Example 2. The adynamic motion of a solid of revolution filled with 
homogeneously rotating inviscid incompressible fluid is essentially periodic, 
and is steady only in particular cases. 

Example 3. The adynamic motion of a free rigid body with three un- 
equal principal moments of inertia is essentially periodic, and is only 
steady in the particular case of rotation round one or other of the three 
principal axes; so also, and according to the same law, is the motion ofa 
rigid body having a hollow or hollows filled with irrotational inviscid 
incompressible fluid, with the three virtual moments of inertia unequal. 

Example 4. The adynamic motion of a hollow rigid body filled with 
rotationally moving fluid is essentially unsteady and non-periodic, except 
in particular cases. Even in the case of an ellipsoidal hollow and homo- 
geneous molecular rotation the motion is non-periodic. The motion, 
whether rotational or irrotational, of fluid in an ellipsoidal hollow is fully 
investigated in a paper under this title published in the Proceedings of 
the Royal Society of Edinburgh for December 7, 1885. Among other 
results it was proved that the rotation, if initially given homogeneous, 
remains homogeneous, provided the figure of the hollow be never at any 
time deformed from being exactly ellipsoidal. 

+ Communicated by the Author, having been read before the Royal 
Society of Edinburgh on April 18, 1887. 


: 


460 Sir William Thomson on the Stability of 


fluid will sometimes be considered as absolutely free in space 
undisturbed by gravity or other force; and sometimes we 
shall suppose it to be held absolutely fixed. But more fre- 
quently we may suppose it to be held by solid supports of 
real, and therefore viscously elastic, material ; so that it will 
be fixed only in the same sense as a real three-legged table 
resting on the ground is fixed. The fundamental philoso- 
phic question, What is fixity? is of paramount importance 
in our present subject. Directional fixedness is explained in 
Thomson and Tait’s ‘ Natural Philosophy,’ 2nd edition, Part I. 
§ 249, and more fully discussed by Prof. James Thomson in 
a paper “On the Law of Inertia, the Principle of Chrono- 
metry, and the Principle of Absolute Clinural Rest and of 
Absolute Rotation.”’ For our present purpose we shall cut 
the matter short by assuming our platform, the earth or the 
floor of our room, to be absolutely fixed in space. 

3. The object of the present communication, so far as it 
relates to inviscid fluid, is to prove and to illustrate the proof 
of the three following propositions regarding a mass of fluid 
given with any rotation in any part of it :— 

(I.) The energy of the whole motion may be infinitely in- 
creased by doing work in a certain systematic manner on the 
containing-vessel and bringing it ultimately to rest. 

(II.) If the containing-vessel be simply continuous and be 
of natural viscously elastic material, the fluid given moving 
within it will come of itself to rest. 

(III.) If the containing-vessel be complexly continuous and 
be of natural viscously elastic material, the fluid will lose 
energy ; not to zero, however, but to a determinate condition 
of irrotational circulation with a determinate cyclic constant 
for each circuit through it. 

4. To prove 3 (I.) remark, first, that mere distortion of the 
fluid, by changing the shape of the boundary, can increase 
the kinetic energy indefinitely. For simplicity, suppose a 
finite or an infinitely great change of shape of the containing- 
vessel to be made in an infinitely short time; this will distort 
the internal fluid precisely as it would have done if the fluid 
had been given at rest, and thus, by Helmholtz’s laws of vor- 
tex motion, we can calculate, from the initial state of motion 
supposed known, the molecular rotation of every part of the 
fluid, after the change. For example, let the shape of the 
containing-vessel be altered by homogeneous strain ; that is 
to say, dilated uniformly in one, or in each of two, directions, 
and contracted uniformly in the other direction or directions, 
of three at right angles to one another. The liquid will be 
homogeneously deformed throughout ; the axis of molecular 


Steady und of Periodic Fluid Motion. 461 


rotation in each part will change in direction so as to keep 
along the changing direction of the same line of fluid par- 
ticles ; and its magnitude will change in inverse simple pro- 
portion to the distance between two particles in the line of the 
axis. 

5. But, now, to simplify subsequent operations to the utmost, 
suppose that anyhow, by quick motion or by slow motion, the 
containing-vessel be changed to a circular cylinder with per- 
forated diaphragm and two pistons, as shown in fig. 1. In 
the present circumstances the motion of the liquid may be 
supposed to have any degree of complexity of molecular rota- 
tion throughout. It might chance to have no moment of 
momentum round the axis of the cylinder, but we shall sup- 
pose this not to be the case. If it did chance to be the case 
(which could be discovered by external tests), a motion of the 
cylinder, round a diameter, to a fresh position of rest would 
leave it with moment of momentum of the internal fluid round 
the axis of the cylinder. Without further preface, however, 
we shall suppose the cylinder to be given, with the pistons as 
in fig. 1, containing fluid in an exceedingly irregular state of 
motion, but with a given moment of momentum M round the 
axis of the cylinder. The cylinder itself is to be held absolutely 
fixed, and therefore whatever we do to the pistons we cannot 
alter the whole moment of momentum of the fluid round the 
axis of the cylinder. 

6. Suppose, now, the piston A to be temporarily fixed in 
its middle position CC, and the 
whole containing-vessel of cylinder 
and pistons to be mounted on a 
frictionless pivot, soas to be free to 
turn round A A’ the axis of the 
cylinder. If the vessel be of ideally 
rigid material, and if its inner sur- 
face be an exact figure of revolu- 
tion, it will, though left free to turn, 
remain at rest, because the pressure 
of the fluid on it is everywhere in 
plane with the axis. But now, in- 
stead of being ideally rigid, let the 
vessel be of natural viscous-elastic 
solid material. The unsteadiness 
of the internal fluid motion will cause deformations of the 
containing-solid with loss of energy, and the result finally 
approximated to more and more nearly as time advances is 
necessarily the one determinate condition of minimum energy 
with the given moment of momentum; which, as is well 


Figs 2, 


eno 3 aes 


462 Sir William Thomson on the Stability of 


known and easily proved, is the condition of solid and fluid 
rotating with equal angular velocity. If the stiffness of the 
containing-vessel be small enough and its viscosity great 
enough, it is easily seen that this final condition will be 
closely approximated to in a very moderate number of times 
the period of rotation in the final condition. Still we must 
wait an infinite time before we can find a perfect approxima- 
tion to this condition reached from our highly complex or 
irregular initial motion. We shall now, therefore, cut the 
affair short by simply supposing the fluid to be given rotating 
with uniform angular velocity, like a solid within the con- 
taining-vessel, a true figure of revolution, which we shall now 
again consider as absolutely rigid, and consisting of cylinder 
with perforated diaphragm and two movable pistons, as repre- 
sented in fig. 1. : 

7. Give A a sudden pull or push and leave it to itself; 
it will move a short distance in the direction of the impulse 
and then spring back*. Keep alternately pulling and push- 


* The subject of this statement receives an interesting experimental 
illustration in the following passage, extracted from the Proceedings of the 


Fig. 2. 


Royal Institution of Great Britain for March 4, 1881; being an abstract 
of a Friday-evening discourse on “ Elasticity viewed as possibly a Mode of 


Steady and of Periodic Fluid Motion. 463 


ing it always in the direction of its motion. It will not 
thus be brought into a state of increasing oscillation, but the 
work done upon it will be spent in augmenting the energy of 
the fluid motion: so that if, after a great number of to-and- 
fro motions of the piston with some work done on it during 
each of them, the piston is once more brought to rest, the 
energy of the fluid motion will be greater than in the begin- 
ning, when it was rotating homogeneously like a solid. It 
has still exactly the same moment of momentum and the same 
vorticity* in every part; and the motion is symmetrical 
round the axis of the cylinder. Hence it is easily seen that 
the greater energy implies the axial region of the fluid being 
stretched axially, and so acquiring angular velocity greater 
than the original angular velocity of the whole fluid mass. 

8. The accompanying diagram (fig. 3) represents an easily 
performed experimental illustration, in which rotating water 
is churned by quick up-and-down movement of a disk carried 
on a vertical rod guided to move along the axis of the con- 
taining-vessel which is attached to a rotating vertical shaft. 
The kind of churning motion thus produced is very different 
from that produced by the perforated diaphragm ; but the 
ultimate result is so far similar, that the statement of § 7 is 
equally applicable to the two cases. In the experiment, a 
little air is left under the cork, in the neck of the containing- 
vessel, to allow something to be seen of the motions of the 
water. When the vessel has been kept rotating steadily for 
some time with the churn-disk resting on the bottom, the sur- 
face of the water is seen in the paraboloidal form indicated 
(ideally) by the upper dotted curve (but of course greatly 
distorted by the refraction of the glass). Now, by finger and 
thumb applied to the top of the rod, move smartly up and down 
several times the churn-disk. A hollow vortex (or column of 


Motion,” and now in the press for republication along with other lectures 
and addresses in a volume of the ‘ Nature Series.’ “A little wooden 
ball, which when thrust down under still water jumped up again ina 
moment, remained down as if imbedded in jelly when the water was caused 
to rotate rapidly, and sprang back as if the water had elasticity like that 
of jelly when it was struck by a stiff wire pushed down through the 
a of the cork by which the glass vessel containing the water was 
ed’ 

* The vorticity of an infinitesimal volume dv of fluid is the value of 
dv. w/e, where w is its molecular rotation, and e the ratio of the distance 
between two of its particles in the axis of rotation at the time considered, 
to the distance between the same two particles at a particular time of 
reference. The amount of the vorticity thus defined for any part of a 
moving fluid depends on the time of reference chosen. Helmholtz’s fun- 
damental theorem of vortex motion proves it to be constant throughout 
all time for every small portion of an inviscid fluid. 


464 Stability of Steady and of Periodic Fluid Motion. 


air bounded by water), ending irregularly a little above the disk, 
1s seen to dart down from the neck of the vessel. If, now, the 


Fig. 3. 


a  - CS, 


b 
4 
3 
3 
y 


QW 


churn-disk is held at rest in any position, the ragged lower 
end of the air-tube becomes rounded and drawn up, the free 
surface of the water taking a succession of shapes, like that 
indicated by the lower dotted curve, until after a few seconds 
(or about a quarter of a minute) it becomes steady in the 
paraboloidal shape indicated by the upper dotted curve. 

9. We have supposed the piston brought to rest after having 
done work upon the fluid during a vast but finite number of 
to-and-fro motions. But if left to itself it will not remain at 
rest ; it will get into a state of irregular oscillation, due to 
superposition of oscillations of the fluid according to an infi- 
nite number of fundamental modes, of the kind investigated 
in my article “ Vibrations of a Columnar Vortex,” Proc. Roy. 
Soc. Hdinb., March 1, 1880, but not, as there, limited to being 
infinitesimal! If the motion of the piston be viscously resisted 
these vibrations will be gradually calmed down ; and if time 
enough is allowed, the whole energy that has been imparted 
to the liquid by the work done on the pistons will be lost, and 
it will again be rotating uniformly like a solid, as it was in 
the beginning. 

[To be continued. ] 


, 465 J 


LIU. Notices respecting New Books. 


A Treatise on Algebra. By Profs. OLIVER, Wart, and Jones. 
(Ithaca, N. Y.: Dudley Finch, 1887; pp. viii+412.) 


HIS is not an Elementary Textbook, and so is not a work for 
ordinary school-use. It is a work very much of the same 

high character as that by Prof. Chrystal which we had occasion 
lately to notice in these columns, and, like it, this also is only a first 
volume. With points of similarity there are numerous points of 
dissimilarity. The motto of both is “Thorough.” Our present 
Authors—an unusual combination, a triple chord—‘‘assume no 
previous knowledge of Algebra, but lay down the primary definitions 
and axioms, and, building on these, develop the elementary principles 
in logical order; add such simple illustrations as shall make 
familiar these principles and their uses.” Then as to form: 
“ Make clear and precise definition of every word and symbol used 
in a technical sense; make formal statement of every general 
principle, and, if not an axiom, prove it rigorously; make formal 
statement of every general problem, and give a rule for its solution, 
with reasons, examples, and checks; add such notes as shall 
indicate motives, point out best arrangements, make clear special 
cases, and suggest extensions and new uses.” It will be gathered 
from this outlme, and our Authors, we think, have kept close to 
this chart, that here is about the same departure from ordinary 
textbooks as in the case we have referred to above. Indeed, to 
our mind we have almost too much logic and careful detail, but for 
college students and mathematical teachers this elaboration is of 
great service. Indeed the book has been written for the classes 
which have been and are under the authors’ training. They them- 
selves admit that the Work has so grown under their hands as to 
embrace many topics quite beyond the range of ordinary college 
instruction. The book fulfils their desire that it should be a 
stepping-stone to the higher analysis. Having indicated the 
nature of the work we give now some of the matters discussed in 
the twelve chapters. ‘The first is on primary definitions and signs ; 
the second is on primary operations (a valuable chapter); the third 
on Measures, Multiples, and Factors ; the fourth on Permutations 
and Combinations ; the fifth on Powers and Roots of Polynomials ; 
the sixth on Continued Fractions; the seventh on Incom- 
mensurables, Limits, Infinitesimals, and Derivatives ; the eighth 
on Powers and Roots; the ninth on Logarithms; the tenth on 
Imaginaries (with graphic representation and preparation for 
Quaternions); the eleventh on Equations (Bezout’s method, 
graphic representation of quadratic equations, application of 
continued fractions to the same class of equations, maxima and 
minima); and the last on Series (the elementary ones, convergence 
and divergence, indeterminate coefficients, finite differences, inter- 
polation, Taylor’s theorem, and the computation of logarithms). 
We have come across much that is new to us and much of interest. 


Phil. Mag. 8. 5. Vol. 23, No. 144. May 1887. 21 


“ > ' See 23 7 es SAID Ne Pm aoe & 2 
5 ETE RE A RE A 


466 Geological Society :— 


The work requires rather close reading in parts, and the arrange- 
ment of the text, too crowded, militates in our opinion against an 
enjoyable perusal of the text. But our view on these points must 
go for what itis worth. The appearance of the work externally 
and the type and apparently great accuracy in printing are all Al. 
In an extra volume the Authors promise to treat of theory of 
equations, integer analysis, symbolic methods, determinants and 
groups, probabilities, and insurance, with a full index. Examples 
accompany the text and conclude each chapter. 


LIV. Proceedings of Learned Societies. 


GEOLOGICAL SOCIETY. 
[Continued from p. 222.] 


January 26, 1887.—Prof. J. W. Judd, F.R.S., President, 
in the Chair. 


pee following communications were read :— 
1. “On the Correlation of the Upper Jurassic Recs of the Jura 
with those of England.” By Thomas Roberts, Esq., M.A., F.G.S. 
The author described at length his observations on the rocks of 
the Jurassic system, from the Callovian to the Purbeckian inclusive, 
first in the Canton of Berne and then in the more southerly Cantons 
of Neuchatel and Vaud. The sections in the former differed ma- 
terially from those in the latter, and the following stages and sub- 
stages were observed :— 


Nort District. Souru District. 


Purbeckian. Purbeckian. 
Portlandian. Portlandian. 
Virgulian. 

Pterocerian. Pterocerian. 
Astartian. 


Astartian. 

Calcaire a Nérinées. —— 

Corallian. Oolithe Corallienne. ee 
Terrain 4 chailles siliceux. Corallian. 


Oxford Terrain 4 chailles marno-caleaire. Pholadomian. Oxford; 
xoraian-*| Calcaire 4 Seyphies inférieur. Spongitian. On re 

Callovj Le fer sous-Oxfordien. Supérieur. Gales, 
aNovian. | Zone of Amm. macrocephalus. Inférieur. a 


Dalle naerée, &c. Dalle nacrée. 


Bathonian. { 


Some of the lithological and paleontological differences between 
these rocks and the English Oolites were noticed, and the views of 
Oppel, Marcou, Waagen, Blake, and Renevier, as to the relations of 
the beds in the two countries, were commented upon. The Author 
then proceeded to compare the fossils of the Swiss Jurassic beds 
with those of their English representatives, stage by ane and 
finally suggested the following correlation :— 


Upper Jurassic Rocks of the Jura and England. 


ENGLAND. 


Upper. 


PURBECK: Middle. 


Lower. 


Portland stone. 
sand, &e. 


72 


Upper Kimeridge Clay. 


Clays with Exogyra virgula. 


re 3 Ammonites alternans. 


LOwER 


KIMERIDGE. 
Clays with Astarte supracorallina. 


“ Ostrea deltoidea. 


Kimeridge Passage-beds. 


9? 


467 


SWISS JURA. 


| Valangien. 


Purbeckien. 
| 


| 
| 
} 


Porilandien. 


Ptérocérien. 


Astartien. 


{ Supracoralline. 


Coral Rag. 


| Calcaire & Nérinées. 


s 


Oolithe Corallienne. 


Coralline Oolite. 


CoRALLIAN. { Terrain 4 chailles siliceux. | 


Middle Caleareous Grit. 


aa SS 
Hambleton Oolite. Ee Se eee aa Pholadomien. } ben Cee 
\ Lower Calcareous Grit. SELES 


eee | 


Clays with cordatt Ammonites. | Le fer sous-Oxfordien. 


ornati Ammonites. 


Ee j 
at i | 


iZone of Amm. macroce- 
| phalus. 


33 39 


\ Kelloway Rock. 


Cornbrash. | Bathonien. 


2. “The Physical History of the Bagshot Beds of the London 
Basin.” By the Rev. A. Irving, B.Sc., B.A., F.G.S. 

The Author, in reviewing the position taken up by him, attempted 
to estimate the value of such palzontological evidence as exists, and 
insisted on the importance of the physical evidence in the first place. 
He gave reasons for considering the evidence of pebbles, pipe-clay, 
derived materials, irony concretions, percentages of elementary 
carbon (ranging in the more carbonaceous strata up to nearly 23°/.) 
taken together with the evidence of carbon in combination, as ad- 
duced in former papers, freshwater Diatoms (now, perhaps recorded 
for the first time in the Middle and Lower Bagshot), and the micro- 
scopic structure of the sands and clays, as furnishing such a cumu- 
lative proof of the fluviatile and deita origin of the majority of the 
Middle and Lower Bagshot Beds, as can hardly be gainsaid ; while 
he regarded the wide distribution of the Sarsens, taken along with 
the absence of such evidence as is quoted above, as indicating, along 
with the fauna, a much greater areal range formerly of the Upper 
Bagshot than of the strata below them. 


‘NAITIVIOD 


"NUTAOTTIVY 


468 Intelligence and Miscellaneous Articles. 


He referred to the evidence furnished by the Walton section (Q. J. 
G. S. May, 1886), the Brookwood deep well (Geol. Mag. August, 
1886), the contemporaneous denudation of the London Clay (Geol. 
Mag. September, 1886) as affording further support to the view 
which he has advocated; gave six new sections on the northern 
side of the area, showing (1) the attenuation of the Lower Bag- 
shots beneath the Middle Bagshot ciays, (2) the greater development 
of clays towards the margin at the expense of the sands, (3) con- 
temporaneous transverse erosion of the London Clay, (4) cases of 
overlap, (5) the occurrence of massive pebble-beds at nearly the 
same altitude along the northern flank underlying (as at Hast- 
hampstead and Bearwood) Upper Bagshot sands, and resting either 
immediately upon, or in near proximity to, the London Clay; and 
added an account of his observations on the flank of St. Anne’s Hill, 
Chertsey, which he takes to be nothing more than an ancient river- 
valley escarpment, subsequently eroded by rain-water, the hollows 
thus formed having been subsequently filled up and covered over by 
pebbles and other débris of the beds in the higher part of the hill, — 
these assuming the character of ordinary talus material. The con- 
sideration of the southern margin of the Bagshot district is reserved 
for a future paper. 

The Author considered that his main position, resting as it does 
upon physical evidence, remains untouched by the attempt of later 
writers to disprove it; while the disproof breaks down even on 
its own lines (the stratigraphical), the paper in which this dis- 
proof is insisted upon being characterized by (1) an incomplete 
grasp of the problem on the part of its authors, (2) equivocal data, 
(3) omission of important evidence, (4) inconsistencies, (5) erro- 
neous statements. 3 

The Author (while correcting some errors of stratigraphical detail, 
which appeared in his former paper, from insufficiency of data) 
maintained that (though occasional intercalated beds with marine 
fossils may be met with, as is commonly the case in a series of 
delta- and lagoon-deposits) the view he has put forward is, in the 
main, established ; and he proposed the following classification of 
the Bagshot Beds of the London Basin :— 


Old Reading. New Reading. 
1. Upper Bagshot Sands =1. Marine-estuarine Series. 
2. Middle Bagshot Sands 
and Clays =2. Freshwater Series. 
3. Lower Bagshot Sands 


LV. Intelligence and Miscellaneous Articles. 


ON THE INERT SPACH IN CHEMICAL REACTIONS. 
BY OSCAR LIEBREICH. 
A CCORDING to all previous observations, it has been assumed 


that a chemical reaction in liquids which are perfectly mixed 
takes place uniformly and simultaneously in all parts unless cur- 


‘a i 
~ 


Intelligence and Miscellaneous Articles. 469 


rents are produced in consequence of inequalities of temperature. 
In the reduction of copper sulphate by grape-sugar, on heating, the 
suboxide is first perceived in the upper part. We know also that 
in reducing liquids which contain certain metallic salts, the products 
of reduction are deposited on the surfaces opposed to them. It has 
never, however, been observed that in liquids, perfect mixture 
being presupposed, certain parts are withdrawn from the reaction, 
or show some retardation in the change. 

J have succeeded in demonstrating the existence of a space in 
mixtures in which a chemical reaction is not visible. I have called 
it the ¢nert space (todter Raum). In introducing this idea as the 
result of my experiments, I would define it as that space in a 
uniformly mixed liquid in which the reaction occurs either not at 
all, or is retarded, or takes place to a less extent than in the 
principal liquid. 

Reaciion-space and inert space can be most sharply separated 
from each other in the experiments which Ladduce. The occurrence 
of such an inert space is best demonstrated with hydrate of chloral, 
which, when treated with sodium carbonate, decomposes into 
chloroform, according to the following equation : 


C,Cl,0,H, +Na,CO,=CHCl, + NaHCO, +NaHCo,,. 


With a suitable concentration, and mixture in proper equivalents, 
the chloroform separates not in thick oily drops, but as a fine mist 
which gradually collects in drops at the bottom. The reaction 
does not start at once, but depends on concentration and tempera- 
ture. The concentration proper for the observations can be so 
arranged that the commencement of the reaction varies between 1 
and 25 minutes. ‘This time may even be considerably prolonged *. 

If the reaction is made in an ordinary test-tube, there is a space 
of 1 to 3 mm. below the meniscus, which is not affected by the 
reaction ; that is, it remains perfectly clear; and the reaction-space 
is bounded above with the sharpness of a hair, by a surface curved 
in the opposite direction to that of the meniscus. 

The upper space in the liquid which thus remains clear is the 
inert space in the hydrate-of-chloral reaction. 

Even after the tube has been left still for 24 hours this space is 
visible ; for the boundary of the mert space can still be distinctly 
recognized by minute spherules of chloroform which have not sunk. 
Tf the test-tube is gently agitated, so that the chloroform-mist 
passes into the inert space, after a few minutes the chloroform 
settles to its former boundary, and the separation between the 
inert space and the reaction-space is again reproduced. 

Careful observation showed that the clear layer of liquid was 
diminished by the ascent of the chloroform-mist, and was not 
Increased by sinking. 

I have observed the inert space in this reaction in differently 


* I used equal volumes of aqueous solutions of 331 gr. hydrate of 
chloral and 212 gr. sodium carbonate in the litre, which were diluted to 
a corresponding extent, so as to prolong the duration of the reaction. 


470 Intelligence and Miscellaneous Articles. 


shaped vessels. If we take a glass box with parallel sides which 
are ata distance of a centimetre apart, it is seen that the mert 
space presents itself as a surface curved in the opposite direction to 
the meniscus. It can moreover be observed that at the positions of 
greatest curvature, a gradual equalization or a fresh reaction-zone 
is formed. Ii a horizontal glass cylinder closed by parallel glass 
plates is taken, the curvature of the active space is seen in great 
sharpness and beauty. 

If the reaction takes place between two glass plates which are 
inclined to each other at an acute angle so that their line of contact 
is vertical, the height of the meniscus is represented by a deeper 
position of the inert space. 

In capillary tubes which, after being filled, are placed horizontally, 
the inert space is met with on each side. ven if the capillary 
tubes are taken so fine, that the lumen must be examined by a 
magnifying-power of 300 times, the active and the inert space can 
be separately observed. The reaction occurs with separation of 
small molecular drops of chloroform in the middle of the liquid 
cylinder, while it remains clear at each end. With very small 
drops in capillary tubes there is no reaction *. 

If tubes closed at the top are filled with the active mixture so 
that there is no air-bubble, the decomposition is uniform throughout 
the entire liquid. If, however, tubes open atthe top are filled with 
the liquid, and are closed by a small transparent animal membrane ~ 
stretched in a lead frame, it is possible by carefully raising it to 
show here also the inert space. 

If a glass tube open at both ends is placed on a fine membrane, 
and is closed at the top also by a membrane, it is seen that when 
the tube is held vertically an inactive space can be observed below, 
in which the chloroform gradually settles as a cloud. I have not 
been able to ascertain whether the reaction in this case is also 
limited at the sides of the tube. 

If a specimen of the liquid be taken from the inert space by 
means of a capillary tube, and it be warmed, decomposition at once 
sets in. ‘This isa proof that the two substances contain unaltered 
hydrate of chloral and sodium carbonate. It is of course im- 
portant to observe the phenomena of the inert space by other 
reactions which take place slowly. The reaction which takes 
place between iodic and sulphurous acids according to the following 
equations : 

380,+ H1IO,=380,+1H 
51H +HI0,=3H,0+ 61 


was found to be particularly suitable, since it has been found by 
Landolt ? that by suitable dilution, and variation of the quantities, 
it can be delayed at pleasure and in accordance with a definite law. 

The occurrence of the iodine reaction is made manifest by the 


* For this experiment it is necessary to free the liquid from absorbed 
air by boiling. | 
+ Berliner Sttzungsberichte, 1885, xvi., and 1886, x. 


Intelligence and Miscellaneous Articles. 471 


addition of soluble starch, which by the sudden blue coloration 
indicates the liberation of iodine. 

