ee Dy Sg, gh, er pt pa maa ety a] Pak i : " PD. Ss ee a SNL WES te ooh z ny Ee POA 10 iat Ai alk “ ‘ S 7 ft Pi Staak Ap Y Y Md ea Bec lisaeaiadiocray en itt te pea ‘ ‘ 4 p Moog? Be betel wei = tai witty LePadinty hist: Nie RE TE ; } “*r7 ; 2 ‘ OTS Ve White A RY TIEN ONS + 7 q , ys te £4 GRY a ie fh nopbe aat ding: «Rsv : 4 fp tie dap ma IS bass sat han ; My ; Paci NP Deakehepetion badd ene : ee nt Sie 2k ion ak a nity ’ < oteg A i and rif An bird ean pane ener neve ax i) vs i) Wire; Beef) i 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 \ | i a, f “> re) LONIAN DE pes LONDON: TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. SOLD BY LONGMANS, GREEN, AND CO.; KENT AND CO.; SIMPKIN, MARSHALL, AND Cov. ; AND WHITTAKER AND CO.;—AND BY ADAM AND CHARLES BLACK, AND T, AND T. CLARK, EDINBURGH; SMITH AND SON, GLASGOW ;—— HODGES, FOSTER, AND CO., DUBLIN ;—-PUTNAM, NEW YORK ;— VEUVE J. ROYVEAU, PARIS ;—AND ASHER AND CO., BERLIN. “Meditationis est perscrutari occulta; contemplationis est admirari perspicua..... Admiratio generat queestionem, queestio investigationem, nvestigatio inventionem.”—Hugo de S. Viciore. ——‘“ (Cur spirent venti, cur terra dehiscat, Cur mare turgescat, pelago cur tantus amaror, Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat flagrare cometas, Quid pariat nubes, veniant cur fulmina ceelo, Quo micet igne Iris, superos quis conciat orbes 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. ; de po ee i re t. } ’ a ; - a é é 1 ‘ ~ ~ ‘ tu i tn hs . + 4 a uh ; . pe F ; ‘ ) LS \ h ‘ : “SS = $0 Hei Reetite MESO ss ZIS0%. CSOs CaCl. @ b, i Che 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 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 - 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 = 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. . 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. 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. Lxv 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

+ 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. =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 o Ss s > B = oy aa g i o0&G OOLES o01IG OOLFE o01G OY6gE oG0G OSIFE of0G OFGIE 0006 OSTLE oL6T OOGLE of61 OOGOP ‘| HAV, 0G0G OVGCE oG0G 09666 0006 OSeI& 0006 OS8GE oL61 OOGFE oG61 O0EGE 8-661 0098¢ oG61 OF9LZ oG61 00826 of61 OLGLG 0661 OF89G o(6T 0G6LZE oGSI OG6SG CE-o8L1 OGGIG 6-oGLT OSP0G C0-.EL41 O9S6L G:o0LT OOL8T 6-.99T OOLLT L-oG9T OOg9T 8-ofFl OFEGL Ei ees pie as Xe) = nq . . . . ee ae we 10 —N a QQ . « . . OAH aa ae of N =) nN a sey veg rey Falk es fa ES cow ecce 7 ay 6T "yUa0 *qno “OUIN]O A CLG } OLG 439 