’ Solutions were used containing 0°25 gr. of iodic acid in a litre 
of water, or the same quantity in the litre of a mixture of equal 
parts of glycerine and water. 

The sulphurous acid was used of such concentration that 5 cub. 
cent. of its solution in water just decolorized 2 cub. cent. of a 
one-per-cent. solution of potassium permanganate. 

On mixing 10 cub. cent. of solution of iodic acid with 3 cub. 
cent. of sulphurous acid, the reaction sets in in about 5 minutes, and 
in the various glass vessels shows an inert space above, which lasts 
for a time depending on the temperature. 

The iodine reaction presents a phenomenon to which I shall 
afterwards recur; that is, the occurrence of this reaction in the 
centre of the tube. If a vertical glass tube 4 millim. in the clear 
is filled by aspiration, and subsequent closing by an indiarubber 
tube and clamp, trom the active liquid which is contained in a wide 
glass cylinder, a fine blue thread is seen to form in the tube, while 
the surrounding liquid remains clear and colourless. The blue 
coloration extends gradually from the thread thronghout the entire 
liquid column. 

It could be observed in this phenomenon that the reaction in the 
wider vessel set in sooner than in the narrow tube. 

If either the hydrate-of-chloral or the iodic-acid mixture is placed 
in a vessel in which the liquids can be drawn through fine glass 
beads, no chemical reaction at all is produced. 

It follows thus from these experiments :— 

1. That in liquids the space of chemical action is bounded by an 
inactive zone (the wert space), where the liquid is in contact with 
the air, or is separated from it by a fine membrane. 

2. That the reactions take place more slowly in narrow than in 
wide tubes. 

3. That capillary spaces can entirely suspend chemical reactions. 

As lam engaged in continuing this investigation, I hope soon, 
after a further extension of the experiments and the use of other 
chemical reactions, to be able to report fresh results.—Berliner 
Stizungsberichte, November 4, 1886. 


— 


APPARATUS FOR THE CONDENSATION OF SMOKE BY STATICAL 
ELECTRICITY. BY H. AMAURY. 


A glass cylinder is placed on a tripod perforated in the centre, 
and below it a tin-plate box with an opening in the side and at 
the top, in which touch-paper, tinder, or tobacco can be burned, and 
thus the cylinder be filled with smoke. To the top of the cylinder 
is fitted a small lid in which is a vertical tube. At half the height 
of the cylinder are two diametrically opposite tubuli, through which 
pass metal rods; these are connected with vertical rods parallel to 
the sides and provided with points. If these combs are connected 
with the conductors of an electrical machine, and the latter is 
worked, the smoke is condensed.— Beiblatter der Physil:, No. 2, 1887. 


FPF ee eae ee aan 
Pri iecelels 48 Sa: 


472 Intelligence and Miscellaneous Articles. 


THE HEATING OF THE GLASS OF CONDENSERS BY INTERMITTENT 
ELECTRIFICATION. BY J. BORGMANN. 


The author takes two bundles of 30 cylindrical condensers, each 
consisting of a glass tube 46 cm. in length and 5 mm. in diameter ; 
each tube was coated externally with tinfoil, and filled with copper 
filings, and a copper wire inserted, the ends being closed with 
paratiin or shellac. Hach thirty tubes are formed into bundles, all 
the outsides and insides being severally connected. One bundle was 
also coated on the outside with tinfoil to improve the conductivity. 
These two bundles of condensers were placed respectively in two 
large air-thermometers. Hach reservoir consisted of a glass tube 


- of about 50 cm. length and 4:5 cm. internal diameter, which was 


surrounded by another tube of the same length and 7 cm. diameter. 
Through the brass ends of the reservoirs passed on the one hand 
the electrodes, and on the other the limb of the manometer. The 
manometer filled with naphtha consisted of three limbs, of which 
two were connected with the two reservoirs of the air-thermometer. 

The charging was effected by means of a Kuhmkorff, and was 
measured by a Siemens electrodynamometer. Notwithstanding its 
better external conductivity, the bundle C was more heated than 
the other, A. 

If ¢ is the deflection of the electrodynamometer in divisions of 
the scale, Aa and Ac the displacement of the naphtha in the mano- 
meter in millimetres, which measure the quantities of heat, it was 
found that 

€ 345 280 147 101 je 343 159 
Ac 11:3» 9:84... 4°84 2:9. f, Aa LOS aie 
e/A 30° 284 82:4 348 e¢/A 317 306 

It follows from this that the heatings of the condensers are 
approximately proportional to the square of the difference of 
potential of the coatings.—Beiblatter der Physik, 1887, p. 55. 


ON THE CHEMICAL COMBINATION OF GASHS. 


To the Editors of the Philosophical Magazine and Journal. 


GENTLEMEN, Riga, April 8, 1887. 
In the April Number of the Philosophical Magazine for this 
year Prof. J. J. Thomson complains that [ have misunderstood his 
theory of the Chemical Combination of Gases. After a repeated 
study of the paper, I must confess that Prof. Thomson is in the 
main right. As in my criticism I have done Prof. Thomson an 
injustice which I am not able entirely to repair, I will not dwell 
upon the injustice which he in the heat of his defence has done me 
in his answer, the more so as it has no scientific, but a mere per- 
sonal interest. 
Have the kindness to insert the above explanation in the next 
Number of your Magazine. Yours truly, 
W. Ostwatp. 


THE 
LONDON, EDINBURGH, ann DUBLIN 


PHILOSOPHICAL MAGAZINE 


AND 


JOURNAL OF SCIENCE. 


[FIFTH SERIES.] 
JUNE 1887. 


LVI. The Laws of Motion. By Ropert FRaNKLIN MurrHeEaD, 
B.A., of St. Catharine’s College, Cambridge*. 


Preface. 


HE aim of this Essay is to state in the clearest manner 
possible the best eaisting conception of dynamical 
science. The writer believes that the statement of dyna- 
mical principles here given is to be found implicitly in 
the reasonings of the best modern masters of the science, but 
that it has never hitherto been stated explicitly. The general 
statement indeed has sometimes been made that the proof of 
a hypothesis or theory is its agreement with the facts, or that 
the whole Principia is the proof of the Laws of Motion. 
But I have pointed out in detail that the very conceptions 
and definitions of Dynamics are unintelligible when taken 
singly. I have endeavoured to free the science of Dynamics 
from survivals from its childhood, in the shape of extra- 
kinetic definitions of dynamical concepts, and @ priori 

assumptions. 

The Laws of Motion. 


In view of the enormous development to which the science 
of Dynamics has attained in modern times, of the simplicity 
of its fundamental conceptions, and of the unquestioned 


* Communicated by Professor James Thomson; being the Essay to 
which the second Smith’s Prize was awarded in 1886. 


Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 2K 


474 Mr. R. F. Muirhead on the Laws of Motion. 


validity ot its processes and results, it may appear somewhat 
strange that much difficulty has been found in stating its 
principles in a satisfactory form. 

,- “In the preface to the second edition of Tait and Steele's 
_ ‘Dynamics of a Particle’ we read (referring to the chapter 
on the Laws of Motion) :—“ These five pages, faulty and even 
erroneous as I have since seen them to be, cost me almost as 
much labour and thought as the utterly disproportionate 
remainder of my contributions to the volume; and I cannot 
but ascribe this result in part, at least, to the vicious system 
of the present day, which ignores Newton’s Third Law, dc.” 

And when we read Clerk Maxwell’s notice of the 2nd 
edition of Thomson and Tait’s treatise in ‘ Nature,’ * we feel 
that the reform introduced by Thomson and Tait, in ** return- 
ing to Newton,” still leaves something to be desired. This 
feeling is strengthened when we learn from the late Prof. 
Clifford +, that “‘no mathematician can attach any meaning 
to the language about force, mass, inertia, &c. used in current 
text-books of Mechanics.” 

It will then be worth while to clear up the logic of the 
science, and, if possible, to state the laws of motion in a form 
that shall be free from all ambiguity and confusion. 

Let us cast a brief and partial glance over the history of 
the development of dynamical first principles. 

Though one region of the science of Dynamics, namel 
Statics, was cultivated by the ancients, it was left for Galileo 
Galilei to become the pioneer of dynamical science in its full 
extent. 

‘Before Galileo, the idea of force as something measurable 
was attained to. The causes tending to disturb rest were 
perceived to have a common kind of effect, so that for the 
purposes of Statics they could be represented by the tension 
of cords produced by suspending from them weights of 
determinate magnitude. Galileo paved the way for the intro- 
duction of the kinetic idea of force, %.e. that of the cause of 
the acceleration of the motion of bodies. It is noteworthy, 
however, that he approached the subject from a kinematical 
standpoint. In his ‘ Dialogues,’ he treats of the science of 
“ Local Motion,” not of the science of Force; and in his 
investigations on the motion of Projectiles in that work, his 
aim is to describe the properties of their motion, not to 
speculate on causes. 

Another stage was reached when Newton published the 


* ‘Nature,’ yol. xx. p. 213, ff. 
+ Ibid. vol. xxii. June 10th. 


Mr. R. F. Muirhead on the Laws of Motion. A75 


Principia. The Definitions and Axioms therein propounded 
include all the principles underlying the modern science of 
Dynamics. Subsequent progress has been either in the 
direction of mathematical development or application to special 
problems, or in attempts to improve the form of statement. 

Let us now inquire whether Newton’s scheme of Definitions 
and Axioms is satisfactory. 

We are struck at once by the fact (noticed by many 
writers) that the First Law of Motion is previously stated or 
implied in the Definition of Inertia. This, however, may be 
passed over as a mere awkwardness of arrangement. 

Another defect which has been pointed out by several 
writers, is the absence of any definition of equal times, which 
renders the expression “ uniformiter” in Law I. perfectly 
indefinite. 

Of course the law implies that all bodies unacted on by 
force pass through spaces in any interval of time whatever, 
which are in the same proportion, so that taking any one such 
body as chronometer, the First Law of Newton may be 
affirmed of all the rest. We may, however, object to a form 
of statement which does not directly state, but implies the 
physical fact. 

Again, “uniform rectilinear motion”? has no meaning 
unless with reference to some base of measurement. And 
the Law is not true except with reference to bases of a certain 
type. [For instance, the “fixed stars describe not straight 
lines, but circles, taking the Earth as base of measurement.” 

Newton’s own statement is that the Laws of Motion are to 
be understood with reference to absolute position and absolute 
time. 

The only explanation given of absolute time, is that in 
itself and of its own nature, without reference to anything 
else, it flows uniformly. 

In explanation of the expressions “ absolute motion” and 
“absolute position,” we have the statement that “ Absolute 
and relative motion and rest are distinguished from one 
another by their properties, causes, and effects. Itis a property 
of rest that bodies truly at rest are at rest among themselves, 
but true rest cannot be defined by the relative positions of 
bodies we observe........ The causes by which true 
and relative motion are distinguished from one another are 
the forces impressed on the bodies to produce motion. ‘True 
motion cannot change except by forces impressed. 

“The effects by which absolute motion is distinguished 
from relative are the centrifugal forces of rotation. For 


2K2 


a eS 


— 


Le 


=. oh 


—— 


476 Mr. R. F. Muirhead on the Laws of Motion. 


merely relative rotation these forces are zero ; in true rotation 
they exist in greater or less degree.” * 

Thereafter comes the well-known experiment of the rotating 
vessel of water. 

Now the first criterion helps us only in a negative way, by 
enabling us to deny the attribution of true rest to both of 
two systems when they are moving relatively to each other. 

The second criterion involves reasoning in acircle. Force is 
defined as that which produces change of motion ; hence to 
define unchanged or uniform motion as that which takes place 
when no force acts does not carry us beyond the previous 
definition, and is nugatory. 

The third criterion, taken along with the first, implies a 
physical fact, viz. that when two bodies severally show no 
centrifugal force, they have no rotation relative to one 
another. 

Consider now Law II. It amounts merely to a definition 
of force, specifying how it is to be measured. 

This has been recognized by several writers. Some, how- 
ever, have expanded it into the further assertion that when 
two forces act simultaneously on a body, each produces its 
own effect independently of the other, in accelerating the body’s 
motion. But such a statement is entirely nugatory if we 
keep by the kinetic definition of force. It is then simply an 
identical proposition like “‘ A is A,” as will be seen by sub- 
stituting in the statement “ acceleration of mass ” for ‘ force.” 

We now perceive that even the residuum of meaning which 
remained after our criticism of Law I. and the statements 
regarding Absolute Motion seems to disappear. For we 
were supposed to recognize a body absolutely at rest by the 
absence of centrifugal force. But force is recognizable only 
by its accelerative effect, while the acceleration must be 
reckoned relative to a body absolutely at rest, which rest, 
again, we cannot recognize until we know absolute motions. 
We are thus reasoning in a circle. 

Law III. This law at first sight undoubtedly seems to 
express an experimental fact. We may therefore be sur- 
prised to find that Newton deduces one case of it (viz. that of 
two mutually attracting bodies) from Law I. (see Scholium 
to the Awiomata). 

This seeming paradox arises from the fact that in this 
Scholium Newton makes Law I. apply to a body or system 
of finite size, and not necessarily without rotation. This 
assumes that there is some one point (centre of Inertia) 


* Newton’s Principia, Scholium to the Definttiones. 


Mr. R. F. Muirhead on the Laws of Motion. 477 


whose motion may be taken to represent that of the system, 
which implies that the 3rd Law is true so far as the parts of 
such a system are concerned. Now it seems difficult to draw 
a valid distinction between such a system and any mass- 
system whatever ; in fact it seems quite as legitimate to 
assume that every mass-system has a centre of Inertia. 

But if this assumption were made, then clearly the first 
Law could be deduced from the third in all its generality, 
and vice versa. 

We see that in this respect again Newton’s arrangement. 
is defective. We find that the experimental fact is not 
stated directly, but enplied in the assumption of the existence 
of a mass-centre. In fact, strictly read, Newton’s Definitions 
and Axioms abound in logical circles, nugatory statements, 
and illusory definitions ; and what real meaning they imply 
is not at all explicit. 

The need for the removal of many obscurities which pertain 
to the science of Dynamics as set forth in the Principia of 
Newton, and in the writings of his successors, has been clearly 
perceived by Professor James Thomson. In his paper on the 
“ Law of Inertia, &c.,” * he propounds the following Law of 
Inertia :— 

“For any set of bodies acted on each by any force, a 
Reference-Frame and a Reference Dial-traveller are kine- 
matically possibie, such that relatively to them conjointly the 
motion of the mass-centre of each body undergoes change 
simultaneously with any infinitely short element of the dial- 
traveller progress, or with any element during which the 
force on the body does not alter in direction nor in magnitude, 
which change is proportional to the intensity of the force act- 
ing on that body, and to the simultaneous progress of the 
dial-traveller, and is made in the direction of the force.” 

For explanations of the terms used I refer to the paper 
itself. At the end of this paper we have the assertion : ‘‘ The 
Law of Inertia here enunciated sets forth all the truth which 
is either explicitly stated, or is suggested by the First and 
Second Laws in Sir Isaac Newton’s arrangement.” 

Professor Thomson’s Law is doubtless, so far as order and 
logic are concerned, an immense advance on the Newtonian 
arrangement. Let us inquire whether it can be accepted as 
absolutely satisfactory. 

How are we to measure the “forces” referred to? If 
kinematically, then we are again involved in a logical circle, 
as may be seen by substituting in the Law, for the words 


* Proc, R, S. E, 1883-4, p. 668, 


478 Mr. R. F. Muirhead on the Laws of Motion. 


“force acting on that body” the words ‘‘ rate of change of 
motion of that body,’’ and for the words “ direction of force ” 
the words “ direction of change of motion.”’ And we cannot 
entertain any other measure of force, for reasons which will 
be adduced later on. | 

Again, Prof. Thomson, by not restricting his statement 
to infinitesimal particles, has to assume the existence of 
mass-centres. How is a mass-centre to be defined? We 
shall give reasons later for rejecting any but a kinetic defini- 
tion of mass and mass-centre. But it is impossible to arrive 
at a kinetic definition when we start by assuming a know- 
ledge of the measurement of mass in the Fundamental Law 
of Motion, as is done by Professor Thomson. 

While noting therefore that Professor Thomson has adopted 
the right method of defining chronometry and “true rest,’ 
we cannot accept his Law as a satisfactory statement of the 
fundamental principle of Dynamical science. 

Let us endeavour to frame, after the manner of Professor 
Thomson, a statement which shall be satisfactory. Taking 
the definitions of dial-traveller and reference-frame, aS given 
in the paper referred to, let us proceed thus :— 

Let a material system be conceived divided into an infinite 
number of particles whose greatest linear dimensions are all 
infinitesimal. To each particle let us attribute a certain value 
called its provisional-mass. Let us adopt a reference-frame 
and dial-traveller. Let the acceleration of any particle multi- 
plied into its provisional-mass be called the apparent-force on 
the particle. Then it is possible so to choose the provisional- 
masses, the dial-traveller, and the reference-frame, so that the 
provisional-masses and the apparent-forces shall, within the 
limits of error of observation, have relations expressible by 
the laws of physical science, 7. e. the law of the Indestructi- 
bility of Matter, the law of Hquality of Action and Reac- 
tion the law of Universal Gravitation, the laws of electric, 
magnetic, elastic, and capillary action, &., &e. Such a 
system being chosen, the provisional-masses in it are masses 
and the apparent-forces, forces. The dial-traveller indicates 
“ absolute time,” and the reference-frame is absolutely without 
rotation or acceleration. 

We have thus kinetic definitions of force, mass, absolute 
time-measurement, and of absolute rest so far as that is possible. 

It is evident kinematically that any other reference-frame 
which has no rotation or acceleration relatively to one chosen 
as above would lead to exactly the same results; and that 
this would not be the case if any reference-frame not fulfilling 
this condition were chosen. 


Mr. R. F. Muirhead on the Laws of Motion. 479 


The above statement includes all in the First and Second 
Laws of Newton that can concewwably be tested by experiment or 
observation. 

We observe that Newton’s Third Law appears classed 
along with other laws of physics, and along with that of the 
Indestructibility of Matter, which must be assumed as a 
preliminary to the ordinary statement of Dynamical Laws 
before the measurement of matter has received its definition. 

In our statement of the fundamental principle of Dynamics, 
neither of these Laws is assumed, and it could be modified so 
as to be equally definite and intelligible were they untrue. 

By dealing with infinitesimal particles, we have avoided 
the necessity of assuming a priori the existence of mass- 
centres ; for on the supposition that the angular motion of no 
element is infinite (or, more generally, that there is no finite 
relative acceleration or velocity between the parts of any 
particle), the motion of any point of a particle might be taken 
to represent the motion of that particle. 

To define the expression force acting on a body, used in 
Dynamics, we would require simply to define the centre of 


mass by the usual analytical equations of the type pe 


=m’ 
where the summation extends over all the particles of the body, 
and then to define the mass of the body by =m, and the force 
on the body as that acting on its whole mass supposed con- 
centrated at its centre of mass. 

What would be the meaning of “a force acting on a body 
at a certain point’’? ‘This expression is appropriate only to 
rigid bodies, or at least to such as retain their shape unaltered 
while under consideration. The meaning would be that this 
force, acting on the particle at the point referred to, together 
with the forces between particles determined by the kinema- 
tical conditions of rigidity, are the actual forces on the body. 

One objection might be raised to the fundamental Law of 
Dynamies, as above stated by us ; it seems awkward to imply 
a knowledge of the whole of physical science in stating that 
fundamental principle. 

‘This objection leads us to cast aside Prof. James Thomson’s 
type of statement, and to adopt another, which states exactly 
the same thought in a different form. We shall propound as 
preliminary a science of Abstract Dynamics, which shall be a 
pure science to the same extent as Kinematics is a pure science. 

It is as follows :-— | 

In a dynamical system, each particle is credited with a certain 
mass, and by coordinates with reference to a system of coordinate 
anes its position and motion are determined. When a particle 


a ~ A an oer ms* Oe rs SS ee 4 5 An ~ er: Sy hy Single ae is Oe _ -=. >< 2s >. 
I Oa ST I SERRE NMS anomie eee SS +s 2 4 
~ RSs RS SSS Seat Sas Se —s" me —eSee aoe “3 SRNL SME METS 5 no 
Ss Se Ee Sea SS SS Ee EEE Oe eee 


= ease 


480 Mr. R. F. Muirhead on the Faws of Motion. 


as accelerated, tt is said to have a force acting upon it in the 
direction of the acceleration and of magnitude proportional to 
the acceleration and mass conjointly. 

The system of chronometry is arbitrary, as well as the 
system of coordinate axes. 

The expressions, mass of a body, centre of mass of a body, 
force on a body, and force acting on a body at a point, are 
defined in the same way as before. 

This forms the subject of ‘‘ Abstract Dynamics,” which deals 
only with mental conceptions, and which is a sort of Kine- 
matics, but Kinematics enriched by the conceptions of force 
and mass. 

This being premised, then, in place of Newton’s Definitions 
and his First and Second Laws of Motion, we have the 
Physical Law or Theory that we can so choose the masses to be 
assigned to our material particles, our coordinate axes, and our 
system of chronometry, that the forces may be resolved by the 
parallelogram of forces into such as are expressed by our 


~ Physical Laws. 


Perhaps we should keep more faithfully to the historical 
conception of Dynamics were we to state our Law of Hxperi- 
mental Dynamics as follows :— 

It is possible to choose the masses of the solar system, the 
axis, and the chronometry, so that the masses shall correspond 
with those of Astronomy, and the forces shall be resolvable into 
such as will be eapressed by the Law of Universal Gravitation, 
and conformable to Newton’s 3rd Law of Motion and to the 
Law of the Indestructibility of Matter (Conservation of Mass). 

Then true time, absolute velocity, and mass-measurement 
being defined from this system, there would be the further Law 
of Physics, that the forces on the various particles composing 
the different members of the solar system and others are expres- 
sible by our various Physical Laws or Theories. 

We have now arrived at the conclusion that the attempt to 
state the Laws of Motion by means of a set of detached defini- 
tions and axioms is futile. We have found that Newton’s 
First Law of Motion cannot be stated until we have the con- 
ception of a certain system of reference, whose definition 
involves the knowledge of the First Law, as well as the defini- 
tion of force, &e. We have therefore seen that the Experi- 
mental Principle of Dynamics should be stated as an organic 
theory or hypothesis. We have found it convenient to 
formulate a science of Abstract Dynamics, which is an ex- 
tended Kinematics, depending only on space and time-measure- 
ments, but including the ideas of force and mass (abstract). 

By means of this we can state in a succinct form the 


Mr. R. F. Muirhead on the Laws of Motion. 481 


experimental Law or Hypothesis of Dynamics (applied), 
which enables us to give to time-measurement such a specifi- 
cation that durations of time, as well as other dynamical 
magnitudes, are made to depend ultimately for their measure- 
ment solely on space-measurement and observations of coinci- 
dences in time. 

These conclusions we have arrived at by assuming that only 
kinetic specifications for the measurement of force, mass, and 
tune, and only a kinetic definition of “ true rest” are admissible. 

Before attempting to justify these assumptions, it may be 
expedient to devote a few paragraphs to a general considera- 
tion of the idea of our method. A theory is an attempt to 
dominate our experience ; it is a conception which may enable 
us, with as little expenditure of thought as possible, to 
remember the past and forecast the future. 

The theory of Universal Gravitation is an example of a 
very successful attempt, perfectly successful so far as it has 
been tested. So with the Huclidean Geometry. 

On the other hand, we have theories which have been found 
useful to enable us to dominate one region of experience, 
while they break down in certain directions. The Newtonian 
Hmission Theory of Light is an example. There are others 
which, if they do not break down absolutely, involve the mind 
in difficulties hitherto unsolved; e. g. the ‘ elastic-solid” 
Wave Theory of Light. 

_ Theories which are found to break down when applied to 
their full extent, as well as theories which have not been 
sufficiently tested, are often called ‘ working hypotheses.” 

The only merits or demerits a theory can have arise from 
these two desiderata : (1) it must not be contradicted by any 
part of our experience ; (2) it must be as simple as possible *. 

Thus, for example, consider the two rival theories : (1) 
that the earth has a certain amount of rotation about its 
axis ; (2) that it has norotation. The latter will be found to 
agree perfectly with our experience, provided we assume asa 
new physical law that there is a repulsive force of magnitude 
wr away from the Harth’s axis at every point of space, and 
~ at right angles to the axis 
of the Earth, and to the shortest distance of the point from 


also at every point a force 2 


* Since physical theories form an organic whole, of course these quali- 
ties must be considered with reference to the body of physical theory as 
a whole. Thus, of two theories, one may, taken by itself, be less simple 
than another, and yet be preferred to it, because the whole body of 
physical theory becomes simpler when it is adopted. Sometimes, too, a 
theory may be preferred because it seems to promise better for the future. 


i 


482 Mr. R. F. Muirhead on the Laws of Motion. 


the axis, where o is the angular velocity in the first theory, 
and 7 is the distance of any point from the axis*. But we 
reject the latter theory on account of its greater complexity. 
It is incorrect to say that the one is true and the other false. 

It follows that there is no essential difference between a 
hypothesis and a theory, or what is called a law of nature. 
One may be less exact than another, or less simple, or less 
sufficiently tested, but the difference is one of degree. 

Now there are two opposite methods of stating dynamical 
principles ; the one employing independent definitions of the 
various conceptions, the other that adopted in this Hssay. 
Both, so far as observation has tested them, correspond equally 
to the facts. The question is, then, Which is the simpler? 
Which comprehends the various relations with the least 
expenditure of mental energy ? 

_ According to the former method, force, mass, time measure- 

ment, and “true rest’ would be defined as preliminaries to 
the science of Dynamics, and independently of that science. 
According to the latter, these conceptions are defined by 
means of one Law or Hypothesis. 

Probably to learners unaccustomed to abstract reasoning, 
who do not probe the processes of proof employed to the 
bottom, the former method may be preferable because its 
conceptions are more concrete ; but to one who has mastered 
the essential relations of the subject, the latter will be found 
superior. 

Let us discuss the idea of force. What are the alternatives 
to the kinetic definition of force and force-measurement ? 
We might take some arbitrary standard, such as a spring- 
balance having a graduated scale. ‘This would cbviously have 
the disadvantage of want of permanence, or, to speak more 
accurately, that of liability to invalidate all our other methods 
of reckoning force, by reason of some physical change occur- 
ring in the standard balance. Further, such a method would 
be incapable of accuracy sufficient for many of our physical 
problems, where we deal with forces so small as to be insen- 
sible to our present observing powers on such a standard; 
forces whose magnitude, therefore, we could not define, even 
theoretically. And, besides, any such arbitrary definition of 
force would be contrary to our whole tendency in modern 
science. Suppose, for instance, experiment were to disclose 
that Newton’s Second Law was untrue, the forces being thus 


* We might either suppose these new forces not conformable to the 
law of the equality of Action and Reaction, which would then have to be 
modified ; or we might suppose the reactions to observed actions to exist 
in the fixed stars, and to be beyond our present means of observation. 


Mr. R. F. Muirhead on the Laws of Motion. 483 


measured, should we hesitate between rejecting the law or 
rejecting the method of force-measurement ? And it is certain 
that we cannot find a spring-balance which would render this 
event unlikely to happen. 

A more promising method would be the definition of unit 
of force as the weight of a certain piece of matter at a certain 
place on the Harth’s surface. The force F would then be de- 
fined as being equal to the weight of a body whose mass was 
F times the standard mass. ‘This would involve an inde- 
pendent method of mass-measurement, which we shall con- 
sider later. In treating questions of the secular changes of 
the Earth such a definition would be useless, unless we were 
also to specify the date as well as the place of the weighing 
supposed to be at the base of force-measurement; and this 
could not be brought into connexion with measurements at 
any other date without employing the whole science of Dyna- 
mics, which would thus involve reasoning in a circle. 

A modification of this method would be one in which force- 
measurement would be made to depend on the gravitational 
or astronomical unit of mass, as well as the theory of the 
force of gravitation. But this also would be a system of 
force-measurement, involving for its conception the whole 
science of Dynamics, of which it would not be independent. 

When Statics is treated as a science, independent of Kine- 
tics, force is sometimes left undefined at first, while the mode 
of procedure is as follows:—We are supposed to have a 
certain idea of the nature of force, partly based on the sensa- 
tions we experience when our body forms one of the two 
bodies which exert force on one another”, and starting from 
this, by the aid of &@ priori reasoning the idea of the measure- 
ment of force is evolved. Then, with the help of certain 
physical axioms and constructions (“ transmissibility of force,” 
“superposition of forces in equilibrium,” &c.), the parallelo- 
gram of forces is proved. 

All this has a very artificial character, and would lead us 
to prefer the simpler kinetic conception of force; but still 
further argument is required before we get to Kinetics. The 
“Second Law of Motion” is proved by means of experiments 
which could not be accurately performed, and whose inter- 
pretation generally involves a knowledge of the science whose 
foundations we are laying. Then the proportionality of force 
to mass is thus proved:— 

Suppose two equal masses acted on by equal and parallel 

* The so-called ‘“ sense of force” should be called ‘sense of stregs.”’ 


Our bodies subjected to forces, however great, if the force on each part is 
proportional to its mass and in a common direction, feel nothing. 


484 Mr. R. F. Muirhead on the Laws of Motion. 


forces; they have the same acceleration. Next, suppose they 
form parts of a single body; the acceleration will “ evidently ” 
be the same as before, &e. (Third Law of Motion assumed.) 
Hence accelerations being equal, force varies as mass. 

This method has been discredited of late, chiefly through 
the influence of Thomson and Tait’s ‘ Natural Philosophy,’ so 
that we may omit further discussion upon it. 

It may be remarked, however, that those who have most 
emphatically declared against the statical measure of force do 
not seem to perceive what is logically implied in that course. 
(Cf. Professor Tait’s Lecture on Force.) 

Consider next the idea of mass. 

The definition based on the weight of bodies is open to the 
same objections as the corresponding method in the case of 
force. 

If we define mass by reference to chemical affinity *, or to 
volumetric observations, we in the first place lose the sim- 
plicity of the kinetic method, and secondly we adopt a con- 
ception of mass which is different from the actual conception 
of modern science. This is demonstrated if we ask ourselves: 
Supposing experiment to show a discrepancy between the 
mass as measured kinetically and as measured otherwise, 
which method should we call inexact? If the former, Kine- 
tics could no longer be considered an exact science. 

Consider next the reference system, and the idea of true rest. 
The most obvious arbitrary definition of the system to which 
the motions of bodies in Dynamics are to be referred is to look 
on the centre of gravity of the Solar system as the fixed 
point, and the directions of certain fixed stars as fixed direc- 
tions. The objections are, first, this would be a very incon- 
venient system in discussing the cosmical Dynamics ; second, 
it is not the actual conception of the science of the present day. 
If one of the stars chosen were found to have a motion com- 
pared with the average position of neighbouring stars, we 
should certainly conclude that its direction was not “ fixed ”’ 
in the dynamical sense. 

It has been suggested to take as a fixed direction that of 
the perpendicular to the ‘‘ invariable plane of the Solar system.” 
This really is not an independent definition, and is open to 
the objections we previously urged against such, when isolated 
from the fundamental law of Hxperimental Dynamics. 

The foregoing methods have been well criticised by 
Streintz t, who propounds in their stead a method of re- 


* See Maxwell’s ‘ Matter and Motion,’ art. xlvi. 
+ Die physikalischen Grundlagen der Mechantk. Leipzig, 1883. 


Mr. R. F. Muirhead on the Laws of Motion. 485 


ference to a “ Fundamental Korper,”’ which is any body not 
acted on by external forces and having no rotation. The 
absence of rotation is to be determined by observations of 
centrifugal force (as in Newton’s experiment of the rotating 
bucket of water). Now as Streintz takes the kinetic definition 
of force, it involves reasoning in a circle to speak at this stage 
of a body “not acted on by forces.”’ Further, if the observa- 
tions of centrifugal force are to be made with the whole re- 
sources of Dynamics, and our knowledge of the laws of nature, 
this is virtually the kinetic definition of force, but stated 
in a form which involves reasoning in a circle. If, on the 
other hand, want of rotation is to be defined as existing when 
the surface of a bucket of water does not appear to deviate 
from planeness, then our stock objections to such definitions 
of dynamical ideas reappear. 

A most instructive discussion relating to this subject is 
given by Professor Mach in his book Die Mechanik in ihrer 
Enitwickelung, historisch-kritisch dargestellt, pp. 214-222. Let 
us quote a sentence on p. 218:— 

“Instead of saying ‘the direction and velocity of a mass 
# in space remain constant,’ we can say ‘the mean accelera- 
tion of the mass u with reference to the masses m, m’, m"... 
ge urge SPN spy 1 
1Is=VU, or dz Sip =U. € 
latter expression is equivalent to the former, so soon as 
we take into consideration masses which are great enough, 
numerous enough, and distant enough.”’ 

On the previous page, referring to Newton’s bucket ex- 
periment, he remarks that no one can say how the experiment 
would come out were we to increase the mass of the bucket 
continually ; and, further, that we should be guilty of dis- 


at the distances 7, 7’, 7’... 


honesty, were we to maintain that we know more of the motion 


of bodies than that their motion relative to the very distant 
stars appears to follow the same laws as Galileo formulated for 
terrestrial bodies relative to the Harth. | 

Of course this charge of dishonesty cannot be urged against 
the method of this Essay, as explained in our paragraphs on 
the nature of theories. And our definition of “ true rest”’ 
being based entirely on experiment and observation, is not 
affected by Prof. Mach’s strictures on the use of the terms 
absolute rest, absolute space, Kc. 

Though on the principles of this Essay no exception in 
principle can be taken to Prof. Mach’s substitute for the 
“First Law of Motion above quoted,” we reject it because it 
is not the actual conception which has been historically evolved 
in Dynamics. 


486 Mr. R. F. Muirhead on the Laws of Motion. 


Lastly, let us consider the conception of time-measurement. 

The only rival definition of equal times that need be con- 
sidered is that adopted by Streintz, and ascribed by him to 
D’Alembert and Poisson, viz. ‘Times are equal in which 
identical processes take place.’’ The difficulty here would be 
to distinguish when we have identical processes going on. 
We find that practically this will reduce to assuming each 
rotation of the Harth with reference to the fixed stars a pro- 
cess identical with all the others. For the ‘‘ processes ’’ must 
consist in movements of matter, of which the Earth’s rotations 
are the most ‘ identical ’’? we have experience of. 

But even these we know are not absolutely identical, so 
that our definition is not practicable. With this definition, 
what should we mean by saying that the rotation period of 
the Harthis altering? We should mean that if identical pro- 
cesses happened at different dates, their durations measured by 
sidereal time would differ. But the only identical processes 
actually available are wrapped up in the general dynamical 
theory of the Solar system; so that this theoretically inde- 
pendent definition of time turns out to involve all our Dynamics 
implicitly when we try to give it physical meaning. 

In seeking to justify our preference of kinetic definitions 
over non-kinetic definitions of our fundamental dynamical 
conceptions, we have found that the latter, besides being 
theoretically inconvenient, very often have only an illusory 
independence of Dynamics. 

In fact no one has ever built up a science of Dynamics 
from independently formed conceptions ; and to do so in a 
strictly logical manner would require expositions whose length 
would render them tedious in the extreme. 

We have hitherto made no reference to any scheme of 
dynamical principles apart from that of Newton, and those 
various modifications of it proposed by later writers. This 
course has been adopted in order to concentrate attention upon 
the principle at issue. 

Systems of Dynamics founded on such principles as Mau- 
pertius’s “ Principle of Least Action,” or Gauss’s “ Principle 
of Least Coercion ” (Kleinsten Zwanges), may be treated from 
exactly the same point of view, and will not be further re- 


ferred to. 


Note A.—On Theories and Hypotheses, 
In the preceding Essay we have assumed as known the science of 
Geometry ; but of course the views put forward in this Essay con- 
cerning the nature of physical theories apply equally to geometri- 


NS = ks 


Mr. R. F. Muirhead on the Laws of Motion. 487 


eal theories. This is the standpoint adopted by Riemann in his 
epoch-making paper, “ Ueber die Hypothesen welche der Geometrie 
zu Grunde liegen.” That space is infinite and that one and only 
one parallel to a straight line can be drawn through any point, are, 
it is true, the simplest hypotheses which serve to express our ex- 
perience ; but, as Helmholtz points out in his tract Ueber che 
Erhaltung der Kraft, at page 7, the task of theoretical science is 
only completed when we have proved that our theories are the 
only ones by which the phenomena can be explained. “Dann 
ware dieselbe als die nothwendige Begriffsform der Naturauffas- 
sung erwiesen; es wirde derselben alsdann also auch objective 
Wabrheit zuzuschreiben sein.” 

In his critique of the second edition of Thomson and Tait’s 
treatise on Natural Philosophy (‘ Nature,’ vol. xx. p. 213), Clerk 
Maxwell clearly indicates the hypothetical nature of abstract Dy- 
namics. On p. 214 we read :—‘‘ Why, then, should we have any 
change of method when we pass on from Kinematics to abstract 
Dynamics? Why should we find it more difficult to endow moving 
figures with mass than to endow stationary figures with motion? 
The bodies we deal with in abstract Dynamics are just as completely 
known to us as the figures in Euclid. They have no properties 
whatever, except those which we explicitly assign to them...... 
We have thus vindicated for figures with mass, and, therefore, for 
force and stress, impulse and momentum, work and energy, their 
place in abstract science beside form and motion.” 

“The phenomena of real bodies are found to correspond so 
exactly with the necessary laws of dynamical systems that we can- 
not help applying the language of Dynamics to real bodies,” &c. 

It will be seen that, so far as they go, the above extracts are in 
complete harmony with the views in this Essay. It is to be re- 
eretted that these views are not consistently followed out in Clerk 
Maxwell’s book ‘ Matter and Motion.’ In that book, while there are 
very many clear expositions of particular points, the arrangement is 
in many parts highly illogical. This has been pointed out to a 
certain extent by Streintz in his aforementioned book, and the 
reader of the foregoing Essay will have little difficulty in making 
further criticisms. 

One point in Maxwell’s book (‘ Matter and Motion’) calls for 
special notice, viz., his @ prior proof of the first law of Motion. 
This proof rests on the assumption of the impossibility of defining 
absolute rest. ‘‘ Hence,” he says, “the hypothetical law is with- 
out meaning unless we admit the possibility of defining absolute 
rest and absolute velocity.” But itis obvious that if the “ hypo- 
thetical law” spoken of (velocity diminishing at a certain rate) 
corresponded with experience, we should then have, by that very 
fact, a conception of absolute rest and absolute velocity which 
would be perfectly intelligible, so that the assumption “absolute 
rest unintelligible” would not be justified. Thus, Maxwell’s con- 
clusion, “‘ It may thus be shown that the denial of Newton’s law is 
in contradiction to the only system of consistent doctrine about 


488 Mr. R. F. Muirhead on the Laws of Motion. 


space and time which the mind has been able to form” is unwar- 
ranted. 

Kirchhoff in his Mechamk appears to adopt a view somewhat 
similar to that set forth in this Essay. In his preface we find him 
stating as the problem of Mechanik, “die in der Natur vor sich 
gehenden Bewegungen vollstandig und zwar auf die einfachste 
Weise zu beschreiben.” 

This author uses the term force only as a convenient means of 
expressing equations shortly in words. Mass appears as a coeffi- 
cient in the equations of motion, and thus receives a kinetic defi- 
nition. Butno explanations are given as to time-measurement, or 
as to the axes of reference. 


Nore B.—Newton’s Absolute Space and Time. 


My criticisms of the Newtonian scheme of Definitions and Axioms 
have been directed not so much against what I suppose to be 
Newton’s meaning, as against the form in which it is put, especially 
as against that form on the supposition that force is to be measured 
kinetically. 

Thus, instead of looking on the Second Law as a mere definition 
of force-measurement, we might suppose that Newton had in his 
mind some non-kinetic conception of force-measurement ; in which 
case the Second Law would be a real and not an illusory statement 
of physical fact, though imperfect through the want of any speci- 
fication of how force was to be measured. 

Again, take the question of absolute space and time, with respect 
to which Newton’s laws are stated. 

There are three ways of looking at it. Some characterize these 
terms as mere metaphysical nonsense (Mach, p. 209). Streimtz* 
quotes the Hypothesis I. from the third Book of Newton’s Prin- 
cypia to show that by absolute rest Newton means rest relative to 
the centre of gravity of the universe. But Newton evidently places 
this Hypothesis in a different category from his laws of motion. 

I think the meaning of the terms amounts simply to this, that 
Newton looked on Dynamics as an abstract science. ‘“ In rebus 
philosophicis abstrahendum est a sensibus” 7, “loca primaria moveri 
absurdum est” +. And an abstract science is one which deals 
with a certain body of conceptions, every relation in which holds 
with absolute exactness. ‘The point at which considerations as to 
degree of exactitude may arise, is its application to experience. » 

If this be the correct view of Newton’s meaning, then the fore- 
going Hssay has been simply the explicit and developed statement of 
that meaning. 

Thomson and Tait, while in various ways improving the form in 
which they state the Newtonian theory, entirely ignore his idea of 
“absolute space and time,” which, as I have tried to show, is the 
germ of the true theory. 


* Physikalische Grundlagen, p. 10. 
+ Scholium to Definitiones. 


ah ih 


Production, Properties, and Uses of the Finest Threads. 489 


The late C. Neumann, in his pamphlet Ueber die Principien der 
Galilei-Newtonschen Theorie (Leipzig, 1870), like Newton, postu- 
lates an “absolute rest.” He does so by assuming that there is a 
*“Korper Alpha,” an ideally existing body which is absolutely at 
rest and absolutely rigid, with respect to which the First Law of 
Newton holds good. 

Streintz criticises this rather unintelligently, I think, for it is 
evident in reading Neumann’s essay that this is merely an. awk- 
ward and metaphorical way of stating the theory of an “ Abstract 
Dynamics.” 


Nore C.—The Parallelogram of Force. 

Force being defined kinetically, it is hardly necessary to demon- 
strate this proposition. It follows as easily from the parallelogram 
of accelerations as that does from the parallelogram of velocities, or 
the parallelogram of velocities from the parallelogram of steps. 

This applies primarily to forces acting on a particle, but it is easy 
to extend the theorem to “ forces acting on a body,” as defined in 
the Essay. 


LVII. On the Production, Properties, and some suggested Uses 
of the Finest Threads. By ©. V. Boys, Demonstrator of 
Physics at the Science Schools, South Kensington”. 


HAVE lately required for a variety of reasons to have 
fibres of glass or other material far finer than ordinary 
spun glass; I have therefore been compelled to devise means 
for producing with certainty the finest possible threads. As 
these methods may have some interest, and as some results 
already obtained are certainly of great importance, I have 
thought it desirable to bring this subject under the notice of 
the Physical Society, even though at the present time any 
account must of necessity be very incomplete. 
The subject may be naturally divided, as in the title, into 


; three parts. 


1. Production. 


The results of the natural methods of producing fibres by 
living things, as spiders, caterpillars, and some other creatures, 
are well known ; but it is useless to attempt to improve on 
Nature in this direction by our own methods. 

Fibres are also produced naturally in volcanoes by the 
rushing of steam or compressed gases past melted lava, which 
is carried off and drawn out into the well-known Pelés hair. 
The same process is employed in making wool from slag, for 


* Communicated by the Physical Society : read March 26, 1887. 


Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 2 L 


490 Mr. 0. V. Boys on the Production, Properties, 


clothing boilers, &c. ; but in each of these cases the fibres are 
matted together, they are not adapted to the requirements of a 
Physical Laboratory. By drawing out glass softened by heat 
by a wheel we obtain the well-known spun glass. 

There is a process by which threads may be made which is 
natural in that natural forces only are employed, and the thread 
is not in any way touched during its production. This is the 
old, but now apparently little-known experiment of electrical 
spinning. Ifa small dish be insulated and connected with an 
electrical machine and filled with melted rosin, beeswax, pitch, 
shellac, sealing-wax, Canada balsam, guttapercha, burnt india- 
rubber, collodion, or any other viscous material, the contents 
will, if they reach one edge of the dish, at once be shot out in 
the most extraordinary way in one, two, or it may be a dozen 
threads of extreme tenuity, travelling at a high speed alon 
“lines of force.” If the material is very hot, the liquid 
cylinders shot out are unstable and break into beads, which 
rattle like hail on a sheet of paper a few feet off. As the 
material cools, the beads each begin to carry long slender 
tails, and at last these tails unite the beads in twos and threes ; 
but the distance between the beads is far greater than that 
due to the natural breaking of a cylinder into spheres, as 
after the first deformation of the surface occurs which deter- 
mines the ultimate spheres the repulsive force along the thread 
continues, and drags them apart many times their natural 
distance. As the temperature continues to fall and the 
material to become more viscid, the beads become less 
spherical, and the tails less slender, and at last a perfectly 
uniform cylindrical thread is formed. If sealing-wax is 
employed, and a sheet of paper laid for it to fall on, the paper 
becomes suffused in time with a delicate rosy shade produced 
by innumerable fibres separately almost invisible. On placing 
the fingers on the paper, the web adheres and can be raised 
in a sheet as delicate and intricate as any spider’s-web. 

It is interesting to see how these fibres fly to any conduct- 
ing body placed in their path. If the hand is held there it is 
quickly surrounded by a halo of the finest threads. If a 
lighted candle is placed in the way, the fibres are seen by the 
light of the candle to be rushing with the greatest velocity 
towards it, but when a few inches off they are discharged by 
the flame, they stop, turn round, and rush back as fast into 
the saucer whence they came. ‘The conditions for the success 
of this beautiful experiment are not very easily obtained*. 

Fibes spun by the electrical method are so brittle that they 


* If the wick of the candle is connected with the opposite pole of the 
machine, the threads at one stage are sure to return to the saucer. 


Se 
= _ 


——— a 


and some suggested Uses of the Finest Threads. 491 


do not seem to be’of any practical use. It is possible, how- 
ever, that this method might be available for reducing to a 
fine state of division such of the rosins or other easily fusible 
bodies as cannot readily be powdered mechanically. 

On returning to bodies which, like glass, require a high 
temperature for their fusion, to which the electrical method is 
inapplicable, we find that the only method practically available 
is that of drawing mechanically. It would seem that if finer 
threads than can be formed by the ordinary process of glass- 
spinning were required, it would be necessary to obtain a 
higher speed, to have the glass hotter, and to have as small a 
quantity as possible hot. I put this idea to a test by mounting 
at the back of a blowpipe-table a pair of sticks which could be 
suddenly moved apart by a violent pull applied to each near 
their axes. By these means the upper ends were separated 
about 6 feet, and the motion was so rapid that it was impos- 
sible to follow it. A piece of glass drawn out fine was 
fastened to the end of each stick, and the ends of these heated 
by a minute blowpipe-flame. They were immediately made to 
touch and allowed to fly apart. In this way I obtained threads 
of glass about 6 feet long, finer than any spun glass I have 
examined. By using the oxyhydrogen jet with the same 
apparatus, still finer threads were produced. It was evident 
then that the method was right ; but some more convenient 
device which also would make long threads would be prefer- 
able. 

There are several ways of obtaining a high speed, the most 
usual depending on an explosive ; but it would be difficult to 
arrange in a short time a gun which could be used to shoot 
a projectile carrying the thread which would not also destroy 
the thread by the flash. It is possible that an air-gun could be 
so:arranged. Rockets when at the period of most rapid com- 
bustion have an acceleration which is enormous. ‘Thus a well- 
made 2-oz. rocket is at one part of its flight subject to a 
force of over 3 lb. in gravitative measure. ‘This force, acting 
on such a body for 10 seconds only, would, neglecting atmo- 
spheric resistance, starting from rest, carry it more than 6 miles. 
The acceleration is about 28 times that due to gravity on the 
earth, or about the same as that on the sun. Anyone who 
will stay in a room with a lighted two-ounce rocket, having 
no stick or head, will obtain a more vivid notion of the value 
of gravity on the sun than in any other way I know. 

A rocket is perhaps more available for thread-drawing than 
a gun, but it does not seem altogether convenient. One 
other method, however, is so good in every respect, that there 
seems no occasion to try a better. The bow and arrow at 

2L2 


- 


492 Mr. C. V. Boys on the Production, Properties, 


once supply a ready means of instantly producing a very high 
velocity, which the arrow maintains over a considerable dis- 
tance. For the special purpose under consideration, the 
lightest possible arrow is heavy enough. I have made arrows 
of pieces of straw, which may be obtained from wool-shops, 
a few inches long, having a needle fastened to one end for 
a point. Arrows made in this way travelled the length of 
the two rooms in which I made these experiments—about 
90 feet—in what seemed to be under half a second. They 
completely pierced a sheet of card at that distance, which I 
put up thinking that a yielding target might damage them 
less than the wall, and were then firmly stuck unharmed in the 
wall behind; in every way they behaved so well that I do 
not think a better make of arrow possible. 

The bow I used was a small cross-bow held in a vice 
with a trigger that could be pulled with the foot. The first 
bow was made of oak, the first wood that came to hand. I 
then made some bows of what was called lance-wood (it was 
unlike any lance-wood I have seen) ; but the trajectory was 
at once more curved, the arrow took perceptibly longer to 
travel, and the threads produced were thicker. As the arrow 
is so light, the only work practically that the bow has to do 
is to move itself; that wood then which has the highest 
elasticity along the fibres for its mass is most suitable ; in 
other words, that wood which has the greatest velocity of 
sound is best. I therefore made bows of pine, and obtained 
still higher velocities and finer threads than I could obtain 
with oak bows. 

With a pine bow and an arrow of straw I have obtained a 
glass thread 90 feet long and j5} 9 inch in diameter, so 
uniform that the diameter at one end was only one sixth more 
than that at the other. Pieces yards long seemed perfectly 
uniform. 

A fragment of drawn-out glass was attached to the tail of 
the arrow by sealing-wax, and heated to the highest possible 
temperature in the middle, the end heing held in the fingers. 
With every successful shot the thread was continuous from 
the piece held in the hand to the arrow 90 feet off. The 
manipulation is, however, difficult, but another plan equally 
successful has the advantage of being quite easy. It is not 
necessary to hold the tail of glass at all; if the end of the 
tail only be heated with the oxyhydrogen jet until a bead 
about the size of a pin’s head is formed, and the arrow shot, 
this bead will remain behind on account of its inertia, and 
the arrow go on, and between them will be pulled out the 
thread of glass. 


, 


and some suggested Uses of the Finest Threads. 493 


Prof. Judd has kindly given me a variety of minerals 
which I have treated in this manner. Some behave like glass 
and draw readily into threads, some will not draw until below 
a certain temperature, and others will not draw at all, being 
_ perfectly fluid like water, or when a little cooler perfectly 

ard. 

Among those that will not draw at all may be mentioned 
Sapphire, Ruby, Hornblende, Zircon, Rutile, Kyanite, and 
Fluorspar. 

Hmerald and Almandine will draw, but care is required to 
obtain the proper temperature. In the case of the Garnet 
Almandine, if the temperature is too high, the liquid cylinder, 
if formed, breaks up, and a series of spheres fall on the table 
in front of the bow. At a slightly lower temperature the 
thread is formed, but it is beaded at nearly regular intervals 
for part of its length. 

Several minerals, especially complex silicates as Orthoclase, 
draw very readily, but that which surpasses all that I have 
tried at present is Quartz, which, though troublesome in many 
ways at first, produces threads with certainty. It required 
far more force to draw quartz threads than had been previously 
experienced. The arrow, instead of continuing its flight, 
hardly disturbed by the drag of the thread, invariably fell very 
low, and was not in general able to travel the whole distance. 
So great is the force required that I split many arrows before 
I succeeded at all. I have obtained threads of quartz which 
are so fine that I believe them to be beyond the power of any 
possible microscope. Mr. Howes has lent me a +4,-in. Zeiss of 
excellent definition, and though, on looking at suitable objects, 
definite images appear to be formed on which are marks 
corresponding according to the eyepiece-micrometer to toq/o90 
inch, yet these threads are hopelessly beyond the power of the 
instrument to define atall. On taking one that tapers rapidly 
from a size which is easily visible, the image may be traced 
until it occupies a small fraction of one division, of which 13°4 
correspond to yo/9 inch on the stage ; then the diffraction 
bands begin to overlap the image until it is impossible to say 
what is the edge of the image. Having reached this stage, 
the thread may be traced on and on round the most marvellous 
convolutions, the diffraction-fringe now alone appearing at all, 
but getting fainter and apparently narrower until the end is 
reached. ‘That a real thing is being looked at is evident, for 
if the visible end is drawn away the convolutions of fringes 
travel away in the same direction. It is impossible to say 
what is the diameter of these threads ; they seem to be certainly 
less than yg !990 inch for some distance from the end. 