Evaporation and Dissociation. GE-086 O0GZE GE-.086 OOFLE GE: 0086 OGLOF GE: 0086 00996 G8-0086 00686 CE-0()8G OGO00E o0GG OLT8z o0GZ O9TGE 00GG OSSts 0096 OLLES 0046 OGOGS 00GG O1G9G o&6G O0GGSG GE-0086 O6SFS o8GG O€E8S o66G OOTOS GE-.086 O89rP o&GG OSLIG o8GG OFLEG o66G OLSEG o0GG OG8EG o0GZ 0000E o0GG O18LzG 0066 00966 o09G OséLé GE- 086 OZ96P o0GG OFGIG o0GG OFEGS NGG OFSES oG0G O0CESG o8GG 01996 oO1G O0GIG o01G O&6L6 o6GG OG6LE o0GG O8S0P GE-0086 O& L6G GE- 0086 O88EG GE- 0086 O61GG oG0G OGCOZG oG0G OOLTG oG0G OOLGG oL6T OLGES o0GG OVGIG oG0G OG9GG of0G O&OLG o0GG OFETE o6GG OG6E6 o0G6 OLFOG oOUG O9F1G 00GG O8S6G oG6T O6IGI of61 O0GO00G of61 06606 of61 06616 0G0G OOFFG 0006 OC6FG 0006 00G9G o01G OSF6G o0GG O9GES o0GG OGOPV 00GG OFSST o0GG OSé61 o0GG OGG0G of61 OGc6I of61 6106 oG61 O90TG oG6T OGO0GSG oL61 OGEES oL6T OUTEG oL61 OF9GG of0G OOFSG o01G OG60E o6GG OFI9S o0GG 09G0¢ of 61 O689T of 61 OL9LT of61 O8SsTt 8-o861 OOG6T 8-061 O€00G 8-.€61 06606 80861 OL8IG of61 OO00EG of 61 OSOFG oG61 O61SZ o00G OGELG of 0G 00166 o06G OFFS of6G 00688 of6T OSOLT of61 OFLLI of6L OOG8T of61 OOL6T oS61. 09661 oS61 OG80G oS6L OO8TZ of61 O&0ES of61 OS LPG oG61 OSVAST A oL61 OL99G 0006 OOP8G o0IG OOFGE o06G OS8Lé 8-o861 OF89T 8-o86T OOGLT 80861 O8Z8T oG6T O968T oG6T OF86T o¢6I OFLOG oG6L OOLTZ 8-o861 OL8ZSG 8-861 O0S666 8-661 09096 of61 OG19G oLG6T O9GLE oG0G 00806 o0 1G OSOFE o6T O&G9T o06T O8TLI o06T OLGLT o0 61. OFL8T o06T O996T o06T OSG0G o06T OSFIG oS61 OLLGG oc 61 O&S8&G oS61 OL8FG of6T O1G9G of61 01696 o00G 09666 oG0G OPsIE of81 O00c9T oG8T Og89T of8T OLGLT oG81 OOS8T of8T 06061 oG8T 06661 oG8T 09806 oG6T OG9GG oc61 OG9EG oG61 O99VG 8-o861 OG696 of 6T O000LG oL61 OS186 0006 006626 oG81 OST9TL oG8T O€sS9L oG8T OLGLT of8T OOE8T of8T OS I6T oG8T OG66T oG8I 00806 o(6T OGEGS o06T OOFEG 0061. OOEFS oS61 OGLGS 8-61 OF99G oG6T O9FLS oLb61 OFF8S of LT OGSST of LT OOT9T oG LT OSLOT oGhT OOFLT of LT OLO8T of LT OO88T of LT OSG6T oG8I O6LIG of8T1 009GG oGsT OSEES oG6L O0¢&G 0861 OOF9S oG61 O9FLG oG61 06916 oGLI OFFET of LI 0009T of LT OL99T oGLT OSEL1 oGLT 0908T oGLI O8L8T oGLT OSc6T oG81 OG9TG © of8T OPSEG ofS OSES o06T OF 1s oG6L OOT9G 80861 000LZ of61 00926 L eer suey eees eae | oe eee rowel ae Drs. Ramsay and Young on 440 GE-0086 Orrel GE: 0086 O086T CE-0086 01906 GE: 0086 OFITS 0&6 O96PT 0096 O008T 0096 O8L8T 0096 00961 006 OG8ET 006 OOr9T o0GG OPOLT 0066 O8LLT of61 O9LZL of61 OGOGT of6L O19S1 of61 OIZ9T of 61 OO8éI oG61 OS TST of61 OGLE of6T ogeoT oCGG OS6E o&6G O1GS o&6G 0094 0866 O€S6 o66G OL6TT 80861 OSLGL 8-o661 OO0gT 80661 OLSST 8-o661 OLZIT of61 OLLE of61 O98P of 61 0G0L of61 0098 of61 OgOTT 0061 OO0gGT o061 O6LP1 o06T OSéST 0061 00691 o06T 0896 o061 OO8P 8:0861 O10L 8-o661 09g8 80661 09601 oG81 OSE6l oG81 O8Vr1 of8T OGOgT of81 O09gT “(panunquos) J e[qeL of81 0096 of81 OSL 0061 O69 o061 09F8 o06T O8LOT oG81 O8GcT of8T OOSTI of8T OZOET of SL OsgeT oG LT 0096 oG kT O09F oS81 O89 of81 OFE8 oG8I O090T oGkT O9sIT oGLI Ossét oGLI ObErI oGLT OF6rT o0G 098 o0G 0cOT 009 O9GT L064 691 o00T 0686 o00L 0¢S9g oT OST9 00ST OStL o0ST 0SE6 o0¢T 0O0LOT o0ST OFccT o0GT Orscl 00ST OgesT oo) i) oO S Yer} N . . . ° e . ete mm . . ° > i=) rm 16 Le . . . i=) ba) MA Meee ae ae ae oS oe =) ie) te) | j=) N for) pea Baa Ae S Yen) a) @ ro "qu90 *qno ‘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 LGI8T 60681 SPL8T 8h906 9G91G PrLos C8686 LOEGS 68196 SEP8G T1806 LOVE VI6VE 90116 PGOIF CO6PP FLE0E 6968 DGS SLGL F866 Gr6L O&681 90991 O8cLT 60081 T6L8T 66961 6806 TE9IG LELGS OF GEG PLEGG 66196 69886 89 10€ L90GE VOLPE GLEE FOG6E SE8IT OO8ET PEP9T OOT AT 9I8LT 98981 80F61 EPE0G OLETS 1V¥GS PE9ES GG6VG GEEIG GT6L6 IV966 OFFIE V6EEE GoPSé PO6LE GOOGG LOP9G OL8Lz 99666 PS806 G6GGE PG8EE GPEVG SP9GG 68696 TTE8z P8966 Gg90& PSLIE SLOEG I88FG 80196 99GLZ PIE8G E062 8966 O8ZEG GEPEG 08996 96996 PaGLG {8086 PVHP8G 1698 FOSP Se0L Z8c8 9601 SILZI EZ0GT S6EGT OIZ9I 19891 G6SGL1 LFE8T 6161 2006 98606 E8616 GLOSS 6S1FS 8ZacG FEE FFOLG CEPLG FZOLG 9896 6687 6002 6998 1G60T 999GT Gcorl GEST 6ET9T P8LOT OLPLT GPG8I 8R06T LE661 69806 IP8IG GS866 LYGEG 910G6 GL6IG PELIG CGOLG PCILS OLIE 918P F169 9098 8801 88gc1 PEStl LIPST LTO9T O999T LEELT OOTST Té681 GOL6T 89906 66916 9196S PLIES 869FG 96996 FLEIG G8P9G PLEITG TS9E I6L¥ G69 9GF8 88L0T GOGGT IPLPI LGOGGT 688¢T G6G9T 68TLT [P6LT. LGL8T PLC6I LGFOG POELG IGEGG ILEEG IVETG 6L1GG PLLGS CP8G6 VPSGS 9098 6GLP 8$89 I€é8 S901 98z61 69VF1 L66V1 S9gGT 6LT9T OG89T LPGLT GGE8T L606T O&66T 10806 8891G C19GG LIFES IETF VOSEG CSGhS P8PES g1gé FOOF 6F99 0808 69C01 69811 C68ET G6ETI GC6FI 16rS1 08091 OPL9T ISPLT Grist CL881 I§96T T9606 OOTIG GOLIG 6F0GG VI61G G901G PPE6I SLES ) Ba Ig€é9 POLL 8616 FOGIT SrOsl FOPET 696ET O9FFT TL6F1 68SS1 PFI9T 60L9T G6GLI S9SLT 14681 66881 69061 ST681 PPT GLC9T vPIET L8CE 16GP LGC9 EorL 8886 TLLOL SLFGL S066T 6ISET GLLET [&cv1 SELF ELEET EgLeT LEGIT G699T FPOLT FIELT LOELT OS89T Oost G8081 4006 TE86 L996 €81G 6619 6694 8096 6996 1886 LOTOT PEsOl PEg0l 96101 66601 91601 09601 VI80T OLFOT €PL6 G6F8 €689 L686+ GL86— 6161 LIVG LECE 0696 OFIP G8GP 610P 9L8E 6896 SohE 6816 O€LG 6066 GoPl Lov + Choe 8286— 187S$— 6116— O8FFI— OGGZB— €8LhE — 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. | ‘ i i i , F f 4a) a i i" fy 4 rt ¥ i bi i : Se es ee) a ee ee) 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, i) im Win | iat Mt i a thas At wt | 4 { i i 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 «\ — 25) so 25 10 90 80 70 60 50) 40. 