494 Mr. C. V. Boys on the Production, Properties, 


It might be possible to calculate what would be the appear- 
ance presented by a cylinder of given refractive power, and 
1, 2, 3, &e. tenths of a wave-length of any kind of light 
in diameter, when seen with a particular microscope. By no 
other means does it seem possible to find out what the true 
size of the ends of these threads really is. 


2. Properties. 


I can at present say very little of the properties of these 
very fine fibres ; I am now engaged with Mr. Gregory and 
Mr. Gilbert in investigating their elasticity. The strength 
goes on increasing as they become finer, that is, when due 
allowance is made for their reduced sectional area, and it 
seems to reach that of steel, about 50 tons to the inch in 
ordinary language ; but on this point I have not yet, made 
any careful experiments. 

The most obvious property of these fibres is the production 
of all the colours of the spider-line when seen in a brilliant 
hight. The most magnificent effect of this sort I have seen, 
was produced by a thread of almandine. One of these the 
length of the room, even though illuminated with gas-light 
only, was glistening with every colour of the rainbow. In 
attempting, however, to wind it up, it vanished before me. 
It is of course only visible in certain directions. 

The chief value of threads to the physicist lies in their 
torsion. Spun glass, as is now well known, cannot be used 
for instruments of precision, because its elastic fatigue is so 
great that, after deflection, it does not come back to the original 
position of rest, but acquires a new position which perpetually 
changes with every deflection. If left alone, this position 
slowly works back towards a definite place more rapidly as it 
is further from it. | 

To compare threads made of different materials, I made a 
flat cell in which a galvanometer-mirror, made by Elliot Bros., 
might hang, being attached to the lower end of the thread. 
The upper end was secured to a fixed support, and a fixed tube 
protected the length of the fibre from draught. The cell, 
which could be moved independently of the rest, was protected 
by a cover. By means of a lamp and scale, the exact position 
of rest of the mirror could be determined with great accuracy. 
On turning the cell round as many times as might be desired, 
the mirror was turned with it, and could be left any time in any 
position. On turning the cell back again, the mirror was 
allowed to come to its new position of rest, air-resistance of 
the cell bringing about this result in a few swings. By this 
means I hoped quickly and accurately to determine the fatigue 


— ee 


and some suggested Uses of the Finest Threads. , 495 


in a variety of threads, but an unforeseen difficulty arose which 
I cannot yet explain. When the cell was moved round slightly 
so as not to touch the mirror, the mirror moved at first in the 
same direction as was to be expected, but it came to rest in a 
new position, to reach which it had to move in the opposite 
direction to the movement of the cell. Whichever way the 
cell was shifted, the mirror always went the other to find its 
position of rest. Thinking that it or the cell were electrified, 
I damped both by breathing on them, but with no result, and 
the next day the same effect was observable. So great was this 
effect that I could set the cell with greater accuracy by 
watching the spot of light than by the pointer carried by the 
cell working over a 4-inch circle. 

Thinking that magnetism might have something to do with 
this effect, I brought a horseshoe-magnet near the mirror, 
when it was instantly deflected through a large angle. An 
examination of the cement used (Loudon’s bicycle cement) 
showed that it was magnetic. Of many cements examined, 
sealing-wax was more nearly neutral than any other. Bicycle 
cement and electrical cement were strongly magnetic; all 
others except sealing-wax strongly diamagnetic. The appa- 
ratus was therefore taken to pieces and carefully cleaned. It 
was put together with as small a quantity of sealing-wax as 
possible, and the mirror was attached to a fragment of thin 
pure copper wire, which again was fastened by a speck of 
sealino-wax to the thread. Hven then the same kind of 
effect as that already described occurred. Still a magnet 
deflected the mirror, but not so much, and the cell was 
practically neutral; yet, when the cell was turned a little, 
the mirror changed its position of rest. 

Without pursuing this question further, I put a window in 
the protecting tube and turned the mirror by means of a 
small instrument passed up from below. Thus neither window 
nor support were moved. A piece of spun glass nearly 9 
inches long gave a period of oscillation to the mirror of 2°3 
seconds about. A lamp and millimetre-scale were placed 50 
inches from the mirror. As all the observations were 
expressed in tenths of a millim., to about which extent they 
can be trusted, it is convenient to employ one scale of numbers 
of which one tenth millim. is the unit. One complete turn of 
the mirror is very nearly 160,000 on thisscale. If the mirror 
is moved through 160,000 in either direction and held for one 
minute, and then allowed to take its new position, the change 
in the position of rest is as soon as it can be read about 370. 
This is reduced in about three minutes to 110. If the mirror is 
moved through three turns, 480,000 of the scale, and held one 


496 Mr. C. V. Boys on the Production, Properties, 


minute, the position of rest is at first moved about 1100, which 
falls in three minutes to about 400. I have given these figures, 
not because the effect is not perfectly well known, but to serve 
as comparison figures to those that are to follow. They can — 
only be properly represented on a time-diagram. 

A piece of the same fibre that was used in the last experi- 
ment was laid in a box of charcoal and heated in a furnace 
to a dull red heat and allowed to cool slowly. This was 
examined in the same way as the last. The effect of a 
movement of 160,000 for one minute was now only about 60, 
which was reduced to about 45 inthree minutes. The change 
for 480,000 lasting one minute was at first about 250, which 
fell to about 180 in three minutes. 

Annealed spun glass then shows far less of this effect than 
spun glass not annealed, but it is slower in recovering. It is 
possible that if time were given, it would show as great an 
effect as plain glass. The only mineral from which at the 
present time I have obtained any valuable results in this 
direction, is quartz. Here the effect of the usual minute at 
160,000 was only 7, in the place of 370 for glass, at 820,000 
only 17, and 640,000 only 32, which in four minutes fell to 
22. This fibre was as usual fastened at each end by sealing- 
wax. When this experiment was made, the thread had only 
just been fastened. The same fibre treated previously in the 
same way, but some days after fastening, did not even show 
this effect; but as this was before I had completed the 
proper cell, the observations cannot so well be trusted. After 
a complete turn, there was not a movement of one tenth of a 
millim., nor had the position changed this much in 16 hours. 
It is as yet too soon to be sure, but this seems to point to the 
possibility of the very slight effect observed being largely due 
. to the sealing-wax. Whether this is so or not does not much 
matter, the behaviour of the quartz thread approaches suffi- 
ciently near to that of an ideal thread, to make it of the 
utmost value as a torsion-thread. I hope shortly to be able to 
bring results of carefully conducted experiments on the 
elastic fatigue of quartz and other fibres before the notice of 
this Society. } 

A thread of annealed quartz behaves like a thread of quartz 
not annealed. That it was affected by the process of annealing 
is evident, because in the first place it was very rotten and 
difficult to handle, and in the second a piece of quartz fibre, 
which was wound up, retained its form. By this test, quartz 
can only be partly annealed in a copper box, as any form is not 
retained perfectly ; at a temperature above that of melting 
copper, quartz seems to perfectly retain any form given to it. 


and some suggested Uses of the Finest Threads. A97 
It is probable that a body hung by a fibre of quartz and 


vibrating in a perfect vacuum would remain twisting back- 
wards and forwards for a far longer time than a similar body 
hung by a glass thread, also that the most perfect balance- 
spring for a watch would be one of quartz. I have a piece of 
quartz drawn out to a narrow neck which just cannot hold up 
its head ; this keeps on nodding in all directions for so long a 
time, even in the air, as to make it evident that the material 
has very unusual properties. 


3. Uses. 


As torsion-threads these fibres of quartz would seem to be 
more perfect in their elasticity than any known ; they are as 
strong as steel, and can be made of any reasonable length 
perfectly uniform in diameter, and, as already explained, 


exceedingly fine. The tail ends of those that become invisible 


must have a moment of torsion 100 million times less than 
ordinary spun glass ; and though it is impossible to manipulate 
with those, there is no difficulty with threads less than yody5 
inch in diameter. 

I have made a spiral spring of glass of about 30 turns 
which weighs about one milligram; this, examined by a 
microscope, would show a change in weight of a thing hung 
by it of one 10 millionth of a gram. Since this has been 
annealed its elastic fatigue is that of annealed glass, and 
therefore very small. J have succeeded in doing the same 
thing with a quartz fibre, but the difficulties of manipulation 
are very great in consequence of the rottenness of annealed 
quartz. The glass spring can be pulled out straight, and 
returns perfectly to its proper form. 

Since these fibres can be made finer than any cobweb, it is 
possible that they may be preferable to spider-lines in eye- 
pieces of instruments ; they would in any case be permanent, 
and not droop in certain kinds of weather. 

Those who have experienced the trouble which the shifting 
zero of a thermometer gives, might hope for a thermometer 
made of quartz. When made, it would probably be more 
perfect in this respect than a glass thermometer, but the 
operation of making would be difficult. 

These very fine fibres are convenient for supporting small 
things of which the specific gravity is required, for they weigh 
nothing, and the line of contact with the surface of the water 
is so small, that they interfere but little with the proper swing 
of the balance. 

It seemed possible that a diffraction-grating made of fibres 
side by side in contact with one another would produce 


498 Production, Properties, and Uses of the Finest Threads. 


spectra which would be brighter than those given by a 
corresponding grating of ordinary construction, because not 
only is all the light which falls on the surface brought to a 
series of linear foci forming the bright lines instead of being 
half removed, as is usually the case, but the direction of the light 
on reaching these lines is not normal to the grating as usual, 
and therefore in the direction of the central image, but 
spreading, and thus in the direction of all the spectra. I 
picked out a quantity of glass fibre not varying in diameter 
more than one per cent., and made a grating in this way 
covering about:one eighth of an inch in breadth. This not 
only showed three spectra on each side, and a quantity of 
scattered light, but all the spectra were closely intersected by 
interference-bands, such as are seen when a Newton’s ring of 
a high order is seen ina spectroscope. This is probably due to 
a cumulative error in the position of the fibres, for they . 
were spaced by being pushed up to one another with a 
needle-point, or to light passing between the fibres in a few 
places where dust particles keep them apart. 

A. diffraction-grating made of these fibres, spaced with a 
screw to secure uniformity, and of a thickness equal to the 
spaces between them (and one of 1000 lines to the inch could 
be easily made) would be far more perfect for the number of 
lines than any scratched on a surface ; that is, for investigation 
on the heat of a spectrum, sucha grating would be preferable 
to a scratched one, as there is no uncertainty as to the grating 
_ or to the substance of which it is made*. Ifthe transparency 
“of the fibres interfered they could be rendered opaque by 
metallic deposit without visibly increasing their diameter. 

There is one use to which the fibres of quartz tailing-off to 
a mere nothing might be applied, namely as a test-object for 
a microscope. Theory shows that no microscope can truly 
show any structure much less than +5,/559 inch, or divide two 
lines much less than this distance apart. Natural bodies such 
as Diatoms &c. have this advantage, that they can be ob- 
tained in any quantity alike, but no one knows what the real 
structure of these may be. Nobert’s bands,are good in that | 
we know the number of lines in any band, but as to the indi- 
vidual appearance of the lines and spaces it is impossible to 
say anything. These fibres have the advantage that we have 
a single thing of known form, which tapers down from a 
definite size to something too small even to be seen. Though 
it may be possible to calculate the size from the appearance 
of the fringes, yet whether the size is known or not, at each 


* See ‘ Heat,’ by Prof. Tait, p. 268. 


Electrical Resistance of Vertically-suspended Wires. 499 


point we have a definite thing of known form which can be 
examined by a series of microscopes, and the point up to 
which it can be clearly seen observed for each. 

I have thought it worth while to bring this subject forward 
in this very incomplete form, because there are already 
results of interest and there is so much prospect of more, that 
it is likely that Members may be glad to investigate some of 
the questions raised. 


LVIII. On the Electrical Resistance of Vertically-suspended 


Wires. By SHELFORD BipweE tt, J.A., F.R.S.* 


ROM the experiments to be described in this paper, it 
appears probable that the electrical resistance of verti- 
cally-suspended copper and iron wires alters to a small extent 


with the direction of the current traversing them. Ifthe wire 


is of copper, the resistance is slightly greater when the cur- 
rent goes upwards than when it goes downwards ; while, on 
the other hand, the resistance of an iron wire is apparently 
greater for downward than for upward currents. 


1 eg 


The arrangement employed for exhibiting this effect is 
shown in the annexed diagram. A wire, A B, of the material 


* Communicated by the Physical Society: read March 12, 1887. 

T Venturing to imitate the fanciful analogy used by Sir William 
Thomson, who, in discussing the thermoelectric effect now universally 
associated with his name, speaks of the “specific heat” of electricity, we 
may perhaps also speak of the “specific gravity” of electricity, and say 
that (like its specific heat) it is positive in copper and negative in iron. 


500 — Mr. S. Bidwell on the Poaticdl Resistance 


to be tested is suspended at its middle point, P, from a support 
10-5 metres above a metre-bridge, to the terminals, TT’, of 
which the ends of the wire are connected. Another wire, C, 
is soldered at one end to P, and connected through the gal- 
vanometer, G, with the slider, 8. A resistance of 100 ohms 
is inserted in each of the gaps, R R’, and a commutator, K, is 
interposed between the two-cell battery, D, and the bridge. 

With this arrangement, supposing that the two halves of 
the wire A B are of uniform sectional area and in the same 
physical condition, and that the various parts of the apparatus 
are in fair order and adjustment, there will be a balance when 
the slider is near the middle division of the scale. And if 
the resistances in the circuit are independent of the direction 
of the current, it is clear that the balance will be maintained 
notwithstanding that the commutator K be reversed. But 
this is found not to be the case. 

A series of experiments was made with a copper wire 


‘A millim. in diameter (No. 28 B.W.G.), and having a total 


- resistance of 2:11 ohms. The commutator was first set so . 


that the current through the wire passed up the portion B 
and down the portion A (7. e. in the direction BPA), and a 
balance was obtained by adjusting the slider. The commu- 
tator was then reversed and the current made to pass up A 
and down B. This at once destroyed the balance, and in 
order to restore it, it was necessary to move the slider several 
divisions towards the right. Assuming that the total resist- 
ance of the wire remains constant, this result may be explained 
by supposing that the reversal of the current is accompanied 
by increased resistance in the portion A, and diminished re- 
sistance in the portion B. Owing to its vertical suspension, 
the resistance of that portion of the wire in which the current 
travels upwards is greater than it would be if the wire were 
placed in a horizontal position, while the resistance of the 
portion in which the current travels downwards is less. 

The experiment was repeated with an iron wire of larger 
size, its diameter being ‘8 millim. (No. 22 B.W.G.). With 
this the effect of reversal was smaller ; but it was well marked, 
and of the opposite nature to that observed in the former case. 

The readings obtained in the two series of experiments are 
given in the following Table :— 


of Vertically-suspended Wires. 501 


Copper Wire. 


Scale-readings. 
Number of 


experiment. Difference. 
Current direct. | Current reversed. 
1 569 633 — 64 
2 567 | 637 | —70 
3 595 | 651 —56 
Mean difference ...... —63'°3 


| 
| 
| 
| 


| 
1 780 770 | 410 
2 760 | 748 | p12 
5 759 | 748 Naess 
Mean difference ...... +11 | 


I believe these effects are associated with certain thermo- 
electric phenomena discovered by Sir William Thomson. In 
his famous Bakerian lecture, published in the Philosophical 
Transactions for 1856, he showed that if a stretched copper 
wire is connected with an unstretched wire of the same metal 
and the junction heated, a thermoelectric current will flow 
from the stretched to the unstretched wire through the hot 
junction ; while, if the wires are of iron, the direction of the 
current will be from unstretched to stretched. It follows, 
therefore, from the laws of the Peltier effect, that if a battery- 
current is caused to flow from a stretched to an unstretched 
wire, heat will be absorbed at the junction when the metal is 
copper, and will be developed at the junction when the metal 
is iron: and if the direction of the current is reversed the 
thermal effects will also be reversed. 

Now a vertically suspended wire is unequally stretched by 
its own weight, the stress gradually increasing from zero at 
the lowest point to a maximum at the highest. Any small 
element of the wire is more stretched than a similar element 
immediately below it, and less stretched than a neighbouring 


502 «= Mr. S.. Bidwell on the Electrical Resistante 


element just above it. Thus a current of electricity, in pass- 
ing from the lowest to the highest point of such a wire, 
is always flowing from relatively unstretched to relatively 
stretched portions. If, then, the wire were of copper, heat 
would be evolved throughout its whole length ; the tempera- 
ture of the wire would rise, and its resistance would conse- 
quently be increased. With a current flowing from top to 
bottom, the temperature of the wire would fall and its resist- 
ance diminish. So also an iron wire would be cooled and 
and have its resistance lowered by an upward current, while 
a downward current would heat it and increase its resistance. 
The changes of resistance are thus, as I believe, proximately 
due to changes of temperature. | 

The resistance of the bridge-wire used in my experiments 
was ‘244 ohm, and, as already mentioned, an additional resist- 
ance of 100 ohms was placed in each of the gaps adjoining 
the bridge-wire. Denoting the resistance of the half A of 
the suspended wire by a, and that of B by 6, we have, from 
the first experiment with the copper wire (the result of which ~ 
agrees closely with the mean) :— 


For direct current, 


a 100° + °569 x *244° 
O 100% AS 1 x c244e 
Dis 1001389 
~ 100105 
Also 
a ab — alien 
Hence 


aia 2: 
b=1:0548208°. 
For reversed current, 
a_ 100°+°633 x :244° 


b 100° 4-367 x 244° 
_ 100154 
— 100090 


* Ofcourse the resistances are not really measured to the high degree 
of accuracy suggested by these figures; but any small error of excess or 
defect would be approximately the same for the two values of a (with 
direct and reversed currents) and would not materially affect their differ- 
ence, to which alone importance is attached. 


a 


| of Vertically-suspended Wires. 508 

And, as before, | 
a+b=2:11". 

Hence 

a=1-0553372° *, 


b=1°0546628". 


When therefore the current was reversed, the value of a 
was increased by 


1:0553372 —1:0551792 ohm 
='(00158 ohm. 


This is equivalent to about 16 thousandths per cent. 

Assuming that a change of temperature of 1° C. produces 
an alteration of -4 per cent. in the resistance, it follows that 
the temperature of the copper wire was = degree C. higher 
with an upward than with a downward current. 

The current traversing the wire was not measured, but it 
was probably about 1 ampere. 

It will be seen from the figures in the Table, that the 
changes which occurred in the resistance of the iron wire 
were considerably smaller than those observed in the case of 
copper. This was unexpected, since the thermoelectric effects 
are, I believe, somewhat greater withiron. But the apparent 
anomaly is obviously to be accounted for, at least in part, by 
the higher specific resistance of iron. With the same electro- 
motive force the current per unit of sectional area would be 
six or seven times greater in copper than in iron, and the 
Peltier effect is proportional to the current. To render the 
results in the two cases strictly comparable, other less impor- 
tant differences, such as those of specific heat and radiating- 
power, would have to be taken into account. 

If a convenient opportunity offered it would be satisfactory 
to repeat the experiments with much longer wires, such as 
might be suspended in the shaft of a coal-pit or in a shot- 
tower. The effects hitherto observed are so small that they 
might possibly be due to accidental causes, and I publish this 
account of them with some diffidence. 


* See note in preceding page. 


Fo goa 4 


LIX. The Evolution of the Doctrine of Affinity. 
By Professor Lornar Meyer, of Tiibingen”. 


aL may not be amiss, on the issue of a Journal { specially 

devoted to the theoretical and physical aspects of 
Chemistry, to take a rapid survey of the development of the 
doctrine of chemical affinity, a correct knowledge of which is, 
and must ever remain, the most important object of the theory 
of our science. 

The former doctrines of affinity, conceived without know- 
ledge of the laws of chemical combination, reached their acme 
in Berthollet’s teaching, which united all previous investiga- 
tions and speculations into a compact theory. 

The basis of Berthollet’s conception was his statement that 
the chemical action of every substance must be proportional 
to its active mass and to a constant depending on its nature, 
and named by him Affinity, except in so far as external con- 
ditions (e.g. temperature, state of aggregation, solubility, 
volatility, and so on) acted as retarding or accelerating causes. 
The doctrine of Berthollet is now fully recognized, although 
it “was for long ignored or forgotten. This unfortunate 
neglect is explicable when it is remembered that, along with 
the most illustrious of his contemporaries, he committed the 
error of supposing that the capacity for saturation was a mea- 
sure of affinity. Sir Humphrey Davy had, indeed, shown that 
this assumption led to not a few difficulties ; but it was first dis- 
proved by the brilliant experimental development by Berzelius 
of Richter’s ‘‘ Stoichiometry ”’ and Dalton’s Atomic Theory. 

That, in consequence of this disproof of an unimportant and 
incidental addition to the experimentally correct doctrine of 
Berthollet, his doctrine should have almost been forgotten, and 
have been completely neglected, would appear inconceivable, 
if we did not consider the enormous influence exercised by 
Berzelius on the growth of Chemistry. He united to an acute 
perception of the most minute peculiarities in the behaviour of 
chemical substances, and the most refined choice of analytical 
and synthetical methods, a special talent for systematic 
arrangement of facts discovered from day to day by himself 
and by his students. All theoretical views were employed by 
him in support of his system ; indeed, he accepted none unless 
it proved of assistance in his endeavour to perfect his mar- 
vellous arrangement of the chemical elements and their com- 
pounds. He was indifferent to theoretical speculations which 

* Translated and communicated by Professor William Ramsay. 

' + Zeitschrift fiir physikalische Chenue, edited by W. Ostwald and 
J. van’t Hoff (Riga and Leipsig). 


Evolution of the Doctrine of Affinity. 505 


did not seem to further his great work; while he offered a 
most strenuous opposition to all those which he conceived 
would bring disorder into his classification. He even disputed 
for half a generation Davy’s discovery of the elementary cha- 
racter of chlorine, simply because he could not reconcile it 
with his views. 

But the discovery in the earlier part of this century of the 
relations between the electrical and the chemical behaviour of 
elements and compounds appeared to him to afford great 
assistance in the development of his system; and hence he 
based his whole classification on positive and negative cha- 
racters of substances, manifested electrically ; and for a time, 
at least, he identified affinity with electrical attraction. 

Alongside of this electrochemical hypothesis, every other 
doctrine of affinity appeared superfluous; and, as a con- 
sequence, Berthollet’s teachings were forgotten, although they 
were by no means contrary to the newer views. The electro- 
chemical theory of Berzelius, however, was never fully deve- 
loped in detail. Hven though he laid great stress on it, though 
he often referred to it, and insisted on its fundamental nature, 
yet there is not to be found in any of his numerous memoirs 
in which it is mentioned, nor in his Jahresbericht, in which 
he criticised the electrochemical theories of other investi- 
gators, nor even in any one of the numerous editions of his 
Textbook, an attempt at a complete exposition of his theory. 
In actual fact, the electrochemical theory never rose above 
the general conception that the chemical and electrical 
behaviour of bodies are closely connected. The explanation 
was only an apparent one: it consisted only in ascribing to 
electrical causes observed chemical facts. An attempt to 
measure affinities on such a basis failed, owing either to the 
lack of experimental data or to its being contradicted by 
them. 

Hrroneous deductions from his theory misled Berzelius, 
not only in causing him to disbelieve Davy’s proof of 
the elementary nature of chlorine, but also in leading him 
vigorously and persistently to dispute Faraday’s electrolytic 
law. While he withdrew from his opposition to Davy after 
a sixteen years’ struggle (1826), when the analogy between 
hydrogen chloride and hydrogen sulphide had been fully 
recognized, he continued to reject until his death that most 
important of all electrochemical discoveries, Faraday’s law. 
These two facts serve sufficiently to show that Berzelius’s 
theory was unable to yield a thorough explanation of affinity. 
That in spite of such weak points, ‘sufficiently evident today, 
a man of Berzelius’s great power could hold fast to them 


Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 2M 


506 Prof. L. Meyer on the 


throughout his whole life, and, moreover, impress others with | 


their truth for many years, was a consequence of the enormous 
benefit which his systematic arrangement conferred on che- 
mical science. Hven his system died, as soon as its incom- 
petence to classify organic compounds became manifest ; and 
the chemical world looked on its departure with indifference. 

From this period, however, the speculations of chemists 
ran in quite a new channel. It was not so necessary to 
investigate the mode of action of the forces of affinity, as to 
prepare and examine the wonderful forms of combination 
which by its influence the atoms could be induced to assume 
in organic compounds. _ This work for long absorbed the 
attention of chemists. The obstinate battles fought over the 
laws governing the linkage of atoms are still so fresh in the 
minds of at least the older of the present generation of 
chemists, that they need not here be more than mentioned. 

During this contest regarding the constitutional formule 
of organic compounds, a complete revolution of the doctrines 
of affinity was in progress, which was prompted chiefly by 
facts in inorganic Chemistry. Chemists had persistently clung 
to the assumption, long before proved untenable, that heat 
was a form of matter capable of entering into combination with 
other forms of matter to produce chemical compounds. Al- 
though Rumford, at the end of last century, had proved that 
heat is a mode of motion (a view held even in the 16th and 
17th centuries*), and Davy had furnished a brilliant con- 
firmation of his proof, yet even up to the middle of this century 
it was stated in the most widely-read textbooks of Chemistry 
that heat, light, and electricity are to be regarded as impon- 
derable forms of matter. It must be noted that, though 
rejected by all journals of Physics, Julius Robert Mayer’s 
treatise received compensation, oddly enough, by finding 
a resting-place in Liebig’s ‘ Annals of Chemistry’. With 
the recognition of the importance of the mechanical theory 
of heat arose the hope that, by its help, our knowledge of the 
doctrine of affinity might be materially advanced. 

The view was at once suggested that, just as a heavy body, 
in consequence of the mutual attraction between it and the 
earth, moves towards the earth with accelerated velocity, 
thereby converting potential energy, due to its elevated 
position, into kinetic energy, so the atoms, as a consequence 
of their affinity, move towards each other, converting their 
affinity into energy of motion, which, as a rule, is manifested 
in the form of heat. According to this doctrine, the heat 


* Bacon, Novum Organum, Lib. ii. Aph. xx. 
+ Ann. Chem. Pharm, 1842, vol. xlii. p. 233. 