30 2 10 O DRYER: ERE EN Per centage diffused . See 12 = aS ‘S i tom : oe ee an eee : Zaasihe Tepe a ear : f P =“ ie Ss sed cael ae es - = RE Bap ER eee + ey t Se ae ree + > 2 a ion : aoe sites : ay - = Bere ~ ieee os « ee ee aoa S . : > ; 5 3 F , - ~ a Faanel : : Baar ri - . x . 1 ™ > «e : eu! : ; : - < 5 ) . S: ne se S U a ~ : a x a » « 2 z , = - = - z = ca hows ‘i ae 2 nae = ave - 5 ¢. -) % : : * ¥ r Saar ‘ = . oi ie = = i ‘ 2 : ~ - 2 z = 2 ed % a & ve 2 pes " . = : m= rs 5 a 2 - oa ‘ nN i - a 1 i ~ " = i a F aS 4 + - S r Z . 5 5 ‘ f ' e z : t 2 e y 5 x x = 4 < oy t \ i: is a & is c 5 3 sii : é 2 ‘ 4 s = 3 ahs - 3 : L =4 3 4 = r kg 2 or - z - ” < = é = Ms . cae he - < ~ ‘ a i Fs ‘ 3 ; ‘ ; Phil. Mag. 5.0. Vol. 28. PI. IL. - ’ } ae, : ‘ ” t , ¢ r if ig “6 wy . > a J \s Lad ; A re a P y } ‘ i, ag 7 + » | b ’ ‘ : j 4 ey i 4 iy ak d ¥ hk i J ; - ’ x. & z i .< i 8 | bay i ry. ‘ eon . Vas ae. ‘ x . | py a ; y ae eis + rae | c * od 3.3. Vol. 23, Pl. il Ny" 11/1914 A a NS V0 000 00 LE 1 “BML Souq wuaquy N\A a QD Corer) ————) QUT UT) Ut al HHH Micrel cf I aD ii pad gs | ei a TH | HH I. a wa ¥ 2 i fe ds e ‘ ~ ’ ri * ) é \e (( 5 i +3) Sj i! ij i ; —> | Se Roe = AY ae) N a O = r ei] “ | 2) ~~ ee " . * e Phil, Mag, S.5.Vol. 23. PLIV. " a Fig.5. eee ry LOR Fal ‘il mc | ql> Sti stark ty *, > i r ¥ Phil, Mag. 5.5. Vol. 23. PIV. b. Va NO; ? NaNO; 0NaNO3 2 NaNO; LR 03 Water it Solutions. ; | 60° 80° 100° . ith. f~ mh aw fe mms. 54.000. mms: +- ) 52000 60000 EOO00: 38000 56000 48000) SLO00 (32000. 30000 412000) 40000) + load 26000 38000 86000 2000: 22000 34000 82000 300v0/ 6000 28000; E000. 26000}— 24.000) 22000 2000 maf ee eo 2 Se - OBE OLE S096 OSS OLS OES pee eg ry 9.5. Vol.23. Pl. VIL. 40000 $8000) = i 86000)|— 34000) = = at 32000 = 80000) =] aa 28000) 26000 2000 22000) 1 el 20000) - i 18000 16000] Bile. fairy 07) ee We ay VZ Ve / E - i 7 % f \, 2 a & ine ca - ; x 7 a Isochorte Lines for Ether. Phil. Mag. S.5. Vol. 23. Pl. IX = Riereun = ict creeG teas PA ; aol. Tans. 8 io | a 0 ui 30000 H Ze ie i Co te i is 16 7 1s 20000 1» 5 | als al ae 10000) = 50 a eee = ee 00 [i Pa ‘ra oi) y a Cae == | a eee | =10000 20) b B tj D 16) 15) =20000 a) B hk =30000 ‘|| I ~4ggo0ie ai 70" 450" Bon +n" * O*Abochate. 5 5 5 200) 250" 300" 0° 7 0 Bo Mintern Bros. ith - Pl TY me NAV PN VW] IN AAA] oe Ki 66000 Vol 23. JANUARY 1897. . No. 140. Published the First Day of every Month.—Price 2s. 6d. i : _fHE } LONDON, EDINBURGH, anp DUBLIN PHILOSOPHICAL MAGAZINE, AND JOURNAL OF SCIENCE. Being a Continuation of Tilloch’s ‘Philosophical Magazine,’ Nicholson’s ‘Journal, and Thomson's ‘Annals of Philosophy.’ CONDUCTED BY SIR ROBERT KANH, LL.D. F.R.S? M.R.LA. F.C.S. SIR WILLIAM THOMSON, Knr. LL.D. F.RB.S. &. 3 AND WILLIAM FRANCIS, Pu.D. F.LS. F.R.A.S. F.C.8. Re, Anis Pg ~ FIFTH SERIES. (2°) = © t UW fe \ VO ae N° 140.—JANUARY 1887. — iy WITH ONE PLATE. oe Illustrative of Mr. J. J. 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Published the First Day of every Month.—Price 2s. 6d. THE LONDON, EDINBURGH, anp DUBLIN PHILOSOPHICAL MAGAZINE, AND ‘JOURNAL OF SCIENCE. Being a Continuation of Tilloch’s ‘Philosophical Magazine,’ Nicholson’s ‘Journal, and Thomson’s ‘Annals of Philosophy.’ CONDUCTED BY SIR ROBERT KANE, LL.D. F.R.S. M.R.I.A. F.C.S. SIR WILLIAM THOMSON, Kyr. LL.D. F.R.S. &c. AND WILLIAM FRANCIS, Pu.D. F.L.S. F.R.A.S. F.C.S. FIFTH SERIES. Ne 142.—MARCH 1887. WITH ONE PLATE. Illusirative of Lord Rayieten’s Paper on the Behaviour of Tron and” Steel under the Operation of Feeble Magnetic Forces. LONDON: PRINTED BY TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. Sold by Longmans, Green, and Co.; Kent and Co.; Simpkin, Marshall, and Co.; and Whittaker and Co.;—and by A.and G. Black, and T. and T. Clark, Edin- burgh; Smith and Son, Glasgow :—Hodges, Foster, and Co., Dublin :—Putnam, New York :—Veuve J. Boyveau, Paris:—and Asher and Co., Berlin. sixth Hdition, hevised and Enlarged. Crown ovo, 1Us. 0d, — ~ AN ELEMENTARY TREATISE a ; ON ;. THE DIFFERENTIAL CALCULUS, CONTAINING 3 THE THEORY OF PLANE CURVES, WITH NUMEROUS EXAMPLES. By BENJAMIN WILLIAMSON, M.A, F.B.S., Fellow of Trinity College, Dublin, &c. London: Lonemans, GREEN, and Co. Royal 8vo, with numerous Woodcuts, price 7s. 6d. A MANUAL FOR TIDAL OBSERVATIONS, By Mayor A. W. BAIRD, R.E., F.RS., &e. TAYLOR and F RANCIS, Red Lion Court, Fleet Street, E.C. Now ready, Price £1 1s. Od. ae Demy 8vo, 360 pages, illustrated with 34 single and 8 folding Plates, including numerous drawings of Mountain Structure and Scenery, by the Author, from Nature. THE ORIGIN OF MOUNTAIN RANGES; Cueaercd EXPERIMENTALLY, STRUCTURALLY, DYNAMICALLY and in relation to their GEOLOGICAL HISTORY. By T. MELLARD READE, C.E., F.G.S., F.B.LB.A., Past President of the Liverpool Geological Society. ? . Tayionr and Francts, Red Lion Court, Fleet Street, E.C. Price £6 10s. Od. TAYLOR'S SCIENTIFIC MEMOIRS. Selected from the Transactions of Foreign Academies of Science and Learned Societies, and from Foreign Journals. First Series, August 1836 to