Evolution of the Doctrine of Affinity. 507 


evolved by the sum of the impacts should bea measure of the 
affinity. But, from the first, difficulties have to be surmounted 
in accepting this view. One of the greatest is that it seldom 
occurs that a compound is formed through the union of isolated 
atoms ; in almost all cases the atoms themselves have to be 
liberated from compounds iu which they have existed in a 
state of combination. As this liberation must be accompanied 
by gain of energy (i.e. by absorption of heat), while the 
formation of the new compound gives rise to loss of energy 
(i.e. a heat-evolution), it happens that, as a rule, only the 
difference between the absorption and emission is manifested 
externally. Hence this doctrine of affinity can deduce from 
observations, not the absolute value of either affinity, but only 
the amount by which the one is greater than the other. 
Many other difficulties present themselves; especially the 
fact that, along with chemical changes, physical changes 


(alteration of the state of aggregation, of the volume, and so 
‘on) occur simultaneously, and are themselves accompanied by 


an emission or by an absorption of heat. Moreover, thermo- 
chemical experiments are by no means easy of execution, and 
are subject to many sources of error; hence it is not to be 


wondered that slow progress was made in developing the 


thermochemical doctrine of affinity. The more numerous 
the observations, and the greater their accuracy, the greater 
the number of instances in which theory and experiment 
failed to display coincidence. It has been frequently observed 
that a much larger evolution of heat occurs on neutralizing an 
evidently weak acid than a strong one capable of expelling 
the weak one more or less completely from its compounds 
with bases. ‘The expulsion, in such cases, is attended by an 
absorption, not an evolution, of heat. Similar facts have also 
been noticed in nota few other chemical reactions, which must 
be, and have been, regarded as produced by the action of 
affinity; e.g., the formation of the ethereal salts of organic 
acids by their action on the alcohols*. Many attempts have 
been made to explain reactions which are accompanied by 
negative heat-changes, and to bring them into unison with 
the thermal doctrine of affinity. But as all such attempts 
have been unsuccessful, the fundamental hypothesis of the 
doctrine loses much of its probability. And consequently its 
most ardent supporter, M. Berthelot, has relinquished his 
assertion that, by the action of the affinities of all the substances 


_ partaking in a reaction, those compounds are always formed 


which are accompanied by the greatest evolution of heat. He 
has modified his statement to this: that there is present “a 


* J. Thomsen, Thermoch. Untersuch. iv. p. 388, 
2M 2 


508 Prof. L. Meyer on the 


tendency towards (tends vers la) the production of such states 
of combination.””* But even in order to reconcile this state- 
ment with reactions distinctly accompanied by a negative 
heat-change, far-fetched and artificial, explanations of an 
unsatisfactory nature are necessary. Nevertheless, the funda- 
mental hypothesis—that the heat of combination is, in reality, 
affinity transformed into kinetic energy—might have passed 
as true for a much longer time, had not the progress of 
thermo-chemical research shown it to be thoroughly un- 
tenable. 

It will be remembered that Julius Thomsen} made use of 
the positive or negative heat-change accompanying a chemical 
reaction to determine the extent to which the reaction had 
proceeded ; and this was legitimate, owing to the fact that the 
heat-change is proportional to the quantity of matter which 
has altered its form of combination. While investigating 
the expulsion of acids from their salts in dilute aqueous solution 
by other acids, the very remarkable observation was made that 
that acid is by no means always the stronger which evolves 
the greatest heat on neutralization. For example, although 
sulphuric acid when neutralized in dilute aqueous solution 
gives rise to an evolution of heat surpassing by three thousand 
units that furnished by an equivalent amount of nitric or 
hydrochloric acid, yet it is only half as strong an acid as the 
latter ; that is, if equivalent amounts of nitric and sulphuric 
acids be mixed with an amount of caustic soda equivalent to 
one of them, the sulphuric acid enters into combination with 
only half as much soda as the nitric acid ; so that one third of 
the nitric acid remains in the free state, while two thirds of 


the sulphuric acid is free. There can be absolutely no doubt — 


that nitric acid is by far the stronger acid, although, judging 
from the thermal theory of affinity, sulphuric acid should be 
the stronger. 

While Thomsen was prosecuting his researches, it was 
generally held that the evolution of heat was an absolute mea- 
sure of affinity ; hence Thomsen devised the term “ avidity ”’ 
to express the “ tendency of an acid towards neutralization.” 
But this is nothing else than the real affinity of the acid 
towards the base, labelled with a special name to avoid con- 
fusion. Ostwald t, who confirmed and extended Thomsen’s 
researches by wholly different methods, named this quantity 
“relative affinity.” : 

It appeared, from the investigation of a great number of 


* M. Berthelot, Essaz de mécunique chimique, i. p. 421. 
+ Thermoch. Unters. 1. p. 97 et seg. 
{ T. pr. Chem. 1877, xvi. p. 385; xviii. p. 328, 


Ewolution of the Doctrine of Affinity. 509 


acids, that there was absolutely no connexion between avidity 
or relative affinity and heat of neutralization. Hven the order 
of magnitude of the two, when a number of instances is com- 
pared, is entirely different. The strongest of all acids—nitric 
acid—occupies only the nineteenth place among forty acids, 
when they are arranged in the order of their heats of neu- 
tralization ; while hydrofluoric acid evolves most heat on 
neutralization, although its avidity is only one twentieth of 
that of nitric acid ; and so with other instances. As it would 
be absurd to ascribe the greatest affinity to a base to an acid 
which is in great part expelled by another, it must be acknow- 
ledged that the fundamental hypothesis of the thermal doctrine 
of affinity is not justified by fact. 

But a further conclusion follows from Thomsen’s investi- 
gations, namely, that, while the heat of formation of compounds 
depends on the nature of their constituents, it does not, at 
least in many cases, depend on any special change, attraction, 
or affinity in which both constituents are concerned. This is 
most easily seen when the heat of formation of salts from strong 
bases and acids is considered. If, as Thomsen has experimen- 
tally shown, the total amount of heat evolved during the process 
of formation of a salt from the elements which it contains, and 
its solution in a large quantity of water be measured, the 
extremely remarkable fact is to be noticed that a definite dif- 
ference in composition involves a similarly definite difference 
in heat of formation, varying only within very narrow limits*. 

The heat of formation of a salt of lithium is, for example, in 
round numbers, always 11,400 calories greater than that of a 
salt of sodium with the same acid, and about 2000 calories 
greater than that of a salt of potassium; and so for other 
metals. This has been proved for the chlorides, bromides, 
iodides, hydrates, hydrosulphides, sulphates, dithionates, and 
nitrates of nineteen metals. If, on the other hand, the metal 
remain the same, but the acid radical be varied, a definite 
difference in the heat of formation is again to be observed for 
each acid radical. That of the bromides is always about 
21,800 calories less than that of the chlorides, and that of the 
iodides 52,300 less; while the chlorides invariably evolve 
during their formation about 200,000 calories less than the 
corresponding sulphates. The heat of formation of every salt 
may therefore be represented as the sum of certain numbers, 
each of which numbers is peculiar to one constituent group or 
element, and remains constant into whatever form of combi- 
nation that element or group enters. It is therefore possible, 


* Thomsen, Thermoch. Untersuch. iii. pp. 290, 456, 545; see also 
Lothar Meyer’s Moderne Theorien der Chemie, 5th edition, p. 448. 


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510 Prof. L. Meyer on the 


so soon as these constants have been definitely determined for 
each constituent group or element, to calculate the heat of 
formation of the salt in a manner similar to that by which the 
molecular weight of a compound may be deduced from the 
atomic weights of its constituents. 

This simple relation could not hold were the heats of for- 
mation of salts affected by the affinity of the constituents for 
each other ; for in such a case each constituent would contri- 
bute so much the more to the heat of formation the greater 
the affinity between it and the other constituent with which 
it combined. We must therefore believe that the evolution of 
heat, developed by the formation of a compound, results solely 
from the change of condition of the constituent elements. Similar 
laws, as Ostwald has shown, apply also to other changes of 
condition, e. g. to expansion or contraction, or to change of 
optical properties which accompany the formation of salts. 
These also can be calculated by simple addition; for their 
constituent numbers are constants belonging to each one of the 
reacting substances, and are independent of the nature of the 
other. 

The thermal theory of affinity has a changed complexion 
owing to this discovery. What was formerly attributed to 
the mutual action of several substances must now be regarded 
as change of condition of each individual substance, each being 
wholly independent of all others with which it may combine. 
The heat-change accompanying a chemical change must no 
longer be regarded as the conversion of potential energy into 
kinetic energy, owing to the mutual attraction of the atoms ; 
but it must be concluded that each substance, each atom, each 
compound, possesses its own peculiar store of available energy, 
capable of being increased or diminished by its entering into 
reaction, or by any change of condition. But this store of 
energy and its changes must in nowise be confounded with 
affinity—that is, with the reason of chemical change. For 
the amount of energy lost by a substance during reaction 
depends solely on its own nature, and on the kind of change 
which it undergoes; not on the nature of the substance 
through whose influence this change is produced. ‘To employ 
an old simile, affinity acts like the spark to the powder ; like 
the trigger which releases the weight by which, under the 
action of gravity, the pile is driven in. Just as in these and 
similar cases there is no proportionality between the effective 
cause and the final result, so also with chemical changes. 

Such considerations lead naturally to the old question re- 
garding the necessity of imagining any special affinity or 
attracting-power between atoms. And the more proofs are 


Evolution of the Doctrine of Affinity. 511 


furnished that not only the supposed affinity, but even the 
actually measured avidity, is an inherent property of each 
separate kind of matter, independent of any reaction with any 
other kind of matter, the more doubtful is the necessity. The 
more recent investigations of Ostwald* have shown that the 
most heterogeneous actions of acids, in influencing the chemical 
changes of various substances,—as, for example, in decompo- 
sing the amides, in forming the ethereal salts, in inverting the 
sugars, and moreover in influencing the electrical conduc- 
tivity,—are all dependent on the same constant, the affinity or 
avidity. If the ability of acids to act remain the same in rela- 
tion to so many different phenomena, the assumption appears 
justified that it is caused, not by mutual action, by attraction 
of one kind or another, but is in reality something peculiar to 
the nature of the acids themselves. 

There might be a temptation to believe that, in relinquish- 
ing the hypothesis of an attractive force between the atoms, 
we must also relinquish the possibility of any definite con- 
ception of the influences of the nature of reacting bodies in 
determining chemical changes. But this is by no means 
the case. For just as it was formerly supposed that the heat 
liberated during an act of combination was the equivalent in 
kinetic energy of so much potential energy due to the attrac- 
tion of the atoms, it is open to regard the atoms as particles 
in rapid motion, but devoid of attracting-power, the whole of 
whose store of energy consists in this motion, and is therefore 
kinetic; and it may therefore be assumed that such atoms 
may unite to form molecules, or that such molecules may 
otherwise react, owing to some as yet undiscovered relation 
between their modes of motion and velocities. Itis of course 
unnecessary to picture to ourselves attractive forces. They 
may or may not be conceived, but they are of no great im- 
portance to science. For my part I believe that a less 
restricted and prejudiced view of the facts is to be attained 
by abandoning the hypothesis of mutual attraction between 
atoms, and avoiding all reference to the unnecessary distinc- 
tion between the potential and the kinetic energy of the atoms. 

It may fall hard on many who have devoutly believed in 
the thermal theory of affinity, exalting it high above all facts, 
to see it dethroned ; perhaps here and there some will refuse 
to abandon it, like Berzelius with his electrochemical theory, 
chiefly prompted by the fear that when it is gone the kingdom 
of chaos, so painfully conquered, may againarise. Yet things 

* “Studien zur chemischen Dynamik,” Journ. prakt. Chem. xxvii. p.1; 


xxvii. p. 449; xxix. p. 385; xxxi. p. 807. ‘ Hlectrochemische Studien,” 
Ibid. xxx. p. 225; xxxi, p. 483; xxxil. p. 300; xxxiii. p. 352, 


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512 Ewolution of the Doctrine of Affinity. ‘ 


are not in so dangerous a plight as might at first appear. 
The thermal doctrine of affinity has had no more real influence 
on the steady experimental evolution of Chemistry than had 
the electrochemical theory of Berzelius. Both long held 
honourable places as great general truths ; attempts have been 
made with both to apply them to experimental details ; but, 
as frequently happens, theory and experiment did not agree, 
the theory has been calmly ignored, and we must trust to the 
future to make things plain. If, once again, a theory has 
unexpectedly proved untenable, once again the old course of 
events will be repeated ; attempts at generalization have been 
too soon made: “gestit enim mens exsilire ad magis gene- 
ralia ut acquiescat’’ (Bacon). Theory has attempted to 
precede fact; it has pursued a false path, and must wait 
until fact with quiet progress shows the way. 

Such is our present state of knowledge. But long before 
the thermochemical doctrine of affinity became untenable, the 
efforts of investigators had been directed to ascertaining the 
conditions on which chemical reactions depend, such as the 
influence of time, of temperature, of mass, and of solvents ; 
and to the measurement of the resulting changes in volume, 
in evolution of heat, and similar phenomena. Now that ex- 
periment has shown the fallacy of an attempt to deduce 
chemical change from the fundamental principles of thermo- 
chemistry, we hail with joy the appearance of a new, really 
kinetic, doctrine of affinity, which, quietly and unostenta- 
tiously making its way along the road of induction, holds out 
to us the prospect of a real knowledge of the essential nature — 
of chemical change. By its help also those numerous thermo- 
chemical observations, which were unable to lend support to a 
onesided theory, for which they furnished the sole basis, 
acquire for the first time their true meaning when viewed in 
connexion with all other phenomena accompanying chemical 
change. Thus, although one illusion more has been dissi- 
pated by this new evolution of the doctrine of affinity, yet 
science is enriched by the acquisition of a less hypothetical 
and more far-reaching and inclusive conception of the nature 
of chemical combination.* | 


* A short paper by Mr. Clarence A. Seyler, ‘On the Thermal Equi- 
valents” of some elements and groups, has been published in the ‘Chemical 
News’ of April 1, vol. lv. p. 147. 


[ 513 ] 


LX. Contributions to the Theory of the Constitution of the 
Diazoamido-Compounds. By RapHaEL Meupota, F.R.S., 
F.I.C., F.C.S., Professor of Chemistry in the City and Guilds 
of London Institute, Finsbury Technical College™. 


ie a series of investigations upon the diazoamido-compounds 

which have been carried out by me in conjunction with 
Mr. F. W. Streatfeild{, certain facts have been brought to 
light which are quite inexplicable by any of the formule at 
present in use; and it has therefore become necessary to 
reconsider the whole question of the chemical constitution of 
these interesting derivatives, which have taxed to the utmost 
the ingenuity of all those chemists who have concerned them- 
selves with their investigation. 

The formula which up to the present time has been generally 
adopted is due to Kekulé{, and is thus expressed in general 
terms :— | 


x) N,ANE CY, 


X and Y being similar or dissimilar radicals. The chief 
objection to this formula has hitherto been its asymmetrical 
character, which renders it incapable of explaining the remark- 
able observation of Griess§, which is now confirmed from 
many sides, viz. that the mized diazoamido-compounds, in 
which X and Y are dissimilar, are always identical whichever 
radical is first diazotized. This difficulty has been to some 
extent met by the suggestion of Victor Meyer||; and ina 
former paper by Mr. Streatfeild and myself{] it was shown 
that the results of our investigations, as far as these had been 
carried, could be explained by means of this hypothesis of the 
formation of intermediate additive compounds with a consi- 
derable show of reason. 

The extension of our work has, however, forced me to the 
conclusion that Kekulé’s formula does not adequately express 
all the known facts concerning the diazoamido-compounds ; 
and if this formula is, as I believe it must be, abandoned, the 
supplementary hypothesis is also rendered unnecessary. 

The evidence which has led to the present theoretical 


* Communicated by the Author. 
+ Journ. Chem. Soc., Trans. 1886, p. 624; and 1887, p. 102. 


{ Lehrbuch d. org. Chem. vol. ii. pp. 689,715, and 741; Zeit. f. Chem. 
1866, pp. 308, 689, and 700. 


§ Ber. deut. chem. Gesell. vii. (1874), p. 1619. 
| Ibid. xiv. (1881), p. 2447, note. 
4] Journ, Chem, Soc., Trans. 1887, p. 116. 


514 Prof. R. Meldola on the Theory of the 


discussion is briefly as follows :—By the action of diazotized 
metanitraniline upon paranitraniline an unsymmetrical diazo- 
amido-compound is obtained, which has a melting-point of 
211°. The same compound is obtained by reversing the order 
of combination, % e. by acting upon metanitraniline with 
diazotized paranitraniline. According to Kekuleé’s view, this 
substance could have only one of the two formule :-— 


I, 
(p) NO, e C,H, ° N, .NH. C,H, ° NO, (m), 


II. 
(m) NO,.C,H,.N,.NH.C,H,. NO, (p). 


By replacing the H-atom of the NH-group by ethyl an 
ethyl-derivative (of m.p. 148°) is formed; and this, on the 
same theory, could have only one of the two corresponding 
formulee :-— 


TI, ‘ 
(p) NO,.C,H,.N,.N(C,H;) . C,H, . NO, (m), 


IV. 
(m) NO, ° C,H, ° N, . N(C.H;) ° C,H, ° NO, (p). 


If the unsymmetrical compound had the formula L., its 
ethyl-derivative (II1.) might have been expected to be iden- 
tical with the compound produced by the action of diazotized 
paranitraniline upon ethylmetanitraniline ; if it had the for- 
mula II., its ethyl-derivative (1V.) might have been expected 
to be identical with the compound produced by the action of 
diazotized metanitraniline upon ethyl-paranitraniline. As - 
a matter of fact, it has been found that the ethyl-derwative of 
the unsymmetrical compound is identical with neither of the 
compounds prepared by the action of the diagotized nitranilines 
upon the ethyl-nitranilines. We have therefore to allow the 
existence of three isomeric ethyl-derivatives containing para- 
and metanitraniline residues, a fact which cannot be repre- 
sented by Kekulé’s formula. The properties of these and all 
the allied compounds prepared by us in the course of the 
inquiry are summarized in the following Table :— 


515 


Constitution of the Diazoamido-Compounds. 


Compound. Melting-point. Decomposed by cold HCl into 

1. Action of diazotized p-nitraniline A mixture of p- and m-nitrodiazo- 
upon m-nitraniline, or of diazotized 211° benzene-chlorides and m- and 
m-nitraniline upon p-nitraniline ...... | p-nitranilines. 
2. Action of diazotized p-nitraniline 993° p-nitrodiazobenzene-chloride and 
WPOM! P=OUCAMUMMNG ws ene ence scse ieriell p-nitraniline. 
8. Action of diazotized m-nitraniline 194° m-nitrodiazobenzene-chloride and 
upon m-nitraniline.........ceceeseeeees és m-nitraniline. 

; : “( A-mixture of p- and m-nitrodiazo- 
4, Prepared by the ethylation of com- 148° bézené-chlorides “acd sp-” andl 
POCOUL CHUN Cpa leenrancgrererrr rrr pricnatce aiden ; na 

m- ethylnitranilines. 

5. Prepared by the ethylation of com- 
pound No. 2, or by the action of di- 191°-192° p-nitrodiazobenzene-chloride and 
azotized p-nitraniline upon p-ethyl- p-ethylnitraniline. 
valli halla: Bea Sneha pe Un Habe hoen ase nee 
6. Prepared by the ethylation of com- 
pound No. 38, or by the action of 119° m-nitrodiazobenzene-chloride and 
diazotized m-nitraniline upon m- m-ethylnitraniline. : 
ethylnitraniline ......... eaigthinwsyuveeiees 
7. Prepared by the action of diazotized 11749-1759 { m-nitrodiazobenzene-chloride and. 
m-nitraniline upon p-ethylnitraniline i p-ethyInitraniline. 
8, Prepared by the action of diazotized 187° p-nitrodiazobenzene-chloride and 
p-nitraniline upon m-ethylnitraniline* m-ethylnitraniline. 


Decomposed by hot HCl into 


A mixture of p- and m-nitrochlor- 
benzenes and p- and m-nitranilines, 


p-nitrochlorbenzene and p-nitrani- 
line, 

m-nitrochlorbenzene and m-nitrani- 
line. 

A mixture of p- and m-nitrochlor- 
benzenes and p- and m-ethylnitra- 
nilines. 


— 


p-nitrochlorbenzene and p-ethylni- 
traniline. 


m-nitrochlorbenzene and m-ethylni- 
traniline. 


—_—— —_— 


m-nitrochlorbenzene and p-ethylni- 
traniline. 


. ee 


p-nitrochlorbenzene and m-ethylni- 
traniline. 


* Owing to the fact that this compound was distinct in appearance from the other ethyl-derivatives (Nos. 4, 5, 6, and 7) we were at first 


led to suppose that it was an amidoazo-compound, 


and that the unsymmetrical compound (No. 1) accordingly had 


the formula I., the iso- 


meric transformation when the metauitraniline was first diazotized being explained by Victor Meyer’s hypothesis (Journ. Chem. Soc. ‘Trans. 


1887, p. 116). A more searching investigation has, however, shown that the ethyl-derivative of m.p. 


diazo-compound. 


187° has all the characters of a true 


516 Prof. R. Meldola on the Theory of the 


If the ethyl-derivatives Nos. 7 and 8 are formulated on 
Kekulé’s type they would have the formule IV. and III. 
respectively, and thus no other expression is left for the ethyl- 
derivative No. 4. 

The conditions to be fulfilled by any formula proposed for 
the diazoamido-compounds are, therefore, (1) that it should 
be symmetrical so as to represent the identity of mixed diazo- 
amido-compounds, and (2) that it should be capable of repre- 
senting more than two isomeric alkyl-derivatives of mixed 
compounds. These conditions are certainly not met by the 
formula now in use ; and the objections which apply to this 
apply also to the alternative formula proposed by Strecker*:-— 


X.N.NH.Y 


This formula fails to explain the existence of more than two 
isomeric alkyl-derivatives of the unsymmetrical (mixed) com- 
pounds ; and is even less able than Kekulé’s of representing 
the identity of mixed compounds, since it is incapable of the 
rearrangement suggested by Victor’ Meyer. 

The first symmetrical formula proposed to explain the iden- 
tity of mixed diazoamido-compounds is due to Griesst, the 
discoverer of these compounds, who suggested that diazoamido- 
benzene and its analogues should be written according to the 
type :— 
uf C,H. —N—N—N—C,H, 


lobe oleae el 


This formula certainly explains the identity of mixed com- 
pounds, but is otherwise open to certain objections; since in 
the first place it represents diazoamidobenzene as a phenylene 
derivative, and in the next place it shows the presence of 
three N H-groups containing three replaceable hydrogen atoms. 
All our experiments upon the salts and alkyl-derivatives of 
the dinitrodiazoamido-compounds have shown, however, that 
only one replaceable H-atom is presentt. This formula, 
moreover, is not capable of explaining the easy resolution of 


* Ber. deut. chem. Gesell. iv. (1871), p. 786 ; Erlenmeyer, 2bed. vii. (1874), 
p- 1110, and xvi. (1883) p. 1457. Also Blomstrand, ibzd. viii. tasty 

Ol, 
, + Ber. deut. chem. Gesell. x. (1877), p. 528. 

{ These compounds give only monalkyl-derivatives; and the same 
appears to be the case with diazoamidobenzene, according to Messrs. 
Friswell and Green (Journ. Chem. Soc., Trans. 1886, p. 748), to whom I 
communicated the method of alkylization in the course of conversation, 
and who applied it to this compound successfully. 


Constitution of the Diazoamido- Compounds. 517 


diazoamido-compounds by acids, nor the production of mixed 
products from mixed compounds (see the foregoing Table). 
Another symmetrical formula has been proposed by Victor 
Meyer™, viz.:— 
X—N——N—Y 
AN 
H 


but this was abandoned by him as having but little probability. 
One of the greatest objections to this formula is that it fails to 
represent the N-atom which is attached to the replaceable 
H-atom as being also directly attached to one or the other of 
the aromatic radicals. The decomposition of the ethyl- 
derivatives of the dinitrodiazoamido-compounds by acids 
shows that this mode of attachment of the NH-group certainly 
exists (see the foregoing Table). 

In the course of the present investigations another symme- 
trical formula has suggested itself, which may be here given:— 


H ui 

KX _N—y or %—N—-Y 
aS N 
N 


This formula does not, however, appear to me to have any 
probability, as it fails to explain the decomposition of the 
diazoamido-compounds by acids, or the existence of isomeric 
alkyl-derivatives. Moreover, the formula of diazoamidoben- 
zene written on this type :— 


H 
C,H; e N e C,H; 
eX 
N=N 


would indicate a close relationship between this substance and 
the remarkably stable diphenylamine. The latter is not 
found, however, among the reduction-products of diazoamido- 
benzene ; and there is no experimental evidence of any kind 
in favour of such a relationship. 


Before proceeding to put forward my own views upon the 
constitution of these compounds it will be desirable to take a 


* Ber. deut. chem. Gesell. xiv. (1881), p. 2447, note. 


518 Prof, R. Meldola on the Theory of the 


general view of their characters, so as to gain a clear notion 
of all the conditions whicb have to be fulfilled by any proposed 
formula. These characters are summarized below, those com- 
pounds containing similar radicals being spoken of as “normal” 
compounds, and those containing dissimilar radicals as “‘mixed”’ 
compounds :— 

(1) Normal compounds are prepared by diazotizing an 
amine, X.NHb., and acting with the diazo-salt upon another 
molecule of the same amine, X . NH4, or, what amounts to the 
same thing, one molecule of nitrous acid may be made to act 
upon two molecules of X . N Hg. 


(2) Mixed compounds are obtained by diazotizing an amine, 
X .NH,, and acting with the diazo-salt upon one molecule of 
another amine, Y.NH,. The same compound results if the 
order of combination is reversed. 


(3) The diazoamido-compounds, both mixed and normal, 
contain one atom of hydrogen easily replaceable by metals 
and alkyl radicals. If the aromatic radicals contain strongly 
acid groups (such as NO,), the resulting diazoamido-com- 
pounds may be distinct monobasic acids. 


(4) Normal compounds are resolved by acids into their 
constituents, the diazo-salt and amine. 


(5) Mixed compounds are resolved by acids into a mixture 
of the two bases from which they are derived, and a mixture 
of the two diazo-salts corresponding to these two bases. 


(6) Alkyl derivatives of normal compounds may be pre- 
pared in two ways:— 

a. By the action of a diazotized amine, X.NH,, upon 
the alkylamine of the same base, X. NHR. 

8. By the direct alkylization of the normal diazoamido- 
compound. 

(7) The alkyl-derivatives of normal compounds are decom- 
posed by acids into their constituents, the diazo-salt and 
alkylamine. 

(8) Alkyl-derivatives of mixed diazoamido-compounds are 
formed by the direct alkylization of these compounds (see 
group 2). 

(9) Another group of mixed alkyl-derivatives can be pre- 
pared by the action of a diazotized amine, X . NHg, upon the 
alkyl-derivative of a dissimilar amine, Y. NHR. These com- 
pounds are isomeric with those of the preceding group. 

(10) Mixed alkyl-derivatives of group (8) are resolved by 
the action of acids into a mixture of the two diazo-salts and 
the two alkylamines. 


pe I ce tsi ci 


Constitution of the Diazoamido- Compounds. 519 
(11) Mixed alkyl-derivatives of group (9) are resolved by 


acids into their constituents, the diazo-salt and the alkylamine, 
but not into a mixture of diazo-salts and alkylamines, as is 
the case with the compounds of group 9*. 

(12) Normal compounds, by the action of weak reducing 
agents, are reduced to the original amine X.NH,, and the 
hydrazine X.N.H;. Mixed compounds give, on reduction, 
the base X .NH, and the hydrazine Y.N.,H;, or the base 
Y .NH, and the hydrazine X . N,H3f. 

(13) Alkyl-derivatives of normal compounds reduce to the 
hydrazine X.N.H; and the alkylamine X.NHRf. Alkyl- 
derivatives of mixed compounds give, on reduction, the 
hydrazine X- or Y . N.H3;, and the alkylamine Y- or X . NHR. 
This production of alkylamines indicates that the N-atom 
which is in combination with the alkyl-radical is also attached 
to the aromatic nucleus ||. 

From the foregoing summary it will be seen that the mixed 
diazoamido-compounds and their alky!-derivatives display the 
most striking characters, and are of special importance to the 
present discussion, because it is in the attempt to formulate 
these compounds on Kekulé’s plan that the greatest difficulties 
are encountered. In view of the objections which apply to 
all the formulz hitherto proposed it has been no easy matter 
to suggest any alternative formula ; but I believe that the true 
solution of the problem will be arrived at by regarding phenyl 
as a triatomic radical, C,H;!, instead of monatomic, as has 
always been assumed in previous formule. This suggestion 
is in accordance with Fittig’s theory of the constitution of 


* To this class, in addition to the ethyl-derivatives of m.p. 174°-175° and 
187° (Nos. 7 and 8 in the Table), belong the two following compounds :— 
(1) produced by the action of diazotized p-toluidine upon ethylaniline, 
and (2) prepared by the action of diazotized aniline on ethyl-p-toluidine. 
These two compounds are zsomeric; the first being decomposed by acids 
(hot) into p-cresol and ethylaniline, and the second into phenol and 
ethyl-p-toluidine (Nolting and Binder, Bull. Soc. Chim. vol. xlii. p. 341 ; 
Gastiger, id. p. 342). . This pair of isomerides is completely analogous 
to our two ethyl-derivatives (Nos. 7 and 8), which they resemble in their 
mode of decomposition. 

t+ Thus the compound produced by the action of diazotized aniline 
upon p-toluidine or the reverse gives, on reduction, phenylhydrazine and 
p-toluidine (Nolting and Binder, loc. cit. p. 336). 

¢ Thus the compound obtained by the action of diazotized aniline on 
methylaniline reduces to phenylhydrazine and methylaniline (doc. ctt.). 

§ Thus the compound prepared by Gastiger by the action of diazotized 

aniline on ethyl-p-toluidine reduces to phenylhydrazine and ethyl-p- 
toluidine (Joc. cit. p. 342). 
__ || The presence of substituents in one or both aromatic radicals may 
interfere with the formation of hydrazines; in such cases the correspond- 
ing substituted amines are formed, or, if the substituent is NO,, the cor- 
responding diamines. 


520 Prof. R. Meldola on the Theory of the 


quinone, this compound being regarded by him as a double 
ketone :— 


Phenylene, according to this view, must be regarded as a 
tetratomic radical, and a slight extension of the same view 
enables us to consider phenyl as triatomic:— 


H 
ue ee 
& 
HC ‘cH HCO ‘OH 
lene ens 
HC OH HC OH 
Ww NA 


If this assumption be made, it then becomes possible to 
construct formule for the mixed diazoamido-compounds which 
meet all the requirements of the case, and which must, there- 
fore, commend themselves to the notice of all chemists who, 
like myself, have been puzzled to explain the behaviour of 
these compounds in accordance with the existing theoretical 
notions. The formula now proposed may be written in two 
ways:— 

N:N N.N 


a 
mei 


Of these two formule I am disposed to attach the greater 
weight to the first because it indicates the presence of the 
very stable azo-group, —N : N—; and this is in accordance 
with the general character of the compounds, which decom- 
pose under the influence of acids or of reducing agents in 
such a manner that the N-atoms of the azo-group always 
remain in combination, either in the form of a diazo-salt or a 
hydrazine. On the other hand, the second formula indicates 
the presence of the group =—C—N . N=C=; and this might 


Constitution of the Diazoamido-Compounds. 521 


be expected to split asunder between the N-atoms more 
readily on reduction or on decomposition by acids than is 
shown to be the case by experiment. 

According to the proposed formula the unsymmetrical 
compound cf m.p. 211° (No. 1 in the table) and its ethyl- 
derivative of m.p. 148 (No. 4 in the table) would be thus 


written :— 


vee : aN 
(p) NO, . CoH, C.H,.NO, (m), 
Sy 
H 
v7 2 NK 
(p) NO,. CeHy’ >C.Hy . NO, (m). 
<3 Va 
CH, 


It may now be pointed out how far these formule are in 
harmony with the known characters of the mixed diazoamido- 
compounds. In the first place it is obvious that the formula 
is symmetrical, and thus explains the identity of the com- 
pounds irrespective of the order of combination. Putting P 
for the p-nitraniline residueand M for the m-nitraniline residue, 
this fact may be thus represented without assuming the for- 
mation of any intermediate additive compound:— 


ae : tine 
Reh 


H 
Consider in the next place the decomposition by hydro- 
chloric acid :— 


N:N a cc et 
B x Si eg ee M 
oN N ye N d 
H H 


If separation took place along the line ab, the products would 
Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 2N 


522 —-~ Prof, R. Meldola on the Theory of the 


be p-nitrodiazobenzene-chloride and m-nitraniline ; along cd, 
the products would be m-nitrodiazobenzene-chloride and p- 
nitraniline. As a matter of fact all four products are obtained, 
so that decomposition must take place in both directions. The 
same explanation obviously applies to the mixed decompo- 
sition-products of the ethyl-derivative. Again, the formula 
shows the presence of the one replaceable H-atom in combi- 
nation with the N-atom which is attached to the aromatic 
radicals. The question of the existence of more than two 
isomeric alkyl-derivatives will be considered subsequently. 
The formula which has now been suggested for the mixed 
diazoamido-compounds derived from p- and m-nitraniline 
can be applied with equal success to all other mixed com- 
pounds. Thus, to take examples of those compounds whose 


products of decomposition have been studied :— 


N:N. 


Both these compounds were discovered and their decompo- 
sition products studied by Griess : the first is obtained by the 
action of diazotized aniline upon p-toluidine, or of diazotized 
p-toluidine upon aniline; the second is similarly produced 
from aniline and amidobenzoic acid. When heated with acids 
the first compound gives a mixture of aniline, p-toluidine, 
phenol, and p-cresol (Nolting and Binder)* ; and the second 
compound under similar circumstances gives aniline, phenol, 
oxybenzoic, and amidobenzoic acid (Griess). The products 
in these cases indicate separation along both planes of decom- 
position ab and cd. 

The explanation of the existence of more than two isomerie 
alkyl-derivatives of mixed diazoamido-compounds is closely 
connected with the question whether the new formula can be 
applied to the normal diazoamido-compounds. The following 
considerations will show that these compounds cannot be for- 
mulated on the new type :— 

As a type of the normal compounds let us consider that 
derived from p-nitraniline (No. 2 in the table). If this had 
the new formula it (and its ethyl-derivative) would have to 
be written :— 


* This compound reduces to phenyl-hydrazine and p-toluidine, thus 
indicating a preferenco of the N,- (and therefore the NH-NH,) group to 
remain attached to the more positive radical. The separation is in this 
case along ab only. 


a —_ 


Constitution of the Diazoamido- Compounds. 523 
N ° oN 
(p) NO,. C,H, 


Ne 


H 


Ne N 
(p) NO,. on Nott . NO; (p). 
ie. 
CoH; 

Now the ethyl-derivative (No. 5 in the table) is prepared 
by the direct ethylation of the compound itself, and also by 
the aetion of diazotized p-nitraniline upon p-ethylnitraniline. 
Analogy would therefore lead us to suppose that if the ethyl- 
derivative had the above constitution, the other ethyl-deriva- 
tives, prepared by the action of diazotized p-nitraniline upon 
m-ethylnitraniline (m.p. 187°) and of diazotized m-nitraniline 
upon p-ethylnitraniline (m.p. 174°-175°), would have a similar 


C,H. NOs (p), 


constitution :— 
N:N 
(p) NO, Hewes eet NO,\ (mn), 
Se i seh 
C,H, 
N 


OH... 


But these two formule are identical with one another and 
with that of the ethyl-derivative of m.p. 148°, whereas their 
melting-points and mode of decomposition show most con- 
clusively that the three compounds are isomeric and not 
identical. It must, therefore, be concluded that the formula 
how proposed does not apply to the normal compounds, and 
the suggestion at once arises whether these may not be the 

2N2 


524 Prof. R. Meldola on the Theory of the 


true representatives of Kekulé’s type. In answer to this I 
may point out that, as far as the experimental evidence at 
present goes, the normal compounds and the analogous ethyl- 
derivatives may be written on Kekulé’s type:— 


(p) NO,. CoH. Np. NH . OgH, . NO, (p), 
m.p. 223°, 


(m) NO, . C,H, .Ny..NH. C,H, . NO, (m), 
m.p. 194°, 


(p) NO, . C5H,. Nz .N(C,H,) . C,H, . NO, (p), 
m.p. 191°-192°, 


(m) NO, . CsH, . N;./N(C,H;) . CoH, . NO; (m), 
m.p. 119°. 


(p) NO, . C,H, . N; - N(C2H;) . C,H, . NO, (m), 
mp. VSi?; 


(m) NO, . CH, .. Ny ..N(Q,Hg) . CH. NOz (p). 
m.p. 174°-175°. 


The modes of decomposition of these compounds are ex- 
plained by the above formule by supposing the planes of 
separation to be along the dotted lines ; and it further appears 
that the isomerism of the three ethyl-derivatives of m.p’s 148°, 
174°-175°, and 187° (Nos. 4, 7 and 8 in the table) may be 
explained by the different formule ascribed to these com- , 
pounds respectively. | 

But although Kekulé’s formula may pass muster for the 
normal compounds, we are not necessarily reduced to this as 
a final expression ; and I am strongly inclined to the belief — 
that it will have to be abandoned also in the case of these 
compounds. In the first place, as there is a great resemblance 
in character between the normal and the mixed compounds, 
analogy leads me to suppose that their constitutions are not 
so widely different, as appears from the two modes of formu- 
lation :— 


N:N 
Lo ae XN Na 
Pe and X.N>,. New 


H 


Constitution of the Diazoamido- Compounds. 525 


In the next place the group —N=N—NH— assumed to 
be present, according to the prevailing view, has always 
seemed to me to be a most improbable arrangement of N- 
atoms, and without any analogy among chemical compounds. 
Those compounds in which three combined N-atoms are pre- 
sent are only stable when the N-atoms form a closed chain, 
as in Griess’s benzeneimide :— 


or in the azimidobenzene of this same author*:— 


N 


H 


From these considerations I am led to conclude that an 
open chain of three nitrogen atoms does not exist in any of 
the diazoamido-compounds, and the remarkable stability of 
the dinitrodiazoamido-compounds in the presence of alkalis Tf 
certainly supports this view. 

The formula which I now venture to suggest for the normal 
compounds is, as far as I can see, at any rate as equally 
capable as Kekulé’s of representing the characters of these 
compounds, and at the same time indicates the analogy of 
these to the normal compounds. It has moreover the advan- 
tage of doing away with the assumption of the open chain of 


N-atoms :— 


< 
NH.X 


According to this formula the preceding compounds would 
be written :— 


* I have accepted this formula rather than the alternative one 


N 
eu | \NH, because, according to Boessneck (Ber. xix. 1886, p. 1757), 

\n~Z 
acetorthotoluylene-diamine yields acetazimidotoluene by the action of 
nitrous acid. The N.C,H,O-group must, therefore, be attached to the 
aromatic nucleus, and this acetyl compound gives azimidotoluene by 
hydrolysis, so that the NH-group must also be attached to the aromatic 
nucleus. 

Tt These compounds can be boiled with strong potash solution for days 
without undergoing any alteration (Journ. Chem. Soc., Trans. 1886, 
p- 627). Even the simpler compounds like diazoamidobenzene are much 
more stable in neutral or alkaline solutions than is generally supposed. 


526 Prof. R. Meldola on the Theory of the 
N:N N:N 


(p or m) x0.00n Kf (p or m) nose. 


SS NE.C,H,.NO, (p or m) 


N 
N:N N:N 
(p) No.0 (m) NO,.C,H, < 
SSC ee alle / CH; 
N\C-H..NO, (m) N\C,Hi-NO, (p) 
m.p. 187°. m.p. 1749-175°. 
[The planes of decomposition are represented by the dotted 


lines. 

Kt sl readily be seen that these formule are in harmony 
with the characters of the compounds which they represent. 
Thus, taking the products of decomposition of the last pair of 
isomeric ethyl-derivatives, the 187° m.p. modification is re- 
solved into p-nitrodiazobenzene-chloride and m-ethy]nitrani- 
line, while the other modification yields m-nitrodiazobenzene- 
chloride and p-ethylnitraniline. 

The corresponding pair of isomeric ethyl-derivatives con- 
taining aniline and toluidine residues, prepared by Noélting 
and Binder and by Gastiger, may be similarly formulated :— 


N:N N:N 


[e- (p) ome 
ni/ Ooi = é C,H, 


\C_H, (p) MY C,H, f 


These compounds, which have already been referred to, are 
decomposed by acids (hot); the first into phenol and p-ethyl- 
toluidine, and the second into p-cresol and ethylaniline. 

The mixed compounds containing both aromatic and fatty 
radicals are in all respects analogous to the normal compounds, 
and, according to Wallach*, behave like these on decompo- 
sition. Thus the typical compounds of this group, first pre- 
pared by Baeyer and Jager} by the action of diazobenzene 
salts upon ethylamine, dimethylamine, and piperidine, may be 
written :— 


N:N N 


N:D N:N 
Hs: we C,H; <| C,H; Gees - 
7 NE.C,H, i N(CH). = Ne 


* Ineb. Ann, vol. ccxxxv. p. 2388. 
+ Ber. deut. chem. Gesell. viii. (1875) pp. 148, 898. 


Constitution of the Diazoamido-Compounds. 527 


A few remarks may here be made in connexion with the 
transformation of diazoamido- into amidoazo-compounds. It 
has always been supposed hitherto that this transformation is 
preceded by a resolution of the diazoamido-compound into its 
constituents*. This may be the case in the presence of excess 
of acid; but it is doubtful whether such a resolution occurs 
when the diazoamido-compound (say diazoamidobenzene) is 
allowed to stand in the presence of excess of aniline containing 
only a small quantity of aniline hydrochloride. Itis well known, 
however, that such a mixture will effect the complete conver- 
sion of diazoamido- into amidoazobenzene in the course of a 
few hours, especially if aided by heat. If the formula of 
diazoamidobenzene is written according to the present view, 
it will be seen ‘that a slight rearrangement of the ‘ bonds” 
would convert it into a symmetrical compound of the type 
already proposed for the mixed diazoamido-compounds ; 
thus :— 


N:N N:N. 
C,H; in | C,H; a > C,H; 
NH.O,H; it 


It seems not improbable that such a symmetrical compound 
may precede the formation of amidoazobenzene, the separation 
(accompanied by the migration of the H-atom to the NH- 
group ){ occurring along one of the dotted lines. 

Although the formule now proposed for the normal diazo- 
amido-compounds appear capable of meeting all the require- 
ments of the case, it will be of interest to point out that other 
molecular arrangements in which phenyl functions as a tri- 
atomic radical are possible :— 


at Be 

xX ue 

SN ye SN :N.X 
iF i; Kil. 


* See the last contribution to this question by Wallach (Zeb. Ann. 
vol. eexxxy. p. 238). Iam bound to express the opinion, however, that 
the suggestion there thrown out does not materially add to the solution 
of the problem (Proc. Chem. Soc. 1887, p. 27). 

+ Such a transference of hydrogen is analogous to that which takes 
place when hydrazobenzene, C,H;. NH.NH.C,H.,, is converted into ben- 
zidine, NH,.C,H,.C,H,.NH,, by the action of acids. No previous 
resolution into constituents has ever been supposed in this case. The 
transformation appears rather to be of the nature of a rotation of the ben- 
zene rings, and there is reason for believing that a similar rotation takes 
place in the decomposition of mixed diazoamido-compounds by acids. 
This point cannot be discussed, however, until the evidence is more 
complete. 


928 Constitution of the Diazoamido- Compounds. 


All these formule are, however, more or less open to objec- 
tion, and need not be further discussed at present. It will 
suffice to mention that No. ILI., which at first sight might 
appear the most probable of the three, is incapable of repre- 
senting such compounds as diazobenzenedimethylamide. 


The views now advanced concerning the constitution of this 
interesting group of compounds open up suggestive lines of 
investigation i in the direction of isomerism as connected with 
position. In the formula representing phenyl as a triad 
radical previously given, the free bonds have been represented 
in the para-position, because the ortho-quinone of the benzene 
series does not appear to be capable cf existence. But the 
formula obviously allows the possibility of such an ortho- 
arrangement :— 


a 
hes Cus 
aN 
HO cs. He e2 
ip a | 
HC CH HO CH 
WA 77, 
(‘ | ( 
i tl 


The para-position of the substituents is, however, in har- 
mony with the known behaviour of diazoamidobenzene, the 
isomeric amidoazobenzene having its substituents in the para- 
position. It seems probable, therefore, if there is anything 
in the previously expressed view concerning this isomeric 
transformation, that at least in this diazoamido-compound 
the substituents —N:N— and =NH= are in the para- 
position. 

Summing up ihe general results of the present discussion, 
it appears to me that the formula now proposed for the mixed 
diazoamido-compounds is the only one that has hitherto been 
in harmony with all the known characters of these compounds, 
and as such it is at any rate worthy of serious consideration 
by chemists. Analogy leads to the belief that the normal 
compounds have a similar, or at least a partly similar con- 
stitution ; but the evidence is not so satisfactory in these 
cases, owing to the similarity of the two halves of the mole- 
cule, which renders it impossible to follow the course of 
decomposition with the same certainty as in the mixed com- 
pounds. Before, however, the whole question of the consti- 
tution of the diazoamido-compounds can be completely worked 
out on the lines here suggested, it will be necessary to have a 


Maximum and Minimum Energy in Vortex Motion. 529 


much larger body of experimental evidence. Further inves- 
tigations in the required direction are now in progress in the 
laboratory of the Finsbury Technical College. 

Postscript.—Since writing the foregoing paper, the detailed 
evidence which has led to the conclusion that the ethyl- 
derivative of m.p. 187° is a diazo-compound has appeared in 
a communication to the Chemical Society (Journ. Chem. Soc., 
Trans. 1887, p. 434). Additional evidence of the production of 
mixed compounds on the decomposition of mixed diazoamido- 
derivatives is given in a recent paper by Heumann and Oeco- 
nomides (Ber. 1887, p. 904). These authors find that the 
mixed compound C,H;.N;H.C,H;, on being heated with 
phenol, gives a mixture of aniline and p-toluidine together 
with oxyazobenzene and p-tolueneazophenol. 


LXI. On the Stability of Steady and of Periodic Fluid Motion 
(continued from May number).—Mazimum and Minimum 
Energy in Vortex Motion*. By Sir Wituiam Taomson, 
1A ia ep 


10. ge condition for steady motion of an incompressible 

inviscid fluid filling a finite fixed portion of space 
(that is to say, motion in which the velocity and direction of 
motion continue unchanged at every point of the space within 
which the fluid is placed) is that, with given vorticity, the 
energy is a thorough maximum, or a thorough minimum, or 
aminimax. The further condition of stability is secured, by 
the consideration of energy alone, for any case of steady 
motion for which the energy is a thorough maximum or a 
thorough minimum ; because when the boundary is held fixed 
the energy is of necessity constant. But the mere consi- 
deration of energy does not decide the question of stability 
for any case of steady motion in which the energy is a 
minimax. 

11. It is clearf that, commencing with any given motion, 
the energy may be increased indefinitely by properly-designed 
operation on the boundary (understood that the primitive 
boundary is returned to). Hence, with given vorticity, but 
with no other condition, there is no thorough maximum 
of energy in any case. There may also, except in the case 
of irrotat‘onal circulation in a multiplexly continuous vessel 


* Being a communication read before the British Association, Section A, 
at the Swansea Meeting, Saturday, August 28, 1880, and published in the 
Report for that year, p.473; and in ‘ Nature,’ Oct. 28, 1880. Reprinted 
now with corrections, amendments, and additions. . 

+ See also §§ 3 to 9 above. 


# 


Fae Sir W. Thomson on Maximum and 


referred to in § 3 III. above, be complete annulment of the 
energy by operation on the boundary (with return to the pri- 
mitive boundary), as we see by the following illustrations :— 

(a) Two equal, parallel, and oppositely rotating, vortex 
columns terminated perpendicularly by two fixed parallel 
planes. By proper operation on the cylindric boundary, they 
may, in purely two-dimensional motion, be thoroughly and 
equably mixed in two infinitely thin sheets. In this condition 
the energy is infinitely small. 

(b) A single Helmholtz ring, reduced by diminution of its 
aperture to an infinitely long tube coiled within the enclosure. 
In this condition the energy is infinitely small. 

(c) A single vortex column, with two ends on the boundary, 
bent till its middle infinitely nearly meets the boundary; and 
further bent and extended till it is broken into two equal and 
opposite vortex columns, connected, one end of one to one end 
of the other, by a vanishing vortex lgament infinitely near 
the boundary ; and then further dealt with till these two 
columns are mixed together to virtual annihilation. } 

12. To avoid, for the present, the extremely difficult general 
question illustrated (or suggested) by the consideration of such 
cases, confine ourselves now to two-dimensional motions in a 
space bounded by two fixed parallel planes and a closed 
cylindric, not generally circular cylindric, surface perpen- 
dicular to them, subjected to changes of figure (but always 
truly cylindric and perpendicular to the planes). Also, for 
simplicity, confine ourselves for the present to vorticity either 
positive or zero, in every part of the fluid. It is obvious that, 
with the limitation to two-dimensional motion, the energy 
cannot be either infinitely small or infinitely great with any 
given vorticity and given cylindric figure. Hence, under 
the given conditions, there certainly are at least two stable 
steady motions—those of absolute maximum and absolute 
minimum energy. ‘The configuration of absolute maximum 
energy clearly consists of least vorticity (or zero vorticity, if 
there be fluid of zero vorticity) next the boundary and greater 
and greater vorticity inwards. The configuration of absolute 
minimum energy clearly consists of greatest vorticity next 
the boundary, and less and less vorticity inwards. If there 
be any fluid of zero vorticity, all such fluid will be at rest 
either in one continuous mass, or in isolated portions sur- 
rounded by rotationally moving fluid. For illustration, see 
figs. 4 and 5, where it is seen how, even in so simple a case as 
that of the containing vessel represented in the diagram, there 
can be an infinite number of stable steady motions, each with 
maximum (though not greatest maximum) energy ; and also 


Minimum Energy in Vortex Motion. 531 


an infinite number of stable steady motions of minimum 
(though not least minimum) energy. 

13. That there can be an infinite number of configurations of 
stable motions, each of them having the energy of a thorough 
minimum (as said in § 12), we see, by considering the case 
in which the cylindric boundary of the containing canister 
consists of two wide portions communicating by a narrow 
passage, as shown in the drawings. If such a canister be 
completely filled with irrotationally moving fluid of uniform 
vorticity, the stream-lines must be something like those indi- 


cated in fig. 4. 


Fig. 4. 


- Hence, if a not too great portion of the whole fluid is irro- 
tational, it is clear that there may be a minimum energy, and 
therefore a stable configuration of motion, with the whole of 
this in one of the wide parts of the canister ; or the whole in 
the other ; or any proportion in one and the rest in the other. 


Hig, OD. 


Single intersection of stream-lines in rotational motion 
may be at any angle, as shown in fig. 4. It is essentially 
at right angles in irrotational motion, as shown in fig. 9, 
representing the stream-lines of the configuration of maai- 
mum energy, for which the rotational part of the liquid is 
in two equal parts, in the middles of two wide parts of the 
enclosure. There is an infinite number of configurations of 


532 Sir W. Thomson on Maximum and 


maximum energy in which the rotational part of the fluid is 
unequally distributed between the two wide parts of the 
enclosure. sVeLuM 

14. In every steady motion, when the boundary is cir- 
cular, the stream-lines are concentric circles and the fluid is 
distributed in co-axial cylindric layers of equal vorticity. In 
the stable motion of maximum energy, the vorticity is greatest 
at the axis of the cylinder, and is less and less outwards to the 
circumference. In the stable motion of minimum energy the 
vorticity is smallest at the axis, and greater and greater out- 
wards to the circumference. To express the conditions sym- 
bolically, let T be the velocity of the fluid at distance r from 
the axis (understood that the direction of the motion is per- 
pendicular to the direction of r), and let a be the radius of the 
boundary. ‘The vorticity at distance r is 


(742) 
*\ pr dr} 


If the value of this expression diminishes from r=0 to r=a, 
the motion is stable, and of maximum energy. If it increases 
from r=O0 to r=a, the motion is stable and of minimum 
energy. If it increases and diminishes, or diminishes and 
increases, as 7 increases continuously, the motion is unstable”. 