September 1852: 21 parts. New Series, Natural History: 4 parts. Natural Philosophy: 4 parts, : ; 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, ~ Eneke, Fresnel, Gauss, Goldschmidt, Hansen, Helmholtz, Henry, Hoffmann, Hoffmeister, Holtzmann, Jacobi, Jamin, Knoblauch, Koene, Koren, Krohn, l.amont, Lenz, Le Verrier, Liebig, Lowig, Magnus, Melioni, Menabrea, Meyen, Mitscherlich, Mohl, Moser, Mossotti, Muller, Neumann, Nobili, Ohm, Ostro- gradsky, Pelouze, Peltier, Plateau, Plucker, Poggendorff, Poisson, Pouillet, Provostaye, Redtenbacher, Regnault, Riess, Romer, Rose, Rudberg, Savart, Schleiden, Schmidt, Schultess, Seebeck, Sefstrém, Senarmont, Siebold, Verany, Vogt, Von Baer, Von Wrede, Walckenaer, Wartmann, Wichura, Wiedemann, , Weber. : TayLor and Francis, Red Lion Court, Fleet Street, E.C. [ADVERTISEMENTS continued on 3rd page of Cover. oe -, Je, Ss ‘cetiet > 2 | Vol. 23. APRIL 1887. No. 14% e few wc oheccesererce uy TH : APR 13 168/ LONDON, EDINBURGH IN PHILOSOPHICAL MAGAZINE, AND JOURNAL OF SCIENCE. Being a Continuation of Tilloch’s ‘Philosophical Magazine,’ Nicholson’s ‘Journal,’ and Thomson’s ‘Annals of Philosophy.’ CONDUCTED BY SIR ROBERT KANE, LL.D. F.R.S. M.R.I.A. F.C.S8. SIR WILLIAM THOMSON, Knr. LL.D. F.R.S. &e. | AND ? WILLIAM FRANCIS, Pu.D. F.L.S. F.R.A.S. F.C.S8. FIFTH SERIES. N° 143.—APRIL 1887. WITH TWO PLATES. Illustrative of Prof. 8. P. THompson’s Paper on an Arc-Lamp, and Mr. T. Grav’s on an Improved Form of Seismograph. LONDON: PRINTED BY TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET, Sold by Longmans, Green, and Co.; Kent and Co.; Simpkin, Marshall, and Co.; and Whittaker and Co.;—and by A. und C. Black, and T. and T. Clark, Edin- burgh; Smith and Son, Glasgow :—Hodges, Foster, and Co., Dublin:—Putnam, New York:—Veuve J. Boyveau, Paiis:—and Asher and Co., Berlin. i Now ready, Price £1 1s. Od. Demy 8vo, 360 pages, illustrated with 34 single and 8 folding Plates, including : numerous drawings of Mountain Structure and Scenery, by the Author, from — Nature. a THE ORIGIN OF MOUNTAIN RANGES, Considered EXPERIMENTALLY, STRUCTURALLY, DYNAMICALLY, and in relation to their GEOLOGICAL HISTORY. By T. MELLARD READE, C.E., F.G.S., F.R.1.B.A., Past President of the Liverpool Geological Society. Tay Lor and Francis, Red Lion Court, Fleet Street, E.C. Royal 8vo, with numerous Woodcuts, price 7s. 6d. . A MANUAL FOR TIDAL OBSERVATIONS. By Mason A. W. BAIRD, R.E., F.R.S., &e. Tayor and Francis, Red Lion Court, Fleet Street, E.C. Royal 4to, cloth boards, price £1. FACTOR TABLE FOR THE SIXTH MILLION, CONTAINING THE LEAST FACTOR OF EVERY NUMBER NOT DIVISIBLE BY 2, 3, or 5, _ BETWEEN 5,000,000 and 6,000,000. By JAMES GLAISHER, F.R.S. Uniform with the above, - FACTOR TABLES FOR THE FOURTH AND FIFTH MILLION S. Price £1 each. 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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, Hoffmeister, Holtzmann, Jacobi, J amin, 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, Provostaye, Redtenbacher, Regnault, Riess, Romer, Rose, Rudberg, Savart, Schleiden, Schmidt, Schultess, Seebeck, Sefstrom, Senarmont, Siebold, Verany, Vogt, Von Baer, Von Wrede, Walckenaer, Wartmann, Wichura, Wiedemann, ~ Weber. ; TayLor and Francis, Red Lion Court, Fleet Street, E.C. [ADVERTISEMENTS continued on 3rd page of Cover. Vol. 23. — MAY 1887. No. 144. Published the First Day of every Menth.—Price 2s. 6d. ant © LONDON, EDINBURGH, anp DUBLIN PHILOSOPHICAL MAGAZINE, AND JOURNAL OF SCIENCE. Being a Continuation of Tilloch’s ‘Philosophical Magazine,’ Nicholson’s ‘Journal,’ and Thomson’s ‘Annals of Philosophy.’ CONDUCTED BY | SIR ROBERT KANE, LL.D. F.R.S. M.RB.LA. F.C. SIR WILLIAM THOMSON, Kyr. LL.D. F.BS. &e. AND WILLIAM FRANCIS, Pa.D. F.LS. F.RA.S. F.C.S. FIFTH SERIES. N° 144.—MAY 1887. WITH SIX PLATES. Illustrative of Mr. W. W. J. Nicox’s Paper on the Expansion of Salt- Solutions, and Drs. Ramsay and Youne’s on Evaporation and Dissociation. LONDON: PRINTED BY TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET. Sold by Longmans, Green, and Co.; heat and Co.; Simpkin, Marshall, and Co.; and Whittaker and Co.;—and by A. and ©. Biack, and T. and T. Clark, Hdin- burgh; Smith and Son, Glasgow :—-Hodges, Foster, and Co., Dublin :—Putnam, New York :— Veuve J. Boyveau, Pa.is:—and Asher and Co., Berlin. Now ready, Price £1 1s. Od. Nature. THE ORIGIN OF MOUNTAIN RANGES, Considered EXPERIMENTALLY, STRUCTURALLY, DYNAMICALLY, and in relation to their GHOLOGICAL 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. e . . . . “‘ 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.” —Liverpool 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. Taytor and Francis, Red Lion Court, Fleet Street, E.C. Royal 8vo, with numerous Woodcuts, price 7s. 6d. A MANUAL FOR TIDAL OBSERVATIONS. . By Masor A. W. BAIRD, B.E., F.BS., &e. TayLor and Francis, Red Lion Court, Fleet Street, H.C. Price £6 10s. Od. TAYLOR’S SCIENTIFIC MEMOIRS. Selected from the Transactions of Foreign Academies of Science and Learned Societies, and from Foreign Journals. First Series, August 1836 to September 1852: 21 parts. New Series, Natural History: 4 parts. Natural Philosophy: 4 parts. i 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, 