15. As a simplest subcase, let the vorticity be uniform 
through a given portion of the whole fiuid, and zero through 
the remainder. In the stable motion of greatest energy, the 
portion of fluid having vorticity will be in the shape of a cir- 
cular cylinder rotating like a solid round its own axis, coin- 
ciding with the axis of the enclosure ; and the remainder of 
the fluid will revolve irrotationally around it, so as to fulfil the 
condition of no finite slip at the cylindric interface between 
the rotational and irrotational portions of the fluid. The 
expression for this motion in symbols is 


c= Cro noms — to 2p: 
meee 


r 


and from r=b to r==a. 


* This conclusion I had nearly reached in the year 1875 by rigid mathe- 
matical investigation of the vibrations of approximately circular cylindric 
vertices ; but 1 was anticipated in the publication of it by Lord Rayleigh, 
who concludes his paper “ On the Stability, or Instability, of certain Fluid 
Motions” (‘ Proceedings of the London Mathematical Society,’ Feb. 12, 
1880) with the following statement :—“ It may be proved that, if the fluid 
move between two rigid concentric walls, the motion is stable, provided 
that in the steady motion the rotation either continually increases or 
continually decreases in passing outwards from the axis,’—which was 
unknown to me at the time (August 28, 1880) when I made the com- 
munication to Section A of the British Association at Swansea. 


Minimum Energy in Vortex Motion. 533 


16. In the stable motion of minimum energy the rotational 
portion of the fluid is in the shape of a cylindric shell, en- 
closing the irrotational remainder, whichin this case is at rest. 
The symbolical expression for this motion is 


T=0, when r< /(a?—0?), 


a? — §? 


and T=E(r— ), when r> 4/(a?—6?). 

17. Let now the liquid be given in the configuration (14) 
of greatest energy, and let the cylindric boundary be a sheet 
of a real elastic solid, such as sheet-metal with the kind of 
dereliction from perfectness of elasticity which real elastic 
solids present ; that is to say, let its shape when at rest be a 
function of the stress applied to it, but let there be a resist- 
ance to change of shape depending on the velocity of the 
change. Let the unstressed shape be truly circular, and let 
it be capable of slight deformations from the circular figure 
in cross section, but let it always remain truly cylindrical. 
Let now the cylindric boundary be slightly deformed and left 
to itself, but held so as to prevent it from being carried round 
by the fluid. The central vortex column is set into vibration 
in such a manner that longer and shorter waves travel round 
it with less and greater angular velocity*. These waves cause 
corresponding waves of corrugation to travel round the cylin- 
dric bounding sheet, by which energy is consumed, and 
moment of momentum taken out of the fluid. Let this pro- 
cess go on until acertain quantity M of moment of momentum 
has been stopped from the fluid, and now let the canister run 
round freely in space, and, for simplicity, suppose its material 
to be devoid of inertia. The whole moment of momentum 
was initially— 

mE D(a? 40?) ; 
it is now 


me b? (a? xa 3b”) — 


and continues constantly of this amount as long as the 
boundary is left free in space. The consumption of energy 
still goes on, and the way in which it goes on is this: the 
waves of shorter length are indefinitely multiplied and exalted 
till their crests run out into fine laminz of liquid, and those 
of greater length are abated. Thus a certain portion of the 
irrotationally revolving water becomes mingled with the 
central vortex column. The process goes on until what may 


* See ‘Proceedings of the Royal Society of Edinburgh’ for 1880, 
or ‘ Philosophical Magazine’ for 1880, vol. x. p. 155: “ Vibrations of a 
Columnar Vortex :” Wm. Thomson. 


534 Sir W. Thomson on Maximum and 


be called a vortex sponge is formed; a mixture homogeneous* 
on a large scale, but consisting of portions of rotational and 
irrotational fluid, more and more finely mixed together as 
time advances. The mixture is altogether analogous to the 
mixture of the white and yellow of an egg whipped together 
in the well-known culinary operation.’ Let b’ be the radius 
of the cylindric vortex sponge, and @ its mean molecular 
rotation, which is the same in all sensibly large parts. 
Then, b being as before the radius of the original vortex 
column, we have 


TOs trom res0 tone 


and 
T= C0" ie from r=b' to r=a% 
where 
C= Cb? /b”, 
and 
Pee iD 
Pa cae 


18. Once more, hold the cylindric case from going round 
in space, and continue holding it untilsome more moment of 
momentum is stopped from the fluid. Then leave it to itself 
again. ‘The vortex sponge will swell by.the mingling with it 


* Note added May 13, 1887.—I have had some difficulty in now proving 
these assertions (§§ 17 and 18) of 1880. Here is proof. Denoting for 
brevity 1/2 of the moment of momentum by p, and 1/2m of the energy 
by e, we have | 


a 
=\ Tr.rdr, and e=i("T?. rdr. 
B i ; ai A 
The problem is to make e least possible, subject to the conditions: (1) that 
p has a given value; (2) that 
T =: 
(5+ Z)Ee, and 20 ; 
and (3) that when r=a, T= (6?/a; this last condition being the resultant of 
ms iS ae 
\3 - == =) r di =J Srar, 


which expresses that the total vorticity is equal to that of ¢ uniform within 
the radius 6. The configuration described in the last three sentences of 
§ 17 and the first three of § 18 clearly solve the problem when 


M <3n(b?(a?—b?); or p>2 (67a. 
The fourth sentence of § 18 solves it when 

M = 30 (0°(a? —0?); or p= 1 ¢07a2. 
The second paragraph of § 18 solves it when 

M> 37¢b?(a?— 6"); or p <1 ¢Ba?. 


Minemum Energy in Vortex Motion. 5385 


of an additional portion of irrotational liquid. Continue this 
process until the sponge occupies the whole enclosure. 

After that continue the process further, and the result will 
be that each time the containing canister is allowed to go 
round freely in space, the fluid will tend to a condition in 
which a certain portion of the original vortex core gets filtered 
into a position next to the boundary, (within a distance from 
the axis which we shall denote by c), and the fluid in this space 
tends to a more and more nearly uniform mixture of vortex 
with irrotational fluid. This central vortex sponge, on repe- 
tition of the process of preventing the canister from going 
round, and again leaving it free to go round, becomes more 
and more nearly irrotational fluid, and the outer belt of pure 
vortex becomes thicker and thicker. The resultant motion is 
“ii 2 2 2 

ie FOP IG, 


272 
Tatr—2—, for r>c; 


and the moment of momentum is 
eTCY tae) (ere yt 


The final condition towards which the whole tends is a belt 
constituted of the original vortex core now next the boundary ; 
and the fluid which originally revolved irrotationally round it 
now placed at rest within it, being the condition (16 above) 
of absolute minimum energy. Begin once more with the con- 
dition (15 above) of absolute maximum energy, and leave the 
fluid to itself, whether with the canister free to go round some- 
times, or always held fixed, provided only it is ultimately held 
from going round in space ; the ultimate condition is always 
the same, viz. the condition (16) of absolute minimum energy. 
The enclosing rotational belt, being the actual substance of the 
original vortex, is equal in its sectional area to wb”; and 
therefore c’=a’—l”. The moment of momentum is now 
47fb*, being equal to the moment of momentum of the 
portion of the original configuration consisting of the then 
central vortex. 

19. It is difficult to follow, even in imagination, the very 
fine—infinitely fine—corrugation and drawing-out of the 
rotational fluid; and its intermingling with the irrotational 
fluid; and its ultimate re-separation from the irrotational 
fluid, which the dynamics of §§ 17, 18 have forced on our 
consideration. This difficulty is obviated, and we substitute 


536 Sir W. Thomson on Maximum and 


for the “ vortex sponge”’ a much easier (and in some respects 
more interesting) conception, vortex spindrift, if (quite arbi- 
trarily, and merely to help us to understand the minimum- 
energy-transformation of vortex column into vortex shell) we 
attribute to the rotational portion of the fluid a Laplacian* 
mutual attraction between its parts “‘insensible at sensible 
‘distances’? and between it and the plane ends of the con- 
taining vessel of such relative amounts as to cause the inter- 
face between rotational and irrotational fluid to meet the end 
planes at right angles. Let the amount of this Laplacian 
attraction be exceedingly small—so small, for example, that 
the work required to stretch the surface of the primitive 
vortex column to a million million times its area is small in 
comparison with the energy of the given fluid motion. 
Everything will go on as described in §§ 17, 18 if, instead of 
“run out into fine lamine of liquid” (§ 17, line 31) we sub- 
stitute break off into millions of detached fine vortex columns ; 
and instead of “sponge”? (passim) we substitute “ spin- 
drift.” 

20. The solution of minimum energy for given vorticity 
and given moment of momentum (though clearly not unique, 
but infinitely multiplex, because magnitudes and orders of 
breaking-off of the millions of constituent columns of the 
spindrift may be infinitely varied) is fully determinate as to 
the exact position of each column relatively to the others ; and 
the cloud of spindrift revolves as if its constituent columns 
were rigidly connected. The viscously elastic containing 
vessel, each time it is left to itself, as described in §§ 17, 18, 
flies round with the same angular velocity as the spindrift 
cloud within ; and so the whole motion goes on stably, without 
loss of energy, until the containing vessel is again stopped or 
otherwise tampered with. 

21. It might be imagined that the Laplacian attraction 
would cause our slender vortex columns to break into detached 
drops (as it does in the well-known case of a fine circular jet 
of water shooting vertically downwards from a circular tube, 
and would do for a circular column of water given at rest in 
a region undisturbed by gravity), but it could not, because the 
energy of the irrotational circulation of the fluid round the 
vortex column must be infinite before the column could 
break in any place. The Laplacian attraction might, how- 
ever, make the cylindric form unstable; but we are excluded 


* So called to distinguish it from the “ Newtonian ” attraction, because, 
I believe, it was Laplace who first thoroughly formulated “ attraction in- 
sensible at sensible distances,” and founded on it a perfect mathematical 
theory of capillary attraction. 


Minimum Energy in Vortex Motion. 537 


from all such considerations at present by our limitation (§ 12) 
to two-dimensional motion. 

22. Annul now the Laplacian attraction and return to our 
purely adynamic system of incompressible fluid acted on only 
by pressure at its bounding surface, and by mutual pressure 
between its parts, but by no “applied force’’ through its 
interior. For any given momentum between the extreme 
possible values {b7(a?—ib”) and 47€0*, there is clearly, 
besides the §§ 17, 18 solution (minimum energy), another 
determinate circular solution, viz. the configuration of circular 
motion, of which the energy is greater than that of any other 
circular motion of same vorticity and same moment of 
momentum. This solution clearly is found by dividing the 
vortex into two parts—one a circular central column, and the 
other a circular cylindric shell lining the containing vessel ; 
the ratio of one part to the other being determined by the con- 
dition that the total moment of momentum have the prescribed 
value. But this solution (as said above, § 14 and footnote) 
may be proved to be unstable. 

I hope to return to this case, among other illustrations of 
instability of fluid motion—a subject demanding serious con- 
sideration and investigation, not only by purely scientific 
coercion, but because of its large practical importance. 

23. For the present I conclude with the complete solution, 
or practical realization of the solution (only found within ‘the 
last few days, and after §§ 10-18 of the present article were 
already in type) of a problem on which I first commenced trials 
in 1868: to make the energy an absolute maximum in two- 
dimensional motion with given moment of momentum and given 
vorticity in a cylindric canister of given shape. ‘The solution is, 
in its terms, essentially unique ; “absolute maximum ” mean- 
ing the greatest of maximums. But the same investigation 
includes the more extensive problem : To find, of the sets of 
solutions indicated in § 12, different configurations of the 
motion having the same moment of momentum. For each 
of these the energy is a maximum, but not the greatest 
mavinuum, for the given moment of momentum. The most 
interesting feature of the practical realization to which I have 
now attained is the continuous transition from any one steady 
or periodic solution, through a series of steady or periodic 
_ solutions, to any other steady or periodic solution, produced by 
a simple mode of operation easily understood, and always under 
perfect control. The operating instrument is merely a stirrer, 
a thin round column, or rod, fitted perpendicularly between 
the two end plates, and movable at pleasure to any position 

Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 20 


538 Sir W. Thomson on Maximum and 


parallel to itself within the enclosure. Itis shown, marked §, 
in figs. 6, 7,8, 9: representing the solution of our problem 


Fig. 6. Fig. 7. 


C 


Still water 


Still water. 


Fig. 6.2Dotted circle with arrowheads refers to the velocity of the stirrer 
and of the dimple, not to the velocity of the fluid. 

Fig, 7. Arrowheads in the vortex refer to velocity of fluid. Arrowheads 
in the irrotational fluid refer to the stirrer and dimple. Arrow- 
heads in abcirefer to motion of irrotational fluid relatively to the 
dimple. 


Fig, 8. 


weocso oe 


a 


44 


e S 2 
Almost motionless 
\ 
\ 
XN 


Fig. 8. Arrowheads refer to motion of the stirrer, and of the vortex as a 
whole. 

Fig. 9. Arrowheads on dotted circle refer to orbital motion of ec, the 
centre of the vortex. Arrowheads on full fine curves refer to 
absolute velocity of fluid. 


Minimum Energy in Voriex Motion. 539 


for the case of a circular enclosure and a small part of its 
whole volume occupied by vortex, to which exigency of time 
limits the present communication. 

24. Commence with the vortex lining uniformly the en- 
closing cylinder, and the stirrer in the centre of the still 
water within the vortex. The velocity of the water in the 
vortex increases from zero at the inside to &?/a at the out- 
side, in contact with the boundary ; according to the notation 
of §§ 14 and 15. Now move the stirrer very slowly from its 
central position and carry it round with any uniform angular 
velocity <¢b/a and >4$¢b/a. A dimple, as shown in fig. 6, 
will be produced, running round a little in advance of the 
stirrer, but ultimately falling back to be more and more nearly 
abreast of it if the stirrer is carried uniformly. If now the 
stirrer is gradually slowed till the dimple gets again in advance 
of it as in fig. 6, and is then carried round in a similar relative 
station, or always a little behind the radius through the middle 
of the dimple, the angular velocity of the dimple will decrease 
gradually and its depth and its concave curvature will increase; 
till, when the angular velocity is }¢b/a, the dimple reaches the 
bottom (that is, the enclosing wall) with its concavity a right 
angle, as shown in fig. 7, and the angular velocity of propa- 
gation becomes 4 ¢b/a. 

25. The primitively endless vortex belt now becomes divided 
at the right angle, and the two acquired ends become rounded ; 
provided the stirrer be carried round always a little rearward, 
or considerably rearward, of abreast the middle of the gap. 
Figs. 8 and 9 show the result of continuing the process till 
ultimately the vortex becomes central and circular (with only 
the infinitesimal disturbance due to the presence of the stirrer, 
with which we need not trouble ourselves at present). 

26. Suppose, now, atany stage of the process, after the for- 
mation of the gap, the stirrer to be carried forward to a station 
somewhat in advance of abreast of the middle of the gap ; or 
somewhat rearward of the rear of the vortex (instead of some- 
what in advance of the front as shown in fig. 8). The velocity 
of propagation will be augmented (by rearward pull!), the 
moment of momentum will be diminished : the vortex train 
will be elongated till its front reaches round to its rear, each 
then sharpened to 45° and brought into absolute contact with 
the enclosing wall: the front and rear unite in a dimple 
gradually becoming less; and the process may be continued 
till we end as we began, with the vortex lining the inside of 
the wall uniformly, and the stirrer at rest in the middle of 
the central still-water. 

[To be continued. | 


[40.4 


LXII. On the Variations in the Electrical Resistance of Anti- 
mony and Cobalt in a Magnetic Field. By Dr. G. Fas, 
Assistant in the Physical Institute of the Royal Unwersity of 
Padua*. : 


N a series of researches, still in progress, on the variations 
in the electrical resistance of different bodies when 
brought into a magnetic field, I arrived at some results with 
antimony and cobalt which I believe to be new and interesting. 
Reserving for a future occasion a fuller account of my inves- 
tigations and also of the methods and instruments employed, 
I think it may not be useless to publish a preliminary notice. 

It is well known, particularly from the experiments of Sir 
W. Thomson and M. A. Righi, that magnetism has a distinct 
influence on the electrical resistance of iron and nickel, and 
much more upon that of bismutht. Looking at the coefii- 
cients of rotation found by Hall, Righi, and others, and at the 
explanations given, the idea suggests itself that a connexion 
may exist between these coefficients and the variation of the 
electrical resistance in a magnetic field. On the other hand, 
the difference in the behaviour of iron and bismuth in a mag- 
netic field seems to be connected with the fact that the first of 
these two metals is paramagnetic and the second diamagnetic. 
From these considerations and from others, which I shall not 
now enter upon, | have undertaken to examine various sub- 
stances. The results of my experiments agree with my 
previsions. 

First of all I thought that cobalt and antimony would in par- 
ticular be worthy of investigationt. Cobalt, asis well known, 
occupies the third place in the list of paramagnetic metals, 
whilst antimony is found immediately after bismuth in the 
list of diamagnetic metals. 

I have examined antimony in the form of very small cylin- 
ders, which I prepared by melting the metal in a crucible 
and drawing it into thin glass tubes. The glass was after- 
wards broken and taken away by alternately cooling and 
heating. Two thick wires of copper were soldered in the 
ends of the cylinders in order to connect them up in the elec- 
trical circuit. As I shall describe on another occasion the 


* Communicated by the Author. 

+ From experiments repeated by myself, I have obtained results agree- 
ing with those of Sir W. Thomson for nickel and with those of M. Righi 
for bismuth. 

{t And manganese also; but I have not yet been able to get it pure and 
in a convenient form, 


Electrical Resistance of Antimony and Cobalt. 5A 


method of measurement employed by me, I will only observe 
that it was like that of Matthiessen and Hockin, with the 
exception of some modifications suggested by the special cir- 
cumstances and object of my investigation. The magnetic 
field was formed by a large Ruhmkorff electromagnet, ex- 
cited by a number of Bunsen’s elements or by a dynamo 
machine. 

My experiments on antimony showed that, when brought 
into a magnetic field, there was an increase in its electrical 
resistance, both across and along the lines of force. It ap- 
peared to me moreover that, with the same intensity of the 
magnetic field, the increase across was greater than that along 
the lines of force. 

Cobalt I investigated in the form of a small thin plate, pre- 
pared by electrolysis of the chloride, or by depositing the metal 
on a plate consisting of a mixture of graphite and stearine, as 
indicated by M. Righi*. I soldered two thick copper wires 
at the two ends of this small plate of cobalt, before detaching 
it from the plate of graphite and stearine. ‘These wires were 
rigidly connected by means of a piece of ebonite, and served 
to make the connexions in the circuit. By a movable support 
I could very easily adjust the small plate in any position in 
respect to the lines of force. By a long series of observations 
I found that :— 

(a) When the plate of cobalt was arranged in the magnetic 
field with its plane perpendicular to the lines of force, a dimi- 
nution of its electrical resistance was observed. 

(6) When the plate was arranged parallel to the lines of force, 
and the current also had the same direction, an increase in its 
electrical resistance was observed. 

Judging therefore from the intensity of the effects, the 
behaviour of antimony is the same as that found by M. 
Righi for bismuth ; and the behaviour of cobalt the same as 
that found by Sir W. Thomson for iron and nickel. 

I will not dwell on similar experiments on other substances, 
because the results are not yet definitive. 

In the meantime I must express my obligations to Prof. 
Righi for his encouragement in these experiments, and for 
also giving me the means of making them in the Physical 
Institute under his direction. 


Padua, December 12, 1886. 


* Mem. dell’ Ace. di Bologna, (4) v. 1883, p. 122; WN. Cimento, (3) 
xv. 1884, p. 140, 


a sy LF 


LXII. The Differential Equation of the most general Substi- 
tution of one Variable. By Captain P. A. MacManon, &.A.* 


[ the Philosophical Magazine for February 1886, Dr. T. 
Muir considers the differential equations of the general 
conic and cubic curves by a perfectly general method. 
The general linear substitution 


__(a,0)(@,1) 
2 EVV (@,1) 
leads, as is well known, to the differential equation 


dy dy genet. 
2 oe db ( dat) =0 


wherein the expression on the left has been called the 
Schwarzian derivative : this is a reciprocant; but it isalsoan 
invariant, as may be seen by writing 


dy d’y d’y 
~ —|]! —~ =9!) —_% —3! 
dig ORT TOT gna aan ae 
when it assumes the form : 

12(tb—a’). 


In the case of the general substitution of order n, the 
resulting expression is no longer a reciprocant, but it is an 
invariant (catalecticant) of a certain binary quantic fT. 

For, writing 

jo See ES _ Un 

(agp. ical) | Na 

we have 
Nn; 

Differentiating this equationn + 1,n+2,n+3,...2n+1 times 
successively by Leibnitz’s theorem, and putting 

CN av, 

dae 9” dex 


there results the set of equations :— 


= Vo, 


* Communicated by the Author. 

+ The formation of the differential equation was recently set as a ques- 
tion in an examination for Fellowship at Trinity College, Dublin; but 
I am not aware that its connexion with the theory of Invariants has been 
before noticed. 


The Differential Equation of the general Substitution. 543 


. | ,, Va) (n+ 1)! (2) Gays 
mtiVat (n+1)ynVn + QW (n—1)! 7") aN, +s s Bete Al T! YWVn =(0, 
: } n+2)! n+2)! x 
ntaVnt (W+2)YnsiVa + 4 2 Yn VO 4. + i a yoVS=0, 
n+)! n+3)! A 
na V nt (0+ 8)ynaVO tone) YntiVn beeet a 2 ysVn=0, 
a, (n+)! 5 (2n+1)! a 
ee ones tt ald 


or writing 


Y=llt, y=2!a, yy=d!la,,... ¥,=p! 
— are 
= (n) (n—1) (m—2) 
cs ~ V5 gee eae + ay a a Va. +.- 
lL yam (n=1) (n—2) 
do Vn +1 V + Ag (n—D)! 7 V; +.. 
i VO + 1 Vey Na 
oa? "GT cee 06 = iit 
1 (n) 1 (n—1) 1 (n—2) 
Gn Vn iene + dnt1 Tm —9yI Vm +.. 
Hliminating the n+1 Sil 
| = (n) 2), 
pVn A Gain “i Mi Sas (cae 


(n+1)! 


A599 


+ On-1V in =. 
bd, oN =O 


oor OnsiVn =0, 


sick: Gon—1Vn= 0. 


between aie n-+1 equations, we find that the desired differ- 


ential equation is 


Ao ay Ag eee an— 1 Qn 
On—1 Un OAntierrs Agn—2 Aen-1 


=(). 


This determinant is the catalecticant of the binary quantic 


Gi \CR YT 


(& digs G5 «\<s 


and by counting the constants, we see that the general substi- 
tution is the complete primitive of the differential equation. 


Royal Military Academy, Woolwich, 
ine 21st, 1887. 


[ 544 J 


LXIV. Intelligence and Miscellaneous Articles. 


CURIOUS CONSEQUENCES OF A WELL-KNOWN DYNAMICAL 
THEOREM. BY G, JOHNSTONE STONEY, M.A., D.SC., FR See 


HERE is a well-known theorem in the science of Dynamics, re- 
lating to a system of bodies mm motion, which may act on each 
other, but are not acted on by any external force. The theorem 
in question is, that if at any instant the velocities of the several 
bodies of the system be reversed, without any other change being 
made (7. ¢. without altering either their masses or the laws accord- 
ing to which they attract or otherwise act on one another), then 
will all the bodies of the system retrace their steps, traversing in 
the reverse direction the same paths which they had previously 
described, and in such manner that any position through which 
any one of these bodies had passed in its onward progress, at a 
certain time before the reversal, will be repassed with the same 
velocity, but in the opposite direction, at the same interval of time 
after the reversal. 

Now, if we regard the universe as a dynamical system, it is 
exactly such a dynamical system as this theorem presupposes. 
Its several parts act on one another, but are not subjected to any 
other forces. And it is of interest to study what would be the 
result if such a reversal as the theorem supposes were to take place 
throughout the whole universe. Wemust, of course, suppose that 
the reversal affects all the motions of the universe, not only its 
molar motions, but its molecular motions also; and not only the 
motions of its ponderable matter, but also the motions of the ether. 

In order to be in a position to study the effects, let us first 
suppose that we are spectators of this far-reaching change, without 
being ourselves affected by it—that we are, from an intellectual 
standpoint, as it were outside the great system whose future 
history we want to trace, simply observing everything that takes 
place, and not in any way interfering with it, nor ourselves in any 
way transformed by the change. 

To such a spectator the past history of the universe would 
repeat itself in reverse order, and many of the conditions under 
which it would do so would appear to him very strange. The 
bird which was shot to-day by the sportsman, and which is now 
lying in his kitchen, will, if the reversal of the universe were to 
take place at this instant, be restored by the keeper to the game- 
bag, will be carried by him, walking backwards, to the place 
where the pointer had fetched it in, where he will take it out, and 
lay it on the ground. Thence the dog will lft it m his mouth, 
and, trotting backwards, will reach the spot where the bird fell, 
where, however, it will now rise to the height at which it was shot, 
from which it will fly away backwards unharmed. Meanwhile, 
the vapours into which the powder had been dissipated will stream 
back into the barrel of the fowling-piece, and condense themselves 


', * Reprinted, by permission, from the “ Scientific Proceedings” of the 
Royal Dublin Society of the 19th January, 1887. 


Intelligence and Miscellaneous Articles. 545 


again into gunpowder, while the grains of shot will rush towards 
the muzzle of the gun, and crowd into its breach. 

It is of importance to observe that, under the new conditions 
of the universe, all true dynamical laws will remain the same as at 
present, but many quasi-dyna‘nical laws will be reversed. Thus, 
the first law of thermodynamics—the law of the equivalence of 
energy—will remain unaltered, but the second law will become its 
converse. Instead of a warmer body tending to impart heat to a 
cooler body, as at present, the new condition of things will tend to 
make their temperatures more divergent. Heat wiil become 
mechanical energy directly, and without requiring the accom- 
panying degradation of energy which now takes place. Friction, 
instead of retarding the progress of bodies, will help them forward. 
The air, instead of impeding a missile passing through it, will 
urge it on. And, when reviewing a system so divergent from 
what we find in the actual universe about us, it is very instructive 
to bear in mind that the universe, under the new conditions that we 
suppose, would be as perfect a dynanucal system as the actual unwerse 
is. This places before the mind in a very strong light the grave 
error which is too often made when such laws as I ine referred — 
to—the second law of thermodynamics, &c.—are supposed to be 
true dynamical laws. 