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, Provostaye, Redtenbacher, Regnault, Riess, Romer, Rose, Rudberg, Savart, Schleiden, Schmidt, Schultess, Seebeck, Sefstrom, Senarmont, Siebold, Verany, Vogt, Von Baer, Von Wrede, Walckenaer, Wartmann, Wichura, Wiedemann, Weber. TayLor and Francis, Red Lion Court, Fleet Street, E.C. [ADVERTISEMENTS continued on 3rd page of Cover. Demy 8vo, 860 pages, illustrated with 34 single and 8 folding Plates, including a numerous drawings of Mountain Structure and Scenery, by the Author, from — Vol. 23. | JUNE 1887. No. 145. Published the First Day of every Month.—Price 2s. 6d. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE, AND JOURNAL OF SCIENCE. Being a Continuation of Tilloch’s ‘Philosophical Magazine,’ Nicholson’s ‘Journal, and Thomson’s ‘Annals of Philosophy.’ CONDUCTED BY SIR ROBERT KANE, LL.D. F.R.S. M.R.LA. F.C.8,, SIR WILLIAM THOMSON, Knr. LL.D. F.R.S. &c. AND : WILLIAM FRANCIS, Pu.D. F.L.S. F.R.A.S. F.C.S. FIFTH, SERIES. PRINTED BY TAYLOR AND FRANCIS, RED LION COURT, FLEET STREET, Sold by Longmans, Green, and Co.; Kent and Co.; Simpkin, Marshall, and Co.; and Whittaker and Co.;—and by A. and C. Black, and. T. and T. Clark, Edin- burgh; Smith and Son, Glasgow :—-Hodges, Foster, and Co., Dublin: :—Putnam, New York: —Veuve J. Boyveau, Paris :—and ge and Co., Berlin. Vg Now ready, Price £1 Is. Od. Demy 8vo, 360 pages, illustrated with 34 single and 8 folding Plates, including numerous drawings of Mountain Structure and Scenery, by the Author, from Nature. "THE ORIGIN OF MOUNTAIN RANGES, | 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. | Royal 8vo, with numerous Woodcuts, price 7s. 6d. A MANUAL FOR TIDAL OBSERVATIONS. By Mason A. W. BAIRD, B.E., F.R.S., &e. Taytor and Francis, Red Lion Court, Fleet Street, E.C. Price £6 10s. Od. TAYLOR’S SCIENTIFIC MEMOIRS. Selected from the Transactions of Foreign Academies of Science and Learned Societies, and from Foreign Journals. First Series, August 1836 to September 1852: 21 parts. New Series, Njgural History: 4 parts. Natural Philosophy : 4 parts. 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, 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, Provostaye, Redtenbacher, Regnault, Riess, Romer, Rose, Rudberg, Savart, Schleiden, Schmidt, Schultess, Seebeck, Sefstrém, Senarmont, Siebold, Verany, Vogt, Von Baer, Von Wrede, Walcken al yan, Wichura, Wiedemann, Weber. $ ik ee 7? 7, TayLor and Francis, Red Ijon Court, Fleet Street, E.C. [ADVERTISEMENTS continued on 3rd page of Cover. eee ae ee ee ee gt ee oe ee a ‘ Vs t if \ a i aia, | wa 3 9088 012 rASS TAS SF