This naturally leads up to the consideration whether the laws 
of causation would be affected. Those relating to true causes 
would not be affected: those relating to quasi-causes would all be 
inverted. ‘True causes never precede their effects ; they are always 
strictly simultaneous with them. The science of Dynamics recog- 
nizes true causes only. All change of the motion of a body is in 
that science attributed to forces acting while the change is taking 
place ; and the persistence of a body in motion while no forces are 
acting on it is due to the inertia of the body, 7. ¢. the body itself is 
the cause of it. It is because the inertia of a body is a sufficient 
cause for its continuing in motion that time can elapse between 
events in nature. Whether the motion changes or does not 
change, the effect and its true cause are accurately simultaneous. 
The dispute as to whether action takes place at a distance does not 
disturb this statement. Every one who does not suppose that the 
sun attracts the earth from a distance and without lapse of time, 
supposes that some medium pervading the intervening space com- 
municates the action; and it is not the distant body, but the sur- 
face of this medium where it touches the body’acted on, that upon 
this view can alone be recognized in the science of Dynamics as the 
true immediate cause of the changes of motion of the second body. 
Thus, in all cases, dynamical effects arise along with, and not 
after, their causes. But in popular language, and indeed in all 
but very carefully strict language, many events are spoken of as 
caused by events that have preceded them. Thus, in the usual 
loose way of talking, we may speak of a ball’s having been re- 
acted on by the ground as the cause why it is now ascending, 
although a moment’s reflection would show that, in strict lan- 
guage, the reaction of the ground has caused only those changes 


Phil. Mag. 8. 5. Vol. 23. No. 145. June 1887. 2P 


546 Intelligence and Miscellaneous Articles. 


of motion that occurred while the ground was pressing against the 
ball, and that the ball’s afterwards continuing to ascend is due to 
its inertia. Sometimes the two classes of causes are distinguished 
as Immediate and remote. Now the change which we have sup- 
posed the universe to undergo would in no way affect immediate, 
that is true, causes; but all that we now recognise as an antecedent 
or quasi-cause would, to the spectator looking on at the universe 
from without, be changed into the effect, and that which is now 
the effect would to his apprehension occur first and become the 
cause. 

These seem the first lessons which the study we have entered 
upon impresses upon us. But it is capable of giving further in- 
struction. Hitherto we have supposed the altered universe looked 
at by a spectator who was himself unaffected by the change. But 
we are all ourselves parts of this universe, and the series of 
thoughts that occur in our minds are quite as much events that 
happen in the universe as the motions we see around us. Such a 
reversal of all the velocities of the universe as I have supposed, if 
it really took place, would affect us and the motions in our brains 
as well as everything else in the universe; and we have now to 
consider what the effect of this would be, and how it would modify 
our observation of what is going on around us. From the instant 
of the supposed reversal, the thoughts which had occupied our 
minds previous to it will recur, repeating themselves backwards, 
just like every other event in the universe. The memory of 
having eaten our breakfast will present itself first; the sensation 
that we are eating 1t will come on afterwards: at least this is the 
order in which we must as yet describe these thoughts in our mind 
as occurring ; it is the order in which they would appear to that 
outsider whom we before supposed to be surveying the universe. 
But the relation of the one thought to the other in our own mind 
—of the memory to the sensations remembered—will be after the re- 
versal exactly the same* as it was when these same thoughts occurred 
before in their right order. Now, TIME Is ONLY AN ABSTRACT TERM 
REFERRING TO ALL SUCH RELATIONS, just as mankind is an abstract 
term referring to the individuals that are men. And just as it is 
individual men who have a real existence, and not mankind in the 
abstract, so is it the individual time-relations occurring between 
real thoughts or real events that have a real existence, and not 
time itself, which is a mere word. But as we have found that the 
time-relations between our thoughts after the supposed reversal 
are absolutely the same as the time-relations between these same 
thoughts when they occurred before the reversal, then to us, if we 
share in the reversal, our thoughts and the events in the world 
about us will seem to occur in-the same order of time as they did 
before the reversal, and the moment of reversal will in both cases 
appear to us to occur last in point of time. In other words, our sup- 


* In fact, the time-relation between the two states of mind amounts to 
this, that a part of the one state of mind is a memory of the whole, or of 
a part, of the other state of mind ; and this is equally the case after as ~ 
before the reversal. . 


Intelligence and Miscellaneous Articles. 547 


position of the reversal of all the motions of the universe, when it 
embraces the whole universe, ourselves included, does not really 
involve a repetition of the events in reverse order, but only a 
second way of reviewing the past history of the world. 

These considerations do not seem altogether unfruitful. They 
emphasize the distinction between true and quasi-dynamical laws, 
they clear our thoughts with reference to the relation of cause and 
effect, and, above all, they help to dispel from our minds the 
prevalent error that time has an existence in itself independently 
of the particular time-relations that prevail between the thoughts 
that really occupy our mind, or between events * that actually occur 
in the universe about us, or between those events and our thoughts. 
In reality the ageregate of these individual time-relations is the 
whole of what exists in nature as a background for our conceptions 
about time. 


ON THE GASEOUS AND LIQUID STATES OF MATTER. 


To the Editors of the Philosophical Magazine and Journal. 
GENTLEMEN, 


Asa paper recently published by Wroblewski ( Wiener Monatsheft 
fiir Chemie, 1886, p. 383) is at present being abstracted into other 
Journals, and as we dissent from every one of his conclusions on 
the ground that they have no foundation on facts, we beg for per- 
mission to point out the difference between our views and those 
expressed in his paper. 

1. Wroblewski states that lines of equal density (isopyknics) 
ought never to cut one another. On the contrary, they have been 
frequently proved experimentally to cut one another during the 
phenomena of superheated liquid, or “boiling with bumping,” and 
“‘ supersaturated vapour.” 

2. The data have been calculated by means of Sarrau’s equation, 
which, although approximately expressing the results, is certainly 
not true; for the isopyknics should be straight, and not curved. 
This has been pointed out by Amagat and by ourselves for the 
gaseous state, and by ourselves for the liquid state. 

3. The vapour-pressure curve is not continuous with the curve 
expressing the lowest values of p.v, and can hardly be said to be 
identical with it below the critical point, as the vapour-pressure 
curve is independent of volume. We have shown that the mini- 
mum values of p.v lie on the nearly vertical portions of the serpen- 
tine isothermal lines denoting continuous change from the liquid to 
the gaseous state. 

4, Sarrau’s equation, rigorously applied, involves intersection of 
the isopyknic lines; and the author discards it on that account 
where it no longer bears out his conclusions. We accept Sarrau’s 
equation as a close approximation to the truth, although not rigo- 
rously true, and, as such, believe that it correctly represents at any 
rate the general form of the isothermal curves. 


* Thoughts in other people’s minds are some of the events that occur 
in the universe about us; that is, in the rest of the universe, excluding 
ourselves. 


548 Intelligence and Miscellaneous Articles. 


5. Wroblewski states that the isopyknics, on approaching the 
vapour-pressure curve, form a bundle of curves parallel to the 
vapour-pressure curve. We have already pointed out that facts 
contradict this view. . | 

6. Wroblewski restates a view, propounded by one of us in 
the Proc. Roy. Soc. xxx. p. 323. This view must, in consequence 
of our more accurate and recent researches, be abandoned. The 
distinction between liquid and gas above the critical pointis wholly 
illusory ; and we believe that such distinction would also disappear 
below the critical point, were it possible to follow experimentally 
the continuous change from liquid to gas. 

7. Wroblewski also asserts that heat is required to convert liquid 
into gas at the critical temperature. We have only to refer to our 
published papers on the thermal properties of alcohol and ether 
(Trans. Roy. Soc. 1886, part i. plate 5, and 1887, part i. plate 5) 
to disprove this assertion. Moreover all our experimental obser- 
vations tend to show that no abrupt change of state occurs at any 
point on the isothermal ; in fact that heat of vaporization vanishes, 
its place being taken by heat due to compression or expansion. 

8. Andrews does not make the statement attributed to him by 
Wroblewski, but gives a qualified definition of the words “ liquid” 
and “‘gas.” 

It will thus be seen that our results contradict Wroblewski’s 
views in every particular; and that there is no reason to reject the 
views of Dr. Andrews as regards the liquid and gaseous states. 

We are, Gentlemen, 
Your obedient servants, 
Witiiam RAMSAY, 
SypNEY Youn«. 


LECTURE EXPERIMENTS ON THE CONDUCTIVITY OF SOUND. 
BY M. HESEHUS. 


Rods of the size of an ordinary pencil are prepared of steel, 
glass, wood, guttapercha, cork, and caoutchouc. They are con- 
nected in threes by means of a caoutchouc rod, tubes of the same 
material being interposed between them. The rods are held in 
one hand, with their lower ends on a sounding-box, and the free 
ends are successively touched by the stem of a vibrating tuning- 
fork, which is held in the other hand. ‘The sound is not perceptible 
when the caoutchouc rod is touched, but it becomes more so as the 
other rods are successively touched. It can thus be shown that the 
intensity remains constant, if one rod is replaced by another of the 
same material, but the dimensions of which vary in the same pro- 
portion. By varying the length alone the intensity is changed ; in 
like manner it is changed also by varying the section while the 
length is constant. The method is sufficiently sensitive to show 
the difference between the conductivity of wood parallel and per- 
pendicular to the fibres, and even to determine the numerical ratio 
of these conductivities.—Journal de Physique, April 1887. 


[ 549 ] 


INDEX to VOL. XXIII. 


———e—-——— 


ABBOTT (Rev. T. K.) on the order 
of lever to which the oar belongs, 
58. 

Abercromby (Hon. R.) on the sun- 
rise-shadows of Adam’s Peak in 
Ceylon, 29. 

Acetic acid, on the specific heat of 
the vapour of, 223. 

Affinity, on the doctrine of, 504. 

Air, on the passage of the electric 
current through, 384. 

Aluminium, on the galvanic polari- 
zation of, 304. 

Amaury (H.), apparatus for the con- 
densation of smoke by statical elec- 
tricity, 471. 

Antimony, on the variations in the 
electrical resistance of, in a mag- 
netic field, 540. 

Are-lamp for use with the Duboscq 
lantern, on an, 333. 

Armstrong (Prof. H. E.) on the con- 
stitution of carbon compounds, 73. 

Atmospheric phenomena, on some 
remarkable, 29. 

Avogadro’s law, on the theoretical 
proof of, 805, 433. 

Battery, on a simple form of water-, 
303. 

Bell (L.) on the absolute wave-length 
of light, 265. 

Bidwell (S.) on the electrical. resist- 
ance of vertically-suspended wires, 
499, 

Boltzmann (Prof. L.) on the theore- 
tical proof of Avogadro’s Law, 305. 

Books, new :—Reade’s Origin of 
Mountain-Ranges, 213; Descrip- 
tive Catalogue of a Collection of 
the Economic Minerals of Canada, 
216; Journal and Proceedings of 
the Royal Society of New South 
Wales for 1885, 216 ; Capt. Noble’s 
Hours with a Three-Inch Tele- 
scope, 218; Chrystal’s Algebra, 


219; Peirce’s Elements of the 
Theory of the Newtonian Potential 
Function, 220; Annual Companion 
tu the ‘Observatory,’ 299; Carr’s 
Synopsis of Elementary Results in 
Pure Mathematics, 300; Oliver, 
Wait, and Jones’s Treatise on Al- 
gebra, 465. 

Borgmann (J.) on the passage of the 
electric current through air, 384; 
on the heating of the glass of con- 
densers by intermittent electrifi- 
cation, 472. 

Bosanquet (R. H. M.) on silk v. wire 
in galvanometers, 149; on the law 
of the electromagnet and the law 
of the dynamo, 338; on the deter- 
mination of coefficients of mutual 
induction, 412. 

Bottomley (J. T.) ona nearly perfect 
simple pendulum, 72. 

Boys (C. V.) on the production, pro- 
perties, and uses of the finest 
threads, 489. 

Brown (W.) on the effects of per- 
cussion on the magnetic moments 
of steel magnets, 293, 420. 

Carbon compounds, on the constitu- 
tion of, 73, 109. 

Cauchy’s theory of reflection and re- 
fraction of light, on, 151. 

Chemical reactions, on the inert space 
in, 468. 

Cobalt, on the variations in the elec- 
trical resistance of, in a magnetic 
field, 540. 

Coleman (J. J.) on liquid diffusion, 1. 

Condensers, on the heating of the 
glass of, by intermittent electrifi- 
cation, 472, 

Conductivity, on electric, 339. 

Copper, on the atomic weight of, 138. 

Dessau (B.) on metallic layers which 
result from the volatilization of a 
kathode, 384. 


550 


Diazoamido-compounds, on the con- 
stitution of the, 518. 

Differential equation of the most 
general substitution of one vari- 
able, on a, 542. 

Gee on liquid, 1; on gaseous, 

5. 

Discordant observations, on, 264. 

Dissociation, on, 435. 

Duboseq lantern, on an arc-lamp for 
use with the, 333. 

Dynamical theorem, on some curious 
consequences of a well-known, 544. 

Dynamo, on the law of the, 338. 

Harth, on the contraction during cool- 
ing of a solid, 145. 

Edgeworth (F. Y.) on discordant 
observations, 364. 

Elasticity, on the longitudinal and 
torsional, of iron, 249. 

Electric current, on the passage of 
the, through air, 384. 

Electrical resistance of vertically- 
suspended wires, on the, 499. 

Electricity, notes on, 225; on the 
action of the discharge of, of high 
potential on solid particles sus- 
pended in the air, 301; on an ap- 
paratus for the condensation of 
smoke by statical, 471. 

Electrification, on the heating of the 
lass of condensers by intermittent, 
472. 

Electromagnet, on the law of the, 
308. 

Electromagnetic disturbances into 
wires, on the propagation of, 10. 

Error, on the law of, 364. 

Extensometers, on various, 289. 

Evaporation, on, 435. 

Faé (Dr. G.) on the electrical resist- 
ance of antimony and cobalt in a 
magnetic field, 540. 

Fibres, on very fine glass and quartz, 
489, 

Fisher (Rev. O.) on the amount of 
the elevations attributable to com- 
pression through the contraction 
during cooling of a solid earth, 145. 

Fluid motion, on the stability of 
steady and of periodic, 459. 

Foster (Prof. G. C.) on a method of 
determining coefficients of mutual 
induction, 121. 

Galvanometer, ballistic, on the deter- 
mination of coefficients of mutual 
induction by means of the, 412. 


INDE. 


Galvanometers, on silk v. wire sus- 
pensions in, 46, 149. 

Gas, on the equilibrium of a, under 
its own gravitation only, 287. 

Gaseous state, on the transition from 
the liquid to the, of matter, 435, 547, 

Gases, on the foundations of the ki- 
netic theory of, 141; on the che- 
mical combination of, 379, 472. 

Geological Society, proceedings of 
the, 69, 221, 466. 

Gibson (H.) on the tenacity of spun 
glass, 351. 

Glass, on the tenacity of spun, 351; 
on the finest fibres of, 489. 

Gray (T.) on silk v. wire suspensions 
in galvanometers and on the ri- 
gidity of silk fibre, 46; on an im- 
proved form of seismograph, 353. 

Gregory (R. A.) on the tenacity of 
spun glass, 351. 

Heat, on the action of, on potassic 
chlorate and perchlorate, 375. 
Heaviside (O.) on the self-induction 

of wires, 10, 173. 

Hesehus (M.) on the conductivity of 
sound, 548, 

Hughes (Prof. T. M‘K.) on the drifts 
of the Vale of Clwyd, 69. 

Induction, on a method of determi- 
ning coefficients of mutual, 121, 
412. 

Induction-balances, on, 184. 

Inductor, on the determination of 
coefficients of mutual induction by 
means of the ballistic galvanometer 
and earth-, 412. 

ee on a form of spherical, 
381. 

Iron, on the behaviour of, under the 
operation of feeble magnetic forces, 
225; on some of the physical pro- 
perties of, 245. 

Irving (Rey. A.) on the physical his- 
tory of the Bagshot beds of the 
London basin, 467. 

Kathode, on metallic layers which 
result from the volatilization of a, 
384. 

Lever, on the order of, to which the 
oar belongs, 58, 222. 

Liebreich (O.) on the inert space in 
chemical reactions, 468. 

Light, on Cauchy’s theory of reflec- 
tion and refraction of, 151; on the 
relative wave-lengths of, 257; on 
the absolute wave-length of, 265, 


INDEX. 


Liquid diffusion, on, 1. 

surfaces of revolution, on the 

critical mean curvature of, 35. 

, on the transition from the, to 
the gaseous state of matter, 435,547. 

Liquids, on the nature of, 129. 

MacMahon (Capt. P. A.) on the dif- 
ferential equation of the general 
substitution, 542.. 

Magnetic field, on the strength of the 
terrestrial, in buildings, 381; on 
the variations in the electrical re- 
sistance of antimony and cobalt in 
a, 040. 

moments, on the effect of per- 
cussion on, 295. 

Magnetism, notes on, 225. 

Magnets, on the effects of percussion 
on the magnetic moments of steel, 
293, 420. 

Matter, on the gaseous and liquid 
states of, 435, 547. 

Measuring-instruments used in me- 
chanical testing, 282. 

Meldola (Prof. R.) on the constitu- 
tion of the diazoamido-compounds, 
515. 

Meyer (Prof. L.) on the doctrine of 
affinity, 504. 

Mills (Dr. E. J.) on the action of 
heat on potassic chlorate and per- 
chlorate, 575. 

Motion, on the laws of, 475. 

Muirhead (R. F.) on the laws of 
motion, 475. 

Nicol (Dr. W. W. J.) on the expan- 
sion of salt-solutions, 585. 

Nitrogen tetroxide, on the specific 
heat of the vapour of, 225. 

Oar, on the order of lever to which 
the, belongs, 58, 222. 

Obermayer (A. von) on the action of 
the discharge of electricity of high 

otential on solid particles sus- 
pended in the air, 501. 

Ostwald (Prof. W.) on the chemical 
combination of gases, 379, 472. 

Pendulum, on a nearly perfect simple, 
72. 

Percussion, on the effects of, in 
changing the magnetic moments 
of steel magnets, 293. 

Permeability, on electric, 339. 

Pichler (M. von) on the action of the 
discharge of electricity of high po- 
tential on solid particles suspended 
in the air, 301. 


551 


Pickering (Prof. S. U.) on the con- 
stitution of carbon compounds, 109; 
on delicate thermometers, 401; on 
the effect of pressure on thermo- 
meter-bulbs and on some sources 
of error in thermometers, 406. 

Polarization, on the galvanic, of alu- 
minium, 504. 

Potassic chlorate and perchlorate, on 
the action of heat on, 375. 

Quartz fibres, on the finest, 493. 

Ramsay (Dr. W.) on the influence of 
change of condition from the liquid 
to the solid state on vapour-pres- 
sure, 61; on the nature of liquids, 
129; on evaporation and dissocia- 
tion, 455; on the gaseous and 
liquid states of matter, 547. 

Rayleigh (Lord) on electricity and 
magnetism, 225. 

Roberts (T.) on the correlation of the 
Upper Jurassic rocks of the Jura 
with those of England, 466. 

Rowland (Prof. H. A.) on the rela- 
tive wave-lengths of the lines of 
the solar spectrum, 257; on a 
simple and convenient form of 
water-hattery, 303. 

Rucker (A. W.) on the critical mean 
curvature of liquid surfaces of re- 
volution, 35, 

Rutley (F.) on the metamorphic 
rocks of the Malvern Hills, 70, 

Salt-solutions, on the expansion of, 
385. 

Seismograph, on an improved form 
of, 353, 

Shadows, on peculiar sunrise-, 29, 

Shaw (W.N.) on the atomic weights 
of silver and copper, 138. 

Silk fibre, on the rigidity of, 46. 

Silver, on the atomic weight of, 
158. 

Smoke, apparatus for the condensa- 
tion of, by statical electricity, 471. 

Solar spectrum, on the relative wave- ~ 
lengths of the lines of the, 257. 

Sound, on the velocity of, in iron, 250; 
on the conductivity of, 548. 

Spectrum, on the relative wave- 
lengths of the lines of the solar, 
257. 

Steel, on the behaviour of, under 
feeble magnetic forces, 225, 

Stoney (Dr. G. J.) on some curious 
consequences of a well-known dy- 
‘namical theorem, 544, 


552 


Streinz (Dr. F.) on the galvanic po- 
larization of aluminium, 304. 

Tait (Prof.) on the foundations of 
the kinetic theory of gases, 141; 
on the assumptions required for the 
proof of Avogadro’s law, 433. 

Tarleton (IF. A.) on the order of lever 
to which the oar belongs, 222. 

Thermal properties of stable and dis- 
sociable bodies, on the, 129. 

Thermometer-bulbs, on the effect of 
pressure on, 406. 

Thermometers, on delicate, 401; on 
some sources of error in, 406. 

Thompson (Prof. 8. P.) on an are- 
lamp for use with the Duboscq 
lantern, 335. 

Thomson (J. J.) on the chemical 
combination of gases, 379. 

Thomson (Sir W.) on stationary 
waves in flowing water, 52; on the 
front and rear of a free procession 
of waves in deep water, 113; on 
the waves produced by a single 
impulse in water of any depth, or 
in a dispersive medium, 252; on 
the formation of coreless vortices 
by the motion of a solid through 
an inviscid incompressible fluid, 
255; on the equilibrium of a gas 
under its own gravitation only, 
287 ; on the stability of steady and 
of periodic fluid motion, 459; on 
maximum and minimum energy 
in vortex motion, 529. 

Threads, on the production, proper- 
ties, and uses of the finest, 489. 
Threlfall (R.) on the specific heats of 
the vapours of acetic acid and ni- 

trogen tetroxide, 223. 

Tomlinson (H.) on the effects of the 

physical properties of iron, pro- 


* 


INDEX. 


duced by raising the temperature 
100° C., 246. 

Unwin (Prof. W. C.) on measuring- 
instruments, 282. 

Vapour-pressure, on the influence of 
change of condition from the liquid 
to the solid state on, 61. 

Ventosa (V.) on a form of spherical 
integrator, 381. 

Vortex motion, on maximum and 
minimum energy in, 529. 

Vortices, on the formation of core- 
less, 255. 

Walker (J.) on Cauchy’s theory of 
reflection and refraction of light, 
151. 

Water-battery, on a simple form of, 
303. 

Wave-length, on the absolute, of 
light, 265. 

Wave-lengths, on the relative, of the 
solar spectrum, 257. 

Waves, on stationary, in flowing 
water, 52; in deep water, on the 
front and rear of a free procession 
of, 113; on, produced by a single 
impulse in water of any depth, 252. 

Whitaker (W.) on some deep borings 
in Kent, 222. 

Wires, on the self-induction of, 10,173. 

Witz (A.) on the strength of the 
terrestrial magnetic field in build- 
ings, 381. 

Wroblewski (M.) on the gaseous and 
liquid states of matter, 547. 

Young (Dr. §.) on the influence of 
change of condition from the liquid 
to the solid state on vapour-pres- 
sure, 61; on the nature of liquids, 
129; on evaporation and dissocia- 
tion, 435; on the gaseous and 
liquid states of matter, 547. 


END OF THE TWENTY-THIRD VOLUME, 


Printed by T’aytor and Francis, Red Lion Court, Fleet Street. 


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Vol 23. JANUARY 1897. . No. 140. 


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e . . . . 

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Encke, Fresnel, Gauss, Goldschmidt, Hansen, Helmholtz, Henry, Hoffmann, 
Hoffmeister, Holtzmann, Jacobi, Jamin, Knoblauch, Koene, Koren, Krohn, 
Lamont, Lenz, Le Verrier, Liebig, Lowig, Magnus, Melloni, Menabrea, Meyen, 
Mitscherlich, Mohl, Moser, Mossotti, Muller, Neumann, Nobili, Ohm, Ostro- 
gradsky, Pelouze, Peltier, Plateau, Plucker, Poggendorff, Poisson, Pouillet, 
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| Considered EXPERIMENTALLY, STRUCTURALLY, DYNAMICALLY, 
i and in relation to their GEOLOGICAL HISTORY. 


By T. MELLARD READE, C.E., F.G.S., F.R.LB.A., 
Past President of the Liverpool Geological Society. 


Extracts from Reviews (up to April 7, 1887). 
‘We very heartily recommend this valuable work to the attention of geologists, as 
an important contribution to terrestrial dynamics.”— Philosophical Magazine. 


“Tt is long since geological literature has been enriched by so able, so philosophical, 
and so profound a work.” —Knowledge. 


“The work marks a distinct advance, and is a valuable contribution to physical 
geology, and must take its rank accordingly.” —The Builder. 


“Nothing could be more suited to entice the student into further research than 
so charmingly-written and clearly-reasoned a treatise.”— Christian World. 


“Mr. Reade’s work is a valuable contribution to the perplexing subject of mountain 


making.”—J. D. Dana (‘American Journal of Science’). 

: “The author has had the advantage of being his own artist, and has embellished the 
volume with a wealth of illustration rarely to be found in scientific books.”—Liverpool 
Courier. 


: ; ‘¢ By his title he perhaps unconsciously courts comparison with Darwin’s celebrated 
| ‘Origin of Species,’ and there is some analogy between them in the immense area of 
fact and speculation covered by the subject in either case.”—Lzverpool Daily Post. 


“The book has two merits: it takes nothing for granted, and it does not err on the 
side of assuming too much knowledge on the part of its readers.” Nature, 


Tayior and Francis, Red Lion Court, Fleet Street, E.C. 


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A complete set of this valuable Work, containing memoirs by Arago, Balard, 
Becquerel, Berzelius, Bessel, Biot, Botto, Bunsen, Burmeister, Carus, Cauchy, 
Chevreul, Clausius, Clapeyron, Cruger, Danielssen, Dove, Ehrenberg, Emmerich, 
Encke, Fresnel, Gauss, Goldschmidt, Hansen, Helmholtz, Henry, Hoffmann, 
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